# Who Gave You the Epsilon?: and Other Tales of Mathematical History

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*Marlow Anderson; Victor Katz; Robin Wilson*

MAA Press: An Imprint of the American Mathematical Society

This book picks up the history of mathematics from where Sherlock Holmes in Babylon left it. The forty articles of Who Gave You the Epsilon? continue the story of the development of mathematics into the nineteenth and twentieth centuries. The articles have all been published in the Mathematical Association of America journals and are in many cases written by distinguished mathematicians such as G. H. Hardy and B. van der Waerden. The articles are arranged thematically to show the development of analysis, geometry, algebra and number theory through this period. Each chapter is preceded by a Foreword, giving the historical background and setting and the scene, and is followed by an Afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book is ideal for anyone wanting to explore the history of mathematics.

#### Reviews & Endorsements

As a collection of interesting articles on the history of 19th- and 20th-century mathematics, the present volume is hard to beat. The 41 papers, covering many diverse areas, not just calculus, are mostly accessible to undergraduate mathematics majors, yet their professors will also likely enjoy them and learn quite a bit as well. Highly Recommended.

-- C. Bauer, Choice

The present volume is a sequel to Sherlock Holmes in Babylon and other tales of mathematical history, MAA Spectrum, Math Assoc. America, Washington, DC, 2004. The earlier book treated the period before 1800, while this book describes developments in the 19th and 20th centuries. It is an anthology of over 40 papers previously published in journals of the Mathematical Association of America, the majority in the American Mathematical Monthly, about a third in Mathematics Magazine and two in the College Mathematics Journal. Except for seven .Monthly papers from the years 1900 (2), 1913, 1920, 1934, 1937, and 1951, all the papers appeared between 1972 and 2000 inclusive. Many of the authors are respected historians of mathematics. Each of the four chapters is bracketed by a Foreword that gets forth the themes and an Afterward that provides a guide for further reading. There is a good mixture of material that focuses on mathematical developments and that treats the personalities and sociology of the mathematical community. For some topics, the treatment is quite detailed. In such a collection as this, the choice of topics is of necessity unbalanced; the papers are sorted into three chapters under the broad themes of analysis, geometry and axiomatics, and algebra and number theory. The final chapter includes three papers that survey the state of mathematics at the beginning, the midpoint and the end of the 20th century. This collection can be read with profit and enjoyment by both professional mathematicians and undergraduate students specializing in mathematics.

-- E.J. Barbeau, Mathematical Reviews

# Table of Contents

## Who Gave You the Epsilon?: and Other Tales of Mathematical History

Table of Contents pages: 1 2

- Cover cover11
- copyright page ii3
- Title page iii4
- Introduction vii8
- Contents ix10
- Analysis 112
- Foreword 314
- Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus JUDITH V. GRABINER 516
- Evolution of the Function Concept: A Brief Survey ISRAEL KLEINER 1425
- Introduction 1425
- Precalculus developments 1425
- Euler’s Introductio in Analysin Infinitorum 1526
- The vibrating-string controversy 1627
- Fourier and Fourier series 1829
- Dirichlet’s concept of function 1930
- Pathological functions 2132
- Baire’s classification scheme 2233
- Debates about the nature of mathematical objects 2334
- Recent developments 2435
- References 2637

- S. Kovalevsky: A Mathematical Lesson KAREN D. RAPPAPORT 2738
- Highlights in the History of Spectral Theory L. A. STEEN 3647
- 1 Principal axes theorem 3647
- 2 Infinite systems of linear equations 3748
- 3 Integral equations 3849
- 4 David Hilbert 3849
- 5 Hilbert-Schmidt spectral theory 3950
- 6 The Lebesgue integral 4152
- 7 Quantum mechanics 4152
- 8 John von Neumann 4354
- 9 Von Neumann–Stone spectral theory 4455
- 10 Gelfand-Naimark theorem 4556
- 11 Unfinished business 4657
- References 4859

- Alan Turing and the Central Limit Theorem S. L. ZABELL 5263
- Why did George Green Write his Essay of 1828 on Electricity and Magnetism? I. GRATTAN-GUINNESS 6172
- 1 Honor to Green 6172
- 2 Three strands in eighteenth-century mechanics 6172
- 3 Poisson and the appearance of divergence theorems 6273
- 4 Green and the place of surface integrals 6374
- 5 Sources and influences 6475
- 6 Options for publication 6576
- 7 On Green’s second period 6576
- 8 Recognition 6677
- References 6778

