A Biography of theWorld'sMostMysteriousNumberAlfreds. Posamentier& IngmarLehmannAfterwordbyDr. Herbert A. Hauptman,Nobel Laureate@Prometheus Books59 John GlennDriveAmherst, NewYork 14228-2197Published 2004 by Prometheus BooksPi: A Biography of theWorlds Most Mysterious Number. Copyright 2004 by Alfred S.Posamentier and Ingmar Lehmann. All rights reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmitted in any form or by any means, dig-ital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via theInternet or a Web site without prior written permission of the publisher, except in the caseof brief quotations embodied in critical articles and reviews.Inquiries should be addressed toPrometheus Books59 John Glenn DriveAmherst, New York14228-2197VOICE: 716-691-0133, ext. 207FAX: 716-564-2711WWW.PROMETHEUSBOOKS.COM08 07 06 05 04 54 32 ILibrary of Congress Cataloging-in-Publication DataPosamentier, Alfred S.Pi: a biography of the world's most mysterious number / Alfred S. PosamentierandIngmar Lehmann.p. cm.Includes bibliographical references and index.ISBN1-59102-200-2 (hardcover:alk.paper)I. Pi. I. Lehmann, Ingmar. II. Title.QA484.P67 2004512.7'3---dc222004009958Printedin the United States of America on acid-free paperContentsAcknowledgments7Preface9Chapter 1 What Is1t?13Chapter 2 The History of 1t41Chapter 3 Calculating the Value of 1t79Chapter 4 1t Enthusiasts117Chapter 5 1t Curiosities137Chapter 6 Applications of 1t157Chapter 7 Paradox in1t21756EpilogueContents245Afterword by Dr. Herbert A. Hauptman 275Appendix A A Three-Dimensional Example of aRectilinear Equivalent to a CircularMeasurement 293Appendix B Ramanujan's Work 297Appendix C Proof ThateTr >1re301Appendix 0 A Rope around the Regular Polygons 305ReferencesIndex309313AcknowledgmentsThe daunting task of describing the story of 1t for the general readerhad us spend much time researching and refreshing the many tidbitsof this fascinating number that we encountered in our many decadesengagedwithmathematics. It wasfunandenriching. Yetthemostdifficult part was to be able topresent thestory of 1tinsucha waythat thegeneral readerwouldbeabletosharethewondersof thisnumber withus. Therefore,it wasnecessary tosolicit outside opin-ions. We wish to thank Jacob Cohen and Edward Wall, colleagues atthe City College of New York, for their sensitive reading of the entiremanuscript and for making valuable comments in our effort to reachthe general reader. Linda Greenspan Regan, who initially urged us towrite thisbook,did a fine job in critiquing the manuscript fromtheviewpoint of a general audience. Dr. Ingmar Lehmann acknowledgesthe occasional support of Kristan Vincent in helping him identify theright English words to best express his ideas. Dr. Herbert A.78 AcknowledgmentsHauptmanwishestothankDeannaM. Hefner fortypingtheafter-word and Melda Tugac for providing some of the accompanying fig-ures. Special thanks is due to Peggy Deemer for her marvelous copy-editing and for apprisingus of the latest conventions of our Englishlanguage while maintaining the mathematical integrity of the manu-script.It goes without sayingthat thepatienceshownbyBarbaraand Sabine during the writing of this book was crucial to its suc-cessfulcompletion.PrefaceSurelythetitlemakesit clearthat thisisa bookabout1t, but youmaybe wonderinghowabookcouldbewrittenaboutjust onenumber. We will hope to convince youthroughout this book that1tisnoordinarynumber. Rather, it isspecial andcomes upinthemost unexpected places. You will also find how useful this numberis throughout mathematics. We hopetopresent1ttoyouina very"reader-friendly" way-mindful of the beauty that is inherent in thestudy of this most important number.You may remember that in the school curriculum the value that1t took on was either 3.14, 3 ~ , o r 2; . For a student's purposes, thiswas more than adequate. It might have even been easier tosimplyuse1t =3. But what is 1t? What is the realvalue of 1t? Howdowedetermine the value of 1t? Howwasit calculated inancient times?Howcanthevaluebefoundtodayusingthemost moderntech-910 Prefacenology? How might 1t be used? These are just some of the questionsthat we will explore as you embark on the chapters of this book.We will begin our introduction of 1t by telling you what it is androughlywhereitcame from. Just aswithanybiography(and thisbook is no exception), we will tell youwho named it and why, andhowit grewuptobewhat it istoday. Thefirst chaptertellsyouwhat1t essentially is and how it achieved its current prominence.In chapter 2 we will take you through a brief history of the evo-lutionof1t. Thishistorygoesbackaboutfourthousandyears. Tounderstandhowold the concept of 1t is,compare it to our numbersystem, the place value decimal system, that has only been used intheWesternworldforthepast802years! I Wewillrecall the dis-covery of the 1t ratio as a constant and the many efforts to determineits value. Along the way we will consider such diverse questions asthevalue of 1t asit ismentionedintheBible and itsvalue in con-nectionwith thefield of probability. Once the computer enters thechase for finding the "exact" value of 1t,the story changes its com-plexion. Now it is no longer a question of finding the mathematicalsolution, but rather how fast and how accurate can the computer bein giving us an ever-greater accuracy for the value of 1t.Now that we have reviewed the history of the development of thevalue of 1t, chapter 3 provides a variety of methods for arriving at itsvalue. Wehavechosenawidevarietyofmethods, someprecise,someexperimental, andsome just goodguessing. Theyhavebeenselected so that the average reader can not only understand them butalso independentlyapply them to generate the value of 1t. There aremanyvery sophisticated methods to generate the value of 