RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

RePoSS: Research Publications on Science Studies 

RePoSS #11: 

The Mathematics of Niels Henrik 

Abel: Continuation and New 

Approaches in Mathematics 

During the 1820s 

Henrik Kragh Sørensen 

October 2010 

Centre for Science Studies, University of Aarhus, Denmark 

Research group: History and philosophy of science

Please cite this work as: 

Henrik Kragh Sørensen (Oct. 2010). The Mathematics of Niels 

Henrik Abel: Continuation and New Approaches in Mathematics 

During the 1820s. RePoSS: Research Publications on Science 

Studies 11. Aarhus: Centre for Science Studies, University of 

Aarhus. url: ��. 

Copyright c○ Henrik Kragh Sørensen, 2010

The Mathematics of 

NIELS HENRIK ABEL 

Continuation and New Approaches 

in Mathematics During the 1820s 

HENRIK KRAGH SØRENSEN

For Mom and Dad 

who were always there for me 

when I abandoned 

all good manners, 

good friends, 

and common sense 

to pursue my dreams.

The Mathematics of 

NIELS HENRIK ABEL 

Continuation and New Approaches 

in Mathematics During the 1820s 

HENRIK KRAGH SØRENSEN 

PhD dissertation 

March 2002 

Electronic edition, October 2010 

History of Science Department 

The Faculty of Science 

University of Aarhus, Denmark

This dissertation was submitted to the Faculty of Science, 

University of Aarhus in March 2002 for the purpose of ob- 

taining the scientific PhD degree. It was defended in a public 

PhD defense on May 3, 2002. A second, only slightly revised 

edition was printed October, 2004. 

The PhD program was supervised by associate professor 

KIRSTI ANDERSEN, History of Science Department, Univer- 

sity of Aarhus. 

Professors UMBERTO BOTTAZZINI (University of Palermo, 

Italy), JEREMY J. GRAY (Open University, UK), and OLE 

KNUDSEN (History of Science Department, Aarhus) served 

on the committee for the defense. 

c○ Henrik Kragh Sørensen and the History of Science De- 

partment (Department of Science Studies), University of 

Aarhus, 2002–2010. 

The dissertation was typeset in Palatino using pdfLATEX. 

First edition (March 2002, 7 copies) was copied and bound 

by the Printshop at the Faculty of Science, Aarhus. 

Second edition (October 2004, 5 copies) was printed and 

bound by the Printshop at Agder University College, Kris- 

tiansand. 

This electronic edition was compiled on October 28, 2010. 

For further information, additions, corrections, and 

contact to the author, please refer to the website 

��. 

The picture on the front page is a painting of NIELS HENRIK 

ABEL performed by the Norwegian painter JOHAN GØRB- 

ITZ during ABEL’s time in Paris 1826. It is the only authentic 

depiction of ABEL and is reproduced from (Ore, 1957). 

The picture on the reverse shows a curlicue frequently used 

by ABEL in his notebooks to mark the end of manuscripts. It 

is reproduced from (Stubhaug, 1996).

Contents 

Contents i 

List of Tables vii 

List of Figures ix 

List of Boxes xi 

List of Theorems etc. xiii 

Summary xv 

Preface to the 2004 edition xvii 

Abel’s mathematics in the context of traditions and changes . . . . . . . . . . xvii 

Recent literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 

Preface to the 2002 edition xxi 

Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii 

I Introduction 1 

1 Introduction 3 

1.1 The historical and geographical setting of ABEL’s life . . . . . . . . . . . 4 

1.2 The mathematical topics involved . . . . . . . . . . . . . . . . . . . . . . . 4 

1.3 Themes from early nineteenth-century mathematics . . . . . . . . . . . . 7 

1.4 Reflections on methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2 Biography of NIELS HENRIK ABEL 17 

2.1 Childhood and education . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.2 “Study the masters” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 

2.3 The European tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

2.4 Back in Norway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

i

3 Historical background 39 

3.1 Mathematical institutions and networks . . . . . . . . . . . . . . . . . . . 39 

3.2 ABEL’s position in mathematical traditions . . . . . . . . . . . . . . . . . 41 

3.3 The state of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 

3.4 ABEL’s legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

II “My favorite subject is algebra” 47 

4 The position and role of ABEL’s works within the discipline of algebra 49 

4.1 Outline of ABEL’s results and their structural position . . . . . . . . . . . 50 

4.2 Mathematical change as a history of new questions . . . . . . . . . . . . . 53 

5 Towards unsolvable equations 57 

5.1 Algebraic solubility before LAGRANGE . . . . . . . . . . . . . . . . . . . . 59 

5.2 LAGRANGE’s theory of equations . . . . . . . . . . . . . . . . . . . . . . . 65 

5.3 Solubility of cyclotomic equations . . . . . . . . . . . . . . . . . . . . . . . 72 

5.4 Belief in algebraic solubility shaken . . . . . . . . . . . . . . . . . . . . . . 80 

5.5 RUFFINI’s proofs of the insolubility of the quintic . . . . . . . . . . . . . . 84 

5.6 CAUCHY’ theory of permutations and a new proof of RUFFINI’s theorem 90 

5.7 Some algebraic tools used by GAUSS . . . . . . . . . . . . . . . . . . . . . 95 

6 Algebraic insolubility of the quintic 97 

6.1 The first break with tradition . . . . . . . . . . . . . . . . . . . . . . . . . 99 

6.2 Outline of ABEL’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 

6.3 Classification of algebraic expressions . . . . . . . . . . . . . . . . . . . . 101 

6.4 ABEL and the theory of permutations . . . . . . . . . . . . . . . . . . . . . 108 

6.5 Permutations linked to root extractions . . . . . . . . . . . . . . . . . . . . 110 

6.6 Combination into an impossibility proof . . . . . . . . . . . . . . . . . . . 112 

6.7 ABEL and RUFFINI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 

6.8 Limiting the class of solvable equations . . . . . . . . . . . . . . . . . . . 124 

6.9 Reception of ABEL’s work on the quintic . . . . . . . . . . . . . . . . . . . 125 

6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 

7 Particular classes of solvable equations 141 

7.1 Solubility of Abelian equations . . . . . . . . . . . . . . . . . . . . . . . . . 142 

7.2 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

7.3 The concept of irreducibility at work . . . . . . . . . . . . . . . . . . . . . 157 

7.4 Enlarging the class of solvable equations . . . . . . . . . . . . . . . . . . . 160 

8 A grand theory in spe 163 

8.1 Inverting the approach once again . . . . . . . . . . . . . . . . . . . . . . 163 

8.2 Construction of the irreducible equation . . . . . . . . . . . . . . . . . . . 165 

ii

8.3 Refocusing on the equation . . . . . . . . . . . . . . . . . . . . . . . . . . 171 

8.4 Further ideas on the theory of equations . . . . . . . . . . . . . . . . . . . 176 

8.5 General resolution of the problem by E. GALOIS . . . . . . . . . . . . . . 181 

IIIInterlude: ABEL and the ‘new rigor’ 189 

9 The nineteenth-century change in epistemic techniques 191 

10 Toward rigorization of analysis 193 

10.1 EULER’s vision of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 

10.2 LAGRANGE’s new focus on rigor . . . . . . . . . . . . . . . . . . . . . . . 197 

10.3 Early rigorization of theory of series . . . . . . . . . . . . . . . . . . . . . 198 

10.4 New types of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 

11 CAUCHY’s new foundation for analysis 207 

11.1 Programmatic focus on arithmetical equality . . . . . . . . . . . . . . . . 207 

11.2 CAUCHY’s concepts of limits and infinitesimals . . . . . . . . . . . . . . . 209 

11.3 Divergent series have no sum . . . . . . . . . . . . . . . . . . . . . . . . . 210 

11.4 Means of testing for convergence of series . . . . . . . . . . . . . . . . . . 212 

11.5 CAUCHY’s proof of the binomial theorem . . . . . . . . . . . . . . . . . . 214 

11.6 Early reception of CAUCHY’s new rigor . . . . . . . . . . . . . . . . . . . 218 

12 ABEL’s reading of CAUCHY’s new rigor and the binomial theorem 221 

12.1 ABEL’s critical attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 

12.2 Infinitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 

12.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 

12.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 

12.5 ABEL’s “exception” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 

12.6 A curious reaction: Lehrsatz V . . . . . . . . . . . . . . . . . . . . . . . . . 241 

12.7 From power series to absolute convergence . . . . . . . . . . . . . . . . . 246 

12.8 Product theorems of infinite series . . . . . . . . . . . . . . . . . . . . . . 251 

12.9 ABEL’s proof of the binomial theorem . . . . . . . . . . . . . . . . . . . . 254 

12.10Aspects of ABEL’s binomial paper . . . . . . . . . . . . . . . . . . . . . . . 260 

13 ABEL and OLIVIER on convergence tests 265 

13.1 OLIVIER’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 

13.2 ABEL’s counter example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 

13.3 ABEL’s general refutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 

13.4 More characterizations and tests of convergence . . . . . . . . . . . . . . 272 

14 Reception of ABEL’s contribution to rigorization 277 

14.1 Reception of ABEL’s rigorization . . . . . . . . . . . . . . . . . . . . . . . 277 

iii

14.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 

IVElliptic functions and the Paris mémoire 283 

15 Elliptic integrals and functions: Chronology and topics 285 

15.1 Elliptic transcendentals before the nineteenth century . . . . . . . . . . . 286 

15.2 The lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 

15.3 LEGENDRE’s theory of elliptic integrals . . . . . . . . . . . . . . . . . . . . 292 

15.4 Left in the drawer: GAUSS on elliptic functions . . . . . . . . . . . . . . . 296 

15.5 Chronology of ABEL’s work on elliptic transcendentals . . . . . . . . . . 297 

16 The idea of inverting elliptic integrals 299 

16.1 The importance of the lemniscate . . . . . . . . . . . . . . . . . . . . . . . 299 

16.2 Inversion in the Recherches . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 

16.3 The division problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 

16.4 Perspectives on inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 

17 Steps in the process of coming to “know” elliptic functions 321 

17.1 Infinite representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 

17.2 Elliptic functions as ratios of power series . . . . . . . . . . . . . . . . . . 325 

17.3 Characterization of ABEL’s representations . . . . . . . . . . . . . . . . . 328 

17.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 

18 Tools in ABEL’s research on elliptic transcendentals 331 

18.1 Transformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 

18.2 Integration in logarithmic terms . . . . . . . . . . . . . . . . . . . . . . . . 339 

18.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 

19 The Paris memoir 347 

19.1 ABEL’s approach to the Paris memoir . . . . . . . . . . . . . . . . . . . . . 347 

19.2 The contents of ABEL’s Paris result and its proof . . . . . . . . . . . . . . 351 

19.3 Additional, tentative remarks on ABEL’s tools . . . . . . . . . . . . . . . . 371 

19.4 The fate of the Paris memoir . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 

19.5 Reception of the Paris memoir . . . . . . . . . . . . . . . . . . . . . . . . . 376 

19.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 

20 General approaches to elliptic functions 379 

20.1 ABEL’s version of a general theory of elliptic functions . . . . . . . . . . . 379 

20.2 Other ways of introducing elliptic functions in the nineteenth century . . 381 

20.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 

iv

V ABEL’s mathematics and the rise of concepts 385 

21 ABEL’s mathematics and the rise of concepts 387 

21.1 From formulae to concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 

21.2 Concepts and classes enter mathematics . . . . . . . . . . . . . . . . . . . 393 

21.3 The role of counter examples . . . . . . . . . . . . . . . . . . . . . . . . . 395 

21.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 

A ABEL’s correspondence 403 

B ABEL’s manuscripts 407 

Bibliography 411 

Index of names 431 

Index 435 

v

List of Tables 

2.1 Time of publications of CRELLE’s Journal 1826–29 . . . . . . . . . . . . . . . . 32 

2.2 List of most productive authors in CRELLE’s Journal 1826–29 . . . . . . . . . 33 

2.3 Summary of NIELS HENRIK ABEL’s biography . . . . . . . . . . . . . . . . . 37 

5.1 RUFFINI’s classification of permutations . . . . . . . . . . . . . . . . . . . . . 86 

5.2 The N m 

circles formed by applying (As 

At ) to A1, . . . , AN. . . . . . . . . . . . . . 94 

6.1 The order and degree of some expressions in CARDANO’s solution to the 

general cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

10.1 LAGRANGE’s monographs on his algebraic analysis . . . . . . . . . . . . . . 197 

11.1 CAUCHY’s textbooks on the calculus . . . . . . . . . . . . . . . . . . . . . . . 208 

15.1 LEGENDRE’s publications on elliptic transcendentals . . . . . . . . . . . . . . 293 

15.2 LEGENDRE’s classification of elliptic integrals . . . . . . . . . . . . . . . . . . 294 

15.3 ABEL’s early unpublished works on elliptic integrals and related topics . . . 298 

18.1 Important dates in the ABEL-JACOBI-rivalry . . . . . . . . . . . . . . . . . . . 346 

A.1 Correspondence sorted by sender . . . . . . . . . . . . . . . . . . . . . . . . . 403 

A.1 Correspondence sorted by sender (cont.) . . . . . . . . . . . . . . . . . . . . . 404 

A.1 Correspondence sorted by sender (cont.) . . . . . . . . . . . . . . . . . . . . . 405 

B.1 Abel manuscript collections in Oslo . . . . . . . . . . . . . . . . . . . . . . . . 408 

B.2 Abel manuscripts in Oslo, MS:592 . . . . . . . . . . . . . . . . . . . . . . . . . 408 

B.3 Abel manuscripts in Oslo, MS:434 . . . . . . . . . . . . . . . . . . . . . . . . . 409 

B.4 Abel notebooks in Oslo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 

B.5 Abel manuscripts in the Mittag-Leffler Institute, Djursholm . . . . . . . . . . 410 

vii

List of Figures 

2.1 NIELS HENRIK ABEL (1802–1829) . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.2 The southern part of Norway with Christiania and Finnøy marked . . . . . 19 

2.3 BERNT MICHAEL HOLMBOE (1795–1850) . . . . . . . . . . . . . . . . . . . . 20 

2.4 CARL FERDINAND DEGEN (1766–1825) . . . . . . . . . . . . . . . . . . . . . . 25 

2.5 AUGUST LEOPOLD CRELLE (1780–1855) . . . . . . . . . . . . . . . . . . . . . 28 

5.1 JOSEPH LOUIS LAGRANGE (1736–1813) . . . . . . . . . . . . . . . . . . . . . . 65 

5.2 EDWARD WARING (1734–1798) . . . . . . . . . . . . . . . . . . . . . . . . . . 71 

5.3 CARL FRIEDRICH GAUSS (1777–1855) . . . . . . . . . . . . . . . . . . . . . . 73 

5.4 PAOLO RUFFINI (1765–1822) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 

5.5 AUGUSTIN-LOUIS CAUCHY (1789–1857) . . . . . . . . . . . . . . . . . . . . . 91 

6.1 Limiting the class of solvable equations . . . . . . . . . . . . . . . . . . . . . 125 

6.2 WILLIAM ROWAN HAMILTON (1805–1865) . . . . . . . . . . . . . . . . . . . 129 

7.1 N. H. ABEL’S (1802–1829) drawing of the lemniscate in one of his notebooks 153 

7.2 Last page of ABEL’S manuscript for Mémoire sur une classe particulière . . . . 156 

7.3 Extending the class of solvable equations: Abelian equations . . . . . . . . . 160 

8.1 Page from ABEL’S notebook manuscript on algebraic solubility . . . . . . . 177 

8.2 EVARISTE GALOIS (1811–1832) . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 

10.1 BERNARD BOLZANO (1781–1848) . . . . . . . . . . . . . . . . . . . . . . . . . 202 

10.2 JEAN BAPTISTE JOSEPH FOURIER (1768–1830) . . . . . . . . . . . . . . . . . . 205 

12.1 Comparison of CAUCHY’s and ABEL’s structures of the basic theory of in- 

finite series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 

12.2 Graphical representation of ABEL’s “Exception” . . . . . . . . . . . . . . . . 240 

13.1 L. OLIVIER’S geometrical argument . . . . . . . . . . . . . . . . . . . . . . . 268 

15.1 EULER’s rectification of an ellipse by infinite series . . . . . . . . . . . . . . . 288 

15.2 LEONHARD EULER (1707–1783) . . . . . . . . . . . . . . . . . . . . . . . . . . 291 

15.3 ADRIEN-MARIE LEGENDRE (1752–1833) . . . . . . . . . . . . . . . . . . . . . 293 

16.1 Stamp depicting Gauss and the construction of the regular 17-gon. . . . . . 300 

ix

16.2 ABEL’S extension to the complex rectangle . . . . . . . . . . . . . . . . . . . 305 

18.1 CARL GUSTAV JACOB JACOBI (1804–1851) . . . . . . . . . . . . . . . . . . . . 332 

19.1 GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866) . . . . . . . . . . . . 374 

21.1 The equality of the concepts of explicit algebraic expressions and ABEL’s 

normal form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 

21.2 Delineating the border between a concept and its super-concept. . . . . . . . 396 

x

List of Boxes 

1 The algebraic reduction of the cubic equation . . . . . . . . . . . . . . . . . . 60 

2 Proof of Lehrsatz I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 

3 A modern reconstruction of DIRICHLET’s proof. . . . . . . . . . . . . . . . . 239 

4 ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem . . . . . . . . 249 

5 Proof that the trigonometric coefficients cannot approach zero . . . . . . . . 256 

6 Rectification of the ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 

7 Rectification of the lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 

8 A.-M. LEGENDRE’S (1752–1833) reduction of elliptic integrals . . . . . . . . 295 

9 An important lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 

10 Linear interdependence of q0, . . . , qn−1 . . . . . . . . . . . . . . . . . . . . . . 359 

xi

List of Theorems etc. 

Problem 1 Division Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

Problem 2 Division Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 

Theorem 12 DU BOIS-REYMOND . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 

Theorem 13 Generalized Cauchy product theorem . . . . . . . . . . . . . . . . . 253 

Proof 2 Proof of lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 

Theorem 16 Main Theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 

Theorem 17 Main Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 

xiii

Summary 

The present PhD dissertation uses the mathematics of the Norwegian N. H. ABEL 

(1802–1829) as a framework for describing and analyzing trends in the development 

of mathematics during the first half of the nineteenth century. ABEL’S mathematics 

is read and interpreted in its context and used to describe a fundamental change in 

mathematics in the early nineteenth century in which concepts replaced formulae as 

the basic objects of mathematics. 

The dissertation is structured into five parts: 1) an introductory part consisting of 

biographical and other historical framework, 2) three descriptive parts each devoted 

to a particular theme analyzed from a particular discipline in ABEL’S mathematical 

production, and 3) a part comprising the syntheses of a general transition in mathe- 

matics in the early nineteenth century as seen from the perspective of ABEL’S works. 

Introduction. In the introductory part, ABEL’S biography is described to point out 

some of the formative instances in the creation of one of the important mathematicians 

of the first half of the nineteenth century. Because ABEL’S biography has been written 

repeatedly — and recently in an excellent cultural biography — the biography is only 

intended to locate ABEL’S production in its contexts of his life and the mathematics of 

his time. 

New questions: Algebraic solubility. The first of the three studies of ABEL’S math- 

ematics deals with his contributions to the theory of equations. It is illustrated how 

ABEL was led to ask a new kind of question of the solubility of equations which 

would have seemed both counter intuitive and futile to mathematicians a few gen- 

erations before. With the foundation in works of J. L. LAGRANGE (1736–1813) and 

A.-L. CAUCHY (1789–1857), ABEL was able to prove that the algebraic solution of 

general quintic equations was impossible. This result restricted the class of solvable 

equations and separated it from the class of all polynomial equations. In another line 

of research, ABEL proved that an extensive class of equations — later called Abelian 

equations — were algebraically solvable. Compared with the previous result, the sol- 

ubility of the Abelian equations showed that the extension of the concept of solvable 

equations did not collapse. Subsequently, this branch of the theory of equations be- 

came a question of delineating the extension of solvable equations, i.e. of drawing 

xv

the border between solvable and unsolvable equations by some other criterion. ABEL 

commenced research on this issue but had to leave it incomplete. In part II, ABEL’S 

research on all these issues is carefully analyzed based on the works of his main pre- 

decessors and contemporaries. The reception of ABEL’S research and the subsequent 

development of the theory is also addressed. 

New epistemic standards: Rigorization. ABEL’S devotion to and adaption of the 

rigorization movement spearheaded by CAUCHY is the topic of part III. By describing 

ABEL’S critical attitude towards the existing practices of rigor and his publications on 

the binomial theorem and a certain type of criteria of convergence, it is illustrated how 

the new epistemic standards were manifesting themselves in a period of rapid transi- 

tion in analysis. Starting with a description of the Eulerian focus on algebraic equality, 

it is described how CAUCHY’S new emphasis on arithmetical equality effected the cen- 

tral concepts of continuous functions and converget series. Furthermore, the so-called 

exception which ABEL presented to a theorem by CAUCHY is treated in some detail 

because it will later prove to be an important example in the description of the change 

from formula based to concept based mathematics. 

New objects: Elliptic and higher transcendentals. The third and final pillar of ABEL’S 

mathematics which is treated in the present dissertation concerns his works on ellip- 

tic and higher transcendentals. Selected aspects of his work are again presented and 

discussed from a diachronical viewpoint. In this case, special emphasis is given to the 

way ABEL was led to his formal inversion of elliptic integrals into elliptic functions. 

Furthermore, ABEL’S means of obtaining workable representations of the formally de- 

fined object is described. Thereafter, special attention is paid to the techniques which 

he employed in studying transcendentals and it is illustrated how algebraic methods 

figured prominently in his toolbox. In the process, it is also described how his style of 

argument often relied essentially on manipulations of formulae in ways which could 

sometimes lead to results which were true “in general”. Finally, the changing inter- 

nal relationships between definitions and results are illustrated by describing various 

ways towards a general theory of elliptic functions. 

Syntheses. In the ultimate part, the preceding descriptions and discussions of ABEL’S 

mathematics are thrown into perspective by arguing that the development of mathe- 

matics in the early nineteenth century can be understood as a change of paradigms: 

An Eulerian, formula based paradigm is contrasted with a concept based paradigm. 

Various aspects of ABEL’S works — including delineation problems, ABEL’S exception, 

and the nature of arguments which are only true “in general” — are then all inter- 

preted based on this transition in paradigms. 

xvi

Preface to the 2004 edition 

For this second edition, some minor changes have been made to the initial version, 

which was handed in on March 27, 2002 and defended on May 3 the same year. The 

changes include a number of corrections of misprints, a revision of the layout and the 

figures, and the addition of a name index, a subject index, and a list of boxes. 1 Fur- 

thermore, I have included a new preface below, which elaborates on the general theme 

of dissertation, namely the analysis of NIELS HENRIK ABEL’S (1802–1829) mathemat- 

ics within a transition from formula based to concept based mathematics. This new 

“introduction” is an updated and distilled version of the lecture which I gave at the 

defence in May 2002. 

Funded by grants from the Faculty of Science (Aarhus) and from the Netherlands’ 

Organization for Scientific Research (NWO), I have continued to elaborate on this 

transitional framework, producing two papers focusing on particular aspects such 

as critical revision, exceptions, and habituation — processes which are discussed in 

the present work. One of these papers is currently accepted for publication in His- 

toria Mathematica, (Sørensen, 2005); the other remains in the pipeline and should be 

submitted soon. In the future, I will continue to elaborate on the general analytical 

framework and its impact on analysing ABEL’S mathematics. 

Abel’s mathematics in the context of traditions and 

changes 

The aim of the dissertation was twofold: to describe and analyse ABEL’S mathematics 

within its historical context and to draw perspectives on the general development of 

mathematics in the early nineteenth century from the Abelian corpus of mathematics. 

The analyses which are drawn from ABEL’S mathematics are — obviously — re- 

lated to it, and it has been my main ambition to point to general developments which 

shed light on ABEL’S mathematics. That is to say, I do not claim that these trends are 

independent of ABEL’S mathematics and could (or should) apply to other periods of 

time, other topics of mathematics, or other mathematicians. However, it is my convic- 

1 I am grateful for the corrections which were pointed out to me by OLE HALD and by many other 

friendly people. However, if some misprints have endured, I would not be too surprised and therefore 

beg the forgiveness of the reader. 

xvii

tion that they are genuine themes, and I have sought to develop and document this 

position in this dissertation and in my subsequent research. 

The main theme, which I have idenfied and used to structure and analyse ABEL’S 

mathematics is one of a transition from a predominantly formula based approach to 

mathematics towards a more concept based one. 

The formula based approach to mathematics can be found most clearly in the 

works of such mathematicians of the eighteenth century as L. EULER (1707–1783) or 

A.-M. LEGENDRE (1752–1833). These mathematicians approached analysis (including 

the theory of equations) in a formula based way, in which formulae were the central 

objects of mathematics. Their mathematics — to a large extend — consisted of manip- 

ulations of formulae which produced new formulae as results. 

As a counterpart to the formula based approach, I introduce concept based ap- 

proach of the nineteenth century. This way of thinking about and doing mathemat- 

ics was championed by such mathematicians as G. P. L. DIRICHLET (1805–1859) and 

G. F. B. RIEMANN (1826–1866). They thought mainly in terms of concepts, and their 

mathematics consisted of researching the relations between concepts, including repre- 

sentations of concepts, extensions of concepts, and the precise delineation of concepts. 

This framework of formula based and concept based mathematics has helped me 

to organise my study of ABEL’S mathematics because it explains several particular as- 

pects and provides an organising scheme. Thus, in the second part — on the theory 

of algebraic solubility — the framework provides an explanation and structuring of 

what I term the delineation problem: finding the precise extension of the quality of solv- 

able equations. In the third part (rigorisation of analysis), the framework suggests that 

mathematicians were struggling with changing conceptions about their objects, not just 

the ways of manipulating them. This led to the necessity for critical revision which is 

also better understood within a framework of transitions. And in the fourth part (el- 

liptic and higher transcendentals), the framework finally suggests that habituation — 

coming to know new objects — was a prominent problem which played a part in the 

shaping of concept based mathematics. 

This extremely brief outline of the suggestive and explanatory powers of the frame- 

work is meant to justify and explain the organising principle of the dissertation. In my 

subsequent research, I have worked to expand more on the framework and its analyt- 

ical powers. 

Recent literature 

Since the dissertation was submitted in 2002, new literature on ABEL and his mathe- 

matics has emerged. In particular, ABEL’S Paris memoir was located — first partially, 

then completely; see (Del Centina, 2002; Del Centina, 2003). I am grateful to NILS 

VOJE JOHANSEN for providing me with a xerox copy of the famous treatise. Later, 

xviii

in connection with the Abel centennial in Oslo 2002, the proceedings were published 

(Laudal and Piene, 2004). These contain a mathematical introduction by CHRISTIAN 

HOUZEL which elaborates on the previous publications of this scholar. Besides these, 

a number of other publications deal with ABEL’S mathematics and his topics; these 

include (Radloff, 2002) which sheds interesting light on ABEL’S theory of solubility 

and (Pesic, 2003). In my future research and publishing on ABEL’S mathematics, these 

will provide good possibilities for discussions. Although this list is not complete and 

no use of these publications has been made in the present work, they are cited here for 

the convenience of the reader. 

xix

Preface to the 2002 edition 

The present PhD dissertation grew out of long held curiosity towards the multifaceted 

transformation which mathematics underwent during the nineteenth century. In this 

respect, the project is about how mathematics came to have a form recognizable to us 

as modern mathematics. As a pragmatic and useful tool, KIRSTI ANDERSEN — who 

supervised the project — suggested studying the mathematics of ABEL in order to get 

a firmer hold of the transition which was thereby also restricted to a shorter period 

in the first half of the nineteenth century. Based on detailed and often cumbersome 

studies of ABEL’S mathematics, some aspects of the transition stand out which are 

here described and analyzed from the perspective of concept based mathematics. 

Layout 

Quotations. Quotations are used extensively to convey the authentic arguments and 

thoughts. All quotations are presented in English in the text with the original included 

in a footnote. This method has been chosen because it increases the flow of the main 

text. The translations are sometimes based on existing translations; in such cases refer- 

ences are given. Otherwise, the translations have been made by the author. Through- 

out, small-caps have been reserved for names mentioned in the text. For this reason, 

small-caps found in quotations have been replaced by bold-face. 

References and footnotes. References to the bibliography are given in the footnotes 

and consist of the name(s) of the author(s) and the year of (original) publication. Some 

items are referred to through collected works or other compilations; in such cases, the 

bibliography contains both the publication in the collection (primary) and the original 

means of publication (secondary). For papers published in A. L. CRELLE’S Journal für 

die reine und angewandte Mathematik, the original publication is always the primary one. 

In all cases, page references relate to the primary method of publication mentioned in 

the bibliography. Thus, for instance, (N. H. Abel, 1826f, 311) refers to the first page of 

ABEL’S binomial paper as published in CRELLE’S Journal für die reine und angewandte 

Mathematik whereas (L. Euler, 1760, 585) refers to the first page of EULER’S paper as 

printed in the Opera. In the bibliography, authors are listed alphabetically ordered 

by their last name. Items by the same author are listed chronologically and potential 

xxi

items published in the same year are separated by letters. There are a few exceptions 

to these rules — mainly concerning ABEL’S publications. All collected works and a 

few other items have been given more illustrative names, e.g. (N. H. Abel, 1839; N. 

H. Abel, 1902e) which denote the first edition of ABEL’S Œuvres and the Norwegian 

Festschrift of 1902, respectively. Some of ABEL’S manuscripts have been dated and 

published posthumously; for these, the year they were written is included in brackets 

as in (N. H. Abel, [1828] 1839). References to ABEL’S manuscripts and notebooks are 

of the form (Abel, MS:351:A). Letters are refered to as (Abel→Holmboe, Kjøbenhavn, 

1823/06/15. In N. H. Abel, 1902a, 3–4), which denotes the letter from ABEL to B. M. 

HOLMBOE (1795–1850) sent from Copenhagen on June 15, 1823. I have only used 

published letters, and the references are given. 

Mathematics and notation. It has been my general ambition to unwrap and disen- 

tangle the mathematics presented in the dissertation to such a degree that the reader 

who holds no particular knowledge of the topics discussed but is familiar with math- 

ematical reasoning and mathematical notation should be able to benefit from the ar- 

guments and analyses. At the same time, it has been a high priority of mine to present 

the mathematics produced in the early nineteenth century in a way which respects and 

represents the way its creators thought about it. However, I have introduced a mini- 

mum of notational advances, in particular combining sums into the modern notation 

using the summation sign. I have also occasionally renumbered indices or replaced 

symbols to ease the notation. Throughout, I write Σn for the symmetric group on n 

symbols, which is elsewhere frequently referred to as Sn. Slightly off-topic mathemat- 

ical themes have been placed in boxes shaded gray. 

Names and portraits. Upon first mention, historical actors are listed with their full 

Christian names and years of birth and death according to the Dictionary of Scientific 

Biography. 2 In situations where the person is not included in the Dictionary of Scien- 

tific Biography, other sources are employed and explicitly referred to. 3 Full names and 

dates of important persons are sometimes repeated in various parts. Unless other- 

wise noticed, all pictures stem from the history of mathematics internet archives at St. 

Andrews, Scotland. 4 

Acknowledgments 

My utmost gratitude extends towards my supervisor, KIRSTI ANDERSEN. For more 

than four years, KIRSTI has always kindly guided me and put up with my chang- 

ing moods. KIRSTI has read most of the chapters of the present dissertation while 

2 (Gillispie, 1970–80). 

3 Mostly (Biermann, 1988; Poggendorff, 1965; Stubhaug, 1996). 

4 ��. 

xxii

they were under production. The endless effort which she put into her constructive 

criticism has made my presentation more easily understandable and sharpened my 

arguments considerably. For this, I am infinitely grateful. 

Benefitting from my supervisor’s elaborate network, I have had the opportunity to 

discuss aspects of my project with experts who have all been extremely forthcoming. 

During my visit with professor H. J. M. BOS in Utrecht in November and Decem- 

ber 1998, the foundation for the overall structure of the present work was laid and I 

am greatly indebted to professor BOS for his always timely suggestions and contin- 

ued kind interest. In May and June of 2000, I had the opportunity to visit professor 

UMBERTO BOTTAZZINI in Palermo. Discussing my progress with one of the great- 

est experts in the field was an unforgettable experience and I owe a lot of inspiration 

to the kind counselling of professor BOTTAZZINI. Closer to home, professor JESPER 

LÜTZEN has critically and constructively sharpened my research through a number 

of discussions. In particular, LÜTZEN’S insightful and inspiring examination of my 

progress report (1999) helped me improve its contents and made me aware of inter- 

esting parallels also in need of consideration. 

In Norway, historians of mathematics have taken a kind interest in my work. In 

particular, I wish to thank professor REINHARD SIEGMUND-SCHULTZE for his com- 

ments on a paper of mine; 5 the present dissertation has also benefitted from these 

comments. The unsurpassable biography of ABEL had been written by ARILD STUB- 

HAUG less than two years before I embarked on my project. 6 I deeply admire STUB- 

HAUG’S cultural biography and enjoyed meeting with him to discuss our common 

obsession. However, the greatest of my debts to him was that his work made it easier 

for me to focus on my project: the (almost) exclusive study of ABEL’S mathematics. 

For many years, I have enjoyed being part of the History of Science Department. 

I have had the chance to discuss my project, mathematics, science, and life in general 

with some very inspiring and kind people. I wish to thank the entire staff and various 

generations of students of the Department for their openness and support. I particu- 

lar, I wish to thank ANITA KILDEBÆK NIELSEN, LOUIS KLOSTERGAARD, TERESE M. 

O. NIELSEN, and BJARNE AAGAARD for interesting discussions from which this dis- 

sertation has benefitted, directly or indirectly. I am also indebted to THOMAS BRITZ 

for comments and corrections on part II and to my mother, KIRSTEN STENTOFT, for 

proof reading and help in translating quotations into English. 

My thanks also go to SIGBJØRN GRINDHEIM at the manuscript collection of the 

university library in Oslo for providing me with copies of ABEL’S manuscripts, and 

to HANS ERIK JENSEN, Statsbiblioteket Aarhus University for tracing the Vienna re- 

view of ABEL’S impossibility proof. 7 I also wish to thank KLAUS FROVIN JØRGENSEN 

for kindly taking the time to provide me with a list of the ABEL-manuscripts held in 

5 (Sørensen, 2002). 

6 (Stubhaug, 1996), translated into English (Stubhaug, 2000). 

7 Used in section 6.7. 

xxiii

the Mittag-Leffler archives in Djursholm, Sweden, 8 to UDAI VENEDEM for providing 

me with a list of items reviewed in BARON DE FERRUSAC’S Bulletin, and to OTTO B. 

BEKKEN for his interest in my work and for providing me references to the announcements 

by BRIOT and BOUQUET. 9 

It goes almost without saying that despite the generous help which I have received, 

any remaining mistakes, misprints, or misunderstandings are solely my responsibility. 

The material presented in part II constitutes the reworking of a 140-page chapter 

appended to my progress report (Sørensen, 1999). Aspects of the research leading up 

to the dissertation have been presented in various colloquia, courses and meetings. 

8 (I. Grattan-Guinness, 1971, 372–373). 

9 Touched upon in chapter 14. 

xxiv

Part I 

Introduction 

1

Chapter 1 

Introduction 

In the aftermath of the French Revolution of 1789, the political and scientific scenes in 

Paris and throughout Europe underwent radical changes. Social and educational re- 

forms introduced the first massive instruction in mathematics at the newly established 

École Polytechnique in Paris; and mathematics, itself, changed and developed into a 

form recognizable to modern mathematicians. In the first decades of the nineteenth 

century, the neo-humanist movement greatly influenced Prussian academia and as an 

effect, mathematics was promoted into a very prominent position in the curriculum 

of secondary schools. At the university level, mathematics gained a certain autonomy 

and started to evolve along a distinctly theoretical line with less focus on applications 

and mathematical physics. 

The present work centers on one of the main innovative figures in mathematics in 

the 1820s, the Norwegian NIELS HENRIK ABEL (1802–1829), and describes his con- 

tribution to and influence on the fermentation of the mathematical discipline in the 

early nineteenth century. Born at the periphery of the mathematical world and with 

a life-span of less than 27 years, ABEL nevertheless contributed importantly to the 

disciplines which he studied. The overall outline of this presentation is recapitulated 

in the following three sections which introduce ABEL’S professional background and 

training, the mathematics of his works, and the treated themes of development in 

mathematics in the first half of the nineteenth century. Throughout, ABEL’S mathe- 

matics is seen in its mathematical context, and the influences of mathematicians such 

as A.-L. CAUCHY (1789–1857), C. F. GAUSS (1777–1855), J. L. LAGRANGE (1736–1813), 

and A.-M. LEGENDRE (1752–1833) is traced and described. This approach provides 

a background for discussing aspects of continuity and transformation in mathematics 

as can be envisioned from ABEL’S works. 

3

4 Chapter 1. Introduction 

1.1 The historical and geographical setting of ABEL’s 

life 

ABEL lived in a politically turbulent time during which his birthplace, Finnøy, be- 

longed to three different monarchies. When ABEL was born in 1802, Finnøy belonged 

to the Danish-Norwegian twin monarchy but in the wake of the Napoleonic Wars, the 

province of Norway was ceded to Sweden after a short spell of independence. Edu- 

cation in the twin monarchy was centered in Copenhagen, and only in 1813 was the 

university in Christiania (now Oslo) opened. The scientific climate was beginning to 

ripe, but mathematics was not studied at a high level. 

As was common practice for the sons of a minister, ABEL attended cathedral school 

in Christiania and soon got the young B. M. HOLMBOE (1795–1850) as a mathematics 

teacher. HOLMBOE was the first to notice ABEL’S affinity for and skills in mathe- 

matics and they began to study the works of the masters in special private lessons. 

In 1821, after graduating from the cathedral school, ABEL enrolled at the university 

but continued his private studies of the masters of mathematics. In 1824, he applied 

for a travel grant to go to the Continent and he embarked on his European tour in 

1825. It brought him to Berlin and Paris where he had the opportunities to meet some 

of the most prominent mathematicians of the time and frequent the well equipped 

continental libraries. More importantly, ABEL came into contact with A. L. CRELLE 

(1780–1855) in Berlin. CRELLE became ABEL’S friend and published most of ABEL’S 

works in the Journal für die reine und angewandte Mathematik, which he founded in 1826. 

When ABEL returned to Norway in 1827 he found himself without a permanent job 

and with no family fortune to cover his expenses, he took up tutoring in mathematics. 

He had suffered from a lung infection during his tour, and in 1829 he succumbed to 

tuberculosis. 

ABEL’S geographical background thus dictated his approach to mathematics; it 

forced him to study the masters and advance in isolation to do original work. In his 

short life span he carefully studied works of the previous generation and went be- 

yond those. During the months abroad, he came into contact with the newest trends 

in mathematics, and was immediately engaged in new research. Almost all his pub- 

lications were written during or after the tour. The presentation of ABEL’S historical 

and biographical background serves to provide a framework for tracing ideas, influ- 

ences, and connections in his work. 

1.2 The mathematical topics involved 

ABEL’S mathematical production span a wide range of topics and theories which were 

important in the early nineteenth century. His primary contributions are universally 

considered to be in the theory of algebraic solubility of equations, in the rigorization of

1.2. The mathematical topics involved 5 

the theory of series, and in the study of elliptic functions and higher transcendentals. 

However, some of ABEL’S other works (published or unpublished) also have their 

place in the contexts of other disciplines, e.g. in the solution of particular types of 

differential equations, in the prehistory of fractional calculus, in the theory of integral 

equations, or in the study of generating functions. However, to keep the focus of the 

present dissertation, these “minor” topics have not been included and emphasis is put 

on equations, series, and elliptic and higher transcendentals. 

Theory of equations. The essentials of mathematics in the eighteenth century come 

down to the work of a single brilliant mind, L. EULER (1707–1783). Through a lifelong 

devotion to mathematics which spanned most of the century preceding the French 

Revolution, EULER reformulated the core of mathematics in profound ways. Inspired 

by his attempts to demonstrate that any polynomial of degree n had n roots (the so- 

called Fundamental Theorem of Algebra), EULER introduced another important math- 

ematical question: Can any root of a polynomial be expressed in the coefficients by 

radicals, i.e. by using only basic arithmetic and the extraction of roots? This ques- 

tion concerned the algebraic solubility of equations and to EULER it was almost self- 

evident. However, mathematicians strove to supply even the evident with proof, and 

LAGRANGE developed an elaborated theory of equations based on permutations to 

answer the question. Though a believer in generality in mathematics, LAGRANGE 

came to recognize that the effort required to solve just the general fifth degree equa- 

tion might exceed the humanly possible. In LAGRANGE’S native country, Italy, an 

even more radical perception of the problem had emerged; around the turn of the cen- 

tury, P. RUFFINI (1765–1822) had made public his conviction that the general quintic 

equation could not be solved by radicals and provided his claim with lengthy proofs. 

ABEL’S first and lasting romance with mathematics was with this topic, the theory 

of equations; his first independent steps out of the shadows of the masters were un- 

successful ones when in 1821 he believed to have obtained a general solution formula 

for the quintic equation. Provoked by a request to elaborate his argument, he realized 

that it was in err, and by 1824 he gave a proof that no such solution formula could 

exist. The proof, which was based on a detailed theory of permutations and a clas- 

sification of possible solutions, reached world (i.e. European) publicity in 1826 when 

it appeared in the first volume of CRELLE’S Journal für die reine und angewandte Math- 

ematik. But as so often happens, solving one question only leads to posing another. 

Realizing that the general fifth degree equation could not be solved by radicals, ABEL 

set out on a mission to investigate which equations could and which equations could 

not be solved algebraically. Despite his efforts — which were soon distracted to an- 

other subject — ABEL had to leave it to the younger French mathematician E. GALOIS 

(1811–1832) to describe the criteria for algebraic solubility.


Elliptic functions. Since the emergence of the calculus toward the end of the sev- 

enteenth century, the mathematical discipline of analysis had been able to treat an 

increasing number of curves. In his textbook Introductio in analysin infinitorum of 1748, 

EULER elevated the concept of function to the central object of analysis. Concrete func- 

tions were studied through their power series expansions and the brilliant calculator 

EULER obtained series expansions for all known functions including the trigonomet- 

ric and exponential ones. However, EULER did not stop there but ventured into the 

territory of unknown functions of which he tried to get hold. One important type of 

function which analysis had struggled to treat on a par with the rest was the so-called 

elliptic integrals which can measure the length of an arc of an ellipse. 

Mathematicians such as EULER and LEGENDRE felt and spoke of an unsatisfactory 

restriction of analysis because it was only able to treat a limited set of elementary tran- 

scendental functions. Admitting new functions into analysis meant obtaining the kind 

of knowledge about these functions that would allow them to be given as answers. If a 

function today is nothing more than a mapping of one set into another, the knowledge 

of a function then included tabulation of values, series expansions and other represen- 

tations, differential and integral relations, functional relations, and much more. 

When ABEL made elliptic integrals his main research topic, much knowledge con- 

cerning these objects had already been established. An algebraic approach which had 

profound influence on ABEL was GAUSS’ study of the division problem for the circle 

(construction of regular n-gons) in the Disquisitiones arithmeticae. 1 GAUSS had hinted 

that his approach could be applied to the lemniscate integral, a particularly simple 

case of elliptic integrals, and ABEL took it upon himself to provide the claim with 

a proof. By a new idea, soon to be praised as one of the greatest in analysis, ABEL 

inverted the study of elliptic integrals into the study of elliptic functions: Instead of 

considering the value of an integral to be a function of its upper limit, he considered 

the upper limit to be a function of the value of the integral (compare arcsin and sin). 

Through formal substitutions and certain addition formulae, ABEL obtained elliptic 

functions of a complex variable. By this inversion of focus, ABEL managed to place 

the entire theory of elliptic integrals on a new and much more fertile footing. Fueled 

by a fierce competition between ABEL and the German mathematician C. G. J. JA- 

COBI (1804–1851), the new theory gained almost immediate momentum and became 

one of the central pillars of and main motivations for nineteenth century advances in 

mathematics. 

Although ABEL had presented the crucial idea of inverting elliptic integrals into 

elliptic functions, his impact on the further development of the theory stemmed as 

much from a vast generalization of the addition formulae presented in a paper which 

he handed in to the Parisian Académie des Sciences in 1826 (not published until 1841). 

In this paper, ABEL treated an even broader class of integrals generalizing the elliptic 

1 (C. F. Gauss, 1801).

1.3. Themes from early nineteenth-century mathematics 7 

ones and — again using primarily algebraic methods — proved more general versions 

of the addition theorems. The quest of later mathematicians to reapply ABEL’S dar- 

ing inversion of elliptic integrals to this broader class of integrals led to much of the 

important development in complex analysis and topology in the nineteenth century. 

Rigor. Although the theory of equations was closest to ABEL’S heart, and the the- 

ory of elliptic functions brought him fame in the nineteenth century, his mathemat- 

ical legacy remembered in the twentieth century is just as much about his intense 

reception of CAUCHY’S new rigor. Picking up from LAGRANGE’S theory of functions, 

CAUCHY had placed concepts such as continuity and convergence in the foreground 

and founded these concepts on a new interpretation of limits. Equally importantly, 

CAUCHY had shown a way of working with these concepts to deduce properties of 

classes of objects (e.g.. continuous functions or convergent series) rather than explicit, 

often lengthy, studies of specific objects. 

In a memorable and often quoted letter dated 1826 (first published 1839), ABEL 

expressed his conversion to Cauchy-ism and gave the new rigor its dogmatic mani- 

festo. Apparently more radical than CAUCHY himself, ABEL helped determine the 

formulation of the new rigor through his interpretative readings of CAUCHY. In the 

process of re-founding analysis on rigorous grounds, central concepts were specified 

and changed (stretched) to an extent where they included elements whose behavior 

was deemed abnormal. The encounter and resolution of these abnormalities, excep- 

tions as they were often called, was an integrated part of the rigorization process; such 

exceptions — which a modern reader would consider counter examples — shed inter- 

esting light on the role and use of concepts in mathematics in the early nineteenth 

century. 

1.3 Themes from early nineteenth-century mathematics 

The early nineteenth century marks a period of transition and fermentation in mathe- 

matics which involves most layers of the discipline, external as well as internal. With 

the boundaries fixed, say, between 1790 and 1840, a definite change in the way mathe- 

matics was performed and presented is evident; research mathematicians began work- 

ing in institutions set up for instruction in mathematics and started presenting their 

results in professional periodicals with substantial circulation. However, the change 

even effected the internal core of the discipline: how mathematics was done, what 

mathematics was, and which mathematical questions were interesting. Gradually, 

concepts and relations between concepts took an increasingly central position in math- 

ematics research; although the concern for concrete objects never ceased completely.


Concept based mathematics. Concepts such as function, continuity of functions, irre- 

ducibility of equations, and convergence of series attained central importance in math- 

ematical research in the transitional period. CAUCHY’S contribution to the rigoriza- 

tion of the calculus laid as much in applying technical definitions of concepts to prove 

theorems as with providing the definitions, themselves. Generalization in the 1820s 

turned the attention from specific objects to classes of objects, which were then in- 

vestigated. This shift of attention toward collections of individual objects had a very 

direct influence on the style of presenting mathematical research. In the ‘old’ tradition, 

mathematical papers could easily be concerned with explicit derivations (calculations) 

pertaining to single mathematical objects. Although this presentational style far from 

ceased to fill periodicals, a less explicit style gained impetus in the first half of the 

nineteenth century. By deriving properties of classes instead of individual objects, 

the arguments became more abstract and often more comprehensible by lowering the 

load of calculations and simplifying the mathematical notation. The transition is ev- 

ident in ABEL’S works which show deep traces of the calculation based approach to 

doing mathematics as well as being markedly conceptual at times; his 1826 paper on 

the binomial theorem is a fascinating mixture of both approaches. 

Abstract definitions and coming to know mathematical objects. In many develop- 

ing fields of mathematics in the early nineteenth century, new concepts were specified 

by the use of abstract definitions based on previous proofs, intentions, and intuitions. 

In the approach which I term concept based mathematics, the concepts were defined in 

the modern sense that there is nothing more to a concept than its definition. However, 

when abstract definitions determine the extent of a concept, representations and de- 

marcation criteria are required in order to get hold of properties of objects, and this 

quest for understanding, coming to know, the objects is an important aspect of early 

nineteenth century mathematics. In many ways, analogies may be drawn to the ef- 

fort of coming to know geometrical objects, e.g. curves, in the seventeenth century. To 

mathematicians of the seventeenth century, a curve meant more than any single given 

piece of information. In particular, an equation (or a method of constructing any num- 

ber of points on the curve) was not considered sufficient to accept the curve as known. 

Similarly, in the nineteenth century, knowledge of an elliptic function meant more 

than just a formal definition and included various representations, basic properties, 

and even tabulation of values. 

The question of coming to know a mathematical object relates to the problem of 

accepting the object as solutions to problems. The reduction of properties of curves to 

questions pertaining those basic curves which were considered well known was im- 

portant in the seventeenth century. However, certain properties were not expressible 

in basic curves (or functions) but required higher transcendentals such as elliptic inte- 

grals. Thus, much of EULER’S research on elliptic integrals in the eighteenth century 

can be seen as an effort to make these integrals basic in the sense of acceptable so-

1.4. Reflections on methodology 9 

lutions to problems. This research program was continued and reformulated in the 

nineteenth century during which the foundations, definitions, and framework of el- 

liptic functions underwent repeated revolutions. 

Critical revision. The critical mode of thought, rooted in the Enlightenment, had a 

profound impact on mathematics. Together with the demand for wider instruction in 

mathematics, the critical attitude brought about a deeply sceptical reading of the mas- 

ters which focused on the foundations. In geometry, some mathematicians began to 

believe in the possibility of a non-Euclidean version, and in analysis, the long-standing 

problem of the foundation of the calculus was made an important mathematical research 

topic. 2 

CAUCHY’S definition of the central concept of limits was itself a novelty, but of 

equal importance was the outlook for a concept based version of the calculus. CAUCHY’S 

new foundation for the calculus was arithmetical and introduced the arithmetical con- 

cept of equality. In the wake of the change of foundations of the calculus, certain 

objects and methods could no longer be allowed into analysis, and it became a quest 

to prop up parts of the mathematical complex recently made insecure. In particular, 

CAUCHY had to abolish from analysis all divergent series which had formerly been 

interpreted by a formal concept of equality. However, divergent series had provided 

new insights to mathematicians which they were reluctant to abandon and it became 

a legitimate, albeit difficult, mathematical problem to investigate how problematic 

or outright unjust procedures had led to correct results. Resolution of this problem 

laid in further specification of concepts involved and a heightened awareness of the 

procedures employed in arguments. For instance, by the mid-nineteenth century, the 

unreflective interchange of orders of limit processes had been identified as problem- 

atic and concepts such as absolute and uniform convergence had been introduced and 

put to use in theorems and proofs. 

1.4 Reflections on methodology 

The present study aims at illustrating important conceptual developments in math- 

ematics which took place in the first decades of the nineteenth century. In order to 

introduce a focus on the many-faceted aspects of these developments, the mathemat- 

ical production of ABEL has been taken as a starting point. In the present section, 

some of the methodological choices and considerations involved in the project are 

briefly discussed. It is not my ambition to present a coherent theoretical framework 

for historical enquiries but rather to make some considerations explicit and open for 

discussion. 

2 See e.g. (Grabiner, 1981b).


1.4.1 Diachronic descriptions 

It is within the framework of mathematics developed by EULER and cultivated in the 

eighteenth century that ABEL’S production is rightfully seen. Although his mathe- 

matics has been perceived as a very important step forward in a linear development, 

ABEL’S mathematical ideas were rooted in the previous attitude and style; and many 

of the famed new trends are only barely recognizable in his work. Therefore, anachro- 

nisms and teleological conceptions have to be dismissed in favor of a diachronic, more 

hermeneutic approach. In the process of tracing and describing this historical evolu- 

tion of mathematical content, it is of the utmost importance that ABEL’S works be 

studied within their contemporary framework — their mathematical context. 

Each of the main theories outlined above with which ABEL was involved is re- 

viewed with the purpose of illustrating how they were effected by currents of mathe- 

matical change in the early nineteenth century. To do so, ABEL’S mathematics is pre- 

sented and discussed based on a contextualized reading which emphasizes ABEL’S 

own methods and tools. To place these in their proper historical contexts, the theories 

and results will be traced back into the eighteenth century in search of the inspirations 

and their progressions in the nineteenth century will be followed. In the theory of 

equations, for instance, the works of EULER on the fundamental theorem of algebra 

will be briefly introduced; more emphasis will be given to the works of LAGRANGE 

and GAUSS which served as the direct inspirations for ABEL and most of his contem- 

poraries in the field. Then, in the nineteenth century, special emphasis is given to 

those works which share their inspirations with the works of ABEL, in this case the 

works of RUFFINI and GALOIS. For each disciplinary theme, a sketch of the further 

development after the initial decades of the nineteenth century is then given in or- 

der to illustrate how the ideas and currents which were barely discernible in the first 

decades came to play very important roles in the conceptions of mathematicians. The 

descriptions of the ensuing histories also serve to illustrate how ABEL’S works were 

valued and received by the following generations. 

Besides, it has been a secondary aim of the present study to make ABEL’S authentic 

mathematical thought available to the mathematically trained reader who is not famil- 

iar with early nineteenth century technicalities. In order to understand and evaluate 

ABEL’S role in the formation of modern mathematics, this presentation will always 

favor the original source over any modern approach. 

The setting of ABEL’S mathematics within the general view of mathematics ex- 

pressed by EULER will also be manifest in another way as a chronological mark. It has 

been necessary to trace many of the ideas and methods of ABEL’S mathematics back 

to the middle of the eighteenth century, but they might be even older. However, as 

this is a work on ABEL’S mathematics, such hypothesis will rarely be made explicit 

and EULER will be attributed things, which he did do — perhaps not as the first.


1.4.2 Philosophical theories and their applicability 

Philosophical theories enter the framework of the present study only rather implicitly. 

Written with the utopian goal of being an “account of things which happened”, the 

outlines of a certain perception of concepts such as change and transformation is never- 

theless discernible. The internal (and external) structure of scientific change has been 

subjected to many philosophical investigations over the past decades. In the present 

context, however, two of the founding theories have served as inspirations; those of 

Kuhnian paradigms and revolutions developed for the sciences in general and those of 

Lakatosian dialectics which were developed explicitly for mathematics and illustrated 

by examples from the nineteenth century. These two philosophical positions are so 

well established within the history of science community that only a brief presenta- 

tion is given with an emphasis on their applicability to the present study. 

T. S. KUHN (1922–1996), paradigms, crises, and revolutions. In his very influential 

monograph The structure of scientific revolutions of 1962, 3 KUHN advocated an explana- 

tion of the dynamics of scientific change. The total mental and physical entourage of 

a science at a given time was encompassed in the notion of paradigms. Paradigms are 

abruptly replaced through revolutions which are the responses to crises brought about 

by a compilation of anomalies inexplicable within the ruling paradigm. Once a revolu- 

tion has taken place, a new paradigm is introduced and communication between two 

distinct paradigms (e.g. over which paradigm to prefer) becomes irrational or extra- 

scientific. KUHN’S original model — although deeper than the present description — 

was a simplistic one which was amended and extended by numerous studies follow- 

ing its publication. Here, on the other hand, it will not be taken to serve as a complete 

model but rather as inspiration and terminological framework used to capture impor- 

tant aspects. 

As a model of mathematical change in the early nineteenth century, the Kuhnian 

system offers some obvious advantages; the important position given to anomalies in 

bringing about crises and revolutions was further extended in the works of I. LAKA- 

TOS (1922–1974) (see below). However, as has been emphasized by many philosophers 

and historians of mathematics, no overthrow of knowledge seems to occur in mathematics; 

thus no truly Kuhnian revolutions seem possible in the mathematical realm. 4 

The remaining notions of the Kuhnian conceptual framework such as paradigms, 

anomalies and crises are, however, applicable and useful in the description and anal- 

ysis of mathematical change, even if KUHN’S dynamics are not always appropriate. 

In the first sections of part II, for instance, it is illustrated how a mathematical theory 

came into being by a change of focus (a paradigmatic change) which shifted emphasis 

to questions of solubility. The theme will recur even more distinctly in part III which 

3 (Kuhn, 1962). 

4 See (Gillies, 1992).


documents ABEL’S role and position in the most Kuhnian of changes in mathematics 

during the early nineteenth century: the complete reformulation of analysis according 

to CAUCHY’S new program of arithmetical rigor. 

LAKATOS and the extension of concepts. Further philosophical inspiration is taken 

from LAKATOS’ Proofs and Refutations published as a series of articles in 1963–64 and as 

a book in 1976. 5 LAKATOS described the dynamics of mathematical change in terms of 

a dialectic between proofs and counter examples by means of proof revisions. In the main 

part of the Proofs and Refutations, LAKATOS explained his theory by exhibiting a ratio- 

nally reconstructed development of the Eulerian polyhedral formula; in appendices, 

he further illustrated the theory by exhibiting applications to other concepts including 

the development of the concept of uniform convergence (see part III). 

LAKATOS saw the process of proof as central to the mathematical endeavor. Incor- 

porating into mathematics a version of K. R. POPPER’S (1902–1994) falsificationism, 

LAKATOS described mathematical change as a continued revision of proofs to reflect 

objections raised by counter examples. LAKATOS classified counter examples as either 

local (refuting only part of a proof, but not the overall statement) or global (refuting 

the overall statement, but not necessarily any identifiable part of the proof). 

Counter examples could, in LAKATOS’ description, be constructed from existing 

proofs by a process of concept stretching by which a partially defined concept was rede- 

fined in an extended version which — although possibly more precise — encompassed 

instances (objects) not covered by the previous — often more intuitive — version of the 

concept. 

In response to such falsifications (refutations) by counter examples, LAKATOS sug- 

gested various strategies for refining the proofs. A naive approach would try to ex- 

plain the counter examples away, either by arguing that they were too pathological 

to be taken seriously or (more interestingly) by restricting the theorem to a narrower 

domain for which it was believed to surely valid; the latter approach was named ex- 

ception barring by LAKATOS. A more fruitful response to the refutation by counter 

examples — and the one which LAKATOS’ philosophy dogmatized — was the method 

of proof analysis which took the counter examples more seriously. By carefully ana- 

lyzing the counter example and the proof which it refuted, proof analysis produced a 

new proof in which a refuted lemma was replaced by an unrefuted one which might 

cause an alteration of the overall statement. Thus, LAKATOS suggested, theorems were 

produced which had very explicit assumptions and were very hard to refute. 

Just as was the case with the Kuhnian model, LAKATOS’ model — in all its general- 

ity — is often found inadequate to describe the actual historical development of math- 

ematics. On the other hand, LAKATOS’ model offers some further concepts which 

often ease the description and analysis of past events. Most importantly, LAKATOS’ 

5 (Lakatos, 1976). A good description of LAKATOS’ life and philosophy can be found in (Larvor, 1998).


description of counter examples and the role that they play in mathematical change 

elaborates the role played by anomalies in the Kuhnian model and suggests a more 

refined view on the status of a mathematical theory in crisis. 

The Lakatosian theory of mathematical evolution is present as background through- 

out; it will surface sporadically in parts II–IV and become important again in the final, 

more analytical part V. 

M. EPPLE’S epistemic configurations. Quite recently, EPPLE has suggested the no- 

tion of epistemic configurations in order to be able to discuss change in mathematics 

in another context. 6 In EPPLE’S analysis, epistemic configurations consist of epistemic 

objects and epistemic techniques and are manipulated in mathematical workshops. The 

concept of epistemic objects encompasses the immaterial objects with which mathe- 

matics deals. These are manipulated and investigated by a number of methods of 

producing (or obtaining) mathematical knowledge; these methods are the epistemic 

techniques. The precise applicability and range of EPPLE’S concepts and their use- 

fulness in historical analysis is not the primary objective here. Instead, as with the 

inspirations of KUHN and LAKATOS, I have taken the liberty of using EPPLE’S terms 

to ease the analysis and discussion of what I believe to be a fundamental change in 

mathematics in the early nineteenth century: the change from formula based to concept 

based mathematics which is addressed in chapter 21. 

1.4.3 Existing literature 

Being one of the important mathematicians of the nineteenth century, ABEL’S person 

and his mathematics have been subjected to study for a multitude of different rea- 

sons. A few general trends of the literature on ABEL can profitably be identified at 

this point. 7 At the relevant places in the subsequent parts, references are given to the 

secondary literature which is listed in the bibliography. 

ABEL in the history of mathematics literature. In the professional literature in the 

history of mathematics, ABEL is often mentioned in order to illustrate one or more of 

the following aspects: 

1. ABEL’S life story is invoked to illustrate the conditions of young mathematicians 

two centuries ago. This aspect is closely related to the biographies treated below. 

2. ABEL’S letters from Paris are used to illuminate how the confrontation with 

CAUCHY’S new rigor brought about a radical change. For instance, U. BOTTAZZ- 

INI (⋆1947) quotes in extenso from these letters in his comprehensive account of 

the evolution of analysis in the nineteenth century. 8 

6 (Epple, 2000). 

7 For a thematic listing of the ABEL literature, see also (Sørensen, 2002). 

8 (Bottazzini, 1986).


3. The modern highlights of ABEL’S production, e.g. the binomial theorem or the 

insolubility of the quintic, are described to shed some light on the evolution of 

the theories and the involved concepts. 

4. ABEL’S mathematics is described per se in order to give a presentation of his 

production. Very good examples include articles by P. L. M. SYLOW (1832–1918) 

in the ABEL centennial memorial volume and the second edition of the collected 

works. 9 

The present study incorporates all these approaches to give a comprehensive overview 

of ABEL’S mathematical production as well as positioning it within a broader frame 

describing themes of mathematical change in the period. 

Two other types of studies treating the life and works of ABEL delineate them- 

selves: biographies and interpretations. 

Biographies — scientific or not. As should become clear in the next chapter, ABEL’S 

biography includes all the components of a truly romantic biography of a misun- 

derstood genius who rose from the dust to become a nobility of mathematics. Such 

biographies have been written; 10 but more interestingly, biographies have also been 

written which serve a purpose of their own. The first biographies appeared as obituar- 

ies written by ABEL’S friends soon after his death. Of primary importance in describ- 

ing ABEL’S mathematics are the obituaries written by HOLMBOE and CRELLE which 

include first hand descriptions of ABEL’S mathematical work. 11 Although a larger 

number of biographies could be listed, the most widely circulated and very well re- 

searched twentieth century biography was written by Ø. ORE (1899–1968); 12 it has 

been used mainly to help set the chronology straight. The human and cultural as- 

pects of ABEL’S life has most recently been very carefully researched and described 

by A. STUBHAUG (⋆1948) who meticulously sets the cultural scene of early nineteenth 

century Norway and Europe. 13 STUBHAUG’S biography has relieved me of any obli- 

gation to produce biographical news concerning ABEL’S person; the biography which 

is provided in chapter 2 serves merely to set the framework of the subsequent chap- 

ters. It is my hope that the present study of ABEL’S mathematics will complement 

STUBHAUG’S book on his life and environment to produce a picture of ABEL’S person 

and his mathematics. 

Renderings of ABEL’S work in modern theories. By the very nature of mathemat- 

ics, mathematical knowledge seems to accumulate and only change its presentational 

9 (N. H. Abel, 1881; L. Sylow, 1902). 

10 E.g. (Bell, 1953). 

11 (A. L. Crelle, 1829b; Holmboe, 1829). 

12 (Ore, 1954; Ore, 1957). 

13 (Stubhaug, 1996; Stubhaug, 2000).


form or its internal relations within mathematical structures. For this reason, mathe- 

maticians often hope to find inspiration in the works of their predecessors. Frequently, 

this leads to the publication of modernized versions of historical proofs. By itself, this 

practice is very good as long as the author and the community recognize that it is pre- 

cisely a revisited proof or theorem and precisely not a diachronic description of that 

proof or theorem within its contemporary structure. 

Such revisits to ABEL’S production are most frequently made to his theory of alge- 

braic solubility of equations, more precisely to his proof of the insolubility of the quin- 

tic equation. 14 ABEL’S other main contributions attract less attention; the binomial 

theorem because it has become an integral part of basic mathematical knowledge, and 

the Abelian Theorem (see part IV) because its original form has been surpassed and the 

result has been recast in a different theory. 

As should now be clear, the methodology of the present approach can be summa- 

rized thus: A diachronic reading of the original sources of ABEL’S mathematics with 

the purpose of analyzing themes of mathematical change in the early nineteenth cen- 

tury, in particular the rise of concept based mathematics. 

14 See e.g. (R. Ayoub, 1982; Radloff, 1998).

Chapter 2 

Biography of NIELS HENRIK ABEL 

The life of NIELS HENRIK ABEL (1802–1829) was not always a happy one. Born in 

a time of national upheaval and into a family with few provisions against the hard 

times, his actions were always restricted by pecuniary concerns. A melancholic, he 

preferred to be surrounded by people, but due to his shy and modest nature he felt se- 

cure only with a score of friends including his elder brother, his sister, his mathematics 

teacher B. M. HOLMBOE (1795–1850), Mrs. C. A. B. HANSTEEN (1787–1840) — ABEL’S 

benefactor and the wife of his university professor, and his mentor in Berlin A. L. 

CRELLE (1780–1855). ABEL fell in love with C. KEMP (1804–1862) in 1823 and they 

were engaged the following year. Unfortunately, ABEL’S position never became se- 

cure enough for them to marry. The last years of ABEL’S life were spent in uncertainty 

with hopes of a more stable future either at home or abroad. When he died, ABEL’S 

mathematical star was still rising, and years would pass before the world knew exactly 

how bright it had been. 

The short and yet very creative life of ABEL has caught the interest of many bi- 

ographers. Confined within the romantic period and exhibiting distinctly romantic 

features itself, the biographers have often focused on ABEL’S poverty and contempo- 

rary lack of acknowledgment; 1 both features frequently found in the romanticization 

of mathematicians and scientists. Another genre of biography has constructed and 

researched a controversy with C. G. J. JACOBI (1804–1851) over the priority of the 

inversion of elliptic integrals. 2 The most recent and excellent biography written by 

A. STUBHAUG (⋆1948) has taken a different angle, describing and bringing to life the 

cultural and political context in which ABEL lived and which is so important for Norwegian 

self-image. 3 

As STUBHAUG’S book is such a convincing description of the cultural and bio- 

graphical background of ABEL’S life, the present biography serves only to provide a 

self-contained presentation of the temporal framework in which ABEL’S mathemati- 

1 See e.g. (Bell, 1953) or, more soberly, (Ore, 1950; Ore, 1954; Ore, 1957). 

2 E.g. (Bjerknes, 1880; Bjerknes, 1885; Bjerknes, 1930). This debate was also the subject of (Koenigsberger, 

1879). 

3 (Stubhaug, 1996), translated into English in (Stubhaug, 2000). 

17

18 Chapter 2. Biography of NIELS HENRIK ABEL 

Figure 2.1: NIELS HENRIK ABEL (1802–1829) 

cal production was localized. All facts presented here have been taken from existing 

literature, primarily HOLMBOE’S obituary, C. A. BJERKNES’ (1825–1903) biographies, 

and STUBHAUG’S contextual biographical study. 4 References to these works will not 

always be made explicit. I will deliberately desist from giving detailed analyses or 

speculations concerning the personality and private life of ABEL except where sup- 

ported by the biographies and ABEL’S correspondence. 

2.1 Childhood and education 

ABEL was born as the second son into an incumbent’s family on 5 August 1802. 

ABEL’S father, S. G. ABEL (1772–1820), himself the son of a minister, had been ed- 

ucated in Copenhagen and received a call to the rural parish of Finnøy in 1800 (see 

figure 2.2). 

Also in 1800, SØREN GEORG married A. M. SIMONSEN (1781–1846), the daughter 

of a wealthy merchant, and together they had six children; five boys and a girl. In 

1804, SØREN GEORG took over the more lucrative parish of Gjerstad from his father 

who had died the year before. Nevertheless, due to the family increase, the costs of 

educating the children, a nationalist sentiment to contribute to the founding of the 

university, and the troubled times for the nation, the ABEL-family remained without 

4 (Holmboe, 1829), (Bjerknes, 1880; Bjerknes, 1885; Bjerknes, 1930), (Stubhaug, 1996).

2.1. Childhood and education 19 

Figure 2.2: The southern part of Norway with Christiania and Finnøy marked. See 

also (Stubhaug, 2000, 136). 

fortune — a situation which only deteriorated after SØREN GEORG died in 1820 leav- 

ing his widow a thirty year commitment of financial donations to the university in 

Christiania. 

During his years in Copenhagen, SØREN GEORG had been influenced by the re- 

formist education and he took care of the primary education of his sons himself. Ac- 

cordingly, the focus was on the catechism and skills in reading, writing, and basic 

arithmetic. In 1815, when NIELS HENRIK was 13, he and his elder brother H. M. ABEL 

(1800–1842) were sent to the Cathedral School in Christiania. There, they were taught 

classical and modern languages, as well as arithmetic and geometry. ABEL passed his 

examen artium in 1821 with first grades in the mathematical disciplines, second grade 

in French, and only third grades in German, Latin, and Greek. 5 In the lower classes, 

ABEL demonstrated no particular affinity for the mathematical disciplines but was a 

fair student in all subjects. However, this situation dramatically changed to the better 

due to an unfortunate event which took place in 1817–18. 

In November of 1817, H. P. BADER (1790–1819), ABEL’S mathematics teacher, phys- 

ically molested one of his pupils who later died from this hands-on approach to math- 

ematics education. BADER, who had a record of hot temper and violent teaching 

methods, was excused from his teaching obligations, and HOLMBOE stepped in as 

5 (N. H. Abel, 1902d, 3)


Figure 2.3: BERNT MICHAEL HOLMBOE (1795–1850) 

a replacement in 1818. 6 HOLMBOE, who was ABEL’S senior by only 7 years, had been 

educated at the cathedral school himself and had attended mathematics courses un- 

der S. RASMUSSEN (1768–1850) at the university. HOLMBOE soon noticed a special 

talent for mathematics in ABEL and they began reading extra-curricular mathematics 

together. 

2.2 “Study the masters” 

Soon, HOLMBOE began to realize what mathematical talent he had at hand and by 

the autumn of 1818, HOLMBOE urged ABEL to study the important works in mathe- 

matics on his own. In one of his notebooks, ABEL pursued calculations inspired by 

P.-S., MARQUIS DE LAPLACE’S (1749–1827) use of generating functions. Between all 

the calculations, he noted in the margin: 

“If one wants to know what one should do to obtain a result in more conformity 

with Nature one should consult the works of the famous Laplace where this 

theory is exposed with the most clarity and to an extent in accordance with the 

importance of the subject. It is also easy to see that a theory written by M. Laplace 

must be much superior to any other written by less bright mathematicians. By the 

way it seems to me that if one wants to progress in mathematics one should study 

the masters and not the pupils.” 7 

6 For details, see (Stubhaug, 1996, 172–174).

2.2. “Study the masters” 21 

HOLMBOE’S list of the masters whom they studied together sheds interesting light 

on the mathematical literature of the early nineteenth century as seen from the pe- 

riphery. 8 In the eyes of the two Norwegians, the two most influential writers were 

L. EULER (1707–1783) and J. L. LAGRANGE (1736–1813); but the list also included 

S. F. LACROIX (1765–1843), L. B. FRANCOEUR (1773–1849), S.-D. POISSON (1781– 

1840), C. F. GAUSS (1777–1855), and J. G. GARNIER (1766–1840). In the following 

paragraphs, the influences from these authors are outlined. 

EULER. EULER and HOLMBOE studied algebra and calculus from EULER’S works; 

ABEL’S first independent adventures into creative mathematics were greatly inspired 

by the great Swiss mathematician. Although the sources are not very explicit, it is be- 

yond doubts that ABEL studied EULER’S Introductio in analysin infinitorum [Introduc- 

tion to the infinite analysis] of 1748. 9 To what extent ABEL also knew of EULER’S other 

publications including his papers in the transactions of the St. Petersburg Academy 

is left for speculation; we shall return to the question when we see examples of EU- 

LER’S — possibly indirect — influence on ABEL in parts II and IV. 

The big four. LAGRANGE, LACROIX, POISSON, and GAUSS all belong to the heavy- 

weight division of mathematics in the late eighteenth century with massive and im- 

portant works on the calculus and algebra. Although a writer of very influential text- 

books on the calculus, 10 LAGRANGE mainly inspired ABEL through his work on the 

theory of equations which redefined the viewpoint from which this theory was to be 

attacked. 11 LACROIX’ effort laid more in organization and presentation than in cre- 

ative research; his three volume textbook on the calculus, Traité de calcul différentiel et 

intégral [Treatise on differential and integral calculus], ran multiple editions beginning 

in 1797–1800. 12 In the Traité, LACROIX presented an survey of the calculus based on 

the research of his contemporaries and picking up a variety of approaches and foun- 

dations from different authors. 

The lesser souls. The two authors in HOLMBOE’S list who today are lesser known, 

FRANCOEUR and GARNIER, both wrote textbooks on mathematics which found wide 

circulation toward the end of the eighteenth century. ABEL almost certainly studied 

7 “Si l’on veut savoir comment on doit faire pour parvenir à un resultat plus conforme à la nature il 

faut consulter l’ouvrage du celebre Laplace où cette theorie est exposée avec la plus grande clarté 

et dans une extension convenable à l’importance de la matière. Il est en outre aisé de voir que une 

theorie ecrite par M. Laplace doit être bien superieure à toute autre donnée des geometres d’une 

claire inferieure. Au reste il me parait que si l’on veut faire des progres dans les mathématiques il 

faut étudier les maitres et non pas les écoliers.” (Abel, MS:351:A, 79, marginal note). 

8 (Holmboe, 1829, 335). 

9 (L. Euler, 1748). 

10 E.g. (Lagrange, 1813). 

11 (Lagrange, 1770–1771). 

12 (Lacroix, 1797; Lacroix, 1798; Lacroix, 1800).


FRANCOEUR’S Cours complet de mathematiques pures [Complete course on pure mathe- 

matics], 13 which in two volumes dedicated to the emperor of Russia, introduced arith- 

metic, geometry, algebra, and differential and integral calculus. The textbook had been 

translated into German by one of ABEL’S mentors, C. F. DEGEN (1766–1825), 14 but 

since ABEL only came to master German during his tour 1825–27, he probably studied 

the French original. 

Besides writing textbooks on algebraic analysis (LAGRANGE’S approach to the cal- 

culus), in 1807, GARNIER translated EULER’S Vollständige Einleitung zur Algebra [Com- 

plete introduction to algebra] of 1770 into French. 15 With his limited knowledge 

of German, it is doubtful whether ABEL read EULER in the original language, but 

through the translations by LAGRANGE or GARNIER or even through FRANCOEUR’S 

complete course on pure mathematics, ABEL became acquainted with the elementary 

parts of contemporary mathematical knowledge, in particular the solution of cubic 

and bi-quadratic equations. 

2.2.1 An alleged solution formula 

While still in grammar school, ABEL approached one of the most prestigious prob- 

lems of contemporary mathematics: the search for an algebraic formula expressing 

the solution of the quintic equation. Since the Western Renaissance, similar formulae 

for equations of the first four degrees had been known. In 1821, ABEL believed to 

have found a closed algebraic expression solving the next case: the general quintic. 

He wrote down his result and showed it to his teacher HOLMBOE, who took it to C. 

HANSTEEN (1784–1873), one of the two professors in science at the Christiania Univer- 

sity. 16 HANSTEEN, who together with HOLMBOE were among the few people in Nor- 

way competent enough to have a chance of understanding ABEL’S tedious argument, 

took it to the University’s collegium academicum. The collegium took note of ABEL’S ar- 

gument and wanted to make it public to a broader mathematical audience. However, 

as the young Norwegian state was itself without means of such a publication with a 

wide circulation, ABEL’S paper was sent to professor DEGEN in Copenhagen with the 

hope that it be published in the transactions of Royal Danish Academy of Sciences and 

Letters. DEGEN’S assessment proved to have a profound influence on ABEL’S career. 

Upon reception of the paper, DEGEN scrutinized ABEL’S solution, and DEGEN’S re- 

sponse to HANSTEEN is the only existing written source of ABEL’S adventure. There, 

DEGEN requested an elaboration, a numerical example, and a rewriting of the manu- 

script for the other members of Videnskabernes Selskab to be able to read it. To ABEL, 

the refusal to immediately publish his result must have been disappointing. However, 

13 (L.-B. Francœur, 1809). 

14 (L.-B. Francœur, 1815). In 1839, it was again translated into German by KÜLP. 

15 It has been translated into English as (L. Euler, 1972). 

16 Very unfortunately, ABEL’S supposed solution has not survived and only speculative reconstructions 

can be suggested.


as he sat down to provide the details, he must have realized that DEGEN had spared 

him a humiliating entry onto the mathematical scene. Before 1824, ABEL realized that 

no algebraic solution formula could be found for the general quintic, and thus that his 

solution had been flawed and his search in vain. ABEL never sent an elaboration to 

DEGEN, never published in the Transactions of the Royal Danish Academy of Science, and 

when ABEL and DEGEN eventually met in person two years after their initial corre- 

spondence, ABEL had other things on his mind. 

2.2.2 A student at the young university 

When ABEL enrolled at the university in 1821, the university was still in its constitu- 

tional phase. Founded in 1811 and opened in 1813 as only the third university in the 

twin monarchy (after Copenhagen and Kiel), the Christiania university initially only 

offered degrees in theology, law, medicine, and philosophy. The study of science and 

mathematics was subsumed under the philosophical faculty and no course of studies 

led to any degree in the sciences. Thus, when ABEL enrolled, his determination to 

study mathematics defied the existing structure of academic qualification. He must 

have hoped that his extraordinary talents alone would be enough to secure him a fu- 

ture in academia. 

During his years at the university, ABEL attended lectures by the two professors in 

mathematics and astronomy, RASMUSSEN and HANSTEEN. The mathematical lectures 

were primarily on elementary mathematics, spherical geometry, and applications to 

astronomy, and ABEL had soon learned all he could from these courses. As a comple- 

ment, he continued studying the works of the masters of mathematics. In 1823, ABEL 

came across the Disquisitiones arithmeticae of GAUSS, 17 which provided him with a rich 

source of inspiration and problems for his own research. Itself an immensely impor- 

tant work in the theory of numbers, the Disquisitiones arithmeticae influenced ABEL in 

two other fields: the theory of equations and the rectification of the lemniscate. 

During his years as a student, ABEL held a free room and board at the Regentsen, 

a student residence for the most needy students. Until he was given a stipend from 

the State in 1824, he was financially supported by some of the University professors, 

including RASMUSSEN and HANSTEEN. 18 

First publications in Magazin for Naturvidenskaberne. In 1823, professor HAN- 

STEEN, together with two fellow professors at the university, tried to amend the lack of 

Norwegian periodicals in natural science with the creation of the Magazin for Naturv- 

idenskaberne [Magazine for the natural sciences]. Its aim was to convey Norwegian 

research in the sciences to the educated lay audience and provide an emerging group 

of young scientists with a forum for publication. 

17 (C. F. Gauss, 1801). 

18 (Stubhaug, 1996, 244–245).


In the first issue of the Magazin, the 21 year old ABEL had his first publication. 

Inspired by EULER’S Institutiones calculi differentialis and A.-M. LEGENDRE’S (1752– 

1833) Exercises de calcul integral, ABEL solved two problems in the integral calculus. 19 

In the same year, a second publication by ABEL dealing with the theory of elimina- 

tion was published in the Magazin. 20 On this occasion, HANSTEEN found it necessary 

to add an introduction in which he — summoning G. GALILEI (1564–1642), I. NEW- 

TON (1642–1727), and C. HUYGENS (1629–1695) — argued that mathematics even in 

its purest form was within the scope of the magazine devoted to the natural sciences. 

Taking into account the limited circulation of the Magazin and HANSTEEN’S efforts to 

make ABEL’S work acceptable to the audience, it is doubtful how much ABEL gained 

from these publications. But to a young man — still nothing but a studiosus — getting 

his name on printed paper must have been a great satisfaction. 

When ABEL’S publications in the Magazin were first noticed, it was for all the 

wrong reasons. In 1824, ABEL published some computations pertaining to the influ- 

ence of the Moon on the movement of a pendulum. 21 This problem fitted nicely into a 

research project concerning the magnetic field of the Earth with which HANSTEEN was 

immensely involved. Upon HANSTEEN’S request, the paper was sent to H. C. SCHU- 

MACHER (1784–1873) in Altona, who edited the journal Astronomische Nachrichten [As- 

tronomical intelligencer], for possible republication therein. However, SCHUMACHER 

realized that ABEL had made a computational error which had led him to estimate the 

influence of the Moon to be ten times stronger than was rightfully supported. SCHU- 

MACHER refused to publish the paper, and a correction was subsequently inserted in 

the Magazin. 22 

2.2.3 Visiting Copenhagen 

In 1823, ABEL for the first time left Norway to go on his first educational tour. Sup- 

ported privately by professor RASMUSSEN, ABEL traveled to Copenhagen to visit and 

discuss with the mathematicians there. 

Mathematics in the capital. The mathematical milieu in Copenhagen was not com- 

pletely different from the one in Christiania. 23 In 1823, the mathematical profession 

was centered around the university and the academy of science; six years later, Den 

Polytekniske Læreanstalt [The polytechnic college] 24 was opened. Despite the university 

19 (N. H. Abel, 1823) 

20 (ibid.) 

21 (N. H. Abel, 1824c). 

22 (N. H. Abel, 1824a). 

23 The mathematical milieu in Copenhagen in the first half of the nineteenth century has been the subject 

of a subsequent study by the author in the Danish History of Science project undertaken at the 

History of Science Department at the University of Aarhus. The interested reader may also wish to 

consult the dissertation (in Danish) by KURT RAMSKOV (Ramskov, 1995, chapter 1) or STUBHAUG’S 

book (Stubhaug, 2000, chapter 31) for information. 

24 Today Danmarks Tekniske Universitet [Technical University of Denmark].


Figure 2.4: CARL FERDINAND DEGEN (1766–1825) 

in Copenhagen being older and larger than the one in Christiania, the faculty of phi- 

losophy had only two professorships in mathematics (and one in astronomy) which 

in 1823 were held by DEGEN and E. G. F. THUNE (1785–1829). 

Together, DEGEN and THUNE had meant a change of generations in the mathe- 

matical milieu at the university when they were appointed 1813 and 1815. 25 DEGEN, 

being the more creative of the two, had gained international reputation with a publi- 

cation of tables for the solution of Pellian equations in 1817. 26 To the mathematicians 

in Christiania — as to the general Norwegian public — Copenhagen was still to some 

extent considered the intellectual and cultural capital of the country even after the 

separation in 1814. Thus, by virtue of its former colonial power, the mathematical 

talents of DEGEN, and the circulation and status of the publications of the Royal Dan- 

ish Academy of Sciences and Letters, Copenhagen was the obvious choice for a first 

foreign trip for a young and promising Norwegian mathematician. 

The maturing mathematician. ABEL’S first two works in the sciences estimated im- 

portant enough to receive wider circulation by Norwegian scholars, the alleged solu- 

tion of the quintic and his computations concerning the magnetic field of the Earth, 

were both caught in the review system of the time. In the incipient scientific milieu in 

25 THUNE was originally appointed as professor of astronomy 1815 before he transferred to the professorship 

of mathematics 1819. 

26 (C. F. Degen, 1817)


Norway, the means of publication were limited and advanced knowledge of the sci- 

ences was confined to a few men. Therefore, foreign experts were invited to judge — 

and possibly publish — ABEL’S first papers. DEGEN made reservations concerning the 

completeness of ABEL’S research on the solution of the quintic and refused to present 

ABEL’S paper to the Danish Academy until he had seen the method applied to a nu- 

merical example. The numerical example was explicitly requested as a lapis lydius — 

a test of correctness — and it is not improbable that the confrontation with an explicit, 

difficult problem was what later led ABEL to realize his being in error. 

The HANSTEEN-DEGEN period. In his historical introduction to the centennial memo- 

rial volume, 27 E. B. HOLST (1849–1915) has emphasized the influence of HANSTEEN 

and DEGEN on ABEL’S research 1821–24. P. L. M. SYLOW’S (1832–1918) analysis seems 

applicable on at least two levels: topics and methods. On the topical level, the two pro- 

tagonists exerted contrary influences. Responding to HANSTEEN’S suggestion, ABEL 

had briefly worked on a physical problem; either because of this failed encounter or 

because of a personal inclination, he subsequently focused exclusively on working 

within pure mathematics. Following an advice of DEGEN, ABEL ventured into the 

theory of integration in the tradition of EULER and LEGENDRE. 28 Being an impor- 

tant theory of the eighteenth century left open for further developments, DEGEN had, 

himself, spent some time studying elliptic integrals. But there is no real evidence to 

suggest that DEGEN could have foreseen what his new disciple would do for the disci- 

pline. Although their interests differed, both HANSTEEN and DEGEN were trained in 

the typical eighteenth century mathematical literature including the men whom ABEL 

considered his masters at the time. Formal manipulations and physical applicability 

were considered positive aspects of the approaches of EULER and his mid-eighteenth 

century contemporaries. The DEGEN-HANSTEEN period marks the end of ABEL’S 

youthful encounters with the formal approach to analysis and at the same time marks 

the beginning of a period of intense study of the theory of higher transcendentals 

which would be ABEL’S masterpiece when judged by his contemporaries. 

2.3 The European tour 

After ABEL returned to Christiania from his first trip to Copenhagen, he soon real- 

ized that there was little more for him to gain while isolated in the limited Norwegian 

mathematical community. In 1824, ABEL applied with the support of the professors 

HANSTEEN and RASMUSSEN for a travel grant from the university. ABEL’S primary 

aim was to visit to the mathematical capital of his time, Paris. There, in Paris, math- 

ematics had been institutionalized and cultivated to the highest level in the wake of 

27 (Holst, 1902, 22). 

28 In part IV, the influence on ABEL of these mathematicians will be traced, documented, and analyzed.

2.3. The European tour 27 

the French Revolution. Later in the application procedure, Göttingen, the seat of the 

German champion of mathematics GAUSS was added to the travel plan. 

In his first period of creative mathematical production, 1823–24, while inspired 

by HANSTEEN and DEGEN (see above), ABEL devoted his attention to the theory of 

integration in the tradition of EULER’S Institutiones calculi integralis and LEGENDRE’S 

Traité de calcul integral. By 1824, ABEL had documented his “exceptional abilities in the 

mathematical sciences” 29 an example of which HANSTEEN presented to the collegium 

in the form of a manuscript. ABEL had hoped that — besides the travel grant — the 

collegium would support publication of the result which he believed had international 

importance and could serve as a door opener on his tour. However, the collegium 

decided to only support a travel grant for ABEL to go to the Continent. There was only 

one condition; the grant was only to begin in 1825; until then ABEL had to prepare by 

studying the “learned languages”, which in particular meant French. 30 

2.3.1 Objectives and plans 

In the application to the collegium, sources to the contemporary Norwegian rank- 

ing of the mathematical centres may be found. The choices were mainly made by 

ABEL’S benefactors, the mathematics professors HANSTEEN and RASMUSSEN. When 

they first proposed sending ABEL abroad they suggested that he should go to “the 

places abroad where the most distinguished mathematicians of our time are located, 

perhaps primarily to Paris”. 31 In ABEL’S official application to the King, he wrote: 

“After I have thus in this country by the use of the available tools sought to 

approach the erected goal, it would be very beneficial to me to acquaint myself, 

during a stay abroad at different universities in particular in Paris where so many 

distinguished mathematicians are located, with the newest creations in the science 

[mathematics] and enjoy the guidance of those men who in our time have brought 

it [mathematics] to such a remarkable height.” 32 

Only of July 4th 1825 did the idea of sending ABEL to Göttingen enter into the ap- 

plication when the collegium applied for the grant to be made effective. ABEL submit- 

ted a more detailed travel plan which RASMUSSEN was supposed to comment upon 

to the collegium; this plan is no longer extant. 33 Two months later, on September 7, 

ABEL embarked on his European tour. 

29 HANSTEEN’S opinion as expressed in the accommodating letter from the collegium academicum to 

the department of the church (N. H. Abel, 1902d, 7). 

30 (ibid., 12). 

31 (ibid., 7). 

32 “Efter at jeg saaledes her i Landet ved de her forhaanden værende Hjelpemidler har stræbet at 

nærme mig det foresatte Maal, vilde det være mig særdeles gavnligt ved et Ophold i Udlandet ved 

forskjellige Universiteter, især i Paris hvor saa mange i udmærkede Mathematikere findes, at blive 

bekjendt med de nyeste Frembringelser i Videnskaben og nyde de Mænds Veiledning som i vor 

Tidsalder have bragt den til en saa betydelig Høide.” (ibid., 13). 

33 (ibid., 20, footnote).


2.3.2 ABEL in Berlin 

Figure 2.5: AUGUST LEOPOLD CRELLE (1780–1855) 

After a brief stop to visit DEGEN in Copenhagen and another to visit SCHUMACHER in 

Hamburg, 34 ABEL’S first extended stay was in Berlin. He had not had any real plans 

of going to Berlin, but went there together with a group of friends, all of whom belong 

to the founding generation of Norwegian scientists, in particular in geology, and who 

were all on educational tours of Europe. ABEL wrote to HANSTEEN in his first report 

from the European tour where he obviously had to defend spending time in Berlin: 

“You may have wondered why I first traveled to Germany; I did so partly 

because I could thereby stay with friends and partly because I would be less likely 

not to make the most of my time since I can leave Germany at any time to go to 

Paris which should be the most important place for me.” 35 

In the same letter, ABEL described the acquaintance which he had made with one of 

the local mathematicians, the professional administrator Geheimrat [Privy Councilor] 

CRELLE whom he had been told about by the Copenhagen mathematician H. G. V. 

34 Part of ABEL’S obligation was to carry out experiments for HANSTEEN measuring the Earth’s magnetic 

field at different locations. 

35 “De har maaskee forundret Dem over hvorfor jeg først reiste til Tyskland; men dette gjorde jeg deels 

fordi jeg da kom til at leve sammen med Bekjendtere deels fordi jeg da var mindre udsat for ikke at 

anvende Tiden paa den bedste Maade, da jeg kan forlade Tyskland hvert Øjeblik det skal være for 

at reise til Paris, som bør være det vigtigste Sted for mig.” (Abel→Hansteen, Berlin, 1825/12/05. 

N. H. Abel, 1902a, 9–10).


SCHMIDTEN (1799–1831). Introducing himself in his stuttering German, ABEL called 

upon the busy Geheimrath soon after his arrival in Berlin. Initially, their conversation 

was staggering; only when CRELLE enquired what ABEL had already read, did he re- 

alize that he was facing a young man quite versed in the modern mathematics. ABEL 

presented CRELLE with a copy of his pamphlet on the insolubility of the quintic equa- 

tion, and CRELLE expressed his difficulty understanding the argument. In due time, 

ABEL would present CRELLE with an elaborated argument which would gain world 

wide circulation. 

“I am extremely pleased that I happened to go to Germany and in particular 

Berlin before I came to Paris; since — as you may have learned from my letter to 

Hansteen — I have made the splendid acquaintance with Geheimrath Crelle.” 36 

The founding of the Journal für die reine und angewandte Mathematik. Communi- 

cation of mathematics in the nineteenth century underwent rapid change. In the days 

of EULER, mathematics had been confined to professional amateurs and academicians 

who communicated their results either privately in correspondence, in monographs, 

or in the periodicals of the academies. Only in the beginning of the nineteenth century 

did professional, independent periodicals devoted to mathematics come into being; 

ABEL was instrumental in the creation of the first major German journal of mathemat- 

ics, which CRELLE founded in 1826. 

When ABEL first called upon CRELLE in Berlin, they discussed the relatively low 

status of mathematics in Germany (Prussia). When ABEL happened to mention his 

astonishment at the lack of German periodicals devoted to mathematics, he struck a 

nerve with CRELLE. For years, CRELLE had been engaged in an effort to promote 

mathematics in Prussia. In France, the first journal (Annales de mathématiques pures 

et appliquées) devoted entirely to mathematics had been initiated by J. D. GERGONNE 

(1771–1859) in 1810. 37 In 1822, CRELLE was forced to abandon plans for a German lan- 

guage journal of mathematics due to lack of contributors. 38 However, with the advent 

ABEL and other promising young mathematicians — all looking for a way of pub- 

lishing their results — the time was ripe for another attempt. Following an intensive 

and continuing campaign to secure funding and with substantial personal investment, 

CRELLE had the first volume of his Journal für die reine und angewandte Mathematik pub- 

lished in the spring of 1826. 

CRELLE’S initial idea for the Journal für die reine und angewandte Mathematik was to 

provide a broad German speaking audience with an instrument for presenting and 

keeping up to date with recent research in pure and applied mathematics — possibly 

36 “Overmaade vel fornøiet er jeg fordi jeg kom til at reise til Tyskland og navnligen til Berlin førend jeg 

kom til Paris; thi som Du maaskee har erfaret af mit Brev til Hansteen har jeg her gjort et fortræffeligt 

Bekjendtskab med Geiheimrath Crelle.” (Abel→Holmboe, 1826/01/16. ibid., 13). 

37 (Otero, 1997). 

38 (W. Eccarius, 1976, 233).


through translations. Soon, however, the attention devoted to applied mathematics 

declined and the Journal für die reine und angewandte Mathematik became the mouth- 

piece for a limited group of pure mathematicians. The change in CRELLE’S conception 

of his Journal für die reine und angewandte Mathematik is evident by comparing the in- 

troductions with which he prefaced the two first volumes. In the very first volume, 

CRELLE described the domain of the journal to include both pure mathematics (analy- 

sis, geometry, mechanics), and applied mathematics including optics, theories of heat, 

sound, and probability, and geography and geodesy. 39 This changed quickly, though, 

and in the introduction to the second volume, CRELLE stood down — both on nation- 

alistic and disciplinary ambitions. 

For the first volume, CRELLE took it upon himself to translate the French manu- 

scripts of ABEL and others into German before publication. 40 Despite having been 

taught German for four years at the Cathedral School (written German for two years), 41 

ABEL’S marks in written German were quite inconsistent: 1 and 4 on a scale from 1 

to 5 (1 best); for the Artium, he scored a 3. 42 ABEL was reluctant to write in that language. 

When he eventually prepared a paper in German, ABEL was very proud. 43 

However, CRELLE soon succumbed to the pressure of internationalizing his journal 

and accepted publishing papers in foreign languages. In response, after just a single 

paper prepared in German, ABEL returned to writing exclusively in French. 

CRELLE’S library. In Christiania, ABEL had access to a large section of French lit- 

erature on pure mathematics written by the masters and some of the servants of the 

subject in the eighteenth and early nineteenth century. However, circulation of re- 

sults to the geographical periphery was far from instant, and many of the products 

of the French reorganization of mathematics had not yet been brought to Norway. 

Therefore, it was an explicit motivation for ABEL’S European tour to go to the largest 

libraries and bookstores on the Continent which he expected to find together with the 

rest of the mathematical milieu in Göttingen and Paris. However, one of his most in- 

fluential encounters with the libraries of the Continent took place in Berlin, probably 

in the private library of Geheimrath CRELLE. 

“The afore-mentioned Crelle also has a perfectly splendid mathematical library 

which I use as if it had been my own and from which I benefit particularly 

as it contains all the latest material which he gets as soon as possible.” 44 

39 (A. L. Crelle, 1826). 

40 (W. Eccarius, 1976, 236). CRELLE also occasionally edited the manuscripts. 

41 (Stubhaug, 1996, 520). 

42 (N. H. Abel, 1902d, 3). 

43 (Abel→Holmboe, Wien, 1826/04/16. N. H. Abel, 1902a, 27). 

44 “Den samme Crelle har ogsaa et aldeles fortræffeligt mathematisk Bibliothek, som jeg benytter som 

mit eget og som jeg har særdeles Nytte af da det indeholder alt det nyeste, som han faaer saa snart 

det er mueligt.” (Abel→Hansteen, Berlin, 1825/12/05. ibid., 11).


A.-L. CAUCHY’S (1789–1857) new program of founding analysis on the notion of 

limits as expressed by inequalities had not reached Norway before ABEL left. CAUCHY 

had expressed his thoughts most influentially in the textbook Cours d’analyse intended 

(but never used) for instruction at the École Polytechnique which was printed in 1821. In 

a review of CAUCHY’S Exercises de mathématiques, CRELLE spoke highly of CAUCHY’S 

insights and his other works, and there can be little doubt that during ABEL’S time in 

Berlin, CAUCHY’S new analysis was discussed by a circle of mathematicians around 

CRELLE. One member of the circle, the mathematician M. OHM (1792–1872) 45 reacted 

to CAUCHY’S new rigorization by devising his own approach to algebraic analysis 

which was kept the formal aspect which CAUCHY had rejected. 46 In a letter, ABEL 

records how the circle discontinued its meetings because of G. S. OHM’S (1789–1854) 

arrogant mentality. 47 It is possible, that mathematical topics may also have played a 

role. 

“The work [Exercises de mathématiques] is full of deep analytical investigations 

as it would be expected from the acute and inventive author of Cours d’analyse, 

Leçons sur le calcul infinitésimal, Leçons sur l’application du calcul infinitésimal à la 

géométrie, etc.” 48 

ABEL discovered the works of CAUCHY in 1826. CAUCHY’S new approach to the 

theory of infinite series took ABEL by storm, and soon ABEL became one of CAUCHY’S 

most devoted missionaries (see part III). In print, ABEL first disclosed his sympathies 

in his work on the binomial theorem, which was printed in the early spring of 1827 

(see table 2.1), but ABEL’S letters allow us to date his encounter with the new Cauchian 

rigor in analysis more precisely. In a famous letter dated 16 January 1826, i.e. while in 

Berlin, ABEL wrote to HOLMBOE: 

“Taylor’s theorem, the foundation for the entire higher mathematics, is equally 

ill founded. I have found only one rigorous proof which is by Cauchy in his Resumé 

des leçons sur le calcul infinitesimal.” 49 

It is likely, that it was also in CRELLE’S library that ABEL came across CAUCHY’S 

famous textbook Cours d’analyse, a work which had tremendous consequences for 

ABEL’S attitude toward rigor. 50 

45 For the years of birth and death, see (Jahnke, 1987, 103). OHM was the younger brother of the famous 

physicist OHM. 

46 (Jahnke, 1987; Jahnke, 1993). 

47 (Abel→Hansteen, Berlin, 1825/12/05. N. H. Abel, 1902a, 11). 

48 “Das Werk ist voller tiefer analytischer Untersuchungen, wie sie sich von dem scharfsinningen und 

an neuen Ideen reichen Verfasser des “Cours d’analyse”, “Leçons sur le calcul infinitésimal,” der “Leçons 

sur l’application du calcul infinitésimal à la géométrie, etc.” erwarten lassen.” (A. L. Crelle, 1827, 400). 

49 “Det Taylorske Theorem, Grundlaget for hele den høiere Mathematik er ligesaa slet begrundet. Kun 

eet eneste strængt Beviis har jeg fundet og det er af Cauchy i hans Resumé des leçons sur le calcul 

infinitesimal.” (Abel→Holmboe, 1826/01/16. N. H. Abel, 1902a, 16). 

50 (I. Grattan-Guinness, 1970b, 79). Again, this influence will be documented in part III.


Volume Number Time of publication 

1 1 February–March 1826 

1 2 June 1826 

1 3 

1 4 February–March 1827 

2 1 5 June 1827 

2 2 20 September 1827 

2 3 

2 4 12 January 1828 

3 1 25 March 1828 

3 2 26 May 1828 

3 3 

3 4 3 December 1828 

4 1 25 January 1829 

4 2 28 March 1829 

4 3 

4 4 10 June 1829 

Table 2.1: Time of publications for CRELLE’S Journal für die reine und angewandte Mathematik 

1826–1829. Compiled from SYLOW’S notes in (N. H. Abel, 1881, vol. 2). 

The Berlin mathematical scene. In Berlin, mathematics was cultivated in three dis- 

tinct — and largely disjoint — circles: the academy, the university, and the circle around 

CRELLE. The Academy had played its role in the era of academies when it had housed 

such eminent mathematicians as EULER and LAGRANGE. However, after LAGRANGE 

had moved to Paris in 1784 without being suitably replaced, mathematics at the Academy 

inevitably and irreversibly declined. 51 

In the first ordinary mathematics chair at the university — which had only opened 

in 1810 — J. G. TRALLES (1763–1822) had resided. The brothers A. VON HUMBOLDT 

(1769–1859) and W. VON HUMBOLDT (1767–1835), who had been instrumental in bring- 

ing the university into being, had made efforts to call GAUSS to the chair, but he had 

to settle for less; TRALLES’ academic record shows a marked bias for applied mathe- 

matics, and during his reign pure mathematics was not well off in Berlin not was it 

elsewhere in Germany except for Göttingen. 52 Besides the ordinary professor, a num- 

ber of extraordinary professors and Privatdozenten 53 offered mathematics courses. Af- 

ter TRALLES’ death in 1822, E. H. DIRKSEN (1788–1850), who together with OHM had 

previously served as Privatdozenten, was appointed to the ordinary professorship. 

The real forum for pure mathematics in Berlin in the 1820s centered around CRELLE 

and condensed around the Journal für die reine und angewandte Mathematik once it was 

initiated in 1826. CRELLE organized weekly meetings of a group of young mathemati- 

51 (Knobloch, 1998). 

52 (Biermann, 1988, 20–21), (Rowe, 1998) 

53 These Privatdozenten include EYTELWEIN, GRUSON, LEHMUS, and LUBBE.


Author Vol. 1 Vol. 2 Vol. 3 Vol. 4 1826–29 

ABEL 97 (7) 100 (4) 56 (6) 125 (6) 378 (23) 

STEINER 93 (5) 45 (7) 11 (3) 149 (15) 

PONCELET 11 (2) 60 (1) 71 (1) 142 (4) 

JACOBI 7 (1) 60 (9) 32 (7) 29 (2) 128 (19) 

OLIVIER 66 (7) 54 (4) 1 (1) 121 (12) 

DIRICHLET 62 (4) 18 (2) 80 (6) 

MÖBIUS 36 (2) 36 (2) 72 (4) 

HILL 3 (1) 59 (1) 62 (2) 

CRELLE a 43 (3) 4 (1) 47 (4) 

RAABE 44 (5) 44 (5) 

PLÜCKER 13 (1) 22 (1) 35 (2) 

CLAUSEN 2 (1) 13 (5) 15 (5) 30 (11) 

HUMBOLDT 27 (1) 27 (1) 

OLTMANNS 10 (1) 13 (1) 23 (2) 

DIRKSEN 10 (2) 9 (1) 19 (3) 

HORN 18 (2) 1 (1) 19 (3) 

GRUNERT 17 (3) 1 (1) 18 (4) 

a Only papers explicitly authored by “Crelle” or “The editor” are included. Besides 

these, many anonymous papers must be attributed to CRELLE. 

Table 2.2: List of most productive (in pages, number of papers in parentheses) authors 

in CRELLE’S Journal 1826–29. 

cians, mainly contributors to his Journal für die reine und angewandte Mathematik; but in 

the year of the founding of the journal, the meetings had to be discontinued due to 

personal conflicts within the group. However, the group reorganized under CRELLE’S 

initiative and met each Monday for a social event combining music and mathematics, 

and in smaller groups they discussed while strolling the city. 

The members of CRELLE’S circle can only be identified indirectly from the list of 

authors of the early volumes of the Journal and from the correspondence of the iden- 

tifiable central actors. 54 In ABEL’S letters to his friends in Norway, explicit mention is 

only made of OHM, the extraordinary professor at the university who by an arrogant 

nature caused the weekly seances to be discontinued. However, ABEL wrote about a 

few unnamed young mathematicians with whom he entertained himself. 

2.3.3 Why never Göttingen? 

Already in ABEL’S first letter to HANSTEEN from Berlin, a certain ambiguity concern- 

ing his obligation to go to Göttingen can be found: 

“My winter quarters will be here in Berlin and I have not yet decided when I 

am going to leave. For the sake of Crelle and the Journal, I would like to stay here 

54 CRELLE’S Nachlass appears to have gone on auction shortly after CRELLE’S death in 1855 and must 

be considered lost. (W. Eccarius, 1975, 49, footnote)


as long as possible, and as I gather there is no other place in Germany where I 

would be better off. There is certainly a good library in Göttingen, but that is about 

all because Gauss who is the only capable one there is completely inapproachable. 

However, I must go to Göttingen.” 55 

ABEL’S devotion toward CRELLE and the Journal für die reine und angewandte Ma- 

thematik seems to overpower his ambition and obligation to go to Göttingen in accord 

with the stipend. A month later, a further reservation was expressed in a letter to 

HOLMBOE, where ABEL referred to the personality of the great GAUSS: 

“I am probably going to remain here in Berlin until the end of February or 

March; then I will travel via Leipzig and Halle to Göttingen, not to see Gauss because 

he is said to be intolerably reserved, but for the library which is apparently 

excellent.” 56 

The reservation concerning GAUSS was communicated to HANSTEEN a fortnight 

later, in a letter in which ABEL outlined his plans to accompany CRELLE on a trip to 

Göttingen. Furthermore, ABEL had already his eyes firmly set on Paris; he wrote the 

following to HANSTEEN after describing how he planed to go to Leipzig and Freiburg 

with one of his Norwegian friends: 

“Then I [will] travel back to Berlin in order to join Crelle on the tour to Göttingen 

and the Rhine area. In Göttingen, I shall only stay for a short period of time 

as there is nothing to be gained. Gauss is unapproachable and the library cannot 

possibly be better than those in Paris.” 57 

ABEL’S impression that GAUSS was not easily accessible seems to have been in- 

spired by rumors nourished in Berlin at the time. With his few but ripe policy concern- 

ing publications, GAUSS’ image was largely built from his network of corresponding 

mathematicians. Modern biographers of GAUSS use the published correspondence 

with e.g. the F. BOLYAI (1775–1856) s, ABEL, and others to support the picture of 

the great mathematician as remote and even hostile. However, these events were 

only taking place, and the letters were not publicly known in the 1820s. The Berlin 

mathematicians certainly held GAUSS’ mathematics in high respect but thought less 

of the master’s personal qualities and openness. HUMBOLDT had — with the help of 

CRELLE — tried to call GAUSS to the Berlin polytechnic and later the university, but 

GAUSS had declined all such offers and remained in Göttingen. 

55 “Mit Vinterquarteer kommer jeg til at holde her i Berlin og jeg er endnu ikke ganske enig med mig 

selv naar jeg skal reise herfra. For Crelles og Journalens Skyld vilde jeg gjerne være her saalænge 

som muelig og eftersom jeg hører er der vel intet andet Sted i Tyskland som vil være mig gavnligere. 

Göttingen har rigtignok et godt Bibliothek, men det er ogsaa det eneste; thi Gauss som er 

den eneste der der kan noget, er aldeles ikke tilgjængelig. Dog til Göttingen maae jeg det forstaaer 

sig.” (Abel→Hansteen, Berlin, 1825/12/05. N. H. Abel, 1902a, 12). 

56 “Jeg kommer formodentlig til at blive her i Berlin til Enden af Februar eller Marts, og reiser da 

over Leipzig og Halle til Göttingen (ikke for Gauss Skyld, thi han skal være utaalelig stolt men for 

Bibliothekets Skyld som skal være fortræffeligt).” (Abel→Holmboe, 1826/01/16. ibid., 18). 

57 “Siden reiser jeg tilbage til Berlin for i Følge med Crelle at tage Touren til Göttingen og Rhin-Egnene. 

I Göttingen bliver jeg kun kort da der ikke er noget at hole. Gauss er utilgjængelig og Bibliotheket 

kan ikke være bedre end i Paris.” (Abel→Hansteen, Berlin, [1826]/01/30. ibid., 20).


2.3.4 The European detour 

During February 1826, CRELLE’S possibilities to go to Göttingen crumbled and ABEL’S 

plans for the rest of the tour changed. Complaining repeatedly of his melancholic 

nature, ABEL hesitated and stalled at the prospect of travelling to Paris alone. Instead, 

his Norwegian friends presented an inviting alternative. With their interests largely 

in geology, the Norwegian travelling entourage made it for the Alps, and ABEL chose 

to join them. His defence of this less-than-obvious decision can be read in a very 

charming letter to HANSTEEN: 

“Can I then be blamed for wanting to see some of the southern life. During my 

journey I can work quite hard. Once I am in Vienna and on the road to Paris, the 

straight route almost passes through Switzerland. Why should I not also see a bit 

of that country. My lord! I am not completely without feelings for the beauty of 

Nature. The entire trip will postpone my arrival in Paris by two months and that 

does no harm. I will catch up. Do you not think that I would benefit from such a 

journey?” 58 

The reaction of the sponsor HANSTEEN can only be imagined. 

On his detour through Europe, ABEL did continue working on his mathematics, 

and he called upon the local mathematicians where he could. In Vienna, ABEL brought 

a letter of introduction from CRELLE to the mathematician K. L. VON LITTROW (1811– 

1877) at the observatory. His encounter with LITTROW was perhaps the only strictly 

mathematical benefit gained from the detour itself; mediated by LITTROW, ABEL man- 

aged to circulate his result on the insolubility of the quintic equation in an even wider 

(albeit still German speaking) audience. 59 

2.3.5 Isolation in Paris 

When ABEL eventually arrived in Paris in the summer of 1826, he found the mathe- 

matical scene abandoned: Most of the Paris mathematicians had left the city for the 

countryside. However, ABEL made a brief call to LEGENDRE, whom he described as “a 

really excellent old man”. 60 ABEL later met with CAUCHY without having more than a 

brief and non-technical conversation with him. Besides the opportunity to meet with 

the Parisian mathematicians, ABEL saw in the famous Académie des Sciences a possibil- 

ity for presenting his research. Before he left Norway, he had prepared a paper on the 

58 “Kan man da fortænke mig i at jeg ønsker ogsaa at see lidt af Sydens Liv og Færden. Paa min Reise 

kan jeg jo arbeide temmelig brav. Er jeg nu engang i Wien og jeg skal derfra til Paris saa gaaer jo 

den lige Vei næsten igjennem Schweitz. Hvorfor skulde jeg da ikke ogsaa see lidt deraf? Herre Gud! 

Jeg er dog ikke uden al Sands for Naturens Skjønheder. Den hele Reise vil gjøre at jeg kommer to 

Maaneder senere til Paris end ellers og det gjør da ikke noget. Jeg skal nok hente det ind igjen. Troer 

de ikke at jeg vil have godt af en saadan Reise?” (Abel→Hansteen, Dresden, 1826/03/29. ibid., 24). 

59 ABEL had his article from CRELLE’S Journal reviewed anonymously in the newly founded Vienna 

based journal Zeitschrift für Physik und Mathematik (Anonymous, 1826). The review will be described 

in part II. 

60 (ABEL to HOLMBOE, Paris 1826.8.12, (N. H. Abel, 1902a, 40))


insolubility of the quintic equation which had, however, already served its purpose as 

a door-opener with CRELLE in Berlin where it had been published. Therefore, that re- 

sult was not eligible for the academy and neither did ABEL submit the manuscript on 

the integration of differentials which he had also prepared for the collegium in Chris- 

tiania. Instead, on October 10, ABEL presented a paper, the so-called Paris memoir, 

which he had worked out during the tour and his stay in Paris and which extended 

the collegium manuscript. As is described in much more detail in part IV, the Paris 

mémoire was an algebraic approach to the general theory of integration of algebraic 

differentials and extended the approach to elliptic transcendentals which ABEL had 

already embarked upon while in Germany. 

ABEL’S entire production in Paris was directly influenced by the works of LEGEN- 

DRE on elliptic integrals which had recently appeared. 61 ABEL took a new approach to 

that theory and extended and advanced the program which LEGENDRE had begun to 

include higher transcendentals into analysis. Where the influence of LEGENDRE was 

thus clearly reflected in ABEL’S interests, the influence from CAUCHY was less direct. 

We know from his letters that ABEL bought and read the first issues of CAUCHY’S 

Exercises de mathématiques which include the first presentation of CAUCHY’S theory of 

complex integration. This particular theory was, however, completely without influ- 

ence on ABEL’S inversion of elliptic integrals as is described in part II. 

2.4 Back in Norway 

When ABEL ultimately left Paris around the New Year 1826/27, he headed for Berlin 

where he worked with CRELLE on editing the Journal for some months. There was 

no permanent position for ABEL to return to in Norway; the only possibility, a teach- 

ing position at the University to replace professor RASMUSSEN, had been given to 

HOLMBOE while ABEL was in Paris. During ABEL’S second visit to Berlin, CRELLE 

intensified his efforts to convince ABEL to come work for the Journal für die reine und 

angewandte Mathematik on a permanent level and CRELLE began lobbying for a posi- 

tion for ABEL at one of the new institutions of higher education in Berlin. In the spring 

of 1827, ABEL finally headed back north, back to uncertainty in Christiania. 

Back in his native country, ABEL made a living tutoring in mathematics and sub- 

stituting for HANSTEEN while he went on an expedition to Siberia. Throughout, 

ABEL continued his mathematical research and his production was intensified when 

he learned that the young German mathematician JACOBI was producing astonishing 

results on an important problem in the theory of elliptic functions. A hectic compe- 

tition ensued during which the two mathematicians advanced the theory far beyond 

its previous horizon. Initially, ABEL had been able to provide proofs of some of the 

claims which JACOBI had advanced without giving the proofs. Later, JACOBI’S results 

61 (A. M. Legendre, 1811–1817).

2.4. Back in Norway 37 

1802, Aug. 5 NIELS HENRIK ABEL was born at Finnøy 

1815 ABEL moved to Christiania to attend grammar 

school 

1818 BERNT MICHAEL HOLMBOE took over 

ABEL’s mathematics classes 

1821 ABEL graduated from grammar school and 

was immatriculated at the university in 

Christiania 

1823 ABEL visited DEGEN in Copenhagen 

1825, Sept. 7 ABEL left for the European tour making the 

first stop in Berlin 

1826, Jul. 10 ABEL arrived in Paris 

1826, Dec. ABEL left Paris and returned to Berlin 

1827, May 20 ABEL returned to Christiania 

1829, Apr. 6 NIELS HENRIK ABEL died at Froland 

Table 2.3: Summary of NIELS HENRIK ABEL’s biography 

provided inspiration for problems which ABEL tackled. The competition was fierce 

and only ended when ABEL fell ill around Christmas 1828. Even from his sick-bed, 

ABEL signed the last papers which he sent for publication in CRELLE’S Journal. ABEL 

died at Froland on April 6 1829 at the age of 26. A few dates later good news arrived 

from Berlin: CRELLE had finally secured a position for ABEL in Berlin.