- Connectivity and Smoke-Rings: Green’s Second Identity in its First Fifty Years THOMAS ARCHIBALD 6980
- The History of Stokes’s Theorem VICTOR J. KATZ 7889
- The Mathematical Collaboration of M. L. Cartwright and J. E. Littlewood SHAWNEE L. MCMURRAN AND JAMES J. TATTERSALL 8899
- Dr. David Harold Blackwell, African-American Pioneer NKECHI AGWU, LUELLA SMITH, AND AISSATOU BARRY 98109
- Afterword 109120

- Geometry, Topology and Foundations 111122
- Foreword 113124
- Gauss and the Non-Euclidean Geometry GEORGE BRUCE HALSTED 115126
- History of the Parallel Postulate FLORENCE P. LEWIS 120131
- The Rise and Fall of Projective Geometry J. L. COOLIDGE 125136
- Notes on the History of Geometrical Ideas DAN PEDOE 133144
- A note on the history of the Cantor Set and Cantor function JULIAN F. FLERON 137148
- Evolution of the Topological Concept of “Connected” R. L. WILDER 142153
- A Brief, Subjective History of Homology and Homotopy Theory in this Century PETER HILTON 148159
- The Origins of Modern Axiomatics: Pasch to Peano H. C. KENNEDY 157168
- C. S. Peirce’s Philosophy of Infinite Sets JOSEPH W. DAUBEN 161172
- On the Development of Logics between the two World Wars I. GRATTAN-GUINNESS 172183
- 1 Introduction 172183
- 2 Logicism and its critics 173184
- 3 Formalism and its fate 174185
- 4 Recursion and computability: American logic 175186
- 5 The emergence of intuitionism 177188
- 6 Beyond first-order and finitary logics 177188
- 7 The rise of Polish logic 179190
- 8 Conclusions and comparisons 180191
- References 181192

- Dedekind’s Theorem:√2 ×√3 =√6 DAVID FOWLER 185196
- Afterword 192203

- Algebra and Number Theory 195206
- Foreword 197208
- Hamilton’s Discovery of Quaternions B. L. VAN DER WAERDEN 200211
- Hamilton, Rodrigues, and the Quaternion Scandal SIMON L. ALTMANN 206217
- The men involved: Hamilton and Rodrigues 206217
- The discovery of quaternions 208219
- In praise of Hamilton: the algebra of quaternions 209220
- The trouble starts 210221
- Quaternions and rotations: the first steps 211222
- An optical illusion: the rectangular rotation 211222
- The comical transformation (this heading contains a misprint) 212223
- The Rodrigues programme 213224
- The resolution of the paradoxes 215226
- The decline 216227
- Epilogue 217228
- References 218229

- Building an International Reputation: The Case of J. J. Sylvester (1814–1897) KAREN HUNGER PARSHALL AND EUGENE SENETA 220231
- The Foundation Period in the History of Group Theory JOSEPHINE E. BURNS 230241
- The Evolution of Group Theory: A Brief Survey ISRAEL KLEINER 237248
- The Search for Finite Simple Groups JOSEPH A. GALLIAN 254265
- 1 The alternating groups and the classical linear groups 255266
- 2 Range problem 1–660 255266
- 3 PSL(m, pn) 256267
- 4 Range problem to 1092 256267
- 5 Permutation representations and character theory 257268
- 6 Odd order problem 257268
- 7 Dickson’s simple groups 258269
- 8 The Mathieu groups 259270
- 9 Range problem to 6232 259270
- 10 Burnside’s p^a q^b theorem 260271
- 11 The Chevalley groups 260271
- 12 Groups of Lie type 260271
- 13 Sporadic simple groups 261272
- 14 Thompson’s N-paper 262273
- 15 The p^a q^b r^c problem 263274
- 16 The range problem to1,000,000 263274
- References 267278
- A Simple Song 269280

- Genius and Biographers: The Fictionalization of Evariste Galois TONY ROTHMAN 271282

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