A TO ZOF

MATHEMATICIANS

NOTABLE SCIENTISTS

A TO ZOF

MATHEMATICIANS

TUCKER MCELROY, Ph.D.

A TO Z OF MATHEMATICIANS

Notable Scientists

Copyright © 2005 by Tucker McElroy, Ph.D.

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CONTENTS

List of Entries viiAcknowledgments ix

Introduction xi

Entries A to Z 1

Entries by Field 260Entries by Country of Birth 262

Entries by Country of Major Scientific Activity 264Entries by Year of Birth 266

Chronology 269Bibliography 273

Index 291

vii

Abel, Niels HenrikAdelard of BathAgnesi, Maria GaetanaAlembert, Jean d’Apollonius of PergaArchimedes of SyracuseAristarchus of SamosAryabhata IBabbage, CharlesBacon, RogerBaire, René-LouisBanach, StefanBarrow, IsaacBayes, ThomasBernoulli, DanielBernoulli, JakobBernoulli, JohannBessel, Friedrich WilhelmBetti, EnricoBhaskara IIBirkhoff, George DavidBolyai, JánosBolzano, BernhardBoole, GeorgeBorel, ÉmileBrahmaguptaBrouwer, Luitzen Egbertus JanCantor, GeorgCardano, GirolamoCarnot, LazareCartan, Élie

Cauchy, Augustin-LouisCavalieri, BonaventuraChebyshev, Pafnuty LvovichCh’in Chiu-ShaoChu Shih-ChiehDedekind, RichardDemocritus of AbderaDe Morgan, AugustusDesargues, GirardDescartes, RenéDiophantus of AlexandriaDirichlet, Gustav Peter

LejeuneEratosthenes of CyreneEuclid of AlexandriaEudoxus of CnidusEuler, LeonhardFatou, Pierre-Joseph-LouisFermat, Pierre deFerrari, LudovicoFerro, Scipione delFibonacci, LeonardoFisher, Sir Ronald AylmerFourier, Jean-Baptiste-JosephFréchet, René-MauriceFredholm, IvarFrege, Friedrich Ludwig

GottlobFubini, GuidoGalilei, GalileoGalois, Evariste

Gauss, Carl FriedrichGermain, SophieGibbs, Josiah WillardGödel, Kurt FriedrichGoldbach, ChristianGosset, WilliamGrassmann, Hermann

GünterGreen, GeorgeGregory, JamesHamilton, Sir William RowanHardy, Godfrey Haroldal-Haytham, Abu AliHeaviside, OliverHermite, CharlesHilbert, DavidHipparchus of RhodesHippocrates of ChiosHopf, HeinzHuygens, ChristiaanIbrahim ibn SinanJacobi, CarlJordan, Camilleal-Karaji, Abual-Khwarizmi, AbuKlein, FelixKovalevskaya, SonyaKronecker, LeopoldKummer, ErnstLagrange, Joseph-LouisLaplace, Pierre-Simon

LIST OF ENTRIES

viii A to Z of Mathematicians

Lebesgue, Henri-LéonLegendre, Adrien-MarieLeibniz, Gottfried Wilhelm

vonLeonardo da VinciLévy, Paul-PierreLi ChihLie, SophusLiouville, JosephLipschitz, RudolfLobachevsky, NikolaiLovelace, Augusta Ada ByronMaclaurin, ColinMadhava of SangamagrammaMarkov, AndreiMenelaus of AlexandriaMinkowski, HermannMöbius, AugustMoivre, Abraham deMonge, GaspardNapier, John

Navier, Claude-Louis-Marie-Henri

Newton, Sir IsaacNoether, EmmyOresme, NicolePappus of AlexandriaPascal, BlaisePeano, GiuseppePearson, Egon SharpePeirce, CharlesPoincaré, Jules-HenriPoisson, Siméon-DenisPólya, GeorgePoncelet, Jean-VictorPtolemy, ClaudiusPythagoras of SamosRamanujan, Srinivasa AiyangarRegiomontanus, Johann

MüllerRheticus, GeorgRiemann, Bernhard

Riesz, FrigyesRussell, BertrandSeki Takakazu KowaSteiner, JakobStevin, SimonStokes, George GabrielTartaglia, NiccolòThales of MiletusTsu Ch’ung-ChihVenn, JohnViète, FrançoisVolterra, VitoWallis, JohnWeierstrass, KarlWeyl, HermannWiener, NorbertYang HuiYativrsabhaYule, George UdnyZeno of EleaZermelo, Ernst

ix

Ithank God, who gives me strength and hopeeach day. I am grateful to my wife, Autumn,

who encouraged me to complete this book.Thanks also go to Jim Dennison for title trans-lations, as well as Diane Kit Moser and Lisa Yount

ACKNOWLEDGMENTS

for advice on photographs. Finally, my thanks goto Frank K. Darmstadt, Executive Editor, for hisgreat patience and helpfulness, as well as the restof the Facts On File staff for their work in mak-ing this book possible.

xi

A to Z of Mathematicians contains the fasci-nating biographies of 150 mathematicians:

men and women from a variety of cultures, timeperiods, and socioeconomic backgrounds, all ofwhom have substantially influenced the historyof mathematics. Some made numerous discov-eries during a lifetime of creative work; othersmade a single contribution. The great CarlGauss (1777–1855) developed the statisticalmethod of least squares and discovered count-less theorems in algebra, geometry, and analysis.Sir Isaac Newton (1643–1727), renowned as theprimary inventor of calculus, was a profound re-searcher and one of the greatest scientists of alltime. From the classical era there is Archimedes(287 B.C.E.–212 B.C.E.), who paved the way forcalculus and made amazing investigations intomechanics and hydrodynamics. These three areconsidered by many mathematicians to be theprinces of the field; most of the persons in thisvolume do not attain to the princes’ glory, butnevertheless have had their share in the un-folding of history.

THE MATHEMATICIANS

A to Z of Mathematicians focuses on individualswhose historical importance is firmly estab-lished, including classical figures from the an-cient Greek, Indian, and Chinese cultures aswell as the plethora of 17th-, 18th-, and 19th-century mathematicians. I have chosen to

INTRODUCTION

exclude those born in the 20th century (withthe exception of Kurt Gödel), so that the likesof Dame Mary Cartwright, Andrey Kolmogorov,and John Von Neumann are omitted; this choicereflects the opinion that true greatness is madelucid only through the passage of time. The ear-lier mathematicians were often scientists as well,also contributing to astronomy, philosophy, andphysics, among other disciplines; however, thelatter persons, especially those of the 19th cen-tury, were increasingly specialized in one partic-ular aspect of pure or applied mathematics.Modern figures who were principally known forfields other than mathematics—such as AlbertEinstein and Richard Feynman—have beenomitted, despite their mathematical accom-plishments. Being of the opinion that statisticsis one of the mathematical sciences, I have in-cluded a smattering of great statisticians. Severalsources were consulted in order to compile a di-verse list of persons—a list that nevertheless de-livers the main thrust of mathematical history.

I have attempted to make this material ac-cessible to a general audience, and as a result themathematical ideas are presented in simpleterms that cut to the core of the matter. In somecases precision was sacrificed for accessibility.However, due to the abstruse nature of 19th- and20th-century mathematics, many readers maystill have difficulty. I suggest that they refer toFacts On File’s handbooks in algebra, calculus,

xii A to Z of Mathematicians

and geometry for unfamiliar terminology. Itis helpful for readers to have knowledge ofhigh school geometry and algebra, as well ascalculus.

After each entry, a short list of additionalreferences for further reading is provided. Themajority of the individuals can be found inthe Dictionary of Scientific Biography (New York,

1970–90), the Encyclopaedia Britannica (http://www.eb.com), and the online MacTutor Historyof Mathematics archive (http://www-gap.dcs.st-and.ac.uk/~history); so these references havenot been repeated each time. In compilingreferences I tried to restrict sources to thosearticles written in English that were easily ac-cessible to college undergraduates.

A

1

� Abel, Niels Henrik(1802–1829)NorwegianAlgebra

The modest Norwegian mathematician NielsAbel made outstanding contributions to the the-ory of elliptic functions, one of the most popu-lar mathematical subjects of the 19th century.Struggle, hardship, and uncertainty character-ized his life; but under difficult conditions he stillmanaged to produce a prolific and brilliant bodyof mathematical research. Sadly, he died young,without being able to attain the glory and recog-nition for which he had labored.

Niels Henrik Abel was born the son ofSören Abel, a Lutheran pastor, and Ane MarieSimonson, the daughter of a wealthy merchant.Pastor Abel’s first parish was in the island ofFinnöy, where Niels Abel was born in 1802.Shortly afterward, Abel’s father became in-volved in politics.

Up to this time Abel and his brothers hadreceived instruction from their father, but in1815 they were sent to school in Oslo. Abel’sperformance at the school was marginal, but in1817 the arrival of a new mathematics teacher,Bernt Holmboe, greatly changed Abel’s fate.Holmboe recognized Abel’s gift for mathemat-ics, and they commenced studying LEONHARD

EULER and the French mathematicians. SoonAbel had surpassed his teacher. At this time hewas greatly interested in the theory of algebraicequations. Holmboe was delighted with his dis-covery of the young mathematician, and he en-thusiastically acquainted the other faculty withthe genius of Abel.

During his last year at school Abel at-tempted to solve the quintic equation, an out-standing problem from antiquity; but he failed(the equation has no rational solutions).Nevertheless, his efforts introduced him to thetheory of elliptic functions. Meanwhile, Abel’sfather fell into public disgrace due to alcoholism,and after his death in 1820 the family was leftin difficult financial circumstances.

Abel entered the University of Sweden in1821, and was granted a free room due to his ex-treme poverty. The faculty even supported himout of its own resources; he was a frequent guestof the household of Christoffer Hansteen, theleading scientist at the university. Within thefirst year, Abel had completed his preliminarydegree, allowing him the time to pursue his ownadvanced studies. He voraciously read every-thing he could find concerning mathematics,and published his first few papers in Hansteen’sjournal after 1823.

In summer 1823 Abel received assistancefrom the faculty to travel to Copenhagen, in

2 Abel, Niels Henrik

order to meet the Danish mathematicians. Thetrip was inspirational; he also met his future fi-ancée, Christine Kemp. When he returned toOslo, Abel began work on the quintic equationonce again, but this time, he attempted to provethat there was no radical expression for the so-lution. He was successful, and had his result pub-lished in French at his own expense. Sadly, therewas no reaction from his intended audience—even CARL FRIEDRICH GAUSS was indifferent.

Abel’s financial problems were complicatedby his engagement to Kemp, but he managed tosecure a small stipend to study languages inpreparation for travel abroad. After this, hewould receive a modest grant for two years offoreign study. In 1825 he departed with somefriends for Berlin, and on his way through

Copenhagen made the acquaintance of AugustCrelle, an influential engineer with a keen in-terest for mathematics. The two became lifelongfriends, and Crelle agreed to start a German jour-nal for the publication of pure mathematics.Many of Abel’s papers were published in the firstvolumes, including an expanded version of hiswork on the quintic.

One of Abel’s notable papers in Crelle’sJournal generalized the binomial formula, whichgives an expansion for the nth power of a bino-mial expression. Abel turned his thought towardinfinite series, and was concerned that the sumshad never been stringently determined. The re-sult of his research was a classic paper on powerseries, with the determination of the sum of thebinomial series for arbitrary exponents.Meanwhile, Abel failed to obtain a vacant po-sition at the University of Sweden; his formerteacher Holmboe was instead selected. It is note-worthy that Abel maintained his nobility ofcharacter throughout his frustrating life.

In spring 1826 Abel journeyed to Paris andpresented a paper to the French Academy ofSciences that he considered his masterpiece: Ittreated the sum of integrals of a given algebraicfunction, and thereby generalized Euler’s relationfor elliptic integrals. This paper, over which Abellabored for many months but never published,was presented in October 1826, and AUGUSTIN-LOUIS CAUCHY and ADRIEN-MARIE LEGENDRE wereappointed as referees. A report was not forth-coming, and was not issued until after Abel’sdeath. It seems that Cauchy was to blame for thetardiness, and apparently lost the manuscript.Abel later rewrote the paper (neither was thiswork published), and the theorem describedabove came to be known as Abel’s theorem.

After this disappointing stint in France,Abel returned to Berlin and there fell ill withhis first attack of tuberculosis. Crelle assistedhim with his illness, and tried to procure a po-sition for him in Berlin, but Abel longed to re-turn to Norway. Abel’s new research transformed

Niels Abel, one of the founders of the theory ofelliptic functions, a generalization of trigonometricfunctions (Courtesy of the Library of Congress)

Adelard of Bath 3

the theory of elliptic integrals to the theory ofelliptic functions by using their inverses.Through this duality, elliptic functions becamean important generalization of trigonometricfunctions. As a student in Oslo, Abel had al-ready developed much of the theory, and this pa-per presented his thought in great detail.

Upon his return to Oslo in 1827, Abel hadno prospects of a position, and managed to sur-vive by tutoring schoolboys. In a few monthsHansteen went on leave to Siberia and Abel be-came his substitute at the university. Meanwhile,Abel’s work had started to stimulate interestamong European mathematicians. In early 1828Abel discovered that he had a young Germancompetitor, CARL JACOBI, in the field of ellipticfunctions. Aware of the race at hand, Abel wrotea rapid succession of papers on elliptic functionsand prepared a book-length memoir that wouldbe published posthumously.

It seems that Abel had the priority of dis-covery over Jacobi in the area of elliptic func-tions; however, it is also known that Gauss wasaware of the principles of elliptic functions longbefore either Abel or Jacobi, and had decidednot to publish. At this time Abel started a cor-respondence with Legendre, who was also inter-ested in elliptic functions. The mathematiciansin France, along with Crelle, attempted to se-cure employment for Abel, and even petitionedthe monarch of Sweden.

Abel’s health was deteriorating, but he con-tinued to write papers frantically. He spent sum-mer 1828 with his fiancée, and when visiting herat Christmastime he became feverish due to ex-posure to the cold. As he prepared for his returnto Oslo, Abel suffered a violent hemorrhage, andwas confined to bed. At the age of 26 he diedof tuberculosis on April 26, 1829; two days later,Crelle wrote him jubilantly that he had securedAbel an appointment in Berlin. In 1830 theFrench Academy of Sciences awarded its GrandPrix to Abel and Jacobi for their brilliant math-ematical discoveries.

Abel became recognized as one of the great-est mathematicians after his death, and he trulyaccomplished much despite his short lifespan.The theory of elliptic functions would expandgreatly during the later 19th century, and Abel’swork contributed significantly to this develop-ment.

Further ReadingBell, E. Men of Mathematics. New York: Simon and

Schuster, 1965.Ore, O. Niels Henrik Abel, Mathematician Extraordi-

nary. Minneapolis: University of MinnesotaPress, 1974.

Rosen, M. I. “Niels Henrik Abel and the Equation ofthe Fifth Degree,” American Mathematical Monthly102 (1995): 495–505.

Stander, D. “Makers of Modern Mathematics: NielsHenrik Abel,” Bulletin of the Institute of Mathemat-ics and Its Applications 23, nos. 6–7 (1987): 107–109.

� Adelard of Bath(unknown–ca. 1146)BritishArithmetic

Little is known of the personal life of Adelardof Bath, but his work has been of great impor-tance to the early revival of mathematics andnatural philosophy during the medieval period.His translation of Greek and Arabic classicsinto Latin enabled the knowledge of earlier so-cieties to be preserved and disseminated inEurope.

Adelard was a native of Bath, England, buthis exact birth date is not known. He traveledwidely in his life, first spending time in France,where he studied at Tours. For the next sevenyears he journeyed afar, visiting Salerno, Sicily,Cilicia, Syria, and perhaps even Palestine; it isthought that he also dwelt in Spain. His lattertravels gave him an acquaintance with Arabic

4 Agnesi, Maria Gaetana

language and culture, though he may havelearned Arabic while still in Sicily. By 1130 hehad returned to Bath, and his writings from thattime have some association with the royal court.One of his works, called Astrolabe, was appar-ently composed between 1142 and 1146; this isthe latest recorded date of his activity.

Adelard made two contributions—De eo-dem et diverso (On sameness and diversity) andthe Questiones naturales (Natural questions)—to medieval philosophy, written around 1116and 1137, respectively. In De eodem et diverso,there is no evidence of Arabic influence, andhe expresses the views of a quasi-Platonist. TheQuestiones naturales treats various topics in nat-ural philosophy and shows the impact of hisArabic studies. Adelard’s contribution to me-dieval science seems to lie chiefly in his trans-lation of various works from Arabic.

His early endeavors in arithmetic, publishedin Regule abaci (By rule of the abacus), were quitetraditional—his work reflected current arith-metical knowledge in Europe. These writingswere doubtlessly composed prior to his familiar-ity with Arabic mathematics. Adelard also wroteon the topics of arithmetic, geometry, music, andastronomy. Here, the subject of Indian numer-als and their basic operations is introduced as offundamental importance.

Many scholars believe that Adelard was thefirst translator to present a full Latin version ofEUCLID OF ALEXANDRIA’s Elements. This beganthe process whereby the Elements would come todominate late medieval mathematics; prior toAdelard’s translation from the Arabic, therewere only incomplete versions taken from theGreek. The first version was a verbatim tran-scription from the Arabic, whereas Adelard’ssecond version replaces some of the proofs withinstructions or summaries. This latter edition be-came the most popular, and was most commonlystudied in schools. A third version appears to bea commentary and is attributed to Adelard; itenjoyed some popularity as well.

All the later mathematicians of Europewould read Euclid, either in Latin or Greek; in-deed, this compendium of geometric knowledgewould become a staple of mathematical educationup to the present time. The Renaissance, and theconsequent revival of mathematical discovery,was only made possible through the rediscoveryof ancient classics and their translations. For hiswork as a translator and commentator, Adelardis remembered as an influential figure in the his-tory of mathematics.

Further ReadingBurnett, C. Adelard of Bath: An English Scientist and

Arabist of the Early Twelfth Century. London:Warburg Institute, University of London, 1987.

� Agnesi, Maria Gaetana(1718–1799)ItalianAlgebra, Analysis

Maria Gaetana Agnesi is known as a talentedmathematician of the 18th century, and indeedwas one of the first female mathematicians inthe Western world. A mathematical prodigywith great linguistic talents, Agnesi made hergreatest contribution through her clear exposi-tion of algebra, geometry, and calculus; her col-leagues acknowledged the value of her workwithin her own lifetime.

Born the eldest child of Pietro Agnesi andAnna Fortunato Brivio, Agnesi showed early in-terest in science. Her father, a wealthy professorof mathematics at the University of Bologna, en-couraged and developed these interests. He estab-lished a cultural salon in his home, where hisdaughter would present and defend theses on a va-riety of scientific and philosophical topics. Someof the guests were foreigners, and Maria demon-strated her talent for languages by conversing withthem in their own tongue; by age 11 she was fa-miliar with Greek, German, Spanish, and Hebrew,

Agnesi, Maria Gaetana 5

having already mastered French by age five. Atage nine she prepared a lengthy speech in Latinthat promulgated higher education for women.

The topics of these theses, which were usu-ally defended in Latin, included logic, ontology,mechanics, hydromechanics, elasticity, celestialmechanics and universal gravitation, chemistry,botany, zoology, and mineralogy. Her secondpublished work, the Propositiones philosophicae(Propositions of philosophy, 1738), included al-most 200 of these disputations. Agnesi’s mathe-matical interests were developing at this time;at age 14 she was solving difficult problems inballistics and analytic geometry. But after thepublication of the Propositiones philosophicae, shedecided to withdraw from her father’s salon,since the social atmosphere was unappealing toher—in fact, she was eager to join a convent,but her father dissuaded her.

Nevertheless, Agnesi withdrew from the ex-troverted social life of her childhood, and devoted

the next 10 years of her life to mathematics.After a decade of intense effort, she producedher Instituzioni analitiche ad uso della gioventù ital-iana (Analytical methods for the use of youngItalians) in 1748. The two-volume work won im-mediate praise among mathematicians andbrought Agnesi public acclaim. The objective ofthe thousand-page book was to present a com-plete and comprehensive treatment of algebraand analysis, including and emphasizing the newconcepts of the 18th century. Of course, the de-velopment of differential and integral calculuswas still in progress at this time; Agnesi wouldincorporate this contemporary mathematics intoher treatment of analysis.

The material spanned elementary algebraand the classical theory of equations, coordinategeometry, the differential and integral calculus,infinite series, and the solution of elementarydifferential equations. Many of the methods andresults were due solely to Agnesi, although herhumble nature made her overly thorough in giv-ing credit to her predecessors. Her name is of-ten associated with a certain cubic curve calledthe versiera and known more commonly as the“witch of Agnesi.” She was unaware that PIERRE

DE FERMAT had studied the equation previouslyin 1665. This bell-shaped curve has many in-teresting properties and some applications inphysics, and has been an ongoing source of fas-cination for many mathematicians.

Agnesi’s treatise received wide acclaim forits excellent treatment and clear exposition.Translations into French and English from theoriginal Italian were considered to be of greatimportance to the serious student of mathemat-ics. Pope Benedict XIV sent her a congratula-tory note in 1749, and in 1750 she was appointedto the chair of mathematics and natural philos-ophy at the University of Bologna.

However, Agnesi’s reclusive and humblepersonality led her to accept the position onlyin honor, and she never actually taught at theuniversity. After her father’s death in 1752, she

Maria Agnesi studied the bell-shaped cubic curve calledthe versiera, which is more commonly known as the“witch of Agnesi.” (Courtesy of the Library of Congress)

6 Alembert, Jean d’

began to withdraw from all scientific activity—she became more interested in religious studiesand social work. She was particularly con-cerned with the poor, and looked after the ed-ucation of her numerous younger brothers. By1762 she was quite removed from mathemat-ics, so that she declined the University ofTurin’s request that she act as referee for JOSEPH-LOUIS LAGRANGE’s work on the calculus of vari-ations. In 1771 Agnesi became the director ofa Milanese home for the sick, a position sheheld until her death in 1799.

It is interesting to note that the sustainedactivity of her intellect over 10 years was ableto produce the Instituzioni, a work of great ex-cellence and quality. However, she lost all in-terest in mathematics soon afterward and madeno further contributions to that discipline.Agnesi’s primary contribution to mathematics isthe Instituzioni, which helped to disseminatemathematical knowledge and train future gen-erations of mathematicians.

Further ReadingGrinstein, L., and P. Campbell. Women of Mathematics.

New York: Greenwood Press, 1987.Truesdell, C. “Correction and Additions for Maria

Gaetana Agnesi,” Archive for History of ExactScience 43 (1991): 385–386.

� Alembert, Jean d’ (Jean Le Rond d’Alembert)(1717–1783)FrenchMechanics, Calculus

In the wave of effort following SIR ISAAC NEW-TON’s pioneering work in mechanics, manymathematicians attempted to flesh out themathematical aspects of the new science. Jeand’Alembert was noteworthy as one of these in-tellectuals, who contributed to astronomy, fluidmechanics, and calculus; he was one of the first

persons to realize the importance of the limit incalculus.

Jean Le Rond d’Alembert was born in Parison November 17, 1717. He was the illegitimateson of a famous salon hostess and a cavalry offi-cer named Destouches-Canon. An artisan namedRousseau raised the young d’Alembert, but hisfather oversaw his education; he attended aJansenist school, where he learned the classics,rhetoric, and mathematics.

