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

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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
Page 1 and 2: RePoSS: Research Publications on Sc
Page 3 and 4: The Mathematics of NIELS HENRIK ABE
Page 5 and 6: The Mathematics of NIELS HENRIK ABE
Page 7 and 8: Contents Contents i List of Tables
Page 9: 8.3 Refocusing on the equation . .
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
Page 28 and 29: items published in the same year ar
Page 30 and 31: the Mittag-Leffler archives in Djur
Page 33 and 34: Chapter 1 Introduction In the after
Page 35 and 36: 1.2. The mathematical topics involv
Page 37 and 38: 1.3. Themes from early nineteenth-c
Page 39 and 40: 1.4. Reflections on methodology 9 l
Page 41 and 42: 1.4. Reflections on methodology 11
Page 43 and 44: 1.4. Reflections on methodology 13
Page 45: 1.4. Reflections on methodology 15
Page 48 and 49: 18 Chapter 2. Biography of NIELS HE
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30 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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19.2. The contents of ABEL’s 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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