1t that arewell beyondthescope of thisbook. We havethe general reader inmind with the book's level of difficulty.I. The first publication in we,tern Europe, where the Arabic numerals appeared. wasFibonacci's book Liber abaciin1202.Preface 11With all this excitement through the ages centered on1t, it is nowonder that it has elicited a cultlike following in pursuit of this eva-sive number. Chapter 4 centers on activities and findingsbymath-ematiciansandmathhobbyistswhohaveexploredthevalueof1tandrelatedfields inwaysthat theancient mathematicians wouldnever have dreamed of. Furthermore, withtheadvent of the com-puter, theyhavefoundnewavenues toexplore. Wewill lookatsome of these here.As an offshoot of chapter 4, we have a number of curious phe-nomena that focus on the value and concept of 1t. Chapter 5 exhibitssome of these curiosities. Here we investigate how 1t relates to otherfamous numbers and to other seemingly unrelated concepts such ascontinuedfractions. Again, we have limitedour presentation tomaterial that would require no more mathematical knowledge thanthat of highschool mathematics. Not onlywill youbeamused bysome of the 1t equivalents, but you may even be inspired to developyour own versions of them.Chapter 6 is dedicated to applications of 1t. We begin this chapterwith a discussion of another figurethat is very closely related to thecirclebut isn'tround. ThisReuleauxtriangleistrulyafascinatingexample of how1t just getsaroundtogeometrybeyondthecircle.From here we move on to some circle applications. You will see how1t isquiteubiquitous-it alwayscomesup! Therearesomeusefulproblem-solvingtechniquesincorporatedintothischapterthat willallow you to look at an ordinary situation from a very different pointof view-which may prove quite fruitful.In our final chapter, we present some astonishing relationships in-volving1t and circles. The situation that we will present regarding aropeplacedaroundthe earth will surelychallengeeveryone'sintu-ition. Though a relatively short chapter, it will surely surprise you.It is our intention to make the general reader aware of themyriadof topics surrounding1t that contributetomakingmathe-12 Prefacematicsbeautiful. Wehaveprovideda bibliographyof this famousnumber andmanyofits escapades throughthe fields ofmathe-matics. Perhaps youwill feel motivatedtopursuesomeof theseaspects of IT further, and some of you may even join the ranks of theIT enthusiasts.Alfred S. Posamentierand Ingmar LehmannApril 18, 2004Chapter 1What IsIt?Introduction to1tThisisa book about themysteriousnumber we call n (pronounced"pie," while in much of Europe it ispronounced "pee"). What mostpeople recall about n is that it was often mentioned in school mathe-matics. Conversely, one of the first things that comes to mind,whenaskedwhatwelearnedinmathematicsduringourschool years, issomething about n. We usually remember the popular formulasattached to n, such as 2nr or nr.(To this day, there areadultswholove to repeat the silly response to nr: "No, pie are round!"). But dowe remember what these formulas represent or what this thing calledn is? Usually not. Why, then, write a book about n? It just so happensthat there is almost a cultlike following that has arisen over the con-cept of n. Other books have been written about n. Internet Web sites1314 1treport about its "sightings," clubs meet to discuss its properties, andevenadayonthecalendaris set asidetocelebrateit, this beingMarch14, which coincidentally just happensalsoto be Albert Ein-stein's birthday (in 1879). You may be wondering how March 14 wasselected as 1t day. For those who remember the common value (3.14)that 1t took on in the schools, the answer will become obvious. IIt surely comes as no surprise that the symbol1t is merely a letterin the Greek alphabet. While there is nothing special about this par-ticular letterintheGreek alphabet, itwaschosen, forreasonsthatwe will explore later, to represent a ratio that harbors curiousintrigue and stories of all kinds. It found its way froma member ofthe Greek alphabet to represent a most important geometric constantandsubsequentlyhas unexpectedlyappearedinavarietyof otherareas of mathematics. It has puzzled generations of mathematicianswhohave beenchallengedtodefine it, determine its value, andexplain themanyrelated areasinwhich it sometimes astoundinglyappears. Ubiquitous numbers, such as 1t, make mathematics theinterestingandbeautiful subject that manyfindit tobe. It isourintent to demonstrate this beauty through an acquaintance with 1t.Aspects of1tOur aim here is not to decipher numerous complicated equations, tosolve difficult problems, or to try toexplainthe unexplainable.Rather, it is toexplore the beauty andevenplayfulness ofthisfamous number, 1t, andtoshowwhyit has inspiredcenturiesofmathematicians and math enthusiasts to further pursue and investi-gateitsrelatedconcepts. Wewill seehow1t takesonunexpectedroles, comesupinthemost unexpectedplaces, andprovidestheI.In theUnited States we write the date as 3/14.What Is 1t? 15never-ending challenge to computer specialists of finding ever-more-accurate decimal approximations for the value of 1t. Attemptsat getting further accuracy of the value of 1t may at first seem sense-less. Butallowyourselvestobeopentothechallengesthathaveintrigued generations of enthusiasts.The theme of this book is understanding1t and some of its mostbeautiful aspects. Soweshouldbeginourdiscussionandexplo-ration of 1t by defining it. While for some people1t is nothing morethana touchof thebutton ona calculator, where thena particularnumber appears onthereadout, for others this number holds anunimaginable fascination. Dependingonsizeofthe calculator'sdisplay, the number shown will be3.1415927,3.141592654,3.14159265359,3.1415926535897932384626433832795,orevenlongeLThis