D’Alembert decided on a career as a math-ematician, and began communicating with theAcadémie des Sciences in 1739. During the nextfew years he wrote several papers treating the in-tegration of differential equations. Although hehad no formal training in higher mathematics,

Jean d’Alembert formulated several laws of motion,including d’Alembert’s principle for decomposingconstrained motions. (Courtesy of the National Libraryof Medicine)

Alembert, Jean d’ 7

d’Alembert was familiar with the works ofNewton, as well as the works of JAKOB BERNOULLI

and JOHANN BERNOULLI.In 1741 he was made a member of the

Académie, and he concentrated his efforts onsome problems in rational mechanics. The Traitéde dynamique (Treatise on dynamics) was thefruit of his labor, a significant scientific work thatformalized the new science of mechanics. Thelengthy preface disclosed d’Alembert’s philoso-phy of sensationalism (this idea states that senseperception, not reason, is the starting point forthe acquisition of knowledge). He developedmechanics from the simple concepts of space andtime, and avoided the notion of force.D’Alembert also presented his three laws of mo-tion, which treated inertia, the parallelogramlaw of motion, and equilibrium. It is noteworthythat d’Alembert produced mathematical proofsfor these laws.

The well-known d’Alembert’s principle wasalso introduced in this work, which states thatany constrained motion can be decomposed interms of its inertial motion and a resisting (orconstraining) force. He was careful not to over-value the impact of mathematics on physics—he said that geometry’s rigor was tied to its sim-plicity. Since reality was always more complicatedthan a mathematical abstraction, it is more diffi-cult to establish truth.

In 1744 he produced a new volume calledthe Traité de l’équilibre et du mouvement des fluides(Treatise on the equilibrium and movement offluids). In the 18th century a large amount ofinterest focused on fluid mechanics, since fluidswere used to model heat, magnetism, and elec-tricity. His treatment was different from that ofDANIEL BERNOULLI, though the conclusions weresimilar. D’Alembert also examined the waveequation, considering string oscillation problemsin 1747. Then in 1749 he turned toward celestialmechanics, publishing the Recherches sur la pré-cession des équinoxes et sur la nutation de l’axe de laterre (Research on the precession of the equinoxes and

on the nodding of the earth’s axis), which treatedthe topic of the gradual change in the positionof the earth’s orbit.

Next, d’Alembert competed for a prize atthe Prussian Academy, but blamed LEONHARD

EULER for his failure to win. D’Alembert publishedhis Essai d’une nouvelle théorie de la résistance desfluides (Essay on a new theory of the resistance offluids) in 1752, in which the differential hydro-dynamic equations were first expressed in termsof a field. The so-called hydrodynamic paradoxwas herein formulated—namely, that the flowbefore and behind an obstruction should be thesame, resulting in the absence of any resistance.D’Alembert did not solve this problem, and wasto some extent inhibited by his bias toward con-tinuity; when discontinuities arose in the solu-tions of differential equations, he simply threwthe solution away.

In the 1750s, interested in several nonsci-entific topics, d’Alembert became the scienceeditor of the Encyclopédie (Encyclopedia). Laterhe wrote on the topics of music, law, and reli-gion, presenting himself as an avid proponent ofEnlightenment ideals, including a disdain formedieval thought.

Among his original contributions to math-ematics, the ratio test for the convergence of aninfinite series is noteworthy; d’Alembert vieweddivergent series as nonsensical and disregardedthem (this differs markedly from Euler’s view-point). D’Alembert was virtually alone in hisview of the derivative as the limit of a function,and his stress on the importance of continuityprobably led him to this perspective. In the the-ory of probability d’Alembert was quite handi-capped, being unable to accept standard solutionsof gambling problems.

D’Alembert was known to be a charming,witty man. He never married, although he livedwith his lover Julie de Lespinasse until her deathin 1776. In 1772 he became the secretary of theAcadémie Française (the French Academy), andhe increasingly turned toward humanitarian

8 Apollonius of Perga

concerns. His later years were marked by bitter-ness and despair; he died in Paris on October 29,1783.

Although he was well known as a preemi-nent scientist and philosopher, d’Alembert’smathematical achievements deserve specialrecognition. He greatly advanced the theory ofmechanics in several of its branches, by con-tributing to its mathematical formulation and byconsideration of several concrete problems.

Further ReadingGrimsley, R. Jean d’Alembert, 1717–83. Oxford: Claren-

don Press, 1963.Hankins, T. Jean d’Alembert: Science and the

Enlightenment. Oxford: Clarendon Press, 1970.Pappas, J. Voltaire and d’Alembert. Bloomington:

Indiana University Press, 1962.Wilson, C. “D’Alembert versus Euler on the

Precession of the Equinoxes and the Mechanicsof Rigid Bodies,” Archive for History of ExactSciences 37, no. 3 (1987): 233–273.

� Apollonius of Perga(ca. 262 B.C.E.–190 B.C.E.)GreekGeometry

Greek mathematics continued its developmentfrom the time of EUCLID OF ALEXANDRIA, and af-ter ARCHIMEDES OF SYRACUSE one of the greatestmathematicians was Apollonius of Perga. He ismainly known for his contributions to the the-ory of conic sections (those plane figures ob-tained by slicing a cone at various angles). Thefascination in this subject, revived in the 16thand 17th centuries, has continued into moderntimes with the onset of projective geometry.

Little information on his life has been pre-served from the ravages of time, but it seems thatApollonius flourished sometime between thesecond half of the third century and the earlysecond century B.C.E. Perga, a small Greek city

in the southern portion of what is now Turkey,was his town of birth. Apollonius dwelt for sometime in Alexandria, where he may have studiedwith the pupils of Euclid, and he later visitedboth Pergamum and Ephesus.

His most famous work, the Conics, was com-posed in the early second century B.C.E., and itsoon became recognized as a classic text.Archimedes, who died around 212 B.C.E., ap-pears to be the immediate mathematical prede-cessor of Apollonius, who developed many of theSyracusan’s ideas. The Conics was originally di-vided into eight books, and had been intendedas a treatise on conic sections. Before Apollonius’stime, the basics of the theory of conic sectionswere known: Parabolas, hyperbolas, and ellipsescould be obtained by appropriately slicing a conewith right, obtuse, or acute vertex angles, re-spectively. Apollonius employed an alternativemethod of construction that involved slicing adouble cone at various angles, keeping the ver-tex angle fixed (this is the approach taken inmodern times). This method had the advantageof making these curves accessible to the “appli-cation of areas,” a geometrical formulation ofquadratic equations that in modern time wouldbe expressed algebraically. It is apparent thatApollonius’s approach was refreshingly origi-nal, although the actual content of the Conicsmay have been well known. Much terminology,such as parabola, hyperbola, and ellipse, is due toApollonius, and he generalizes the methods forgenerating sections.

The Conics contains much material that wasalready known, though the organization was ac-cording to Apollonius’s method, which smoothlyjoined together numerous fragments of geomet-rical knowledge. Certain elementary results wereomitted, and some few novel facts were included.Besides the material on the generation of sections,Apollonius described theorems on the rectanglescontained by the segments of intersecting chordsof a conic, the harmonic properties of pole andpolar, properties of the focus, and the locus of

Apollonius of Perga 9

three and four lines. He discusses the formationof a normal line to a conic, as well as certain in-equalities of conjugate diameters. This work,compared with other Greek literature, is quitedifficult to read, since the lack of modern nota-tion makes the text burdensome, and the contentitself is quite convoluted. Nevertheless, persistentstudy has rewarded many gifted mathematicians,including SIR ISAAC NEWTON, PIERRE DE FERMAT,and BLAISE PASCAL, who drew enormous inspi-ration from Apollonius’s classic text.

In the work of PAPPUS OF ALEXANDRIA is con-tained a summary of Apollonius’s other mathe-matical works: Cutting off of a Ratio, Cutting offof an Area, Determinate Section, Tangencies,Inclinations, and Plane Loci. These deal with var-ious geometrical problems, and some of them in-volve the “application of an area.” He uses theGreek method of analysis and synthesis: Theproblem in question is first presumed solved, anda more easily constructed condition is deducedfrom the solution (“analysis”); then from the lat-ter construction, the original is developed (“syn-thesis”). It seems that Apollonius wrote stillother documents, but no vestige of their contenthas survived to the present day. Apparently, hedevised a number system for the representationof enormous quantities, similar to the notationalsystem of Archimedes, though Apollonius gen-eralized the idea. There are also references to theinscribing of the dodecahedron in the sphere,the study of the cylindrical helix, and a generaltreatise on the foundations of geometry.

So Apollonius was familiar with all aspectsof Greek geometry, but he also contributed tothe Euclidean theory of irrational numbers andderived approximations for pi more accuratethan Archimedes’. His thought also penetratedthe science of optics, where his deep knowledgeof conics assisted the determination of variousreflections caused by parabolic and sphericalmirrors. Apollonius was renowned in his owntime as a foremost astronomer, and he evenearned the epithet of Epsilon, since the Greek

letter of that name bears a resemblance in shapeto the Moon. He calculates the distance of Earthto Moon as roughly 600,000 miles, and madevarious computations of the orbits of the plan-ets. In fact, Apollonius is an important player inthe development of geometrical models to ex-plain planetary motion; HIPPARCHUS OF RHODES

and CLAUDIUS PTOLEMY, improving upon his the-ories, arrived at the Ptolemaic system, a feat ofthe ancient world’s scientific investigation pos-sessed of sweeping grandeur and considerablelongevity.

There was no immediate successor toApollonius, though his Conics was recognized asa superb accomplishment. Various simple com-mentaries were produced, but interest declinedafter the fall of Rome, and only the first fourbooks continued to be translated in Byzantium.Another three books of the Conics were trans-lated into Arabic, and Islamic mathematiciansremained intrigued by his work, though theymade few advancements; the final (eighth) bookhas been lost. In the late 16th and early 17thcenturies, several translations of Apollonius’sConics appeared in Europe and were voraciouslystudied by French mathematicians such as RENÉ

DESCARTES, Pierre de Fermat, GIRARD DESARGUES,and Blaise Pascal. When Descartes propoundedhis analytic geometry, which took an algebraic,rather than constructive or geometrical, ap-proach to curves and sections, interest inApollonius’s classic treatise began to wane.However, later in the 19th century, the Conicsexperienced a resurrection of curiosity with theintroduction of projective geometry.

Further ReadingHeath, T. Treatise on Conic Section. New York: Barnes

& Noble, 1961.———. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.Hogendijk, J. “Arabic Traces of Lost Works of

Apollonius,” Archive for History of Exact Sciences35, no. 3 (1986): 187–253.

10 Archimedes of Syracuse

� Archimedes of Syracuse(ca. 287 B.C.E.–212 B.C.E.)GreekGeometry, Mechanics

Of the mathematicians of Greek antiquity,Archimedes should be considered the greatest.His contributions to geometry and mechanics,as well as hydrostatics, place him on a higherpedestal than his contemporaries. And as hisworks were gradually translated and introducedinto the West, he exerted as great an influencethere as his thought already had in Byzantiumand Arabia. In his method of exhaustion can beseen a classical predecessor of the integral cal-culus, which would be formally developed byBLAISE PASCAL, GOTTFRIED WILHELM VON LEIBNIZ,SIR ISAAC NEWTON, and others in the 17th cen-

tury. His life story alone has inspired manymathematicians.

As with many ancient persons, the exact de-tails of Archimedes’ life are difficult to ascertain,since there are several accounts of variable qual-ity. His father was the astronomer Phidias, andit is possible that Archimedes was a kinsman ofthe tyrant of Syracuse, King Hieron II. Certainlyhe was intimate with the king, as his work TheSandreckoner was dedicated to Hieron’s sonGelon. Born in Syracuse, Archimedes departedto Alexandria in order to pursue an educationin mathematics; there he studied EUCLID OF

ALEXANDRIA and assisted the development ofEuclidean mathematics. But it was in Syracuse,where he soon returned, that he made most ofhis discoveries.

Although renowned for his contributions tomathematics, Archimedes also designed numer-ous mechanical inventions. The water snail, in-vented in Egypt to aid irrigation, was a screwlikecontraption used to raise water. More impressiveare the stories relating his construction and ap-plication of the compound pulley: Hieron had re-quested Archimedes to demonstrate how a smallforce could move a large weight. The mathe-matician attached a rope to a large merchant shipthat was loaded with freight and passengers, andran the line through a system of pulleys. In thismanner, seated at a distance from the vessel,Archimedes was able to effortlessly draw the boatsmoothly off the shore into the harbor.

Similar to the pulley, Archimedes discoveredthe usefulness of the lever, noting that the longerthe distance from the fulcrum, the more weight thelever could move. Logically extending this prin-ciple, he asserted that it was feasible to move theworld, given a sufficiently long lever. Anotherpopular story relates that Hieron gave Archi-medes the task of ascertaining whether a certaincrown was made of pure gold, or whether it hadbeen fraudulently alloyed with silver. AsArchimedes pondered this puzzle, he came uponthe bath, and noticed that the amount of water

Archimedes is the great Greek mathematician whoformulated the principles of hydromechanics andinvented early techniques of integral calculus.(Courtesy of the National Library of Medicine)

Archimedes of Syracuse 11

displaced was equal to the amount of his bodythat was immersed. This immediately put him inmind of a method to solve Hieron’s problems,and he leapt out of the tub in joy, running nakedtoward his home, shrieking “Eureka!”

His skill in mechanical objects was un-equaled, and Hieron often put him to use in im-proving the defenses of the city, insisting thatArchimedes’ intellect should be put to somepractical application. When Marcellus and theRomans later came to attack Syracuse, theyfound the city impregnable due to the multi-plicity of catapults, mechanical arms, burningmirrors, and various ballistic devices thatArchimedes had built. Archimedes wrote a bookentitled On Spheremaking, in which he describeshow to construct a model planetarium designedto simulate the movement of Sun, Moon, andplanets. It seems that Archimedes was familiarwith Archytas’s heliocentrism, and made use ofthis in his planetarium.

According to Plutarch, Archimedes wasdedicated to pure theory and disdained the prac-tical applications of mathematics to engineering;only those subjects free of any utility to societywere considered worthy of wholehearted pursuit.Archimedes’ mathematical works consist mainlyof studies of area and volume, and the geomet-rical analysis of statics and hydrostatics. In com-puting the area or volume of various plane andsolid figures, he makes use of the so-calledLemma of Archimedes and the “method of ex-haustion.” This lemma states that the differenceof two unequal magnitudes can be formed intoa ratio with any similar magnitude; thus, the dif-ference of two lines will always be a line and nota point. The method of exhaustion involves sub-tracting a quantity larger than half of a givenmagnitude indefinitely, and points to the idea ofthe eternal divisibility of the continuum (thatone can always take away half of a number andstill have something left). These ideas border onnotions of the infinitesimal—the infinitelysmall—and the idea of a limit, which are key

ingredients of integral calculus; however, theGreeks were averse to the notion of infinity andinfinitesimals, and Archimedes shied away fromdoing anything that he felt would be regarded asabsurd.

The method of exhaustion, which was usedrarely in Euclid’s Elements, will be illustratedthrough the following example: In On theMeasurement of the Circle, Archimedes assumes,for the sake of contradiction, that the area of aright triangle with base equal to the circumfer-ence and height equal to the radius of the circleis actually greater than the area of the circle.Then he is able, using the Lemma of Archimedes,to inscribe a polygon in the circle, with the samearea as the triangle; this contradiction shows thatthe area of the triangle cannot be greater thanthe circle, and he makes a similar argument thatit cannot be less.

The basic concept of the method of ap-proximation, which is similar to the method ofexhaustion, is to inscribe regular figures withina given plane figure and solid such that the re-maining area or volume is steadily reduced; thearea or volume of the regular figures can be eas-ily calculated, and this will be an increasinglyaccurate approximation. The remaining area orvolume is “exhausted.” Of course, the modernway to obtain an exact determination of meas-ure is via the limit; Archimedes avoided this is-sue by demonstrating that the remaining area orvolume could be made as small as desired by in-scribing more regular figures. Of course, onecould perform the same procedure with circum-scribing regular figures.

He also applied these methods to solids,computing the surface area and volume of thesphere, and the volume of cones and pyramids.Archimedes’ methods were sometimes purelygeometrical, but at times used principles fromstatics, such as a “balancing method.” His knowl-edge of the law of the lever and the center ofgravity for the triangle, together with his ap-proximation and exhaustion methods, enabled

12 Aristarchus of Samos

him to improve the proofs of known theoremsas well as establish completely new results.

Archimedes also made some contributions inthe realm of numerical calculations, producingsome highly accurate approximations for pi andthe square root of three. In The Sandreckoner hedevises a notation for enormous numbers and es-timates the number of grains of sand to fill theuniverse. In On the Equilibrium of Planes he provesthe law of the lever from geometrical principles,and in On Floating Bodies he explains the con-cept of hydrostatic pressure. The so-calledPrinciple of Archimedes states that solids placedin a fluid will be lighter in the fluid by an amountequal to the weight of the fluid displaced.

His influence on later mathematics was ex-tensive, although Archimedes may not have en-joyed much fame in his own lifetime. LaterGreeks, including PAPPUS OF ALEXANDRIA andTheon of Alexandria, wrote commentaries onhis writings, and later still, Byzantine authorsstudied his work. From Byzantium his texts cameinto the West before the start of the Renaissance;meanwhile, Arabic mathematicians were familiarwith Archimedes, and they exploited his meth-ods in their own researches into conic sections.In the 12th century translations from Arabic intoLatin appeared, which LEONARDO FIBONACCI

made use of in the 13th century. By the 1400sknowledge of Archimedes had expandedthroughout parts of Europe, and his mathematicslater influenced SIMON STEVIN, Johannes Kepler,GALILEO GALILEI, and BONAVENTURA CAVALIERI.

Perhaps the best-known story concerningArchimedes relates his death, which occurred in212 B.C.E. during the siege of Syracuse by theRomans. Apparently, he was not concerned withthe civic situation, and was busily making sanddiagrams in his home (at this time he was at least75 years old). Although the Roman generalMarcellus had given strict orders that the famousSicilian mathematician was not to be harmed, aRoman soldier broke into Archimedes’ houseand spoiled his diagram. When the aged math-

ematician vocally expressed his displeasure, thesoldier promptly slew him.

Archimedes was an outstanding mathemati-cian and scientist. Indeed, he is considered bymany to be one of the greatest three mathemati-cians of all time, along with CARL FRIEDRICH GAUSS

and Newton. Once discovered by medievalEuropeans, his works propelled the discovery ofcalculus. It is interesting that this profound intel-lect was remote in time and space from the greatclassical Greek mathematicians; Archimedesworked on the island of Syracuse, far from Athens,the source of much Greek thought, and he workedcenturies after the decline of the Greek culture.

Further ReadingAaboe, A. Episodes from the Early History of

Mathematics. Washington, D.C.: MathematicalAssociation of America, 1964.

Brumbaugh, R. The Philosophers of Greece. New York:Crowell, 1964.

Dijksterhuis, E. Archimedes. Copenhagen: E.Munksgaard, 1956.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Hollingdale, S. “Archimedes of Syracuse: A Tributeon the 22nd Century of His Death,” Bulletin ofthe Institute of Mathematics and Its Applications 25,no. 9 (1989): 217–225.

Laird, W. “Archimedes among the Humanists,” Isis82, no. 314 (1991): 629–638.

Neugebauer, O. “Archimedes and Aristarchus,” Isis34 (1942): 4–6.

Osborne, C. “Archimedes on the Dimension of theCosmos,” Isis 74, no. 272 (1983): 234–242.

� Aristarchus of Samos(ca. 310 B.C.E.–230 B.C.E.)GreekTrigonometry

Renowned as the first person to propose a he-liocentric theory (that the planets revolve

Aristarchus of Samos 13

around the Sun) of the solar system, Aristarchuswas both an important astronomer and a first-rate mathematician. Little is known of his life,but his works have survived, in which he calcu-lates various astronomical distances millenniabefore the invention of modern telescopes.

Apparently, Aristarchus was born on the is-land of Samos, which lies in the Aegean Seaclose to the city of Miletus, a center for scienceand learning in the Ionian civilization. He stud-ied under Strato of Lampsacos, director of theLyceum founded by Aristotle. It is thought thatAristarchus was taught by Strato in Alexandriarather than Athens. His approximate dates aredetermined by the records of CLAUDIUS PTOLEMY

and ARCHIMEDES OF SYRACUSE. Aristarchus’s onlywork still in existence is his treatise On the Sizesand Distances of the Sun and Moon.

Among his peers, Aristarchus was known as“the mathematician,” which may have beenmerely descriptive. At that time, the disciplineof astronomy was considered part of mathemat-ics, and Aristarchus’s On Sizes and Distancesprimarily treats astronomical calculations. Accor-ding to Vitruvius, a Roman architect, Aristarchuswas an expert in all branches of mathematics,and was the inventor of a popular sundial con-sisting of a hemispherical bowl with a verticalneedle poised in the center. It seems that his dis-coveries in On Sizes and Distances of the vast scaleof the universe fostered an interest in the physi-cal orientation of the solar system, eventuallyleading to his heliocentric conception of the Sunin the center.

Heliocentrism has its roots in the earlyPythagoreans, a religious/philosophical cult thatthrived in the fifth century B.C.E. in southernItaly. Philolaus (ca. 440 B.C.E.) is attributed withthe idea that the Earth, Moon, Sun, and planetsorbited around a central “hearth of the universe.”Hicetas, a contemporary of Philolaus, believed inthe axial rotation of the Earth. The ancient his-torians credit Heraclides of Pontus (ca. 340B.C.E.) with the Earth’s rotation about the Sun,

but Aristarchus is said to be the first to developa complete heliocentric theory: The Earth orbitsthe Sun while at the same time spinning aboutits axis.

It is interesting that the heliocentric theorydid not catch on. The idea did not attract muchattention, and the philosophical speculations ofthe Ionian era were already waning, to be re-placed by the increasingly mathematical feats ofAPOLLONIUS OF PERGA, HIPPARCHUS OF RHODES,and Ptolemy. Due to trends in intellectual andreligious circles, geocentrism became increas-ingly popular. Not until Nicolaus Copernicus,who lived 18 centuries later, resurrectedAristarchus’s hypothesis did opinion turn awayfrom considering the Earth as the center of theuniverse.

Living after EUCLID OF ALEXANDRIA and be-fore Archimedes, Aristarchus was able to producerigorous arguments and geometrical construc-tions, a distinguishing characteristic of the bettermathematicians. The attempt to make variousmeasurements of the solar system without a tele-scope seems incredible, but it involved the sim-ple geometry of triangles. With the Sun (S),Earth (E), and Moon (M) as the three verticesof a triangle, the angle EMS will be a right an-gle when the Moon is exactly half in shadow.Through careful observation, it is possible tomeasure the angle MES, and thus the third an-gle ESM can be deduced. Once these angles areknown, the ratio of the length of the legs, thatis, the distance to the Moon and the distance tothe Sun, can be determined. Of course, this pro-cedure is fraught with difficulties, and any slighterror in estimating the angles will throw off thewhole calculation. Aristarchus estimated angleMES to be approximately 87 degrees, when it isactually 89 degrees and 50 minutes. From this,he deduces that the distance to the Sun is about20 times greater than the distance to the Moon,when in actuality it is 400 times greater. His the-ory was sound, but Aristarchus was inhibited byhis crude equipment.

