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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. ⃝∞ 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 : 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 , 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-, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Wrought, by Underwood Dudley Out of the Mouths of , 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.) about right triangles; irrationality of V2 12 A.3 (ca. 460-370 B.C.) Atoms in physics and mathematics; volume of a cone 21 A.4 (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 (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.) 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 (A.D. 370?-415) The firstwoman mathematician 62 Appendix: A Proof of ' 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, continuity, derivatives, integrals, series; complex analysis 185 A.27 Abel(1802-1829) Binomial series; fifthdegree equation; integral calculus; elliptic functions 187 A.28 Dirichlet (1805-1859) Convergence of Fourier series; modem definition of a function; analytic number theory 191 A.29 Liouville (1809-1882) Integrals of elementary functions; transcendental numbers 193 A.30 Hermite(1822-1901) Transcendence of e; Hermitian matrices; elliptic functions 195 A.31 Chebyshev(1821-1894) Probability; distribution of prime numbers 197 A.32 Riemann (1826-1866) Riemann integral; Riemann rearrangement theorem; Riemannian geometry; Riemann zeta function; complex analysis 200 A.33 Weierstrass (1815-1897) Foundations of calculus; complex analysis; continuous nowhere-differentiablefunction; Weierstrassian rigor 207

Part B Memorable Mathematics B.1 The Pythagorean Theorem 217 Appendix: The Formulas of Heron and Brahmagupta 222 B.2 More about Numbers: Irrationals, PerfectNumbers, and Mersenne Primes 226 B.3 Archimedes' Quadrature of the Parabola 233 B.4 The Lunes of 237 xii CONTENTS

B.5 Fermat's Calculation of H xn dx forPositive Rational n 240 B.6 How Archimedes Discovered Integration 242 B.7 A Simple Approach to E = Mc2 246 B.8 Rocket Propulsion in Outer Space 249 B.9 A Proof of Vieta's Formula 252 B.10 An Elementary Proof of Leibniz's Formula � = 1 - ½ + ¼ - ½ + · · · 254 B.11 The Catenary, or Curve of a Hanging Chain 256 B.12 Wallis's Product 260 B.13 How Leibniz Discovered His Formula � = 1 - ½ + ¼ - ½ + · · 264 2 : B.14 Euler's Discovery of the Formula !:';" 2 = : 267 2 B.15 A Rigorous Proof of Euler's Formula !:';" \ n: 270 n = 6 B.16 The Sequence of Primes 272 B.17 More About Irrational Numbers. n:Is Irrational 282 Appendix: A Proof that e Is Irrational 285 B.18 Algebraic and Transcendental Numbers. e Is Transcendental 286 B.19 The Series !: _!_of the Reciprocals of the Primes 294 Pn B.20 The Bernoulli Numbers and Some Wonderful Discoveries of Euler 298 B.21 The Cycloid 303 B.22 Bernoulli's Solution of the Brachistochrone Problem 314 B.23 Evolutes and Involutes. The Evolute of a Cycloid 318 1 2 B.24 Euler's Formula !:';" = n: by Double Integration 323 n 2 6 B.25 Kepler's Laws and Newton's Law of Gravitation 326 B.26 Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem 336

Answers to Problems 345

Index 347 PREFACE

On coming to the end of my work on this book and thinking again about its nature and purpose, I am reminded of W. H. Fowler's Preface to his great Modern English Usage: "I think of it as it should have been, with its prolixities docked, its dullnesses enlivened, its fads eliminated, its truths multiplied." And also of W. H. Auden's rueful admission: "A poem is never finished, only abandoned." Some readers will recognize that this book has been reconstructed but of two massive appendices in my 1985 calculus book, with many additions, rearrangements and minor adjustments. Its direct practical purpose is to provide auxiliary material for students taking calculus courses, or perhaps courses on the . There have been a number of requests that this material be made separately available, and I have been happy to take advantage of this occasion to fill in some gaps and reconsider my opinions. I had a friend who said to me once, "I should probably spend about an hour a week revising my opinions." I treasure the remark and value the opportunity to act upon it. 1 My overall aims are bound up with the question, "What is mathematics for?" and with its inevitable answer, "To delight the mind and help us understand the world." I hold the naive but logically impeccable view that there are only two kinds of students in our colleges and universities: those who are attracted to mathematics; and those who are not yet attracted, but might be. My intended audience embraces both types. Part A. This half of the book, entitled Brief Lives, amounts to a biographical history of mathematics from the earliest times to the late nineteenth century. It has two main purposes.

