## Mathematics of the 19th Century: Vol. II: Geometry, Analytic Function Theory (Google eBook)This book is the second volume of a study of the history of mathematics in the nineteenth century. The first part of the book describes the development of geometry. The many varieties of geometry are considered and three main themes are traced: the development of a theory of invariants and forms that determine certain geometric structures such as curves or surfaces; the enlargement of conceptions of space which led to non-Euclidean geometry; and the penetration of algebraic methods into geometry in connection with algebraic geometry and the geometry of transformation groups. The second part, on analytic function theory, shows how the work of mathematicians like Cauchy, Riemann and Weierstrass led to new ways of understanding functions. Drawing much of their inspiration from the study of algebraic functions and their integral, these mathematicians and others created a unified, yet comprehensive theory in which the original algebraic problems were subsumed in special areas devoted to elliptic, algebraic, Abelian and automorphic functions. The use of power series expansions made it possible to include completely general transcendental functions in the same theory and opened up the study of the very fertile subject of entire functions. This book will be a valuable source of information for the general reader, as well as historians of science. It provides the reader with a good understanding of the overall picture of these two areas in the nineteenth century and their significance today. |

### What people are saying - Write a review

### Contents

1 | |

3 | |

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 |

273 | |

274 | |

279 | |

280 | |

283 | |

### Common terms and phrases

### Popular passages

### References from web pages

JSTOR: Mathematics of the 19th Century: Mathematical Logic **...**

Edged by Darrel Haile Indiana University Blooming IN47405 Mathematics of the 19th Century: Mathematical Logic Algebra, Number Theory, Probability Theory. ...

links.jstor.org/ sici?sici=0002-9890(199404)101%3A4%3C369%3AMOT1CM%3E2.0.CO%3B2-X

Mathematics of the 19th Century, Vol. 1 (2nd Revised Edition) an **...**

Sankhy¯a : The Indian Journal of Statistics. 2002, Volume 64, Series B, Pt.1, pp 103-105. Mathematics of the 19th Century, Vol. 1 (2nd Revised Edition) ...

sankhya.isical.ac.in/ search/ 64b1/ 64b1rev2.pdf

Bookreview: Mathematics of the 19th Century: Geometry analytic **...**

Meccanica 32: 261, 1997. Mathematics of the 19th Century: geometry analytic function theory. an Kolmogorov, ap Yushkevich (eds.) Birkhauser Verlag, Basel ...

www.springerlink.com/ index/ M6526670K7W4J681.pdf

Mathematics and Computer Education: Mathematics of the 19th **...**

bnet. findarticles > Mathematics and Computer Education > Spring 2001 > Article > Print friendly. Mathematics of the 19th Century: Constructive Function ...

findarticles.com/ p/ articles/ mi_qa3950/ is_200104/ ai_n8932952/ print

Mathematics of the 19th century

Mathematics of the 19th century: function theory according to Chebyshev, ordinary differential equations, calculus of variations, theory of finite ...

portal.acm.org/ citation.cfm?id=295324&

MATHEMATICS OF THE 19TH CENTURY: GEOMETRY, ANALYTIC FUNCTION **...**

MATHEMATICS OF THE 19TH CENTURY: GEOMETRY, ANALYTIC FUNCTION THEORY Edited by AN Kolmogorov and AP Yushkevich: 291 pp., DM. 118. ...

journals.cambridge.org/ abstract_S0024609397243614

ingentaconnect Bookreview: Mathematics of the 19th Century **...**

Bookreview: Mathematics of the 19th Century: Geometry analytic function theory (Claudio Procesi). Authors: Kolmogorov an; Yushkevich ap ...

www.ingentaconnect.com/ content/ klu/ mecc/ 1997/ 00000032/ 00000003/ 00133560;jsessionid=1pm6qis685pq.alice?format=print

K authors

Mathematics of the 19th century : mathematical logic, algebra, number theory, probability theory (Basel, 1992). M Korsunsky, The atomic nucleus (New York, ...

www-groups.dcs.st-and.ac.uk/ ~history/ Bibliography/ K.html

Sergei Sergeevich Demidov (on his 60th birthday)

Russian Math. Surveys 58:6 1241–1244. c 2003 RAS(dom) and LMS. Uspekhi Mat. Nauk 58:6 189–192. DOI 10.1070/rm2003v058n06abeh000694 ...

www.iop.org/ EJ/ article/ 0036-0279/ 58/ 6/ M22/ RMS_58_6_M22.pdf

KOLMOGOROV BOOKS

Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory Edited by an Kolmogorov and ap Yushkevich ...

www.kolmogorov.com/ books1.html