push of a button still doesn't tell us what 1t actually is. We merelyhave a slick way of getting the decimal value of 1t.Perhaps this is allstudents need to know about 1t: that it represents a specific number thatmight be useful to know.However, here students would be making acolossal mistake to dismiss the importance of the topic, by justfocusing on the application of 1t in particular formulasand getting itsvalue automatically just by the push of a button.The Symbol 1tThe symbol1t is the sixteenth letter of the Greek alphabet, yet it hasgainedfame because of its designation in mathematics. In theHebrew and the Greek languages of antiquity, there were no numer-16 Itical symbols. Hence, theletters of therespectivealphabetsservedasnumerical symbols. Since the Greekalphabet had only twenty-four letters, though twenty-seven were needed, they used three let-ters of Semitic origin, namely, F[digamma] (for 6), 9 [qoph](for90),and~ [san](for 900).The Greeks at the beginning of the fifth century BeE then usedthe notation represented in the following table:2a~y()E(, E)F ~ 1161 2 3 4 5 6 7 8 9tl(A11v~0 It910 20 30 40 50 60 70 80 90p 0' 't U X\jI(J)~100 200 300 400 500 600 700 800 900,a, ~,y ,(),E(, E),F , ~ ,11,61,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000Thus in the old Greek texts It was used to represent the number 80.By coincidence, the Hebrew letter) (pe) has the same value.RecoLLectionsof1tPerhaps by coincidence or by some very loose associations, the letter1t was later chosen by mathematicians to represent a very importantconstant value related to the circle. Remember, the circle is the mostsymmetric plane geometricfigureand one that goesback in historyto prehistoric times. Specifically, It was chosen to represent the ratio2. A comma at the left indicates thousands. The ten thousands are indicated with anMbelowthenumber symbol. Table fromGeorges Ifrah, Universal History of Numerals(New York: Campus, 1986), p. 289.What Isre? 17of the circumferenceof acircle to its diameter.3This wouldbeexpressedsymbolicallyasre = ~ , whereC representsthelength ofthe circumferenceand drepresentsthelengthof thediameter. Thediameter of a circle is twice the length of the radius, d =2r, where ris the length of theradius. If wesubstitute 2r for d, we get re = fr,whichleads us tothefamous formulafor thecircumferenceofacircle:C =2rer, an alternative of which is C =red.The other familiar formula containingre is that the area of a circleis rer2. This formula is more complicated to establish than that for thecircumference of the circle, which followed directly from the defini-tion of re.Formula for theAreaof a CircleLet's consider a relatively simple "derivation" for the formula (A =v)fortheareaof a circlewithradius r. Webeginbydrawingaconvenient-sizecircleonapieceof cardboard. Dividethecircle(which consists of 360) into sixteen equal arcs. This may be doneby marking off consecutive arcs of 22.5 or by consecutivelydividingthecircleintotwoparts, thenfour parts, thenbisectingeach of these quarter arcs, and so on.3. A purist might ask:how do we know that this ratio is the same for all circles? Wewill assume this constancy for now.18 ItFig. 1-1The sixteensectors we haveconstructed(shownabove) arethento be cut apart and placed inthemanner showninthefigurebelow.Fig. 1-2What IsTt? 19This placement suggests that wehave a figurethat approximates aparallelogram.4That is, were the circle cut intomoresectors, thenthefigurewouldlook evenmorelikeatrueparallelogram. Letusassumeit isaparallelogram. Inthiscase, thebasewouldhavealength of half the circumference of the original circle, since half ofthe circle'sarcsareused foreach of thetwosides of theapproxi-mate parallelogram. In other words, we formed something thatresembles a parallelogram where one pair of opposite sides are notstraight lines, rather they are circle arcs. We will progress as thoughthey were straight lines, realizing that we will have lost some accu-racyintheprocess. Thelengthof thebaseis ~ C. SinceC =2Ttr,the base length is, therefore, Ttr. The area of a parallelogram is equalto the product of itsbase andaltitude. Here thealtitude isactuallythe radius, r, of the original circle. Therefore, thearea of the "par-allelogram" (whichis actuallythe areaofthecircle wejust cutapart) is (Ttr)(r) =Ttrz, which gives us the commonly known formulafor the area of a circle. For some readers this might be the first timethat the famous formula for the area of a circle, A =Ttrz, actually hassome real meaning.TheSquareand theCircleWithout taking the reader'sattention too far afield, it might also beinterestingtopoint out that Tt hastheuniquedistinction of takingtheareaof asquare, whosesidehasthelength of theradiusof acircle, and converting its area to that of the circle. It is the constantvalue connector inthiscase. The area of thesquare (fig. 1-3) isrzand, when multiplied byTt,gives usthe area of the circle: Ttrz.4. A parallelogram is a quadrilateral (a four-sided polygon) with opposite sides parallel.20 It,TheValue of ItFig. 