1 The friend was George S. McCue, late of the Colorado College English Department.

xiii DV PREFACE

First, I hope in this way to "humanize" the subject, to make it transparently clear that great human beings created it by great efforts of genius, and thereby to increase students' interest in what they are studying. Science-and in particular mathematics-is something that men and women do, and not merely a mass of observed and abstract theory. The minds of most people turn away from problems-veer off, draw back, avoid contact, change the subject, think of something else at all costs. These people-the great majority of the human race-find solace and comfort in the known and the familiar, and avoid the unknown and unfamiliar as they would deserts and jungles. It is as hard for them to think steadily about a difficultproblem as it is to hold together the north poles of two strong magnets. In contrast to this, a tiny minority of men and women are drawn irresistibly to problems: their minds embrace them lovingly and wrestle with them tirelessly until they yield their secrets. It is these who have taught the rest of us most of what we know and can do, from the wheel and the lever to metallurgy and the theory of relativity. I have written about some of these people from our past in the hope of encouraging a few in the next generation. My purpose is connected with the fact that many students from the humanities and social sciences are compelled against their will to study calculus as a means of satisfying academic requirements. The profound connections that join mathematics to the history of philosophy, and also to the broader intellectual and social history of Western civilization, are often capable of arousing the passionate interest of these otherwise indifferent students. Part B. In teaching calculus over a period of many years, I have collected a considerable number of miscellaneous topics from number theory, geometry, science, etc., which I have used for the purpose of opening doors and forging links with other subjects . . . and also for breaking the routine and lifting the spirits. Many of my students have found these "nuggets" interesting and eye-opening. I have collected most of these topics in this part in the hope of making a few more converts to the view that mathematics, while sometimes rather dull and routine, can often be supremely interesting. The English mathematician G. H. Hardy said, "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." Part B of this book contains a wide variety of these patterns, arranged in an order roughly corresponding to the order of the ideas in most calculus courses. Some of the sections even have a few problems, to give additional focus to the efforts of students who may read them: Sections A.14, B.l, B.2, B.16, B.21, B.25. I repeat the fervent hope I have expressed in other books, that any readers who detect flaws or errors of fact or judgment will do me the great kindness of letting me know so that repairs can be made. George F. Simmons ABOUT TIIEAUTIIOR

George Simmons has the usual academic degrees (Caltech, Chicago, Yale), and taught at several colleges and universities before joining the faculty of Colorado College in 1962, where he was Professor of Mathematics. He is the author of Introduction to Topology and Modern Analysis (1963), Differential Equations with Applications and Historical Notes (1972, Second Edition, 1991), Precalculus Mathematics in a Nutshell (1981), Calculus with Analytic Geometry (1985, Second Edition 1996), and with Steven Krantz, Differential Equations: Theory, Technique, Practice (2006). When not working, eating, drinking or cooking, Professor Simmons is likely to be traveling (Western and Southern Europe, Turkey, Israel, Russia, China, Southeast Asia), playing pool, or reading (literature, history, biography and autobiography, science, and enough thrillersto achieve enjoyment without guilt.)

ANSWERS TO PROBLEMS

Section B.2 2. Because ( 4n + 2)2 is a multiple of 4. 6. If "Vm= a/ b where a and b are positive integers with no common factor > 1, then an = mb" and every prime factor of b must be a prime factor of a (why?), sob= 1. 7. All n's except 1, 10, 1()2, 1()3, .... 8. Assume that xis a rational number a/b (with b > 0) in lowest terms, and show that b = 1 as in Problem 6.

Section B.16 2. (c) 100=4· 25 = 10 · 10. 3. (a) If N is not prime, then since (4m -1)(4n-1) = 4(4mn -m -n) + 1, there is no guarantee that N has a prime factorof the form 4n + 1.

Section 8.21 '\/2ay-y2 1. a sin-1----= '\/2ay -y2 +x. a

4. (a) := v0(1 -cos 8), := Vo sin 8;

(b) 2v0, when Pis at the top of the cycloid; (c) v0 , when 8 = n/2 + 2nn for some integer n.

345 346 MEMORABLE MATIIEMATICS

2 6. }.1ra • 7. 6a. 2 8. ¥na • 312 312 10. x = aG) , y = a(�) • 16 11. x =2b cos 8 +b cos 28, y =2b sin 8-b sin28; 3 a. a b a b 12. x = (a + b) cos 8 - b cos ; 8, y = (a + b) sin 8 - b sin ; 8. 14. x = 3b cos 8 - b cos 38, y = 3b sin 8 - b sin 38; 12a.

