AMS / MAA SPECTRUM VOL 38

AMS / MAA SPECTRUM VOL 38

In M

athem

atical Circles &

Qu

adran

ts I, II, III, IV

How

ard W

. Eves

VOL38AMS / MAA SPECTRUM

166 pages on 50lb stock • Spine 5/16" • Trim size 6 x 94-Color Process

AMS / M

AA PRESS

Mathematical Circles

The 360 different anecdotes compiled in these delightful volumes will add zest to every teacher's mathematics classes. There are appropriate selections for all levels of students. They are short, succinct, and at the same time given in simple settings which enable the reader to identify with the story and its implications. Here is presented the kind of material that makes the difference to the undecided student. I highly recommend you putting a copy next to your worktable.

—The Mathematics Teacher

For many years, famed mathematics historian and master teacher Howard Eves collected stories and anecdotes about mathematics and mathematicians, gath-ering them together in six Mathematical Circles books. Thousands of teachers of mathematics have read these stories and anecdotes for their own enjoyment and used them in the classroom-to add spice and enter tainment, to introduce a human element, to inspire the student, and to forge some links of cultural history. Through a special arrangement with Professor Eves, the Mathematical Association of America (MAA) is proud to reissue all six of the Mathematical Circles books in a three-volume edition.

The first two books were published to acclaim in 1969, as In Mathematical Circles (Volumes 1 and 2). They are bound together as Volume I of the Mathematical Circles Collection. Mathematical Circles Revisited and Mathematical Circles Squared are bound together as Volume 2 of the Collection, and Mathematical Circles Adieu and Return to Mathematical Circles as Volume 3.

This three-volume set is a must for all who enjoy the mathematical enterprise, especially those who appreciate the human and cultural aspects of mathematics.

Howard Eves spent most of his teaching career at the University of Maine at Orono, and more recently at Central Florida University. For 25 years, he edited the Elementary Problems Section of the American Mathematical Monthly. His books include: Great Moments in Mathematics Before 1650, Great Moments in Mathematics After 1650, Mathematical Reminiscences (all for the MAA), Introduction to the History of Mathematics, and his two-volume Survey of Geometry.

Volume IHoward W. Eves

In Mathematical Circles

Quadrants I, II, III, IV

Vol I

In Mathematical CirclesQuadrants I, II, III, IV

Vol I

Howard Eves

(I\

\

In Mathematical CirclesQuadrants I, II, III, IV

Published and Distributed byThe Mathematical Association ofAmerica

In Mathematical Circles: A Selection ofMathematical Stories and Anecdotes, Quadrants Iand II and In Mathematical Circles: A Selection ofMathematical Stories and Anecdotes,Quadrants III and IV were previously published by Prindle, Weber & Schmidt,Incorporated in 1969.

© 2003 byThe Mathematical Association ofAmerica, Inc.

Library of Congress Catalog Card Number 2002116306

ISBN 0-88385-542-9

Printed in the United States ofAmerica

Current Printing (last digit):10 9 8 7 6 5 4 3 2 1

To THE MATHEMATICS TEACHERS OF AMERICA

with so many ofwhom I have

had the pleasure ofworking

SPECTRUM SERIES

Published by

THE M..f\THEMi\.TIC.AL ASSOCIATION OF AMERICA

Committee on PublicationsGerald L. Alexanderson, Chair

Spectrum Editorial BoardGerald L. Alexanderson, Editor

Robert Beezer ] effrey L. Nunemacher\t\Tilliam Dunham ] ean PedersenMichael Filaseta J. D. PhillipsWilliam]. Firey ] ennifer J. Quinn

Erica Flapan Harvey]. Schmidt, ] r.Dan Kalman Sanford L. Segal

Eleanor Lang Kendrick Franklin SheehanEllen]. Maycock Francis Edward SuRussell L. Merris ] ohn E. Wetzel

The Spectrum Series of the Mathematical Association of America was so named toreflect 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 interestthat will appeal to a broad range of readers, including students and teachers ofmathematics, mathematical amateurs, and researchers.

777 Mathematical Conversation Starters, by John de PillisAll the Math That's Fit to Print, by Keith DevlinCircles: A Mathematical View, by Dan PedoeComplex Numbers and Geometry, by Liang-shin HahnCryptology, by Albrecht BeutelspacherFive Hundred Mathematical Challenges, EdwardJ. Barbeau, Murray S. Klamkin, and

William O. J. MoserFrom Zero to Infinity, by Constance ReidThe Golden Section, by Hans Walser. Translated from the original German by Peter

Hilton, with the assistance ofJean Pedersen.I Want to Be a Mathematician, by Paul R. HalmosJourney into Geometries, by Marta SvedJULIA: a life in mathematics, by Constance ReidThe Lighter Side ofMathematics: Proceedings of the Eugene Strens Memorial Conference

on Recreational Mathematics & Its History, edited by Richard K. Guy and RobertE. Woodrow

Lure of the Integers, by Joe RobertsMagic Tricks, Ca-rd Shuffling, and Dynamic Computer Memories: The Mathematics ofthe

Perfect Shuffle, by S. Brent MorrisThe Math Chat Book, by Frank MorganMathematical Apocrypha, by Steven G. KrantzMathematical Carnival, by Martin GardnerMathe1natical Circles Vol I: In Mathematical Circles Quadrants I, II, III, IT{ by Howard

,Y. EYe~

Jvlathelnatical Circles Vol II: Mathematical Circles Revisited and Mathematical CirclesSquared, by Howard W. Eves

l\Jathematical Circles Vol-1II: Mathematical Circles Adieu and Return to MathematicalCircles, by Howard W. Eves

A1athematical Circus, by Martin GardnerJVIathe1natical Cranks, by Underwood DudleyMathematical Fallacies, Flaws, and Flimflam, by Edward J. BarbeauMathematical Magic Show, by Martin GardnerMathematical Reminiscences, by Howard EvesMathematics: Queen and Servant of Science, by E.1: BellMemorabilia Mathematica, by Robert Edouard MoritzNew Mathematical Diversions, by Martin GardnerNon-Euclidean Geometry, by H. S. M. CoxeterNumerical Methods That Work, by Forman ActonNumerology or What Pythagoras Wrought, by Underwood DudleyOut of the Mouths ofMathematicians, by Rosemary SchmalzPenrose Tiles to Trapdoor Ciphers ... and the Return ofDr. Matrix, by Martin GardnerPolyominoes, by George MartinPower Play, by Edward J. BarbeauThe Random Walks of George P6lya, by Gerald L. Alexanderson"The Search for E. T. Bell, also known as John Taine, by Constance ReidShaping Space, edited by Marjorie Senechal and George FleckStudent Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D.

Gaughan, Douglas S. Kurtz, and David PengelleySymmetry, by Hans Walser. Translated from the original German by Peter Hilton,

with the assistance ofJean Pedersen.The Trisectors, by Underwood DudleyTwenty ~ars Before the Blackboard, by Michael Stueben with Diane SandfordThe Words ofMathematics, by Steven Schwartzman

MAA Service Center~O. Box 91112

Washington, DC 20090-1112800-331-1622 FAX 301-206-9789

PUBLISHER'S NOTE

For many years Howard Eves, famed historian of mathematics and masterteacher, collected stories and anecdotes about mathematics and mathematicians and gathered them together in six Mathematical Circles books.Thousands of teachers of mathematics have read these stories and anecdotes for their own enjoyment and used them in the classroom to addspice and entertainment, to introduce a human element, to inspire thestudent, and to forge some links of cultural history. Through a specialarrangement with Professor Eves, the Mathematical Association ofAmerica (MAl\.) is proud to reissue all six of the Mathematical Circlesbooks in this three-volume edition.