Chapter 3 

Historical background 

The social and institutional conditions of mathematics resemble the general social con- 

text in undergoing abrupt transitions in the late eighteenth and early nineteenth cen- 

tury. The period is one of only a handful of instances where the influences of politi- 

cal and social changes can be witnessed directly in mathematical institutions and re- 

search. The main argument of the present chapter will be to establish that not only the 

external, outer shell of mathematics was transformed; the very core of mathematics, 

the ways mathematicians thought about their subject, was also influenced by social 

and political upheaval. 1 

3.1 Mathematical institutions and networks 

Mathematical circles. The metaphor of center and periphery has aptly been applied 

to describe science conducted in geographically “remote” regions such as Scandinavia 

during periods when the most prominent contributions were made at larger centers in 

Central Europe. 2 However, as described below, in the case of mathematics in the early 

nineteenth century a curious picture emerges with a very strong mathematical center 

in Paris and two lesser centers in Germany: the Gaussian ivory tower in Göttingen 

and the emerging mathematical scene in Berlin centered around the university, A. L. 

CRELLE (1780–1855), and his Journal. In the following section, N. H. ABEL’S (1802– 

1829) position within the two mathematical traditions unfolding at these centers — 

for short denoted the French and German traditions — is described. 

Norway. Knowledge of the Norwegian national history during the early nineteenth 

century is of importance for understanding the conditions under which ABEL grew 

1 Some of the historical facts and circumstances have already been touched upon in the preceding 

chapters but are taken up here again from a slightly different perspective. 

2 The terms were first employed in (Shils, 1961) and have since found their way into the history of 

science. 

39

40 Chapter 3. Historical background 

up and evolved into a prominent mathematician. 3 In the middle of the nineteenth 

century, national states appeared as the natural units of political and military power. 

Before that, in the early nineteenth century, unitary states had been the most obvi- 

ous ways of organizing power. In Scandinavia, two unitary states had coexisted as 

rivals for centuries, the Danish-Norwegian monarchy and the Swedish monarchy. In 

1814, in the aftermath of the Napoleonic Wars, Norway was separated from the twin- 

monarchy and ceded to Sweden. During the war, a nationalist sentiment had grown 

strong in Norway; this national pride led to a brief period of independence before the 

transition to Swedish rule. 

Receiving all of his formal education within Norwegian institutions, ABEL belongs 

to the first generation of truly Norwegian scientists. Before the creation of the university 

in 1813, Norwegians who wanted any kind of higher education had to go to Copen- 

hagen; quite a number of them went and later returned to fill administrative or clerical 

positions in Norway. 

At the Christiania Cathedral School, ABEL received qualified and personal tutor- 

ing from B. M. HOLMBOE (1795–1850); and at the University he became the prodigy of 

the professors S. RASMUSSEN (1768–1850) and C. HANSTEEN (1784–1873). HOLMBOE 

was ABEL’S senior by only seven years and had been among the very first students 

to attend the Christiania university where he sat in the mathematics classes of RAS- 

MUSSEN. Such relations between prodigies and benefactors may well originate in the 

fact that the scientific community in Christiania was rather small and yet led by a few 

men with international relations. 

France. In France (in the early nineteenth century that almost exclusively meant 

Paris), by contemporary standards, the scientific community was anything but small. 

After the Revolution, educational reforms were introduced to develop the military 

and civil engineering in France. In order to achieve this goal, large-scale and very 

advanced instruction in mathematics was set up at two newly founded educational 

institutions, the École Normale and the École Polytechnique. A mathematical milieu of 

substantial size and quality established itself in the French capital in the decades fol- 

lowing the Revolution. Teaching at either the École Polytechnique or the École Normale 

provided a means of living for mathematicians; and the Académie des Sciences pro- 

vided a possibility to communicate mathematical research. The focus on the teaching 

of engineers and the sheer volume of classes exerted influences on the contents of 

the mathematics taught and researched; the eighteenth century prevalence of appli- 

cable mathematics continued to dominate, but the communication of the calculus to 

the previously un-initiated helped provoke research on the foundations of the topic. 

The liberal ideas of the Revolution meant a dramatic increase in the numbers of publi- 

cations. This general trend also influenced mathematics; mathematicians could more 

3 The cultural framework of Norwegian society in the first decades of the nineteenth century has been 

aptly described in (Stubhaug, 1996; Stubhaug, 2000); here only a few aspects need repetition.

3.2. ABEL’s position in mathematical traditions 41 

easily have their works either printed or published in one of the journals which were 

set up. 4 However, perhaps as a consequence of its size, the milieu in Paris was a 

very competitive one in the first decades of the nineteenth century; 5 teachers proba- 

bly seemed to a Norwegian more reserved there than in Christiania, and mathematical 

cooperation was actually rarely seen. 

German states. In the early nineteenth century, the German speaking region was 

organized in a multitude of sovereign states. One of the most influential and ambitious 

ones, Prussia, was dominated from its capital Berlin where a national awareness was 

spreading to the sciences and mathematics in the 1820s. 

The German reaction to the events in France around the turn of the century found 

a philosophical grounding in the neo-humanistic movement which sought to reintro- 

duce classical ideals of humanism and learning through educational reforms. These 

reforms — promoted in Prussia by W. VON HUMBOLDT (1767–1835) and others — im- 

proved the position of mathematics within the curriculum and promoted a particular 

view of mathematics. 6 In the opinion of the neo-humanists, mathematics was not to be 

cultivated for its applicability in the sciences; instead, a “pure” form of mathematics 

was promoted with its own set of motivations and ideologies. The important mathe- 

matical branch of algebraic analysis found a unique form with these philosophically 

inspired mathematicians in the form of the German combinatorial school. 7 

In order to implement the educational reforms, new institutional constructions 

were devised. In Berlin, a university was opened in 1810 which included mathe- 

matics; plans for a polytechnic school — where mathematics should also be taught — 

had to be postponed but were carried out in the 1820s. Thus, in Prussia, instruction 

in mathematics of the future teachers — with its focus on pure thought and individ- 

ual contemplation and research — became institutionalized in the first decades of the 

nineteenth century. 8 Outside the realm of the university, a group of mathematicians 

gathered around CRELLE’S Journal für die reine und angewandte Mathematik; in the 1820s, 

this group constituted an alternative mathematical forum in Berlin and a place for 

younger mathematicians to meet. 

3.2 ABEL’s position in mathematical traditions 

Within the Continental tradition in mathematics. ABEL’S interactions with Euro- 

pean mathematics were almost exclusively confined to the centers in Berlin and Paris. 

In the 1820s, a few other peripheral communities of mathematics existed; in particu- 

4 (J. Dhombres, 1985; J. Dhombres, 1986). 

5 (I. Grattan-Guinness, 1982) and (I. Grattan-Guinness, 1990, e.g. 1227). 

6 (Pyenson, 1983). 

7 (Jahnke, 1990; Jahnke, 1996). The combinatorial school is briefly treated upon in Part III. For a fuller 

account, consult (Jahnke, 1987; Jahnke, 1993). 

8 (Begehr et al., 1998; Biermann, 1988; Mehrtens, 1981).


lar on the British Isles, in the Austrian-Hungarian Empire, and in Russia. There are, 

however, nearly no traces of any kinds of interactions with these mathematicians to 

be found in ABEL’S works or letters. 

In the first decades of the nineteenth century, British mathematicians (both outside 

and within the Analytical Society) were consciously trying to adapt the Continental cal- 

culus. 9 Only once — in an 1823-letter to HOLMBOE — did ABEL mention the two con- 

temporary British mathematicians T. YOUNG (1773–1829) and J. IVORY (1765–1842), 

neither of whom were associated with the Analytical Society. 10 From a single sentence 

in one of his letters, we know that ABEL was aware of the existence of the Czech 

theologian, philosopher, and mathematician B. BOLZANO (1781–1848). In one of his 

notebooks, ABEL makes the following rather sudden remark, which initially confused 

the P. L. M. SYLOW (1832–1918), who by 1902 still only knew of Bolzano as a town in 

the Alps: 11 

“Bolzano is a clever fellow from what I have studied” 12 

This remark made SYLOW speculate that ABEL had probably read BOLZANO’S Rein 

analytischer Beweis during the European tour. 13 There is nothing to suggest that ABEL 

met personally with BOLZANO during his stop in Prague during the European de- 

tour. 14 In chapter 12.1, ABEL’S contribution to the rigorization of analysis is described 

in further detail, and a few more details concerning his relation to BOLZANO are dis- 

cussed. 

Although interested in topics central to the endeavors of the Analytical Society 

and BOLZANO (certainly independently), ABEL’S inspirations thus seem to come from 

somewhere else: the centers of Berlin and Paris. 

Between the German and the French traditions. ABEL’S mathematical production 

was confined to the discipline of algebra, the foundations of analysis, and the the- 

ory of higher transcendentals. Thus, his interests did not include geometry and — 

except for a youthful work — also excluded applied mathematics. Although these 

topics constitute important parts of the French and German traditions in mathemat- 

ics in the period, 15 ABEL’S work is nevertheless rightfully interpreted within these 

9 (Craik, 1999). 

10 (N. H. Abel, 1902a, 4–8). IVORY had studied mathematics in Scotland before taking up the subject as 

his profession. In 1819, he retired to become a private mathematician living in London. YOUNG was 

an autodidact natural philosopher with an interest in mathematics. He was elected into the French 

Académie des Sciences in 1827; ABEL and JACOBI had also been nominated for the election. 

11 (L. Sylow, 1902, 12). 

12 “Bolzano er en dygtig Karl i hvad jeg [. . . ]” (Abel, MS:351:A, 61). The rest is crossed out and difficult 

to decipher. See (Stubhaug, 2000, 505, fig. 44). 

13 (Bolzano, 1817), see (L. Sylow, 1902, 12). 

14 Historical speculations based on the similarities of results and the possible personal rendezvous of 

the authors have been found inadequate, as the responses to GRATTAN-GUINNESS’ provocative 

suggestion that CAUCHY plagiarized BOLZANO (I. Grattan-Guinness, 1970a); for a sober review of 

the ensuing controversy, see (Bottazzini, 1986, 123–124 (note 10)). 

15 See e.g. (Dauben, 1981) or (Jahnke, 1994).

3.3. The state of mathematics 43 

traditions. The new French approach to rigor and the German focus on pure math- 

ematics (i.e. mathematics without the justification of applicability) were important 

backgrounds for ABEL’S mathematics; no expressed concerns for applicability can be 

found in ABEL’S writings; he was — in every respect — a pure mathematician. On 

the other hand, as will become evident in chapters 16 and 19, ABEL was no dogmatic 

rigorist when he aimed at producing new mathematical results. 

3.3 The state of mathematics 

Some tendencies in the mathematicians’ thoughts about mathematics just prior to the 

period of main interest merit attention. In particular, prominent mathematicians to- 

ward the end of the eighteenth century expressed the belief that mathematics was just 

about to reach its pinnacle of cultivation. From the eighteenth century perspective, 

where mathematical praxis was to a large part made up of formal and explicit ma- 

nipulations of known algebraic or analytic dependencies, these methods seemed of 

limited scope. For instance, J. L. LAGRANGE (1736–1813) — a few years after his in- 

novative paper on polynomial equations from which our story will commence in the 

next chapter — wrote to J. LE R. D’ALEMBERT (1717–1783) in 1781, 

“It appears to me also that the mine [of mathematics] is already very deep and 

that unless one discovers new veins it will be necessary sooner or later to abandon 

it.” 16 

Similar dark visions seem to be recurring at intervals — often in the form of fin- 

de-siècle pessimism. Even into the nineteenth century, a similar view was expressed 

by J.-B. J. DELAMBRE (1749–1822). In 1810, DELAMBRE delivered a commissioned 

review of the progress made in the mathematical sciences after the French Revolution. 

He expressed his concern over the future of mathematics in the following way: 

“It would be difficult and rash to analyze the chances which the future offers 

to the advancement of mathematics; in almost all its branches one is blocked by 

insurmountable difficulties; perfection of detail seems to be the only thing which 

remains to be done. [. . . ] All these difficulties appear to announce that the power 

of our analysis is practically exhausted in the same way as the power of the ordinary 

algebra was with respect to the geometry of transcendentals at the time of 

Leibniz and Newton, and it is required that combinations are made which open a 

new field in the calculus of transcendentals and in the solution of equations which 

these [transcendentals] contain.” 17 

16 “Il me semble aussi que la mine est presque déjà trop profonde, et qu’à moins qu’on ne découvre 

de nouveaux filons il faudra tôt ou tard l’abandonner.” (Lagrange→d’Alembert, Berlin, 1781. Lagrange, 

1867–1892, vol. 13, 368); English translation from (Kline, 1990, 623). 

17 “Il seroit difficile et peut-être téméraire d’analyser les chances que l’avenir offre à l’avancement des 

mathématiques: dans presque toutes les parties, on est arrêté par des difficultés insurmontables; 

des perfectionnemens de détail semblent la seule chose qui reste à faire; [. . . ] Toutes ces difficultés 

semblent annoncer que la puissance de notre analyse est à-peu-près épuisée, comme celle de


Taken together, the two quotations hint at two possible ways out of the apparently 

imminent stagnation of mathematics: discovery of new questions (veins) and fusions 

of existing theories. After the evidence has been presented in the following three parts, 

it will be illustrated how the mathematical community of the early nineteenth century 

invoked precisely these approaches in a period of such mathematical creativity that 

the remarks of LAGRANGE and DELAMBRE afterwards seem well off the mark. 

3.4 ABEL’s legacy 

It is well known that after ABEL’S death, his name was tied to a romantic picture of a 

neglected mathematical genius. His case was used as fuel for arguments ranging from 

nationalistic awareness to revolutionary issues. His influence on the ensuing century 

was vast — both as mathematical and even personal inspiration. Here, only ABEL’S 

mathematical legacy will be discussed, although certain aspects of his personality and 

destiny are present in the quotations given. More importantly than contributing new 

results to mathematics, ABEL’S programmatic approach caught the attention of the 

leading figures in mathematics in the nineteenth century, in particular C. G. J. JACOBI 

(1804–1851) and K. T. W. WEIERSTRASS (1815–1897). 

The judgement on ABEL’S legacy passed by A.-M. LEGENDRE (1752–1833) and 

JACOBI is legendary, itself. In a letter to CRELLE, LEGENDRE is reported to have said: 

“After having worked by myself, I have felt a great satisfaction paying homage 

to Mr. Abel’s talents, feeling all the merits of the beautiful theorem the discovery 

of which is due to him and to which the qualification monumentum aere perennius 

can be applied.” 18 

JACOBI, who reported LEGENDRE’S homage to ABEL in his review of the third sup- 

plement of LEGENDRE’S Théorie des fonctions elliptiques, at another place qualified and 

generalized the praise which should be given to ABEL’S contribution to mathematics: 

“The vast problems which he [ABEL] had proposed to himself — i.e. to establish 

sufficient and necessary criteria for any algebraic equation to be solvable, for 

any integral to be expressible in finite terms, his admirable discovery of the theorem 

encompassing all the functions which are the integrals of algebraic functions, 

etc. — marks a very special type of questions which nobody before him had dared 

to imagine. He has gone but he has left a grand example.” 19 

l’algèbre ordinaire l’étoit par rapport à la géométrie transcendante au temps de Leibnitz et de Newton, 

et qu’il faut des combinaisons qui ouvrent un nouveau champ au calcul des transcendantes et à 

la résolution des équations qui les contiennent.” (Delambre, 1810, 131); translation based on Kline, 

1990, 623. 

18 “En travaillant pour mon propre compte, j’ai éprouvé une grande satisfaction, de rendre un éclatant 

hommage au génie de Mr. Abel, en faisant sentir tout le mérite du beau théorème dont l’invention 

lui est due, et auquel on peut appliquer la qualification de monumentum aere perennius.” Legendre 

quoted in C. G. J. Jacobi, 1832a, 413.

3.4. ABEL’s legacy 45 

As can be seen from the quote, to JACOBI, ABEL’S legacy laid more in the way he 

asked questions than in the solutions and answers which he provided. ABEL’S ques- 

tions were, in the mind of JACOBI, questions of necessary and sufficient conditions 

for certain properties to hold. This interpretation highlights two aspects which find 

instances in the present work: rigorization as the process of making clear, useful, and 

precise the conditions of theorems, and the new, concept based questions which are 

guaranteed to be answerable, although the answers might defy contemporary intu- 

itions. 

In his youth, WEIERSTRASS was deeply influenced in his career by works written 

by ABEL. Throughout his life, WEIERSTRASS thought highly of ABEL; the following 

statements testify to WEIERSTRASS’ devotion which at times almost resembles envy. 

“The fortunate Abel; he has contributed something of lasting value! — He [ABEL] 

was used to always taking the most elevated point of view. — Abel was distinguished 

by the all-embracing vision directed at the highest position, the ideal.” 20 

These quotations are, of course, equally good sources to WEIERSTRASS’ views on 

mathematics in the second half of the nineteenth century as to ABEL’S mathemati- 

cal production in the 1820s. However, the quotations touch upon the same themes as 

the quotation from JACOBI above; in due time it will be clearer, on which basis WEIER- 

STRASS could claim that ABEL had produced lasting results by taking the most general 

approach toward the idealistic goal of mathematics. 

19 “Les vastes problèmes qui’il s’était proposés, d’établir des critères suffisants et nécessaires pour 

qu’une équation algébrique quelconque soit résoluble, pour qu’une intégrale quelconque puisse être 

exprimée en quantités finies, son invention admirable de la propriété générale qui embrasse toutes 

les fonctions qui sont des intégrales de fonctions algébriques quelconques, etc., etc., marquent un 

genre de questions tout à fait particulier, et que personne avant lui n’a osé imaginer. Il s’en est allé, 

mais il a laissé un grand exemple.” (Jacobi→Legendre, Potsdam, 1829. Legendre and Jacobi, 1875, 

70–71); for a German translation, see (Pieper, 1998, 153). 

20 “Abel der Glückliche; er hat etwas Bleibendes geleistet! — Er [ ABEL] war gewohnt, überall den 

höchsten Standpunkt einzunehmen. — Abel zeichnete der allumfassende, auf das höchste, das Ideale 

gerichtete Blick aus.” The quotes are all taken from (Biermann, 1966, 218).

Part II 

“My favorite subject is algebra” 

47

Chapter 4 

The position and role of ABEL’s works 

within the discipline of algebra 

In the nineteenth century, the theory of equations acquired its status as a mathematical 

discipline with its own set of problems, methods, and legitimizations. In the process, 

N. H. ABEL (1802–1829) played an important role. His works on the algebraic in- 

solubility of the general quintic equation and his penetrating studies of the so-called 

Abelian equations belong to the first results established within this incipient discipline. 

Although ABEL’S investigations raised new questions and answered some of them, 

his methods and his approach was deeply rooted in the works of mathematicians be- 

longing to the previous generations. In particular, ABEL drew upon the algebraic re- 

searches of L. EULER (1707–1783). Therefore, in the following chapter 5, these works, 

similar approaches taken by A.-T. VANDERMONDE (1735–1796), and the even more 

influential works by J. L. LAGRANGE (1736–1813) and C. F. GAUSS (1777–1855) are 

described and analyzed. In the ensuing chapters 6–8, ABEL’S algebraic researches are 

described and their role and impact are analyzed. Focus in this part II will be on de- 

scribing the change in asking and answering questions pertaining to mathematical ob- 

jects; more precisely questions concerning the algebraic solubility of equations. Such 

questions have been central to mathematical development since the Renaissance, but 

starting in the second half of the eighteenth century, they gave rise to a new mathe- 

matical theory. Once this theory-building has been described, the attention is directed 

toward ABEL’S approach to algebraic questions. ABEL’S studies of the quintic equa- 

tion provide an example of how a change in the process of asking questions led to 

unexpected answers. Then, because of the similarity in methods and inspirations, 

ABEL’S questions concerning the geometric division of the lemniscate are treated to 

illustrate how an algebraic topic emerged within an apparently non-algebraic realm. 

Finally, the quest — taken up by ABEL and slightly later by E. GALOIS (1811–1832) — 

to completely describe solvable equations is outlined to provide a first illustration of 

the new and more abstract kind of questions which C. G. J. JACOBI (1804–1851) saw 

49

50 Chapter 4. The position and role of ABEL’s works within the discipline of algebra 

as ABEL’S greatest legacy. 1 

4.1 Outline of ABEL’s results and their structural 

position 

In the penultimate year of the eighteenth century, the Italian P. RUFFINI (1765–1822) 

had published the first proof of the impossibility of solving the general quintic alge- 

braically. Working within the same tradition as ABEL, RUFFINI published his investi- 

gations on numerous occasions; however, his presentations were generally criticized 

for lacking clarity and rigour. Not until 1826 — after ABEL had published his proof of 

this result 2 — did ABEL mention RUFFINI’S proofs, and there is reason to believe that 

ABEL obtained his proof independently of RUFFINI, yet from the same inspirations. 

From LAGRANGE’S comprehensive study of the solution of equations 3 originated 

the idea of studying the numbers of formally distinct values which a rational func- 

tion of multiple quantities could take when these quantities were permuted. The idea 

was cultivated and emancipated into an emerging theory of permutations by A.-L. 

CAUCHY (1789–1857) who in 1815 provided the theory of permutations with its ba- 

sic notation and terminology. 4 CAUCHY also established the first important theorem 

within this theory when he proved a generalization of one of RUFFINI’S results to the 

effect that no function of five quantities could have three or four different values under 

permutations of these quantities. 

Insolubility of the quintic. ABEL combined the results and terminology of CAUCHY’S 

theory of permutations with his own innovative investigations of algebraic expressions 

(radicals). ABEL’S proof is a representation of his approach to mathematics. Once he 

had realized that the quintic might be unsolvable, he was led to study the “extent” 

of the class of algebraic expressions which could serve as solutions: the “expressive 

power” of algebraic solutions. Following a minimalistic definition of algebraic expres- 

sions, ABEL classified these newly introduced objects in a way imposing a hierarchic 

structure in the class of radicals. The classification enabled ABEL to link algebraic ex- 

pressions — formed from the coefficients — which occur in any supposed solution for- 

mula to rational functions of the roots of the equation. By the theory of permutations, 

which ABEL had taken over from CAUCHY, he reduced such rational functions to only 

a few standard forms. Considering these forms individually, ABEL demonstrated — 

by reductio ad absurdum — that no algebraic solution formula for the general quintic 

could exist. 

1 See p. 44, above. 

2 (N. H. Abel, 1824b; N. H. Abel, 1826a). 

3 (Lagrange, 1770–1771). 

4 (A.-L. Cauchy, 1815a).

4.1. Outline of ABEL’s results and their structural position 51 

In the first part of the nineteenth century, the century-long search for algebraic 

solution formulae was brought to a negative conclusion: no such formula could be 

found. To many mathematicians of the late eighteenth century such a conclusion had 

been counter-intuitive, but owing to the work and utterings of men like E. WARING 

(∼1736–1798), 5 LAGRANGE, and GAUSS the situation was different in the 1820s. 

ABEL’S proof was also met with criticism and scrutiny. By and large, though, the 

criticism was confined to local parts of the proof. The global statement — that the gen- 

eral quintic was unsolvable by radicals — was soon widely accepted. 

Abelian equations. In his only other publication on the theory of equations, Mémoire 

sur une classe particulière d’équations résolubles algébriquement 1829, ABEL took a different 

approach. The paper was inspired by ABEL’S own research on the division problem 

for elliptic functions and GAUSS’ Disquisitiones arithmeticae. In it, ABEL demonstrated 

a positive result that an entire class of equations — characterized by relations between 

their roots — were algebraically solvable. 

For his 1829 approach, ABEL seamlessly abandoned the permutation theoretic pil- 

lar of the insolubility-proof. Instead, he introduced the new concept of irreducibility 

and — with the aid of the Euclidean division algorithm — proved a fundamental the- 

orem concerning irreducible equations. 

The equations which ABEL studied in 1829 were characterized by having rational 

relations between their roots. 6 Using the concept of irreducibility, ABEL demonstrated 

that such irreducible equations of composite degree, m × n, could be reduced to equa- 

tions of degrees m and n in such a way that only one of these might not be solvable 

by radicals. Furthermore, he proved that if all the roots of an equation could be writ- 

ten as iterated applications of a rational function to one root, 7 the equation would be 

algebraically solvable. 

The most celebrated result contained in ABEL’S Mémoire sur une classe particulière 

was the algebraic solubility of a class of equations later named Abelian by L. KRO- 

NECKER (1823–1891). These equations were characterized by the following two prop- 

erties: (1) all their roots could be expressed rationally in one root, and (2) these ratio- 

nal expressions were “commuting” in the sense that if θ i (x) and θ j (x) were two roots 

given by rational expressions in the root x, then 

θ iθ j (x) = θ jθ i (x) . 

By reducing the solution of such an equation to the theory he had just developed, 

ABEL demonstrated that a chain of similar equations of decreasing degrees could be 

constructed. Thereby, he proved the algebraic solubility of Abelian equations. 

5 1734 is a more qualified guess for Waring’s year of birth than (Scott, 1976) giving “around 1736”. 

See (Waring, 1991, xvi). 


7 I.e. an equation in which the roots are x, θ (x) , θ (θ (x)) , . . . , θ n (x) for some rational function θ.


ABEL planned to apply this theory to the division problems for circular and elliptic 

functions. However, only his reworking of GAUSS’ study of cyclotomic equations was 

published in the paper. 

Together, the insolubility proof and the study of Abelian equations can be inter- 

preted as an investigation of the extension of the concept of algebraic solubility. On 

one hand, the insolubility proof provided a negative result which limited this exten- 

sion by establishing the existence of certain equations in its complement. On the other 

hand, the Abelian equations fell within the extension of the concept of algebraic solu- 

bility and thus ensured a certain power (or volume) of the concept. 

Grand Theory of Solubility. In a notebook manuscript — first published 1839 in the 

first edition of ABEL’S Œuvres — ABEL pursued further investigations of the exten- 

sion of the concept of algebraic solubility. In the introduction to the manuscript, he 

proposed to search for methods of deciding whether or not a given equation was solv- 

able by radicals. The realization of this program would, thus, have amounted to a 

complete characterization of the concept of algebraic solubility. 

ABEL’S own approach to this program was based upon his concept of irreducible 

equations. In the first part of the manuscript — which appears virtually ready for the 

press — ABEL gave his definition of this concept. Arguing from the definition, he 

proved some basic and important theorems concerning irreducible equations. 

In the latter part of the manuscript — which is less lucid and toward the end con- 

sists of nothing but equations — ABEL reduced the study of algebraic expressions sat- 

isfying a given equation of degree µ to the study of algebraic expressions which could 

satisfy an irreducible Abelian equation whose degree divided µ − 1. However, ABEL’S 

researches were inconclusive. When ABEL’S attempt at a general theory of algebraic 

solubility eventually was published in 1839, another major player in the field, GA- 

LOIS, had also worked on the subject. Inspired by the same tradition and exemplary 

problems as ABEL had been, GALOIS put forth a very general theory with the help of 

which the solubility of any equation could — at least in principle — be decided. 

GALOIS’ writings were inaccessible to the mathematical community until the mid- 

dle of the nineteenth century. His style was brief and — at times — obscure and unrig- 

orous. Many mathematicians of the second half of the nineteenth century — starting 

with J. LIOUVILLE (1809–1882) who first published GALOIS’ manuscripts in 1846 — 

invested large efforts in clarifying, elaborating, and extending GALOIS’ ideas. In the 

process, the theory of equations finally emerged in its modern form as a fertile subfield 

of modern algebra. Part of this evolution concerned mathematical styles. The highly 

computational mathematical style of the eighteenth century, to which ABEL had also 

adhered, was superseded. The old style had been marked by lengthy, rather concrete, 

and painstaking algebraic manipulations. In the nineteenth century, this was replaced 

by a more conceptual reasoning, early glimpses of which can be seen in ABEL’S works 

on the algebraic solubility of equations.

4.2. Mathematical change as a history of new questions 53 

4.2 Mathematical change as a history of new questions 

A permeating theme of the present work is the emergence of new questions in the 

early nineteenth century. The description and analyses of ABEL’S algebraic works 

serve to illustrate three aspects of this process: 

1. New questions may have unexpected answers which push mathematics for- 

ward. 

2. New and fertile questions may arise from importing methods or inspirations 

from one theoretical complex into another; entirely new theories may develop. 

3. A deliberate reformulation of hard but improperly formulated questions may 

transform them into forms more open to mathematical treatment. The process of 

reformulating the question may involve a process of scrutinizing mathematical 

intuitions. 

New questions with unexpected answers. Ever since procedures to algebraically 

compute the roots of cubic and bi-quadratic equations were discovered in the mid- 

dle of the sixteenth century, the search had been on for a generalization to quin- 

tic equations. Once R. DU P. DESCARTES’ (1596–1650) new notational system trans- 

lated the problem into purely algebraic manipulations of symbols, the belief became 

widespread that such a generalization had to be obtainable. Although the goal defied 

even the greatest mathematicians for centuries, the belief remained intact as late as 

the second half of the eighteenth century. EULER, for instance, felt assured enough 

about the general algebraic solubility of equations to utilize it as the basis for proofs 

of another almost self-evident result: the fundamental theorem of algebra. 

In 1770, LAGRANGE decided to study carefully the reasons behind the solubility 

of equations of degrees 1,2,3, and 4 with the hope of obtaining some kind of general 

procedure which could subsequently be applied to the fifth degree equation. LA- 

GRANGE’S investigations were important in two respects: firstly, they provided a the- 

orization of the problem into problems of permutations of the roots — a mathemat- 

ical tool which would become immensely important for the problem, and secondly, 

LAGRANGE envisioned that the powers of his analysis were not powerful enough to 

deduce the desired result. This second observation can be taken as the first hint that 

such solutions were beyond the reach of humans. 

In the last years of the eighteenth century, the full consequences of the failure to 

obtain algebraic solutions to the general quintic were realized and published indepen- 

dently by two mathematicians located at opposite ends of the professional spectrum: 

the German “prince of mathematics” GAUSS and the much lesser known Italian RUF- 

FINI. In 1799, GAUSS remarked as a criticism of EULER that the algebraic solubility 

of equations should not be taken for granted. The same year, RUFFINI published the


first of a series of proofs that the general fifth degree equation could not be solved al- 

gebraically. RUFFINI’S proofs were, as noted, difficult and had little impact, although 

RUFFINI communicated with some of the Parisian mathematicians. Instead, math- 

ematicians took some notice of GAUSS’ 1801 claim in the prestigious Disquisitiones 

arithmeticae to possess a rigorous proof of the insolubility of the quintic equation. 

During the third decade of the 19 th century, the question was finally resolved by 

ABEL and — more generally — by GALOIS. In some respect, RUFFINI had already ob- 

tained the answer in 1799, and comparing the proofs of RUFFINI, ABEL, and GALOIS 

sheds interesting light on the intra- and extra-mathematical mechanisms behind the 

establishment of mathematical knowledge. 

ABEL’S proof of the insolubility of the quintic is a fascinating combination of previ- 

ously established results and an approach designed to make the question addressable. 

Despite lacking in certain respects, ABEL’S proof and its conclusion soon gained wide 

acceptance among the experts. However, for many years to come, some mathemati- 

cians found the conclusion so counter-intuitive that they had to doubt the result. It is 

in this respect that the question led to an unexpected answer. 

Asking algebraic questions of transcendental objects By 1823, ABEL had carefully 

studied GAUSS’ Disquisitiones arithmeticae, although no explicit reference to it was 

made in the insolubility-proofs. When ABEL reacted upon a suggestion by C. F. DE- 

GEN (1766–1825) to turn his attention toward the study of higher transcendentals, he 

found ample inspiration from GAUSS’ work. In the Disquisitiones, GAUSS applied al- 

gebraic studies to the problem of constructing regular polygons with the help of ruler 

and compass. GAUSS suggested that the method could be carried over to the divi- 

sion problem for curves whose rectification depended on a simple elliptic integral, the 

lemniscate integral. In a large paper, which included the foundation of the new objects 

elliptic functions, ABEL provided the details supporting GAUSS’ claim and was led to a 

new class of polynomial equations which were always solvable. 

Thus, in the midst of a realm apparently inherited by highly transcendental objects, 

ABEL focuses upon algebraic relations pertaining to and existing among these objects. 

The theory of higher transcendentals was in a phase of transition in the period, and 

ABEL’S algebraic focus influenced the future developments during the second quarter 

of the nineteenth century. 

The art of asking answerable questions An important ingredient in bringing about 

the change in attitude toward the solubility of the quintic had been ABEL’S way of 

asking questions. In a passage in one of his notebooks, ABEL emphasized that any 

mathematical problem, when formulated properly, is decidable — be it affirmatively 

or not. 8 Thus, for goals which had remained unattainable for years, ABEL suggested 

8 The belief is also present in HILBERT’S “Wir müssen wissen, wir werden wissen.” However, as the 

development in the twentieth century showed, such a belief has to take into account the accepted

4.2. Mathematical change as a history of new questions 55 

a reformulation of the problem to a question of the form “is this goal achievable?” In 

the case of the quintic equation, the search for an algebraic solution was reformulated 

to a question whether such a solution existed at all. 

Such change of attitude toward mathematical goals signal — as JACOBI soon real- 

ized — a change toward more general and abstract mathematics. In order to answer 

questions concerning possibility of existence, ABEL used implicit quantification over 

all possible solutions to the question. His approach was based upon the classification 

and normalization of these objects which were therefore studied — not individually — 

but as items belonging to a collection defined by a concept. Thus, a concept based ap- 

proach to doing mathematics was intimately connected to the kinds of questions asked 

and addressed. 

In the theory of equations, having established the existence of both algebraically 

solvable and unsolvable exemplars, ABEL raised the question of determining directly 

whether a given equation would be solvable or not. In a notebook manuscript, ABEL 

set out to address this question. For certain types of equations, he made some progress; 

however, it was left to GALOIS to outline a theory, based on the same inspirations as 

ABEL, which — when elaborated — was powerful enough to answer the question. 

rules of mathematical reasoning and the system of primitive truth from which deductions are made.

Chapter 5 

Towards unsolvable equations 

By the dawn of the nineteenth century, the theory of equations addressed a wide range 

of questions. For the present purpose, the main question is the one of algebraic solubil- 

ity, but in the eighteenth century, a multitude of other questions concerning existence 

and characterization of roots were intertwined with it. Therefore, in order to broaden 

the perspective, aspects of the history of these approaches are briefly outlined. 

The existence of roots. When R. DU P. DESCARTES (1596–1650) in 1637 claimed that 

any equation of degree n possessed n roots an important theorem of algebra was for- 

mulated whose proof became central to subsequent development. 1 His way out was 

a rather evasive one which consisted of distinguishing the real ones (real meaning “in 

existence”) from the imaginary ones which were products of human imagination. To 

DESCARTES the assertion that any equation of degree n had n roots took the form of a 

general property possessed by all equations and the trick of introducing the imagined 2 

roots saved him from further argument. 3 

“Neither the true nor the false roots are always real; sometimes they are imaginary; 

that is, while we can always conceive of as many roots for each equation as 

I have already assigned; yet there is not always a definite quantity corresponding 

to each root so conceived of.” 4 

To the next generations of mathematicians the character of the core of the theorem 

changed slightly. Where DESCARTES had not dealt with the nature of the imagined 

roots, they did. Soon the problem of demonstrating that all (imagined) roots of a 

1 In fact it had been formulated by GIRARD in 1629 (Gericke, 1970, 65). 

2 I shall use the term “imagined” to distinguish it from the current technical term “imaginary”. The 

word “complex” will be used to denote “imaginary” in the historical sense, i.e. numbers of the form 

a + b √ −1 where a, b are real and b �= 0. 

3 Since the time of CARDANO, negative roots had been called false or fictuous roots. The true roots of 

which DESCARTES spoke were the positive ones. 

4 “Au reste tant les vrayes racines que les fausses ne sont pas tousiours reelles; mais quelquefois 

seulement imaginaires; c’est a dire qu’on peut bien tousiours en imaginer autant que iay dit en 

chasque Equation; mais qu’il n’y a quelquefois aucune quantité, qui corresponde a celles qu’on 

imagine.” (Descartes, 1637, 380); English translation from (Smith and Latham, 1954, 175). 

57

58 Chapter 5. Towards unsolvable equations 

polynomial equations were complex, i.e. of the form a + b √ −1 for real a, b, was raised; 

and around the time of C. F. GAUSS (1777–1855), the theorem acquired the name of 

the Fundamental Theorem of Algebra. 

When G. W. LEIBNIZ (1646–1716) doubted that the polynomial x 4 + c 4 could be 

split into two real factors of the second degree, 5 the validity of the result seemed for a 

moment in doubt. L. EULER (1707–1783) demonstrated in 1749 (published 1751) that 

the set of complex numbers was closed under all algebraic and numerous transcendental 

operations. 6 Thus, at least by 1751 it would implicitly be known that √ i = 1+i 

√ 2 . 

This made LEIBNIZ’S supposed counter-example evaporate, since he factorized his 

polynomial as 

x 4 + c 4 � 

= x 2 � 

− ic 

2� 

x 2 + ic 2� 

� 

= x − √ � � 

ic x + √ � � 

ic x + √ � � 

−ic x − √ � 

−ic . 

Numerous prominent mathematicians of the eighteenth century — among them no- 

tably J. LE R. D’ALEMBERT (1717–1783), EULER, and J. L. LAGRANGE (1736–1813) — 

sought to provide proofs that any real polynomial could be split into linear and quadratic 

factors which would prove that any imagined roots were indeed complex. In the half- 

century 1799–1849 GAUSS gave a total of four proofs 7 which, although belonging to an 

emerging trend of indirect existence proofs, were considered to be superior in rigour 

when compared to those of his predecessors. 

Characterizing roots. The proofs of the Fundamental Theorem of Algebra were mostly 

existence proofs which did not provide any information on the computational aspect. 

Other similar, nonconstructive results were also pursued. An important subfield of the 

theory of equations was developed in order to characterize and describe properties of 

the roots of a given equation from a priori inspections of the equation and without 

explicitly knowing the roots. 

LAGRANGE’S study of the properties of the roots of particular equations was an 

offspring from his attempts to solve higher degree equations through algebraic ex- 

pressions (see below). 8 LAGRANGE’S interest in numerical equations, i.e. concrete 

equations in which some dependencies among the coefficients can exist, can be di- 

vided into three topics: the nature and number of the roots, limits for the values of 

these roots, and methods for approximating these. LAGRANGE made use of analytic 

geometry, function theory, and the Lagrangian calculus in order to investigate these 

topics. 9 

5 (K. Andersen, 1999, 69). 

6 (ibid., 70). 

7 GAUSS’ proofs can be found in (C. F. Gauss, 1863–1933, vol. 3) and have been collected in German 

translation in (C. F. Gauss, 1890). 

8 (Hamburg, 1976, 28). 

9 (ibid., 29–30).

5.1. Algebraic solubility before LAGRANGE 59 

Elementary symmetric relations. A different example of a priori properties of the 

roots of an equation was conceived of by men as G. CARDANO (1501–1576), F. VIÈTE 

(1540–1603), A. GIRARD (1595–1632), and I. NEWTON (1642–1727) in the sixteenth and 

seventeenth centuries. From inspection of equations of low degree they obtained (gen- 

erally by analogy and without general proofs) by a tacit theorem on the factorization 

of polynomials the dependency of the coefficients of the equation 

on the roots x1, . . . , xn given by 

x n + an−1x n−1 + an−2x n−2 + · · · + a1x + a0 = 0 

an−1 = − (x1 + · · · + xn) 

an−2 = x1x2 + · · · + xn−1xn 

. 

a1 = ± (x1x2 . . . xn−1 + · · · + x2x3 . . . xn) 

a0 = ∓x1x2 . . . xn. 

(5.1) 

These equations established the elementary symmetric relations between the roots and 

the coefficients of an equation. When proofs of these relations first emerged, they were 

obtained through formal manipulations of the tacitly introduced factors and were, 

thus, firmly within the established algebraic style. 

The relations (5.1) were to become a central tool in the theory of equations once 

NEWTON and E. WARING (∼1736–1798) realized that they were the basic, or ele- 

mentary, ones upon which all other symmetric functions of the roots depended rationally. 

10 

5.1 Algebraic solubility before LAGRANGE 

Among the multitude of possible questions concerning the unknown roots, one is 

particularly linked to the question of solving equations algebraically. It arose when 

mathematicians began investigating the form in which the roots can be written and is 

thus a first step in the direction of asking general solubility questions. 11 

The general approach taken in solving equations of degrees 2, 3 or 4 had since the 

first attempts been to reduce their solution to the solution of equations of lower degree. 

The example of the third degree equation solved by S. FERRO (1465–1526) around 

1515, by N. TARTAGLIA (1499/1500–1557) in 1539, and by CARDANO, who published 

the solution in 1545, might be illustrative 12 . Although CARDANO’S arguments and 

style were geometric, its algebraic content is presented in algebraic notation in box 1. 

10 See section 5.2.4. 

11 This aspect shall be dealt with below (see page 62ff) and section 8.4. 

12 In the present form, revised to expose central concepts, CARDANO’S solution closely resembles the 

young school-boy’s notes found in the section Ligninger af tredje Grads Opløsning (af Cardan) in 

ABEL’S notebook (Abel, MS:829, 139–141).


The algebraic reduction of the cubic equation When the general third degree equa- 

tion 

was subjected to the transformation 

x 3 + ax 2 + bx + c = 0 

x ↦−→ y − a 

3 

it took the canonical form (in which the term of the second highest degree did not 

appear) 

Letting y = u + v, CARDANO obtained 

and the equation could be satisfied if 

y 3 + ny + p = 0. (5.2) 

0 = (u + v) 3 + n (u + v) + p 

= u 3 + v 3 + (3uv + n) (u + v) + p, 

� u 3 + v 3 + p = 0, and 

3uv + n = 0. 

This system of equations could easily be reduced to the quadratic system (by letting 

U = u 3 , V = v 3 ) 

or 

� U + V = −p 

27UV = −n 3 

U 2 + Up − n3 

27 

= 0, (5.3) 

the solution of which was well known. Thus, U and V could be found, and finding 

u, v was only a matter of extracting 3 rd roots 

giving one of the roots y of (5.2) as 

u = 3√ U and v = 3√ V, 

y = u + v = 3√ U + 3√ V. 

Box 1: The algebraic reduction of the cubic equation 

✷


Purely formal methods were used in reducing the problem of the third degree 

equation to one of solving an equation of lower degree, here (5.3). A similar approach 

was adopted by L. FERRARI (1522–1565) in 1545 and by R. BOMBELLI (1526–1572) be- 

tween 1557 and 1560 to solve the general fourth degree equation. By the seventeenth 

century, the search for reductions of the general fifth degree equation into equations of 

lower degrees was establishing itself as a prestigious mathematical problem. LEIBNIZ 

and E. W. TSCHIRNHAUS (1651–1708) worked on the problem. In 1683 TSCHIRNHAUS 

published a procedure which, if applied to the general fifth degree equation, would 

reduce it to a binomial one with the help of a polynomial equation of degree 4. How- 

ever, as LEIBNIZ soon demonstrated, determining the coefficients of that polynomial 

unavoidably involved solving an equation of degree 24 which rendered TSCHIRN- 

HAUS’S reduction useless for solving the fifth degree equation algebraically. 13 Another 

independent and unsuccessful attempt at reducing the fifth degree equation was made 

by J. GREGORY (1638–1675), whose proposed reduction was based on a sixth degree 

auxiliary (resolvent) equation. 14 

The procedure of reduction to lower degree equations — so naturally suggested 

by incomplete induction from low degree equations — thus failed to give results for 

higher degree equations. The search had largely been conducted in an empirical way 

by proposing different reducing functions. It was wanting of a general and theoretical 

investigation; this was initiated around 1770. 

The search for resolvent equations conducted throughout the sixteenth, seven- 

teenth, and eighteenth centuries is properly seen as the quest to find algebraic solu- 

tions for all polynomial equations, thereby explicitly and constructively demonstrat- 

ing their algebraic solubility. A polynomial equation of degree n such as 

x n + an−1x n−1 + an−2x n−2 + · · · + a1x + a0 = 0 

is said to be algebraically solvable if its roots x1, . . . , xn can all be expressed by algebraic 

expressions in the coefficients a0, . . . , an−1 — the roots must be expressible as finite com- 

binations of the coefficients and constants using the five algebraic operations addition, 

subtraction, multiplication, division, and root extraction. 

From the second half of the eighteenth century, the diverse and largely empirical 

attempts to provide concrete reductions was superseded by theoretical and general 

investigations, mainly by LAGRANGE 1770–1771. In the work of LAGRANGE, the incli- 

nation towards general investigations was accompanied by the idea of studying per- 

mutations. 15 Both parts were essential in finally establishing that the long sought-for 

algebraic solution of the quintic equation was impossible. 

13 (Kracht and Kreyszig, 1990, 27–28) and (Kline, 1990, 599–600). 

14 (Whiteside, 1972, 528). 

15 For LAGRANGE’S focus on the general, see (Grabiner, 1981a, 317) and (Grabiner, 1981b, 39).


LEONHARD EULER. In his paper (L. Euler, 1732b), read to the St. Petersburg Academy 

and published in 1738, EULER gave his solutions to the equations of degree 2, 3, and 4 

and demonstrated that they could all be written in the form 16 

√ A for the second degree equation, 

3√ A + 3 √ B for the third degree equation, and 

4√ A + 4 √ B + 4√ C for the fourth degree equation, 

(5.4) 

where the quantities A, B, C were roots in certain resolvent equations of lower degree 

which could be obtained from the original equation. 17 EULER appears to have been 

the first to introduce the term “resolvent” and to attribute to it the central position it 

was to take in the future research on the solubility of equations. 18 

Extending these results, EULER conjectured that the resolvents also existed for the 

general equation of the fifth degree — and more generally for any higher degree equation 

— and that the roots could be expressed in analogy with (5.4). 19 

“Although this emphasizes the three particular cases [of equations of degrees 

2, 3, and 4], I, nevertheless, think that one could possibly, not without reason, conclude 

that also higher equations would possess similar solving equations. From 

the proposed equation 

x 5 = ax 3 + bx 2 + cx + d, 

I expect to obtain an equation of the fourth degree 

z 4 = αz 3 − βz 2 + γz − δ 

the roots of which will be A, B, C, and D, 

In the general equation 

x = 5√ A + 5√ B + 5√ C + 5√ D. 

x n = ax n−2 + bx n−3 + cx n−4 + etc. 

the resolvent equation will, I suspect, be of the form 

z n−1 = αz n−2 − βz n−3 + γz n−4 − etc., 

whose n − 1 known roots will be A, B, C, D, etc., 

z = n√ A + n√ B + n√ C + n√ D + etc. 

If this conjecture is valid and if the resolvent equations, which can obviously be 

said to have assignable roots, can be determined, I can obtain equations of lower 

degrees, and in continuing this process produce the true root of the equation.” 20 

16 (ibid., 7). 

17 The resolvent equation in the example of the third degree equation is (5.3). 

18 (F. Rudio, 1921, ix, footnote 2). 

19 According to (Eneström, 1912–1913, 346) already LEIBNIZ seemed conviced that the root of the general 

equation of the 5 th degree could be written in the form 

x = 5√ A + 5√ B + 5√ C + 5√ D.


The quotation illustrates how EULER’S conjecture amounted to the algebraic solu- 

bility of all polynomial equations. Returning to the problem, EULER sought to provide 

further evidence for his conjecture. 21 

EULER was led to a related problem concerning the multiplicity of values of rad- 

icals. By calculating the number of values of the multi-valued function consisting of 

n − 1 radicals 

n√ A + n √ B + n√ C + n√ D + . . . , 

EULER found that the function had n n−1 essentially different values, which apparently 

contradicted the fact that the equation of degree n should only have n roots. In a paper 

written in 1759, EULER refined his hypothesis of 1732 and conjectured that the roots 

of the resolvent A, B, C, D were dependent. EULER’S new conjecture was that the root 

would be expressible in the form 

x = ω + A n√ v + B n√ v 2 + C n√ v 3 + · · · + D n√ v n−1 , 

where the coefficients ω, A, B, C, . . . , D were rational functions of the coefficients, and 

the n − 1 other roots would be obtained by attributing to n√ v the n − 1 other values 

a n√ v, b n√ v, c n√ v . . . where a, b, c were the different n th roots of unity. 22 As will be illus- 

trated in chapter 7.1.2, N. H. ABEL (1802–1829) used a similar kind of argument. 

20 “8. Ex his etiamsi tribus tantum casibus tamen non sine sufficienti ratione mihi concludere videor 

superiorum quoque aequationum dari huiusmodi aequationes resolventes. Sic proposita aequatione 

coniicio dari aequationem ordinis quarti 

cuius radices si sint A, B, C et D, fore 

Et generatim aequationis 

x 5 = ax 3 + bx 2 + cx + d 

z 4 = αz 3 − βz 2 + γz − δ, 

x = 5√ A + 5√ B + 5√ C + 5√ D. 

x n = ax n−2 + bx n−3 + cx n−4 + etc. 

aequatio resolvens, prout suspicor, erit huius formae 

z n−1 = αz n−2 − βz n−3 + γz n−4 − etc. 

cuius cognitis radicibus omnibus numero n − 1, quae sint A, B, C, D etc., erit 

x = n√ A + n√ B + n√ C + n√ D + etc. 

Haec igitur coniectura si esset veritati consentanea atque si aequationes resolventes possent determinari, 

cuiusque aequationis in promtu foret radices assignare; perpetuo enim pervenitur ad 

aequationem ordine inferiorem hocque modo progrediendo tandem vera aequationis propositae 

radix innotescet.” (L. Euler, 1732b, 7–8); for a German translation, see (L. Euler, 1788–1791, vol. 3, 

9–10). 

21 (F. Rudio, 1921, ix–x). 

22 (ibid., x–xi).


ALEXANDRE-THÉOPHILE VANDERMONDE. Another very important component of 

the theory of equations in the early nineteenth century was the turn towards focusing 

on the expressive powers of algebraic expressions. This approach can be traced back 

to VANDERMONDE who in 1770 presented the Académie des Sciences in Paris with a trea- 

tise entitled Mémoire sur la résolution des équations. 23 There, he described the purpose 

of his investigations: 

“One seeks the most simple general values which can conjointly satisfy an 

equation of a certain degree.” 24 

As H. WUSSING has remarked, this weakly formulated program only gained importance 

through VANDERMONDE’S use of examples from low degree equations. 25 

VANDERMONDE’S aim was to build algebraic functions from the elementary symmet- 

ric ones which could assume the value of any root of the given equation. His approach 

was very direct, constructive, and computationally based. For example the elementary 

symmetric functions in the case of the general second degree equation 

(x − x1) (x − x2) = x 2 − (x1 + x2) x + x1x2 = 0 

are x1x2 and x1 + x2. The well known solution of the quadratic is 

� � 

1 

x1 + x2 + (x1 + x2) 

2 

2 � 

− 4x1x2 , 

which gives the two roots x1 and x2 when the square root is considered to be a two- 

valued function. Similarly, although with greater computational difficulties, VANDER- 

MONDE treated equations of degree 3 or 4. In those cases, he also constructed algebraic 

expressions having the desired properties. When he attacked equations of degree 5, 

however, he ended up with having to solve a resolvent equation of degree 6. Simi- 

larly, his approach led from a sixth degree equation to resolvent equations of degrees 

10 and 15. Having seen the apparent unfruitfulness of the approach, VANDERMONDE 

abandoned it. Later, the idea of studying the algebraic expressions formed from the 

elementary symmetric functions became central to ABEL’S research. 

Both EULER’S and VANDERMONDE’S approaches are, in spite of their apparently 

unsuccessful outcome, interesting in interpreting ABEL’S work on the theory of equa- 

tions. Firstly, ABEL’S proof of the impossibility of solving the general quintic by 

radicals (see chapter 6) is a fusion of ideas advanced by LAGRANGE and VANDER- 

MONDE, although there is no evidence that ABEL was familiar with VANDERMONDE’S 

work. Secondly, ABEL’S attempted general theory of algebraic solubility (see chapter 

8) bears resemblances to paths followed by EULER, VANDERMONDE and LAGRANGE. 

23 (Vandermonde, 1771). This paragraph on VANDERMONDE is largely based on (Wussing, 1969, 52– 

53). 

24 “On demande les valeurs générales les plus simples qui puissent satisfaire concurremment à une 

Équation [sic] d’un degré déterminé.” (Vandermonde, 1771, 366). 

25 (Wussing, 1969, 53)

5.2. LAGRANGE’s theory of equations 65 

Figure 5.1: JOSEPH LOUIS LAGRANGE (1736–1813) 

In section 8.4, I demonstrate how ABEL rigorized the assumptions of EULER’S conjec- 

ture and provided the conjecture with a proof. Before going into ABEL’S impossibility 

proof, it is necessary to present important results obtained by ABEL’S predecessors (in- 

cluding LAGRANGE) of which he made use, and demonstrate the change in approach 

and belief that made ABEL’S demonstration possible. 

5.2 LAGRANGE’s theory of equations 

Nobody influenced ABEL’S work on the theory of equations more than LAGRANGE. 

The present section briefly outlines the parts of LAGRANGE’S large and very influen- 

tial treatise Réflexions sur la résolution algébrique des équations which were of particular 

importance to ABEL’S work. 26 LAGRANGE’S work is well studied and has often — 

and rightfully so, I think — been seen as one of the first major steps towards linking 

the theory of equations to group theory. 27 However, with the focus mainly on ABEL’S 

approach, emphasis is given only to points of direct relevance for this. 

When LAGRANGE in 1770–1771 had his Réflexions sur la résolution algébrique des 

équations published in the Mémoires of the Berlin Academy, he was a well established 

mathematician held in high esteem. The Réflexions was a thorough summary of the 

26 (Lagrange, 1770–1771). 

27 See for instance (Wussing, 1969, 49–52, 54–56), (Kiernan, 1971), (Hamburg, 1976), or (Scholz, 1990, 

365–372).


nature of solutions to algebraic equations which had been uncovered until then. Like 

EULER and VANDERMONDE had done, LAGRANGE investigated the known solutions 

of equations of low degrees hoping to discover a pattern feasible to generalizations to 

higher degree equations. Where EULER had sought to extend a particular algebraic 

form of the roots, and VANDERMONDE had tried to generalize the algebraic func- 

tions of the elementary symmetric functions, LAGRANGE’S innovation was to study 

the number of values which functions of the coefficients could obtain under permu- 

tations of the roots of the equation. Although he exclusively studied the values of the 

functions under permutations, his results marked a first step in the emerging indepen- 

dent theory of permutations. In turn, this permutation theory was soon, through its 

central role in E. GALOIS’S (1811–1832) theory of algebraic solubility, incorporated in 

an abstract theory of groups which grew out of progress made in the nineteenth and 

twentieth centuries. 28 

The work Réflexions sur la résolution algébrique des équations was divided into four 

parts reflecting the structure of LAGRANGE’S investigation. 

1. “On the solution of equations of the third degree” (Lagrange, 1770–1771, 207– 

254) 

2. “On the solution of equations of the fourth degree” (ibid. 254–304) 

3. “On the solution of equations of the fifth and higher degrees” (ibid. 305–355) 

4. “Conclusion of the preceding reflections with some general remarks concerning 

the transformation of equations and their reduction to a lower degree” (ibid. 355– 

421) 

Of these the latter part is of particular interest to the following discussion. Its aim 

was to provide a link between the number of values a function could obtain under 

permutations and the degree of the associated resolvent equation. Most accounts of 

LAGRANGE’S contribution in the theory of equations emphasize the 100 th section deal- 

ing with the rational dependence of semblables fonctions, a topic which became central 

after the introduction of GALOIS theory. 29 

5.2.1 Formal values of functions 

Central to LAGRANGE’S treatment of the general equations of all degrees were his 

concepts of formal functional equality and formal appearance of expressions. 30 LA- 

GRANGE considered two rational functions (formally) equal only when they were 

given by the same algebraic formulae, in which xy and yx were considered equal 

28 (Wussing, 1969). 

29 For instance (J. Pierpont, 1898, 333–335) and (Scholz, 1990, 370). 

30 (Kiernan, 1971, 46).


because both multiplication (and addition) were implicitly assumed to be commuta- 

tive, associative, and distributive. This concept of formal equality was intertwined 

with LAGRANGE’S focus on the formal appearance of expressions which made the 

form and not the value the important aspect of expressions. Denoting the roots of the 

general µ th degree equation by x1, . . . , xµ, LAGRANGE considered the variables (roots) 

to be independent symbols. For example, in LAGRANGE’S view, the two expressions 

x1 − x2 and x2 − x1 were always (formally) different, although particular values could 

be given to x1 and x2 such that the values of the two expressions were equal. The 

independence of the symbols x1, . . . , xµ reflected the fact that in a general equation, the 

coefficients were considered independent; to treat special, e.g. numerical equations, a 

modified approach had to be adapted. 

LAGRANGE’S formal approach reflects a general eighteenth century conception of 

polynomials not as functional mappings but as expressions combined of various sym- 

bols: variables and constants, either known or unknown. LAGRANGE was not par- 

ticularly explicit about this notion of formal equality which occurs throughout his 

investigations; however, he emphasized that 

“it is only a matter of the form of these values and not their absolute [numerical] 

quantities.” 31 

The focus on formal values was lifted when GALOIS saw that in order to address 

special equations in which the coefficients were not completely general — some or all 

of them might be restricted to certain numerical values — he had to consider the nu- 

merical equality of the symbols in place of LAGRANGE’S formal equality. 

5.2.2 The emergence of permutation theory 

An important part of LAGRANGE’S approach was the introduction of symbols denot- 

ing the roots which enabled him to treat them as if they had been known. 32 This al- 

lowed him to focus his attention on the action of permutations on formal expressions 

in the roots. LAGRANGE set up a system of notation in which 

f �� x ′� � x ′′� � x ′′′�� 

meant that the function f was (formally) altered by any (non-identity) permutation 

of x ′ , x ′′ , x ′′′ . 33 For instance, the expression x ′ + αx ′′ + α 2 x ′′′ would be altered by any 

non-identity permutation if α was an independent symbol (or a number, say, α = 2). 

If the function remained unaltered when x ′ and x ′′ were interchanged, LAGRANGE 

wrote it as 

f �� x ′ , x ′′� � x ′′′�� . 

31 “il s’agit ici uniquement de la forme de ces valeurs et non de leur quantité absolute.” (Lagrange, 

1770–1771, 385). 

32 (Kiernan, 1971, 45). Reminiscences of this can also be found with EULER. 

33 (Lagrange, 1770–1771, 358).


For example, x ′ x ′′ + x ′′′ would remain unaltered by interchanging x ′ and x ′′ , but any 

permutation involving x ′′′ would alter it. Finally, if the function was symmetric (i.e. 

formally invariant under all permutations of x ′ , x ′′ , x ′′′ ), he wrote 

f �� x ′ , x ′′ , x ′′′�� . 

The most important examples of such functions were the elementary symmetric func- 

tions, 

x ′ + x ′′ + x ′′′ , 

x ′ x ′′ + x ′ x ′′′ + x ′′ x ′′′ , and 

x ′ x ′′ x ′′′ . 

With this notation and his concept of formal equality, LAGRANGE derived far- 

reaching results on the number of (formally) different values which rational func- 

tions could assume under all permutations of the roots. With the hindsight that the 

set of permutations form an example of an abstract group, a permutation group, one 

may see that LAGRANGE was certainly involved in the early evolution of permutation 

group theory. As we shall see in the following section, he was led by this approach 

to Lagrange’s Theorem, which in modern terminology expresses that the order of a sub- 

group divides the order of the group. However, since LAGRANGE dealt with the ac- 

tions of permutations on rational functions, he was conceptually still quite far from the 

concept of groups. LAGRANGE’S contribution to the later field of group theory laid in 

providing the link between the theory of equations and permutations which in turn 

led to the study of permutation groups from which (in conjunction with other sources) 

the abstract group concept was distilled. 34 More importantly, LAGRANGE’S idea of in- 

troducing permutations into the theory of equations provided subsequent generations 

with a powerful tool. 

5.2.3 LAGRANGE’s resolvents 

Another result found by LAGRANGE, of which ABEL later made eminent and frequent 

use in his investigations, concerned the polynomial having as its roots all the different 

values which a given function took when its arguments were permuted. Starting with 

the case of the quadratic equation having as roots x and y 

z 2 + mz + n = 0, (5.5) 

LAGRANGE studied the values f [(x) (y)] and f [(y) (x)] which were all the values a 

rational function f could obtain under permutations of x and y. He then demonstrated 

that the equation in t 

34 (Wussing, 1969). 

Θ = [t − f [(x) (y)]] × [t − f [(y) (x)]] = 0


had coefficients which depended rationally on the coefficients m and n of the original 

quadratic (5.5). 35 This may not be so surprising because today this is easily realized by 

observing that the coefficients are symmetric in f [(x) (y)] and f [(y) (x)]. However, 

this was precisely the result, which LAGRANGE was about to prove. 

Subsequently, LAGRANGE carried out the rather lengthy argument for the general 

cubic. Thereby, he proved that the equation which had the six values of f under all 

permutations of the three roots of the cubic as its roots would be rationally expressible 

in the coefficients of the cubic. 

Based on these illustrative cases of equations of low (second and third) degrees, 

LAGRANGE could state the following two results as a general theorem generalizing 

the argument sketched for the quadratic above. 36 In the general case, the degree of the 

equation was denoted µ, and the polynomial having all the values which the given 

function f assumes under permutations of the µ roots was denoted Θ and its degree 

ϖ. LAGRANGE then stated: 

1. The degree ϖ of Θ divides µ! where µ is the degree of the proposed equation, 

and 

2. The coefficients of the equation Θ = 0 depend rationally on the coefficients of 

the original equation. 

In his proof of this general theorem, LAGRANGE’S notation and machinery re- 

stricted his argument slightly. Because he worked with permutations acting on func- 

tions and had no way of clarifying the underlying sets of permutations, his argu- 

ments — which contain all the necessary ideas — may seem to rely on analogies with 

the cases of low degrees. 37 Be that as it may, by any contemporary standards, LA- 

GRANGE’S argument must have been a convincing proof and LAGRANGE’S general 

theorem became an immensely important tool in the investigations of future alge- 

braists. 

“From this it is clear that the number of different functions [i.e. different values 

obtained by permuting the arguments] must increase following the products of 

natural numbers 

1, 1.2, 1.2.3, 1.2.3.4, . . . , 1.2.3.4.5 . . . µ. 

Having all these functions one will have the roots of the equation Θ = 0; thus, if 

it is represented as 

Θ = t ϖ − Mt ϖ−1 + Nt ϖ−2 − Pt ϖ−3 + · · · = 0, 

35 (Lagrange, 1770–1771, 361). 

36 (ibid., 369–370). 

37 Since LAGRANGE’S proof can easily be adapted to newer frameworks of proof, this interpretation 

may be a matter of personal taste. However, I do see a major difference between LAGRANGE’S proof 

by analogy and pattern and the proof later given by CAUCHY (see below).


one will have ϖ = 1.2.3.4 . . . µ and the coefficient M will equal the sum of all the 

obtained functions, the coefficient N will equal the sum of all products of these 

functions multiplied two by two, the coefficient P will equal the sum of all products 

of the functions multiplied three by three, and so on. [. . . ] 

And since we have demonstrated above that the expression Θ must necessarily 

be a rational function of t and the coefficients m, n, p, . . . of the proposed equation, 

it follows that the quantities M, N, P, . . . are necessarily rational functions 

of m, n, p, . . . which one can find directly as we have seen done in the preceding 

sections.” 38 

Expressed in modern mathematical language, the first part of the above result is 

the equivalent of the Lagrange’s Theorem of group theory, which states that the order 

of any subgroup divides the order of the group. As we shall see in section 5.6, the 

first general proof was given by A.-L. CAUCHY (1789–1857) based on his approach to 

working with permutations. 

The second part of the result was used extensively by ABEL, although he never 

gave references when applying it. ABEL used the result in a form equivalent to the 

following theorem, formulated in a compact notation. 

Theorem 1 If φ � � 

x1, . . . , xµ is a rational function which takes on the values φ1, . . . , φϖ under 

all permutations of its arguments x1, . . . , xµ and the equation 

Θ = 

ϖ 

ϖ 

∏ (v − φk) = ∑ Akv k=1 

k=0 

k 

is formed, then all the coefficients A0, . . . , Aϖ are symmetric functions of x1, . . . , xµ. ✷ 

(5.6) 

The link between the above theorem as used by ABEL and LAGRANGE’S second 

result can be obtained through a result which I denote Waring’s formulae. These for- 

mulae, obtained by NEWTON and WARING by different routes and described in the 

next section, were incorporated by LAGRANGE in his work and must have been ac- 

cepted as common knowledge in LAGRANGE’S era. As quoted above, LAGRANGE’S 

38 “D’où l’on voit clairement que le nombre des fonctions différentes doit croître suivant les produits 

des nombres naturels 

1, 1.2, 1.2.3, 1.2.3.4, . . . , 1.2.3.4.5 . . . µ. 

Ayant toutes ces fonctions on aura donc les racines de l’équation Θ = 0; de sorte que, si on la 

représente par 

Θ = t ϖ − Mt ϖ−1 + Nt ϖ−2 − Pt ϖ−3 + · · · = 0, 

on aura ϖ = 1.2.3.4 . . . µ; et le coefficient M sera égal à la somme de toutes les fonctions trouvées, 

le coefficient N égal à la somme de tous les produits de ces fonctions multipliées deux à deux, le 

coefficient P égal à la somme de tous les produits des mêmes fonctions multipliées trois à trois, et 

ainsi de suite. [. . . ] 

Et comme nous avons démontré ci-dessus que l’expression de Θ doit être nécessairement une fonction 

rationnelle de t et des coefficients m, n, p, . . . de l’équation proposée, il s’ensuit que les quantités 

M, N, P, . . . seront nécessairement des fonctions ratinnelles de m, n, p, . . . qu’on pourra trouver 

directement, comme nous l’avons pratiqué dans les Sections précédentes.” (Lagrange, 1770–1771, 

369).


Figure 5.2: EDWARD WARING (1734–1798) 

theorem stated that the coefficients, here A0, . . . , A ¯ω, were rational functions of the co- 

efficients of the given equation. By Waring’s formulae, any such rational function of the 

coefficients was a symmetric function of the roots. 

5.2.4 Waring’s formulae 

The elementary symmetric functions of the roots of an equation, which since the times 

of VIÈTE and NEWTON had been known to agree with the coefficients (see section 5), 

was seen by the little known British mathematician WARING to provide a basis for the 

study of all symmetric functions of the equation’s roots. In his Miscellanea analytica of 

1762, WARING demonstrated that all rational symmetric functions of the roots could 

be expressed rationally in the elementary symmetric functions. 39 In his other more 

influential work Meditationes algebraicae, 40 to which LAGRANGE referred, 41 the result 

was contained in the first chapter. There, WARING dealt with the determination of the 

power sums of the roots x1, . . . , xµ (modern notation) 

39 (Waerden, 1985, 76–77). 

40 (Waring, 1770). 

41 (Lagrange, 1770–1771, 369–370). 

µ 

∑ x 

k=1 

m k 

for integer m


from the coefficients of the equation. 42 The solution was the so-called Waring’s Formu- 

lae giving a procedure alternative to one given earlier by NEWTON. From this, WAR- 

ING proceeded to show how any function of the roots of the form (modern notation 

writing Σµ for the symmetric group) 

∑ x 

σ∈Σµ 

a1 σ(1) xa2 . . . xaµ 

σ(2) σ(µ) with a1, . . . , aµ non-negative integers (5.7) 

could be expressed as an integral function of the power sums of the roots. 43 Thus, 

WARING had demonstrated that all rational and symmetric functions of x1, . . . , xµ de- 

pended rationally on the power sums and thus on the coefficients of the equation by 

the preceding result. 44 

Although this important theorem was stated and proved by WARING, it entered 

the mathematical toolbox of the early nineteenth century mainly through LAGRANGE’S 

adaption of it in his Réflexions (which is the reason for treating it at this place). While 

WARING’S notation and letter-manipulating approach had hampered his presentation, 

LAGRANGE dealt with it in a clear and integrated fashion in the Réflexions. 45 There, he 

observed that if the function f had the form 

f 

��x ′ ′′ 

, x � � � 

x 

′′′� 

x iv� 

. . . 

indicating that x ′ and x ′′ appeared symmetrically, the roots of the equation Θ = 0 

(5.6) would be equal in pairs, whereby the degree could be reduced to µ! 

2 . After briefly 

studying a few other types of functions f , LAGRANGE concluded that if f had the form 

f 

� 

, 

�� 

x ′ , x ′′ , x ′′′ , . . . , x (µ)�� 

, 

i.e. was a symmetric function of the roots, the degree of the equation Θ = 0 (5.6) could 

be reduced to one and f would be given rationally in the coefficients of the original 

equation. 

5.3 Solubility of cyclotomic equations 

Thirty years after LAGRANGE’S creative studies on known solutions to low degree 

equations, and in particular properties of rational functions under permutations of 

their arguments, another great master published a work of profound influence on 

early nineteenth century mathematics. In Göttingen, GAUSS was located at a physical 

distance from the emerging centers of mathematical research in Paris and Berlin. By 

1801, the Parisian mathematicians had for some time been publishing their results in 

42 (Waring, 1770, 1–5). 

43 (ibid., 9–18). 

44 By formal equality, all terms of the same degree would have to have identical coefficients, and thus 

any rational symmetric function could be decomposed into functions of the form (5.7). 

45 (Lagrange, 1770–1771, 371–372).

5.3. Solubility of cyclotomic equations 73 

Figure 5.3: CARL FRIEDRICH GAUSS (1777–1855) 

French — and, within a generation, the German mathematicians would also be writ- 

ing in their native language, at least for publications intended for A. L. CRELLE’S 

(1780–1855) Journal für die reine und angewandte Mathematik. But GAUSS published his 

fundamental work Disquisitiones arithmeticae as a Latin monograph as was still cus- 

tomary for his generation of German scholars. 

The book cosisted of seven sections, although allusions and references were made 

to an eighth section which GAUSS never completed for publication. 46 The main part 

was concerned with the theory of congruences, the theory of forms, and related num- 

ber theoretic investigations. Together, these topics provided a new foundation, em- 

phasis, and disciplinary independence — as well as a wealth of results — for nine- 

teenth century number theorists — in particular G. P. L. DIRICHLET (1805–1859) — 

to elaborate. In dealing with the classification of forms and describing primitive roots, 

GAUSS made use of “implicit group theory”. 47 Despite the fact that both LAGRANGE 

and GAUSS worked with particular instances of groups, neither of them introduced 

an abstract concept of groups. 

One of the new tools applied by GAUSS in the theory of congruences was that of 

primitive roots. In the articles 52–57, GAUSS gave his exposition of EULER’S treatment 

of primitive roots. A primitive root k of modulus µ is an integer 1 < k < µ such that 

46 (C. F. Gauss, 1863–1933, vol. 1, 477). It is, however, included among the Nachlass in the second 

volume of the Werke (ibid.). 

47 (Wussing, 1969, 40–44).


the set of remainders of its powers k 1 , k 2 , . . . , k µ−1 modulo µ coincides with the set 

{1, 2, . . . , µ − 1}, possibly in a different order. A central result obtained was the existence 

of the p − 1 different primitive roots 1, 2, . . . , 

p − 1 of modulus p if p were assumed to be prime. 

5.3.1 The division problem for the circle 

In the last section of his Disquisitiones arithmeticae, 48 GAUSS turned his investigations 

toward the equations defining the division of the periphery of the circle into equal 

parts. He was interested in the ruler-and-compass constructibility 49 of regular poly- 

gons and was therefore led to study in detail how, i.e. by the extraction of which roots, 

the binomial equations of the form 

x n − 1 = 0 (5.8) 

could be solved algebraically. If the roots of this equation could be constructed by ruler 

and compass, then so could the regular p-gon. It is evident from GAUSS’ mathematical 

diary that this problem had occupied him from a very early stage in his mathemati- 

cal career and had been a deciding factor in his choice of mathematics over classical 

philology. 50 The very first entry in his mathematical progress diary from 1796 read: 

“[1] The principles upon which the division of the circle depend, and geometrical 

divisibility of the same into seventeen parts, etc. [1796] March 30 Brunswick.” 51 

In his introductory remarks, GAUSS noticed that the approach which had led him 

to the division of the circle could equally well be applied to the division of other tran- 

scendental curves of which he gave the lemniscate as an example. 

“The principles of the theory which we are going to explain actually extend 

much farther than we will indicate. For they can be applied not only to circular 

functions but just as well to other transcendental functions, e.g. to those 

which depend on the integral � � 1/ √ � 1 − x 4�� dx and also to various types of 

congruences.” 52 

48 (C. F. Gauss, 1801). For historical studies, see for instance (Wussing, 1969, 37–44), (Schneider, 1981, 

37–50), or (Scholz, 1990, 372–376). 

49 Throughout, I refer to Euclidean construction, i.e. by ruler and compass when I speak of constructions 

or constructibility. 

50 (Biermann, 1981, 16). 

51 “[1.] Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim 

partes etc. [1796] Mart. 30. Brunsv[igae]” (C. F. Gauss, 1981, 21, 41); English translation from (J. J. 

Gray, 1984, 106). 

52 “Ceterum principia theoriae, quam exponere aggredimur, multo latius patent, quam hic extenduntur. 

Namque non solum ad functiones circulares, sed pari successu ad multas functiones transscendentes 

applicari possunt, e.g. ad eas, quae ab integrali � 

√ 

dx pendent, praetereaque etiam ad 

(1−x4 ) 

varia congruentiarum genera.” (C. F. Gauss, 1801, 412–413); English translation from (C. F. Gauss, 

1986, 407).


However, as he was preparing to write a treatise on these topics, GAUSS chose to 

leave this extension out of the Disquisitiones. GAUSS never wrote the promised treatise, 

and after ABEL had published his first work on elliptic functions culminating in the 

division of the lemniscate, 53 GAUSS gave him credit for bringing these results into 

print. 54 

A first simplification of the study of the constructibility of a regular n-gon was 

made when GAUSS observed that he needed only to consider cases in which n was a 

prime since any polygon with a composite number of edges could be constructed from 

the polygons with the associated prime numbers of edges. Equations expressing the 

sine, the cosine, and the tangent were well known, but none of those were as suitable 

for GAUSS’ purpose as the equation x n − 1 = 0 of which he knew that the roots were 55 

cos 2kπ 2kπ 

+ i sin = 1 when 0 ≤ k ≤ n − 1. 

n n 

Inspecting these roots, GAUSS observed that the equation xn − 1 = 0 for odd n 

had a single real root, x = 1, and the remaining imaginary roots were all given by the 

equation 

X = xn − 1 

x − 1 = xn−1 + x n−2 + · · · + x + 1 = 0, (5.9) 

the roots of which GAUSS thought of as forming the complex Ω. When GAUSS used 

the term “complex” (Latin: complexum) he thought of it as a collection of objects 

(here roots) without any structure imposed. 56 Initially, GALOIS used the French term 

groupe in a similar (naive) way before it later gradually acquired its status as a mathe- 

matical term. 57 This evolution of everyday words into mathematical concepts appears 

to be a recurring feature of mathematics in the early nineteenth century when so many 

terms became precisely defined and re-defined. 58 GAUSS demonstrated that if r desig- 

nated any root in Ω, all roots of (5.8) could be expressed as powers of r, thereby saying 

that any root in Ω was a primitive n th root of unity. 

5.3.2 Irreducibility of the equation xn −1 

x−1 

= 0 

An interesting feature of GAUSS’ approach was his focusing on the complex or system of 

roots instead of the individual roots. This slight shift in the conception of roots enabled 

GAUSS (as it had enabled LAGRANGE) 59 to study properties of the equations which 

53 (N. H. Abel, 1827b). 

54 (Crelle→Abel, 1828/05/18. N. H. Abel, 1902a, 62). 

55 GAUSS wrote P (periphery) for 2π; however the use of i for √ −1 is his. 

56 Later, in 1831, GAUSS introduced the term complex numbers to denote numbers which had hitherto 

been designated imaginary; (Gericke, 1970, 57). I fail to see any connection between the term complexum 

as used here and the later, technical term. 

57 (Wussing, 1969, 78). 

58 See also section 21.2. 

59 In the preface, GAUSS briefly described his debt to the number theoretic investigations of the “modern 

authors” FERMAT, EULER, LAGRANGE, and LEGENDRE. If GAUSS had read LAGRANGE’S Réflexions, 

he did not refer explicitly to it.


could only be captured in studies of the entire system of roots. To GAUSS, the most 

important properties were those of decomposability and irreducibility. GAUSS demon- 

strated through an ad hoc argument that the function X (5.9) could not be decomposed 

into polynomials of lower degree with rational coefficients. In modern terminology, 

he proved that the polynomial X was irreducible over Q. 

GAUSS’ proof assumed that the function 

X = x n−1 + x n−2 + · · · + x + 1 

was divisible by a function of lower degree 

P = x λ + Ax λ−1 + Bx λ−2 + · · · + Kx + L, (5.10) 

in which the coefficients A, B, . . . , K, L were rational numbers. Assuming X = PQ, 

GAUSS introduced the two systems of roots P and Q of P and Q respectively. From 

these two systems GAUSS defined another two consisting of the reciprocal roots60 � 

ˆP = r −1 � � 

: r ∈ P and ˆQ = r −1 � 

: r ∈ Q . 

Although GAUSS consistently termed the roots of ˆP and ˆQ reciprocal roots, it is easy for 

us to see that they are what we would term conjugate roots since any root in P has unit 

length. 

GAUSS split the subsequent argument into four different cases. The opening one 

is the most interesting one, namely the case in which P = ˆP, i.e. when all roots of 

P = 0 occur together with their conjugates. It may be surprising that GAUSS consid- 

ered other cases as we would expect him to know that in any polynomial with real 

coefficients the imaginary roots occur in conjugate pairs. K. JOHNSEN has argued that 

this apparently unnecessary complication in GAUSS’ argument can be traced back to 

a more general concept of irreducibility over fields different from Q, for instance the 

field Q (i). 61 If so, there are no explicit hints at such a concept in the Disquisitiones, 

and the result which GAUSS proved only served a very specific purpose in his larger 

argument, and did not give a general concept, general criteria, or a body of theo- 

rems concerning irreducibility over Q or any other field. The proof relies more on 

number theory (higher arithmetic) than on general theorems and criteria concerning 

irreducible equations, let alone any general concept of fields distinct from the rational 

numbers Q. 

After observing that P was the product of λ 2 

(x − cos ω) 2 + sin 2 ω, 

paired factors of the form 

GAUSS concluded that these factors would assume real and positive values for all real 

values of x, which would then also apply to the function P (x). He then formed n − 1 

60 The notation ˆP and ˆQ for these is mine. 

61 (Johnsen, 1984).


auxiliary equations 62 

P (k) = 0 where 1 ≤ k ≤ n − 1 

defined by their root systems P (k) consisting of k th powers of the roots of P = 0, 

P (k) = 

� 

r k � 

: r ∈ P , 

P (k) (x) = ∏ (x − s) = ∏ 

s∈P (k) 

r∈P 

� 

x − r k� 

. 

Following the introduction of the numbers pk defined by 

pk = P (k) � 

(1) = ∏ (1 − r) = ∏ 1 − r 

s∈P (k) 

r∈P 

k� 

, 

GAUSS used properties derived in a previous article to establish 

Furthermore, 

n−1 

∏ P 

k=1 

(k) (x) = 

n−1 

∏ pk = 

k=1 

n−1 

∏ 

∏ 

k=1 r∈P 

n−1 

∑ pk = 

k=1 

� 

x − r k� 

= ∏ r∈P 

n−1 

∑ P 

k=1 

(k) (1) = nA. (5.11) 

n−1 � 

∏ x − r 

k=1 

k� = ∏ X = X 

r∈P 

λ , and 

n−1 

∏ P 

k=1 

(k) (1) = X λ (1) = n λ since X (1) = n. 

From the article describing the construction of an equation with the k th powers 

of the roots of a given equation as its roots, GAUSS knew that the coefficients of 

P (1) , . . . , P (n−1) would be rational numbers if the coefficients of P were rationals. Much 

earlier, in article 42, he had furthermore demonstrated that the product of two poly- 

nomials with rational but not integral coefficients could not be a polynomial with 

integral coefficients. Since X had integral coefficients and P had rational coefficients 

by assumption, it followed that the coefficients of P (1) , . . . , P (n−1) would indeed be in- 

tegers, since any P (k) was a factor of X λ with rational coefficients. Consequently, the 

quantities p k would have to be integral, and since their product was n λ and there were 

n − 1 > λ of them, at least n − 1 − λ of the quantities p k would have to be equal to 1 

and the others would have to equal n or some power of n since n was assumed to be 

prime. But if the number of quantities equal to 1 was g it would follow that 

n−1 

∑ pk ≡ g 

k=1 

( mod n) , 

which GAUSS saw would contradict (5.11) since 0 < g < n. 

62 The notation P (k) and P (k) is mine.


The other cases, which in the presently adopted notation can be described as 

2.P �= ˆP and P ∩ ˆP �= ∅, 

3.Q ∩ ˆQ �= ∅, and 

4.P ∩ ˆP = ∅ and Q ∩ ˆQ = ∅, 

could all be brought to a contradiction, either directly or by referring to the first case 

described above. 

The fruitfulness of the proof of the irreducibility of X was that it demonstrated 

that if X was decomposed into factors of lower degrees (such as 5.10) some of these 

had to have irrational coefficients. Thus any attempt at determining the roots would 

have to involve equations of degree higher than one. The purpose of the following 

investigation was to gradually reduce the degree of these equations to minimal values 

by refining the system of roots. 

5.3.3 Outline of GAUSS’s proof 

Continuing from the result above that any root r in Ω was a primitive n th root of unity, 

GAUSS wrote [1] , [2] , . . . , [n − 1] for the associated powers of r. He introduced the con- 

cept of periods by defining the period ( f , λ) to be the set of the roots [λ] , [λg] , . . . , � λg f −1� , 

where f was an integer, λ an integer not divisible by n, and g a primitive root of the 

modulus n. Connected to the period, he introduced the sum of the period, which he also 

designated ( f , λ), 

( f , λ) = 

f −1 � 

∑ λg 

k=0 

k� 

, 

and the first result, which he stated concerning these periods, was their independence 

of the choice of g. 

Throughout the following argument, GAUSS let g designate a primitive root of 

modulus n and constructed a sequence of equations through which the periods (1, g), 

i.e. the roots in X = 0 (5.9), could be determined. Assuming that the number n − 1 

had been decomposed into primes as 

n − 1 = 

u 

∏ pk, k=1 

GAUSS partitioned the roots of Ω into n−1 

p 1 periods, each of p1 terms. From these, he 

formed p1 equations X ′ = 0 having the n−1 

p 1 sums of the form (p1, λ) as its roots. By 

a central theorem proved using symmetric functions, 63 he proved that the coefficients 

of these latter equations depended upon the solution of yet another equation of de- 

gree p1. Thus the solution of the original equation of degree n had been reduced to 

solving p1 equations X ′ = 0 each of degree n−1 

p 1 and a single equation of degree p1. 

63 (C. F. Gauss, 1801, §350).


By repeating the procedure, GAUSS could solve the equation X ′ = 0 by solving p2 

equations of degree n−1 

p 1p2 and a single equation of degree p2. Similarly, the procedure 

could be iterated further until the solution of the equation X = 0 of degree n − 1 had 

been reduced to solving u equations of degrees p1, p2, . . . , pu since the other equations 

would ultimately have degree 1. 

A special case emerged if n − 1 was a power of 2. It was well known that square 

roots could always be constructed by ruler and compass. Therefore, if n had the form 

n = 1 + 2 k , 

the construction of the roots of (5.8) could be carried out by ruler and compass. By ap- 

plying this to k = 4, GAUSS demonstrated that the regular 17-gon could be constructed 

by ruler and compass giving the first new constructible regular polygon since the time 

of EUCLID (∼295 B.C.) (∼295BC). 64 

By the same argument he had devised to consider only prime values of n, GAUSS 

could also conclude that the construction of the regular n-gon was possible by ruler 

and compass when n had the form 

h 

m 

n = 2 ∏ (1 + 2 

k=1 

uk) when {u k} was a set of distinct integers such that {1 + 2 u k} were primes, the so-called 

Fermat primes. The converse implication, that only such n-gons were constructible, 

was claimed without detailed proof by GAUSS: 

“Whenever n − 1 involves prime factors other than 2, we are always led to 

equations of higher degree, namely to one or more cubic equations when 3 appears 

once or several times among the prime factors of n − 1, to equations of the 

fifth degree when n − 1 is divisible by 5, etc. We can show with all rigor that 

these higher-degree equations cannot be avoided in any way nor can they be 

reduced to lower-degree equations. The limits of the present work exclude this 

demonstration here, but we issue this warning lest anyone attempt to achieve geometric 

constructions for sections other than the ones suggested by our theory (e.g. 

sections into 7, 11, 13, 19, etc. parts) and so spend his time uselessly.” 65 

64 GAUSS, himself, was very aware of the progress he had made, see (C. F. Gauss, 1986, 458) and 

(Schneider, 1981, 38–39). As always, the date given for EUCLID is taken from the Dictionary of Scientific 

Biography. 

65 “Quoties autem n − 1 alios factores primos praeter 2 implicat, semper ad aequationes altiores deferimur; 

puta ad unam pluresve cubicas, quando 3 semel aut pluries inter factores primos ipsius 

n − 1 reperitur, ad aequationes quinti gradus, quando n − 1 divisibilis est per 5 etc., omnique rigore 

demonstrare possumus, has aequationes elevatas nullo modo nec evitari nec ad inferiores reduci 

posse, etsi limites huius operis hanc demonstrationem hic tradere non patiantur, quod tamen monendum 

esse duximus, ne quis adhuc alias sectiones praeter eas, quas theoria nostra suggerit, e.g. 

sectiones in 7, 11, 13, 19 etc. partes, ad constructiones geometricas perducere speret, tempusque 

inutiliter terat.” (C. F. Gauss, 1801, 462); English translation from (C. F. Gauss, 1986, 459). Bold-face 

has been substituted for the original small-caps.


The class of equations (the cyclotomic ones), which GAUSS had demonstrated had 

constructible roots, was also interesting from the point of algebraic solubility of equa- 

tions. In his proof, GAUSS had demonstrated that they were indeed solvable by radi- 

cals including only square roots, whereby the first new non-elementary class of solv- 

able equations of high degrees had been established. By the time GAUSS wrote his 

Disquisitiones, he had come to suspect that not all equations were solvable by radicals. 

A few years later, ABEL could consider this newly found class to be a special example 

of equations having the nice property of being algebraically solvable. 

5.4 Belief in algebraic solubility shaken 

In the seventeenth century, the belief in the algebraic solubility (in radicals) of all poly- 

nomial equations seems to have been in little dispute. The question of solubility was 

not an issue when the prominent mathematicians such as TSCHIRNHAUS searched for 

a general solution. Half-way through the eighteenth century the problem had taken 

a slight turn when EULER in 1732 proposed to investigate the hypothesis that the roots 

of the general n th degree equation could be written as a sum of n − 1 root extractions 

of degree n − 1. Although he advanced this as a hypothesis and his search for definite 

proof was in vain, he based his 1749 “proof” of the Fundamental Theorem of Algebra on 

the belief that any polynomial equation could be reduced to pure equations. 66 Towards 

the end of the eighteenth century, the outspoken beliefs of the most prominent math- 

ematicians had changed, though. Mathematicians with a keen interest in the subject 

started to suspect that the reduction to pure equations was beyond — not only their 

grasp — but the power of their existing tools. At the turn of the century one of the most 

influential mathematicians, GAUSS, declared the reduction to be outright impossible. 

The belief in the algebraic solubility of general equations did not vanish completely 

with GAUSS’ proclamation of its impossibility. In chapter 6.9, where I discuss the 

reception of ABEL’S work on the theory of equations, I shall also discuss the “inertia” 

of the mathematical community in this respect. 

5.4.1 “Infinite labor” 

On the British Isles, WARING had recognized patterns which led him to the known 

solutions of low degree equations. Based on analogies, he thought that solutions to 

all equations could be formed but that the amount of involved computations would 

explode beyond anything practical. 

66 By pure equations EULER (and with him GAUSS) meant equations describing explicit functions, i.e. a 

pure equation for x is of the form 

x = some expression.

5.4. Belief in algebraic solubility shaken 81 

“From the preceding examples and earlier observations, we may compose resolutions 

appropriate to any given equation; but in equations of the fifth and higher 

degree the calculations require practically infinite labor.” 67 

The inner tension in WARING’S statement — that the solution was possible in prin- 

ciple but perhaps not in practice — is confusing. It remains unclear exactly what it 

meant to him that he could construct solutions but that the effort required would be 

infinite. 

In France, LAGRANGE felt strong confidence in his approach to the study of poly- 

nomial equations. His detailed studies of low degree equations led him to the conclu- 

sion that each root x1, . . . , xn of the general equation of degree n could be expressed 

through a resolvent equation of degree n − 1 which had the roots 

n 

∑ ω 

k=1 

k j xk (for j = 1, . . . , n − 1) 

where ω1, . . . , ωn−1 were the imaginary n th roots of unity. When LAGRANGE sought 

to prove this result for the fifth degree equation, however, he had to accept that his 

effort was inconclusive. If such a reduction was to be possible at all, other resolvents 

were required. 

“It thus appears that from this one can conclude by induction that every equation, 

of whatever degree, will also be solvable with the help of a resolvent [equation] 

whose roots are represented by the same formulae 

x ′ + y ′′ + y 2 x ′′′ + y 3 x iv + . . . . 

But, as we have demonstrated in the previous section in connection with the methods 

of MM. Euler and Bezout, these lead directly to the same resolvent equations, 

there seems to be reason to convince oneself in advance that this conclusion is defective 

for the fifth degree. From this it follows, that if the algebraic solution of 

equations of degrees higher than four is not impossible, it must depend on certain 

functions of the roots, which are different from the preceding ones.” 68 

Although his investigations had not led to the goal of generalizing known solu- 

tions of low degree equations to a solution to the general fifth degree equation, LA- 

GRANGE was confident that he had presented and founded a true theory — based 

67 (Waring, 1770, 162). The Latin original has not been available. Therefore, reference is given to the 

English translation (ibid.). 

68 “Il semble donc qu’on pourrait conclure de là par induction que toute équation, de quelque degré 

qu’elle soit, sera aussi résoluble à l’aide d’une réduite dont les racines soient représentées par la 

même formule 

x ′ + yx ′′ + y 2 x ′′′ + y 3 x iv + . . . . 

Mais, d’après ce que nous avons démontré dans la Section précédente à l’occasion des méthodes de 

MM. Euler et Bezout, lesquelles conduisent directement à de pareilles réduites, on a, ce semble, lieu 

de se convaincre d’avance que cette conclusion se trouvera en défaut dès le cinquième degré; d’où il 

s’ensuit que, si la résolution algébrique des équations des degrés supérieurs au quatrième n’est pas 

impossible, elle doit dépendre de quelques fonctions des racines, différentes de la précédente.” (Lagrange, 

1770–1771, 356–357).


upon combinations, i.e. permutations — inside which the solution could be investi- 

gated. However, for equations of the fifth and higher degrees the required number of 

calculations and combinations would be exceeding practical possibilities. 

“These are, if I am not mistaken, the true principles of the solution of equations, 

and the most appropriate analysis leading to it. As one can see, it all comes 

down to a sort of calculus of combinations, by which one finds à priori the results 

for which one should be prepared. It should, by the way, be applicable to 

equations of the fifth degree and higher degrees, of which the solution is until 

now unknown. But this application demands a too great number of researches 

and combinations, of which the success is still in serious doubt, for us to follow 

this path in the present work. We hope, though, to be able to follow it at another 

time, and we content ourselves by having laid the foundations of a theory which 

appears to us to be new and general.” 69 

LAGRANGE never wrote the definitive work which he had reserved the right to 

do. By the time GALOIS had substantiated LAGRANGE’S claim for generality and ap- 

plicability of his theory of combinations (see chapter 8.5), LAGRANGE was no longer 

around to celebrate the ultimate vindication of his research in the field of algebraic 

solubility. 

Both WARING and LAGRANGE believed by 1770 that their theories were the nec- 

essary stepping stones towards the study of solutions to general equations. However, 

they both acknowledged that the amount of work required to apply these theories to 

the quintic equation was beyond their own limitations. Before the end of the century, 

even more radical opinions were voiced in print. 

5.4.2 Outright impossibility 

In the introduction to his first proof (published 1799 but constructed two years earlier) 

of the Fundamental Theorem of Algebra, GAUSS gave detailed discussions and criticisms 

of previously attempted proofs. In EULER’S attempt dating back to 1749, GAUSS found 

the implicit assumption that any polynomial equation could be solved by radicals. 

“In a few words: It is without sufficient reason assumed that the solution of 

any equation can be reduced to the resolution of pure equations. Perhaps it would 

not be too difficult to prove the impossibility for the fifth degree with all rigor; I 

will communicate my investigations on this subject on another occasion. At this 

place, it suffices to emphasize that the general solution of equations, in this sense, 

69 “Voilà, si je me ne trompe, les vrais principes de la résolution des équations et l’analyse la plus 

propre à y conduire; tout se réduit, comme on voit, à une espèce de calcul des combinaisons, par 

lequel on trouve à priori les résultats auxquels on doit s’attendre. Il serait à propos d’en faire 

l’application aux équations du cinquième degré et des degrés supérieurs, dont la résolution est 

jusqu’à présent inconnue; mais cette application demande un trop grand nombre de recherches et 

de combinaisons, dont le succés est encore d’ailleurs fort douteux, pour que nous puissions quant 

à présent nous livrer à ce travail; nous espérons cependant pouvoir y revenir dans un autre temps, 

et nous nous contenterons ici d’avoir posé les fondements d’une théorie qui nous paraît nouvelle et 

générale.” (Lagrange, 1770–1771, 403).

5.4. Belief in algebraic solubility shaken 83 

remains very doubtful, and consequently that any proof whose entire strength 

depends on this assumption in the current state of affairs has no weight.” 70 

In 1799, GAUSS’S aim was to scrutinize EULER’S proof of the Fundamental Theo- 

rem of Algebra. For this purpose, it was sufficient for GAUSS to express his suspicion 

that the algebraic solution of general equations was not established with the necessary 

rigor. Thus — at least as it stands — GAUSS’S criticism seems to confront the founda- 

tions and not the validity of this hidden assumption in EULER’S proof. Although it 

is doubtful whether or not GAUSS possessed a demonstration that the validity could 

also be questioned, he certainly suggested the possibility. Two years later in his influ- 

ential Disquisitiones 1801, GAUSS addressed the problem again in connection with the 

cyclotomic equations (see quotation below). Possibly alluding to LAGRANGE’S “very 

great computational work” GAUSS described the solution of higher degree equations 

not merely beyond the existing tools of analysis but outright impossible. 

“The preceding discussion had to do with the discovery of auxiliary equations. 

Now we will explain a very remarkable property concerning their solution. Everyone 

knows that the most eminent geometers have been unsuccessful in the search 

for a general solution of equations higher than the fourth degree, or (to define the 

search more accurately) for the reduction of mixed equations to pure equations. 

And there is little doubt that this problem is not merely beyond the powers of contemporary 

analysis but proposes the impossible (cf. what we said on this subject 

in Demonstrationes nova, art. 9 [above]). Nevertheless it is certain that there are 

innumerable mixed equations of every degree which admit a reduction to pure 

equations, and we trust that geometers will find it gratifying if we show that our 

equations are always of this kind.” 71 

While GAUSS was voicing his opinion on the insolubility of higher degree equa- 

tions in Latin from his position in Göttingen, the support for the solubility of the 

quintic was shaken even more radically by an Italian. P. RUFFINI (1765–1822) had 

published his first proof of the impossibility of solving the quintic in 1799, the same 

year GAUSS had first uttered his doubts about its possibility. But while GAUSS had 

70 “Seu, missis verbis, sine ratione sufficienti supponitur, cuiusvis aequationis solutionem ad solutionem 

aequationum purarum reduci posse. Forsan non ita difficile foret, impossibilitatem iam pro 

quinto gradu omni rigore demonstrare, de qua re alio loco disquisitiones meas fusius proponam. 

Hic sufficit, resolubilitatem generalem aequationum, in illo sensu acceptam, adhuc valde dubiam 

esse, adeoque demonstrationem, cuius tota vis ab illa suppositione pendet, in praesenti rei statu 

nihil ponderis habere.” (C. F. Gauss, 1799, 17–18); for a translation into German, see (C. F. Gauss, 

1890, 20–21). 

71 “Disquisitiones praecc. circa inventionem aequationum auxiliarium versabantur: iam de earum solutione 

proprietatem magnopere insignem explicabimus. Constat, omnes summorum geometrarum 

labores, aequationum ordinem quartum superantium resolutionem generalem, sive (ut accuratius 

quid desideretur definiam) affectarum reductionem ad puras, inveniendi semper hactenus irritos 

fuisse, et vix dubium manet, quin hocce problema non tam analyseos hodiernae vires superet, quam 

potius aliquid impossibile proponat (Cf. quae de hoc argumento annotavimus in Demonstr. nova 

etc. arg. 9). Nihilominus certum est, innumeras aequationes affectas cuiusque gradus dari, quae 

talem reductionem ad puras admittant, geometrisque gratum fore speramus, si nostras aequationes 

auxiliares semper huc referendas esse ostenderimus.” (C. F. Gauss, 1801, 449); English translation 

from (C. F. Gauss, 1986, 445). Bold-face has been substituted for the original small-caps.


only alluded to a proof without communicating it, RUFFINI had taken the step of pub- 

lishing his arguments. 

5.5 RUFFINI’s proofs of the insolubility of the quintic 

In the 1820s, the search for algebraic solutions to equations of higher degree was 

proved to be in vain when ABEL demonstrated the algebraic insolubility of the quintic. 

However, ABEL was not the first to claim the insolubility; more than 25 years before 

him, the Italian RUFFINI had published his investigations which led him to the same 

conclusion and a proof thereof. RUFFINI’S works were not widely known, and during 

his investigations ABEL was unaware of their existence (see section 6.7). ABEL based 

his investigations on the analysis by LAGRANGE and works of CAUCHY on the theory 

of permutations. Although not directly inspired by RUFFINI’S works, RUFFINI played 

an indirect role in fertilizing the ground for ABEL’S work. The indirect influence of 

RUFFINI through the very direct influence of CAUCHY on the development leading 

to ABEL’S work is two-fold. Firstly, these men smoothed the transition from the be- 

liefs described in the previous section to the rigorous knowledge of the insolubility of 

the quintic. Secondly, their investigations took the still young theory of permutations 

to a more advanced level; and in doing so, they provided an important characteriza- 

tion of the number of values a rational function can obtain under permutations of its 

arguments. 

5.5.1 Insolubility proved 

Although GAUSS had proclaimed his belief that the insolubility of the quintic might 

not be difficult to prove with all rigor, the Italian RUFFINI, in 1799, was the first math- 

ematician to state the insolubility as a result and provide the claim with a proof. RUF- 

FINI’S style of presentation was long, cumbersome, and at times not free of errors; and 

his initial proof was met with immediate criticism for these reasons. But convinced of 

the result and his proof, RUFFINI kept elaborating and clarifying his theory in print for 

the next 20 years, producing a total of five different versions of the proof. The proofs 

were published in Italian as monographs in Bologna and in the mathematical mem- 

oirs of the Società Italiana delle Scienze, Modena. Although published and distributed, 

the impact of RUFFINI’S work was limited; among the few non-Italians to take a view- 

point on RUFFINI’S work was CAUCHY (see section 5.5.3). In 1810, J.-B. J. DELAMBRE 

(1749–1822) was aware of RUFFINI’S 1802-proof which he described as “difficult” and 

“not suited for inclusion in works meant as a first introduction” to the subject. 72 I shall 

mainly deal with RUFFINI’S initial proof given in his textbook, 73 which he elaborated 

72 (Delambre, 1810, 86–87). 

73 (Ruffini, 1799).

5.5. RUFFINI’s proofs of the insolubility of the quintic 85 

Figure 5.4: PAOLO RUFFINI (1765–1822) 

on numerous occasions, and his final proof published 1813. 74 

5.5.2 RUFFINI’s first proof 

The writings of RUFFINI were deeply inspired by LAGRANGE’S analysis of the solu- 

bility of equations described in section 5.2. 75 LAGRANGE’S ideas, concepts, and nota- 

tion permeate RUFFINI’S works; and on numerous occasions RUFFINI openly acknowl- 

edged his debt to LAGRANGE. 76 As LAGRANGE had done, RUFFINI studied equations 

of low degrees in order to establish patterns subjectable of generalization. Prior to 

applying his analysis to the fifth degree equation, RUFFINI propounded the corner 

stone of his investigation. Central to his line of argument was his classification of per- 

mutations. Founded in LAGRANGE’S studies of the behavior of functions when their 

arguments were permuted, RUFFINI set out to classify all such permutations of argu- 

ments which left the function (formally) unaltered. RUFFINI’S concept of permutation 

(Italian: “permutazione”) differed from the modern one, and can most easily be un- 

derstood if translated into the modern concept introduced by CAUCHY in the 1840s of 

74 (Ruffini, 1813). The presentation of RUFFINI’S proofs will largely rely on secondary sources, primarily 

(Burkhardt, 1892), (Wussing, 1969, 56–59), and (Kiernan, 1971, 56–60). In (R. G. Ayoub, 1980), his 

proofs are interpreted using concepts from GALOIS theory. 

75 (Lagrange, 1770–1771). 

76 See for instance his preliminary discourse in (Ruffini, 1799, 3–4).


simple permutations 

composite permutations 

� 

⎧ 

⎨ 

⎩ 

powers of a cycle 

powers of a non-cycle 

intransitive ones 

transitive, imprimitive ones 

transitive, primitive ones 

Table 5.1: RUFFINI’s classification of permutations 

systems of conjugate substitutions. 77 Thus, a permutation for RUFFINI corresponded to a 

collection of interchangements (i.e. transitions from one arrangement of symbols e.g. 

123 . . . n to another e.g. 213 . . . n) which left the given function formally unaltered. 78 

Classification of permutations. RUFFINI divided his permutations into simple ones 

which were generated by iterations (i.e. powers) of a single interchangement 79 and 

composite ones generated by more than one interchangement. His simple permutations 

consisting of powers of a single interchangement were subdivided into two types dis- 

tinguishing the case in which the single interchangement consisted of a single cycle 

from the case in which it was the product of more than one cycle. 

RUFFINI’S composite permutations were subsequently subdivided into three types. 80 

A permutation (i.e. set of interchangements) was said to be of the first type if two 

arrangements existed which were not related by an interchangement from the permu- 

tation. 81 Translated into the modern terminology of permutation groups, this type 

corresponds to intransitive groups. RUFFINI defined the second type to contain all per- 

mutations which did not belong to the first type and for which there existed some 

non-trivial subset of roots S such that, in modern notation, σ (S) = S or σ (S) ∩ S = ∅ 

for any interchangement σ belonging to the permutation. Such transitive groups were 

later termed imprimitive. The last type consisted of any permutation not belonging to 

any of the previous types, and thus corresponds to primitive groups. 

Building on this classification of all permutations into the five types (table 5.1), 

RUFFINI introduced his other key concept of degree of equivalence (Italian: “grado di 

uguaglianza”) of a given function f of the n roots of an equation as the number of dif- 

ferent permutations not altering the formal value of f . Denoting the degree of equiva- 

lence by p, RUFFINI stated the result of LAGRANGE (see section 5.2) that p must divide 

n!. 

77 (Burkhardt, 1892, 133). 

78 In modern notation: With f the given function of n quantities, a permutazione to RUFFINI was a set 

G ⊆ Σn such that f ◦ σ = f for all σ ∈ G. 

79 RUFFINI’S simple permutations correspond to the modern concept of cyclic permutation groups. 

80 (Ruffini, 1799, 163). 

81 I.e. there exists two arrangements a and b such that σ (a) �= b for all σ in the set of interchangements.


Possible numbers of values. RUFFINI at this point turned towards the fifth degree 

equation. By an extensive and laborious study, helped by his classification, RUFFINI 

was able to establish that if n = 5 the degree of equivalence p could not assume any 

of the values 

15, 30, or 40. 

Since the number of different values of the function f could be obtained by dividing n! 

by p, he had therefore demonstrated that no function f of the five roots of the quintic 

could exist which assumed 

5! 

15 

= 8, 5! 

30 

= 4, or 5! 

40 

different values under permutations of the five roots. 

Although still embedded in the Lagrangian approach to permutations, RUFFINI’S 

main result can be viewed as a determination of the index (corresponding to his degree 

of equivalence, p) of all subgroups in Σ5. 

Degrees of radical extractions. In order to prove the impossibility of solving the 

quintic algebraically, RUFFINI assumed without proof that any radical occurring in 

a supposed solution would be rationally expressible in the roots of the equation. He 

never verified this hypothesis, which ABEL later independently formulated and proved. 

Based on the assumption and the result that no function of the roots x1, . . . , x5 could 

have 3, 4, or 8 values, RUFFINI could prove the insolubility by a nice and short argu- 

ment which ran as follows. 

He first considered a situation in which among two functions Z and M of x1, . . . , x5 

there existed a relationship of the form 

Z 5 − M = 0 

corresponding to the extraction of a fifth root of a rational function. The situation was 

drawn from the study of a possible solution to the quintic equation where it corre- 

sponded to the inner-most root extraction being a fifth root. By implicitly assuming 

that Z was altered by some interchangement Q which left M unaltered, RUFFINI first 

observed that Q would have to be a 5-cycle. If Z was unaltered by a non-identity in- 

terchangement P, it would also be unaltered by Q −1 PQ which belonged to the same 

permutation. By reference to a result, which he had previously established by exam- 

ining each of the different types of permutations, RUFFINI found (art. 273) that Q 

under these conditions would belong to the same permutation as Q −1 PQ and there- 

fore could not alter Z, contradicting the assumptions made about Q. Thus, no such 

non-identity interchangement P could exist, and the 120 values of Z corresponding to 

different arrangements of x1, . . . , x5 were necessarily distinct. Consequently, the first 

radical to be extracted could not be a fifth root, and since no function of the five roots 

having three or four values existed, it could not be a third or a fourth root, neither. 

Therefore, it had to be a square root. 

= 3


At this point, RUFFINI focused on the second radical to be extracted and the above 

argument applied equally well to rule out the case of a fifth root. Similarly, it could 

not be a square root or a fourth root since these would lead to functions having four 

(2 × 2) or eight (2 × 4) values, which were proved to be non-existent. RUFFINI had 

thus established that any supposed solution to the quintic equation would have to 

begin with the extraction of a square root followed by the extraction of a third root. 

However, as he laboriously proved by considering each case in turn, the six-valued 

function obtained by these two radical extractions did not become three-valued after 

the initial square root had been adjoined. 

The proof which RUFFINI gave for the insolubility of the quintic was thus based on 

three central parts: 

1. The classification of permutations into types (table 5.1) 

2. A demonstration, based on (1), that no function of the five roots of the general 

quintic could have 3, 4, or 8 values under permutations of the roots. 

3. A study of the two inner-most (first) radical extractions of a supposed solution 

to the quintic, in which the result of (2) was used to reach a contraction. 

The mere extent of the classification and the caution necessary to include all cases 82 

combined with RUFFINI’S intellectual debt to LAGRANGE may serve to view RUFFINI’S 

work as filling in some of the “infinite labor” described by WARING and LAGRANGE 

in expressing their doubts about the solubility of higher degree equations (see section 

5.4.1 above). However, RUFFINI’S investigations led to the complete reverse result: 

that the solution of the quintic was impossible. 

One of RUFFINI’S friends and critical readers, P. ABBATI (1768–1842), gave sev- 

eral improvements of RUFFINI’S initial proof. The most important one was that he 

replaced the laborious arguments based on thorough consideration of particular cases 

by arguments of a more general character. 83 These more general arguments greatly 

simplified RUFFINI’S proofs that no function of the five roots of the quintic could have 

3, 4, or 8 different values. ABBATI was convinced of the validity of RUFFINI’S result 

but wanted to simplify its proof, and RUFFINI incorporated his improvements into 

subsequent proofs, from 1802 and henceforth. 

Others, however, were not so convinced of the general validity of RUFFINI’S re- 

sults. Mathematicians belonging to the “old generation” were somewhat stunned 

by the non-constructive nature of the proofs, which they described as “vagueness”. 

For instance, the mathematician G. F. MALFATTI (1731–1807) severely criticized RUF- 

FINI’S result since it contradicted a general solution which he, himself, previously had 

82 According to (Burkhardt, 1892, 135), RUFFINI actually missed the subgroup generated by the cycles 

(12345) and (132). 

83 (ibid., 140).


given. 84 RUFFINI responded with another publication of a version of his proof answer- 

ing to MALFATTI’S criticism; but before the discussion advanced further, MALFATTI 

died. 

5.5.3 RUFFINI’s final proof 

In his fifth, and final, publication of his insolubility theorem 1813, RUFFINI recapitu- 

lated important parts of LAGRANGE’S theory, in which he emphasized the distinction 

between numerical and formal equality, before giving the refined version of his proof. 

According to (Burkhardt, 1892, 155–156), the proof can be dissected into the following 

parts comparable to the parts of the 1799 proof (see point 3 above): 

1. If two functions y and P of the roots x1, . . . , x5 of the quintic are related by 

y p − P = 0 

(for any p) and P remains unaltered by the cyclic permutation (12345), there 

must exist a value y1 of y which in turn changes into y2, y3, y4, and y5. Conse- 

quently, 

where β is a fifth root of unity. 

y k = β k y1 

2. If P is furthermore unaltered by the cyclic permutation (123), then y1 must 

change into γy1 where γ is a third root of unity. 

3. The permutation (13452) is comprised of the two cycles (12345) (123) and y must 

remain unaltered. Therefore, β 5 γ 5 = 1 which in turn implies that γ = 1, demon- 

strating that y cannot be altered by any of the permutations (123), (234), (345), 

(451), or (512). By combining these 3-cycles the 5-cycle (12345) can be obtained, 

and thus y cannot be altered by the 5-cycle, neither. 

4. Consequently, it is impossible by sequential root extractions to describe func- 

tions which have more than two values, and the insolubility is demonstrated. 

5.5.4 Reactions to RUFFINI’s proofs 

In a paper published 1845, 85 P. L. WANTZEL (1814–1848) gave a fusion argument in- 

corporating the permutation theoretic arguments of RUFFINI’S final proof into the setting 

of ABEL’S proof. 86 

84 (Malfatti, 1804). 

85 (Wantzel, 1845). 

86 See also (Burkhardt, 1892, 156).


RUFFINI corresponded with CAUCHY, who in 1816 was a promising young Parisian 

ingenieur. 87 CAUCHY praised RUFFINI’S research on the number of values which a 

function could acquire when its arguments were permuted, a topic CAUCHY, himself, 

had investigated in an treatise published the year before 1815 with due reference to 

RUFFINI (see below). Following this exchange of letters CAUCHY wrote RUFFINI an- 

other letter in September 1821, in which he acknowledged RUFFINI’S progress in the 

important field of solubility of algebraic equations: 

“I must admit that I am anxious to justify myself in your eyes on a point which 

can easily be clarified. Your memoir on the general solution of equations is a work 

which has always appeared to me to deserve to keep the attention of geometers. 

In my opinion, it completely demonstrates the algebraic insolubility of the general 

equations of degrees above the fourth. The reason that I had not lectured on it 

[the insolubility ] in my course in analysis, and it must be said that these courses 

are meant for students at the École Royale Polytechnique, is that I would have 

deviated too much from the topics set forth in the curriculum of the École.” 88 

At least by 1821, the validity of RUFFINI’S claim that the general quintic could 

not be solved by radicals was propounded, not only by a somewhat obscure Italian 

mathematician and the allusions of GAUSS, but also one of the most promising and 

ambitious French mathematicians of the early nineteenth century. However, it should 

take further publications, notably by the young ABEL, before this validity would be 

accepted by the broad international community of mathematicians. 

5.6 CAUCHY’ theory of permutations and a new proof of 

RUFFINI’s theorem 

In November of 1812, CAUCHY handed in a memoir on symmetric functions to the In- 

stitut de France which was published three years later as two separate papers in the 

Journal d’École Polytechnique. 89 The first of the two papers is of special interest in the 

history of solubility of polynomial equations. It bears the long but precise title Mé- 

moire sur le nombre des valuers qu’une fonction peut acquérir, lorsqu’on y permute de toutes 

manières possibles les quantités qu’elle renferme. 90 Although CAUCHY’S issue was not 

the solubility-question, his paper was to become extremely important for subsequent 

research. It was primarily concerned with a more general version of RUFFINI’S result 

87 (Ruffini, 1915–1954, vol. 3, 82–83). 

88 “Je suis impatient, je l’avous, de me justifier à Vos yeux sur un point qui peut être facilement éclairi. 

Votre mémoire sur la résolution générale des équations est un travail qui m’a toujours paru digne de 

fixer l’attention des géomètres, et qui, à mon avis, démontre complètement l’insolubilité algébrique 

des équations générales d’un dégré supérieur au quatrième. Si je n’en ai pas parlé dans mon cours 

d’analyse, c’est que, ce cours étant destiné aux élèves d’École Royale Polytechnique, je ne devois 

pas trop m’écarter des matières indiquées dans les programmes de l’école.” (ibid., vol. 3, 88–89). 

89 (A.-L. Cauchy, 1815a; A.-L. Cauchy, 1815b). 

90 (A.-L. Cauchy, 1815a).

5.6. CAUCHY’ theory of permutations and a new proof of RUFFINI’s theorem 91 

Figure 5.5: AUGUSTIN-LOUIS CAUCHY (1789–1857) 

that no function of five quantities could have three or four different values when its ar- 

guments were permuted (see above). Before going into this particular result, however, 

CAUCHY devised the terminology and notation which he was going to use. Precisely 

in formulating exact and useful notation and terminology, CAUCHY advanced well 

beyond his predecessors and laid the foundations upon which the nineteenth-century 

theory of permutations would later build. 

Notational advances. With CAUCHY, the term “permutation” came to mean an ar- 

rangement of indices, thereby replacing the “arrangements” of which RUFFINI spoke. 

A “substitution” was subsequently defined to be a transition from one permutation 

to another (which is the modern meaning of “permutation”), and CAUCHY devised 

writing it as, for instance, 

� � 

1.2.4.3 

. (5.12) 

2.4.3.1 

CAUCHY’S convention was that in the expression K, to which the substitution (5.12) 

was to be applied, the index 2 was to replace the index 1, the index 4 to replace 2, 3 

should replace 4, and 1 should replace 3. More generally, CAUCHY wrote 

� A1 

A2 

�


for the substitution which transformed the permutation A1 into A2 in the above- 

mentioned way. 91 He then defined ( A1) to be the product of two substitutions (A2 

A6 A3 ) 

and ( A4 A5 ) if it gave the same result as the two applied sequentially,92 in which case 

CAUCHY wrote � A1 

A6 

� 

= 

� �� 

A2 A4 

Furthermore, he defined the identical substitution and powers of a substitution to have 

the meanings we still attribute to these concepts today. 93 The smallest integer n such 

that the n th power of a substitution was the identity substitution, CAUCHY called the 

degree of the substitution. 94 All these notational advances played a central part in 

formalizing the manipulations on permutations and were soon generally adopted. 

