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

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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
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Page 15 and 16: List of Figures 2.1 NIELS HENRIK AB
Page 17: List of Boxes 1 The algebraic reduc
Page 21 and 22: Summary The present PhD dissertatio
Page 23 and 24: Preface to the 2004 edition For thi
Page 25: in connection with the Abel centenn
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Page 30 and 31: the Mittag-Leffler archives in Djur
Page 33 and 34: Chapter 1 Introduction In the after
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26 Chapter 2. Biography of NIELS HE
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Chapter 3 Historical background The
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3.2. ABEL’s position in mathemati
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3.3. The state of mathematics 43 tr
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3.4. ABEL’s legacy 45 As can be s
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Chapter 4 The position and role of
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4.1. Outline of ABEL’s results an
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4.2. Mathematical change as a histo
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4.2. Mathematical change as a histo
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58 Chapter 5. Towards unsolvable eq
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98 Chapter 6. Algebraic insolubilit
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100 Chapter 6. Algebraic insolubili
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Chapter 7 Particular classes of equ
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7.1. Solubility of Abelian equation
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7.2. Elliptic functions 153 Figure
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7.2. Elliptic functions 155 7.2.1 T
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7.3. The concept of irreducibility
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7.3. The concept of irreducibility
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7.4. Enlarging the class of solvabl
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164 Chapter 8. A grand theory in sp
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Part III Interlude: ABEL and the
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192 Chapter 9. The nineteenth-centu
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194 Chapter 10. Toward rigorization
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208 Chapter 11. CAUCHY’s new foun
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Chapter 12 ABEL’s reading of CAUC
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12.1. ABEL’s critical attitude 22
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12.3. Convergence 229 12.3 Converge
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12.3. Convergence 231 and {εm} was
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12.4. Continuity 233 it followed th
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12.4. Continuity 235 Actually, if t
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12.4. Continuity 237 DIRICHLET intr
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12.5. ABEL’s “exception” 239
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12.6. A curious reaction: Lehrsatz
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12.7. From power series to absolute
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12.7. From power series to absolute
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12.8. Product theorems of infinite
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12.8. Product theorems of infinite
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12.9. ABEL’s proof of the binomia
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12.10. Aspects of ABEL’s binomial
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12.10. Aspects of ABEL’s binomial
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266 Chapter 13. ABEL and OLIVIER on
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Chapter 14 Reception of ABEL’s co
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14.1. Reception of ABEL’s rigoriz
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14.2. Conclusion 281 of basic notio
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Part IV Elliptic functions and the
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286 Chapter 15. Elliptic integrals
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300 Chapter 16. The idea of inverti
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Chapter 17 Steps in the process of
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17.1. Infinite representations 323
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17.2. Elliptic functions as ratios
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17.2. Elliptic functions as ratios
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17.3. Characterization of ABEL’s
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Chapter 18 Tools in ABEL’s resear
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18.1. Transformation theory 333 The
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18.1. Transformation theory 335 Obv
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18.1. Transformation theory 337 Sum
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18.2. Integration in logarithmic te
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18.3. Conclusion 345 Summary. ABEL
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Chapter 19 The Paris memoir N. H. A
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19.1. ABEL’s approach to the Pari
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19.3. Additional, tentative remarks
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19.3. Additional, tentative remarks
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19.4. The fate of the Paris memoir
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19.6. Conclusion 377 hyperelliptic
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Chapter 20 General approaches to el
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20.2. Other ways of introducing ell
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20.3. Conclusion 383 All these four
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Chapter 21 ABEL’s mathematics and
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21.1. From formulae to concepts 389
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21.1. From formulae to concepts 391
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21.2. Concepts and classes enter ma
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21.3. The role of counter examples
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21.4. Conclusion 401 It is beyond t
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404 Appendix A. ABEL’s correspond
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Appendix B ABEL’s manuscripts Man
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1. Précis d’une théorie des fon
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Bibliography Abel (MS:351:A). “M
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Bibliography 413 in: Journal für d
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Bibliography 415 — (1990). “Geo
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Bibliography 417 — (1830). “Mé
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Bibliography 419 — (1768). “Ins
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Bibliography 421 Grattan-Guinness,
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Bibliography 423 Jahnke, H. N. (198
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Bibliography 425 Legendre, A. M. (1
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Bibliography 427 Rosen, M. (1981).
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Bibliography 429 Toti Rigatelli, L.
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432 Index of names Frobenius, Georg
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Index École Normale, 40, 187 Écol
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Index 437 Lagrange interpolation, 3
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