In Mathematical Circles, the first two books, were published to acclaim in1969. They are bound together here as Volume I of the MathematicalCircles Collection. Mathematical Circles Revisited and Mathematical CirclesSquared are bound together as Volume 2 of the Collection, andMathematical Circles Adieu and Return to Mathematical Circles as Volume 3.

This three-volume set is a must for all who enjoy the mathematicalenterprise, especially those who appreciate the human and culturalaspects of mathematics.

IX

Ancient mazes consisted of a tortuous path confined to a small area ofground and leading to a tree or shrine in the center, with no chance oftaking a wrong turn. Shown here is the circular maze constructed for theMinotaur. This labyrinth was delineated on the coins of Cnossus, specimens of which are not uncommon.


Quadrants I and II

Howard Eves


10.1090/spec/038/01

PREFACE

Somehow or other, over the years and without any particular effort on mypart, a large number of stories and anecdotes about mathematics andmathematicians. have fallen my way and remained stuck in my mind.These stories and anecdotes have proved very useful in the classroom-aslittle interest-rousing atoms, to add spice and a touch of entertainment, tointroduce a human element, to inspire the student, to instill respect andadmiration for the great creators, to yank back flagging interest, to forgesome links of cultural history, or to underline some concept or idea. Manystudents and teachers have begged me to write up these stories and anecdotes, and a number of publishers have hounded me for them. At last Ihave given in and here offer a sample of the material.

Problems arose from the start. First of all, on marshalling the material Ifound I had far too much for a modest-sized venture; so I decided to selectsome three hundred to four hundred items as a test of reader interest. Nextarose the problem of how to order the material; I decided to present it inrough chronological order with an accompanying index that would lenditself to other types of useful classification. And then there was the problemof the authenticity of some of the material; I decided not to make any effortat documentation, but simply to offer possibly doubtful items as part of theinteresting accumulated folklore of our subject.

Undoubtedly many of the personal stories and anecdotes told hereactually took place, but it is equally certain that some originally true stories have been embroidered over the year.s and ages, and that others havesimply been made up as being apposite to the subjects concerned. Thusthere are· stories that have come down to us about some great men so lostin the mythical haze of the past that really nothing certain can be toldabout them; there are identical anecdotes that have been told about different persons; there are amusing tales that have circulated but have beendenied by the principals involved; there are many cases where the same

XUl

PREFACE

basic story has been told in varying and sometimes conflicting versions.One is reminded of Abraham Lincoln. There are literally hundreds ofanecdotes that have been told about Lincoln; many of these have a realbasis, but there can be no doubt that some of them were embroidered,nvisted, or simply devised to fit the interesting and colorful character ofLincoln.

Particularly difficult is the matter of anecdotes about contemporarypeople. My collection contains a large number of such stories, but I haveheard some of them denied or at least made much less interesting by theprincipals themselves. So, in this first round of stories, I shall stick to thepast wherein the dramatis personae cannot rise up and defend themselves,and I shall refrain from narrating any anecdotes about the living.

The bulk of the material can be read with very little, and most oftenwith no, mathematical background. But here and there do occur items abit more demanding, and there are even a few challenging elementaryproblems. The historical comments and capsules are largely adapted frommy book, An Introduction to the History ofMathematics (Holt, Rinehart andWinston, third edition, 1969). There the interested reader can find fullerhistorical treatments. I am very grateful to The Mathematics Teacher, one ofthe fine official journals of the National Council of Teachers of Mathematics, for permitting me to reproduce in essentially the original formcertain items which appeared there in the Historically Speaking section,a department of the journal that for some years I have had the pleasureof editing.

It is hoped that the general reader may find the potpourri sufficientlysavory, that the teacher may find it useful to serve on occasion, and that thepartaking student may enjoy some of the historical tidbits and (in a noncannibalistic. way) the human flavor.

With sufficient encouragement, I may decide to travel around theMathematical Circle, or at least the more modern part of it, again in thefuture. Toward this possibility, interested readers are cordially invited tocontribute any favorite stories they would like to see in an expanded collection.

HOWARD W EVES

xiv

CONTENTS

PREFACE X'lU

QUADRANT ONE

THE ANIM.AL WORLD, REAL AND IMAGINARY

1° A Scotch crow, 32° The solitary wasp, 43° The Harvard katydid, 44° The ingenious honey bees, 45° A classification of mathematicians, 56° Logarithms and multiplication, 57° Good induction versus bad induction, 68° The mathematical horse, 6

PRIMITIVE MANgo Two plus two, 7

10° Addition of vectors, 711° The great size of three, 812° Gog and Gug, 8

PRE-HELLENIC MATHEMATICS

13° Problem 79 of the Rhind papyrus, 1014° The pyramid of Gizeh, 1115° The greatest Egyptian pyramid, 1216° Squaring the circle, 1317° Plimpton 322, 1418° .The angular degree, 171go .. Magic squares, 1820° The 3-4-5 triangle problem, 19

xv

CONTENTS

A FEW LATER CHINESE STORIES

21 ° The Drunken Dragon loses his hair, 2022° Huai-Wen calculates the dates on a tree, 2023° I-Hsing finds his teacher, 21

THALES

24° How to become rich, 2225° The recalcitrant mule, 2326° Why Thales never married, 2327° Thales as a stargazer, 2328° Credit where credit is due, 2429° Moral advice, 2430° An incongruity, 2431° The Thales puzzle, 2432° Thales, the engineer, 2533° Thales, the astronomer, 2534° Thales, the statesman, 26

PYTHAGORAS

35° The lure of geometry, 2636° The first recorded facts in mathematical physics, 2737° A hecatomb of oxen, 2738° A play on words, 28.~9° A philosopher, 2840° Friendship, 2841° The ~arriage of Pythagoras, 2942° Pythagorean teaching, 2943° Pythagoras's golden thigh, 2944° The end of Pythagoras, 3045° Pythagoras's proof of his theorem, 30

THE PYTHAGOREAN BROTHERHOOD

46° Motto of the Pythagorean Brotherhood, 3147° Himself said it, 3148° Brotherhood loyalty, 3149° Damon and Phintias, 3250° The three questions, 33

XVI

CONTENTS

PYrHAGOREANISJ\f

51° The transmigration of souls, 3352° Number rules the universe, 3353° Amicable numbers, 3454° Deficient, perfect, and abundant numbers, 3555° Pythagorean philosophy and geometry at stake, 3656° Pythagoras justified, 3757° The case for Pythagoreanism, 37

PLATO

58° Plato's motto and the transfer of training, 3859° Michel Chasles and the forged autograph letters, 3960° Some particularly elusive Platonic numerology, 4161° The five Platonic solids, 4262° Kepler's explanation of the Timaeus associations, 4363° The Platonic solids in nature, 4364° The most extraordinary application of the Platonic solids to a

scientific problem, 4365° Some problems concerning the Platonic solids, 4466° The Delian problem, 45

EUCLID

67° The royal road in geometry, 4668° Euclid and the student, 4769° Euclid's Elements compared with Newton's Principia, 4770° The most famous single utterance in the history of science, 4771° The most fruitful single utterance in the history of science, 48

ARCHIMEDES

72° Archimedes' boast, 1873° Archimedes' defense of Syracuse, 4874° The fraudulent goldsmith, 4975° The Archimedean screw, 5076° The stomach of Archimedes, 5177° The death of Archimedes, 5178° The questionable mosaic, 5279°. The tomb of Archimedes, 52

xv11

CONTENTS

ERATOSTHENES AND APOLLONIUS

80° Eratosthenes' measurement of the earth, 5481° The death of Eratosthenes, 5582° The nicknames of Eratosthenes and Apollonius, 5583° The names "ellipse," "parabola," and "hyperbola," 56

DIOPHANTUS

84° Diophantus's personal life, 5985° The syncopation of algebra, 6086° A diophantine riddle, 6187° The Greek meaning of "arithmetic," 62

