LIVES IN MATHEMATICS John Scales Avery February 8, 2021 INTRODUCTION1 I hope that this book will be of interest to students of mathematics and other disciplines related to mathematics, such as theoretical physics and the- oretical chemistry. An apology I must apologize for the fact that the level of the book is uneven. Chapters 1-8, as well as Appendices A and B, are suitable for students who would like to learn calculus and differential equations. However, the remainder of the book is more demanding, and is suitable for more advanced students. Human history as cultural history We need to reform our teaching of history so that the emphasis will be placed on the gradual growth of human culture and knowledge, a growth to which all nations and ethnic groups have contributed. This book is part of a series on cultural history. Here is a list of the other books in the series that have, until now, been completed: Lives in Education • Lives in Poetry • Lives in Painting • Lives in Engineering • Lives in Astronomy • Lives in Chemistry • Lives in Medicine • Lives in Ecology • Lives in Physics • Lives in Economics • Lives in the Peace Movement • 1This book makes use of chapters and appendices that I have previously written, but most of the material in the book’s 19 chapters is new. My son, Associate Professor James Emil Avery of the Niels Bohr Institute, University of Copenhagen, is the co-author of Appendices D, E, F and G. The pdf files of these books may be freely downloaded and circulated from the following web addresses: https://www.johnavery.info/ http://eacpe.org/about-john-scales-avery/ https://wsimag.com/authors/716-john-scales-avery 4 Contents 1 PYTHAGORAS 11 1.1 The Pythagorean brotherhood . 11 1.2 Pythagorean harmony . 11 1.3 Geometry as a part of religion . 16 2 EUCLID 19 2.1 Alexandria . 19 2.2 The Museum and the Great Library of Alexandria . 19 2.3 Euclid is called to the Museum . 20 2.4 The eight books of Euclid's Elements ..................... 22 2.5 Euclid's Book I, On basic plane geometry ................... 22 3 ARCHIMEDES 35 3.1 Heiron's crown . 35 3.2 Invention of differential and integral calculus . 37 3.3 Statics and hydrostatics . 40 3.4 Don't disturb my circles! . 41 4 AL-KHWARIZMI 45 4.1 Al-Khwarizmi's life . 51 4.2 The father of algebra . 51 4.3 Contributions to astronomy . 52 4.4 Contributions to geography . 52 5 OMAR KHAYYAM 55 5.1 Omar's family and education . 55 5.2 Invited to Isfahan . 55 5.3 Linking algebra and geometry . 56 5.4 Omar Khayyam anticipates non-Euclidean geometry . 56 5.5 The Rub´aiy´at . 56 6 RENE´ DESCARTES 71 6.1 Uniting geometry and algebra . 71 5 6 CONTENTS 6.2 Descartes' work on Optics, physiology and philosophy . 77 6.3 Descartes' tragic death . 77 7 NEWTON 79 7.1 Newton's early life . 79 7.2 Newton becomes a student at Cambridge . 79 7.3 Differential calculus . 82 7.4 Optics . 88 7.5 Integral calculus . 92 7.6 Halley visits Newton . 99 7.7 The conflict over priority between Leibniz and Newton . 103 7.8 Political philosophy of the Enlightenment . 108 7.9 Voltaire and Rousseau . 111 8 THE BERNOULLI'S AND EULER 119 8.1 The Bernoullis and Euler . 119 8.2 Linear ordinary differential equations . 121 8.3 Second-order differential equations . 122 8.4 Partial differentiation; Daniel Bernoulli's wave equation . 124 8.5 Daniel Bernoulli's superposition principle . 126 8.6 The argument between Bernoulli and Euler . 126 9 FOURIER 133 9.1 A poor taylor's son becomes Napoleon's friend . 133 9.2 Fourier's studies of heat . 133 9.3 Fourier analysis . 134 9.4 Fourier transforms . 137 9.5 The Fourier convolution theorem . 141 9.6 Harmonic analysis for non-Euclidean spaces . 142 9.7 Fourier's discovery of the greenhouse effect . 142 10 JOSEPH-LOUIS LAGRANGE 145 10.1 A professor at the age of 19! . 145 10.2 Successor to Euler at the Berlin Academy . 145 10.3 Lagrange is called to Paris . 146 10.4 The calculus of variations . 146 10.5 Cyclic coordinates . 152 11 CONDORCET 157 11.1 Condorcet becomes a mathematician . 157 11.2 Human rights and scientific sociology . 158 11.3 The French Revolution . 164 11.4 Drafting a new constitution for France . 164 CONTENTS 7 11.5 Hiding from Robespierre's Terror . 165 11.6 Condorcet writes the Esquisse ......................... 165 12 HAMILTON 171 12.1 Uniting optics and mechanics . 174 12.2 Professor of Astronomy at the age of 21 . 174 12.3 Hamilton's unified formulation . 174 12.4 Quaternions . 180 13 ABEL AND GALOIS 183 13.1 Group theory . 194 13.2 Abel's family and education . 196 13.3 Abel's travels in Europe . 197 13.4 A list of mathematical topics to which Abel contributed . 199 13.5 The life and work of Everiste´ Galois . 199 13.6 Mathematical contributions of Galois . 200 14 GAUSS AND RIEMANN 205 14.1 Gauss contributed to many fields . 205 14.2 Normal distributions in probability theory . 207 14.3 Bernhard Riemann's life and work . 207 14.4 Functions of a complex variable . 220 15 HILBERT 227 15.1 David Hilbert's life and work . 227 15.2 Hilbert space . 228 15.3 Generalized Fourier analysis . 229 15.4 Projection operators . 229 15.5 Some quotations from David Hilbert . 230 16 EINSTEIN 247 16.1 Family background . 247 16.2 Special relativity theory . 259 16.3 General relativity . 261 16.4 Metric tensors . 262 16.5 The Laplace-Beltrami operator . 266 16.6 Geodesics . 270 16.7 Einstein's letter to Freud: Why war? . 271 16.8 The fateful letter to Roosevelt . 273 16.9 The Russell-Einstein Manifesto . 277 8 CONTENTS 17 ERWIN SCHRODINGER¨ 299 17.1 A wave equation for matter . 299 17.2 Felix Bloch's story about Schr¨odinger. 301 17.3 Separation of the equation . 302 17.4 Solutions to the radial equation . 303 17.5 Fock's momentum-space treatment of hydrogen . 305 17.6 The Pauli exclusion principle and the periodic table . 309 17.7 Valence bond theory . 321 17.8 What is life? . 321 18 DIRAC 337 18.1 Dirac's relativistic wave equation . 337 18.2 Some equations . 340 18.3 Lorentz invariance and 4-vectors . 341 18.4 The Dirac equation for an electron in an external electromagnetic potential 343 18.5 Time-independent problems . 344 18.6 The Dirac equation for an electron in the field of a nucleus . 345 19 SHANNON 349 19.1 Maxwell's demon . 349 19.2 Statistical mechanics . 351 19.3 Information theory; Shannon's formula . 352 19.4 Entropy expressed as missing information . 357 19.5 Cybernetic information compared with thermodynamic information . 360 19.6 The information content of Gibbs free energy . 360 A TABLES OF DIFFERENTIALS, INTEGRALS AND SERIES 367 B THE HISTORY OF COMPUTERS 373 B.1 Pascal and Leibniz . 373 B.2 Jacquard and Babbage . 375 B.3 Harvard's sequence-controlled calculator . ..