Section 8.25 1. Since 2 3 = ) M (4,r G )(aT2 ' determine the ratio a 3/T2 for any particular planet (for instance, the earth) and proceed with the arithmetic. 2. (a) 2v'28e2.8 years; (b) 3¥35!!5.2 years; (c) 125 years. INDEX

Abel,Niels Henrik, 1n,183, 187,208,211 Area quoted on Gauss, ln of a cycloid, 115,304 Abelian integrals and functions,212 of an ellipse, 41 Adams,J. C., 334 Aristarchus, 43 Adamson, I. T., 287 , 4, 6, 17, 19, 23, 43, 152 Adler, Mortimer J.,48, 76 worthless physics of, 85 Aleksandrov, A. D., 344 Arithmetic,fundamental theorem of, 273 Alexander, H. G., 153 Arithmeticprogression, primes in,274 Alexandrian Museum and Library,29, 275 Armenian proverbs, 208,344 Algebra,337 Artin, E., 58 Cayley,341 Associative law, 337 division,338 Astroid, 307 fundamental theorem of, 178 Astrology, 71 nonassociative,340 Astronomical unit, 334 Algebraic number,286 Atomic theory, Democritus',24 theory, 273 Attalus I,48 Algebras, Lie, 340 Aubrey,John, 30 Analysis, 275 Augustine, Saint,18, 88 Analytic number theory,280, 294 ,13 Andrews, G. E.,165 Andronicus of ,43 Bacon, Francis, 21 Apollonius, 47,63, 82 Ball, W. W. Rouse, 134 problem of, 50 Barclay, Dr.,2 Apostol,T. M., 280 Barometer, 114, 121 Aquinas, empty doctrines of, 85 Barrow, Isaac, 132 Arago,F., 161 Baseless superstitions, 85 Archimedes, 21, 35,47, 52, 56, 79, 80, 81, Behold!, style of proof, 8,220 108,109,222,233,242,245,275 Belhoste, B., 186 screw, 36 Bell,E. T., 69, 95,96, 103, 105, 164, 171, 181 spiral,40 Bentley,Richard, 135 347 348 INDEX

Bernoulli Champollion,J., 173 James,156, 158,163,258,309 Characteristic,Euler, 168 John,126, 136,156,157,158,162,163, Chastity, St. Augustine on, 18 258,308,309,314 Chebyshev, Pafnuty Lvovich, 197, 277 numbers,299 Child,J. M., 129, 154 Bernoulli's theorem, 159 Christina, Queen of Sweden, 91 Bessel,F. W.,182, 183 Churchill,Winston, quoted on George I, 154 Bhaskara, 220 Cicero,Marcus Tullius, 22, 38 Bochner,Salomon, 4, 69,169, 200 Circular error,127, 310 Boeotians, 183 Circumferenceof earth, 275 Boil on faceof Europe, Louis XIV, 144 Clark Bolyai Arthur,A., 177 Johann, 183 G. N.,126 Wolfgang, 182 Dr. Samuel, 152 Bolza, Oskar, 209 Cohen,I. Bernard,134, 140 Bolzano,Bernard, 213 Colbert,Louis XIV's minister,127 Book-burning,'s, 22 Cole, F. N., 95 Boole, George,148 Commutative law, 337 Bott, R., 341 Complex numbers, 338 Boyer, C. B., 162 Composite number, 272 Brachistochrone, 158, 316 Conant,James B., 114 problem, 136, 160,308, 314 Conic section, 59 Brahe,Tycho, 69, 73, 327 Constant, Euler's, 164 golden nose of, 74 Constructible regular polygons,31, 176 Brahmagupta's formula,224 Constructions,geometric, 287 Bray, Vicar of, 172 Coolidge,J. L., 263 Brouncker, Lord, 164 Copernicus, Nicolaus, 70,327 Brouncker's continued fraction, 164,263 Cornford, F. M.,18 Brun, Viggo, 276 Cotangent,partial fractions expansion of,301 Bulls and cows, Chebyshev quoted on, 198 Courant,R., 31, 168,280,287 Bury,J. B.,65 Couturat, L., 150 Butterfield,H., 127 Crelle,A. L.,188 ,189 Curvature,center of, 318 Cairns, H.,144 Curves,geodesic, 203, 204 Cajori, F., 156,163 Cycloid, 94, 114, 303, 317, 318, 322 Calculus,invention of, 102 area of, 115, 304 Canfora, Luciano,63 length of,115, 304 Cantor,G., 202,213 tangent to, 115,305,312 Catenary, 126, 256 Cycloidal pendulum, Huygens' construction Cauchy, Augustin-Louis,104, 185, 189 of,322 Cavalieri,Bonaventura, 82,102, 106, 113, Cyril, Archbishop of , 65 155 Cavalieri's principle, 108 Declaration of Independence,quoted, 34 Cavendish, Henry, 335 Dedekind, Richard,179, 213 Cayley Deltoid,312 Arthur, 215, 341 Democritus,21, 80, 193, 243 algebra, 341 Democritus' numbers, 341 atomic theory, 24 Celestial mechanics,334 formula, 26 Center of curvature, 318 materialism, 25 Central force,330 Descartes, Rene,78 , 84,97, 115,305,312 Central gravitational force,331 and Snell's Jaw, 100 Century of Genius, 85, 130 erroneous laws of impact, 127 INDEX 349