THE END OF THE GREEK PERIOD

88° Some famous inequalities, 6289° Pappus's extension of the Pythagorean Theorem, 6390° The first woman mathematician, 66

QUADRANT TWO

HINDU MATHEMATICS

91 ° King ASoka, 7192° Inversion, 7193° The rule of three, 7294° Hindu syncopation of algebra, 7295° Bhaskara's daughter, 7396° Behold! 7397° Hindu embroidery, 7498° Buddha's examination, 7699° False" position in the Bakshali manuscript, 76

100° Contrast between Greek and Hindu mathematics, 77101° Srinivasa Ramanujan, 78

ARABIAN MATHEMATICS

102° Arabian names in astronomy, 79103° The origin of our word "algebra," 79104° The origin of our word "algorithm," 80105° The origin of our word "zero," 80106° The origin of our word "sine," 80

XVIll

CONTENTS

1070 Alhazen's madness, 811080 The three students, 821090 Omar's roses, 83

THE RETURN OF M ..t\THEMATICS TO WESTERN EUROPE

1100 Gerbert, Pope Sylvester II, 84III 0 The century of translators, 851120 The Norman kingdom of Sicily, 861130 The Italian commercial centers, 861140 From rabbits to sunflowers, 861150 A mathematical tournament, 901160 The blockhead, 9111 70 Finger numbers, 911180 The eulogist of mathematics, 921190 Submathematical analysis, 93

THE FOURTEENTH, FIFTEENTH, AND SIXTEENTH CENTURIES

1200 The mechanical eagle, 94121 0 Introduction of + and -, 951220 The cossic art, 961230 Leonardo da Vinci's proof of the Pythagorean Theorem, 961240 The stone upon which one may sharpen his wits, 961250 The origin of our equal sign, 981260 The death of Robert Recorde, 981270 Adam Riese, 991280 Nicolaus Copernicus, 991290 Michael Stifel, 1001300 The art ofbeasting, 100

THE EPISODE OF CUBIC AND QUARTIC EQUATIONS

131 0 The story of the algebraic solution of cubic equations, 1021320 Girolamo Cardano, 1021330 Tartaglia, 103·1340 The story of the algebraic solution of quartic equations, 104

FRAN~OIS VIETE

1350 The origin ofa friendship, 1041360 Christian versus unchristian, 1051370 Work unfit for a Christian, 105

XIX

CONTENTS

SI~100T STEVIN, JOHN NAPIER, AND HENRY BRIGGS

1380 A multiple reputation, 1061390 Napier's misjudgment of himself, 1061400 The science fiction writer of his day, 107141 0 Exposing a thief, 1071420 Impounding pigeons, 1071430 The meeting, 1081440 Some terminology, 1081450 Laplace's statement, 1091460 A historical curiosity, 1091470 Napierian logarithms versus natural logarithms, 109

THOMAS HARRIOT AND WILLIAM OUGHTRED

1480 Harriot in America, 1111490 On the origin of > and <, 1111500 The teacher of giants, 113151 0 The invention of the slide rule, 1131520 Oughtred's longevity, 114

GALILEO GALILEI AND JOHANNES KEPLER

1530 The oscillating lamp, 1151540 Falling bodies, 1151550 The telescope, and further trouble, 1161560 The inquisition and the unhappy end of a great scholar, 1171570 Authority versus reasoning in science, 1181580 Galileo's reconciliation of science and Scripture, 1181590 Some Galileo-Kepler correspondence, 1191600 Tycho Brahe's golden nose, 120161 0 Kepler's pertinacity, 1201620 The rarity of problem solvers, 1211630 Pure versus applied mathematics, 1211640 A life of misfortune, 1221650 Numerology and theology, 122

GERARD DESARGUES AND BLAISE PASCAL

1660 Desargues' forgotten book, 1231670 The precocity of Pascal, 1231680 The greatest" might-have-been" in the history of mathematics, 1251690 Pascal's" mystic hexagram" theorem, 125

xx

CONTENTS

1700 Lovis de Montalte and }\mos Dettonville, 126171 0 Two very practical contributions, 1271720 A specious use of probability, 127

RENE DESC.ARTES .AND PIERRE DE FERMAT

1730 A challenge problem and a friendship, 1281740 The birth of an idea, 1291750 Descartes' advice, 1291760 Two significant contributions to mathematical notation, 1301770 The death of Descartes, 1301780 The Fermat numbers, 1311790 Fermat's method of infinite descent, 1321800 The most tantalizing marginal note in the history of mathe

matics, 134

XXI


Quadrants III and IV

10.1090/spec/038/02

Magic circle of circles

The "magic circle of circles," shown in this volume, was created byBenjamin Franklin, and the intricacies of this construction are unfoldedin Item 317°.


Quadrants III and IV

Howard Eves


CONTENTS

QUADRANT THREE

SOME MINOR STORIES ABOUT SOME MINOR MEN

181 0 Gilles Persone de Roberval and arguments of priority, 3182 0 John Wallis and the invention of the infinity symbol, 31830 A facetious remark about the symbol for infinity, 41840 Nicolaus Mercator and the loss of a fee, 41850 Isaac Barrow: prankster, wit, strong man, generous teacher, 41860 Sir Christopher Wren and the Great London Fire, 5

PRE-NE\VTONLJ\N VERSUS POST-NEvVTONIAN MATHEMATICS

1870 Mathematics as a rising continent, 61880 Mathematics as a rock, 71890 The finite and the infinite in mathematics, 7

ISAAC NE,f\,TON AND GOTTFRIED WILHELi\f LEIBNIZ

1900 An ingenious young inventor, 9191 0 Genius is not always bright, 91920 Spurred on by a bully, 91930 Introduction to mathematics, 101940 Newton's absent-mindedness, 101950 Newton's first purely scientific experiment, 111960 Some tributes paid to Newton, 111970 Newton's estimate of himself, 121980 De Morgan's anagrams on Newton, 121990 Newton's dislike of controversy, 122000 The challenge problems, 13201 0 Leibniz's unusual mind, 142020 Leibniz and religion, 14

v

CONTENTS

THE BERNOULLIS

2030 The Bernoulli family, 162040 A blow to primogeniture, 19205 0 Eadem mutata resurgo, 192060 Invito patre sidera verso, 19207 0 Johann Bernoulli's nastiness, 202080 The origin of ~Hopital's rule, 202090 Daniel Bernoulli's two little adventures, 232100 The apple of discord, 23

THE SMALL INITIAL UNDERSTANDING OF THE CALCULUS

211 0 Newton's shift in the hypothesis, 272120 Ghosts of departed quantities, 28213 0 Address to an infidel mathematician, 282140 Johann Bernoulli's postulate, 29215 0 Guido Grandi's mysticism, 292160 Euler's formalism, 30217 0 J ean-Ie-Rond d'Alembert, 32

BONAVENTURA CAVALIERI, YOSHIDA KaYO, AND SEKI KaWA

2180 Cavalieri's method ofindivisibles, 332190 Some applications of Cavalieri's principle, 352200 An "a" for an "i," 35221 0 A synonym for "arithmetic," 362220 The Japanese Newton, 362230 Seki Kowa as a prodigy, 372240 Seki Kowa and his employer, 37

SOME LESSER SEVENTEENTH- AND EIGHTEENTH-CENTURY

BRITISH MATHEMATICIANS

225 0 Up the ladder of success, 382260 Halley's magnanimity and perseverance, 392270 "According to Cocker," 402280 Man of promise, 402290 Death in an arithmetic progression, 402300 An inspiring achievement, 41231 0 An inspiring character, 42232 0 A noble epitaph, 42233 0 A hidden name, 43

VI

CONTE_ TS

SOi\1£ LESSER SE\7E:\TEEj\;TH- ~-\~D EIGHTEEl ~TH-CE TURY

CO TI:\E:\TAL ~t-\THEi\1ATICIA.TS

2340 The \Tan Schooten family, 432350 The Clairaut family, 43236 0 The "great flattener," 44237 0 .4£\ fall from the cold heights, 45238 0 The death of de Lagny, 452390 Probabilite des jugements, 45