LAGRANGE’S Theorem. In order to demonstrate LAGRANGE’S theorem, CAUCHY 

let K denote an arbitrary expression in n quantities, 

A3 

A5 

K = K (x1, . . . , xn) . 

With N = n!, he labelled the N different permutations of these n quantities 

A1, . . . , AN. 

The values which K would acquire when the corresponding substitutions of the form 

( A1 Au ) were applied were correspondingly labelled K1, . . . , KN, 

� � 

A1 

Ku = K for 1 ≤ u ≤ N. 

Au 

If these were all distinct, the expression K would obviously have N different values 

when its arguments were interchanged. In the contrary case, CAUCHY assumed that 

for M indices the values of K were equal 

� 

. 

Kα = K β = Kγ = . . . . 

The core of the proof was CAUCHY’S realization that if the permutation A λ was fixed 

and the substitution ( Aα 

A β ) was applied to A λ giving Aµ, i.e. 

� 

Aα 

Aµ = 

A β 

� 

A λ, 

the corresponding values K λ and Kµ would be identical. Consequently, the different 

values of K came in bundles of M and CAUCHY had deduced that M had to divide 

n!. The central concept of degree of equivalence, which RUFFINI had introduced to mean 

91 (A.-L. Cauchy, 1815a, 67). 

92 (ibid., 73). 

93 (ibid., 73, 74) 

94 (ibid., 76). ABEL was later to change this term to the now standard order.

5.6. CAUCHY’ theory of permutations and a new proof of RUFFINI’s theorem 93 

the number of substitutions which left the given function unaltered, was renamed the 

indicative divisor (French: “diviseur indicatif”) by CAUCHY and was exactly what he 

had denoted by M. Terming the number of different values of K under all possible 

substitutions the index of the function K and denoting it by R, CAUCHY had obtained 

the formula 

n! = R × M. (5.13) 

The RUFFINI-CAUCHY Theorem. After explicitly providing the function of n quan- 

tities a1, . . . , an given by 

� � 

∏ ai − aj 1≤i


Each circle of permutations is represented by a row in the following table: 

A2 = ( As 

At )A1, . . . , Am = ( As 

At )Am−1, A1 = ( As 

At )Am, 

Am+2 = ( As 

At )Am+1, . . . , A2m = ( As 

At )A2m−1, Am+1 = ( As 

At )A2m, 

. . . . . . . . . . . . 

AN−m+2 = . . . , . . . , AN = ( As 

At )AN−1, AN−m+1 = ( As 

At )AN−m. 

These permutations can be reordered when (5.14) is taken into account: 

A1, A2 = ( As 

At )A1, . . . , Am = ( As 

At )m−1 A1, 

Am+1, Am+2 = ( As 

At )Am+1, . . . , A2m = ( As 

At )m−1 Am+1, 

. . . . . . . . . . . . 

AN−m+1, AN−m+2 = ( As 

At )AN−m+1, . . . , AN = ( As 

At )m−1 AN−m+1. 

The notation ( As 

At )A1 indicates that the substitution ( As) 

be applied to the permutation 

At 

A1. 

Table 5.2: The N m 

circles formed by applying (As 

At ) to A1, . . . , AN. 

power of ( As 

At ) to Ax. Consequently, the substitution ( Ax) 

was equal to a power of (As 

Ay At ). 

If m were a prime, the converse would also be true, since if 

� �k � � 

As Ax 

= 

and (k, m) = 1, there existed α, β such that αk + βm = 1, i.e. 

� � � �αk+βm � �αk As As 

As 

= 

= = 

At 

At 

At 

Ay 

At 

� Ax 

The details of this argument were left out by CAUCHY, but were later provided by 

ABEL. 98 Since Ax and Ay corresponded to the same value of K, the function would 

not change if the substitution ( Ax) 

were applied. Consequently, K would also remain 

Ay 

unaltered when the substitution ( As) 

was applied, and the number of different values 

At 

of K, which CAUCHY had denoted M, could not be greater than N m , whereby he had 

reached a contradiction. Setting m = p, CAUCHY had obtained the desired result. 

In the second part, CAUCHY demonstrated by decomposing p-cycles into 3-cycles 

that if the value of K remained unaltered by all substitutions of degree p it would also 

be unaltered by any circular substitution of order 3. The important step was obtained 

Ay 

� α 

. 

by realizing that the product of the two circular substitutions of order p 

� � � � 

αβγδ . . . ζη βγδε . . . ηα 

and 

βγδε . . . ηα γαβδ . . . ζη 


(5.15)

5.7. Some algebraic tools used by GAUSS 95 

was the 3-cycle 

� � 

αβγ 

. (5.16) 

γαβ 

Thus, given any 3-cycle (5.16), the two p-cycles (5.15) could be formed. Under the hy- 

pothesis, these p-cycles left K unaltered, whereby the same was true of their product, 

i.e. the 3-cycle (5.16). 

In the third and final part of the proof CAUCHY established that if the value of 

K was unaltered by all 3-cycles, the function K would either be symmetric or have 

two different values. In his proof, analogous to the second part described above, he 

decomposed the 3-cycle 

� � 

αβγ 

γαβ 

into the product of the two transpositions 

� �� 

αβ βγ 

βα γβ 

which he wrote as (αβ) (βγ). This step of the proof corresponds to proving that the 

alternating group An is generated by all 3-cycles. 

In the remaining part of the paper, CAUCHY demonstrated for functions of six 

arguments, if R < 5 the function would necessarily be symmetric or have two values. 

Generally, CAUCHY noted, for n > 4 no functions of n quantities were known which 

had less than n values without this number being either 1 or 2. After these two early 

papers on the theory of permutations, CAUCHY would let the topic rest for 30 years 

being preoccupied with his many other research themes and his teaching. When he 

finally returned to the theory of permutations in the 1840s, CAUCHY demonstrated the 

following generalization of his 1815 result: That no function of n quantities could take 

on less than n values without either being symmetric or taking on exactly 2 values. 99 

With his paper, 100 CAUCHY founded the theory of permutations by providing it 

with its principal objects: the permutations. He introduced terms and notation which 

enabled him to grasp the substitutions as objects abstracted from their action on the 

formal values of a function, and he provided an important theorem in this new theory 

which he based on an elegant, non-computational proof. 

5.7 Some algebraic tools used by GAUSS 

GAUSS’ first proof of the Fundamental Theorem of Algebra had, in a central way, de- 

pended on geometrical (topological) intuitions. In 1815, GAUSS published a second 

proof of the theorem, 101 this time applying algebraic methods. In the process, GAUSS 

99 (Dahan, 1980, 281–282). 

100 (A.-L. Cauchy, 1815a). 

101 (C. F. Gauss, 1815). Eventually, GAUSS would publish two further proofs (one in 1816 and one in 

1849) bringing his total to four.


spelled out some of the most important algebraic tools of the early 19 th century; there- 

fore some of his tools are briefly sketched in the present context. During the proof, 

GAUSS dealt with results such as the Euclidean algorithm applied to polynomials and 

the “elementariness” of the elementary symmetric functions, both of which will be- 

come immensely important in ABEL’S theory of algebraic solubility as described in 

subsequent chapters. Whether ABEL studied any of GAUSS’ proofs of the fundamen- 

tal theorem of algebra is not clear; there are no explicit references to these proofs in 

ABEL’S writings, nor is ABEL anywhere concerned with the existence of roots. 102 Thus, 

the similarity of methods in GAUSS’ proof and ABEL’S subsequent algebraic research 

may equally well be attributed to their belonging to the same common framework and 

mathematical tradition. 

Explicitly stressing the connection to the procedure used to determine the greatest 

common divisor of integers, GAUSS applied the Euclidean algorithm to polynomials. 

Besides producing the greatest common divisor, the procedure also proved that two 

polynomials Y, Y ′ have no (non-trivial) common divisor if and only if there exists an- 

other pair of polynomials Z, Z ′ such that 

ZY + Z ′ Y ′ = 1. 

The second tool which GAUSS introduced concerned symmetric functions, and 

amounts to the central theorem on symmetric functions. By firstly decomposing any 

symmetric function of a, b, c, . . . in a sum of terms 

Ma α b β c γ . . . 

and secondly imposing an ordering on such terms, GAUSS was able to prove that any 

symmetric function could be realized as an entire function of the elementary symmet- 

ric functions. 

Besides these tools, GAUSS’ argument rested upon central properties of the quan- 

tity which he termed the determinant (today called the discriminant) of Y (x) = ∏ (x − x k), 

∏ i�=j 

� � 

xi − xj . 

GAUSS was able to demonstrate that the determinant vanishes if and only if Y and 

d 

dx Y have a common divisor, i.e. a common root. 

102 Without references, KLINE writes as if ABEL had given a proof of the fundamental theorem of algebra 

(Kline, 1990, 599). I have not been able to identify such a proof, nor have I any idea how KLINE 

had come to believe that ABEL had even worked on it.

Chapter 6 

ABEL on the algebraic insolubility of 

the quintic: limiting the class of 

solvable equations 

In spite of the efforts of P. RUFFINI (1765–1822) and C. F. GAUSS (1777–1855), the 

search for an algebraic solution of the quintic remained an attractive problem to a 

generation of young and aspiring mathematicians. In Norway, N. H. ABEL (1802– 

1829) thought he had solved it, but soon realized that he had been misled. In Germany, 

C. G. J. JACOBI (1804–1851) worked on the problem, 1 and in France E. GALOIS (1811– 

1832), too, thought he had found a solution, only to be disappointed. 2 All of them 

attacked the problem while they still attended pre-university education. The easy 

formulation and yet century-long history of the problem, and a general belief that its 

solution should be possible and not too difficult, made it appear as a good opening 

into doing creative mathematics. 

Inspired by the stimulation of his new, and young, mathematics teacher B. M. 

HOLMBOE (1795–1850), ABEL studied the masters and began to engage in creative 

mathematics of his own. In 1821, he thought he had produced a solution to the gen- 

eral fifth degree equation. In the incipient intellectual atmosphere of Christiania, few 

authorities capable of determining the validity of ABEL’S reasoning could be found. 

But more importantly, the scientific milieu of Norway was still without a means of 

publication of technical mathematical results deserving international recognition. For 

these reasons, professor C. HANSTEEN (1784–1873) sent ABEL’S manuscript to profes- 

sor C. F. DEGEN (1766–1825) in Copenhagen for evaluation and possibly publication 

in the transactions of the Royal Danish Academy of Sciences and Letters. The accompa- 

nying letter which HANSTEEN must have written and the paper, itself, are no longer 

preserved. Our only primary source of information is the letter which DEGEN wrote 

back to HANSTEEN, in which he asked for an elaborated version of the argument and 

1 (G. L. Dirichlet, 1852, 4). 

2 (Toti Rigatelli, 1996, 33). 

97

98 Chapter 6. Algebraic insolubility of the quintic 

an application to a specific numerical example. 

“As for the talented Mr. Abel, I will be happy to present his treatise to the 

Royal Academy of Science. It shows, even if the goal has not been reached, an extraordinary 

head and extraordinary insights, especially for someone his age. Nevertheless, 

I excuse myself to require the condition that Mr. A. sends an elaborated 

deduction of his result together with a numerical example, taken from, for instance, 

an equation such as x 5 − 2x 4 + 3x 2 − 4x + 5 = 0. I believe that this will be a rather 

necessary lapis lydius [Lydian stone] for him, as I recall what happened to Meier 

Hirsche 3 and his ενρηκα [Eureka]; item [furthermore] I would, since the latter part 

of the communicated manuscript would not be easily readable to the majority of 

the members of the Academy, ask for another copy of it.” 4 

We have no indication that ABEL ever produced an elaborated deduction; appar- 

ently the numerical examples worked their part — as the probes of truth — as DEGEN 

had suggested and led ABEL to a radically new insight. In 1824, he published, at his 

own expense, a short work in French entitled Mémoire sur les équations algébriques ou 

l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré. 5 It 

demonstrated the impossibility of solving the general equation of the fifth degree by 

algebraic means — ABEL had left the last essential requirement out of the title. ABEL 

intended the memoir to be his best introduction on his planned tour of the Continent. 

Since he had to pay for the publication himself, he compressed the proof to cover 

only six pages and his style of presentation suffered accordingly. In numerous points 

he was unclear or left advanced arguments out. When ABEL came into contact with 

A. L. CRELLE (1780–1855) in Berlin, he found himself in a position to make his dis- 

covery available to a broader public. He rewrote the argument elaborating the ideas 

of the 1824 proof, and had CRELLE translate it into German for publication in the very 

first issue of Journal für die reine und angewandte Mathematik which appeared in 1826. 6 

Through this paper — and the French report of it, 7 which ABEL wrote for the Bulletin 

des sciences mathématiques, astronomiques, physiques et chimiques edited by BARON DE 

FERRUSAC (1776–1836) 8 — the world gradually came to know that a young Norwe- 

gian had settled the question of solubility of the general quintic in the negative. 

3 M. HIRSCHE (1765–1851) was a teacher of mathematics in Berlin who in 1809 published a collection 

of exercises. There, he thought he had given the general solution to all equations. He quickly discovered 

his error, perhaps by a Lydian probe as DEGEN recommends. (N. H. Abel, 1902e, Oplysninger 

til Brevene, p. 125) 

4 “Hvad den talentfulde Hr. Abel angaar, da vil jeg med Fornøielse fremlægge hans Afhandling for 

det Kgl. V. S. Den viser, om end ikke Maalet skulde være opnaaet, et ualmindeligt Hoved og ualmindelige 

Indsigter, især i hans Alder. Dog maatte jeg som Bøn tilføie den Betingelse: At Hr. A. sender 

en udførligere Deduction af sit Resultat og tillige et numerisk Exempel, tagen f. Ex. af en Ligning 

som denne: x 5 − 2x 4 + 3x 2 − 4x + 5 = 0. Dette vil efter min Overbevisning være en saare nødvendig 

lapis lydius for ham Selv, da man veed, hvorledes det gik Meier Hirsche med hans ενρηκα; item 

maatte jeg, da den sidste Deel af den mig communicerede Afh. ikke vilde være ret læselig for de fleeste 

af S.’s Medlemmer, udbede mig en anden Afskrift af samme.” (Degen→Hansteen, Kjøbenhavn, 

1821/05/21. N. H. Abel, 1902b, 93). 




8 Dates from (Stubhaug, 1996, 580).

6.1. The first break with tradition 99 

In this chapter, I give a presentation of ABEL’S proof using the tools and methods 

available to him. As described in the introduction, 9 this approach allows me to place 

ABEL’S proof in a historical context within mathematics. For expositions of ABEL’S 

proof involving the modern concepts introduced in Galois theory, see for instance (R. 

Ayoub, 1982; M. I. Rosen, 1995; Skau, 1990). 

6.1 The first break with tradition 

In the opening paragraph of the paper in CRELLE’S Journal für die reine und angewandte 

Mathematik, ABEL described the approach he had taken. In order to answer the ques- 

tion of solubility of equations, he proposed to investigate the forms of all algebraic 

expressions in order to determine if they could “solve” the equation. Although ABEL 

throughout spoke of algebraic functions, I use the term algebraic expressions to avoid any 

confusion with the modern concept of a function as a mapping between sets. The 

algebraic expressions which ABEL considered were algebraic combinations of the co- 

efficients of the given equation, and thus his approach was in line with the one taken 

earlier by A.-T. VANDERMONDE (1735–1796) (see section 5.1). 10 

“As is known, the algebraic equations up to the fourth degree can be solved 

in general. Equations of higher degrees, however, only in particular cases, and if I 

am not mistaken, the question: 

Is it possible to solve equations of higher than the fourth degree in general? 

has not yet been answered in a satisfactory manner. The present treatise is concerned 

with this question. 

To solve an equation algebraically is but to express its roots by algebraic functions 

of its coefficients. Therefore, one must first consider the general form of algebraic 

functions and subsequently investigate whether it is possible that the given 

equation can be satisfied by inserting the expression of an algebraic function in 

place of the unknown quantity.” 11 

In the quote, ABEL also introduced an important notion of satisfiability. An equation 

was said to be satisfied by an algebraic expression if the expression was a root of the 

equation. Consequently, an equation was said to be satisfiable if an algebraic expres- 

sion existed which satisfied it. This differed from the notion of algebraic solubility 

which required that all the roots of the equation could be expressed algebraically. 

9 See section 1.4. 

10 (Kiernan, 1971, 67). 

11 “Bekanntlich kann man algebraische Gleichungen bis zum vierten Grade allgemein auflösen, Gleichungen 

von höhern Graden aber nur in einzelnen Fällen, und irre ich nicht, so ist die Frage: 

Ist es möglich, Gleichungen von höhern als dem vierten Grade allgemein aufzulösen? 

noch nicht befriedigend beantwortet worden. Der gegenwärtige Aufsatz hat diese Frage zum Gegenstande. 

Eine Gleichung algebraisch auflösen heißt nichts anders, als ihre Wurzeln durch eine algebraische 

Function der Coefficienten ausdrücken. Man muß also erst die allgemeine Form algebraischer Functionen 

betrachten und alsdann untersuchen, ob es möglich sei, der gegebenen Gleichung auf die 

Weise genug zu thun, daß man den Ausdruck einer algebraischen Function statt der unbekannten 

Größe setzt.” (N. H. Abel, 1826a, 65).


This shift from the trial-and-error based search for solutions toward a theoretical 

and general investigation of the class of algebraic expressions marks ABEL’S first break 

with the traditional approach to the theory of equations. ABEL investigated the extent 

to which algebraic expressions could satisfy given polynomial equations and was led 

to describe necessary conditions. By this choice of focus, ABEL implicitly introduced 

a new object, algebraic expression, into the realm of algebra, and the first part of his 

paper can be seen as an opening study of this object, devised in order to obtain a firm 

description of it and to prove the first central theorem concerning it. 12 In section 19.3, 

another aspect of ABEL’S concept of algebraic expression is taken up. 

6.2 Outline of ABEL’s proof 

The paper in CRELLE’S Journal für die reine und angewandte Mathematik can be divided 

into four sections reflecting the overall structure of ABEL’S proof. In the first section, 

ABEL introduced his definition of algebraic functions and classified these by their or- 

ders and degrees. He used this definition to study the restrictions imposed on the 

form of algebraic expressions if they had to be solutions to a given solvable equation. 

In doing so, he proved the result — which RUFFINI had failed to see — that any radical 

(algebraic sub-expression) contained in a supposed solution would depend rationally 

on the roots of the equation (see section 6.3). 

In the second section, ABEL reproduced the elements of A.-L. CAUCHY’S (1789– 

1857) theory of permutations from 1815 needed for his proof. 13 These included CAUCHY’S 

notation and the result described above as the CAUCHY-RUFFINI theorem (section 5.6) 

demonstrating that no function of the five roots of the general quintic could take on 

three or four different values under permutations of these roots (see section 6.4). 

The third part contained detailed and highly explicit investigations of functions 

of five quantities taking on two or five different values under all permutations of the 

roots. Through an explicit theorem, which linked the number of values under permu- 

tations to the degree of the root extraction (see section 6.5), ABEL demonstrated that 

all non-symmetric rational functions of five quantities could be reduced to two basic 

forms. 

Finally, these preliminary sections were combined to provide ABEL’S impossibil- 

ity proof by discarding each of a number of cases ad absurdum (section 6.6). ABEL’S 

argument can be outlined in the following steps: 

1. ABEL introduced a classification of algebraic expressions to obtain a standard 

form, rational in the roots, which all possible solutions to the general quintic 

equation had to possess. 

12 Studying algebraic expressions as objects has been seen as a first step in what later became the introduction 

of functions as mappings (especially automorphisms) into algebra and separating functions 

from their ties with analysis. (Kiernan, 1971, 70) 

13 (A.-L. Cauchy, 1815a).

6.3. Classification of algebraic expressions 101 

2. The classification also enabled ABEL to link the number of values under permu- 

tations to the exponent of the involved root extraction. 

3. By adapting CAUCHY’S theory of permutations, a restriction of the possible num- 

ber of values under permutations to 2 or 5 was achieved. 

4. Finally, ABEL reduced each of the possible cases by indirect proofs. 

In general, ABEL used references in accordance with the nineteenth century tradi- 

tion. Throughout, ABEL’S approach to the question of solubility of the quintic was 

based on counting the number of values which a rational function took when its ar- 

guments were permuted. Thus, he clearly worked in the tradition initiated by J. L. 

LAGRANGE (1736–1813), and it is a little remarkable that no reference to — or even 

mention of — LAGRANGE was ever made in ABEL’S published works on the theory 

of equations. I take this as an indication that during the half-century elapsed since 

LAGRANGE’S trend-setting research, 14 his results and approach had become common 

practice in the field. On the other hand, ABEL made explicit reference to CAUCHY’S 

work on the theory of permutations, 15 from which he had borrowed the CAUCHY- 

RUFFINI theorem without proof in his original 1824 version. 16 In the proof published 

two years later in CRELLE’S Journal für die reine und angewandte Mathematik, 17 ABEL 

provided the theorem with his own shorter proof, keeping the reference. Thus, by the 

same argument as above, CAUCHY’S much younger theory had not yet been as widely 

established. 

6.3 Classification of algebraic expressions 

The objects which ABEL called algebraic functions — and which I term algebraic expres- 

sions — were explicit algebraic functions: finite combinations of constant and variable 

quantities obtained by basic arithmetical operations. If the operations included only 

addition and multiplication, the expression was said to be entire; if, furthermore, di- 

vision was involved, it was called rational; and if, additionally, root extractions were 

allowed, the expression was denoted an algebraic expression. Subtraction and extraction 

of roots of composite degree were explicitly reduced to addition and the extraction of 

roots of prime degree, respectively, in order to be contained in the above operations. 

In the subsequent classification, ABEL benefited from the simplicity introduced by this 

minimal definition in which only root extractions of prime degree were considered. 

The purpose of ABEL’S investigations of algebraic expressions was to obtain an im- 

portant auxiliary theorem for his impossibility proof. Based on a definition which in- 

14 (Lagrange, 1770–1771). 

15 (A.-L. Cauchy, 1815a). 


17 (N. H. Abel, 1826a).


troduced algebraic expressions as objects, ABEL derived a standard form for these ob- 

jects. Applying it to algebraic expressions which satisfied a given equation, he found 

that these could always be given a form in which all occurring components depended 

rationally on the roots of the equation. 

In his effort to obtain a classification of algebraic expressions, ABEL introduced 

a hierarchy based on the concepts of order and degree. These concepts introduced a 

structure in the class of algebraic expressions allowing ordering and induction to be 

carried out. 

In dealing with the proof which ABEL gave of his auxiliary theorem, we are in- 

troduced to two other concepts which are even more fundamental to his theory of 

algebraic solubility. These are the Euclidean division algorithm and the concept of ir- 

reducibility. In section 6.3.3, the proof is presented in quite some detail to demonstrate 

how ABEL made use of these concepts. They were to become even more important in 

his unpublished general theory of solubility (see chapter 8). 

6.3.1 Orders and degrees 

ABEL’S classification of algebraic functions (expressions) was hierarchic; his means to 

obtain structure were the two concepts of order and degree. The order was introduced 

to capture the depth of nested root extractions, whereas the degree kept track of root 

extractions at the same level by imposing a finer structure. ABEL defined rational 

expressions to be of order 0, and the order concept was thereafter defined inductively. 

Thus, if f was a rational function of expressions of order µ − 1 and root extractions 

of prime degree of such expressions, f would be an algebraic expression of order µ. 

With this idea, ABEL obtained the following standard form of algebraic expressions of 

order µ: 

f (g1, . . . , gk; p√ 1 r1, . . . , pm √ rm) , (6.1) 

where f was a rational expression, the expressions g1, . . . , g k and r1, . . . , rm were alge- 

braic expressions of order µ − 1, and p1, . . . , pm were primes. 

Thus, as indicated, ABEL’S concept of order counted the number of nested root 

extractions of prime degree. For instance, if R was a rational function (i.e. of order 0), 

√ R was of order 1, 3 �√ R of order 2, and similarly 3 �√ R + √ R was of order 2. Also 

4√ R was of order 2, since it would have to be decomposed as two nested square roots, 

�√ R. 

Within each order, ABEL described another hierarchy controlled by the concept 

of degree. While the order served to denote the number of nested root extractions 

of prime degree, ABEL’S concept of the degree of an algebraic expression counted 

the number of co-ordinate root extractions at the top level. Thus in (6.1), it was the 

minimal value of m which would suffice to write the expression in this form. In table 

6.1, I have illustrated the concepts by listing the orders and degrees of one of the


Expression 

� 

� 

Order Degree 

3 

� 

3 

R + � Q 3 + R 2 + 3 

R − � Q 3 + R 2 2 2 

R + � Q 3 + R 2 2 1 

R + � Q 3 + R 2 1 1 

Q 3 + R 2 0 0 

Table 6.1: The order and degree of some expressions in CARDANO’S solution to the 

general cubic x 3 + a2x 2 + a1x + a0 = 0. R and Q are assumed to be certain rational 

functions of the given quantities, here the coefficients a0, a1, a2. 

G. CARDANO (1501–1576) solutions to the general cubic. Any rationally related root 

extractions were, ABEL said, to be combined and any algebraic expressions of order µ 

and degree 0 were to be simplified as algebraic expressions of order µ − 1. 

ABEL never considered whether his definitions of order and degree were total, i.e. 

whether any algebraic expression could (uniquely) be ascribed an order and a degree; 

throughout his investigations of algebraic expressions, ABEL tacitly used that to any 

such object corresponded a unique order and a unique degree. It is obvious that these 

concepts introduced a hierarchy on the class of algebraic expressions (see table 6.1). 

6.3.2 Standard form 

Based on his hierarchy of algebraic expressions, ABEL demonstrated a central theorem 

concerning these newly defined objects. It was to serve as a concrete standard form 

for algebraic expressions. First, ABEL found a slightly modified standard form (6.1) 

by writing an algebraic expression v of order µ and degree m as 

� 

v = f r1, . . . , rk, p√ � 

R , (6.2) 

where f was rational, r1, . . . , rk were expressions of order µ but degree at most m − 1, 

whereas R was an expression of order µ − 1 such that 

p√ 

R could not be expressed 

rationally in r1, . . . , r k, and p was a prime. ABEL obtained this alternative standard 

form (6.2) from (6.1) by allowing the arguments r1, . . . , r k to be of the same order as 

v, but of lower degree. The two standard forms were equivalent and the hierarchic 

structure in the class of algebraic expressions was preserved. 

Writing the rational expression f as the ratio of two entire expressions, 

v = T 

� 

r1, . . . , rk, p√ � 

R 

� 

V r1, . . . , rk, p√ �, 

R 

ABEL specified the form of v as the ratio of two polynomials in p√ R of degree at most 

p − 1, 

v = T 

. (6.3) 

V


After denoting by V1, . . . , Vp−1 the values of V by inserting α k p √ R for p√ R in V (α a p th 

root of unity), ABEL multiplied numerator and denominator of (6.3) by V1V2 . . . Vp−1. 

The denominator thereby became a rational function of r1, . . . , r k “as it is known”. 18 

The conclusion can be seen as an application of ABEL’S implicit version of LAGRANGE’S 

theorem 1. 19 

By this analogous of multiplying the denominator by conjugates, 20 ABEL had shown 

that the expression v could be written as a polynomial in p√ R, 

v = f 

� 

r1, . . . , rk, p√ � p−1 

R = 

∑ 

u=0 

quR u p , 

where R was of order µ − 1 and all the coefficients q0, . . . , qp−1 were functions of order 

µ and degree at most m − 1 such that R 1 p could not be expressed rationally in the coeffi- 

cients. ABEL also stated that the coefficient q1 could be assumed equal to 1. In this last 

step, ABEL’S conclusions concerning the orders and degrees of the other coefficients 

were too bold, as W. R. HAMILTON (1805–1865) and L. KÖNIGSBERGER (1837–1921) 

in 1839 and 1869, respectively, were to point out (see section 6.9.1). 21 In general, this 

step — obtained by dividing each coefficient by q1 — might effect the order of R which 

could now be µ. However, as KÖNIGSBERGER also noticed, the mistake was not an 

essential one and has no consequences for the rest of the proof (see section 6.9.1). 

In ABEL’S version, the standard form of algebraic expressions can be described by 

theorem 2. 

Theorem 2 Let v be an algebraic expression of order µ and degree m. Then 

v = q0 + p 1 n + q2p 2 n + · · · + qn−1p n−1 

n , (6.4) 

where n is a prime, q0, q2, . . . , qn−1 are algebraic expressions of order µ and degree at most 

m − 1, and p is an algebraic expression of order µ [ABEL stated µ − 1, see below] such that 

p 1 n cannot be expressed as a rational function of q0, q2, . . . , qn−1. (N. H. Abel, 1826a, 70) ✷ 

In his modified version, KÖNIGSBERGER only concluded that the algebraic expression 

p was of order µ and degree at most m − 1, and that the order of p 1 n was µ. 

18 (N. H. Abel, 1826a, 69). 

19 The function V can be interpreted as depending upon all the roots of the equation X p = R, i.e. 

�√p p 

V = V R, α √ R, . . . , αp−1 p √ � 

R although only the first argument is actually involved. The values 

V0, . . . , Vp−1 are then obtained by transposing the first argument with any other argument, and 

the theorem 1 states that the product ∏ p−1 

u=0 Vu is a rational function of 

p√ 

R, . . . , αp−1 p √ R and the 

coefficients of V. 

20 In order to obtain a real denominator of the fraction 

a + ib 

c + id 

its numerator and denominator are both multiplied by c − id. 

21 (W. R. Hamilton, 1839; Königsberger, 1869).


Once ABEL had reduced the algebraic expressions to their standard forms (6.4), he 

devoted an entire section to demonstrate the central description of algebraic expres- 

sions which could satisfy a given equation. 

6.3.3 Expressions which satisfy a given equation 

ABEL began with the assumption that the given equation 

k 

∑ cuy 

u=0 

u = 0, (6.5) 

in which the coefficients were rational functions of some quantities x1, . . . , xn, would 

be satisfied by inserting for y an algebraic expression of the form (6.4). He deduced 

that (6.5) would be transformed into an equation in p 1 n and found that he could write 

it as 

n−1 

∑ rup 

u=0 

u n = 0, (6.6) 

in which r0, . . . , rn−1 were rational functions of p, q0, q2, . . . , qn−1. 

The central result which ABEL obtained in this connection was that for this equa- 

tion to be satisfied the coefficients r0, . . . , rn−1 all had to vanish (lemma 1). His proof 

is a beauty and clearly reflects the central methods involved in his approach to the 

theory of equations. 

Lemma 1 If the equation (6.6) is satisfied, the coefficients r0, . . . , rn−1 all vanish. ✷ 

ABEL transformed the assumption that (6.6) could be satisfied into the assumption 

that the two equations 

⎧ 

⎪⎨ 

⎪⎩ 

z n − p = 0 

n−1 

∑ 

u=0 

ruz u = 0 

had one or more common roots. If some of the coefficients r0, . . . , rn−1 did not vanish 

the latter equation would have degree at most n − 1 . Thus, the two equations could 

at most share n − 1 roots, and ABEL denoted the number of common roots by k. When 

he formed the equation having precisely these k roots as its roots, 

k 

∏ (z − zu) = 

u=1 

k 

∑ suz 

u=0 

u = 0 (6.7) 

he realized that the coefficients s0, . . . , s k−1 depended rationally on r0, . . . , rn−1. ABEL 

gave no details at this point, but I assume that he obtained the result applying the 

Euclidean division algorithm to polynomials and considered this procedure to be well


established. C. SKAU considers the Euclidean algorithm among the central pillars of 

ABEL’S impossibility proof. 22 In section 7.3.1, I illustrate how it, indeed, — together 

with the concept of irreducibility — played an important role in ABEL’S theory of 

equations. 

In the very same paragraph, ABEL let 

µ 

∑ tuz 

u=0 

u = 0 (6.8) 

denote the factor of (6.7) of lowest degree with rational coefficients and continued with 

the following statement implicitly introducing irreducibility which ABEL had not used 

or defined thus far: 

and let 

“Let that equation [here (6.7)] be 

s0 + s1z + s2z 2 · · · + s k−1z k−1 + z k = 0 

t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ 

be a factor of its first term [left hand side], where t0, t1 etc. are rational functions 

of p, r0, r1 . . . rn−1; then also 

t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ = 0 

and it is clear, that it can be assumed to be impossible to find an equation of the 

same form of lower degree.” 23 

Thus, certain roots of (6.7) would also be roots of (6.8), ABEL argued, and the µ 

roots of (6.8) would also be roots of z n − p = 0. In the case µ = 1, it would be easy to 

write z, i.e. p 1 n , as a rational function of t0 and t1, and thereby as a rational function of 

p, r0, . . . , rn−1 from (6.6), contrary to the assumption imposed by theorem 2. 

Since µ ≥ 2, ABEL let z and αz denote two distinct common roots of (6.8) and 

z n − p = 0. When he inserted them into (6.8), he obtained 

22 (Skau, 1990, 54). 

23 “Die Gleichung sei 

und 

µ−1 

∑ tu (α 

u=0 

u − α µ ) z u = 0 (6.9) 

s0 + s1z + s2z 2 · · · + s k−1z k−1 + z k = 0 

t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ 

ein Factor ihres ersten Gliedes, wo t0, t1 etc. rationale Functionen von p, r0, r1 . . . rn−1 sind, so ist 

auch 

t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ = 0 

und es ist klar, daß man es als unmöglich annehmen kann, eine Gleichung von niedrigerem Grade 

von der nemlichen Form zu finden.” (N. H. Abel, 1826a, 71).


which was an equation of degree at most µ − 1 having some of the roots of the irre- 

ducible (6.8) as its roots. In this connection, ABEL actually used the word “irreducible” 

for the first time (see the quotation below). Consequently, the polynomial of (6.9) 

would have to be the zero polynomial and a contraction had been reached: 

“But since the equation z µ + tµ−1z µ−1 · · · = 0 is irreducible, it must, since it is 

of the µ − 1’st degree give 

which is impossible.” 24 

α µ − 1 = 0, α − α µ = 0 . . . α µ−1 − α µ = 0; 

The contradicted assumption was that at least one coefficient among r0, . . . , rn−1 

was non-zero, and thus the result (lemma 1) had been demonstrated. 

When ABEL considered n different values y1, . . . , yn of y resulting from substitut- 

ing α k p 1 n for p 1 n in the expression (6.4) for y, he found that these all constituted roots 

of the equation when it was assumed to be algebraically solvable. Through labori- 

ous, albeit not very difficult, algebraic manipulations including a tacit application of 

LAGRANGE’S theorem 1 on resolvents, ABEL then demonstrated that if the equation 

was solvable, the coefficients q0, q2, . . . , qn−1 as well as p 1 n would all depend rationally 

on these roots (and certain roots of unity, such as α). Thereby, he demonstrated that 

all components of a top-level algebraic expression solving a solvable equation were ra- 

tional functions of the equation’s roots. By considering any of these components and 

working downward in the hierarchy, ABEL demonstrated that this applied equally 

well to any component involved in the solution. Thus, he had proved the following 

explicitly formulated and very important auxiliary theorem, corresponding to RUF- 

FINI’S open hypothesis. 25 

Theorem 3 “When an equation can be solved algebraically, it is always possible to give to 

the root [solution] such a form that all the algebraic functions of which it is composed can be 

expressed by rational functions of the roots of the given equation.” 26 

The study of algebraic expressions which ABEL had conducted as a preliminary 

to his impossibility proof had produced two central results for the proof. Firstly, it 

had provided a hierarchy on the algebraic expressions based on the nesting of root 

extractions. Secondly, it had resulted in the auxiliary theorem stated just above, which 

24 “Da nun aber die Gleichung z µ + tµ−1z µ−1 · · · = 0 irreducibel ist, so muß sie, weil sie vom µ − 1 ten 

Grade ist, einzeln 

α µ − 1 = 0, α − α µ = 0 . . . α µ−1 − α µ = 0 

geben; was nicht sein kann.” (ibid., 72). 

25 ABEL carried out his deductions in ignorance of RUFFINI’S work (see section 6.7). 

26 “Wenn eine Gleichung algebraisch auflösbar ist, so kann man der Wurzel allezeits eins solche Form 

geben, daß sich alle algebraische Functionen, aus welchen sie zusammengesetzt ist, durch rationale 

Functionen der Wurzeln der gegebenen Gleichung ausdrücken lassen.” (ibid., 73).


ensured ABEL that any expression which he was to encounter in the hierarchy of a 

solvable equation, would depend rationally on the roots of the given equation. 

6.4 ABEL and the theory of permutations: the 

CAUCHY-RUFFINI theorem revisited 

The second preliminary pillar of ABEL’S impossibility proof was made up of his stud- 

ies of permutations and his proof of the CAUCHY-RUFFINI theorem describing the 

possible numbers of values of rational functions under permutations of their argu- 

ments. Prior to giving his proof of this central result, ABEL summarized much of what 

CAUCHY had done in his 1815-paper, 27 and in doing so ABEL took over CAUCHY’S 

notation and much of his terminology. But while CAUCHY had begun the process of 

liberating the substitutions from the expressions on which they acted, ABEL contin- 

ued the tradition of LAGRANGE. Although he occasionally spoke of the “substitution” 

[Vervandlung] as an independent object, all his deductions concerned their actions on 

expressions. 

“Now let 

A1 

A2 

� � 

αβγδ . . . 

v 

abcd . . . 

be the value, which an arbitrary function v takes, when therein xa, xb, xc, xd etc. are 

inserted instead of xα, xβ, xγ, xδ etc.; then it is clear that, when by A1, A2 . . . Aµ one 

denotes the different forms which 1, 2, 3, 4 . . . n can possibly take by interchanges 

of the exponents 1, 2, 3 . . . n, the different values of v can be expressed as 

� � � � � � � � 

A1 A1 A1 A1 

v , v , v . . . v .” 28 

A3 

With this notation, ABEL proved LAGRANGE’S theorem that the number of differ- 

ent values of the function v would be a divisor of n!. Next, he introduced the concept 

of recurring substitutions [wiederkehrende Verwandlungen] of order p, thereby re- 

placing the word degree chosen by CAUCHY. In the 1840s, CAUCHY was to take up 

ABEL’S terminology on this point. 29 Through a counting argument based on what 

27 (A.-L. Cauchy, 1815a). 

28 “Nun sei 

� � 

αβγδ . . . 

v 

abcd . . . 

der Werth, welchen eine beliebige Function v bekommt, wenn man darin xa, x b, xc, x d etc. statt 

xα, x β, xγ, x δ etc. setzt, so ist klar, daß wenn man durch A1, A2 . . . Aµ die verschiedenen Formen bezeichnet, 

deren 1, 2, 3, 4 . . . n durch Verwechselung der Zeiger 1, 2, 3 . . . n fähig ist, die verschiedenen 

Werthe von v durch 

v 

� A1 

A1 

� 

, v 

� A1 

A2 

� 

, v 

� A1 

ausgedrückt werden können.” (N. H. Abel, 1826a, 74). 

29 (Wussing, 1969, 67). 

A3 

� 

. . . v 

Aµ 

� A1 

Aµ 

�

6.4. ABEL and the theory of permutations 109 

later was termed the pigeon hole principle, 30 ABEL proved that if v took fewer than p 

different values, and ( A1 ) was a recurring substitution of order p, some two among 

Am 

the p values 

� �0 � �p−1 A1 A1 

v , . . . , v 

had to be identical, 

Am 

v 

� �R A1 

Am 

for some R. At this point, the argument was hampered by a typographical error, which 

might have rendered it unintelligible to some readers (see section 6.9). By tacit use of 

the Euclidean algorithm, ABEL found that it would be possible to determine integers 

α, β such that 

proving 

Am 

= v 

Rα = 1 + pβ 

v 

� A1 

Am 

� 

= v. 

The argument thus amounted to proving that if v took fewer than p values under 

permutations, v would be invariant under any substitution of order p (p a prime). All 

these steps had been taken by CAUCHY, and ABEL simply filled in the last details and 

supplied a proof in his shorter presentational style. 

As CAUCHY had done, ABEL subsequently proved that any 3-cycle was the prod- 

uct of two recurring p-cycles and that any 3-cycle could be decomposed into 2-cycles. 

Thereby, he had demonstrated that if the number of values of v was less than the 

largest prime p ≤ n it had to be either 1 or 2. In the process, he also found that if 

the function had two values these would correspond to odd and even numbers of 

transpositions. The result can be summarized in the following theorem. 

Theorem 4 Let v be a function of n quantities x1, . . . , xn. Let the number of values which v 

takes under all permutations of x1, . . . , xn be denoted by λ and let p denote the largest prime 

which is less than or equal to n. If λ 

sults: 

In his paper, ABEL had — thus far — obtained the following two preliminary re- 

1. Based on a hierarchic classification of algebraic expressions, the concept of irre- 

ducibility, and the Euclidean algorithm, ABEL had found that any radical occur- 

ring in a supposed algebraic solution of an equation depended rationally on the roots of 

that equation (see theorem 3). 

30 Also known as the Dirichlet boxing-in principle.


2. ABEL had inherited a result, the CAUCHY-RUFFINI theorem, which limited the 

possible numbers of values of rational functions under permutations of their arguments 

(see theorem 4). Applied to the quintic, he observed that the result proved that 

no function of five quantities could exist which took on three or four different 

values when its arguments were interchanged. ABEL proceeded by exploring 

the remaining cases, i.e. function of five quantities, which took on two or five 

different values. 

Although ABEL chose to present his detailed studies of particular cases before linking 

these two preliminaries, I have chosen to provide this logical link in the following 

section. 

6.5 Permutations linked to root extractions 

A very central link between the two preliminaries described above was provided to- 

ward the end of ABEL’S argument. 31 There, he linked the number of values taken by 

a function v under all permutations of its arguments to the minimal degree of a poly- 

nomial equation which had v as a root and symmetric functions as coefficients. This 

equation is the irreducible equation corresponding to v and was later to take a very 

central position in his general theory of solubility (see chapter 8). 

ABEL let v designate any rational function of x1, . . . , xn which took on m different 

values v1, . . . , vm under permutations of the quantities x1, . . . , xn. By this, he meant 

that the function v had the m different formal appearances v1, . . . , vm when its argu- 

ments were permuted. Of course, v itself was identical to one of these values but as the 

typesetting suggests, ABEL distinguished the values from the function. ABEL formed 

the equation 

m 

∏ (v − vk) = 

k=1 

m 

∑ qkv k=0 

k = 0, 

and claimed that the coefficients q0, . . . , qm were symmetric functions of the quantities 

x1, . . . , xn. ABEL gave no reference and no proof of this assertion, which is now an easy 

consequence of one of LAGRANGE’S theorems concerning resolvents (theorem 1). 

ABEL also maintained that it was impossible to express v as a root of any equation 

of lower degree with symmetric coefficients. He proved this through a reductio ad 

absurdum by assuming that 

µ 

∑ tkv k=0 

k = 0 (6.10) 

was such an equation where the t k were symmetric, and µ < m. If v1 was a root of 

(6.10) it would be possible to divide the polynomial in (6.10) by (v − v1) and obtain 

31 (N. H. Abel, 1826a, 81–82)

6.5. Permutations linked to root extractions 111 

another polynomial P1, 

0 = 

µ 

∑ tkv k=0 

k = (v − v1) P1. 

When the quantities x1, . . . , xn were permuted, it followed that the equation (6.10) 

would be transformed into 

µ 

∑ tkv k=0 

k u = 0 

for some u since the t k’s were symmetric. Since vu was therefore a root of (6.10), di- 

vision in (6.10) by (v − vu) was possible. Thus, ABEL could decompose (6.10) in m 

different ways corresponding to each of the values of v 

0 = 

µ 

∑ tkv k=0 

k = (v − vu) Pu for 1 ≤ u ≤ m. 

Because the formal values v1, . . . , vµ were distinct, it followed that µ = m and ABEL 

had reached a contradiction. 

The corner stone of ABEL’S argument was the demonstration that if v was a root 

of the equation (6.10), any value vu which v might take on under permutations of 

x1, . . . , xn would also be a root of that equation. He summarized the connection thus 

provided in the following way: 

“When a rational function of multiple quantities has m different values, then 

it will always be possible to find an equation of degree m, the coefficients of which 

are symmetric functions, and which has all the values [of v] as roots; but it is not 

possible to find an equation of the described form of lower degree which has one 

or more of these values as roots.” 32 

In this way, ABEL linked the rather new concept of number of values under permu- 

tations to the older one of number of values of expressions of the form n√ y. It had long 

been accepted that square roots were two-valued, cubic roots three valued etc., and 

ABEL thus connected these two apparently different ways of counting the number of 

values of an algebraic expression. The following points summarize ABEL’S important 

applications of this correspondence: 

1. If v = v (x1, . . . , xn) is a rational function which takes the m different values 

v1, . . . , vm under permutations of x1, . . . , xn, an irreducible equation with symmet- 

ric functions t0, . . . , tm of x1, . . . , xn as coefficients can be associated with v, 

m 

∏ (v − vk) = 

k=1 

m 

∑ tkv k=0 

k = 0. 

32 “Wenn eine rationale Function mehrerer Größen m verschiedene Werthe hat, so läßt sich allezeit eine 

Gleichung vom Grade m finden, deren Coefficienten symmetrische Functionen sind, und welche jene 

Werthe zu Wurzeln haben; aber es ist nicht möglich eine Gleichung von der nämlichen Form von 

niedrigerem Grade aufzustellen, welche einen oder mehrere jener Werthe zu Wurzeln hat.” (ibid., 

82).


2. On the other hand, if a rational function v = v (x1, . . . , xn) satisfies an equation 

of degree m with symmetric functions of x1, . . . , xn as its coefficients, the func- 

tion v must have at most m different values under permutations of x1, . . . , xn. If 

the equation is furthermore known to be irreducible, v must take on exactly m 

values. Thus, to the relation v = m√ R corresponds an equation of degree m with 

symmetric coefficients. 

6.6 ABEL’s combination of results into an impossibility 

proof 

The fourth component of ABEL’S impossibility proof concerned detailed and highly 

explicit, “computational” investigations of functions of five quantities having two or 

five values. ABEL sought to reduce all such functions to a few standard forms, an 

approach completely in line with the classification which opened his paper. These 

investigations have been subjected to quite a lot of criticism, rethinking, and eventu- 

ally incorporation into a broader theory, all of which will be dealt with in subsequent 

chapters. 

6.6.1 Careful studies of functions of five quantities 

The CAUCHY-RUFFINI theorem described in sections 5.6 and 6.4 had ruled out the ex- 

istence of functions of five quantities which had three or four different values when 

their arguments were permuted. The remaining relevant (non-symmetric) cases were 

concerned with functions having two or five values. In the case of two-valued func- 

tions, ABEL reduced all such functions to the alternating one which CAUCHY had also 

studied; and when the function had five values, ABEL could write it as a fourth degree 

polynomial in one of the variables with coefficients symmetric in the remaining four. 

Two-valued functions. In order to describe functions of five quantities having two 

values under permutations, ABEL let v denote such a function of x1, . . . , x5 having 

the two values v1, v2. Furthermore, he let v ′ denote a second such function (with the 

values v ′ 1 and v′ 2 ) and formed two further functions 

t1 = v1 + v2, and 

t2 = v1v ′ 1 + v2v ′ 2 .

6.6. Combination into an impossibility proof 113 

ABEL claimed that the functions t1 and t2 were both symmetric. 33 The two functions 

v and v ′ were related through these symmetric functions by 

Then, ABEL chose for v ′ 1 

and concluded that v ′ 2 = −v′ 1 

v1 = t1v ′ 2 − t2 

v ′ 2 − v′ 1 

the alternating function s 

v ′ 1 = s = ∏ 

1≤i


First, ABEL tacitly applied an equivalent to Waring’s formulae (see section 5.2.4) to 

express v rationally in x1 and the elementary symmetric functions A0, . . . , A3 occur- 

ring as coefficients in the equation 

0 = 

5 

∏ (x − xk) = x 

k=2 

4 + A3x 3 + A2x 2 + A1x + A0. 

To ABEL, the calculations to obtain this were straightforward and not worth mention- 

ing. When ABEL factorized the general quintic as 

0 = 

5 

∏ 

k=1 

(x − x k) = (x − x1) 

4 

∑ 

k=0 

A kx k = 

5 

∑ akx k=0 

k , 

he found that the coefficients A0, . . . , A4 depended rationally on a0, . . . , a5. Conse- 

quently, v could also be expressed rationally in x1 and a0, . . . , a5 as 

v = t 

φ (x1) , 

where both t and φ (x1) were entire functions of x1, a0, . . . , a5. By inserting the other 

roots x2, . . . , x5 for x1 in φ (x1), ABEL obtained another four entire functions in which 

the coefficients were symmetric functions of x1, . . . , x5. When ABEL multiplied both 

numerator and denominator by ∏ 5 k=2 φ (x k), 35 tacitly used LAGRANGE’S theorem (1) 

on resolvents, and reduced the degree according to the relationship imposed by the 

quintic equation, he obtained v in the desired form of a fourth degree polynomial in 

x1. 

Five-valued functions in general. In order to obtain a standard form of all functions 

of five quantities having five values, ABEL relied on an extensive investigation of par- 

ticular cases. Denoting by v any function of five quantities, which took on the five 

values v1, . . . , v5 when all its arguments were permuted, ABEL introduced an inde- 

terminate m and formed the function x m 1 v. When only x2, . . . , x5 were permuted, this 

function would attain its values from the list 

x m 1 v1, . . . , x m 1 v5. (6.12) 

ABEL let µ denote the number of different values of x m 1 v when x2, . . . , x5 were per- 

muted in all possible ways. He then considered the different cases corresponding to 

different values of µ in detail and either eliminated them through a reductio ad absur- 

dum or reduced them to the standard form (6.11). Throughout this procedure, it is 

important to keep in mind which quantities are permuted, and ABEL was not always 

very explicit. 

35 A similar argument resembling multiplying the denominator by its conjugate is described in section 

6.3.2.


The first case, in which µ = 5, was eliminated, ABEL said, since that assumption 

would require all the values (6.12) to be different. Considering transpositions of the 

form ( 1k 

k1 ), ABEL found that xm 1 v would take on another 20 different values, which 

would also be distinct from those in (6.12). 36 Thus in total, xm 1 v would take on 25 different 

values, and since 25 did not divide 5! = 120 a contradiction had been obtained. 

Secondly, ABEL assumed µ = 1 and found that the function v would only take on 

one value under all permutations of x2, . . . , x5 and thus the case had been reduced to 

the one above, giving v in the form (6.11). 

Thirdly, for µ = 4, the function x m 1 v would take on the different values xm 1 v1, . . . , x m 1 v4, 

and the function v would take on the values v1, . . . , v4 under permutations of x2, . . . , x5. 

Thus, the function 

v1 + v2 + v3 + v4 

was a symmetric function of x2, . . . , x5, and therefore of the form (6.11). Writing v5 as 

v5 = (v1 + · · · + v5) − (v1 + · · · + v4) , 

ABEL concluded that the symmetric function v1 + · · · + v5 could be incorporated in 

the constant term of (6.11), and therefore v5 itself was of the form (6.11). 

These first three cases were not very difficult to follow. However, the remaining 

two cases were subjected to much criticism from his contemporaries (see section 6.9). 

In a letter to the Swiss mathematician E. J. KÜLP (⋆1801), 37 who in a private corre- 

spondence had asked for clarifications, ABEL described a refined argument, which I 

have incorporated in the present description. 

The fourth case, in which µ = 2, reduced to the well known situation of a function 

having only two values under permutations. ABEL concluded that since x m 1 

v took 

on the two values x m 1 v1 and x m 1 v2 under all permutations of x1, . . . , x5, the function v 

would take on only two values, say v1 and v2, when only x2, . . . , x5 were permuted. 

Letting 

φ (x1) = v1 + v2, (6.13) 

ABEL found that φ (x1) was symmetric under permutations of x2, . . . , x5 and thus 

of the form (6.11). The expression φ (x1) had to take on the five different values 

φ (x1) , . . . , φ (x5) under all permutations of x1, . . . , x5 since only transpositions of the 

form ( 1k 

k1 ) effected the value of φ. 

36 To see this, it suffices to realize that any permutation σ of five quantities can be written as a product 

of a permutation ˜σ fixing the symbol 1 and a transposition τ of the form (1k). Then, if an application 

of σ to v gives vu, it follows that 

(x m 1 v) ◦ σ = (xm 1 ◦ ˜σ ◦ τ) (v ◦ σ) = xm ˜σ(k) vu. 

37 (Abel→Külp, Paris, 1826/11/01. In Hensel, 1903, 237–240).


In the published paper, ABEL involved himself in a difficult reductio ad absurdum 

to rule out this case. However, because the proof given in the letter to KÜLP is more 

detailed, it is presented before the differences between the two proofs are sketched. 

Besides the symmetric function (6.13), there is another obvious symmetric function 

under permutations of x2, . . . , x5, 

f (x1) = v1v2. 

The function f (x1) is of the form (6.11). ABEL introduced 

and found that it must divide 

(z − v1) (z − v2) = z 2 − φ (x1) z + f (x1) = R, (6.14) 

5 

5 

∏ (z − vk) = ∑ pkz k=1 

k=0 

k = R ′ , 

in which p0, . . . , p5 were symmetric functions of x1, . . . , x5 by the theorem 1 on LA- 

GRANGE resolvents. Since R ′ was unaltered by transpositions ( 1u 

u1 ) it followed that all 

the polynomials derived from (6.14) through this transposition, 

z 2 − φ (xu) z + f (xu) = ρu for 1 ≤ u ≤ 5, 

would divide R ′ . However, as R ′ was a polynomial of the fifth degree, some polyno- 

mials among ρ1, . . . , ρ5 had to share a common factor. Assuming that ρ1 and ρ2 had a 

factor in common ABEL concluded 

z = f (x1) − f (x2) 

φ (x1) − φ (x2) . 

This value of z must be one of the values of v and thus the left hand side had five differ- 

ent values. However, the right hand side had 10 different values, and a contradiction 

had been reached, ruling out the case µ = 2. 

The published argument in Beweis der Unmöglichkeit 38 followed the one given in 

the letter to KÜLP until ABEL had demonstrated that 

φ (x1) = v1 + v2 = 

4 

∑ rkx k=0 

k 1 

and had recognized that φ had five different values under permutations of 

x1, . . . , x5. Whereas the proof in the letter then explicitly constructed the polynomi- 

als R and R ′ , the original argument was much more roundabout. Substituting any one 

x k of x2, . . . , x5 for x1, ABEL obtained the value φ (x k) as the sum of two of the five 

values of v. 



“When x1 is sequentially interchanged with x2, x3, x4, x5 one obtains 

v1 + v2 = φ (x1) 

v2 + v3 = φ (x2) 

. 

vm−1 + vm = φ (xm−1) 

vm + v1 = φ (xm) , 

where m is one of the numbers 2, 3, 4, 5.” 39 

The m here is not the indeterminate introduced earlier, but a number introduced 

for this particular purpose. It is unclear to me, and probably also a point of concern 

to ABEL’S contemporaries, how this set of equations could be put on the circular form 

above. But once it had been done (assuming it could be done) it was a simple matter 

of contradicting the different assumptions for m. If m = 2, it followed that φ (x1) = 

φ (x2) and φ could not have five values after all. If m = 3, ABEL deduced that 

2v1 = φ (x1) − φ (x2) + φ (x3) , 

whereby a contradiction was reached because the left hand side had 5 values, whereas 

the right hand side had 5×4 

2 × 3 = 30 values. In a similar way, ABEL claimed he could 

prove that m = 4 or m = 5 could be ruled out as well, 40 which in turn proved that µ 

could not be equal to 2. 

ABEL’S argument presented in the paper depended on a rather obscure sequence 

of functions and was severely criticized. The proof which ABEL gave in his letter to 

KÜLP avoided this central step and was much clearer. I conjecture that KÜLP had 

questioned the sequence of equations, and that ABEL had subsequently developed 

his new proof which he presented as an answer; I have no indication that ABEL had 

possessed the proof presented to KÜLP when he wrote his paper. 

The final case, µ = 3, was ruled out in the same way as µ = 2 above. ABEL found 

that if µ = 3, the function 

v1 + v2 + v3 

would be symmetric under permutations of x2, . . . , x5 and therefore 

v4 + v5 = (v1 + · · · + v5) − (v1 + · · · + v3) 

39 “Vertauscht man der Reihe nach x1 mit x2, x3, x4, x5, so erhält man 

v1 + v2 = φ (x1) 

v2 + v3 = φ (x2) 

. 

vm−1 + vm = φ (xm−1) 

vm + v1 = φ (xm) , 

wo m eine der Zahlen 2, 3, 4, 5 ist.” (ibid., 80). 

40 HOLMBOE supplied the expressions (see section 6.9.1).


could be written in the form (6.11) as he had done in the case µ = 4 above. However, 

ABEL had just demonstrated in the case µ = 2 that no sum of two values of v could 

have five values under permutations of x1, . . . , x5, whereby he reached a contradiction. 

The core of ABEL’S description of functions of five quantities having five values 

under permutations of these consisted of two parts: 

1. A direct manipulation based on LAGRANGE’S theorem 1 on resolvents, resulting 

in a proof that any function of five quantities x1, . . . , x5 which is unaltered by per- 

mutations of four of these, x2, . . . , x5, has the form of a fourth degree polynomial 

(6.11). 

2. A meticulous study of the particular cases in which any function of five quanti- 

ties which has five values under permutations of x1, . . . , x5 is either contradicted 

or proved to be of the form (6.11), too. 

At the conclusion of his investigations, ABEL had added a complete description 

of functions of five quantities having five values to the one he had obtained, in case 

the function had only two values. Thereby, he had obtained workable standard forms 

for all non-symmetric rational functions which could be involved in a supposed solu- 

tion to the general quintic. All he lacked was to put the pieces together to obtain the 

impossibility proof. 

6.6.2 The goal in sight 

To combine his preliminary results into a proof of the algebraic insolubility of the 

general quintic 

x 5 + a4x 4 + a3x 3 + a2x 2 + a1x + a0 = 0, (6.15) 

ABEL assumed that an algebraic solution could be obtained. The auxiliary theorem 3 

obtained earlier ensured him that he could assume that any subexpression occurring 

therein would be a rational function of the roots x1, . . . , x5 of the equation (6.15). Since 

the quintic could not be solved by a rational expression alone, some root extraction 

had to occur. ABEL focused his attention on the algebraic expression of the first order 

in the supposed solution. Thus, he dissected the solution from the inside by focus- 

ing on this innermost non-rational function. According to ABEL’S classification, an 

algebraic expression of the first order contained only rational functions of the coeffi- 

cients a0, . . . , a4 and roots of the form m√ R where m was a prime and R was a rational 

function of a0, . . . , a4. Such roots would satisfy the equation 

v m − R = 0, (6.16) 

and v would have to be a rational function of the roots x1, . . . , x5. His earlier results 

showed that it was impossible to diminish the degree of the equation. Therefore, the


link between root extractions and permutations ensured him that the function v would 

take on m values under all permutations of x1, . . . , x5. Since m was a prime and had 

to divide 5! by LAGRANGE’S theorem, ABEL argued, the only possibilities were that 

m equaled 2, 3, or 5. And since no function of five quantities could have three values 

under permutations by the CAUCHY-RUFFINI theorem, ABEL ruled out this possibility. 

The two remaining cases were subsequently both brought to contradictions. 

The innermost root extraction could not be a fifth root. In the simplest case, corre- 

sponding to m = 5, the function v had to have the form of a fourth degree polynomial, 

as ABEL had demonstrated: 

v = 5√ R = 

4 

∑ rkx k=0 

k 1 . 

Through a process of inversion of polynomials in which the quintic equation (6.15) was 

used to lower the degree, ABEL found that 

x1 = 

4 

∑ skR k=0 

k 5 , 

where s0, . . . , s4 were symmetric functions of x1, . . . , x5. Furthermore, by use of basic 

properties of primitive roots of unity, he obtained 

s1R 1 5 = 1 

� 

x1 + α 

5 

4 x2 + α 3 x3 + α 2 � 

x4 + αx5 , 

where α was a primitive fifth root of unity. The left hand side of the equation was a so- 

lution to a fifth degree equation, and thus had (at most) five different values, whereas 

the right hand side was formally altered by any permutation of x1, . . . , x5 and thus 

had 5! = 120 different values. This ruled out the case m = 5, and the innermost root 

extraction could not be a fifth root. 

The innermost root extraction could not be a square root, neither. ABEL brought 

the case m = 2 to a contradiction in a similar way, although it involved studying 

expressions of the second order as well. He knew that the root would have to be of 

the form 

√ R = p + qs, 

and the other value under permutations would be 

− √ R = p − qs. 

Subtracting these two, ABEL concluded that √ R was of the form 

√ R = qs,


and he saw that any rational combination of such root extractions would continue to 

be of the same form. Therefore, any algebraic expression of the first order contained 

in the solution would have to be of the form 

α + βs 

where α, β were symmetric functions. ABEL observed that such functions were not 

powerful enough to solve the general quintic (6.15), and found that such a solution 

would necessarily contain root extractions of the form 

m ′� α + βs, 

where m ′ was a prime and β �= 0. ABEL knew that such a root, say v, was a rational 

function of x1, . . . , x5. Among the values of v obtained by permuting x1, . . . , x5 he 

found that two were of particular interest, 

When these two were multiplied, 

v1 = m′� α + βs, and 

v2 = m′� α − βs. 

γ = v1v2 = m′� 

α 2 − β 2 s 2 , 

the expression under the root sign was a symmetric function. 

At this point, ABEL again considered two individual cases: either γ, itself, was a 

symmetric function, or it was not. In case γ was a non-symmetric function, it would be 

a first order algebraic expression, and ABEL had proved that for such expressions the 

value of m ′ would have to be 2. This led to a contradiction, since v then had four values 

under permutations of x1, . . . , x5 because β �= 0. However, by the CAUCHY-RUFFINI 

theorem no such function could exist. Consequently, m′� α 2 − β 2 s 2 would have to be a 

symmetric function. By adding v1 and v2, ABEL obtained a function p, 

p = v1 + v2 = R 1 

m ′ + γ 

R R m′ −1 

m ′ 

with R = α + βs. He studied the values of p resulting from substituting α k R 1 

m ′ for 

R 1 

m ′ and demonstrated that p had to have m ′ values under permutations of x1, . . . , x5. 

But since m ′ = 2 had been ruled out, he concluded that m ′ = 5, and the second root 

extraction counted from the inside had to be a fifth root. This time ABEL obtained 

� 

x1 + α 4 x2 + α 3 x3 + α 2 � 

x4 + αx5 

t1R 1 5 = 1 

5 

in which t1 was a symmetric function of the roots. The left hand side was the root of an 

irreducible equation of the tenth degree, 41 thus having 10 values under permutations. 

41 With y = t1R 1 5 and R = α + βs, the equation was � y 5 − t 5 1 α� 2 − t 10 

1 β2 s 2 = 0 in which the coefficients 

are symmetric functions.


The right hand side had a complete 120 values since none of the roots x1, . . . , x5 could 

be interchanged without altering the value of the expression. Thus, a contradiction 

had again been reached. 

The line of ABEL’S argument in knitting together his preliminary investigations 

can be divided into the following steps: 

1. The innermost root extraction in any supposed solution to the general quintic 

had to be either a fifth root (m = 5) or a square root (m = 2); any other possibili- 

ties were ruled out by the CAUCHY-RUFFINI theorem. 

2. The innermost root extraction could not be a fifth root (m �= 5) since this was 

brought to a contradiction by comparing the number of values of certain expres- 

sions. 

3. Thus, the innermost root extraction had to be a square root (m = 2). Then the 

second innermost root extraction was taken into consideration. Its degree has 

been denoted m ′ . 

4. The second innermost root extraction, too, had to be of degree either two (m ′ = 

2) or five (m ′ = 5). 

5. In case the second innermost root extraction was a square root, a function having 

four values under permutations would be obtained, from which a contradiction 

could be reached. Thus m ′ �= 2. 

6. Therefore the second innermost root extraction had to be a fifth root, but this, 

too, was brought to a contradiction in a way similar to step 2 above. 

7. Consequently, no algebraic solution to the general quintic could exist, and the 

algebraic insolubility had been demonstrated. 

Apparently, the argument carried out applied to the quintic equation alone. How- 

ever, ABEL claimed that it also proved the insolubility of all general higher degree 

equations. 

“From this [the insolubility of the general quintic] it follows immediately that 

it is also impossible generally to solve equations of degrees above the fifth. Therefore 

the equations which can be generally solved are only of the four first degrees.” 42 

Although he produced no further evidence, ABEL probably thought of a proof by 

the following argument. If the roots of the general sixth degree equation 

x 6 + a5x 5 + a4x 4 + a3x 3 + a2x 2 + a1x + a0 = 0 

42 “Daraus folgt unmittelbar weiter, daß es ebenfalls unmöglich ist, Gleichungen von höheren als dem 

fünften Grade allgemein aufzulösen. Mithin sind die Gleichungen, welche sich algebraisch allgemein 

auflösen lassen, nur die von den vier ersten Graden.” (N. H. Abel, 1826a, 84).


could be expressed by any algebraic formula, this formula would also provide the so- 

lution to the general fifth degree equation by inserting a0 = 0 in that formula. Central 

to the argument is that the supposed general solution formula for sixth degree equa- 

tions not only produces a single root, but can somehow be made to produce all the 

roots of the equation. This was a recurring idea in ABEL’S work on the theory of equa- 

tions (see for instance theorem 10), which linked the concepts of satisfiability (a single 

root could be found) and solubility (all roots could be found). 

6.7 ABEL and RUFFINI 

According to ABEL and his commentators, ABEL was unaware of the proofs published 

by RUFFINI when he published his proofs of the impossibility result in 1824 and 1826. 43 

Since questions of priority are a frequently recurring theme in the history of mathe- 

matics, this independence of results is noticed by most biographers of ABEL. 44 It is my 

firm conviction — based on the mathematical contents of his proof — that ABEL devel- 

oped his proof independently of RUFFINI. However, the primary sources of informa- 

tion on ABEL’S independence of RUFFINI are limited. The only mention of RUFFINI 

made by ABEL is in his notebook entry on the theory of solubility (see chapter 8), in 

the introduction to which he described RUFFINI’S proof: 

“The first person, and if I am not mistaken, the only one prior to me, who has 

tried to prove the impossibility of the algebraic solution of the general equations, 

is the geometer Ruffini; but his memoir is so complicated that it is very difficult 

to judge the validity of his reasoning. It seems to me that his reasoning is not 

always satisfying. I think that the proof I gave in the first issue of this journal 

[CRELLE’S Journal] leaves nothing to be desired as to rigor, but it does not have all 

the simplicity of which it is susceptible. I have reached another proof based on the 

same principles, but more simple, in trying to solve a more general problem.” 45 

The answer derived from “trying to solve a more general problem” was never 

made available in print, though. As is documented in chapter 8, such an answer was, 

indeed, indirectly obtainable from ABEL’S more general research on algebraic solu- 

bility which even produced an explicit example of a particular special equation which 

could not be solved. 

43 (N. H. Abel, 1824b; N. H. Abel, 1826a) 

44 See for instance (Bjerknes, 1885, 22–23), (Bjerknes, 1930, 23), (Ore, 1954, 89–90), (Ore, 1957, 125), and 

(Stubhaug, 1996, 352–353). 

45 “Le premier, et, si je ne me trompe, le seul qui avant moi ait cherché à démontrer l’impossibilité 

de la résolution algébrique des équations générales, est le géomètre Ruffini; mais son mémoire est 

tellement compliqué qu’il est très difficile de juger de la justesse de son raisonnement. Il me paraît 

que son raisonnement n’est pas toujours satisfaisant. Je crois que la démonstration que j’ai donnée 

dans le premier cahier de ce journal, ne laisse rien à désirer du côté de la rigueur; mais elle n’a pas 

toute la simplicité dont elle est susceptible. Je suis parvenu à une autre démonstration, fondée sur 

les mêmes principes, mais plus simple, en cherchant à résoudre un problème plus général.” (N. H. 

Abel, [1828] 1839, 218).

6.7. ABEL and RUFFINI 123 

The notebook has been dated to 1828 by P. L. M. SYLOW (1832–1918) — a date 

which implies that ABEL disclosed his knowledge of RUFFINI only after returning to 

Christiania. 46 It is most likely that ABEL learned about RUFFINI during his European 

tour, and two instances are of particular importance. During his stay in Vienna in 

April and May 1826, ABEL became acquainted with the local astronomers K. L. VON 

LITTROW (1811–1877) and A. VON BURG (1797–1882). In the first volume of their 

journal Zeitschrift für Physik und Mathematik, occurring while ABEL was in town, an 

anonymous paper on the theory of equations was published. 47 The author, 48 who was 

inspired by ABEL’S proof and praised it highly, reviewed RUFFINI’S proof. Therefore it 

is not unlikely that ABEL learned of RUFFINI’S proof from his Viennese connections. 49 

Once in Paris, ABEL took on the duty of writing unsigned reviews for FERRUSAC’S 

Bulletin des sciences mathématiques, astronomiques, physiques et chimiques of papers pub- 

lished in CRELLE’S Journal für die reine und angewandte Mathematik. We know from 

one of ABEL’S letters that he, himself, wrote the review of his Beweis der Unmöglichkeit 

which gave a short exposition of the flow of the proof. 50 However, appended to the re- 

view was a short note by the editor, J. F. SAIGEY (1797–1871), 51 which drew attention 

to the works of RUFFINI. 52 SAIGEY mentioned CAUCHY’S favorable review of RUF- 

FINI’S treatise and made it clear that CAUCHY’S view was not universally accepted: 

“Other geometers have not understood this demonstration and some have 

made the justified remark that by proving too much, Ruffini could not prove anything 

in a satisfactory manner; to be sure it was not known how an equation of 

the fifth degree, e.g., could not have transcendental roots, equivalent to infinite series 

of algebraic terms, since one demonstrates that every equation of odd degree 

necessarily has some root. By a more profound analysis, M. Abel proves that such 

roots cannot exist algebraically; but he has not solved the question of the existence 

of transcendental roots in the negative.” 53 

Thus, at two instances in 1826, ABEL had been in close contact with journals, in 

which his result was linked to that of RUFFINI. A third possible source of information 

46 (L. Sylow, 1902, 16). 

47 (Anonymous, 1826). 

48 Or authors? Unlike the review in FERRUSAC’S Bulletin (see below), ABEL is not likely to be the 

author, himself. 

49 See (Ore, 1957, 125). 

50 (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a, 44). The paper reviewed is, of course, (N. 

H. Abel, 1826a). 

51 (Stubhaug, 1996, 589). 

52 (N. H. Abel, 1826c, 353–354). 

53 “D’autres géomètres avouent n’avoir pas compris cette démonstration, et il y en a qui ont fait la 

remarque très-juste que Ruffini en prouvant trop pourrait n’avoir rien prouvé d’une manière satisfaisante; 

en effet, on ne conçoit pas comment une équation du cinquième degré, par exemple, 

n’admettrait pas de racines transcendantes, qui équivalent à des séries infinies de termes algébriques, 

puisqu’on démontre que toute équation de degré impair a nécessairement une racine quelconque. 

M. Abel, au moyen d’une analyse plus profonde, vient de prouver que de telles racines ne 

peuvent exister algébriquement; mais il n’a pas résolu négativement la question de l’existence des 

racines transcendantes.” (Saigey in ibid., 354).


on RUFFINI’S research was, of course, CAUCHY whom ABEL met in Paris without any 

traceable interaction taking place. 54 

Although the primary information on how ABEL came to know of RUFFINI’S proofs 

is rather sparse, I find further support for the assumption of independence in the 

mathematical technicalities as documented in the preceding sections. RUFFINI’S and 

ABEL’S differences in notation and approach to permutations, ABEL’S definition of 

algebraic expressions and his careful proof of the auxiliary theorems describing them 

all suggest to me that ABEL’S deduction was a tailored argument for the impossibility, 

independent of any earlier such proofs. The common inspiration from LAGRANGE, 

which permeated both the works of RUFFINI and ABEL, should be evident enough to 

account for similarities in studying the blend of equations and permutations. 

6.8 Limiting the class of solvable equations 

At a conceptual level, ABEL’S proof that the general quintic could not be solved al- 

gebraically was more than just another proof added to the body of mathematics. For 

centuries, mathematical intuition had suggested that an algebraic solution to the fifth 

degree equation should exist but probably be difficult to find. ABEL had demonstrated 

that any supposed solution carried with it an internal contradiction and thus the re- 

sult not only made the belief in general algebraic solubility tremble, it completely 

destroyed it. 

In negating the existence of an algebraic solution, ABEL provided an instance of 

a negative result — negative in the sense that it contradicted contemporary intuition. 

Similar counter intuitive results abound in mathematics in the period as a result of a 

fundamental transition toward concept based mathematics. 55 

The outspoken reactions of the mathematical community to ABEL’S impossibility 

proofs can be divided in three. Some mathematicians, often belonging to the older 

generation or the loosely institutionalized amateur mathematicians, protested against 

the result and held both the statements and their proofs to be flawed. To these math- 

ematicians, the break with their established intuition forced them into their rejection. 

Others accepted the result but provided refinements of the proofs and their assump- 

tions. And yet others not only accepted the results but saw that the quintic only con- 

stituted one example of an unsolvable equation. Thereby, the more general problem 

of algebraic solubility could be formulated. 

From the perspective of investigating the concept of solubility, the quintic helped 

distinguish the class of algebraically solvable equations within the class of all polyno- 

mial equations (see figure 6.1). On the other hand, in his research on the division of the 

circle, GAUSS had demonstrated that infinitely many equations existed which could 

54 For a discussion of the relationship between CAUCHY and ABEL, see chapter 12. 

55 See chapter 21.

6.9. Reception of ABEL’s work on the quintic 125 

❅ ❅❘ 

❄ 

✲ Solvable equations ✛ 

� ✻ 

Polynomial equations 

�✒ 

Figure 6.1: Limiting the class of solvable equations 

be solved algebraically (see section 5.3). Therefore, the class of solvable equations 

did not collapse to a few low degree equations; soon, further solvable equations were 

found (see chapter 7). The search for a procedure useful in determining whether or 

not a given equation could be solved algebraically soon became an interesting project 

for mathematical research. 

In the following section 6.9, I deal with the first two classes of reactions: the global 

and local criticism, which was advanced by ABEL’S contemporaries. In chapter 8, I 

describe how ABEL worked on the general problem of solubility, which was realized 

to its full extent and attacked shortly afterwards by GALOIS (section 8.5). 

6.9 The reception of ABEL’s work on the quintic 

When RUFFINI published his proof of the algebraic insolubility of the quintic in Ital- 

ian in 1799 the mathematical community of Europe paid little attention. Apart from a 

limited Italian discussion involving mathematicians outside the main stream such as 

P. ABBATI (1768–1842) and G. F. MALFATTI (1731–1807), only CAUCHY seems to have 

taken notice. Twenty-five years later, when ABEL published his proof in a brand new 

German mathematical journal, history could have repeated itself. However, ABEL’S 

proof soon became widespread knowledge and acquired a status within the math- 

ematical community of being rigorous and close to definitive. In this section, I trace 

some of the events which played a role in this development, scientific and non-scientific 

factors, in order to describe the influence which ABEL’S research had on the subse- 

quent evolution of the theory of equations. 

��✠ 

❅❅■


Immediate reception. In his short lifetime, ABEL’S impossibility proof was pub- 

lished on three occasions. ABEL’S first proof — published as a pamphlet in 1824 — 

was, although written in French, only sparsely circulated. 56 According to HANSTEEN 

the copy which ABEL had sent to GAUSS in Göttingen was received with very little 

enthusiasm. 57 When ABEL met CRELLE in Berlin, they discussed the subject and be- 

cause CRELLE and others had a hard time following the arguments, ABEL elaborated 

his proof. Under this procedure to clarify, ABEL produced his second proof which 

CRELLE subsequently found worthy of publication, translated into German, and in- 

serted into the first issue of his Journal für die reine und angewandte Mathematik. 58 The 

immediate impact of ABEL’S paper in the mathematical community was limited. The 

following issue of CRELLE’S Journal carried a paper by the unknown mathematician L. 

OLIVIER on the form of roots of algebraic equations based on LAGRANGE’S research. 59 

In it, OLIVIER voiced reservations concerning the solubility of the general equations 

which indicate that he was not an intimate member of the circle around CRELLE and 

had not learned of ABEL’S result prior to its publication in the Journal für die reine und 

angewandte Mathematik. The reservation might even have been inserted by CRELLE 

who probably also translated OLIVIER’S paper into German. 

“Furthermore, a proof of the impossibility of the solution of algebraic equations 

of higher degrees by radicals, should such a proof be possible, would in no 

way contradict the results of the above investigations on the form of the radicals, 

whose generality has been claimed.” 60 

ABEL tried to improve the distribution of his proof as well as the reputation of 

CRELLE’S Journal für die reine und angewandte Mathematik by publishing — as a report 

on the paper in CRELLE’S Journal für die reine und angewandte Mathematik — a third 

version of his proof in FERRUSAC’S Bulletin. 61 However, in his own lifetime, ABEL 

was mainly known for his later work on elliptic functions (see subsequent chapters) 

and the impossibility proof remained less known. In corresponding with ABEL on the 

subject of elliptic functions, A.-M. LEGENDRE (1752–1833) urged ABEL to make public 

his researches on the solubility of equations which ABEL had announced in an earlier 

letter. 62 

“Sir, you have announced a very beautiful work on algebraic solutions which 

has the purpose of giving the solution of any given numerical equation, whenever 

56 (N. H. Abel, 1824b) 

57 See (Hansteen, 1862, 37) and (Stubhaug, 1996, 291). 

58 (N. H. Abel, 1826a) 

59 Of Mr. LOUIS OLIVIER little is known. He published a total of 11 articles in the first four volumes 

of CRELLE’S Journal für die reine und angewandte Mathematik 1826–1829. OLIVIER’S mathematics and 

his relations to the Berlin mathematicians are the themes of a separate article under preparation. 

60 “Uebrigens würde ein Beweis der Unmöglichkeit der Auflösung höheren algebraischer Gleichungen 

durch Wurzelgrößen, wenn ein solcher gelänge, keinesweges den Resultaten der obigen Untersuchungen 

über die Form der Wurzeln, deren Allgemeinheit behauptet wurde, widersprechen.” 

(Olivier, 1826a, 116). 


62 (Abel→Legendre, Christiania, 1828/11/25. N. H. Abel, 1881, 279)


it can be developed in radicals, and to declare any equation unsolvable in this way 

[by radicals] which does not satisfy the required conditions; from this it follows 

as a necessary consequence that the general solution of the equations beyond the 

fourth degree is impossible. I invite you to publish this new theory as soon as you 

can; it would bring you much honour and generally be regarded as the biggest 

discovery remaining to be made in analysis.” 63 

The investigations to which LEGENDRE alluded were also described in one of ABEL’S 

letters to HOLMBOE, 64 and were partially presented in his notebooks (see chapter 8). 

The above citation seems also to indicate that LEGENDRE was unaware of ABEL’S impossibility 

proof in CRELLE’S Journal für die reine und angewandte Mathematik 1826. 65 

HOLMBOE and the first edition of ABEL’S Œuvres. When the French mathematical 

community learned of ABEL’S death in 1829, the Academy sent baron J. F. T. MAU- 

RICE. 66 to condole with the Swedish envoy in Paris and suggest that the Swedish 

Crown Prince OSCAR undertook the publication of ABEL’S complete works. 67 In 1831, 

MAURICE repeated his suggestion and the editorship was delegated to ABEL’S teacher 

and friend HOLMBOE and the university in Christiania. 68 By 1836, HOLMBOE had 

completed his commentaries on the published works, but intended to include also 

selections from ABEL’S unpublished material in the Œuvres. 69 In his report to the 

Ministry of Ecclesiastic Affairs 70 in 1838 HOLMBOE declared that — except for a ma- 

nuscript which ABEL had handed in to the French Academy 71 — he had finished col- 

lecting and commenting upon ABEL’S unpublished works. 72 Two volumes containing 

most of ABEL’S published works (with the noticeable exclusion of ABEL’S Parisian 

manuscript) 73 and some of the unpublished material from his notebooks and Nach- 

lass appeared in 1839. Since most of ABEL’S papers had originally been published 

in French and most of his mature entries in the notebooks were in French, it had been 

decided that the Œuvres should be in French. An effort was made by HOLMBOE to dis- 

63 “Vous m’annoncez, Monsieur, un très beau travail sur les équations algébriques, qui a pour objet 

de donner la résolution de toute équation numérique proposée, lorsqu’elle peut être développée en 

radicaux, et de déclarer insoluble sous ce rapport, toute équation qui ne satisferait pas aux conditions 

exigées; d’où résulte comme conséquence nécessaire que la résolution générale des équations 

au delà du quatrième degré, est impossible. Je vous invite à publier le plutôt que vouz pourrez, 

cette nouvelle théorie; elle vous fera beaucoup d’honneur, et sera généralement regardée comme 

la plus grande dévouverte qui restait à faire dans l’analyse.” (Legendre→Abel, Paris, 1829/01/16. 

N. H. Abel, 1902a, 88–89). 

64 (Abel→Holmboe, Paris, 1826/10/24. ibid., 44–45). 

65 (Holmboe, 1829, 349). 

66 (Stubhaug, 1996, 587). 

67 After a turbulent period of trembling Danish monarchy and brief independence, Norway was integrated 

in the Swedish monarchy 1814. 

68 (Ore, 1957, 269) 


70 As noted in chapter 2, the University was subsumed in the Ministry of Ecclesiastic Affairs. 

71 The search for ABEL’S Paris mémoire is a fascinating story in its own right. See (Brun, 1949; Brun, 

1953) and section 19.4, below. 


73 ABEL’S Parisian manuscript known as the Paris mémoire is dealt with extensively in chapter 19.


tribute copies to prominent mathematicians. Therefore, ten years after ABEL’S death, 

the mathematical community had the opportunity to follow his arguments through 

HOLMBOE’S careful annotations. 

“During the revision of Abel’s works it has been necessary for me to give numerous 

demonstrations and prove many theorems which are presented without 

proof by the author or whose proof is indicated so briefly that for many readers it 

is impossible and for almost all difficult to understand.” 74 

Apart from the elaborations of vaguely suggested arguments, HOLMBOE also cor- 

rected most of the numerous misprints which had occurred in ABEL’S works pub- 

lished in CRELLE’S Journal. These, too, had served to make ABEL’S writings hard to 

understand. 75 

6.9.1 Local criticism of the quintic proof 

Inspired by I. LAKATOS’ (1922–1974) distinction between global and local counter ex- 

amples, the criticism which mathematicians in the first half of the 19 th century ex- 

pressed toward ABEL’S proof of the insolubility of the quintic can be separated in two 

classes. 76 As noted, a handful of mathematicians continued to doubt or dispute the va- 

lidity of the result that the general quintic was algebraically unsolvable. Their doubt 

was largely founded in an incomplete induction that equations were to be solvable; 

and their attitude toward ABEL’S proof ranged from ignorance to indifference. The 

importance of this global criticism is traced in section 6.9.2. On the other hand, ABEL’S 

proof was scrutinized by some of his contemporaries. Their local criticism picked out 

the vulnerable points of ABEL’S argument and sought to illuminate them or supply 

alternative proofs. Central to these local criticisms was the fact that they were based 

on an acceptance of the overall validity of the result but sought to secure some unclear 

arguments. The central parts of ABEL’S argument which was thought in need of elab- 

oration were the classification of algebraic expressions, ABEL’S proof of the CAUCHY- 

RUFFINI theorem, and in particular ABEL’S study of functions of five quantities having 

five values under permutations. 

EDMUND JACOB KÜLP. From ABEL’S correspondence with EDMUND JACOB KÜLP 

only ABEL’S reply to KÜLP’S questions has been preserved. 77 Therefore, we know 

74 “Under Revisionen af Abels Arbeider har det været mig nødvendigt at optegne en heel Deel Udviklinger 

og at bevise mange Sætninger, som hos Forfatteren ere anførte uden Beviis, eller hvis Beviis er 

saa kort antydet, at det for mange Læsere er umueligt og næsten for alle vanskeligt at fatte.” (N. H. 

Abel, 1902d, 50). 

75 (ibid., 49). 

76 The distinction is inspired by (Lakatos, 1976), However, as discussed in the introduction (section 

1.4.2), LAKATOS’ scheme of mathematical evolution by dialectical dynamics can only be applied 

through largely a-historical reconstructions. 

77 (Abel→Külp, Paris, 1826/11/01. In Hensel, 1903, 237–240)


Figure 6.2: WILLIAM ROWAN HAMILTON (1805–1865) 

nothing of KÜLP’S attitude toward the validity of ABEL’S result. KÜLP’S criticism fo- 

cused on two individual parts of ABEL’S argument. The first question was concerned 

with a misprint which occurred in ABEL’S proof of the CAUCHY-RUFFINI theorem. 

Due to the relatively new character of the theory of permutations and their notation, 

KÜLP apparently had trouble following ABEL’S argument and was halted by the mis- 

print. ABEL’S notation was apparently also a problem for KÜLP; in his answer, ABEL 

proved the claim that any 3-cycle could be decomposed as the product of two p-cycles 

by writing out the substitutions in detail. I mention these objections in order to illus- 

trate the difficulties, conceptual and technical, which nineteenth century mathemati- 

cians had in understanding and accepting ABEL’S proof. 

KÜLP’S other objection concerned ABEL’S descriptive classification of rational func- 

tions of five quantities which have five values. Again, we do not have KÜLP’S formu- 

lation but only ABEL’S reply which ABEL posted from Paris less than a year after his 

paper had appeared in CRELLE’S Journal. The argument given in the letter differed 

substantially from the published one. As I have discussed above (in section 6.6.1), 

the original argument was, indeed, very hard to understand. If ABEL’S refined proof 

communicated to KÜLP had made it into print, ABEL’S conclusion might have been 

accepted at an earlier point.


WILLIAM ROWAN HAMILTON. On the British Isles, the debate over the solubility 

of the general quintic took a different turn. During the years 1832–35, G. B. JER- 

RARD (1804–1863) published his three volume work Mathematical Researches in which 

he claimed to have presented a general method of solving equations algebraically. 

At the 1835 meeting of the British Association in Dublin, WILLIAM ROWAN HAMIL- 

TON was appointed reporter on the paper and was thus led into the theory of equa- 

tions. 78 In May 1836, after having dismissed JERRARD’S claim for a general solution to 

higher degree equations in a paper in the Philosophical Magazine, 79 HAMILTON asked 

his friend J. W. LUBBOCK (1803–1865) to supply him with a copy of ABEL’S paper from 

CRELLE’S Journal für die reine und angewandte Mathematik. Approaching it in his very 

thorough and critical style, HAMILTON found it somewhat unsatisfactory, and began 

to write his own exposition of ABEL’S result. 80 The following year he presented his 

investigations to the Royal Irish Academy, in whose Transactions they were printed. 81 

In his study of ABEL’S proof, HAMILTON noticed two “mistakes”, the first of which 

concerned ABEL’S classification of algebraic expressions (see theorem 2). After hav- 

ing translated ABEL’S proof into his own notation, HAMILTON clearly expressed his 

objection: 82 

“Although the whole of the foregoing argument has been suggested by that 

of Abel, and may be said to be a commentary thereon; yet it will not fail to be 

perceived, that there are several considerable differences between the one method 

of proof and the other. More particularly, in establishing the cardinal proposition 

that every radical in every irreducible expression for any one of the roots of any 

general equation is a rational function of those roots, it has appeared to the writer 

of this paper more satisfactory to begin by showing that the radicals of highest 

order will have that property, if those of lower orders have it, descending thus to 

radicals of the lowest order, and afterwards ascending again; than to attempt, as 

Abel has done, to prove the theorem, in the first instance, for radicals of the highest 

order. In fact, while following this last-mentioned method, Abel has been led to 

assume that the coefficient of the first power of some highest radical can always 

be rendered equal to unity, by introducing (generally) a new radical, which in the 

notation of the present paper may be expressed as follows: 

� 

� 

�⎧ 

�⎨ 

� 

� 

α (m) 

k 

⎩ ∑ 

β (m) 

i


is not, in general, free from the irrationalities of the same order introduced by the 

other radicals a (m) 

1 , . . . of that order; and consequently the new radical, to which 

this process conducts, is in general elevated to the order m + 1; a circumstance 

which Abel does not appear to have remarked, and which renders it difficult to 

judge of the validity of his subsequent reasoning.” 83 

To HAMILTON, the mistake made by ABEL had obscured the validity of ABEL’S 

subsequent reasoning, but the validity of the impossibility result, itself, was not ques- 

tioned since HAMILTON had provided it with a proof not based on ABEL’S hierarchy. 

Later, KÖNIGSBERGER would prove that ABEL’S hierarchy of algebraic expressions 

could still be rescued (see below). By the end of the century, it was eventually realized 

that the hierarchic structure imposed on algebraic expressions was actually superfluous 

for the impossibility proof. 84 

HAMILTON continued his scrutiny of ABEL’S proof by attacking ABEL’S character- 

ization of functions of five quantities having five values under permutations: 

“And because the other chief obscurity in Abel’s argument (in the opinion of 

the present writer) is connected with the proof of the theorem, that a rational function 

of five independent variables cannot have five values and five only, unless it 

be symmetric relatively to four of its five elements; it has been thought advantageous, 

in this paper, as preliminary to the discussion of the forms of functions of 

five arbitrary quantities, to establish certain auxiliary theorems respecting functions 

of fewer variables; which have served also to determine à priori all possible 

solutions (by radicals and rational functions) of all general algebraic equations 

below the fifth degree.” 85 

Thus, HAMILTON pointed his finger directly at the two weak points of ABEL’S ar- 

gument. For ABEL’S flawed proof of the central auxiliary theorem — that all occurring 

radicals were rational functions of the roots — which he had proved by the hierar- 

chic structure of algebraic expressions, HAMILTON substituted an argument descend- 

ing and re-ascending the hierarchy of algebraic expressions. 86 The characterization 

of functions of five variables having five values under permutations was also car- 

ried out at length in an analysis which — following ABEL — reduced it to the study of 

such functions when only four of the arguments were permuted. As ABEL had done, 

HAMILTON completed his analysis of these functions through an extensive investigation 

of particular classes. 87 

HAMILTON employed a detailed style of presentation and extensive use of low 

degree equations as examples; nevertheless, his exposition of ABEL’S result is not par- 

ticularly clear and easy to grasp. 88 The degree of detail and a complicated notation 

might also have obscured the main results to some of HAMILTON’S contemporaries. 

83 (ibid., 248); small-caps changed into italic.. 

84 (J. Pierpont, 1896, 200). 

85 (W. R. Hamilton, 1839, 248–249); small-caps changed into italic.. 

86 (ibid., 194–196). 

87 (ibid., 237–246). 

88 (Dickson, 1959, 179) calls it “a very complicated reconstruction of ABEL’S proof”.


Neither HAMILTON’S exposition of ABEL’S proof nor his more direct criticisms of JER- 

RARD’S works seemed to convince JERRARD of his mistake. 89 JERRARD continued to 

announce his claim in the Philosophical Magazine and in 1858 he published his Essay 

on the Resolution of Equations. By that time it was left to A. CAYLEY (1821–1895) and J. 

COCKLE to refute JERRARD’S claims. 90 

BERNT MICHAEL HOLMBOE. The French mathematical community mainly knew 

of ABEL’S work on the solubility of equations through BERNT MICHAEL HOLMBOE’S 

edition of ABEL’S collected works (see above). 91 HOLMBOE’S extensive annotations 

and elaborations were often supplying explicit calculations in places where ABEL had 

been brief. In terms of criticism and modification of ABEL’S proof, HOLMBOE’S anno- 

tations center on three topics: irreducibility, functions of five quantities, and an explicit 

description of the process of inversion which ABEL had employed (see page 119). 

HOLMBOE opened with a short treatment of reducible and irreducible equations, 

in which he gave examples. He explicitly termed an equation irreducible when no 

root of the equation could be the root of an equation of “the same form”, but of lower 

degree. 92 This definition was implicit in ABEL’S paper; 93 it later took on a more explicit 

and very central role in ABEL’S theory of solubility (see chapters 7 and 8). 

Concerning ABEL’S investigations of functions of five quantities with five values, 

HOLMBOE’S annotations are of another character giving alternative proofs of unclear 

points. Remaining faithful to ABEL’S approach in the case in which µ = 2 (see page 

115), HOLMBOE supplied expressions with 30 and 10 different values to rule out the 

cases m = 4 and m = 5 which ABEL had left out. Thus, HOLMBOE sought to complete 

ABEL’S deduction of a contradiction. But sensing the obscure nature of ABEL’S classi- 

fication of functions of five quantities with five values, HOLMBOE set out to derive his 

own. 94 HOLMBOE applied a general theorem, which he had proved in the Magazin for 

Naturvidenskaberne: 

“In the same way one can demonstrate that if u designates a given function 

of n quantities which takes on m different values when one interchanges these n 

quantities among themselves in all possible ways, the general form of the function 

of n quantities which by these mutual permutations can obtain m different values 

will be 

r0 + r1u + r2u 2 + · · · + rm−1u m−1 , 

r0, r1, r2 . . . rm−1 being symmetric functions of the n quantities.” 95 

89 Actually, HAMILTON thought highly of JERRARD’S results, which he interpreted in a restricted 

frame. Although JERRARD’S claim for solving general equations could not be supported, the method 

which he had employed was nevertheless of great importance since it — if applied to the quintic — 

could reduce it to the normal trinomial form x 5 + px + q = 0. 

90 For instance (Cayley, 1861; Cockle, 1862; Cockle, 1863). 

91 (N. H. Abel, 1839). 

92 (Holmboe in ibid., 409). 

93 (N. H. Abel, 1826a, 71, 82). See quotation on page 106. 

94 (Holmboe in N. H. Abel, 1839, 411–413).


HOLMBOE’S proof implicitly involved LAGRANGE’S notion of semblables functions 

(functions which are altered in the same way by the same permutations), and argued 

directly that any function of five quantities, which took on five different values, must 

have the form of a fourth degree polynomial in which the coefficients were symmetric 

functions of x1, . . . , x5. 

The final noteworthy contribution by HOLMBOE to ABEL’S impossibility proof was 

his calculations relating to the process described as inversion of polynomials. HOLM- 

BOE proved — through manipulations on power sums — that any fourth degree poly- 

nomial v in x 

could be inverted into 

v = 

x = 

4 

∑ rαx 

α=0 

α 

4 

∑ sαv 

α=0 

α . 96 

The proof is a tour de force dealing with symmetric functions, much in the style of E. 

WARING (∼1736–1798), although in a clearer notational setting. 

In his commentary, HOLMBOE did not penetrate to the core of the problems spotted 

by HAMILTON. Instead, he elaborated many of ABEL’S arguments and manipulations 

and supplied proofs of obscure passages. HOLMBOE’S only real reservation against 

ABEL’S proof concerned the classification of functions with five values, and HOLM- 

BOE provided an alternative deduction using methods and concepts introduced by 

LAGRANGE and quite familiar to ABEL. 

KÖNIGSBERGER. While HOLMBOE’S elaboration of ABEL’S classification of func- 

tions with five quantities might have settled HAMILTON’S unease on this objection, 

it took longer before HAMILTON’S other reservation was lifted. The objection raised 

against ABEL’S classification of algebraic expressions was lifted in two steps: In 1869, 

KÖNIGSBERGER demonstrated how ABEL’S classification could be rescued by modi- 

fying the claims concerning the orders and degrees of the coefficients in the represen- 

tation 

v = q0 + p 1 n + q2p 2 n + · · · + qn−1p n−1 

n . 

(see page 104). 97 The slight modification which KÖNIGSBERGER introduced revali- 

dated ABEL’S hierarchy on algebraic expressions, and showed that ABEL’S “mistake” 

was of no real consequence to the proof. KÖNIGSBERGER had been stimulated to make 

95 “De la même manière on peut démontrer que, si u signifie une fonction donnée de n quantités qui 

prend m valeurs différentes lorsqu’on échange ces n quantités entre elles de toutes les manières 

possibles, la forme générale de la fonction de n quantités qui par leurs permutations mutuelles peut 

obtenir m valuers différentes sera 

r0 + r1u + r2u 2 + · · · + rm−1u m−1 , 

r0, r1, r2 . . . rm−1 étant des fonctions symétriques des n quantités.” (Holmboe in ibid., 413). 

97 (Königsberger, 1869).


his remedy public by the fact that such a classification was of importance by itself and 

the circumstance that ABEL’S flawed classification had been reproduced in J. A. SER- 

RET’S (1819–1885) textbook Cours d’algèbre. 98 

The two central arguments in ABEL’S proof, to which HAMILTON raised his objec- 

tions, had also stimulated other mathematicians to give alternative proofs and mod- 

ifications of ABEL’S deduction. The defective classification of algebraic expressions, 

which for ABEL served to demonstrate that any radical in a solution was a rational 

function of the roots, had made HAMILTON doubt the subsequent reasoning and he 

supplied another deduction. With KÖNIGSBERGER, the original deduction was res- 

cued by a slight modification, and the flaw was claimed — without detailed proof — 

to be of no importance in the proof. Subsequently, the classification of algebraic ex- 

pressions was disbanded altogether in the impossibility proof. The other obscurity — 

ABEL’S classification of rational functions of five quantities which take on five dif- 

ferent values under permutations — had been noticed in private correspondence by 

KÜLP, and ABEL had presented him with another more transparent deduction which, 

unfortunately, remained unknown to the larger mathematical public. When HAMIL- 

TON noticed the weakness of the published argument, he recast the deduction within 

his own framework; HOLMBOE provided it with a more general proof along the lines 

of other parts of ABEL’S reasoning. 

For various reasons — doubt and curiosity, debate over the validity of result, and 

concerns for the best presentation of ABEL’S work — these mathematicians took up 

weak parts of ABEL’S proof and provided clearer arguments and proofs. This local 

criticism served to establish the overall validity of the impossibility of algebraically 

solving the quintic by examining and improving the proof. 

6.9.2 Dissemination of the knowledge that the quintic was 

unsolvable 

The controversy which raged on the British Isles concerning the insolubility of the gen- 

eral quintic equation seems to have been largely confined to there, 99 although HAMIL- 

TON was also called upon to refute the claim for solubility made by the Italian P. G. 

BADANO. 100 While HAMILTON’S penetrating local criticism of ABEL’S proof was un- 

dertaken to resolve an ongoing debate over ABEL’S statement, the Continental incor- 

poration of ABEL’S result apparently followed another path. On the Franco-German 

scene, I am not aware of any global rejections of ABEL’S result. The style of later local re- 

workings of ABEL’S proof left little clue as to what, besides refinement and aesthetics, 

had spurred the mathematician to reformulate the argument. 

98 (Königsberger, 1869, 168). 

99 Besides JERRARD, MACCULLAGH also transmitted a claim to have solved the general fifth degree 

equation and was refuted by HAMILTON (Hankins, 1980, 438, note 22). 

100 (W. R. Hamilton, 1843; W. R. Hamilton, 1844). See also (Hankins, 1980, 438, note 22). Personal data 

for Mr. BADANO have proved to be inaccessible.


PIERRE LAURENT WANTZEL. In a short paper published in 1845, PIERRE LAURENT 

WANTZEL refined ABEL’S proof by reversing the succession in which the radicals of 

a supposed solution is studied like HAMILTON had done. 101 Although WANTZEL 

deemed ABEL’S proof to be exact, he also found its presentation vague and compli- 

cated. Nevertheless, WANTZEL gave no detailed reasons for his evaluation. 

“Although his [ABEL’S ] proof is basically exact it is presented in a way so 

complicated and so vague that it would not be generally permissible.” 102 

Through a fusion of ABEL’S proof with the even vaguer and more insufficient proof 

by RUFFINI, WANTZEL arrived at a clear and precise proof which he thought would 

“lift all doubts concerning this important part of the theory of equations”. 103 Unfortu- 

nately, he did not specify his “doubts”. 

In his fusion proof, WANTZEL took over the most important of ABEL’S preliminary 

arguments: the classification of algebraic expressions by orders and degrees and the 

auxiliary theorem derived from it (see page 104). By studying any supposed solution 

of the general n th degree equation and permutations of the roots, WANTZEL deduced 

that the outermost root extraction would have to be a square root. 104 Continuing to 

the radical of second highest order, he found that it had to remain unaltered by any 3- 

cycle, and therefore by any 5-cycle. 105 At this point he reached a contradiction because 

the supposed solution would thus only have two different values under all permuta- 

tions of the five roots. 

WANTZEL’S proof was published in the Nouvelles annales de mathematique and soon 

became the widely accepted simplification of ABEL’S proof. It made no use of ABEL’S 

classification of functions of five quantities, and may thus be seen as an indirect lo- 

cal criticism of this classification. On the other hand, it builds directly upon ABEL’S 

classification of algebraic expressions. 

A. E. G. ANDERSSEN. In 1848, the Königlichen Friedrichs-Gymnasium in Breslau in- 

vited its “protectors, sons, and friends” to be present at the annual exams. Included 

with the invitation was a short essay written by one of the teachers; at the time, this 

was not uncommon practice for German Gymnasien. 106 In the essay, A. E. G. ANDER- 

SSEN 107 sought to illuminate the central arguments of ABEL’S impossibility proof. 

Being largely a reproduction of ABEL’S argument with some elaboration of its briefest 

arguments, the interesting parts of ANDERSSEN’S essay are his evaluations of ABEL’S 

101 (Wantzel, 1845). 

102 “Quoique sa démonstration soit exacte au fond, elle est présentée sous une forme trop compliquée 

et tellement vague, qu’elle n’a pas été généralement admise.” (ibid., 57). 

103 (ibid., 57). 

104 (ibid., 62). 

105 (ibid., 63–64). See also CAUCHY’S proof of the CAUCHY-RUFFINI theorem, section 5.6. 

106 (Anderssen, 1848). 

107 No further personal information concerning this Mr. ANDERSSEN has been accessible.


proof. ANDERSSEN found the proof to be simple, coherent, and not built upon calcu- 

lations but on arguments and deductions; at the same time, and possibly for the same 

reasons, he rated it as being difficult. 

“However simple this proof is, first of all because a single idea serves throughout 

as a decisive criterion, secondly because the truths by which the application 

of the main idea is possible, are communicated not by artificial calculations but 

by conclusions and deductions, it nevertheless (and even therefore) demands the 

most thorough contemplation in order to be understood in its entire clarity. Therefore 

it would not be a superfluous work to present the most important arguments 

of this instructive yet difficult proof by examples and further elaborations in their 

true spirit and full power of proof.” 108 

ABEL’S classification of algebraic expressions according to orders and degrees was 

reproduced in an overly simplified form, in which the concept of degree has com- 

pletely vanished. When it came to the classification of functions with five quantities, 

which HAMILTON had scrutinized, ANDERSSEN found it quite satisfactory: 

“Both these two theorems [no function of five quantities can have two or 

five different values under all possible interchanges of the quantities] have been 

proved in Abel’s treatise with a clarity which cannot be improved.” 109 

ANDERSSEN’S essay contained no criticism of parts of ABEL’S proof nor any origi- 

nal modifications but only simple elaborations and some examples. However, its mere 

existence is evidence that ABEL’S result was becoming known to the broader circle of 

German mathematicians. 

LEOPOLD KRONECKER. The introduction of ABEL’S work on the quintic equation 

into German academic circles is due to LEOPOLD KRONECKER. Much of KRONECKER’S 

work on algebra was inspired by ideas which he got reading ABEL and KRONECKER 

completed and rigorized many parts of ABEL’S research. In KRONECKER’S elegant 

proof of the insolubility of the general fifth (and higher) degree equation, ABEL’S 

proof found its final form. KRONECKER presented his simplified version of ABEL’S 

proof in a paper read to the Akademie der Wissenschaften. 110 There, he presented no 

criticism of ABEL’S proof but simply put forward alternative deductions preferable to 

ABEL’S on account of their simplicity and general nature. The validity of the result 

108 “So einfach dieser Beweis ist, erstens weil ein einziger Gedanke durchgehends zum entscheidenden 

Kriterium dient, zweitens weil diejenigen Wahrheiten, kraft deren die Anwendung des Hauptgedankens 

möglich ist, nicht durch künstliche Rechnungen, sondern durch Urtheile und Schlüsse 

vermittelt werden; so erheischt er dennoch, ja eben deswegen das gesammeltste Nachdenken, um in 

seiner ganzen Klarheit begriffen zu werden. Es dürfte daher keine unnöthige Arbeit sein, die wichtigsten 

Argumente dieses eben so lehrreichen als schwierigen Beweises durch Beispiele und weitere 

Ausführung in ihrem wahren Sinne und in ihrer vollständigen Beweiskraft zur Anschauung zu 

bringen.” (Anderssen, 1848, 3). 

109 “Diese beiden Lehrsätze sind in Abel’s Abhandlung mit einer Klarheit bewiesen, welche durch 

Nichts erhöht werden kann.” (ibid., 14). 

110 (Kronecker, 1879).


was never questioned by KRONECKER; his improvements were local in the sense of 

replacing some of ABEL’S arguments by more apt ones. 

Through a detailed reworking of ABEL’S classification, KRONECKER obtained a 

more precise formulation of ABEL’S auxiliary theorem on the rationality of all radicals 

occurring in any solution. KRONECKER let R, R ′ , R ′′ , etc. denote quantities, which 

were to be considered known, and spoke of the collection of these as “the quantities 

R”. Later, this R evolved into his general concept of domains of rationality. 

“In the described way the explicit algebraic function satisfying an equation 

Φ (x) = 0 can be expressed as an entire function of the quantities 

W1, W2, . . . Wµ 

the coefficients of which are rational functions of the quantities R; the quantities 

W are on the one hand entire integer functions of the roots of the equation 

Φ (x) = 0 and of roots of unity and on the other hand determined through a chain 

of equations 

W n β 

β 

= Gβ 

� � 

Wβ+1, Wβ+2, . . . Wµ 

(β = 1, 2, . . . µ) 

n1, n2, . . . being prime numbers and G1, G2, . . . Gµ designating entire functions of 

the bracketed quantities W in which the coefficients are rational functions of the 

quantities R.” 111 

Although apparently formulated in a slightly different way, this theorem is very 

close to the one of ABEL’S auxiliary theorems ensuring the rationality of the involved 

radicals (theorem 3), and served KRONECKER as its equivalent. The improvements are 

mainly the introduction of the rationally known quantities R and the explicit mention 

of the roots of unity. To ABEL, the roots of unity had been “known” in the common 

language version of this word, because he knew enough of them to handle them as 

simple objects. Consequently, roots of unity were not explicitly mentioned. The pro- 

cess of attributing technical mathematical meaning to a common language term oc- 

curred frequently in the period as is evident, for example, in the way GALOIS’S notion 

of groups was transformed from meaning a “collection of objects” into a term with a 

highly technical meaning. 

111 “In der dargelegten Weise erhält die einer Gliechung Φ (x) = 0 genügende explicite algebraische 

Function als ganze Function von Grössen 

W1, W2, . . . Wµ 

dargestellt, deren Coëfficienten rationale Functionen der Grössen R sind, und die Grössen W sind 

einerseits ganze ganzzahlige Functionen von Wurzeln der Gleichung Φ (x) = 0 und von Wurzeln 

der Einheit andererseits durch eine Kette von Gleichungen 

W nβ β = G � � 

β Wβ+1, Wβ+2, . . . Wµ 

(β = 1, 2, . . . µ) 

bestimmt, in denen n1, n2, . . . Primzahlen und G1, G2, . . . Gµ ganze Functionen der eingeklammerten 

Grössen W bedeuten, deren Coëfficienten rationale Functionen der Grössen R sind.” (ibid., 77).


The succeeding part of KRONECKER’S proof concerned the substitution theoretic 

aspects of ABEL’S proof and consisted of an extended version of the CAUCHY-RUFFINI 

theorem. For n > 4, KRONECKER let f designate a function of quantities x1, . . . , xn and 

studied the conjugate functions f1, . . . , fm; these functions were the analogous of what 

ABEL had called the different values of f under all permutations of x1, . . . , xn. KRO- 

NECKER derived the result that for any non-symmetric function f , some permutation 

would exist which altered the value of one of the conjugate functions. He could even 

demonstrate that if only the n! 

� � 

2 permutations, which left the product ∏i

6.10. Summary 139 

of ABEL’S argument but superior in rigour. The classification of algebraic expressions 

was also a concern of some later 19 th century mathematicians, until it was settled by 

KÖNIGSBERGER and KRONECKER. 

The fact that global criticism of ABEL’S impossibility proof was limited can be taken 

as a sign that the mathematical community soon came to realize the overall validity of 

the result. The change of attitude toward the problem, which had been facilitated by 

the statements of experts such as LAGRANGE and GAUSS (section 5.4) and the proofs 

of RUFFINI, which at least were known in some circles in Paris, had been a prerequisite 

for the quick acceptance. However, local criticism was still conducted in an effort to 

make ABEL’S proof clearer and more powerful. Central lemmata, on which doubt 

could be cast, were reexamined and new proofs were given. 

6.10 Summary 

As described, ABEL’S proof of the insolubility of the general quintic was a curious 

combination of general theorems and investigations of particular cases. Partly be- 

cause of the counter intuitive nature of the result and partly because of legitimate 

local objections to ABEL’S argument, the result was subsequently scrutinized. Inter- 

preted in terms of delineation of concepts, the algebraic insolubility of the general 

quintic distinguished the concepts of polynomial equations and algebraically solvable 

equations.

Chapter 7 

Particular classes of equations: 

enlarging the class of solvable 

equations 

If N. H. ABEL’S (1802–1829) proof of the impossibility of solving the general quin- 

tic algebraically was hampered by its brevity and obscure arguments, his only other 

published work on the theory of equations was more mature, beautifully lucid, and 

rigorous. In the Mémoire sur une classe particulière d’équations résolubles algébriquement, 

written in 1828 and published the following year, ABEL abandoned one of the central 

pillars of the impossibility proof — the theory of permutations — and provided a direct 

and affirmative proof of the algebraic solubility of a particular class of equations. 1 

Focusing instead on the other pillar — the concepts of divisibility, irreducibility, and 

the Euclidean algorithm — this work illuminates central ideas in ABEL’S reasoning 

which permeate his entire work on the theory of equations. 

The 1829-paper has become a classic of mathematics for its proof that the class of 

equations, now called Abelian and defined by certain properties of the roots, are always 

algebraically solvable. When contrasted with the contents of the impossibility proof, 

this result highlights a feature of the new ways of asking questions — the mechanisms 

of limiting and enlarging class of objects which in the nineteenth century provided the 

background for a new, concept based approach to mathematics. However, the paper 

contains more information than just this main result; in this chapter I describe some of 

the connections between this work and other parts of ABEL’S research as well as some 

of the very central concepts which ABEL put to use in it. 


141

142 Chapter 7. Particular classes of solvable equations 

7.1 Solubility of Abelian equations 

The structure of ABEL’S Mémoire sur une classe particulière 2 is a descent from the gen- 

eral to the particular. At the outset, ABEL proposed to study irreducible equations in 

which one of the roots depended rationally on another one. The concept of irreducible 

equations took a central place in this research (see section 7.3). Part of the study was 

especially devoted to circular functions to which ABEL had been led by C. F. GAUSS’ 

(1777–1855) work on the cyclotomic equation. Besides this application to circular func- 

tions, ABEL also worked on another application of the general theory to the division 

problem for elliptic functions. Likewise inspired by GAUSS’ Disquisitiones arithmeti- 

cae (see section 7.2), this application was, however, not contained in the paper but 

had been presented the previous year in a paper on elliptic functions. ABEL was led 

by these two applications to an even more general result — valid for a broader class 

of equations having rationally related roots. In this section, I outline ABEL’S results 

before turning to discussions of his inspirations and methods. 

7.1.1 Decomposition of the equation into lower degrees 

Throughout the paper, ABEL studied polynomial equations of degree µ, 

φ (x) = 0, 

in which two roots x1, x ′ were related by the rational function θ, 

x ′ = θ (x1) . 

The quantities which ABEL considered known in his deductions comprise all coeffi- 

cients occurring in φ or θ. From a modern perspective, it will become clear that he also 

considered any required roots of unity to be known. 

ABEL defined the equation φ (x) = 0 to be irreducible when none of its roots could 

be expressed by a similar equation of lower degree (see section 7.3). 

Employing the Euclidean division algorithm (see section 7.3) and the notation 

θ k (x1) for the k th iterated application of the rational function θ to x1, ABEL found 

that the set of roots of φ (x) = 0 split into sequences (chains). He deduced — using the 

irreducibility of φ (x) = 0 — that because the two roots x1, x ′ of the equation φ (x) = 0 

were rationally related, every iteration θ k (x1) would also be a root of φ (x) = 0. There- 

fore, the entire set of roots of φ (x) = 0 could be collected in sequences of equal length, 

say n, and he wrote the roots as (µ = m × n), 3 

θ k (xu) for 0 ≤ k ≤ n − 1 and 1 ≤ u ≤ m. (7.1) 


3 For brevity, I have added to ABEL’S notation the convention θ 0 (x1) = x1.

7.1. Solubility of Abelian equations 143 

After ABEL had divided the roots into sequences, he proceeded to reduce the so- 

lution of the equation of degree µ to equations of lower degrees. To the first sequence 

x1, θ (x1) , . . . , θ n−1 (x1), ABEL assigned an arbitrary rational and symmetric function 

y1 of these quantities. Since θ was also a rational function, y1 was actually a rational 

function of x1, 

y1 = f 

and using the symmetry of y1, ABEL found 

and more generally 

� 

x1, θ (x1) , . . . , θ n−1 � 

(x1) = F (x1) , 

y ν 1 

y1 = 1 

n−1 

n 

∑ 

k=0 

� 

F θ k � 

(x1) 

n−1 

1 

� � 

= 

n ∑ F θ 

k=0 

k ��ν (x1) 

(7.2) 

for any non-negative integer ν. In the same way as ABEL formed the function y1 

from x1, he formed an additional m − 1 functions y2, . . . , ym corresponding to the other 

chains, 

yu = f 

� 

xu, θ (xu) , . . . , θ n−1 � 

(xu) = F (xu) for 1 ≤ u ≤ m. (7.3) 

Each of these produced the equivalent of (7.2) 

ABEL added these (over u) as 

y ν u = 1 

n−1 � � 

n ∑ F θ 

k=0 

k ��ν (xu) 

rν = 

for 1 ≤ u ≤ m and ν ≥ 0. 

m 

∑ y 

u=1 

ν u for ν ≥ 0 (7.4) 

and obtained rational and symmetric functions of all the roots of φ (x) = 0. These 

could, he noticed, therefore be expressed rationally in the coefficients of the known 

functions φ and θ. Once these power sums (7.4) were known, ABEL could determine 

any rational and symmetric function of y1, . . . , ym by the solution of an equation of 

degree m by E. WARING’S (∼1736–1798) result (see section 5.2.4). In particular, ABEL 

found that each of the coefficients of the equation 

m 

∏ (y − yu) = 0 (7.5) 

u=1 

could be determined by solving an equation of the m th degree. 

