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

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346 Chapter 18. Tools in ABEL’s research on elliptic transcendentals 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
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RePoSS: Research Publications on Sc
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The Mathematics of NIELS HENRIK ABE
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The Mathematics of NIELS HENRIK ABE
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Contents Contents i List of Tables
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8.3 Refocusing on the equation . .
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V ABEL’s mathematics and the rise
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List of Figures 2.1 NIELS HENRIK AB
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List of Boxes 1 The algebraic reduc
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Summary The present PhD dissertatio
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Preface to the 2004 edition For thi
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in connection with the Abel centenn
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items published in the same year ar
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the Mittag-Leffler archives in Djur
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Chapter 1 Introduction In the after
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1.2. The mathematical topics involv
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1.3. Themes from early nineteenth-c
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1.4. Reflections on methodology 9 l
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1.4. Reflections on methodology 11
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18 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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Page 363 and 364: 18.1. Transformation theory 333 The
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21.3. The role of counter examples
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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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