LEONHARD EULER

2400 Tributes to Euler, 47241 0 Euler's blindness, 48242 0 Euler's memory, 48243 0 Euler's concentration, 482440 Euler's productivity, 492450 Euler's universality, 502460 The origin of a paper in mechanics, 502470 Euler, the supreme calculator, 512480 Euler joins the Prussian Academy, 512490 Euler returns to the Russian Academy, 522500 A narro,v escape, 52251 0 A formalist and his pencil, 522520 The Euler-Diderot anecdote, 522530 Euler's death, 54

LAGRANGE

2540 ,tVho was the most eminent mathematician of the eighteenthcentury? 55

255 0 Frederick the Great invites Lagrange to Berlin, 57256 0 A scientific poem, 57257 0 A lofty pyramid, 572580 A very short paper, 572590 Wealth versus mathematics, 572600 A great waste, 58261 0 The marriage of September and April, 58

LAPLt\.CE

2620 Laplace seeks employment, 582630 The Newton of France, 59

VB

CONTENTS

2640 An unneeded hypothesis, 592650 II est aise avoir, 592660 The faith of a mechanist, 59267 0 Laplace's "stepchildren," 60

NAPOLEON BONAPARTE

268 0 Mathematics and the welfare of the state, 602690 Napoleon's problem, 612700 Napoleon's theorem, 63

QUADRANT FOUR

ABEL AND AGNESI

271 0 Honored on a postage stamp, 672720 Crelly and Keilhau, 672730 A precocious and versatile somnambulist, 682740 The witch of Agnesi, 69

CHARLES BABBAGE

2750 An unheralded prophet, 722760 The Analytical Society, 72277 0 Babbage and Tennyson, 73278 0 The origin of miracles, 742790 The scientific gadfly, 752800 The cowcatcher and electric telegraph, 75281 0 Legibility of mathematical tables, 762820 Organ grinders, 77

SOME B's2830 John Taine's review of a book by Eric Temple Bell, 782840 In memory of three apples, 782850 Mathematician, violinist, fencer, 792860 Euclid as a physician, 792870 A precocious arithmetician, 792880 With an interest in ships, 812890 A warning, 812900 Pi by probability, 81

Vlll

CONTENTS

CARLYLE AND LEGENDRE

291 0 Thomas Carlyle's geometrical solution of quadratic equations, 832920 Legendre, Thomas Carlyle, and America, 84

MATHEMATICIANS AND NATURE LOVERS

293 0 Driven to the insane asylum, 862940 Mathematics and theology, 87295 0 The neglect and loss of manuscripts, 882960 The well-balanced mathematician, 892970 An appreciator rather than a creator, 90

CLIFFORD AND DODGSON

2980 Strong man and entertainer of children, 912990 Gentle man and entertainer of children, 92

CALCULATING PRODIGIES

3000 Son of a small farmer, 94301 0 Son of a stone mason, 953020 Lightning calculator, 96

AUGUSTUS DE MORGAN

3030 De Morgan's useless eye, 983040 Conscientious champion and great teacher, 983050 The London Mathematical Society, 993060 De Morgan and the actuary, 993070 De Morgan and the solidus notation, 993080 Ten quotes from De Morgan, 100

ALBERT EINSTEIN

3090 Einstein and his hat, 1023100 Einstein's early public address in America, 102311 0 Einstein and his blind friend, 1033120 An Einstein legend, 1043130 Some Einstein quotes, 104

SKIPPING THROUGH THE F's3140 A lethal factor table, 1043150 What became of Karl Feuerbach? 1053160 Enraptured over heat, 107

IX

CONTENTS

31 70 Benjamin Franklin and mathematics, 1083180 The "Little Giant," 111

CARL FRIEDRICH GAUSS

3190 Gauss's precocity, 1123200 The near loss of a genius, 113321 0 How Gauss was won to mathematics, 1133220 Gauss and his chaffinch, 1133230 Gauss and languages, 1143240 Gauss's scientific diary, 1143250 A discredited story, 1153260 Gauss's seal and motto, 1163270 The greatest mathematician in the world, 1163280 Gauss and the extortionists, 1163290 Gauss's declaration, 1173300 The greatest calamity in the history of science, 117331 0 Gauss and Sir Walter Scott, 117

SOME LITTLE MEN

3320 President Garfield and the Pythagorean Theorem, 1183330 Niches in the hall of fame, 1193340 A belated recognition, 1193350 A famous conjecture, 1203360 An instance of misplaced credit, 121

HAMILTON AND HARDY

3370 A patriot, a prodigy, and a lover of animals, 1213380 A visit to a mathematical shrine, 1223390 The case of the two Sir William Hamiltons, 1243400 A trivial relation, 125341 0 Never criticize the sonatas of archdukes, 1253420 The most gigantic gambit conceivable, 126

TEN MISCELLANEOUS STORIES

3430 A double anticipation, 1263440 The Jacobi brothers, 1273450 From the lips of C. G. J. Jacobi, 1283460 L. Kronecker and J. ~ Morgan, 128

x

CO~TENTS

3470 Kronecker's toast, 1283480 Monge's romantic marriage, 1293490 What a released PO-VV brought back, 1293500 What Poisson learned by hanging around, 129351 0 Poisson as a keeper of money, 1303520 Why there is no Nobel Prize in mathematics, 130

J. J. SYL\lESTER ~~D NORBERT WIENER

353 0 Sylvester's memory, 1313540 Sylvester at the University of Virginia, 1313550 Sylvester at Johns Hopkins University, 1323560 Sylvester on music and mathematics, 133357 0 Wiener and his car, 1333580 Wiener and the student, 1343590 Wiener at a colloquium, 1343600 Wiener in the classroom, 134

INDEX, 135

Xl

INDEX

References are to items, not to pages. A number followed by the letter p refers tothe historical capsule just preceding the item of the given number (thus 275prefers to the historical capsule immediately preceding Item 275°).

Abacus, 349Abbott, J. S. C., 268Abel, N. H., 271, 272, 297, 345"Able was I ere I saw Elba," 270Abu Kamil, 114Abundant numbers, 54Academie des Sciences (See French Academy)"According to Cocker," 227Acta eruditorum, 190p, 203Actual infinite, 293Adelard of Bath, IIIAdler, A., 343Aeneid (Virgil), 242, 246Agnesi, M. G., 273, 274Agnesi, M. '1:, 273Ahmespapyrus, 13Al-Battani, IIIAlcuin, 117Alexander the Great, 59, 67Algebra, 6, 10, 55-56, 84-86, 88, 97, 100,

103, 113, 115, 121-122, 124-126, 131p,131-135,137, 148p, 149, 176,217,252,291, 333, 338-339, 359

etymology, 103rhetorical, 85symbolic, 85syncopated, 85

Algebra (Saunderson), 231Algebraic symbolism, 94, 121, 122, 125,

148p, 149, 176, 182, 307, 333Algorithm (etymology), 104Alhazen, 107

problem of, 107Alice in Wonderland (Carroll), 99

Alice Through the Looking Glass (Carroll),299

Al-Khowarizmi, 103-104, Ill, 114, 221Almagest (Ptolemy), 102, Ill, 120, 263Alp Arslan, Sultan, 108AmericanJournal ofMathematics, 355Amicable numbers, 40, 53Ampere, A. M., 300pAnalyse des infini1nent petits (l'Hopital), 208Analyst, The (Berkeley), 212-213Analytical engine, 275pAnalytical Society, 276Analytic geometry, 83, 166, 173p, 174, 274,