A central topic of ABEL’S paper is the detailed study of this decomposition of the 

equation of degree µ = m × n into equations of degrees m and n. His next step


was to focus attention on the equation connected with the first sequence of roots 

x1, θ (x1) , . . . , θ n−1 (x1), i.e. 

n−1 � 

∏ x − θ 

k=0 

k � 

(x1) = 0. (7.6) 

ABEL proved that any coefficient ψ (x1) of this equation would depend rationally on 

y1 and known quantities of φ and θ by the following nice and typical argument. 

Denoting by ψ (x1) any coefficient of (7.6), ABEL formed the expressions 

tν = 

m 

∑ y 

u=1 

ν u · ψ (xu) for ν ≥ 0, 

which he proved to be rational and symmetric functions of all the roots of φ (x) = 0. 

Thereby, tν could be expressed rationally in the known quantities. 

From a set of linear equations equivalent to the matrix equation 

⎡ 

⎢ 

⎣ 

11 . . .1 

y1y2 

.. 

y m−1 

1 y m−1 

2 

. . .ym 

. ... 

. . .y m−1 

m 

⎤ ⎡ ⎤ ⎡ 

ψ (x1) 

⎥ ⎢ ⎥ ⎢ 

⎥ ⎢ 

⎥ ⎢ 

ψ (x2) ⎥ ⎢ 

⎥ ⎢ 

⎥ ⎢ ⎥ = ⎢ 

⎥ ⎢ 

⎦ ⎣ . 

⎥ ⎢ 

⎦ ⎣ 

ψ (xm) 

t0 

⎥ , 

. 

⎥ 

⎦ 

t1 

tm−1 

ABEL deduced that ψ (x1) could be expressed as a rational function of y1, . . . , ym. His 

argument is based on the possibility of attributing a non-vanishing form to the equiva- 

lent of the determinant of the matrix. This was possible because y1 had — up to now — 

been an arbitrary symmetric function, and ABEL gave it the non-vanishing form 

y1 = 

n−1 � 

∏ α − θ 

k=0 

k � 

(x1) , 

where α was unspecified. Furthermore, ABEL continued to show how each of the 

quantities y2, . . . , ym could be replaced by a rational function of y1, and how ψ (x1) 

could be expressed as a rational function of y1 alone. Thus, each coefficient ψ (x1) in 

the equation (7.6) could be determined rationally in y1; and y1 could be determined 

by solving an equation of degree m. ABEL summarized these results in an important 

theorem: 

Theorem 5 “The equation under consideration φx = 0 can thus be decomposed into a num- 

ber m of equations of degree n in which the coefficients are rational functions of a fixed root of 

a single equation of degree m, respectively.” 4 

4 “L’équation proposée φx = 0 peut donc être décomposée en un nombre de m d’équations du degré 

n; donc[!] les coëfficiens sont respectivement des fonctions rationnelles d’une même racine d’une 

seule équation du degré m.” (N. H. Abel, 1829c, 139). The misprint “donc” has been replaced by 

“dont” in both editions of ABEL’S Œuvres. 

⎤


Thus, the original problem of solving the equation φ (x) = 0 of degree µ had 

been reduced to solving certain equations, (7.5) and (7.6), of lower degrees. Gener- 

ally, the equation of degree m would not be solvable by radicals, but as ABEL went on 

to demonstrate, the m equations of degree n could always be solved algebraically. 

7.1.2 Algebraic solubility of Abelian equations 

If all the roots of the equation φ (x) = 0 fell into the same orbit of θ (one chain), i.e. are 

of the form 

x1, θ (x1) , θ 2 (x1) , . . . , θ n−1 (x1) , 

the situation was equivalent to assuming m = 1 above. In this case, ABEL let α denote 

a primitive µ th root of unity and formed the rational expression 

ψ (x) = 

� 

µ−1 

∑ α 

k=0 

k θ k �µ 

(x) . (7.7) 

Through direct calculations, he proved that 

� 

ψ θ k � 

(x) = ψ (x) for all k = 0, 1, . . . , µ − 1, 

which showed that ψ was a symmetric function of the roots of φ (x) = 0. Thus, ψ (x) 

could be expressed rationally in known quantities. Next, ABEL introduced µ radicals 

of (7.7), 

µ√ vu = 

µ−1 

∑ α 

k=0 

k uθ k (x) for 0 ≤ u ≤ µ − 1, (7.8) 

by attributing to αu the different µ th roots of unity 1, α, α 2 , . . . , α µ−1 . From these radi- 

cals, it was a routine procedure for ABEL to obtain the expression 

where A was a constant. 

θ k (x) = 1 

� 

µ−1 

−A + 

µ 

∑ 

u=1 

α uk µ√ vu 

� 

, k = 0, 1, . . . , µ − 1, (7.9) 

The expression (7.9), however, contained µ − 1 extractions of roots with exponent 

µ which seemed to indicate that a total of µ µ−1 different values could be obtained al- 

though the degree of φ (x) = 0 was only µ. ABEL resolved this apparent contradiction, 

similar to one noticed by L. EULER (1707–1783) (see section 5.1), by an elegant argu- 

ment prototypic of his approach to the theory of equations. In the deduction, ABEL 

proved that all the root extractions depended on one of them by considering 

µ√ vk ( µ√ v1) µ−k = 

� 

µ−1 

∑ α 

u=0 

ku θ u � 

(x) × 

� 

µ−1 

∑ α 

u=0 

u θ u �µ−k 

(x) .


Obviously, the form of the right hand side shows that this expression was a rational 

function of x. ABEL stated that it was unaffected by substituting θ m (x) for x and 

considered it so obvious that he did not provide the details. 5 Thus, the expression 

was a rational function of the coefficients of φ (x) = 0; ABEL denoted this function by 

a k, 

µ√ 

vk = ak ( µ√ v1) k . 

v1 

ABEL stated the conclusion of this investigation by giving an algebraic formula for 

the root x, 6 

x = 1 

� 

−A + 

µ 

µ√ v1 + a2 

( 

v1 

µ√ v1) 2 + a3 

( 

v1 

µ√ v1) 3 + · · · + aµ−1 

( 

v1 

µ√ v1) µ−1 

� 

. (7.10) 

All the other roots were contained in this formula by giving µ√ v1 its µ different values 

α k µ √ v1. ABEL expressed the implications for solubility in two theorems capturing the 

essence of this research. If the set of roots fell into one “orbit” of the rational expres- 

sion, θ, ABEL found the equation to be solvable by radicals: 

Theorem 6 “If the roots of an algebraic equation can be represented by: 

x, θx, θ 2 x, . . . θ µ−1 x, 

where θ µ x = x and θx denotes a rational function of x and known quantities, this equation 

will always be algebraically solvable.” 7 

Applying this result to the particular case of irreducible equations of prime degree, 

which always had only one chain, ABEL found that such equations were algebraically 

solvable: 

“If two roots of an irreducible equation, of which the degree is a prime number, 

have such a relation that one can express the one rationally in the other, this 

equation will be algebraically solvable.” 8 

Subsequently, ABEL refined the hypothesis that all the roots could be expressed as 

iterations of a rational function. That hypothesis had ensured algebraic solubility of 

the equation, but the same conclusion could also be established for a broader class of 

equations. Under the general assumption that every root of an equation, χ (x) = 0, 

5 The details can easily be provided by inserting into the right hand side and rearranging terms. 

6 (N. H. Abel, 1829c, 142). 

7 “Si les racines d’une équation algébrique peuvent être représentées par: 

x, θx, θ 2 x, . . . θ µ−1 x, 

où θ µ x = x et θx désigne une fonction rationelle de x et de quantités connues, cette équation sera 

toujours résoluble algébriquement.” (ibid., 142–143). 

8 “Si deux racines d’une équation irréductible, dont le degré est un nombre premier, sont dans un 

tel rapport, qu’on puisse exprimer l’une rationnellement par l’autre, cette équation sera résoluble 

algébriquement.” (ibid., 143).


could be expressed rationally in a single root x, ABEL went on to assume “commuta- 

tivity” of these rational dependencies, i.e. if θ (x) and θ1 (x) were any two roots of the 

equation χ (x) = 0, written as rational expressions in x, the assumption was that 

θ (θ1 (x)) = θ1 (θ (x)) . 

ABEL’S method of proving the algebraic solubility of χ (x) = 0 under this hypothesis 

was to reduce the situation to the one solved above. Since all roots were known ra- 

tionally once x was considered known, it sufficed to search for the root x. In order to 

study an irreducible equation, ABEL focused on the irreducible factor φ of χ having x 

as a root, repeating his concept of irreducibility (see section 7.3). 

“If the equation 

is not irreducible, let 

χx = 0 

φx = 0 

be the equation of lowest degree which the root x satisfies such that the coefficients 

of this equation contain nothing but known quantities.” 9 

Thus, ABEL assumed that φ (x) = 0 was the irreducible factor which had x as 

a root. By the deductions carried out above, the roots were thus expressed as (7.1), 

where for simplicity I write x0 for x: 

The coefficients of the equation 

θ k (xu) for 0 ≤ k ≤ n − 1 and 0 ≤ u ≤ m − 1. 

n−1 � 

∏ z − θ 

k=0 

k � 

(x0) = 0 (7.11) 

could all be expressed rationally in a single quantity y0 (above denoted y1) which was 

a root of an equation (7.5) of degree m. In a footnote, ABEL demonstrated that the 

latter equation was irreducible. Thereby, he had reduced the determination of x to the 

solution of two equations of degrees n and m. Of these, he knew that the former was 

9 “Si l’équation 

n’est pas irréductible, soit 

χx = 0 

φx = 0 

l’équation la moins élevée, à laquelle puisse satisfaire la racine x, les coëfficiens de cette équation ne 

contenant que des quantités connues.” (ibid., 149–150).


algebraically solvable if y0 was considered known. Although the equation of degree 

m 

m−1 

∏ (z − yu) = 0 (7.12) 

u=0 

giving the coefficients of (7.11) would generally not be algebraically solvable, ABEL 

next proved that equation (7.12) ‘inherited’ the property of commutative rational de- 

pendence among its roots, which φ (x) = 0 possessed. Thus, a ‘descent’ down a string 

of equations was made possible. 

ABEL’S proof of this ‘inheritance’, the commutative rational dependence among 

the roots of (7.12), ran as follows. The hypothesis was that all the roots were given 

rationally in a single root, i.e. 

xu = θu (x0) for 0 ≤ u ≤ m − 1. (7.13) 

The expression for yu which in the previous argument was given by (7.3), 

� 

yu = f xu, θ (xu) , . . . , θ n−1 � 

(xu) for 0 ≤ u ≤ m − 1, 

under the current hypothesis became 

� 

y1 = f θ1 (x0) , θ (θ1 (x0)) , . . . , θ n−1 � 

(θ1 (x0)) . 

Combining this with the hypothesis of commutativity of the functions θ and θ1, ABEL 

found 

y1 = f 

� 

� 

θ1 (x0) , θ1 (θ (x0)) , . . . , θ1 θ n−1 �� 

(x0) . 

Therefore, y1 was a rational and symmetric function of the sequence of roots (7.13) and 

could therefore be expressed rationally in y0 and known quantities. Obviously, ABEL 

could carry out the same argument for any other y2, . . . , ym−1. When he let λ (y0) and 

λ1 (y0) denote any two among the quantities y0, . . . , ym−1, he found that, without loss 

of generality, 

y1 = λ (y0) = f 

y2 = λ1 (y0) = f 

� 

θ1 (x0) , θ (θ1 (x0)) , . . . , θ n−1 � 

(θ1 (x0)) and 

� 

θ2 (x0) , θ (θ2 (x0)) , . . . , , θ n−1 � 

(θ2 (x0)) . 

Inserting θ2 (x) for x0 in λ (y0), which transformed y0 into y2, ABEL obtained 10 

λλ1 (y0) = λ (y2) = f 

while inserting θ1 (x) for x0 in λ1 (y0) produced 

λ1λ (y0) = λ1 (y1) = f 

� 

θ1θ2 (x0) , θθ1θ2 (x0) , . . . , θ n−1 � 

θ1θ2 (x0) 

� 

θ2θ1 (x0) , θθ2θ1 (x0) , . . . , θ n−1 � 

θ2θ1 (x0) 

10 Here I deviate from my usual notation by writing the composition of functions in multiplicative 

mode, i.e. θ1θ2 (x0) instead of θ1 (θ2 (x0)). 

, 

.


Since θ1θ2 (x0) = θ2θ1 (x0), ABEL concluded 

λλ1 (y0) = λ1λ (y0) , 

and any two roots of the equation (7.12) would thus also commute. Therefore, the equa- 

tion (7.12) determining the coefficients of (7.11) inherited this property from φ (x) = 0 

and could be treated in the same way as above. Since the degree was reduced by this 

argument, a chain of equations of strictly decreasing degrees could be constructed. At 

some point, where the procedure would have to terminate, the degree had to be 1 and 

the final equation would amount to a rational dependency. 

ABEL had thus established the following important theorem on the solubility of 

this class of equations: 

Theorem 7 The equation φ (x) = 0 is algebraically solvable if the following two requirements 

are met: 

1. All roots of φ (x) = 0 are rational expressions θ1 (x) , . . . , θµ (x) of one root 

2. The rational expressions satisfy a requirement of commutativity θ iθ j (x) = θ jθ i (x). 11 ✷ 

Since the time of L. KRONECKER (1823–1891), equations with these properties have 

been named abelian [abelsche]; 12 in 1932, the Springer-Verlag decided to change the 

first letter into a capital: Abelian. 13 Later, the term Abelian was also adopted to denote 

groups corresponding to Abelian equations, i.e. commutative groups. 

In two theorems, ABEL summarized the implications for the degrees of the equa- 

tions involved in the algebraic solution of the equation φ (x) = 0 in which two roots 

were rationally related. The following theorem completely describes these degrees: 

“Supposing that the degree µ of the equation φx = 0 is decomposed as follows: 

µ = ε v1 

1 

· εv2 

2 

· · · · · εvα 

α , 

where ε1, ε2, . . . , εα are prime numbers, the determination of x can be effected with 

the help of the solution of v1 equations of degree ε1, v2 equations of degree ε2, etc., 

and all these equations will be algebraically solvable.” 14 

11 It is remarkable and unfortunate that (Toti Rigatelli, 1994, 717) got the logic of ABEL’S reasoning 

wrong, reproducing the result as “he [ABEL in (N. H. Abel, 1829c)] showed that, in those equations 

which were solvable by radicals, all roots could be expressed as rational functions of any other root, 

and that these functions were permutable with respect to the four arithmetical operations. That is, 

if F1 and F2 are any two corresponding functional operations, then F1F2x = F2F1x.” 

12 (Kronecker, 1853, 6). 

13 The decision prompted a brief discussion among mathematicians, see 

the letters (Noether→Hasse, 1932.10.29 and 1932.12.09, described at 

��) 

14 “Supposant le degré µ de l’équation φx = 0 décomposé comme il suit: 

µ = ε v 1 

1 

· εv2 

2 

· · · · · εvα 

α , 

où ε1, ε2, . . . εα sont des nombres premiers, la détermination de x pourra s’effectuer à l’aide de la 

résolution de v1 équations du degré ε1, de v2 équations du degré ε2, etc., et toutes ces équations 

seront résolubles algébriquement.” (N. H. Abel, 1829c, 152).


The resemblance to GAUSS’ investigation of the cyclotomic equation is more than 

accidental. In multiple ways, GAUSS’ work was the direct inspiration for this research. 

Part of the purpose of ABEL’S paper was to reproduce GAUSS’S result in this more 

general framework. 

7.1.3 Application to circular functions and the link with GAUSS’ 

Disquisitiones arithmeticae 

ABEL was led to the study of Abelian equations by his in-depth studies of the divi- 

sion problem for elliptic functions (see section 7.2), which in turn were motivated by 

the division problem for circular functions treated by GAUSS in his Disquisitiones arith- 

meticae. 15 In the last part of the paper, ABEL incorporated GAUSS’ division of the circle 

into his broader theory of Abelian equations by the following approach. 

The central result of the paper Mémoire sur une classe particulière was contained in 

the second theorem (here theorem 5) on the reduction of equations of degree m × n to 

m solvable equations of degree n and a single equation of degree m. Originating from 

this theorem, ABEL deduced more particular results in various directions. Assuming 

that all the known quantities (i.e. coefficients) of φ and θ were real numbers, he studied 

the constructions required for the solution of the equation φ (x) = 0. Considering real 

and imaginary parts of the radical µ√ v1 (7.8), ABEL found the (non-algebraic) solution 

formula 

x = 1 

µ 

� 

−A + 

µ−1 � √ � 

∑ fk + gk −1 ( 

k=1 

√ � 

k 

ρ) cos k(δ+2mπ) 

µ 

+ √ −1 sin k(δ+2mπ) 

� 

µ 

� 

where the quantities ρ, A, f1, . . . , fµ−1, g1, . . . , gµ−1 were rational functions of cos 2π µ , 

sin 2π µ 

and the coefficients of φ and θ. From this, he drew the following conclusion 

which was intimately linked to one of GAUSS’ results: 

Theorem 8 “In order to solve the equation φx = 0 it suffices: 

1) to divide the circumference of the circle into µ equal parts, 

2) to divide an angle δ, which can then be constructed, into µ equal parts, 

3) to extract a square root of a single quantity ρ.” 16 

ABEL, himself, remarked that his result was an extension of one of the key results 

found in GAUSS’ Disquisitiones, stating the equivalent conclusion for cyclotomic equa- 

tions: That the solution of the equation x n = 1 could be reduced to the following three 

steps: 17 

15 (C. F. Gauss, 1801). 

16 “[Q]ue pour résoudre l’équation φx = 0, il suffit: 

1) de diviser la circonférence entière du cercle en µ parties égales, 

2) de diviser un angle δ, qu’on peut construire ensuite, en µ parties égales, 

3) d’extraire la racine carrée d’une seule quantité ρ.” 

(N. H. Abel, 1829c, 144). 

17 (C. F. Gauss, 1801, 454) and (C. F. Gauss, 1986, 450).


1) The division of the whole circle into n − 1 parts (n − 1 because the irreducible 

= 0), 

equation in GAUSS’ research was xn −1 

x−1 

2) The division into n − 1 parts of another arc which could be constructed after step 

1 had been completed, and 

3) The extraction of a square root. 

The final step, the extraction of a square root, could be assumed to equal the construc- 

tion of √ n, GAUSS claimed without providing any proof. Later, ABEL adopted and 

proved the assertion. 

In the fifth section of the Mémoire sur une classe particulière, ABEL applied his theory 

directly to the cyclotomic equation and the circular functions related to the division of 

the circle. By the addition formulae for cosine, ABEL could express cos ma rationally 

in cos a, and assuming θ (cos a) = cos ma and θ1 (cos a) = cos m ′ a, he obtained 

θθ1 (x) = θ � cos m ′ a � = cos � mm ′ a � 

= cos � m ′ ma � = θ1 (cos ma) = θ1θ (x) . 

From a previously established result (here theorem 7), ABEL found that cos 2π µ could 

be determined algebraically — which was a well known result. 

ABEL, however, did not stop his investigations of the circular functions at this 

point, as he might have done had he only been interested in the algebraic solubility of 

the division. Assuming that µ = 2n + 1 was prime, ABEL studied the equation 

and used the rational dependency established above 

to write 

n � 

∏ X − cos 

k=1 

2kπ 

� 

= 0, (7.14) 

2n + 1 

θ (cos a) = cos ma 

θ k (cos a) = cos m k a. 

By an argument based on GAUSS’ primitive roots of the module 2n + 1, ABEL demon- 

strated that the roots of (7.14) were 

x, θ (x) , θ 2 (x) , . . . , θ n−1 (x) where θ n (x) = x. 

Therefore, the equation (7.14) was algebraically solvable by ABEL’S third theorem 

(here theorem 6), and ABEL adapted theorem 8 to this particular equation, obtain- 

ing the same result as GAUSS had found. Furthermore, ABEL presented a proof of 

the result, which GAUSS had only announced, that the square root extracted in step 3 

could always be made to equal √ 2n + 1 (in ABEL’S variables). 

The contents of ABEL’S Mémoire sur une classe particulière can be summarized in the 

following five points depicting a descent from the general to the particular:


1. A general study of equations in which one root depended rationally on another. 

2. A restriction to irreducible equations and an application of the concept of ir- 

reducibility to prove that if x and θ (x) were roots of the irreducible equation 

φ (x) = 0, then so was θ k (x) for all integers k. 

3. A study of equations of degree µ = m × n in which the result was obtained 

that the solution of such equations could be reduced to solving m algebraically 

solvable equations of degree n and a single (generally unsolvable) equation of 

degree m. 

4. An application of these — and other — results to the class of Abelian equations, 

and a demonstration that these were always solvable by radicals. 

5. A further application of this result to the circular functions by which GAUSS’ 

results on the cyclotomic equation were reproduced. 

ABEL had further ideas for applications of this new theory to elliptic functions, 

but these were not printed on this occasion (see below). In his research on Abelian 

equations, KRONECKER much later came to the conclusion that “these general Abelian 

equations in reality are nothing but cyclotomic equations.” 18 ABEL’S paper contains, 

however, more than just the solubility-result for Abelian equations, and the general 

theory of the class of equations with rationally dependent roots sprung from — and 

had quite interesting implications for — ABEL’S approach to the theory of elliptic func- 

tions. 

7.2 Elliptic functions 

In his very first publication on elliptic functions entitled Recherches sur les fonctions 

elliptiques, 19 ABEL made several interesting innovations. 20 ABEL devoted a large por- 

tion of the first part of the Recherches to the inversion of elliptic integrals into elliptic 

functions, the extension of these functions into the complex domain, and the study 

of algebraic relations involving these functions. He derived addition formulae and 

studied the singularities of elliptic functions in order to address the central problem, 

which can be summarized in the following way: 

Problem 1 (Division Problem) Given an integer m and the value φ (mβ) of an elliptic 

function of the first kind, φ, at mβ, express φ (β) by radicals. ✷ 

18 “[. . . ] so daß dise allgemeinen Abelschen Gleichungen im Wesentlichen nichts Anderes sind, als 

Kreistheilungs-Gleichungen.” (Kronecker, 1853, 11). 


20 The history of these elliptic functions and ABEL’S works on them is studied in much greater depth 

in part IV. For the present discussion, I am only concerned with the ideas behind ABEL’S result on 

the solubility of Abelian equations.

7.2. Elliptic functions 153 

Figure 7.1: ABEL’S drawing of the lemniscate in one of his notebooks. (Stubhaug, 1996, 

270) 

ABEL’S inspirations for this problem were twofold. The case in which m = 2 and φ 

was the lemniscate function useful in measuring the arc length of the lemniscate curve 

(see figure 7.1) 

φ (x) = 

� x 

0 

dx 

√ 1 − x 4 

had been settled in the eighteenth century by G. C. FAGNANO DEI TOSCHI (1682– 

1766). 21 In his study of the equivalent problem for circular functions GAUSS had 

expressed his conviction that his approach would apply equally well to other tran- 

scendentals, for instance the lemniscate integral (see the quotation in section 5.3.1, p. 

74). 

ABEL had learned of FAGNANO DEI TOSCHI’S work and the tradition in research 

on elliptic integrals through his studies of the much more advanced works on the 

subject by EULER and A.-M. LEGENDRE (1752–1833). 22 Complementary to his gen- 

eralization of FAGNANO DEI TOSCHI’S result to the bisection of elliptic functions of 

the first kind, ABEL gave a detailed investigation of the division of such functions 

into 2n + 1 parts. Reformulated in the light of the addition formulae, which he had 

previously developed, ABEL obtained a different version of the problem, summarized 

in: 

Problem 2 (Division Problem) Given n, solve the equation 

φ ((2n + 1) β) = P2n+1 (φ (β)) 

Q2n+1 (φ (β)) 

which has degree (2n + 1) 2 . ✷ 

ABEL’S central insight was that the equation of degree (2n + 1) 2 could be reduced 

to lower degree equations which were always solvable if the divisions of the periods 

21 (Houzel, 1986, 298). 

22 See chapter 15.


of the elliptic function were known. Addressing this division of the complete periods, 

ABEL demonstrated, directly inspired by GAUSS, that the roots 

φ 2 

� kω ′ 

2n + 1 

� 

for 1 ≤ k ≤ n 

could be found by solving an equation of degree 2n + 2 which might not, however, be 

solvable by radicals. 

In the second part of the Recherches sur les fonctions elliptiques which appeared in 

1828, 23 ABEL applied the preceding investigation to the lemniscate integral. In com- 

plete correspondence with GAUSS’ result for the division of the circle, ABEL stated his 

result, using n in two different meanings: 

“The value of the function φ � � 

mω [the lemniscate function] can be expressed 

n 

by square roots whenever n is a number of the form 2n or 1 + 2n , the latter number 

being prime, or a product of multiple numbers of these two forms.” 24 

Therefore, the division of the lemniscate into n equal parts could always be con- 

structed by ruler and compass if n was a number of the described form. 

In the Recherches sur les fonctions elliptiques, ABEL used direct methods to reduce the 

degrees and prove the solubility of the involved equations. However, as he soon re- 

alized, these properties depended on a deeper relation between the roots of the equa- 

tions, and in his letters he considered the division of the lemniscate as a by-product 

of his research in the theory of equations. 25 As ABEL indicated in the introduction to 

the Mémoire sur une classe particulière, he had planned to apply the theory concerning 

these equations to elliptic functions: 

“After having presented this theory in its generality, I will apply it to circular 

and elliptic functions.” 26 

Although no explicit application ever appeared in print (see section 7.2.1), it is not 

hard to see that for instance the equation 

n � 

∏ X − φ 

k=1 

2 

� 

kω ′ �� 

= 0 

2n + 1 

falls into the category studied in the general theory because of the rational dependency 

expressed by the addition formulae for φ. 


24 “La valeur de la fonction φ � � 

mω 

n peut être exprimée par des racines carrées toutes les fois que n 

est un nombre de la forme 2n ou 1 + 2n , le dernier nombre étant premier, ou même un produit de 

plusieurs nombres de ces deux formes.” (ibid., 168). 

25 (Abel→Holmboe, Paris, 1826/12. N. H. Abel, 1902a, 52) and (Abel→Holmboe, Berlin, 1827/03/04. 

ibid., 57). 

26 “Après avoir presenté généralement cette théorie, je l’appliquerai aux fonctions circulaires et elliptiques.” 

(N. H. Abel, 1829c, 132).

7.2. Elliptic functions 155 

7.2.1 The lost sections 

The paper Mémoire sur une classe particulière, 27 which was published in the second is- 

sue of the fourth volume of A. L. CRELLE’S (1780–1855) Journal appearing on March 

28 th 1829, i.e. a few days before ABEL’S death, was not complete. At the end of the 

published part, following the application to circular functions, CRELLE added a foot- 

note: 

“The author of this treatise will, on another occasion, present applications to 

elliptic functions.” 28 

At the end of ABEL’S manuscript for the Mémoire sur une classe particulière, 29 the 

opening page of a sixth — not printed — section entitled “Application aux fonctions 

elliptiques” can still be found (see figure 7.2). In the limited space of this one page, 

ABEL outlined the link with the Recherches sur les fonctions elliptiques. Its purpose was 

to facilitate the application of his newly developed theory to the division problem. 

From a letter to CRELLE — which ABEL wrote in October 1828 — it becomes clear that 

ABEL had sent a manuscript including the application to elliptic functions to CRELLE 

for publication in the Journal. 30 Because of his intense competition with C. G. J. JA- 

COBI (1804–1851) on elliptic functions, 31 ABEL urged CRELLE to rush publication of his 

sketch of his general theory of elliptic functions, the Précis d’une théorie des fonctions el- 

liptiques. 32 ABEL wanted CRELLE to delay the publication of the Mémoire sur une classe 

particulière, which had been scheduled for publication in the first issue of the fourth 

volume, and to leave out the part concerning the application to elliptic functions. 33 

CRELLE followed ABEL’S desire and published the Mémoire sur une classe particulière 

in the second issue. The Précis d’une théorie des fonctions elliptiques was published in 

the third and fourth issues of the fourth volume of the Journal, concluding a volume 

in which ABEL had published repeatedly on elliptic functions. 

Unfortunately, CRELLE’S correspondence and Nachlass appears to have been lost. 34 

Therefore, little hope remains of finding the lost sections of ABEL’S paper. Neverthe- 

less, some information on their contents can be reconstructed from two sources: a 

notebook entry and the paper Précis d’une théorie des fonctions elliptiques. 

In one of ABEL’S notebooks, a brief list of contents of the manuscript Mémoire sur 

une classe particulière was found. 35 It was intended for ABEL’S own use and carried cor- 

rectly numbered references to central formulae and results, both in the published and 

27 (ibid.), described above. 

28 “L’auteur de ce mémoire donnera dans une autre occasion des applications aux fonctions elliptiques.” 

(ibid., 156, footnote). 

29 (Abel, MS:592, 64). 

30 (Abel→Crelle, Christiania, 1828/10/18. Biermann, 1967, 28–29) 

31 See part IV. 

32 (N. H. Abel, 1829d) 

33 This letter published in 1967 thus settles the speculation of SYLOW as to why these parts were not 

published (L. Sylow, 1902, 18). 

34 See footnote 54 on 33. 

35 (Abel, MS:351:C, 52). See also (N. H. Abel, 1881, II, 310–311) and (L. Sylow, 1902, 7–8).


Figure 7.2: The last page of ABEL’S manuscript for Mémoire sur une classe particulière 

(Abel, MS:592, 94) with the crossed out beginning of a sixth section. See also (N. H. 

Abel, 1881, II, 313).

7.3. The concept of irreducibility at work 157 

in the missing sections. Thus it must have been produced shortly before the manu- 

script was sent to CRELLE. Apparently, the manuscript contained two further sections 

besides the five published ones. In the sixth section, concerning the application to 

elliptic functions, ABEL listed the result that if 

was integral, the complex division 

m 2 + 2n + 1 

2µ + 1 

φ (m − αi) ω 

2n + 1 

could be effected by solving a µ th degree equation. 36 This is a generalized version 

of the division problem treated above. The seventh section concerned the transfor- 

mations of elliptic functions; a topic which also constituted a major part of ABEL’S 

competition with JACOBI. In the notebook, ABEL listed a number of formulae captur- 

ing central results. In order to produce a reliable interpretation, ABEL’S works in the 

transformation theory of elliptic functions have to be taken into consideration. 37 

7.3 The concept of irreducibility at work 

Of central importance to ABEL’S research in the theory of equations was his use of 

the concepts of irreducibility and divisibility. In his Disquisitiones arithmeticae, GAUSS 

devoted a paragraph to the following result concerning the equation X = 0 which 

corresponded to the system Ω of imaginary n th roots of unity: 

“Theory of the roots of the equation x n − 1 = 0 (where n is assumed to be 

prime). 

Except for the root 1, the remaining roots contained in (Ω) are included in the 

equation X = x n−1 + x n−2 +etc.+x + 1 = 0. 

The function X cannot be decomposed into lower factors in which all the coefficients 

are rational.” 38 

GAUSS demonstrated the indecomposibility of X by an ad hoc argument and did 

not put it to central use later in the proof (section 5.3). In ABEL’S impossibility proof, 

numerous allusions to irreducibility had been made; however, they all served as sim- 

plifications and not as central concepts (see section 6.3.3). 39 By 1829 ABEL promoted 

the concept into a fundamental one on which theorems could be built. ABEL’S defini- 

tion of irreducibility was intended to capture the same property as GAUSS had demon- 

strated for X, although ABEL spoke of irreducible equations where GAUSS had spoken 

36 (Abel, MS:351:C, 52). 

37 This theme is taken up in part IV. 

38 “Theoria radicum aequationis x n − 1 = 0 (ubi supponitur, n esse numerum primum). 

Omittendo radicem 1, reliquae (Ω) continentur in aequatione X = x n−1 + x n−2 + etc. +x + 1 = 0. 

Functio X resolvi nequit in factores inferiores, in quibus omnes coefficients sint rationales.” (C. F. 

Gauss, 1801, 417); English translation from (C. F. Gauss, 1986, 412). 

39 (N. H. Abel, 1826a, 71).


of indecomposable functions. This switch from polynomial functions to their associated 

equations was not uncommon, and is mainly a distinction in terms. ABEL gave his first 

definition of irreducibility in a footnote in the paper on Abelian equations: 

“An equation φx = 0, in which the coefficients are rational functions of a 

certain number of known quantities a, b, c, . . . , is called irreducible when it is impossible 

to express any of its roots by an equation of lower degree, in which the 

coefficients are also rational functions of a, b, c, . . . .” 40 

The first — and highly useful — theorem which ABEL demonstrated with this defi- 

nition was that no equation could share a root with an irreducible one without having 

all the roots of the irreducible equation as roots. In the following section, I describe 

ABEL’S proof and use of this important theorem. 

7.3.1 EUCLID’s division algorithm 

Formulated in the terminology of the Mémoire sur une classe particulière, the central 

theorem on irreducible equations was the following one expressing the property de- 

scribed above: 

Theorem 9 “If one of the roots of an irreducible equation, φx = 0, satisfies another equation, 

f x = 0, where f x denotes a rational function of x and known quantities which are supposed 

contained in φx, this latter equation will also be satisfied if instead of x any other root of the 

equation φx = 0 is inserted.” 41 

ABEL gave a proof of this theorem — again relegated to a footnote — which is a 

beautiful application of the division algorithm much along the lines of a modern ar- 

gument. Because f was a rational function, ABEL could write it as 

f = M 

, (7.15) 

N 

where M and N were entire functions of x. But, as ABEL noticed, “any [polynomial] 

function of x can always be put on the form P + Q · φx where P and Q are entire func- 

tions such that the degree of P is less than that of the function φx.” 42 This application 

of the division algorithm with remainder was well known to ABEL and received no 

40 “Une équation φx = 0, dont les coefficients sont des fonctions rationnelles d’un certain nombre 

de quantités connues a, b, c, . . . s’appelle irréductible, lorsqu’il est impossible d’exprimer aucune 

de ses racines par une équation moins élevée, dont les coefficiens soient également des fonctions 

rationnelles de a, b, c, . . . .” (N. H. Abel, 1829c, 132, footnote). 

41 “Si une des racines d’une équation irréductible φx = 0 satisfait à une autre équation f x = 0, où f x 

désigne une fonction rationnelle de x et des quantités connues qu’on suppose contenues dans φx; 

cette dernière équation se trouvera encore satisfaite en mettant au lieu de x une racine quelconque 

de l’équation φx = 0.” (ibid., 133). 

42 “mais une fonction de x peut toujours être mise sous la forme P + Q · φx, ou P et Q sont des fonctions 

entières, telles, que le degré de P soit moindre que celui de la fonction φx.” (ibid., 132–133, footnote).

7.3. The concept of irreducibility at work 159 

further comment. 43 By inserting into (7.15), ABEL found 

f (x) = 

P + Q · φ (x) 

. (7.16) 

N 

Next, he let x denote a common root of φ and f and concluded that x would also be 

a root of P = 0. However, if P were not identically zero, “this equation gives x as a 

root of an equation of degree less than that of φx = 0; which is a contradiction of the 

hypothesis. Therefore, P = 0 and it follows that f x = φx · Q 

N .”44 Thus, it was obvious 

that f would vanish whenever φ did and, therefore, that any root of φ (x) = 0 would 

also be a root of f (x) = 0. 

ABEL put this important theorem to use in the very first description of the equa- 

tions treated in the Mémoire sur une classe particulière. If x ′ and x were two roots of the 

irreducible equation φ (x) = 0 among which a rational dependency existed, 

x ′ = θ (x) , 

then every iterated application of θ to x would also be a root of this equation. ABEL’S 

demonstration followed directly from the theorem above. He argued that since it fol- 

lowed from the hypothesis that the equations 

φ (θ (x)) = 0 and φ (x) = 0 

had a root, x, in common, theorem 7.3 stated that for any root, y, of φ (x) = 0, θ (y) 

would also be a root of that equation. Once he had established this result, the argu- 

ment of ABEL’S paper was on its way, and the complex of conclusions described above 

could be obtained. 

ABEL turned the concept of irreducibility of equations, which had existed as an 

ad hoc tool before into a central foundation upon which a building of theorems could 

be established. 45 The irreducibility in ABEL’S sense was defined as minimality of the 

equation expressing the roots under the restriction that the coefficients must depend 

rationally on the same quantities as the original equation. From this definition, gen- 

eralizations were later made toward the general concept of domain of rationality. But 

working with this definition — and the division algorithm of EUCLID (∼295 B.C.) — 

ABEL demonstrated the important theorem 9 of divisibility, which in turn established 

the basic property of the class of equations studied in the Mémoire sur une classe particulière. 

46 

43 It had been explicitly employed in GAUSS’ second proof of the Fundamental Theorem of Algebra (see 

section 5.7). 

44 “cette équation donnera x, comme racine d’une équation d’un degré moindre que celui de φx = 0; 

ce qui est contre l’hypothèse; donc P = 0 et par suite f x = φx · Q 

45 

N .” (ibid., 133,footnote). 

See also (L. Sylow, 1902, 23–24). 

46 (N. H. Abel, 1829c).


❅❅■ 

✻ 

✛ Solvable equations ✲ 

��✠ 

❄ 

Polynomial equations 

Figure 7.3: Extending the class of solvable equations: Abelian equations 

7.4 Enlarging the class of solvable equations 

ABEL considered the positive result demonstrating the solubility of certain equations 

as a counterpart to the insolubility of higher degree general equations. In the intro- 

duction to the Mémoire sur une classe particulière, ABEL wrote: 

“It is true that the algebraic equations are not generally solvable, but there is a 

particular class of each degree for which the algebraic solution is possible.” 47 

To this class of solvable equations belonged the equations of the form x n − 1 = 0 

studied by GAUSS and the generalizations of these obtained by ABEL in the paper. 

Only few other equations were explicitly known to be solvable, and ABEL’S result 

can thus be seen to provide a demonstration that the total class of solvable equations 

had a certain range. In the limitation-enlargement model suggested in section 6.8, 

the situation can be described by figure 7.3 and much of ABEL’S research to describe 

the precise extent of solubility can be interpreted in this context. In a letter to B. M. 

HOLMBOE (1795–1850) written during his stay in Paris, ABEL described the problem 

and his progress: 

“I am currently working on the theory of equations, which is my favorite 

theme, and have finally reached a point where I see a way to solve the following 

general problem: To determine the form of all algebraic equations which can 

be solved algebraically. I have found an infinitude of the fifth, sixth, seventh, etc. 

degree which had never been smelled before.” 48 

47 “Il est vrai que les équations algébriques ne sont pas résolubles généralement; mais il y en a une 

classe particulière de tous les degrés dont la résolution algébrique est possible.” (N. H. Abel, 1829c, 

131). 

� �✒ 

❅ ❅❘

7.4. Enlarging the class of solvable equations 161 

Thus, ABEL’S two publications on the theory of equations which appeared during 

his lifetime contributed a negative, limiting result of insolubility of the general higher 

degree equations and a positive, enlarging result of the solubility of a certain class of 

equations of all degrees. The program set out above in the letter to HOLMBOE was 

pursued by ABEL from his time in Paris, and traces of it can be found in his note- 

books. However, his correspondence also announced further and far-reaching results 

for which no detailed studies or proofs have been recovered. The determination of 

the exact extension of the concept of algebraic solubility was approached by ABEL 

through a theory largely based on the same tools as his published works, but never 

completed nor published in his lifetime. Therefore, the solution to this fundamental 

problem is rightfully attributed to E. GALOIS (1811–1832). In the next chapter, ABEL’S 

steps toward a general theory of solubility are analyzed against the background of his 

other works and GALOIS’ contemporary ideas. 

48 “Jeg arbeider nu paa Ligningernes Theorie, mit Yndlingsthema og er endelig kommen saa vidt at jeg 

seer Udvei til at løse følgende alm: Problem. Determiner la forme de toutes les équation algébriques 

qui peuvent être resolues algebriquement. Jeg har fundet en uendelig Mængde af 5te, 6te, 7de etc. 

Grad som man ikke har lugtet indtil nu.” (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a, 

44).

Chapter 8 

A grand theory in spe: algebraic 

solubility 

In his correspondence with A. L. CRELLE (1780–1855) and B. M. HOLMBOE (1795– 

1850), N. H. ABEL (1802–1829) announced numerous results in the theory of equa- 

tions beyond the impossibility of solving the quintic and the study of Abelian equa- 

tions. Some concerned the form of solutions to algebraically solvable equations of the 

fifth degree, 1 others dealt with solubility results for broader classes of equations, 2 and 

yet others testify to ABEL’S general progress in his program of determining the form 

of solvable equations. 3 The information provided in the letters is complemented by 

a notebook entry dating from 1828 which has been included in both editions of the 

Œuvres under the title Sur la résolution algébrique des équations. 4 The entry begins as a 

manuscript almost ready for press, but after some introductory remarks, a few theo- 

rems and some deductions it turns from its initial thoroughness and clarity to nothing 

but bare calculations. Nevertheless, when considered together, these sources give an 

impression of the methods and extent of the general theory of algebraic solubility 

which ABEL set out to develop in the last years of his life. 

8.1 Inverting the approach once again 

The notebook manuscript dealt with the general form of algebraically solvable equa- 

tions. In one of the two lengthy introductions which ABEL wrote for this work the 

problem was clearly set out: 

“Given an equation of any given degree, to determine whether or not it could 

be satisfied algebraically.” 5 

1 (Abel→Crelle, Freyberg, 1826/03/14. N. H. Abel, 1902a, 21–22). 

2 (Abel→Crelle, Christiania, 1828/08/18. ibid., 72–73). 

3 (Abel→Holmboe, Paris, 1826/10/24. ibid., 44–45). 

4 (N. H. Abel, [1828] 1839). 

5 “Une équation d’un degré quelconque étant proposée, reconnaître si elle pourra être satisfaite algébriquement, 

ou non.” (N. H. Abel, 1881, vol. 2, 330). 

163

164 Chapter 8. A grand theory in spe 

ABEL’S initial step in solving this general problem was to reformulate it in the 

following program which he described in the introduction to the other version of the 

manuscript. 

“From this, the following two problems stem naturally whose complete solution 

comprises the entire theory of the algebraic solution of equations, namely: 

1) To find all equations of any determinate degree which are algebraically solvable. 

2) To decide whether or not a given equation is algebraically solvable.” 6 

Thus, the problem of determining the algebraic solubility of equations had been 

inverted once again. In principle, ABEL’S program amounted to listing — by some de- 

scriptive form — all equations of a certain degree which could be solved algebraically 

and then deducing whether any given equation was in this list. In pursuing this prob- 

lem, ABEL focused on a given algebraic expression, a radical, and sought to describe 

the irreducible equation which it satisfied. In doing so, the concept of irreducibility ac- 

quired its second importance in ABEL’S research as a means of obtaining the equation 

linked to a given radical. This shift from working with equations of which some rep- 

resentation was known, either as a fifth degree polynomial or as relations among its 

roots, to general equations which were only characterized by their external structure 

as being irreducible is what I consider a second inversion of approach. 

ABEL’S inversion was intimately connected to a general consideration on mathe- 

matical methodology. In his introduction, he described this inversion of approach in 

a much quoted paragraph: 

“To solve these equations [of the first four degrees], a uniform method was 

discovered which, it was thought, was applicable to an equation of any degree; 

but in spite of all the efforts of a Lagrange and other distinguished geometers, the 

proposed goal could not be reached. This led to the assumption that the solution of 

the general equation was algebraically impossible; but this could not be decided 

since the adopted method had only been able to lead to reliable conclusions in 

the case in which the equations were solvable. In fact, one proposed to solve the 

equations without knowing if that was possible. In this case, one might come to 

the solution although that was not certain at all; but if by misfortune the solution 

was impossible, one might search an eternity without finding it. To infallibly reach 

anything in this matter, it is necessary to follow another route. One should give the 

problem such a form that it will always be possible to solve it, which can always 

be done for any problem. Instead of demanding a relation, of which the existence 

is unknown, one should ask whether such a relation is possible at all.” 7 

6 “De là dérivent naturellement les deux problèmes suivans, dont la solution complète comprend 

toute la théorie de la résolution algébrique des équations, savoir: 

1) Trouver toutes les équations d’un degré déterminé quelconque qui soient résolubles algébriquement. 

2) Juger si une équation donnée est résoluble algébriquement, ou non.” 

(N. H. Abel, [1828] 1839, 218–219). 

7 “On découvrit pour résoudre ces équations une méthode uniforme et qu’on croyait pouvoir appliquer 

à une équation d’un degré quelconque; mais malgré tous les efforts d’un Lagrange et d’autres

8.2. Construction of the irreducible equation 165 

As previously noted, ABEL’S belief that any problem could be converted into a 

solvable one was held by most mathematicians throughout the 19 th century. It be- 

came prominent in the so-called Hilbert Programme before the development of axiomat- 

ics stressed that decidability could only be asked and answered relatively to the (ax- 

iomatic) system in which the problem was embedded. 

With ABEL’S new driving question, modified from the ones motivating the impos- 

sibility proof and the study of Abelian equations, it was his intention to explore the 

grey area between the entire set of equations and the ones known to be solvable (see 

figures 6.1 and 7.3). ABEL’S hope was to delineate the border line between solvable 

and unsolvable equations by some external characteristic. 

8.2 The construction of the irreducible equation 

satisfied by a given expression 

The general program of ABEL’S research was to construct a list of all irreducible solv- 

able equations and subsequently match any given equation against this list. His at- 

tempt at implementing this scheme consisted of a construction of the irreducible equa- 

tion satisfied by a given algebraic expression. After establishing certain properties of 

this equation from the expression which satisfies it, ABEL returned to the problem of 

determining whether a given equation was solvable or not. 

The first part of ABEL’S notebook manuscript contained theorems and results pre- 

sented in a clear and deductive manner. Their contents showed frequent similarities 

with the opening studies of the form of algebraic expressions satisfying an equation as 

carried out in the impossibility proof (see section 6.3.3). If anything, the 1828 notebook 

lacked — by comparison to the impossibility proof — the clear, albeit defective, classi- 

fication of algebraic expressions of which only reminiscences were given. The clas- 

sification established in the notebook was insufficient to cover some of the required 

deductions, and it is possible that ABEL, himself, had noticed this deficiency (see be- 

low). 

Basic concepts. In the opening section of the manuscript proper (following a lengthy 

introduction), ABEL outlined his own characterization of algebraic expressions which 

géomètres distingués on ne put parvenir au but proposé. Cela fit présumer que la résolution des 

équations générales était impossible algébriquement; mais c’est ce qu’on ne pouvait pas décider, 

attendu que la méthode adoptée n’aurait pu conduire à des conclusions certaines que dans le cas 

où les équations étaient résolubles. En effet on se proposait de résoudre les équations, sans savoir 

si cela était possible. Dans ce cas, on pourrait bien parvenir à la résolution, quoique cela ne fût 

nullement certain; mais si par malheur la résolution était impossible, on aurait pu la chercher une 

éternité, sans la trouver. Pour parvenir infailliblement à quelque chose dans cette matière, il faut 

donc prendre une autre route. On doit donner au problème une forme telle qu’il soit toujours possible 

de le résoudre, ce qu’on peut toujours faire d’une problème quelconque. Au lieu de demander 

une relation dont on ne sait pas si elle existe ou non, il faut demander si une telle relation est en effet 

possible.” (ibid., 217).


could occur in the solution of a solvable equation. This characterization had been one 

of the points of objection to his impossibility proof of 1826 (see section 6.9.1). However, 

only the objections raised by E. J. KÜLP (⋆1801) were known to ABEL and ABEL did 

not react directly to them in the notebook. The characterization which ABEL presented 

in the notebook was only a limited version of the one found in the impossibility proof. 

In the notebook, ABEL described the radicals from the outer-most one inward in the 

following form 

1 

µ 1 

2 

µ 1 

y = P0 + P1 · R1 

+ P2 · R1 

+ · · · + Pµ 1−1 · R1 

, (8.1) 

in which P0, . . . , Pµ 1−1 and R1 were rational expressions in known quantities and the 

1 1 

µ 2 µ 3 

other radicals R2 

, R3 

, . . . . In relation to the route he had taken in the impossibility 

proof, he abandoned the concept of degree of algebraic expressions and imposed only 

the hierarchy from the concept of order which counted the number of nested root 

extractions of prime degree. 

ABEL introduced three notational concepts which he used throughout the prelimi- 

nary part of the manuscript to simplify his notation: 

1. He chose to denote algebraic expressions by writing their order as subscripts, for 

µ 1 −1 

µ 1 

instance writing Am for an algebraic expression A of order m. 

2. With y being of the form (8.1) and φ (y) = 0 an equation satisfied by y, ABEL 

chose to write the equation as φ (y, m) = 0 if all the coefficients of φ (y) were 

algebraic expressions of order m. Furthermore, he denoted the degree of the 

equation by δφ (y, m). 

3. Most importantly, he introduced a symbol ∏ Am for the product of all values of 

Am obtained from attributing to the outermost radical in Am, R 1 µ , all its possible 

values, R 1 µ , ωR 1 µ , . . . , ω µ−1 R 1 µ (ω a µ th root of unity). Thus, if 

Am = 

the new symbol denoted the expression 

∏ Am = 

µ−1 

∑ pkR k=0 

k µ , 

� 

µ−1 µ−1 

∏ ∑ pkω u=0 k=0 

uk R k � 

µ . 

Using these concepts and a number of immediate consequences derived from them, 

ABEL constructed and characterized the irreducible equation associated with a given 

algebraic expression.


Lemmata. In the first lemma, ABEL obtained a result which had played a central role 

in his impossibility proof. It stated that if the equation 

µ−1 

∑ 

u=0 

tuy 

u 

µ 1 

1 

= 0 (8.2) 

could be satisfied, in which the coefficients t0, . . . , tµ−1 were rational functions of ω, 

known quantities (i.e. coefficients of the equation φ (y) = 0), and lower order radicals, 

then all the coefficients had to vanish, i.e. t0 = t1 = · · · = tµ−1 = 0 (cmp. lemma 1). 

By and large, the proof resembled the one given in 1826 (see section 6.3.3) but differed 

when ABEL had to eliminate the possibility of a first degree irreducible factor. Letting 

z = y 1 

µ 1 , ABEL assumed that the irreducible factor (corresponding to an irreducible 

equation) ∑ κ u=0 snz n divided (8.2) and excluded the possibility of k ≥ 2. The case of a 

first degree irreducible factor was briefly dismissed by the following argument: 

“Thus, it is necessary that k = 1, but that gives 

from which 

which is similarly impossible.” 8 

s0 + z = 0 

z = µ √ 

1 y1 = −s0, 

As P. L. M. SYLOW (1832–1918) has remarked, the conclusion that z = −s0 is 

impossible is essentially correct, 9 it can be supported if an improved hierarchy is im- 

posed on the radicals. 10 Again, ABEL’S notebook does not contain all the technical 

details of his deductions. 

ABEL put forward another important proposition when he claimed that the roots 

of satisfiable equations come in “bundles”. He stated that if the equation 

was satisfied by an algebraic expression of order n 

8 “Il faut donc que k = 1, or cela donne 

d’où 

φ (y, m) = 0 (8.3) 

y = 

µ−1 

∑ 

k=0 

s0 + z = 0 

p ky 

k 

µ 1 

1 , 

z = µ √ 

1 y1 = −s0, 

ce qui est de même impossible.” (N. H. Abel, [1828] 1839, 229). 

9 (Sylow in N. H. Abel, 1881, vol. 2, 332). 

10 As shown by (Holmboe in N. H. Abel, 1839, vol. 2, 289) and (Maser in Abel and Galois, 1889, 149).


it would also be satisfied if ωu 1 

µ 

y1 

1 

µ 

were inserted for y1 

(ω a µth root of unity). He gave 

no explicit proof of this result, which is a simple consequence of the vanishing of the 

coefficients of (8.2). 11 The result provided the important connection that any root of 

∏ φ (y, m) = 0 would also be a root of φ (y, m) = 0. 

The manuscript also contains the fundamental characterization of irreducible equa- 

tions that no equation can share a root with an irreducible equation without the latter 

dividing the former (cmp. theorem 7.3). ABEL derived this along the lines described 

in section 7.3 but applied the terminology developed in the manuscript. By implicit 

application of the Euclidean division algorithm, ABEL demonstrated that if the equa- 

tions 

φ (y, m) = 0 and φ1 (y, n) = 0 

had a common root, φ was assumed to be irreducible, and n ≤ m, then 

φ1 (y, n) = φ (y, m) · f (y, m) . 

Properties of ∏ φ (y, m). In his subsequent argument, ABEL sought to describe the 

irreducible equation satisfied by a given algebraic expression. The central tool employed 

was his construction of this equation based on the construction of ∏ φ (y, m) 

and the demonstration of its properties. The construction which ABEL gave was 

mainly existential; it amounted to proving the existence of an equation having spe- 

cific useful properties. 

Continuing to build upon the fundamental result on irreducible equations, ABEL 

proved the following theorem. 

Theorem 10 If 

then for some m ′ 

φ1 (y, n) = f (y, m) · φ (y, m) , 

φ1 (y, n) = f1 

� y, m ′ � · ∏ φ (y, m) . ✷ 

ABEL’S proof was elegant and made prototypical usage of the previously estab- 

lished theorems and the concept of the outer-most radical. Denoting by µ√ y1 the outer- 

most root extraction of φ (y, m) = 0, c.f. (8.1), this equation would also be satisfied 

if ω k µ √ y1 were substituted for µ√ y1 where ω was a µ th root of unity. Consequently, 

ω k µ √ y1 was a root of φ and, therefore, also of φ1. Thus, φ1 would have the different 

roots of φ corresponding to different values of k as roots. As ABEL noticed, if these 

factors corresponding to different values of k had no common factors (were relatively 

prime), their product would also be a factor of φ and the proof had been completed. 

In the impossibility proof of 1826, ABEL had stated this result, which translated into 

the notation of the manuscript concludes that “it is clear that the given equation must 

11 See for instance (Holmboe in N. H. Abel, 1839, vol. 2, 289).


1 

µ 

be satisfied by all values of y which are obtained by attributing to y1 

1 

µ 

ωy1 

, ω2 1 

µ 

y1 

, . . . , ωn−1 1 

µ 

y 

all the values 

1 ”12 (see section 6.3.3). In 1826, it had been given no proof, but 

in the notebook, ABEL provided the proof as an easy and elegant application of the 

fundamental concepts and tools. 

ABEL proceeded by establishing a central link between the irreducibility of φ (y, m) = 

0 and that of ∏ φ (y, m) = 0. 

Theorem 11 If the equation 

is irreducible, then so is the equation 

φ (y, m) = 0 

φ1 (y, m) = ∏ φ (y, m) = 0. ✷ 

ABEL argued for this theorem by a reductio ad absurdum proof against which SYLOW 

later raised well founded objections. ABEL assumed that φ1 was reducible and that 

φ2 (y, m ′ ) was an irreducible 13 factor of ∏ φ (y, m) = 0. Under these assumptions, 

φ2 and φ would have a common root since all the roots of ∏ φ were also roots of 

φ. The assumed irreducibility of φ then enabled ABEL to conclude that because the 

irreducible φ and φ2 had a root in common, φ would be a factor of φ2, 

φ2 

This in turn implied (by theorem 10) 

φ2 

� y, m ′ � = f (y, m) · φ (y, m) . 

� y, m ′ � = f1 

� ′′ 

y, m � · ∏ φ (y, m) . (8.4) 

� �� 

=φ1(y,m) On the other hand, φ2 had been assumed to be an irreducible factor of φ1 implying 

deg φ2 < deg φ1, which contradicted (8.4). 

SYLOW’S objections concerned the properties of ∏ φ. Besides certain points, at 

which ABEL left out assumptions of irreducibility, SYLOW noticed that ABEL tacitly 

assumed that φ (y, m) did not have factors in which all the coefficients were rational 

expressions in inner radicals and known quantities. If such factors were involved, the 

equation ∏ φ (y, m) = 0 might turn out to be a power of an irreducible equation. 14 

SYLOW repaired ABEL’S argument by refining his hierarchy of algebraic expressions. 

12 “[. . . ] so ist klar, daß der gegebenen Gleichung durch alle die Werthe von y genug werden muß, 

welche man findet, wenn man der Größe p 1 n alle die Werthe αp 1 n , α 2 p 1 n , . . . , α n−1 p 1 n beilegt.” (N. H. 

Abel, 1826a, 72). 

13 Actually, ABEL did not, presumably inadvertently, state the condition of irreducibility of φ2. 

14 (Sylow in N. H. Abel, 1881, vol. 2, 332).


Construction of the irreducible equation. With the first theorems and the lemmata 

described above, ABEL was in a position to give a construction of the irreducible equa- 

tion which a given algebraic expression satisfied. More importantly, this construction 

allowed him to demonstrate that central properties of this equation could be deduced 

from properties of the initially given algebraic expression. ABEL let 

am = f ( µm √ ym, µ √ 

m−1 ym−1, . . . ) 

denote a given algebraic expression and constructed the irreducible equation ψ (y) = 0 

which would have am as a root in the following way. 

Since am was to satisfy ψ (y) = 0, it would be necessary that y − am was a factor of 

ψ. By the theorem 10, it followed that 

φ1 (y, m1) = ∏ (y − am) 

would also be a factor. Because y − am was a first degree polynomial and, therefore, 

irreducible, it followed that φ1 was also irreducible (by theorem 11). Consequently, 

φ1 was an irreducible factor of ψ (y) and the procedure could be repeated yielding a 

sequence of irreducible factors 

φn (y, mn) = ∏ φn−1 (y, mn−1) , 

in which the radicals of am were sequentially removed by the analogue of multiplying 

with the complex conjugate (c.f. section 6.3.2). 

ABEL claimed that the sequence of positive integers m1, m2, . . . was decreasing but 

gave no explicit argument. However, by J. L. LAGRANGE’S (1736–1813) theorem (sec- 

tion 5.2.3) it is not hard to see that ∏ am is a rational function of ym and the inner 

radicals involved. Therefore, the order of ∏ am is less than the order of am. Thus, 

at a certain point (after, say, u steps) the sequence m1, m2, . . . had to vanish, and an 

equation would be obtained in which all the coefficients were rationally known. This 

equation was the sought-for ψ (y) = 0, 

ψ (y) = φu (y, 0) = ∏ φu−1 (y, mu−1) . 

Directly from this construction, ABEL deduced his characterization of the irre- 

ducible equation satisfied by a given algebraic expression, laying the foundations for 

his further reasoning. He summarized the properties in the following four points: 15 

Proposition 1 The following four results link properties of the irreducible equation ψ (y) = 0 

satisfied by a given algebraic expression am to properties of the expression itself: 

1. The degree of ψ is the product of certain exponents of root extractions occurring in am. 

Among these exponents, the one of the outer-most root extractions is always present. 

15 (N. H. Abel, [1828] 1839, 232–233)

8.3. Refocusing on the equation 171 

2. The exponent of the outer-most root extraction divides the degree of ψ [actually contained 

in 1, above]. 

3. If ψ can be algebraically satisfied, it is also algebraically solvable. All its roots are ob- 

tained by attributing to the root extractions y 

1 

µu 

mu 

all their possible values. 

4. If the degree of ψ is µ, the expression am may have µ, and no more than µ, values. ✷ 

ABEL’S deduction of these properties was straightforward from considerations on 

the exponents of involved root extractions and the construction described. A formal 

consideration of the uniqueness of the irreducible equation constructed was not car- 

ried out, but must have seemed obvious to ABEL. 

8.3 Refocusing on the equation 

The first theorems and the construction of the irreducible equation connected to a 

given algebraic expression are fascinating pieces of mathematics revealing traces of 

ABEL’S profound ideas. Whereas the presentation of these fundamental results was 

lucid — and basically acceptable to present day mathematicians — ABEL’S following 

investigations in the notebook took another form. As he progressed farther from the 

well established results founded in the theory of LAGRANGE, his explanatory remarks 

and general narrative became ever more sparse until they finally ceased altogether. 

However, ABEL’S notebook is the only source illustrating how he planned to proceed, 

and I will try to reconstruct the central result of these investigations, which was never 

presented in a form intended for publication. 

Because ABEL’S argument, from this point onward, consists of little but equations, 

I have reconstructed how he could, with his tools and methods, have argued. In lim- 

iting myself to ABEL’S argument for the reduction of the general problem to Abelian 

equations, I remain close to the sources. ABEL’S unfinished manuscript inspired math- 

ematicians of the nineteenth century — such as C. J. MALMSTEN (1814–1886), SYLOW, 

and L. KRONECKER (1823–1891) — to elaborate and extend the investigation; 16 recently, 

L GÅRDING and C. SKAU have taken up the problem anew. 17 

SYLOW has speculated that ABEL recognized the insufficiency of his description 

of algebraic expressions. In response to his realization, ABEL should, according to 

SYLOW, have abandoned his attempt at presenting a manuscript ready for printing 

and instead recorded his further findings in the order and form in which he came to 

them. 18 As also noted by SYLOW, this change in style of presentation was not uncom- 

mon. In his notebooks, ABEL frequently started out writing coherent manuscripts, 

16 (Malmsten, 1847), (Sylow, 1861), (L. Sylow, 1902, 18–22), and (Kronecker, 1856). 

17 (Gårding, 1992) and (Gårding and Skau, 1994). 

18 (L. Sylow, 1902, 19).


which gradually turned into a sequence of formulae. 19 At a later time, when the ideas 

had matured and proofs had been improved, the results emerged in another manu- 

script or in print. 

In the third section of the Sur la résolution algébrique des équations, which was enti- 

tled “On the form of algebraic expressions which can satisfy an irreducible equation 

of a given degree”, ABEL reverted his approach once again. In the impossibility proof, 

he had fixed the equation (the general quintic) and sought to describe any algebraic 

expression which could satisfy it. In the opening part of the notebook manuscript, he 

had reversed this approach in order to describe the simplest equation which a given 

algebraic expression could satisfy; ABEL’S concept of simplicity was, of course, that of 

irreducibility. But in this third section, ABEL once again fixed the equation 

φ (y) = 0 (8.5) 

of degree µ and tried to analyse the form of any algebraic expression am of order m 

which could satisfy it. 

ABEL’S attention restricted to equations of prime degree. ABEL’S ambition had 

been to treat — in all its generality — all degrees µ. From his correspondence there 

is some indication that he made some progress in solving this general problem. 20 

However, the notebook manuscript only contains conclusive arguments concerning 

the simpler case in which µ was a prime. The pivotal tools in ABEL’S investigations 

were the results on the constructed irreducible equation, summed up in the proposi- 

tion 1 above, and his penetrating knowledge of properties of Abelian equations (see 

chapter 7). 

For ABEL, the first — and most important — consequence of assuming µ prime was 

to rewrite am in accordance with proposition 1:2. Writing s in place of ym above, he 

found 

am = 

µ−1 

∑ pks k=0 

k µ , 

which follows from the fact that the exponent of the outer-most root extraction in am 

had to divide µ. The proposition 1:3 furthermore stated that the other roots of (8.5) 

could be obtained by inserting ω u s 1 µ for s 1 µ . ABEL denoted 21 these µ roots z0, . . . , zµ−1, 

zu = 

µ−1 

∑ pkω k=0 

uk s k µ for 0 ≤ u ≤ µ − 1. 

Since each of these was a root of the equation (8.5), they had to remain unaltered when 

all the root extractions in p0, . . . , pµ−1, s were given all their respective possible values, 

ABEL argued from proposition 1:3. 

19 (L. Sylow, 1902, 8). 

20 (Abel→Holmboe, Berlin, 1827/03/04. N. H. Abel, 1902a, 57). 

21 I have chosen to enumerate them starting from zero, whereas ABEL began with the number 1. The 

benefit of my enumeration is simplicity of the subsequent formulae.


The coefficients p0, . . . , pµ−1 depended rationally upon s. In the following, ABEL 

investigated the dependency of the coefficients p0, . . . , pµ−1 upon s. ABEL linked the 

choice of other root extractions 22 in the expressions for p0, . . . , pµ−1 to permutations 

of the roots z0, . . . , zµ−1 in a way resembling the auxiliary theorem 3 of the impos- 

sibility proof. There, ABEL had used results obtained from permuting the roots to 

demonstrate that any radical occurring in a supposed solution formula would have 

to be a rational function of the roots of the equation. A similar result was needed in 

this context which was more general than the quintic studied in 1826. Although his 

explicit calculations took another form, the underlying ideas of the reworking remain 

the same. 

ABEL based his argument on letting 23 ˆp0, . . . , ˆpµ−1, ˆs denote any set of values of 

p0, . . . , pµ−1, s corresponding to choosing other roots of unity in the algebraic expres- 

sions for p0, . . . , pµ−1, s. The above argument ensuring that z0 was unaltered by other 

choices of root extractions, was summarized by ABEL as 

µ−1 

∑ pkω k=0 

uk s k µ = 

µ−1 

∑ 

k=0 

ˆp k ˆω uk ˆs k µ for 0 ≤ u ≤ µ − 1. 

Through a simple interchange of the order of summation, ABEL found that the first 

coefficient p0 was unaltered if another root extraction ˆs of s was chosen. Turning his at- 

tention to the quantities s and ˆs, he then — by a sequence of formulae — demonstrated 

that there existed an integer ν such that these quantities were related by the equation 

ˆs = p µ ν s v . (8.6) 

In the course of his deductions, ABEL introduced the further simplification p1 = 1 

which earlier led him into the mistaken assumptions on the degrees and orders of 

the coefficients in the impossibility proof (see sections 6.3.2 and 6.9.1). In the present 

situation, it had no negative implications, though. With this simplification, the roots 

z0, . . . , zµ−1 could be expressed as 

zu = p0 + ω u s 1 µ + 

µ−1 

∑ pkω k=2 

uk s k µ for 0 ≤ u ≤ µ − 1. 

Summing over the roots and using basic properties of primitive roots of unity, ABEL 

obtained 

s 1 µ−1 

1 

µ = 

µ ∑ 

k=0 

pus u µ−1 

1 

µ = 

µ 

∑ 

k=0 

ω −k z k, and 

ω −ku z k for 2 ≤ u ≤ µ − 1. 

22 By “choosing another root extraction”, I mean (in a general setup) choosing α n√ y for n√ y where α is 

an n th root of unity. 

23 ABEL wrote s ′ , w ′ , p ′ 0 , . . . , p′ µ−1 for the quantities I have denoted ˆs, ˆω, ˆp0, . . . , ˆpµ−1. I have altered his 

notation to make powers such as ˆs k µ more readable.


For any u > 1, ABEL had, therefore, explicitly demonstrated that pus was a rational 

function of the roots z0, . . . , zµ−1. 

The irreducible equation for s was Abelian. The ultimate result of ABEL’S studies 

of the solubility of equations amounted to a characterization of the irreducible equation 

P = 0 which the quantity s satisfied. By arguments founded in C. F. GAUSS’ (1777– 

1855) theory of primitive roots, ABEL found that P = 0 had the property of having all 

its roots representable as the “orbit” of a rational function (see page 145) whereby the 

equation fell into the category studied in the Mémoire sur une classe particulière. 24 

Denoting the degree of the irreducible equation P = 0 by ν, ABEL could express its 

ν roots in one of the two forms 

s or p µ m k1 s m k 1 for 1 ≤ k1 ≤ ν − 1 

where m k1 ∈ {2, 3, . . . , µ − 1}. He deduced this from (8.6) described above, since 

choosing any other root extraction would give an ˆs of the form p µ 

θ sθ . Fixing some m, a 

sequence could be constructed, possibly renumbering the coefficients p0, . . . , p k1−1, 

s1 = p µ 

0 sm , 

s2 = p µ 

1 sm 1 , 

. 

s k1 = p µ 

k 1−1 sm k 1−1 . 

At some point, the sequence would stabilize because only finitely many different roots 

of P = 0 could be listed. Assuming this to have occurred after the k th 

1 

which point the value could be assumed to be s again, ABEL wrote 

s = sk1 = p µ 

k1−1sm k1−1 = smk k1−1 1 

∏ p 

u=0 

µmu 

k1−(u+1) . 

Dividing this equation by s and extracting the µ th root, he obtained the relation 

s mk1 −1 

µ 

k1−1 ∏ p 

u=0 

mu 

k = 1. 

1−u−1 

iteration, at 

Since the product was a rational function of s by the previous result, ABEL concluded 

that the exponent of s would have to be integral 

24 (N. H. Abel, 1829c) 

m k 1 − 1 

µ 

= integer, 

or m k 1 ≡ 1 (mod µ) .


The central question of this part of the paper was whether k1 = ν, i.e. whether all 

the roots of P = 0 were found in the sequence above. ABEL answered this important 

question by a nice application of GAUSS’ primitive roots, although his presentation 

in the notebook becomes increasingly obscure (see figure 8.1). Eventually, nothing 

but a sequence of equations can be found. However, ABEL’S intended argument can 

be inferred and reconstructed. In the following, I add some explanation to ABEL’S 

equations based on arguments by HOLMBOE and SYLOW. 25 

In order to demonstrate that s 

1 1 

µ µ 

1 , . . . , sk 

were rational functions of s 

1 1 µ , ABEL let m 

denote a primitive root of the modulus µ and recast the procedure described above as 

1 

µ 

s1 

1 

µ 

s2 

= p0s mα 

µ , 

m 

= p1s 

α 

µ 

1 , 

1 

µ 

sk 

= pk−1s mα 

µ . 

At some point, say after the k th iteration, the procedure would stabilize and give 

s 1 m 

µ = s αk k−1 

µ × ∏ p 

u=0 

muα 

k−u−1 . 

By the same argument as above, ABEL could write 

m αk − 1 

µ 

. 

(8.7) 

= integer, (8.8) 

and he concluded that k divided µ − 1. This conclusion can be seen to impose a mini- 

mality condition upon k with respect to (8.8). However, in ABEL’S equations no men- 

tion of such a minimality requirement can be found. The congruence (8.8) 

led ABEL to introduce n such that 

m αk ≡ 1 (mod µ) 

αk = (µ − 1) n. 

In subsequent reasoning, ABEL repeatedly used the fact that (k, n) = 1 without going 

into details. However, it is a consequence of the minimality of k mentioned above. 

Through a sequence of deductions based on primitive roots and congruences inspired 

by GAUSS, ABEL could link a number β to the sequence (8.7) such that 

βk = µ − 1. 

25 (Holmboe in N. H. Abel, 1839, vol. 2, 288–293), (Sylow in N. H. Abel, 1881, vol. 2, 329–338), and (L. 

Sylow, 1902, 18–22).


If any root existed outside the sequence (8.7), a sequence could be based on this 

root, and a similar deduction would produce another pair of integers β ′ , k ′ related by 

β ′ k ′ = µ − 1. 

However, as ABEL demonstrated, from two such sequences a third one corresponding 

to β ′′ = gcd (β, β ′ ) could also be constructed with the same property 

β ′′ k ′′ = µ − 1. 

ABEL knew that if β = β ′ , the two initial sequences were not distinct. If the two 

initial sequences were assumed to be maximal, a contraction was obtained, since the 

sequence corresponding to β ′′ was longer than both the initial sequences. 

Thus, ABEL had demonstrated that the assumption of a root existing outside the 

maximal sequence (8.7) led to a contradiction, and therefore all the roots were located 

in a single chain. Using the same notation as in the Mémoire sur une classe particulière, 

ABEL wrote the set of roots of P = 0 as 

s, θ (s) , θ 2 (s) , . . . , θ ν−1 (s) , where θ ν (s) = s, 

and the equation P = 0 was seen to be a specimen of the class of equations which have 

become known as Abelian equations (see chapter 7). 

The first result of ABEL’S research had been to reduce the search for algebraic ex- 

pressions satisfying an arbitrary equation to the search for expressions satisfying an 

irreducible one. As SYLOW remarks, 26 the present investigation had led to the fur- 

ther restriction to studying only the possible solutions to irreducible Abelian equations 

whose degree divided µ − 1. The desired complete characterization of expressions 

solving irreducible Abelian equations was, however, not undertaken in the notebook 

study. 

8.4 Further ideas on the theory of equations 

Besides the described reduction to Abelian equations, the notebook manuscript and 

ABEL’S letters contain other interesting results one of which addressed the form of 

roots of solvable equations. This result can be seen as an elaboration and rigorization 

of one of L. EULER’S (1707–1783) claims. 

The form of roots of solvable equations: rigorizing EULER. At the end of the in- 

vestigation of possible solutions in the notebook, ABEL found that if an equation was 

solvable by radicals, its solution would be based on the relationship 

26 (L. Sylow, 1902, 21). 

1 

µ 

sk 

= A ν−1 

i ∏ 

u=0 

m 

a 

kuα 

µ 

u 

for 0 ≤ k ≤ ν − 1,

8.4. Further ideas on the theory of equations 177 

Figure 8.1: One of the last pages from ABEL’S notebook manuscript on algebraic solubility 

(Abel, MS:696, 66). Reproduced from (N. H. Abel, 1902e, facsimile III)


where a0, . . . , aν−1 were roots of an irreducible Abelian equation of degree ν and the 

coefficients A i were rational expressions in s. The root z0 of the initial equation was in 

turn given from the sequence s 

1 

µ 

0 

, . . . , s 

z0 = p0 + 

1 

µ 

ν−1 

ν−1 

∑ 

u=0 k=0 

by a relationship of the form 

ν−1 

∑ 

m 

φu (sk) · s 

u 

µ 

k , 

where φ0, . . . , φν−1 were rational functions. In a letter to CRELLE dated 1826, ABEL 

had announced a result for equations of the fifth degree which was a particular case 

of the above. 

“When an equation of the fifth degree, whose coefficients are rational numbers, 

is algebraically solvable, one can always give its roots the following form: 

x = c + A · a 1 5 · a 2 5 

1 · a 4 5 

2 · a 3 5 

3 + A1 · a 1 5 

1 · a 2 5 

2 · a 4 5 

3 · a 3 5 

where 

� 

a = m + n 1 + e2 + 

� 

a1 = m − n 

� 

a2 = m + n 

+ A2 · a 1 5 

2 · a 2 5 

3 · a 4 5 · a 3 5 

1 + A3 · a 1 5 

3 · a 2 5 · a 4 5 

1 · a 3 5 

2 

� 

h 

� 

1 + e2 + h 

� 

1 + e2 − h 

� 

� 

a3 = m − n 1 + e2 + 

A = K + K ′ a + K ′′ a2 + K ′′′ aa2, 

A1 = K + K ′ a1 + K ′′ a3 + K ′′′ a1a3, 

A2 = K + K ′ a2 + K ′′ a + K ′′′ aa2, 

A3 = K + K ′ a3 + K ′′ a1 + K ′′′ a1a3. 

h 

� 

1 + e2 + � 

1 + e2 � 

, 

� 

1 + e2 − � 

1 + e2 � 

, 

� 

1 + e2 + � 

1 + e2 � 

, 

� 

1 + e2 − � 

1 + e2 � 

, 

The quantities c, b [h], e, m, n, K, K ′ , K ′′ , K ′′′ are all rational numbers. 

In this way, however, the equation x 5 + ax + b = 0 cannot be solved as long as 

a and b are arbitrary quantities.” 27 

Probably from his realization that all quantities involved in the solution are ratio- 

nals, square roots of rationals, or fifth roots of rationals, ABEL concluded that there 

were values of a and b for which the equation x 5 + ax + b = 0 could not be solvable by 

27 “Wenn eine Gleichung des fünften Grades, deren Coëfficienten rationale Zahlen sind, algebraisch 

auflösbar ist, so kann man immer den Wurzeln folgende Gestalt geben: 

x = c + A · a 1 5 · a 2 5 

1 · a 4 5 

2 · a 3 5 

3 + A1 · a 1 5 

1 · a 2 5 

2 · a 4 5 

3 · a 3 5 

+ A2 · a 1 5 

2 · a 2 5 

3 · a 4 5 · a 3 5 

1 + A3 · a 1 5 

3 · a 2 5 · a 4 5 

1 · a 3 5 

2

8.4. Further ideas on the theory of equations 179 

radicals. In this way, the insolubility of fifth degree equations of the standard form 28 

x 5 + ax + b = 0 was demonstrated directly: If the equation had been solvable, ABEL 

possessed a solution formula, which he saw was not powerful enough to give the 

solution of arbitrary equations. 

In a letter to HOLMBOE from the same year, the result on the form of roots was 

given another twist. 

“Concerning equations of the 5 th degree I have found that whenever such an 

equation can be solved algebraically, the root must have the following form: 

x = A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′ 

where R, R ′ , R ′′ , R ′′′ are the 4 roots of an equation of the 4 th degree and have the 

property that they can be expressed with help of only square roots. — It has been 

a difficult task for me with respect to expressions and notation.” 29 

In this form, the statement is a refined version of EULER’S “conjecture” that the 

solution of the fifth degree equation should be of the form 

A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′ (8.9) 

where R, R ′ , R ′′ , R ′′′ were solutions to an equation of the fourth degree (see section 

5.1). 

wo 

� 

a = m + n 1 + e2 + 

� 

a1 = m − n 

� 

a2 = m + n 

� 

a3 = m − n 1 + e2 + 

� 

� 

h 1 + e2 + � 

1 + e2 � 

, 

1 + e2 � � 

+ h 1 + e2 − � 

1 + e2 � 

, 

1 + e2 � � 

− h 1 + e2 + � 

1 + e2 � 

, 

� � 

h 1 + e2 − � 

1 + e2 � 

, 

A = K + K ′ a + K ′′ a2 + K ′′′ aa2, 

A1 = K + K ′ a1 + K ′′ a3 + K ′′′ a1a3, 

A2 = K + K ′ a2 + K ′′ a + K ′′′ aa2, 

A3 = K + K ′ a3 + K ′′ a1 + K ′′′ a1a3. 

Die Grössen c, b, e, m, n, K, K ′ , K ′′ , K ′′′ sind alle rationale Zahlen. 

Auf diese Weise lässt sich aber die Gleichung x 5 + ax + b = 0 nicht auflösen, so lange a und b 

beliebige Grössen sind.” (Abel→Crelle, Freyberg, 1826/03/14. N. H. Abel, 1902a, 21–22). 

28 If formulated in positive way, the researches of JERRARD (see section 6.9.1) demonstrated that every 

fifth degree equation could be transformed to this normal trinomial form. (W. R. Hamilton, 1839, 

251) 

29 “Med Hensyn til Ligninger af 5th Grad har jeg faaet at naar en saadan Ligning lader sig løse algebraisk 

maa Roden have følgende Form: 

x = A + 5√ R + 5√ R ′ + 5√ R ′′ + 5√ R ′′′ 

hvor R, R ′ , R ′′ , R ′′′ ere de 4 Rødder af en Ligning af 4de Grad, og som lade sig udtrykke blot ved 

Hjelp af Qvadratrødder. — Det har været mig en vanskelig Opgave med Hensyn til Udtryk og 

Tegn.” (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a, 45).


ABEL had turned the argument around and demonstrated that, although not all 

fifth degree equations were algebraically solvable, those which were all had solutions 

of the form (8.9). A particular instance of this result had already been obtained for 

Abelian equations in the Mémoire sur une classe particulière as recorded in (7.10). As a 

result of this inversion of argument, EULER’S hypothesis can be seen as a bold conjec- 

ture which ABEL later turned into a proof through a restriction on the class of objects 

dealt with. Where EULER had been concerned with the class of all fifth degree equa- 

tions, ABEL restricted (barred) his results on the form of roots to only those equations 

which were algebraically solvable. 

An extension of the class of Abelian equations. In a later letter to CRELLE, writ- 

ten around the same time as the notebook entry, i.e. 1828, ABEL announced further 

results in the theory of equations. Generalizing the assumptions on the rational corre- 

spondences between roots of an irreducible equation sufficient to guarantee solubility, 

ABEL had found: 

“If three roots of an irreducible equation of a certain prime degree have such a 

relation between them that one can express one of the roots rationally in the two 

others, the equation under consideration will always be solvable by radicals.” 30 

As SYLOW has noticed, the assumption on the rational relationship among the 

three roots is not quite clear: The mathematical correct assumption is that all the roots 

of the equation can be expressed rationally if any two among them are considered 

known. 31 In the form of a corollary to his result, ABEL gave the result contained in 

the Mémoire sur une classe particulière that if two roots of an irreducible equation of 

prime degree were rationally related, the equation would be algebraically solvable. 

Although this indicates that ABEL had, at the time of writing the Mémoire sur une 

classe particulière, the result on the solubility of irreducible equations of prime degree 

in which any root can be written as 

x i = θ i (x0, x1) 

at his disposal, he never made the more general result public in print. 

This class of equations, which ABEL saw contained the so-called Abelian ones, was 

taken up by E. GALOIS (1811–1832) after whom they are now named. Within his the- 

ory (see chapter 8.5), GALOIS stated the theorem that it was a necessary and sufficient 

condition for algebraic solubility that “if some two of the roots of an irreducible equation 

of prime degree are considered known, the others can be expressed rationally.” 32 

30 “Si trois racines d’une équation quelconque irreductible d’un degré marqué par un nombre premier 

sont liées entre elles de la manière que l’on pourra exprimer l’une de ces racines rationellement en 

les deux autres, l’équation en question sera toujours resoluble à l’aide de radicaux.” (Abel→Crelle, 

Christiania, 1828/08/18. N. H. Abel, 1902a, 73). 

31 (L. Sylow, 1902, 17). 

32 “Théorème. Pour qu’une équation irréductible de degré premier soit soluble par radicaux, il faut et 

il suffit que deux quelconques des racines étant connues, les autres s’en déduisent rationnellement.” 

(Galois, 1831c, 69).

8.5. General resolution of the problem by E. GALOIS 181 

The major parts of ABEL’S research on equations which can be rendered intelligible 

have been presented above. Nevertheless, ABEL’S notebooks are filled with notes and 

scribbles for additional research which he never translated into a finished form suit- 

able for presentation. During the few remaining years of his life, ABEL became pre- 

occupied with other mathematical topics. Thus, we can only wonder what he might 

have achieved, had he returned to the theory of solubility per se. 

8.5 General resolution of the problem by E. GALOIS 

ABEL’S attempt at a general theory of the algebraic solubility of equations was not 

published until the first edition of the Œuvres 1839. Hence, it is most likely that GA- 

LOIS was unaware of ABEL’S general research when he wrote down his theory in the 

early 1830s. GALOIS knew the published works of LAGRANGE and A.-L. CAUCHY 

(1789–1857), and he had probably read ABEL’S two publications on the theory of equa- 

tions — the impossibility proof of 1826 and the Mémoire sur une classe particulière pub- 

lished 1829 33 — as well as ABEL’S more widely known works on the theory of elliptic 

functions, the Recherches sur les fonctions elliptiques 34 and the Précis d’une théorie des 

fonctions elliptiques 35 . 36 GALOIS “vehemently denied” 37 dependence on ABEL as can 

be seen from the fragmentary Note sur Abel, 38 but undeniably they share many of their 

inspirations. In section 8.5.1, I briefly describe GALOIS’ unified theory before I com- 

ment upon the common inspiration and central problems shared in the works of ABEL 

and GALOIS (section 8.5.2). 

The turbulent life of EVARISTE GALOIS as well as the interplay between his life 

and the fate of his mathematics have been studied intensively. 39 GALOIS’ theory of 

algebraic solubility was not made public to the mathematical community except for a 

small group of members of the Institut de France until J. LIOUVILLE (1809–1882) pub- 

lished selections from GALOIS’ mathematical manuscripts in the Journal de mathéma- 

tiques pures et appliquées in 1846. 40 Subsequently, many mathematicians in the sec- 

ond half of the nineteenth century invested great efforts in incorporating GALOIS’ 

at times fragmentary and non-rigorous mathematics into the new standards of clar- 

ity and rigour. The process made mathematicians like KRONECKER return to ABEL’S 

works and manuscripts (see section 6.9.2), but was largely an enterprise of digesting 

GALOIS’ work. Therefore, the reception of GALOIS’ theory is not the primary concern 

33 (N. H. Abel, 1826a; N. H. Abel, 1829c) 

34 (N. H. Abel, 1827b; N. H. Abel, 1828b) 

35 (N. H. Abel, 1829d) 

36 (Wussing, 1969, 75). 

37 (Kiernan, 1971, 90). 

38 (Galois, 1831b). 

39 For instance (Wussing, 1975), (Rothman, 1982), or (Toti Rigatelli, 1996). 

40 (Lützen, 1990, 559–580).


Figure 8.2: EVARISTE GALOIS (1811–1832) 

in the present context, 41 which focuses on the differences and similarities between the 

almost concurrent works of ABEL and GALOIS. 

8.5.1 The emergence of a general theory of solubility 

In a sequence of manuscripts, GALOIS attacked the same two problems as ABEL had 

suggested in order to describe the extension of algebraic solubility (see section 8.1). 

ABEL had attempted to solve the first problem — that of finding all solvable equations 

of a given degree — in his notebook manuscript described in this chapter. ABEL’S sec- 

ond question concerning the determination of whether a given equation was algebra- 

ically solvable or not was the direct purpose of GALOIS’ theory. GALOIS intended to 

give characterizations of solubility which could, at least in principle, be used to decide 

the solubility of any given equation, but the machinery needed for actually determining 

the solubility of given equations was of lesser interest to him. 42 

The important feature of GALOIS’ theory was to associate a structure called a group 

to any given equation such that the question of solubility of equations could be trans- 

lated into questions concerning these structures. Although the concept of group only 

saw its first instances and was not a developed abstract concept in the works of GA- 

41 It has been dealt with extensively in the literature, for instance (J. Pierpont, 1898), (Kiernan, 1971), 

(Hirano, 1984), (Scholz, 1990), or (Martini, 1999). 

42 (Kiernan, 1971, 83).


LOIS, he was instrumental in bringing about the structural approach to mathematics, 

which came to dominate much of 20 th century mathematics. 43 

GALOIS’ work was, as he himself somewhat laconically remarked, 44 founded in 

the theory of permutations most of which he had taken over from CAUCHY. GALOIS 

considered an equation of degree m 

φ (x) = 0 

having the roots x1, . . . , xm, and claimed that the group of the equation G — later called 

the Galois group — could always be found, which had the following two properties: 

1. that every function of the roots x1, . . . , xm which was (numerically) invariant un- 

der the substitutions of G was rationally known, and conversely, 

2. that every rational function of the roots x1, . . . , xm was invariant under the sub- 

stitutions of G. 

GALOIS took over the concept of rationally known from LAGRANGE but changed 

the notion of invariant to stress numerical invariance instead of LAGRANGE’S formal in- 

variance in order to deal with special (i.e. non-general) equations. However, GALOIS’ 

proof of the existence of the group of the equation suffered from the unclear character 

of his concept of invariance. 45 

Although the concepts of permutation and substitution underwent some uncom- 

pleted changes in GALOIS’ manuscripts, he clearly perceived the multiplicative nature 

of substitutions — understood as transitions from one arrangement (permutation) to 

another — as well as the multiplicative closure of the GALOIS group. 

“It is clear in the group of permutations under consideration, the arrangement 

of letters is not important, but only the substitutions on the letters, by which we 

move from one permutation to another. Thus, if in similar group one has the 

substitutions S and T, one is also certain to have the substitution ST.” 46 

The second component of GALOIS’ theory addressed the reduction of the group 

of an equation by the adjunction of quantities to the set of rationally known quantities. 

By adjoining to the rationally known quantities a single root of an irreducible aux- 

iliary equation, GALOIS could decompose the group of the equation into a number, 

p, of subgroups. These had the remarkable property that applying a substitution to 

43 These aspects of GALOIS’ work have been studied by, for instance, (Wussing, 1969) and (Kiernan, 

1971). 

44 (Galois, 1830, 165). 

45 (Kiernan, 1971, 80–81). 

46 “Comme il s’agit toujours de questions où la disposition primitive des lettres n’influe en rien, dans 

les groupes que nous considérons, on devra avoir les mêmes substitutions quelle que soit la permutation 

d’où l’on sera parti. Donc si dans un pareil groupe on a les substitutions S et T, on est sûr 

d’avoir la substitution ST.” (Galois, 1831c, 47). I have extended the translation found in (Kiernan, 

1971, 80).


permutations in one of the subgroups gave the permutations of another subgroup. 47 

When GALOIS adjoined the entire set of roots of the irreducible auxiliary equation, he 

obtained an even more remarkable result: 

“Theorem. If one adjoins to an equation all the roots of an auxiliary equation, 

the groups in question in theorem II [i.e. the p subgroups mentioned above] 

will furthermore have the property that the substitutions are the same in each 

group.” 48 

Of this important theorem GALOIS gave no proof, but hastily remarked “the proof 

will be found.” 49 The contents of the theorem is GALOIS’ characterization of the defin- 

ing property of what was be called normal subgroups, since GALOIS’ statement corresponds 

to saying that all the conjugate classes of a subgroup U are identical. 50 

The link between properties of the decomposition into normal subgroups of the 

group of the equation and the algebraic solubility of the equation was provided in the 

far-reaching fifth problem of the manuscript. Using modern concepts and terms, it 

can be summarized as follows. Assuming that the equation under consideration had 

the group G, and that p was the smallest prime divisor of the number of permutations 

in G, GALOIS argued that the equation could be reduced to another equation having 

a smaller group G ′ whenever a normal subgroup N existed in G with index p. Fur- 

thermore, the link with algebraic solubility was provided when GALOIS stated that the 

equation would be solvable in radicals precisely when its group could be decomposed 

into the trivial group by iterated applications of the preceding principle. 51 

GALOIS applied the general result on algebraic solubility in two ways to obtain 

important characterizations of solubility of equations. First, he sought criteria for sol- 

ubility of irreducible equations of prime degree and found the following: 

“Thus, for an irreducible equation of prime degree to be solvable by radicals it 

is necessary and sufficient that any function which is invariant under the substitutions 

x k x ak+b 

[a and b are integer constants] is rationally known.” 52 

47 (Galois, 1831c, 55). 

48 “Théorème. Si l’on adjoint à une équation toutes les racines d’une équation auxiliaire, les groupes 

dont il est question dans le théorème II jouiront de plus de cette propriété que les substitutions sont 

les mêmes dans chaque groupe.” (ibid., 57). 

49 “On trouvera la démonstration.” (ibid., 57). 

50 (Scholz, 1990, 384). 

51 (ibid., 384–385). 

52 “Ainsi, pour qu’une équation irréductible de degré premier soit soluble par radicaux, il faut et il 

suffit que toute fonction invariable par les substitutions 

x k x ak+b 

soit rationnellement connue.” (Galois, 1831c, 69).


Thus, GALOIS had characterized solvable irreducible equations of prime degree p 

by the necessary and sufficient requirement that their GALOIS group contained nothing 

but permutations corresponding to the linear congruences 53 

i → ai + b (mod p) where p ∤ a. (8.10) 

From this characterization of solubility, GALOIS deduced a second one which ABEL 

had also hit upon (see section 8.4), when he demonstrated his eighth proposition: 

“Theorem. For an equation of prime degree to be solvable by radicals it is necessary 

and sufficient that any two of its roots being known, the others can be deduced 

rationally from them.” 54 

The character of GALOIS’ reasoning often left quite a lot to be desired. When LI- 

OUVILLE eventually published GALOIS’ manuscripts, he accompanied them with an 

evaluation of GALOIS’ clarity and rigour: 

“Clarity is indeed an absolute necessity. [. . . ] Galois too often neglected this 

precept.” 55 

In making GALOIS’ new ideas available to the mathematical community and in 

providing proofs and elaborations of obscure points, mathematicians of the second 

half of the nineteenth century invested much effort in the theory of equations, per- 

mutations, and groups. Although GALOIS had found out how the solubility of a 

given equation could be determined by inspecting the decomposability of its asso- 

ciated group into a tower of normal subgroups, a number of points were left open 

for further research. To mathematicians around 1850, three problems were of primary 

concern: GALOIS’ construction of the group of an equation was considered to be unrig- 

orous, no characterization of the important solvable groups had been carried out, and a 

certain arbitrariness of the order of decomposition also remained. These matters were 

cleared, one by one, until the theory ultimately found its mature form in the abstract 

field theoretic formulation of H. WEBER (1842–1913) and E. ARTIN (1898–1962). 56 

8.5.2 Common inspiration and common problems 

As mentioned earlier (p. 181), GALOIS and ABEL drew extensively on common sources. 

The ideas of invariance under permutations of the roots, founded in LAGRANGE’S 

work, 57 were important to both of them; and they both relied on the general theory 

53 (Scholz, 1990, 385). 

54 “Théorème. Pour qu’une équation de degré premier soit soluble par radicaux, il faut et il suffit que 

deux quelconques des racines étant connues, les autres s’en déduisent rationnellement.” (Galois, 

1831c, 69). 

55 (Liouville quoted from Kiernan, 1971, 77). 

56 (ibid.) and (Scholz, 1990, 392–398). 

57 (Lagrange, 1770–1771)


of permutations and notations which CAUCHY had developed. 58 GALOIS’ investiga- 

tions, however, took a different approach from the one ABEL had employed even in 

his attempt at a general theory of solubility. GALOIS’ decisive step of relating a certain 

group to an equation and transforming the investigation of solubility of the equation 

into investigating properties of the group was as distant from ABEL as it was from 

LAGRANGE. But although ABEL’S research on algebraic solubility did not appear un- 

til 1839, it might have helped the mathematical community understand the purposes 

and intentions of GALOIS’ difficult manuscripts. As H. WUSSING has put it, ABEL 

would probably have been the only person with the capacity to immediately under- 

stand GALOIS’ works, but unfortunately ABEL had died before GALOIS ever wrote 

down his manuscripts. 59 

A common component of central importance to both ABEL and GALOIS was the 

concept of irreducibility (see section 7.3). In one of his manuscripts, GALOIS defined an 

equation to be reducible whenever it had rational divisors, and irreducible otherwise. 60 

This definition closely resembles the one given by ABEL, who had been more explicit 

about the rationality of the divisor, though. The first theorem on irreducible equations, 

which ABEL proved, can also be found in GALOIS’ manuscripts: 

“Lemma I. An irreducible equation cannot have any root in common with another 

rational equation without dividing it.” 61 

Of this lemma GALOIS gave no proof, but he used the concept and lemma ex- 

tensively. Through GALOIS the concept of irreducibility in its present sense finally 

entered algebra as a central concept upon which deductions could be built. 

ABEL’S investigations had led to abstract and not easily applicable results con- 

cerning solubility. The same is true for GALOIS’ approach — even to a larger extent. 

ABEL’S positive criteria of solubility of, for instance, Abelian equations concerned cer- 

tain relationships existing among the unknown roots of an equation. In case nothing 

but the coefficients of the equation was known, this approach had no chance of pro- 

ducing an answer to the question of the solubility of the equation. In GALOIS’ concept 

of the group of an equation, this non-constructive approach is carried to an extreme. 

GALOIS had tried to prove that such a group always existed, but did not address the 

question of how to construct it. He had presented his thoughts in a sequence of mem- 

oirs, one of which he had handed in to the Institut de France in January 1831. The 

reviewers, S. F. LACROIX (1765–1843) and S.-D. POISSON (1781–1840), immediately 

noticed this “deficiency” and allowed it to play a role in their refusal: 

“[. . . ] it should be noted that [the theorem] does not contain, as the title would 

have the reader believe, the condition of solubility of equations by radicals. [. . . ] 

58 (A.-L. Cauchy, 1815a; A.-L. Cauchy, 1815b) 

59 (Wussing, 1975, 397). 

60 (Galois, 1831c, 45). 

61 “Lemme I. Une équation irréductible ne peut avoir aucune racine commune avec une équation 

rationnelle sans la diviser.” (ibid., 47).


This condition, if it exists, should have an external character, that can be tested 

by examining the coefficients of a given equation, or, at most, by solving other 

equations of lesser degree than that proposed. We made all possible efforts to 

understand M. Galois’ evidence. His thesis is neither clear enough, nor sufficiently 

developed to enable us to judge its rigour.” 62 

The interplay between the theory of equations and the flourishing theory of ellip- 

tic functions had been essential in ABEL’S approach (see section 7.2). The division of 

elliptic functions had given rise to certain classes of equations described by relations 

among the roots, and ABEL had pursued his favorite subject, the theory of equations 

(see the quotation on page 160), in investigating the question of algebraic solubility of 

these equations. Although not to the same extent engaged in research on elliptic func- 

tions, GALOIS also saw the modular equations of elliptic functions as an important 

application of and inspiration for his theory of solubility. After GALOIS was expelled 

from the École Normale in 1831, he offered classes on, among other subjects of algebra, 

“elliptic functions treated as pure algebra”, 63 presumably dealing with the subject in 

a way similar to ABEL’S approach. In the 1831-manuscript, 64 GALOIS gave a general 

solution to the division problem concerning the division of an elliptic function of the 

first kind into p n equal parts, where p was a prime. The central step of the proof was 

given by his result that any rational function which is unaltered by linear congruence 

substitutions of the form (8.10) is known. Just as ABEL had generalized his interest in 

elliptic functions into the integration theory of algebraic functions, GALOIS’ investi- 

gations took a similar turn, and a large part of his manuscripts concerned this theory. 

The creation of Galois Theory in many ways marked the transition into modern 

mathematics. The concept of group was implicitly introduced by GALOIS, and he ex- 

plicitly gave it its name; but more importantly, GALOIS’ revolutionary attitude toward 

explicit arguments in mathematics marked a transition from arguments based on ma- 

nipulations of formal expressions to more concept based deductions. To many nine- 

teenth century mathematicians this transition — together with the fragmentary and 

hasty character of GALOIS’ arguments — rendered the new results “vague”, faulty, or 

at least in need of elaboration and proof. 65 The transition proved to be irreversible, 

though, and concept based mathematics was the mathematics of the future. 

62 (Toti Rigatelli, 1996, 90). 

63 (ibid., 79–80). 

64 (Galois, 1831a) 

65 (Kiernan, 1971, 59).

Part III 

Interlude: ABEL and the ‘new rigor’ 

189

Chapter 9 

The nineteenth-century change in 

epistemic techniques 

Of the numerous transitions in epistemic techniques — changes in how mathematics 

was conducted — which took place in the 1820s, few were as far reaching as the ini- 

tiation of the movement aiming at rigorizing analysis through arithmetization. The 

rigorization of analysis involved fundamental changes in the basic concepts of the 

discipline and also manifested itself on the technical level. The causal events leading 

to the rigorization are varied and span both external and internal factors. However, 

it is no coincidence that the rigorization was originally promoted in textbooks which 

were needed for the large-scale instruction in mathematics brought about by external 

events. 

Critical revision: A change in epistemic techniques. Central to the replacement 

of existing practice was the prominence given by leading research mathematicians 

to the rigorization program and the critical revision. J. L. LAGRANGE’S (1736–1813) 

textbooks marked a new awareness concerning the foundations of the calculus — later 

rigorization built upon the Lagrangian program. 

In the 1820s, A.-L. CAUCHY (1789–1857) presented his revision of the foundations 

of analysis which meant ingeniously revising the basic notions of the discipline. At 

the core of the change, CAUCHY discarded the eighteenth century conception of for- 

mal equality between expressions in favor of a new concept of arithmetical equality 

between functions. This change had implications for most of the other basic notions: 

limits, convergence, continuity, and differentiability to name but a few. For instance, 

CAUCHY was led by his new rigor to abandon attributing meaning to sums of di- 

vergent series and to promote tests of convergence into central positions within his 

theoretical framework. 

By the mid-1820s, N. H. ABEL (1802–1829) expressed severe concerns for the con- 

temporary state of the calculus: he felt that it lacked system and rigor. Simultaneously, 

ABEL revealed his interest in finding out how the previous generations could have ob- 

191

192 Chapter 9. The nineteenth-century change in epistemic techniques 

tained correct results from their “unrigorous” foundations. This question represents 

another aspect of the critical revision; an aspect which is intimately tied to the percep- 

tion of cumulativity in mathematics. 

Toward a concept based version of analysis. One of the main achievements of CAUCHY’S 

new rigor was the new internal relationship between definitions, theorems and proofs. 

Not only did CAUCHY promote notions such as limits into the core concepts of analysis 

he also put his definitions into direct use in forming new concepts, e.g. convergence 

and continuity, and in proving theorems about these concepts. Thus, CAUCHY’S defi- 

nitions in certain senses continued existing trends but were more concrete and appli- 

cable in stating and demonstrating theorems. Furthermore, CAUCHY’S critical revi- 

sion forced him to restructure the network of definitions and theorems changing the 

internal fabric of the theory. 

Interpreted in the framework of the nineteenth century transition toward concept 

based mathematics, the rigorization of analysis thus provides an important example in 

which both the structure and the techniques of a discipline underwent deep changes. 

In the following chapters, the general transition is described and analyzed from 

the perspective of ABEL. ABEL’S impact on the rigorization program mainly consists 

of three themes: 

1. A very critical attitude which was mainly expressed in letters. 

2. A proof of the binomial theorem which surpassed its predecessors in generality 

and rigor. 

3. A discussion on the existence of general criteria of convergence. 

All three themes fall in the changing standard of analysis which was brought about 

by CAUCHY’S revision of the discipline. Therefore, important aspects of this context 

must first be described.

Chapter 10 

Toward rigorization of analysis 

From the time the calculus emerged in the 17 th century until the end of the 18 th cen- 

tury, mathematicians and philosophers were wary when confronted with questions 

concerning its foundations. To some extent ignoring foundational questions, mathe- 

maticians focused on creating new results which could be useful in answering inter- 

esting questions, for instance in the field of mathematical physics. To some mathe- 

maticians working toward the end of the 18 th century, rigorously founding the calcu- 

lus remained one of the few open problems; but one of relatively lesser importance 

than the development of new analytical results. 1 To others, primarily J. L. LAGRANGE 

(1736–1813), the foundations of the calculus became a prestigious mathematical research 

problem. 2 

The transformation of concepts, theorems, and proofs in the process of rigorization 

in analysis have been subject to a variety of historical enquiries; in the following, em- 

phasis is given to establishing and illustrating certain ideas and developments which 

are of importance in subsequent chapters. 3 

10.1 EULER’s vision of analysis 

To understand the revision and the contents of the refocus on rigor, some aspects of 

eighteenth century analysis are of key importance. In particular, the results and tech- 

niques of L. EULER (1707–1783) dominated the way mathematicians worked in the 

field for half a century. 

Focus on functions and formal equality. Beginning with his influential monograph 

Introductio in analysin infinitorum, 4 EULER promoted functions to become the basic ob- 

1 See for instance the quotations in section 3.3 frequently invoked to document a belief in the stagnation 

of the mathematical sciences. 

2 For LAGRANGE’S algebraic approach to the calculus, see e.g. (Grabiner, 1990); for its influence on 

CAUCHY, see (Grabiner, 1981b). The best general presentation of the development of analysis in the 

nineteenth century is, I think, (Bottazzini, 1986). 

3 For the evolution of rigorization in analysis, see e.g. (Bottazzini, 1986; Jahnke, 1999; Lützen, 1999). 

4 (L. Euler, 1748). 

193

194 Chapter 10. Toward rigorization of analysis 

jects of analysis. EULER’S definitions and use of functions have attracted the interest 

of historians of mathematics. 5 In the present context, the two most important aspects 

of EULER’S approach are: 

1. EULER’S variable quantities were universal in the sense that they would “com- 

prise all determinate values” including positive and negative, rational and irra- 

tional, and real and imaginary values. 

2. EULER defined a function of a variable quantity to be an “analytic expression 

composed in any way from the variable quantity and numbers or constant quan- 

tities”. The operations allowed to form analytic expressions were algebraic opera- 

tions, both finite and infinite. 

Together, these two aspects entail an important interpretation of the concept of 

equality between functions. To EULER, two analytic expressions were considered 

equal if one could be transformed into the other by a sequence of (formal) manip- 

ulations. For instance, in developing methods for expanding rational functions into 

power series, EULER described — in the Introductio — a method by which the two ex- 

pressions 

1 

1 − x and 

∞ 

∑ x 

n=0 

n 

should be considered equal because the latter could be obtained by (formally) carrying 

out the division. 6 Of course, EULER was aware that peculiar results would emerge if 

certain numerical values were inserted for x and the equality was believed to apply to 

this numerical case as well. The proper interpretation of the sum 

1 − 1 + 1 − 1 + . . . 

had been a controversial subject throughout the first half of the eighteenth century. 

To EULER, its sum would be 1 2 by the formal equality above. Generally, EULER chose 

to focus on the formal aspect of functional equalities ignoring the “paradoxes” which 

might occur if numerical values were inserted. 

To a modern reader, EULER’S disregard for numerical convergence may seem odd. 

However, it corresponds to a paradigm in analysis — the Euclidean paradigm — which 

focused on the fruitful manipulations of finite or infinite expressions; the un-problematic 

transition from one such representation to another constituted a cornerstone of EU- 

LER’S skillful investigations in analysis. 

5 See e.g. (Jahnke, 1999; Lützen, 1978; Youschkevitch, 1976). 

6 (L. Euler, 1748, §60–61).

10.1. EULER’s vision of analysis 195 

10.1.1 The binomial theorem 

In its various forms and various degrees of specialization, i.e. various restrictions on 

m and x, the binomial theorem asserts the equality 

(1 + x) m = 1 + m 

1 

m (m − 1) 

x + x 

1 · 2 

2 + 

m (m − 1) (m − 2) 

x 

1 · 2 · 3 

3 + . . . . 

The theorem became one of the pivotal points of analysis since it was first employed as 

a heuristic tool by I. NEWTON (1642–1727) to obtain series expansions for expressions 

such as (1 + x) 1 2 . For integral exponents (m ∈ N), the binomial theorem reduced to 

the well known — and firmly established — binomial formula 

(1 + x) m = 

m 

∑ 

n=0 

� m 

n 

� 

x n for m ∈ N. 

NEWTON used extrapolation from the cases of integral exponents to obtain the equal- 

ity of the finite and infinite expressions in situations corresponding to fractional ex- 

ponents (e.g. m = 1 2 , above). In the eighteenth century, the binomial theorem was 

provided with various proofs. 7 

To further illustrate the Eulerian paradigm in analysis, the role played by the bino- 

mial theorem within EULER’S structuring of analysis provides many interesting hints; 

furthermore, that theorem is of direct importance in understanding the way analysis 

was reorganized in the early nineteenth century. 

EULER’S first proof of the binomial theorem: the link with Taylor series. In the In- 

troductio, EULER gave no general proof of the binomial theorem but repeatedly used 

a particular version in which he let n → ∞ in the binomial formula. Later, he pre- 

sented two different proofs of this highly important tool. The first proof, published in 

his sequel textbook Institutiones calculi integralis, 8 highlighted the intimate connection 

between the binomial theorem and the Taylor expansion theorem which in modern 

notation stated that any function f (later with certain restrictions) could be expanded 

as 

f (x + a) = f (x) + f ′ (x) 

a + 

1 

f ′′ (x) 

1 · 2 a2 + . . . . 

For EULER, the binomial theorem was a rather easy consequence of the Taylor expan- 

sion provided the relation 

d 

dx xµ = µx µ−1 

had previously been established for exponents µ. However, as EULER later realized, 

the binomial theorem was central to the differentiation of such monomials if µ was 

not an integer. Therefore, proving the binomial theorem from the Taylor series expan- 

sion had created a vicious circle in the argument. Nevertheless, proofs of the binomial 

7 The history of the binomial theorem has attracted the interest of many scholars, see e.g. (Dhombres 

and Pensivy, 1988; Pensivy, 1994). 

8 (L. Euler, 1755, 276–279).


theorem from Taylor theorem recurred throughout the century and even into the nineteenth 

century. 9 

EULER’S second proof of the binomial theorem based on functional equations. In 

his second proof of the binomial theorem, published in 1775, 10 EULER devised his 

proof following an outline which would recur in most subsequent “rigorous” proofs. 

EULER introduced the notation 

[m] = 1 + m 

1 

m (m − 1) 

x + x 

1 · 2 

2 + . . . 

to denote the binomial series associated with the exponent m. Thus, proving the bino- 

mial theorem thus amounted to proving the equality [m] = (1 + x) m . The central step 

in the proof was the realization that the brackets satisfied a functional equation 11 

[m + n] = [m] · [n] . 

EULER’S proof of the functional equation was based on formally multiplying the corre- 

sponding infinite series. Once EULER had obtained the above functional equation and 

the binomial formula secured the equality [m] = (1 + x) m for integral m, he extended 

the domain for m by the computation 

[1] = 

� 

m · 1 

� � �m 1 

= 

m m 

⇒ 

m,n∈N 

� 

n 

� 

= [n] 

m 

1 m = (1 + x) n m . 

Thus, EULER proved the binomial theorem for all fractional exponents and claimed — 

without giving any proof — that it extended to all real exponents by way of continuity 

(see below). In summary, the central steps of EULER’S second proof of the binomial 

theorem are: 

1. The binomial formula, [m] = (1 + x) m for m ∈ N. 

2. The functional equation [m + n] = [m] · [n] proved by manipulating the associ- 

ated power series. 

3. An extension to rational exponents. 

4. A further extension to real exponents by continuity arguments. 

In complete correspondence with his views on formal equality, EULER did not ven- 

ture into considerations of the convergence of the infinite expression contained in the 

binomial theorem. To him, the theorem simply stated a formal equivalence of two 

different representations of the same function (expression). 

9 On the proof by WALLACE, see (Craik, 1999, 252–253). 

10 (L. Euler, 1775). 

11 For the history of functional equations, mainly with CAUCHY, see (J. Dhombres, 1992).

10.2. LAGRANGE’s new focus on rigor 197 

1797 Théorie des fonctions analytiques 

1806 Leçons sur le calcul des fonctions 

1813 Théorie des fonctions analytiques, nouvelle 

édition 

Table 10.1: LAGRANGE’s monographs on his algebraic analysis 

10.2 LAGRANGE’s new focus on rigor 

The Eulerian approach to analysis based on functions and series representations proved 

highly productive for mathematicians with right kinds of intuitions and understand- 

ing. Toward the end of the century, a number of events — in particular the external 

influence of mass instruction in mathematics and the change of generations — intro- 

duced a different view on the status of analysis. Its fruitfulness was admired but its 

lack of strict logical order was realized by some of its most distinguished practitioners. 

More so than anybody else, JOSEPH LOUIS LAGRANGE was instrumental in fertilizing 

the ground for a fundamental revision of the Eulerian paradigm. 

LAGRANGE presented his new algebraic theory of functions in three important 

monographs (see table 10.1). In what follows, references are made to the second edi- 

tion of the Théorie des fonctions analytiques which was the latest of the three and was 

included in LAGRANGE’S collected works. 12 

Importantly, LAGRANGE believed he could prove that any function could be ex- 

panded “by the theory of series” 13 into a series of the form 

f (x + i) = f (x) + ip (x) + i 2 q (x) + i 3 r (x) + . . . . 

The functions p, q, r, . . . were called the ‘derived’ functions of f , and it was the crux 

of the theory to show that they corresponded to the ordinary differentials obtained in 

the usual — less rigorous — way. 

Thus, at the very center of the Lagrangian system laid the expansion of a function 

into a power series. As J. V. GRABINER has convincingly described in her thesis, the 

expansion into power series was not an assumption in the Lagrangian system but was 

provided with an algebraic proof using one of EULER’S ideas. 14 As a consequence, the 

general expansion of any function into power series was made into a general principle 

replacing the important tool for obtaining such expansions which EULER had used to 

such a high effect, the binomial theorem. 

LAGRANGE’S contribution to the rigorization of the calculus was at least twofold: 

1. The mere fact that LAGRANGE — “a most illustrious mathematician” — devoted 

12 (Lagrange, 1813). 

13 (ibid., 7–8). 

14 (Grabiner, 1990, 93ff).


so much attention to the foundational questions raised the prestige of such ques- 

tions; rigorization became a legitimate mathematical research topic. 

2. Just as importantly, LAGRANGE’S work on rigorization provided a revolutionary 

new synthesis of the formal interpretation of series at the heart of the calculus. 

Furthermore, the binomial theorem and the expansion into power series (Taylor 

series) changed their internal relationship and dependency (see below). 

10.3 Early rigorization of theory of series 

In the first decades of the 19 th century, a number of mathematicians responded to 

the call for rigorization in the theory of infinite series. Two of the most interesting 

reactions to the state of rigor in the theory of infinite series were made by C. F. GAUSS 

(1777–1855) and B. BOLZANO (1781–1848). 

To GAUSS, BOLZANO, and their contemporaries, analysis was a conglomerate of 

various methods, key results, and foundations. An interesting illustration of the con- 

currently existing approaches to the discipline can be found in the textbooks written 

by S. F. LACROIX (1765–1843) just before the turn of the century. 15 In three volumes, 

LACROIX presented much of the key material of analysis adapting various approaches 

and foundations to suit his needs. 

Both GAUSS and BOLZANO reacted inspired partly by philosophical arguments; in 

the following, some of the relevant aspects of their contributions are outlined. 

10.3.1 GAUSS’ hypergeometric series 

GAUSS’ main contribution to the rigorization of the theory of series consisted of a 

paper concerning the so-called hypergeometric series . 16 The paper was presented in 

1812 and published the following year. 17 To GAUSS, the hypergeometric series 

F (α, β, γ, x) = 1 + 

∞ 

x 

∑ 

n=1 

n n 

(α + m) (β + m) 

n! ∏ 

m=0 (γ + m) 

constituted a preferred representation of a vast range of functions including logarith- 

mic, elliptic, and other transcendental functions. By studying this series in its gen- 

erality, GAUSS obtained knowledge of the functions which it could represent. GAUSS 

never conducted a full investigation of which functions it could represent but the study 

of the series remained an interesting topic, in particular for Göttingen mathematicians. 

GAUSS’ research represents an intermediate between the old direct and more special- 

ized representations and the modern concept based approach to analysis. 18 More im- 

15 (Lacroix, 1797; Lacroix, 1798; Lacroix, 1800). 

16 The name hypergeometric series is a later invention, see (Wussing, 1982, 299). 

17 (C. F. Gauss, 1813). 

18 See also chapter 21.

10.3. Early rigorization of theory of series 199 

portantly, GAUSS’ investigations of the hypergeometric series also contained a number 

of very interesting results. In particular, GAUSS’ attitude toward convergence of series 

and criteria for deciding the convergence is relevant to the present analyses. 