14 Aryabhata I

This is discussed in On Sizes and Distances,where he states several assumptions and fromthese proves the above estimate on the dis-tance to the Sun and also states that the di-ameter of Sun and Moon are related in thesame manner (the Sun is about 20 times aswide across as the Moon). He also computesthat the ratio of the diameter of the Sun to thediameter of the Earth is between 19:3 and 43:6,an underestimate.

It is noteworthy that trigonometry had notyet been developed, and yet Aristarchus devel-oped methods that essentially estimated thesines of small angles. Without precise means ofcalculation, Aristarchus was unable to attain ac-curate results, although his method was brilliant.Because heliocentrism was not accepted at thetime, Aristarchus failed to achieve much famein his own lifetime. Nevertheless, he was one ofthe first mathematicians to obtain highly accu-rate astronomical measurements.

Further ReadingHeath, T. Aristarchus of Samos, the Ancient Copernicus.

Oxford: Clarendon Press, 1966.———. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.Neugebauer, O. A History of Ancient Mathematical

Astronomy. New York: Springer-Verlag, 1975.———. “Archimedes and Aristarchus,” Isis 34

(1942): 4–6.

� Aryabhata I(476–550)IndianAlgebra, Geometry

Little is known of the life of Aryabhata, who iscalled Aryabhata I in order to distinguish himfrom another mathematician of the same namewho lived four centuries later. Aryabhata playeda role in the development of the modern cur-rent number system and made contributions to

number theory at a time when much of Europewas enveloped in ignorance.

He was born in India and had a connectionwith the city Kusumapura, the capital of theGuptas during the fourth and fifth centuries; thisplace is thought to be the city of his birth.Certainly, his Aryabhatiya was written inKusumapura, which later became a center ofmathematical learning.

Aryabhata wrote two works: the Aryabhatiyain 499, when he was 23 years old, and anothertreatise, which has been lost. The former workis a short summary of Hindu mathematics, con-sisting of three sections on mathematics, timeand planetary models, and the sphere. The sec-tions on mathematics contain 66 mathematicalrules without proof, dealing with arithmetic, al-gebra, plane trigonometry, and sphericaltrigonometry. However, it also contains more ad-vanced knowledge, such as continued fractions,quadratic equations, infinite series, and a tableof sines. In 800 this work was translated intoArabic, and had numerous Indian commentators.

Aryabhata’s number system, the one he usedin his book, gives a number for each of the 33letters of the Indian alphabet, representing thefirst 25 numbers as well as 30, 40, 50, 60, 70, 80,90, and 100. It is noteworthy that he was famil-iar with a place-value system, so that very largenumbers could easily be described and manipu-lated using this alphabetical notation. Indeed, itseems likely that Aryabhata was familiar withzero as a placeholder. The Indian place-valuenumber system, which would later greatly influ-ence the construction of the modern system, fa-cilitated calculations that would be infeasibleunder more primitive models, such as Romannumerals. Aryabhata appears to be the origina-tor of this place-value system.

In his examination of algebra, Aryabhatafirst investigates linear equations with integercoefficients—apparently, the Aryabhatiya is thefirst written work to do so. The question arosefrom certain problems of astronomy, such as the

Aryabhata I 15

computation of the period of the planets. Thetechnique is called kuttaka, which means “to pul-verize,” and consists of breaking the equationinto related problems with smaller coefficients;the method is similar to the Euclidean algorithmfor finding the greatest common divisor, but isalso related to the theory of continued fractions.

In addition, Aryabhata gave a value for pi thatwas accurate to eight decimal places, improvingon ARCHIMEDES OF SYRACUSE’s and APOLLONIUS OF

PERGA’s approximations. Scholars have arguedthat he obtained this independently of theGreeks, having some particular method for ap-proximating pi, but it is not known exactly howhe did it; Aryabhata also realized that pi was anirrational number. His table of sines gives ap-proximate values at intervals of less than fourdegrees, and uses a trigonometric formula to ac-complish this.

Aryabhata also discusses rules for summingthe first n integers, the first n squares, and thefirst n cubes; he gives formulas for the area of tri-angles and of circles. His results for the volumesof a sphere and of a pyramid are incorrect, butthis may have been due to a translation error.Of course, these latter results were well knownto the Greeks and might have come to Aryabhatathrough the Arabs.

As far as the astronomy present in the text,which the mathematics is designed to elucidate,there are several interesting results. Aryabhatagives an excellent approximation to the circum-ference of the Earth (62,832 miles), and explainsthe rotation of the heavens through a theory of

the axial rotation of the Earth. Ironically, this(correct) theory was thought ludicrous by latercommentators, who altered the text in order toremedy Aryabhata’s mistakes. Equally remark-able is his description of the planetary orbits asellipses—only highly accurate astronomical dataprovided by superior telescopes allowedEuropean astronomers to differentiate betweencircular and elliptical orbits. Aryahbhata gives acorrect explanation of the solar and lunareclipses, and attributes the light of the Moon toreflected sunlight.

Aryabhata was of great influence to laterIndian mathematicians and astronomers. Perhapsmost relevant for the later development of math-ematics was his place-number system. His theo-ries were exceedingly advanced considering thetime in which he lived, and the accurate compu-tations of astronomical measurements illustratedthe power of his number system.

Further ReadingGupta, R. C. “Aryabhata, Ancient India’s Great

Astronomer and Mathematician,” MathematicalEducation 10, no. 4 (1976): B69–B73.

———. “A Preliminary Bibliography on AryabhataI,” Mathematical Education 10, no. 2 (1976):B21–B26.

Ifrah, G. A Universal History of Numbers: FromPrehistory to the Invention of the Computer. NewYork: John Wiley, 2000.

van der Waerden, B. “The ‘Day of Brahman’ in theWork of Aryabhata,” Archive for History of ExactSciences 38, no. 1 (1988): 13–22.

B

16

� Babbage, Charles(1792–1871)BritishAnalysis

The name of Charles Babbage is associated withthe early computer. Living during the industrialage, in a time when there was unbridled opti-mism in the potential of machinery to improvecivilization, Babbage was an advocate of mecha-nistic progress, and spent much of his lifetimepursuing the invention of an “analytic engine.”Although his ambitious project eventually endedin failure, his ideas were important to the subse-quent develop of computer logic and technology.

Born on December 26, 1792, in Teignmouth,England, to affluent parents, Babbage exhibitedgreat curiosity for how things worked. He waseducated privately by his parents, and by thetime he registered at Cambridge in 1810, he wasfar ahead of his peers. In fact, it seems that heknew more than even his teachers, as mathe-matics in England had lagged far behind the restof Europe. Along with George Peacock and JohnHerschel, he campaigned vigorously for the re-suscitation of English mathematics. Together withPeacock and Herschel, he translated Lacroix’sDifferential and Integral Calculus, and became anardent proponent of GOTTFRIED WILHELM VON

LEIBNIZ’s notation over SIR ISAAC NEWTON’s.

Upon graduating, Babbage became involvedin many diverse activities: He wrote several pa-pers on the theory of functions and applied math-ematics and helped to found several progressivelearned societies, such as the AstronomicalSociety in 1820, the British Association in 1831,and the Statistical Society of London in 1834.He was recognized for his excellent contributionsto mathematics, being made a fellow of theRoyal Society in 1816 and Lucasian professor ofmathematics at Cambridge in 1827; he held thislatter position for 12 years without teaching, be-cause he was becoming increasingly absorbed bythe topic of mechanizing computation.

Babbage viewed science as an essential partof civilization and culture, and even thoughtthat it was the government’s responsibility to en-courage and advance science by offering grantsand prizes. Although this viewpoint is fairlycommon today, Babbage was one of its first ad-vocates; before his time, much of science andmathematics was conducted in private researchby men of leisure. He also advocated pedagogi-cal reform, realizing that great teaching was cru-cial for the future development of mathematics;however, he did little with his chair at Cambridgetoward realizing this goal.

His interests were remarkably diverse, in-cluding probability, cryptanalysis, geophysics,astronomy, altimetry, ophthalmoscopy, statistical

Babbage, Charles 17

linguistics, meteorology, actuarial science, light-house technology, and climatology. Babbage de-vised a convenient notation that simplified thedrawing and reading of engineering charts. Hisliterature on operational research, concernedwith mass production in the context of pin man-ufacture, the post office, and the printing trade,has been especially influential.

Babbage was, as a young man, lively and so-ciable, but his growing obsession with con-structing computational aids made him bitterand grumpy. Once he realized the extent of er-rors in existing mathematical tables, his mindturned to the task of using machinery to accom-plish faultless calculations. Initially, he imagineda steam-powered calculator for the computationof trigonometric quantities; he began to envisiona machine that would calculate functions andalso print out the results.

The theory behind his machine was themethod of finite differences—a discrete analog

of the continuous differential calculus. Anypolynomial of nth degree can be reduced, throughsuccessive differences, to a constant; the inverseof this procedure, taking successive sums, wouldbe capable of computing the values of a polyno-mial, given some initial conditions. In addition,this concept could be extended to most nonra-tional functions, including logarithms; thiswould allow the mechanistic computation of thevalue of an arbitrary function.

Unfortunately, Babbage did not succeed. Hecontinually thought up improvements for the sys-tem, becoming more ambitious for the final“Difference Engine Number One.” This machinewould handle sixth-order differences and 20 dec-imal numbers—a goal more grandiose than feasi-ble. He never completed the project, though aSwedish engineer, in Babbage’s own lifetime, builta modest working version based on a magazineaccount of the Englishman’s dream. It seems thatthe principal reason for Babbage’s failure was theprohibitive cost, though another cause is foundin his new design to build an “analytical engine.”

The analytical engine, in its design and plan-ning, was a forerunner of the modern computer.Based on Joseph-Marie Jacquard’s punch cardsused in weaving machinery, Babbage’s machinewould be run by inserting cards with small holes;springy wires would move through the holes tooperate certain levers. This concept described amachine of great versatility and power. The mill,the center of the machine, was to possess 1,000columns with 50 geared wheels apiece: up to1,000 50-digit numbers could be operated onwith one of the four main arithmetic operations.Data, operation, and function cards could be in-serted to provide information on variables, pro-grams, and constants to the mill. The outputwould be printed, and another part of the ma-chine would check for errors, store information,and make decisions. This corresponds to thememory and logic flow components of a moderncomputer. However, in one important aspectBabbage’s analytical engine differs from the digital

Charles Babbage, inventor of an early mechanicalcomputer and founder of computer science (Courtesyof the Library of Congress)

18 Bacon, Roger

computer: His was based on a decimal system,whereas computers operate on a binary system.

Although the plans for this machine im-pressed all who viewed them, Babbage did notreceive any financial support for its construction.He died on October 18, 1871, in London, with-out seeing the completion of his mechanisticprojects. However, his son later built a small milland printer, which is kept in the ScienceMuseum of London.

Babbage was a highly creative mathemati-cian whose ideas foreshadowed the major thrustof computer science in the second half of the20th century. His work in pure mathematics hashad little impact on successive generations ofmathematicians, but his ideas on the analyticalengine would be revisited over the next century,culminating in the design of early computers inthe mid-1900s.

Further ReadingBabbage, H. Babbage’s Calculating Engines. Los

Angeles: Tomash, 1982.Buxton, H. Memoir of the Life and Labors of the Late

Charles Babbage Esq., F.R.S. Los Angeles:Tomash, 1988.

Dubbey, J. The Mathematical Work of Charles Babbage.Cambridge: Cambridge University Press, 1978.

Hyman, A. Charles Babbage: Pioneer of the Computer.Princeton, N.J.: Princeton University Press, 1982.

Morrison, P., and E. Morrison. Charles Babbage andHis Calculating Engines. New York: DoverPublications, 1961.

Wilkes, M. “Babbage as a Computer Pioneer,” HistoriaMathematica 4, no. 4 (1977): 415– 440.

� Bacon, Roger(ca. 1214–ca. 1292)BritishArithmetic

In 13th-century Europe, there was no pursuit ofscience as there is today: the medieval church,

having gone so far as to make reason irrelevantin matters of faith and knowledge, substitutingthe unmitigated authority of papal decree andcanon law, reigned over a stifling intellectual cli-mate. However, the use of reason and empiricism,when coupled with the knowledge of a rationalGod’s creation of a rational world, would proveto be the epistemology of science for the nextseveral centuries, which resulted in numerousdiscoveries. Roger Bacon was an early figure inthis paradigm shift, vigorously acting as a keyproponent of the utility of mathematics andlogic within the spheres of human knowledge.Natural philosophy, which in his view was sub-servient to theology, could serve toward the ad-vancement of the human task generally speaking

Roger Bacon proposed that mathematical knowledgeshould be arrived at through reason rather thanauthority. (Courtesy of the National Library of Medicine)

Bacon, Roger 19

(the dominion and ordering of the Earth and,more specifically, the development of thechurch). Later scientific endeavor, starting inthe 18th and 19th centuries, would abandonthese theistic roots in favor of reason as the soleauthority in man’s pedagogical quest; but Bacon’spromotion of the use of mathematics in part-nership with faith in God was to remain theguiding epistemology for several centuries.

Bacon’s birth has been calculated to be ap-proximately 1214, though scholars differ on thisdetail since there is no exact record. ThisEnglishman came of a family that had sufferedpersecution from the baronial party, due to theirfailed support of Henry III. His early instructionin the Latin classics, including Seneca andCicero, led to his lifelong fascination with nat-ural philosophy and mathematics, further incul-cated at Oxford. After receiving his M.A. degreein about 1240, he apparently lectured in theFaculty of the Arts at Paris from 1241 to 1246.He discussed various topics from Aristotle’sworks, and he was a vehement advocate of com-plete instruction in foreign languages. Bacon un-derwent a drastic change in his conception ofknowledge after reading the works of RobertGrosseteste (a leading philosopher and mathe-matician of the region) when he returned toOxford in 1247; he invested considerable sumsof money for experimental equipment, instru-ments, and books, and sought out acquaintancewith various learned persons. Under Grosseteste’sinfluence, Bacon developed the belief that lan-guages, optics, and mathematics were the mostimportant scientific subjects, a view he main-tained his whole life.

By 1251 he had returned to Paris, and he en-tered the Franciscan order in 1257. The chapterof Narbonne was presided over by Bonaventure,who was opposed to inquiries not directly relatedto theology; he disagreed sharply with Bacon onthe topics of alchemy and astrology, which heviewed as a complete waste of time. Bacon, onthe other hand, while agreeing that they had no

discernible or predictable impact on the fates ofindividuals, thought it possible for the stars to ex-ert a generic influence over the affairs of theworld; he also experimented in alchemy, thequest to transmute lead into gold. Due to thesepolitical difficulties, Bacon made various propos-als on education and science to Cardinal Guy deFolques, who was soon elected Pope Clement IVin 1265. As pope he formally requested Bacon tosubmit his philosophical writings, and theEnglishman soon produced three famous works:Opus maius (Great work), Opus minus (Smallerwork), and Opus tertium (Third work) within thenext few years.

The Opus maius treated his opinions on nat-ural philosophy and educational reform. Authorityand custom were identified as impediments tolearning; although Bacon submitted to the au-thority of the Holy Scriptures, he believed thewisdom contained therein needed to be devel-oped by reason, rightly informed by faith. In thisone sees some early seeds of Protestant thoughtabout the proper balance of authority and rea-son. However, Bacon was not a believer in purededuction detached from the observed world,like the Greek philosophers and mathematiciansof antiquity; rather, he argued for requisition ofexperience. Information obtained through theexterior senses could be measured and quantifiedthrough instruments and experimental devicesand analyzed through the implementation ofmathematics. By studying the natural world, itwas possible, Bacon argued, to arrive at some un-derstanding of the Creator of that natural world.Thus, all of human knowledge was conceived ina harmonious unity, guided and led by theologyas the regent of science. Hence it was necessaryto deepen the understanding of languages, math-ematics, optics, experimental science, alchemy,metaphysics, and moral philosophy.

Bacon’s view on authority was somewhatprogressive: without moderation, authoritywould prevent the plowing of intellectual fur-rows given provenience by rational disputation.

20 Bacon, Roger

However, it must not be thought that a prede-cessor of nihilism, moral relativism, or other an-tiauthoritative systems can be found in Bacon—he believed in one truth (Christianity), but soughtto use reason as a fit tool for advancing the inter-ests of the kingdom of God and the civilization ofman. The heathen should be converted by argu-ment and persuasion, never by force.

Mathematics was to play an important rolein Bacon’s entire system. Of course, he under-stood the term in a broad sense, as inclusive ofastronomy and astrology, optics, physical causa-tion, and calendar reform, with even applicationsto purely religious matters. His work in optics re-lied heavily on geometry, and stood on the shoul-ders of EUCLID OF ALEXANDRIA, CLAUDIUS

PTOLEMY, and ABU ALI AL-HAYTHAM, as well asGrosseteste. Along with Grosseteste, he advo-cated the use of lenses for incendiary and visualpurposes. Bacon’s ideas on refraction and reflec-tion constituted a wholly new law of nature. Hiswork on experimental science laid down threemain goals: to certify deductive reasoning fromother subjects, such as mathematics, by experi-mental observation; to add new knowledge notattainable by deduction; and to probe the secretsof nature through new sciences. The last pre-rogative can be seen as an effort toward attain-ing practical magic—the requisitioning of naturetoward spectacular and utilitarian ends.

Bacon lists four realms of mathematical ac-tivity: human business, divine affairs (such aschronology, arithmetic, music), ecclesiastical tasks(such as the certification of faith and repair of thecalendar), and state works (including astrologyand geography). Mathematics, the “alphabet ofphilosophy,” had no limits to its range of applica-bility, although experience was still necessary inBacon’s epistemology. Despite his glowing praiseof “the door and key of the sciences,” it appearsthat Bacon’s facility in mathematics was not great.Although he has some original results in engi-neering, optics, and astronomy, he does not fur-nish any proofs or theorems of his own devising.

He also made some contributions in the ar-eas of geography and calendar reform. He statedthe possibility of journeying from Spain to India,which may have influenced Columbus centurieslater. Bacon’s figures on the radius of the Earthand ratio of land and sea were fairly accurate,but based on a careful selection of ancient au-thorities. His map of the known world, now lost,seems to have included lines of latitude and lon-gitude, with the positions of famous towns andcities. Bacon discussed the errors of the Juliancalendar with great perspicuity, and recom-mended the removal of one day in 125 years,similar to the Gregorian system.

Certainly, after his death, Bacon had manyadmirers and followers in the subsequent cen-turies. He continued writing various communi-cations on his scientific theories, but sometimeafter 1277 he was condemned and imprisonedin Paris by his own Franciscan order, possiblyfor violating a censure. His last known writingwas published in 1292, and he died sometimeafterward.

Bacon contributed generally to the advanceof reason and a rational approach to knowledgein Europe; his efforts influenced not only thecourse of mathematics but also the history of sci-ence more generally. The writings of Baconwould be familiar to later generations of math-ematicians working in the early 17th century.

Further ReadingBridges, J. The Life and Work of Roger Bacon: An

Introduction to the Opus Majus. Merrick, N.Y.:Richwood Publishing Company, 1976.

Easton, S. Roger Bacon and His Search for a UniversalScience; a Reconsideration of the Life and Work ofRoger Bacon in the Light of His Own Stated Purposes.Westport, Conn.: Greenwood Press, 1970.

Lindberg, D. Roger Bacon’s Philosophy of Nature: ACritical Edition. Oxford: Clarendon Press, 1983.

———. “Science As Handmaiden: Roger Bacon andthe Patristic Tradition,” Isis 78, no. 294 (1987):518–536.

Baire, René-Louis 21

� Baire, René-Louis(1874–1932)FrenchAnalysis

In the late 19th century some of the ideas onthe limits of sequences of functions were stillvague and ill formulated. René Baire greatly advanced the theory of functions by consider-ing issues of continuity and limit; his effortshelped to solidify the intuitive notions then incirculation.

René-Louis Baire was born in Paris onJanuary 21, 1874, one of three children in a mid-dle-class family. His parents endured hardship inorder to send Baire to school, but he won ascholarship in 1886 that allowed him to enterthe Lycée Lakanal. He completed his studieswith high marks and entered the École NormaleSupérieure in 1892.

During his next three years, Baire becameone of the leading students in mathematics,earning first place in his written examination.He was a quiet, introspective young man ofdelicate health, which would plague himthroughout his life. In the course of his oralpresentation of exponential functions, Bairerealized that the demonstration of continuitythat he had learned was insufficient; this realiza-tion led him to study the continuity of functionsmore intensely and to investigate the general na-ture of functions.

In 1899 Baire defended his doctoral the-sis, which was concerned with the propertiesof limits of sequences of continuous functions.He embarked on a teaching career at local ly-cées, but found the schedule too demanding;eventually he obtained an appointment as pro-fessor of analysis at the Faculty of Science inDijon in 1905. Meanwhile, Baire had alreadywritten some papers on discontinuities of func-tions, and had also suffered a serious illness in-volving the constriction of his esophagus. In 1908he completed a major treatise on mathematical

analysis that breathed new life into that sub-ject. From 1909 to 1914 his health was incontinual decline, and Baire struggled to ful-fill his teaching duties; in 1914 he obtained aleave of absence and departed for Lausanne.Unfortunately, the eruption of war preventedhis return, and he was forced to remain therein difficult financial circumstances for the nextfour years.

His mathematical contributions weremainly focused around the analysis of functions.Baire developed the concept of semicontinuity,and perceived that limits and continuity offunctions had to be treated more carefully thanthey had been. His use of the transfinite num-ber exercised great influence on the Frenchschool of mathematics over the next severaldecades. Baire’s most lasting contributions areconcerned with the limits of continuous func-tions, which he divided into various categories.He provided the proper framework for studyingthe theory of functions of a real variable; pre-viously, interest was peripheral, as mathemati-cians were only interested in real functions thatcame up in the course of some other investiga-tion. Thus, Baire effected a reorientation ofthought.

Baire’s illness made him incapable of re-suming his grand project, and after the war hefocused instead on calendar reform. He later re-ceived the ribbon of the Legion of Honor andwas elected to the Academy of Sciences; sadly,his last years were characterized by pain and fi-nancial struggles. As a result, he was able todevote only limited amounts of time to math-ematical research. He died in Chambéry,France, on July 5, 1932.

Baire’s work played an important role in thehistory of modern mathematics, as it representsa significant step in the maturation of thought.His ideas were highly regarded by ÉMILE BOREL

and HENRI LEBESGUE, and exerted much influ-ence on subsequent French and foreign mathe-maticians.

22 Banach, Stefan

� Banach, Stefan(1892–1945)PolishAnalysis

Stefan Banach is known as the principal founderof functional analysis, the study of certain spacesof functions. He influenced many students dur-ing his intense career as a research mathemati-cian, and many of the most important results offunctional analysis bear his name.

