## Mathematics of the 19th Century: Geometry, Analytic Function TheoryAndrei N. Kolmogorov, Adolf-Andrei P. Yushkevich 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 |

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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...