foliumof, 101 lemma,280 jealous of Pascal,120 theorem on the infinityof primes,273 quoted on Eudemus,8, 48, 50 Constantijn Huygens,126 Euler, Leonhard,63, 103, 104, 105, 160, 161, curves,117 170,198,228,230,276,281,285,294, Devil,and Fermat's Last Theorem, 105 298, 325,341 Dickson, L. E.,340 characteristic,168 Differentialgeometry, intrinsic, 203 Euler's Dijksterhuis,E. J.,36 constant,164 Laertius,5, 22 formula ,64 fore; 8, 162 Diophantus,63, 96,165 forpolyhedra, 167 theorem, 66 for E 1/n2, 267, 270,323 Dirichlet,P. G. L.,105, 174,200 for E l/n2\ 302 principle,192, 201 identity, 204,295 theorem on primes,274 infinite productfor sine, 163, 269 Disbelief, importanceof, 9 Even perfect numbers, theorem of,228 Distance,mean, of planet,333 Evolute,318 Distributive laws,337 Eynden, C. V., 297 Division algebra, 338 Dobell, C.,129 Factorization theorem, unique,227, 273 Donne,John, 78 Fadiman, Clifton,105 Drachmann,A. G., 52,54 Fauvel,John,91,182 Dudley, Underwood,105 Fermat, Pierre, 63, 110, 121,165 , 240 Duncan,A. M.,73 Fermat's Dunnington,G. Waldo,179 principle of least time, 101, 314 theorem, 103 e last,105, 288 irrationality of,285 on polygonalnumbers, 104 transcendence of,290 two squares, 227, 344 2 E=Mc , 247 Fifth-degree equation, 188, 195 Earth,circumference of, 275 Figurate numbers, 15 Eddington, Sir Arthur,19 Folium of Descartes, 101 Edwards,H. M.,105, 280 Fontenelle, 8.,142 Einstein,Albert, 131,204,246 Foolish legacies of Plato, 39 quoted on Riemanniangeometry, 204 Force Einstein's law of motion, 247 central, 330 Elementary functions,integrals of, 194 central gravitational,331 Elements, Euclid's, 15 Formula Ellipse,59, 61 Brahmagupta's,224 area of, 41 Brouncker's, 263 Elliptic Democritus',26 functions,189, 195, 211 Euler's integrals, 211 fore; 8, 162 ,24 forpolyhedra, 167 2 Energy, potential,259 for E 1/n , 267, 270, 323 ,25 for E 1/n2\ 302 Epicycloid, 305 Heron's,52, 222 , 35, 41, 242, 275 Leibniz'sfor :rr: I 4, 254, 264 sieve of,275 Vieta's, 252 Euclid,9, 29, 94, 100,228, 230,294 Four squares theorem, Lagrange's, 227, 341 Euclid's Fourier, Joseph, 173, Elements, 15 series, 192,202 3S0 INDEX