Concepts of convergence and a criterion of convergence. Before he could advance 

to deeper questions, GAUSS emphasized that the convergence or divergence of the 

hypergeometric series had to be investigated. In his paper on the hypergeometric 

series, GAUSS gave no explicit definition of convergence. However, by the following 

argument, GAUSS claimed that the series converged for |x| < 1 and diverged for |x| > 

1. As was customary, these requirements were stated verbally without the notation of 

numerical values (see, e.g., quotation below). 

GAUSS compared the coefficients of two sequential powers of x, say the coefficients 

of x m and x m+1 , and found that their ratio 

1 + γ+1 

m 

1 + α+β 

m 

+ γ 

m 2 

+ αβ 

m 2 

approached the value 1 when m was taken to be increasingly large. GAUSS then con- 

cluded that for any complex value of x with |x| < 1, the series would be convergent “at 

least from some point onward” and lead to a determinate finite sum. In case |x| > 1, 

the series would necessarily diverge and it could not have a sum. 19 GAUSS summa- 

rized his position: 

“Since our function is defined as the sum of a series, it is obvious, that our 

investigations are naturally confined to the cases in which the series actually converges 

and that it is absurd to ask for the value of the series whenever x has a 

value greater than unity.” 20 

Of the cases with |x| = 1, GAUSS only investigated x = 1 and found that under the 

condition α + β − γ < 0, the series would have a finite sum. 21 

In the above context, GAUSS appears to have employed a concept of series conver- 

gence which corresponded to the partial sums approaching a finite limit. We are easily 

led to believe that GAUSS’ familiar looking notions such as convergent and sum meant 

the same to him as they do to us. However, another concept of convergence was also 

in use at GAUSS’ time and even appeared later in his manuscripts (see below). There- 

fore, it is worth re-examining the evidence to see if it appears different with this added 

information. 

Originating with J. LE R. D’ALEMBERT (1717–1783) in the mid-eighteenth century, 

the term convergent was used by mathematicians within the formal paradigm to denote 

19 (ibid., 126). 

20 “Patet itaque, quatenus functio nostra tamquam summa seriei definita sit, disquisitionem natura 

sua restrictam esse ad casus eos, ubi series revera convergat, adeoque quaestionem ineptam esse, 

quinam sit valor seriei pro valore ipsius x unitate maiori.” (ibid., 126). For a German translation, see 

(C. F. Gauss, 1888, 10). 

21 (C. F. Gauss, 1813, 139, 142–143).


series in which the numerical value of the general term vanished monotonically, i.e. 

series ∑ an for which the sequence |an| was monotonically decreasing and approached 

zero. 22 The vanishing of terms, clearly contrasted to the convergence of the partial 

sums, can be found in an unpublished manuscript written by GAUSS probably after 

1831. 23 

“By convergence of an infinite series, I will simply understand nothing but the 

infinite approaching of its terms toward 0 when the series is infinitely continued. 

The convergence of a series in itself is thus to be distinguished from the convergence 

of its summation toward a finite limit; however, the latter implies the former 

but not the other way around.” 24 

Exactly which concept of convergence, GAUSS had in mind in his research on the 

hypergeometric series can seem unclear. From a modern perspective, we are tempted 

to assume that GAUSS interpreted convergence as convergence of the partial sums and 

interpret GAUSS’ comparison of subsequent terms as an implicit quotient criterion. 

However, GAUSS’ reasoning can equally well be interpreted within the older concept 

of D’ALEMBERT-convergence. 25 

In terms of the development described in the next chapter, GAUSS’ investigation 

on the hypergeometric series is important in three respects: 

1. GAUSS’ investigation was confined to a particular series, albeit one with three 

parameters which enabled GAUSS to model a number of transcendental func- 

tions using it. 

2. GAUSS insisted on establishing the convergence of the series before speaking of 

its sum. He used an implicit theorem — apparently equivalent to the ratio test 26 

(see subsequent chapters) — to determine restrictions on the variable x. 

3. Despite aiming at “the rigorous methods of the ancient geometers” 27 , GAUSS’ 

theory of infinite series as expressed in the paper on the hypergeometric series 

was rudimentary and not spelled out in much detail. For instance, it is not com- 

pletely clear precisely what his basic notions meant. 

22 (Grabiner, 1981b, 60). 

23 (Schneider, 1981, 55–56). 

24 “Ich werde unter Convergenz, einer unendlichen Reihe schlechthin beigelegt, nichts anders verstehen 

als die beim unendlichen Fortschreiten der Reihe eintretende unendliche Annäherung ihrer 

Glieder an die 0. Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz 

ihrer Summirung zu einem endlichen Grenzwerthe; letztere schliesst zwar die erstere ein, 

aber nicht umgekehrt.” (C. F. Gauss, Fa, Kapsel 46a, A1–A13, 400). 

25 In most of the (earlier) secondary literature, e.g. (Pringsheim, 1898–1904, 79), GAUSS’ emphasis on 

establishing the convergence of the hypergeometric series and his use of the quotient comparison 

have been taken as precursors of the rigorization program (see next chapter). SCHNEIDER has aptly 

interpreted GAUSS’ concept of convergence in terms of the sequence of terms (Schneider, 1981, 56). 

26 The ratio test is also sometimes called the quotient test but I will use the term ratio test, throughout. 

27 “Ostendemus autem, et quidem, in gratiam eorum, qui methodis rigorosis antiquorum geometrarum 

favent, omni rigore.” (C. F. Gauss, 1813, 139).


To summarize the debate, we have to emphasize three dates. In 1812, GAUSS’ 

presented his research on hypergeometric series in which his concept of convergence 

remains undefined; in 1821, A.-L. CAUCHY (1789–1857) promoted the convergence 

of the partial sums into the only acceptable definition of convergence; but as late as 

1831, GAUSS employed a D’ALEMBERT-like concept of convergence which entailed the 

vanishing of the terms and did not provide convergence of the series. To believe that 

GAUSS had anticipated CAUCHY’S notion of convergence and the ratio test in 1812 

thus seems to be the least efficient interpretation. GAUSS may very well have held the 

same conceptions about convergence in 1812 as he evidently did in 1831. Instead, it 

seems that until CAUCHY’S work, different notions of convergence were co-existing 

and the position of definitions and tests of convergence within the structure of the 

theory of series floated. 

10.3.2 BOLZANO’s rigorization of the binomial theorem 

Contrary to GAUSS, the Czech priest and mathematician BOLZANO did not have the 

ear of the international mathematical community although his ideas and visions for 

the foundation of the calculus reached even further than GAUSS’. To promote inter- 

est in his work, BOLZANO published critical investigations and new proofs of key 

theorems of analysis. He hoped that mathematicians would pay more attention to a 

broader philosophical program which he was developing. 

In 1816 and in Prague, BOLZANO published a book entitled Der binomische Lehrsatz 

which is of particular relevance to the current purpose. 28 In that book, BOLZANO 

scrutinized existing derivations of the binomial theorem before going on to present his 

own proof. As noted, N. H. ABEL (1802–1829) once praised BOLZANO’S cleverness 

(see p. 42); important aspects of ABEL’S criticism may well have their origins with 

BOLZANO. 

BOLZANO’S critical attitude. In the introduction of his book, BOLZANO reviewed 

the structures of previous proofs of the binomial theorem. In the process, BOLZANO 

developed a penetrating criticism of the accepted methods of reasoning with infinite 

series. Soon, others would repeat BOLZANO’S criticism — at least, ABEL’S judgement 

of eighteenth century epistemic techniques in analysis resembled some of BOLZANO’S 

points. 

A number of interesting themes were raised in BOLZANO’S introduction. BOLZANO 

observed that the foundation of the entire “higher analysis” (calculus) rested on Tay- 

lor’s Theorem and that this theorem in turn relied on the binomial theorem. Conse- 

quently, the obscure status of the proof of the latter theorem had severe implications 

for the entire discipline. 

28 (Bolzano, 1816). BOLZANO’S titles are often very precise and very long; here the abridged version is 

used throughout.


Figure 10.1: BERNARD BOLZANO (1781–1848) 

Indeed, from B. TAYLOR’S (1685–1731) days, proofs of Taylor’s Theorem had relied 

on an analogy between repeated differences and the binomial formula. 29 However, the 

relevant step in the proof of Taylor’s Theorem seems to have been a limit process based 

on the binomial formula in which the exponent n increased to infinity and thus did not 

rely on the full binomial theorem. This distinction between the binomial theorem and 

the indicated limit process does not seem to have been undertaken by eighteenth and 

nineteenth century mathematicians, though. 

Next, BOLZANO criticized previous proofs for operating with (completed) infinite 

series, i.e. working with series as if they were polynomials. Instead, he proposed a 

concept of numerical limit processes based on (variable) quantities (Größen) ω which 

could be assumed positive but less than any given value. He also described these 

quantities as “quantities which can be made as small as one desires.” 30 Importantly, 

BOLZANO’S ω was not a completed infinitesimal but a variable quantity which de- 

pended on a limit process. 

In continuation of the previous point, BOLZANO insisted that restrictions be im- 

posed on the binomial such that the series was (arithmetically) convergent. He claimed 

that previous proofs had “proved to much” by not taking such restrictions on x into 

account and forbade application of the theorem outside the domains of convergence 

of the series. In the argument, BOLZANO employed a counter example which based 

29 See e.g. (Jahnke, 1999, 139–142). 

30 “Größen, welche so klein werden können, als man nur immer will.” (Bolzano, 1816, v).


on uncritical use of the binomial theorem 

√ 1 

−1 = (1 − 2) 2 = 1 − 1 1 1 

· 2 − · 4 − · 8 − . . . (10.1) 

2 8 16 

exhibited the imaginary unit as an infinite sum of real numbers. 31 

Following his program, BOLZANO rejected all previous proofs: NEWTON’S proof 

because it had been based on extrapolation and not “everything which corresponds 

to known truths is necessarily true”; 32 the proofs based on the expansion into Taylor 

series because they introduced a vicious circle and the binomial theorem should be the 

more basic of the two theorems; and even EULER’S second proof because it operated 

with completed infinite series and did not consider the convergence of the series. 

Revision of EULER’S proof. Subsequent to his critical remarks, BOLZANO presented 

his own new rigorized proof of the binomial theorem. He based it on the outline 

of EULER’S second proof but replaced the way in which EULER handed infinite series 

with his new concept of numerical equality and limits. Overturning EULER’S manipu- 

lations of completed infinite series, BOLZANO worked with the partial sums and limit 

arguments. Expressed in EULER’S notation, BOLZANO proved by multiplying the first 

s terms of [m] with the first t terms of [n], that the first min (s, t) terms of the product 

corresponded to the first min (s, t) terms of [m + n]. 33 If we introduce the notation [m] t 

s 

to denote the sum of the terms ranging from s to t in the series [m], 34 the result can be 

expressed as 

� 

[m] s 

1 [n]t 

�min(s,t) 1 = [m + n] 

1 

min(s,t) 

1 . 

However, the terms after min (s, t) would not always be equal in the two expressions 

but BOLZANO found that the difference 

� 

[m] s 

1 [n]t 

�r+s 1 

min(s,t) 

− [m + n]s+t 

min(s,t) 

could be made smaller than any given positive value by taking min (s, t) sufficiently 

large provided that |x| < 1. 35 Thus, BOLZANO obtained his proof of the functional 

equation 

under the important assumption |x| < 1. 

[m] · [n] = [m + n] 

Extension to real exponents. EULER’S (second) proof of the binomial theorem had 

focused on rational exponents. At the end of the argument, he suggested that other 

(positive) exponents could also be considered: 

31 (ibid., vi). 

32 (ibid., xi). 

33 (ibid., §38). 

34 This notation has been adapted from (Hauch, 1997). 

35 (Bolzano, 1816, §40).


“[. . . ] and thus it shows that 

� � 

i 

= (1 + x) 

a 

i a , 

which demonstrates that our theorem is true when for the exponent n any fraction 

i 

a is taken, from this its truth is evident for all positive numbers taken in place of 

the exponent n.” 36 

BOLZANO was slightly more specific on binomial expansion for irrational expo- 

nents. As a consequence of the meaning of (1 + x) i for i an irrational number, BOLZANO 

claimed, (1 + x) i could be approached as closely as desired by (1 + x) m n for m, n inte- 

gers. Inserting m n for i everywhere in the series and letting m n 

approach i, the sum 

would approach (1 + x) i as closely as desired. Thus, BOLZANO alluded to his concept 

of continuity applied to the exponentiation and to power series in order to obtain the 

binomial theorem for all real exponents. 37 

10.4 New types of series 

The series discussed thus far have all been power series but in the early nineteenth 

century, this situation changed. Series which were not power series had emerged 

in various contexts in the eighteenth century but became very important in the first 

decades of the nineteenth century, mainly through investigations in the theory of heat 

conducted by J. B. J. FOURIER (1768–1830). 38 

FOURIER’S term-wise integration. From the first decade of the nineteenth century, 

FOURIER had begun representing physical phenomena — mainly heat conduction — 

by trigonometric series. In 1822, his investigations were published as a monograph. 39 

One of FOURIER’S central tricks was the term-wise integration of an infinite series em- 

ployed to obtain the Fourier coefficients in the following way. Assuming that a function 

φ (x) could be expanded as 

φ (x) = 

36 “[. . . ] atque hinc in genere manifestum fore 

∞ 

∑ ai sin ix, (10.2) 

i=1 

� � 

i 

= (1 + x) 

a 

i a , 

ita ut iam demonstratum sit theorema nostrum verum esse, si pro exponente n fractio quaecunque 

i 

a accipiatur, unde veritas iam est evicta pro omnibus numeris positivis loco exponentis n accipiendis.” 

(L. Euler, 1775, 215–216). 

37 (Bolzano, 1816, §46). 

38 FOURIER and his works leading to Fourier series have been widely studied, see e.g. (Bottazzini, 

1986; I. Grattan-Guinness, 1972). 

39 (Fourier, 1822).

10.4. New types of series 205 

Figure 10.2: JEAN BAPTISTE JOSEPH FOURIER (1768–1830) 

FOURIER multiplied both sides of the equation by sin nx and integrated from 0 to π, 

� π 

0 

φ (x) sin nx dx = 

∞ � 

∑ ai i=1 

sin ix sin nx dx, 

where the integration of the sum was carried out term-wise. By the orthogonality, 

⎧ 

� 

⎨ 

π 

if n = i, 

sin ix sin nx dx = 2 

⎩ 

0otherwise, 

FOURIER found the coefficients of the expansion (10.2) to be 

π 

2 a � π 

i = φ (x) sin ix dx. 

0 

The interchange of summation and integration would soon become a point of objec- 

tion against FOURIER’S rigor. 

SIMÉON-DENIS POISSON’S peculiar example. Almost simultaneous with FOURIER’S 

first investigations, a problem arose which also involved series which were not power- 

series. The problem was raised by SIMÉON-DENIS POISSON in 1811 and was inten- 

sively debated for the next decades, in particular in the French journal Annales de 

mathématiques pures et appliquées. 40 

40 This is well described in (Jahnke, 1987, 105–117) and (Bottazzini, 1990, lx–lxiii).


It all began when POISSON noticed that a peculiar situation arose from letting m = 

1 3 and x = π in the binomial expansion of (2 cos x) m , 41 

(2 cos x) m = 

∞ 

∑ 

n=0 

∏ n−1 

k=0 (m − k) 

cos ((m − 2n) x) . 

n! 

In a short paper, POISSON observed that the left hand side had the three values 

− 3√ 2, 3√ � 

1 + 

2 

√ � 

−3 

, and 

2 

3√ � 

1 − 

2 

√ � 

−3 

2 

although the right hand side was a single-valued function, 

cos 

� 

π 

� ∞ ∏ 

3 ∑ 

n=0 

n−1 

k=0 

� � 

13 − k 

n! 

= 1 

2 × (1 + 1) 1 3 = 

Thus, the sum of the series on the right hand side corresponded to neither of the values 

of the expression on the left hand side but was the average of its two complex values. 

Poisson’s example is a particular example of the kind of strange relations which 

could result by interpreting formal equalities in situations outside the domain of nu- 

3√ 2 

2 . 

merical equality. In this sense, it is similar to the peculiar formal equality 

1 

2 

= 1 − 1 + 1 − 1 + . . . 

which had puzzled mathematicians in the eighteenth century. However, Poisson’s ex- 

ample was a convergent series and the problem was that it did not agree with its true 

value. 

Upon POISSON’S publication, mathematicians sought to understand how and why 

this peculiarity emerged, and the debate also spread to Berlin. In Berlin, A. L. CRELLE 

(1780–1855) and M. OHM (1792–1872) became interested in the explanation of this 

phenomenon — as did the mysterious L. OLIVIER who will appear prominently in 

chapter 13. 42 ABEL also became acquainted with the problem and it provoked him 

into producing a new proof of the binomial theorem. Before attention is focused on 

ABEL’S work in the theory of series, the following chapter is devoted to his greatest 

inspiration in the field: CAUCHY’S Cours d’analyse. 

41 (Poisson, 1811). POISSON wrote x = 200 ◦ and thus adhered to the new radian system. 

42 See e.g. (Jahnke, 1987, 105–117).

Chapter 11 

CAUCHY’s new foundation for analysis 

Against the background of J. L. LAGRANGE’S (1736–1813) algebraic foundation for the 

calculus, another and radically different program of rigorization emerged when A.-L. 

CAUCHY (1789–1857) set out to write a textbook suitable for his courses at the École 

Polytechnique. In a sense, CAUCHY’S famous textbook Cours d’analyse continues the 

Lagrangian program as its subtitle Analyse algébrique testifies, 1 but its contents consti- 

tuted a remarkable break with the Lagrangian system. In the Cours d’analyse, CAUCHY 

reformulated and revised the theory of infinite series from his novel viewpoint based 

on a shift in the conception of equality (see below). 2 Later, CAUCHY continued the 

reworking of the foundations of the calculus in two further textbooks dealing with the 

differential and integral calculus (see table 11.1). 

The Lagrangian foundation for the calculus relied on a notion of equality between 

functions (expressions) which was largely formal and had been inherited from L. EU- 

LER (1707–1783) (see chapter 10). In CAUCHY’S hands, the concept of equality shifted 

toward focusing on numerical or arithmetical equality: to CAUCHY, two functions 

were only equal if they produced equal numerical results for equal numerical values 

of the arguments. By way of a few central examples, I will document how this change 

of approach was implemented and what its consequences were. 

11.1 Programmatic focus on arithmetical equality 

Generality of algebra. Describing the methods used in the Cours d’analyse, CAUCHY 

stressed the way in which he had fought to obtain the standard of rigor which is char- 

acteristic of geometry by denouncing arguments relying on the “generality of alge- 

bra”. Such arguments could be, CAUCHY admitted, suitable inductions for obtaining 

1 (A.-L. Cauchy, 1821b). 

2 CAUCHY’S Cours d’analyse marks a turning point in the history of the calculus and has been given 

due attention by historians of mathematics. The most comprehensive presentation is probably BOT- 

TAZZINI’S introduction to a photographic reproduction of the Cours d’analyse (Bottazzini, 1990). In 

particular, the section entitled The “Generality of Algebra” (ibid., xliv–xcvii) is of direct relevance to 

the present discussion. 

207

208 Chapter 11. CAUCHY’s new foundation for analysis 

1821 Cours d’analyse d’École Royale Polytechnique. 

Première partie. Analyse algébrique 

1823 Résumé des leçons données a l’École 

Royale Polytechnique sur le calcul infinitésimal 

1829 Leçons sur le calcul différentielle 

Table 11.1: CAUCHY’s textbooks on the calculus 

the truth but should never be allowed to act as exact proofs. In particular, CAUCHY 

mentioned how arguments by the generality of algebra had been leading mathemati- 

cians into unfounded passages “from convergent to divergent series, from real quan- 

tities to imaginary expressions”. 3 CAUCHY continued, 

“Similarly, one should realize that they [arguments by the generality of algebra] 

tend to attribute to algebraic formulae an indefinite extension whereas in 

reality, the majority of these formulae only subsists under certain conditions and 

for certain values of the quantities which they contain.” 4 

Important examples of the problems which this requirement addressed was the 

relationship between formulae such as 

1 

1 − x and 

∞ 

∑ x 

n=0 

n 

(11.1) 

which have been described above. Mathematicians unknowingly adhering to the for- 

mal concept of equality had been aware that counter-intuitive results could emerge 

if numerical values were inserted into the two expressions and their equality was ex- 

tended to cover numerical values. For instance, 

1 

1 − (−1) 

1 

= , but 

2 

∞ 

∑ (−1) 

n=0 

n = 1 − 1 + 1 − 1 + . . . 

and the sum did certainly not represent the value 1 2 in any numerical sense. 

CAUCHY removed these anomalies by dismissing the concept of formal equality 

and carefully analyzing the conditions under which a numerical equality between ex- 

pressions such as (11.1) would hold. Thus, he found and emphasized that for numer- 

ical equality it was required that |x| < 1, and consequently the peculiar results were 

all explained away. 

3 (A.-L. Cauchy, 1821a, iii). 

4 “On doit même observer qu’elles tendent à faire attribuer aux formules algébriques une étendue 

indéfinie, tandis que, dans la réalité, la plupart de ces formules subsistent uniquement sous certaines 

conditions, et pour certaines valeurs des quantités qu’elles renferment.” (ibid., iii).

11.2. CAUCHY’s concepts of limits and infinitesimals 209 

11.2 CAUCHY’s concepts of limits and infinitesimals 

In the preliminaries, CAUCHY defined what he understood under the terms limit and 

infinitely small. The two concepts are closely related in CAUCHY’S book although their 

interrelation is not trivial. The proper interpretation of the concepts has sparked some 

controversy in the historical literature which has sometimes chosen to focus on one 

of the concepts and neglecting the other. To be fair to the source and to facilitate a 

discussion of N. H. ABEL’S (1802–1829) reading of it, both CAUCHY’S definitions are 

reproduced and translated here. 5 

First, CAUCHY defined his concept of limit: 

“Whenever the values successively attributed to one and the same variable 

approach a fixed value indefinitely in such a way that it eventually differs from it 

by as little as one desires, the latter is called the limit of all the others.” 6 

A few sentences later, CAUCHY gave his definition of infinitely small quantities which 

he based on the notion of limits: 

“Whenever the successive numerical values of one and the same variable decrease 

indefinitely in such a way that they become less than any given number, 

this variable becomes what is called infinitely small or an infinitely small quantity. A 

variable of this kind has zero for limit.” 7 

Various conceptions of limits and infinitesimals had previously been suggested as 

the foundations for the calculus and EULER called the calculus the “algebra of zeros” 

because of his prolific use of infinitesimals. However, CAUCHY gave a process-based 

definition of limits and — more importantly — showed how to work with it. Corre- 

spondingly, to CAUCHY, infinitesimals were variable quantities which were involved 

in limit processes and could be made as small as desired by particular choices of the 

variable of the limit process. 

It has puzzled certain historians of mathematics why CAUCHY simultaneously 

employed limits and retained the older concept of (completed) infinitesimals. 8 The 

fact remains that CAUCHY employed both concepts in different proofs and probably 

thought of them as equivalent but suited for different purposes. 9 

Importantly, CAUCHY sometimes used symbols to denote infinitely small quanti- 

ties which were really (according to the definition) variables which tended toward the 

5 A good interpretation is provided in (Grabiner, 1981b, 80–81). 

6 “Lorsque les valeurs successivement attribuées à une même variable s’approchent indéfiniment 

d’une valeur fixe, de manière à finir par en différer aussi peu que l’on voudra, cette dernière est 

appelée la limite de toutes les autres.” (A.-L. Cauchy, 1821a, 4). 

7 “Lorsque les valeurs numériques successives d’une même variable décroissent indéfiniment, de 

manière à s’abaisser au-dessous de tout nombre donné, cette variable devient ce qu’on nomme 

un infiniment petit ou une quantité infiniment petite. Une variable de cette espèce a zéro pour 

limite.” (ibid., 4). 

8 See discussion in (Lützen, 1999, 198–211). 

9 See e.g. (Grabiner, 1981b, 87).


limit zero. However, by introducing symbols, the process under which the variable 

vanished was obscured and the order in which limit processes were conducted was 

not explicit. 

11.3 Divergent series have no sum 

When CAUCHY extended the procedure of analyzing the requirements for numerical 

equality involving series, he was led to a conclusion which he knew would be painful 

for his contemporaries to accept. 

“It is true that in order to always remain faithful to these principles, I see myself 

forced to accept multiple propositions which may appear a bit harsh at first 

sight. For instance, in the sixth chapter, I announce that a divergent series has no 

sum.” 10 

CAUCHY’S treatment of series began with series of positive real terms (section 

VI.2), was then extended to series of general real terms (section VI.3), before he went 

on to treat series with complex terms (chapter IX). In the sixth chapter, CAUCHY elab- 

orated his definition of convergence and his attitude toward divergent series. 

“Let 

sn = u0 + u1 + u2 + · · · + un−1 

be the sum of the first n terms where n designates any integer. If, for ever increasing 

values of n, the sum sn approaches a certain limit s indefinitely, the series is 

said to be convergent and the above mentioned limit is called the sum of the series. 

In the contrary case, if the sum sn does not approach any fixed limit when n 

increases indefinitely, the series is divergent and no longer has a sum.” 11 

As the quotations demonstrate, CAUCHY sought to limit the concept of “sum of a 

series” to apply only to convergent series. This position was radicalized by ABEL in 

his correspondence as we will see in section 12.3: ABEL wanted an outright ban on 

divergent series and saw them as the creation of the Devil. To CAUCHY, who was also 

a very creative mathematician outside fundamental issues, divergent series remained 

of interest in asymptotic mathematics; only in questions of foundational nature, they 

were not attributed any sum. 

10 “Il est vrai que, pour rester constamment fidèle à ces principes, je me suis vu forcé d’admettre plusieurs 

propositions qui paraîtront peut-être un peu dures au premier abord. Par exemple, j’énonce 

dans le chapitre VI, qu’un série divergente n’a pas de somme [. . . ]” (A.-L. Cauchy, 1821a, iv). 

11 “Soit 

sn = u0 + u1 + u2 + · · · + un−1 

la somme des n premiers termes, n désignant un nombre entier quelconque. Si, pour des valeurs 

de n toujours croissantes, la somme sn s’approche indéfiniment d’une certaine limite s; la série sera 

dite convergente, et la limite en question s’appellera la somme de la série. Au contraire, si, tandis 

que n croît indéfiniment, la somme sn ne s’approche d’aucune limite fixe, la série sera divergente, et 

n’aura plus de somme.” (ibid., 123).

11.3. Divergent series have no sum 211 

For the moment, focus will be given to another intrinsic aspect of CAUCHY’S defini- 

tion: the position of certain concepts and theorems within the theoretical framework. 

If only convergent series were allowed to have a sum, and knowledge of the value of 

the sum was required to apply the definition to determine whether the series was con- 

vergent or not, a problem emerged. Phrased in modern notation, it is not practical to 

attempt establishing that some s exists such that 

|sn − s| → 0 as n → ∞ 

without knowing which s could be a candidate; for complicated or general series, such 

candidates might not be available. Therefore, CAUCHY’S theoretical complex — in 

an essential way — required a means of investigating convergence without any prior 

knowledge of the purported sum. This important problem was met by a number 

of criteria of convergence which were also among the chief innovations in the Cours 

d’analyse. 

After giving his definition of convergence, CAUCHY gave a first characterization of 

his new concept, which was, however, of little use in practically establishing conver- 

gence (see below). 

“Thus, for the series u0 + u1 + u2 + · · · + un + . . . to be convergent, it is first 

necessary that the general term un decreases indefinitely when n grows. However, 

this condition does not suffice and it must also be so that for increasing values of 

n the different sums 

i.e. the sums of the quantities 

un + un+1, 

un + un+1 + un+2, 

etc., 

un, un+1, un+2, etc. 

taken starting from the first and to whatever number one may wish, eventually always 

produce numerical values less than any assignable limit. Conversely, whenever 

these two conditions are fulfilled, the convergence of the series is assured.” 12 

12 “Donc, pour que la série (1) soit convergente, il est d’abord nécessaire que le terme général un 

décroisse indéfiniment, tandis que n augmente; mais cette condition ne suffit pas, et il faut encore 

que, pour des valeurs croissantes de n, les différentes sommes 

c’est-à-dire, les sommes des quantités 

un + un+1, 

un + un+1 + un+2, 

&c. . . 

un, un+1, un+2, &c. . . 

prises, à partir de la première, en tel nombre que l’on voudra, finissent par obtenir constamment 

des valeurs numériques inférieures à toute limite assignable. Réciproquement, lorsque ces diverses 

conditions sont remplies, la convergence de la série est assurée.” (ibid., 125–126).


In the second half of the nineteenth century, CAUCHY’S characterization of conver- 

gence by means of the so-called Cauchy criterion became even more important when it 

was realized, that no proof of CAUCHY’S last assertion could be given. The solution 

devised by mathematicians such as J. W. R. DEDEKIND (1831–1916) and G. CANTOR 

(1845–1918) to secure the validity of CAUCHY’S second claim was to construct the real 

numbers in such ways that they possessed this property of completeness. 

11.4 Means of testing for convergence of series 

With his new emphasis on convergence, CAUCHY’S theory needed criteria of conver- 

gence which would operate simply from the general terms without any information 

about the sum. Based on previous observations, CAUCHY established three important 

such tests which are still all important today. First, he proved the root test to the effect 

that if n√ un has a limit k as n → ∞, the series ∑ un will be convergent if k < 1 and 

divergent if k > 1. In case k = 1, nothing could be said of the convergence by this 

criterion. CAUCHY then transformed the root test to obtain the ratio test (see below) 

and a logarithmic criterion by comparison with the harmonic series. 

It is a general feature of these criteria of convergence that there are cases for which 

they do not provide any information concerning convergence. As we will see in chap- 

ter 13, the search for a complete test of convergence which could always determine the 

convergence or divergence of series from its general term was also actively pursued 

in the 1820s. 

CAUCHY’S proof of the ratio test. CAUCHY’S proved his criteria of convergence by 

an ingenious route albeit complicated. 13 First, he proved the root test directly from 

the assumptions and the previously established convergence of the geometric pro- 

gression. Next, he referred to a previously established theorem to the effect that if 

the sequences { n√ � � 

un+1 

un} and were both convergent, their limits would be equal. 

un 

Ultimately, this theorem provided the proof of the ratio test. To get a grasp of the way 

CAUCHY reasoned with his concepts and the way in which he obtained his criteria, 

the details of his proof are considered. 14 Later, after ABEL’S way of commencing the 

theory of series has been described, the role of the ratio test in the two theories can be 

discussed. 

CAUCHY proved the convergence part of the root test by letting U denote a number 

k Then, he observed that a large integer n had to exist such that for any larger 

number, say N ≥ n, 

N√ uN < U, i.e. uN 

13 (A.-L. Cauchy, 1821a, 132–135). 

14 Today, this theorem is standard textbook material in basic calculus courses. The modern proof 

closely resembles CAUCHY’S proof.

11.4. Means of testing for convergence of series 213 

Thus, the tail of the series was term-wise less than a convergent geometric progression, 

and the convergence of the series ∑ un was concluded. Similarly, CAUCHY proved the 

divergence part of the root test by comparing with a divergent geometric progression. 

CAUCHY based his proof of the ratio test on the following result which he had 

previously obtained. 

“2nd theorem. If the function f (x) is positive for large values of x and the 

ratio 

f (x + 1) 

f (x) 

converges toward the limit k when x increases indefinitely, the expression 

[ f (x)] 1 x 

will converge at the same time toward the same limit.” 15 

In his proof, CAUCHY distinguished between two cases namely k finite or not; we 

shall here only be concerned with the case where k is finite. CAUCHY let ε denote an 

as yet unspecified number which was presumably very small. By assuming that for 

x ≥ h, 

f (x + 1) 

f (x) 

∈ [k − ε, k + ε] , 

CAUCHY found by the theory of means which he developed in a note 16 that the geo- 

metric mean 17 of 

f (h + 1) 

, 

f (h) 

f (h + 2) 

, . . . , 

f (h + 1) 

f (h + n) 

f (h + n − 1) , 

would also belong to this interval, i.e. 

� 

n f (h + n) 

= k + α, α ∈ [−ε, ε] . 

f (h) 

Then, CAUCHY found by inserting x = h + n 

which meant 

f (x) = f (h) · (k + α) x−h 

f (x) 1 x = f (h) 1 x · (k + α) 1− h x → 

x→∞ k + α. 

15 “2.e Théorème. Si, la fonction f (x) étant positive pour de très-grandes valeurs de x, le rapport 

f (x + 1) 

f (x) 

converge, tandis que x croit indéfiniment, vers la limite k, l’expression 

[ f (x)] 1 x 

convergera en même temps vers la même limite.” (ibid., 53–54). 

16 (ibid., note II). 

17 The geometric mean of the quantities a1, . . . , an was the quantity n� ∏ n k=1 a k.


Thus, the limit of f (x) 1 x belonged to the same arbitrarily small interval as the limit 

of 

f (x+1) 

f (x) , and thus the two limits were equal. Now, the ratio test followed by letting 

f (n) = un and observing that the limits of n√ un and u n+1 

un coincided. 

11.5 CAUCHY’s proof of the binomial theorem 

CAUCHY agreed with his predecessors in considering the binomial theorem a corner 

stone of the calculus. His proof of it relied on and promoted two of his new techniques 

and concepts in analysis: those of functional equations and continuous functions. 18 As 

EULER had done, CAUCHY considered the functional equation 

of which he knew that the binomial 

f (m) f (n) = f (m + n) (11.2) 

f (m) = (1 + x) m 

(11.3) 

was a continuous solution for all m provided x was fixed. On the other hand, the 

function defined by the infinite series 

∞ 

∑ 

k=0 

� � 

m 

x 

k 

k 

(11.4) 

was also a solution to the functional equation (11.2) under the assumptions that it 

converged and m was a rational number. 

To demonstrate that the series (11.4) satisfied the functional equation, CAUCHY 

had to be able to multiply infinite series. He invented a way of multiplying absolutely 

convergent series which rigorously established the convergence of the product. 19 Based 

on the argument which EULER had also used, CAUCHY then knew that the series (11.4) 

coincided with f (m) for all rational values of m. Therefore, the general equality of 

(11.3) and (11.4) would be proved if the series was a continuous function of m. In order 

to prove the continuity of the series (11.4), CAUCHY devised and proved a general 

theorem to the effect that a convergent sum of continuous functions was always a 

continuous function. Later, this theorem would arouse much controversy (see below). 

CAUCHY’S way of multiplying infinite series. In the Cours d’analyse, CAUCHY in- 

vented a way of multiplying two absolutely convergent series such that the product 

would be a new convergent series. As he had done throughout, CAUCHY developed 

his theory of infinite series in three steps: 

1. Series of real, positive terms (section VI.2) 

18 For CAUCHY’S theory of functional equations, see (J. Dhombres, 1992). 

19 (A.-L. Cauchy, 1821a, 157).

11.5. CAUCHY’s proof of the binomial theorem 215 

2. Series of general real terms (section VI.3) 

3. Series of complex terms (chapter IX) 

In each of these three theories, CAUCHY developed a product theorem, 20 and CAUCHY’S 

proofs will be relevant when compared with ABEL’S subsequent proofs. The multipli- 

cation theorems applying to series of real terms (the first and second) are the most in- 

teresting for the present study. In his theory of series of positive terms, CAUCHY had 

stated and proved that if two series ∑ un and ∑ vn were convergent and converged 

toward s and s ′ , the series whose general term was 

wk = ∑ 

n+m=k 

unvm 

(11.5) 

would be convergent and converge toward the product ss ′ . When he wanted to gener- 

alize this theorem to general series of real terms, CAUCHY imposed the restriction that 

each of the series ∑ un and ∑ vn was to be convergent when their terms were replaced 

by their absolute values, i.e. both factors were to be absolutely convergent, although the 

term and an elaborate concept was only invented some years later (see section 12.7). 

In the first case, in which all terms were positive quantities, CAUCHY proved the 

theorem by a direct argument. He let s ′′ 

n designate the sum of the first n terms of the 

purported product series (11.5) and defined 

to obtain 

He had thus obtained 

⎧ 

⎪⎨ 

n − 1 

if n is odd 

m = 2 

⎪⎩ n − 2 

if n is even 

2 

n−1 

∑ wk < 

k=0 

n−1 

∑ wk > 

k=0 

� 

n−1 

∑ uk k=0 

� 

m 

∑ uk k=0 

sm+1s ′ m+1 

� � n−1 

∑ 

vk k=0 

� � 

m 

∑ vk k=0 

� 

� 

< s′′ 

n < sns ′ n, 

. 

and 

and by letting m grow beyond all bounds, the theorem was established. 

When he came to generalize this theorem to the case of general real terms, CAUCHY 

wanted to apply the simpler case of series with positive terms. With the same notation 

as above, he obtained the formula 

20 (ibid., 141–142,147–149,283–285). 

s ′ nsn − s ′′ 

2n−2 

n = 

∑ 

t=n 

∑ 

m+k=t 

umv k, (11.6)


and if the terms um and v k were positive, the difference (11.6) would converge to zero 

as a consequence of the theorem for series of positive terms. If the terms were not all 

positive, CAUCHY could still apply the theorem to the corresponding series of numer- 

ical values ρn = |un| and ρ ′ n = |vn|. Therefore, in this case, the difference (11.6) would 

still converge to zero and yet majorize the series of the original terms. Thus, CAUCHY 

had established the convergence and the product equation for absolutely convergent 

series of real terms. 

CAUCHY’S concept of continuous functions. CAUCHY’S concept of continuous func- 

tions (see below) was among the main innovations of his new calculus. There, in the 

definition and in the proofs, his new foundation on an algebraic concept of limits 

played its most important role. In the eighteenth century, EULER had used the term 

continuous to indicate that the function was defined by the same analytic expression 

throughout its domain. 21 However, in the Cours d’analyse, CAUCHY took it upon him- 

self to completely redefine the concept of continuous functions in order to capture a 

different property of the functions: their continuous, unbroken variation. 

“Let f (x) be a function of the variable x and suppose that for every value of 

x between two given boundaries this function always takes a unique and finite 

value. If, starting from a value of x contained between these boundaries, one attributes 

to x an infinitely small increment α, the function will receive the increment 

f (x + α) − f (x) 

which depends simultaneously on the new variable α and the value of x. Between 

the two boundaries assigned to the variable x, the function f (x) will be a continuous 

function of this variable if for every value of x between these boundaries, the 

numerical value of the difference 

f (x + α) − f (x) 

decreases indefinitely with that [numerical value] of α. In other words, the function 

f (x) remains continuous with respect to x between the given boundaries if, between 

these boundaries, an infinitely small increment of the variable produces an infinitely small 

increment of the function.” 22 

21 (L. Euler, 1748). See (Lützen, 1978) and (Youschkevitch, 1976). 

22 “Soit f (x) une fonction de la variable x, et supposons que, pour chaque valeur de x intermédiaire 

entre deux limites données, cette fonction admette constamment une valeur unique et finie. Si, en 

partant d’une valeur de x comprise entre ces limites, on attribue à la variable x un accroissement 

infiniment petit α, la fonction elle-même recevra pour accroissement la différence 

f (x + α) − f (x) , 

qui dépendra en même temps de la nouvelle variable α et da la valeur de x. Cela posé, la fonction 

f (x) sera, entre les deux limites assignées à la variable x, fonction continue de cette variable, si, 

pour chaque valeur de x intermédiaire entre ces limites, la valeur numérique de la différence 

f (x + α) − f (x) 

décroit indéfiniment avec celle de α. En d’autres termes, la fonction f (x) restera continue par rapport 

à x entre les limites données, si, entre ces limites, un accroissement infiniment petit de la va-

11.5. CAUCHY’s proof of the binomial theorem 217 

CAUCHY’S novel definition defines continuity not at a point (as is customary to- 

day, see below) but on an entire interval enclosed by two boundary points. 23 Thus, 

CAUCHY’S implicit choice of infinitesimal ω such that 

| f (x + α) − f (x)| = ω 

could seem to be independent of x on the interval and the definition would actually 

be of what is now known as a uniformly continuous function. The doubt over the proper 

interpretation is introduced by the fact CAUCHY’S use of the symbol ω to designate 

the infinitesimal: it “hides” the order in which the limit processes are to be carried out. 

CAUCHY and series of functions. In a famous theorem which provided an impor- 

tant step in CAUCHY’S proof of the binomial theorem, CAUCHY sought to link the 

concepts of convergence and continuity. Since we will discuss the theorem in details 

in chapter 12, its entire wording and CAUCHY’S proof of it are reproduced here. 

“1st theorem. Whenever the different terms of the series u0 + u1 + u2 + · · · + un + 

. . . are functions of one and the same variable x and continuous with respect to this variable 

in the neighborhood of a particular value for which the series is convergent, the sum 

s of the series is also a continuous function of x in the neighborhood of that particular 

value.” 24 

As was customary, CAUCHY actually presented the proof before he made the theo- 

rem explicit. The proof which he gave proceeded along the following lines. If the sum 

is split after n terms 

s = sn + rn = 

n−1 

∑ un + 

k=0 

∞ 

∑ un, (11.7) 

k=n 

the partial sum sn is a polynomial and therefore continuous and the remainder rn can 

be made less than any given quantity by the convergence of the series. In consequence, 

the difference s (x + α) − s (x) could be made less than any assignable quantity and 

the sum was therefore continuous. As we are well aware today, with our common 

interpretations of the basic concepts of continuity, limits, and convergence, the theo- 

rem is false as stated. In section 14.1.2, its future history through the works of ABEL, 

P. L. VON SEIDEL (1821–1896) (and G. G. STOKES (1819–1903)) and CAUCHY again 

is outlined to understand how ABEL’S contribution to rigorization was accepted and 

interpreted. 

riable produit toujours un accroissement infiniment petit de la fonction elle-même.” (A.-L. Cauchy, 

1821a, 34–35). 

23 See also (Bottazzini, 1990, lxxxi–lxxxiii) and (Giusti, 1984). 

24 “1.er Théorème. Lorsque les différens termes de la série (1) sont des fonctions d’une même variable 

x, continues par rapport à cette variable dans le voisinage d’une valeur particulière pour laquelle la 

série est convergente, la somme s de la série est aussi, dans le voisinage de cette valeur particulière, 

fonction continue de x.” (A.-L. Cauchy, 1821a, 131–132).


For now, we have to stress the importance this theorem played in CAUCHY’S proof 

of the binomial theorem. The argument was, that since both the binomial (1 + x) m and 

the infinite power series (11.4) satisfied the same functional equation for all rational m, 

and since they were both continuous functions of m, they would coincide for all real 

values of m. Thus, CAUCHY proved the binomial theorem for all real exponents (here 

designated m). 

In CAUCHY’S theory of rigorously restructuring calculus, the binomial theorem 

played an extremely important role. 25 It provided one of the basic, theoretical bricks 

which could be used to rebuild the existing theory of real analysis. Historians have 

argued that CAUCHY’S rigorization and his proof of the binomial theorem constitute 

a veiled attack on Fourier series. 26 However, although CAUCHY was critical toward 

J. B. J. FOURIER’S (1768–1830) reasoning, I find such a hypothesis largely unnecessary 

as CAUCHY’S rigorization program makes good sense from its own premises. 27 

11.6 Early reception of CAUCHY’s new rigor 

CAUCHY’S Cours d’analyse dealt exclusively with the theory functions from the per- 

spective of infinite series. Later in the 1820s, he also published lectures pertaining to 

rigorously founding the theory of differentiation and integration. 28 

As a textbook for the École Polytechnique, the Cours d’analyse was not successful. 

Because of internal animosities among the teachers, CAUCHY’S textbook was never 

used as a textbook but it may have served as inspiration for students preparing for 

the entrance exams of the school. Among his fellow mathematicians, CAUCHY’S pro- 

gram also received mixed reactions. In Germany, A. L. CRELLE (1780–1855) men- 

tioned CAUCHY’S textbooks in very positive terms, 29 and a German translation of the 

Cours d’analyse appeared in 1828. 30 However, a distinct German reaction also existed 

which sought to continue the formal, algebraic approach to foundations of analysis in 

the so-called combinatorial school initiated by C. F. HINDENBURG (1741–1808), and M. 

OHM (1792–1872) pursued his own rigorization program. 31 

Thus, in the 1820s, the mathematical community could be divided into three camps 

reflecting their attitudes toward rigor: 

1. Some had picked up CAUCHY’S vision of a rigorization of analysis; both its 

theme and its tools. They joined in the restriction to arithmetical equality and 

adopted CAUCHY’S redefinition of central concepts in terms of limits. 

25 (Grabiner, 1981b, 111). 

26 See e.g. (Bottazzini, 1986, 110). 

27 (Grabiner, 1981b, 111). 

28 (A.-L. Cauchy, 1823; A.-L. Cauchy, 1829). 

29 (Crelle, 1827; Crelle, 1828). 

30 (A. L. Cauchy, 1828). 

31 See (Jahnke, 1992).

11.6. Early reception of CAUCHY’s new rigor 219 

2. Others, primarily Germans, felt a similar need for a new rigorous foundation 

of the calculus. Often inspired by needs stimulated by the German educational 

reforms, they sought to distill a combinatorial theory from the approaches of 

LAGRANGE and others. 

3. The rest, and the vast majority, were mostly concerned with contributing new 

mathematical knowledge. Although they probably also sometimes worried about 

the foundations of their discipline, they left it to the teachers and experts to 

straighten it out. 

In the 1820s, the need for rigorization had entered the agenda and at least two 

programs had been suggested for resolving the need. In the course of the century, 

the need became even more urgent and CAUCHY’S conception became the preferred 

solution — after it had undergone the interpretations of ABEL, G. P. L. DIRICHLET 

(1805–1859), G. F. B. RIEMANN (1826–1866), K. T. W. WEIERSTRASS (1815–1897) and 

others.

Chapter 12 

ABEL’s reading of CAUCHY’s new rigor 

and the binomial theorem 

N. H. ABEL (1802–1829) was one of the first converts to A.-L. CAUCHY’S (1789–1857) 

new program of rigorizing analysis. Alone and together with B. M. HOLMBOE (1795– 

1850), ABEL had studied L. EULER’S (1707–1783) works on analysis and other text- 

books on the calculus from the late eighteenth century. 1 But during his time in Berlin, 

ABEL learnt of CAUCHY’S textbooks Cours d’analyse and Resumé des leçons. 2 His ac- 

quaintance with these works left clear traces in his publications and letters. Although 

CAUCHY’S rigorization program comprised all of analysis (at least in principle), ABEL 

was particularly interested in the theory of series. ABEL’S interest in the theory of 

series manifested itself in two publications, numerous remarks in letters, and an inter- 

esting draft in one of his notebooks. 

ABEL’S most prestigious contribution to the rigorization of analysis was a paper 

published in 1826 which contained a new proof of the binomial theorem. 3 ABEL had 

previously employed a complete induction argument to deduce the binomial formula. 4 

The method of complete induction had previously been used by B. BOLZANO (1781– 

1848) in his Der binomische Lehrsatz, 5 and together with ABEL’S curious and flattering 

remark concerning BOLZANO (see page 42), this might suggest that ABEL was familiar 

with this work. However, there is no direct evidence to support this speculation. 

When it came to proving the binomial theorem, ABEL followed the path set out by 

CAUCHY’S Cours d’analyse, which he praised highly, recommending it to anybody who 

loved the rigor of mathematics. 6 However, in certain details, ABEL’S deduction dif- 

fered slightly from the guideline of the Cours d’analyse. In particular, ABEL presented 

his own way of deducing the important ratio tests and when he found that Cauchy’s 

1 See the sections 3.2 and 2.2, above. 

2 (A.-L. Cauchy, 1821a; A.-L. Cauchy, 1823). 

3 (N. H. Abel, 1826f). 


5 (Bolzano, 1816, §7–10). 

6 (N. H. Abel, 1826f, 313). 

221

222 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem 

Theorem “suffered exceptions”, he replaced it with a tailored — but also problematic — 

theorem which he found sufficient to carry through his argument. The description and 

analysis of all these aspects and the differences between ABEL’S and CAUCHY’S con- 

cepts and proofs are the purposes of the present chapter. 

The Cours d’analyse was certainly ABEL’S main inspiration for his research on the 

theory of series. From another one of his letters, 7 we learn that ABEL had bought 

and read the first nine issues of CAUCHY’S Exercises des mathématiques. Although they 

proved highly interesting in another context, 8 these installments contained nothing 

with an explicit bearing on the rigorization of the theory of series. Another one of 

CAUCHY’S publications — the Resumé des leçons — did address infinite series and, as 

noticed, we know from one of his letters that ABEL was familiar with it. Continu- 

ing the quotation on page 31 taken from a letter to HOLMBOE, ABEL expressed his 

concerns over the state of analysis (see below) and wrote: 

“The Taylorian Theorem, the foundation for all higher mathematics is equally 

ill founded. Only one rigorous proof have I found and that is by Cauchy in his 

Resumé des leçons sur le calcul infinitesimal. There, he proves that 

φ (x + α) = φx + αφ ′ x + α2 

2 φ′′ x + . . . 

whenever the series is convergent (but it is frequently used in all cases).” 9 

Thus, ABEL claimed that CAUCHY had proved in the Resumés that any convergent 

Taylor series expansion represents the function. Certainly, CAUCHY considered the 

theorem of B. TAYLOR (1685–1731) of major importance in his lectures and repeated 

his programmatic criticism of working with divergent series. However, the statement 

which ABEL attributed to CAUCHY is exactly what CAUCHY criticized by ways of a 

counter example in the Resumé: In the Résume, CAUCHY considered the function de- 

fined by f (x) = e 

1 − 

x2 whose derivatives all vanished at the origin and whose Maclau- 

rin series therefore was the zero function. This counter example and the use which 

CAUCHY made of it will be discussed further in section 21.3. 

Thus, it appears that ABEL misread or misunderstood CAUCHY. How could a 

smart mathematician who was — in the words of I. GRATTAN-GUINNESS — was “more 

Cauchyian than Cauchy” be led to such a statement? 10 A hint may be taken from 

7 See the quotation p. 306. 

8 See section 16.2.3. 

9 “Det Taylorske Theorem, Grundlaget for hele den høiere Mathematik er ligesaa slet begrundet. Kun 

eet eneste strængt Beviis har jeg fundet og det er af Cauchy i hans Resumé des leçons sur le calcul 

infinitesimal. Han viser der at man har: 

φ (x + α) = φx + αφ ′ x + α2 

2 φ′′ x + . . . 

saa ofte Rækken er convergente, (men man bruger den rask væk i alle Tilfælde).” (Abel→Holmboe, 

1826/01/16. N. H. Abel, 1902a, 16–17). 

10 (I. Grattan-Guinness, 1970b, 80).

12.1. ABEL’s critical attitude 223 

from a manuscript entitled Sur les séries which ABEL worked on and hoped to present 

in A. L. CRELLE’S (1780–1855) Journal. However, the publication never materialized 

and ABEL’S manuscript was left in the form of an interesting draft. It was eventu- 

ally published in the Œuvres. 11 In the draft, ABEL expanded an otherwise unspecified 

function 

and rearranged its terms to find 

f (x + ω) = a0 + a1 (x + ω) + a2 (x + ω) 2 + . . . (12.1) 

f (x + ω) = a0 + a1x + a2x 2 + · · · + (a1 + 2a2x + . . . ) ω + . . . . 

From this, ABEL concluded 

f (x + ω) = f (x) + f ′ (x) 

1 ω + f ′′ (x) 

2 ω2 + . . . (12.2) 

“if this series is convergent”. 12 The argument was followed by remarks to the effect 

that the series of (12.2) was indeed always convergent! This serves to illustrate that de- 

spite the extensive criticism which ABEL raised against the unrigorous reasoning with 

series, his own reasoning was constantly at risk of making the same mistakes. Fur- 

thermore, the example shows how the reordering of terms was an unrealized problem 

in the 1820s. This becomes interesting when we consider the emergence of a concept 

of absolute convergence (see below). 

12.1 ABEL’s critical attitude 

ABEL’S name is frequently mentioned in the same sentence as CAUCHY and K. T. W. 

WEIERSTRASS (1815–1897) when historians of mathematics attempt to pin-point the 

movement within mathematics known as rigorization or — more specifically — arith- 

metization. 13 And certainly, after an almost religious conversion, ABEL became an ar- 

dent follower of a version of CAUCHY’S new rigor; a version which ABEL to a large 

extent helped form, himself. On the other hand, rigorizing the calculus meant re- 

founding the entire domain of analysis on a completely new system, and ABEL’S 

mathematical contribution to the rigorization was limited to a single sub-discipline, 

the theory of infinite series. But of equal importance, ABEL’S written testimony of his 

conversion to Cauchyism and his hearted, public interpretation of some of its doctrines 

helped shape the movement in the nineteenth century. In this and the following chap- 

ter, ABEL’S critical attitude as well as his contributions to the theory of series will be 

investigated and analyzed. 


12 (ibid., 204). 

13 See e.g. (Kline, 1990, 948).


ABEL’S critical attitude was expressed both in his letters and in the opening para- 

graphs as well as in the overall structure of his important paper on the binomial the- 

orem. ABEL attacked the usual way of reasoning about infinite series which he saw 

as an induction from the permissible ways of reasoning about finite series, i.e. poly- 

nomials. This distinction between finite and infinite arguments was one of the core 

components of ABEL’S critical attitude. 

A direct, outspoken ban on divergent series. When ABEL announced the contents 

of his binomial paper, he explicitly pointed to the fact that convergence of a series was 

a requirement to be established before anything like an equality between the series 

and other (possibly infinite) expressions were to be asserted. He wrote, combining his 

distinction between finite and infinite expressions and his criticism of divergent series, 

“This equation [the CAUCHY product of infinite series] is completely correct 

when both of the series 

u0 + u1 + . . . and v0 + v1 + . . . 

are finite. If they are infinite, they must firstly necessarily converge — because a divergent 

series has no sum — and then the series in the second term [the CAUCHY 

product] must also converge. Only with these restrictions is the statement above 

correct. If I am not mistaken, this restriction has hitherto not been taken into 

account.” 14 

Thus, even in the case of the CAUCHY multiplication of infinite series, ABEL found 

reason to criticize current practice on the same grounds as stated above; different rea- 

soning applied to finite and infinite series, and divergent series have no sum. When 

ABEL’S proof of the multiplication theorem has been described, a further analysis of 

its relations to CAUCHY’S original version can be described (see section 12.8, below). 

In some of his letters, ABEL was even more outspoken about the status of divergent 

series and the implications they had had on the development of rigorous mathematics. 

ABEL’S notion of rigorous proof and critical revision. In another frequently quoted 

letter, written shortly after leaving Berlin in 1826 and directed to C. HANSTEEN (1784– 

1873), ABEL spoke of turning more of his attention toward the study of analysis, and 

again commented on the status of the field. 

“I will commit all my strength to shedding some light on the immense darkness, 

which incontestably covers analysis. It [analysis] completely lacks all plan 

14 “Diese Gleichung ist vollkommen richtig, wenn die beiden Reihen 

u0 + u1 + . . . und v0 + v1 + . . . 

endlich sind. Sind sie aber unendlich, so müssen sie erstlich nothwendig convergiren, weil eine 

divergirende Reihe keine Summe hat, und dann muß auch die Reihe im zweiten Gliede ebenfalls 

convergiren. Nur mit dieser Einschränkung ist der obige Ausdruck richtig. Irre ich nicht, so ist diese 

Einschränkung bis jetzt nicht berücksichtigt worden.” (N. H. Abel, 1826f, 311).


and coherence, so it is highly remarkable that it can be studied by so many — and 

now worst of all that it is not treated rigorously. [. . . ] Whenever one proceeds in 

the ordinary fashion, it is probably all right; but I have had to be very cautious because 

the theorems which have been accepted without rigorous proof (i.e. without 

proof) have struck such deep roots with me that I constantly run the risk of using 

them without further probing.” 15 

Here, we learn of another distinction which ABEL saw between ordinary proofs and 

rigorous proofs. He was acutely aware that a fundamental change in the techniques of 

proving mathematical theorems was required. At the same time, ABEL also expressed 

the opinion that it would be interesting to investigate how unrigorous reasoning had 

led to correct results in almost all cases. This idea of critically revising the existing 

structure of a mathematical theory is as old as the rigorization movement and had 

first been stated in the Berlin Academy prize problem for 1784. 16 ABEL also suggested 

an answer to the question when he emphasized that analysis until most recently had 

only worked with power series and for these, the established methods of reasoning 

did not lead to false results. A completely different situation could arise if other series 

were included in the study, 17 ABEL remarked thereby alluding to both Fourier series 

and Poisson’s example described above. 

ABEL and the paradoxes of analysis. On two occasions, in letters to HOLMBOE and 

HANSTEEN written in 1826, ABEL described some of the paradoxes to which unrig- 

orous reasoning had led. 18 Of the two, the letter to HOLMBOE is the most detailed. 

There, ABEL ridiculed anybody willing to claim equalities such as 

0 = 1 − 2 n + 3 n − 4 n + . . . . (12.3) 

ABEL gave no references as to where he had picked up this absurd equality but we 

get a hint from another much more famous example which he described. In the letter 

and in an intriguing footnote in the binomial paper — which is discussed below, see 

12.6 — ABEL called attention to the series 

x 

2 = 

∞ (−1) 

∑ 

n=1 

n−1 sin nx 

n 

(12.4) 

15 “Alle mine Kræfter vil jeg anvende paa at bringe noget mere Lys i det uhyre Mørke som der uimodsigelig 

nu findes i Analysen. Den mangler saa ganske al Plan og System, saaat det virkelig er 

høist forunderlig at den kan studeres af saa mange og nu det værste at den aldeles ikke er stræng 

behandlet. [. . . ] Naar man blot gaaer almindelig tilværks saa gaaer det nok; men jeg har maattet 

være særdeles forsigtig, thi de engang uden strængt Bevis (c : uden Bevis) antagne Sætninger har 

slaaet saa dybe Rødder hos mig at jeg hvert Øjeblik staaer Fare at bruge dem uden nøiere Prøvelse.” 

(Abel→Hansteen, Dresden, 1826/03/29. N. H. Abel, 1902a, 22–23). 

16 (Grabiner, 1981b, 40–43), the passage on revision in the prize proposal is translated (ibid., 41). 

17 (Abel→Hansteen, Dresden, 1826/03/29. N. H. Abel, 1902a, 22–23). 

18 (Abel→Holmboe, 1826/01/16. In ibid., 13–19) and (Abel→Hansteen, Dresden, 1826/03/29. In 

ibid., 22–26).


which is the Fourier series corresponding to the function f (x) = x 2 on the interval 

]−π, π[. However, as ABEL noted in the letter, by inserting e.g. x = π he would be led 

to the absurd equality 

π 

2 = 

∞ (−1) 

∑ 

n=1 

n−1 sin nπ 

= 0. 

n 

In his letter, ABEL applied this example to illustrate that although an equality held 

for x < π it could fail in the limit x = π. This criticism is thus an elaboration of 

CAUCHY’S dismissal of the generality of algebra. ABEL continued, 

“Operations are applied to infinite series as if they were finite but is that permissible? 

I doubt it. — Where has it been proved that one obtains the differential 

of a series by differentiating each term? It is easy to present examples for which 

this it not true.” 19 

Here, ABEL indirectly criticized J. B. J. FOURIER’S (1768–1830) interchange of the limit 

processes involved in term-wise integration. Differentiating the series (12.4), ABEL 

obtained 

1 

2 = 

∞ 

∑ (−1) 

n=1 

n−1 cos nx 

in which the series was divergent. Now, the example (12.3) can be seen to result if this 

procedure of differentiation is repeated and either x = π or x = π 2 

is inserted. 

ABEL’S reaction to Poisson’s example. A strong connection between ABEL’S research 

on the theory of series and Poisson’s example is clearly discernible from his letter to 

HOLMBOE. 20 There, ABEL explained how he had undertaken to find the sum of the 

series 

m (m − 1) 

cos mx + m cos (m − 2) x + cos (m − 4) x + . . . 

2 

which was an important open problem at the time. ABEL mentioned that a large num- 

ber of mathematicians (including S.-D. POISSON (1781–1840) and CRELLE) had failed 

to solve the problem but that he, himself, had found a complete answer in the form 

m (m − 1) 

cos mx + m cos (m − 2) x + cos (m − 4) x + · · · = (2 + 2 cos 2x) 

2 

m 2 cos mkπ 

� 

in which m > −1, k an integer and k − 1 � � 

2 π < x < k + 1 � 

2 π; for m < −1, the series 

was divergent and this led to ABEL’S outburst against the use of divergent series: 

“Divergent series are the creations of the Devil and it is a shame that anybody 

dare construct a demonstration upon them.” 21 

19 “Man anvender alle Operationer paa uendelige Rækker som om de vare endelige, men er dette 

tilladt? Vel neppe. — Hvor staar det beviist at man faaer Differentialet af en uendelig Række ved at 

differentiere hvert Led? Det er let at anføre Exempler hvor dette ikke er rigtigt.” (Abel→Holmboe, 

1826/01/16. N. H. Abel, 1902a, 18). 

20 (Abel→Holmboe, 1826/01/16. In ibid., 13–19).


Thus, ABEL’S criticism of divergent series was intimately tied to his interest in Poisson’s 

example and the other paradoxes in the theory of infinite series. 

The core components of ABEL’S criticism. ABEL’S critical position toward the con- 

temporary conceptions of rigor in analysis can thus be divided into three parts. The 

first part was primarily rhetorical emphasizing his belief that the accepted standards 

were utterly insufficient and ABEL supported his argument by enlisting a number of 

“paradoxes”, in particular (12.3). 

The second — and more substantial part — consisted of an attempt at locating the 

points where the customary reasoning was led astray. Among the critical points, ABEL 

repeated and radicalized CAUCHY’S dogma that divergent series have no sum and 

should not be treated in analysis. Beside his ban on divergent series, ABEL also re- 

peated CAUCHY’S concern for numerical equality and stressed that even if two expres- 

sions were numerically equal in the interior of an interval they needed not coincide in 

the endpoints. This led him to explicitly question specific practices such as the passing 

to the limit in power series and the term-wise differentiation (and integration). 

The third component of ABEL’S critical position was more constructive. In his 

letters and publications, he suggested that most of the unfortunate paradoxes and 

malpractices arose out of considering series which were not power series. This led 

him to focus attention on power series which he saw as some sort of safe haven where 

the commonly used methods would still apply. 

The structure of ABEL’S proof of the binomial theorem. The path which ABEL took 

in his publication on the binomial theorem makes a lot of sense when seen from a per- 

spective integrating Poisson’s example and the components of ABEL’S criticism listed 

above. In the binomial paper, ABEL separated two distinct problems concerning the 

binomial theorem. First, he wanted to find the precise set of assumptions on m and x 

for which the binomial series 

∞ 

∑ 

n=0 

� � 

m 

x 

n 

n 

converged. In order to do so, he developed and revised some important theorems 

in the theory of series. And second, he wanted to investigate whether — in the cases 

where the series converged — the sum of the series agreed with (one of the values of) 

the binomial 

(1 + x) m . 

This approach was adapted to overcome the problems of multi-valued functions which 

lie at the core of the Poisson’s example. 

21 “Divergente Rækker ere i det Hele noget Fandenskab, og det er en Skam at man vover at grunde 

nogen Demonstration derpaa.” (Abel→Holmboe, 1826/01/16. ibid., 16).


12.2 Infinitesimals 

ABEL’S reading and rendering of CAUCHY’S fundamental concepts came to influence 

the development of analysis in the nineteenth century. At certain points, his interpre- 

tations were clearer and more specific and some of them would eventually coincide 

with the standardized interpretations laid down by men such as G. P. L. DIRICHLET 

(1805–1859) and WEIERSTRASS. 

One of the basic notions which ABEL introduced in his binomial paper was that of 

infinitesimals. In a footnote, ABEL explained, 

“For brevity, in this paper ω denotes a quantity which can be less than any 

given arbitrarily small quantity.” 22 

Despite the awe which ABEL felt for CAUCHY’S work, this definition is not truly in 

accord with CAUCHY’S notion of infinitesimals. As discussed above, CAUCHY had 

interpreted infinitesimals as variables with limit zero but in ABEL’S definition, the 

infinitesimals seem to reenter as completed quantities less than any finite quantity 

but different from zero. The limit process has seemingly faded into the background. 

To illustrate this way of designating infinitesimals by symbols, we may reconsider 

CAUCHY’S proof of the Cauchy Theorem (see page 217) interpreted in ABEL’S notation. 

ABEL did not undertake this proof, but the arguments are directly to those which he 

employed in proving the Lehrsatz IV (see below). By continuity of the finite polyno- 

mial sn, 

and by the convergence of s, 

Therefore 

sn (x + α) − sn (x) = ω, 

rn (x + α) = ω, (12.5) 

rn (x) = ω. 

s (x + α) − s (x) = ω 

and the continuity has been “proved”. This way of designating infinitesimals by the 

same symbols regardless of the way in which they depend on other variables and 

infinitesimals hid and obscured the basic problems of the above argument. In the ar- 

gument, the n which appears in (12.5) must depend on α and ω and can be unbounded 

as α and ω vanish. However, it took quite some time and a detailed analysis of these 

dependencies to clear out the proof (see section 14.1.2, below). 

22 “Die Kürze wegen soll in dieser Abhandlung unter ω eine Größe verstanden werden, die kleiner 

sein kann, als jede gegebene, noch so kleine Größe.” (N. H. Abel, 1826f, 313, footnote).

12.3. Convergence 229 

12.3 Convergence 

While ABEL’S definition and use of infinitesimals were not completely in the line of 

CAUCHY’S new rigor, his concept of convergence — and the importance which he at- 

tributed to it — closely resembled CAUCHY’S. 

“Definition. An arbitrary 

v0 + v1 + v2 + · · · + vm etc. 

will be called convergent if the sum v0 + v1 + · · · + vm steadily approaches a certain 

limit for ever increasing values of m. This limit will be called the sum of the 

series. In the contrary case, the series is called divergent and therefore has no sum. 

From this definition follows that for a series to converge, it will be necessary and 

sufficient that the sum vm + vm+1 + · · · + vm+n steadily approach zero for ever 

increasing values of m whatever value n may have.” 23 

Just as CAUCHY had done, ABEL quickly related the convergence of a series to the 

Cauchy criterion and claimed that it constituted a necessary and sufficient condition 

for convergence. As described above, the assertion that convergence followed from 

the Cauchy criterion was later realized to be non-trivial, but in the 1820s it was con- 

sidered obvious. Although both CAUCHY and ABEL drew the connection between 

convergence and the Cauchy criterion, ABEL gave the criterion a much more central 

position in his theory of series as will be described below (see page 231). 

Immediately following his definition of convergence, ABEL made the rather curious 

remark that “in every arbitrary series, the general term vm will approach zero.” 24 

Judging from the context, an omission of the word “convergent” must have crept in at 

this point. 25 

An extended ratio test: Lehrsätze I&II. The first theorem of ABEL’S binomial paper 

is most remarkable because of its conceptual contents. Without proof (see a modern- 

ized proof in box 2), ABEL observed that for any series of positive terms 

23 “Erklärung. Eine beliebige Reihe 

∞ 

∑ ρm 

m=0 

v0 + v1 + v2 + · · · + vm u.s.w. 

soll convergent heißen, wenn, für stets wachsende Werthe von m, die Summe v0 + v1 + · · · + vm sich 

immerfort eine gewisse Gränze nähert. Diese Grenze soll Summe der Reihe heißen. Im entgegengesetzten 

Falle soll die Reihe divergent heißen, und hat alsdann keine Summe. Aus dieser Erklärung 

folgt, daß, wenn eine Reihe convergiren soll, es nothwendig und hinreichend sein wird, daß, für 

stets wachsende Werthe von m, die Summe vm + vm+1 + · · · + vm+n sich Null immerfort nähert, 

welchen Werth auch n haben mag.” (ibid., 313). 

24 “In irgend einer beliebigen Reihe wird also das allgemeine Glied vm sich Null stets nähern.” (ibid., 

313). 

25 This omission has therefore also been silently corrected in the French translation found in (N. H. 

Abel, 1839; N. H. Abel, 1881).


Proof of Lehrsatz I In order to present a proof of Lehrsatz I, we write ε = α−1 

2 

and chose n ∈ N such that 

Then, by iteration, 

ρm+1 ≥ (α − ε) ρm for all m ≥ n. 

ρm+1 ≥ (α − ε) m−n ρn. 

The choice of ε ensures that ρm+1 increases beyond all bounds. Now assume that the 

series in (12.6), ∑ εmρm, is to be convergent. Then, in particular, a k ∈ N exists such 

that 

If m ≥ max (k, n), 

meaning 

|εm+1ρm+1| < ε for all m ≥ k. 

ε > |εm+1| · ρm+1 ≥ |εm+1| (α − ε) m−n ρn 

|εm+1| ≤ ε 

ρn 

1 

m−n → 0 for m → ∞ 

(α − ε) 

because α − ε > 1. In conclusion, if ∑ εmρm is be convergent, the sequence {εm} has 

to converge toward zero. And since this is not the case, the sum (12.6) cannot be 

convergent. ✷ 

Box 2: Proof of Lehrsatz I 

for which the ratio of consecutive terms converges toward α > 1, 

any linear combination 

ρm+1 

ρm 

→ α > 1, 

∞ 

∑ εmρm 

m=0 

will be divergent, provided the sequence {εm} does not converge to zero. 

> 0 

(12.6) 

The contents of this Lehrsatz I is thus a generalization of one part of CAUCHY’S ra- 

tio test of convergence. It is remarkable from a conceptual viewpoint that ABEL’S first 

theorem would be one of divergence when his entire theory was so focused on conver- 

gent series. The Lehrsatz I is thus — as it stands — a negative demarcation criterion. 

This apparent imbalance was leveled by the second theorem. ABEL’S Lehrsatz II 

is a counterpart to the Lehrsatz I describing analogous — but this time sufficient — 

conditions for convergence. ABEL found that if 

ρm+1 

ρm 

→ α < 1

12.3. Convergence 231 

and {εm} was a sequence of terms which do not exceed 1 (ABEL was not explicit about 

requiring εm to be positive; if this assumption is not made, the requirement would be 

|εm| < 1), the series 

∞ 

∑ εmρm 

m=0 

was necessarily convergent. Throughout, ABEL’S use of numerical values was spo- 

radic. At times, he noticed that numerical values had to be taken, at other times such 

a remark might be inferred from his language, and at yet other times — as will be 

shown with respect to Lehrsatz VI — it seems to have evaded his attention. For in- 

stance, in the example given above, ABEL actually required that εm did not surpass 

unity which could also mean −1 < εm < 1. 

ABEL’S proof of Lehrsatz II had similarities with the modernized proof of Lehrsatz 

I given in box 2 although he chose to use the characterization of convergence by the 

Cauchy criterion given above. He argued that for m sufficiently large, 

and consequently 

m+n 

∑ ρk < ρm 

k=m 

ρ m+k < α k ρm 

n 

∑ α 

k=0 

k 1 − α 

= ρm 

n+1 

1 − α 

< ρm 

1 − α . 

Since εm < 1, a similar conclusion would hold for the series εmρm, 

m+n 

∑ εkρk < 

k=m 

ρm 

1 − α . 

(12.7) 

ABEL concluded the argument by observing that ρm → 0 followed from (12.7), and 

thus the convergence of the series was secured by the characterization. 

Although ABEL’S way to the two theorems might seem obvious to a modern reader, 

it is interesting to compare ABEL’S theorems including their proofs with CAUCHY’S 

original deduction of the ratio test as given in the Cours d’analyse (see page 212, above). 

Such a comparison reveals that CAUCHY and ABEL devised different structural sys- 

tems for their theories of infinite series. Although the results were the same, the the- 

orems and proofs played slightly different roles in the two systems. In CAUCHY’S 

theory, the characterization of convergent series by means of the Cauchy criterion was 

noticed but was of little use in obtaining the other — more important — tests of con- 

vergence. Instead, these tests were derived from the convergence of geometric pro- 

gressions which was proved directly. In ABEL’S theory, however, the Cauchy criterion 

was made into a central tool which he used to deduce his slightly generalized version 

of the ratio test (see figure 12.1). This made the proof of the ratio test much simpler in 

ABEL’S framework than it had been in CAUCHY’S Cours d’analyse. In the manuscript 

Sur les séries, ABEL placed the Cauchy criterion in an equally central position.


CAUCHY’s structure ABEL’s structure 

Definition of convergence 

✟ 

✟❍ 

✟✟✙ 

❍❍❍❥ 

Cauchy Geometric 

criterion progression 

❄ 

Root test 

❄ 

Ratio test 

Definition of convergence 

❄ 

Cauchy criterion 

❄ 

Ratio test 

Figure 12.1: Comparison of CAUCHY’s and ABEL’s structures of the basic theory of 

infinite series. 

An auxiliary theorem: Lehrsatz III. As his third theorem, 26 ABEL presented an aux- 

iliary result which — although not difficult — was put to great use in the proofs to 

follow. He demonstrated that if {tn} denoted a sequence whose partial sums were 

bounded, 

m 

∑ tk < δ for all m ∈ N, 

k=0 

and {εn} denoted a decreasing sequence of positve terms, then 

rm = 

m 

∑ εktk < δε0 for all m ∈ N. 

k=0 

ABEL’S proof consisted of a rather simple manipulation, in which he observed that 

with 

each term could be written as 

and, thus, 

rm = 

= 

m 

∑ εktk = 

k=0 

Since {εn} was decreasing, 

26 (N. H. Abel, 1826f, 314) 

pm = 

m 

∑ tk, k=0 

t k = p k − p k−1, 

m 

∑ εk (pk − pk−1) = 

k=0 

m−1 

∑ pk (εk − εk+1) + εmpm. 

k=0 

0 < ε k − ε k+1 < ε k < ε0, 

m 

∑ εkp k − 

k=0 

m−1 

∑ εk+1p k 

k=0

12.4. Continuity 233 

it followed that 

rm < 

and the inequality had been obtained. 

12.4 Continuity 

m−1 

∑ δ (εk − εk+1) + εmδ = δε0, 

k=0 

Just as had been the case with CAUCHY’S proof of the binomial theorem, the concept 

of continuity played an important role in ABEL’S proof. In his paper on the binomial 

theorem, ABEL gave rudiments of a different rendering of the theory of interaction 

between the concepts of continuity and convergence. ABEL’S definition of continuity 

seems to closely resemble CAUCHY’S (see page 216), although it may be noticed that 

ABEL’S definition is only formulated in the terminology of limits. 

“Definition. A function f (x) shall be called a continuous function of x between 

the boundaries x = 0 and x = b when for any arbitrary value of x between these 

limits, the quantity f (x − β) for ever decreasing values of β approach the limit 

f (x).” 27 

Although their definitions of continuity are almost identical, ABEL and CAUCHY 

attributed slightly different meaning to their concepts when they were employed. The 

apparent ambiguity concerning the order in which quantification is to be made in 

CAUCHY’S definition was resolved in ABEL’S persistent insistence on point-wise def- 

initions. ABEL’S definition as stated seems just as susceptible to the ambiguity as 

CAUCHY’S, but ABEL throughout interpreted it to mean that a function is continuous 

at a point x ∈ [0, b] if f (x − β) → f (x) as β → 0. 

Combining continuity, convergence, and power series. The fourth and fifth theo- 

rems of ABEL’S binomial paper provided two important combinations of the concepts 

of continuity and convergence. The fourth theorem, Lehrsatz IV, stated and proved the 

continuity of a power series in the interior of its interval of convergence, while Lehrsatz 

V attempted to provide a rigorous replacement for what Cauchy’s Theorem (see page 

217) had promised but not rigorously delivered. At this point, Lehrsatz IV together 

with its proof will be described first, and Lehrsatz V will be postponed to be discussed 

in its proper context of ABEL’S famous Ausnahme or counter example to CAUCHY’S 

theorem (see section 12.6). At that point, the strong internal relations between the two 

theorems will also be described and explained. 

As his fourth theorem, ABEL stated and proved the following result which has 

become a classic of the theory of series and is often associated with ABEL’S name. 

27 “Erklärung. Eine Function f (x) soll stetige Function von x, zwischen den Grenzen x = 0, x = b 

heißen, wenn für einen beliebigen Werth von x, zwischen diesen Grenzen, die Größe f (x − β) sich 

für stets abnehmende Werthe von β, der Grenze f (x) nähert.” (ibid., 314).


“Lehrsatz IV. When the series 

f (α) = v0 + v1α + v2α 2 + · · · + vmα m + . . . 

converges for a certain value δ of α, it will also converge for every smaller value of 

α. Furthermore it will be of the sort that f (α − β) approaches the limit f (α) for 

ever decreasing values of β provided α is less than or equal to δ.” 28 

In most modern presentations, the variable α is interpreted as a complex variable, 

and the theorem states the continuity of a power series in the interior of its disc of 

convergence. However, in this part of the paper, ABEL was exclusively interested in 

power series with real terms. 

In order to facilitate comparison with ABEL’S proof of the fifth theorem of the paper 

and in order to exemplify ABEL’S use of infinitesimals in his arguments, a presentation 

of ABEL’S proof is worth giving. It should be remarked even before embarking on 

a tour of ABEL’S proof, that at a number of points its argument diverges from the 

modernized version of the proof; these occasions will be noticed below and elaborated 

in the following section. 

ABEL began his proof by splitting the power series after m terms, 

φ (α) = 

Then, he rewrote the tail of the series as 

m−1 

∑ vnα 

n=0 

n ∞ 

and ψ (α) = ∑ vnα 

n=m 

n . 

ψ (α) = 

∞ 

∑ 

n=m 

and obtained from Lehrsatz III the inequality 

ψ (α) < 

� 

α 

�n vnδ 

δ 

n 

� 

α 

�m · p (12.8) 

δ 

“where p denotes the largest number among the quantities vmδ m , vmδ m + vm+1δ m+1 , 

vmδ m + vm+1δ m+1 + vm+2δ m+2 etc.” 29 This definition of p might seem strange or dan- 

gerous to the modern reader. When translated into modern notation, ABEL’S p corre- 

sponds to the following supremum 

28 “Lehrsatz IV. Wenn die Reihe 

p = sup 

k 

m+k 

∑ 

n=m 

f (α) = v0 + v1α + v2α 2 + · · · + vmα m + . . . 

vnδ n . (12.9) 

für einen gewissen Werth δ von α convergirt, so wird sie auch für jeden kleineren Werth von α 

convergiren, und von der Art seyn, daß f (α − β), für stets abnehmende Werthe von β, sich der 

Grenze f (α) nähert, vorausgesetzt, daß α gleich oder kleiner ist als δ.” (N. H. Abel, 1826f, 314). 

29 “wenn p die größte der Größen vmδ m , vmδ m + vm+1δ m+1 , vmδ m + vm+1δ m+1 + vm+2δ m+2 u.s.w. bezeichnet.” 

(ibid., 315).


Actually, if taken literally, ABEL’S p would be the maximum of the sums in (12.9) — not 

the supremum — but this distinction is beyond the point because for ABEL, the entire 

discussion on the definition of p was a non-issue. The quantity p was simply (and 

un-problematically) defined to be the greatest one among a sequence of numbers. The 

nature of the number p will be pursued in section 12.6, where Lehrsatz V will shed even 

more light on this non-issue. In the present case, the sequence of numbers is bounded 

since the series is assumed to converge at δ, but this was not explicitly remarked by 

ABEL. 

In order to follow ideas of ABEL’S proof, it suffices to take either ABEL’S naïve 

definition of p or the modernized one expressed in (12.9). From the equality (12.8), 

ABEL then concluded, that 

“for any value of α which is less than or equal to δ, m can be taken sufficiently 

large that 

ψ (α) = ω.” 30 

Next, ABEL observed that φ (α) was an entire function of α, i.e. a polynomial, and 

thus β could be taken small enough that 

φ (α) − φ (α − β) = ω. 

By combining these two results, ABEL concluded 

f (α) − f (α − β) = ω. 

Here we encounter ABEL’S way of operating with infinitesimals. The internal de- 

pendencies among ω, m, and α have been completely obscured by the notation and 

the argumentative style. 

A simple observation, inspired by comparing ABEL’S proof with modern exposi- 

tions of the calculus, concerns the use of infinitesimals. Today, infinitesimals have been 

completely abandoned from “rigorous” presentations of the calculus, and to a person 

trained within this program, ABEL’S usage of infinitesimals and even CAUCHY’S dual 

definitions involving both limits and infinitesimals can be repulsive. But to ABEL they 

were legitimate means of proving theorems. 

ABEL’S Lehrsatz IV and the paradoxes of analysis. In the binomial paper, the fourth 

theorem is given without further comments, but in his letters, ABEL had related it to 

one of the strongest ongoing discussions among analysts. As already described in 

30 “Mithin kann man für jeden Werth von α, der gleich oder kleiner ist, als δ, m groß genug annehmen, 

daß 

ist.” (ibid., 315). 

ψ (α) = ω


section 12.1, ABEL observed that a common practice for evaluating infinite sums, say 

∑ an had been to transform the series into a power series ∑ anx n , obtain an expression 

for the sum and then insert x = 1. Commenting on this practice, ABEL wrote, 

“This is probably right, but it appears to me that one cannot assume it without 

proof; just because 

φ (x) = a0 + a1x + a2x 2 + . . . 

for all values of x less than 1, it is not thereby said that the same conclusion holds 

for x = 1.” 31 

In order to illustrate his point of criticism, ABEL remarked that his claim was cer- 

tainly to the point if the power series failed to converge for x = 1 in which case it had 

no sum. However, when he gave explicit examples, he took them from the emerging 

theory of trigonometric series and not from within the realm of power series. And 

there is very good reason why he did not give a power series as a counter example; 

his fourth theorem states that for power series, the procedure of passing to the limit 

(inserting x = 1) can only fail in case the series is divergent for x = 1. Thus, Lehrsatz IV 

is the assurance needed to justify this procedure for the class of power series provided 

the resulting series is assumed to converge. 

DIRICHLET’S modification of ABEL’S proof. In 1862, J. LIOUVILLE (1809–1882) re- 

ported having discussed ABEL’S fourth theorem with his friend DIRICHLET, who had 

died just a few years before. LIOUVILLE had expressed his concern about the orig- 

inal proof of ABEL’S very important theorem which he found difficult to present in 

courses and even to understand. On the spot, DIRICHLET gave an alternative proof of 

Lehrsatz IV, which LIOUVILLE felt would remove all such difficulties. It was this new 

proof by DIRICHLET which LIOUVILLE reproduced in verbatim in a short note in his 

Journal de mathématiques pures et appliquées. 32 Similarly, a page in G. F. B. RIEMANN’S 

(1826–1866) Nachlass contains his reworking of ABEL’S proof which also supports the 

impression that ABEL’S original proof was not universally accepted. 33 

Before the differences between the ABEL’S and DIRICHLET’S proofs are discussed 

and analyzed, a presentation of DIRICHLET’S new proof is required. A modern recon- 

struction of DIRICHLET’S proof is given in box 3. 

For the infinite series 

A = 

∞ 

∑ am, 

m=0 

31 “Dette er vel rigtigt; men mig synes at man ikke kan antage det uden Beviis, thi fordi man beviser 

at 

φ (x) = a0 + a1x + a2x 2 + . . . 

for alle Værdier af x som er mindre end 1, saa er det ikke derfor sagt at det samme finder Sted for 

x = 1.” (Abel→Holmboe, 1826/01/16. N. H. Abel, 1902a, 17). 

32 (G. L. Dirichlet, 1862). The proof is also described in (I. Grattan-Guinness, 1970b, 108). 

33 (Laugwitz, 1999, 207).


DIRICHLET introduced the partial sums 

sn = 

n 

∑ am. 

m=0 

He assumed that the numerical values of the partial sums were always bounded by a 

constant k and that they converge toward the limit A as n grows beyond all bounds. 

Then DIRICHLET introduced the associated power series 

S = 

∞ 

∑ amρ 

m=0 

m , 

where 0 < ρ < 1 and thus wanted to prove that S (ρ) → A when ρ → 1. He observed 

that since am = sm+1 − sm, it could be rewritten as 

into 

S = s0 + 

∞ 

∑ (sm − sm−1) ρ 

m=1 

m . (12.10) 

DIRICHLET subsequently transformed this expression for the power series (12.10) 

S = (1 − ρ) 

∞ 

∑ 

m=0 

smρ m 

by the finite argument, i.e. by considering only the first n + 1 terms of (12.10) 

Sn+1 = s0 + 

n 

∑ (sm − sm−1) ρ 

m=1 

m n−1 

= (1 − ρ) ∑ smρ 

m=0 

m + snρ n 

(12.11) 

and the observation that snρ n “vanishes for n = ∞”, i.e. snρ n → 0 as n → ∞. Since the 

two expressions correspond for any finite n, their limits also correspond, 

S = lim 

n→∞ Sn+1 = (1 − ρ) lim 

∞ 

sn = (1 − ρ) ∑ 

n→∞ 

m=0 

smρ m . 

Now, DIRICHLET wanted to prove that the expression (12.11) converged to A as 

ε = 1 − ρ converged to zero. To do so, he split the series (12.11) into two parts 

S = (1 − ρ) 

n−1 

∑ smρ 

m=0 

m ∞ 

+ (1 − ρ) ∑ 

m=n 

smρ m , (12.12) 

and observed that the first sum was bounded by εnk, since |smρ m | < |sm| < k. Thus, 

he claimed that it converged to zero with ε → 0, which is true, provided n is kept 

fixed. As for the second sum, DIRICHLET claimed it could be written as 

P (1 − ρ) 

∞ 

∑ 

m=n 

ρ m = Pρ n = P (1 − ε) n , (12.13) 

provided P be chosen as a number between the smallest and the largest among the 

quantities sn, sn+1, . . . . There is no explicit explanation for this claim, but it could be 

obtained in a number of ways, either from CAUCHY’S theory of means or as an easy 

consequence of the intermediate value theorem of the integral calculus. Because the 

partial sums sn, sn+1, . . . all converge to A, DIRICHLET had proved his claim that S 

converges to A when ρ converges to 1.


Comparison of ABEL’S and DIRICHLET’S proofs. It is interesting to note how the 

attitude toward infinitesimals and limit arguments evolved over the first 40 years after 

CAUCHY’S Cours d’analyse and we can get an indication of this by comparing ABEL’S 

and DIRICHLET’S proofs. Contrary to ABEL’S proof, DIRICHLET completely avoided 

the use of infinitesimals in his proof. Instead, he argued completely within the process 

based interpretation of limits when he reduced the series to finitely many terms, ma- 

nipulated the polynomials and applied the limit process n → ∞. However, DIRICH- 

LET’S notation still hid the order in which limit processes are to be sequenced. 

In the ultimate step of DIRICHLET’S proof, two limit processes were involved; both 

expressions (12.12) and (12.13) involved both ε and n which were intended to converge 

toward zero and infinity, respectively. A modern reconstruction of the limit processes 

of DIRICHLET’S argument could proceed along the lines suggested by the proof in box 

3. In the box, it is illustrated how the limit processes can be straightened by first fixing 

a value of n such that |sm − A| is sufficiently small for all m ≥ n and then specifying 

the ε that will make the power series differ from A by as little as had been required. 

The order of DIRICHLET’S proof does not reflect the order in which the limit processes 

are to be carried out, and neither does his notation. Therefore, we are still faced with a 

line of argument in which limit processes are not as clearly identified and sequentially 

ordered as it is required today. 

12.5 ABEL’s “exception” 

The “exception” in the binomial paper. In proofs of the binomial theorem which 

follow EULER’S method of extending the binomial formula through the use of the 

functional equation, some argument based on continuity has to be applied to get from 

rational to real exponents. To meet this demand in his proof, CAUCHY deduced and 

stated the so-called Cauchy’s Theorem (see section 11.5). At the corresponding point of 

his binomial paper, ABEL discarded CAUCHY’S version of the theorem because he had 

discovered that it “suffered exceptions”: 34 

“Remark. In the above-mentioned work of Mr. Cauchy (on page 131) the following 

theorem can be found: 

»Whenever the different terms of the series 

u0 + u1 + u2 + u3 + . . . etc. 

are functions of one and the same variable quantity and moreover continuous 

functions with regard to this variable in the vicinity of a particular value for which 

the series is convergent, then the sum s of the series will also be a continuous 

function of x in the vicinity of that particular value.« 

34 For CAUCHY’S original formulation, which is authentically translated in ABEL’S paper, see page 

217.

12.5. ABEL’s “exception” 239 

A modern reconstruction of DIRICHLET’s proof. Let δ > 0 be given and choose (by 

convergence) n such that 

Then chose ε1 > 0 and ε2 > 0 such that 

|sm − A| < δ 

for all m ≥ n. 

6 

nεk < δ 

3 for ε < ε1, and 

|ρ n − 1| < δ 

for all ε = 1 − ρ < ε2. 

3k 

Then, if ε < min {ε1, ε2}, the equation (12.12) becomes 

n−1 

|S − A| ≤ (1 − ρ) ∑ |sm| ρ m � 

� 

� 

+ �(1 

− ρ) 

� 

m=0 

∞ 

∑ 

m=n 

In the first sum, the interesting reconstructed inequality is 

(1 − ρ) 

n−1 

∑ 

m=0 

When the second sum is rewritten as 

(1 − ρ) 

|sm| ρ m ≤ ε 

n−1 

∑ 

m=0 

smρ m − A 

|sm| ≤ εnk < δ 

3 . 

∞ 

∑ smρ 

m=n 

m ∞ 

− A = P (1 − ρ) ∑ ρ 

m=n 

m − A = Pρ n − A 

� 

� 

where P ∈ infm≥n sm, supm≥n sm , the inequalities of interest obtained from the requirements 

are 

|Pρ n − A| ≤ |Pρ n − P| + |P − A| 

≤ k × δ δ 

+ 2 × 

3k 6 

= 2δ 

3 . 

Combining the inequalities show that for ε < min {ε1, ε2}, 

|S − A| ≤ δ 2δ 

+ 

3 3 

Thus, as this modern reconstruction illustrates, it can be proved along the lines of 

DIRICHLET’S proof that for ε → 0 (i.e. ρ = 1 − ε → 1), the power series S (ρ) converges 

to A. The interrelation among the limit processes is contained in the specification of ε1 

and ε2. ✷ 

= δ. 

Box 3: A modern reconstruction of DIRICHLET’s proof. 

� 

� 

� 

� 

� .


❜ 

✟ ✟✟✟✟✟✟✟✟✟ 

� � 

❜ 

✟ ✟✟✟✟✟✟✟✟✟ 

� � 

❜ 

✟ ✟✟✟✟✟✟✟✟✟ 

� � 

−3π −2π −π π 2π 3π 

❜ 

Figure 12.2: Graphical representation of ABEL’s “Exception”, ∑ ∞ n=1 

The series 

π 2 

− π 2 

❜ 

❜ 

(−1) n−1 sin nx 

n . 

However, it appears to me that this theorem admits [or suffers] exceptions. 

For instance, the series 

sin φ − 1 1 

sin 2φ + sin 3φ − . . . etc. 

2 3 

is discontinuous for every value (2m + 1) π of x where m is an integer. As is well 

known, a multitude of series with similar properties exist.” 35 

∞ (−1) 

∑ 

n=1 

n−1 sin nx 

n 

(12.14) 

is a particularly simple trigonometric series: it is the Fourier series expansion of the 

function f (x) = x 2 on the interval ]−π, π[ (see figure 12.2). As such, it can be found 

in FOURIER’S works, for instance in the Théorie analytique de la chaleur, and even as 

a side result in one of EULER’S papers. 36 Possibly because EULER’S and FOURIER’S 

arguments for the convergence of the series (12.14) might have been wanting from 

the perspective of the new rigor, ABEL explicitly derived it as a result of some of the 

formulae proved in the binomial paper (see page 259, below). 

Aspects of ABEL’S “exception”. For subsequent reference, a few points concerning 

ABEL’S “exception” must be brought to attention. First, the exception was one of the 

35 “Anmerkung. In der oben angeführten Schrift des Herrn Cauchy (Seite 131) findet man folgende 

Lehrsatz: 

»Wenn die verschiedenen Glieder der Reihe 

u0 + u1 + u2 + u3 + . . . u.s.w. 

Functionen einer und derselben veränderlichen Größe sind, und zwar stetige Functionen, in Beziehung 

auf diese Veränderliche, in der Nähe eines besonderen Werthes, für welchen die Reihe 

convergirt, so ist auch die Summe s der Reihe, in der Nähe jenes besonderen Werthes, eine stetige 

Function von x.« 

Es scheint mir aber, daß dieser Lehrsatz Ausnahmen leidet. So ist z. B. die Reihe 

sin φ − 1 1 

sin 2φ + sin 3φ − . . . u.s.w. 

2 3 

unstetig für jeden Werth (2m + 1) π von x, wo m eine ganze Zahl ist. Bekanntlich giebt es eine 

Menge von Reihen mit ähnlichen Eigenschaften.” (N. H. Abel, 1826f, 316, footnote). 

36 (Fourier, 1822, 182, 241) and (L. Euler, 1754, 584); see also (I. Grattan-Guinness, 1970b, 84–85).

12.6. A curious reaction: Lehrsatz V 241 

“new series” which — according to ABEL’S opinion (see page 250, below) — had only 

recently entered analysis and brought so many paradoxes with it. Second, from a 

modern perspective, it is curious that ABEL called the series an “exception” and not a 

counter example or even a paradox. Although the binomial paper was translated from 

a French manuscript by CRELLE (see page 30), this choice of words appears not to have 

been merely accidental. Furthermore, it appears to have mattered to ABEL that the 

exception was not “singular” — if required, a multitude of similar exceptions could 

be devised. These points will enter into the argument in chapter 21. Finally, it should 

be observed that the “exception” was actually a recurring item in ABEL’S works on 

rigorization. Above (see page 225), it has been described how ABEL employed the 

same series to criticize the practice of differentiating a series by differentiating each 

term. Similarly, it appeared in one of ABEL’S drafts when he wanted to probe the 

limits of the theorem — Lehrsatz V — which was his tailored replacement for Cauchy’s 

Theorem. 37 

12.6 A curious reaction: Lehrsatz V 

ABEL’S fifth theorem: a revision of CAUCHY’S theorem. The fifth theorem of ABEL’S 

binomial paper plays a central role in a story to be told in a subsequent chapter. For 

the present, the theorem is mainly of interest because it, like Lehrsatz IV, provides an 

important combination of the three concepts currently under consideration: conver- 

gence, continuity, and power series. In his fifth theorem, ABEL found that the binomial 

series was a continuous function; a result which was inherently important in the ap- 

proach to the binomial theorem chosen by CAUCHY and adapted by ABEL. Again, the 

statement of the theorem is worth quoting, 38 

“Lehrsatz V. Let 

v0 + v1δ + v2δ 2 + . . . etc., 

be a convergent series in which v0, v1, v2, . . . are continuous functions of one and 

the same variable quantity x between the boundaries x = a and x = b. Then the 

series 

f (x) = v0 + v1α + v2α 2 + . . . , 

where α < δ is convergent and a continuous function of x between the same 

boundaries.” 39 

37 (N. H. Abel, [1827] 1881, 202); see page 245, below. 

38 The statement of the theorem as given in the paper is sloppy in a couple of respects. First, the β 

introduced at the end should obviously be a δ, and the convergence of the initial series 

∞ 

∑ vnδ 

n=0 

n 

must obviously also be assumed. Both corrections have been made by the editors in ABEL’S collected 

works (N. H. Abel, 1881, I, 223–224).


ABEL’S proof of Lehrsatz V. ABEL’S proved the fifth theorem by an approach closely 

resembling his proof of the preceding theorem. He first did away with the first claim 

of convergence by referring to the fourth theorem. The fourth theorem also told him 

that the sum function was continuous with respect to α; and he then moved on to 

prove the continuity of the sum function considered to be a function of x. As was 

common practice, ABEL split the sum function into two 

φ (x) = 

n−1 

∑ vm (x) α 

m=0 

m ∞ 

and ψ (x) = ∑ vm (x) α 

m=n 

m . 

Just as he had done for the fourth theorem, he rewrote ψ (x) as 

ψ (x) = 

∞ 

∑ 

m=n 

� 

α 

�m vm (x) δ 

δ 

m 

and introduced a quantity θ (x) to denote “the greatest among the quantities vmδ m , 

vmδ m + vm+1δ m+1 , vmδ m + vm+1δ m+1 + vm+2δ m+2 .” 40 Thus, θ (x) in the fifth theorem 

took the place of the quantity p used in the proof of the fourth theorem (see also be- 

low). In his further argument, ABEL used Lehrsatz III to write 

ψ (x) < 

� 

α 

�m θ (x) 

δ 

just as he had done previously and his proof of the continuity of f (x) followed ex- 

actly the same arguments as had been used in the fourth theorem. Again, infinitesi- 

mals were used instead of explicit limit processes when ABEL claimed that m could be 

chosen such that ψ (x) = ω, which allowed him to write 

f (x) − f (x − β) = φ (x) − φ (x − β) + ω. 

Then it was a simple matter to observe that φ was a polynomial and therefore β could 

be chosen small enough that 

and the theorem had been proved. 

39 “Lehrsatz V. Es sei 

φ (x) − φ (x − β) = ω 

v0 + v1δ + v2δ 2 + . . . u.s.w., 

eine Reihe, in welcher v0, v1, v2 continuierliche Functionen einer und derselben veränderlichen Größe 

x sind, zwischen den Grenzen x = a und x = b, so ist die Reihe 

f (x) = v0 + v1α + v2α 2 + . . . , 

wo α < β [α < δ], convergent und eine stetige Function von x, zwischen denselben Grenzen.” (N. 

H. Abel, 1826f, 315). 

40 “wenn man durch θ (x) die größte unter den Größen vmδ m , vmδ m + vm+1δ m+1 , vmδ m + vm+1δ m+1 + 

vm+2δ m+2 u.s.w. bezeichnet.” (ibid., 315).


An objection to Lehrsatz V: is θ uniformly bounded? The major problem with 

ABEL’S proof of the fifth theorem is closely tied to its connection with his proof of the 

preceding theorem. In his proof of Lehrsatz IV, ABEL had introduced the quantity p to 

denote the largest quantity among the partial tails of the series. In the proof of Lehrsatz 

IV, ABEL’S reasoning can be ‘saved’ by the observation that since the series ∑ vmδ m is 

convergent, its tails are bounded. Therefore, an upper bound — if not an outright max- 

imum — will exist which can be used for p. Such an argument is nowhere to be found 

in ABEL’S proof, and there are two points indicating that it was neither at his disposal 

nor of his concern. First, ABEL spoke of “the largest among” an infinite collection of 

quantities, i.e. of a maximum. If he had had anything but a naïve intuition about this 

step in his argument he might well have expressed himself differently using phrases 

analogous to “bounded by”. Second, in the proof of the fifth theorem which is mod- 

elled precisely over the proof of the fourth theorem, this exact step in the argument 

falls apart. 

When, in the proof of Lehrsatz V, ABEL introduced θ (x) analogous to the quantity 

p above 

θ (x) = largest quantity among 

n+k 

∑ vm (x) δ 

m=n 

m for k ≥ 0, (12.15) 

it seems to be a point-wise definition of the function θ (x). ABEL clearly thought of 

θ (x) as a quantity which, given x represented the largest among an infinite collection 

of quantities each depending on x. When, in the proof, ABEL claimed that 

ψ (x) = ψ (x − β) = ω, 

he implicitly used a supposed property of the function θ (x) — that the choice of n could 

be made uniformly throughout a small region surrounding x. However, in ABEL’S 

argument, there is no way of assuring that θ satisfies this requirement. 

As P. L. M. SYLOW (1832–1918) has remarked, 41 ABEL’S argument is sound if one 

further restriction is imposed on the convergence. If a constant M exists which uni- 

formly bounds the general term around x0 

|vm (x) δ m | ≤ M for all m and for all x ∈ � x0 − x ′ , x0 + x ′′� , 

ABEL’S reasoning can be applied by observing that both 

|ψ (x)| and |ψ (x − β)| will be less than M 

� � 

αδ m 

1 − α δ 

However, as observed, this is a reconstruction and certainly not part of ABEL’S argu- 

ment. 

41 (N. H. Abel, 1881, II, 303). 

.


Another proof of the Lehrsatz V from ABEL’S notebook. In ABEL’S notebooks, re- 

sults similar to the Lehrsatz V can be found treated in the manuscript Sur les séries 

which was presumably written in 1827. 42 Thus, ABEL returned to the theorem after 

the binomial paper had been published and attacked it from a slightly different per- 

spective. In his notes on the Sur les séries, M. S. LIE (1842–1899) interprets this fact 

as clear evidence that ABEL had become dissatisfied with the version printed in the 

Journal. 43 The manuscript was never completed for printing and its contents exhibit 

the characteristics of a draft. In particular, the precise assumptions and some of the 

notation are not made explicit and interpretation is slightly difficult. I interpret the 

relevant part of the manuscript as presenting two new deductions of the Lehrsatz V. 

In his manuscript, ABEL dealt with a function f (y) introduced as a power series 

in x with coefficients which vary continuously in y, 

f (y) = 

∞ 

∑ φn (y) x 

n=0 

n 

(12.16) 

and assumed that the series was convergent for x < α and all values of y near β. 

ABEL’S first “proof” of the continuity of f at y = β consisted in interchanging the 

limit processes, and he wrote it as 

lim f (y) = 

y=β−ω 

∞ � 

∑ 

n=0 

lim 

y=β−ω φn (y) 

� 

x n = 

∞ 

∑ Anx 

n=0 

n = R 

with the convention An = lim y=β−ω φn (y). The practice of interchanging limit pro- 

cesses had been among the points which attracted ABEL’S criticism, in particular when 

it came to term-wise differentiation (see above). Accordingly, ABEL did not simply in- 

terchange the two processes but made the additional restriction that the series R had 

to be convergent. 

However, as ABEL’S own “exception” could have illustrated, even the convergence 

of the resulting series was sufficient to warrant the general interchange of limits. Thus, 

ABEL had to make some use of the particular form of the series (12.16). However, ABEL 

made no remarks on this argument and due to the style of the notebook, its role and 

status — was it an observation? a theorem? a hypothesis? — remains my suggestive 

interpretation. 

ABEL’S second “deduction” is much more interesting and is presented here based 

on ABEL’S original argument and LIE’S reconstruction of it. 44 In this deduction, ABEL 

studied the differences between the corresponding terms of the series for f (β − ω) 

and f (β). 

42 (N. H. Abel, [1827] 1881, 201–202). 


44 (ibid., II, 326). 

(φn (β − ω) − An) x n .


Assuming that x1 was a value such that x < x1 < α, ABEL introduced a bound by 

assuming that the m’th term was the maximum of these differences, 

(φm (β − ω) − Am) x m 1 

= max 

n≥0 {(φn (β − ω) − An) x n 1 } . (12.17) 

This step resembles the introduction of the problematic quantity θ (x) in the bino- 

mial paper and the existence (i.e. finiteness) of such a maximum was apparently un- 

problematic to ABEL. Accordingly, LIE has suggested the same method of saving 

ABEL’S argument as SYLOW had done for the Lehrsatz V, i.e. by turning its existence 

into an explicit assumption (see above). ABEL concluded that 

f (β − ω) − R = ξ 

1 − x x 1 

(φm (β − ω) − Am) x m 1 

for some ξ ∈ [−1, 1]. When he let ω vanish, ABEL observed that the term 

φm (β − ω) − Am 

also vanished by the continuity of φm. Therefore, ABEL concluded, the function f was 

continuous. 

As described, ABEL’S two notebook proofs of the Lehrsatz V are slightly different 

from the printed version. However, they share the same structure and many of the 

methods which they apply, in particular concerning the belief in the existence of uni- 

form bounds (12.15 and 12.17). It is tempting to speculate with LIE that ABEL had 

realized that his original proof of Lehrsatz V was problematic — perhaps seizing on 

the same objection as SYLOW did and proposing the solution which amounts to uni- 

form convergence (see above). However, despite the new proofs, ABEL’S treatment 

of Lehrsatz V continued to suffer from essentially the same problems and such an in- 

terpretation is not compelling. If ABEL had become uneasy about his proof, it was 

probably for another reason or perhaps he just wanted another proof of a well estab- 

lished result? 

Probing the extent of Lehrsatz V. Following his new proof of Lehrsatz V in the note- 

book, ABEL observed that the theorem demonstrated the continuity of the function 

f (y) = 

∞ 

x 

∑ 

n=1 

n sin ny 

n 

for all x < 1, although for x = 1, the function — which was the “exception” of his 

binomial paper — had certain discontinuities. Under similar assumptions, the series 

corresponding to x = 1 could also fail to be divergent, altogether, ABEL observed and 

exemplified. These remarks again illustrate ABEL’S repeated criticism of the unwar- 

ranted passage to the limit in series.


12.7 From power series to absolute convergence 

As indicated in his letter to HANSTEEN and in the general approach of the binomial 

paper, ABEL put a lot of emphasis on power series in his attempt to rebuild the theory 

of series. In particular, ABEL’S replacement for the invalidated Cauchy Theorem was 

based on a particular kind of series which were power series in one variable with 

coefficients which were continuous functions of another variable. Well into the second 

half of the nineteenth century, this particular argument was found to be better recast 

within a concept of absolute convergence which had emerged over the century. 

Emergence of a concept of absolute convergence. During the 19 th century, numer- 

ical (absolute) values of real numbers and the moduli of complex numbers entered 

ever more explicitly in arguments of analysis. As described in the examples from 

CAUCHY’S Cours d’analyse and ABEL’S binomial paper, the only way mathematicians 

could describe numerical values in the first decades of the century was through verbal 

formulations. Generally, CAUCHY was quite careful about these in stating his theo- 

rems on series; but proper concern for numerical values was often lacking in ABEL’S 

formulations. 45 It appears that notation such as |x| was only invented by WEIER- 

STRASS in unpublished papers of the 1840s and did not become customary until the 

1870s. 46 

In the first decades of the 19 th century, series of numerical values mostly entered 

the picture in connection with the multiplication theorem. In the Cours d’analyse, when 

CAUCHY generalized his multiplication theorem for series of positive terms to more 

arbitrary series, he based his argument on the assumption of convergence of the series 

of absolute terms. However, despite its use in proving theorems, CAUCHY’S implicit 

concept of absolute convergence still lacked most of its later structural position. 

Immediately following the proof of the multiplication theorem, CAUCHY took an 

interesting step in investigating the consequences of relaxing the assumptions. He 

proved, based on squaring the series 

∞ 

∑ 

n=1 

(−1) n−1 

√ , (12.18) 

n 

that the assumption of absolute convergence was indeed necessary: Because the terms 

of the alternating series (12.18) are decreasing in absolute value, the series was conver- 

gent as CAUCHY had proved. 47 However, it was not absolutely convergent and when 

CAUCHY produced the square of the series, he obtained another divergent series. 48 

Here, CAUCHY took a rather modern step of using a counter example to a fictitious 

45 See e.g. the editors’ remark in (Lakatos, 1976, 134). 

46 (K. Weierstrass, 1876, 78) and e.g. (K. Weierstrass, [1841] 1894). 

47 (A.-L. Cauchy, 1821a, 144). 

48 (ibid., 149–150). The divergence of the series ∑ 1 

√ n can be obtained by comparing with the harmonic 

series.

12.7. From power series to absolute convergence 247 

more general theorem in order to illustrate that the requirements of his own theorem 

were necessary. However, as F. MERTENS (1840–1927) was later to show, 49 if scru- 

tinized more carefully, the example illustrated that the multiplication theorem could 

fail if both factors were non-absolutely convergent. This led P. L. WANTZEL (1814– 

1848) to prove a version of the multiplication theorem which only assumed absolute 

convergence of one of the factors (the other factor just being assumed convergent). 

The real beginning of a concept of absolute convergence came with DIRICHLET’S 

paper on primes in arithmetic progression which was published in 1837. 50 In that 

paper, DIRICHLET introduced a separation of convergent series into two classes based 

on the convergence of the series which resulted when the terms were replaced by 

the absolute values: Either the series of absolute values remained bounded or it was 

unbounded. For series of the first class (absolutely convergent series), DIRICHLET 

stated that their convergence and sum remained unaffected if the order of terms was 

altered. In particular, in double (and multiple) sums, the order of summation wold 

not effect the result. Dirichlet observed that these properties — which were certainly 

nice and expected — could fail to hold for series of the second class, and he gave two 

examples of what could happen: a convergent series could either become divergent 

or alter its sum if its terms were rearranged. 51 

For his Habilitation in 1854, RIEMANN presented a paper on the representability of 

functions by trigonometric series. 52 The paper is a milestone in the theory of trigono- 

metric series and the theory of integrals and also contains interesting remarks on the 

concept of absolute and non-absolute convergence. In the historical preface, RIEMANN 

outlined the previous developments in the field and claimed that DIRICHLET’S impor- 

tant 1829 paper on the convergence of trigonometric series was directly inspired by 

DIRICHLET’S discovery of the distinction between absolute and non-absolute conver- 

gence. 53 RIEMANN expressed his belief that the prevalence of power series in analysis 

was the reason why those concepts had not previously been separated. 54 There are no 

obvious traces of the alleged inspiration visible in DIRICHLET’S paper of 1829 but — 

as mentioned — the distinction became very explicit in a paper with a different topic 

in 1837. 

RIEMANN advanced a step beyond DIRICHLET’S observation of the differences be- 

tween absolutely and non-absolutely (conditionally) convergent series when he de- 

scribed a very simple method by which the partial sums of a conditionally convergent 

series could be made to approach any given value by proper rearrangement of the 

terms of the series. Central to RIEMANN’S argument was the realization that if a se- 

ries ∑ an was conditionally convergent, the series of its positive and negative terms 

49 (Mertens, 1875). 

50 (G. L. Dirichlet, 1837), see also e.g. (I. Grattan-Guinness, 1970b, 94–95). 

51 See also (ibid., 94–95). 

52 (B. Riemann, 1854). 

53 (ibid., 235) DIRICHLET’S paper is (G. L. Dirichlet, 1829). 

54 (B. Riemann, 1854, 235). See below.


would both have to converge to infinity. This could be used to prescribe a procedure 

by which examples such as those given by DIRICHLET could be constructed. As D. 

LAUGWITZ (1932–2000) remarks, 55 the Riemann arrangement theorem is not a particu- 

larly deep mathematical result in its own right but exemplifies the new approach to 

the concepts of convergence which was developing in the mid-nineteenth century. 

P. D. G. DU BOIS-REYMOND (1831–1889) on a generalization of Lehrsatz V. A con- 

cept of absolutely convergent series was thus being established in the nineteenth cen- 

tury and it was gradually emerging as a very central and useful tool in doing analysis. 

In 1871, DU BOIS-REYMOND published a short note which treated CAUCHY’S theorem 

on the continuity of an infinite sum of continuous functions which had also been the 

subject of ABEL’S Lehrsatz V. 56 DU BOIS-REYMOND was an active participant in the 

restructuring of analysis which — flooding from WEIERSTRASS’ lectures in Berlin — 

took place in the last part of the nineteenth century. In the note which falls into this 

Weierstrassian tradition of rigorizing analysis, DU BOIS-REYMOND proved a theorem 

to the following effect. 

Theorem 12 (DU BOIS-REYMOND) If a series 

is considered, for which the series 

∞ 

∑ wn (x) µn 

n=1 

∞ 

∑ µn 

n=1 

(12.19) 

converges absolutely and for which the functions wn are continuous functions on the interval 

[a, b] and all the wn as well as limn→∞ wn remain finite on that interval, then the series (12.19) 

is a continuous function of x on the interval [a, b]. ✷ 

As well as providing a proof of this theorem, DU BOIS-REYMOND also gave ex- 

amples in which the theorem would warrant continuity and examples in which no 

such conclusion could be drawn from the theorem. The theorem which DU BOIS- 

REYMOND presented was a generalization of ABEL’S fifth theorem; the latter could be 

deduced from the former (see box 4). It is interesting to compare the two theorems 

and the motivating problems which inspired them. 

The focus on power series: Comparing theorems. ABEL’S fifth theorem sought to 

avoid the over-generality of Cauchy’s Theorem by focusing on some particular form 

of convergence similar to the convergence of power series. Compared to Cauchy’s 

55 (Laugwitz, 1999, 211). 

56 (Bois-Reymond, 1871); DU BOIS-REYMOND referred to both CAUCHY and ABEL.

12.7. From power series to absolute convergence 249 

ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem Actually, ABEL’S 

fifth theorem is a consequence of DU BOIS-REYMOND’S theorem. 

Corollary 1 Assume that 

∞ 

∑ vn (x) δ 

n=1 

n 

(12.20) 

is convergent for some δ > 0 and that the functions vn are continuous functions of x on some 

interval I. Then, for any 0 < α < δ, the function 

f (x) = 

∞ 

∑ vn (x) α 

n=1 

n 

(12.21) 

is a continuous function of x on the interval I. ✷ 

PROOF We wish to use DU BOIS-REYMOND’S theorem to prove the theorem stated 

above. For this, we write 

and denote 

f (x) = 

∞ 

∑ vn (x) δ 

n=1 

n � α 

�n δ 

wn (x) = vn (x) δ n and 

� 

α 

�n µn = . 

δ 

Now, we are ready to test the requirements of DU BOIS-REYMOND’S theorem. The 

first requirement, that ∑ µn converges absolutely is obviously satisfied since α < δ. 

Secondly, the functions wn are obviously finite and continuous since this was required 

of vn. Lastly, we have to show that limn→∞ wn (x) is also finite. However, this is 

an easy consequence to draw from the convergence of (12.20) which ensures us that 

limn→∞ wn (x) = 0 for all x ∈ I. Thus, the continuity of (12.21) follows from DU 

BOIS-REYMOND’S theorem. � 

Box 4: ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem


Theorem, one further assumption was introduced in ABEL’S fifth theorem to the effect 

that the series 

f (x, δ) = 

∞ 

∑ vn (x) δ 

n=1 

n 

(12.22) 

was convergent for some δ > 0; and the conclusion of continuity only applied to 

f (x, α) where 0 < α < δ, i.e. on the interior of the interval of convergence. Apparently, 

this further requirement did away with ABEL’S own exception to Cauchy’s Theorem; the 

series 

∞ (−1) 

∑ 

n=1 

n sin nx 

n 

could not easily be transformed into a power series with radius of convergence sharply 

greater than one. However, it was hardly this barring of known exceptions which 

prompted ABEL to formulate his fifth theorem in the way he did. Instead, two other 

factors are most likely to have contributed to the formulation of the theorem. First, 

ABEL’S fifth theorem was closely modelled over Lehrsatz IV, and their proofs were 

almost identical. At the crucial step of the proof, a generalization was made from 

the constant p to the function θ (x) which were both designed to serve as uniform 

bounds. Second, the focus on power series was introduced by ABEL as a heuristic 

which “saved” ordinary intuition and many previously established results. 

In the letter to HANSTEEN quoted above, ABEL expressed his concern over a dark- 

ness which he saw persisting in the field of mathematical analysis. He even described 

some ideas concerning the reason for the relatively few paradoxes which these obscure 

and ill-founded procedures had created. 