278, 291Angular degrees, 18Anna, Empress, 248Anne, Queen, 224Annuities upon Lives (De Moivre), 229Apol10nius, 80p, 82, 83, 88p, 90, 120, 226problem of, 135"Apple of discord," 210Aquinas, '1:,119Arabian mathematics, 102pArago, F., 240Archimedean screw, 75Archimedes, 72p, 72-79, 80p, 88p, 120,

131p, 133,205,210, 319p, 320, 321,324, 330

Archytas, 58pAristotle, 59, 154, 155Arithmetic, 1-3,9, 11, 13, 19,22,23,85,86,

91-93,95,97-101,104-105,110,113,114,116,124,127,130, 138p, 138,223,227,242,287,300-302,319, 349

135

INDEX

Arithmetica (Diophantus), 179-180Arithmetica infinitorum (Wallis), 182, 193Arithmetica integra (Stifel), 129Arithmetick, being a Plain and Easy Method

(Cocker), 227Arithmetic mean, 88Arithmology, 130Ars conjectandi aacob Bernoulli), 182, 203Ars magna (Cardano), 131, 134Artis analyticaepraxis (Harriot), 148pAryabhata, 92, 106ASoka, King, 91Assassin (etymology), 108Astronomer-mathematicians, 153pAstronomical Society, 276Astronomy, 27, 33, 64, 102, 124, 128, 155,

160-161, 226, 263Aubrey,.]., 152Autobiography (Franklin), 31 7Ayton, J., 338

Babbage, C., 275p, 275-282Babylonian mile, 18Bacon, E, 5Bacon, R., 118Bakshali manuscript (anonymous), 99Ball, W. W. R., 82, 185, 325Ballard, W. H., 354Barberini, Cardinal, 156Barlow, ~, 54Barrow, 1., 185, 190p, 333Beasting, 130Beeckman, I., 173Beethoven, L. von, 356Bell, E. T, 15, 119, 202, 283Benedict XIV: Pope, 273Berkeley, Bishop G., 211-213, 230Berlin Academy, 235-237, 24OP, 248-249,

254pBernard, E., 226Bernoulli, Christoph, 203Bernoulli, Daniel, 203, 207, 209, 245Bernoulli, Daniel II, 203Bernoulli distribution, 203Bernoulli equation, 203Bernoulli family, 203p, 203, 234, 237Bernoulli, Jakob, 182,203-207, 240pBernoulli, Jakob II, 203Bernoulli, Johann, 200, 203-204, 207-208,

210, 214

Bernoulli, Johann II, 203Bernoulli, Johann III, 203Bernoulli, Johann Gustav, 203Bernoulli, Nicolaus, 203, 245Bernoulli numbers, 203Bernoulli polynomials, 203Bernoulli theorem, 203Bertrand, J., 59BESK digital computer, 54Beta, 82Beta functions, 216Bhaskar~ 93, 95-97, 101Bidder, G. ~, 301Biot, J. B., 265, 267Birds and number sense, 1Birkhoffs (father and son), 203pBode, G. H., 322Bolyai, E, 284, 285Bolyai, J., 83, 284-285Bolyai, W. (See Bolyai, F.)Bolzano, B., 286Bonaparte, N. (See Napoleon Bonaparte)Book ofRevelations, 130Boole, G., 190p, 289Bouvelles, C., 210Bowditch, H., 287Bowditch, N., 265, 287-288Brachistochrone, 200, 203Brahe, T, 160, 161Brahmagupta, 93, 94, 97Brief and True Report of the New Found, Land

of Virginia, of the Commodities, and of theNature. and Manner of the NaturalInhabitants, A (Harriot), 149

BriefLives (Aubrey), 152Briggs, H., 138p, 143-144, 147, 225Briggsian logarithms, 147British Association for the Advancement of

Science, 276, 296Brownell, G., 31 7Buddha, 98Budget of Paradoxes, A (De Morgan), 252,

300pBuffon, Comte de, 290Burckhardt, H., 314Biittner, Master, 319Caesar, Julius, 59Cajori, F., 269, 338Calculating machines, 54, 167, 275p, 275,

278

136

INDEX

Calculating prodigies, 300p, 300-302, 319,337

Calculation of pi, 332pCalculus, 181, 190p, 200, 211p, 210-216,

218, 220, 222, 230, 276, 308integralis, 203summatorius, 203

Calculus (Lacroix), 276Cambuston, H., 307Cantor, G., 293, 294

aphorism, 293Cantor, M., 13Cardano, G., 131-132, 134Caritat, A.. N. (See Condorcet, Marquis de)Carlyle, T, 291, 292Carroll, L., 37, 298-299Cassini family, 203pCassini, G. D., 236Cassini, J., 236Castle ofKnowledge, The (Recorde), 124Cauch~A. L., 217, 244, 295, 296Cavalieri, B., 181,218-220

principle, 218-219Cayley, A., 244, 296, 307, 353

line, 169Cellini, B., 13lpChandler, A., 31 7Characteristic, 144Characteristica generalis, 190pCharles, M., 59, 166Charles II, 152, 185Chebyshev, PL., 334Cheng Chhu-Hui, 23Chernac, L., 314Christ, 132Christina, Queen, l73p, 177Chrystal, G., 333Cicero, 79Cipher (etymology), 105Circle principle, 222Circles ofproportion, The (Oughtred), 151Clairaut, C. A., 235Clairaut family, 203p, 235Clavis mathematicae (Oughtred), 148p, 193Clement IX, Pope, 220Cleopatra, 59Clifford, WK., 228, 298-299Cocker, E., 227, 317Colburn, Z., 300-301Collinson, P, 31 7

Colombini, Blessed John, 220Colson, J., 274Commentary on Euclid, Book I (Proclus), 336Common logarithms, 147Compendium Euclidis Curiosi (Mohr), 343Condorcet, Marquis de, 239,247Congruency by subtraction, 123Conic sections, 83, 166-167Copernican theory, 155-156, 159, 161Copernicus, N., 128, 155, 236Cossic art, 122Cotes, R., 228, 231Coulomb's law, 57Cowcatcher, 280Crelle, A. L., 297, 314Crelle's Journal, 297Croesus, King, 32Cromwell, 0., 195Crowd of numbers, 53Cubic and quartic equations, 131p, 131-134Cusa, N., 210Cycloid, 168, 181, 186,200,203,210

Da Coi, Z., 134D'Alembert,Jean-le-Rond, 217, 235, 262

principle, 21 7Dalton's law of multiple proportion, 57Damaras,9Damon, 49Darwin, C. R., 158Dase, Z., 302, 314Davies, C., 292De Bry, T, 149Decimal fractions, 138De diuina proportione (Pacioli), 123Deductive mathematics, 13pDeficient numbers, 54De Lagny, T E, 238Delamain, R., 151Del Ferro, S., 131Delian problem (See Duplication of the

cube)Democritus, 218De Moivre, A., 229, 231De Morgan, A., 86,130,152,198,252, 30ap,

303p, 303-307, 308, 339De numervsa potestatum resolutione (Viete), 137Desargues, G., 166p, 166, 173pDescartes, R., 8, 53, 167, 173p, 173-177,

193,210,234

137

INDEX

Destouches, General, 217De Sua, F., 294Determinants, 222De triangulis omnimodis (Regiomontanus),

120Dettonvi1le, A., 170Diatribe du Dr. Akakis (Voltaire), 237Diderot, D., 217, 252Die Coss (Rudolf£), 122Difference engine, 275pDifferential and Integral Calculus (De

Morgan), 304Differential geometry, 235Digit (etymology), 117Dionysius, 49Diophantine problems, 83-84, 89Diophantus, 84p, 84-85, 90, 94,179-180Dirichlet, ~ G. Lejeune-, 180, 356Discorsi e dimostrazioni matematiche intorno a

due nuove scienze (Galileo), 155, 156Discours de la Methode pour bien conduire sa rai

son et chercher la verite dans les sciences(Descartes), 174, 176

Disquisitiones arithmeticae ( Gauss) , 325Doctrine and Application of Fluxions, The