1-3Now that we have anunderstanding of what1t meant inthe contextoftheseoldfamiliar formulas, weshall explorewhat theactualvalue is of this ratio rt. One wayto determine this ratio would be tocarefully measure the circumference of a circle and its diameter andthenfindthe quotient of these two values. Thismight be done witha tapemeasure or witha piece of string. Anextraordinarily carefulmeasurement might yield3.14, but suchaccuracy is rare. As amatter offact, toexhibit the difficultyofgettingthis two-placeaccuracy, imagine twenty-five peoplecarryingout this measure-ment experimentwithdifferent-sizecircular objects. Imaginethentakingtheaverage of their results(i.e.,each of their measured cir-cumferences divided by their measured diameters). You wouldlikelybe hardpressedto achieve the accuracy of 3.14.Youmay recall that inschool the commonlyused value for 1t is3.14or . Either is onlyanapproximation. Wecannot get theWhat Is 1t? 21exact value of 1t.So how does one get a value for1t? We will nowlook at some of the many ingenious ways that mathematicians overthe centuries have tried to get ever-more-precise values for 1t. Someare amusing; others are baffling. Yet most had significance beyondjust getting closer approximations of 1t.One of the more recent attempts to get a closer approximation of1t took placein Tokyo. Inhislatest effort, inDecember 2002, Pro-fessor Yasumasa Kanada (a longtime pursuer of 1t) and nine others attheInformationTechnologyCenterat TokyoUniversitycalculatedthe value of 1t to1.24 trillion decimal places,whichis six times thepreviously known accuracy,calculated in1999. Theyaccomplishedthis featwith a Hitachi SR8000 supercomputer,which is capable ofdoing2trillioncalculationspersecond. Youmayask, whydoweneed such accuracy for the value of 1t? We don't. The methods of cal-culation are simply used to check the accuracy of the computer andthe sophistication of the calculating procedure (sometimes referred toas an algorithm), that is, how accurate and efficient it is. Another wayof looking at this is how long will it take the computer to get an accu-rate result? In the case of Dr. Kanada, it took his computer over sixhundred hours to do this record-setting computation.It might be worthwhile to consider the magnitude of 1.24 tril-lion. Howolddoyouthinka personwhohaslived1.24trillionseconds might be? The question may seemirksome since itrequires having to consider a very small unit a verylarge numberof times. However, weknowhowlonga secondis. But howbigisonetrillion?A trillionis 1,000,000,000,000, oronethousandbillion. Thus, tocalculate howmany seconds there are inoneyear: 365 x 24 x 60 x 60= 31,536,000 seconds. Therefore,1,000,000,000,000 =31709791983764586504312531709792"" 3171031,536,000 ,. ,years, or one would have to be in his 31,710th year of life to haveIi ved one trillionseconds!22 7tThevalueof1t continuestofascinateus. Whereasacommonfractionresultsina periodicdecimal, 1tdoesnot. A periodicdec-imal is a decimal that eventually repeats its digits indefinitely. Con-sider thecommonfraction ~ . Bydividing1 by3, wegetitsdec-imalequivalentas0.3333333.5Thisdecimal hasa period of one,whichmeans that theonedigit, 3, repeats indefinitely. Herearesome other periodic decimals:I - 2 - 22 =.50000, "3 =.6666, and "7=0.285714285714285714.We place a bar over the last repeating period to indicate its con-tinuous repetition. The decimal ~ has a period of six, since there aresix places continuously repeating.Thereisnoperiodicrepetitionin the decimal value of 1t.Asamatter of fact, although some would use the decimal approximationof 1t tomany places asa table of random numbers-usefulin ran-domizing a statistical sample-there is even a flaw there. When youlook at, say, the first 1,000 decimal places of 1t, you will not see thesame number ofeachofthe tendigits represented. Shouldyouchoosetocount, youwill findthat thedigitsdonot appearwithequal frequency even in the first150 places. For example, there arefewer sevens (10 in the first 150 places) than threes (16 in the first150 places). We will examine this situation later.1t PecuLiaritiesTherearemanypeculiaritiesinthislist of digits. MathematicianJohnConwayhasindicatedthat if youseparate the decimal value5. The bar over the 3 indicates that the 3 repeats indefinitely.What IsTt? 23ofTt intogroupsof tenplaces, theprobabilityof eachof thetendigitsappearinginany of these blocksisabout oneinfortythou-sand. Yet he shows that it does occur in theseventh such group often places, as you can see from the grouping below:Tt =3.1415926535 8979323846 2643383279 502884197169399375105820974944159230781641062862089986280348253421170679 8214808651 32823066470938446095 50582231725359408128...Another wayof saying thisis that every other grouping of ten hasat least onerepeatingdigit. Thesumsofthesedigits alsoshowsome nice results: the sum of the first 144 placesis 666, a numberwith some curious properties as we shall see later.Onoccasion, we stumble uponphenomenainvolving Tt thathave nothing whatsoever to do with a circle. For example, the prob-abilitythat a randomlyselectedinteger(wholenumber)hasonlyunique prime divisors6is ..;.Clearlythisrelationshiphasnothing11:to dowitha circle,yet it involvesthe circle'sratio, Tt. Thisis justanother feature that adds to the centuries-old fascinationwithTt.TheEvoLution of theVaLueof1tThere is much to be said for the adventures of calculating the valueofTt. Wewillconsider someunusualeffortsinthe next fewchap-ters. However, it is interesting to note that Archimedes of Syracuse6. "Uniqueprime divisors"referstodivisorsof anumberthat areprimenumbersand not used more than once. For example, the number 105 is a number with unique primedivisors: 3, 5, and 7, while 315is a number that doesnot haveunique prime divisors: 3,3,5, and 7, since the prime divisor 3 is repeated.24 7t(287-212 BCE)showed the value of1t to lie between3 ~ and3~ .That is,223 < O!XXllXJO()OOOOOO{)OOO(IOOIXl 1414211'i6217W9'i0488016H8724209'i2 91X9262h 14tl21f,'i64'iX41"i297711954 1 612712MOO26H04429477111787401[ 1 2H'iXI94'i0744'iH'i017660411780299,()(X)()(X)()()(J()(J(JU(KX)(J(J()()()(J())()()()()() 1 4h41016151177'i45870548neFrompage 146weprovidefor themathematicsenthusiast someproofs of the fact that eTC > ne.Proof IY = f(x)=ex is monotonously growing in R (R is the set of realnumbers)Xl < x2~ f(x,) < f(x2)Supposed we know:e Inn < n, then we can conclude:e Inn < n ~f(e Inn) 1teProof IIIIY =!