Little is known of Banach’s personality,other than that he was hardworking and dedi-cated to mathematics. Born in Krakow on March30, 1892, to a railway official, Banach was turnedover to a laundress by his parents; this woman,who became his foster mother, reared him andgave him his surname, Banach. At the age of 15he supported himself by giving private lessons.He graduated from secondary school in 1910.After this he matriculated at the Institute ofTechnology at Lvov, in the Ukraine, but did notgraduate. Four years later he returned to hishometown. There he met the Polish mathe-matician Hugo Steinhaus in 1916. From thistime he became devoted to mathematics; itseems that he already possessed a wide knowl-edge of the discipline, and together withSteinhaus he wrote his first paper on the con-vergence of Fourier series.

In 1919 Banach was appointed to a lecture-ship at the Institute of Technology in Lvov,where he taught mathematics and mechanics. Inthis same year he received his doctorate in math-ematics, even though his university educationwas incomplete. His thesis, said to have signaledthe birth of functional analysis, dealt with inte-gral equations; this is discussed in greater lengthbelow. In 1922 Banach was promoted in con-sideration of an excellent paper on measure the-ory (measures are special functions that computethe lengths, areas, and volumes of sets). Afterthis he was made associate professor, and thenfull professor in 1927 at the University of Lvov.

Also, in 1924 he was elected to the PolishAcademy of Sciences and Arts.

Banach made contributions to orthogonalseries and topology, investigating the propertiesof locally meager sets. He researched a more gen-eral version of differentiation in measure spaces,and discovered classic results on absolute conti-nuity. The Radon-Nikodym theorem was stimu-lated by his contributions in the area of measureand integration. He also established connectionsbetween the existence of measures and ax-iomatic set theory.

However, functional analysis was Banach’smost important contribution. Little had beendone in a unified way in functional analysis: VITO

VOLTERRA had a few papers from the 1890s onintegral equations, and IVAR FREDHOLM andDAVID HILBERT had looked at linear spaces. From1922 onward, Banach researched normed linearspaces with the property of completeness—nowcalled Banach spaces. Although some other con-temporary mathematicians, such as Hans Hahn,RENÉ-MAURICE FRÉCHET, Eduard Helly, and NOR-BERT WIENER, were simultaneously developingconcepts in functional analysis, none performedthe task as thoroughly and systematically asBanach and his students. His three fundamentalresults were the theorem on the extension ofcontinuous linear functionals (now called theHahn-Banach theorem, as both Banach andHahn proved it independently); the theorem onbounded families of mappings (called theBanach-Steinhaus theorem); and the theoremon continuous linear mappings of Banach spaces.He introduced the notions of weak convergenceand weak closure, which deal with the topologyof normed linear spaces.

Banach and Steinhaus founded the journalStudia mathematica (Mathematical studies) butBanach was often distracted from his scientificwork due to his writing of college and second-ary school texts. From 1939 to 1941 he servedas dean of the faculty at Lvov, and during thistime was elected as a member of the Ukrainian

Barrow, Isaac 23

Academy of Sciences. However, World War IIinterrupted his brilliant career; in 1941 theGermans occupied Lvov. For three years Banachwas forced to research infectious diseases in aGerman institute, where he fed lice. When theSoviets recaptured Lvov in 1944, Banach re-turned to his post in the university; unfortu-nately, his health was shattered by the poorconditions under the German army, and he diedon August 31, 1945.

Banach’s work later became more widelyknown to mathematicians laboring in the fieldof functional analysis. His name is attached toseveral mathematical objects and theorems, giv-ing evidence to his importance as one of theprincipal founders of functional analysis.

Further ReadingHoare, G., and N. Lord. “Stefan Banach (1892–1945):

A Commemoration of His Life and Work,” TheMathematical Gazette 79 (1995): 456–470.

Kauza, R. Through a Reporter’s Eyes: The Life of StefanBanach. Boston: Birkhäuser, 1996.

Ulam, S. Adventures of a Mathematician. Berkeley:University of California Press, 1991.

� Barrow, Isaac(1630–1677)BritishCalculus

Isaac Barrow was the first to discover certain as-pects of differential calculus. There is some con-troversy about this, and also about the extent ofhis influence on SIR ISAAC NEWTON, who was hissuccessor at Cambridge. However, Barrow’s lec-tures on geometry contain some of the first the-orems of calculus, and for this he is renowned.

Barrow was born in October 1630 (the ex-act date is unknown) to Thomas Barrow, a pros-perous linen draper and staunch royalist. Hismother, Anne, died in childbirth. A rebel in hisyounger days, Barrow later became disciplined

and learned Greek, Latin, logic, and rhetoric. In1643 he entered Trinity College, where he wouldremain for 12 years. Barrow, like his father, wasa supporter of the king, but at Trinity the at-mosphere became increasingly antiroyalist. Heearned his B.A. degree in 1648, was elected col-lege fellow in 1649, and received his M.A. degreein mathematics in 1652. With these credentials,he entered his final position as college lecturerand university examiner.

It is likely that his next appointment wouldhave been a professorship of Greek, but Barrowwas ejected from his position by Cromwell’s gov-ernment in 1655. Barrow sold his books and em-barked on a tour of Europe, which lasted for fouryears. When he returned from his travels,Charles II had just been restored to power; Barrowtook holy orders and thereby obtained the Regiusprofessorship. In 1662 he also accepted the

Isaac Barrow, early discoverer of certain rules andresults of calculus (Courtesy of the Library ofCongress)

24 Barrow, Isaac

Gresham professorship of geometry in London,and the next year was appointed as first Lucasianprofessor of mathematics at Cambridge. Duringthe next six years Barrow concentrated his ef-forts on writing the three series of Lectiones, acollection of lectures that are discussed below.

Barrow’s education had been quite tradi-tional, centered on Aristotle and Renaissancethinkers, and on some topics he remained veryconservative. But he was greatly intrigued by therevival of atomism and RENÉ DESCARTES’s natu-ral philosophy—his master’s thesis studiedDescartes in particular. By 1652 he had readmany commentaries of EUCLID OF ALEXANDRIA,as well as more advanced Greek authors such asARCHIMEDES OF SYRACUSE. His Euclidis elemento-rum libri XV (Euclid’s first principles in 15 books),written in 1654, was designed as an undergradu-ate text, stressing deductive structure over con-tent. He later produced commentaries on Euclid,Archimedes, and APOLLONIUS OF PERGA.

Apparently, Barrow’s scientific fame was dueto the Lectiones (Lectures), though they have notsurvived. The first Lucasian series, the Lectionesmathematicae (Mathematical lectures)—givenfrom 1664 to 1666—is concerned with the foun-dations of mathematics from a Greek viewpoint.Barrow considers the ontological status of mathe-matical objects, the nature of deduction, spatialmagnitude and numerical quantity, infinity and theinfinitesimal, proportionality and incommensura-bility, as well as continuous and discrete entities.His Lectiones geometricae (Geometrical lectures)were a technical study of higher geometry.

In 1664 he found a method for determiningthe tangent line of a curve, a problem that was tobe solved completely by the differential calculus;his technique involves the rotation and transla-tion of lines. Barrow’s later lectures are a general-ization of tangent, quadrature, and rectificationprocedures compiled from his reading ofEvangelista Torricelli, Descartes, Frans vanSchooten, Johann Hudde, JOHN WALLIS,Christopher Wren, PIERRE DE FERMAT, CHRISTIAAN

HUYGENS, BLAISE PASCAL, and JAMES GREGORY.The material of these lectures was not totally orig-inal, being heavily based on the above authors,especially Gregory, and Barrow’s Lectiones geomet-ricae were not widely read.

Barrow also contributed to the field of op-tics, though his Lectiones opticae (Lectures on op-tics) was soon eclipsed by Newton’s work. Theintroduction describes a lucid body, consisting of“collections of particles minute almost beyondconceivability,” as the source of light rays; coloris a dilution of thickness. The work is developedfrom six axioms, including the Euclidean law ofreflection and sine law of refraction. Much ofthe material is taken from ABU ALI AL-HAYTHAM,Johannes Kepler, and Descartes, but Barrow’smethod for finding the point of refraction at aplane interface is original.

Much has been hypothesized of the rela-tionship between Barrow and Newton; some saythat Newton derived many of his ideas about cal-culus from Barrow, but there is little evidence ofthis. By late 1669 the two collaborated briefly,but it is not clear if they had any interaction be-fore that time. In that year Barrow had resignedhis chair, being replaced by Newton, in order tobecome the Royal Chaplain of London, and in1675 became university vice-chancellor.

Barrow never married, being content withthe life of a bachelor. His personality was blunt,and his theological sermons were extremely lu-cid and insightful, although he was not a popu-lar preacher. Barrow was also one of the firstmembers of the Royal Society, incorporated in1662. He was small and wiry, and enjoyed goodhealth; his early death on May 4, 1677, was dueto an overdose of drugs.

Barrow’s mathematical contribution seemssomewhat marginal compared with the prodi-gious output of his contemporary Newton.However, he was an important mathematicianof his time, earning fame through his popularLectiones, and was the first to derive certainpropositions of differential calculus.

Bayes, Thomas 25

Further ReadingFeingold, M. Before Newton: The Life and Times of

Isaac Barrow. New York: Cambridge UniversityPress, 1990.

––––––. “Newton, Leibniz, and Barrow Too: AnAttempt at a Reinterpretation,” Isis 84, no. 2(1993): 310–338.

Hollingdale, S. “Isaac Barrow (1630–1677),” Bulletinof the Institute of Mathematics and Its Applications13, nos. 11–12 (1977): 258–262.

Malet, A. “Barrow, Wallis, and the Remaking ofSeventeenth Century Indivisibles,” Centaurus39, no. 1 (1997): 67–92.

� Bayes, Thomas(1702–1761)BritishProbability, Statistics

The field of statistics is split between two fac-tions: Bayesians and Frequentists. The lattergroup, sometimes known as the Orthodox, main-tains a classical perspective on probability,whereas the former group owes its genesis toThomas Bayes, a nonconformist preacher andamateur statistician. Though his writings werenot copious, in distinction to many of the fa-mous mathematicians of history, the extensiveinfluence of one remarkable essay has earnedBayes no small quantity of fame.

Born in 1702 to a dissenting theologian andpreacher (he opposed certain doctrines and tra-ditions of the established Anglican Church),Bayes was raised in his father’s nontraditionalviews. With a decent private education, Bayesassisted his father in his pastoral duties inHolborn, London, and later became the minis-ter at Tunbridge Wells. He never married, butpossessed a wide circle of friends.

Apparently, Bayes was familiar with thecurrent mathematics of the age, including thedifferential and integral calculus of SIR ISAAC

NEWTON and the well-laid ideas of classical

probability. Bayes’s mathematical work, Intro-duction to the Doctrine of Fluxions, was publishedin 1736. Newton’s work on calculus, which in-cluded the concept of infinitesimals, sometimescalled fluxions, was controversial, as many sci-entists abhorred the concept of infinitely smallquantities as intellectually repugnant. In fact,Bishop Berkeley—a contemporaneous philoso-pher—had written the Analyst, a thorough cri-tique of Newton’s work; Bayes’s Doctrine ofFluxions was a mathematical rebuttal of Berkeley,and was appreciated as one of the soundestapologies for Newton’s calculus.

But Bayes acquired some fame for his paper“Essay Towards Solving a Problem in theDoctrine of Chances,” published posthumouslyin 1763. Although probability theory was al-ready well founded with recent texts by JAKOB

BERNOULLI and ABRAHAM DE MOIVRE, theoreticalbastions of a similar ilk were lacking for thebranch of statistics. The task that Bayes set forhimself was to determine the probability, orchance, of statistical hypotheses’ truth in lightof the observed data. The framework of hy-pothesis testing, whereby scientific claims couldbe rejected or accepted (technically, “not re-jected”) on the basis of data, was vaguely un-derstood in some special cases—SIR RONALD

AYLMER FISHER would later formulate hypothesistesting with mathematical rigor, providing pre-cision and generality. Of course, to either rejector not reject a claim gives a black or white de-cision to a concept more amenable to shades ofgray (perhaps to a given statistical hypothesis aprobability could be attached, which would in-dicate the practitioner’s degree of confidence,given the data, of the truth of the proposition).This is the question that Bayes endeavored toanswer.

The basic idea is that prior notions of theprobability of an event are often brought to a sit-uation—if biasing presuppositions exist, they colorthe assessment of the likelihood of certain un-foreseen outcomes, and affect the interpretation

26 Bernoulli, Daniel

of observations. In the absence of prior knowl-edge, one could assume a so-called noninforma-tive prior distribution for the hypothesis, whichwould logically be the uniform probability dis-tribution. Bayes demonstrated how to computethe probability of a hypothesis after observationshave been made, which was designated by theterm posterior distribution of the hypothesis. Hismethod of calculation involved a formula thatexpressed the posterior probability in terms ofthe prior probability and the assumed distribu-tion of the data; this was subsequently calledBayes’s theorem.

Whereas the mathematics involved is fairlyelementary (many students learn Bayes’s theo-rem in the first two weeks of a course on prob-ability and statistics), the revolutionary conceptwas that scientific hypotheses should be as-signed probabilities of two species—the priorand the posterior. It seems that Bayes was notsatisfied with his argument for this formulation,and declined to publish the essay, even thoughthis theoretical work gave a firm foundation forstatistical inference. A friend sent the paper tothe Royal Society after Bayes’s death, and thework was popularized by the influential PIERRE-SIMON LAPLACE. Bayes was a wealthy bachelor,and spent most of his life performing religiousduties in the provinces. He was honored by in-clusion to the Royal Society of London in 1742,perhaps for his Doctrine of Fluxions. He died onApril 17, 1761, in Tunbridge Wells, England.

Much controversy has arisen over Bayes’smethodology. The Bayesians show the logicalfoundation of the theory, which agrees with thegeneral practice of science. The Frequentist op-position decries the variation in statistical re-sults, which will be contingent upon the sub-jective choice of prior. It is appropriate to pointout that, not only the analyses of classical sta-tistics (especially nonparametric statistics) andmathematics, but the results of scientific en-deavor more generally, are always contingentupon presuppositional assumptions that cannot

be completely justified. Some Bayesians conceiveof probabilities as objective degrees of confidence,whereas others conceive of purely subjective be-liefs—the Bayesian framework corresponds to theupdating of belief structures through the accu-mulation of empirical information. It seems thatBayes himself was indifferent or at a median be-tween these two philosophical extremes.

Further ReadingBarnard, G. “Thomas Bayes—a Biographical Note,”

Biometrika 45 (1958): 293–315.Dale, A. “Thomas Bayes: A Memorial,” The

Mathematical Intelligencer 11, no. 3 (1989): 18–19.Gillies, D. “Was Bayes a Bayesian?” Historia Mathematica

14, no. 4 (1987): 325–346.Savage, L. The Foundations of Statistics. New York:

Dover Publications, 1972.Smith, G. “Thomas Bayes and Fluxions,” Historia

Mathematica 7, no. 4 (1980): 379–388.Stigler, S. The History of Statistics: The Measurement

of Uncertainty before 1900. Cambridge, Mass.:Belknap Press of Harvard University Press, 1986.

———. “Thomas Bayes’s Bayesian Inference,” Journalof the Royal Statistical Society. Series A. Statisticsin Society 145, no. 2 (1982): 250–258.

� Bernoulli, Daniel(1700–1782)SwissMechanics, Probability

The 18th century was relatively bereft of math-ematical talent in comparison with the intellec-tual wealth of the 1600s; however, DanielBernoulli was among the few rare geniuses ofthat time, making significant contributions tomedicine, mathematics, and the natural sci-ences. In particular, his labors in the mechani-cal aspects of physiology, infinite series, rationalmechanics, hydrodynamics, oscillatory systems,and probability have earned him great renownas an outstanding scientist.

Bernoulli, Daniel 27

Daniel Bernoulli was born on February 8,1700, in Groningen, the Netherlands, into thewell-known Bernoulli family: his father was thefamous mathematician JOHANN BERNOULLI, whowas then a professor at Groningen, and hismother was Dorothea Falkner, member of an af-fluent Swiss family. Daniel Bernoulli was closeto his older brother Nikolaus, but later fell vic-tim to his father’s jealous competitiveness. In1705 Johann Bernoulli relocated the family inBasel, occupying the chair of mathematics re-cently held by his deceased brother Jakob.Daniel Bernoulli commenced the study of logicand philosophy in 1713 and passed his bac-calaureate in 1716. Meanwhile he studied math-ematics under the supervision of his father andNikolaus. Daniel Bernoulli was not destined forbusiness, as a failed apprenticeship in commerce

testified; instead, he continued his Basel studiesin medicine, later journeying to Heidelberg(1718) and Strasbourg (1719) to pursue knowl-edge. The next year he returned to Basel, andhe earned his doctorate in 1721 with the dis-sertation De respiratione (Of respiration).

His application for the professorship ofanatomy and botany was denied, and neither washe able to obtain the chair of logic. In 1723 hetraveled to Venice to continue his medicalstudies under Michelotti. His 1724 publicationof Exercitationes mathematicae (Mathematicalexercises) earned him enough fame that he re-ceived an offer from the St. Petersburg Academy,and he stayed in Russia from 1725 to 1732, mak-ing the acquaintance of LEONHARD EULER. Hisdear brother Nikolaus suddenly died, and the se-vere climate was not to Bernoulli’s liking; thesefactors encouraged Bernoulli to return home.After three failed applications to Basel, he ob-tained the chair of anatomy and botany in 1732.

The Russian period was quite fruitful forBernoulli. During this time he accomplished im-portant work in hydrodynamics, the theory ofoscillations, and probability. His return to Baselevolved into a tour of Europe, where he was cor-dially received by numerous scholars. At thistime his father competed with Bernoulli over thepriority of the work on hydrodynamics calledHydrodynamica (Hydrodynamics); completed in1734 and published in 1738, his father’s ownHydraulica (Hydraulics) was predated to 1732.

In the field of medicine, in which he wasforced to work for some periods of his life,Bernoulli turned his intellect toward mechani-cal aspects of physiology. His 1721 dissertationwas a review of the mechanics of breathing, anda 1728 paper addressed the mechanics of musclecontraction, dispensing with the notion of fer-mentation in the blood corpuscles. Bernoullialso determined the shape and location of theentrance of the optic nerve into the bulbus, andlectured on the calculation of work done by theheart; he later established the maximum amount

Daniel Bernoulli, known for his outstandingcontributions to hydrodynamics and the theory ofoscillations (Courtesy of the National Library ofMedicine)

28 Bernoulli, Daniel

of work (activity over a sustained period) that ahuman could perform in a day.

However, Bernoulli’s interests were absorbedby mathematical problems motivated by scien-tific questions. His four-volume Exercitationesmathematicae treats a variety of topics: the gameof faro, the outflow of water, differential equa-tions, and the lunulae (figures bounded by twocircular arcs). He later investigated divergent se-ries, such as the infinitely continued alternatingsum and subtraction of the number one, whichEuler and GOTTFRIED WILHELM VON LEIBNIZ

thought summed to one-half. Bernoulli obtainedsums for trigonometric series and investigatedthe theory of infinite continued fractions.

His contribution to mechanics lay in the ar-eas of oscillations of rigid bodies and the me-chanics of flexible and elastic bodies; these newareas were thoroughly addressed by the collabo-rative efforts of Bernoulli and Euler. Bernoulliexplains the principle of gravity and magnetism,dispensing with the vortex theory of RENÉ

DESCARTES and CHRISTIAAN HUYGENS. The the-ory of rotating bodies, the center of instanta-neous rotation, and the conservation of liveforce are some of his other contributions, as wellas the friction of solid bodies. He obtained widefame from Hydrodynamica, which gives a historyof hydraulics, formulas for the outflow of a fluid,oscillations of water in a tube, theory for hy-draulic machinery (such as pumps, including thescrew of ARCHIMEDES OF SYRACUSE), motions of“elastic fluids” (gases), and the derivation of theBernoulli equation for stationary currents. Thisbook also contains the determination of pres-sure on a container caused by a fluid, and thepressure of a water jet on an inclined plane—put into practice to propel boats many yearslater.

Together with Euler, Bernoulli dominated themechanics of elastic bodies, deriving equilibriumcurves for such bodies in 1728. He determinedthe curvature of a horizontal elastic band fixedat one end, and defined the “simple modes” and

frequencies of oscillation of a system with morethan one body. After leaving St. Petersburg,Bernoulli’s ongoing correspondence with Eulerresulted in more literature: the small vibrationsof both a plate immersed in water and a rod sus-pended from a flexible thread. Here he stressedthe difference between simple and composite vi-brations. In papers written between 1741 and1743, Bernoulli treats the transversal vibrationsof elastic strings, considering a horizontal rod af-fixed to a vertical wall. To derive the equationfor vibration, he implemented the relation be-tween curvature and moment. His 1753 treatiseon oscillations resulted in a description of thegeneral motion as the superposition of numer-ous single vibrations, given by an infinitetrigonometric series. Later Bernoulli consideredthe oscillations of organ pipes and the vibrationsof strings of uneven thickness.

Bernoulli also advanced the theory of prob-ability and statistics; his most novel work in thisarea was De mensura sortis (Concerning themeasure of chance), which addresses a problemin capital gains, and introduces the concept ofa utility function—described by Bernoulli as themoral value of a quantity of capital. In 1760 heexamined a problem of mortality in medical sta-tistics, giving a differential equation relating therelevant variables. He later used an urn modelin applications to population statistics, attempt-ing to determine the average duration of mar-riage for each age group. It is interesting thatBernoulli uses the infinitesimal calculus in prob-ability, taking an early step toward the notion ofa continuous random variable and the statisticaltheory of errors. The latter subject he viewed aspart of probability, using a semicircle as an ap-proximation to the distribution of errors; this issimilar to the modern theory, which uses CarlFriedrich Gauss’s probability curve.

In 1743 Bernoulli switched to lecturing inphysiology, and in 1750 he finally obtained thechair of physics; he continued lecturing until1776, displaying fascinating physics experiments

Bernoulli, Jakob 29

that attracted a large audience at Basel. For ex-ample, he was able to conjecture Coulomb’s lawof electrostatics as a result of experimental evi-dence from his lectures. He died on March 17,1782, having received numerous prizes and hon-ors in his lifetime, for example winning the GrandPrize of the Paris Academy in 1734 and 1737. Infact, Bernoulli won 10 prizes for essays entered inthe competitions of the Paris Academy, whichwere usually given on topics of public interest,such as the best form of an anchor and the rela-tionship between tides and lunar attraction. Hewon two prizes on the topic of magnetism and im-proved the construction of the compass.

Bernoulli was an outstanding scientist andmathematician. His principal mathematicalcontributions lay in differential equations, me-chanics, and probability. Bernoulli’s efforts,along with the work of Euler, would influencesubsequent mathematicians of the 19th century.

Further ReadingCannon, J., and S. Dostrovsky. The Evolution of

Dynamics: Vibration Theory from 1687 to 1742.New York: Springer-Verlag, 1981.

Gower, B. “Planets and Probability: Daniel Bernoullion the Inclinations of the Planetary Orbits,”Studies in History and Philosophy of Science 18, no.4 (1987): 441–454.

� Bernoulli, Jakob (Jacques Bernoulli)(1654–1705)SwissDifferential Equations, Probability

The Bernoulli family produced many mathe-maticians who contributed to diverse branchesof mathematics such as probability, calculus, andnumber theory, and Jakob Bernoulli was the firstmember of that impressive congregation. His ge-nius lay in the clever solution of certain highlyspecific problems, many of which possessed a rel-evancy to the external world.