(Simpson), 291Doctrine ofChances (De Moivre), 229Dodgson, C. L. (See Carroll, L.)Duke of Brunswick, 190p, 319pDulong-Petit law for specific heats, 57Duplication of the cube, 66, 249

Ecole Normale, 254p, 268, 316Ecole'Polytechnique, 254p, 268, 295, 316Economy of Manufactures and Machinery

(Babbage), 275Edison, T, 160Eells, W C., 254Einstein, A., 237, 309p, 309-313Electric telegraph, 280Elements (Euclid), 108, Ill, 124, 167, 184,

193, 230, 286, 292, 332, 336, 343Elements de geomitrie (Legendre), 291, 292Elements ofGeometry (Leslie), 291Eliot, G., 298Elizabeth, Empress, 249Ellipse, 83Elliptic functions, 324Elliptic geometry, 83Ellis, H., 196

Empirical mathematics, 13pEncyclopaedia Britannica, 339Encyclopedia Metropolitana, 307Encyclopedie 21 7Engineering, 14, 32Epsilon, 82Equiangular spiral, 205Eratosthenes, 80p, 80-82, 185Ergodic theory, 359Etymology, 102-106, 108, 117, 144, 182,225Euclid, 54, 61, 65, 67p, 67-70, 72p, 80p, 88p,

89-90,108, Ill, 124, 133,167,184,193,221, 230, 258, 286, 292, 332, 336, 343

Euclidean algorithm, 114Euclides Danicus (Mohr), 343Eudoxian theory, 286Eudoxus, 66, 286Euler, J. A., 240pEuler, L., 53, 180, 203, 216, 237, 240p,

240-253, 254p, 254, 295-296, 335"Eureka,"324Eves, H., 83, 89, 179, 339

Factor table, 314Faraday's law of electrolysis, 57Farrar, J., 292Farrer,J. A., 59Father Bongus, 130Father Charlet, 175Feldmann,Jr., R. W., 317Felkel, 314, 332pFenn, J., 336Fermat, C. S., 179Fermat, P de, 53, 168, 173p, 178-180, 210,

274last theorem, 180numbers, 178

Feuerbach, K. W, 315, 332ptheorem, 315

Fibonacci, 13,114-116,122sequence, 114

Fibonacci Quarterly, 114Figurate numbers, 324Finger numbers, 117Fior, A., 131Fitzgerald, E., 108, 109Flamsteed, J., 226, 233Fourier, J. B. J., 268, 316

series, 316Franklin, B., 317

138

INDEX

Franklin, F., 355Fraunhofer's diffraction grating, 57Frederick II, 112, 115Frederick the Great; 236, 237, 240p, 248,

249, 252, 255French Academy, 59, 167, 217, 235, 239,

241, 247, 267, 295, 325Fubini, C. G., 318Fubini, G., 318

Galileo, 59, 120, 153p, 153-159, 203, 210,218

law of falling bodies, 57Galois, E., 228, 254p, 295Galton, Sir F., 9Galvanoplastics, 344Gamma function, 216Garfield,]. A., 332Gauss, C. F., 178, 196, 284, 300p, 319p, 319-

331, 356Gazetas de Mexico, 307Geminus,61Geometria del compasso (Mascheroni), 343Geometric mean, 88Geometrie (Descartes), 193, 234Geometry, 12, 14-16, 18,20,31,35, 37,45,

55, 61-70, 76-77, 79-80, 83, 89, 96,100, 107-108, 123-124, 167-168, 178,181, 184,218-219,230,246,269-270,274,278,286,291,306,308,315,321,324, 332, 334, 343, 349

Gerbert, 110, 120Gherardo of Cremona, 106, IIIGlaisher, ]. W L., 314Godel, K., 294Gog and Gug, 12Goldbach, C., 335

conjecture, 335Golden ratio, 114Golden rectangle, 114"Grand aplatisseur," 236Grandi, L. G., 215,274Graphomath, 308Gravitational field, 312"Greatest Egyptian pyramid," 15"Great Geometer, The," 82Greek and Hindu mathematics contrasted,

100Greenville, Sir R., 148Gregory, D., 203p, 333

Gregory, ]., 203p, 274Gresham, Sir L, 138pGrimaldi, 202Grimm, P, 250Ground ofAries, The (Recorde), 124Gunter, E., 151, 225

Hakim, Caliph, 107Halley, E., 190p, 199,213,226,233,235Halsted, G, B., 285Hamilton, Sir W, 339Hamilton, Sir W. R., 256, 337-339Hanseatic League, 127Hardy, G. H., 101, 340-342Harmonic mean, 88Harmony of the Worlds (Kepler), 161Harriot, L, 125, 148p, 148-149Hart, H., 334Hasan Ben Sabbah, 108Haiiy's law of rational indices, 57Hawkins, ].,227"Helen of geometry," 210Helmholtz, H. L. F. von, 356Henry I\T, 135Heraclitus, 43Hermes, Professor, 178Hermite, C., 271Heron of Alexandria, 120Herschel, Sir 1., 275p, 276Hiero, King, 72-74Hieronymus, 31Hilbert, D., 293, 294Hill, L, 278Hindu and Greek mathematics contrasted,

100Hindu-Arabic numerals, 91, 110,116Hindu mathematics, 91p, 91-101Hipparchus, 102Hippasus, 47Hisab al-jabn w'al-muqabalah (al-Khowarizimf) ,

103Histoire natwrelle (Buffon), 290History ofMathematics, A (Cajori), 269, 338Hjelmslev, 1., 343Hoggatt,]r., \Z, 114Holzmann, W, 179Hoof, 219Horace, 351Horbon, Madame, 348Huai-Wen, 22

139

INDEX

Humboldt, A. von, 327Hutton, C., 209Huygens, C., 120,179, 19OP, 203, 210,234,236Hydrodynamica (Daniel Bernoulli), 203Hydrostatics, 74, 138Hypatia, 90, 203pHyperbola, 83Hyperbolic geometry, 83

IBM 7090 digital computer, 54I-Hsing, 23Induction, 7Infinite series, 215-216Infinity, 183, 189

symbol for, 182Inquisition, 156Insects and geometry, 4Insects and number sense, 2-3Inversion, 92Involution, 83Isochrone, 203

jacobi, C. G.j., 344, 345.jacobi, M. H., 344james I, 138pjames, R. C., 83"japanese Newton," 222jefferson, T, 332Jinko-ki (Yoshida K6yii), 221, 223john of Palermo, 115john of Seville, IIIjohnson, M., 10jones, W., 333J oumal fUr die reine und angewandte Mathematik,

297Journal of the Franklin Institute, 317justinian, Emperor, 58pjuvenal, 117

Keilhau, B. M., 272Kelvin, Lord, 316Kemp, C., 272Kepler, j., 62, 64-65, 153p, 159-160, 163

164, 166, 193laws of planetary motion, 57, 161

Keyser, C.j., 70Khayyam, Omar (See Omar Khayyam)Kirkman points, 169Klein, F., 83Knox,j., 138p

Koenig (Konig), S., 209, 237Koshu, Lord of, 224Kramp, C., 333Kronecke~ L., 293-294, 346-347Kulik,j.~, 314Kummer, E., 180

Lacroix, S. F., 276La Disme (Stevin), 138pLa geometrie (Descartes), 176Lagrange, j. L., 203, 254p, 254-261La Hire, ~ de, 166Lambert,j. H., 314Lame, G., 114, 180Laplace, ~ S., 145, 262p, 262-267, 287,