“In my opinion, it [the reason for the few paradoxes] lies in the fact that the 

functions which analysis has dealt with have mostly been expressible by powers. 

As soon as others [functions] enter, which certainly does not happen often, it often 

does not go well and from false conclusions a string of connected false theorems 

flow.” 57 

According to ABEL, it was the introduction of new kinds of series which had pro- 

duced problems for theorems which implicitly relied on properties of power series 

although they were often expressed so as to apply to all series. As RIEMANN’S sim- 

ilar remarks seems to indicate, this was a generally held — and valid — belief in the 

nineteenth century. 

Half a century after ABEL’S solution to the problem raised by his counter example 

to Cauchy’s Theorem, DU BOIS-REYMOND devised another answer to the same prob- 

lem. Set in a different time and inspired by the system of analysis which WEIERSTRASS 

57 “Efter mine Tanker ligger den deri at de Functioner som Analysen hidentil har beskjæftiget sig med 

mestendels lade sig udtrykke ved Potenser. — Saasnart der komme andre imellem hvilket rigtig nok 

ikke ofte er Tilfældet saa gaaer det gjerne ikke godt og af falske Slutninger opstaae da en Mængde 

med hinanden forbundne urigtige Sætninger.” (Abel→Hansteen, Dresden, 1826/03/29. N. H. Abel, 

1902a, 22–23).

12.8. Product theorems of infinite series 251 

taught in Berlin, DU BOIS-REYMOND’S solution differed from ABEL’S at a conceptual 

level. When compared with ABEL’S theorem and its proof, DU BOIS-REYMOND’S the- 

orem differed in three respects. First, when he focused on the purpose which ABEL’S 

use of power series had served in the original proof, DU BOIS-REYMOND could relax 

the assumptions and only assume that the series ∑ µn converged (absolutely). Second, 

during the semi-century, a more rigid concept of absolute convergence had emerged 

which made the use of numerical values in series explicit and consistent. In the 

process, absolute convergence had become a concept about which theorems could be 

proved. Finally, in DU BOIS-REYMOND’S proof, the uniformity requirement discussed 

above was explicitly taken into account by the assumption that limn→∞ wn (x) remain 

finite. 

The example of DU BOIS-REYMOND’S revision of Cauchy’s Theorem serves to illus- 

trate how the concept of absolute convergence became a very central and powerful 

concept in the theory of series. Formulating theorems using absolute convergence of- 

ten led to more functional assumptions which were directly usable in the proofs. In 

this example, this was contrasted with the older focus on formal assumptions which 

stressed the particular formal appearance of the objects — here the particular form of 

the series (12.22) — under consideration. 

12.8 Product theorems of infinite series 

Besides the theorems discussed above, another basic, important theorem on infinite 

series received renewed interest in ABEL’S paper on the binomial theorem; and again, 

this theorem goes back to CAUCHY’S Cours d’analyse. 

In the binomial paper, ABEL’S sixth and final preliminary theorem dealt with the 

product of two infinite series. In its presentation, it reveals an intriguing transition in 

the understanding of the concept of absolute convergence and reads as follows, 

“Lehrsatz VI. When by ρ0, ρ1,ρ2 etc., ρ ′ 0 , ρ′ 1 , ρ′ 2 etc one designates the numerical 

values of the respective terms of two convergent series 

then the series 

v0 + v1 + v2 + . . . = p and 

v ′ 0 + v ′ 1 + v ′ 2 + . . . = p ′ , 

ρ0 + ρ1 + ρ2 + . . . and 

ρ ′ 0 + ρ ′ 1 + ρ ′ 2 + . . . 

are likewise convergent. Similarly, the series 

whose general term is 

r0 + r1 + r2 + · · · + rm 

rm = v0v ′ m + v1v ′ m−1 + v2v ′ m−2 + · · · + vmv ′ 0,


and whose sum is 

will also be convergent.” 58 

(v0 + v1 + v2 + . . . ) × � v ′ 0 + v ′ 1 + v ′ 2 + . . . � 

As it appears, the theorem consists of two halves each contributing a distinct con- 

clusion: 

1. Convergence of terms implies convergence of numerical terms. 

2. Convergence of the Cauchy product toward the correct sum. 

The first of these two conclusions is wrong, and it is difficult to explain the mishap 

in ABEL’S presentation. In the collected works, it has been corrected without com- 

ments by replacing “so sind die Reihen [. . . ] ebenfalls noch convergent” with “si les 

séries [. . . ] sont de même convergente”. 59 In the proof, ABEL does not give argu- 

ments for the first part of the supposed theorem; instead it is used as an assumption 

in proving the Cauchy product theorem. 

Thus, based on ABEL’S proof, SYLOW and LIE attributed the mishap to a slip of the 

pen or perhaps a slight incompetence on the part of the translator. This is probably 

the best interpretation available, but a little more may perhaps be inferred from the 

fact that such a misprint found its way into a professional journal — as did a number 

of others. First, this fact suggests that conceptual handling of series and series of nu- 

merical values was not very well established among the mathematical class to which 

CRELLE belonged. And secondly, we may also be tempted to infer something on the 

58 “Lehrsatz VI. Bezeichnet man durch ρ0, ρ1, ρ2 u.s.w., ρ ′ 0 , ρ′ 1 , ρ′ 2 u.s.w. die Zahlenwerthe der resp. Glieder 

zweier convergenten Reihen 

so sind die Reihen 

v0 + v1 + v2 + . . . = p und 

v ′ 0 + v′ 1 + v′ 2 + . . . = p′ , 

ρ0 + ρ1 + ρ2 + . . . und 

ρ ′ 0 + ρ′ 1 + ρ′ 2 

ebenfalls noch convergent, und auch die Reihe 

deren allgemeines Glied 

und deren Summe 

+ . . . 

r0 + r1 + r2 + · · · + rm 

rm = v0v ′ m + v1v ′ m−1 + v2v ′ m−2 + · · · + vmv ′ 0 , 

(v0 + v1 + v2 + . . . ) × � v ′ 0 + v′ 1 + v′ 2 

ist, wird convergent seyn.” (N. H. Abel, 1826f, 316–317). 

59 (N. H. Abel, 1881, 225). 

+ . . . �

12.8. Product theorems of infinite series 253 

standards of the newly established journal, which was hampered by some similar and 

less grave misprints in the first years, although its standards of technical printing were 

quite high. 

ABEL’S proof of Cauchy product theorem followed a path similar to those taken by 

CAUCHY in his proofs (see section 11.5). ABEL let pm and p ′ m denote the partial sums 

of the factors p and p ′ and wrote 

After introducing the notation 

he found 

2m 

∑ rk = pmp 

k=0 

′ m−1 

m+ ∑ 

k=0 

u = 

p kv ′ 2m−k 

� �� 

=t 

+ 

∞ 

∑ ρk and u 

k=0 

′ ∞ 

= ∑ ρ 

k=0 

′ k , 

m−1 

∑ v2m−kp k=0 

′ k 

� �� 

=t ′ 

m−1 

t 

k=0 

′ 2m−k and t′ m−1 

′ 

. (12.23) 

“without reference to the sign”, i.e. for the numerical values of t and t ′ . ABEL then 

employed the Cauchy sequence characterization of convergence (for its prominent posi- 

tion in the Abelian framework, see above) to ensure that since the series ∑ ρ k and ∑ ρ ′ k 

were convergent, the sums 

m−1 

∑ ρ 

k=0 

′ 2m−k and 

m−1 

∑ ρ2m−k k=0 

would both tend to zero as m grew to infinity. Thus, ABEL claimed, by setting m equal 

to infinity, the equation (12.23) became 

∞ 

∑ rk = 

k=0 

� 

∞ 

∑ vk k=0 

� 

× 

� 

∞ 

∑ v 

k=0 

′ � 

k . 

At this point, the theorem was proved, but ABEL continued his argument by gen- 

eralizing the theorem through the use of power series. ABEL now abandoned the 

assumptions that both factors had to be absolutely convergent in favor of the assump- 

tion that both the factors and the Cauchy product were (simply) convergent. In his 

notes on ABEL’S binomial paper, SYLOW wrote of this generalization: “The theorem 

VI is due to Cauchy but the new form which it is given [. . . ] originates with ABEL.” 60 

The generalized version of the Cauchy product theorem can thus be stated as follows. 

Theorem 13 (Generalized Cauchy product theorem) If the three series 

∞ 

∑ tk, k=0 

∞ 

∑ t 

k=0 

′ k , and 

∞ 

∑ 

k=0 

∑ 

tnt 

n+m=k 

′ m 

60 “Le théorème VI est dú à Cauchy, mais la forme nouvelle qu’il a reçue page 226 appartient à Abel.” 

(ibid., II, 303).


are all convergent, then 

� � � 

∞ 

∞ 

∑ tk × ∑ t 

k=0 

k=0 

′ � 

k = 

∞ 

∑ 

∑ 

k=0 n+m=k 

tnt ′ m. 

ABEL proved this theorem through elegant application of the previously established 

theorems. First he simply assumed that {tk} and � t ′ � 

k were two sequences 

converging to zero. Then, by his Lehrsatz II, the two power series 

∞ 

∑ tkα k=0 

k and 

∞ 

∑ t 

k=0 

′ kαk would be convergent for 0 < α < 1, even if numerical values were taken. Thus, be the 

version of the Cauchy product theorem expressed in Lehrsatz VI, 

� 

∞ 

∑ tkα k=0 

k 

� � 

∞ 

× ∑ t 

k=0 

′ kαk � 

= 

� 

∞ 

∑ 

k=0 

∑ tnt 

n+m=k 

′ m 

and by letting α → 1, Lehrsatz IV supplied the desired conclusion. 

12.9 ABEL’s proof of the binomial theorem 

After having established his six preliminary theorems, ABEL proceeded to the bino- 

mial theorem. Throughout the binomial paper, ABEL never equated the expressions 

(1 + x) m and 

∞ 

∑ 

µ=0 

∏ µ−1 

k=0 

(m − k) 

x 

µ! 

µ 

directly because the former could be multi-valued whereas the latter had only a single 

value as a function of x (see page 227). Instead, ABEL began his argument by asking 

for which values of m and x the binomial series 

φ (x) = 

∞ 

∑ mµx 

µ=0 

µ 

converged, where he let mµ represented the binomial coefficient 

mµ = 

m (m − 1) (m − 2) . . . (m − µ + 1) 

µ! 

� 

α k , 

= ∏µ−1 s=0 (m − s) . 

µ! 

Transformation into real series. ABEL wanted to include complex values of m and 

x, which he introduced by letting 

x = a + ib and m = k + ik ′ , 

✷

12.9. ABEL’s proof of the binomial theorem 255 

where the notation i has been adopted for ABEL’S 

binomial coefficients in polar form, 

which meant 

δµ 

m − µ + 1 

µ 

= δµ 

� cos γµ + i sin γµ 

√ −1. ABEL wrote the factors of the 

� � 

cos γµ + i sin γµ , 

� k + ik 

= ′ − µ + 1 

, 

µ 

and for each given µ, the values of δµ and γµ could be found. When these factors were 

multiplied to produce the binomial coefficients, ABEL found 

� � 

� 

mµ = 

With the conventions 

� 

µ 

∏ δn 

n=1 

× 

cos 

x = α (cos φ + i sin φ) , λµ = 

� 

µ 

∑ γn 

n=1 

+ i sin 

� 

µ 

∑ γn 

n=1 

µ 

∏ δn, and θµ = µφ + 

n=1 

�� 

. 

µ 

∑ γn, 

n=1 

ABEL had thus decomposed the general term of the binomial series into the form 

mµx µ = λµ 

� � µ 

cos θµ + i sin θµ α , 

thereby reducing the binomial series to its real and imaginary parts, 

φ (x) =1 + 

∞ 

∑ 

µ=1 

λµα µ cos θµ +i 

� �� 

=p 

∞ 

∑ 

µ=1 

λµα µ sin θµ . (12.24) 

� �� 

=q 

Convergence of the binomial series. Having obtained the decomposition of the bi- 

nomial series into real and imaginary parts (12.24), ABEL claimed that it converged if 

α < 1 and diverged if α > 1. In order to prove this claim, he applied his version of the 

ratio test, observing that because 

� 

� �2 � � 

k − µ k ′ 2 

δµ+1 = 

+ → 1 as µ → ∞, 

µ + 1 µ + 1 

the ratio of consecutive terms converged to α, 

λµ+1α µ+1 

λµα µ 

= δµ+1α → α for µ → ∞. 

ABEL took care of the trigonometric factors of the general terms, cos θµ and sin θµ 

by applying his own version of the ratio test as expressed in his Lehrsätze I&II. How- 

ever, he did not provide any details of the argument. In the simplest case, α < 1, the 

absolute convergence of both the series p and q can be obtained directly from Lehrsatz


Proof that the trigonometric coefficients cannot approach zero First, we consider 

the series 

The calculation 

p − 1 = 

∞ 

∑ λµα 

µ=1 

µ cos θµ. 

cos θµ+1 = cos � � 

θµ + φ + γµ+1 

= cos φ cos � � � � 

θµ + γµ+1 − sin φ sin θµ + γµ+1 

= cos φ � � 

cos θµ cos γµ+1 − sin θµ sin γµ+1 

− sin φ � � 

sin θµ cos γµ+1 + cos θµ sin γµ+1 

shows that if � � 

cos θµ is convergent, T = lim cos θµ, then 

� 

T = −T cos φ + sin φ 1 − T2 because lim cos γµ = −1 and lim sin γµ = 0. 

Consequently, if � � 

cos θµ is convergent, its limit has to be given by the equation 

� 

1 − cos φ 

T = ± 

, 

2 

which is only zero if cos φ = 1, i.e. if x is on the real axis. 

Similarly, for the series 

q = 

∞ 

∑ λµα 

µ=1 

µ sin θµ, 

the situation is completely analogous and the conclusion remains the same. 

Box 5: Proof that the trigonometric coefficients cannot approach zero 

II, because the trigonometric coefficients never surpass 1, numerically. On the other 

hand, if α > 1, the divergence of the series p and q rested on the observation that nei- 

ther of the trigonometric coefficients approached zero. The details of this observation 

which unless x is real are provided in box 5. 

In the case α < 1, ABEL proceeded to utilize his previously established theorems. 

First, he showed by Lehrsatz VI, that provided φ (m), φ (n), and the Cauchy product 

φ (m) φ (n) were all convergent series, the product was equal to φ (m + n). And since 

φ (m + n) was assumed to be convergent, ABEL had showed that φ (m) was a solution 

to the functional equation. 

In order to express everything in real variables, ABEL next introduced 

φ (m) = p + qi = r (cos s + i sin s)


which he wrote as 

φ � k + k ′ i � = f � k, k ′� � cos ψ � k, k ′� + i sin ψ � k, k ′�� . 

With n = l + l ′ i, ABEL found the analogous of the above and proceeded to express 

φ (m + n) in the same way to find internal relations of f and ψ. He expressed the 

functional relation φ (m + n) = φ (m) φ (n) in terms of the real arguments 

f � k + l, k ′ + l ′� = f � k, k ′� f � l, l ′� , and 

ψ � k + l, k ′ + l ′� = 2Mπ + ψ � k, k ′� + ψ � l, l ′� , 

where M denoted an integer. Now, ABEL wanted to find the functions f and ψ which 

satisfied these equations. He first proved that f was a continuous function, basically 

because it was composed of continuous functions. Similarly, he claimed ψ could be 

assumed to be continuous by choosing a constant value for M. 

ABEL then obtained the equation 

ψ � k, k ′ + l ′� + ψ � l, k ′ + l ′� = 2Mπ + ψ � 0, k ′� + ψ � 0, l ′� + ψ � k + l, k ′ + l ′� . 

This equation helped him determine the way ψ depended upon its first argument. 

ABEL let θ (k) = ψ (k, k ′ + l ′ ) which rendered the equation as 

θ (k) + θ (l) = a + θ (k + l) (12.25) 

and he proceeded to solve this equation. He did so by first proving directly that for 

integer ρ and any k, 

In particular, for k = 1, ABEL found the solution 

ρθ (k) = (ρ − 1) a + θ (ρk) . (12.26) 

θ (ρ) = ρ (θ (1) − a) + a 

for integer values of ρ. He then proved that this result extended first to rational values 

of ρ and then to any positive or negative real value of ρ by the continuity of θ. His 

extension to rational values was classical: By (12.26), 

� � 

µ 

ρθ = (ρ − 1) a + θ (µ) = (ρ − 1) a + µ (θ (1) − a) + a, i.e. 

ρ 

� � 

µ 

θ = a + 

ρ 

µ 

(θ (1) − a) . 

ρ 

ABEL next investigated the second argument of ψ by similar methods, and he found 

ψ � k, k ′� = βk + β ′ k ′ − 2Mπ.


Having solved the functional equation of the angular function ψ, ABEL next re- 

duced the functional equation of the modular function f to the same equation. He 

observed that if 

the equation 

was reduced to 

which he had just solved to find 

Therefore, he found the solution 

f � k, k ′� = e F(k,k′ ) , 

f � k + l, k ′ + l ′� = f � k, k ′� f � l, l ′� 

F � k + l, k ′ + l ′� = F � k, k ′� + F � l, l ′� 

F � k, k ′� = δk + δ ′ k ′ . 

φ � k + k ′ i � = e δk+δ′ k ′ � cos � βk + β ′ k ′ � + i sin � βk + β ′ k ′�� 

and reduced it to its real and imaginary terms. Still, the constants β, β ′ , δ, δ ′ were not 

more precisely determined. ABEL returned to this issue and provided formulae for 

determining these constants, 

α sin φ 

β = arctan 

1 + α cos φ and 

δ = 1 

2 log 

� 

1 + 2α cos φ + α 2� 

. 

Finally, ABEL employed his Lehrsatz IV to treat the case α = 1 as the limit case of 

the previously considered cases. He summarized the results of the entire investigation 

in the following way: 

Theorem 14 I. Whenever the series 

1 + 

m + ni 

1 

is convergent, it has the sum 

� 

(1 + a) 2 + b 2� m 2 

(a + bi) + 

b 

−n arctan 

e 1+a × 

� 

cos 

� 

m arctan b 

1 + a 

(m + ni) (m − 1 + ni) 

1 · 2 

+ i sin 

(a + bi) 2 + . . . 

� 

(1 + a) 2 + b 2�� 

n 

+ 

2 log 

� 

m arctan b n 

+ 

1 + a 2 log 

� 

(1 + a) 2 + b 2�� 

II. The series is convergent for every value of m and n whenever the quantity √ a 2 + b 2 is less 

than one. If √ a 2 + b 2 is equal to one, the series is convergent for every value of m comprised 

between −1 and +∞ if one does not simultaneously have α = −1. If α = −1, m must be 

positive. In every other case, the series is divergent. 61 

✷ 

61 (N. H. Abel, 1826f, 333–334)


This characterization contained the complete two-part answer to the questions which 

ABEL had raised: the sum of the binomial series when it is convergent and the condi- 

tions of its convergence. The cumbersome form of the sum of the binomial series arises 

partly from the fact that ABEL expressed its complex variables separated into real and 

imaginary parts, and partly from the answer it gives to the problem of multivalued 

answers: ABEL’S expression for the sum of the series only has a single value because 

the bracket is a positive number and the extraction of roots of positive numbers results 

in a canonical, positive value. 

An example relating to ABEL’S “exception”. At the very end of the paper, ABEL 

used the results which he had found to carry out the summation of certain interesting 

series. In particular, the first example is of interest in connection with ABEL’S famous 

exception. 

In the first example, 62 ABEL proposed to sum the series 

α sin φ − 1 

2 α2 sin 2φ + 1 

3 α3 sin 3φ + . . . 

which he found was convergent for |α| < 1 where it converged toward the value β 

above, 

β = arctan 

α sin φ 

1 + α cos φ = 

∞ 

∑ 

n=1 

(−1) n−1 sin nφ 

α 

n 

n . 

To determine the value for α = 1, it sufficed to let α approach the limit 1 provided the 

resulting series remained convergent (Lehrsatz IV). Thus, for φ between −π and π, 

1 

φ = arctan 

2 

sin φ 

1 + cos φ = ∑ (−1)n−1 sin nφ 

. 

n 

For φ = ±π, the situation was different because the series vanished and the expression 

for β degenerated. ABEL observed: 

“It follows, that the function 

sin φ − 1 1 

sin 2φ + sin 3φ − . . . 

2 3 

has the remarkable property of being discontinuous for the values φ = π and 

φ = −π.” 63 

Thus, in this case, ABEL used the same object as in the exception for another purpose. 

This time, he wanted to illustrate the same point as in the notebook (see section 12.6): 

that although the series of the form ∑ vm (x) α m was continuous for α < 1, it needed 

not be continuous for α = 1. 

62 (ibid., 336–337). 

63 “Hieraus folgt, daß die Function: 

sin φ − 1 1 

sin 2φ + sin 3φ − u.s.w. 

2 3 

die merkwürdige Eigenschaft hat, für die Werthe φ = π und φ = −π unstetig zu seyn.” (ibid., 

336–337).


The solution to Poisson’s example. The ultimate result in ABEL’S binomial paper 

concerned the series which had probably inspired him to work on the binomial theo- 

rem in the first place. As a result of applying his characterization of the convergence 

of the binomial series, ABEL found precise conditions for the convergence of the bino- 

mial series corresponding to (2 cos x) m . The result — an analogy of which ABEL also 

communicated to HOLMBOE in a letter (see page 226) — was that the identity 

(2 cos x) m = cos mx + m 

1 

cos (m − 2) x + m (m − 1) 

1 · 2 

cos (m − 4) x + . . . 

was valid when m was positive and x belonged to the interval � − π 2 , π 2 

� . Thus, ABEL 

ruled out validity of this formula in the situation of Poisson’s example which had in- 

volved setting x = π. For general values of x, ABEL obtained an identity which in- 

volved a correction term, 

� (2ρ−1)π 

2 

(2 cos x) m cos 2ρmπ = 

∞ 

∑ 

k=0 

� � 

m 

cos (m − 2k) x 

k 

, (2ρ+1)π 

� 

2 . Thus, ABEL’S resolution to Poisson’s example consisted of 

for x ∈ 

two steps deriving from his general proof of the binomial theorem. First, he divided 

the values of x into smaller intervals in which the value of cos x had a constant sign. 

And second, he considered all expressions as single valued and introduced an addi- 

tional term to provide the correction. 

12.10 Aspects of ABEL’s binomial paper 

Having presented and investigated the contents of ABEL’S binomial paper, I believe 

that three slightly broader aspects of it also merit attention: ABEL’S use of complex 

numbers, his use of functional equations, and the style of the binomial paper. 

12.10.1 ABEL’s understanding of complex numbers 

Compared with CAUCHY’S Cours d’analyse, ABEL’S proof of the binomial theorem ex- 

celled by including complex values of the exponent. For complex values of the ar- 

gument x, CAUCHY had reduced the study of the binomial series corresponding to 

(1 + x) m to the study of two real series by writing out the real and imaginary parts. 

By diligent use of polar representations of complex numbers, ABEL succeeded in re- 

ducing the functional equations for complex exponents m to the simple, additive one. 

Thus, in the binomial paper, ABEL worked with complex numbers which he always 

reduced to pairs of reals either as real and imaginary parts or in polar representation. 

In his inversion of elliptic integrals into elliptic functions (see chapter 16, below), ABEL 

also worked with complex numbers as arguments of functions. Again, complex num- 

bers were reduced to real and imaginary parts. From the rather scarce evidence, it

12.10. Aspects of ABEL’s binomial paper 261 

seems justified to say that ABEL held a strictly algebraic view of complex numbers 

and that he considered these numbers rather unproblematic. 

Because some of ABEL’S initial theorems — in particular Lehrsatz IV — dealt with 

power series, they have subsequently been interpreted as pertaining to complex vari- 

ables. However, there is no absolute indication that this was the interpretation which 

ABEL held. ABEL’S original formulations were not very explicit about these issues 

and often neglected taking numerical values into consideration. However, I find very 

little reason to suspect that ABEL would develop his theorems for complex variables 

and afterwards go through rather cumbersome arguments to reduce complex series to 

series of real terms. 

Going through ABEL’S loans from the Christiania University library, V. BRUN (1885– 

1978) discovered that ABEL had — in 1822 — borrowed the volume of the transactions 

of the Danish Academy in which his fellow Norwegian C. WESSEL (1745–1818) had 

published his geometric interpretation of complex numbers as directed line segments 

in the plane. 64 However, as Ø. ORE (1899–1968) also pointed out, ABEL was prob- 

ably much more interested in a paper on equations which C. F. DEGEN (1766–1825) 

published in the same volume. 65 Even if ABEL read WESSEL’S paper — which seems a 

reasonable assumption given the limited amount of Danish mathematical literature — 

he certainly never did anything to adopt its idea or promote it in any other way. This 

just supports K. ANDERSEN’S (⋆1941) hypothesis that the geometrical interpretation 

of complex numbers was not a hot topic in the first decades of the nineteenth century. 66 

12.10.2 ABEL on functional equations 

As had been the case in both EULER’S and CAUCHY’S proof of the binomial theorem, 

functional equations played a central part in ABEL’S proof. In his Cours d’analyse, 

CAUCHY had developed the topic into a theory of its own and he studied multiple 

types of functional equations. 67 

In a paper published in 1827 in CRELLE’S Journal, 68 ABEL presented some results 

on functional equations, which when applied to the functional equation 

φ (x) + φ (y) = φ (x + y) (12.27) 

gave the solution φ (x) = Ax as the unique (continuous) solution to this equation. 

It was precisely this functional equation which was central to ABEL’S proof of the 

binomial theorem. Whereas EULER and CAUCHY had used the equation 

f (x) f (y) = f (x + y) 

64 (Brun, 1962, 110–111). For an analysis of WESSEL’S work, see (K. Andersen, 1999). 

65 (C. F. Degen, 1799). 

66 (K. Andersen, 1999, 94). 

67 (J. Dhombres, 1992). 



as the foundation for their proofs, ABEL chose to focus on the other equation (12.27) 

because it was better suited for his investigations of complex exponents. When he had 

to address the multiplicative functional equation expressing the modulus of the series, 

he transformed it into the form (12.27) by way of exponentiation. 

The additive functional equation (12.27) had also been studied by CAUCHY in his 

Cours d’analyse, and it can be interpreted as testimony to ABEL’S familiarity with that 

book that he was able to replace the basic tool of the previous proofs with one more 

suited for his slightly more general situation. 

Two general problems concerning functional equations. As mentioned, ABEL pub- 

lished two papers in 1826 and 1827 in which he addressed questions concerning func- 

tional equations. The problems which he attacked were within the immediate scope 

of CAUCHY’S approach to the theory although they may appear a little odd. Thus, 

ABEL’S results can be seen as contributing to the early growth of the theory of func- 

tional equations. 

In paper published in 1826, 69 ABEL dealt with functions f such that f (z, f (x, y)) 

was symmetric in x, y, z. For such functions, ABEL obtained the characterization: 

“Whenever a function f (x, y) of two independent variable quantities x and 

y has the property that f (z, f (x, y)) is a symmetric function of x, y, z, there will 

always be a function ψ for which 

ψ ( f (x, y)) = ψx + ψy.” 70 

Furthermore, ABEL found that the stipulated function ψ could be determined by the 

differential equation 

ψ (x) = ψ ′ � ∂ f 

∂x (x, y) 

(y) 

dx. 

∂ f 

∂y (x, y) 

In the process, ABEL also integrated the equation 

to find 

∂r (x, y) 

φ (y) = 

∂x 

�� 

r = ψ 

� 

φ (x) dx + 

∂r (x, y) 

φ (x) 

∂y 

� 

φ (y) dy , 

a result which will resurface in section 16.2.2 where ABEL’S deduction of the addition 

theorems of elliptic functions are described. 

69 (N. H. Abel, 1826e). 

70 “Sobald eine Function f (x, y) zweier unabhängig veränderlichen Größen x und y die Eigenschaft 

hat, daß f (z, f (x, y)) eine symmetrische Function von x, y und z ist, so muß es allemal eine Function 

ψ geben, für welche 

ist.” (ibid., 13). 

ψ f (x, y) = ψ (x) + ψ (y)

12.10. Aspects of ABEL’s binomial paper 263 

The following year, ABEL treated another problem concerning the form of func- 

tions satisfying a functional relation. 71 In that paper, he studied the form of functions 

φ which satisfy 

φ (x) + φ (y) = ψ (x f (y) + y f (x)) . (12.28) 

ABEL found his main result which stated that the most general way of satisfying the 

equation (12.28) was with 

φ (x) = φ ′ � 

(0) f (0) 

dx 

f (x) f ′ � � 

x 

and ψ (x) = φ (0) + φ . 

(0) x f (0) 

One very simple application of this result is particularly interesting and a reconstruction 

of it is given below. If we let f (x) = 1, we obtain α = f (0) = 1 and α ′ = f ′ (0) = 

0. Therefore, the function φ which satisfies the equation (ψ = φ) 

φ (x) + φ (y) = φ (x + y) (12.29) 

must satisfy the requirement 

� 

dx 

φ (x) = φ (0) = xφ (0) + C, 

1 

where C = 0 by (12.29). Consequently, ABEL’S result leads to the result that the only 

(continuous) solution to the functional equation 

φ (x) + φ (y) = φ (x + y) 

is the linear function φ (x) = Ax for some constant A. As observed, this paper — 

which was published in 1827 — therefore contains a generalization of the additive 

functional equation (12.27) which had been so important to his proof of the binomial 

theorem. However, in the binomial paper, ABEL probably relied directly on CAUCHY’S 

Cours d’analyse. 

12.10.3 Concepts and calculations in the binomial paper 

ABEL’S paper on the binomial series shows a remarkable blend of concepts and ex- 

plicit calculations. A superficial, textual analysis of ABEL’S paper reveals a division 

of the paper: First, six preliminary theorems were presented which were applicable 

to classes of series. These were cast in a strictly Euclidean presentational style with 

definitions of convergence and continuity, statements of the six theorems and proofs 

following each theorem. Second, detailed and explicit considerations of the conver- 

gence of the binomial series as well as formulae for its sum were given. These in- 

vestigations relied extensively on explicit manipulations of the formulae in forms as 

described above. When analyzed from this perspective, ABEL’S binomial paper shows 

traits of both concept based and formula based mathematics as discussed in chapter 

21, below. Thus, the binomial paper is an example of the transitional status of ABEL’S 

mathematics which exhibited similarities with both paradigms. 


Chapter 13 

ABEL and OLIVIER on convergence 

tests 

Besides his proof of the binomial theorem, N. H. ABEL’S (1802–1829) only other pub- 

lication on analysis — in which his main interest was the theory of infinite series — is a 

curious little argument against another mathematician named L. OLIVIER. 1 In the first 

issue of the second volume of the Journal für die reine und angewandte Mathematik, A. L. 

CRELLE (1780–1855) had accepted a paper by OLIVIER in which the latter claimed to 

have obtained a general — yet very simple — test of convergence. Despite OLIVIER’S 

claims, the test was not generally applicable and in the next (i.e. third) volume of the 

Journal, ABEL published his refutation which consisted of a counter example to the 

original claim made by OLIVIER as well as a proof that no such criterion could ever be 

found: it was an utopian dream. 2 

13.1 OLIVIER’s theorem 

In the article entitled Remarques sur les séries infinies et leur convergence, 3 OLIVIER worked 

with a distinction between convergent, indeterminate, and divergent series (see below) 

and stated a criterion to distinguish convergent and non-convergent series. This cri- 

terion, which OLIVIER called a “criterion of convergence of infinite series” 4 was the 

following: 

“Thus, if one finds that in an infinite series the product of the n th term – or the 

n th group of terms which keep the same sign — by n is zero for n = ∞, one can 

regard this single circumstance as a sign that the series is convergent. Reciprocally, 

the series cannot be convergent unless the product n · an is zero for n = ∞.” 5 

1 Very little is known about LOUIS OLIVIER. However, based on his pattern of publication, I believe 

that he was a non-professional mathematician with ties to Berlin. I hope to be able to present my 

analyses of OLIVIER’S mathematical production in the near future. 

2 For an analysis of OLIVIER’S criterion and ABEL’S response, see also (I. Grattan-Guinness, 1970b, 

139–143) and (Goar, 1999). 

3 (Olivier, 1827). 

4 “criterium de la convergence des séries infinies” (ibid., 34). 

265

266 Chapter 13. ABEL and OLIVIER on convergence tests 

OLIVIER’S curious inserted remark — that the n th group of terms with the same sign 

could be considered instead of an — probably derived from the fact that within such a 

group, reordering of the terms could not effect the convergence or sum of the series. 

However, both in OLIVIER’S paper and in the present analysis, the important case 

arises by considering individual terms. 

For the historian, OLIVIER’S theorem requires an interpretation which is by no 

means easy or unambiguous; for instance, what does it mean that “this circumstance 

is a sign”? The main question in interpreting the theorem lies in this phrase, since 

it could be read to mean that if nan → 0 then the series will always be convergent. 

This was certainly the way it was interpreted by some of OLIVIER’S readers; however, 

the phrasing is sufficiently weak to call for further investigation. In the following, 

OLIVIER’S argument is outlined in order to illustrate how mathematicians in the early 

19 th century still argued about infinite series. Then, to supplement the theorem and 

its proof, a consideration of the examples to which it was applied is necessary before 

a weighed interpretation of the theorem can be given. 

13.1.1 OLIVIER’s first proof 

OLIVIER gave two arguments which illustrate how he came to believe in his theorem. 

The first argument was given immediately before the theorem was stated, whereas 

the second one was prompted by ABEL’S objection to the theorem and printed as a 

response to ABEL’S note. 6 

In 1827, OLIVIER divided infinite series into three categories: convergent, indeter- 

minate, and divergent. His definitions and the ensuing proofs are difficult to represent 

fairly, because his concepts are different and vague and his style of reasoning is rather 

verbal and leaves few hints on the unclear points. OLIVIER’S definition of convergent 

series consisted of two requirements: 

“One calls a series convergent which has the following two properties, namely: 

that one finds its numerical value ever more exactly when one calculates successively 

more terms and that by continuing the calculation indefinitely, one can approach 

the true value of the entire series to any degree one wishes.”” 7 

In OLIVIER’S definition, we see a curious and obscure mixture of the old and the 

new concepts of convergence. At the same time, OLIVIER speaks of numerical approx- 

5 “Donc si l’on trouve, que dans une série infinie, le produit du n me terme, ou du n me des groupes 

de termes qui conservent le même signe, par n, est zéro, pour n = ∞, on peut regarder cette seule 

circonstance comme une marque, que la série est convergente; et réciproquement, la série ne peut 

pas être convergente, si le produit n.an n’est pas nul pour n = ∞.” (Olivier, 1827, 34). 

6 (Olivier, 1828) 

7 “On appelle convergente une série, qui a les deux propriétés suivantes, savoir: qu’on trouve sa valeur 

numérique d’autant plus exactement, qu’on calcule successivement plusieurs termes, et qu’en 

continuant indéfiniment ce calcul, on peut se rapprocher de la vraie valeur de la série totale à tel 

degré qu’on voudra.” (Olivier, 1827, 31).

13.1. OLIVIER’s theorem 267 

imation in language similar to A.-L. CAUCHY’S (1789–1857) and of the “true value” 

of the series which resembles the Eulerian formal equality between functions. 

OLIVIER separated non-convergent series into indeterminate and divergent ones: 

“On the contrary, one calls a series indeterminate if continuing the calculation 

of terms does not make it approach anything. 

And one calls a series divergent in which the successive terms, added together, 

produces results which differ more and more from the true value of the series.” 8 

This distinction between two types on non-convergent series was probably inspired by 

the discussion of Poisson’s example in which OLIVIER also participated without making 

any noticeable contributions. 9 

OLIVIER proceeded to express the two criteria of his definition of convergence in a 

slightly different form. First, he observed, the terms of the series (or groups of terms 

with the same sign) had to constantly decrease. Second, the sum of terms after the n th 

term, i.e. the tail of the series, had to be zero for n = ∞. He gave similar translations 

of the concepts of indeterminate and divergent series. 

To obtain his theorem, OLIVIER first investigated the first condition concerning the 

vanishing of the terms. He stated that this condition would always be satisfied if the 

ratio of consecutive terms was always less than one. Thus, he apparently missed out 

on cases in which an+1 an < 1 but lim an+1 an 

= 1. For the second condition to also be 

fulfilled, OLIVIER noted that it would be necessary and sufficient that nan → 0 for 

n → ∞. 

Under hypothesis nan → 0 and the further assumption that the terms an vanish as 

n increases, OLIVIER claimed that 

if the series was written as 

R ≤ nan 

a1 + a2 + · · · + an + R. 

And thus, the vanishing of the tail R followed. How OLIVIER came to this [false] belief 

will be clearer below. 

On the other hand, the tail R could not vanish without nan also vanishing, OLIVIER 

claimed. Using the constantly decreasing nature of the terms, OLIVIER found 

na2n ≤ R ≤ nan 

where R suddenly meant “the sum of n terms which follows after the n th term.” 10 

Consequently, if nan vanished, so did R and the convergence of the series was secured. 

8 “Au contraire, on appelle indéterminée une série, qui ne donne aucun rapprochement, en continuant 

le calcul des termes. 

Et on appelle divergente une série, dont les termes suivants, ajoutés aux précédents, ne donnent 

que des résultats, qui s’éloignent plus en plus de vraie valeur de la série.” (ibid., 31). 

9 (Olivier, 1826b). For a contemporary evaluation, see ([Saigey], 1826, 112). 

10 “[. . . ] R, ou la somme des n termes qui suivent le n me terme.” (Olivier, 1827, 34).


Figure 13.1: OLIVIER’S geometrical argument 

Thus, OLIVIER’S proof rested on some uncertain principles. Firstly, OLIVIER’S di- 

vision of series into three classes and his definition of convergent series was less sharp 

and less useful than other existing concepts. Secondly, he assumed that in any (nu- 

merically) convergent series, the general terms were monotonically decreasing. As 

indicated, this assumption was central to his proof. Finally, OLIVIER switched from 

considering R as the tail of the series having infinitely many terms to cutting it off after 

n terms. This transition between infinite and finite objects represented limit processes 

which were never spelled out. 

13.1.2 OLIVIER’s second proof 

Reacting to ABEL’S criticism (see below), OLIVIER gave a short indication of the in- 

tuition behind his original proof. 11 There, he showed by way of a geometrical figure 

(see figure 13.1) how he had reasoned. OLIVIER led Bb denote the term an and ob- 

served that the area of the parallelogram Bbuv which represents the value nan would 

be greater than the smaller parallelograms, e.g. Bc, Cd, etc. The inequality was ob- 

tained by the constant decreasing of the terms an. The equality between the parallelo- 

gram Bbvu and nan represented OLIVIER’S different uses of infinite values for n. Thus, 

the above interpretation of OLIVIER’S first proof seems to be confirmed. 

OLIVIER reacted explicitly to ABEL’S criticism by observing that although his the- 

orem seemed to be well founded and the deduced examples were correct, ABEL had 

nevertheless observed that it was not “generally applicable”. OLIVIER saw ABEL’S 

criticism as an indication of the care which should be observed when dealing with 

infinite quantities and locate the mistake to his working indifferently with finite and 

infinite quantities. 

Finally, OLIVIER revised his theorem in order to make it correct by substituting the 

11 (Olivier, 1828).

13.2. ABEL’s counter example 269 

assumption 

n 

� � 

∞ 

∑ amn → 0 

m=1 

for the original nan → 0. Here, we see how OLIVIER approached the convergence of 

Cauchy sequences. 

13.1.3 The application to examples 

When OLIVIER came to apply his theorem to particular series, he did so as a complete 

test of convergence, i.e. as the bi-implication 

nan → 0 ⇔ ∑ an convergent. 

There are, however, no definite tests of this interpretation, because OLIVIER did not 

apply his criterion to any divergent series for which nan → 0. 

One of the most interesting examples, which OLIVIER did treat, was the binomial 

theorem. After long manipulations of the binomial coefficients, OLIVIER used his the- 

orem to state that the binomial series 

(1 + c) m m (m − 1) 

= 1 + mc + c 

1 · 2 

2 + . . . 

was convergent for any exponent if c < 1 and for m ≥ 0 if c = 1. The binomial series 

was divergent for c > 1 or for c = 1 if m < 0. In this way, OLIVIER obtained the 

convergence of the binomial series for real arguments and exponents. 

13.2 ABEL’s counter example 

Soon after the publication of OLIVIER’S paper, ABEL responded with a short note in 

the Journal. 12 There, ABEL commented on OLIVIER’S theorem in the following words, 

“The latter part of this theorem is very true but the first [part] does not seem 

to be so. For example, the series 

1 1 1 

1 

+ + + · · · + + . . . 

2 log 2 3 log 3 4 log 4 n log n 

is divergent although nan = 1 

log n is zero for n = ∞.”13 

Here, ABEL politely suggested that the first part of OLIVIER’S theorem “did not 

seem to be true”. In one of his notebooks, ABEL’S draft for the paper can be found; 

there he was more dramatic, remarking “Thus, Mr. Olivier is seriously mistaken.” 14 In 


13 “La dernière partie de ce théorème est très juste, mais la première ne semble pas l’être. Par exemple 

la série 

1 1 1 

1 

+ + + · · · + + . . . 

2 log 2 3 log 3 4 log 4 n log n 

est divergente, quoique nan = 1 

log n soit zéro pour n = ∞.” (ibid., 79). 

14 “Donc M. Olivier s’est trompé sérieusement.” (N. H. Abel, [1827] 1881, II, 199).


section 21.3, ABEL’S comments on OLIVIER’S theorem will serve as one among a class 

of cases where counter examples were employed for different ends and with differing 

confidence in the early nineteenth century. 

Central to ABEL’S proof was the inequality 

log (1 + x) < x (13.1) 

which he claimed was valid for all positive x. For x ≥ 1, it was obvious to ABEL and 

he gave no argument. It can be easily obtained by observing that 

x − log (1 + x) 

is increasing for x ≥ 1 and positive for x = 1. For x < 1, ABEL gave an argument 

employing the expansion of the logarithm into power series as 

log (1 + x) = 

∞ 

∑ 

n=1 

(−1) n−1 

n 

x n = x − 

∞ 

∑ 

n=1 

� 

1 x 

− 

2n 2n + 1 

� 

x 2n . 

From this, he observed that the parentheses were always positive which produced the 

desired inequality. Here, ABEL thus rearranged the terms of the logarithmic series 

without further ado. 15 

ABEL employed the inequality (13.1) for x = 1 n 

1 

n 

or written differently 

> log 

� 

1 + 1 

� 

n 

= log 

log (1 + n) 

log n 

< 

n + 1 

n 

to produce 

= log (n + 1) − log n, 

� 

1 + 1 

� 

. 

n log n 

Taking logarithms and using the inequality (13.1) again, ABEL obtained 

� � � 

log (1 + n) 

log log (1 + n) − log log n = log 

< log 1 + 

log n 

1 

� 

< 

n log n 

ABEL had thus produced the inequality 

which when summed from 2 to n gave 

log log (1 + n) < log log n + 1 

n log n , 

log log (1 + n) < log log 2 + 

n 

∑ 

k=2 

1 

k log k . 

1 

n log n . 

Since the left hand side obviously became infinite for n = ∞, the series on the right 

→ 0. 

hand side was divergent contradicting OLIVIER’S theorem since nan = 1 

log n 

ABEL’S conclusion was again remarkably reserved and apparently underplayed, 

“The theorem announced in the above citation is thus at fault in this case.” 16 

15 For more on the history of absolute convergence, see section 12.7. 

16 “Le théorème énoncé dans l’endroit cité est donc en défaut dans ce cas.” (N. H. Abel, 1828a, (400)).

13.3. ABEL’s general refutation 271 

13.3 ABEL’s general refutation 

After he had given his counter example to OLIVIER’S theorem, ABEL might have been 

expected to leave the matter. However, he had more to say on the issue. Creating some 

procedures to obtain from one divergent series another one which diverged much 

slower, ABEL could prove that the quest which OLIVIER had undertaken was bound 

to result in frustration. 

ABEL observed that if the series 

∞ 

∑ an 

n=0 

was divergent, then so was the series where each term had been divided by the re- 

spective partial sums, 

∞ 

an 

∑ sn n=1 

= 

∞ 

∑ 

n=1 

an 

∑ n−1 

k=0 an 

. 

The proof was easily obtained from arguments resembling the proof above. ABEL 

observed for n ≥ 1, inequality (13.1) produced 

and therefore, by summation, 

log sn − log sn−1 < an−1 

, 

sn−1 

log sn − log a0 < 

Since sn → ∞, the left hand side diverged, which implied the divergence of the right 

n 

∑ 

k=1 

hand side. We may express this result in modern language as lemma 2. 

Lemma 2 If ∑ ∞ n=0 an is a divergent series of positive terms, then the series defined as 

∞ 

an 

∑ sn n=1 

will be divergent as well. ✷ 

Now, in order to generally refute the theorem proposed by OLIVIER, ABEL as- 

sumed that a function φ taking integer arguments existed such that the series ∑ an 

was convergent if and only if φ (n) an → 0 as n → ∞. The series obtained as 

∞ 

∑ 

n=1 

1 

φ (n) 

ak . 

sk (13.2) 

would then produce a general counter example to this generalized theorem. The series 

(13.2) was divergent by the criterion, because φ (n) an = 1. On the other hand, when 

the procedure of obtaining a derived divergent series was applied to (13.2), a new 

series was obtained 

∞ 

∑ 

n=2 

1 

φ (n) ∑ n−1 

k=1 

1 

φ(k) 

. (13.3)


Since the series (13.2) was divergent, the series (13.3) would have to be divergent as 

well (by the procedure above). On the other hand, the generalized criterion, when 

applied to (13.3) gave 

φ (n) an = 

1 

∑ n−1 

k=1 

, 

1 

φ(k) 

which by the very divergence of (13.2) converged to zero for n → ∞. Thus, the se- 

ries (13.3) produced a general counter example to ABEL’S generalization of OLIVIER’S 

proposed convergence criterion. 

By this very elegant proof, ABEL turned OLIVIER’S proposed criterion against itself 

and it imploded. Thus, ABEL proved that no simple test of convergence of series could 

be devised. Interpreted as a question of delineation of concepts, ABEL’S result thus 

meant that the extent of the concept of convergent series was not easily determined by 

external criteria. 

13.4 More characterizations and tests of convergence 

In its published form, ABEL’S answer to OLIVIER’S paper was a negative one, in the 

sense that it refused a proposed theorem. However, in his notebooks, ABEL elabo- 

rated some of the ideas found therein to such a degree as to produce new, positive 

knowledge in the form of new characterizations and tests of convergence. 17 

In his notebook draft, ABEL obtained his own version of a limit comparison theorem 

which provided a necessary criterion for convergence. He claimed that if ∑ φ (n) 

was a divergent series, and ∑ an was a convergent one, it would be necessary that 

“the smallest among the limits of 

an 

φ(n) be zero.”18 ABEL’S proof was indirect: Under 

the contrary assumption, he wrote un = pnφ (n) where pn ≥ α. Then 

∑ un > ∑ αφ (n) = α ∑ φ (n) → ∞. 

From this, ABEL obtained the second part of OLIVIER’S theorem which he had not 

objected to: Because ∑ 1 n was known to be divergent, if ∑ an was to be convergent, it 

would be necessary that nan vanished as n became infinite. 

Pairs of convergent and divergent series. Also in the notebook, we find a general- 

ization of the lemma 2 to the effect that the divergence of ∑ an implied the divergence 

of the series ∑ an 

s α n 

where 0 ≤ α ≤ 1 (lemma 2 results from setting α = 1). A converse to 

this result was also obtained when ABEL proved that if the series ∑ an was divergent, 

then the series 



∞ 

an 

∑ 

n=1 s 1+α 

n

13.4. More characterizations and tests of convergence 273 

would be convergent if α > 0. Thus, from a divergent series, ABEL had prescribed 

means of obtaining two derived series, one of which was divergent, the other conver- 

gent. 

In another section of the note, ABEL devised another way of obtaining a divergent 

series, which would lead him to a new test of convergence. ABEL found that for any 

continuous function φ (n) which increased without bounds for n → ∞, the series of 

derived terms, 

would be divergent. 

∞ 

∑ φ 

n=1 

′ (n) , (13.4) 

ABEL’S proof proceeded from the Taylor series expansion of φ (to the second term 

and with remainder), 

φ (n + 1) = φ (n) + φ ′ (n) + φ′′ (n + θ) 

, for some 0 < θ < 1. 

2 

At this point, ABEL’S draft style made the precise assumptions of the ensuing de- 

ductions difficult to interpret. However, if ABEL’S requirements interpreted to mean 

φ ′′ (n) < 0, we obtain what was his next line, 

φ (n + 1) − φ (n) < φ ′ (n) . 

Then, the divergence of (13.4) followed by summation, 

φ ′ (n) > φ (n + 1) − φ (0) → ∞ as n → ∞. 

Subsequently, ABEL applied this procedure to prove the divergence of the series 

∞ 

∑ 

n=2 

1 

n ∏ m k=1 logk for m integral, 

n 

where log k n = log log k−1 n. ABEL did so by defining 

and differentiating it to obtain 

φ ′ m (n) = 

φm (n) = log m (n + a) 

1 

(n + a) ∏ m−1 

k=1 logk (n + a) . 

As a consequence of the theorem stated above, the series (corresponding to a = 0) 

was divergent. 

∞ 

∑ φ 

n=2 

′ m (n) = 

∞ 

∑ 

n=2 

1 

n ∏ m−1 

k=1 logk n


On the other hand, ABEL next turned to a function intimately related to the one 

studied above, 19 

ψ (n) = φm (n) 1−α 

. 

1 − α 

This time, ABEL’S calculations produced the inequality 

ψ (n + 1) − ψ (n) > ψ ′ (n + 1) 

corresponding to the fact that ψ ′ was a decreasing function. Consequently, through a 

number of calculations, ABEL was led to a series 

∞ 

∑ 

n=1 

1 

n log m (n) α+1 ∏ m−1 

k=1 logk n 

which was convergent if α > 0 and another one (corresponding to α = −1) 

which was divergent. 

∞ 

∑ 

n=1 

1 

n ∏ m−1 

k=1 logk n 

A logarithmic test of convergence. These methods of constructing convergent and 

divergent series led ABEL to a new test of convergence. The underlying idea of ABEL’S 

argument starts from the two series, one convergent and the other divergent, and 

compares a given series with these two typical ones. He found by simple arguments 

based on the results above, that if 

� 

log 

lim 

log m+1 n 

1 

unn ∏ m−1 

k=1 logk n 

� 

> 1, (13.5) 

the series ∑ un was convergent. ABEL’S criterion also indicated, that if the limit in 

(13.5) was < 1, the series ∑ un would be divergent. In its polished form, ABEL’S 

criterion thus became the following: 

Theorem 15 For a series of positive terms ∑ un, the limit 

� 

1 log un 

k = lim 

n→∞ 

d 

dn logm � 

n 

log m+1 n 

is considered. If k > 1, the series will be convergent; if k < 1, it will be divergent; and if k = 1, 

nothing can be said of the convergence or divergence of the series by this test. ✷ 

This result was later rediscovered by J. L. F. BERTRAND (1822–1900). 20 

19 ABEL actually also denoted this function by φ, but to avoid confusion, I have chosen to label it ψ. 

20 (Bertrand, 1842).

13.4. More characterizations and tests of convergence 275 

ABEL’S continued interest in the theory of series and — in particular — in obtain- 

ing new tests of convergence for series are indications of a continued interest in this 

topic. Due to CAUCHY’S re-founding of analysis, tests of convergence were becoming 

increasingly important, and a number of new tests were discovered in the nineteenth 

century. 21 

21 See e.g. (I. Grattan-Guinness, 1970b, 131–151).

Chapter 14 

Reception of ABEL’s contribution to 

rigorization 

In the course of the nineteenth century, analysis underwent an elaborate program of 

rigorization which effected both the techniques, results, and questions of the dis- 

cipline. Basic notions such as real numbers, continuous functions, integrals, and 

trigonometric functions were revised and deeply changed as reflections of a funda- 

mental transition in the ways mathematicians thought about their subject. The chang- 

ing concepts and attitudes have been studied intensively by historians. 1 In the twen- 

tieth century, N. H. ABEL’S (1802–1829) critical attitude and his part in the revision of 

analysis have received some interest but in the first decades after his death, these were 

not issues which attracted the most interest to his mathematics. The following section 

briefly discusses the reception of ABEL’S work on rigorization and certain aspects of 

the subsequent development. 

14.1 Reception of ABEL’s rigorization 

Because of the overwhelming development of analysis in the nineteenth century and 

ABEL’S apparently limited direct impact, the reception of ABEL’S contribution to the 

rigorization movement is only briefly described from two different perspectives. 2 

14.1.1 Binomial theorem 

The most immediate reaction to ABEL’S binomial paper was actually a non-reaction. 

In 1829 and 1830, A. L. CRELLE (1780–1855) published two papers on the binomial theorem 

demonstrating that the subject had not been closed by ABEL’S paper of 1826. 3 

CRELLE’S proofs were based on his previous research on the so-called analytical facul- 

1 See e.g. (Bottazzini, 1986; Hawkins, 1970). 

2 I hope to subsequently substantiate the analysis of the reception of this part of ABEL’S research 

through detailed studies of the works of selected, later mathematicians. 

3 (A. L. Crelle, 1829a; A. L. Crelle, 1830). 

277

278 Chapter 14. Reception of ABEL’s contribution to rigorization 

ties and were only partially within the Cauchyian approach. CRELLE only considered 

real arguments and exponents and divided his research into two parts correspond- 

ing to the two papers. First, he showed by formal arguments from the trivial identity 

1 = 1 that the binomial and its series had to be identical. The argument involved finite 

differences which had been such a key component of his research within the German 

combinatorial school. Second, CRELLE investigated the convergence of the binomial 

series dividing into separate cases corresponding to various assumptions on a and k 

(he wrote his binomial as (1 + a) k ). In each case, CRELLE considered the remainder 

terms of the series and established conditions of convergence or divergence. 

Concerning CRELLE’S publications on the binomial theorem, two remarks can be 

made. First, the fact that CRELLE published on a particular case of the binomial the- 

orem (real arguments and exponents) after ABEL’S more general result testifies to the 

debate between the Cauchyian program and the German algebraic school. CRELLE 

wrote in his introduction that he considered his proof to fulfill all requirements in- 

cluding being truly rigorous and general and simultaneously clear and elementary. 

These positive attributes were obtained through the use of algebraic manipulations. 4 

Second, CRELLE did not initially consider or even mention the necessity of conver- 

gence of the binomial series. ABEL’S critical attitude may have provoked CRELLE to 

take up the issue in the second paper. Thus, at least in Germany, A.-L. CAUCHY’S 

(1789–1857) new program of numerical equality was not immediately accepted — not 

even in CRELLE’S Journal after ABEL’S publication and the translation of the Cours 

d’analyse. 5 

Later in the nineteenth and the twentieth century, when the German combinato- 

rial school eventually lost ground, ABEL’S proof of the binomial theorem was recog- 

nized as the first rigorous and general proof. 6 The local criticism and scrutiny did not 

severely impair the evaluation of ABEL’S proof — primarily because the fundamen- 

tal notions and knowledge of power series developed immensely over the nineteenth 

century. 

14.1.2 From ABEL’s “exception” to uniform convergence 

As already indicated and cited, one of the major historical interests in ABEL’S contri- 

bution to the rigorization movement was the “exception” which he presented against 

Cauchy’s Theorem (see section 12.6). 7 The Fourier series representation of the function 

f (x) = x 2 on an interval such as ]−π, π[ provided an example that not every convergent 

sum of continuous functions was itself a continuous function as the Fourier series 

was periodically discontinuous at the end-points of the interval. 

4 (A. L. Crelle, 1829a, 305). 

5 (A. L. Cauchy, 1828). 

6 See e.g. (Stolz, 1904). 

7 This history is particularly well described in (Bottazzini, 1986). LAKATOS’ reconstruction (Lakatos, 

1976) is also extremely interesting.

14.1. Reception of ABEL’s rigorization 279 

ABEL’S “exception” met with little response in the 1820s and 1830s. Actually, it 

does not seem to have been quoted as influential in the first half of the 19 th century. In 

the 1840s, however, other and probably independent events put Cauchy’s Theorem (see 

page 217) back on the agenda. Independently and simultaneously in 1847, the math- 

ematicians G. G. STOKES (1819–1903) and P. L. VON SEIDEL (1821–1896) published 

investigations of the conditions under which a convergent sum of continuous func- 

tions would not result in a continuous function. 8 In both cases, their research led to 

the realization that a particular mode of convergence was involved and SEIDEL gave 

it the name of “arbitrarily slow convergence”; 9 STOKES developed a refined hierarchy 

of modes of convergence on intervals. 10 Of the two publications, I find SEIDEL’S par- 

ticularly interesting because it was set up in the form of a proof analysis and stressed 

the importance of keeping focus on the relations between limit processes. SEIDEL 

even proposed notational advances which would help clarify the interdependence of 

nested limit processes. Such thoughts were important in completely separating limit 

processes from infinitesimals (see below). 

CAUCHY’S eventual reaction. Apparently, even these researches of British and Ger- 

man mathematicians did not directly prompt any reaction from the French mathemati- 

cians, in particular CAUCHY. Eventually, CAUCHY did address the Cauchy Theorem 

again in an address to the Paris Academy of 1853. 11 In a paper, prompted by remarks 

made by French colleagues earlier that year, 12 CAUCHY described how the theorem of 

the Cours d’analyse could be amended so that it no longer suffered any exceptions. The 

fix which he proposed was the uniform convergence of the series in a form similar to 

the modern requirement. CAUCHY refined the assumptions of the theorem by requir- 

ing that a number N existed such that the difference |sm (x) − sn (x)| was less than ε 

for all values of x in the interval I under consideration when m, n ≥ N. CAUCHY’S 

requirement can easily be read as the modern definition of uniform convergence on 

the interval I, 

∀ε > 0 ∃N > 0 ∀m, n ≥ N ∀x ∈ I : |sm (x) − sn (x)| < ε. 

With this stricter assumption, the original proof of the theorem carried through even 

without a more elaborate notation to handle the two limit processes. In the paper, 13 

CAUCHY considered the series 

∞ 

sin nx 

∑ n n=1 

8 (Seidel, 1847; Stokes, 1847). GRATTAN-GUINNESS has pointed to the works of BJØRLING and considered 

him the “fourth man” in this development besides STOKES, SEIDEL, and CAUCHY (who was 

also involved, see below). See (I. Grattan-Guinness, 1986). 

9 (Seidel, 1847, 37). 

10 See (Stokes, 1847). 

11 (A.-L. Cauchy, 1853). 

12 (Briot and Bouquet, 1853a; Briot and Bouquet, 1853b). 

13 (A.-L. Cauchy, 1853, 31).


which represents the function f (x) = π−x 

2 

on the interval ]0, 2π[. Thus, CAUCHY 

made use of a function similar to Abel’s exception and spoke of his job as removing the 

possibility of such exceptions. However, judging from his subsequent proof revision, 

CAUCHY seems to have adapted a post-1820 use of counter examples (see chapter 21). 

CAUCHY’S attitude toward the status of the Cauchy Theorem have been debated 

among historians of mathematics and different conclusions have been reached, in particular 

depending on which parts of CAUCHY’S 1853-paper have been emphasized. 14 

With the advent of non-standard analysis in the twentieth century, 15 interpretations of 

various radicalism have proposed which render CAUCHY’S use of infinitesimals cor- 

rect within an enlarged system of real numbers. Although it is difficult to argue that 

CAUCHY was a modern non-standard analyst, 16 the debate over re-interpretations of 

his works inspires consideration of the developments which led the modern concep- 

tion of the basic notions to be fixed they way they were in the century before non- 

standard analysis. 

The notions of convergence and continuity. ABEL’S reading of CAUCHY’S original 

definitions of continuity and convergence became standardized in the course of the 

nineteenth century century, noticeably through the works and teachings of G. P. L. 

DIRICHLET (1805–1859) and K. T. W. WEIERSTRASS (1815–1897). Whereas CAUCHY 

had possibly been unclear or ambiguous about his definitions, ABEL certainly read 

point-wise convergence and point-wise continuity into them. 17 This reading was en- 

forced by the path subsequently taken in analytical research, in particular concerning 

trigonometric series. For example, a multiplicity of different modes of convergence 

were introduced as the century unfolded. Some of the new modes of convergence such 

as absolute convergence and the ones introduced by SEIDEL and STOKES have been 

touched upon above. Similar to the creation of the concept of uniform convergence, a 

deliberate distinction between point-wise and uniform continuity was eventually intro- 

duced by H. E. HEINE (1821–1881) in 1872. 18 Toward the end of the nineteenth cen- 

tury, the concepts of convergence were even complemented (or stretched) by concepts 

of summability which could also treat non-convergent series. In the form of summabil- 

ity introduced by G. F. FROBENIUS (1849–1917), a class of non-convergent series such 

as ∑ (−1) n x n was ascribed a sum in a new sense. FROBENIUS considered the class of 

series for which the average of the partial sums converged and called this limit the 

sum of the series. His central result generalized ABEL’S Lehrsatz IV and proved that 

his new definition of sum was a conservative extension of the Cauchyian definition. 

The variety of different concepts was the result of the explicit and precise formulation 

14 See e.g. the different interpretations in (Giusti, 1984), (Laugwitz, 1987), and (Bottazzini, 1990, xci). 

15 See e.g. (Cleave, 1971; Laugwitz, 1988–89). 

16 SPALT has come close to making this claim in (Spalt, 1981), though. More recently, he has retracted 

and debated this claim, see (Spalt, 2002, 326). 

17 See chapter 12. 

18 See (Dugac, 1989, 91–94).

14.2. Conclusion 281 

of basic notions in the Cauchyian sense to which ABEL had also contributed. 

14.2 Conclusion 

As described, ABEL’S contribution to the rigorization movement in analysis consisted 

of three aspects. Firstly, ABEL’S initial publication in the field on a general and rig- 

orous proof of the binomial theorem was an impressive and early adoption of the 

Cauchyian program in the theory of series. In six theorems, ABEL presented results per- 

taining to series which were tailored for his proof of the binomial theorem. Compared 

with CAUCHY’S original proof, ABEL generalized the binomial theorem to include 

complex exponents by solving a slightly different functional equation. However, the 

six introductory theorems and the definitions with which they operated also revealed 

a development from CAUCHY’S Cours d’analyse. Most importantly, ABEL consistently 

read CAUCHY’S definitions of convergence and continuity as point-wise definitions. 

These interpretations led him to an exception to Cauchy’s theorem and he subsequently 

replaced the effected theorem with his own version. Later, the exception would lead to 

the concept of uniform convergence. Secondly, ABEL partook in the debate over criteria 

of convergence which had become very important in CAUCHY’S reformulation of the 

theory of series. By means of a counter example and a very general argument, ABEL 

showed that no criteria of a particular form could complete delineate the concept of 

convergent series. The methods which he employed to this end also led him to a new 

test of convergence which, however, he did not publish. Eventually, ABEL’S private 

criticism and scrutiny of the existing methods in analysis was expressed in his let- 

ters but only indirectly in his publications. This critical attitude may have influenced 

some of his contemporaries but — I believe — it was generally not considered among 

his most important contributions until more historical enquiries made it a central in- 

dication of the historical development of rigor in analysis. 

ABEL’S contribution to the rigorization movement has simultaneously been inter- 

preted in terms of changing epistemic standards. It has been described how ABEL was 

aware that a new set of standards were being deployed and that arguments should be 

modified to conform to these new norms. Moreover, ABEL advocated a critical re- 

vision which aimed at investigating how true results and only few paradoxes could 

arise from standards of argument which were no longer deemed to be rigorous. These 

aspects as well as ABEL’S famous exception and the growth of new concepts will be- 

come important issues when the transition of paradigms is discussed in chapter 21. 

There, the notion of critical revision is invoked as an explanation of the apparent cu- 

mulative nature of mathematics and the a rather literal interpretation of the exception 

is suggested. 

In conclusion, I find it fair to say that ABEL’S work on the rigorization of the theory 

of series was an interlude — albeit a quite passionate one. ABEL’S contributions were


slightly marginal and not universally appreciated. Mathematicians who subsequently 

adhered to the rigorization program may have included ABEL among their heroes and 

have certainly adopted some of his notions and results but ABEL’S direct role was far 

from the role of the key initiator of the rigorization — CAUCHY.

Part IV 

Elliptic functions and the Paris 

mémoire 

283

Chapter 15 

Elliptic integrals and functions: 

Chronology and topics 

After the calculus was invented in the seventeenth century, it was quickly applied 

to classical problems concerning curves. One of the main achievements of the new 

tool was the ability to treat curves which had previously been outside the reach of 

geometry. For instance, the quadrature of the hyperbola provided an analytical way 

of describing and treating logarithmic functions. After the calculus had conquered 

such basic curves as the logarithmic and trigonometric ones, the determination of the 

length of an ellipse became a major obstacle on the path to generality. 

In the eighteenth century, L. EULER’S (1707–1783) new vision of the calculus as 

founded upon functions also transformed the way in which curves were approached. 1 

The way in which the elliptic transcendentals enter into the realm of analysis touches 

upon a number of points which will be described below and — primarily — connected 

to N. H. ABEL’S (1802–1829) work: 

1. During the eighteenth century, a need came to be felt to include higher transcen- 

dentals (i.e. functions different from the algebraic, logarithmic, and trigonomet- 

ric ones) into analysis on a par with the well established elementary functions. In 

order to accept these new objects into analysis, they had to undergo a process of 

becoming known. This process manifested itself in various ways, e.g. in the search 

for acceptable analytical representations of the new objects. 

2. Because the new functions were in some senses generalizations of the elementary 

functions, their study opened possibilities of generalization of existing results. 

In the process, insights into the new objects were obtained which also helped 

making them known. 

3. Ultimately, the consensus on how to introduce particular new higher transcen- 

dentals — as primitive functions of algebraic differentials — led to a research pro- 

1 For EULER’S approach to analysis, see section 10.1. 

285

286 Chapter 15. Elliptic integrals and functions: Chronology and topics 

gram aimed at describing in a general way properties of larger classes of tran- 

scendentals. 

By the end of the 1820s, the theory of elliptic (and even higher) transcendentals was 

establishing itself as one of the major research fields in mathematics in the century. 

Thus, much effort was put into the field and many connections with and implications 

for other fields were discovered. 

Evidently, the above themes represent aspects of the fundamental change toward 

concept based mathematics. Based on a description of the background and contents of 

ABEL’S work with these objects, the transition becomes quite evident and susceptible 

of further qualification. 

The theory of transcendental functions constitutes a major part of ABEL’S works 

and provide material for further analysis of some of the characterizations of his work. 

For instance, it will become clear that he employed one standard of rigor — differ- 

ent from the sharp manifest of his foundational research on infinite series — studying 

these new objects, even when the theory of infinite series was involved. Infinite series 

(and products) were used to represent the new objects of analysis by better established 

ones. This process of coming to know new objects can be traced in ABEL’S works and 

merits attention. 

Furthermore, interesting aspects of ABEL’S general inclination toward algebraic 

methods is evident in many of his researches on transcendental objects. In order to 

analyze this algebraic approach to the theory of transcendental objects, an understand- 

ing of the questions which ABEL wanted to answer is required. Only when questions 

and methods are viewed together can sense be made of the statement that “ABEL’S 

approach was algebraic”. 

15.1 Elliptic transcendentals before the nineteenth 

century 

Ellipses were known to and treated by mathematicians since the times of the Greeks. 

They knew that the ellipse can be obtained by a central projection from a circle and de- 

duced numerous results concerning these objects in their mathematical investigations 

of conic sections. However, with the advent of symbolic notation, the calculus, and I. 

NEWTON’S (1642–1727) theory of gravitation, the study of curves such as the ellipse 

changed. 

15.1.1 Rectification of the ellipse 

When the creators and early practitioners of the calculus sought to promote their new 

tool, they often attacked problems which belonged to the classical realm of analysis of 

curves. Traditionally, the study of curves included such problems as the construction

15.1. Elliptic transcendentals before the nineteenth century 287 

of points on the curve, its rectification and quadrature and the determination of its 

tangents, centres of curvature, involutes, and evolutes. A number of these properties 

together constituted the knowledge required for a curve to be known and it was one 

of the greatest achievements of the calculus to devise a method for obtaining most of 

them from a single defining equation. 2 

A variety of different curves was treated in the first decades of the calculus, and 

many problems were reduced to “simpler” problems, primarily to the construction of 

points on algebraic curves, the rectification of the circle (inverse trigonometric func- 

tions) and the quadrature of the hyperbola (logarithmic functions). A classification of 

construction problems was established based on the simpler problems to which the 

solution of the given problem could be reduced. 

Despite the efforts of the mathematicians, certain problems defied the accepted 

known means of solution; for instance, when asked to compute the arc lengths of 

ellipses and some other curves, mathematicians found that the known simple integrals 

did not suffice. 

In the year 1675, G. W. LEIBNIZ (1646–1716) directed two enquiries concerning the 

rectification of the ellipse to the British mathematicians J. GREGORY (1638–1675) and 

NEWTON. LEIBNIZ received the answer that the British mathematicians could only 

compute the length of an ellipse by approximation, i.e. with the help of infinite series, 

and did not possess any closed expression for the length. LEIBNIZ, himself, at the time 

believed that he could reduce this problem (and the rectification of the hyperbola) to 

the quadratures of the circle and the hyperbola. Later, LEIBNIZ realized that he had 

been misled by a computational error. 3 

As can be seen from box 6, the rectification of the ellipse involved the computation 

of an integral of the form � R(x) dx 

√ P4(x) in which R and P4 were polynomials such that 

deg P4 ≤ 4. Such integrals (with the relaxed assumption that R be only a rational function) 

were soon called elliptic integrals by G. C. FAGNANO DEI TOSCHI (1682–1766). 4 

LEIBNIZ’ question reflects the search for simpler, finite representations of elliptic inte- 

grals. 

In a paper written in 1732 but not published until six years later, 5 EULER deduced 

a series representation of a quarter of the circumference of an ellipse. Based on a figure 

(see figure 15.1) in which M represented a point on the ellipse with center C and semi- 

axes CA = a and CB = b, EULER expressed the differential of the arc-length �AM 

as 

ds = b2√ b 2 + t 2 + nt 2 dt 

(b 2 + t 2 ) 3 2 

2 For information on these aspects of curves, see e.g. (Loria, 1902). For a general discussion on the 

conceptions of curves before the advent of the calculus, see (H. J. M. Bos, 2001). 

3 (Hofmann, 1949, 75,118). 

4 (Natucci, 1971, 516). For more on FAGNANO DEI TOSCHI’S work on elliptic integrals, see below. 

5 (L. Euler, 1732a).


Rectification of the ellipse Consider the ellipse with major axis 2a and minor axis 

2b given by the Cartesian equation 

Obviously, from this equation 

x 2 

a 

b 

y2 

+ = 1. 

2 2 

x = a cos θ and y = b sin θ 

which means 

dx 

dy 

= −a sin θ and = b cos θ. 

dθ dθ 

To compute the arc length, we find 

� 

s (θ) = 

� � 

dx 

dθ 

� � 

= b 

� 2 

+ 

� �2 � 

dy � 

dθ = a 

dθ 

2 sin2 θ + b2 cos2 θ dθ 

1 − k 2 sin 2 θ dθ with k 2 = b2 − a 2 

This is precisely the form of A.-M. LEGENDRE’S (1752–1833) second kind of elliptic 

integrals with the modulus k (denoted F (θ, k) in table 15.2). 

In order to reduce the integral to the form � R(x) dx 

, we substitute z = sin θ and get 

and thus 

dz = cos θ dθ = 

√ P4(x) 

b 2 

� 

1 − sin 2 θ dθ ⇒ dθ = 

� � 

1 − k2 sin2 � 

θ dθ = 

√ 1 − k2z2 � 

√ dz = 

1 − z2 Box 6: Rectification of the ellipse 

. 

1 

√ 1 − z 2 dz 

1 − k 2 z 2 

� (1 − k 2 z 2 ) (1 − z 2 ) dz. 

Figure 15.1: EULER’S rectification of an ellipse by infinite series (L. Euler, 1732a, 2)

15.2. The lemniscate 289 

in which the semi-axes were related by 

a 2 = (n + 1) b 2 . 

EULER then expanded the square root by use of the binomial theorem 

� 

(b 2 + t 2 ) + nt 2 = � 

b 2 + t 2 + 

∞ 

∑ 

µ=1 

(−1) µ−1 ∏ µ−1 

k=1 (2k − 1) 

∏ µ 

k=1 (2k) 

n µ t 2µ 

(b2 + t2 ) 2µ−1 

2 

Thus, EULER obtained the differential in the form (A0, A1, . . . were specified constants) 

∞ 

2 

ds = b ∑ 

µ=0 

Aµn µ t 2µ dt 

(b2 + t2 . 

µ+1 

) 

which he next integrated term-wise from 0 to ∞ to obtain the rectification of a quarter 

of the ellipse in the form 

�AMB = π 

2 

15.2 The lemniscate 

∞ 

∑ 

µ=0 

∏ µ−1 

k=0 (2k + 1)2 

∏ µ 

k=1 (2k)2 (2µ − 1) n µ . 

Another curve which received the attention of mathematicians starting with the broth- 

ers JAKOB I BERNOULLI (1654–1705) and JOHANN I BERNOULLI (1667–1748) was the 

so-called lemniscate. 6 The curve was defined by the Cartesian equation 

� 

x 2 + y 2� 2 � 

2 = a x 2 − y 2� 

, 

and both brothers recognized that the arc length of the curve depended on an integral 

of the form � 

(see box 7). 

dz 

√ 1 − z 4 

In Italy, the autodidact nobleman FAGNANO DEI TOSCHI took up the study of the 

lemniscate. 7 By a set of theorems, FAGNANO DEI TOSCHI was able to prove that the 

division of the quadrant of the lemniscate into k parts could be constructed by ruler 

and compass if k was of one of the forms 2 × 2 m , 3 × 2 m , or 5 × 2 m . By elimination of 

the intermediate variable x in the substitutions 

x = 

� 

1 − √ 1 − z 4 

FAGNANO DEI TOSCHI obtained that 

dz 

√ 1 − z 4 

z 

and x = 

du 

= 2√ 

1 − u4 √ 2u 

√ 1 − u 4 , 

and he had obtained the duplication of any segment of the lemniscate arc. 

6 See (H. J. M. Bos, 1974). Mostly, discovery of the curve is attributed to BERNOULLI alone as he holds 

priority of publication and gave the curve its name. 

7 Unfortunately, I not have had access to FAGNANO DEI TOSCHI’S original works. Instead, the short 

outline is based on (R. Ayoub, 1984; Siegel, 1959). 

.


Rectification of the lemniscate To find the arc length of the lemniscate given by the 

polar equation 

we compute 

i.e. 

r 2 = a 2 cos 2θ, 

2r dr = −2a 2 sin 2θ dθ, 

ds 2 = r 2 dθ 2 + dr 2 = a 2 cos2 2θ + sin2 2θ 

dθ 

cos 2θ 

2 , 

a dθ a dθ 

ds = √ = � 

cos 2θ 

� 

s (θ) = a 

dθ 

� 

1 − 2 sin2 θ . 

1 − 2 sin 2 θ , 

This is an example of an elliptic integral of the first kind. With the substitution z = 

sin θ, we find (see box 6) 

� 

s = a 

1 dz 

√ √ 

1 − 2z2 1 − z2 � 

= a 

dz 

√ 1 − 3z 2 + 2z 4 . 

Box 7: Rectification of the lemniscate 

15.2.1 Addition of lemniscatic arcs 

EULER took his inspiration directly from the works of FAGNANO DEI TOSCHI. In 1750, 

after FAGNANO DEI TOSCHI had published his collected works, the author sent a copy 

to the Berlin Academy of Sciences of which he was a member. The following year, 

on 23 December 1751, the work came into the hands of EULER who was given the 

assignment of commenting upon it. 8 C. G. J. JACOBI (1804–1851) has called this date 

the birthday of elliptic functions. 

In the process of preparing an answer for FAGNANO DEI TOSCHI, EULER became 

very interested in the topic of lemniscate integrals. EULER commented on his new 

research in a letter to C. GOLDBACH (1690–1764): 

“Recently, I have come across a curious integration. Just as the integral of 

√ is yy + xx = cc + 2xy 

1−yy √ 1 − cc, the integral of the 

the equation dx 

√ 1−xx = dy 

equation dx 

√ 1−x 4 

= dy 

√ 1−x 4 is 

� 

yy + xx = cc + 2xy 1 − c4 − ccxxyy.” 9 

8 (Siegel, 1959). 

9 “Neulich bin ich auch auf curieuse Integrationen verfallen. Dann gleich wie von dieser Äquation 

√ dx = 

(1−xx) 

dy 

√ (1−yy) das integrale ist yy + xx = cc + 2xy � (1 − cc), also ist von dieser Äquation

15.2. The lemniscate 291 

Figure 15.2: LEONHARD EULER (1707–1783) 

In the letter, EULER applied the theorem to demonstrate how the difference be- 

tween two segments of the arc of an ellipse could be rectified. There is no proof of the 

theorem in the letter but EULER published a proof in 1756/56. 10 The proof progressed 

by direct differentiation of the purported integral 

to obtain 

y 2 + x 2 = c 2 � 

+ 2xy 1 − c4 − c 2 x 2 y 2 

dx 

√ 1 − x 4 = 

dy 

� 1 − y 4 . 

(15.1) 

The theorem founded a particular branch of the theory of elliptic integrals as it con- 

tained the so-called addition theorem for lemniscate integrals. If x and y were related 

by (15.1), the equation 

� dx 

(1−x4 ) = 

dy 

� 

(1−y4 ) 

� x 

0 

das integrale: 

dt 

√ 

1 − t4 = 

� y 

0 

dt 

√ + C 

1 − t4 � 

yy + xx = cc + 2xy (1 − c4 ) − ccxxyy.” 

(Euler→Goldbach, 1752. Euler and Goldbach, 1965, 347–348); also (Fuss, 1968, I, 567). 

10 (L. Euler, 1756/57).


holds where C is constant (independent of x and y). However, if we let y = 0, we find 

x 2 = c 2 , and thus 

C = 

� c 

0 

dt 

√ 1 − t 4 . 

In this form, the addition theorem for lemniscate integrals is apparent, 

� x 

0 

dt 

√ 

1 − t4 + 

� y dt 

√ 

0 1 − t4 = 

� z 

0 

dt 

√ , if 

1 − t4 x 2 + y 2 + z 2 x 2 y 2 = z 2 � 

+ 2xy 

15.2.2 EULER’s rectification of the lemniscate 

1 − z 4 . 

At least twice, EULER deduced expressions for the arc length of the lemniscate by 

infinite series. In the third part of his trilogy, the Institutiones calculi integralis, 11 EULER 

found the relation 

� 1 

0 

x m+1 dx 

√ 1 − x 2 

� 

m 1 

= 

m + 1 0 

xm−1 dx 

√ . (15.2) 

1 − x2 Subsequently, 12 EULER expressed the length of the first quadrant of the lemniscate 

based on the relation 

� 1 

Using the binomial theorem, he wrote 

0 

� 

1 + x 2� − 1 2 

0 

dx 

√ 

1 − x4 = 

� 1 � 

1 + x 

0 

2� − 1 2 

dx 

√ 1 − x 2 . 

= 1 − 1 

2 x2 1 · 3 

+ 

2 · 4 x4 1 · 3 · 5 

− 

2 · 4 · 6 x6 + . . . 

and when the term-wise integration was carried out, EULER found the expression 

� � 

1 dx π 

√ = 1 − 

1 − x4 2 

12 

22 + 1232 2242 − 123252 2242 � 

+ . . . 

62 using the relation (15.2), above. 

The deduction of the result in the Institutiones is less complicated than the similar 

result for the ellipse given by EULER in 1732 and described above. However, the basic 

tools of the two approaches are the same: expansion by use of the binomial theorem 

and term-wise integration of the power series which was thus obtained. 

15.3 LEGENDRE’s theory of elliptic integrals 

Toward the end of the eighteenth century, LEGENDRE gave the theory of elliptic inte- 

grals a new twist with his contributions. In a number of lengthy papers and mono- 

11 (L. Euler, 1768, XI, 208). 

12 (ibid., XI, 211).

15.3. LEGENDRE’s theory of elliptic integrals 293 

1793 Mémoire sur les transcendantes elliptiques 

1811,1817,1816 Exercises de calcul intégral sur divers 

ordres de transcendantes et sur les 

quadratures 

1825,1826,1828 Traité des fonctions ellipitiques et des 

intégrales eulériennes 

Table 15.1: LEGENDRE’s publications on elliptic transcendentals 

Figure 15.3: ADRIEN-MARIE LEGENDRE (1752–1833) 

graphs, LEGENDRE developed his theory of elliptic and other higher integrals. 13 Even- 

tually, in the 1820s toward the end of his life, LEGENDRE decided to publish his re- 

search on elliptic integrals in the form of a number of monographs. The first two vol- 

umes of the Traité des fonctions elliptiques et des intégrales eulériennes, laid the foundation 

and presented the state of the art in the field. 

Generally, LEGENDRE’S theory of elliptic integrals concerned the transformation 

and numerical approximation of these integrals. An important position in LEGEN- 

DRE’S approach to the theory was taken by the classification of such integrals into 

a small number of canonical forms. LEGENDRE worked with three canonical forms 

which he termed the elliptic functions of the first, second, and third kinds [espèce] (see 

table 15.2 and box 8). Later, ABEL reserved the word elliptic function for the inverse 

13 (A. M. Legendre, 1811–1817; A.-M. Legendre, 1793).


Kind Integral in x Integral in φ Symbol 

� 

� 

dx 

dφ 

First √ √ F (φ, k) 

(1−x2 )(1−k2 x2 ) 

1−k2 sin2 φ 

Second � � 1−k2x2 1−x2 � 

dx 

� 

1 − k2 sin2 φ dφ E (φ, k) 

� 

� dφ 

Third 

Π (φ, n, k) 

dx 

(1+nx 2 ) √ (1−x 2 )(1−k 2 x 2 ) 

(1+n sin 2 φ) √ 1−k 2 sin 2 φ 

The variables φ and x are related by x = sin φ. 

In accordance with 18 th century practice, the integrations 

are to be performed from 0 to the x and φ, respectively. 

Table 15.2: LEGENDRE’s classification of elliptic integrals 

function of the integrals and to avoid confusion, I will often refer to LEGENDRE’S el- 

liptic functions as elliptic integrals. 

LEGENDRE introduced the notation 

∆ (φ) = ∆ (φ, k) = 

� 

1 − k 2 sin 2 φ. 

LEGENDRE’S approach based on reducing elliptic integrals to a number of basic 

forms was not entirely new. In the 1780s, J. L. LAGRANGE (1736–1813) had made 

similar attempts reducing all elliptic integrals to the basic form 

� 

M � x 2� dx 

� (1 ± p 2 x 2 ) (1 ± q 2 x 2 ) 

in which M was a rational function. 14 Later, ABEL took up the reduction and sug- 

gested another set of basic forms. Thus, from the last decades of the 18 th century and 

into the 1820s, the theory of elliptic integrals was still in its formative process and the 

basic representations had not been decided upon yet. With the advent of LEGENDRE’S 

Traité des fonctions elliptiques, its author hoped to settle the foundations once and for 

all and his categorization into three kinds of integrals was successful. Soon thereafter, 

however, the theory changed dramatically with the novel ideas of ABEL and JACOBI, 

and instead it was JACOBI’S Fundamenta nova which became the foundation of the theory 

and established its notation. 15 

Complete integrals and the reduction program. LEGENDRE introduced the notation 

E1 , F1 , Π1 for the complete integrals which corresponded to φ = π 2 . These particular 

numbers (functions of the modulus k) received some special attention and a number 

of remarkable relations were discovered among them. LEGENDRE developed the com- 

plete integrals E 1 and F 1 into series. LEGENDRE also devoted quite some effort to the 

problems of comparison of elliptic integrals of the three kinds. 

14 (Lagrange, 1784–1785, 264). 

15 (C. G. J. Jacobi, 1829). See also chapter 20.

15.3. LEGENDRE’s theory of elliptic integrals 295 

LEGENDRE’S reduction of elliptic integrals In his reduction, LEGENDRE first considered 

the integral � P dx 

R in which P was a polynomial and R was the square root of 

a fourth degree polynomial, 

� 

� 

� 

R = 

� 4 

∑ 

m=0 

αmx m . 

Obviously, such integrals could be studied by studying the simpler ones of the form 

� x k dx 

R which LEGENDRE denoted Πk (this is distinct from the Π which denotes integrals 

of the third kind). LEGENDRE found 

x k−3 � 

R = 

= 

� 

d x k−3 � 

� 

R = (k − 3) 

4 � 

∑ 

m=0 

k − 3 + m 

2 

x k−4 � 

R dx + 

x k−3 R ′ dx (15.3) 

� 

αmΠ m+k−4 . (15.4) 

This meant, that for any k ≥ 4, the integral Π k depended algebraically on the previous 

integrals Π 0 , Π 1 , . . . , Π k−1 . By writing out (15.4), LEGENDRE observed that also the 

integral Π 3 only depended algebraically on Π 0 , Π 1 , and Π 2 . Therefore, the integral 

� P dx 

R in which P was a polynomial could be reduced algebraically to the integrals Π 0 , 

Π 1 , and Π 2 . Furthermore, knowledge of Π 0 and Π 2 would also entail knowledge of 

Π1 by a linear transformation. 

Thus, elliptic integrals � P(x) dx 

R in which P was a polynomial had been taken care of 

and had been reduced to the two integrals � dx 

R and � x2 dx 

R . For more general, rational 

functions P, LEGENDRE expanded into partial fractions, considered � 

dx , and 

(1+nx)R 

applied a similar line of argument. 

Initially, LEGENDRE chose and ordered his basic forms according to the conic sec- 

tions whose rectification they described. Thus, the integral E = � ∆ dφ was considered 

the most basic as it described the rectification of the ellipse. The second class was ini- 

tially represented by Υ = ∆ tan φ − � ∆ dφ + b 2 � dφ 

∆ 

hyperbola. This class was, however, soon replaced by F = � dφ 

∆ 

represented by the integral Π = � dφ 

(1+n sin 2 φ)∆ 

involved a third parameter, n. 

which represented the arc of the 

. The third class was 

which, contrary to the two first kinds, 

Box 8: LEGENDRE’S reduction of elliptic integrals


In order to facilitate computation of numerical values, LEGENDRE developed a the- 

ory by which a sequence of moduli could be constructed which allowed the numerical 

approximation of elliptic integrals to be reduced. 

15.4 Left in the drawer: GAUSS on elliptic functions 

When C. F. GAUSS (1777–1855) was informed of ABEL’S first publication on elliptic 

functions, the Recherches, his answer was extraordinary. 16 GAUSS was impressed with 

ABEL’S work and was happy to see that the young Norwegian had relieved him of the 

obligation to publish a third of his own knowledge concerning these elliptic functions. 

Furthermore, GAUSS was surprised to see that ABEL had followed almost exactly the 

same route as he, himself, had taken to the point where their symbols were the same. 

As GAUSS never published any of the monographs on elliptic functions which he had 

intended, historians have had to look in his diary and in some of his manuscripts for 

hints concerning his results and methods. 

From 1797, GAUSS’ diary documents an increasing interest in the lemniscate inte- 

gral. 17 Despite a lasting interest and many connections to other parts of his research, 

GAUSS never published on the theory of lemniscate integrals. Thus, GAUSS’ ideas 

only indirectly influenced the development of the theory of elliptic functions in the 

1820s. In order to illustrate how GAUSS arrived at some of his insights and to under- 

stand his remarks on ABEL’S Recherches, a brief discussion of important points in his 

manuscripts and in his mathematical diary is given. Emphasis is here put on the in- 

version of the lemniscate integral into GAUSS’ lemniscate function, the periods of the 

lemniscate function, and GAUSS’ representation of the lemniscate function by various 

infinite expressions. 

Whereas GAUSS’ diary obviously provides a strict chronological frame, his manu- 

scripts are less clearly ordered. The manuscripts contained in the Werke have been 

compiled and put into an order which fit the editor. Therefore, and because GAUSS’ 

ideas had no direct impact on his immediate successors, I have taken the liberty to 

treat his production in its thematic contexts. 

The role of GAUSS’ knowledge. As mentioned, GAUSS deferred publication on the 

subject of lemniscate functions. According to SCHLESINGER, GAUSS had hoped to 

publish on his research on higher transcendentals in a form which would combine his 

three greatest interests in the field: the lemniscate function, the arithmetic-geometric 

means and the hypergeometric series. 18 GAUSS did publish on the hypergeometric 

series, 19 and there is a brief description of arithmetic-geometric means in his work De- 

16 (Gauss→Bessel, 1828.03.30. In Gauss and Bessel, 1880). 

17 (C. F. Gauss, 1981; J. J. Gray, 1984). 

18 (Schlesinger, 1922–1933, 27). 

19 (C. F. Gauss, 1813).

15.5. Chronology of ABEL’s work on elliptic transcendentals 297 

terminatio attractionis. 20 Thus, when GAUSS spoke of the third of his research which 

ABEL had anticipated, he probably referred to the theory of lemniscate functions. Con- 

cerning the lemniscate function, GAUSS had performed the inversion, extended to 

complex variables, found the resulting function to be doubly periodic, expressed its 

addition formulae, and obtained infinite representations of it. The division problem of 

the lemniscate had played an important role (together with the arithmetic-geometric 

means) in motivating his research. Concerning all these results and aspects, GAUSS 

was certainly correct in observing the similarity with ABEL’S approach. It is possible, 

but not a necessary assumption, that rumors of GAUSS’ investigations and their meth- 

ods had spread to ABEL through H. C. SCHUMACHER (1784–1873) and C. F. DEGEN 

(1766–1825); certain of GAUSS’ letters to SCHUMACHER suggest that GAUSS for a short 

while considered it a possibility. 21 

15.5 Chronology of ABEL’s work on elliptic 

transcendentals 

Little is known of ABEL’S first encounters with elliptic functions. Presumably, ABEL 

took up DEGEN’S suggestion in the letter to C. HANSTEEN (1784–1873) and began 

studying the higher transcendentals possibly through the works of EULER and LEG- 

ENDRE. A letter from his stay in Copenhagen 1823 indicates that he had shown DEGEN 

a small paper in which “inverse functions of elliptic transcendentals” played a role. 22 

In both editions of ABEL’S Œuvres, a number of manuscripts are included which were 

among the papers destroyed in a fire in B. M. HOLMBOE’S (1795–1850) house in 1849. 23 

According to HOLMBOE, the manuscripts date from before ABEL’S European tour, i.e. 

they were written before 1825. 24 Apparently based on these manuscripts and the let- 

ter from Copenhagen (as there are no other primary sources), 25 some historians have 

credited ABEL with possessing the key results and methods around 1823. 

It was, however, not until during and after the European tour that ABEL developed 

and published his research on elliptic functions and higher transcendentals which 

would merit so much attention. ABEL’S mature research on the topics can be seper- 

ated into three categories. His first publication on the subject was the Recherches, which 

introduced the crucial idea of inverting elliptic integrals of the first kind into elliptic 

functions and established the latter as doubly periodic functions of a complex variable. 

Simultanously with the publication of ABEL’S Recherches, the German mathemati- 

cian CARL GUSTAV JACOB JACOBI announced some results on the transformation of 

20 (C. F. Gauss, 1818). 

21 (Schlesinger, 1922–1933, 167). I hope to have more to say on this at a later stage in connection with 

future research on the mathematical milieu in Copenhagen in the early nineteenth century. 

22 (Abel→Holmboe, Kjøbenhavn, 1823/08/04. N. H. Abel, 1902a, 5). 

23 (N. H. Abel, 1881, II, 324) and (Stubhaug, 1996, 560). 

24 (Holmboe in N. H. Abel, 1839, i) 

25 (Abel→Holmboe, Kjøbenhavn, 1823/08/04. In N. H. Abel, 1902a, 4–8).


VII Propriétés remarquables de la fonction 

y = φx déterminée par l’équation f y.dx − 

f x � (a − y) (a1 − y) (a2 − y) . . . (am − y) = 

0, f étant une fonction quelconque de y 

qui ne devient pas nulle ou infinie lorsque 

y = a, a1, a2, . . . , am 

VIII+IX Sur une propriété remarquable d’une classe très 

étendue de fonctions transcendantes 

X Sur la comparaison des fonctions transcendantes 

XIII Théorie des transcendantes elliptiques 

Table 15.3: ABEL’S early unpublished works on elliptic integrals and related topics. 

The manuscripts are no longer extant but HOLMBOE dated them all to the period before 

ABEL’S European tour. The roman numerals indicate the position of the manuscript 

in (N. H. Abel, 1881, II). 

elliptic integrals which were astounding to ABEL. JACOBI had obtained results which 

were special cases of ABEL’S own findings, and ABEL was surprised by the sudden 

element of competition. For a period of time, ABEL devoted himself to explaining and 

elaborating the results of JACOBI within his own framework and this constituted a 

second topic in his research on elliptic functions. 

ABEL’S last approach to elliptic functions was the most general. Applying the 

theory which he developed in the Paris memoir concerning integration of algebraic 

differentials (see chapter 19) to the special case of elliptic functions, ABEL could sketch 

a very general approach to elliptic functions which — based on functions of the first 

kind — introduced all kinds of elliptic functions. 

These aspects — which far from exhaust the discipline of elliptic and higher tran- 

scendentals in the early nineteenth century — will be addressed in the subsequent 

chapters. A complete description of the history of these transcendental objects is way 

outside the scope of the present work as their study was one of the most important 

and widely studied mathematical topics in the period. Instead, selections have been 

made to illustrate how new objects were being introduced and how new tools — and 

primarily algebraic tools — were being put to use in the investigation of these new 

objects.

Chapter 16 

The idea of inverting elliptic integrals 

As noted, N. H. ABEL (1802–1829) probably started developing his interest in ellip- 

tic integrals as a direct response to C. F. DEGEN’S (1766–1825) suggestion. Before his 

European tour, he was already well acquainted with the comprehensive treatment the 

theory had been given by A.-M. LEGENDRE (1752–1833) and he had begun develop- 

ing his own ideas. Soon, the theory of elliptic functions would occupy most of his 

resources. As already stated, ABEL’S first publication on the theory of elliptic tran- 

scendentals was his Recherches sur les fonctions elliptiques which appeared in two parts 

in A. L. CRELLE’S (1780–1855) Journal in 1827 and 1828. 1 

16.1 The importance of the lemniscate 

ABEL’S Recherches were designed to address particular questions pertaining to elliptic 

integrals of the first kind because these integrals had “the most remarkable and simple 

properties”. 2 Later, ABEL’S penetrating knowledge of elliptic integrals of the first kind 

would also allow him to attack all elliptic functions from a general perspective (see 

chapter 20). 

In the Recherches, ABEL studied integrals of the form 

� 

dx 

� (1 − c 2 x 2 ) (1 + e 2 x 2 ) 

(16.1) 

which do not immediately belong to either of the kinds classified by LEGENDRE. ABEL 

argued that his choice of representation made the obtained formulae more simple and 

stressed that in (16.1), the integrand was more symmetric than in LEGENDRE’S stan- 

dard form. 

The simplicity of central formulae which ABEL emphasized is particularly clear in 

one specific application of the theory which ABEL developed in the second part of the 

Recherches. 3 There, ABEL solved the division problem for the lemniscate integral which 

1 (N. H. Abel, 1827b; N. H. Abel, 1828b). 

2 (N. H. Abel, 1827b, 102). 


299

300 Chapter 16. The idea of inverting elliptic integrals 

Figure 16.1: Stamp depicting Gauss and the construction of the regular 17-gon. 

is undoubtedly the simplest integral of the form (16.1) corresponding to e = c = 1. 4 

Thus, there is a suggestion that ABEL’S choice of representation of the integrals is a 

direct reflection of one of the main purposes of the Recherches, the adaption of C. F. 

GAUSS’ (1777–1855) methods from the Disquisitiones arithmeticae to the division of the 

lemniscate. 5 

16.2 Inversion in the Recherches 

The issue of CRELLE’S Journal which contained ABEL’S inversion of elliptic integrals 

into elliptic functions was published on 20 September 1827; 6 the date is of importance 

in analyzing the internal relations between ABEL’S and C. G. J. JACOBI’S (1804–1851) 

approaches (see below and section 18.1, below). 

In the introduction to the Recherches, ABEL described his idea: 

“In this memoir, I propose to study the inverse function, i.e. the function φα 

determined by the equations 

4 See (Glaisher, 1902). 

5 See also chapter 7. 


� 

dθ 

α = � 

1 − c2 sin 2 θ and 

sin θ = φ (α) = x.” 7

16.2. Inversion in the Recherches 301 

Thus, from ABEL’S own description of it, the idea of considering the inverse func- 

tions appears quite natural. However, by this simple step, an entirely new class of 

functions was introduced, and they certainly looked different from anything known 

so far. 

With ABEL’S choice of representation of the integral, the inversion became 

α = 

� x 

0 

dx 

� (1 − c 2 x 2 ) (1 + e 2 x 2 ) � φ (α) = x. (16.2) 

First, ABEL introduced a special name for the integral from 0 to 1 c : 

“By thus letting 

ω 

2 = 

� 1 

c 

0 

∂x 

√ [(1 − c 2 x 2 ) (1 + e 2 x 2 )] , 

it is evident that φ (α) is positive and increasing from α = 0 to α = ω 

2 .”8 

This remark seems to indicate that ABEL was well aware that for the inversion to be 

meaningful, the integral had to be a strictly monotonous function. 

ABEL’S next step consisted in the observation that the integral was an odd function 

of x, and thus φ (−α) = −φ (α). At this point, ABEL had thus obtained the function φ 

� 

. 

for a segment of the real axis � − ω 2 , ω 2 

16.2.1 Going complex 

ABEL’S study of the inverse functions of elliptic integrals relied importantly on the 

extension of these inverse functions to allow for imaginary and complex arguments. 

As discussed below, this aspect is extremely interesting in connection with the creation 

of a (rigorous) theory of complex integration. 

In analogy with the substitution of −x for x used above, ABEL observed: 

“By inserting into (1) xi instead of x (where i for short represents the imaginary 

quantity √ −1) and designating the value of α by βi, it gives 

xi = φ (βi) and β = 

� x 

0 

∂x 

√ [(1 + c 2 x 2 ) (1 − e 2 x 2 )] .” 9 

7 “Je me propose, dans ce mémoire, de considérer la fonction inverse, c’est-à-dire la fonction φα, 

déterminée par les équations 

(N. H. Abel, 1827b, 102). 

8 “En faisant donc 

� 

α = 

sin θ = φ (α) = x.” 

ω 

2 = 

� 1 

c 

0 

∂θ 

√ � 1 − c 2 sin 2 θ � et 

∂x 

√ [(1 − c 2 x 2 ) (1 + e 2 x 2 )] , 

il est évident, que φα e[s]t positif et va en augmentant depuis α = 0 jusqu’à α = ω 2 [. . . ]” (ibid., 104).


The formal substitution of an imaginary value thus seemed to preserve the form 

of the function with the sole exception that the roles of the quantities c 2 and e 2 were 

interchanged. To ABEL this was fully sufficient, as he simple wrote: 

“thus, one sees by supposing c instead of e and e instead of c, 

φ (αi) 

i 

changes into φ (α) .” 10 

Thus, when I let φ (c,e) (α) denote the function in (16.2), ABEL’S formal imaginary 

substitution gave 

φ (c,e) (αi) = iφ (e,c) (α) 

and he had found the function φ (c,e) for a section of the imaginary axis � − ¯ω 2 , ¯ω � 

2 i, 11 in 

which 

¯ω 

2 = 

� 1 

e 

0 

16.2.2 Addition theorems 

dx 

� (1 + c 2 x 2 ) (1 − e 2 x 2 ) . 

Of central importance to ABEL’S approach to the inversion was the use which he made 

of addition formulae for elliptic functions. 

Auxiliary functions f and F. ABEL introduced two auxiliary functions which he 

named f and F, derived from φ (α), which played central parts in his deductions and 

were treated analogous to φ, 

f (α) = 

� 

1 − c2φ2 � 

(α) and F (α) = 1 + e2φ2 (α). 

Obviously, the product of these functions equals φ ′ (α) and the functions f and F were, 

themselves, doubly periodic functions corresponding to JACOBI’S cn and dn, respec- 

tively, which will be introduced and discussed later. 

9 “En mettant dans (1.) xi au lieu de x (ou i, pour abréger, représente la quantité imaginaire √ − 1) et 

désignant la valeur de α par βi, il viendra 

xi = φ (βi) et β = 

� x 

0 

∂x 

√ [(1 + c 2 x 2 ) (1 − e 2 x 2 )] .” 


10 “[. . . ] donc on voit, qu’en supposant c au lieu de e et e au lieu de c, 

φ (αi) 

i 

se changera en φα.” 

(ibid., 104). 

11 I write [a, b] i as a short-hand for the segment of the imaginary axis which can also be written as 

{xi : x ∈ [a, b]}.


ABEL’S derivation of the addition formulae. ABEL’S way of obtaining addition for- 

mulae for elliptic functions resembles L. EULER’S (1707–1783) argument (see section 

15.2.1) because both proceeded from a suggested formula. In ABEL’S case, he sought 

to establish the identity 

φ (α + β) = 

φ (α) f (β) F (β) + φ (β) f (α) F (α) 

1 + e 2 c 2 φ 2 (α) φ 2 (β) 

and similar formulae for the auxiliary functions f and F, 

f (α + β) = f (α) f (β) − c2 φ (α) φ (β) F (α) F (β) 

1 + e 2 c 2 φ 2 (α) φ 2 (β) 

(16.3) 

and (16.4) 

F (α + β) = F (α) F (β) + e2φ (α) φ (β) f (α) f (β) 

1 + e2c2φ2 (α) φ2 . (16.5) 

(β) 

ABEL denoted the right hand side of (16.3) by r = r (α, β) and proceeded to dif- 

ferentiate r with respect to α. The expression which he obtained after inserting the 

values of f and F proved to be symmetric in α and β. Therefore, and because r itself 

was symmetric in α and β, ABEL concluded that 

∂r 

∂α 

∂r 

= . (16.6) 

∂β 

This differential equation, ABEL claimed, 12 showed that r was a function of α + β, 

r = ψ (α + β) . (16.7) 

Upon inserting β = 0, ABEL immediately recognized ψ = φ, and the addition formula 

for φ had been obtained. 

To understand how ABEL concluded that the solution to the differential equation 

(16.6) must be of the form (16.7), we can get a hint from one of his earlier papers, 

published in the Journal. 13 In that paper, ABEL had established that the solution of the 

equation 14 

has the solution 

� � 

∂r 

σ (y) = 

∂x 

�� 

r = ψ 

� 

σ (x) dx + 

� � 

∂r 

σ (x) 

∂y 

� 

σ (y) dy 

where ψ was arbitrary. 15 To apply to the situation of the addition formulae, take σ = 1 

to obtain (16.7). 

12 The validity of this claim will be discussed below. 

13 (N. H. Abel, 1826e). 

14 ABEL’S use of d has been replaced by ∂; and ABEL wrote φ where I have substituted σ. 

15 (ibid., 12–13).


Periods of φ. With the addition formulae in place, ABEL inserted β = ± ω 2 and β = ¯ω 2 

to find by direct computation 

� 

φ α ± ω 

� 

= ±φ 

2 

Similarly, for the auxiliary functions 

� 

f α ± ω 

� 

2 

� 

F α ± ω 

� 

2 

� 

ω 

� � 

f (α) 

and φ α ± 

2 F (α) ¯ω 

2 i 

� � 

¯ω 

= ±φ 

2 i 

� 

F (α) 

f (α) . 

� 

φ � φ (α) 

� 

ω and f 

F (α) 2 � � 

and F α ± 

F (α) ¯ω 

2 i 

� 

= ∓ F � ω 2 

= F � ω 2 

� 

α ± ¯ω 

2 i 

� 

= f � ¯ω 2 i � 

f (α) ; 

= ∓ f � ¯ω 

2 i � 

φ � φ (α) 

� 

¯ω 

2 i f (α) . 

When he combined these and inserted e.g. α = α + ω 2 and β = ω 2 , ABEL found 

� 

φ (α + ω) = φ α + ω 

2 

= φ 

� 

ω 

� − 

2 

F( ω 2 ) 

φ( ω 2 ) 

ω 

� 

+ = φ 

2 

F( ω 2 ) 

F(α) 

φ(α) 

F(α) 

� 

ω 

� 

� � 

ω f α + 2 

2 F � α + ω � 

2 

= −φ (α) . 

In other words, φ (α + 2ω) = φ (α), and ABEL had discovered that 2ω was a period of 

φ. Similarly, 2 ¯ωi was also found to be a period of φ. 

The value of φ for any complex value α + βi of its argument could thus be found, 

ABEL emphasized, from the values φ (α) , f (α) , F (α) and φ (iβ) , f (iβ) , F (iβ). Fur- 

thermore, if 

α + βi = � mω ± α ′� + � n ¯ω ± β ′� i 

such that α ′ ∈ � 0, ω � � � 

2 and β ′ ¯ω ∈ 0, 2 , the values of these six functions could be obtained 

from the values of φ (α ′ ) , f (α ′ ) , F (α ′ ) and φ (β ′ i) , f (β ′ i) , F (β ′ i) by formulae 

such as 

φ (α) = φ � mω ± α ′� = ± (−1) m φ � α ′� . 

Consequently, the value of φ (and of f and F) at any complex argument was determined 

by the values of φ (α) (and f and F) in which α ∈ � 0, ω � � � 

¯ω 

2 or α ∈ 0, 2 i. 

ABEL’S extension of the elliptic function φ to the entire complex plane may thus be 

summarized in the following steps (see figure 16.2): 

1. The elliptic function φ (α) was obtained by inversion of the elliptic integral on a 

� 

. Because the function was odd, it was simultane- 

segment of the real axis � 0, ω 2 

ously found for α ∈ � − ω 2 , 0� . 

2. By a formal, imaginary substitution the function φ (iβ) was found for β ∈ � 0, ¯ω � 

2 i 

and consequently for β ∈ � − ¯ω 2 , 0� i. The value of φ (c,e) (iβ) was obtained from 

the inversion of a related elliptic function φ (e,c) (β) on a segment of the real axis.


¡ ! 

2 

¹! 

2 i 

¡ ¹! 

2 i 

Figure 16.2: ABEL’S extension to the complex rectangle 

3. The important addition formulae was demonstrated by differentiation. 

4. The two periods of φ, 2ω and 2 ¯ω, were direct results of the addition formulae 

which also provided a way of reducing φ (α + iβ) to the values φ (α ′ ) and φ (β ′ i) 

in which α ′ ∈ � 0, ω � � � 

2 and β ′ ¯ω ∈ 0, 2 i. 

Zeros and poles of φ. Solution of φ (x) = φ (a). After having established the ad- 

dition formulae, ABEL proceeded to investigate the singular points of φ, i.e. its zeros 

and poles. He found that every zero of φ was of the form 

mω + n ¯ωi for m, n ∈ Z 

and that every pole of φ was of the form 

� 

m + 1 

� � 

ω + n + 

2 

1 

� 

¯ωi for m, n ∈ Z. 

2 

ABEL applied a formula — which he had derived directly from the addition formu- 

lae — to the equation 

φ (x) − φ (y) = 0 

and concluded that the complete solution to this equation was of the form 

x = (−1) m+n y + mω + n ¯ωi for m, n ∈ Z. (16.8) 

This determination of all the roots of the (transcendental) equation φ (x) − φ (a) = 0 

would soon become very important for ABEL’S main objective, the solution of the 

division problem (see below). 

C 

! 

2


16.2.3 The question of complex integration 

Before going into the division problem, we pause to discuss and comment on ABEL’S 

inversion and extension of elliptic functions to include complex variables. 

In a letter to B. M. HOLMBOE (1795–1850), ABEL praised A.-L. CAUCHY (1789– 

1857) highly and described how he had struggled to understand the nine issues of the 

Exercises de mathématiques which he had bought and studied: 

“Cauchy is mad and nothing is to be obtained from him although at present, 

he is the mathematician who knows how mathematics should be conducted. His 

things are excellent but he writes very obscurely. At first, I almost did not understand 

a thing of his works but now it is better. He is having a series of papers 

printed under the title Exercises des mathématiques. I buy them and read them carefully. 

Nine issues have appeared since the beginning of this year.” 16 

A large part of CAUCHY’S Exercises de mathématiques from the year 1826 concerns 

the introduction of CAUCHY’S revolutionary new idea of residues. The year before, in 

1825, CAUCHY had laid the foundation for his theory of complex integration with a 

brochure entitled Mémoire sur les intégrales définies, prises entre des limites imaginaries. 17 

Although ABEL never referred directly to the memoir, P. L. M. SYLOW (1832–1918) 

has found evidence in certain calculations in one of ABEL’S notebooks from the time 

in Paris that ABEL knew of it. 18 

ABEL’S concept of functions. In the Recherches, ABEL spoke of finding the values of 

the function φ (α) for given α. Thus, ABEL’S deductions do not seem to be defining 

this function but rather to be manipulations which make the value of the function 

available to the observer. This concept of functions resembles EULER’S (see section 

10.1) in the sense that the function is tacitly supposed to exist for all values of the 

variable although it is only strictly meaningful for a subset of the arguments, in this 

case a segment of the real axis. 19 

ABEL on integration between imaginary limits. As noted, SYLOW found some sug- 

gestion in ABEL’S Notebook A — which dates from 1826 — that ABEL actually thought 

of his elliptic functions as defined by complex integration. For instance, one finds in 

that notebook the formula 

f (x + yi) = 

� (x+yi) 

0 

dp 

� (1 − p 2 ) (1 − c 2 p 2 ) 

16 “Cauchy er fou, og der er ingen Udkomme med ham, omendskjøndt han er den Mathematiker som 

for nærværende Tid veed hvorledes Mathematiken skal behandles. Hans Sager ere fortræffelige 

men han skriver meget utydelig. I Førstningen forstod jeg næsten ikke et Gran af hans Arbeider nu 

gaar det bedre. Han lader nu trykke en Række Afhandlinger under titel Exercises des Mathematiques. 

Jeg kjøber og læser dem flittig. 9 Hefter ere udkomne fra dette Aars Begyndelse.” (Abel→Holmboe, 

Paris, 1826/10/24. N. H. Abel, 1902a, 43). 

17 (A.-L. Cauchy, 1825). 


19 See also (Nørgaard, 1990).


which is nowhere found in the published version in the Recherches. 20 Later in the same 

notebook, ABEL wrote 

and deduced the differential equations 

� � 

dp 

= 

dx 

� 

φ x + y √ � 

−1 = p + q √ −1 

� � 

dq 

dy 

and 

� � 

dp 

= − 

dy 

� � 

dq 

dx 

which are the important Cauchy-Riemann equations. 21 Thus, as SYLOW concludes, 22 

there is good reason to believe that ABEL had studied CAUCHY’S works on integra- 

tion between imaginary limits. However, there is still no direct indication that ABEL 

allowed any of these studies or considerations to have an impact on the way he pre- 

sented his inversion of elliptic integrals. 

Complex integration or formal substitution in the Recherches? As the evidence 

seems to be inconclusive, the interpretation of ABEL’S inversion must be left to the his- 

torian and depends on the temper of the interpretor. I believe that ABEL’S inversion 

was formal in the sense that he employed a formal, imaginary substitution to obtain 

the extension to imaginary arguments. Whether or not, he found any reassurance of 

his method in CAUCHY’S theory of integration remains an undecidable question. 

16.2.4 GAUSS’ unpublished results on lemniscate functions 

The idea of inverting elliptic integrals into elliptic functions did not belong uniquely 

to ABEL. Actually, contrary to beliefs expressed throughout the secondary literature, 

the idea had occurred to LEGENDRE. 23 What LEGENDRE did not fully realize, though, 

was that the inverted functions should most naturally be considered as functions of a 

complex variable. This idea is most frequently attributed to GAUSS in whose drawer it 

remained, however. We may learn a bit more of the idea of inverting elliptic integrals 

by considering extracts from GAUSS’ unpublished works and by comparing with the 

approach taken by JACOBI after ABEL’S inversion had been published. 

In what appears to be GAUSS’ first manuscript on the lemniscate function, we get 

an impression of his approach. GAUSS wrote: 

“We designate the value of the integral from x = 0 to x = 1 by 1 

2 ¯ω. We denote 

the variable x of the respective integral by the sign sin lemn and its complementary 

integral to 1 

2 ¯ω by cos lemn. Thus, 

� 

� � � 

dx 

1 dx 

sin lemn √ = x, cos lemn ¯ω − √ = x.” 

1 − x4 2 1 − x4 24 

20 (Abel, MS:351:A, 64). 

21 (ibid., 100). 


23 See (Krazer, 1909, 55) and (J. J. Gray, 1984, 103).


Thus, the function sin lemnα produces the upper limit of the lemniscate integral 

whose value is α. This is the inverse function of the lemniscate integral and the direct 

counterpart to (special case of) the elliptic function φ which ABEL later, independently, 

introduced in his Recherches. 

Was GAUSS’ sin lemn a complex function? Clearly, GAUSS had taken the step of 

considering the inverse function of the lemniscate integral. He invested extensive 

effort in developing representations by infinite series, infinite products, and ratios of 

infinite series. In the literature, GAUSS is universally credited with the discovery of the 

doubly periodic nature of the lemniscate function. 25 This claim is generally supported 

by GAUSS’ consideration of the degree of the division problem for the lemniscate. 

GAUSS found, and noted in his diary, that the division problem for the lemniscate into 

n parts led to an equation of degree n 2 . 26 This may well have been GAUSS’ motivation 

for considering complex values of the argument. In a manuscript, in which GAUSS 

wrote the lemniscate function as the ratio of two infinite products 

he stated formulae such as 

� 

4√ 

2P φ + 1 

2 ω 

� 

which amounted to 

sin lemnφ = 

P (φ) 

Q (φ) , 

� 

= pφ and p φ + 1 

2 ω 

� 

= − 4√ 2P (φ) 

P (φ + ω) = 1 

� 

4√ p φ + 

2 1 

2 ω 

� 

= −P (φ) . 

This demonstrated the periodic nature of P and a similar result was obtained for Q. 

More interestingly, GAUSS also wrote 

which would indicate that 

P (iψω) = ie πψ2 

P (ψω) and Q (iψω) = e πψ2 

Q (ψω) 

sin lemn (iψ) = i sin lemn (ψ) 

and therefore produce the second period of sin lemnφ. GAUSS’ manuscripts also con- 

tain numerous formulae expressing the addition and multiplication of the lemniscate 

function. 27 

24 “Valorem huius integralis ab x = 0 usque ad x = 1 semper per 1 2 ¯ω designamus. Variabilem x 

respectu integralis per signum sin lemn denotamus, respectu vero complementi integralis ad 1 2 ¯ω 

per cos lemn. Ita ut 

� 

sin lemn 

dx 

√ = x, 

1 − x4 � � 

1 

cos lemn ¯ω − 

2 

� 

dx 

√ = x.” 