Originally from Amsterdam, the Bernoulliswere a thriving family of drug merchants whohad immigrated to Basel. Jakob Bernoulli wasborn on December 27, 1654, in Basel, to NikolausBernoulli, a city magistrate, and MargarethaSchönauer, a banker’s daughter. Jakob Bernoulliwas intended for a mercantile career as well, buthis proclivities for scientific investigation wouldmark his destiny for another path. After attain-ing the master of arts degree in philosophy in1671, he went on to receive a licentiate in the-ology five years later. However, it seems thatBernoulli had little interest or predilection to-ward evangelical ministry; he has been describedas self-willed, stubborn, and aggressive, with aninferiority complex. During this time he studiedmathematics and astronomy, although his fatherattempted to dissuade him from it. In 1676 hecame to Geneva as a tutor, and there started hisscientific diary called Meditationes; next he jour-neyed to France, where he spent two years learn-ing the methodologies of Cartesian scientificphilosophy. A second educational journey to theNetherlands and England in 1681 put him incontact with contemporary mathematicians. Asa result, Bernoulli soon formulated a theory ofcomets (1682) and gravity (1683). Returning toBasel, Jakob began to lecture on the mechanicsof solid and liquid bodies; he sent reports ofhis investigations to scientific journals, andmeanwhile worked through RENÉ DESCARTES’sGéométrie (Geometry). His contributions ingeometry and algebra (he showed how a trian-gle could be divided into four equal parts by twostraight perpendicular lines) were placed in anappendix to the fourth edition of the Géométrie.

Bernoulli next presented four studies in for-mal logic, published in the media of a disputa-tion, from 1684 to 1686, and his first work inprobability appeared in 1685. He was also fa-miliar with the writings of JOHN WALLIS andISAAC BARROW on infinitesimals in optical andmechanical problems, and in this way was in-troduced to calculus.

30 Bernoulli, Jakob

In 1684 Bernoulli married Judith Stupanus,the daughter of a wealthy pharmacist. One of hisyounger brothers, JOHANN BERNOULLI, started to at-tend the University of Basel; as a respondent toJakob Bernoulli’s logical debates, JohannBernoulli earned his master of arts degree in1685. Formally he studied medicine, but in se-cret pursued mathematics under the tutelage ofJakob Bernoulli. The relationship between thetwo brothers would prove fraught, as their sim-ilar personalities led to implacable friction andrivalry.

In 1687 Bernoulli was appointed professor ofmathematics at the University of Basel, and atthis time he studied and mastered the differen-tial calculus of GOTTFRIED WILHELM VON LEIBNIZ;as a result, in 1689 Bernoulli produced a theoryof infinite series, established the law of largenumbers from probability theory, and brought at-tention to the importance of complete induction.The analysis of CHRISTIAAN HUYGENS’s solutionof the problem of the curve of constant descentin a gravitational field furnishes an excellent ex-ample of Bernoulli’s mastery of Leibnizian calcu-lus—it was in this context that the term integralfirst appeared. He later investigated elasticitythrough a simple differential equation (1694),and also researched the parabolic and logarith-mic spirals (1691). His procedure of evolutes fordetermining the focal line of incident parallelrays of light on a semicircular mirror consists ofgenerating an algebraic curve through the en-velope of its circles of curvature. This later ledto a differential equation that described the formof a sail that was inflated by the wind (1692,1695). Bernoulli worked carefully on a widerange of ancient as well as modern problems, in-cluding the so-called Bernoulli differential equa-tion, using the tools of differential calculus withexpert facility.

Jakob Bernoulli and Johann Bernoulli’s af-fections became increasingly sundered, mainlydue to their mutual conflict of personality.Though inferior to his younger brother in terms

of intuition and speed of thought, JakobBernoulli’s mind could more deeply penetrate asubject. A famous 1696 problem proposed byJohann Bernoulli, called the brachistochrone,was concerned with the determination of acurve of quickest descent between two points.Jakob Bernoulli solved this in 1697, and alsocorrected Johann Bernoulli’s solution of theisoperimetric problem in 1701, which the latterrefused to recognize until long after JakobBernoulli’s death. Their mutual antipathy soonled to criticism of each other’s work, and theycontinued the debate in print from 1699 to1700.

Bernoulli’s main achievements lie in hisclever analysis of particular problems of mathe-matical, classical, and mechanical interest. Hedeveloped a theory of natural phenomena basedon the collision of ether particles, discussed thecenter point of oscillation, and discovered prop-erties of the resistance of elastic bodies. The cen-ter of gravity of two bodies in uniform motion,the shape of a stretched cord, centrally accel-erated motion, and the collective impulse ofmany shocks are some of the mechanical prob-lems that he considered. In engineering, hetreated the drawbridge problem in 1695, whichwas to determine the curve of a sliding weighthanging on a cable that holds the drawbridgein balance.

In the Theory of Series (published in five dis-sertations from 1682 to 1704), he develops se-ries for pi and the logarithm of 2, investigatescompound interest, exponential series, and theharmonic series. The Ars conjectandi (Art of con-jecturing), published posthumously in 1713, isBernoulli’s most original work: the theory ofcombinations, exponential series, Bernoulliannumbers, expected profit from various games ofchance, probability as measure of confidence, apriori and a posteriori probability, and the lawof large numbers are among the outstandingelements of this work. He died in Basel, onDecember 27, 1705, from tuberculosis.

Bernoulli, Johann 31

Perhaps his contribution to probability is hismost significant legacy, as this field has beenextensively developed from his early efforts.Certainly, he advanced algebra, infinitesimalcalculus, the calculus of variations, mechanics,and infinite series as well. Bernoulli was widelyread by later generations of mathematicians, andis recognized today for his contributions to cal-culus and probability.

Further ReadingBoswell, T. “The Brothers James and John Bernoulli

on the Parallelism between Logic and Algebra,”History and Philosophy of Logic 11, no. 2 (1990):173–184.

Hacking, I. “Jacques Bernoulli’s ‘Art of Conjecturing,’”British Journal for the Philosophy of Science 22(1971): 209–229.

Stigler, S. The History of Statistics: The Measurementof Uncertainty before 1900. Cambridge, Mass.:Belknap Press of Harvard University Press, 1986.

� Bernoulli, Johann (Jean Bernoulli)(1667–1748)SwissDifferential Equations

The second of the famous Bernoulli brothers,Johann Bernoulli was part of a remarkable fam-ily of mathematicians. It was his fate to spendhis early career under the shadow of his ac-complished brother JAKOB BERNOULLI, but heeventually became renowned for his own ge-nius. A leading proponent of Leibnizian differ-ential calculus in later life, Bernoulli was at onepoint the most eminent mathematician ofEurope.

Johann Bernoulli was born on August 6,1667, in Basel, the 10th child of a wealthy mer-cantile family. The Bernoullis were originallyfrom Holland, but Johann Bernoulli’s father,Nikolaus Bernoulli, had settled in Switzerland asa druggist and married the affluent Margaretha

Schönauer. Originally, Johann Bernoulli was in-tended for a career in business, but after a failedapprenticeship as a salesman, he was permittedin 1683 to enroll at the university. His olderbrother Jakob Bernoulli was lecturing there onexperimental physics, and Johann Bernoullibenefited from his elder’s tutelage in mathemat-ics. Responding to one of Jakob Bernoulli’s 1685logical disputations, Johann Bernoulli was ele-vated to magister artium and commenced thestudy of medicine. His first publication of fer-mentation processes appeared in 1690, and heearned his doctorate in 1694 with a mathemat-ical dissertation in the field of medicine.

Meanwhile, Johann Bernoulli was avidlypursuing the study of mathematics (without hisfather’s approval), and together with JakobBernoulli mastered GOTTFRIED WILHELM VON

LEIBNIZ’s differential calculus. Johann Bernoulli’ssolution of the catenaria problem, posed by JakobBernoulli in 1691, demonstrated his talent and

Johann Bernoulli contributed to the development ofcalculus and used the method of integration as aninverse operation of differentiation to solve differentialequations. (Courtesy of the National Library ofMedicine)

32 Bernoulli, Johann

marked him as a leading mathematician ofEurope. At that time he was in Geneva, but soonhe moved to Paris, where he won recognitionthanks to his “golden theorem”: the determina-tion of a formula for the radius of curvature ofan arbitrary curve. Bernoulli met Guillaume deL’Hôpital, and was employed by the latter to tu-tor him in infinitesimal calculus, for whichBernoulli was handsomely rewarded. WhenBernoulli later returned to Basel, their corre-spondence continued, and became the source fora first calculus textbook titled Analyse des infin-iment petits (Analysis of the infinitely small).Bernoulli was a faithful and eager communica-tor, writing 2,500 letters with 110 scholars overthe course of his life; among these persons wasLeibniz, with whom Bernoulli exchanged his sci-entific views starting in 1693.

During this period, a hiatus from his med-ical studies, Bernoulli obtained several mathe-matical results, which were published as shortpapers. Of principal significance is his work onexponential functions and the series develop-ment of such by integration. Integration wasviewed as the inverse operation to differentia-tion, and thus could be used to solve differentialequations. This idea was borne out by his solu-tion of several outstanding problems, includingJakob Bernoulli’s “Bernoulli equation”––JohannBernoulli’s piercing intuition allowed an ele-gance of solution that Jakob Bernoulli’s more bru-tal techniques could not attain, which illustratedthe contrast between the two brothers’ intel-lects. Johann Bernoulli’s formulation of expo-nential calculus, which is simply the applicationof Leibniz’s differential calculus to exponentialfunctions, further extended the applicability ofinfinitesimal methods. In 1695 he summed theinfinite harmonic series, developed additiontheorems for trigonometric and hyperbolicfunctions, and described the geometric genera-tion of pairs of curves. The summation of re-ciprocal squares remained impervious to bothBernoullis’ efforts, and was later to be computed

by LEONHARD EULER, Johann Bernoulli’s ableststudent.

Having completed his degree in medicine,Bernoulli accepted the chair of mathematics atthe University of Groningen. He had alreadymarried Dorothea Falkner when he departed forHolland, and was brimming with resentment to-ward Jakob Bernoulli. The relationship with hisbrother had already begun to disintegrate: bothmen had quarrelsome, pugnacious personalities,and Johann Bernoulli was an avid debater andpolemicist. However, Johann Bernoulli’s feisti-ness extended beyond his brother; in 1702 hewas involved in theological quarrels withGroningen professors, and was labeled a followerof Spinoza.

In June 1696 Bernoulli posed the followingproblem, known as the brachistochrone: to de-termine the path of quickest descent betweentwo fixed locations. Dedicating the problem “tothe shrewdest mathematicians of all the world,”Bernoulli gave a half-year time limit to find thesolution; Leibniz, who solved the problem im-mediately, accurately predicted that only fivepersons in the world were capable of success—SIR ISAAC NEWTON, Leibniz himself, theBernoulli brothers, and L’Hôpital. The brachis-tochrone provides another contrast of the broth-ers’ abilities: Jakob Bernoulli’s cumbersomeanalysis laid the foundations for the calculus ofvariations, whereas Johann Bernoulli’s approachingeniously reduced the problem to a questionin optics, and he deduced the correct differen-tial equation from the law of refraction. JakobBernoulli subsequently posed the isoperimetricproblem, whose solution required the new cal-culus of variations, which had been characteris-tically underestimated by Johann Bernoulli. Hispublished solution was therefore inadequate, re-sulting in Jakob Bernoulli’s unbridled disparage-ment in published critique. It was not until manyyears after Jakob Bernoulli’s death that JohannBernoulli would admit the supremacy of the cal-culus of variations—so deep ran the enmity

Bessel, Friedrich Wilhelm 33

aroused by wounded ego and controversy. In1718 Johann Bernoulli produced an elegant so-lution of the isoperimetric problem utilizingJakob Bernoulli’s methodology, and this workcontained the early notions for the modern cal-culus of variations.

Johann Bernoulli’s work on the cycloid, inhis description the “fateful curve of the 17thcentury,” promulgated his development of inte-gration of rational functions via the method ofpartial fractions. A formal algebraic approach tosuch calculations was typical of Johann Bernoulli,and his influence in the common techniquesof calculus has been felt through moderntimes.

After Jakob Bernoulli’s death in 1705,Johann Bernoulli succeeded him at the chair ofmathematics in Basel, apparently a decision mo-tivated by his family. He soon became involvedin the politically fraught priority dispute be-tween Newton and Leibniz, and he openly crit-icized Taylor’s support of the method of fluxions(the Newtonian calculus). In later debates andcompetitions, Bernoulli was able to successfullyanalyze certain problems, such as the trajectoryof the ballistic curve in the general case, towhich the Newtonian calculus was insufficient.After Newton’s death in 1727, Bernoulli wouldbe recognized as the leading mathematician ofEurope. At Basel he researched theoretical andapplied mechanics, and in 1714 he published hisonly book, Théorie de la manoeuvre des vaisseaux(Theory of the maneuver of vessels). In this workhe criticizes French navigational theories anddeveloped the principle of virtual velocities,with applications to conservative mechanicalsystems. In other papers he investigated thetransmission of momentum, the motion of plan-ets, and the phenomenon of the luminousbarometer.

Bernoulli was greatly honored during hislifetime, being granted membership of the acad-emies of Paris, Berlin, London, St. Petersburg,and Bologna. He benefited from a high social

status in Basel, due to his marital connectionsand family wealth, and held various civic officesthere. He died on January 1, 1748, in Basel. Hisgenius in solving particular mathematical prob-lems made him one of the top mathematiciansof his time. In terms of legacy, he was not as suc-cessful as his brother Jakob Bernoulli, but nev-ertheless left influential work on mechanics anddifferential equations behind.

Further ReadingBoswell, T. “The Brothers James and John Bernoulli

on the Parallelism between Logic and Algebra,”History and Philosophy of Logic 11, no. 2 (1990):173–184.

� Bessel, Friedrich Wilhelm(1784–1846)GermanAnalysis

The field of astronomy had developed quicklyby the 19th century, and mathematics retainedits vital importance to this sister science.Friedrich Bessel not only became one of thegreatest astronomers, accurately calculating var-ious astronomical distances and labeled as thefounder of the German school of practical as-tronomy, but also developed outstanding math-ematical theories to explain the perturbations ofplanetary orbits.

On July 22, 1784, Friedrich Bessel was bornin Minden, Germany. His father was a civil ser-vant of that town, and his mother was a minis-ter’s daughter. Bessel had a large family, consist-ing of six sisters and two brothers. Besselattended the Gymnasium (German high school)in Minden, but after four years he departed tobecome a merchant’s apprentice. While inschool, he had an inclination toward mathe-matics and physics, but did not exhibit any note-worthy degree of ability until he was 15 years ofage. In 1799 he commenced his apprenticeship

34 Bessel, Friedrich Wilhelm

with Kulenkamp, a famous mercantilist firm; hequickly demonstrated his facility with calcula-tions and accounting, and as a result was pro-vided a meager salary, emancipating him fromdependence on his parents.

Meanwhile, Bessel spent evenings studyingvarious subjects in preparation for his future ca-reer as a cargo officer. He soon mastered geog-raphy, Spanish, and English, as well as the art ofnavigation—this discipline first aroused his fas-cination for astronomy. Not content simply toknow the technology of his trade, Bessel setabout researching the deeper aspects of astron-omy and mathematics, considering this founda-tional knowledge to be essential. Among his firstachievements in the field of astronomy was thedetermination of the longitude of Bremen, us-ing a sextant that he had built. He also beganto peruse the astronomical literature, and in this

manner discovered the astronomer ThomasHarriot’s 1607 observations of Halley’s comet.After completing the reduction of Harriot’s ob-servations (a process that involves compensatingfor the refraction of light caused by the Earth’satmosphere and generally freeing the observa-tions of errors), he presented the astronomerHeinrich Olbers with his own calculation of theorbit in 1804. The result was in close agreementwith Halley’s work, and Olbers encouragedBessel to supplement these reductions with someadditional observations; the fruit of this laborwas an article printed in the MonatlicheCorrespondenz (Monthly correspondence). Indepth of material worthy of a doctoral disserta-tion, this paper attracted the notice of manyreaders and marked a transition in Bessel’s life.

In early 1806, before the termination of hisapprenticeship, Bessel became an assistant at aprivate observatory near Bremen, which wasowned by a wealthy civil servant with an inter-est in astronomy who had contacts with manyscientists. At the observatory Bessel acquired athorough schooling in the observation of plan-ets and comets, and meanwhile made furthercontributions toward the calculation ofcometary orbits. In 1807 he commenced the re-duction of James Bradley’s observations on 3,222stars, which marked one of Bessel’s greatestachievements. Friedrich Wilhelm III of Prussiaconstructed a new observatory in Königsberg,and Bessel was appointed as its director and asprofessor of astronomy in 1809. Since he did nothave a doctorate, the University of Göttingengave him one at the suggestion of CARL

FRIEDRICH GAUSS, who had earlier met Bessel in1807.

During construction of the observatory,Bessel continued his work in the reduction ofBradley’s data; for his resulting tables of refraction,he was awarded the Lalande Prize in 1811 by theInstitut de France. In 1813 he began his observa-tions in the completed observatory, and remainedin Königsberg as a teacher and researcher for the

Friedrich Bessel, a great astronomer, invented Besselfunctions to study the perturbations of planetary orbits.(Courtesy of the Library of Congress)

Betti, Enrico 35

rest of his life. In 1812 he married JohannaHagen, by whom he had two sons and threedaughters. This felicitous marriage was shad-owed by illness and his sons’ early deaths, andBessel found distraction in walking and hunting.

Bessel accomplished much in the field of as-tronomy. The reduction of Bradley’s data al-lowed a proper determination of the stars’ posi-tions and motions, but Bessel’s own program ofobservation and immediate reduction resultedin highly accurate data. He also gave the firstaccurate estimate of the distance of a fixed star,using triangulation techniques and a heliometer.He was also involved in geodesy, the measure-ment of the Earth, completing an 1830 trian-gulation of East Prussia with a new measuringapparatus and Gauss’s method of least squares.Bessel’s resulting estimate of the parameters ofthe Earth’s dimensions earned him internationalfame.

Bessel was interested in mathematicsthrough its close connection to astronomy. Theproblem of perturbation in astronomy wasamenable to analysis using certain special con-fluent hypergeometric functions, later calledBessel functions. There were two effects of anintruding planet on the elliptical orbit of a givenplanet: the direct effect of the gravitational per-turbation and the indirect effect arising from themotion of the sun caused by the perturbingplanet. Bessel separated the two influences, andBessel functions appear as coefficients in the se-ries expansion of the indirect effect. In his studyof the problem, Bessel made an intensive studyof these special functions, which are describedin his Berlin treatise of 1824. Special cases ofthese functions had been known for more thana century, having been discovered by JOHANN

BERNOULLI and GOTTFRIED WILHELM VON LEIBNIZ;DANIEL BERNOULLI (1732) and LEONHARD EULER

(1744) had also investigated Bessel coefficients.But Bessel’s motivation arose from their appli-cation to astronomy, not as a detached study inpure mathematics.

His health was in decline from 1840, andhis last major journey to England was in 1842;as a result of his participation in the Congressof the British Association in Manchester, Besselwas encouraged to complete and publish someremaining research. After two agonizing yearsbattling cancer, he died on March 17, 1846, inKönigsberg.

Although Bessel is principally known as anastronomer, like Gauss he made outstandingcontributions to pure mathematics that could beapplied to astronomy. His name is attached tothe special functions mentioned above, as wellas to an inequality that is today used in Fourieranalysis and the theory of Hilbert spaces. Boththe Bessel functions and the Bessel inequalityhave enduring relevance for modern mathe-maticians.

Further ReadingClerke, A. A Popular History of Astronomy during the

Nineteenth Century. Decorah, Iowa: Sattre Press,2003.

Fricke, W. “Friedrich Wilhelm Bessel (1784–1846),”Astrophysics and Space Science: An InternationalJournal of Cosmic Physics 110, no. 1 (1985):11–19.

Hamel, J. Friedrich Wilhelm Bessel. Leipzig: BSB B.G.Teubner, 1984.

Klein, F. Development of Mathematics in the 19thCentury. Brookline, Mass.: Math Sci Press, 1979.

Lawrynowicz, K. Friedrich Wilhelm Bessel, 1784–1846.Basel: Birkhäuser Verlag, 1995.

Williams, H. The Great Astronomers. New York:Simon and Schuster, 1930.

� Betti, Enrico(1823–1892)ItalianTopology, Algebra

Enrico Betti is known for his contributions toGalois theory (an abstract algebraic theory used

36 Betti, Enrico

to solve algebraic equations, developed byEVARISTE GALOIS) and the theory of elliptic func-tions. His work on the analysis of hyperspace wasto later inspire JULES-HENRI POINCARÉ in thefoundation of algebraic geometry.

Betti was born on October 21, 1823, inPistoia, Italy, and his father died when he wasquite young. As a result, his mother oversaw hiseducation, and he later matriculated at theUniversity of Pisa, receiving a degree in thephysical and mathematical sciences. Afterwardhe became involved in the war for Italian inde-pendence, participating as a soldier in the bat-tles of Curtatone and Montanara. His subsequentprofession was as a high school mathematicsteacher in Pistoia, though he continued his ownresearches into pure mathematics concurrently.

Much of Betti’s work was in the field of al-gebra. The work of Evariste Galois, which re-ceived little recognition during its author’s brieflife, was largely summarized in a personal letterof 1832 that was later published by JOSEPH LI-OUVILLE in 1846. Since that time, Betti furtheredGalois’s work on the solubility of algebraic equa-tions by radicorational operations (the issue ofdetermining which equations could have theirsolutions expressed in terms of radicals and ra-tional numbers). By connecting Galois’s workwith the prior researches of NIELS HENRIK ABEL

and Paolo Ruffini, Betti bridged the gap betweenthe new methods of abstract algebra and theclassical problems (such as the quintic) consid-ered previously. Many at the time viewed thelabors of Galois as irrelevant and sterile, butBetti’s elaborations in two papers of 1852 and1855 constitute an important step toward re-versing those adverse opinions; today, Galoistheory is seen as a fruitful and lovely componentof abstract algebra.

He also investigated the theory of ellipticfunctions, a popular topic in the 19th century;Betti described this branch of mathematics by re-lating it to the construction of certain transcen-dental functions in 1861, and KARL WEIERSTRASS

developed these ideas further in the ensuingyears. Taking another, nonalgebraic look at thesame subject, Betti investigated elliptic func-tions from the perspective of mathematicalphysics. Under the guidance of BERNHARD

GEORG FRIEDRICH RIEMANN, whom Betti had metin Göttingen in 1858, Betti researched the pro-cedures used in electricity and mathematicalanalysis.

In 1865 Betti accepted a professorship at theUniversity of Pisa, which he retained for the re-mainder of his life. Later he became rector of theuniversity and director of the teachers’ college inPisa. From 1862 he was a member of the Italianparliament, briefly served as undersecretary ofstate for public education in 1874, and becamea senator in 1884. However, his principal inter-ests were not in politics or administration, but inpure mathematical research; Betti desired only tohave solitude for intellectual reflection, and an-imated intercourse with his close friends.