Francis,Sir Frank C.,147 Newton's law of, 78, 326 Frankfurt,H. G.,150, 153 inverse square,328 Frederick the Great,162, 170 Gravitational force,central, 331 Freeman,Kathleen, 22 Gray,J., 182 Frobenius,G., 339 Great Sulk, 138 Function Greek geometry, 15 Abelian,212 Green, George,181 concept of,192 elementary,integrals of,194 Hadamard,Jacques,205,206,277 elliptic,189, 195, 211 Hadlock,C. R.,287 gamma, 164 Hahn,Hans, 214 nowhere-differentiable,214 Hall,A. Rupert, 145 Riemann zeta,205, 294 Hall, Marie Boas,145 Fundamental theorem Halley,Edmund, 48, 133, 134, 169 of algebra,178 Hamilton,Sir William,170, 336,338 of arithmetic,273 Hardy, G. H.,19, 35, 103, 165,231,306 Harvey,William, and Descartes,90 Hawkins, David,280 Galileo,69, 73, 78, 94, 106, 111, 113, 114, Hazard,P., 126 126,258,305 Heaslet, M. A.,103 Galileo's Heath,T. L.,5, 6, 19, 36, 40, 41, 42,44, 46, net of awareness,114 50,58,242 paradox,213 Hegel, G. W., 178 Galton, Francis,158 Heiberg,J. L.,45, 242 Games, 50 Heisenberg,Werner, 196 Gamma function, 164 Heliocentric system, 327 Gardner,Martin, 1 Helmholtz, Hermann von,210 Gauss, Carl Friedrich,31, 104, 105, 161, 175, Herivel,John, 174 188,191,198,200,201,203,212,277, Hermite, Charles,187, 189, 194, 195, 207, 336,338 284,290 quoted on lawyers, 177,212 , 5, 217 quoted on Riemann,201 Heron ,52 Gauss, Helen W.,175 Heron's Gelfond,A. 0., 194,290 automatic temple door-opener,55 Genus of surface,168 formula,52, 222 Geocentricsystem, 326 fountain,55 Geodesic curves,203, 204 jet engine,54 Geometric constructions,287 principle of reflection,52 Geometry steam turbine,54 Greek,15 Herschel,Sir William,73, 135 intrinsic differential,203 Herstein,I. N., 340 non-Euclidean, 182 Hieron, King of Syracuse, 36 Riemannian, 203 Highet, , 43 George I, 153,154 Hilbert,David, 58,105, 181, 192,290 Gibbon, Edward,65 Hiltebeitel,A., 180 Gibbs, Willard,210 ,237 Giesy, D. P.,270 Hobbes,Thomas, 30,93, 117, 263 Golden nose of Tycho,74 Hofmann,J. E.,129, 145,146, 154 Goldstine, H. H., 145 Holmes, Oliver Wendell, 3,10, 84 Grains of sand in visible universe,232 Hooke,Robert, 133 Graph theory, 166 Hopf, H., 340 Gravitation Horoscopeof Jesus, 73 Newton's force of, 76 Hospital, G. F. A. de I',159 INDEX 3S1

