Mathematics of the 19th Century: Vol. II: Geometry, Analytic Function Theory

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Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich
Springer Science & Business Media, Apr 30, 1996 - Mathematics - 291 pages
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The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century [in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers).
 

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Contents

Geometry
1
1 ANALYTIC AND DIFFERENTIAL GEOMETRY
3
The Differential Geometry of Monges Students
5
Gauss Disquisitiones generates circa superficies curvas
7
Minding and the Formulation of the Problems of Intrinsic Geometry
12
The French School of Differential Geometry
17
Differential Geometry at Midcentury
21
Differential Geometry in Russia
24
Mobius Theorie der elementaren Verwandschaft
101
The Topology of Surfaces in Riemanns Theorie der Abelschen Funktionen
102
The Multidimensional Topology of Riemann and Betti
103
Jordans Topological Theorems
104
The Klein Bottle
105
7 GEOMETRIC TRANSFORMATIONS
106
Helmholtz Paper Uber die Thatsachen die der Geometrie zu Grunde liegen
107
Kleins Erlanger Programm
109

The Theory of Linear Congruences
26
2 PROJECTIVE GEOMETRY
27
Poncelets Traite des proprietes projectives des figures
29
The Analytic Projective Geometry of Mobius and Plucker
31
The Synthetic Projective Geometry of Steiner and Chasles
36
Staudt and the Foundation of Projective Geometry
40
Cayleys Projective Geometry
43
3 ALGEBRAIC GEOMETRY AND GEOMETRIC ALGEBRA
44
Algebraic Surfaces
45
Geometric Computations Connected with Algebraic Geometry
47
Hamiltons Vectors
51
4 NONEUCLIDEAN GEOMETRY
53
Gauss Research in NonEuclidean Geometry
56
Janos Bolyai
57
Hyperbolic Geometry
58
J Bolyais Absolute Geometry
61
The Consistency of Hyperbolic Geometry
62
Propagation of the Ideas of Hyperbolic Geometry
65
Beltramis Interpretation
67
Cayleys Interpretation
69
Kleins Interpretation
71
Elliptic Geometry
73
5 MULTIDIMENSIONAL GEOMETRY
75
Cayleys Analytic Geometry of n Dimensions
76
Grassmanns Multidimensional Geometry
77
Pluckers Neue Geometrie des Raumes
78
The Multidimensional Geometry of Klein and Jordan
81
Riemannian Geometry
83
Riemanns Idea of Complex Parameters of Euclidean Motions
87
The Work of Christoffel Lipschitz and Suvorov on Riemannian Geometry
89
The Multidimensional Theory of Curves
90
Multidimensional Surface Theory
94
Multidimensional Projective Geometry
96
6 TOPOLOGY
97
Generalizations of Eulers Theorem on Polyhedra in the Early Nineteenth Century
98
Listings Vorstudien zur Topologie
99
Transference Principles
111
Cremona Transformations
113
CONCLUSION
115
Analytic Function
119
Development of the Concept of a Complex Number
121
Complex Integration
125
The Cauchy Integral Theorem Residues
128
Elliptic Functions in the Work of Gauss
132
Hypergeometric Functions
138
The First Approach to Modular Functions
145
Power Series The Method of Majorants
148
Elliptic Functions in the Work of Abel
153
CGJ Jacobi Fundamenta nova functionum ellipticarum
158
The Jacobi Theta Functions
162
Elliptic Functions in the Work of Eisenstein and Liouville The First Textbooks
166
Abelian Integrals Abels Theorem
173
Quadruply Periodic Functions
178
Results of the Development of Analytic Function Theory over the First Half of the Nineteenth Century
183
V Puiseux Algebraic Functions
189
Bernhard Riemann
198
Riemanns Doctoral Dissertation The Dirichlet Principle
201
Conformal Mappings
215
Karl Weierstrass
220
Analytic Function Theory in Russia Yu V Sokhotskii and the SokhotskiiCasoratiWeierstrass Theorem
227
Entire and Meromorphic Functions Picards Theorem
236
Abelian Functions
245
Abelian Functions Continuation
249
Automorphic Functions Uniformization
257
Sequences and Series of Analytic Functions
264
Conclusion
270
Literature
273
Collected Works and Other Original Sources
274
Auxiliary Literature to Chapter 1
279
Auxiliary Literature to Chapter 2
280
Index of Names
283
Copyright

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Page 287 - History of ! and Technology, Moscow, Russia (Eds) Mathematics in the 19th Century Mathematical Logic, Algebra, Number Theory, Probability Theory 1992. 322 pages. Hardcover ISBN 3-7643-2552-6 The history of nineteenth-century mathematics has been much less studied than that of preceding periods. The historical period covered in this book extends from the early nineteenth century up to the end of the 1930s, as neither 1801 nor 1900 are, in themselves, turning points in the history of mathematics...

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History of Topology
I.M. James
Limited preview - 1999

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