(x) =x-;=if;IIn y =InXXIIn y =- In xx303y'= 0: Left side = 0=>Right side =0 (numerator) I - In x =0 =>In x =I, therefore, x =ey"(e) < 0and so on as in proof IImaximum at x =e304Proof IVAppendix Cx 2x3For x > 0,ex =1 + x + +- + ..., i.e., ex >1, ex >1 + x2! 3!Tr1C 1C --I 1C1t > e ~ - > 1and x = - -1> 0; therefore, e e > 1+(-- 1)e e e1C 1C1 + x =1 + (-- 1) = - ~e ee(Trle) 1C> -, then multiply by e to gete eTree >1t306 AppendixDThe length of the rope is 35 +I. The perimeter of thelarger tri-angleis 3(s + 2b) = 3s + 6b.With 35 + 1= 35 +6b, it follows immediately thatI =6b and b=~ .Weknow thattan 600 (or tan !!.) = !!., sowe get3 "b I J3a =~ . , . =6Ji = " =0.09622504486 ... =0.096, or the lengthof a is about 9.6 em.For a regular pentagon:,"\ (j \ bFig. 0-2The length of the ropeis 55 + I.The perimeter of the larger regular pentagon is 5(s + 2b) =55 +lOb.With 55 + 1 = 55 + lOb, it follows that1 = lOb and b = ~ . SinceA Rope around the Regular Polygonstan 360(or tan!!') = ~ , we get5 "307a= ~ , . . . =JJ5+ I =0.1376381920... =0.138,whichindicatesWI"" 250 100that the distance between the pentagons, a,is about13.8 cm.For a regular hexagon:s.aabs +2b bFig. 03The length of the ropeis 6s + I.The perimeter of the larger regular pentagon is 6(s +2b) =6s + 12b.With 6s + 1 = 6s +12b, it follows that1=12b and b =-12.Since tan300(ortan%) = ~ , we getb ) .Jja =;;;;W; =iM =i2 =0.1443375672... = 0.144,308 A Rope around the Regular Polygonswhich indicates that the distance, a, between the hexagons is about14.4 em.The distance a between the respective parallel sides of the ropeand initial polygons is also shown by these four regular polygons tobe independent of the side lengths of the initial polygons.For a regular polygon of n sides (called ann-gon:)The length of the rope is ns +1.The perimeter of the larger regular polygon is n(s + 2b) =ns + 2nb.With ns + 1 =ns + 2nb, it followsthat1 =2nb and b = ...!.....n b 2nBecause tan- = -,we getn ab 1na==ora =cot-n n'__n_tan- 2ntan-n n2nReferencesAndersen, David G. "Pi Search." http://www.angio.net/pi/piquery.Badger, L. "Lazzarini's Lucky Approximation of n." MathematicsMag-azine 67, no.2 (1994):83-91.Ball, W. W. Rouse, and H.S. M. Coxeter. Mathematical Recreations andEssays. 13th ed.New York:Dover, 1987, pp. 55, 274.Beckmann, Petr. A History of n.New York: St. Martin's, 1971.Berggren, Lennart, JonathanBorwein, andPeter Borwein. Pi:ASourceBook. New York: Springer Verlag, 1997.Blatner, David. The Joy of n. New York: Walker, 1997. Seehttp://www.joyofpi.com.Boyer, Carl B. A History ofMathematics. NewYork: John Wiley & Sons, 1968.Castellanos, Dario. "The Ubiquitous Pi." Mathematics Magazine 61(1988):67-98,148-63.Dorrie, Heinrich."Buffon's Needle Problem."100 Great Problems of309310 ReferencesElementary Mathematics: Their History and Solutions. New York: Dover,1965, pp. 73-77.Eves, H. An Introduction to the History of Mathematics. 6th ed. Philadel-phia:Saunders, 1990.Gardner, Martin. "Mathematical Games: Curves of Constant Width." Sci-entific American (February1963): 148-56.---. "Memorizing Numbers." The Scientific American Book ofMath-ematical Puzzles andDiversions. NewYork: SimonandSchuster,1959, p. 103.---. "The Transcendental Number Pi." Chap. 8 in Martin Gardner:sNewMathematicalDiversions fromScientific American. NewYork:Simon and Schuster, 1966, pp. 91-102.Gridgeman, N. T. "GeometricProbabilityandtheNumber n." ScriptaMathematica 25 (1960): 183-95.Hatzipolakis, A. P. "Pi Philology." http://www.cilea.itl-bottoni/www-cilea/F90/piphil.htm.Havermann, H. "Simple Continued Fraction Expansion of Pi."http://odo.ca/-haha/cfpi.html.Kaiser, Hans, and Wilfried Nobauer. Geschichte der Mathematik. Vienna:HOider-Pichler-Tempsky, 1998.Kanada Laboratory home page. http://www.super-computing.org.Keith, Michael. World of Words and Numbers. http://users.aol.comls6sj7gtlmikehome.htm.Olds, C. D. Continued Fractions. Washington, DC: Mathematical Asso-ciation of America, 1963.Peterson, Ivars"APassionfor Pi."Mathematical Treks: FromSurrealNumberstoMagicCircles. Washington, DC: Mathematical Associa-tion of America, 2001.Posamentier, AlfredS. Advanced EuclidianGeometry. Emeryville, CA:Key College Publishing, 2002.Rajagopal, C. T., and T.V. Vedamurti Aiyar. "A Hindu Approximation toPi." Scripta Mathematica18 (1952):25-30.References 311Ramanujan, Srinivasa. "ModularEquationsandApproximationston."Quarterly Journal of Mathematics 45(1914): 350-72. Reprinted inS. Ramanujan:Collected Papers, ed. G. H. Hardy, P. V. Seshuaigar,and B. M. Wilson, 22-39.New York:Chelsea, 1962.Roy, R. "The Discovery of the Series Formula forJr by Leibniz, Gregory,and Nilakantha." Mathematics Magazine 63, no. 5 (1990):291-306.Singmaster, David, "The Legal Values of Pi." Mathematicallntelligencer7, no. 2 (1985):69-72.Stem, M. D. "A Remarkable Approximation toJr." Mathematical Gazette69, no.449 (1985): 218-19.Volkov, Alexei. "Calculation of Jr in Ancient China:From Liu Hui to ZuChongzhi." Historia Scientiarum 2nd ser. 4, no.2 (1994): 139-57.Index2m, 13See also circumference of a circle22/7asapproximation of 1t,9,20,138,14924-gon,91defined, 91 n5See also polygon48-gon,91See also polygon355/133asapproximationof1t,138,149360 degrees in a circle, 137-38666 and1t, 152-54768 and1t, 155-56accuracy of 1t calculations, 99n13See also value of 1tAhmes, 42--44, 42n IAlbanian mnemonics forremem-bering1t, 123algorithms, 74, 115dartboard algorithm, 105-108AI-Kashi, Ghiyath,75AI-Kowarizmi, Abu, 75Almagest (Ptolemy), 60alphabetGreek, 15-16Hebrew, 28Semitic, 16American Mathematical Monthly(journal), 37"AMERICAN 1t" (Lesser), 135-36ancient Egypt and1t, 42--44angles, trisecting of, 42n2, 92, 92n8annulus, 173n8Anthoniszoon, Adriaen, 115n22, 149313314 IndexApery, Roger, 286, 286nIApollonius of Perge, 56apothem, 80defined, 80nlapple, circumference of, 226-27applications of 1t,II,157-216approximate