Humanity's perception of itself,335 Knopp, K., 146,298 Hutchins,Robert M.,48, 76 Koestler,Arthur, quoted, 1, 71 Huxley,Aldous, 221 Kolmogorov,A. N., 344 Huygens,Christiaan, 90, 125, 133, 145, 146, Konigsberg bridge problem,165 154,158, 258, 309 Kovalevsky, Sonia, 209 construction of the cycloidal pendulum,322 Koyre,Alexandre, 153 Huygens, Constantijn,126 Kronecker,L., 175 Hypatia,62 Kummer,E. E.,191 Hyperbola, 59, 61 Kurosh,A. G., 340 Hypocycloid,305 Kuzmin,R., 290 Hypothesis,Riemann, 205 Lagrange,Joseph Louis,104, 169 Identity,Euler's, 204, 295 Lagrange's foursquares theorem,227, 341 Index,infamous, 76 Lanczos,C., 127, 180 Induction,mathematical, 122 Laplace, Pierre Simon de,161, 171 Infiniteproduct forsine, Euler's,163, 269 Lavrent'ev,M. A.,344 Infinityof primes,Euclid's theorem on, 273 Law Inquisition, John Milton quoted on,113 associative,337 Integrals commutative,337 Abelian,212 of gravitation,Newton's, 78, 326 elliptic,211 inverse square, 328 logarithmic,277 of large numbers,Bernoulli's, 159 of elementary functions,194 of motion Intellectual architecture of Elements, 34 Einstein's, 247 Intellectual civilization,3 Newton'ssecond, 246,249, 330 Intrinsic differentialgeometry, 203 of refraction, Snell's, 90,100, 314 Involute,322 Laws Irrational numbers,282 distributive, 337 Irrationality of planetary motion, Kepler's,74, 75, 76, of e, 285 327,330,333,334 of .ir, 284 Lawyers,Gauss quoted on,177, 212 Leeuwenhoek,Antony van,129, 146 Jacobi, C. G. J., 183, 184, 189, 200 Legacies, Plato's foolish,39 Jefferson,Thomas, 20 Legendre,A. M.,105 Jesus,horoscope of, 73 Lehmer,D. H.,95, 276 Jet engine,Heron's, 54 Leibniz,Gottfried Wilhelm, 47, 90,122, 126, Jones,Burton W.,287 128,132,137,140,141,159,160,258, Jones, David E. H., 29 309 Leibniz's formula for .ir/4, 254, 264 Kac, M., 253 Lemma,Euclid's, 280 Kant,Immanuel, 34, 135, 183, 213 Leverrier,Jean,334 Kayser, R., 146 Library ofAlexandria, 29,275 Kepler,Johannes, 47, 49,69, 106,327 Lie algebras,340 Kepler's laws, 133, 134, 327 Lindemann,Ferdinand, 194,287,288,290 first,74, 333 Liouville, Joseph,193 second,75, 330 Liouville's theorem third, 76, 334 on elementary integrals,194 Kervaire,M., 341 on transcendental numbers, 194, 288 King of Persia,23, 27, 193 Lobachevsky,Nikolai, 183, 197 Kingsley,Charles, 64 Locke,John,34, 135 Kirk, G. S., 6, 24 Loemker, L. E., 142,146, 150,152 Klein,F., 178 Logaritl\plic integral,277 Kneale, W. and M., 148 Lost illusions,34 352 INDEX

Louis XIV,boil on faceof Europe,144 quoted on giants,118 Lowell, J. R.,12 Newton's Lucretius,25 forceof gravitation,76 Lune of Hippocrates,237 inverse square law of gravitation,328 Lutzen, Jesper,193 law of gravitation,78, 326 Optic/cs, 136 Machiavelli, Niccolo,208 Principia, 134, 328 Magie, W. F.,114 second law of motion,246, 249,330 Mahan,A. T.,142, 144 Niven,I., 284 ,290,297,306 Manuel,Frank E., 139 Nonassociativealgebra, 340 Marcellus,35, 37 Non-Euclidean geometry, 182 Mars,74, 126 Nonsense,ridiculous, 13 Marx, Karl,25 Nowhere-differentiablefunction, 214 Materialism of Democritus,25 Number Mathematical induction,122 mysticism,17 Maxwell,James Clerk,168, 181 theory Mean distance of planet,333 algebraic, 273 Mechanics,celestial, 334 analytic, 280,294 Mersenne, Marin, 78,87, 93, 97,113, 126, Numbers 231,305 algebraic, 286 primes, 95, 231 Bernoulli,299 Mill,John Stuart,141 Cayley, 341 Milnor,J., 341 complex,338 Milton,John, 113 composite, 272 Mittag-Leffler,G., '2ff/ figurate,15 Momentum, 246 irrational,282 Monads,147 perfect, 94, 227 Monsters of modernanalysis, 214 even, theorem of, 228-231 More, L. T.,98 prime, 227,272 Morehead,J.,180 theorem,199, 278 Mortimer, Ernest,119, 123 transcendental,194, 286 Motion Numerology,17, 19 Einstein's law of, 247 Newton's second law of,246, 249,330 Oldenburg,Henry, 145 planetary, Kepler's laws of, 74, 75 76,326, Optic/cs, Newton's,136 327,334 Orcibal,Jean, 119 Mueller,Ian,62,64 Ore,Oystein, 185, 189 Museum of Alexandria, 29 Orestes, 65 Music of the spheres,16, n Oriental mysticism, 13 Mysticism Orwell, George, 214 number,17 oriental,13 Palter,Robert, 140 ,57 Napoleon,172, 173 Pappus' theorem, 58,221 Needham, Joseph, 11 Parabola,59, 61 Nelson, H., 231 Paradox, Galileo's,213 Nephroid, 312 Paroxysms of piety, Pascal's,1 02 Net of awareness, Galileo's 114 ,34 Neuenschwander,E., 186,212 Partial fractions expansion of cotangent,301 Neugebauer, Otto, 53 Partitions,165 Newman, James R., 50,166, 334 Pascal,Blaise, 102, 103, 119, 154 Newton,Isaac,34,35,47, 76,91,98,131, Pascal's 145,153,160,171,309,326 Pensees, 122, 123 INDEX 353

