AMS / MAA SPECTRUM VOL 95 CALCULUSGems Brief Lives and Memorable Mathematics George F. Simmons 10.1090/spec/095 Calculus Gems Brief Lives and Memorable Mathematics Originally published in 1992 by McGraw Hill, Inc. Published by The Mathematical Association of America, 2007. ISBN: 978-1-4704-5128-8 LCCN: 2006939070 Copyright © 2007, held by the American Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2020 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 25 24 23 22 21 20 AMS/MAA SPECTRUM VOL 95 Calculus Gems Brief Lives and Memorable Mathematics George F. Simmons Coordinating Council on Publications James Daniel, Chair Spectrum Editorial Board Gerald L. Alexanderson, Chair Robert Beezer Jeffrey L. Nunemacher William Dunham J. D. Phillips Michael Filaseta Ken Ross Erica Flapan Marvin Schaefer Michael A. Jones Sanford Segal Keith Kendig Franklin Sheehan SPECTRUM SERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers. 777 Mathematical Conversation Starters, by John de Pillis 99 Points of Intersection: Examples-Pictures-Proofs,by Hans Walser. Translated fromthe orig- inal German by Peter Hilton and Jean Pedersen All the Math Thats Fit to Print, by Keith Devlin Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons Carl Friedrich Gauss: Titan of Science, by G. Waldo Dunnington, with additional material by Jeremy Gray and Fritz-Egbert Dohse The Changing Space of Geometry, edited by Chris Pritchard Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher The Early Mathematics of Leonhard Euler, by C. Edward Sandifer The Edge of the Universe: Celebrating JO Years of Math Horizons, edited by Deanna Haunsperger and Stephen Kennedy Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William 0. J. Moser The Genius of Euler: Reflectionson his Life and Work, edited by William Dunham The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. I Want to Be a Mathematician, by Paul R. Halmos Journeyinto Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Rec- reational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Numbers of the Professor, by Owen O'Shea and Underwood Dudley Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris Martin Gardner s Mathematical Games: The entire collection of his ScientificAmerican columns The Math Chat Book, by Frank Morgan Mathematical Adventures for Students and Amateurs, edited by David Hayes and Tatiana Shubin. With the assistance of Gerald L. Alexanderson and Peter Ross Mathematical Apocrypha, by Steven G. Krantz Mathematical ApocryphaRedux, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circles Vol I: In Mathematical Circles Quadrants I, II, III, IV,by Howard W. Eves Mathematical Circles Vol II: Mathematical Circles Revisited and Mathematical Circles Squared, by Howard W. Eves Mathematical Circles Vol III: Mathematical Circles Adieu and Return to Mathematical Circles, by Howard W. Eves Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences, by Howard Eves Mathematical Treks: From Surreal Numbers to Magic Circles, by lvars Peterson Mathematics: Queen and Servant of Science, by E.T. Bell Memorabilia Mathematica, by Robert Edouard Moritz Musings of the Masters: An Anthology of Mathematical Reflections, edited by Raymond G. Ayoub New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouths of Mathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers ... and the Return of Dr. Matrix, by Martin Gardner Po(vominoes, by George Martin Power Play, by Edward J. Barbeau The Random Walks of George Po(va, by Gerald L. Alexanderson Remarkable Mathematicians, from Euler to \'On Neumann. loan James The Search/or E.T. Bell. also known as John Taine. by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Sherlock Holmes in Babylon and Other Tales of Mathematical History. edited by Marlow Anderson, Victor Katz, and Robin Wilson Student Research Projects in Calculus. by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assis- tance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford The Words of Mathematics, by Steven Schwartzman For Hope and Nancy, my wife and daughter, who still make it all worthwhile CONTENTS Preface xiii Part A Brief Lives The Ancients A.I Thales (ca. 625-547 B.C.) Invented geometry and the concepts of theorem and proof; discovered skepticism as a tool of thought 3 A.2 Pythagoras (ca. 580-500 B.C.) Pythagorean theorem about right triangles; irrationality of V2 12 A.3 Democritus (ca. 460-370 B.C.) Atoms in physics and mathematics; volume of a cone 21 A.4 Euclid (ca. 300 B.C.) Organized most of the mathematics known at his time; Euclid's theorems on perfect numbers and the infinity of primes 28 A.5 Archimedes (ca. 287-212 B.C.) Determined volumes, areas and tangents, essentially by calculus; found volume and surface area of a sphere; centers of gravity; spiral of Archimedes; calculated 1r 35 Appendix: The Text of Archimedes 43 A.6 Apollonius (ca. 262-190 B.C.) Treatise on conic sections 47 Appendix: Apollonius' General Preface to His Treatise 50 A.7 Heron (first century A.D.) Heron's principle; area of a triangle in terms of sides 52 A.8 Pappus (fourth century A.D.) Centers of gravity linked to solids and surfaces of revolution 57 Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections 59 ix X CONTENTS A.9 Hypatia(A.D. 370?-415) The firstwoman mathematician 62 Appendix: A Proof of Diophantus' Theoremon Pythagorean Triples 66 The Forerunners A.10 Kepler (1571-1630) Founded dynamical astronomy; started chain of ideas leading to integral calculus 69 A.11 Descartes(1596-1650) Putative discoverer of analytic geometry; introduced some good notations; firstmodern philosopher 84 A.12 Mersenne(1588-1648) Lubricated the flow of ideas; cycloids; Mersenne primes 93 A.13 Fermat(1601-1665) Actual discoverer of analytic geometry; calculated and used derivatives and integrals; founded modernnumber theory; probability 96 A.14 Cavalieri (1598-1647) Developed Kepler's ideas into an early form of integration 106 A.15 Torricelli (1608-1647) Area of cycloid; many calculus problems, even improper integrals; invented barometer; Torricelli's law in fluid dynamics 113 A.16 Pascal(1623-1662) Mathematical induction; binomial coefficients; cycloid; Pascal's theorem in geometry; probability; influencedLeibniz 119 A.17 Huygens(1629-1695) Catenary; cycloid; circular motion; Leibniz's mathematics teacher (what a pupil! what a teacher!) 125 The Early Moderns A.18 Newton (1642-1727) Invented his own version of calculus; discovered Fundamental Theorem; used infiniteseries; virtually created physics and astronomy as mathematical sciences 131 Appendix: Newton's 1714(?) Memorandum of the Two Plague Years of 1665 and 1666 139 A.19 Leibniz (1646-1716) Invented his own better version of calculus; discovered Fundamental Theorem; invented many good notations; teacher of the Bernoullibrothers 141 A.20 The Bernoulli Brothers (James 1654-1705, John 1667-1748) Learned calculusfrom Leibniz, and developed and applied it extensively; infiniteseries; John was teacher of Euler 158 A.21 Euler (1707-1783) Organized calculus and developed it very extensively; codified analytic geometry and trigonometry; introduced symbols e, n, CONTENTS xi i, f(x), sinx, cosx; infiniteseries and products; calculus of variations; number theory; topology; mathematical physics; etc. 161 A.22 Lagrange(1736-1813) Calculus of variations; analytical mechanics 169 A.23 Laplace(1749-1827) Laplace's equation; analytic probability; celestial mechanics 171 A.24 Fourier (1768-1830) Fourier series; the heat equation 173 The Mature Modems A.25 Gauss (1777-1855) Initiated rigorous analysis with convergence proofs forinfinite series; number theory; complex numbers in analysis, algebra and number theory; differential geometry; non-Euclidean geometry; etc. 175 A.26 Cauchy (1789-1857) Careful treatment of limits,