value of 1t, 66n28, 114,115n22, 142-45, 149See also value of 1tArabic numerals, IOn1arbelos, 206n13, 211-13, 215Archimedes ofSyracuse, 23-24, 58,59, 75, 149,211-15,283-84,288methodofdeterminingvalueof1t,52-56,80-91,99,276-80arcs inside a square, 218area of a circle, 17-19, 51ratios, 179n9and right triangles, 52-53and squares, 53-54See also1t[2area of a sphere, 283-84area of a square, 19area of irregular shape, 186-89yin-yang, 194area of ring within circle, 204-206Aristotle, 24, 69Arithmetica infinitorum (Wallis), 24,64-65arithmetic series, 68n31Arybhata, 75AutomaticSequence ControlledCalculator, 85-86n3Babbage, Charles, 85-86n3Babylonia, 44n6Babylonians and1t, 44, 75Bailey, David H., 77Baker, H. F., 113base e and naturallogarithms, 30, 68,68n33, 146Basel problem, 284Bernoulli, Jakob, 31, 284Bernoulli, Johann,31Bible, 10, 149, 153value of 1t, 27, 45, 60, 75Big Dipper, 190-91Blaschke, Wilhelm, 165Bottomley, S., 123Bouyer, Martine, 72, 76Brahmagupta,75breadthof circle, 159nlof Releaux triangle, 159-68Brouncker, William, 65, 65n26Buffon, GeorgesLouisLeclercComte de, 38Buffon needleproblem, 38-39,70-71,137asexample of Monte-Carlomethod, 105Bulgarian mnemonics for remem-bering 1t, 123Cajori, Florian, 29calculating value of 1t. See history of1t; value of 1tcalculator, mechanical, 85-86n3, 280calipers, 162defined, 162n2Castellanos, Dario, 142n*, 152,152n15, 153-54Catherine the Great, 31Cavalieri, (Francesco) Bonaventura, 293Cavalieri congruent, 294Ceulen, Ludolphvan. SeevanCeulen, LudolphChartres, R., 70-71Index 315China and 1t, 61, 75ChiShona mnemonics for remem-bering 1t, 123chord, 204, 206,281, 297-99Chudnovsky, David v., 72, 73, 77, 115Chudnovsky, Gregory v., 72, 73, 77, 1I5circle, squaringthe. Seesquaringthecirclecircles, II, 17, 137-38, 157,222nn3--4arcs inside a square, 181-86,218area of, 52-55area within a ring, 204-206breadth of, 159-68circumference vs. height. Seespheresconcentric, 198-99,220-22,222-29,237--43congruent, 175-78,217-18inscribing and circumscribing apolygon, 91-98and parallel pieces, 202-203ratio of circumference to diameter,10,54-55,220-22,245,275ratios of area, 179n9seven-circles arrangement, 175-77trisecting of, 197-203unusual relationships, 211-15Seealsocircumferenceof acircle;quarter circles; semicirclescircumference of a circle, 13, 17, 80and diameter, 17n3, 54, 220-22,245, 275. See also 2mvs. height of a cylinder. See spheres1t established as a constant, 51See also circles; value of 1tcircumference of spheroids, 222-29Claudius Ptolemaeus, 59-60Clausen, Thomas, 70coincidences and1t, 142--46Columbian Exposition (Chicago,1893),37complex numbers, 68computers, 10, II, 116supercomputer and value of 1t, 21concentric circles, 198-200, 220-22,222-29congruent, Cavalieri, 294congruent circles, 175-78,217-18Conon of Samos, 52constant ring, 204-206constants, I04n 14circumference to diameter of acITcle,10,54-55,220-22,245,2751t as a, 10,51, 51nl2construction of a cube, 42n2continued fractions, 11,65,65nn25-26, 146-52, 150n IIconvergents, 64-66,110,147--48,147n9, 149, 150,284defined, 64n24, IIOnl8converging series, 64n21Conway, John, 22-23counting lattice points to determine1t,101-103counting squares to determine 1t,99-101cube, construction of, 42n2cubit, 27nl4curiosities about 1t, 137-56Cusanus,91-98cylinderheight vs. circumference, 26rolling, 218-20and sphere, 57-59da Architectura (Vitruvius), 59Dahse, Zacharias, 69316 Indexdartboard algorithm, 105-108Dase, Johann.See Dahse, Zachariasdates found in1t, 133decimal, periodic, 22, 22n4,65n27, 142n6decimal numberingsystem, 60n 17,280De Lagny, Thomas Fantet, 76denominator, rationalizing, 95n I0diameter of a circle, 221 n2and circumference, 17, 17n3, 54,220-22, 245,275Dichampt, Michele, 72, 76digitsfrequency of digits in1t, 129-31repetition of digits in1t, 132distribution of digits in1t, 129-31divisors, prime, 23, 23n6, 139dodecagon, 54,91See also polygondolphin shape, 186-89Dudeney,Henry Ernest, 223-24Durer, Albrecht, 63Dutchmnemonicsfor remembering1t, 123e (base of natural logarithms). See basee and naturallogarithmse", 146,301-304em =-I, 30, 68, 68n34earthrailway track around the equator,236-38rope around the equator, 222-29,240-43walk around the equator, 238-39Egypt, ancient and1t, 42-44, 75Einstein, Albert, 14, 118, 120, 140Elements (Euclid), 50-51Elements de Geometrie (Legendre), 69Elijah of Vilna, 27-28Energon (Vim, Germany), 169English mnemonics for remembering1t, 122-23, 124ENIAC computer, 72, 76equals sign, first used, 43n5equatorrailway track around, 236-38rope around, 222-29, 240-43walk around, 238-39equilateral triangles,231rope around, 305-306Eratosthenes of Cyrene, 52"Essai d'arithmetique morale"(Buffon), 38Euclidean constructions, 25nIIEuclid of Alexandria, 50-51Euler, Leonhard, 25, 25n I0, 29,30-33, 67-68, Illnl9, 151-52,276,284-86Euler series, used in calculation of 1t,110-12Eves, Howard, 123, 293Exploratorium (San Francisco),118-20extreme values, 227fallaciesin geometry, 34-36Felton, G. E., 72, 76Ferguson, D. F., 71, 76Fibonacci, Leonardo Pisano, IOn I, 62,75,113Fibonacci numbers, 62n21Filliatre, J., 72, 76fractionalapproximations of 1t2217,9,20,138,149355/133, 138, 149See also value of 1tIndex 317fractionscontinued, 11, 65, 65nn25-26,146-52, 150nllimproper, 146French mnemonics for rememberingn, 124frequency of digits in n, 129-31Gahaliya, 60Galileo Galilei, 293nlGaon of Vilna, 27-28Gardner, Martin, 122-23Gauss, Carl Fredrich, 69, 101-103gematria, 28Genuys,72, 76geoid, 224n6geometry, fallacies,34-36Germanmnemonicsfor rememberingn, 124-25Goldbach, Christian, 152n15Goldbach's conjecture, 73Golden Ratio, 146defined, 146n7Goodwin, Edward Johnson, 36-37Gosper, William, 77Goto, Hiroyuki, 128Greek alphabet, 15-16Greek mnemonics for remembering n,125Greek numbering system, 16, 16n2Greeks and n, 