principle,121 of reflection,Heron's, 52 ProvincialLetters, 122 Pascal's,121 theorem,120 Problem Pasteur,Louis, 129 brachistochrone,136, 160,308,314 Pauling,Linus, 78 Konigsberg bridge, 165 Peano,Giuseppe,214 of Apollonius,SO Pedersen,H., 151 ,29 Pendulum clock,126, 127 Product Pepys,Samuel,135 Euler's,for the sine,163, 269 Pedect numbers,94, 227 Wallis's 163,260, 269 theorem of even,228-231 Property ,239 tautochrone, 310 Periodic time,334 topological, 168 Pi (.1r),irrationality of,284 Pseudo-primes,280 Planck,Max, 209 Ptolemaic system, 326 Plato,S, 13, 18, 22, 31, 39 of Alexandria, 29, 62,326 Plato's Pythagoras, 7, 12,33,226 Academy,31 Pythagorean book-burning,22 theorem,58, 218 foolish legacies,39 triples,68 numerology,19 Platonic Quadrature,233 realism,19 Quaternions,338 solids,32 ,36, 37, 39 Poincare,Henri, 195 Rademacher,H., 68, 104 Point,rational, 284 Rational point,284 Pollock,F., 146 Raven,J. E.,6, 24 Polya,G., 122, 165 Rearrangement theorem, Riemann,202 Polygon, regular,30 Reciprocals of primes,series of, 295 constructible,31, 176 Recorde, Robert, 157 Polygonal numbers,Fermat's theorem on, Refraction, Snell's law of, 90,100, 314 104 ,30 Polyhedra,Euler's formulafor, 167 constructible, 31, 176 Polyhedron,regular, 31, 72 Regular polyhedron,31, 72 Popper, Karl,10, 20 Reid,C., 290 Porges,Arthur, 105 Rest mass, 248 Potential energy, 259 Richelieu, Cardinal, 93 Poussin,Charles de la Vallee, 278 Ridiculous nonsense, 13 Power series for tangent,299 Riemann, Bernhard,161, 173, 191,200, 2n Prime number, 227,272 Gauss quoted on,201 theorem, 199,278 Riemann Primes hypothesis,205 Euclid's theorem on the infinityof, 273 rearrangement theorem, 202 in arithmetic progression,274 zeta function,205, 294 Mersenne,95, 231 Riemannian geometry,203 series of reciprocals of, 295 Einstein,on, 204 twin, 276 Robbins, H., 31,168,280,287 Principia, Newton's,134, 328 Roberval,Gilles Persone, 113,115 Principle Roemer, Ole,145 Cavalieri's, 108 Rope-stretchers, 217 Dirichlet, 192, 201 Ruler-and-compass constructions,287 of least time, Fermat's, 101, 314 Ruling Yahoo, 154 3S4 INDEX