1 − x4 (C. F. Gauss, 1863–1933, III, 404). 

25 (Schlesinger, 1922–1933) and see (J. J. Gray, 1984, 102–103). 

26 (ibid., 102). 

27 [Ref]


16.2.5 JACOBI’s inversion in the Fundamenta nova 

As will be described in section 18.1, a third inversion of elliptic integrals was per- 

formed by CARL GUSTAV JACOB JACOBI who in 1829 published the first book entirely 

devoted to the study of the new elliptic functions. 28 As will also be illustrated in 

section 18.1, JACOBI’S main objective with his research on elliptic integrals and func- 

tions was the development of transformation theory. After having devised the first set 

of theorems concerning the transformation of elliptic integrals, JACOBI presented his 

version of the inversion: 

“Letting � φ 

0 

dφ 

√ 1−k 2 sin 2 φ = u, geometers have accustomed themselves to call 

the angle φ the amplitude of the function u. In the following, this angle is denoted 

by amplu or shorter by 

Thus, if 

then 

φ = amu. 

u = 

� x 

0 

dx 

� (1 − x 2 ) (1 − k 2 x 2 ) 

x = sin amu.” 29 

JACOBI then introduced the complete integrals already stressed by LEGENDRE, 

K = 

K ′ = 

28 (C. G. J. Jacobi, 1829). 

29 “Posito � φ 

0 

� 1 

0 

� π 2 

0 

� π 

dx 

2 

� = 

(1 − x2 ) (1 − k2x2 ) 0 

� 

dφ 

1 − k ′ k ′ sin2 φ 

where k ′ k ′ + kk = 1. 

� 

dφ 

1 − k2 sin2 φ 

and 

dφ 

√ 1−k 2 sin 2 φ = u, angulum φ amplitudinem functionis u vocare geometrae consueverunt. 

Hunc igitur angulum in sequentibus denotabimus per amplu seu brevius per: 

Ita, ubi 

erit: 

(ibid., 81). 

φ = amu. 

u = 

� x 

0 

x = sin amu.” 

dx 

� (1 − x 2 ) (1 − k 2 x 2 ) ,


Next, JACOBI stated the addition formulae which were presented as well known re- 

sults concerning elliptic integrals. Only then were complex values of the variable in- 

troduced through a substitution sin φ = i tan ψ in the integrand: 

� 

dφ 

1 − k2 sin2 = � 

i dψ 

φ cos2 ψ + k2 sin2 = � 

i dψ 

ψ 1 − k ′ k ′ sin2 . 

ψ 

Finally, JACOBI obtained the doubly periodic nature of the function sin amu from the 

addition formulae. 

As described, JACOBI’S inversion is quite similar to ABEL’S approach. Based on 

two different elliptic integrals (corresponding to the complementary moduli k and k ′ ), 

JACOBI could obtain the value of sin amiu on the imaginary axis. Then, by the addition 

formulae which were apparently assumed to be valid for these complex values of u 

and v, the two independent periods were deduced. JACOBI was aware that the doubly 

periodic nature was a new and important feature of these new functions: 

“elliptic functions have two periods, one real and one imaginary whenever 

the modulus k is real. Both [periods] will be imaginary when the modulus itself is 

imaginary. We call this the principle of double periodicities.” 30 

JACOBI’S book became the corner stone of the research on elliptic functions in the 

following generation, and his notation and ways of introducing elliptic functions be- 

came standard for a while until he changed it by introducing elliptic functions by 

certain infinite series (see chapter 20). In that respect, JACOBI’S works surpassed LEG- 

ENDRE’S effort to update his monographs with the newest developments by ABEL and 

JACOBI which resulted in a supplement to his Traité des fonctions elliptiques published 

in 1828. 31 

16.2.6 Comparison: An earlier idea on inversion 

As indicated, ABEL’S inversion in the Recherches was the first inversion of elliptic inte- 

grals into elliptic functions of a complex variable to appear in print. However, prior to 

his departure on the European tour, ABEL had written a manuscript which also dealt 

with the inversion of functions and which is interesting in the discussion of whether 

ABEL used complex integration or not. 

The result. In a manuscript which bears the lengthy but very accurate title Propriétés 

remarquables de la fonction y = φx déterminée par l’équation f y.dx − f x � (a − y) (a1 − y) (a2 − y) . . . (am 

0, f étant une fonction quelconque de y qui ne devient pas nulle ou infinie lorsque y = 

30 “functiones ellipticas duplici gaudere periodo, altera reali, altera imaginaria, siquidem modulus k 

est realis. Utraque fit imaginaria, ubi modulus et ipse est imaginarius. Quod principium duplicis 

periodi nuncupabimus.” (C. G. J. Jacobi, 1829, 87). 

31 (A. M. Legendre, 1825–1828, III).


a, a1, a2, . . . , am, 32 ABEL considered a problem also bearing on the inversion of ellip- 

tic integrals. There, he studied the function y = φ (x) given by a differential equation 

(which occurs in the title of the paper) 

dy 

dx = 

� 

ψ (y) 

, in which ψ (y) = 

f (y) 

m 

∏ (ak − y) 

k=1 

and f (a1) , . . . , f (am) were finite and non-zero. Of such functions φ (x), ABEL proved 

that they have m(m−1) 

2 

(possibly non-distinct) periods 2 (α k − αm) determined by 

� ak 

αk = 

0 

f (y) dy 

� ψ (y) . 

One of ABEL’S applications of the result. To recognize the connection with the in- 

version of elliptic integrals, consider first the case (given by ABEL) of trigonometric 

functions, 

i.e. � � 

f (y) dy 

� = 

ψ (y) 

This gave 

f (y) = 1 and ψ (y) = (1 − y) (1 + y) 

α1 = 

� 1 

0 

dy 

� 1 − y 2 

dy 

� 1 − y 2 

= arcsin y. 

= π 

2 and α2 = −π 

2 , 

φ (x + 2nπ) = φ (x) . 

A speculative application of the same result. There is no explicit restrictions on the 

roots a1, . . . , am mentioned by ABEL. If we, extending ABEL’S example, allow the roots 

to be imaginary and consider the lemniscate integral 

we find periods 

f (y) = 1 and ψ (y) = 1 − y 4 = (±1 − y) (±i − y) , 

� 1 dy 

α1 = −α2 = � , and 

0 1 − y4 � i dy 

α3 = −α4 = � 

0 1 − y4 = iα1. 

In the last integral, the imaginary integration could be performed via the formal sub- 

stitution iy = z which ABEL employed in the Recherches. 33 Thus, the two periods of 

the elliptic functions were immediate generalizations of the results obtained in the 

manuscript. 

32 (N. H. Abel, [1825] 1839a). 

33 (N. H. Abel, 1827b, 104). See above.


ABEL’S deduction. ABEL’S way of obtaining the described results was to expand the 

function φ in a Taylor series 

φ (x + v) =φ (x) 

� �� 

=y 

+ 

∞ 

∑ v 

k=1 

2k Q2k + 

� 

ψ (y) 

With y = a k and α k equal to the corresponding value of x, 

ABEL found 

� ak 

αk = 

0 

φ (α k + v) = a k + 

f (y) dy 

� ψ (y) , 

∞ 

∑ 

k=0 

∞ 

∑ v 

k=1 

2k Q2k and thus in this case, φ (α k + v) was an even function of v, 

By inserting v ′ = α k − v, ABEL obtained 

Therefore, 

φ (α k + v) = φ (α k − v) . 

φ � 2α k − v ′� = φ � v ′� . 

φ (2α k − 2αm + v) = φ (v) , 

and the function was therefore periodic. In general 

� 

� 

φ 

v + 2 

m 

∑ 

k,k ′ =1 

n k,k ′ (α k − α k ′) 

In particular, by taking for v a zero of φ, the values 

were also zeros of φ. 

v + 2 

16.2.7 Conclusion 

m 

∑ 

k=1 

n kα k for n1, . . . , nm ∈ Z with 

= φ (v) . 

v 2k+1 Q 2k+1. 

m 

∑ nk = 0 

k=1 

Because ABEL’S general inversion — which admittedly did not explicitly concern el- 

liptic integrals — was written before he embarked on the European tour in 1825, it 

cannot rely on any knowledge of the new theory of complex integration which was 

presented that same year. Also admittedly, the manuscript does not contain any com- 

plex integrals or complex periods but I find the suggested application of the result 

plausible. I do so, because I read ABEL’S inversion of elliptic integrals in the Recherches 

rather literally and see in it a formal substitution without any justification in complex 

integration. This theme will surface again when the need and means of representa- 

tions for elliptic functions are discussed in chapter 17.

16.3. The division problem 313 

16.3 The division problem 

As already indicated, one of the main objectives of ABEL’S Recherches was the so-called 

division problems which encompass deducing and solving the equations which deter- 

mine the division of the function φ (mα) into m parts, i.e. the equations which deter- 

mine φ (α) from φ (mα). 

ABEL’S starting point for these investigations was the addition formula (16.3). 

Based on these formula, he found expressions such as 

φ ((n + 1) β) = −φ ((n − 1) β) + 

2φ (nβ) f (β) F (β) 

1 + c 2 e 2 φ 2 (nβ) φ 2 (β) . 

Consequently, ABEL observed that the functions φ (nβ), f (nβ), and F (nβ) depended 

rationally on φ (β), f (β), and F (β) and he wrote, e.g. 

φ (nβ) = Pn 

Qn 

in which Pn and Qn were polynomial functions of φ (β), f (β), and F (β). ABEL then let 

x = φ (α), y = f (β), z = F (β) and manipulated the equation to obtain the relations 

in which 

Qn+1 = Qn−1Rn and 

Pn+1 = −Pn−1Rn + 2yzPnQnQn−1 

Rn = Q 2 n + e 2 c 2 x 2 P 2 n. 

Obviously, Rn was a polynomial function in x 2 , and from the basic formulae 

i.e. 

φ (β) = P1 

Q1 

and φ (2β) = 

2φ (β) f (β) F (β) 

1 + e 2 c 2 φ 4 (β) 

Q0 = 1, Q1 = 1, P0 = 0, P1 = x, 

P2 

= , 

Q2 

ABEL found that Qn was always an entire function of x2 . Furthermore, P2n 

xyz and P2n+1 x 

were also entire functions of x2 . ABEL merely provided the first few particular cases 

but the argument is easily completed by induction: 

P2n+2 −P2n 

= 

xyz xyz Q2 � 

P2n+1 c2e2x 2P2nP2n+1 2n+1 − 

x yz 

= −P2n 

xyz Q2 P2n+1 

2n+1 − 

x 

+ 2Q2n+1Q2n 

� 

c 2 e 2 x 2 2 P2n P2n+1 

× x + 2Q2n+1Q2n 

xyz x 

and all parts are seen to be entire functions of x 2 by the induction hypothesis. Simi- 

larly, 

P2n+1 

x 

−P2n−1 

= R2n + 

x 

2yzP2nQ2nQ2n−1 

x 

= −P2n−1 

R2n + 2y 

x 

2 2 P2n 

z 

xyz Q2nQ2n−1 

and the same conclusion holds because y 2 = 1 − c 2 x 2 and z 2 = 1 + e 2 x 2 . 

� 

�


Bisection by algebraic means. The bisection of the function φ was the easiest example 

of the program which ABEL was developing. In order to express x = φ � � 

α 

2 by 

φ (α), ABEL employed the addition formula for f and F (16.4 and 16.5) to get 

� 

α α 

� 

f (α) = f + = 

2 2 

y2 − c2x2z2 1 + e2c2 � 

1 − c2x2 = 

x4 � − c2x2 � 1 + e2x2� 1 + e2c2x 4 

and 

� 

α α 

� 

F (α) = F + = 

2 2 

z2 + e2x2y2 1 + e2c2 � 

1 + e2x2 = 

x4 � + e2x2 � 1 − c2x2� 1 + e2c2x 4 

. 

When ABEL formed the ratio 

he found that 

F (α) − 1 

1 + f (α) = 

φ 

2e 2 x 2 (1−c 2 x 2 ) 

1+e 2 c 2 x 4 

2(1−c 2 x 2 ) 

1+e 2 c 2 x 4 

� 

α 

� 

= 

2 

1 

� 

F (α) − 1 

e 1 + f (α) . 

= e 2 x 2 , 

Thus, ABEL had proved that if the value of φ (α) was known, φ � � 

α 

2 could be expressed 

algebraically and actually using only the extraction of square roots. 

Division into an odd number of parts. In order to complete the program, ABEL had 

to find a way of algebraically obtaining φ � � 

α 

2n+1 from φ (α) and this proved both more 

difficult and much more fruitful. In chapter 7, we have already met the Mémoire sur 

une classe particulière where ABEL later introduced Abelian equations as generalizations 

of these investigations. 34 

ABEL wanted to solve the equation which he wrote as 

φ (α) = P2n+1 

Q2n+1 

and similar equations for f (α) and F (α). Although ABEL’S argument was greatly 

simplified by his slightly more general approach of the memoir on Abelian equations, 

the original version of the Recherches is worth a brief description to facilitate a com- 

parison. I have suppressed most of the computational technicalities in the following 

typical step of the proof. 

ABEL let θ denote an imaginary (2n + 1)’th root of unity and introduced three aux- 

iliary functions, 

ψ (β) = 


n 

∑ θ 

µ=−n 

µ � 

φ1 β + 

φ1 (β) = 

2µ ¯ωi 

2n + 1 

n 

∑ 

m=−n 

� 

φ β + 2mω 

� 

, 

2n + 1 

� 

, and ψ1 (β) = 

n 

∑ θ 

µ=−n 

µ φ1 

� 

β − 

� 

2µ ¯ωi 

. 

2n + 1


Based on a direct application of the addition formulae, ABEL obtained an expression 

� � 

2µ ¯ωi 

φ1 β ± = Rµ ± R 

2n + 1 

′ � 

µ (1 − c2x2 ) (1 + e2x2 ) 

in which Rµ and R ′ µ were rational functions of x = φ (β). Consequently, ABEL found 

that both the functions 

ψ (β) ψ1 (β) = λ (β) and ψ (β) 2n+1 + ψ1 (β) 2n+1 = λ1 (β) (16.9) 

were rational in x. By direct calculation, he also found that both functions (16.9) were 

invariant if another root β + kω+k′ ¯ωi 

2n+1 

of the equation 

φ ((2n + 1) β) = P2n+1 

Q2n+1 

(16.10) 

was substituted for β. Thus, ABEL knew that λ (β) and λ1 (β) were rational in the 

coefficients of (16.10), in particular in the quantity φ ((2n + 1) β). When he solved the 

system of equations (16.9), ABEL found 

ψ (β) = 2n+1 

� 

� 

� 

�λ1 (β) 

2 + 

� 

λ1 (β) 2 

4 

− λ (β). (16.11) 

From these, ABEL obtained φ1 (β) and then φ (β) by similar arguments. 

However, as ABEL observed, the formula for φ (β) which he had obtained also 

contained the quantities 

� � 

ω 

φ 

2n + 1 

� � 

¯ωi 

and φ . 

2n + 1 

Thus, in order to completely solve the problem, these two quantities should also be 

determined and ABEL demonstrated how the equation P2n+1 = 0 which determined 

these could be reduced to equations of lower degrees, one of degree 2n + 2 and 2n + 2 

equations of degree n. Furthermore, ABEL also proved that the equations of degree 

n were always solvable by radicals. In the process, ABEL employed tools similar to 

those described above as well as some knowledge of primitive roots of an integer. Im- 

portantly, ABEL knew qualitatively how the roots were interrelated (by 16.8) and used 

this knowledge to investigate the system of roots and prove its reduction to equations 

of lower degree, some of which were proved to be solvable by radicals. 

16.3.1 Division of the lemniscate 

The culmination of ABEL’S research into the division problem was his application of 

the theory to the case of the lemniscate. The symmetry of ABEL’S representation of 

the elliptic integrals became evident when he chose e = c = 1 to obtain the lemniscate 

integral 

φ (α) = x, α = 

� x 

0 

dx 

√ 1 − x 4 .


In light of the previous result, ABEL’S investigations mainly concerned the division 

of the complete integral into an odd number of equal segments. He did so by first 

carrying out the division of the complete integral into 4v + 1 parts. ABEL’S argument 

was explicitly designed to make the case accessible with the Gaussian approach to the 

solution of cyclotomic equations. A brief outline of ABEL’S reasoning will provide a 

few interesting comparisons with GAUSS’ approach and the more general solution of 

the problem found in ABEL’S paper on Abelian equations. 

ABEL first assumed that 4v + 1 was the sum of two squares, 4v + 1 = α 2 + β 2 and 

found that α + β had to be an odd integer. In this case, he found an equation which he 

wrote as 

φ ((α + βi) δ) = x T 

S 

(16.12) 

with α = mδ, β = µδ, x = φ (δ), and T and S two entire functions of x4 . ABEL’S 

real objective was the considerations pertaining to δ = ω 

� 

α+βi for which he obviously 

found that x = φ (δ) = φ was a root of the equation T = 0. It thus became his 

� ω 

α+βi 

objective to solve this equation. 

Expressing the roots of T = 0. First, by his very powerful determination of the roots 

of φ (ξ) = 0, ABEL found that all the roots of T = 0 were related by 

(α + βi) δ = mω + µ ¯ωi = (m + µi) ω 

since ω = ¯ω for the lemniscate integral. Thus, any root was contained in the formula 

� 

m + µi 

x = φ 

α + βi ω 

� 

if m and µ were allowed to assume all integral values. However, in order to count 

each root only once, ABEL found that the set of roots of T = 0 could be listed as 

� � 

ρω 

φ 

α + βi 

for − α2 + β 2 − 1 

2 

≤ ρ ≤ α2 + β 2 − 1 

2 

(16.13) 

and he argued by an application of the Euclidean algorithm for integers: ABEL let λ, λ ′ 

be determined by the equation 

and t denote an integer in order to obtain 

αλ ′ − βλ = 1, 

µ + β (µλ + tα) −α 

� �� 

=k 

� µλ ′ + tβ � 

� �� 

=k ′ 

= 0. 

Then, with ρ = m + αk − βk ′ , ABEL obtained 

m + µi 

α + βi 

= ρ 

α + βi − k − k′ i


and therefore, 

� 

m + µi 

φ 

α + βi ω 

� 

Because of the relation 

the bounds of (16.13) were obtained. 

� 

ρ 

= φ 

α + βi ω − kω − k′ � 

iω = (−1) k+k′ 

� � 

ρ 

φ . 

α + βi 

ρ = m + µ � λα + λ ′ β � � 

+ t α 2 + β 2� 

, 

The expressed roots of T = 0 are all distinct. To realize that all the roots of the 

equation T = 0 were contained in the list (16.13), ABEL first observed that none of 

the roots corresponding to different values of ρ could be identical. He did so using 

the same techniques as above. Then, ABEL found that T had no multiple roots by 

observing that a multiple root of T would be a common root of T = 0 and T ′ = 0. 

However, any root of T ′ = 0 would also be a root of S = 0 which was forbidden by 

assuming that the rational function T S 

(16.12) was expressed in its reduced form. As a 

consequence, ABEL found that all the roots of the original equation reduced to roots 

of an equation R = 0 of degree 2v in which the roots were 

φ 2 

� ω 

α + βi 

� 

, φ 2 

� 2ω 

α + βi 

� 

, . . . , φ 2 

� 2vω 

α + βi 

This equation could, ABEL observed, “easily be solved by the method of GAUSS.” 35 

Actually, ABEL solved it using the same approach as in the general division problem, 

i.e. the approach which led to (16.11), above. 

Geometrical division of the lemniscate. Having solved the equation R = 0, ABEL 

thus had access to the values 

� � 

kω 

φ for k = 1, 2, . . . , 2v, 

α + βi 

and he now proceeded to obtain the value φ � � 

ω 

4v+1 by the following brief argument. 

By the addition theorem, ABEL expressed 

� � � � 

mv mv 2mαω 

φ + = φ 

α + βi α − βi 4v + 1 

� � � � 

mv 

mw 

in terms of φ α+βi and φ α−βi where the latter could be obtained from the former 

“by changing i into −i”. 36 Since 2α and 4v + 1 were relatively prime, ABEL could write 

any integral n as 

n = 2mα − (4v + 1) t, 

35 “Cela posé, on peut aisément résoudre l’équation R = 0, à l’aide de la méthode de M. Gauss.” 


36 (ibid., 166). 

� 

.


i.e. 

φ 

� � � 

2mαω nω 

= φ 

4v + 1 4v + 1 

� 

+ tω = (−1) t � � 

nω 

φ 

4v + 1 

which for n = 1 made φ � � 

ω 

4n+1 accessible (known). 

Throughout, ABEL had explicitly only considered the division into a prime number 

of parts. When ABEL then made the further assumption that 4v + 1 = 1 + 2 n , he found 

by considering the expressions obtained that all the root extractions reduced to square 

roots. In particular, ABEL had to use that the solution of the equation θ2n−1 = 1 could 

be reduced to square roots; this result is precisely the main result of GAUSS’ research 

on the division of the circle. Combining this result with the case of bisection and 

the general integral multiplication, ABEL could summarize his investigations on the 

division of the lemniscate: 

“The value of the function φ � � 

mω can be expressed by square roots whenever 

n 

n is a number of the form 2n or a prime number of the form 1 + 2n or a product of 

multiple numbers of these two forms.” 37 

Two aspects of ABEL’S result merit attention. First, ABEL’S argument hinges on 

the factors 2 m0, 2 n 1 + 1, . . . , 2 n k + 1 of n to be relatively prime because he wanted to 

decompose any number m ′ into its residues modulo these factors, 

� 

m0 m1 

φ + 

2n0 2n1 + 1 + · · · + mk 2n � � 

= φ 

k + 1 

m ′ 

2 n0 (2 n 1 + 1) . . . (2 n k + 1) 

If two of the Fermat primes were identical, the decomposition would no longer be pos- 

sible. 38 ABEL’S deductions immediately leading to the stated theorem contain tacitly 

the distinctness of the Fermat primes, but it could have been explicitly included in the 

statement. 

Second, the result states a sufficient condition of geometrical constructibility and 

says absolutely nothing of the necessity of this condition. The same can be said of 

GAUSS’ stated result on the division of the circle. However, GAUSS also stated that the 

division of the circle would lead to precisely his equations and would therefore not 

be possible with ruler and compass unless the number of parts were of the prescribed 

form. Similarly, ABEL could have stated that division of the lemniscate was only pos- 

sible if n was a product of a power of 2 and distinct Fermat primes. 39 However, the 

proof of such a statement would go beyond the types of questions which ABEL asked 

concerning these classes of equations. 

37 “La valeur de la fonction φ � � 

mω 

n peut être exprimée par des racines carrées toutes les fois que n 

est un nombre de la forme 2n ou un nombre premier de la forme 1 + 2n , ou même un produit de 

plusieurs nombres de ces deux formes.” (N. H. Abel, 1828b, 168). 

38 Consider writing e.g. 3 

39 

a+b 

25 as 5 with a, b ∈ Z. 

For a proof using more modern techniques, see (M. Rosen, 1981). 

� 

.

16.4. Perspectives on inversion 319 

16.4 Perspectives on inversion 

The present chapter has documented ABEL’S inversion of elliptic integrals into elliptic 

functions and the intimately related step of extending the resulting function to allow 

complex values of the argument. 

This extension to the complex domain which laid the foundation for ABEL’S re- 

search on elliptic functions was based on a formal substitution. Although a rigor- 

ous foundation for complex integration was being undertaken in the period, the ev- 

idence suggests that ABEL based his inversion on an imaginary substitution in the 

Eulerian tradition which presumed the existence of the function and sought to dis- 

cover/construct its values. 

The step toward considering the inverse function of general elliptic integrals (of 

the first kind) was probably motivated by the division problem for the lemniscate 

to which ABEL was led by GAUSS’ remark in the Disquisitiones arithmeticae. In the 

process of solving the division problem for the lemniscate, ABEL solved the associated 

equation using a technical and rather ad hoc approach. Later, in the Mémoire sur une 

classe particulière (see chapter 7), ABEL approached the same set of problems but from a 

more conceptual approach in which he had clearly grasped the very essential property 

of the equations. 

ABEL’S tailored deductions relied extensively on manipulations of formulae; in 

particular, ABEL’S addition theorems and his characterization of the roots of the equa- 

tion φ (α) = φ (β) served him as important tools. When it came to the investigations 

on the solubility of the division problems, number theoretic arguments inspired by 

GAUSS’ Disquisitiones arithmeticae were also used.

Chapter 17 

Steps in the process of coming to 

“know” elliptic functions 

As already described in the previous chapter, N. H. ABEL (1802–1829) introduced 

his elliptic functions by means of a formal inversion of the elliptic integrals. In or- 

der to make these new objects known, however, this definition appears to have been 

quite insufficient. ABEL, himself, was not explicit about the problem — but a com- 

parison of ABEL’S approach with A.-M. LEGENDRE’S (1752–1833) highly numerical 

approach suggests that ABEL’S purely formal definition based on the formal inver- 

sion was lacking in certain respects. For instance, in ABEL’S approach, it would be 

difficult to compute particular values of ABEL’S elliptic functions purely from the def- 

inition. Therefore, he developed means of representing his new functions by existing 

objects and the objects which he chose were — not surprisingly — infinite series and 

products. 

17.1 Infinite representations 

In the first part of the Recherches, 1 a large portion of the text is occupied with highly 

technical and formula-based manipulations which aim at describing the elliptic func- 

tion φ (and the derived functions f and F) in infinite products and series. Two charac- 

teristic and interesting examples pertaining to the expansion in doubly infinite sums 

are discussed below. 

Based on the multiplication problem described above, ABEL had found that 

φ ((2n + 1) β) = P2n+1 

Q2n+1 

(17.1) 

in which deg P2n+1 = (2n + 1) 2 and deg Q2n+1 = (2n + 1) 2 − 1. Furthermore, P 2n+1 

x 


321

322 Chapter 17. Steps in the process of coming to “know” elliptic functions 

was a polynomial in x 2 and so was Q2n+1. Thus, with 

P2n+1 (x) = ax (2n+1)2 + · · · + bx and 

Q2n+1 (x) = cx (2n+1)2 −1 + · · · + d, 

ABEL found that the sum of the roots of the equation (17.1) would equal c aφ ((2n + 1) β). 

Furthermore, he already knew the complete solution of the equation (17.1), and thus 

he obtained 

φ ((2n + 1) β) = A 

n 

∑ 

m=−n 

n 

∑ 

µ=−n 

Similarly for the products of the roots of (17.1), 

φ ((2n + 1) β) = B 

n 

∏ 

m=−n 

(−1) m+µ � 

φ β + 

n 

∏ 

µ=−n 

� 

φ β + 

� 

mω + µ ¯ωi 

. (17.2) 

2n + 1 

� 


. 

2n + 1 

These formulae invited the limit process n → ∞, and it is interesting to see how ABEL 

carried it out. 

17.1.1 Determination of the coefficient A 

In order to determine the constant A of (17.2), ABEL wanted to insert a particular value 

for β and he chose β = ω 2 + ¯ω 2 i. However, this is a singular value (a pole) of φ, and 

ABEL applied a limit argument in the following form. First, using the relation 

� 

φ α + ω ¯ω 

+ 

2 2 i 

� 

= − i 1 

ec φ (α) 

derivable from the addition formulae, ABEL found that for (m, µ) �= (0, 0), 

� 

mω + µ ¯ωi ω ¯ω 

φ 

+ + 

2n + 1 2 2 i 

� � 

mω + µ ¯ωi ω ¯ω 

+ φ − + + 

2n + 1 2 2 i 

� 

= 0. 

(17.3) 

Consequently, the sum reduced to the term corresponding to (m, µ) = (0, 0). For the 

last term, ABEL found that 

A = lim 

β→ ω 2 + ¯ω 2 i 

φ ((2n + 1) β) 

φ (β) 

= 

by (17.3) lim 

α→0 

φ (α) 

φ ((2n + 1) α) 

φ 

= lim 

α→0 

� (2n + 1) � ω 

2 + ¯ω 2 i + α�� 

φ � ω 

2 + ¯ω 2 i + α� 

= 1 

2n + 1 

where tacit applications of the differentiability of φ and of the Rule of l’Hospital are 

involved.

17.1. Infinite representations 323 

17.1.2 Infinite sums 

In order to express φ (α) by an infinite series, ABEL set β = α 

2n+1 . Thus, 

φ (α) = φ ((2n + 1) β) = 1 

2n + 1 

n 

∑ 

n 

∑ (−1) 

m=−n µ=−n 

m+µ φ 

� 

β + 

� 


. 

2n + 1 

Following a string of manipulations designed to group the terms of the right hand 

side, ABEL reduced the problem to the search for the limit of the double sum 

with 

n−1 n−1 

∑ ∑ 

m=0 µ=0 

ψ (m, µ) = 1 

2n + 1 

(−1) m+µ ψ (m, µ) (17.4) 

2φ � α 

φ 2 � α 

2n+1 

2n+1 

in which ζ (x) = f (x) F (x) and εµ = 

In order to find the limit of (17.4), ABEL remarked, 

and 

� ζ 

� − φ 2 

� 

m + 1 

2 

� � 

εµ 

2n+1 

� 

εµ 

2n+1 

� 

� 

ω + 

� 

µ + 1 

� 

2 

“one attempts to put the preceding quantity [here (17.4)] on the form 

P + v, 

in which P is independent of n and v is a quantity which has the limit zero, because 

then the quantity P is exactly the limit which is sought.” 2 

ABEL had a candidate in mind for the expression P when he defined 

θ (m, µ) = 2α 

α 2 − ε 2 µ 

ψ (m, µ) − θ (m, µ) = 

2α 

(2n + 1) 

2 Rµ. 

His candidate was the double sum ∑ ∞ m=0 ∑ ∞ µ=0 (−1) m+µ θ (m, µ) and he proceeded in 

the following way in obtaining this limit. 

For each value of m, ABEL argued, the difference was 

n−1 

∑ 

µ=0 

(−1) µ n−1 

(ψ (m, µ) − θ (m, µ)) = 2α ∑ 

µ=0 

2 “il faut essayer de mettre la quantité précédente sous la forme 

P + v, 

(−1) µ Rµ 

, 

2 

(2n + 1) 

où P est indépendant de n, et v une quantité qui a zéro pour limité, car alors la quantité P sera 

précisément la limite dont il s’agit.” (N. H. Abel, 1827b, 156). 

¯ωi.


and he claimed and proved that the right hand side was “of the form v 

2n+1 ”. In the 

process, ABEL made use of the result that 

φ (α) = α + Aα 3 + . . . 

because φ was an odd function and φ ′ (0) = f (0) F (0) = 1. 

ABEL’S next step concerned the sum of θ. He found that 

and therefore, for each m, 

∞ 

∑ (−1) 

µ=n 

µ θ (m, µ) = v 

2n + 1 

n−1 

∑ (−1) 

µ=0 

µ ψ (m, µ) = 

When he summed these, ABEL found 

n−1 n−1 

∑ ∑ (−1) 

m=0 µ=0 

m+µ n−1 

ψ (m, µ) = ∑ (−1) 

m=0 

m 

and, as ABEL wrote, 

n−1 

∑ 

m=0 

vm 

2n + 1 

∞ 

∑ (−1) 

µ=0 

µ θ (m, µ) + vm 

2n + 1 . 

� 

∞ 

∑ (−1) 

µ=0 

µ � 

n−1 

θ (m, µ) + ∑ 

m=0 

= nv 

2n + 1 

= v 

2 , 

vm 

2n + 1 

in which “v is a quantity which has zero for its limit”. 3 Consequently, ABEL had found 

a way of expressing φ (α) as a double infinite sum, e.g. 

φ (α) = 1 

ec 

∞ 

∑ 

∞ 

∑ (−1) m+µ 

� 

(2µ+1) ¯ω 

m=0 µ=0 

(α−(m+ 1 2)ω) 2 +(µ+ 1 2) 2 − 

¯ω 2 

(2µ+1) ¯ω 

(α+(m+ 1 2)ω) 2 +(µ+ 1 2) 2 ¯ω 2 

� 

. 

(17.5) 

ABEL did not stop after having obtained the expansion in infinite series (17.5). 

Instead, he used similar methods to search for expressions for φ (α) involving infinite 

products. ABEL also invested an effort in obtaining expressions involving only one 

infinite series and transcendental objects in the terms. Among the formulae which he 

obtained, the following was probably the simplest: 

φ (α) = 2 

∞ 

π 

ec ¯ω ∑ (−1) 

k=0 

k ε2k+1 − ε−(2k+1) r2k+1 − r−(2k+1) where 

� 

ε = exp α π 

� 

� 

ωπ 

� 

and r = exp . 

¯ω 

2 ¯ω 

Thus, as described, ABEL used his multiplication formulae to obtain rather simple 

infinite representations for elliptic functions. When he passed to the infinite limit, his 

arguments did not conform to the strict standards of rigor, which he had advocated 

in the theory of series. Further perspectives on ABEL’S motivations for searching for 

infinite expressions and his methods of obtaining them are described and discussed 

in the subsequent sections. 

3 (N. H. Abel, 1827b, 161).

17.2. Elliptic functions as ratios of power series 325 

17.2 Elliptic functions as ratios of power series 

In the Précis, 4 listed among the established facts of elliptic functions, ABEL observed 

that elliptic functions of the first kind (which he now denoted λ, see below) were 

expressible as the ratio of two convergent power series, 

With the notation 

λ (θ) = θ + a1θ3 + a2θ5 + . . . 

1 + b2θ4 + b3θ6 . (17.6) 

+ . . . 

φ (θ) = θ + a1θ 3 + a2θ 5 + . . . and 

f (θ) = 1 + b2θ 4 + b3θ 6 + . . . , 

(17.7) 

ABEL claimed that the functions φ and f (which are not to be confused with the func- 

tions of the same names in the Recherches) satisfied the linked functional equations 

φ � θ ′ + θ � φ � θ ′ − θ � = φ 2 (θ) f 2 � θ ′� − φ 2 � θ ′� f 2 (θ) and 

f � θ ′ + θ � f � θ ′ − θ � = f 2 (θ) f 2 � θ ′� − c 2 φ 2 (θ) φ 2 � θ ′� . 

(17.8) 

The sign of the first equation is actually wrong, see below. ABEL also mentioned the 

same result in his letter to LEGENDRE but he never published any demonstration of 

it. 5 

In order to see how ABEL came to this expression, the following reconstruction 

may be suggested based on ABEL’S sparse hints. 

In his comments published in the second volume of the Œuvres, M. S. LIE (1842– 

1899) has presented a reconstruction of ABEL’S reasoning based on the same sources, 

papers and manuscripts. LIE indicated how ABEL’S manuscript notes — given the 

power series expansion (17.7) — could be interpreted as steps toward determining the 

remaining coefficients a1, a2, . . . , b2, b3, . . . . However, I interpret the notes slightly dif- 

ferently and infer from them a suggestion of how ABEL came to claim the series ex- 

pansion (17.7), itself, by use of the expansion in two Maclaurin series. I am confident 

that the following reconstruction is close to ABEL’S original argument. 

Derivation of functional equations. In the Précis, ABEL had presented the following 

consequence of the addition formulae, 

Supposing that λ (θ) was written as 

λ � θ ′ + θ � λ � θ ′ − θ � = λ2 (θ ′ ) − λ 2 (θ) 

1 − c 2 λ 2 (θ) λ 2 (θ ′ ) . 

λ (θ) = 

φ (θ) 

f (θ) 

4 (N. H. Abel, 1829d). 

5 (Abel→Legendre, Christiania, 1828/11/25. N. H. Abel, 1902a, 82). 

(17.9)


for some functions φ and f , ABEL obtained 

φ (θ ′ + θ) φ (θ ′ − θ) 

f (θ ′ + θ) f (θ ′ − θ) = 

φ2 (θ ′ ) 

f 2 (θ ′ ) − φ2 (θ) 

f 2 (θ) 

1 − c2 φ2 (θ)φ2 (θ ′ ) 

f 2 (θ) f 2 (θ ′ ) 

= φ2 (θ ′ ) f 2 (θ) − φ 2 (θ) f 2 (θ ′ ) 

f 2 (θ) f 2 (θ ′ ) − c 2 φ 2 (θ) φ 2 (θ ′ ) 

and by comparing numerators and denominators, the functional equations (17.8) re- 

sult with the exception of the change of sign in the numerator. 6 This is evidently a 

mistake in ABEL’S paper — repeated in the Œuvres — as can be seen by setting θ = 0 

and observing that because λ (0) = 0, φ (0) must also be zero. 

Except for this small error, this part of the argument is completely straight-forward 

and fits well with ABEL’S other manipulations and the formulae which he presented 

in the Précis. In order to obtain the series expansions, we may get a clue from one of 

ABEL’S notebooks in which he proceeded to differentiate the equations (17.8). 7 

Coefficients of φ and f . First, we observe from the second functional equation by 

letting θ = θ ′ = 0 that 

f 2 (0) = f 4 (0) , i.e. f 2 (0) = 1. 

To be precise, f (0) = 0 was also a possibility but that would produce 

for all x which is not a relevant case. 

f 2 (x) = 0 

In the functional equations, letting θ = 0 gives 

φ � θ ′� φ � −θ ′� = −φ 2 � θ ′� f 2 (0) , i.e. φ � −θ ′� = −φ � θ ′� , 

and φ is therefore an odd function. Thus, the coefficients of all odd powers in a power 

series expansion must be zero. The same argument applied to the second functional 

equation produces 

f � θ ′� f � −θ ′� = f 2 � θ ′� 

and therefore, f is an even function. 

Now, a few more coefficients may be determined, for instance f (0) and f ′′ (0) 

which can be found by differentiating the relation 

f (2θ) = f 4 (θ) − c 2 φ 4 (θ) 

twice and letting θ = 0 to obtain f 3 (0) = 1 and by the previous result, f (0) = 1. 

To determine f ′′ (0) we differentiate again and insert θ = 0 to obtain 

f ′′ (0) = 0. 

6 As mentioned above. 

7 (Abel, MS:351:C, 193), see also (N. H. Abel, 1881, II, 319).

17.2. Elliptic functions as ratios of power series 327 

The final coefficient which ABEL specified was φ ′ (0). It may be found directly by 

differentiation of (17.9) 

φ ′ (0) = d 

� 

� 

λ (x) f (x) � 

dx 

� x=0 

= λ ′ (0) f (0) = 1. 

Thus, summarizing the results, ABEL could claim by the use of an expansion in 

two Maclaurin series and detailed studies of the differential quotients that 

with the further information 

λ (θ) = ∑∞n=0 anθ2n+1 ∑ ∞ n=0 bnθ2n a0 = 1, b0 = 1, and b1 = 0. 

In the letter to LEGENDRE, ABEL claimed that the coefficients were always poly- 

nomial functions of the modulus c 2 . This claim can also be seen to be an easy conse- 

quence of the defining functional equations and can be proved by induction. 

Convergence of φ and f . Furthermore, ABEL claimed — both in the Précis and in 

the letter — that the two series of (17.6) were always convergent. This is perhaps the 

most difficult of ABEL’S claims to obtain from the approach presented above. Fur- 

thermore, there are no hints of ABEL’S reasoning left neither in the notebooks, nor in 

his publications. He may have obtained some sort of indication of convergence from 

the approach described above, or he may simply have stated the convergence of the 

Maclaurin series as an unproven fact. It is, of course, also possible if not probable that 

ABEL had actually grasped an implicit concept of meromorphic functions as general- 

ized rational functions. 

Later, the expression of elliptic functions as the ratio of convergent power series 

was exactly the point which provoked K. T. W. WEIERSTRASS’ (1815–1897) interest 

in research mathematics and — according to WEIERSTRASS, himself — convinced him 

that he wanted to become a mathematician. 8 WEIERSTRASS’ approach centered on 

differential equations and contained a theorem explicitly stating the convergence of 

the power series equivalent to φ and f . 

Further coefficients of φ and f . Besides the coefficients which were determined 

in the paper, ABEL’S manuscript also contains the series including the coefficients 

a1 = − 1+c2 

6 and b2 = − c2 

12 . On the same sheet of the notebook, the differential quotients 

f ′ (x) = c2 

3 x3 , f ′′ (x) = −c2x2 , and f ′′′ (x) = −2c2x can also be found which are 

only correct provided terms with higher powers of x have been neglected, e.g. by con- 

sidering the local behaviour for small values of x. A further differentiation produces 

f (4) (0) = −2c 2 

8 (Weierstrass→Lie, Berlin, 1882/04/10. N. H. Abel, 1902b, 104).


and 

b2 = f (4) (0) 

c2 

= −2c2 = − 

4! 4! 12 . 

Thus, the calculations appear to be probes of ABEL’S result. The notebook does not 

contradict my interpretation, I believe, that ABEL obtained his representation of λ 

by means of two Maclaurin series. However, as LIE’S interpretation also illustrates, 

ABEL’S publications on the subject are too few and his notes to difficult to interpret to 

present any final opinion on the debate, in particular concerning the convergence of 

the series. 

17.3 Characterization of ABEL’s representations 

Having presented an outline and discussion of the technical details of ABEL’S deriva- 

tion of the representations, a few themes are summarized which will also be relevant 

to the discussion in subsequent chapters in part IV and in section 21.2. 

17.3.1 ABEL’s style of reasoning. 

Characteristic of ABEL’S style in the Recherches, the derivation of the infinite represen- 

tations such as (17.5) relies extensively on manipulations of formulae and is repeated 

afterwards for the functions f and F, although the arguments are highly similar. On a 

related theme, ABEL introduced symbols to denote most of his auxiliary and interme- 

diate calculations. These facts are textual evidence of the formula-manipulating style 

in which ABEL’S Recherches is mainly written. 

Brief comparison with L. EULER (1707–1783). As described, ABEL’S way of obtain- 

ing infinite expressions for the elliptic function φ closely resembles EULER’S methods. 

ABEL transformed an expression for φ (α) into a form which contained a number (re- 

lated to) n terms and then proceeded to split this expression into the sum of a part 

independent of n and a part which vanished with increasing values of n. 

This is comparable to EULER’S derivation of the power series expansion of the ex- 

ponential function in the Introductio ad analysin infinitorum. 9 There, EULER — cloaked 

in his language of infinitesimals — considered 

� 

1 + x 

n 

� n 

, (17.10) 

expanded it by the binomial formula, and let n grow to infinity to get an infinite num- 

ber of terms. Simultaneously, this turned the expression into exp x. 

Compared to EULER’S expansion, ABEL’S original formula was valid for any value 

of n. The number n was only an auxiliary quantity later to disappear when he found 

limit expressions which were independent of n. 

9 (L. Euler, 1748, 85–87).

17.3. Characterization of ABEL’s representations 329 

Convergence of infinite expressions. As LIE commented in the notes, ABEL’S “meth- 

ods in obtaining expressions for the functions φ (α), f (α), F (α) in series and infinite 

products do not seem to us to be satisfactory in all details.” 10 In particular, it is re- 

markable that the ardent rigorist of the binomial paper never even mentioned the 

word convergence in the Recherches. In the Précis, however, the convergence of the se- 

ries (17.6) was stated as a fact without further explanation. 

ABEL’S way of obtaining series expansions of elliptic functions essentially let the 

number of terms in a finite expression grow to infinity. ABEL’S concern for the con- 

vergence of the process was dealt with by the method of writing the expression as one 

part which was independent of n and another part which vanished with increasing n. 

Thus, it appears, two different standards of rigor in the theory of series were in- 

volved in ABEL’S research using infinite series. In foundational issues, a strict adher- 

ence to A.-L. CAUCHY’S (1789–1857) program and the associated theoretical complex 

was advocated by ABEL. However, when it came to research on new groundbreaking 

objects, ABEL used the methods which he had learned from EULER and was content 

with observing that his results were sound. 

17.3.2 The need for multiple representations 

A final aspect which is revealing of the role played by representations of elliptic func- 

tions in ABEL’S works is the necessity of obtaining multiple representations. Under- 

standably, functions introduced in such an indirect way as the inversion of a non- 

elementary integral needed some other means of numerical determination and this is 

one of the roles played by representations in ABEL’S theory. 

Convergence. As already noticed repeatedly, ABEL was not very explicit about the 

convergence of his infinite representations. This may have been a reason for not rely- 

ing on any single representation but deriving multiple and various representations in 

the hope that at least some of them would prove adequate in particular instances. 

Applications. A connected motivation for multiple representations could also be the 

ambition of multiple applications (within pure mathematics). In the following chapter, 

an instance where infinite representations play a central role in the proof of a theorem 

will be described. ABEL and his contemporaries would have hoped and expected to 

find many similar instances. 

Aesthetics. The third aspect of the discussion is less technical and more of a personal 

and contextual nature which can only be appreciated in a broader time scale. In section 

21.2, where the discussion of representations is taken up again, it will also become 

10 (N. H. Abel, 1881, II, 306).


clear that the preferred definitions and representation vary over time and between 

mathematical traditions. 

In ABEL’S theory of elliptic functions, the objects were introduced as inversions of 

elliptic integrals. The largely Eulerian tradition to which ABEL’S Recherches generally 

belongs emphasized representations of transcendental functions by infinite (power) 

series and infinite products. On the way to obtaining these representations, ABEL also 

came across the functional equations (17.8) and proved that elliptic functions were 

doubly periodic and could be written as the ratio of power series. All these represen- 

tations and key results were later assumed as definitions of the concept elliptic function 

depending on the setting and context in which they were introduced. 


One major achievement of the search for representations was, of course, that based on 

formulae such as (17.5), approximations to φ (α) could be computed with any degree 

of accuracy. Another, and equally interesting — but less anticipated — result was that 

infinite expressions, themselves, could play a role in the development of the theory. In 

the Recherches, this aspect remained little cultivated but in subsequent papers, ABEL 

occasionally applied infinite expressions, even to answer algebraic questions (see next 

chapter). 

In the Recherches, ABEL did more than solve the division problem for the lemnis- 

cate. While the lemniscate provided a clear question to which he produced a clear 

answer, the other part of the paper dealt with problems of more intrinsic nature and 

provided answers which are now hardly recognizable as answers because the ques- 

tions which they answer have faded in importance.

Chapter 18 

Tools in ABEL’s research on elliptic 

transcendentals 

In order to illustrate how N. H. ABEL (1802–1829) worked with elliptic transcenden- 

tals, the two most important topics are described in some details below. In order to 

make the presentation coherent, special emphasis is given to some of ABEL’S papers 

which illustrate important points concerning the types of tools involved by ABEL in 

his research on elliptic transcendentals. 

18.1 Transformation theory 

With A.-M. LEGENDRE’S (1752–1833) systematization of the theory of elliptic inte- 

grals, the transformations which he devised to reduce one integral to the simplest one 

in its class were powerful and important tools. ABEL approached the theory of trans- 

formations in his Recherches but his main contributions were spelled out in a number 

of articles written as a reaction to results announced by C. G. J. JACOBI (1804–1851) in 

the journal Astronomische Nachrichten. 

Competition with JACOBI. An important theme in a large part of the ABEL-related 

research has been his rivalry with the contemporary German mathematician JACOBI. 1 

As noted in connection with the idea of inverting elliptic integrals into elliptic func- 

tions, ABEL realized that he was in the middle of a rivalry when he became aware of 

JACOBI’S published announcements in H. C. SCHUMACHER’S (1784–1873) Astronomis- 

che Nachrichten. 2 The present section describes ABEL’S contribution to the theory in 

which JACOBI had initially been the most interested. By studying ABEL’S techni- 

cal machinery in some details, his means of dealing with elliptic functions become 

clearer. Most interestingly, ABEL employed largely algebraic methods in his research 

1 See also chapter 2 and section 16.2.5. 

2 In a letter to SCHUMACHER, HANSTEEN described how ABEL, upon learning of these papers, became 

very pale and had to rush to the nearest bakery for a dram; see (Holst, 1902, 89) and (E. 

Andersen, 1975, 104). 

331

332 Chapter 18. Tools in ABEL’s research on elliptic transcendentals 

Figure 18.1: CARL GUSTAV JACOB JACOBI (1804–1851) 

but also put the machinery of infinite representations to good use in obtaining an im- 

portant theorem in transformation theory. After a spell of shorter papers which com- 

plemented his two major papers, the Recherches and the Précis, 3 ABEL left the theory 

of transformations alone — and he soon died, of course. Transformation theory was 

not ABEL’S major purpose but it was a topic which provoked the interest of some of 

his most important contemporaries. 

In his announcements in the Astronomische Nachrichten, JACOBI communicated only 

the results and not the methods which he had employed to deduce them. The first 

biographies of ABEL have invested an effort in calling attention to ABEL’S priority 

in proving the results. In particular, C. A. BJERKNES (1825–1903) responded to L. 

KÖNIGSBERGER’S (1837–1921) account of the discovery of elliptic functions by arguing 

that JACOBI only obtained his proofs after learning of ABEL’S inversion. 4 However, 

since the turn of the twentieth century, historians and mathematicians have agreed 

that although JACOBI initially obtained his results through an ingenious but unrig- 

orous heuristic, he probably developed the inversion of elliptic integrals on his own, 

possibly inspired by reading through ABEL’S Recherches. 5 To indicate the hectic char- 

acter of the events, some key dates of the rivalry have been presented in table 18.1. 

3 (N. H. Abel, 1827b; N. H. Abel, 1828b) and (N. H. Abel, 1829d), respectively. 

4 (Bjerknes, 1880; Koenigsberger, 1879). 

5 (Mittag-Leffler, 1907; Ore, 1954; Pieper, 1998).

18.1. Transformation theory 333 

The central question of transformation theory. ABEL picked up the theme of trans- 

formations of elliptic integrals from LEGENDRE. In ABEL’S words, the central problem 

was posed with variations on multiple occasions, for instance in the following way: 

“To find all the possible cases in which one can satisfy the differential equation 

dy 

� 

(1 − y2 ) � 1 − c2 = a 

1y2� dx 

� (1 − x 2 ) (1 − c 2 x 2 ) 

by an algebraic equation between the variables x and y and supposing the moduli c 

and c1 less than unity and the coefficient a either real or imaginary.” 6 

Expressed in the notation of LEGENDRE’S differentials, the above question asks 

for every possible way of transforming x algebraically into y in such a way that the 

integral with modulus c1 transformed into the integral with modulus c. 

18.1.1 ABEL’s response to JACOBI’s announcements 

ABEL first published in the Astronomische Nachrichten in 1828; 7 in a lengthy paper, he 

demonstrated how the theory which he had developed in his Recherches could answer 

a question raised by JACOBI. In the paper — which is entitled Solution d’un problème 

général concernant la transformation des fonctions elliptiques — , 8 ABEL began by describ- 

ing key results concerning the inverse of elliptic integrals deduced in the Recherches. 

With the notation 

and the inversion 

∆ (x) = 

� 

θ = 

0 

� 

(1 − c 2 x 2 ) (1 − e 2 x 2 ) 

dx 

and x = λ (θ) , 

∆ (x) 

ABEL presented the highlights of the Recherches in two theorems which summarize 

(16.3) and (16.8), respectively. First, he expressed the addition theorem for the elliptic 

function of the first kind λ, 

λ � θ ± θ ′� = λ (θ) ∆ (θ′ ) ± λ (θ ′ ) ∆ (θ) 

1 − c2e2λ2 (θ) λ2 (θ ′ . 

) 

Second, ABEL described the conditions on the arguments which ensured that the func- 

tion took identical values, 

λ (θ) = λ � θ ′� if and only if θ ′ = (−1) m+m′ 

θ + mω + m ′ ω ′ 

6 “Trouver tous les cas possibles où l’on pourra satisfaire à l’équation différentielle: 

dy 

� 

(1 − y2 ) � 1 − c2 = a 

1y2� dx 

� (1 − x 2 ) (1 − c 2 x 2 ) 

(18.1) 

par une équation algébrique entre les variables x et y, en supposant les modules c et c1 moindre que 

l’unité et le coeffcient a réel ou imaginaire.” (N. H. Abel, 1829a, 33). 

7 (N. H. Abel, 1828d). The paper is dated 27 May 1828. 

8 (ibid.).


in which the semi-periods were determined by 

� � 

ω 1 

= θ and 

2 c 

ω′ 

2 

= θ 

� � 

1 

. 

e 

ABEL expressed the central properties of this theorem: 

“This theorem is generally valid no matter whether the quantities e and c are 

real or imaginary. In the paper cited above [Recherches], I have proved it in the 

case where e 2 is negative and c 2 is positive. [. . . ] The quantities ω, ω ′ always have 

an imaginary ratio. Otherwise, they have the same role in the theory of elliptic 

functions as the number π has in the theory of circular functions.” 9 

Now, in order to address the transformation problem, ABEL observed that the 

method of indeterminate coefficients could be applied. This method amounted to ap- 

proaching the problem by introducing two power series with indeterminate coeffi- 

cients and using the defining equations to obtain relations among the terms. How- 

ever, as ABEL critically remarked, this method would lead to extremely cumbersome 

calculations, and ABEL proposed a simpler and more direct one. Below, this method 

is briefly described. 

Rational transformations. With the notation and basic results set up, ABEL turned to 

a question which he proposed and ascribed great importance for the theory of elliptic 

functions. He was interested in finding all the possible ways in which the differential 

equation 

dy 

��1 − c2 1y2� � 1 − e2 = ±a 

1y2� dx 

� (1 − c 2 x 2 ) (1 − e 2 x 2 ) 

(18.2) 

could be satisfied in which y was an algebraic function of x. In the paper, ABEL limited 

his considerations to rational functions y = ψ (x) because the general question “at first 

seems too difficult”. 10 

ABEL’S first result in this situation was an algebraic one, not so different from re- 

sults obtained in his paper on Abelian equations. 11 By a string of manipulations, ABEL 

found that if the equation (18.2) was satisfied, the roots of the equation ψ (x) = y had 

the remarkable property of being related in a very specific way: if λ (θ) represented 

one of the roots, any other root of the equation would be representable as λ (θ + α) 

where α was a constant, i.e. 

y = ψ (λ (θ)) = ψ (λ (θ + α)) . 

9 “Ce théorème a lieu généralement, quelles que soient les quantités e et c, réelles ou imagainaires. 

Je l’ai démontré pour le cas où e 2 est négatif et c 2 positif dans le mémoire cité plus haut [. . . ]. Les 

quantités ω, ω ′ sont toujours dans un rapport imaginaire. Elles jouent d’ailleurs dans la théorie des 

fonctions elliptiques le même rôle que le nombre π dans celle des fonctions circulaires.” (N. H. Abel, 

1828d, 366). 

10 (ibid., 365). 

11 See chapter 7.


Obviously, only finitely many roots could exist, and the value of α could be obtained 

from (18.1), 

α = µω + µ ′ ω ′ 

(18.3) 

in which µ, µ ′ were rational constants. However, the degree of the equation y = ψ (x) 

might surpass the number of different values produced in this fashion and a new 

group corresponding to a new value α2 of α might be necessary. ABEL found that a 

certain number ν (which he did not describe in any detail) existed such that all the 

roots of the equation y = ψ (x) would be representable by the different values of the 

expression 

λ 

� 

θ + 

� 

ν 

∑ knαn 

n=1 

when k1, . . . , kν took all integer values. However, the different values could also be 

represented as (possibly changing the values of α1, α2, . . . ) 

λ (θ) , λ (θ + α1) , . . . , λ (θ + αm−1) 

in which α1, . . . , αm−1 were still rational linear combinations of ω and ω ′ (as in 18.3). 

ABEL then wrote the rational function ψ (x) = p(x) 

q(x) 

obtained 12 

m−1 

p (x) − q (x) y = A 

∏ 

n=0 

with no common divisors and 

(x − λ (θ + αn)) . 

The constant A was of the form A = f − gy with f , g constants. ABEL’S next step 

was to find an expression for A and he did so by first imposing a limiting assumption 

and gradually relaxing it. First, ABEL considered the case in which both p and q were 

polynomials of the first degree. In this case (e.g. by Euclidean division), 

and ABEL found 

y = f ′ + f x 

g ′ + gx , 

dy = f g′ − f ′ g 

(g ′ dx. 

2 

+ gx) 

When he inserted this into the original differential equation, its dependence on dy 

disappeared. Consequently, ABEL could conclude that the differential equation in this 

case implied either of the three solutions 

I.y = ax, c 2 1 

II.y = a 1 

ec 

= c2 

a 2 ,e2 1 

x , c2 c2 

1 = 

III.y = m 1 − x√ ec 

1 + x √ ec , c1 = 1 

m 

12 Here I write α0 = 0 for brevity. 

a 2 ,e2 1 

e2 

= , or 

a2 = e2 

a 

√ c − √ e 

√ c + √ e ,e1 = 1 

m 

2 , or 

√ √ 

c + e 

√ √ , a = 

c − e im 

(c − e) . 

2


Next, ABEL returned to the results which he had previously established and now 

let f ′ and g ′ denote the coefficients of x m−1 in p and q. Then, by comparing the coeffi- 

cients, ABEL found 

f ′ − g ′ y = − ( f − gy) 

Consequently, when he isolated y, ABEL found 

m−1 

∑ 

n=0 

y = f ′ + f φ (θ) 

g ′ + gφ (θ) 

λ (θ + αn) . 

� �� 

=φ(θ) 

which would serve to determine y as a function of x in all but those particular cases 

where φ (θ) reduced to a constant. 

In order for y to be a rational function of x, ABEL observed that the new function 

φ (θ) must likewise be rational in x. ABEL set out to investigate the circumstances un- 

der which this would be the case. By computing and combining values of the elliptic 

function λ, he found that φ (θ) was always a rational function of x and that it was given 

by 

φ (θ) = (1 − k) x + k′′ − k ′ 

ec 

1 

x + 

m−1 

∑ 

n=1 

2x∆ (αn) 

1 − e 2 c 2 λ 2 (αn) x 2 

in which k, k ′ , k ′′ were constants which were either zero or one. Based on this repre- 

sentation, ABEL separated three cases corresponding to various combinations of the 

values of k, k ′ , k ′′ . 

In the end, after his technical manipulations in the various cases, ABEL found that 

the differential equation under consideration 

dy 

� 

(1 − y2 ) � 1 − e2 = ± 

1y2� was satisfied precisely if a, e1, and y were given by 

a = k 

n−1 

∏ 

m=0 

n−1 

∏ 

m=1 

� 

m 

λ 

n ω 

� 

e1 = e n 

, y = k 

� 

n−1 

∏ λ 

m=0 

a dx 

� (1 − x 2 ) (1 − e 2 x 2 ) = ±a dθ 

n−1 � 

∏ λ θ + 

m=0 

mω 

� 

, and 

n 

� 

2m + 1 

2n ω 

��2 where n was an arbitrary integer and the constants k and ω were given by 

1 = k 

� 

2m + 1 

λ 

2n ω 

� 

dx 

� . 

(1 − x2 ) (1 − e2x2 ) 

and ω 

2 = 

� 1 

0 

Following this characterization of the solutions to the problem, ABEL first trans- 

lated the result into the trigonometric language employed by LEGENDRE and then 

announced a number of “remarkable theorems on elliptic functions” 13 which are of 

less relevance in the present context. 

13 (N. H. Abel, 1828d, 385).


Summary: an algebraic proof. As described, ABEL’S deduction consisted of five 

steps: First, ABEL set up his notation and definitions and introduced important results 

from the Recherches. Second, ABEL found that if λ (θ (x)) is a root, i.e. if y = 

ψ (λ (θ (x))), then any other root has the form λ (θ (x) + α). Next, the constant α could 

be determined and a general representation of the roots can be given — possibly in- 

volving multiple “orbits” corresponding to α1, α2, . . . . Fourth, the relation between y 

and x could be spelled out. Eventually, it could be necessary to consider a number of 

cases in order to describe these relations and deduce formulae of particular interest. 

A few broader points should also be observed. First, ABEL’S proof made central 

use of the properties of elliptic functions which had been deduced in the Recherches. 

In particular, the solution of the equation λ (x) = λ (y) — which originated from the 

double periodicity of the function λ — became very instrumental in the present con- 

text just as it had been in the solution of the division problem (see section 16.3). Sec- 

ond, the approach which ABEL took may well be described as an algebraic one; it relied 

on algebraic tools such as specific knowledge of the roots of certain polynomial equa- 

tions, division of polynomials, and considerations of the rationality of certain func- 

tions. These are tools which were also present in ABEL’S purely algebraic researches 

on solubility (see part II). However, ABEL also adopted another approach to the same 

question. 

Counting the possible numbers of transformations. In another paper — this time 

published in CRELLE’S Journal and motivated by another of JACOBI’S papers — , 14 

ABEL gave the theory of transformation a slightly different turn. He continued the 

path laid out in the Astronomische Nachrichten and made frequent references to the 

paper described above, but in the Journal, ABEL wanted to count and enumerate the 

different transformations. ABEL considered a rational transformation of x into y of 

a certain prime degree 2n + 1 and found — by employing algebraic tools similar to 

those described above — that 12 (2n + 2) different transformations corresponding to 

12 (n + 1) different values of the transformed modulus were generally possible. ABEL 

remarked that for certain particular values of the modulus c, the number of transfor- 

mations might degenerate. This notion of arguments carried out “in general” will be 

discussed further in section 19.3 and chapter 21. 

ABEL’S determination of the number of transformations spurred a reaction from 

LEGENDRE who believed that it was at odds with JACOBI’S determination of the de- 

gree of the so-called modular equation. JACOBI had claimed that for a transformation of 

prime degree 2n + 1, 2n + 2 values of the transformed modulus were possible. Thus, 

ABEL’S value was six times JACOBI’S number of transformations. However, as ABEL 

argued in a letter to LEGENDRE, JACOBI had indeed solved an equation with 2n + 2 

different roots but each root of this equation could also produce five other values for 

14 (N. H. Abel, 1828e); JACOBI’S paper is (C. G. J. Jacobi, 1828).


the transformed modulus and ABEL’S result was valid. 15 

18.1.2 An additional note 

In a subsequent issue of the Astronomische Nachrichten, ABEL inserted a note which 

added a different deduction of the main result of the previous one. 16 Whereas ABEL 

had initially employed direct and detailed manipulations to obtain the characteriza- 

tion of transformations, he now used one of the infinite representations which he had 

also obtained in the Recherches. 

From the Recherches, ABEL imported the expansion of the auxiliary function f in an 

infinite product and manipulated it into the current setting in which it produced 

� 

λ (α) = A × ψ α π 

� 

¯ω 

× 

∞ � � 

∏ ψ 

n=1 

(nω + α) π 

¯ω 

� � 

ψ 

(nω − α) π 

¯ω 

with A a constant and 

1 − e−2x 

ψ (x) = . 

1 + e−2x Furthermore, the periods ω and ¯ω were also related in ABEL’S usual way to the mod- 

ulus c. ABEL now inserted α = θ + m n for m = 0, 1, . . . , n − 1 and multiplied these 

expressions together to obtain the central formula 

with 

n−1 � 

∏ λ θ + 

m=0 

mω 

� 

n 

�� 

= A n � 

ψ δ π 

� ∞ � � 

∏ ψ (ω1 + δ) 

¯ω1 m=0 

π 

� 

ψ (ω1 − δ) 

¯ω1 

π 

� 

¯ω1 

δ = ¯ω1 ω1 

θ and 

¯ω ¯ω1 

= 1 

n 

ω 

. (18.4) 

¯ω 

The important idea which ABEL utilized now was to relate this formula to the sim- 

ilar one for the transformed function λ ′ corresponding to the modulus c1, the periods 

¯ω and ¯ω1, and the constant A1. ABEL found that 

λ ′ 

� 

¯ω1 

¯ω θ 

� 

= A1 

An n−1 � 

∏ λ 

m=0 

θ + mω 

n 

whenever the modulus c1 was such that the periods were related by (18.4). 

This time, we see how the knowledge of an infinite representation of the associated 

function f helped ABEL make statements about the transformation of elliptic func- 

tions. This is particularly interesting because it illustrates a different approach from 

the more algebraic one which he had initially taken. With direct access to a represen- 

tation of the function f , ABEL could employ a mixture of finite and infinite results to 

obtain he characterization of the conditions of rational transformations. 

15 (Abel→Legendre, Christiania, 1828/11/25. N. H. Abel, 1902a, 79). 


�

18.2. Integration in logarithmic terms 339 

18.2 Integration in logarithmic terms 

Another problem which figures significantly in ABEL’S approach to and research on 

higher transcendentals was the question of integration in more elementary forms. In 

chapter 15, it was described how mathematicians attacked the study of elliptic inte- 

grals although these were non-elementary. One of the approaches adopted was to 

relate a number of elliptic integrals by elementary functions or to investigate situa- 

tions in which the integration could indeed be effected in elementary (or finite) terms. 

A similar idea was pursued by ABEL in his investigations on what he called “theory 

of integration”. Thus, ABEL’S understanding of this notion differed from the present 

one in the sense that it was highly formal or algebraic and did not concern a numeri- 

cal interpretation of the integral. Such an interpretation was, of course, part of A.-L. 

CAUCHY’S (1789–1857) complete program of rigorization and became very important 

in the 19 th century mainly in the efforts to answer the challenges raised by Fourier 

series. 

Reminiscences of the Collegium mémoire. The first evidence of ABEL’S interest in 

the theory of integration (in finite terms) originates from descriptions of a paper which 

is no longer extant. As was already described in section 2.3, ABEL had hoped to em- 

bark on his European Tour shortly after the application was sent to the Collegium of 

the University in 1824. Before that time, in March 1823, ABEL presented a manuscript 

to the Collegium academicum through C. HANSTEEN (1784–1873). It concerned “a gen- 

eral presentation of the possibility of integrating all possible differential formulae” 17 . 

The manuscript was given to professors HANSTEEN and S. RASMUSSEN (1768–1850) 

for their professional evaluation. Their review was positive but no means of publish- 

ing the paper were at hand and it was subsequently lost. However, from ABEL’S pub- 

lished research, we may get an impression of what it could have contained. ABEL’S 

notebooks contain a number of entries related to the question of integration in finite 

terms; in particular, a manuscript for a large memoir on the theory of elliptic transcen- 

dentals from this perspective has been included in the Œuvres. 18 Nevertheless, the 

present description focuses on his main publication on the subject which occurred in 

the Journal in 1826. 19 

The local context of ABEL’S work on integration in finite terms was mainly related 

to the same theme as his research in the Paris memoir (see chapter 19, below). How- 

ever, it also included such issues as the reduction of all elliptic integrals to four basic 

kinds which ABEL undertook in his manuscripts and which had been a corner stone 

of LEGENDRE’S theory of elliptic integrals. 20 In the 1840s, mainly through the works 

17 “en almindelig Fremstilling af Muligheden at integrere alle mulige Differential-Formler” (N. H. 

Abel, 1902d, 4). 

18 (N. H. Abel, [1825] 1839b). 


20 (N. H. Abel, [1825] 1839b, 101); for LEGENDRE’S theory, see section 15.3.


of J. LIOUVILLE (1809–1882), the theory of integration in finite terms established itself 

as an independent theory investigated for its own results. In this context, the theory 

and ABEL’S contribution to it have been well described in J. LÜTZEN’S biography of 

LIOUVILLE. 21 Referring to LÜTZEN’S description, a presentation of ABEL’S argument 

and a brief discussion of relevant points of ABEL’S contribution are included below. 

18.2.1 Characterization by continued fractions 

In the paper from 1826, 22 ABEL investigated conditions under which the integral 

� 

ρ dx 

√R 

(18.5) 

could be reduced to the logarithmic expression 

log p + q√R p − q √ . (18.6) 

R 

The article first dealt with this question of reduction, but as ABEL ultimately noticed, 

the answer obtained was actually the answer to a more general question. ABEL noted 

that in case the integral (18.5) could be represented by logarithmic functions in any 

way, it would always have a representation of the form (18.6). ABEL promised a proof 

of this assertion but never published one; it was eventually given by P. L. CHEBYSHEV 

(1821–1894). 23 

A non-empty class. ABEL found by direct differentiation that for 

he would have 

Writing dz in the form 

dz = 

z = log p + q√ R 

p − q √ R , 

pq dR + 2 (p dq − q dp) R 

. 

(p 2 − q 2 R) √ R 

M dx 

dz = 

N √ R with 

M = pq dR 

� 

+ 2 p 

dx dq 

� 

− qdp R and (18.7) 

dx dx 

N = p 2 − q 2 R, (18.8) 

he had thus found that for such values of M and N, 

� M dx 

N √ R = log p + q√ R 

p − q √ R . 

ABEL concluded: 

21 (Lützen, 1990, chapter IX). 


23 (Chebyshev, 1853).


ρ dx 

“From this it follows, that in the differential √ an infinitude of rational func- 

R 

tions ρ can be found which make this differential integrable by 

� 

logarithms; furthermore, 

this is done by an expression of the form log .” 24 

� p+q √ R 

p−q √ R 

Thus, ABEL had proved that the class of differentials which were integrable by loga- 

rithms was non-empty. 

Delineation of the class. It was the converse of this result, that ABEL really wanted 

to investigate in the paper published in CRELLE’S Journal. He formulated the prob- 

ρ dx 

lem of determining all differentials of the form √ which could be integrated in the 

R 

logarithmic form 

log p + q√R p − q √ . (18.9) 

R 

This problem can be interpreted as another instance of a problem of delineation for a class 

of objects, in this case the class of objects integrable in the logarithmic form (18.9). 25 

With ρ = M N an entire function, ABEL took similar steps as above (18.7 and 18.8) 

and found the relation 

dp dN 

M 2 dx − p N dx 

= . 

N q 

Then followed a series of reductions to obtain a simple description of the relations 

between M, N, and R. Because M N was an entire function of x, it followed from this 

equation that p dN 

N dx was also an entire function of x, 

N = 

Reduction into partial fractions implied that 

dN 

N dx = 

n 

∏ (x + ak) k=0 

mk . 

n 

∑ 

k=0 

m k 

x + a k 

and because p dN 

N dx was to be entire, ABEL could write 

p = p1 

n 

∏ (x + ak) k=0 

in which p1 was an entire function. As a consequence of the relation N = p 2 − q 2 R, 

ABEL found 

n 

∏ (x + ak) k=0 

mk=p 2 n 

1 ∏ (a + ak) k=0 

2 

� �� 

=N 

24 “Daraus folgt, daß sich in dem Differential 

� �� 

=p 2 

−q 2 R 

ρ dx 

√ R , für die rationale Function ρ unzählige Formen fin- 

den lassen, die dieses Differential durch Logarithmen integrabel machen, und zwar durch einen 

Ausdruck von der Form log 


� p+q √ R 

p−q √ R 

� 

.” (N. H. Abel, 1826d, 186).


and because R did not contain any square factors and p and q could be assumed rela- 

tively prime, ABEL found m0 = m1 = · · · = mn = 1 and obtained the factorization 

R = R1 

n 

∏ (x + ak) = R1N. 

k=0 

with R1 an entire function. With this result, ABEL had found a reduced characteriza- 

tion in the form 

p 2 1 N − q2 R1 = 1 and M 

N 

= p1q dR 

dx 

� 

+ 2 p dq 

dx 

� 

− qdp R1. 

dx 

Considerations of degrees. ABEL’S next step was to investigate the consequences of 

the first part of the characterization obtained above, 

p 2 1 N − q2 R1 = 1. (18.10) 

As he remarked, the equation could be solved by the method of indeterminate coeffi- 

cients but this approach would be extremely cumbersome and not lead to any general 

conclusion. Instead, he proposed a different approach. Before embarking on his novel 

approach, ABEL introduced the notations δP to denote the degree of the (rational) 

function P and EP to denote the entire part of P, i.e. 

u = Eu + u ′ with δu ′ < 0. 

Judging from the detailed introduction of these concepts, ABEL did not assume them 

to be familiar to his readers. Concerning these new concepts, ABEL easily proved the 

following lemma: 

Lemma 3 If the functions u, v, z are related by 

and δz < δv, then 

u 2 = v 2 + z 

Eu = ±Ev. ✷ 

ABEL now returned to the equation p 2 1 N − q2 R1 = 1 and applied his new result 

(lemma 3). ABEL immediately obtained 

� 

δ p 2 1N � 

and consequently 

� 

= δ q 2 � 

R1 

2δp1 + δN = 2δq + δR1, i.e. 

δ (NR1) = 2 (δq + δR1 − δp1)


and because NR1 = R, ABEL had found that the highest power in R had to be an even 

number. ABEL wrote δN = n − m and δR1 = n + m and generalized the study of the 

equation (18.10) to the equation 

in which v was an entire function with δv < δN+δR 1 

2 

ABEL then wrote 

p 2 1 N − q2 R1 = v (18.11) 

= n. 

R1 = Nt + t ′ with t = E R1 

N and δt′ < 0. 

As a consequence of the assumptions, ABEL found that δt = 2m and he wrote t in the 

form 

t = t 2 1 + t′ 1 

in which δt ′ 1 < m. With these conventions, ABEL had remodelled the equation (18.11) 

into 

After rewriting the equation as 

ABEL observed that 

and he applied lemma 3 to obtain 

v = p 2 1N − q2 � Nt + t ′� = 

� 

= p 2 1 − q2t 2 � 

1 N − q 2 � t ′ 1N + t′� . 

� 

p 2 1 − q2 � 

t N − q 2 t ′ 

� �2 p1 

= t 

q 

2 v 

1 + 

Nq2 + t′ t′ 

1 + 

N , 

� 

v 

δ 

Nq2 + t′ � 

t′ 

1 + < m = δt1, 

N 

E 

� � 

p1 

= ±Et1 = ±t1. 

q 

This meant, that ABEL had found a relation between p1 and q of the form 

p1 = t1q + β in which δβ < δq. 

Through a sequence of similar, very explicit manipulations, ABEL transformed the 

equation (18.11) into the form 

in which 

s1β 2 − 2r1ββ1 − sβ 2 1 = v (18.12) 

δr1 = 1 

2 δR = n, δβ1 < δβ, δs < n, and δs1 < n.


His investigations now turned toward solving the equation (18.12). ABEL did so by 

observing that the process used above could be iterated producing a sequence of rela- 

tions similar to (18.12). After n − 1 iterations, he found the relation 

snβ 2 n−1 − 2rnβn−1βn − sn−1β 2 n = (−1) n−1 v, 

in which δβn < δβn−1. 

Because the sequence of degrees was decreasing, it would eventually produce δβm = 

0, i.e. βm = 0, and the final relation would then become 

smβ 2 m−1 = (−1)m−1 v. 

Using this information, ABEL ascended the chain of β, β1, . . . , βm in the reverse order 

each time finding expressions for βn−1 of the form 

βn−1 = 2µnβn + βn+1. 

By solving these relations for the first term of the chain β, ABEL found an expression 

for β 

β 1 as a finite continued fraction. 

In order to answer the question of logarithmic integration of the original differen- 

tial, ABEL next investigated the consequences for the radical √ R. He found from his 

earlier results that by assuming m infinite, the expansion of √ R would be 

√ 

R = t1 + 

2µ + 

1 

1 

2µ 1+ 1 

2µ 2 +... 

Here, ABEL noticed in a footnote that the equality of √ R and its continued fraction 

should not be interpreted as a numerical equality except in those situations where the 

continued fraction has a value. 

Finally, ABEL translated an earlier assumption that one among the quantities s1, s2, . . . 

should be independent of x into the property that the continued fraction for √ R 

should be periodic. The assumption on s1, s2, . . . had been introduced to ensure the 

solubility of the equations, and it thus amounted to a criterion for the possibility of in- 

tegrating 

ρ dx 

√ R in logarithmic terms. ABEL summarized his investigations as a complete 

criterion of logarithmic integrability stating that for polynomials ρ, the integration 

� ρ dx 

√R = log y + √ R 

y − √ R 

. 

(18.13) 

could be effected if and only if the expansion of √ R into continued fractions was 

periodic. In the affirmative case, the function y was determined by the first period of 

the continued fraction for √ R.


Summary. ABEL’S investigations concerning integration on the logarithmic form (18.13) 

serves to illustrate some interesting aspects. First, ABEL’S interest in the problem of in- 

tegrating differentials in logarithmic forms reveals the position of his research within 

a tradition of reducing complicated integrals to simpler ones. At the same time, the 

way he attacked the problem was rather novel. In his approach, ABEL applied the 

program which he had presented in his notebook research on solubility of equations 

and did not search “by divination” for an integration of the specific form (see section 

8.1). Instead, he took upon himself to establish the precise conditions under which the 

integration would be possible. Second, ABEL employed highly algebraic tools involv- 

ing polynomials, degrees of rational functions, and considerations of dependencies 

among quantities to reach his conclusion. At the point, where his argument came to 

involve an infinite representation in the form of an expansion of √ R into a continued 

fraction, he stressed that the equality should be interpreted as a formal one suited for 

determining the involved quantities. 


In the present chapter, two major examples of ABEL’S work with elliptic transcenden- 

tals have been described in order to illustrate the tools which he employed. In partic- 

ular, ABEL’S recurring use of algebraic methods has been documented. This algebraic 

approach to the theory of higher transcendentals was a general theme in ABEL’S ap- 

proach and it will be described further in the following chapter. At points where he 

involved infinite expressions, they were often regarded as algebraic equalities — the 

precise conditions for convergence were rarely addressed. However, as noted, infinite 

representations could sometimes improve the deductions considerably.


1827.06.12 JACOBI dated his first letter to SCHUMACHER 

1827.08.02 JACOBI dated his second letter to SCHUMACHER 

1827.09 Extracts from JACOBI’S two letters to SCHUMACHER were published 

in the Astronomische Nachrichten (C. G. J. Jacobi, 1827b). 

1827.09.20 The issue of A. L. CRELLE’S (1780–1855) Journal containing the 

first part of ABEL’S Recherches appeared. 

1827.11.18 JACOBI dated his Demonstratio which was published in the Astronomische 

Nachrichten (C. G. J. Jacobi, 1827a). 

1828.01.25 JACOBI dated his one page addition to ABEL’S Recherches. 

1828.02.12 ABEL sent the second part of the Recherches to CRELLE. 

1828.04.02 JACOBI dated his first letter inserted in CRELLE’S Journal. 

1828.05.26 The issue of CRELLE’S Journal with the second part of ABEL’S 

Recherches and its note reacting to JACOBI was published 

1828.07.21 JACOBI dated his second letter inserted in CRELLE’S Journal. 

1828.10.03 JACOBI dated his third letter inserted in CRELLE’S Journal. 

1828.12.03 The issue of CRELLE’S Journal containing ABEL’S investigation 

on the number of transformations appeared. 

1829.01.11 JACOBI dated his fourth and final letter inserted in CRELLE’S 

Journal. 

Table 18.1: Important dates in the ABEL-JACOBI-rivalry

Chapter 19 

The Paris memoir 

N. H. ABEL’S (1802–1829) most famous result was first communicated in a paper 

which he delivered to the Parisian Academy of ScienceAcadémie des Sciences in 1826. 

The result which ABEL obtained in the so-called Paris memoir was a rather technical 

one which dealt with the integration of algebraic differentials. 1 In its original, it was 

formulated in the typical style of ABEL, his predecessors, and many of his contem- 

poraries but during the century ABEL’S result was recast in a quite different language 

and in another mathematical structure. 2 In modern mathematics, ABEL’S result is typ- 

ically considered a part of algebraic geometry; readers who wish to see a presentation 

of the result from such a modern perspective can consult e.g. (Shafarevich, 1974). 

Besides presenting the main results, the present rendering of ABEL’S Paris memoir 

aims at describing the central tools which ABEL employed in his reasoning. The fo- 

cus on tools facilitates a continuation of the comparison with the methods involved in 

ABEL’S purely algebraic works and also serves to support the discussion of the imme- 

diate reception of ABEL’S results taken up in section 19.5. ABEL’S arguments in the 

Paris memoir were conducted in a style heavily dependent on explicit manipulations 

of formulae. In section 19.5.1, his subsequent announcements of the main results and 

the much clearer sketches of proof contained therein are described and discussed. 

19.1 ABEL’s approach to the Paris memoir 

ABEL’S Paris memoir represents a pivotal point in his mathematical production. Many 

investigations which ABEL had previously undertaken for their own sake and brought 

to interesting conclusions were surpassed by the main result of the Paris memoir. The 

Paris memoir was three-fold monumental: it was the culmination of a line of research 

which ABEL had undertaken for years, it contained the result which brought him 

widespread fame in the nineteenth century, and yet it provided this result with an 

incredibly long and cumbersome proof. 

1 ABEL’S result is also discussed in e.g. (Cooke, 1989; J. Gray, 1992). 

2 For a brief discussion of styles, see chapter 21. 

347

348 Chapter 19. The Paris memoir 

19.1.1 Set to work in Paris 

ABEL arrived in Paris on July 10, 1826. For quite some time, he had worked on the 

study of functions whose differentials satisfy certain algebraic conditions. In a letter 

to C. HANSTEEN (1784–1873) written shortly after his arrival, ABEL explained how he 

had postponed introducing himself to the Institut de France until his mastery of the 

French language had improved. He continued the letter: 

“Furthermore, and in particular, I want to complete the memoir I am working 

on and which I intend to present to the Institute. When it is complete, which 

will soon be the case, I will go there. The memoir has come out very well and 

contains many new things which I believe merit attention. It is the first draft of a 

theory of an infinitude of transcendental functions. — I nourish the hope that the 

Academy [Académie des Sciences] will have it printed in the Mémoires des savants 

étrangers.” 3 

Finally, on October 30, 1826, ABEL presented his memoir to the Institut de France. 

Three days later ABEL sent an article for publication in J. D. GERGONNE’S (1771–1859) 

Annales in which he presented his research on simultaneous solutions to two poly- 

nomial equations. 4 This paper contained an elaboration of one of the main tools of 

ABEL’S Paris memoir; it is discussed in section 19.3, below. When he left France on 29 

December 1826, 5 ABEL had still not received any reaction from the Institut concerning 

his memoir and, in fact, he was never to receive one. ABEL’S Paris memoir was mis- 

placed before G. LIBRI (1803–1869) was commissioned with its printing. It eventually 

occurred in the Mémoires présentés par divers savants in 1841. The fate and reception of 

ABEL’S Paris memoir are briefly described in section 19.4. 

19.1.2 Tools in ABEL’s toolbox 

The Paris memoir — when seen together with some of ABEL’S other publications de- 

scribed in chapter 18 — provides new insights into the toolbox of the creative, young 

mathematician. As could be expected of a mathematician devoted to algebra, the 

compartments for results concerning polynomials and equations are remarkably well 

equipped. 

1. ABEL used algebraic deductions and EUCLID’S (∼295 B.C.) algorithm in ways 

similar to those which he had already employed in his research on algebraic 

3 “Desuden vil jeg først og fremst have en Afhandling færdig som jeg arbeider paa og som jeg vil 

forelægge Institutet. Naar denne, hvilket snart skeer, er færdig gaar jeg derhen. Denne Afhandling 

er lykkets mig særdeles godt, og indeholder meget nyt og som jeg troer værdig Opmærksomhed. 

C’est la prémière ébauche d’une théorie d’une infinité de fonctions transcendantes. — Jeg har det 

Haab at Academiet vil lade den trykke i Mémoires des savants étrangers.” (Abel→Hansteen, Paris, 

1826/08/12. N. H. Abel, 1902a, 40). 


5 (Lange-Nielsen, 1927, 65).

19.1. ABEL’s approach to the Paris memoir 349 

solubility. One central example was a tacitly employed result which has been 

presented in lemma 4, below. 

2. Another algebraic result was employed by ABEL to the effect that for any real 

polynomial without multiple roots, such as p (x) = ∏ n k=1 (x − xk), � 

0,if α ≤ n − 2 and 

n x 

∑ 

k=1 

α k 

p ′ (xk) = 

1,if α = n − 1. 

This result, which is a consequence of the so-called Lagrange interpolation is dis- 

cussed in section 19.3. 

3. ABEL also borrowed results concerning primitive roots and congruences from 

C. F. GAUSS’ (1777–1855) Disquisitiones arithmeticae. Again, we have already 

noted how well acquainted ABEL was with this book by GAUSS. 

4. Finally, ABEL used expansions into series of decreasing powers to great effect. 

Such expansions had been forcefully employed in the 17 th century, and ABEL 

certainly considered the procedure well established. In ABEL’S work it was com- 

bined with a particular emphasis on the coefficient of x −1 , the coefficient which 

elsewhere, with A.-L. CAUCHY (1789–1857), became known as the residue. 

Most of these issues are addressed in some details in section 19.3 after ABEL’S use 

of them in the Paris memoir has been described. 

19.1.3 The presentational style of the Paris memoir 

When compared to the other works in ABEL’S corpus, the style of the Paris memoir 

stands out in a number of respects. When compared to the subsequent partial an- 

nouncements of results contained in the Paris memoir (see section 19.5.1), a pattern be- 

comes discernible. At the textual level, ABEL’S papers fell between two traditions, one 

mainly based on algebraic manipulations and derivations of formulae and an emerg- 

ing one returning to the Euclidean norm of definitions, theorem statements, and proofs 

(see chapter 21). In this continuum of styles, the Paris memoir belongs to the manip- 

ulation based tradition with its long, tedious, and very explicit derivations of explicit 

formulae. The theorems which I have extracted (Main Theorems I and II, 16 and 17) 

are reconstructions, and reformulating ABEL’S main results in the “If . . . , then . . . ” 

structure of modern theorem-based mathematics is by no means an easy and trivial 

task; the translation from ABEL’S explicit manipulative style to the structure of theo- 

rems is not a bijection, it requires interpretation. 

19.1.4 ABEL’s notational innovations 

1. ABEL used a summation shorthand 

ΣFx = Fx1 + Fx2 + · · · + Fxn


An important lemma 

Lemma 4 Let χ (y) = 0 be an irreducible equation of degree n and let θ (y) be an equation 

of degree n − 1. Then y can be expressed rationally in χ, θ. ✷ 

PROOF (PROOF OF LEMMA 4) By the Euclidean algorithm, there exist polynomials q, r 

such that 

χ = qθ + r 

where deg r < deg θ. Since χ is irreducible and deg q = deg χ − deg θ = 1 > 0, 

deg r > 0. Thus, there exist numbers s, t such that 

Consequently, 

y = 

q (y) = sy + t. 

q (y) − t 

s 

= 

χ(y)−r 

θ(y) 

− t 

, 

s 

and y has been expressed rationally in χ, θ. � 

Box 9: An important lemma 

which apparently was innovative with him. As is evident from even this exam- 

ple, both the index over which the summation is to be performed and the upper 

summation limit are implicit in the shorthand version. 

2. For a rational function Fx, ABEL let ΠFx denote the coefficient of 1 x in the series 

expansion of Fx in decreasing powers of x. Designating the ‘same’ object as the 

residue which CAUCHY studied from an emerging perspective of his calculus of 

residues, ABEL’S Π corresponds to CAUCHY’S E. 

3. ABEL also introduced the operation h on algebraic functions which represented 

a general degree of algebraic functions. 

4. In the ultimate example of hyperelliptic integrals, ABEL introduced the notation 

EA and εA for A any real number to denote the integer and remaining part, A = 

EA + εA (EA ∈ Z and 0 ≤ εA < 1). It is worth remarking that ABEL did not — in 

the Paris memoir considered as a whole — apply this notation consistently. Until 

the ultimate section, he preferred the verbal formulation ‘the greatest integer 

contained in the number’. 

All these notational innovations enabled ABEL to comprehend, master, and manip- 

ulate objects in a precise way which had hitherto been difficult to obtain.

19.2. The contents of ABEL’s Paris result and its proof 351 

19.2 The contents of ABEL’s Paris result and its proof 

ABEL’S objective in the memoir was to study integrals of the form 

� 

f (x, y) dx 

in which x and y were related by some algebraic equation (19.1) and f was a rational 

function. Such integrals provided a way of generalizing elliptic integrals; any elliptic in- 

tegral could be written in the form above. However, it was not through a direct study 

of one such integral that something new was to be learned, but by studying relations — 

arising from the additional equation (19.3) — among a number of such integrals. 

The contents of the Paris memoir can be structured into several results and their 

primary applications: 

1. The establishment of Main Theorem I on integration of certain sums of algebraic 

differentials by elementary functions, 

2. The establishment of Main Theorem II on the number of independent integrals of 

algebraic differentials, and 

3. Application of Main Theorem II to the simplest case, the case of hyperelliptic 

integrals. 

In the following, the results and methods of first two of these three parts will be 

described; as will the other instances where ABEL presented his findings on related 

issues. ABEL’S reasoning is rather cumbersome and not completely flawless. Some 

of the subsequent objections and comments — primarily by P. L. M. SYLOW (1832– 

1918) — are referred to in the course of the presentation. However, despite the reser- 

vations, ABEL’S original argument is presented to illustrate how he ingeniously used 

the tools at his disposal. Even if the contents and purpose of ABEL’S arguments can 

seem to evade attention, his various tools and the contents of the Paris memoir are 

subsequently summarized. 

To various degrees of authenticity, ABEL’S argument has been described from the 

viewpoint of the application to hyperelliptic integrals, see e.g. (Brill and Noether, 1894; 

Cooke, 1989). However, as will be discussed in section 19.5.1, the chronology and in- 

ternal logical structure suggests that the results of the Paris memoir were indeed prior 

to and to some extent independent of the applications to this (afterwards) immensely 

important special case.


19.2.1 Main Theorem I 

In the Paris memoir, 6 ABEL dealt with two quantities x and y related through an irre- 

ducible polynomial equation such as 

χ (y) = 0, (19.1) 

where χ is a polynomial in y whose coefficients are polynomial functions of x, 

χ (y) = 

n 

∑ pk (x) y 

k=0 

k . (19.2) 

This relation (19.2) implicitly introduced n functions y (1) , . . . , y (n) of x corresponding 

to the n roots the equation would have for any particular value of x. 

Introducing another (later to be specialized) equation in y whose degree was one 

less than χ, 

ABEL formed the product 

θ (y) = 

r = 

n−1 

∑ qk (x) y 

k=0 

k = 0, (19.3) 

n � 

∏ θ y 

k=1 

(k)� 

, (19.4) 

by inserting the n different solutions of (19.2) into (19.3) and multiplying the n results. 

The coefficients q0, . . . , qn−1 could contain some indeterminate quantities a1, . . . , aN, 

and r was found to be an entire function of x and a1, . . . , aN by methods “imported” 

from the theory of equations. 

In order to focus attention, ABEL split the product r into parts dependent on and 

independent of the indeterminate quantities 

r = F0 (x) F (x) , (19.5) 

where only F depended on a1, . . . , aN. ABEL then considered the equation 

F (x) = 0, (19.6) 

which would provide expressions for its roots x1, . . . , xµ in terms of a1, . . . , aN, 

F (x) = 

µ 

∏ (x − xk) . 

k=1 

These roots would become very important in the ensuing deductions. 7 


7 At this point it might be fruitful to summarize ABEL’S results this far. He knew that r, defined from 

the polynomials θ and χ, was an entire function of x and the indeterminates a1, . . . , aN. Any root 

of the equation r (x) = 0 corresponded to a value y for which θ (y) = 0 by obvious inspection. 

However, r (x) = 0 also meant either F0 (x) = 0 or F (x) = 0. The former case represented an 

equation independent of the indeterminates a1, . . . , aN, whereas the latter introduced a relationship 

between x and the indeterminates. Thus, with given indeterminates, r (x) = 0 would mean either 

that x belonged to the set � � 

x1, . . . , xµ or that F0 (x) = 0. To each xk in � � 

x1, . . . , xµ corresponded a 

value of y, which ABEL termed yk, such that θ (yk) = 0.


After these algebraic operations, ABEL now for the first time employed the calculus 

in differentiating the equation (19.6) above. ABEL wrote the differentiation as 

F ′ (x) dx + ∂F (x) = 0, (19.7) 

where F ′ (x) represents the differential of F with respect to x and ∂F (x) represents 

the differential of F with respect to all the indeterminates. This relationship was a 

fundamental one, and ABEL immediately put it to use. He introduced the differential 

dv = 

µ 

∑ f (xk, yk) dxk, k=1 

where f was a rational function. This differential was the real object of concern in these 

investigations. Through a sequence of deductions employing the theory of equations 

(see example below), ABEL reasoned that dv was a rational function of the parameters 

a1, . . . , aN. Therefore, its integral v would have to be expressible by algebraic and 

logarithmic functions of these parameters 8 , 

v = 

µ � 

∑ 

k=1 

f (x k, y k) dx k = algebraic and logarithmic terms. 

This result is what I have termed Main Theorem I. 

Theorem 16 (Main Theorem I) Under the present assumptions, the sum 

µ � 

∑ 

k=1 

f (x k, y k) dx k 

can be expressed by algebraic and logarithmic functions of the parameters a1, . . . , aN. ✷ 

An example of ABEL’S use of the theory of equations. In order to see that dv was 

indeed a rational function of the parameters, ABEL first claimed that the simultaneous 

equations (19.1) and (19.3) expressed y k as a rational function of x k, 9 

y k = ρ (x k) . 

Rearranging the equation (19.7) then produced 

f (x, y) dx = − 

f (x, ρ (x)) 

F ′ ∂F (x) = φ2 (x) . 

(x) 

Of this function φ2, ABEL observed that it was obviously rational in x and the param- 

eters. Thus, dv could be rewritten as 

dv = 

µ 

∑ φ2 (xk) (19.8) 

k=1 

8 The integral of any rational function was of course expressible by rational and logarithmic terms. 

9 In order to see that ρ is rational as claimed, please observe that deg θ = deg χ − 1. See the proof of 

lemma 4 in box 9.


where φ2 was a rational function of the parameters and its explicit argument 10 . How- 

ever, because the right hand side of (19.8) was both rational and symmetric in the roots 

x1, . . . , xµ of the equation (19.6), dv could be expressed rationally in the coefficients of 

F by a basic theorem which ABEL knew from his theory of equations (see section 5.2.4). 

However, the coefficients of F were supposed to depend rationally on the parameters, 

and the claim had been demonstrated. 

19.2.2 An explicit expression for v 

ABEL summarized the results of the Main Theorem I and introduced the way forward 

with these words: 

“Previously, we have demonstrated how it is always possible to form the rational 

differential dv. However, as the indicated method will generally be very 

long and nearly impractical for slightly complicated functions, I will give another 

[method] by which one will immediately obtain the expression of the function v 

in all possible cases.” 11 

The expression for v, which ABEL obtained in “all the possible cases” was of the 

following form (we shall comment on the particulars and the notation below) 

v = C − Πφ (x) + 

α 

∑ 

ν=1 

ν dν−1φ1 (x) 

dxν−1 . (19.9) 

First, we will pay some attention to ABEL’S arguments in order to illustrate how they 

relate to the deduction of Main Theorem I and how they introduce tools which would 

become important in deducing Main Theorem II (see below). 

A sequence of manipulations — in which ABEL again made important use of his 

knowledge from the theory of algebraic equations — led ABEL to express the sought 

after differential in the following form 

dv = − ∑ 

R1 (x) 

A · F ′ (x) · ∏ α k=1 (x − β k) ν k 

, (19.10) 

where R1 was an entire function and A was a constant. The next step was the reduction 

of this expression. ABEL first revised the notation and wrote 

dv = − 

µ 

∑ 

k=1 

R2 (x k) 

F ′ (x k) − 

µ 

∑ 

k=1 

R3 (xk) θ1 (xk) · F ′ , (19.11) 

(xk) 10 ABEL chose to denote this function φ2, although no φ1 had been introduced at this point. This might 

suggest that the logical order in ABEL’S head of these sections had been reversed in the written 

version. See below. 

11 “Nous avons montré dans ce qui précède comment on peut toujours former la différentielle rationelle 

dv; mais comme la méthode indiquée sera en général très-longue, et pour des fonctions un peu 

composées, presque impractible, je vais en donner une autre, par laquelle on obtiendra immédiatement 

l’expression de la fonction v dans tous les cas possibles.” (N. H. Abel, [1826] 1841, 150).


where the auxiliary θ1 had been introduced by 

and 

θ1 (x) = A 

α 

∏ 

k=1 

(x − β k) ν k , 

R1 (x) = θ1 (x) R2 (x) + R3 (x) , with 

deg R3 < deg θ1. (19.12) 

The introduction of the additional auxiliary functions R2 and R3 was made to ease the 

study of the quotient R 1 

θ 1 , which because of (19.10) was at the centre of ABEL’S interest. 

Implicitly using the method of Lagrange interpolation, 12 ABEL reduced the first term 

of (19.11), 

µ 

R2 (xk) ∑ F 

k=1 

′ (xk) = Π R2 (x) 

F (x) 

where the symbol Π was introduced in the following way: 

“Thus, in designating by ΠF1x the coefficient of 1 

x in the development of any 

function F1x according to decreasing powers of x, one will get [. . . ]” 13 

Because of (19.12), the reduction could be written 

µ 

R2 (xk) ∑ F 

k=1 

′ (xk) = Π R1 (x) 

θ1 (x) F (x) . 

ABEL’S reduction of the remaining term of (19.11) was marred by a faulty calculation 

which has been noticed and elaborated by SYLOW in his notes in the Œuvres. 14 

ABEL claimed — by an argument based on expansion into partial fractions — that 

R3 (x) 

θ1 (x) = 

α 

d 

∑ vk k=1 

vk−1 dβvk−1 � 

θ (v k) 

1 

R3 (β) 

(β) · (x − β) 

� 

β=β k 

. (19.13) 

However, during his deductions concerning partial fractions he had mistakenly placed 

the factor (x − β) in the denominator instead of in the numerator. 15 This flaw perme- 

ated the ensuing calculations, and a simple counter example could serve to demon- 

strate that the result claimed in (19.13) is only valid in some very particular cases. 

12 See below. 

13 “En désignant donc par ΠF1x le coefficient de 1 x dans le développement d’une fonction quelconque 

F1x, suivant les puissances descendantes de x, on aura [. . . ]” (ibid., 155). 

14 (Sylow in N. H. Abel, 1881, II, 295–296). 

15 The mistake is indeed ABEL’S which can be seen from the fact that it occurs in the manuscript of the 

Paris memoir.


ABEL and the method of Lagrange interpolation. Without any specific reference, 

ABEL employed a result to the effect that 

n 

∑ 

k=1 

p (xk) χ ′ (xk) = 

� 

0if deg p < n − 1 

1if deg p = n − 1 

for any normed polynomial p where x1, . . . , xn are the roots of the polynomial equation 

χ (x) = 0, i.e. 

χ (x) = 

n 

∏ (x − xk) . (19.14) 

k=1 

The tool behind this result is known today as Lagrange interpolation, and — in various 

forms — it played central roles in ABEL’S arguments in the Paris memoir. Lagrange 

interpolation is used to demonstrate that for any polynomial such as (19.14), 

1 

χ (x) = 

n 

∑ 

k=1 

1 

(x − xk) χ ′ . (19.15) 

(xk) Expansion into partial fractions using Lagrange interpolation. The method of ex- 

panding a quotient of polynomials into partial fractions was well established in the 

18 th century once it was known that the denominator could be decomposed into a 

product of linear and quadratic terms. The generality of the method thus rested es- 

sentially on the Fundamental Theorem of Algebra, and the proof of the latter theorem was 

often seen mainly as a prerequisite in rigorously founding this established practice (cf. 

GAUSS). 

The central trick in expanding a quotient into partial fractions is closely related to 

the method of Lagrange interpolation. If the polynomials are 

f2 (x) = 

f1 (x) and 

n 

∏ (x − xk) , 

k=1 

where the roots of f2 are distinct, Lagrange interpolation (19.15) yields 

f1 (x) 

f2 (x) = 

n 

∑ 

k=1 

f1 (x) 

(x − x k) f ′ 2 (x k) . 

Thus, when applied in the integral calculus, this formula reduces the integration of 

a fraction to the integration of n fractions, the denominator of each of which only 

contains a first degree polynomial.


ABEL’S result is faulty when the denominators have multiple roots. For situations 

in which the polynomial f2 has multiple roots, the procedure can be extended to ac- 

commodate this case as well. The flawed result which ABEL used (see above) was for 

a general rational function 

with 

the fraction could be expanded as 

where 

A m k = 

f2 (x) = 

f1 (x) 

f2 (x) = 

f1 (x) 

f2 (x) 

n 

∏ (x − xk) k=1 

mk , 

n mk A 

∑ ∑ 

k=1 m=1 

m k 

(x − xk) m 

d m k−1−m pk 

Γ (m k + 1 − m) dβ m k−1−m , 

pk = Γ (mk − 1) f1 (βk) . 

(βk) f (m k) 

2 

However, ABEL’S formulae for the coefficients were wrong; they should have been 

A m k = 

1 d 

Γ (mk + 1 − m) 

mk−m dβm � m � 

k (x − β) R3 (x) 

, 

k−m θ1 (x) 

as SYLOW pointed out. 16 Incidentally, this formula was very similar to ABEL’S starting 

point: 

where 

““[. . . ] one will get 

for x = β; [. . . ]” 17 

A1 = dν−1 p 

Γν · dβ ν−1 , A2 = 

p = (x − β)ν R3x 

θ1x 

16 (Sylow in N. H. Abel, 1881, II, 295). 

17 “[. . . ] on aura 

où 

A1 = dν−1 p 

Γν · dβ ν−1 , A2 = 

p = (x − β)ν R3x 

θ1x 

pour x = β; [. . . ]” (N. H. Abel, [1826] 1841, 156). 

d ν−2 p 

Γ (v − 1) dβ ν−2 , . . . , Aν = p, 

d ν−2 p 

Γ (v − 1) dβ ν−2 , . . . , Aν = p, 

β=β k


The ensuing step was, however, unwarranted as ABEL claimed that 

p = Γ (ν + 1) R3 (β) 

θ (ν) 

. 

1 (β) 

One can guess how ABEL came to the latter belief by applying the rule of G.-F.-A. DE 

L’HOSPITAL (1661–1704) v times to the definition of p as both numerator and denomi- 

nator vanish. In the above presentation, the problem which ABEL’S deduction suffered 

from is hidden in the notation. First of all, ABEL’S way of suppressing the subscript 

k has made the β and x appear symbolically similar, although x is a true variable 

whereas β1, . . . , βα are the roots of a certain polynomial. This distinction is at the core 

of SYLOW’S objection to ABEL’S argument. 18 However, with a minor adjustment to 

the definitions, ABEL’S final product (19.9) of the argument could be allowed. 

19.2.3 Main Theorem II 

After the first four sections of the Paris memoir, ABEL had thus obtained a formula 

which was essentially (apart from the corrections indicated above) the following, 

v = 

µ 

µ � 

∑ ψ (xk) = ∑ 

k=1 

k=1 

f (x k, y k) dx k = C − Πφ (x) + 

α 

∑ 

ν=1 

ν dν−1φ1 (x) 

dxν−1 . 

This expression allowed him to commence a study of the number of free parameters 

which would eventually lead to the second main theorem — the celebrated Abelian 

Theorem. To follow his argument, we need to backtrack a little to properly understand 

the use of the eliminant equation r = 0 (see page 352). 

ABEL’S trick was first to study the consequences of one further assumption con- 

cerning the factor F0 of r containing the indeterminate quantities. ABEL assumed that 

F0 had α distinct zeros, 

α 

F0 (x) = ∏ (x − βk) k=1 

µ k ; 

an assumption which — provided F0 is not a constant — introduced α linear interrelations 

among the coefficients q0, . . . , qn−1 of the auxiliary polynomial θ (y) (19.3). 19 

ABEL found the fact that the coefficients of θ (y) formed a non-independent set to be — 

in general — a contraction of the original hypothesis which required nothing of the co- 

efficients q0, . . . , qn−1. Consequently, he concluded that F0 (x) had to be a constant and 

r (x) could not — in general — contain any factor independent of the auxiliary quanti- 

ties. 

Under this assumption, ABEL proceeded to describe the various functions involved. 

The important outcome of these investigations was that a certain function f2 (x) in- 

troduced much earlier in the investigations, reduced to unity, thereby providing the 

result that f (x, y) χ ′ (y) equated the entire function f1 (x, y). 


19 See box 19.2.3.


Linear interdependence of q0, . . . , qn−1 In order to see how ABEL obtained the linear 

interrelations among the coefficients of θ (y), we notice from combining the factoriza- 

tion (19.5) with the definition of r (19.4), 

r (x) = F (x) · F0 (x) 

= 

n 

∏ θ 

k=1 

� 

y (k) (x) 

� 

= 

n n−1 

∏ ∑ 

k=1 m=0 

� 

qm (x) y (k) �m (x) . 

Thus, since r (β1) = F (β1) · 0 = 0, some k must exist for which θ 

n−1 � 

∑ qm (x) y 

m=0 

(k) �m (x) = 0. 

� 

y (k) � 

(β1) = 0, i.e. 

This relation is a linear interdependence among the q0, . . . , qn−1 in which the 

here serving as coefficients, are functions of x. 

Box 10: Linear interdependence of q0, . . . , qn−1 

� 

y (k)� m 

, 

ABEL’S way of proving f (x, y) χ ′ (y) to be an entire function. ABEL’S investiga- 

tions leading to the result that f (x, y) χ ′ (y) was an entire function progressed along 

complicated and tedious arguments. At the very outset of the paper, ABEL had split 

the rational function of x in the following way 

f (x, y) χ ′ (y) = f1 (x, y) 

, (19.16) 

f2 (x) 

where f2 (x) was an entire function of x independent of y. By combining this with the 

important partial differentiation (19.7), ABEL found 

f (x, y) dx = 

f1 (x, y) 

f2 (x) · χ ′ (y) dx 

1 

= − 

F0 (x) · F ′ (x) · f2 (x) 

n 

∑ 

k=1 

f1 (x, y k) 

χ ′ (y k) 

r 

θ (y k) ∂θ (y k) . 

Later, during his investigations leading to the Main Theorem I, ABEL had studied the 

function 

which he had chosen to write as 

f1 (x, y) 

f1 (x, y) 

r 

∂θ (y) 

θ (y) 

r 

θ (y) ∂θ (y) = R′ (y) + R (x) y n−1 , 

where R ′ (y) indicated an entire function of x and y in which no powers of y beyond 

the (n − 2)’nd occur, and R (x) was an entire function of x independent of y. By use of


the method of Lagrange interpolation described above and the factorization of r (19.5), 

ABEL found 

R (x) = F0 (x) · F (x) · 

n 

∑ 

k=1 

f1 (x, y k) 

χ ′ (y k) 

∂θ (y k) 

θ (y k) . 

Under the present assumption F0 (x) = 1, ABEL concluded that 

φ1 (x) = 

1 

f (m) 

n 

∑ 

2 (x) k=1 

f1 (x, y k) 

χ ′ (y k) log θ (y k) , 

and consequently, the sum of integrals took the form (cmp. equation 19.9) 

� 

f1 (x, y) dx 

∑ f2 (x) χ ′ (y) = C − Π ∑ f1 (x, y) 

f2 (x) χ ′ (y) 

+ ∑ m dm−1 

dβ m−1 

log θ (y) 

� 

1 

f (m) 

2 (β) ∑ f1 (β, B) 

χ ′ � 

log θ (B) 

(B) 

. (19.17) 

After inspecting the results of certain simple assumptions concerning f2, ABEL argued: 

“In the equation (19.17), the right-hand-side is in general a function of the 

quantities a, a ′ , a ′′ , etc. If one supposes this function equal to a constant, certain 

relations among these quantities thus generally result; but there are also certain 

cases for which the right-hand-side reduces to a constant no matter what the values 

of the quantities a, a ′ , a ′′ , etc. are. We investigate this case: 

From this it is evident that the function f2x must be constant, because in the 

contrary case the right-hand-side necessarily contains the quantities a, a ′ , a ′′ . . . , 

with respect to the arbitrary values of these quantities.” 20 

ABEL’S argument here seems a little roundabout; in the cause of argument he in- 

troduced an important assumption — that v now reduces to a constant. He argued 

that unless f2 then also reduced to a constant, the right hand side of (19.17) would 

involve the auxiliary quantities, whereas the left hand side — which was nothing but 

v = ∑ � f (x, y) dx by (19.16) — was now a constant. Thus, unless f2 was constant, cer- 

tain relations among the indeterminates a, a ′ , a ′′ , . . . would result — a contradiction. In 

the end, ABEL had obtained a representation of the constant v in the following form 

� 

f1 (x, y) dx 

∑ χ ′ (y) 

= C − ∑ Π f1 (x, y) 

χ ′ (y) 

log θ (y) . 

If we pause for a second to consider what the consequences of this new hypothesis, 

the constancy of v, are, we can suggest some motivation for this at first sight rather un- 

natural assumption which will be elaborated during the uncovering of the remaining 

20 “Dans le formule (43) [here (19.17)], le second membre est en général une fonction des quantités 

a, a ′ , a ′′ , etc. Si on le suppose égal à une constante, il en résultera donc en général certaines relations 

entre ces quantités; mais il y a aussi certaines cas pour lesquels le second membre se réduit à une 

constante, quelles que soient d’ailleurs les valeurs des quantités a, a ′ , a ′′ , etc. Cherchons ces cas: 

D’abord il est évident que la fonction f2x doit être constante, car dans le cas contraire le second 

membre contiendrait nécessairement les quantités a, a ′ , a ′′ . . . , vu les valeurs arbitraires de ces quantités.” 

(N. H. Abel, [1826] 1841, 161).


details. If v is a constant, say v = 0, the basic objects of the inquiry satisfy 

� 

0 = ∑ 

f (x k, y k) dx k. 

The end product of the present section of the paper is the Abelian Theorem (Main Theo- 

rem II) states something about exactly such sums of related integrals. 

ABEL next expanded the relevant function, whose derivative was rational in x by 

(19.8), according to decreasing powers of x, 

∑ f1 (x, y) 

χ ′ (y) 

log θ (y) = R log x + 

∞ 

∑ Akx k=0 

µ0−k 

, (19.18) 

where R was “a function of x independent of a, a ′ , a ′′ , etc.,” 21 A0, A1, . . . were independent 

of x, and µ0 designated an integer. 22 

If expression (19.18) were to express a constant (independent of the indeterminates 

a, a ′ , a ′′ , . . . ), ABEL observed that since these quantities occurred in A0, A1, . . . , the sec- 

ond term corresponding to these had to vanish. And since he was only concerned 

with the coefficient of 1 x , his conclusion was that µ0 < −1. He expressed this using a 

newly introduced notational advance in the following sentence: 

“This done, in designating by the symbol hR the highest exponent of x in the 

development of any function R of this quantity following decreasing powers, it is 

evident that µ0 will be equal to the largest integer contained in [less than or equal 

to] the numbers 

h f1 (x, y ′ ) 

χ ′ y ′ 

, h f1 (x, y ′′ ) 

χ ′ y ′′ 

� 

f1 

, . . . h 

x, y (n)� 

χ ′ y (n) 

. 

It is necessary that all these numbers must be less than the unit taken negatively.” 23 

21 “R étant une fonction de x indépendante de a, a ′ , a ′′ , etc.” (ibid., 161). 

22 The version printed in the Savants étrangers read at this point 

⎧ 

⎨ 

R log x = 

⎩ 

+Aµ0 

A0x µ0 + A1x µ0−1 + . . . 

+ Aµ0+1 

x 

+ Aµ0+2 

x 2 

+ . . . 

In the collected works (Sylow in N. H. Abel, 1881, II, 296), SYLOW commented: “C’est évidemment 

une faute d’écriture, ou d’Abel ou de Libri.” After the original manuscript has been recovered, it 

has become evident that the misprint is indeed due to LIBRI. 

23 “Cela posé, en désignant par le symbole hR le plus haut exposant de x dans le développement d’une 

fonction quelconque R de cette quantité, suivant les puissances descendantes, il est clair que µ0 sera 

égal au nombre entier le plus grand contenu dans les nombres: 

h f1 (x, y ′ ) 

χ ′ y ′ , h f1 (x, y ′′ ) 

χ ′ y ′′ � 

f1 x, y 

, . . . h 

(n)� 

χ ′ y (n) 

; 

il faut donc que tous ces nombres soient inférieurs à l’unité prise négativement.” (N. H. Abel, [1826] 

1841, 161).


Thus, for the individual terms of the sum, which were in general just algebraic func- 

tions of x, the ‘degree’ hR needed not be an integer, whence the conclusion 

h f1 (x, y k) 

χ ′ (y k) 

< −1. (19.19) 

The investigation now turned to algebraic manipulations of these new symbols. 

Determination of the most general form of the function f1 (x, y). ABEL put his new 

tool to immediate use. From the general formula 

h R1 

R2 

he derived from (19.19) the inequalities 

= hR1 − hR2, 

h f1 (x, y k) < hχ ′ (y k) − 1, 

which he claimed made “it easy to deduce the most general form of the function 

f1 (x, y) in each particular case.” 24 ABEL’S argument — but not its result — has been 

found unrigorous at this point, see e.g. SYLOW’S notes, the paper by ELLIOT, and be- 

low. 25 However, it is worth following the steps of his argument to see how he went 

about it. 

Because 

ABEL obtained 

χ ′ (yk) = ∏ (yk − ym) , 

m�=k 

hχ ′ (yk) = ∑ h (yk − ym) , 

m�=k 

and when the y1, . . . , y k were ordered according to decreasing degrees, 

hy k ≥ hym if k ≤ m, 

he found “in general, except for certain particular cases which he did not consider:” 26 

h (y k − ym) = hy min(k,m). 

The analogy with the ordinary degree operator makes the above-mentioned par- 

ticular cases easy to illustrate. For instance, if we have two monic polynomials of the 

same degree, the degree of their difference is strictly less than either of the original 

degrees, 

�� 

deg x 2 � � 

+ x − 1 − x 2 �� 

− x + 2 = deg (2x − 3) = 1. 

24 “De ces inégalités on déduira facilement dans chaque cas particulier la forme la plus générale de la 

fonction f1 (x, y).” (N. H. Abel, [1826] 1841, 162). 

25 (Sylow in N. H. Abel, 1881, II, 296–297) and (Elliot, 1876, 404–406). 

26 “Alors on aura, en général, excepté quelques cas particuliers que je me dispense de considérer:” 

(N. H. Abel, [1826] 1841, 162).


Later, in chapter 21, we shall have more to say on this notion of ‘in general, except for 

certain particular cases.’ 

Writing f1 (x, y), which was by construction an entire function of y, in the way 

ABEL concluded 

Based on the identity 

this evolved into 

f1 (x, y) = 

n−1 

∑ tm (x) y 

m=0 

m , 

htmy m k < hχ′ (y k) − 1 for k = 1, . . . , n and m = 0, . . . , n − 1. 

h (tmy m ) = ht k + mhy, 

htm + mhy < hχ ′ (y k) − 1. 