45-52Gregory, James, 67Guilloud, Jean, 72, 76, 77Hall 31 of Palais de la Decouverte(Paris), 118Hardy, G. H., 113, 113n20harmonic series, 68, 68n31, 151Heron of Alexandria, 56hexagon,44n7rope around, 233-34, 307-308See also polygonhexagonalnumbers, 155defined, 155n20hidden code in the Bible, 27Hindu numbering system, 55, 62, 113Hippocrates of Chios, 45, 51Hippocrates of Cos, 51Histoire naturelle (Buffon), 38history of n, 41-77and ancient Egypt, 42--44and Babylonians, 44and China, 61computers used, 72-74and Greece, 45-52Jewish contributions, 60during nineteenth century, 69-70and Old Testament, 45prehistory, 41--42and Ptolemy, 59-60during Renaissance, 62and Romans, 59during seventeenth century, 64-67during sixteenth century, 63summary of pursuit of value of,75-77during the twentieth century, 70-74See also value of nHitachi SR8000 supercomputer,21Hobson, E. w., 113Hon Han Shu, 75IBM Corporation, 85-86n3i (imaginary unit of the complex num-bers square root of -1), 68, 215-16ii, 215-16imaginary numbers, 215-16improper fractions, 146318 IndexIndiana House of Representatives, 37Indiana Senate, 37Indian Mathematical Society, 112infinite product, 63defined, 63n22infinite series, 284Information Technology Center(Tokyo), 21integers, 284, 285odd, 286positive, 70See also numberslntroductio in analysin infinitorum(Euler), 29, 67-68irrationality of 1t, 60, 70irrational number, 24-25, 69, 70n36,148defined, 25n7irregular shapes and 1t, 178-97,200-201Italian mnemonics for remembering 1t,126iteration method of computation,96-98,98-99defined, 96n IIJeans, James, 123Jeenel, J., 72, 76Jewish contributions to1t,60Jones, William, 29, 67Kamata,76Kanada, Yasumasa, 21, 72, 77,129-33Kasner, Edward, 30Kenko, Takebe, 76King Solomon's temple, 27, 45Klein, (Christian) Felix, 68Kubo, y., 77Lambert, Johann Heinrich, 25n8, 148Laplace, Pierre Simon, 38lattice points, counting to find value of1t, 101-103Lazzarini, Mario, 39Legendre Adrien-Marie, 25, 69legislating 1t,36-37Leibniz, Gottfried Wilhelm, 67,67n29,85-86n3,108-10Leibniz series, 109n17usedincalculationof 1t, 108-109,110-11Lesser, Lawrence, 135-36Liber abaci (Fibonacci), IOn I, 62, 113Lindemann, Carl Louis Ferdinand, 25, 70squaring the circle, 43n3Liu Hui, 61, 75logarithms, natural, 30,68, 68n33, 146Loomof God: Mathematical Tapes-tries at the Edge of Time. The(Pickover), 146Los Numeros (Caro), 127Ludolph's number, 24, 64lunes, 49-52defined, 49nl0MacArthur Foundation, 73Machin, John, 76magic square, 154defined, 154n17Maimon, Moses ben. SeeMaimonidesMaimonides, 60, 60n 18Marcellus (emperor of Rome), 56March14 celebration of 1t, 14, 118-20Marcus Vitruvius Pollio.See Vitruviusmathematical function, and symbolsdeveloped by Euler, 30mathematical symbols. See symbols,mathematicalIndex 319Mathematics and the Imagination(Kasner and Newman), 30Matsunaga, 76meandering rivers, measure of,139-41mean proportional theorems, 98nl2Measurement of the Circle(Archimedes), 52-54measuring river length, 139-41Method of Fluxions (Newton), 38Metius, Adriaen, 115, 115n22Metropolis,N. c., 72Miyoshi, Kazunori, 72, 76mnemonics for remembering 1t,122-28Mohwald, R., 144moltensea(KingSolomon'stemple),27, 27nl3Monte-Carlo method of determining 1t,105-108mushroom shape, 178-81Nakayama, Kazuhika, 72naturallogarithms, 30, 68, 68n33, 146and symbols developed by Euler, 30natural numbers, 121, 132-33, 139, 151Newman, James, 30Newton, Isaac,38, 76, 108n-gon. See polygonNicholas of Cusa. See CusanusNicholson, S. c., 72, 76nonrectilinear figures. See lunesnumber of the beast, 153-54numberscomplex, 68hexagonal, 155, 155n20imaginary, 215-16irrational, 24-25, 25n7, 69, 70n36,148natural, 121, 132-33, 139, 151perfect, 155, 155nl8prime, 23, 23n6, 73, 139rational, 70, 286real, 215-16relatively prime, 71, 71n37, 139,287-88repeatedafter decimal. Seeperiodicdecimalsquare-free, 286--87transcendental, 25, 25n9, 25n 10, 70,70n36triangular, 155, 155n19See also integersnumber series, 121Euler series, 110-12Fibonaccinumbers, 62n21Leibniz series, 108-109, 110-11natural found in1t, 132-33number systemsArabic, IOn Idecimal, 60n17, 280Greek, 16, 16n2Hebrew, 28Hindu, 55, 62, 113sexagesimal, 60, 60n 17Old Testament and1t, 45On the Sphere and the Cylinder(Archimedes), 58Otho, Valethus, 75Oughtred, William, 29Palais de la Decouverte (Paris), 118paradoxes of 1t, 34-36, 217-43"ParadoxParty. A discussion of SomeQueer Fallacies and Brain-Twisters, The" (Dudeney),223-24320 Indexparallel lines dividing a circle,202-203parallelogram, 19defined, 19n4parallel planes, 294parallel tangents, 159, 160Pascal, Blaise, 85-86n3Pascaline, 85-86n3Peirce, Benjamin, 30-31pentagon, rope around, 232, 306-307perfect number, 155defined, 155nl8perimeterofirregular shape, 186-88,194-96periodic decimal, 22, 22n4, 65n27, 142n6periphery and1t, 29, 67Phidias,52physical properties used to calculate n,103-105Pickover, Clifford A., 1461te, 146,301-3041tf2, 13See also area of a circlen Song, 135-36Pisano, Leonardo. See Fibonacci,Leonardo Pisanopizza pie divide into 3, 197-203planes, parallel, 294Plutarch, 284Polaris star, 191Polish mnemonics for remembering n,126polygondefined,44n7inscribing and circumscribing a circle,54-55,61,64,80-91,276-80multisidedusedtocalculaten, 62,63rope around, 230-36, 305-308Portuguese mnemonics for remem-bering n, 126positive integers, 70-71Practicageometriae(Fibonacci),62prehistory and n, 41-42prime numbers, 73as divisors, 23, 23n6, 139relatively prime numbers, 71,71n37, 139,287-88probability, 10,38-39,70-71,288and n, 139of repeatingdigitsinvalue of n,22used in calculation of 1t, 105-108Proceedings of the St. PetersburgAcademy, 32product, infinite, 63, 63n22Prussian Academy,31Ptolemy, 59-60, 75Pythagoreantheorem, 45-50, 95, 98.102, I 64n4, 205, 207-208, 242.295Pythagorean triple, 153quadrilateral, 19n4, 218nIquarter circles, 