Russell,Bertrand, 10, 20, 28, 30, 148, 149 Tangent,power series for,299 Ryle,Gilbert, 20, 150 Tautochrone,128, 158 property,310 Sabra,A. I.,90, 101 Terman,Lewis M., 142 Saint Augustine,18, 88 Thales,4, 12, 13 Saint Paul,150 Theonof Alexandria,62 Saint Thomas Aquinas,empty doctrines of, Theophrastus,43 85 Theorem Sarton, George,169 Bernoulli's,159 Scalar,336 Diophantus',66 Schneider,Th., 290 Dirichlet's,on primes,274 Schofield,M., 6, 24 Euclid's,on the infinityof primes,273 Schrodinger,Erwin, 196 Fermat's, 103 Screw of Archimedes,36 last,105, 288 Second law of motion,Newton's, 246, 249 on polygonal numbers,104 Seidenberg,A., 58 two squares,227, 344 Self-evident truths,34 Lagrange's four squares,227, 341 Series Liouville's of reciprocals of primes,295 on elementary integrals,194 power, fortangent, 299 on transcendentalnumbers, 194, 288 Seven liberal arts, 14-15 of algebra,fundamental, 178 Shakespeare, 168 of arithmetic, fundamental,273 Shipley,Sir Arthur, 6 of even perfect numbers,228-231 Siegel,C. L.,290 Pappus',58, 221 Sieve of Eratosthenes,275 Pascal's,120 Simmons,George F.,36, 53, 114, 123, 132, prime number,199,278 164,213, 258 Pythagorean,58,218 Sine, Euler's infinite product for,163, 269 Riemann rearrangement,202 Sirius, 130 unique factorization,227, 273 Skepticism, 3, 9-10, 88 Theory Slowinski, D., 231 algebraic number,273 Smith, D. E., 97, 145,176,203,309 analytic number,280, 294 Snell, W., 100 graph, 166 law of refraction, 90,100, 314 Thompson,S. P.,129 (church historian), 64 Thrust of a rocket engine,250 Spengler,Oswald, 142 Tietze,Heinrich, 31, 91, 104,176, 191 Sphere,volume of,242 Time, periodic,334 Spinoza, Baruch,34, 146, 149,150 Titan, moon of Saturn,126 Spiral of Archimedes,40 Titchmarsh, E. C.,205, 214 Steam turbine, Heron's,54 Toeplitz,0., 68 Stobaeus, 29 Topological properties,168 Stone, M. H.,213 Topology,201 , 44 Torricelli, Evangelista, 109,113, 121, Struik, D. J., 97, 98,105, 117,122, 128, 148, 305 155,156,160 Torricelli's law, 114 Suidas, 65 Torus,81 Sum of reciprocals of squares, Euler's Toynbee, ArnoldJ., 1,209 formula for,267, 270, 323 Transcendenceof e, 290 Swift, J. D., 63 Transcendentalnumbers, 194,286 Swift, Jonathan, quoted on George I, 154 Truesdell,C., 151, 168, 210 Sylvester, J. J., 197 Truths,self-evident, 34 Synesius of Cyrene, 65 Twin primes, 276 Sz.-Nagy, Bela, 214 Two squares theorem,Fermat's, 227,344 INDEX 3SS

Unique factorization theorem,227, 273 Weil,Andre,68,105 Unit, astronomical, 334 Westfall,Richard S.,138, 139, 151 Unspeakable secret,17 Weyl,H., 32 Usher,A. P.,54 Wheeler,L. P., 210 Uspensky,J. V.,103 Whewell,William, 135 Whipple,Fred C., 77 Vandiver,H. S.,105 Whitehead,Alfred North,38, 130, 152, 156 Vavilov,S. I.,136 Wiener,Philip P.,143 Vectors,337 Wigner,E. P., 196 Vicar of Bray,172 Will to Believe,10 Vieta,Fran�is, 253 Will to Doubt, 10 Vieta's formula,252 Wilson, E. B., 210 Voltaire,35, 89, 122, 144, 149, 162 Wolf, A., 150, 151 Volume of sphere, 242 Wren, Christopher,115, 133,305 Wright,E. M., 103,165,306 Wallis,John, 116,137,221,260,263 Wright,Thomas, 135 Wallis's product, 163, 260,269 Waltershausen,W. Sartorius von, 175 Xenophanes, 14,25 Warner,Seth,287 , Wilhelm, 181 Yutang, Lin, 89 Weierstrass, Karl, 207 misspent youth,207 Zeta function,Riemann, 205, 294 AMS / MAA SPECTRUM

Calculus Gems, a collection of essays written about mathematicians and mathematics, is a spin-off of two appendices (“Biographical Notes” and “Variety of Additional Topics”) found in Simmons’ 1985 calculus book. With many additions and some minor adjustments, the material will now be available in a separate softcover volume. The text is suitable as a supplement for a calculus course and/or a history of mathematics course, The overall aim is bound up in the question, “What is mathematics for?” and in Simmons’ answer, “To delight the mind and help us understand the world”. The essays are independent of one another, allowing the instructor to pick and choose among them. Part A, “Brief Lives”, is a biographical history of mathemat- ics from earliest times (Thales, 625–547 BC) through the late 19th century (Weierstrass, 1815–1897) that serves to connect mathemat- ics to the broader intellectual and social history of Western civiliza- tion. Part B, “Memorable Mathematics”, is a collection of interest- ing topics from number theory, geometry, and science arranged in an order roughly corresponding to the order of most calculus courses. Some of these sections have a few problems for the student to solve. Students can gain perspective on the mathematical experience and learn some mathematics not contained in the usual courses, and instructors can assign student papers and projects based on the es- says. The book teaches by example that mathematics is more than computation. Original illustrations of influential mathematicians in history and their inventions accompany the brief biographies and mathematical discussions.

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