327-328Lavoisier, A. L., 260Law of cosines, 89Laws of Thought, The (Boole),289Laws of Verse, The (Sylvester), 355Lazarus, 59Lazzerini, 290, 332pLegendre, A. M., 180,291-292Lehmer, D. H., 203pLehmer, D. N., 203p, 314Leibniz, G. W, 120, 170, 190p, 196, 199

203, 208, 210, 211p, 211, 222, 254,274,276

Lemniscate of Bernoulli, 203Lemonnier, ~ C., 261Leonardo da Vinci, 123Leonardo Pisano (See Fibonacci)Leo X, Pope, 130Leslie, j., 291Lexell, 253:LHopital, G. F. A. de, 208

rule, 208Liber abaci (Fibonacci), 114Lilavati,95Lilavati (Bhaskara), 95Lincoln, A., 332Linkages, 334Lipkin, 334Lippershay, H., 155Lisle's law of constant angles, 57"Little Giant, The," 318Little People, The (Clifford), 298Littlewood, j. E., 341Lobachevsky, N. I., 83Logarithmic spiral, 205

140

INDEX

Logarithms, 6, 129, 138p, 145-146, 151,184,218, 225, 268, 302

Briggsian, 147common, 147etymology, 144N apierian versus common, 147

Logic, 308, 339Logistic, 86London l\tlathematical Society, 305Lo-shu, 19Lucas, v: D., 59Ludlam, \V, 336Lu Hung, 23Luther, M., 129, 130

Maclaurin, C., 230MacLane, S., 359Magic circle, 317Magic squares, 19, 317Mahavira, 97Manfred, 112Mannheim, A., 151,334Mantissa (etymology), 144Marcellus, 77Mary Magdalene, 59Mascheroni, L., 269, 343Mathematician's Apology, A (Hardy), 342Mathematicians, classification of, 5Mathematics and sciences, 118Mathematics as a rising continent, 187Mathematics as a rock, 188Mathematics, Queen and Servant of Science

(Bell),283Mathematics Teacher, The, 59,83,89, 179,208,

210,254,317,339Maupertuis, ~ L. M. de, 236-238Maurice of Nassau, 155, 173p, 173Maximilian I, 120Maximilian Joseph, 315Maxwell's electromagnetic wave equations,

57Means

arithmetic, 88geometric, 88harmonic, 88subcontrary, 88

Mechanique analytique (Lagrange), 256Mechanique celeste (Laplace), 263-265, 287Memory,23Menaechmus, 66

Mendeleeff's periodic chart, 57~fercator, G., 184

projection, 184Mercator, N., 184L\1.es souvenirs de vingt ans de sejour a Berlin

(Thiebault), 252lVIethod ofFluxions (Newton), 290l\1.ethod ofFluxions (Saunderson), 231Method of indivisibles, 181,218, 220Method of infinite descent, 179Meziriac, B. de, 179Milo, 41Miracles, origin of, 278L\1.irifici logaTithmorum canonis descTiptio

(Napier), 138pl\1.iscellanea analytica (De Moivre), 2291\1itsuyoshi (See Yoshida Koyu)Mittag-Leffler, G. M., 352Mnesarchus, 43Mohr, G., 343Monge, G., 268, 316, 348Montalte, L. de, 170Moral expectation, 203Morgan, J. ~, 346Moscow papyrus, 15Mozart, W. A., 356MTS. MiniveT (Jan Struther), 246Muller, J. (See Regiomontanus)Mysteriurn cosmogTaphicum (Kepler), 159

Nach, Adam Riese, 92, 127,227Napier, J., 120, 129-130, 138p, 139-145,

147bones, 308

Napierian logarithms, 147Napoleon Bonaparte, 257, 264, 268p, 268

270, 316, 328, 349problem, 269theorem, 270

Needle problem, 290Nemorarius, 105Nero, 130Nesselmann, G. H. F., 85Neugebaue~ 0.,17New American Practical NavigatoT (Bowditch),

287New EnglandJournal ofEducation, 332Newton, I., 59, 69, 120, 151, 153p, 156, 174,

182, 185, 190p, 190-200, 203p, 203,

141

INDEX

209-211, 211p, 213, 222, 224, 226,228-231,236,254,276,284,287,290,300p, 31~, 320, 329, 356

law of universal gravitation, 57"Newton of France" 263,Nicolo of Brescia (See Tartaglia)Nine-point circle, 315Nizam ul Mulk, 108Nobel, A., 352

prizes, 352 .:-.Non-Euclidean geometry, 108,284-285Notizenjournal (Gauss), 324Number mysticism, 52, 55-57, 60Number sense, 1-3, 11Number theory, 17, 40, 53, 84, 86, 115, 173p,

178-180

Olbers, H. W M., 328Omar Khayyam, 108-109, 131pOmnibus, 171"On Newton's rule for the discovery ofimag

inary roots" (Sylvester), 356On the Sphere and Cylinder (Archimedes),

131pOpera (Johann Bernoulli), 208Operations research, 275Optics (Alhazen), 107Opus Majus (R. Bacon), 118Oughtred, W, 120, 148p, 150-152, 193, 333Pacioli, L., 123Paganni, N., 53Pappus, 4, 88p, 88-90Parabola, 83Parabolic geometry, 83Parallel postulate, 70, 258, 284, 336Partial differential equations, 21 7Pascal, B., 59, 120, 166p, 167-172, 173p, 175,

210, 228line, 169"mystic hexagram" theorem, 169principle, 167

Pascal, G., 167Pathewaie to Knowledge, The (Recorde), 124Peacock, G., 276Peasant finger multiplication, 117Peaucellier, A., 334Pell, J., 314Penny Cyclopedia, 304Pennsylvania Gazette, 317

Pensees (Pascal), 168, 172Pentathlus, 82Perfect numbers, 54Perier, Madame, 167Peter the Great, 240pPetersburg paradox, 203Peuerbach, G. von, 120Pfaff, J. F., 327Phillip II, 136Phillips, J. ~, 210Philomathic Society, 334Philosophiae naturalis principia mathematica

(Newton), 190p, 199, 226, 287-288Philosophical and Mathematical Dictionary, A

(Hutton), 209Philosophical Transactions of the Royal Society,

210Phintias, 49Phu-Chu,23Physics, 36, 57, 72-75Pi, 290, 306, 333Pierce, G. w., 3Pillow Problems (Carroll), 299Plaine Discovery ofthe whole Revelation ofSaint

John, A (Napier), 139Planck's quanta, 57Plato, 58p, 58, 60-61, 66, 82

Academy, 58p, 58, 66Plato of Tivoli, IIIPlatonic solids, 61-65Playfair, J., 336

postulate, 336Plimpton 322, 17Plucker lines, 169Plume, Dr., 228Plutarch, 31, 72, 77Poisson, S. D., 350-351Poncelet, J. V:, 343, 349Pope, A., 196Postage stamps, 271Potential infinite, 293Predicting end of world, 129Prime numbers, 314, 335Primitive people, 9-12Primitive Pythagorean triples, 17Principia mathematica (Whitehead and

Russell), 190p, 289Probabilite des jugements, 239Probability, 168, 172, 229, 239, 290Problem of Alhazen, 107

142

I DEX

Problem solying~ 296Proclus, 67, 336Projective differential geolnetry, 318Projective geometry, 166p, 166, 169, 173p,