ABEL now combined the information contained in the equations () obtaining 

hχ ′ (y k) − mhy k − 1 = (n − m − k) hy k + 

k−1 

∑ hyu − 1. 

u=1 

Since y1, . . . , yn were assumed to be ordered according to decreasing degrees, the min- 

imal value among these (over k = 1, . . . , n) was obtained for k = n − m, resulting in 

the value 

� ′ 

min hχ (yk) − mhyk − 1 

k=1,...,n 

� = hχ ′ n−m−1 

(yn−m) − mhyn−m − 1 = 

Therefore, because htm was an integer, 

where 0 < εn−m−1 ≤ 1. 

htm = 

∑ 

u=1 

hyu − 1. 

n−m−1 

∑ hyu − 2 + εn−m−1, (19.20) 

u=1 

Grouping of roots according to their degree. The next step in ABEL’S analysis was 

to write 

from which he obtained 

hy1 = m1 

µ1 

with (m1, µ1) = 1 

hy1 = hy2 = · · · = hyµ 1 = m1 

by an argument involving tools from the theory of equations. In his investigations 

on algebraic solubility of equations, ABEL had proved (see e.g. chapter 8) that if an 

equation (here χ (y) = 0) was satisfied by an expression such as y = Ax 

µ1 

m1 µ 1 , an entire


sequence of distinct roots could be obtained by substituting for x 1 

µ 1 in y the result of 

x 1 

µ 1 multiplied with the different µ1’th roots of unity. Therefore, the roots y1, . . . , yn 

fell into sequences with equal degrees, 

τ’th sequence: hy kτ+1 = hy kτ+2 = · · · = hy kτ+nτµτ 

n = 

ε 

∑ nτµτ. 

τ=1 

mτ 

= , (mτ, µτ) = 1, 

µτ 

When focusing his attention on an root ym belonging to the τ’th sequence 

ABEL found from (19.20) 

m = kτ + 1 + β, with 0 ≤ β ≤ kτ+1 − kτ, 

htn−m = ht n−kτ−β−1 = 

= 

= 

= 

τ−1 

∑ 

α=1 

k α+1−kα 

∑ 

τ=1 

kτ+β 

∑ hyu − 2 + εkτ+β u=1 

β 

∑ hykτ+τ − 2 + εkτ+β τ=1 

hy kα+τ + 

τ−1 

∑ (kα+1 − kα) 

α=1 

mα 

+ β 

µα 

mτ 

µτ 

τ−1 

∑ nαmα + β 

α=1 

mτ 

µτ 

− 2 + ε kτ+β 

+ ε kτ+β − 2 (19.21) 

Number-theoretic arguments to determine the form of f1 (x, y). The product ε kτ+β · 

µτ, for which ABEL introduced a special symbol 27 A τ,β, was found to be the least 

positive number ζ for which 

µτ | βmτ + ζ. (19.22) 

In order for us to see that A τ,β has the prescribed property, it suffices to first observe 

from (19.21) that 

β mτ 

µτ 

+ ε kτ+β = βmτ + A τ,β 

µτ 

is an integer because tm is an entire function. Next, if we assume 

we obtain 

βmτ + ζ 

µτ 

= K < K ′ = βmτ + Aτ,β , 

µτ 

� K ′ − K � µτ = A τ,β − ζ = µτε kτ+β − ζ 

27 Actually ABEL would write A (γ) 

β for this quantity, but I have chosen to move indices into subscripts.


and thus 

ε kτ+β = � K ′ − K � + ζ 

which is a contradiction. Thus K = K ′ and A τ,β is the smallest number such that 

(19.22) holds. This condition of minimality, which ABEL just noticed as a matter of 

µτ 

> 1 

fact, was soon (see below) invoked and found to be of great use. 

found 

When ABEL spelled out the results obtained above for the first sequence τ = 1, he 

ht n−β−1 = −2 + β m1 

µ1 

+ A 1,β 

µ1 

= −2 + βm1 + A1,β . 

µ1 

For ‘small’ β (starting with β = 0), the right hand side is obviously negative, because 

A 1,β < µ1 by the definition of ε kτ+β 

A 1,β 

µ1 

= ε β ≤ 1 

and the other term vanishes for β = 0. Consequently, some β ′ ≥ 0 existed such that 

which obviously meant that 

ht n−β−1 < 0 for β = 0, . . . β ′ 

(19.23) 

t n−β−1 = 0 for β = 0, . . . , β ′ . (19.24) 

The general form of the function f1 (x, y) in the light of these results thus became 

f1 (x, y) = 

n−β ′ −1 

∑ tk (x) y 

k=0 

k 

in which β ′ was the largest integer less than µ 1 

m + 1, 

1 

� � 

µ1 

β ′ = 

m1 

+ 1 

. 

(19.25) 

ABEL’S way of obtaining this ultimate description of β ′ was found by SYLOW to miss 

certain particular cases. 28 

Based on the expression (19.25) which ABEL had obtained for the function f1 (x, y), 

he concluded: 

“A function like f1 (x, y) always exists when β does not surpass n − 1.” 29 

It is difficult to see exactly what ABEL meant by this phrase, which seems to infer 

that the existence of the function was deduced from the representation (19.25). How- 

ever, on a logical basis, the existence of the function f1 (x, y) had been presupposed in 

the decomposition of f (x, y) χ ′ (y), see (19.16). 


29 “Une fonction telle que f1 (x, y) existe donc toujours à moins que β ′ ne surpasse n − 1.” (N. H. Abel, 

[1826] 1841, 166).


Investigation of the complementary case β ′ ≥ n. In order to study what happened 

if β ′ ≥ n, ABEL wrote 

µ1 

m1 

+ 1 = n + ε, ε ≥ 0 

and obtained for the inverse fraction only two possibilities 

m1 

µ1 

= 1 m1 

or 

n − 1 µ1 

= 1 

n . 

In both cases, ABEL claimed, the integral � f (x, y) dx would be expressible in alge- 

braic and logarithmic terms. His argument proceeded from claiming that the equation 

χ (y) = 0 was linear in x, 

χ (y) = P (y) + xQ (y) . 

If this was the case, the integrand in � f (x, y) dx was quickly seen to be a rational 

function, and the result was thus well known. However, as SYLOW has observed, 30 

in the case left unnoticed above, the conclusion of linearity does not hold, and the 

deduction thus suffers from this incompleteness. 

Back on track: the case β ′ ≤ n − 1. Returning to the more complicated case, ABEL 

noticed: “Thus, except for this case [β ′ ≥ n], the function f1 (x, y) always exists” 31 and 

he went on to elaborate the consequences of the hypothesis β ′ ≤ n − 1. He began by 

reducing the study of the equation 

to the study of the individual terms 

� n−β 

∑ 

∑ 

′ −1 

m=0 tmym χ ′ dx = C (19.26) 

(y) 

� 

xkym dx 

∑ χ ′ (y) . 

His next and decisive step was to begin considering the htm + 1 coefficients in the 

polynomial tm. He found that the function f1 (x, y) contained 

n−β ′ −1 

∑ (htm + 1) = 

m=0 

n−β ′ −1 

∑ htm + n − β 

m=0 

′ = 

n−2 

∑ htm + n − 1 (19.27) 

m=0 

coefficients and chose to designate this number of coefficients by γ. Once this number 

had been introduced, it became ABEL’S first objective to derive other general formu- 

lae for it and to study certain particular cases. Once these investigations had been 

concluded, ABEL again returned to (19.26), remarking that it was even valid in certain 

cases not included in the deduction: 


31 “Excepté ce cas donc, la fonction f1 (x, y) existe toujours” (N. H. Abel, [1826] 1841, 167).


“The formula (59) [here (19.26)] is generally valid for all values of the quanti- 

ties a, a ′ , a ′′ , . . . whenever the function r does not have a factor of the form F0x; in 

that case, it is also valid if F0x and χ′ y 

f1(x,y) 

vanish for the same value of x.”32 

Algebraic manipulations pertaining to the number γ. ABEL’S tedious manipula- 

tions of the expression for γ made critical use of a result in the line of GAUSS’ theory 

of moduli and primitive roots — referred to by ABEL’S as “the theory of numbers” 33 — 

as presented in GAUSS’ Disquisitiones arithmeticae. 34 ABEL found that because for the 

τ’th sequence mτ and µτ were relatively prime, and A τ,β ≡ −βmτ (mod µτ), 

nτµτ−1 

∑ 

β=0 

A τ,β = nτ 

� � 

µτ−1 

∑ k = nτ 

k=0 

µτ (µτ − 1) 

. 

2 

By combining this with other previously established formulae, ABEL found 

γ = 1 + 

� 

ε τ−1 

∑ nτµτ 

τ=1 

∑ nvmv + 

v=1 

mτnτ − 1 

2 

� 

− 

ε 

∑ 

τ=1 

nτ (mτ + 1) 

. (19.28) 

2 

At this point, two particular cases were noticed mainly as examples of how to calculate 

with the formula (see below). 

ABEL’S examples of calculating γ. After deducing the formula (19.28) by algebraic 

and number theoretic manipulations, ABEL gave some examples of how it could be 

used in particular cases to determine γ. 

1. If all the roots y1, . . . , yn have the same degree (ε = 1) 

the expression for γ reduced to 

hy1 = · · · = hyn = m1 

, 

m1n1 − 1 

γ = 1 + n1µ1 

2 

µ1 

− n1 (m1 + 1) 

. 

2 

If, furthermore, µ1 = n = n1, corresponding to the situation in which all the 

roots y1, . . . , yn involve n’th roots of x, it reduced further into 

γ = (n − 1) m1 − 1 

. 

2 

32 “La formule (59) a généralement lieu pour des valeurs quelconques des quantités a, a ′ , a ′′ , . . . toutes 

les fois que la fonction r n’a pas un facteur de la forme F0x; mais dans ce cas elle a encore lieu, sinon 

χ ′ y 

f1(x,y) 33 

F0x et s’évanouissement pour une même valeur de x.” (ibid., 169). 

(ibid., 168) 

34 (C. F. Gauss, 1801). See also the discussion in section 5.3.


2. If all the roots y1, . . . , yn have integer degrees, 

and 

the formula became (ε = n) 

γ = 1 + 

= 1 + 

� 

n τ−1 

∑ 

τ=1 

n 

∑ 

hy1, . . . , hyn ∈ Z, 

n1 = · · · = nε = 1, 

∑ mν + 

ν=1 

mτ − 1 

2 

τ−1 

∑ 

τ=1 ν=1 

� 

mν − n = 1 − n + 

− 

n 

mτ + 1 

∑ 2 τ=1 

n 

∑ (n − τ) mτ. 

τ=1 

It is interesting to consider the usefulness of these examples. It appears that the 

examples were both chosen because the assumptions made therein corresponded to 

particularly interesting cases and because they illustrate cases, in which the rather 

complicated formula (19.28) — which looked even more complicated in ABEL’S nota- 

tion than in my modern one — reduced to extremely simple forms. The first class of 

equations considered (in which all degrees were equal) contains equations such as 

χ (x, y) = y n − p (x) = 0 

in which p is a polynomial. The second assumption (all roots have integer degrees) 

applies to equations of the form 

χ (x, y) = ∏ k 

(y − p k (x)) = 0. 

The indeterminates a, a ′ , a ′′ , . . . . ABEL chose to designate by α the number of inde- 

terminates a, a ′ , a ′′ , . . . and ventured to investigate the relationships between the roots 

x1, . . . , xµ and the indeterminates a1, . . . , aα. To the α indeterminates corresponded α 

equations 

θ (yτ) = 0 for τ = 1, . . . , α 

which were linear in the indeterminates (see box 19.2.3, above). These equations “in 

general” served to express the indeterminates rationally in x1, . . . , xα and y1, . . . , yα. 

Only in cases of multiple roots would the equations not suffice. In such cases ABEL 

involved the calculus which could be used to produce a set of α independent equations 

for determining a1, . . . , aα. When ABEL divided F (x) by ∏ α τ=1 (x − xτ), he obtained 

another equation 

F1 (x) = 

∏ α τ=1 

F (x) 

= 0 

(x − xτ)


which was of degree µ − α, has as its roots xα+1, . . . , xµ and whose coefficients were 

rational functions of x1, . . . , xα and y1, . . . , yα. Thus, ABEL concluded from Main Theo- 

rem I that any sum such as ∑ α k=1 ψ k (x k) could be expressed by a known function v and 

a similar sum of functions, 

α 

∑ ψk (xk) = v − 

k=1 

µ 

∑ 

k=α+1 

ψk (xk) . (19.29) 

The relation γ = µ − α. The expression (19.29) at first might seem like a mere rep- 

etition of the Main Theorem I, but as ABEL stressed, the number of terms on the right 

hand side (µ − α) shows remarkable features. The stress put on the number γ is cer- 

tainly one of the important aspects of ABEL’S paper, and it has received widespread 

mathematical interest not least after G. F. B. RIEMANN (1826–1866) transformed it 

into a coherent concept of genus, see section 19.3. For now, we focus our attention 

completely on ABEL’S argument and the inner logic of the paper. 

As he had done above, ABEL again divided his argument by whether r has a factor 

independent of the indeterminates or not. He started with the latter case, F0 (x) = 1 

for which he found that all the coefficient functions q0, . . . , qn−1 were arbitrary and 

their hq k + 1 coefficients had to correspond to the indeterminates, 

α = 

n−1 

n 

∑ (hqk + 1) = ∑ hqk + n − 1. 

k=1 

k=1 

Obtaining a corresponding formula for the other case, in which F0 (x) �= 1, proved 

much more tedious. In general, ABEL claimed, the equation 

r (x) = F0 (x) F (x) (19.30) 

would impose hF0 conditions, but particular forms for y can eliminate some of these 

conditions. 35 If the number of conditions imposed by (19.30) is hF0 − A, the number 

of indeterminates could be counted as 

α = 

n−1 

∑ (hqk + 1) − (hF0 − A) . (19.31) 

k=1 

On the other hand, ABEL easily obtained from the definitions 

and 

hr = h 

hr = hF0 + hF = hF0 + µ 

� � 

n 

n 

∏ θ (yk) = ∑ hθ (yk) , 

k=1 

k=1 

35 Here, a remarkably clear juxtaposition of “in general” and “particular cases” was given.


which allowed him to rewrite (19.31) as 

α = 

= 

n−1 

∑ hqk + n − 1 − (hr − µ) + A 

k=1 

n−1 

∑ hqk − 

k=1 

n 

∑ hθ (yk) + n − 1 + A + µ. (19.32) 

k=1 

Therefore, ABEL turned to algebraically manipulating the “degrees” hθ (y k) along the 

same lines as had been followed in describing γ above. 

Obviously, from the inequality 

and the formula 

ABEL obtained 

hθ (y) ≥ h (qmy m ) for each m = 0, . . . , n − 1, 

h (qmy m ) = hqm + mhy, 

hθ (y k) ≥ hqm + mhy k for k = 1, . . . , n. 

Designating by ρτ the index of the maximal value of h (qmy m ) within the τ’th sequence 

of roots, ABEL employed the same machinery which had served him before, although 

this time in a slightly different notational dressing. Summing the excesses ε τ,k within 

the τ’th sequence, 

nτµτ−1 

∑ ετ,k = Cτ, 

k=1 

ABEL found by the manipulations and number theoretic results 

or less specifically (Cτ ≥ 0) 

µ − α ≥ γ − A + 

µ − α ≥ γ − A. 

ε 

∑ Cτ, 

τ=1 

However, the inequality in (19.33) was actually an equality, 

µ − α = γ − A, (19.33) 

as ABEL deduced by another tedious sequence of manipulations. 

Specializing the relation expressed (19.33) to the other case (in which F0 (x) = 1) 

led to the result that if r did not contain any factors independent of the indeterminate 

quantities, then 

µ − α = γ. 

ABEL concluded these investigations by considering situations in which the coef- 

ficients q0, . . . , qn−1 were subjected to some kinds of conditions. Again arguing very 

“generally”, ABEL could claim that the result (19.33) would not generally be substan- 

tially altered, although the constant A should reflect the additional conditions. 

In the paper’s eighth section, ABEL applied the results obtained thus far to calcu- 

late γ in an example for which n = 13.

19.3. Additional, tentative remarks on ABEL’s tools 371 

µ − α independent of α. In the previous argument, ABEL had reached an expression 

such as 

hqm = hqρ 1 + Mm for m = 0, . . . , n − 1 

in which Mm is independent of hqρ 1 . Thus, counting the number of coefficients in 

q0, . . . , qn−1 yields as an upper limit for α 

ABEL could therefore write 

α ≤ nhqρ 1 + 

n−1 

∑ Mm. 

m=0 

α = nhqρ 1 + M, 

in which M was independent of hqρ 1 . Therefore, taken together with (19.32), µ − α 

was found to be independent of the value of α. 

II: 

ABEL summarized these results in an announcement of the central Main Theorem 

“The equation (74) [here (19.29)] enables us to express a sum of any number of 

given function of the form ψx by a sum of a particular number of functions. The 

last number can always be supposed equal to µ which — in general — will be its 

smallest value.” 36 

In summary, the Main Theorem II can be expressed as follows: 

Theorem 17 (Main Theorem II) Under the present assumptions, 

τ � xk 

∑ 

k=1 

f (x k, y k) dx k = 

γ � zk 

∑ 

k=1 

f (x k, y k) dx k + v 

in which γ is independent of τ and x1, . . . , xτ, z1, . . . , zγ are given algebraically in x1, . . . , xτ 

and v is an algebraic and logarithmic function. ✷ 

As his final act before turning toward the applications of this result (see discussion 

in section 19.5.2, below), ABEL generalized the theorem to apply to linear combina- 

tions of integrals with rational coefficients. 

19.3 Additional, tentative remarks on ABEL’s tools 

As mentioned in section 19.1.2 and described at length above, ABEL’S Paris memoir 

is full of interesting applications of algebraic tools. In the present section, some brief 

perspectives on five of the most important tools are offered. 

36 “L’équation (74) nous met donc en état d’exprimer une somme d’un nombre quelconque de fonctions 

données, de la forme ψx, par une somme d’un nombre déterminé de fonctions. Le dernier 

nombre peut toujours être supposé égal à γ , qui, en général, sera sa plus petite valeur.” (N. H. 

Abel, [1826] 1841, 185).


The theory of residues and the expansion in decreasing power series. One of ABEL’S 

in the expansion of the function 

tools was to focus attention on the coefficient of 1 x 

f (x) according to decreasing powers of x (see page 355). Judged by ABEL’S way of 

introducing the operation which he denoted Π f x, the expansion of the function f 

into series of decreasing powers was not considered problematic; my best guess is 

that it was considered a formal operation or a well established fact. However, simul- 

taneously, CAUCHY was attributing new importance to the same object although in a 

completely different theoretical environment when he laid the foundations for his new 

calculus of residues in a series of papers in the Exercises de mathématiques. 37 From the 

quote on page 306, we know that ABEL bought and studied these installments. Thus, 

this mathematical object acquired importance in two distinct theories possibly from 

two very distinct approaches. However, when we include another of ABEL’S tools, we 

may get the impression that ABEL’S Π might stand closer to CAUCHY’S residues. 

Lagrange interpolation and ABEL’S Annales-paper. ABEL’S made repeated use of a 

result which derives from the process of Lagrange interpolation. In its original form, 

Lagrange interpolation concerned the problem of fitting a polynomial of degree n − 1 

through n specified points in the plane {(xk, f (xk))} n 

k=1 . J. L. LAGRANGE (1736–1813) 

had attacked this problem in 1795 and presented the polynomial 

Pn (x) = 

n 

∑ f (xk) k=1 

as the solution; 38 the function ω (x) was defined by 

ω (x) = 

and its derivative consequently satisfied 

ω (x) 

(x − x k) ω ′ (x k) 

n 

∏ (x − xk) k=1 

ω ′ (xk) = ∏ (xk − xm) . 

m�=k 

This result can easily be proved and it immediately leads to the applications which 

ABEL made of it in the Paris memoir. Interestingly, CAUCHY pursued the same re- 

sult with his new theory of residues and explicitly referred to the process as Lagrange 

interpolation. 39 In a short paper which ABEL published in GERGONNE’S Annales de 

mathématiques, 40 he deduced the same result by elimination in two polynomial equa- 

tions. These remarks are meant to indicate that Lagrange interpolation was a powerful 

tool which was being investigated from numerous perspectives in the early nineteenth 

37 See e.g. (A.-L. Cauchy, 1826). 

38 (Kolmogorov and Yushkevich, 1992–1998, III, 262). 

39 (A.-L. Cauchy, 1826, 35–37). 


19.3. Additional, tentative remarks on ABEL’s tools 373 

century. I believe that the problem of elimination can be considered sufficient inspira- 

tion for ABEL to treat Lagrange interpolation but the very central role which it together 

with the operator Π played in the Paris memoir (see above) may also suggest that ABEL 

had actually picked up his tools from CAUCHY’S new theory of residues although he 

did not adapt the full theory. 

The degree-operator. ABEL’S arguments about the number of independent integrals 

γ relied extensively on ways of determining the number of independent (or free) coef- 

ficients. In turn, their determination was based on an extended degree operator which 

could apply to the implicitly defined algebraic functions with which he dealt. ABEL 

introduced the fractional degree operator hR as the highest exponent in a develop- 

ment of R according to decreasing powers. As was the case with the residue-operator, 

no explicit considerations as to the validity of this definition are presented and it ap- 

pears to be a well known, formal trick. ABEL presented the basic rules for the de- 

gree operator and when he wanted to apply it to differences between to expressions 

among y1, . . . , yn, he insisted that these expressions be ordered according to their de- 

gree. However, as observed on page 362, particular cases could still arise in which the 

identity 

h (ym − y k) = max {hym, hy k} (19.34) 

did not hold. Such cases were apparently peculiar situations of little interest, and 

ABEL dismissed them by claiming that the equation (19.34) would hold “in general”. 

In his comments, 41 SYLOW made a considerable effort in clarifying ABEL’S arguments 

and in particular in revising his deduction of the properties of γ by making explicit 

some of the assumptions which ABEL had not made when he simply argued “in gen- 

eral”. The notion of equations being “generally valid” will be addressed further in 

chapter 21 where it was be interpreted and explained based on the notion of formula 

based mathematics. 

The genus. The number which ABEL denoted γ expressed the number of indepen- 

dent integrals related to a particular algebraic differential. As we have seen, ABEL’S 

deduction of the invariance of the number γ was cumbersome and hampered by cer- 

tain points where it was not completely clear and rigorous. Furthermore, although his 

arguments were highly explicit they did not immediately produce a way of generally 

computing the number γ. In the subsequent decades, it became an extremely promi- 

nent mathematical problem to rigorously establish the basis for ABEL’S theorems and 

to investigate the number γ further. Eventually, RIEMANN presented an approach 

based on multi-sheeted surfaces and introduced the name genus and the symbol p for 

ABEL’S γ. 42 The further description of RIEMANN’S theory is, unfortunately, way be- 

41 (N. H. Abel, 1881, II). 

42 (B. Riemann, 1857).


Figure 19.1: GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866) 

yond the present scope. 43 However, in the present context, it serves to demonstrate 

that ABEL’S ideas were pursued and rigorized over the ensuing decades. 

Birth of a new concept: algebraic functions. The final aspect which will be tenta- 

tively discussed here concerns the introduction of a new concept of implicitly defined 

algebraic functions. In his research on the solubility of equations, ABEL had — for obvi- 

ous reasons — only been interested in investigating explicit algebraic functions which 

could serve as solutions for equations. However, in the Paris memoir, implicit algebraic 

functions were the primary objects of study and ABEL developed a number of tools 

for their investigation. Later, implicitly given algebraic functions (or just algebraic 

functions) became very important objects of mathematics and — as BRILL remarks — 

ABEL may be seen as the initiator of the theory of algebraic functions; in particular, he 

proved a very important theorem in the theory. 44 Later, J. LIOUVILLE (1809–1882) also 

adopted — following ABEL but departing from his French precursors — the implicitly 

given algebraic functions into his theory of integration in finite terms. 45 In summary, 

we are faced with an introduction of a new set of objects, and ABEL’S highly formula 

based investigations of these objects and developments of tools for their study can be 

interpreted as a way of getting to know and express results about this new branch. 

43 One rather recent, very brief sketch is given in (Houzel, 1986, 310–313). 

44 (Brill and Noether, 1894, 212). 

45 (Lützen, 1990, 370).

19.4. The fate of the Paris memoir 375 

19.4 The fate of the Paris memoir 

ABEL’S Paris memoir had a strange and adventurous fate which had certain influences 

on ABEL’S career. 46 After ABEL had delivered his memoir to the Académie des Sciences, 

the Academy commissioned A.-M. LEGENDRE (1752–1833) and CAUCHY to present a 

report on it. The manuscript soon landed on CAUCHY’S desk, where it withered until 

1829. After having learned of ABEL’S untimely death from a communication on 22 

June 1829 by LEGENDRE, 47 CAUCHY finally took the time to write the report which 

was dated 29 June 1829. 48 CAUCHY explicitly noticed in his report how ABEL treated 

implicitly defined algebraic functions and that this was a necessary requirement for 

his theorems to be true. 49 This is further indication that the introduction of implicitly 

given algebraic functions was not completely obvious and standard. 

Manuscript lost and found. Based on CAUCHY’S positive report, the Académie des 

Sciences decided to include the ABEL’S Paris memoir in the Mémoires présentés par divers 

savants once a new copy had been prepared. Furthermore, ABEL was — posthumously 

and jointly with C. G. J. JACOBI (1804–1851) — awarded the Grand prix of the Académie 

des Sciences in 1830. However, the manuscript was misplaced — possibly as a result 

of the turbulent events in Paris in 1830 — as a Norwegian enquiry would realize in 

1832 when B. M. HOLMBOE (1795–1850) was beginning to prepare the first edition of 

ABEL’S Œuvres. Consequently, the Œuvres appeared in 1839 without the Paris memoir. 

Apparently, this provoked some reaction from the [Académie des Sciences]Académie 

which commissioned LIBRI with the job of seeing it through print and, as noted, it was 

published in 1841. 

While preparing the second edition of ABEL’S collected works, M. S. LIE (1842– 

1899) in 1874 obtained permission to consult the original manuscript supposedly held 

in the archives of the Académie des Sciences. 50 However, the manuscript was again 

nowhere to be found in the archives and again had to be considered lost. Conse- 

quently, the Paris memoir was included in the second edition of ABEL’S collected works 

but with the version printed by the French Academy in 1841 as the source. In the twen- 

tieth century, a copy of the manuscript was first located in Rome by P. HEEGAARD 

(1871–1948) in 1942 before V. BRUN (1885–1978) 51 succeeded in finding the majority 

of ABEL’S original manuscript in Florence ten years later. Recently, in 2000, the fi- 

nal eight missing pages have been found and the whereabouts of the entire original 

manuscript of ABEL’S Paris memoir are known for the first time in 150 years. 

46 The fate of the Paris memoir is described in most biographies of ABEL. Additionally, information 

for the present sketch is drawn from papers including (Brun, 1949; Brun, 1953; Lange-Nielsen, 1927; 

Lange-Nielsen, 1929). 

47 (Lange-Nielsen, 1929, 14). 

48 (Lange-Nielsen, 1927, 67). 

49 CAUCHY’S report is reproduced in (ibid., 69). 

50 (N. H. Abel, 1881, II, 294) 

51 Information from (Scriba, 1980).


19.5 Reception of the Paris memoir 

Some aspects of the reception of ABEL’S Paris memoir have already been touched upon 

above when it was noted how ABEL introduced a new class of objects into mathemat- 

ical study. In the present section, some attention is paid to the primary application of 

the Paris memoir which dealt with the so-called hyperelliptic integrals . 

19.5.1 ABEL’s announcements of the Paris memoir 

In response to the lacking reaction from the Parisian Académie des Sciences, ABEL went 

ahead and made public some of his results originally contained in the Paris memoir. 

He did so in two papers published in A. L. CRELLE’S (1780–1855) Journal in 1828 and 

1829. 52 The first of these contained the application of the Paris memoir to hyperelliptic 

integrals and will be treated in some detail below. The second one entitled “Démon- 

stration d’une propriété générale d’une certaine classe de fonctions transcendantes” had been 

signed by ABEL January 6, 1829 and was published in the second issue of the fourth 

volume of the Journal which appeared just days before ABEL’S death. In that paper, 

written in haste by an already ill-taken mathematician, ABEL presented the first main 

theorem of the Paris memoir and gave a short and direct proof based on the theorem of 

LAGRANGE (theorem 1) which had already served him well in the theory of equations. 

At the end of the two page proof, ABEL promised to present multiple applications of 

Main Theorem I on a later occasion. 

19.5.2 Application to hyperelliptic integrals 

The application to hyperelliptic integrals has been seen by some historians of mathe- 

matics as the true motivation for ABEL’S deduction of the Abelian Theorem. 53 Certainly, 

the so-called hyperelliptic integrals figured prominently among ABEL’S examples in 

the Paris memoir when he applied the results to integrals of the form 

� � � � 

f x, φn (x) dx 

in which f was a rational function and Pn was a polynomial of degree n. These in- 

tegrals are separated from the greater class considered in the Paris memoir by corre- 

sponding to an equation of the form χ (x, y) = y 2 − φn (x). 

For ordinary elliptic integrals (n = 3 or n = 4), ABEL’S calculations in the Paris 

memoir showed that γ = 1 and thus, any sum of similar elliptic integrals could be 

reduced to a single elliptic integral of the same form and possibly some � algebraic � and 

n−1 

logarithmic terms. For hyperelliptic integrals, ABEL found that γ = 2 where ⌊x⌋ 

denotes the integer value of x. Thus, for instance, if n = 6, any number of similar 

52 (N. H. Abel, 1828c; N. H. Abel, 1829b). 

53 See (Brill and Noether, 1894, 210–211) and the discussion in (Cooke, 1989).


hyperelliptic integrals could be reduced to two such integrals and simpler terms. This 

was also one of the main results of the paper Remarques sur quelques propriétés générales 

d’une certaine sorte de fonctions transcendantes which ABEL published in CRELLE’S Journal 

in 1828. 54 

ABEL’S result was picked up by JACOBI who praised it highly and suggested that 

it could be used as a form of generalized addition theorem for hyperelliptic functions . 

Thus, for hyperelliptic integrals, say, with X of degree 5, 

� x 

0 

dx 

√ X = Φ1 (x) and 

� x 

0 

x dx 

√X = Φ1 (x) , 

the problem became to express the upper limits of the involved integrals as 

such that 

x = λ (u, v) and y = λ1 (u, v) 

Φ (x) + Φ (y) = u and Φ1 (x) + Φ1 (y) = v. 55 

This idea — which is a generalization of the addition theorems for trigonometric and 

elliptic functions — is called the (Jacobian) inversion problem and it attracted a great 

deal of attention in the 1830s and 1840s. Eventually, around 1850, it was solved inde- 

pendently by A. GÖPEL (1812–1847) and J. G. ROSENHAIN (1816–1887) who applied 

a generalization of JACOBI’S theory of theta functions for elliptic functions (see next 

chapter) to this case. 56 In the process, they were forced to accept that the functions 

were depended on two complex variables and had four periods. Thus, the search 

to generalize the theory of basic pattern of the theory of elliptic integrals even further 

forced mathematicians to face even more peculiar functions. The continuing extension 

of the theory of higher transcendentals posed new demands to the rigor and tools of 

the mathematician and thereby influenced the general development of mathematics 

in important ways. 


In the present chapter, ABEL’S Paris memoir has been described in some details to facil- 

itate a discussion of the tools which ABEL employed. Combining the characterization 

of the tools employed here with those described in the previous chapter, we are led to 

see that algebraic methods were extremely important in ABEL’S research on transcen- 

dentals. These methods range from results which he had originally used to study the 

solubility of equations to newly developed tools which, nevertheless, also often were 

based in polynomials, equations, considerations of rational dependence, and the like. 


55 (C. G. J. Jacobi, 1832b, 400); see also (Houzel, 1986, 311). 

56 (Göpel, 1847; Rosenhain, 1851).


As described, ABEL’S deductions in the Paris memoir were extremely cumbersome 

and not always universally permitted. Generally, ABEL’S arguments in the memoir fell 

well inside the formula based paradigm: they were concerned with formulae, were car- 

ried by manipulations of formulae, and arguments which were not universally true 

but only true “in general” were acceptable. In chapter 21, these features will be shown 

to be intimately connected with an “old” paradigm which was gradually being re- 

placed in the period. Thus, and because it was not available for 15 years, it is not 

surprising to find that ABEL’S proof in the Paris memoir was not so widely accepted 

and imitated as the statements or results to which they led.

Chapter 20 

General approaches to elliptic 

functions 

The present chapter serves to present the ideas which matured the theory of elliptic 

functions in the nineteenth century. In particular, attention is called to the variations 

in the ways of introducing elliptic functions because these variations are indicators of 

the changing conceptions of rigorous foundations. Moreover, it is interesting to follow, 

how results are turned into definitions to meet the changing standards of rigorous 

definitions. 

20.1 ABEL’s version of a general theory of elliptic 

functions 

In his ultimate publication, the Précis d’une théorie des fonctions elliptiques, 1 N. H. ABEL 

(1802–1829) addressed the theory of elliptic functions from a more general perspective 

than the Recherches. Since the Recherches which dealt exclusively with elliptic func- 

tions of the first kind, ABEL had published a number of smaller papers on the theory 

of elliptic functions, some investigations on integration in finite form, and various 

announcements of his Paris memoir. ABEL had also been working on a continuation 

of the Recherches which he postponed to devote his energy to completing the Précis. 

The second Recherches mémoire was eventually published by M. G. MITTAG-LEFFLER 

(1846–1927) in 1902. 2 However, to focus on the most general of ABEL’S approaches, we 

shall limit the discussion to the Précis. In the Précis, these threads were woven together 

to present an exposition of the entire theory of elliptic functions which simultaneously 

addressed all three kinds. 



379

380 Chapter 20. General approaches to elliptic functions 

Reiterating established knowledge. ABEL began his Précis — which he privately 

called the “knockout of C. G. J. JACOBI’ (1804–1851)’ — by iterating some of his pre- 

vious results pertaining to elliptic functions of the first kind. With the radical 

� 

∆ (x, c) = (1 − x2 ) (1 − c2x2 ), 

he introduced his three kinds of elliptic integrals as 

� 

dx 

˜ω (x, c) = 

∆ (x) , 

� x 2 dx 

˜ω0 (x, c) = , and 

∆ (x) 

� 

dx 

Π (x, c, a) = � � . 

∆ (x, c) 

1 − x2 

a 2 

One of the major tricks of the Précis was that once the elliptic integral of the first kind 

had been inverted 

θ = 

� λ(θ) 

0 

dx 

∆ (x, c) , 

this elliptic function could be used to describe the elliptic integrals of the second and 

third kind, 

� 

˜ω0 (x, c) = 

λ 2 � 

(θ) dθ and Π (x, c, a) = 

dθ 

1 − λ2 θ 

a 2 

Thus, ABEL had essentially reduced the problem of inverting all three kinds of inte- 

grals and he could combine the study of all elliptic functions in knowledge about the 

elliptic functions of the first kind. 

Obviously, among the key results which ABEL iterated for the elliptic function λ 

was its two periods and the complete solution of the equation λ (θ ′ ) = λ (θ) which 

we have already encountered multiple times. Moreover, ABEL also presented various 

infinite representations of λ and investigated the conditions of transformations. Thus, 

all the key components of his previous approaches were included in the Précis, albeit 

in a more coherent and lucid form. 

General properties of elliptic functions. The major new purpose of ABEL’S Précis 

was to investigate a new program of representation for elliptic functions. In the pro- 

cess, ABEL made important use of the insights which he had developed and presented 

in relation with his Paris memoir. 

With two polynomial functions f (even) and φ (odd), ABEL defined 

ψ (x) = f (x) 2 − φ (x) 2 ∆ (x) 2 

which was an even function and therefore could be split in factors as 

ψ (x) = A 

µ 

∏ 

n=1 

� 

x 2 − x 2 � 

n . 

. 

(20.1)

20.2. Other ways of introducing elliptic functions in the nineteenth century 381 

In this situation, ABEL found by employing Lagrange interpolation that the sum of inte- 

grals of the third kind reduced to a logarithmic expression 

µ 

∑ Π (xn, a) = C − 

n=1 

a 

2∆ (a) 

f (a) + φ (a) ∆ (a) 

log . (20.2) 

f (a) − φ (a) ∆ (a) 

ABEL extended this property of elliptic integrals of the third kind to analogous re- 

sults for elliptic integrals of the first and second kind. For integrals of the second kind, 

ABEL observed that ˜ω (x) = lima→∞ Π (x, a) whereas the logarithmic term vanished 

under this limit process, 

µ 

∑ 

n=1 

˜ω (xn) = C. 

For integrals of the second kind, ABEL considered the expansion of (20.2) according to 

increasing powers of 1 a 

and compared coefficients of 1 

a 2 to conclude 

µ 

∑ 

n=1 

˜ω0 (xn) = C − p 

where p was an algebraic function of x1, . . . , xµ. 

Thus, ABEL used tools similar to those which he employed in the Paris memoir to 

deduce results which also bear similarities with the Main Theorem I (theorem 16). 

A new program of representability. In the second chapter, ABEL suggested a very 

general question: He wanted to describe all integrals of algebraic differentials which 

could be expressed by algebraic, logarithmic, and elliptic functions. Thus, if com- 

pared with the Paris memoir, the elliptic functions were now accepted among the basic 

functions to which other higher transcendentals could be reduced. However, when he 

came to answer the question, he restricted himself to attack transformation problems 

and other relations among elliptic integrals. The more specific contents of his inves- 

tigations are considered outside the present scope, although its presentation would 

probably reiterate the description of ABEL’S methods and tools which have been sug- 

gested in the previous chapters. 

ABEL did not manage to complete his papers before he died. Nevertheless, his 

approach illustrated the fruitful influence which the results of Paris memoir could have 

on the theory of elliptic functions. However, because of his early death, it was left 

to ABEL’S contemporaries and competitors to outline the future development of the 

theory of elliptic functions. 

20.2 Other ways of introducing elliptic functions in the 

nineteenth century 

During the nineteenth century, the definitions of elliptic functions were turned upside 

down a number of times. In chapter 16, it has been described, how ABEL introduced

382 Chapter 20. General approaches to elliptic functions 

elliptic functions (of the first kind) by a formal inversion of the corresponding elliptic 

integral and an extension to the complex domain. However, his contemporaries and 

successors soon found other ways of introducing elliptic functions more preferably. In 

the present context, it suffices to consider three different approaches. 

JACOBI. With ABEL out of the competition, JACOBI’S influence on the theory of el- 

liptic functions toward the end of the 1820s was overwhelming. JACOBI’S notation 

and means of introducing the functions became standardized through a number of 

publications starting with his Fundamenta nova which was the first monograph de- 

voted to the study of the elliptic functions. 3 In the Fundamenta nova, JACOBI de- 

fined elliptic functions as inverses of elliptic integrals but beginning with a course 

taught in 1838, the changed the foundation to the so-called theta-functions. These 

were a particular set of four exponential series the simplest of which can be written 

as ϑ (x) = ∑n∈Z (−1) n qn2e2nix . 4 Based on these series, JACOBI could introduce and 

investigate all parts of the theory of elliptic functions. Thus, JACOBI introduced series 

as the basic objects upon which everything else should be built. Other definitions by 

series, e.g. as ratios of power series were also being suggested and adopted. 

J. LIOUVILLE (1809–1882). A completely different approach to the introduction of 

elliptic functions was taken by LIOUVILLE who chose to develop an entire theory for 

doubly periodic functions. Of such functions, he was able to deduce a number of re- 

sults and eventually prove that they could be used to represent the inverses of elliptic 

integrals. 5 Thus, LIOUVILLE circumvented the approach of ABEL and JACOBI and in- 

vestigated the concept of functions defined by what ABEL had deduced as a property. 

The strength of the approach was that eventually, the classes of elliptic functions and 

doubly periodic, meromorphic functions were found to coincide. 

K. T. W. WEIERSTRASS (1815–1897). The final approach which I wish to mention 

was through differential equations. For instance, WEIERSTRASS introduced his func- 

tion ℘ (u) as the solution to the differential equation 

� �2 dy 

= 4y 

du 

3 − g2y − g3 for g2, g3 constants 

which had a pole at u = 0. 6 Subsequently, WEIERSTRASS found means of obtaining 

more direct representations of his elliptic functions, for instance in the form 

σ (u + v) σ (u − v) 

℘ (v) − ℘ (u) = 

σ2u · σ2v where the function σ could be expanded in an infinite product. 7 

3 (C. G. J. Jacobi, 1829). 

4 (Houzel, 1986, 304–305). 

5 (Lützen, 1990, 555). 

6 (Houzel, 1986, 298). 

7 (ibid., 306).


All these four definitions ultimately define the same objects and it is a major sign 

of strength of the theory over the century that so many different approaches had been 

described. Mathematicians were free to chose which of the definitions satisfied their 

requirements for usability and rigor; subsequently, the other representations could be 

deduced. 


In the present part, ABEL’S approach to the theory of elliptic functions has been de- 

scribed from a number of perspectives. Based on a presentation of aspects of the the- 

ory of elliptic integrals in the eighteenth century, it has been illustrated, how ABEL 

introduced elliptic functions as formal inverses of elliptic integrals and extended the 

resulting function to the complex domain. The main inspiration which ABEL drew 

from C. F. GAUSS’ (1777–1855) suggestion that the division problem from the circle 

could also be attacked for the lemniscate has also been described. With the formal in- 

version as his definition, ABEL sought for ways of representing his elliptic functions, 

and some aspects of this program such as the standards of rigor and the use and re- 

quirements of representations have been addressed. Subsequently, special attention 

has been devoted to illustrating certain aspects of the tools which ABEL applied in the 

theory of transcendentals. In particular, it has been documented, how ABEL made re- 

peated and prolific use of algebraic methods resembling those which he had employed 

in his research on the solubility of equations. Finally, the changing roles of definitions 

and representations have been briefly debated in the present chapter. 

In conclusion, ABEL’S research on elliptic functions and higher transcendentals 

was his most directly influential legacy. Culminating in the later formulation of JA- 

COBI, ABEL’S Paris memoir suggested a problem which attracted mathematicians for 

decades and influenced the development of mathematics in the entire nineteenth cen- 

tury. In the next and final chapter, aspects of ABEL’S own research on elliptic functions 

will serve to illustrate how he sometimes worked within the formula based paradigm.

Part V 

ABEL’s mathematics and the rise of 

concepts 

385

Chapter 21 

ABEL’s mathematics and the rise of 

concepts 

In the preceding parts, I have presented and discussed the main parts of N. H. ABEL’S 

(1802–1829) mathematical production ranging from the theory of equations over the 

installation of rigor in the theory of series to the exploding field of elliptic and higher 

transcendentals. In addition, in each part, I have simultaneously addressed three 

broader themes: the rise of new questions with new kinds of answers, the change 

in the standards of doing mathematics, and a change in the objects and methods of 

mathematics. 

In this concluding part, I unify these themes by arguing that they are signs of a 

rise of concept based mathematics. In four steps, it will be argued that a large part of 

the development in mathematics in the early nineteenth century can appropriately 

be analyzed from the perspective of a change in the objects with which mathematics 

dealt. Firstly, the change is introduced by defining and discussing the “paradigms” 

of formula based mathematics and concept based mathematics. Secondly, these concepts 

are employed to present a brief analysis of how the ways of introducing objects into 

mathematics developed in the early nineteenth century. Thirdly, the changing role 

of counter examples in the two paradigms is discussed in some details. Finally, some 

reservations to the analysis are presented before a conclusion is drawn. Due to the lim- 

ited scope of ABEL’S mathematical production, my frame of interpretation can only be 

preliminary and I hope to develop it further through subsequent research by involving 

the works of other mathematicians. 

21.1 From formulae to concepts 

It seems fair to state that in the early nineteenth century, mathematics changed quite 

dramatically. 1 Any comparative and contextualized reading of the works of, say, L. 

1 Whether or not the early nineteenth century is an apt periodization in the history of mathematics 

has been discussed, though. See e.g. (Mikulinsky, 1982; Otte, 1982). 

387

388 Chapter 21. ABEL’s mathematics and the rise of concepts 

EULER (1707–1783) and K. T. W. WEIERSTRASS (1815–1897) or G. F. B. RIEMANN 

(1826–1866) will reveal that the problems, methods, and styles of analytical mathe- 

matics developed immensely and changed fundamentally during the century from 

the 1750s to the 1850s. In my view, a large part of the change in mathematics in the 

period can be understood by analyzing a fundamental change in the basic objects of 

the combined discipline of algebra and analysis. 

My description of changes in mathematics will — dictated by the scope of the pre- 

vious parts — mostly deal with the development of the algebraic and analytic disci- 

plines. In section 21.4, the applicability of the analyses outside these disciplines is 

briefly addressed. 

It will be argued that mathematics in the eighteenth century was tied to formu- 

lae and that mathematicians worked within a framework which was — in essential 

ways — adapted to these objects. In the early nineteenth century, so it is argued, these 

basic objects were gradually replaced by concepts and the change was so fundamental 

that it influenced all layers of mathematical knowledge and knowledge production. 

To allow for a more precise discussion, tentative definitions of the two styles (paradigms) 

of mathematics are given below. I have adopted the excessively broad Kuhnian term 

“paradigm” to include the entire mental horizon of the group of mathematicians who 

worked in the tradition. At the same time, I have introduced catch-word characteri- 

zations of the paradigms by terming them formula based and concept based . Below, the 

paradigms and their relations will be discussed further and certain relevant aspects of 

the preceding presentation of ABEL’S mathematics will be analyzed. 

21.1.1 The Eulerian paradigm of formula based mathematics 

By formula based mathematics, I mean to indicate a paradigm prevalent in the eighteenth 

century in which formulae were the carriers of mathematical knowledge. Formulae 

were both the results and the methods of mathematics, and mathematicians thought 

about and in terms of formulae. Mathematical results were derived through strings of 

explicit, formal manipulations of representations (formulae) and were stated in terms 

of new formulae. 

The essential notion of formula can be thought of as representations of mathemat- 

ical objects by symbols. However, such interpretations tend to be anachronistic and 

beside the point because — as I shall argue — the formulae were the basic objects of 

mathematics and only gradually became representations of other objects. 2 

In analysis, the primary occurrence of formulae was in the form of functions; the 

study of functions had been based on the study of their algebraic formulae. For these 

reasons, this paradigm could also have been named function based mathematics if it 

2 In the seventeenth and part of the eighteenth century, formulae had also been representations of 

e.g. curves (see section 15.1). However, as described, they became the primary objects in EULER’S 

new version of analysis.

21.1. From formulae to concepts 389 

✬ 

✫ 

All explicit 

algebraic 

expressions 

✩✬ 

= 

✪✫ 

All algebraic 

expressions of the 

form (6.4). 

✩ 

✪ 

Figure 21.1: The equality of the concepts of explicit algebraic expressions and ABEL’s 

normal form. 

was only to apply in analysis. However, in the algebraic discipline such a description 

would be misguided precisely because not functions but formulae were at the centre 

of mathematical reasoning (see e.g. section 5.2). If any other name should be used for 

formula based mathematics, the term Eulerian paradigm might be well suited. 

For the present purpose, formula based mathematics is best thought of in terms of 

e.g. EULER’S introduction and manipulation of various algebraic expressions in analy- 

sis. ABEL’S mathematics also frequently exhibits key characteristics of this paradigm, 

e.g. in his manipulations of formulae in the Recherches or the latter part of the bino- 

mial paper (see chapter 17 and 12, respectively). On both these occasions, ABEL based 

his deductions on sequences of step-wise manipulations of formulae to obtain results 

which were, themselves, formulae. 

21.1.2 A new paradigm of concept based mathematics 

The anti-thesis to formula based mathematics in the present context is termed concept 

based mathematics. In analogy with the formula based version, this paradigm empha- 

sized thought in and about concepts by which I mean classes of objects. The concept 

based mathematics deals primarily with defining, representing, and relating concepts. 

The collection of objects which fall under a concept is called the extension or domain of 

the concept. 

Typically, concept based mathematics could be concerned with e.g. continuous func- 

tions, differentiable functions, or algebraically solvable equations. The mathematical theo- 

rems dealing with concepts would then contain results relating these, e.g. by pointing 

out their differences or their overlaps or by relating one concept to another. In a truly 

concept based approach to mathematics, even representations become theorems relat- 

ing concepts; ABEL’S deduction of the normal form for (explicit) algebraic expressions 

stated that the two concepts were identical (see figure 21.1, section6.3, and below). 

For concept based mathematics to be efficient, specific knowledge of the individual 

objects within a concept has to fade in importance. Individual objects would serve


important roles as examples and counter examples but the definitions of the concepts 

must possess qualities which make them useful and central in the investigation of the 

concept. Such investigations, in turn, can benefit from the shift of focus onto concepts 

and produce results which were impossible (or very difficult) if only individual objects 

were considered. Thus, in order to analyze concept based mathematics, the role of 

definitions, representations, and arguments of relation and delineation of concepts 

become key points of enquiry. 

21.1.3 The shift from formula based to concept based mathematics 

The purpose of introducing the paradigms of formula based and concept based math- 

ematics is to characterize the development in the early nineteenth century as a tran- 

sition from the former to the latter. This transition manifests itself in various ways 

which interact with the changing basic objects of mathematics. The questions asked, 

the tools employed to answer these questions, and the types of answers which are 

possible and expected all change as consequences of this shift. 

In the first half of the nineteenth century, some mathematicians were aware that 

their style of mathematics differed essentially from the standards of their time. ABEL 

expressed how heavy loads of computations could hamper the progress of research, 3 

and E. GALOIS (1811–1832) described his own works as “analysis of analysis” which 

would reduce the hitherto dominating calculations to particular cases. 4 This aware- 

ness of the transition grew stronger during the century and towards the end of the 

nineteenth century, mathematicians became increasingly explicit about it. For in- 

stance, F. RUDIO (1856–1929) wrote: 

“The essential principle of the newer mathematical school, which is established 

by Gauss, Jacobi, and Dirichlet, is that whereas the older one sought to 

reach the goal by lengthy and complicated calculations (as even still in Gauss’ Disquisitiones) 

and deductions — it comprises an entire field by avoiding those and 

applying a genius method in a main idea and simultaneously presents the end 

result in its highest elegance by a single strike. While the former [the older approach] 

after a long sequence eventually reached a fertile ground by progressing 

from theorem to theorem, the latter [the new approach] immediately produces a 

formula in which the complete sphere of truths of an entire field is compactly contained 

and only ought to be extracted and expressed. In the old way, one could 

also — if need be — prove theorems; but only now can the true nature of the entire 

theory be seen, its internal gears and wheels.” 5 

3 (N. H. Abel, [1828] 1839, 217–218). 

4 (Galois, 1831c, 11). 

5 “Das wesentliche Princip der neueren mathematische Schule, die durch Gauss, Jacobi und Dirichlet 

begründet ist, ist im Gegensatz mit der älteren, dass während jene ältere durch langwierige und 

verwickelte Rechnung (wie selbst noch in Gauss’ Disquisitiones) und Deduktionen zum Zweck zu 

gelangen suchte, diese mit Vermeidung derselben durch Anwendung eines genialen Mittels in einer 

Hauptidee die Gesammtheit eines ganzen Gebietes umfasst und gleichsam durch einen einzigen 

Schlag das Endresultat in der höchsten Eleganz darstellt. Während jene, von Satz zu Satz fortschrei-

21.1. From formulae to concepts 391 

RUDIO clearly expressed the transition; the formula which he describes as the prod- 

uct of the new approach is not a formula in the present sense but rather a theorem. 

A number of similar statements can be found by other late-nineteenth century math- 

ematicians, e.g. C. F. KLEIN (1849–1925) who described how G. P. L. DIRICHLET 

(1805–1859) would avoid long computations in favor of acute logical analyses. 6 

This shift in the roles of formulae and concepts has been noticed and investigated 

from slightly different perspectives by historians of mathematics. In particular, it has 

been addressed by H. N. JAHNKE (⋆1948) and by D. LAUGWITZ (1932–2000), who 

pointed to the significant influence which RIEMANN had in bringing about the eventual 

change of paradigms. 7 

The basic objects of mathematics. The definitions of the paradigms suggest that the 

purported shift from formula based to concept based mathematics was a question of 

the size of the domains of mathematical results. Interpreted purely as a change in 

domains, the new approach could be seen as consisting of results which are simul- 

taneously true for a number of objects of the old paradigm (formulae). However, 

there is more to the transition that this; it concerns a real and fundamental change 

from formulae to concepts as the basic objects of mathematics. In the one extreme, a 

manipulation of a particular algebraic formula might produce another algebraic for- 

mula which would then be a mathematical result. At the other end of the spectrum, a 

number of results developed in the nineteenth century pointed out the differences be- 

tween important concepts such as continuous and differentiable functions or proved 

that particular classes of functions could be represented in particular ways. The ability 

to state and prove results for abstractly defined classes of objects is one of the main as- 

pects of the rise of concept based mathematics. Similarly, the issues of relating concepts 

and representing concepts are two of the central topics in a fully fledged concept-based 

version of mathematics. 

The techniques and questions of mathematics. Connected to the transition in the 

basic objects of mathematics, the techniques and questions of mathematics also un- 

derwent fundamental changes. The types of questions asked and the methods for 

answering them were not the same in the two paradigms. In the formula based 

paradigm, mathematical texts could be made up of long sequences of manipulations 

which transformed one formula into others or answered particular questions by de- 

tend, nach einer langen Reihe endlich zu einigem fruchtbaren Boden gelangt, stellt diese gleich von 

vorn herein eine Formel hin, in welcher der vollständige Kreis der Wahrheiten eines ganzen Gebietes 

konzentriert enthalten ist und nur herausgelesen und ausgesprochen zu werden darf. Auf 

die frühere Art konnte man die Sätze zwar auch zur Not beweisen, aber jetzt sieht man erst das 

wahre Wesen der ganzen Theorie, das eigentliche innere Getriebe und Räderwerk.” (F. Rudio, 1895, 

894–895). 

6 (Klein, 1967, 250). 

7 See e.g. (Jahnke, 1987) and (Laugwitz, 1999, 293–340). These are both very interesting works dealing 

with discussions similar to the present one but from slightly different perspectives.


veloping formulae which “solved” them. At times, concept based mathematics could 

apply the manipulation of “representations” provided that some representation result 

made it relevant. But more typically and interestingly, in concept based mathematics, 

statements about the extension of a concept grew to become the key results of math- 

ematics. A particularly illustrative example of these questions has been presented 

in part II where ABEL’S research on the quintic first showed that not all polynomial 

equations were solvable, i.e. the concepts of polynomial equation and algebraically solv- 

able equations were distinct, although related. Later in his research, ABEL’S proof of 

the algebraic solubility of Abelian equations was another almost prototypical concept 

based result, at least in its final formulation. This result showed that the concept of 

Abelian equations was contained in the concept of algebraically solvable equations. When 

he first encountered Abelian equations in connection with the division problem (see 

section 16.3), ABEL’S argument relied extensively on his particular knowledge of the 

individual objects and was thus much more formula based. 

The styles of mathematics. Not surprisingly, the changing techniques of mathemat- 

ics manifested themselves at the textual level. Because formulae had been the carriers 

of knowledge and argument in the formula based paradigm, mathematical publica- 

tions relied extensively on the powers of formulae and mathematical texts could be 

dominated by strings of explicit manipulations of formulae. Eventually, a conclusion 

could be stated in the form of a theorem. In the concept based paradigm, a Euclidean 

style with its emphasis on definitions, theorems, and proofs became the customary 

style of written mathematics. This presentational style emphasized the precise state- 

ment of assumptions and the internal relations between concepts and theorems. 

A revolution? When a change of paradigms is involved, the question of revolutions 

naturally arises. According to the recent debate, revolutions in mathematics appears 

not to be the most apt scheme of interpreting the history of the discipline. 8 In particu- 

lar, the requirements of incommensurability seems to prohibit revolutions appearing 

in mathematics because the truth status of mathematical statements apparently never 

changes. This also seems to apply to the change of paradigms discussed here. During 

and after the transitional period, mathematicians devoted an effort to reconstructing 

and re-interpreting the established knowledge to make it fit into the new system. This 

is particularly visible in analysis where A.-L. CAUCHY’S (1789–1857) deliberate redef- 

inition of basic notions and priorities changed the status of certain results and per- 

ceptions. As a result, men like ABEL sought to refound the theory in such a way that 

absolute truth was retained by making explicit the domains of validity for the state- 

ments. This process can be called critical revision and its general success precludes 

revolutions in mathematics. In section 21.3, the role of counter examples in the early 

nineteenth century is invoked to shed some light on this discussion. 

8 See primarily the articles in (Gillies, 1992).

21.2. Concepts and classes enter mathematics 393 

21.2 Concepts and classes enter mathematics 

As the basic objects of mathematics went from formulae to concepts, new methods 

and standards for introducing the objects were developed and the internal purpose of 

mathematical research also changed. 

21.2.1 Defining concepts 

While an object in the formula based paradigm could be introduced by merely exhibit- 

ing its formula, the introduction of concepts into the concept based paradigm required 

more sophisticated methods. However, these methods were not necessarily new — a 

number of them had been around since the births of the Euclidean style of mathemat- 

ics and Aristotelian logic in Ancient Greece. In the present context, two aspects of the 

new importance given to definitions deserve special attention. First, genetic defini- 

tions and nominal definitions are discussed and their interactions described. Second, 

the introduction of concepts with special properties through careful definitions is em- 

phasized. 

Genetic and nominal definitions. Concepts were often introduced by either genetic 

or nominal definitions. A genetic definition consists of prescribing the way the concept 

is constructed from other, simpler concepts whereas a nominal definition simply as- 

sociates a name to something. Typical examples of a genetic definitions in the present 

material include EULER’S definition of functions and ABEL’S definition of explicit al- 

gebraic expressions (see sections 10.1 and 6.3, respectively). These two examples also 

illustrate a very important difference in defining concepts: EULER’S definition was 

purely nominal whereas ABEL put his definition to essential use in obtaining his nor- 

mal form of explicit algebraic expressions. 9 Nominal definitions were being discussed 

in the early nineteenth century but the debate mainly centered on the ancient question 

whether or not nominal definitions implied the existence of any objects under the con- 

cept being defined. 10 The main objection against nominal definitions from a concept 

based paradigm could have been that they were not useful in obtaining knowledge of 

the concept being defined. 11 

Definition by desired property. The ultimate way of associating knowledge through 

definitions would be to let properties serve as definitions. In a sense, this is the final 

lesson of I. LAKATOS’ (1922–1974) Proofs and Refutations: the polyhedra which satisfy 

the Eulerian formula are collected as a concept and called Eulerian polyhedra and 

9 See e.g. (Laugwitz, 1999, 311) and section 6.3. 

10 Among the mathematicians involved were GERGONNE and OLIVIER. See e.g. (Otero, 1997, 74–81) 

and (Olivier, 1826c). 

11 In (Grabiner, 1981b), GRABINER has similarly emphasized the role which CAUCHY’S new definitions 

played for his foundation for the calculus.


of those, the Eulerian formula is trivially true. 12 Nevertheless, such definitions can 

be extremely useful in order to investigate other properties. With CAUCHY’S funda- 

mental shift towards arithmetic — rather than algebraic — equality, the numerical con- 

vergence of partial sums of series was given prime importance by using it to define 

convergent series. 13 Thus, a property of formal series — which could be numerically 

convergent or not — was used to define a concept which was subsequently promoted 

and investigated. A similar change went on in the theory of elliptic functions where 

ABEL’S original formal inversion of elliptic integrals was replaced by other definitions 

of elliptic functions. Many of the definitions of elliptic functions following ABEL’S 

original one turned properties — which were results in ABEL’S theory — into defini- 

tions. The motivations for this change in the status of properties of elliptic functions 

are many; rigor and theoretical applicability figure prominently among them. 

21.2.2 Relating concepts 

As a result of the transition, theorems about concepts and relating concepts came to 

dominate mathematics. Two types of relations among concepts were of principal im- 

portance: the representation of concepts and the determination of the extension of 

concepts. 

Representing concepts. Mathematical symbolism and formulae had proved to be 

an extremely useful and powerful tool in developing theories in the formula based 

paradigm. In order to be able to continue this line of research into the concept based 

paradigm, representations of concepts became quite important. Central instances in- 

clude ABEL’S classification of explicit algebraic expressions and the multitude of rep- 

resentations of elliptic functions which he developed. A particularly revealing exam- 

ple of the benefits of representations was illustrated in section 18.1 where ABEL’S use 

of infinite representations in the theory of transformation was discussed. The study 

of concepts in their entirety and not the individual objects meant that statements con- 

cerning the impossibility of certain representations could also be made and proved. 

This is particularly true of ABEL’S proof of the insolubility of the quintic (see chapter 

6) in which a representation of all explicit algebraic expressions was proved not to be 

sufficiently powerful to encompass the implicitly defined algebraic expression corre- 

sponding to a solution of the general fifth degree equation. The very same example 

also serves to illustrate the problem of distinguishing concepts. 

Distinguishing concepts. With the focus on concepts, it also became an important 

question to determine whether two concepts were identical or differed in their ex- 

tensions. One of the very best examples is the debate which during the nineteenth 

12 (Lakatos, 1976). 

13 See section 11.1.

21.3. The role of counter examples 395 

century separated the concepts of continuous and differentiable functions by construct- 

ing ever more pathological functions belonging to the former concept but not to the 

latter one. 14 The process of investigating concepts can often be thought of as a dialectic 

effort alternating between limiting and extending the domain of the concept. In point- 

ing out the existence of objects within a concept and differences between concepts, 

examples and counter examples became very important mathematical tools. A num- 

ber of similar uses of examples and counter examples can also be found in ABEL’S 

works. The most conspicuous example is in the theory of equations where ABEL’S 

proof on the quintic interpreted as limiting the class of solvable equations is precisely 

in this line of results. The use of counter examples as limitations on concepts is a quite 

modern one which is only meaningful within the concept based paradigm (see section 

21.3). 

Delineating concepts. One type of questions concerning the relation between con- 

cepts is so important that it deserves special attention; I have called it delineation of 

concepts. This notion refers to a set of questions which concern the precise characteri- 

zation of the extension of a concept by some external and applicable criterion. In other 

words, these questions ask for a (feasible) method of determining whether a given par- 

ticular object falls within the extension of a concept or not. In analogy with the steps 

of limiting and extending the extension of concepts (figures 6.1 and 7.3, respectively), 

a graphical representation of the delineation of concepts is produced in figure 21.2. 

ABEL’S unfinished research on a general theory of algebraic solubility was moti- 

vated by precisely this problem of determining whether or not a given equation could 

be solved algebraically. Similarly, the search for complete criteria of convergence also 

sought to delineate the extension of convergent series once and for all. The search for 

delineation of solvable equations came to a fruitful conclusion when GALOIS’ criterion 

was finally accepted as an answer. The complete determination of the concept of con- 

vergent series was never so successful; the only complete characterization obtained 

was the Cauchy criterion (see page 212) which did not fully meet the demand for being 

external and easy to apply. 

21.3 The role of counter examples 

It has been described how the problem of investigating the extension of concepts 

led to a particular use of examples and counter examples. Inspired by ABEL’S cu- 

rious remarks about his “exception” to Cauchy’s Theorem (see section 12.5), I suggest 

that counter examples played fundamentally different roles in the two paradigms dis- 

cussed here. 

14 See (K. Volkert, 1987; K. Volkert, 1989).


❅❅❘ ✬ 

❅❅■ 

✫ 

��✒ ��✠ 

Super-concept 

❄ 

✻ 

��✠ ✩ 

��✒ ✲✛ Concept ✲✛ 

❄ 

✻ 

❅❅❘ ✪ 

❅❅■ 

Figure 21.2: Delineating the border between a concept and its super-concept. 

21.3.1 Theorems with exceptions 

In his binomial paper, ABEL described how he found Cauchy’s Theorem to “suffer ex- 

ceptions” and I find it puzzling to investigate how theorems could possibly admit 

exceptions in the 1820s. First, however, the very phrasing of ABEL’S statement must 

be considered. Then, by way of recalling arguments carried out “in general”, the con- 

nection between exceptions and the formula based paradigm opens up. 

The authenticity of the wording. One may try to explain ABEL’S wording away as 

a result of his shyness and veneration for CAUCHY. For instance, in his criticism of 

L. OLIVIER, ABEL used the mild phrase “this part does not seem to be true” in the 

printed version rather than the more severe judgement “Mr. Olivier is seriously mis- 

taken” which we find in ABEL’S notebooks. 15 This would suggest that exceptions were 

a milder form of criticism than outright counter examples or even paradoxes which 

were also terms found in ABEL’S vocabulary. Besides, the problem remains that we 

only have A. L. CRELLE’S (1780–1855) translation of ABEL’S original manuscript at 

our disposal and single words in an edited manuscript can easily be over-interpreted. 

Nevertheless, when CAUCHY eventually reacted to ABEL’S exception, he did so explic- 

itly stating that he wanted to correct the statement of his theorem so that “it no longer 

admitted exceptions” 16 (see section 14.1.2). Thus, the word “exceptions” was chosen 

in this connection, and I believe that the following interpretation makes it plausible 

that ABEL actually meant that Cauchy’s Theorem suffered an exception — or rather, a 

number of exceptions. 


16 “Au reste, il est facile de voir comment on doit modifier l’énoncé du théorème, pour qu’il n’y ait 

plus lieu à aucune exception.” (A.-L. Cauchy, 1853, 31–32).


Arguments carried out “in general”. In the formula based paradigm, a situation 

sometimes arose in which the formula carrying the mathematical argument did not 

apply in all (numerical) cases. A number of such examples have been described above 

starting with EULER’S awareness that peculiar numerical results could emerge if specific 

values were inserted in expressions which were formally equal. 17 

As J. V. GRABINER describes, 18 J. L. LAGRANGE (1736–1813) held a strong and 

lifelong belief in the concept of “the general” in mathematics. Not only could formulae 

which were valid “in general” be of high importance — a general approach and system 

in mathematics was also strived for. When LAGRANGE presented his argument that 

“all” functions could be expanded in Taylor series, he was also aware that this might 

indeed fail to be true for particular functions at particular points. 19 However, these 

instances where the general results failed to be true were particular, peculiar, and of 

little interest to mathematicians ascribing to the formula based paradigm. 

In connection with ABEL’S Paris memoir, an even more elaborate case was pre- 

sented. At a crucial point in his argument to determine the number µ of independent 

integrals, ABEL employed a generalized degree operator called h. 20 Just as is the case 

for the ordinary degree operator of polynomials deg P (which ABEL also used), the 

degree of a sum may fail to be the maximum of the two degrees, 

deg (P1 + P2) ? = max {deg P1, deg P2} , (21.1) 

if deg P1 = deg P2. However, in the Paris memoir, ABEL was not interested in peculiar- 

ities and he simply argued that the equality corresponding to (21.1) was true “in gen- 

eral”, i.e. with the exception of some particular cases of little interest (see page 362). 

Once the paradigms had shifted, the precise determination of the number µ (called 

the genus) and the investigation and exposure of the necessary assumptions became 

a hot topic of mathematics. 

In a similar situation, ABEL concluded his summary of well known properties of 

elliptic functions in the Précis by the statement: 

“The formulae which have been presented above uphold with certain restrictions 

if the modulus c is arbitrary, real or imaginary.” 21 

ABEL’S way of obtaining the important formulae — often through tedious manipula- 

tions of infinite representations — could result in particular cases for which the formu- 

lae degenerated or produced false results. However, these cases were few and did not 

constitute an obstacle to presenting the formulae. 


18 See (Grabiner, 1981a, 317) or (Grabiner, 1981b, 39). 

19 See e.g. (Lagrange, 1813, 29–30). 


21 “Les formules présentées dans ce qui précède ont lieu avec quelques restrictions, si le module c est 

quelconque, réel ou imaginaire.” (N. H. Abel, 1829d, 245).


The number of exceptions. As indicated, in the formula based paradigm, results 

which suffered a few exceptions could still be very useful and the existence of ex- 

ceptions did not immediately lead to the overthrow of theorems. This suggests an 

interesting way of interpreting the last part of ABEL’S famous footnote: Besides in- 

troducing his exception, ABEL also claimed that many similar functions existed. This 

indicates that the number of exceptions played a role. A similar remark can also be 

found in connection with CAUCHY’S example of a non-zero function whose Maclaurin 

series is the zero-function, 22 

1 − 

f (x) = e x2 . 

This function represented an exception to the general belief in the expansion in power 

series which laid at the heart of the Lagrangian approach to analysis. 23 In 1822 and 

1829, 24 CAUCHY presented this example and observed how to construct other func- 

tions with the same property of not being represented by their Maclaurin series except 

at a single point. 

Both these examples suggest that if theorems in the formula based paradigm con- 

tained a quantification as “for all . . . ”, it might be necessary to introduce a statistical 

interpretation of the for-all quantification as K. VOLKERT has suggested. 25 Exceptions 

and their numbers were noticed but no clear distinction between refuted (false) the- 

orems and theorems with exceptions can be drawn. Theorems could be valid even if 

they suffered exceptions as long as the known exceptions were not too many or too 

important. 

Exceptions and the formula based paradigm. Thus, the argument is that exceptions 

did have a place in mathematics of the formula based paradigm. The highly compu- 

tational deductions based on long sequences of manipulations with finite and infinite 

representations occasionally led to results which were (only) true “in general”. In- 

stead of discarding such results, they were accepted with the knowledge or intuition 

that they should not be uncritically applied. However, as this intuition and general un- 

derstanding of mathematics shifted towards the concept based paradigm, exceptions 

became oddities — and counter examples became very powerful tools of argument in 

this new paradigm. 

21.3.2 Counter examples and concepts 

In the concept based paradigm, counter examples acquired a position much closer to 

their modern usage. As noted, counter examples are very instrumental in pointing 

out the differences between concepts and thereby helping to determine the extension 

22 Strictly speaking, the function should also be defined at the origin, f (0) = 0. For a good discussion 

on this issue, see (Bottazzini, 1990, lxix). 


24 (A.-L. Cauchy, 1822, 277) and (A.-L. Cauchy, 1829, 394–395). 

25 (K. T. Volkert, 1986, 144–145).


of concepts. Used as tools of criticism, a theorem to which a counter example could 

be presented was certainly false in the concept based approach. There was no room 

for theorems with exceptions. In a sense, the concept based approach adhered to a 

viewpoint similar to the Lakatosian one that theorems with counter examples should 

either be discarded or modified to range over a smaller domain. There is an abun- 

dance of such applications of counter examples in the 1820s. ABEL presented one very 

elaborate example in his refutation of OLIVIER when he showed that no criterion of 

the proposed form could ever be constructed having the properties which OLIVIER 

had sought. However, the young mathematician who made the most use of counter 

examples in the 1820s and 1830s was probably DIRICHLET. 

In 1829, 26 when DIRICHLET presented his famous result on the convergence of 

Fourier series, he started the paper with a scrutiny of an earlier paper by CAUCHY. 27 

In particular, DIRICHLET criticized a point in the proof where CAUCHY had used an 

implicit assumption which DIRICHLET identified as follows: If the series ∑ an was 

convergent, any other series ∑ bn such that lim bn = 1 would also be convergent. 

an 

Against this argument, DIRICHLET presented the counter example 

an = (−1)n 

√ and bn = 

n (−1)n 

� 

√ 1 + 

n 

(−1)n 

� 

√ 

n 

of which the series ∑ an was convergent but the series ∑ bn diverged. DIRICHLET de- 

scribed CAUCHY’S conclusion as “not permissible” 28 because it was easy to construct 

a counter example. 

To DIRICHLET, the existence of one single, local counter example thus seems to 

have rendered the theorem false; in particular, we find none of the above remarks that 

“infinitely many similar counter examples may be found or constructed” in DIRICH- 

LET’S papers. 29 In some instances, a counter example led DIRICHLET to dismiss the 

faulty theorems as false and begin his own deductions from other principles. In other 

situations, DIRICHLET drew inspiration from his counter examples to revise existing 

proofs in ways which later led to proof analysis. 

Later in the nineteenth century, counter examples acquired their modern status as 

complete refutations of theorems. To a mathematician educated at one of the German 

universities in the second half of the nineteenth century, a theorem could absolutely 

not admit exceptions and the precise formulation of theorems and proofs had truly 

become one of the trademarks of mathematics. 

ABEL’S use of counter examples seems to fall in both paradigms. As noted, sense 

can be made of ABEL’S exception to Cauchy’s Theorem if it is interpreted in the formula 

26 (G. L. Dirichlet, 1829, 120). 

27 (A.-L. Cauchy, 1827). Actually, DIRICHLET referred to a paper published in 1823 in the Mémoires de 

l’Académie des Sciences; but no paper with these details can be found in CAUCHY’S Œuvres. Thus, it 

is here assumed that DIRICHLET actually meant (ibid.). 

28 “Mais cette conclusion n’est pas permise” (G. L. Dirichlet, 1829, 158). 

29 (G. L. Dirichlet, 1829; G. L. Dirichlet, 1837).


based paradigm. On the other hand, ABEL’S dismissal of OLIVIER’S criterion of con- 

vergence shows signs of a concept based refutation. There, a single counter example 

was invoked to refute OLIVIER’S claim and an elaborate analysis was employed to 

show that the concept of tests of convergence could not contain a criterion of a slightly 

generalized form. 30 

The role of mathematical intuition. Importantly, the changing status of counter ex- 

amples also reflects a change in a mental entity which may be called mathematical in- 

tuition. 31 This intuition comprises the expertise, prejudices, and expectations of active 

mathematicians who have gained an insight into their objects and is thus part of T. S. 

KUHN’S (1922–1996) disciplinary matrix. 

During the eighteenth century, mathematicians built up a high degree of insight 

into representations of functions, in particular into power series or other algebraic 

expressions. When this insight was formulated, it often took the form of formulae 

relating certain entities by means of algebraic notation and the formulae were consid- 

ered to have aesthetic properties described as simplicity or degrees of symmetry. As 

illustrative examples, consider the solution formulae for general equations of low de- 

gree or the power series expansions of elementary transcendental functions. The art of 

mathematics also consisted of the trained ability to recognize patterns and manipulate 

representations to obtain various generalizations. 

As a result of the change of paradigms, the contents and role of mathematical in- 

tuition also changed. A new kind of intuition emerged which helped mathematicians 

see differences and similarities between concepts and suggested ways of obtaining re- 

lations among concepts. As an indication of this change in intuition, mathematicians 

occasionally brought over intuitions from the old paradigm into the new one. This 

could lead them to generalize results into forms in which they were then no longer 

permitted. Thus, the changing intuitions are intimately connected with the process of 

concept stretching which LAKATOS has discussed as part of interpreting mathematical 

development. 32 


The analytical scheme of a transition from a formula based paradigm to one based on 

concepts has shown its applicability in interpreting events in the disciplines of algebra 

and analysis in the 1820s. In particular, the role of new definitions, the coexistence of 

theorems and exceptions, and the new problems of delineation have contributed to 

throwing ABEL’S mathematical production into perspective. 


31 For a discussion of mathematical intuition, see also (K. T. Volkert, 1986). 

32 E.g. (Lakatos, 1976).


It is beyond the present scope to analyze and speculate as to the causal reasons for 

the purported change of paradigms in analysis and algebra. Neither is it the present 

purpose to discuss at length the general applicability of this frame of interpretation. 

However, it must be noticed that the interpretation might be limited to the disciplines 

described here; in particular, it does not appear to be immediately applicable to geom- 

etry. With some right, one could argue that the transition could be interpreted simply 

as a maturing of the involved theories. Still, I believe that the simultaneous instances 

of the change of style as described above are sufficient to suggest that a general change 

in the modes of thought was involved. 

New questions, new standards, new objects. In the presentation and analysis of 

ABEL’S mathematical production, three local themes were introduced. Based on the 

description of his works in algebra, I have argued that a new type of questions was 

being introduced into mathematics. These new questions were indicative of the fun- 

damental change of paradigms. Concerning ABEL’S works in the foundations of anal- 

ysis, it was illustrated how the change of basic definitions and standards of proof also 

reflected the new focus on concepts. Finally, a cross section of ABEL’S works on new 

transcendentals illustrated how these transcendental objects were being treated with 

the help of algebraic methods and also how the introduction of new objects led to 

important questions of representation. 

Throughout the description and analysis of ABEL’S works, much attention has 

been paid to their mathematical contexts. The inspirations which ABEL drew from 

his predecessors and contemporaries have been described in order to illustrate how 

ABEL’S works grew continuously out of the mathematical contexts. At the same time, 

ABEL’S works were — at a number of points — remarkably novel and due attention 

has been paid to these aspects. To generalize, ABEL’S methods and the problems 

which he attacked were generally well established whereas the questions which he 

raised and the approaches which he took in attacking these problems were often new 

and ground-breaking. In connection with the fundamental transition, this manifested 

itself in the sense that ABEL had one foot firmly placed in each of the two paradigms.

Appendix A 

ABEL’s correspondence 

In 1881, when ABEL’s collected works were published in their second edition, SYLOW 

and LIE included some of ABEL’s correspondence. 1 In 1902, a centennial Festschrift on 

ABEL’s life, work, and correspondence appeared both in Norwegian and in French 

including transcriptions of all known letters to and from ABEL. 2 In the twentieth 

century, however, additional letters have been found, and for the convenience of the 

reader, the table ?? provides a list of all the letters pertaining to ABEL known to the 

author. All the letters appearing in the Norwegian version of the Festschrift were also 

included in the French version. Missing information for year, month, or date indicate 

that that particular information was not available in the letter. 

Table A.1: Correspondence sorted by sender 

1822/01/18 (Abel→Aas, Kristiania, 1822/01/18. In Kragemo, 1929, 49) 

1822/01/25 (Abel→Aas, Kristiania, 1822/01/25. In ibid., 49–50) 

1822/02/06 (Abel→Aas, Christiania, 1822/02/06. In ibid., 50) 

1826/10/16 (Abel→Abel, Paris, 1826/10/16. In N. H. Abel, 1902a, 41–43) 

1827/02/26 (Abel→Boeck, Berlin, 1827/02/26. In ibid., 55–56) 

1827/01/15 (Abel→Boeck, Berlin, 1827/01/15. In ibid., 52–55) 

1826/11/01 (Abel→Boeck, Paris, 1826/11/01. In ibid., 47–48) 

1828/08/18 (Abel→Crelle, Christiania, 1828/08/18. In ibid., 67–73) 

1827 (Abel→Crelle, Christiania, 1827. In ibid., 60–61) 


1826/03/14 (Abel→Crelle, Freiberg, 1826/03/14. In N. H. Abel, 1881, 266) 

1827/11/15 (Abel→Crelle, Christiania, 1827/11/15. In ibid., 268) 


1828/10/18 (Abel→Crelle, Christiania, 1828/10/18. In Biermann, 1967, 27–29) 

1828? (Abel→Crelle, 1828?. In Biermann and Brun, 1958, 85) 

1826/12/04 (Abel→Crelle, Paris, 1826/12/04. In N. H. Abel, 1902a, 50–51) 

1826/03/14 (Abel→Crelle, Freyberg, 1826/03/14. In ibid., 21–22) 

1 (N. H. Abel, 1881) 

2 (N. H. Abel, 1902e; N. H. Abel, 1902f). 

403

404 Appendix A. ABEL’s correspondence 

Table A.1: Correspondence sorted by sender (cont.) 

1826/08/09 (Abel→Crelle, Paris, 1826/08/09. In N. H. Abel, 1902a, 38–39) 

1826/08/09 (Abel→Crelle, Paris, 1826/08/09. In N. H. Abel, 1881, 267) 

1826/12/04 (Abel→Crelle, Paris, 1826/12/04. In ibid., 268) 

1824/03/02 (Abel→Degen, Christiania, 1824/03/02. In P. Heegaard, 1935, 33–37) 

1824/03/02 (Abel→Degen, Christiania, 1824/03/02. In P. Heegaard, 1937, 1–5) 

1828/09/22 (Abel→Fru Hansteen, Christiania, 1828/09/22. In N. H. Abel, 1902a, 67) 

1828/07/29 (Abel→Fru Hansteen, Froland, 1828/07/29. In ibid., 65) 

1828/08 (Abel→Fru Hansteen, Froland, 1828/08. In ibid., 65–66) 

1828/07/21 (Abel→Fru Hansteen, Froland, 1828/07/21. In ibid., 63–64) 

(Abel→Fru Hansteen. In ibid., 62) 

1827/03 (Abel→Fru Hansteen, Berlin, 1827/03. In ibid., 58–59) 

1827/08/18 (Abel→Fru Hansteen, Christiania, 1827/08/18. In ibid., 60) 

1826/01/16 (Abel→Fru Hansteen, Berlin, 1826/01/16. In ibid., 19–20) 

1825?/12/08 (Abel→Fru Hansteen, Berlin, 1825?/12/08. In ibid., 12–13) 

1828/11 (Abel→Fru Hansteen, Christiania, 1828/11. In ibid., 75–78) 

[1826]/01/30 (Abel→Hansteen, Berlin, [1826]/01/30. In ibid., 20–21) 

1826/03/29 (Abel→Hansteen, Dresden, 1826/03/29. In N. H. Abel, 1881, 263–265) 

1826/05/28 (Abel→Hansteen, Grätz, 1826/05/28. In N. H. Abel, 1902a, 32) 

1826/03/29 (Abel→Hansteen, Dresden, 1826/03/29. In ibid., 22–26) 

1826/08/12 (Abel→Hansteen, Paris, 1826/08/12. In ibid., 39–41) 

1825/12/05 (Abel→Hansteen, Berlin, 1825/12/05. In ibid., 9–12) 

1823/06/15 (Abel→Holmboe, Kjøbenhavn, 1823/06/15. In ibid., 3–4) 

1823/08/04 (Abel→Holmboe, Kjøbenhavn, 1823/08/04. In ibid., 4–8) 

1828/06/29 (Abel→Holmboe, Froland, 1828/06/29. In ibid., 64–65) 

1826/01/16 (Abel→Holmboe, 1826/01/16. In ibid., 13–19) 

1823/08/03 (Abel→Holmboe, Copenhague, 1823/08/03. In N. H. Abel, 1881, 254–258) 

1826/10/24 (Abel→Holmboe, Paris, 1826/10/24. In ibid., 259–261) 

1826/12 (Abel→Holmboe, Paris, 1826/12. In ibid., 261–262) 

1827/03/04 (Abel→Holmboe, Berlin, 1827/03/04. In ibid., 262) 

1827/03/04 (Abel→Holmboe, Berlin, 1827/03/04. In N. H. Abel, 1902a, 56–58) 

1826/12 (Abel→Holmboe, Paris, 1826/12. In ibid., 51–52) 

1827/01/20 (Abel→Holmboe, Berlin, 1827/01/20. In ibid., 55) 

1826/04/16 (Abel→Holmboe, Wien, 1826/04/16. In ibid., 26–31) 

1826/10/24 (Abel→Holmboe, Paris, 1826/10/24. In ibid., 43–47) 

1826/06/15 (Abel→Holmboe, Bolzano, 1826/06/15. In ibid., 33–37) 

1825/09/15 (Abel→Holmboe, Kjøbenhavn, 1825/09/15. In ibid., 9) 

1826/07/05 (Abel→Keilhau, Zurich, 1826/07/05. In ibid., 37) 

1826/11/01 (Abel→Külp, Paris, 1826/11/01. In Hensel, 1903, 237–240) 

1828/11/25 (Abel→Legendre, Christiania, 1828/11/25. In N. H. Abel, 1902a, 78–86) 

1828/11/25 (Abel→Legendre, Christiania, 1828/11/25. In N. H. Abel, 1881, 271–279) 

1823 (Abel→Olsen, Christiania, 1823. In Brun and Jessen, 1958, 22–23)

Table A.1: Correspondence sorted by sender (cont.) 

1828/09/10 (Crelle→Abel, 1828/09/10. In N. H. Abel, 1902a, 66) 

1828/05/18 (Crelle→Abel, 1828/05/18. In ibid., 62) 

1829/04/08 (Crelle→Abel, Berlin, 1829/04/08. In ibid., 89–90) 

1826/11/24 (Crelle→Abel, Berlin, 1826/11/24. In ibid., 48–50) 

1829/05/10 (Crelle→Holmboe, Berlin, 1829/05/10. In N. H. Abel, 1902b, 97–98) 

1840/05/15 (Crelle→Holmboe, Berlin, 1840/05/15. In ibid., 102) 

1821/05/21 (Degen→Hansteen, Kjøbenhavn, 1821/05/21. In ibid., 93–96) 

1830/07/24 (Det franske Institut→Abel’s efterladte, Paris, 1830/07/24. In N. H. Abel, 1902a, 101) 

1752 (Euler→Goldbach, 1752. In Euler and Goldbach, 1965) 

1829 (Jacobi→Legendre, Potsdam, 1829. In Legendre and Jacobi, 1875) 

1830/02/22 (Keilhau→Boeck, Froland, 1830/02/22. In N. H. Abel, 1902b, 98–100) 

1781 (Lagrange→d’Alembert, Berlin, 1781. In Lagrange, 1867–1892, vol. 13, 368–370) 

1828/10/25 (Legendre→Abel, Paris, 1828/10/25. In N. H. Abel, 1902a, 74–75) 

1829/01/16 (Legendre→Abel, Paris, 1829/01/16. In ibid., 87–89) 

1832/04/11 (Löwenhielm→Hansteen, Paris, 1832/04/11. In N. H. Abel, 1902b, 101–102) 

1824/08/02 (Schumacher→Hansteen, Altona, 1824/08/02. In ibid., 97) 

1829/04/07 (Smith→Holmboe, Froland, 1829/04/07. In ibid., 97) 

1881/12/09 (Weierstrass→Lie, Berlin, 1881/12/09. In ibid., 103) 

1882/04/10 (Weierstrass→Lie, Berlin, 1882/04/10. In ibid., 103–104) 

1873 (Weierstrass→du Bois-Reymond, 1873. In K. Weierstrass, 1923, 199–201) 

405

Appendix B 

ABEL’s manuscripts 

Manuscripts and drafts constitute an important source for historical inquiry, especially 

when the historian aims at discussing the genesis of certain ideas. In the case of ABEL, 

the extent sources not published within or immediately after his lifetime fall into two 

categories: 

1. Manuscripts — to various degrees of completion — of papers not published in 

ABEL’s lifetime. 

2. Drafts and notebooks documenting the working mathematician but not intended 

for publication. 

Items in the first category was to some extent considered and included in the com- 

pilations for both editions of ABEL’s collected works (N. H. Abel, 1839; N. H. Abel, 

1881). In the second volume of the second edition (1881) collected works, SYLOW also 

included a general presentation of the known items from the second category 1 . 

The present appendix has as its aim to document the whereabouts of archival mate- 

rial concerning ABEL. This aim is achieved by reproducing registrants of the archives 

at the Manuscript Collection, University of Oslo (see table B.1) and the Mittag-Leffler 

Institute, Djursholm, Sweden (table B.5). 

1 (N. H. Abel, 1881, vol. 2, 283–289) 

2 The author gratefully acknowledges the participation of KLAUS FROVIN JØRGENSEN in obtaining 

the information presented in table B.5. 

407

408 Appendix B. ABEL’s manuscripts 

MS:592 Manuscripts and letters (table B.2) 

MS:434 Fragments from manuscripts (table B.3) 

MS:435 Note accompaning the Abel manuscripts 

(fragmentary, 6 pages) 

MS:969-4 Nogle Bemærkninger om Vinkelfunktionerne 


MS:589 Belonging to the Théorie de la résolution 

algébrique des équations (fragmentary, 2 

pages) 

MS:592 Précis de la théorie des fonctions elliptiques 

(168 pages) 

MS:920-4 Niels Abel Berlin-Paris 1825–182? (11 

pages) 

MS:188-8 Note über die Function . . . (2 pages) 

MS:969-4 Paa Froland og ved Abels grav 4. og 5. august 

(4 pages) 

Multiple ABEL’s notebooks (table B.4) 

Table B.1: Abel manuscript collections in the Manuscript Collection, University Library, 

Oslo. 

1. Mémoire sur une classe particulière d’équations résolubles 

algébriquement (64 pages) 

2. Note sur quelques formules elliptiques (18 pages) 

3. Théorèmes sur les fonctions elliptiques (11 pages) 

4. Démonstration d’une propriété générale d’une certaine 

classe de fonctions transcendantes (4 pages) 

5. Matematiske uddrag fra N. H. Abel’s breve (13 pages) 

Table B.2: Abel manuscripts in the Manuscript Collection, University Library, Oslo, 

MS:592.

1. Précis d’une théorie des fonctions elliptiques (fragmentary, 

8 pages) 

2. Om transformationer af elliptiske funktioner, hvorved 

de to perioder divideres med hvert sit tal (fragmentary, 

9 pages) 

3. Divisionsligningers opløsning (fragmentary, 2 pages) 

4. Unidentified (fragmentary, 1 page) 

5. Belonging to the Précis or the Recherches (fragmentary, 

2 pages) 

6. Belonging to the Paris mémoire (fragmentary, 2 pages) 

7. Belonging to the Précis (fragmentary, 7 pages) 

8. Belonging to the Recherches second part (fragmentary, 

2 pages) 

9. Table of contents for a work on elliptic functions (fragmentary, 

4 pages) 

10. Integration ved hjælp af algebraiske, logaritmiske og 

eksponentielle funktioner (fragmentary, 2 pages) 

11. Belonging to the Abelian Theorem for elliptic functions 


Table B.3: Abel manuscripts in the Manuscript Collection, University Library, Oslo, 

MS:434. 

MS:351:A Notebook A Mémoires de Mathématiques par N. 

H. Abel. Paris le 9 Août 1826 (202 

pages fol.) 

MS:436 Notebook B Without title (178 pages fol.) 

MS:351:C Notebook C Without title (215 pages fol.) 

MS:696 Notebook D Remarques sur divers points de 

l’analyse par N. H. Abel, 1er Cahier 

le 3 Sept. 1827 (136 pages 4:o) 

MS:829 Notebook E Mathematiske Udarbeidelser af Niels 

Henrik Abel (192 pages 4:o) 

MS:749 Matematiske Afhandlinger (170 

pages 4:o) 

Table B.4: Abel’s mathematical notebooks in the Manuscript Collection, University 

Library, Oslo. 

409

410 Appendix B. ABEL’s manuscripts 

1. Manuskript af Abel (V.Terquem, Bulletin, T. I, p. 56) 

2. Manuskript i 4:o: Note sur quelques formules elliptiques 

3. Manuskript i 8:o: Théories sur les fonctions elliptiques 

4. P.M. af Phragmén 

5. Acta korrektur af Recherches sur les fonctions elliptiques 

6. Tryckt: Mémoire sur une propriété générale d’une 

classe très-étendue de fonctions transcendantes. 1841. 

4:o. 

7. Mémoire sur les équations algébriques etc. Chr:ia 

1824. 4:o. 

8. Oplösning af et Par Opgaver ved Hjelp af bestemte 

Integraler. 8:o 

9. Almindelig methode til at finde Funktioner af een 

variable Störrelse. . . 8:o. 

Table B.5: Abel manuscripts in the Mittag-Leffler Institute, Djursholm, Sweden. 2

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— (MS:592). “Mémoire sur une classe particulière d’équations résolubles algébriquement”. 

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— (MS:696). “Remarques sur divers points de l’analyse par N. H. Abel, 1er Cahier 

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Index of names 

Abbati, Pietro (1768–1842), 88, 125 

Abel, Hans Mathias (1800–1842), 19 

Abel, Søren Georg (1772–1820), 18 

Andersen, Kirsti (⋆1941), 261 

Artin, Emil (1898–1962), 185 

Badano, P. Gerolamo, 134 

Bader, Hans Peter (1790–1819), 19 

Bernoulli, Jakob I (1654–1705), 289 

Bernoulli, Johann I (1667–1748), 289 

Bertrand, Joseph Louis François (1822–1900), 

274 

Bjerknes, Carl Anton (1825–1903), 18, 332 

Bjørling, 279 

Bolyai, Farkas (1775–1856), 34 

Bolzano, Bernard (1781–1848), 42, 198, 201– 

204, 221 

Bombelli, Rafael (1526–1572), 61 

Bos, Henk J. M., xix 

Bottazzini, Umberto (⋆1947), xix, 13, 207 

Brill, 374 

Brun, Viggo (1885–1978), 261, 375 

Burg, Adam von (1797–1882), 123 

Cantor, Georg (1845–1918), 212 

Cardano, Girolamo (1501–1576), 57, 59, 60, 

102, 103 

Cauchy, Augustin-Louis (1789–1857), xi, xii, 

3, 7–9, 12, 13, 31, 35, 36, 42, 50, 69, 

70, 84, 85, 89–95, 100, 101, 108–110, 

112, 119–121, 123, 125, 128, 135, 137, 

138, 181, 183, 186, 191–193, 196, 201, 

206–219, 221–224, 226–231, 233, 235, 

237, 238, 241, 246, 248, 251, 253, 260– 

263, 267, 275, 278–282, 306, 307, 329, 

339, 349, 350, 372, 373, 375, 392– 

394, 396, 398, 399 

Cayley, Arthur (1821–1895), 132 

Chebyshev, Pafnuty Lvovich (1821–1894), 340 

Cockle, James, 132 

Crelle, August Leopold (1780–1855), 4, 5, 

14, 17, 28–37, 39, 41, 44, 73, 98–101, 

122, 123, 126–130, 155, 157, 163, 178, 

431 

180, 206, 218, 223, 226, 241, 252, 261, 

265, 277, 278, 299, 300, 337, 341, 346, 

376, 377, 396 

d’Alembert, Jean le Rond (1717–1783), 43, 

58, 199–201 

Dedekind, Julius Wilhelm Richard (1831– 

1916), 212 

Degen, Carl Ferdinand (1766–1825), 22, 23, 

25–28, 54, 97, 98, 261, 297, 299 

Delambre, Jean-Baptiste Joseph (1749–1822), 

43, 44, 84 

Descartes, René du Perron (1596–1650), 53, 

57 

Dirichlet, Gustav Peter Lejeune (1805–1859), 

xiv, 73, 219, 228, 236–239, 247, 248, 

280, 391, 399 

Dirksen, Enno Heeren (1788–1850), 32 

du Bois-Reymond, Paul David Gustav (1831– 

1889), 248–251 

Elliot, 362 

Epple, Moritz, 13 

Euclid (∼295 B.C.), 79, 159, 348 

Euler, Leonhard (1707–1783), xiv, 5, 6, 8, 10, 

21, 22, 24, 26, 27, 29, 32, 49, 53, 58, 

62–67, 73, 75, 80, 82, 83, 145, 153, 

176, 179, 180, 193–197, 203, 207, 209, 

214, 216, 221, 238, 240, 261, 285, 287– 

292, 297, 303, 306, 328, 329, 388, 389, 

393, 397 

Eytelwein, Johann Albert (1764–1849), 32 

Fagnano dei Toschi, Giulio Carlo (1682–1766), 

153, 287, 289, 290 

Fermat, Pierre de (1601–1665), 75 

Ferrari, Ludovico (1522–1565), 61 

Ferro, Scipione (1465–1526), 59 

Ferrusac, baron de (1776–1836), 98, 123, 126 

Fourier, Jean Baptiste Joseph (1768–1830), 

204, 205, 218, 226, 240 

Francoeur, Louis Benjamin (1773–1849), 21, 

22

432 Index of names 

Frobenius, Georg Ferdinand (1849–1917), 280 

Galilei, Galileo (1564–1642), 24 

Galois, Evariste (1811–1832), 5, 10, 49, 52, 

54, 55, 66, 67, 75, 82, 84, 97, 125, 137, 

138, 161, 180–187, 390, 395 

Garnier, Jean Guillaume (1766–1840), 21, 22 

Gauss, Carl Friedrich (1777–1855), 3, 6, 10, 

21, 23, 27, 32, 34, 49, 51–54, 58, 72– 

80, 82–84, 90, 95–97, 124, 126, 139, 

142, 150–154, 157–160, 174, 175, 198– 

201, 296, 297, 300, 307, 308, 316– 

319, 349, 356, 367, 383 

Gergonne, Joseph Diaz (1771–1859), 29, 348, 

372, 393 

Girard, Albert (1595–1632), 57, 59 

Goldbach, Christian (1690–1764), 290 

Grabiner, Judith V., 197, 393, 397 

Grattan-Guinness, Ivor, 42, 222, 279 

Gregory, James (1638–1675), 61, 287 

Gruson, Johann Philipp (1768–1857), 32 

Gårding, Lars, 171 

Göpel, Adolph (1812–1847), 377 

Hamilton, William Rowan (1805–1865), 104, 

130, 131, 133–136, 138 

Hansteen, Cathrine Andrea Borch (1787–1840), 

17 

Hansteen, Christopher (1784–1873), 22–24, 

26–28, 33–36, 40, 97, 125, 224, 225, 

246, 250, 297, 331, 339, 348 

Heegaard, Poul (1871–1948), 375 

Heine, Heinrich Eduard (1821–1881), 280 

Hilbert, David (1862–1943), 54 

Hindenburg, Carl Friedrich (1741–1808), 218 

Hirsche, Meier (1765–1851), 98 

Holmboe, Bernt Michael (1795–1850), xviii, 

4, 14, 17–22, 31, 34–36, 40, 42, 97, 

117, 127, 128, 132–134, 138, 160, 161, 

163, 175, 179, 221, 222, 225, 226, 260, 

297, 298, 306, 375 

Holst, Elling Bolt (1849–1915), 26 

Humboldt, Alexander von (1769–1859), 32, 

34 

Humboldt, Wilhelm von (1767–1835), 32, 41 

Huygens, Christiaan (1629–1695), 24 

Ivory, James (1765–1842), 42 

Jacobi, Carl Gustav Jacob (1804–1851), 6, 17, 

36, 42, 44, 45, 49, 55, 97, 155, 157, 

290, 294, 298, 300, 302, 307, 309, 310, 

331–333, 337, 346, 375, 377, 380, 382, 

383 

Jahnke, Hans Niels (⋆1948), 391 

Jerrard, George Birch (1804–1863), 129–132, 

134, 138, 179 

Johnsen, Karsten, 76 

Kemp, Christine (1804–1862), 17 

Klein, Christian Felix (1849–1925), 391 

Kline, Morris (⋆1908), 95 

Kronecker, Leopold (1823–1891), 51, 136–138, 

149, 152, 171, 181 

Kuhn, Thomas S. (1922–1996), 11, 13, 400 

Königsberger, Leo (1837–1921), 104, 131, 133, 

134, 138, 332 

Külp, Edmund Jacob (⋆1801), 22, 115–117, 

128, 129, 134, 138, 166 

l’Hospital, Guillaume-François-Antoine de 

(1661–1704), 358 

Lacroix, Sylvestre François (1765–1843), 21, 

186, 198 

Lagrange, Joseph Louis (1736–1813), xi, 3, 

5, 7, 10, 21, 22, 32, 43, 44, 49–51, 53, 

58, 61, 64–73, 75, 81–86, 88, 92, 101, 

104, 107, 108, 110, 113, 114, 116, 118, 

119, 124, 126, 132, 133, 139, 170, 171, 

181, 183, 185, 186, 191, 193, 197, 198, 

207, 219, 294, 372, 376, 397 

Lakatos, Imre (1922–1974), 11–13, 128, 278, 

393, 400 

Laplace, Pierre-Simon, marquis de (1749– 

1827), 20 

Laugwitz, Detlef (1932–2000), 248, 391 

Legendre, Adrien-Marie (1752–1833), xiv, 3, 

6, 24, 26, 27, 35, 36, 44, 75, 126, 127, 

153, 288, 292–297, 299, 307, 309, 310, 

321, 325, 327, 331, 333, 336, 337, 339, 

375 

Lehmus, Daniel Christian Ludolf (1780–1863), 

32 

Leibniz, Gottfried Wilhelm (1646–1716), 58, 

61, 62, 287 

Libri, Guglielmo (1803–1869), 348, 361, 375 

Lie, Marius Sophus (1842–1899), 244, 245, 

252, 325, 328, 329, 375 

Liouville, Joseph (1809–1882), 52, 181, 185, 

236, 340, 374, 382 

Littrow, Karl Ludwig von (1811–1877), 35, 

123 

Lubbe, Samuel Ferdinand (1786–1846), 32 

Lubbock, John William (1803–1865), 130

Index of names 433 

Lützen, Jesper, xix, 340 

MacCullagh, James (1809–1847), 134 

Malfatti, Gian Francesco (1731–1807), 88, 125 

Malmsten, Carl Johann (1814–1886), 171 

Maurice, Jean Frédéric Théodor, 127 

Mertens, Franz (1840–1927), 247 

Mittag-Leffler, Magnus Gustav (1846–1927), 

379 

Newton, Isaac (1642–1727), 24, 59, 70–72, 

195, 203, 286, 287 

Ohm, Georg Simon (1789–1854), 31 

Ohm, Martin (1792–1872), 31–33, 206, 218 

Olivier, Louis, 126, 206, 265–272, 393, 396, 

399, 400 

Ore, Øystein (1899–1968), 14, 261 

Poisson, Siméon-Denis (1781–1840), 21, 186, 

206, 226 

Popper, Karl Raimund (1902–1994), 12 

Rasmussen, Søren (1768–1850), 20, 23, 24, 

26, 27, 36, 40, 339 

Riemann, Georg Friedrich Bernhard (1826– 

1866), xiv, 219, 236, 247, 250, 369, 

373, 388, 391 

Rosenhain, Johann Georg (1816–1887), 377 

Rudio, Ferdinand (1856–1929), 390, 391 

Ruffini, Paolo (1765–1822), 5, 10, 50, 53, 54, 

83–92, 97, 100, 101, 107, 108, 110, 

112, 119–125, 128, 135, 137, 139 

Saigey, Jacques Frédéric (1797–1871), 123 

Schlesinger, 296 

Schmidten, Henrik Gerner v. (1799–1831), 

29 

Schneider, 200 

Schumacher, Heinrich Christian (1784–1873), 

24, 28, 297, 331, 346 

Seidel, Philipp Ludwig von (1821–1896), 217, 

279, 280 

Serret, Joseph Alfred (1819–1885), 133 

Simonsen, Anne Marie (1781–1846), 18 

Skau, Christian, 106, 171 

Spalt, Detlef, 280 

Stokes, George Gabriel (1819–1903), 217, 279, 

280 

Stubhaug, Arild (⋆1948), xix, 14, 17, 18, 24 

Sylow, Peter Ludvig Mejdell (1832–1918), 14, 

26, 32, 42, 122, 155, 167, 169, 171, 

175, 176, 180, 243, 245, 252, 253, 306, 

307, 351, 355, 357, 358, 361, 362, 365, 

366, 373 

Tartaglia, Niccolò (1499/1500–1557), 59 

Taylor, Brook (1685–1731), 202, 222 

Thune, Erasmus Georg Fog (1785–1829), 25 

Tralles, Johann Georg (1763–1822), 32 

Tschirnhaus, Ehrenfried Walter (1651–1708), 

61, 80 

Vandermonde, Alexandre-Théophile (1735– 

1796), 49, 64, 66, 99 

Viète, François (1540–1603), 59, 71 

Volkert, Klaus, 398 

Wallace, William (1768–1843), 196 

Wantzel, Pierre Laurent (1814–1848), 89, 135, 

247 

Waring, Edward (∼1736–1798), 51, 59, 70– 

72, 80, 82, 88, 133, 143 

Weber, Heinrich (1842–1913), 185 

Weierstrass, Karl Theodor Wilhelm (1815– 

1897), 44, 45, 219, 223, 228, 246, 248, 

250, 280, 327, 382, 388 

Wessel, Caspar (1745–1818), 261 

Wussing, Hans, 64, 186 

Young, Thomas (1773–1829), 42

Index 

École Normale, 40, 187 

École Polytechnique, 3, 31, 40, 207, 218 

a priori properties of roots, 58 

Abel’s Vienna review, xix 

Abel’s exception, 395 

Abel’s manuscripts, xix 

Abelian equations, 49, 51, 52, 141, 142, 145, 

149, 150, 152, 163, 165, 171, 172, 176, 

178, 180, 186, 314, 316, 334, 392 

Abelian groups, 149 

Abelian Theorem, 15, 358, 361, 376 

absolutely convergent series, 215 

Académie des Sciences, 6, 35, 40, 42, 64, 347, 

348, 375, 376 

Akademie der Wissenschaften, 136 

Algebra 

Fundamental Theorem of, 5, 10, 58, 80, 

82, 95 

algebraic expression, 61 

algebraically solvable equations, 392 

amplitude, 309 

anachronism, 10 

Analytical Society, 42 

Annales de mathématiques pures et appliquées, 

29, 205, 348, 372 

anomaly, 11 

arithmetic-geometric mean, 296, 297 

arithmetical foundation of calculus, 9 

arithmetical concept of equality, 9 

Astronomische Nachrichten, 24, 331–333, 337, 

338 

basic curves, 8 

basic functions, 381 

Berlin Academy, 32 

Berlin Academy, 65 

binomial formula, 221 

binomial theorem, 221 

binomial formula, 238 

binomial series, 227, 241, 254, 255, 259, 260, 

263 

435 

binomial theorem, 221, 224, 227, 233, 238, 

241, 251, 254, 260, 261, 263 

British Association, 129 

Bulletin, 123, 126 

Bulletin des sciences mathématiques, astronomiques, 

physiques et chimiques, 98, 123 

Cauchy criterion, 395 

Cauchy’s Theorem, 251, 395, 396, 399 

Cauchy-ism, 7 

Cauchy-Riemann-equations, 307 

Cauchy-Ruffini theorem, 100, 101, 108, 110, 

112, 119–121, 128, 135, 137 

chains of roots, 145, 146 

chains of roots, 142, 143 

change, 11 

Christiania cathedral school, 4 

circle, 287 

circle of permutations, 93 

circular substitution, 94 

Collegium academicum, 339 

coming to know objects, 8 

commutativity of rational dependencies, 147 

complex of roots, 75 

complexum, 75 

composite permutations, 85 

computational mathematical style, 52 

concept 

definition, 8 

concept based mathematics, xiii, xiv, xvii, 8, 

9, 13, 15, 55, 124, 141, 187, 192, 286, 

387–395, 398–400 

concept stretching, 7, 12, 400 

conceptual mathematical style, 52 

conjugate roots, 76 

constructibility, 74 

constructions, 74 

convergence 

d’Alembert, 200, 201 

Gauss, 200 

Olivier, 265 

convergence of Fourier series, 399

436 Index 

convergence tests, 265 

counter example, 7, 12 

crisis, 11 

critical attitude, 9 

critical revision, xiii, xiv, 9, 191, 192, 281, 

392 

cumulative nature of mathematics, 281 

cyclotomic equation, 52, 79 

cyclotomic equation, 72, 83, 142, 150–152, 

316 

cyclotomic equations, 150 

Danmarks Tekniske Universitet, 24 

degree of equivalence, 92 

degree of substitution, 91 

degree of equivalence, 86, 87 

delineation problem, 400 

delineation of concepts, xiv, 139, 272, 390, 

395 

delineation of solvable equations, 395 

delineation problem, xiv, 341 

Den Polytekniske Læreanstalt, 24 

descent down string of equations, 148 

dialectics, 11 

differentiable functions, 395 

Dirichlet boxing-in principle, 109 

disciplinary matrix, 400 

divergent series, 265, 267 

division of the lemniscate, 315 

division problem, 155, 157, 313 

for circular functions, 150 

for the lemniscate, 154 

for elliptic functions, 142, 150, 153 

for the circle, 6, 74, 151, 154 

for the lemniscate, 154 

of the circle, 150 

domain of rationality, 159 

elementary symmetric relations, 59 

epistemic configuration, 13 

epistemic object, 13 

epistemic technique, 13 

equality 

arithmetical, 9 

formal, 9 

Euclid’s division algorithm, 142 

Euclid’s division algorithm, 141, 158, 159 

Euclidean construction, 74 

Euler’s hypothesis, 80 

exception, 7, 396 

exception barring, 12 

exceptions, xiii 

false roots, 57 

falsification, 12 

Fermat primes, 79, 318 

fictuous roots, 57 

form of expression, 67 

formal concept of equality, 9 

formula based mathematics, xiii, xiv, 13, 263, 

373, 374, 378, 387–392 

French Revolution, 3, 40, 43 

functional equations, 261 

Fundamental Theorem of Algebra, 5, 10, 58, 

80, 82, 95, 159 

Galois theory, 99 

Grand prix, 375 

groupe, 75 

habituation, xiii, xiv 

hyperbola, 285, 287 

hyperelliptic functions, 377 

hyperelliptic integrals, 350, 351, 376, 377 

hypergeometric series, 198–201, 296 

identical substitution, 91 

imagined roots, 57 

imprimitive groups, 86 

indecomposable functions, 158 

indeterminate series, 265, 267 

index, 87 

index of a function, 92, 93 

indicative divisor, 92 

inertia of mathematical community, 80 

inherited property, 148 

Institut, 348 

Institut de France, 90, 181, 186, 348 

intransitive groups, 86 

irreducibility, 157 

irreducible equations, 159 

irreducible Abelian equations, 176 

irreducible Abelian equations, 52 

irreducible equations, 52, 141, 142, 146, 158 

Journal, 33, 35–37, 39, 122, 126, 128, 129, 155, 

223, 244, 261, 265, 269, 278, 299, 300, 

303, 337, 339, 341, 376, 377 

Journal d’École Polytechnique, 90 

Journal de mathématiques pures et appliquées, 

181, 236 

Journal für die reine und angewandte Mathematik, 

4, 5, 29, 73, 98, 126 

Königlichen Friedrichs-Gymnasium, 135 

l’Hospital’s rule, 322

Index 437 

Lagrange interpolation, 349, 355, 356, 360, 

372, 373, 381 

Lagrange’s Theorem, 68 

Lagrangian program, 207 

Mémoires, 65 

Mémoires de l’Académie des Sciences, 399 

Mémoires présentés par divers savants, 348, 375 

Maclaurin series, 398 

Magazin, 24 

Magazin for Naturvidenskaberne, 23, 132 

mathematical intuition, 400 

mathematical workshop, 13 

multi-valued function, 63 

Napoleonic Wars, 4 

Napoleonic Wars, 40 

neo-humanist movement, 3, 41 

Nouvelles annales de mathematique, 135 

orbit of theta, 145 

order of substitution, 91 

paradigm, 11 

period of roots, 78 

sum, 78 

permutations, 85, 91, 92 

circle, 93 

composite, 85 

simple, 85 

permutazione, 85 

Philosophical Magazine, 130, 132 

pigeon hole principle, 109 

powers of a substitution, 91 

primitive groups, 86 

primitive roots, 73 

principle of double periodicity, 310 

product of substitutions, 91 

proof analysis, 12 

proof revision, 12, 280 

Prussia, 3, 41 

pure equations, 80 

pure equations, 80 

quadrature of hyperbola, 287 

quadrature of hyperbola, 285, 287 

quadrature of circle, 287 

quadrature of hyperbola, 295 

rationally known quantities, 183 

reciprocal roots, 76 

rectification of hyperbola, 287 

Regentsen, 23 

residues, 306 

resolvent, 62 

resolvent equation, 62 

revolution, 11 

rigorization program, 191 

roots 

false, 57 

imagined, 57 

true, 57 

Royal Danish Academy of Sciences and Letters, 

26 


25 


24 


22, 97 

Royal Irish Academy, 130 

Ruffini’s missed subgroups, 88 

ruler-and-compass constructibility, 74 

Savants étrangers, 361 

semblables fonctions, 66 

simple permutations, 85 

Società Italiana delle Scienze, 84 

St. Petersburg Academy, 21, 62 

substituions 

order, 91 

substitutions, 91 

circular, 94 

degree, 91 

identical, 91 

powers, 91 

product, 91 

sum of period of roots, 78 

symmetric group, 72 

systems of conjugate substitutions, 85 

Taylor series, 312, 397 

teleology, 10 

tests of convergence, 400 

The Enlightenment, 9 

Transactions, 130 

Transactions of the Royal Danish Academy of 

Science, 23 

transformation, 11 

true value of series, 267 

Tuberculosis, 4 

Videnskabernes Selskab, 22 

Zeitschrift für Physik und Mathematik, 35, 123

RePoSS issues 

#1 (2008-7) Marco N. Pedersen: “Wasan: Die japanische Mathematik der Tokugawa 

Ära (1600-1868)” (Masters thesis) 

#2 (2004-8) Simon Olling Rebsdorf: “The Father, the Son, and the Stars: Bengt Strömgren 

and the History of Danish Twentieth Century Astronomy” (PhD dissertation) 

#3 (2009-9) Henrik Kragh Sørensen: “For hele Norges skyld: Et causeri om Abel og 

Darwin” (Talk) 

#4 (2009-11) Helge Kragh: “Conservation and controversy: Ludvig Colding and the imperishability 

of “forces”” (Talk) 

#5 (2009-11) Helge Kragh: “Subatomic Determinism and Causal Models of Radioactive 

Decay, 1903–1923” (Book chapter) 

#6 (2009-12) Helge Kragh: “Nogle idéhistoriske dimensioner i de fysiske videnskaber” 

(Book chapter) 

#7 (2010-4) Helge Kragh: “The Road to the Anthropic Principle” (Article) 

#8 (2010-5) Kristian Peder Moesgaard: “Mean Motions in the Almagest out of Eclipses” 

(Article) 

#9 (2010-7) Helge Kragh: “The Early Reception of Bohr’s Atomic Theory (1913-1915): A 

Preliminary Investigation” (Article) 

#10 (2010-10) Helge Kragh: “Before Bohr: Theories of atomic structure 1850-1913” (Article) 

#11 (2010-10) Henrik Kragh Sørensen: “The Mathematics of Niels Henrik Abel: Continuation 

and New Approaches in Mathematics During the 1820s” (PhD dissertation) 

#12 (2009-2) Laura Søvsø Thomasen: “In Midstream: Literary Structures in Nineteenth- 

Century Scientific Writings” (Masters thesis) 

#13 (2011-1) Kristian Danielsen and Laura Søvsø Thomasen (eds.): “Fra laboratoriet til 

det store lærred” (Preprint) 

#14 (2011-2) Helge Kragh: “Quantenspringerei: Schrödinger vs. Bohr” (Talk) 

#15 (2011-7) Helge Kragh: “Conventions and the Order of Nature: Some Historical Perspectives” 

(Talk) 

#16 (2011-8) Kristian Peder Moesgaard: “Lunar Eclipse Astronomy” (Article) 

#17 (2011-12) Helge Kragh: “The Periodic Table in a National Context: Denmark, 1880- 

1923” (Book chapter) 

#18 (2012-1) Henrik Kragh Sørensen: “Making philosophy of science relevant for science 

students” (Talk)

RePoSS (Research Publications on Science Studies) is a series of electronic 

publications originating in research done at the Centre for Science Studies, 

University of Aarhus. 

The publications are available from the Centre homepage 

(��). 

ISSN 1903-413X 

Centre for Science Studies 

University of Aarhus 

Ny Munkegade 120, building 1520 

DK-8000 Aarhus C 

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RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

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