Tales of Mathematicians and Physicists

Simon Gindikin, Alan Shuchat9780387360263, 0-387-36026-3

This revised and greatly expanded second edition of the Russian text Tales of Mathematicians and Physicists contains a wealth of new information about the lives and accomplishments of more than a dozen scientists throughout five centuries of history: from the first steps in algebra up to new achievements in geometry in connection with physics. The heroes of the book are renowned figures from early eras, such as Cardano, Galileo, Huygens, Leibniz, Pascal, Euler, Lagrange, and Laplace, as well some scientists of last century: Klein, Poincar?, and Ramanujan.

A unique mixture of mathematics, physics, and history, this volume provides biographical glimpses of scientists and their contributions in the context of the social and political background of their times. The author examines many original sources, from the scientists’ research papers to their personal documents and letters to friends and family; furthermore, detailed mathematical arguments and diagrams are supplied to help explain some of the most significant discoveries in calculus, celestial mechanics, number theory, and mathematical physics. What emerges are intriguing, multifaceted studies of a number of remarkable intellectuals and their scientific legacy.

Written by a distinguished mathematician and accessible to readers at all levels, this book is a wonderful resource for both students and teachers and a welcome introduction to the history of science.

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Table of contents :
Preface to theEnglish Edition……Page 7
Preface to theThird Russian Edition……Page 12
Preface to theFirst Russian Edition……Page 16
Ars Magna(The Great Art)……Page 20
Scipione dal Ferro……Page 22
Gerolamo Cardano……Page 25
Cardano and Tartaglia……Page 28
Luigi Ferrari……Page 30
Ars Magna……Page 31
Remarks on Cardano’s Formula……Page 34
The Fourth-Degree Equation……Page 35
Ferrari and Tartaglia……Page 36
The Fate of Our Heroes……Page 37
Appendix……Page 40
I The Discovery of the Laws of Motion……Page 46
II The Medicean Stars……Page 64
Christiaan Huygens and Pendulum Clocks……Page 98
Pendulum Clocks……Page 101
Centrifugal Force and a Clock with a Conical Pendulum……Page 103
The Physical Pendulum……Page 106
Maritime Clocks……Page 107
Appendix……Page 108
I The Cycloid and the Isochronous Pendulum……Page 111
II Roulettes and Their Tangents……Page 127
III The Brachistochrone, or Yet Another Secret ofthe Cycloid……Page 134
Epilogue……Page 142
Blaise Pascal……Page 146
Sticks and Coins……Page 147
Hexagramme Mystique, or Pascal’s Great Theorem……Page 149
Pascal’s Wheel……Page 153
“Abhorring a Vacuum”……Page 154
“The Geometry of Chance”……Page 158
Louis de Montalte……Page 160
Amos Dettonville……Page 162
Pensées……Page 164
The Golden Age of Analysis……Page 168
Leibniz and his Mathematical Journey……Page 175
Mathematics and the “Conquest of the Minds” of Sovereigns……Page 181
Leonhard Euler……Page 188
The Academy’s First Years……Page 190
Euler in Petersburg……Page 192
In the Service of a “Royal Philosopher”……Page 196
In Russia Again……Page 201
A Great Legacy……Page 203
Arithmetic……Page 205
Analytic Number Theory……Page 209
Series and Infinite Products……Page 211
Additive Number Theory……Page 214
Analysis……Page 216
Geometry……Page 220
Mechanics……Page 222
Astronomy……Page 224
“Letters to a Princess”……Page 225
Concluding Remarks……Page 228
A Letter from Turin……Page 230
Giuseppe Luigi……Page 233
The Foundations of Statics……Page 235
The Principle of Least Action……Page 237
First Work in Astronomy……Page 238
Visit to Paris……Page 239
Lagrange in Berlin……Page 240
Analytical Mechanics……Page 241
Celestial Mechanics……Page 247
Arithmetic Works……Page 250
Algebraic Reflections……Page 251
Crisis……Page 254
In Paris……Page 255
Pedagogical Activities……Page 258
The Last Years……Page 261
Pierre-Simon Laplace……Page 263
Beaumont-Paris, 1749–1789……Page 264
Revolution, Empire, Restoration……Page 266
Celestial Mechanics……Page 270
The System of the World……Page 274
“Common Sense Reduced to Calculation”……Page 276
I Gauss’ Debut……Page 279
Braunschweig, 1777–1795……Page 280
Constructions with Straightedge and Compass……Page 283
A Few Words about Complex Numbers……Page 284
Regular -Gons and Roots of Unity……Page 285
Constructing the Regular 17-Gon……Page 286
Detailed Calculations……Page 289
Symmetry in the Set of Roots of Equation……Page 291
Possible Generalizations and Fermat Primes……Page 294
Concluding Remarks……Page 295
Quadratic Residues……Page 297
Fermat’s Theorem and Euler’s Criterion……Page 299
The Law of Quadratic Reciprocity……Page 306
The Favorite Science of the Greatest Mathematicians……Page 308
The Helmstadt Dissertation……Page 309
The Lemniscate and the Arithmetic–Geometric Mean……Page 310
Asteriods……Page 312
The Inner Geometry of a Surface……Page 315
Non-Euclidean Geometry……Page 317
Electrodynamics and Terrestrial Magnetism……Page 319
Appendix: Construction Problems Leadingto Cubic Equations……Page 321
A Knight’s Spurs……Page 326
A “Golden Century” of Geometry……Page 328
The Cayley–Klein Model……Page 330
The Erlanger Program……Page 332
An External Student in Riemann’s School……Page 335
The Last Forty Years……Page 336
The Magic World ofHenri Poincaré……Page 338
A Detour into Physics……Page 339
“Poincaria” and Its Geometry……Page 340
Distance and Displacement……Page 342
Rigid Bodies in Poincaria……Page 349
A Letter to Cambridge……Page 351
The Wonder from Kumbakonam……Page 353
“A Synopsis of Elementary Results in Pureand Applied Mathematics”……Page 355
From Numbers to Formulas……Page 356
Choosing a Profession……Page 357
In Cambridge……Page 359
Remembrance……Page 360
On the Advantages ofCoordinates and the Art ofChaining Hyperboloids……Page 362
The Complex World ofRoger Penrose……Page 381
“The Golden Age of Geometry”……Page 382
Projective Coordinates……Page 385
The Manifold of Lines (Plücker Coordinates)……Page 386
The Complex Picture……Page 387
Interpreting Real quadrics in the Language of Complex Lines(the Case of a Sphere)……Page 388
Realizing a Hyperboloid as a Family of Lines……Page 389
A Metric on the Manifold of Lines……Page 390
Self-Dual Metrics……Page 393