Betti’s work in the area of theoretical physicsled to a law of reciprocity in elasticity theory,known as Betti’s theorem (1878). He first mas-tered GEORGE GREEN’s methods for the integra-tion of PIERRE-SIMON LAPLACE’s equations in thetheory of potentials, and utilized this methodol-ogy in the study of elasticity and heat. He alsoanalyzed hyperspace in 1871; Poincaré wouldlater draw inspiration from Betti to expand thesepreliminary investigations. The term Betti num-bers, coined by Poincaré, would be commonlyused as measurable characteristics of an algebraicvariety (a high-dimensional surface that can beexpressed as the locus of points satisfying an al-gebraic equation).

Betti was an excellent teacher, bringing hispassion and extensive knowledge to the class-room, and he was an ardent proponent of a re-turn to classical education. He regarded EU-CLID OF ALEXANDRIA’s Elements as a model textfor instruction, and strongly advocated its returnto the secondary schools. He influenced severalgenerations of students at Pisa, guiding many

Bhaskara II 37

toward the pursuit of scientific knowledge. Hedied on August 11, 1892, in Pisa.

Betti’s impact on mathematics is still felt to-day. His early research into algebraic topologywas fundamental, as the enduring importance ofthe Betti numbers to questions of classificationtestifies. Perhaps even more important was hisdevelopment of Galois theory, which has be-come a huge component of modern studies inabstract algebra.

Further ReadingWeil, A. “Riemann, Betti and the Birth of Topology,”

Archive for History of Exact Sciences 20 (1979):91–96.

� Bhaskara II(1114–1185)IndianTrigonometry, Geometry, Algebra

Indian mathematicians contributed to the fur-ther development of the digital number system,and also supplemented the wealth of geometri-cal and arithmetical information then available.During the Dark Ages of Europe, mathematicswas slowly progressing in the Middle East andIndia, and Bhaskara II was one of the best-known mathematicians of his time.

Bhaskara II is distinguished from BhaskaraI, an Indian from the seventh century wellknown for his exposition of ARYABHATA I’s as-tronomy. Bhaskara II was recognized for his workin astronomy, but also for his efforts in puremathematics. He was born in 1114 in India, butlittle is known of his life. Apparently, he camefrom a family of Brahmans known as theSandilya gotra, and was born in the city ofVijayapura. Among his contemporaries, Bhaskarawas famous for his scientific talents, as he hadnot only mastered the previous knowledge ofBRAHMAGUPTA and others, but also expanded itthrough his own contributions.

Bhaskara was appointed head of the astro-nomical observatory at Ujjain, the foremost cen-ter of mathematical knowledge in India at thattime. Due to this eminent position, Bhaskararepresented the acme of mathematical knowl-edge in the world, since little of importance wastranspiring in Europe in the 12th century.Bhaskara possessed a deep understanding ofnumber systems and the solution of equations;as a successor to Brahmagupta, he grasped theconcepts of zero and negative numbers. He stud-ied numerous Diophantine problems—equationsin one or more variables with integer coeffi-cients—and often obtained solutions that wereextremely large (this would have been impossi-ble to achieve without an excellent number sys-tem facilitating such calculations).

There are at least six writings that can def-initely be attributed to Bhaskara. The Lilavati(The beautiful), which is addressed to a womanof that name (perhaps his daughter or his wife),contains 13 chapters on mathematics, includingsuch topics as arithmetic, plane geometry, solidgeometry, algebra (called “the pulverizer”), theshadow of a gnomon, and combinations of dig-its. (A gnomon is a geometric shape that hadfascinated the Greeks; it is the L shape remain-ing when one rectangle is removed from a largerone.) Bhaskara is careful to define his terms pre-cisely, and discusses arithmetical and geometri-cal progressions of numbers. His discussion of thecombination of digits, essentially a contributionto modern arithmetic, was perhaps of greatestimportance. This was easily Bhaskara’s most pop-ular work, with almost three dozen commen-taries and numerous translations.

Bhaskara easily manipulates the arithmetic ofnegative numbers, and knows how to multiply byzero. In addition, he avoided Brahmagupta’s mis-take of attempting division by zero, realizing thedifficulty inherent in this operation; in theBijaganita (Root extraction), Bhaskara writes thatany number divided by zero is infinity, which iscloser to the truth. His method of multiplication

38 Birkhoff, George David

for large numbers is somewhat different from themodern technique, but amply effective. Bhaskaraalso demonstrates particular rules for squaringnumbers, even though this is a special case ofmultiplication. He treats inverse proportion bydiscussing the rule of three, the rule of five, therule of seven, and the rule of nine.

The Bijaganita treats algebra: positive andnegative numbers (negative numbers were later“invented” by Fibonacci in Europe), zero, vari-ous types of equations (including the quadratic),and the multiplication of several unknowns.Again, there are several commentaries andtranslations.

Bhaskara’s Siddhantasiromani, written in1150, consists of two parts. The first section,called the Ganitadhyaya, treats mathematical as-tronomy, addressing the mean and true longi-tude of planets, diurnal motion, syzygies, lunarand solar eclipses, planetary latitudes, heliacalrisings and settings of the planets, the lunar cres-cent, and planetary conjunctions. The secondportion, called the Goladhyaya, deals with thesphere and is largely an explication of the for-mer part: the nature of the sphere; cosmography;geography; planetary mean motion; eccentric-epicyclic model of planetary motion; construc-tion of an armillary sphere; spherical trigonom-etry; calculations of the eclipse, visibility of theplanets, and lunar crescent; astronomical in-struments; description of the seasons; and theperformance of astronomical computations. Healso treats the sine function, expressing more in-terest in this function for its own sake, develop-ing the well-known sum and product identities.The Siddhantasiromani also has more than adozen commentaries and many translations.

Next, there is the Vasanabhasya, which isBhaskara’s own commentary on the Siddhantasiro-mani. The Karanakutuhala (Calculation of as-tronomical wonders), written in 1183, givessimpler rules than the Siddhantasiromani for solv-ing problems in astronomy. It discusses the meanand true longitudes of planets, diurnal motion,

lunar and solar eclipses, heliacal risings and set-tings, the lunar crescent, planetary conjunc-tions, and syzygies. Finally, Bhaskara’s Vivaranahas not been studied. Also, the Bijopanaya, writ-ten in 1151, has been attributed by some toBhaskara, though this appears to be a later for-gery.

Bhaskara’s manifold achievements and su-perb talent placed him in a revered positionamong Indian intellectuals, and in 1207 an ed-ucational institution was endowed in order tostudy his writings. He certainly affected laterIndian mathematicians, who were heavily influ-enced by his work on astronomy and mathe-matics.

Further ReadingCalinger, R. Classics of Mathematics. Upper Saddle

River, N.J.: Prentice Hall, 1995.Datta, B. “The Two Bhaskaras,” Indian Historical

Quarterly 6 (1930): 727–736.Ifrah, G. A Universal History of Numbers: From Prehistory

to the Invention of the Computer. New York: JohnWiley, 2000.

Joseph, G. The Crest of the Peacock: Non-European Rootsof Mathematics. London: Penguin Books, 1990.

Patwardhan, K., S. Naimpally, and S. Singh. Lilavati ofBhaskaracarya. Delhi, India: Motilal BanarsidassPublications, 2001.

� Birkhoff, George David(1884–1944)AmericanAnalysis, Geometry

There were few great American mathematiciansuntil the 20th century; Europe had dominatedmathematics for centuries. Birkhoff representsan important step in the reversal of this pattern;his brilliant discoveries in differential equations,geometry, and dynamics led to his recognitionas one of the foremost mathematicians ofAmerica.

Birkhoff, George David 39

George Birkhoff was born in Overisel,Michigan, on March 21, 1884. His father was adoctor, and Birkhoff received his early educationat the Lewis Institute in Illinois. He spent a yearat the University of Chicago before transferringto Harvard, graduating in 1905. He returned tothe University of Chicago, completing his doc-toral thesis two years later.

After Chicago, Birkhoff worked as a lec-turer at the University of Wisconsin, duringwhich time he married Margaret ElizabethGrafius in 1908. He spent a few years atPrinceton before becoming a professor atHarvard, later becoming the dean of the Facultyof Arts and Sciences from 1935 to 1939. Dueto his professorship, he was able to devote most

of his energy to mathematical research and theadvising of graduate students.

Birkhoff’s thesis dealt with boundary-valueproblems from the theory of differential equa-tions, which he extended in later years. His earlyresearch treated linear differential equations, dif-ference equations, and the generalized Riemannproblem. This area of mathematics is relevant tomathematical physics, with applications toquantum mechanics. Birkhoff’s research pro-gram proved to be ambitious: to construct a sys-tem of differential equations given a particularset of “singular points” (points of discontinuityin the solution). This effort has now evolvedinto an extensive field of research; Birkhoff tookthe first steps.

His major interest in the field of analysis wasdynamical systems. Birkhoff attempted to extendthe work of JULES-HENRI POINCARÉ on celestialmechanics, and he proved one of the latter’s lastconjectures involving the fixed points of con-tinuous transformations of an annulus. Birkhoffintroduced the concepts of wandering, central,and transitive motions, and investigated thetopic of transitivity. The main corpus of moderndynamics emerged from Birkhoff’s ideas, includ-ing ergodic theory and topological dynamics. Hisminimax principle and theorem on fixed pointsof transformations provided motivation in theareas of analysis and topology.

Birkhoff also thought deeply about thefoundations of relativity and quantum mechan-ics, and he contributed some theoretical pa-pers to these subjects. Although controversialamong physicists, these works provide originalcritiques and a novel approach to relativity.Birkhoff also contributed to combinatorics,number theory, and functional analysis. His texton geometry has influenced American peda-gogical trends in the teaching of high schoolgeometry.

Birkhoff was highly regarded by his col-leagues, and was viewed as one of the eminentmathematicians of America at that time. He

George Birkhoff, an early great Americanmathematician who studied the mathematicalfoundations of relativity (Courtesy of the Library of Congress)

40 Bolyai, János

was most influenced by Maxime Bôcher ofHarvard and E. H. Moore of the University ofChicago, through whom he learned algebraand analysis. Birkhoff was president of theAmerican Mathematical Society in 1925; hehad many friends and collaborators in Europe,such as Jacques Hadamard, Tullio Levi-Civita,and Sir Edmund Whittaker. He died inCambridge, Massachusetts, on November 12,1944.

Birkhoff’s main contributions lay in dynam-ical systems, but he also stimulated interest intopology and difference equations. Much ofmodern mathematics can trace a connection tothe work of Birkhoff; he also represented the be-ginning of the trend away from European dom-ination of mathematics.

Further ReadingButler, L. “George David Birkhoff,” American National

Biography 2. New York: Oxford University Press,1999.

Etherington, I. “Obituary: George David Birkhoff,”Edinburgh Mathematical Notes 1947 36 (1947):22–23.

Morse, M. “George David Birkhoff and His Mathe-matical Work,” American Mathematical Society.Bulletin. New Series, 52 (1946): 357–391.

� Bolyai, János (Johann Bolyai)(1802–1860)HungarianGeometry

One of the outstanding problems of Greekgeometry was the proof of the fifth postulateof Euclid’s Elements (often referred to as theparallel postulate) from the other, more intu-itive axioms. It was equivalent to the state-ment that through any point separate from agiven line, one could construct a unique par-allel line; from this statement, one can deducethat the sum of the angles of any triangle is

equal to two right angles. Many attempts overthe centuries to establish this axiom rigorouslyhad failed, with the latest and most notable at-tempt by Farkas Bolyai. His son János Bolyaiwould eventually construct a new, consistentgeometry independent of the fifth axiom. Eventhough priority for this discovery is credited toCARL FRIEDRICH GAUSS, János Bolyai performedhis research in ignorance of this, and so is of-ten credited as a cofounder of non-Euclideangeometry.

János Bolyai was born on December 15,1802, in Kolozsvár, Hungary, to Farkas Bolyaiand Susanna von Árkos. The Bolyai family wasdescended from a long line of aristocrats, andFarkas Bolyai had farmed their estates before be-coming a professor of mathematics, physics, andchemistry at the Evangelical-Reformed Collegeat Marosvásárhely. He was also a close friend ofCarl Friedrich Gauss. János Bolyai showed greattalent in many areas, including mathematics andmusic, displaying proficiency in the violin at ayoung age. In 1815 he began study at his father’scollege, and in 1818 entered the imperial acad-emy at Vienna in preparation for a military ca-reer, contrary to Farkas’s desire for him to studyin Göttingen under Gauss.

Young Bolyai graduated in 1822, but mean-while his interest in geometry, especially the par-allel postulate, had been awakened by his father’sown lengthy obsession. Indeed, Farkas Bolyai hadspent many years attempting the fifth axiom’s de-duction, without success; his correspondencewith Gauss on this subject led to the latter’s owndiscovery of non-Euclidean geometry, whichhis embarrassed conservatism never disclosed.Farkas Bolyai even warned his son emphaticallyagainst engaging his intellect with that problemin 1820, wishing to spare him many moments ofanguish, confusion, and despair. However, theimpetuous youth continued to contemplate thequestion.

After several years of vain labor, Bolyaiturned in 1823 toward the construction of a

Bolyai, János 41

geometry that did not require the fifth postu-late—a geometry that in fact dispensed with thataxiom altogether. Meanwhile, he graduated fromthe academy and began his first tour in Romaniaas a sublieutenant. He later visited his father in1825, presenting his manuscript on his theory ofabsolute space—a space where through a givenpoint not on a line, many distinct lines throughthe point could be constructed parallel to thegiven line, in direct refutation of the parallelpostulate. Farkas Bolyai could not accept thisnew geometry, but he sent the manuscript toGauss. The latter responded in 1832, aston-ished that János Bolyai had independentlyreplicated his own work, and claiming his prior-ity by more than three decades. Gauss directedJános Bolyai to explore various questions, suchas the volume of the tetrahedron in absolutespace, but the young Hungarian was not en-couraged. The assertion of Gauss’s priority wasfirst met with apprehension, and then with re-sentment.

Meanwhile, Bolyai finished his military ca-reer in Lvov in 1832: he was often sick withfever, so the army gave him a pension and dis-missed him from service. Apparently, he hadearned a reputation as a dashing officer with apredilection for duels. He came home to livewith his father, and his manuscript was publishedas “Appendix Explaining the Absolutely TrueScience of Space” in Farkas’s Tentamen in 1832,a systematic treatment of geometry, algebra, andanalysis. However, this essay (as well as thebook) received no response from mathemati-cians, and his discouragement over the situationwith Gauss drove Bolyai into a reclusion bothsocial and mathematical.

The relationship between father and sonwas also strained, mainly due to irritation overthe unenthusiastic reception of their work. JánosBolyai withdrew to the small family estate atDomáld, and in 1834 married Rosalie vonOrbán, by whom he had three children. In 1837both Bolyais attempted to retrieve their mathe-

matical reputation through participation in acompetition of the Jablonow Society. The topictreated the rigorous geometric construction ofimaginary numbers, which was a subject of in-terest for many mathematicians, such as Gauss,SIR WILLIAM ROWAN HAMILTON, and AUGUSTIN-LOUIS CAUCHY. János Bolyai’s solution resem-bled Hamilton’s, but failed to gain the desiredrecognition, which only exacerbated his melan-cholic tendencies. He continued sporadic re-search in mathematics, of variable quality; hisbest results concerned absolute geometry, therelation of absolute trigonometry and sphericaltrigonometry, and the volume of the tetrahe-dron in absolute space. Some work by NIKOLAI

LOBACHEVSKY on the same type of geometryreached him in 1848, and acted as an impetusto further his efforts. In his latter efforts, Bolyaibecame more concerned with the consistencyof absolute space—whether logical contradic-tions might arise from his construction; thesewould not be resolved until later in the 19thcentury.

He continued work until 1856, the year thathis father died, and his marriage with Rosaliebroke up at the same time, increasing his isola-tion. Bolyai also worked on a theory of salva-tion, stressing the bond between individual anduniversal happiness. He died on January 27,1860, after a protracted illness.

Bolyai made one solitary contribution tomathematics that was so outstanding in its cre-ativity and importance as to merit him somefame, despite his status as a maverick. Alongwith Gauss and Lobachevsky, Bolyai is consid-ered a cofounder of non-Euclidean geometry.These unusual geometries, initially scorned asugly and useless, have found acceptance in the20th century due to their great relevance to thecurved space of our own universe.

Further ReadingNemeth, L. “The Two Bolyais,” The New Hungarian

Quarterly 1 (1960).

42 Bolzano, Bernhard

� Bolzano, Bernhard (BernardusPlacidus Johann Nepumuk Bolzano)(1781–1848)CzechoslovakianAnalysis, Logic, Topology

An outstanding problem of the early 19th cen-tury, later to result in the radical developmentsof GEORG CANTOR, was to determine the founda-tions of the real number system. Such propertiesas the infinite divisibility of the real numbers andthe density of rational numbers among irrationalshad not been grasped, and as a result the basictheory of functions, including such topics as con-tinuity and differentiability, were not well un-derstood. Bernhard Bolzano, an active advocateof rigorous foundations for science and mathe-matics, made significant contributions to theknowledge of analysis; his emphasis on the ne-cessity of a precise real-number system led to itsdevelopment at the hands of RICHARD DEDEKIND,and his other researches were precursors to anarithmetic of the infinite and modern logic aswell.

Bernardus Placidus Johann NepumukBolzano was the fourth child of the piousCaecilia Maurer and a civic-minded art dealernamed Bernhard Bolzano. He was born onOctober 5, 1781, in an ancient district of Prague,one of 12 children; his father was an Italian im-migrant with an interest in social work who laterestablished an orphanage. As a result of this en-vironment, the younger Bolzano was concernedwith ethics throughout his life, possessing anacute sensitivity to injustice.

In 1791 Bolzano entered the PiaristGymnasium. He studied philosophy at theUniversity of Prague in 1796. His interest inmathematics became stimulated through readingKästner, who took great care to prove proposi-tions that were commonly perceived to be evi-dent. After 1800 Bolzano turned from philosophyto theology, though he had continuing doubts

about the truth of Christianity. Instead, heturned toward moralism and away from super-natural religion, believing the supreme ethic tolie in the action that most benefited society.However, he reconciled this personal perspec-tive with his commitment to Catholicism.

The emperor of Austria had decided to es-tablish a chair in the philosophy of religion atevery university, as part of the Catholic restora-tion movement against the Enlightenment.Much freethinking had spread through Bohemia,and the emperor feared the consequences of suchradical ideas in view of the destruction wroughtby the French Revolution. Bolzano was ap-pointed to the chair at the University of Praguein 1805, despite his own Enlightenment sympa-thies. His lectures on religion were attendedwith enthusiasm, wherein he expounded his per-sonal views without reserve.

Bolzano was respected by his colleagues, andbecame dean of the philosophical faculty in1818. Meanwhile, Vienna brought a chargeagainst him in 1816, since his Enlightenmentviews had made him unpopular with the con-servative government; he was dismissed in 1819,forbidden to publish, and put under police su-pervision. Bolzano stubbornly refused to repentof his heresies, and the ordeal finally ceased in1825 through the intercession of the nationalistleader Dobrovsky.

Although Bolzano was mainly concernedwith social and religious issues, he was alreadyattracted to the methodological precision ofmathematics and logic. This led to some excel-lent contributions to mathematical analysis, al-though these accomplishments rarely met withany significant acknowledgment. Two unsolvedproblems—the proof of EUCLID OF ALEXANDRIA’sparallel postulate and the foundation of analysisthrough clarification of infinitesimals—claimedBolzano’s attention. His 1804 Betrachtungen übereinige Gegenstände der Elementargeometrie (Viewson some articles of elementary geometry) at-tempted to describe a theory of triangles and

Bolzano, Bernhard 43

parallels through a purely linear theory, whichwas never fully fleshed out. Ignorant of the workof NIKOLAI LOBACHEVSKY and JÁNOS BOLYAI onnon-Euclidean geometry, Bolzano developed amethodological critique of Euclid’s Elements inhis manuscript “Anti-Euklid.” For example, herequired a proof of the statement that any closedcurve divides the plane into two disjoint por-tions; this result later became known as theJordan curve theorem, proved by CAMILLE JOR-DAN. Partially through the objections and ques-tions raised by Bolzano, the field of mathematicsknown as topology came into existence in thelate 19th century.

His Rein analytischer Beweis (Pure analyticproof) of 1817 obtained important results rele-vant to the foundation of mathematical analysis,later completed in his 1832–35 Theorie der ReellenZahlen (Theory of real numbers). Many othermathematicians, such as JOSEPH-LOUIS LAGRANGE

and JEAN LE ROND D’ALEMBERT, had attempted toliberate mathematics from the notion of the in-finitesimal introduced by SIR ISAAC NEWTON andGOTTFRIED WILHELM VON LEIBNIZ in the 17th cen-tury, but Bolzano met with the first success inRein analytischer Beweis. Here he gives the defi-nition of a continuous function that is still inuse today, and obtains a result on the propertyof assuming intermediate values. He also intro-duces the notion of a greatest lower bound of aset of real numbers having a given property, aconcept that is a cornerstone in the theory ofreal numbers. Bolzano also discusses the “Cauchyconvergence criterion,” that a sequence of func-tions tends to some limit if the members of thesequence get closer to one another.

Although the proofs are incomplete, thiswas due to the present inadequacy of the con-cept of a real number. In his Functionenlehre(Functions model), a more complete theory offunctions is presented, including several resultslater rediscovered by KARL WEIERSTRASS in the lat-ter half of the 19th century. Bolzano showed thata continuous function over a closed interval must

attain an extremal value, now called theExtreme Value Theorem in calculus; the proofrequires the Bolzano-Weierstrass theorem aboutaccumulation points of bounded sequences. Hedistinguishes between continuity and the prop-erty of assuming intermediate values as strongerand weaker features, respectively. He developsthe connection between monotonicity and con-tinuity and gives the construction of the Bolzanofunction, which was continuous but nowhere dif-ferentiable, significantly predating Weierstrass’sown such example. The Functionenlehre con-tained many errors, including the false notionthat the limit of a sequence of continuous func-tions must necessarily be continuous, and thatterm-wise integration of an infinite series is al-ways possible.

His theory of quantities was completed inthe Theorie der Reelen Zahlen, but this manuscriptwas not published and therefore failed to exertan influence on the subsequent development ofanalysis. Bolzano describes such real numbers asbeing capable of arbitrarily precise approxima-tion by rational numbers. Also, his Paradoxiendes Unendlichen (Paradoxes of infinity) containmany intriguing fragments of set theory, and hetakes the subject to the boundary of cardinalarithmetic, the calculus of infinite sets. Bolzanoobserves that an infinite set can be put into one-to-one correspondence with a proper subset, andthat this actually characterizes infinite sets.However, he does not take the next step indefining cardinals of infinity; Dedekind (1882)would later use this property of infinite sets todefine infinity, and Cantor would develop aranking of infinities.

From 1820 Bolzano worked on the treatiseWissenschaftslehre (Scientific model) of 1837,which was a theory of science grounded in logic.Its four volumes dealt with the proof of the ex-istence of abstract truths, the theory of abstractideas, the condition of the human faculty of judg-ment, the rules of human thought in the questfor truth, and the rules for dividing the sciences.