178-81, 218See also circles; semicirclesrace course and n, 170-72radius of circle, andsymbols devel-oped by Euler, 30railway track around the equator,236-38Ramanujan, Srinivasa, 71, 113n20,144, 297-99methodof determiningvalueof n,112-16Ramanujan's theorem, 112Index 321ratioof circumferenceof circletodiam-eter. See circumference of a circleGolden, 146, 146n7rational coefficients, 25n9, 70n36rationalizing the denominator, 95n I0rationalnumber, 70, 286real numbers, 215-16reciprocals, III n19, 285-86Record, Taylor 1.,36Recorde, Robert, 43n5rectilinear figures, 49, 293-95defined, 49n9Reitwiesner, George, 72, 76relatively prime numbers, 71, 139,287-88defined,7ln37See also circlesRenaissance and1t, 62repeating decimal. See periodic dec-imalrepetition of digits in 1t, 132Reuleaux, Franz, 158Reuleaux triangle, II, 158-70, 159n IRhind, Alexander Henry, 42n IRhind Papyrus, 42-44rhombus, 218defined, 218n Iright trianglesand area of circles, 52-53and semicircles, 49-50SeealsoPythagoreantheorem; tri-anglesring, constant, 204-206rivers, length of and1t, 139-41Romanian mnemonics for remem-bering1t, 126Romanus, Adrianus. See Roomen,Adriaen vanRome and history of 1t, 59Roomen, Adriaen van, 63, 63n23, 75ropearound a square, 229-31, 232around equator, 222-29around other spheroids, 226-29around polygons, 231-36, 305-308Royal Technical University of Berlin, 158Russian Academy (St.Petersburg), 31Rutherford, William, 70, 76Salinon (Archimedes), 213-15Schickardt, Wilhelm, 85-86n3Seki Kowa, Takakazu, 76semi-annulus, 173semicircles, 141 n5and Pythagorean theorem, 48-49and right triangles, 49-50spiral formed, 172-74sum of lengths, 34-36See also circles; quarter circlesSemitic alphabet, 16seriesarithmetic, 68n31converging, 64n24defined, 151nl2harmonic, 68, 68n31, 151infinite, 284summability of, 285-86Serres, Franzose Olivier de, 104seven-circles arrangement, 175-77sexagesimal numbering system, 60,60nl7Shanks, Daniel, 72, 72n41, 76Shanks, William, 70, 71, 72n41, 76, 118shapes using circle arcs, 181-86dolphin shape, 186-89mushroom shape, 178-81teardrop shape, 193-96, 200-20 I322 IndexSharp, Abraham, 76shoemaker's knife.See arbelosSiddhanta, 75similtude, 179n9Sindebele mnemonics for remem-bering 1t, 126sine function,82-83defined, 82n2Singmaster, David, 37Smith, LeviB., 71, 76solid figures. See three-dimensional figuresSolomon (king), 27, 45solstice, 191-93Spanishmnemonics for remembering1t, 127spheres, 293in cylinder, 57-59earth, ropearoundequator, 222-29,240--43and geoids, 224n6plane passing through, 295n3railway track around the equator,236-38volume and surface area, 283-84walk around the equator, 238-39spiral and1t, 172-74sports and1t, 170-72square,218n1and area of circles, 53-54circle arcs inside, 178-86rope around, 229-31, 232square-free numbers, 286-87square rootof minus one, 30, 68of 1t, 145of the square root,71 n39of ten, 142squares, countingtofind value of 1t,99-101squaring the circle, 25-26, 92, 92n7and ancient Egypt, 42--43and Greeks, 45, 49and Lindemann, (CarlLouis)Ferdi-nand, 43n3statistics, used in calculation of 1t,105-108StS'llum, Hans-Henrik, 139Strassnitzky, L. K. Schulz von, 69n35,76summability of series, 285-86L (summation sign), 30summer solstice, 191,193Swedishmnemonicsforremembering1t, 127Sylvester, James Joseph, 150nllsymbols, mathematicaldeveloped by Euler, 30e (base ofnatural logarithms), 30,68, 68n33=sign first used, 43n5(imaginary unit of the complexnumbers square root of -I), 30, 68,215-161t first used, 29, 43n4L (summation sign), 30Synopsis palmariorum matheseos(Jones), 29, 67Takahashi, 77Tamura, Yoshiaki, 72, 77tangent, 87-88, 148defined, 87n4tangents, parallel, 159, 160teardrop shape, 193-96, 200-20 Itennis ball, circumference of,226-27tetrahedron, 293Theorieanalytique des probabilities(Laplace), 39Index 323three-dimensional figures, 49circumference vs. height, 26cylinder, rolling, 218-20cylinder and sphere, 57-59rectilinear equivalent to circularmeasurement, 293-95spheres and geoids, 222-29,224n6,237-43, 293, 295n3used in calculation of 1t, I 04n14three slices of pizza, 197-203Tokyo University, 21track and fieldmeets and1t, 170-72transcendental equation, 203transcendentalnumber, 25, 25n I0, 70defined, 25n9, 70n36trianglesarea of equilateral triangle, 164,l64n4equilateral, 231, 305-306Reuleaux triangle, II, 158-70,159nIand symbols developed by Euler, 30See also Pythagorean theorem; righttrianglestriangular number, 155defined, 155n19trisecting an angle, 42n2, 92, 92n8trisecting of a circle, 197-203true value of 1t, 66n28See also value of 1tTsu Ch'ung Chi, 75, 149"Ubiquitous 1t, The" (Castellanos),152, 152nl5Ukrainian Academy of Sciences, 73unique prime divisors, 23, 23n6University of Tokyo, 77Ushiro, Y., 77value of 1t, 20-22, 66n28, 79-116Archimedes method, 80-91, 276-80Buffon needle problem, 38-39counting lattice points, 10I-I 03counting squares to determine 1t,99-101Cusanus method, 91-99Euler seriesmethod of determining,110-12evolution of, 23-26genius method of determining,112-16,297-99geometric constructions, 297-99Leibniz series method of deter-mining, 108-109, 110-11Monte-Carlo method of deter-mining, 105-108summary of pursuit of value of, 72,75-77using physical properties to calcu-late, 103-105See also approximate value of 1t;history of 1tvalues, extreme, 227van Ceulen, Ludolph, 24, 64, 75Vega, George FreiheIT von, 76Viete,63, 75Vilna, Elijah of, 27-28, 28nl5Vitruvius, 59, 75volume of sphere, 283-84von Neumann, John, 72Wallis, John, 24, 29, 64, 67, 151nl4Wang Fau, 75Wankel, Felix, 169Watt, Henry James, 168Watt, James, 168n6Watt Brothers Tools Factories(Wilmerding, PA), 168"Whetstone of Witte, The"(Recorde),43n5324winter solstice, 191-93Wrench, John w., Jr., 71, 72, 76yin and yang, 190-96Yoshino, S., 77IndexZimbabwe mnemonics for remem-bering1t, 123, 126Zu Chongzhi, 61