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 Mathemat- ics During the 1820s. RePoSS: Research Publications on Science Studies 11. Aarhus: Centre for Science Studies, University of Aarhus. url: http://www.css.au.dk/reposs. 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 http://www.henrikkragh.dk/phd/. 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 Introduction1 1 Introduction3 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