349Proposals Relating to the Education of Youth in

Pennsylvania (Franklin), 317Propositiones philosophicae (Agnesi), 273Prout's law of definite proportion, 57Provincial Letters (Pascal), 168, 170Prussian l\cademy (See Berlin Academy)Pseudomaths, 300p, 308Pseudo-\'\ itch, 274Ptolemy, C., 90, 102, Ill, 156, 163Pyramid of Gizeh, 14

greatest Egyptian, 15Thales puzzle, 31

Pythagoras, 35p,35-45,47,51,53, 56-57,64Pythagorean Brotherhood, 35p, 46-52, 54,

55, 57, 60, 83, 347Pythagorean theorem, 12,20,37,38,45,89,

96, 123, 332Pythagorean triples, 17

primitive, 17Pythias,49

Quadrature ofCurves (1\ewton), 211Quantum theory, 313Quaternions, 338-339Radiolaria, 63Rahn,J. H., 314Raleigh, Sir W, 148Ramanujan, S., 101Rechenmeisters, 127Recorde, R., 124-127, 149Reductio ad absurdum, 342Regiomontanus, 120, 179Regular polygons, 321, 324Regular polyhedra, 61-65Regular sexagesimal numbers, 17Relativity theory, 311Rhaeticus, G.J., 128Rhetorical algebra, 85Rhind, A. H., 13Rhind papyrus, 13, 16Richelot, F J ., 178Riese, A., 127Robert of Chester, IIIRoberval, G. ~ de, 179, 181, 210Romanus, A., 135

Rosenbaum, R..A.. , 59Royal Obseryatory, 279Roval Society, 182, 186, 190p, 200, 226,

, 231, 27Sp, 279Rubai)'at (Gmar Khayyam), 108Rudim.ents ofivlathematics (Ludlam), 336Rudio, F., 240Rudolff, C., 122Rule of false position, 99Rule of three, 93Russell, B., 289Russian .L~cadem) (See St. Petersburg

AL\cademy)Rusty compasses, 343

Saccheri, G., 108Sachs, A. J., 17St. Peter, 59St. Petersburg Academy, 203, 235, 240p,

245, 249Salmon points, 169Saunderson, N., 231Savile, Sir H., 138pSavonarola, 55Schnirelmann, L., 335Schopenhauer, A., 282Schroedinger's wave equation, 57Science ofAbsolute Space (J. Bolyai), 285Scientific mathematics, 13pScott, Sir W, 308, 331Seki Kowa, 222-224Set theory, 293Severi, F, 343Shi Huang-ti, Emperor, 19Si monumentum requiris, circumspice, 232Simpson, T, 291Simpson, \'\., 109Sine (etymology), 106Sixtus IV: Pope, 120Slide rule, 151,225,334Smith, C. L., 149Sociable chain of numbers, 53Society for the Diffusion of C seful Rnnw-

ledge, 304Socrates, 58p, 322Solid geometry, 218, 219Solon, 26"Southern Tycho," 226Space method for determining eminence,

254

143

INDEX

Spherical ring, 219Spherics (Theodosius), IIIStar names (etymology), 102Statics, 138Statistical Society of London, 276Stefan's law of radiation, 57Steiner, ]., 343

points, 169Stevin, S., 120, 138p, 138Stifel, M., 122, 129, 130Stirling's formula, 229Stobaeus, 67, 68Stokes, A. A., 307Struik, D. ]., 208Stukeley, W., 194Subconscious mathematics, 9pSubcontrary mean, 88Subfactorial, 333Submathematical analysis, 119Surya Siddhanta (anonymous), 97SWAC digital computer, 54Sylvester, ]. ]., 353p, 353-356Sylvester II, Pope, 110, 120Symbolic algebra, 85Syncopated algebra, 85, 94Synesius of ayrene, 90Syntaxis mathematica (See Almagest (Ptolemy))Systematic mathematics, 13p, 24p

Tables ofLogarithms (Babbage), 281Taine, ].,283Tangled Tale, A (Carroll), 299Tartaglia, 131, 133-134Tautochrone, 203Tencin, Madame de, 21 7Tennyson, Lord A., 277Tensor analysis, 310Textbook Algebra (Chrystal), 333Thales, 24-34, 35p

puzzle, 31Theaetetus, 61Theano, 41Theodorus of Cyrene, 58pTheodosius, IIITheon of Alexandria, 203pTheorie analytique de la chaleur (Fourier), 316Theorie analytique des probabilites (Laplace),

266Theorie de la figure de la Terre (Clairaut), 235Theorie de la Lune (Clairaut), 235

Theory of equations, 135, 137, 148p, 356Theory of ideals, 180Theory of numbers, 54, 314, 324, 335Thiebault, 252Timaeus of Locri, 61, 62Torricelli, E., 167, 181, 210Traite de Dynamique (D'Alembert), 217Traite du triangle arithmetique (Pascal), 168Trigonometric functions (etymology), 106Trigonometric series, 316Trigonometry, 6, 17-18,89, 100, 106, 225Ts'ai Yung, 21

LTndecidable problems, 294LTrban ~ Pope, 220LTrban VIII, Pope, 156

Valdes, M. A., 307Van Amringe,]. H., 292Van der Hoeke, 121Van Schooten, E, 234Van Schooten family, 203p, 234Van Schooten, ~, 234Vector algebra, 10Victoria, Queen, 299Viiete, F., 135p, 135-137, 193,234Vinogradoff, I. M., 335Virgil, 246"Vision of Sin, The" (Tennyson), 277Viviani, V:, 210Vlacq, A., 144Voltaire, 237, 240p

Wallis, ]., 150, 182, 193,210,226, 300pWard, S., 150Washington, G., 332vVatson, Master, 287-288Weeks,W. E, 354vVheelbarrow, 171Whetstone of Witte, The (Recorde), 124, 126vVhewell, w., 163White, Captain]., 149Whitworth, W. A., 333Widman, J., 121Wiener, N., 353p, 357-360Wilhelm, Kaiser, 130Witch of Agnesi, 274~olfskehl, ~, 180vVren, Sir C., 150, 186, 210, 232

Xylander, G. (See Holzmann, \'\1.)

144

Yenri, 222Yoshida K6Yll, 221, 223"Young, G. C., 60Yu, Emperol~ 19

Zero (etymology), 105

Ir\DEX

145

AMS / MAA SPECTRUM VOL 38

In M

athem

atical Circles &

Qu

adran

ts I, II, III, IV

How

ard W

. Eves

VOL38AMS / MAA SPECTRUM

166 pages on 50lb stock • Spine 5/16" • Trim size 6 x 94-Color Process

AMS / M

AA PRESS

Mathematical Circles

The 360 different anecdotes compiled in these delightful volumes will add zest to every teacher's mathematics classes. There are appropriate selections for all levels of students. They are short, succinct, and at the same time given in simple settings which enable the reader to identify with the story and its implications. Here is presented the kind of material that makes the difference to the undecided student. I highly recommend you putting a copy next to your worktable.

—The Mathematics Teacher

For many years, famed mathematics historian and master teacher Howard Eves collected stories and anecdotes about mathematics and mathematicians, gath-ering them together in six Mathematical Circles books. Thousands of teachers of mathematics have read these stories and anecdotes for their own enjoyment and used them in the classroom-to add spice and enter tainment, to introduce a human element, to inspire the student, and to forge some links of cultural history. Through a special arrangement with Professor Eves, the Mathematical Association of America (MAA) is proud to reissue all six of the Mathematical Circles books in a three-volume edition.

The first two books were published to acclaim in 1969, as In Mathematical Circles (Volumes 1 and 2). They are bound together as Volume I of the Mathematical Circles Collection. Mathematical Circles Revisited and Mathematical Circles Squared are bound together as Volume 2 of the Collection, and Mathematical Circles Adieu and Return to Mathematical Circles as Volume 3.

This three-volume set is a must for all who enjoy the mathematical enterprise, especially those who appreciate the human and cultural aspects of mathematics.

Howard Eves spent most of his teaching career at the University of Maine at Orono, and more recently at Central Florida University. For 25 years, he edited the Elementary Problems Section of the American Mathematical Monthly. His books include: Great Moments in Mathematics Before 1650, Great Moments in Mathematics After 1650, Mathematical Reminiscences (all for the MAA), Introduction to the History of Mathematics, and his two-volume Survey of Geometry.

Volume IHoward W. Eves


Quadrants I, II, III, IV

AMS / MAA SPECTRUM VOL 38

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