44 Boole, George

Although this work went largely unnoticed atthe time, there is a close resemblance to mod-ern logic, especially in Bolzano’s notions of ab-stract proposition, idea, and derivability.

From 1823 Bolzano spent his summers at theestate of his friend Hoffmann in southernBohemia. He later lived there for more than adecade. In 1842 he returned to Prague, where hecontinued his mathematical and philosophicalstudies until his death on December 18, 1848.Bolzano was an important mathematician of the19th century, whose quest for truth led to ex-cellent work on the foundations of the real num-ber line. His name is found in many areas ofanalysis, such as the Bolzano-Weierstrass theo-rem and the Bolzano function; he is regarded asone of the founders of the modern theory of realanalysis.

Further ReadingBerg, J. Bolzano’s Logic. Stockholm: Almqvist and

Wiksell, 1962.George, R. “Bolzano’s Concept of Consequence,” The

Journal of Philosophy 83, no. 10 (1986): 558–564.

———. “Bolzano’s Consequence, Relevance, andEnthymemes,” Journal of Philosophical Logic 12,no. 3 (1983): 299–318.

van Rootselaar, B. “Bolzano’s Theory of RealNumbers,” Archive for History of Exact Sciences 2(1964/1965): 168–180.

� Boole, George(1815–1864)BritishLogic, Analysis

For much of history, the various fields of math-ematics developed separately, or at least wereseen as distinct areas of study. However, variousmathematicians attempted to present a mathe-matical description of the foundations of logic,and to construct a logical arithmetic that would

facilitate the resolution of abstruse philosophicalarguments through verifiable computation. Thefirst thinker to make significant progress towardthese goals was George Boole, an Englishmannoted for his contributions to logic as well as op-erator theory.

George Boole was born on November 2,1815, in Lincoln, England, to a cobbler namedJohn Boole. The latter’s real interest lay in math-ematics and the design of optical instruments,and his business consequently suffered from hisdistraction. George Boole was educated in therudiments of mathematics by his father, but dueto poverty was unable to pursue higher educa-tion. However, encouraged by his father, Booleadvanced in his understanding of mathematicsand soon acquired a familiarity with Latin,Greek, French, and German. Although his skill

George Boole constructed a viable, logical calculus inwhich philosophical arguments could be resolved.(Courtesy of the AIP Emilio Segrè Visual Archives)

Boole, George 45

with literature was exemplary, his primary in-terest was mathematics.

At the age of 15 he started teaching inLincoln. The Mechanics Institute was foundedin 1834; Royal Society publications circulatedthrough the school reading room, of which JohnBoole became curator, and George Boole de-voted his remaining spare moments to the pe-rusal of mathematical literature. In particular, hemade his way through SIR ISAAC NEWTON’sPrincipia with little assistance, and his local rep-utation led to a public speech marking the pres-entation of a Newton bust to the institute. By1840 he contributed regularly to the CambridgeMathematical Journal and the Royal Society; histalents were later recognized by award of a RoyalMedal in 1844 and election to fellowship of theRoyal Society in 1857.

Boole’s scientific writings are comprised ofsome 50 papers on various topics, two textbookssummarizing his research, and two volumes onmathematical logic. The texts, on differentialequations (1859) and finite differences (1860),were used for decades, and display Boole’s keenintellect and fluid use of operators. The materialon differential equations was original, using a dif-ference and forward-shift operator to solve lin-ear difference equations. Papers from 1841 and1843 treated linear transformations, displayingan invariance principle for quadratic forms; thetheory of invariants would be rapidly developedby other mathematicians in the latter half of the19th century. Other work addressed differentialequations, where Boole made much use of thedifferential operator D.

In 1849 Boole applied for the professorshipof mathematics at the newly created Queen’sCollege of Cork, and his appointment despitethe absence of a formal university degree boretestimony to his widely recognized mathematicalabilities. Though burdened with a heavy teach-ing load at Cork, Boole now inhabited an envi-ronment more conducive to research. He was adedicated teacher, believing in the importance of

education—perhaps in consideration of his ownlack. In 1855 he married Mary Everest, the nieceof a professor of Greek in Queen’s College.

After 1850 Boole mainly investigated thetheory of probability, as this was related to hisdeep and abiding interest in the foundations ofmathematical logic. His use of operators greatlyfurthered their power of applicability, but Boolewas cautious about their indiscriminate use, andwas always careful to verify the conditions oftheir implementation; he also stressed the ne-cessity of clear definitions. As a result of theseprecise inquiries, Boole came to realize that avariable representing a nonnumerical quantity,such as a logical statement or other mathemat-ical object, was not only mathematically validbut of great use in many enterprises.

A dispute had arisen between the philoso-pher Sir William Hamilton and the mathemati-cian AUGUSTUS DE MORGAN over whether logicbelonged to the domain of philosophy or math-ematics. De Morgan, who was a friend of Boole,had made several contributions to logic throughhis laws on the theory of sets, but Hamilton wasskeptical that mathematics could be of any ben-efit; Boole defended the validity of a mathe-matical approach to logic in MathematicalAnalysis of Logic (1847), and laid an axiomaticframework for logic much like the foundationof classical geometry. History would later provethat Boole and De Morgan had won the argu-ment, as mathematical logic has since evolvedinto a thriving (and surprisingly convoluted)discipline.

The attempt to reduce logic to a pure cal-culus had been previously attempted by GOT-TFRIED WILHELM VON LEIBNIZ; the dream was toreplace lengthy, quarrelsome philosophical de-bates with an algebraic system capable of re-solving doubtful propositions through simplecomputations. Early efforts drew heavily onEuclidean arithmetic as an analogy for algebraiclogic, but encountered thorny difficulties.Boole’s construction was original and different,

46 Borel, Émile

and essentially was a completely new algebra—different from arithmetic, and yet valid for itsown purpose. The ideas seem to have originatedfrom Boole’s familiarity with operators: Hewould apply an operator with a defining prop-erty to some universe of elements, and therebyobtain all individuals or elements with that par-ticular property. For example, an operator mightbe defined to select carrots from any universe ofobjects under discourse, such as the contents ofone’s garden. The successive application of op-erators to a universe, which was commutative,defined a multiplication for the algebra. Fromthis starting point, Boole developed a notion ofsubtraction (which involved the complement ofa set), addition (associated by Boole to the ex-clusive “or,” though in modern times to the in-clusive “or”), and even division. It is interestingthat this was the first known idempotent alge-bra, which has the property that the square ofany operator is equal to itself—since applying anoperator twice in succession is equivalent to ap-plying it just one time. This situation signals aclear and irrevocable departure from more fa-miliar arithmetic, where the only idempotentsare one and zero.

In Investigation of the Laws of Thought, Booleapplies this calculus to the laws of probability.Using the symbol P(A) for the probability of anevent A, Boole describes the multiplication ofprobabilities in terms of the probability of theintersection of two independent events, the sumof probabilities as the probability of the mutu-ally exclusive union of two events, and so forth.This symbolism allowed him to correct earlierwork in probability. Boole’s health began to de-cline in 1864, and when caught in the rain onthe way to class, he gave his lecture in wetclothes. This event may have hastened hisdeath, which occurred on December 8, 1864, inBallintemple, Ireland.

Investigation of the Laws of Thought is certainlyBoole’s most important legacy; many others wouldlater expand on his work in mathematical logic

and so-called Boolean algebras. Even the flow ofcomputer programs, which implement Booleanvariables (a quantity that either takes the valueof “true” or “false”), utilizes his theory. The de-sign of electric circuits is particularly amenableto the use of a Boolean algebra, due to the bi-nary system of on-off switches.

Further ReadingBarry, P. George Boole: A Miscellany. Cork, Ireland:

Cork University Press, 1969.Bell, E. Men of Mathematics. New York: Simon and

Schuster, 1965.Feys, R. “Boole as a Logician,” Proceedings of the Royal

Irish Academy. Section A. Mathematical andPhysical Sciences 57 (1955): 97–106.

Kneale, W. “Boole and the Revival of Logic,” Mind57 (1948): 149–175.

MacHale, D. George Boole: His Life and Work. Dublin,Ireland: Boole Press, 1985.

Taylor, G. “George Boole, 1815–1864,” Proceedings ofthe Royal Irish Academy. Section A. Mathematicaland Physical Sciences 57 (1955): 66–73.

� Borel, Émile(1871–1956)FrenchAnalysis, Probability

Émile Borel is known as one of the founders ofthe modern theory of analysis. His developmentof measure theory and probability are perhapsthe most important, as these ideas have led to ablossoming of research and activity in his wake;it is also noteworthy that he invented game the-ory a few decades before John Von Neumanntook up that subject.

Born on January 7, 1871, Émile Borel wasthe son of a Protestant pastor in the village ofSaint-Affrique. His mother, Émilie Teisié-Solier,was descended from a local merchant family. Hisfather, Honoré Borel, had recognized his son’sextraordinary talents, and in 1882 Émile Borel

Borel, Émile 47

was sent to a nearby lycée. Later he obtained ascholarship to prepare for university in Paris, andthere became attracted to the life of a mathe-matician; deciding upon this vocation, he put allhis energy toward this goal, exercising great self-discipline. In his entrance exams Borel won firstplace to both the École Polytechnique and theÉcole Normale Supérieur, but in 1889 he chosethe latter institution.

In his first year the young mathematicianpublished two papers and set his lifelong patternof diligent study and determined focus. He hadrejected the religion of his father, and took a ra-tionalistic view toward human life and science.During his time as an undergraduate, he mademany friendships, which would later facilitatehis extensive cultural and political influence.Borel graduated in 1893 as first of his class, andimmediately came to the University of Lille asa teacher, writing his thesis and 22 papers onvarious mathematical subjects within the nextthree years. Upon obtaining his degree fromLille, he returned to the École NormaleSupérieur, where he continued his prodigiousrate of research.

His early research focused on the solution ofcertain problems but, inspired by GEORG CAN-TOR in 1891, Borel’s mind turned to the powerof set theory. His 1894 thesis initiated the mod-ern concept of measure (a precise and generalmathematical formulation of the concept ofmeasuring length, area, and volume), which hassince proved to be fundamental in both the the-ory of real functions and probability. He also ex-plored divergent series, nonanalytic continuation,denumerable probability (in between the finiteand continuous probability theory), Diophantineapproximation, and the distribution theory ofanalytic functions. These concepts are indebtedto the genius of Cantor, in particular the notionof a denumerable set, that is, a set that can beput in one-to-one correspondence with the nat-ural numbers. Two famous results are the Heine-Borel theorem, which concerns the compactness

of closed and bounded subsets of the real line,and the statement that any denumerable set hasmeasure (length) zero. These nonintuitive factsdisplay Borel’s cunning intellect as well as his vi-sion for pure mathematics.

Borel developed measure theory in the en-suing years, and his name is attached to many ofthe mathematical objects of that subject, suchas sigma-algebras and the concept of measura-bility. He also saw that measure theory was anappropriate foundation for probability (1905),since probabilities of events can be viewed asmeasurements of the likelihood that a particu-lar event occurs. As a result, Borel was able tointroduce denumerable probabilities (1909),giving rise to many distributions so useful in thetheory of statistics. However, Borel retained acautious regard for the infinite, rejecting non-constructive definitions as well as Cantor’s hier-archies of infinity beyond the denumerable sets.Later, HENRI LEBESGUE and RENÉ-LOUIS BAIRE

would push set theory and measure theory to lev-els of abstraction that Borel was unwilling toventure into.

Borel also influenced later mathematicsthrough his simple proof of Picard’s theorem in1896, an outstanding problem that had defiedsolution for almost 20 years; his methods wereinstrumental to the subsequent maturation ofcomplex function theory. His 1899 work on di-vergent series filled a notable gap in currentknowledge of infinite series, and his work onmonogenic functions (summarized in 1917) sup-plied a vital link between analytic and extremelydiscontinuous functions. With age, Borel turnedtoward physical and social problems that wereamenable to mathematical techniques, despisinggeneralization for its own sake.

For some time, Borel had been attracted toMarguerite Appell, and they were married in1901; as a novelist and intellectual, she com-plemented her husband’s varied activities.Childless, the pair adopted Fernand Lebeau, theorphaned son of Borel’s sister. Marguerite and

48 Brahmagupta

Émile Borel launched La revue du mois (Themonthly review) in 1906, a widely read period-ical, though the post–World War I economiccrisis would render its continued existence fi-nancially infeasible. Borel continued his re-markable profundity and wealth of publications,and as he aged his interests broadened from puremathematics to applications and public affairs.He wrote texts, edited books, contributed to thedaily press, played a leading role in universityaffairs, and maintained a diverse body of ac-quaintances. This heightened level of activitywas possible due to intensity and efficiency;though generous of his time in the course of hisresponsibilities, he found social interruptions ir-ritating, and would not tolerate those whowasted his time with peripheral matters.

In 1909 Borel obtained the chair of theoryof functions at the Sorbonne, and commenceda 32-year tenure on the University Council as arepresentative of the faculty of science. The nextyear he concurrently served as vice director ofthe École Normale Supérieur, but this office waslater terminated by the advent of World War I.Borel served in the military and later organizedresearch in the War Office. As a result, his in-terests turned increasingly toward applications.After the war Borel moved to the chair of prob-ability and mathematical physics at theSorbonne, and he maintained mere honoraryconnections with the École Normale.

Most of Borel’s more brilliant ideas had al-ready been conceived by the start of World WarI, and during this period of his life he continuedtheir development through applications to sci-ence. However, between 1921 and 1927 hewrote several papers on game theory, which wasentirely original and significantly predates VonNeumann’s work in that area. Borel used gamesof strategy as models of military and economicsituations, assuming the rationality of the com-peting “players.” In this avenue he consideredmixed strategies (which involve chance), sym-metric games, and infinite games (those with

infinitely many actions available), and provedthe minimax theorem for three players.

During this time, Borel’s political career ad-vanced rapidly, as he moved through the officesof mayor of Saint-Affrique, councillor of theAveyron district, Radical-Socialist member ofthe Chamber of Deputies, and minister of thenavy. Through these opportunities, Borel wasable to promote scientific legislation and foundthe Centre National de la Recherche Scientifique,as well as raise funds for the Institut HenriPoincaré. After retiring from politics in 1936and from the Sorbonne in 1940, Borel contin-ued to produce numerous books and papers; healso participated in the French Resistance. Hewas honored by numerous awards during his life-time, and he died on February 3, 1956, shortlyafter a fall during the return from a conferenceof statistics in Brazil.

Borel made many great contributions toanalysis. Most significant of all were his early dis-coveries of measure theory; the study of meas-ures has since grown into a major discipline ofmodern mathematics, with numerous applica-tions to the theory of probability. Borel must alsobe recognized as the first inventor of moderngame theory, which since the development byJohn Von Neumann and John Nash has becomea fruitful field of mathematics with numerous ap-plications.

Further ReadingCollingwood, E. “Emile Borel,” Journal of the London

Mathematical Society 34 (1959): 488–512.

� Brahmagupta(ca. 598–665)IndianAlgebra

Brahmagupta was another early Indian as-tronomer and mathematician who contributedto the advancement of mathematics. Born in

Brouwer, Luitzen Egbertus Jan 49

598, Brahmagupta was the son of Jisnugupta—the name “gupta” may indicate membership inthe Vaisya caste. When he was 30 years old,Brahmagupta composed the Brahmasphutasi-ddhanta, a work on astronomy and mathematics.In this book he is called Bhillamalacarya, or theteacher from Bhillamala. His second work wasthe Khandakhadyaka, composed sometime afterMarch 15, 665.

The Brahmasphutasiddhanta contains 24 chap-ters, treating the following topics: mean and truelongitudes of planets, diurnal motion, lunar andsolar eclipses, heliacal risings and settings, lunarcrescent and shadow, planetary conjunctions, al-gebra, the gnomon, and the sphere. He also dis-cusses measurements and instruments, includingtables of values. The Khandakhadyaka deals mostlywith astronomy, addressing the arddharatrika sys-tem: computation of the tithis and naksatras, lon-gitudes of the planets, diurnal motion, lunar andsolar eclipses, heliacal risings and settings, the lu-nar crescent, and planetary conjunctions. The ap-pendix also discusses the projection of eclipses.

Brahmagupta’s understanding of numbersystems represented significant progress be-yond his contemporaries. The concept of zerowas foreign to Indian mathematicians, andBrahmagupta defined it as the difference of anumber with itself. He then derived its basicproperties; he knew that zero was an additiveidentity, for example, but had difficulty definingdivision by zero, not realizing that this operationis impossible. Negative numbers are introducedby subtracting a (positive quantity) from zero,and are described as “debts”; Brahmagupta thendemonstrates the arithmetic of positive and neg-ative numbers, and shows, for instance, that anegative times a positive is a negative.

He also discusses extended multiplication oflarge numbers, utilizing a place-value numbersystem quite similar to the modern method.Indeed, the current number system is derived,with some modifications, from the Indian math-ematicians. Brahmagupta presents a procedure

for computing square roots that is actually equiv-alent to the iterative Newton-Raphson method.In order to solve certain quadratic equations,Brahmagupta introduced a symbolic algebraicnotation and probably used the method of con-tinued fractions. Some other topics of theBrahmasphutasiddhanta include rules for summingseries—such as the sum of consecutive integers,squares, and cubes—and formulas for areas ofquadrilaterals.

Brahmagupta believed in a stationary Earth,and he computed the length of the year, over-estimating the true period slightly. His secondwork, the Khandakhadyaka, also gives an inter-polation formula for the computation of sines.

Brahmagupta influenced later Indian mathe-maticians, such as BHASKARA II, who improved onthe knowledge of negative numbers and the prop-erties of zero. He died sometime after the publi-cation of his second work, in 665. Consideringhis time period, he was quite advanced; at thetime, no mathematics was being done in Europe.Perhaps he is most notable for his definition ofthe concept of zero, which has had an enormousimpact on civilization ever since its inception.

Further ReadingChatterjee, B. “Al-Biruni and Brahmagupta,” Indian

Journal of History of Science 10, no. 2 (1975):161–165.

Datta, B. “Brahmagupta,” Bulletin of the CalcuttaMathematical Society 22 (1930): 39–51.

Ifrah, G. A Universal History of Numbers: FromPrehistory to the Invention of the Computer. NewYork: John Wiley, 2000.

� Brouwer, Luitzen Egbertus Jan(1881–1966)DutchLogic, Topology

One of the hotly contested topics of 20th-cen-tury mathematics was the logical foundation of

50 Brouwer, Luitzen Egbertus Jan

the discipline; specifically, certain mathemati-cians were laboring to show that the axiomaticformulation of mathematics was consistent (thatany proposition could be proved to be either trueor false, but not both). Brouwer represented anopposition to this agenda, putting forth his intu-itionistic mathematics as a desirable alternative.

Luitzen Brouwer was born on February 27,1881, in the town of Overschie in theNetherlands. He was intellectually precocious,completing his high school education at the ageof 14; in 1897 he entered the University ofAmsterdam, where he studied mathematics for thenext seven years. Brouwer quickly mastered con-temporary mathematics, and he obtained new re-sults concerning continuous motions on manifolds.

Brouwer’s interests were diverse. His math-ematical activity included topology, mappings,and logic, as well as mystic philosophy. His per-sonal view of mathematics as a free mental ac-tivity was constructivist and differed sharplyfrom the formalistic approach espoused by DAVID

HILBERT and BERTRAND RUSSELL. Brouwer partic-ipated in the debate over mathematics’ founda-tions; he rejected the idea that logic should bethe pillar of mathematics—rather, logic was justan expression of noted regularities and patternsin constructed systems. The bizarreness of thisview became apparent when Brouwer attackedthe law of the excluded middle, which states thateither a given statement or its logical negationmust be true (which is used in the “proof by con-tradiction” method).

Brouwer’s doctoral thesis of 1907, On theFoundations of Mathematics, expresses his opin-ions. Out of these ideas was born “intuitionisticmathematics,” which places an emphasis on theability to construct mathematical objects. He re-jected the law of the excluded middle in his sys-tem and criticized Hilbert’s attempt to prove theconsistency of arithmetic.

In the five years from 1907 to 1912, Brouwerdiscovered several valuable results. He studiedHilbert’s fifth problem, the theory of continuous

groups, and in the process discovered the planetranslation theorem and the “hairy ball theo-rem,” which states that a smooth vector field onan even-dimensional sphere must vanish some-where—or, in other words, every hairdo musthave a cowlick.

Brouwer also studied various topologicalmappings, developing the technique of using so-called simplices to approximate the continuousmaps. The associated degree led to the notionof homotopy class, which allowed the topologi-cal classification of many manifolds. As a result,the notion of dimension (in the topologicalsense) was put on a more rigorous footing.

In 1912 he was appointed as a professor ofmathematics at the University of Amsterdam,and he soon resumed his research into the foun-dations of mathematics. In 1918 he published adifferent set theory, which did not rely on thelaw of the excluded middle, followed by similarnotions of measure and function in the follow-ing years. As could be expected, the theoremshe obtained are somewhat different (for exam-ple, real functions are always uniformly contin-uous). For these reasons, his results were not fullyaccepted, and many mathematicians have sim-ply ignored his point of view. Proof by contra-diction is a very powerful, commonly usedmethod of proof; mathematicians are not will-ing to forsake the many theorems they can es-tablish in order to embrace Brouwer’s potentiallymore rigorous system.

From 1923 onward, Brouwer focused on hisintuitionistic agenda, attempting to persuademathematicians to reject the law of the excludedmiddle. In the late 1920s logicians began inves-tigating the connection of Brouwer’s logic to theclassical logic; after KURT GÖDEL’s incompletenesstheorems annihilated David Hilbert’s program,more people became interested in the intuition-istic approach to mathematics.

Brouwer gained international recognitionfrom several societies and academies. He died inBlaricum, the Netherlands, on December 2,

Brouwer, Luitzen Egbertus Jan 51

1966. Although his efforts to persuade mathe-maticians toward his own point of view weremostly unsuccessful (again, this was due in partto the reluctance to give up the powerful toolof proof by contradiction, and also because theintuitionistic framework is rooted in mystic phi-losophy), Brouwer raised awareness about thelimitations of any mathematical system andcorrectly predicted the demise of any attemptto establish the consistency and completenessof an axiomatic system. He is an importantcharacter in the history of mathematical logic,representing the antirationalistic counter-movement of mysticism that arose in the 20thcentury.

Further Readingvan Dalen, D. Mystic, Geometer, and Intuitionist: The Life

of L. E. J. Brouwer. Oxford: Clarendon Press, 1999.Franchella, M. “L. E. J. Brouwer: Toward Intuitionistic

Logic,” Historia Mathematica 22, no. 3 (1995):304–322.

Mancosu, P. From Brouwer to Hilbert: The Debate onthe Foundations of Mathematics in the 1920s. NewYork: Oxford University Press, 1998.

van Stigt, W. “L. E. J. Brouwer: Intuitionism andTopology,” Proceedings, Bicentennial CongressWiskundig Genootschap (1979): 359–374.

———. “L. E. J. Brouwer ‘Life, Art and Mysticism,’ ”Notre Dame Journal of Formal Logic 37, no. 3(1996): 389–429.

52

study of the mathematical infinite to the theo-logical infinity of God.

Cantor attended the Gymnasium inWiesbaden, and later the Realschule inDarmstadt, where his interest in mathematicswas first aroused. His university studies in Zurichcommenced in 1862 and resumed in Berlin in1863 after his father’s sudden death. KARL WEIER-STRASS was the leading mathematician of Berlinat the time, and attracted numerous students asdisciples. Cantor studied different branches ofmathematics, and even wrote a dissertation onnumber theory, but his main interest was thetheory of real numbers and infinite series.

Throughout his life, Cantor had numerousfriendships, some of which (such as his lengthyassociation with RICHARD DEDEKIND) were fueledby scientific correspondence and collaboration.He was president (1864–65) of the MathematicalSociety, an organization that attempted to unifythe work of diverse mathematicians, and laterwas an active promoter of scientific exchange.Later, he founded the Association of GermanMathematicians (1890), becoming the presidentin 1893; the first international congress of math-ematicians, a result of his endeavors, was heldin Zurich in 1897.

In 1867 Cantor obtained his doctorate. Hebecame a teacher at the University of Halle twoyears later. The position was poorly paid, but

C� Cantor, Georg

(1845–1918)GermanLogic, Analysis

Modern mathematics, in all of its branches, isfounded upon set theory; that is, introductorymaterial in such fields as probability and algebrainvariably commences with a discussion of setsand logic. However, prior to the 20th centurythis was not the case, since mathematics was inearlier times conceived in less formal tones;mathematical truth was viewed as inseparablyconnected to metaphysical truth. Current resultsin set theory have demolished this idealism—witness the work of KURT GÖDEL on incom-pleteness and the independence of the contin-uum hypothesis. Georg Cantor was a pivotalfigure in this transition, providing the first stepstoward modern set theory and at the same timeremaining as a final proponent of classicalthought in mathematics before the flood of ax-iomatic formalism.

Georg Cantor was born on March 3, 1845,in St. Petersburg to German parents; his father,also named Georg Cantor, was a wealthyProtestant merchant, and his mother, MarieBöhm, was a Catholic from a line of renownedviolinists. Religion would form an importantcomponent of Cantor’s thought, as he tied his

Cantor, Georg 53

Cantor was able to survive due to the inheri-tance received from his father. He married VallyGuttmann in 1874, and together they built ahappy home with five children—his wife’s goodhumor contrasting with Cantor’s melancholictendencies. He attained full professorship in1879 and continued his labors on set theory un-til his death; Cantor had hoped to obtain a moreprestigious position in Berlin, but LEOPOLD KRO-NECKER continually blocked his efforts. Kroneckerwas one of Cantor’s former teachers who dispar-aged the radical theory of sets.

Cantor’s first significant work lies in the areaof mathematical analysis. The basic concept ofthe real number system was still deficient in somerespects, and Cantor’s early labors in so-called

fundamental series (now called Cauchy se-quences) bolstered the foundations. As a result,one could represent any real number as the limitof a sequence of rational numbers, thoughCantor also described formulations involving in-finite series and infinite products.

After an 1873 exchange with Dedekind,Cantor turned toward the question of whetherthe set of real numbers could be placed in one-to-one correspondence with the natural numbers(any such set would therefore be called “count-able” or “denumerable”). It was already knownthat the rational numbers were countable, but noone had considered this new question; its solu-tion in the negative was to initiate the moderntheory of sets. Cantor’s famous diagonalizationargument shows the real numbers to be un-countable and, as a corollary, the existence ofuncountably many transcendental numbers(those numbers that are not the solution to anyinteger coefficient algebraic equation, such as pi).

In 1874 Cantor turned to more difficultproblems, such as establishing the impossibilityof a one-to-one correspondence between asquare and a line segment. After three years ofeffort, Cantor instead constructed a counterex-ample, giving an explicit invertible function thatmapped the line into the square; this construc-tion seemed to defy all intuitive concepts of di-mension, and the result severely irritated manyconservative mathematicians. Today, the topolog-ical concept of dimension is tied to continuousone-to-one correspondences (with continuous in-verse) called homeomorphisms—Cantor’s strangemap was discontinuous.

The theory of point set topology is deeplyindebted to Cantor’s genius: He addressed suchtopics as point sets, closure, and density, and inmany cases created the definitions himself. Thenotion of perfect set (one that is both closed anddense in itself) has a Cantorian genesis, and themost famous example of such is the “Cantor set”that is formed by removing successive middlethirds from a line segment iteratively. A century

Georg Cantor, founder of modern set theory, as wellas the study of hierarchies of infinity and theirarithmetic (Courtesy of the Library of Congress)

54 Cardano, Girolamo

later, this same set would motivate the study offractal geometry and the metric definition of di-mension. The concept of continuum, a term ofphilosophical parlance in existence since me-dieval times, was given an exact mathematicaldefinition by Cantor: A continuum was a con-tinuous perfect set.

Much of his fundamental work can be foundin “Über eine Eigenschaft des Inbegriffes aller reellenalgebraischen Zahlen” (“About a characteristic ofthe essence of all real algebraic numbers”) of1874. Here he begins the delineation of the hi-erarchies of infinity: those sets that are counta-ble versus those that have the “power” of thecontinuum (and are uncountable)—the lattertype being a higher species of the infinite. Forevery set there is a higher power, obtained bytaking the set of all subsets of a given set (calledthe power set). Cantor later proves that therecan be no one-to-one correspondence betweena set and its power set—the latter is always“larger.” In this fashion, a kingdom of infinitiescan be constructed and studied; one outstandingquestion is whether there is an infinite set withcardinality (level of infinity) between the natu-ral numbers and the continuum. The assertionthat there is no such set is known as the con-tinuum hypothesis, and Cantor was consumedwith its proof, perhaps even contributing to hislater lapse into madness. These cardinal num-bers had their own transfinite arithmetic, inwhich such Trinitarian statements as “one plusone plus one equals one” held some validity.

Many of his peers mocked Cantor’s theories,as they easily upset the classical intuition abouthow mathematical objects must behave. However,the work gained acceptance toward the beginningof the 20th century, and Cantor later became anhonorary member of several mathematical soci-eties. Since 1884 depression had afflicted him,perhaps brought on by his intense effort to solvevarious problems, such as the continuum hy-pothesis. He died on January 6, 1918, in the psy-chiatric clinic of Halle University.

Cantor’s bold work in set theory opened upentire new vistas of mathematical thought, fu-eling 20th (and 21st) century research into fun-damentals, set theory, real analysis, logic, andfractal geometry. The turn toward fundamentalsresulted in the foundation of each branch ofmathematics upon axiomatic set theory, and thisin turn gave rise to the mathematical philoso-phy of formalism—the belief that mathematicsconsists of semantic rules that are carefully ma-nipulated, as in a game, to arrive at new knowl-edge. This differs substantially from Cantor’sPlatonic thought, which conceived of the ab-stract existence of mathematical structures madeconcrete through various realizations in our ownuniverse. For example, the atoms of the universeshould be countable (according to Cantor), aconcrete realization of the notion of denumer-ability. The absolute limit of the power operationresulted in an ultimate infinity, Cantor’s visionof God, in which the concrete and the abstractwere married. It is ironic that one who indirectlycontributed to the advance of formalism shouldbe one of the last great advocates of Platonicthought.

Further ReadingDauben, J. Georg Cantor: His Mathematics and

Philosophy of the Infinite. Cambridge, Mass.:Harvard University Press, 1979.

Grattan-Guinness, I. “Towards a Biography of GeorgCantor,” Annals of Science 27 (1971): 345–391.

Hill, L. “Fraenkel’s Biography of Georg Cantor,”Scripta Mathematica 2 (1933): 41–47.

� Cardano, Girolamo(1501–1576)ItalianAlgebra, Probability

The Renaissance was a time of bold intellectualvoyaging, where the man of learning might in-vestigate all the branches of philosophy;

Cardano, Girolamo 55

Girolamo Cardano epitomized this intrepid,proud, and inquisitive spirit. Cardano’s name issomewhat infamous among mathematicians, dueto his questionable dealings with the mathe-matician NICCOLÒ TARTAGLIA. However, his con-tributions to mathematics and science are nu-merous, including a vague preliminary formulationof imaginary numbers as well as the basic rulesof probability. In his own age he was reputed asa great doctor, and his writings have influenceddiverse areas of science such as geology and me-chanics.

Girolamo Cardano was born on September24, 1501, to an Italian jurist, Fazio Cardano, anda widow, Chiari Micheri. The boy was illegiti-mate, and his mother possessed a nasty temper;

Cardano’s childhood was unpleasant, punctu-ated with illness. His father, a friend ofLEONARDO DA VINCI, encouraged Cardano tostudy the classics, mathematics, and astrology,and he commenced his university studies in 1520at Pavia. Six years later, Cardano had completedhis studies in Padua with a doctorate in medicine.

Medicine was to be Cardano’s foremost ca-reer; he would later acquire great fame for hisremedies. He started his practice in Saccolongo,near Padua, where he remained for six years.Apparently Cardano had been unable to marrydue to impotence, but upon obtaining releasefrom this affliction he wed Lucia Bandareni in1531, and she bore him two sons and a daugh-ter. Three years later, Cardano became a teacherof mathematics in Milan thanks to the interven-tion and encouragement of his father’s aristocraticfriends. Simultaneously, Cardano continued topractice medicine, and he achieved a remarkabledegree of success, such that his colleagues werestricken with envy; soon after 1536, when hepublished his first work De malo recentiorummedicorum usu libellus (A book concerning thebad practice of modern doctors), he was the fore-most physician in Milan. In 1552 he even jour-neyed to Scotland in order to cure the archbishopof Edinburgh of asthma.

Cardano’s 1539 Practica arithmeticae et men-surandi singularis (Practice of mathematics andindividual measurements) was devoted to nu-merical calculation, and in this work he revealshis talent in the manipulation of algebraic ex-pressions. He could solve some equations of thirddegree and higher (the solution of the quadraticwas well known) before encountering Tartaglia.Cardano soon became acquainted with the lat-ter mathematician, who had mastered the solu-tion of the cubic (third degree equation) by ageneral method. After much importuning ofTartaglia, Cardano obtained the secret of thecubic’s solution upon swearing an oath not toreveal it. However, Tartaglia was not the origi-nator of this method, and he had learned it from

Girolamo Cardano developed an early notion ofimaginary numbers and the rules of probability andpublished a method for solving the cubic equation.(Courtesy of the Library of Congress)

56 Cardano, Girolamo

SCIPIONE DEL FERRO; when Cardano learned ofdel Ferro’s prior discovery, he considered his oathto be irrelevant and subsequently published themethod in his Artis magnae sive de regulis alge-braicis liber unus (Book one of the great art orconcerning the rules of algebra) of 1545. Ofcourse, this action infuriated Tartaglia, who feltthat he had been betrayed; his later publicationsaccused Cardano of perjury, and Tartaglia con-tinued to castigate his character.

The Artis magnae sive de regulis algebraicisliber unus presented many new ideas in the fieldof algebra. The so-called Cardano’s rule providesthe solution of cubic equations that lack a sec-ond-degree term, and Cardano also explainedhow to linearly transform an arbitrary cubic intothis reduced form. He observes that an equationof degree more than one must have more thanone root, and that knowing a root is tantamountto reducing the degree of the polynomial by one.These facts would later be formulated as part ofthe fundamental theorem of algebra, proved byCARL FRIEDRICH GAUSS in the 18th century.Cardano also discusses the solution of the quar-tic, or fourth degree, equation, attributed to hisson-in-law LODOVICO FERRARI. Of course, thequintic, or fifth degree, polynomial did not ad-mit of a method of solution, as would be provedby NIELS HENRIK ABEL and EVARISTE GALOIS cen-turies later.

Cardano also investigated the numerical ap-proximation to the solution of an equation, us-ing the method of proportional parts togetherwith an iteration scheme. The idea of approxi-mation (at least for equations) first appears withCardano, and would much later be systemati-cally developed through SIR ISAAC NEWTON’smethod and the subsequent theory of numericalanalysis. He also observed the relationship be-tween the roots of a polynomial and its coeffi-cients, and is therefore considered the father ofthe theory of algebraic equations. In some situ-ations he even used imaginary numbers, whichwould not be formally developed for centuries.

Besides this famous work in algebra, Cardanowas also known for his passion for games ofchance, such as dice, chess, and cards. His Liberde ludo aleae (Book on games of chance), com-pleted in his old age, gives a first treatment of thetheory of probability. This predates the pioneer-ing work of BLAISE PASCAL and PIERRE DE FERMAT,although these two were unaware of Cardano’sprior work and hence were not influenced by it.The important idea in this work is that evenchance follows certain rules—the laws of proba-bility. Probability as a numerical, measurablequantity is introduced as the ratio of outcomes inwhich an event can occur to all possible out-comes; today, this is known as the classical defi-nition of probability, and has been replaced by ameasure theoretic formulation, since Cardano’sconception can only deal with equally likely out-comes. He enunciates the law of large numbers,as well as various other rules of probability, suchas the multiplication law for independent events.

Meanwhile, Cardano advanced in his careeras a physician, in 1543 accepting the chair ofmedicine at the University of Pavia, where hetaught until 1560. In this year, his elder son wasexecuted for poisoning his wife, and Cardanowas meanwhile wearied by the public condem-nation of his enemies as well as the dissolute lifeof his second son. As a result, Cardano left forthe University of Bologna, where he obtainedthe chair of medicine in 1562.

His interests in astrology and magic led toan accusation of heresy, and the Inquisition in1570 imprisoned Cardano; apparently he hadcast the horoscope of Jesus Christ, attributingthe Lord’s eventful life to the influence of thestars. After a few months in prison, Cardano re-pented of his teaching, and obtained the favorof Pope Pius V. In Rome, during the last year ofhis life, Cardano wrote De propria vita (Book ofmy life), a thorough autobiography; he died onSeptember 21, 1576.

Cardano authored some 200 works on med-icine, mathematics, physics, religion, philosophy,

Carnot, Lazare 57

and music, representing the typical Renaissancethinker in the diversity of his thought. Besideshis mathematical writings, he published two en-cyclopedias of natural science, a compendium ofknowledge, superstition, and the occult, con-tributed to mechanics and hydrodynamics, anddeveloped early theories for the formation ofmountains by the erosion of water, as well asconceiving the cycle of evaporation, condensa-tion, and precipitation. He deduced the rising ofthe ocean floor from the presence of marine fos-sils on dry land, experimentally estimated theratio of densities of air and water, and de-scribed numerous mechanical devices, includ-ing “Cardano’s suspension.” He noted that thetrajectory of a projectile was similar to that of aparabolic curve, and he affirmed the impossibil-ity of perpetual motion (excepting the celestialbodies). The numerous editions of his works beartestimony to his enduring and widespread influ-ence on the next generation of thinkers.

Further ReadingFierz, M. Girolamo Cardano, 1501–1576: Physician,

Natural Philosopher, Mathematician, Astrologer,and Interpreter of Dreams. Boston: Birkhäuser,1983.

Kenney, E. “Cardano: ‘Arithmetic Subtlety’ andImpossible Solutions,” Philosophia Mathematica II(1989): 195–216.

Ore, O. Cardano, the Gambling Scholar. Princeton,N.J.: Princeton University Press, 1953.

� Carnot, Lazare(1753–1823)FrenchCalculus

It is rare that a person achieves fame as both apolitician and a scientist, but Lazare Carnot wasrenowned for both his mathematical and me-chanical achievements, and known as the“Organizer of Victory” in the wars of the French

Revolution. His principle of continuity in thetransmission of power is of great historical im-portance to mechanical engineering, and hemade contributions to the foundations of analy-sis and calculus.

Born on May 13, 1753, in Nolay, France,Lazare Carnot was the son of a bourgeois lawyerof Burgundy. He was educated at the OratorianCollege at Autun, and later at Paris, beinggroomed for a military career in the Corps ofEngineers. He graduated after two years of at-tendance at the military school of Mézières; GAS-PARD MONGE was one of his professors, but theyhad little interaction. Carnot’s approach tomathematics was slower, being more concernedwith fundamentals. This attitude would later

Lazare Carnot defined a machine abstractly andformulated the principle of continuous transfer ofpower. (Courtesy of the National Library of Medicine)

58 Carnot, Lazare

prove to have great relevancy for mechanics andphysics.

After various posts, Carnot was assigned toArras, and in 1787 he became acquainted withRobespierre. Meanwhile, Carnot had taken towriting on the topics of mechanics, mathemat-ics, and military strategy, partly in order to easehis frustration over the slow progression of hiscareer. The Académie des Sciences in Paris pro-posed a competition in 1777 over the topic offriction in simple machines; Carnot’s submissionfailed to attain the prize, but he later developedhis memoir into Essai sur les machines en général(Essay on general machines), published in 1783.Although it attracted little attention at the time,this work was an initial step in the French lit-erature of engineering mechanics. Carnot alsocomposed an essay on mathematics, in compe-tition for a 1784 Berlin prize for justification ofthe infinitesimal calculus. He failed to win anyaward, but his paper later evolved into Réflexionssur la métaphysique du calcul infinitesimal(Reflections on the metaphysics of infinitesimalcalculus) published in 1797.

Carnot achieved some literary fame for hiswritings on military strategy; his Éloge de Vauban,a tribute to Vauban (the founder of the RoyalCorps of Engineers and an influential militarytheorist since the time of Louis XIV), won firstprize in a 1784 competition. In this paper, Carnotaffirms the importance of fortifying strategic lo-cations and discusses war’s proper use as the de-fense of civilization. This perspective clashed withthe advocates of gallantry and movement.Ironically, Carnot would later abandon his con-servative theories when commanding a largelyundisciplined and untrained mass of soldiers.

His political career began in 1791 with hiselection to the Legislative Assembly, advocatingcareer advancement for those with talent. In theensuing year he came to distrust the monarchy,and adopted the republicanism common to hissocial class more out of civic commitment thanpolitical conviction. When war broke out in

April 1792, Carnot was employed in militaryservices, and he defeated the loyalists later thatyear. His integrity and organizational ability wasevident, making him a valuable and trustworthyagent of the Republic. Next, he defendedBelgium from attack, rallying the demoralizedtroops to their duty, and he was subsequently ap-pointed to the Committee of Public Safety, al-though his main obligation was to the ongoingwar. Perhaps because of his great skill in militarymatters, Carnot survived both the downfall ofRobespierre in 1794 and the ensuing reaction,becoming a leading member of the Directory.Before being deposed by a leftist insurrection in1797, Carnot had granted Napoleon control ofthe Italian army; the former fled to Switzerlandand Germany, but in 1800 returned afterNapoleon’s seizure of power. As a result Carnotwas named minister of war, an office that he soonresigned due to differences of opinion.

At this point Carnot returned to his formerscientific interests, serving on various commissionsto investigate French inventions of a mechanicaland military nature. Always a committed patriot,Carnot volunteered to aid Napoleon during thelatter’s retreat from Moscow to the Rhine;Carnot received the government of Antwerp,which he successfully defended. During theHundred Days, he again supported the emperor,and he was appointed minister of the interior.However, when the monarchy was subsequentlyreinvested, Carnot was forced to flee Franceonce more; he lived out his remaining years inMagdeburg, dying on August 2, 1823.

Carnot’s Essai sur les machines en général wasa theoretical treatise on mechanical engineer-ing. He gave an abstract definition of a ma-chine, and developed the concept that powercould be transmitted without perturbation orturbulence. His principle of perfect conversionsaid that motion could be smoothly translatedwithout loss, that live force results in the mo-ment of activity; or, in modern terms, inputequals output.

Cartan, Élie 59

His contributions to pure mathematics re-ceived public recognition, for his Réflexions surla métaphysique du calcul infinitesimal was pub-lished in several languages and enjoyed a widedistribution. Carnot explains that the genius ofcalculus lay in the introduction of certain errors,which were later, through the procedures of thecalculation, exactly compensated; hence, the re-sults were exact, even though an approximationis initially introduced. AUGUSTIN-LOUIS CAUCHY,BERNHARD BOLZANO, and CARL FRIEDRICH GAUSS

would later place calculus on a rigorous founda-tion, but this explanation was intended for thegeneral public. Carnot’s other geometric writingsfocused on negative and imaginary numbers,which he attempted to purge from mathematics,claiming that they were not reasonable quanti-ties. He demonstrates the inadequacy of illus-trating negative numbers as signed numbers,with a location on the left part of an axis;Carnot’s approach is far from the formalism ofthe modern era, and he shunned the senselessshuffling of mathematical symbols. After severalpublications on his concept of geometric quan-tity, Carnot published a book on the science ofgeometric motion, an attempt to connect hismechanical and geometrical studies.

Further ReadingBoyer, C. “Carnot and the Concept of Deviation,”

The American Mathematical Monthly 61 (1954):459–463.

Dupre, H. Lazare Carnot, Republican Patriot. Philadel-phia: Porcupine Press, 1975.

� Cartan, Élie(1869–1951)FrenchGeometry, Topology

A prominent theme of 20th-century mathemat-ics was the study of differentiable manifolds(smooth high-dimensional surfaces). Cartan

exerted a profound influence over the develop-ment of this topic and contributed greatly to theshape of modern mathematics.

Élie Cartan was born in Dolomieu, France,on April 9, 1869, to a village blacksmith. Dueto his poverty, it was unlikely that the youngCartan would ever study at a university, but hisearly talent in mathematics was recognized by apassing school inspector; as a result of this in-tervention, Cartan was able to study at the ly-cée in Lyons and later joined the École NormaleSupérieure in Paris. After graduation, he com-menced his research in the area of Lie groups,which are a special type of differentiable mani-fold. Cartan held various teaching positions un-til he secured a professorship at the Universityof Paris in 1912.

Cartan’s doctoral thesis was concerned withproviding a rigorous foundation to WilhelmKilling’s results on Lie algebras. Later, Cartancompleted the classification of simple Lie algebrasand determined their irreducible representations.For some three decades following his dissertation,Cartan was practically alone in his chosen field,as most French mathematicians were more inter-ested in function theory. Due to his isolation,Cartan was forced to develop entirely originalmethods.

In 1913 Cartan discovered the spinors, amathematical object of relevance in quantummechanics. After 1925 he became more inter-ested in topology, and started studying globalproperties of Lie groups, discovering several im-portant results. His methods in differential sys-tems were revolutionary, providing general solu-tions independent of a particular choice ofvariables. One of his chief tools, which he de-veloped in the decade following his thesis, wasthe calculus of exterior differential forms. Hethen applied his technique masterfully to a se-ries of problems from differential geometry, Liegroups, and general relativity.

Cartan also made substantial contributionsto differential geometry, and it is said that he

60 Cauchy, Augustin-Louis

revitalized the subject after the initial work ofGEORG FRIEDRICH BERNHARD RIEMANN. His con-cept of “moving frames” gave great power for es-tablishing new theorems, and has become one ofthe most important techniques in modern math-ematics. Cartan’s concept of a connection helpeddefine a more general type of geometry suitableto modeling the universe according to the pre-cepts of general relativity. His connections wereable to completely describe the symmetricRiemannian spaces, which play an important rolein number theory.

It was a long time before Cartan obtainedthe recognition warranted by his genius and cre-ativity. This was in part due to his humble na-ture, but also because the originality of his ideaseluded the grasp of his contemporaries. Hetaught at Paris until his retirement in 1940, andwas elected to the French Academy of Sciencesin 1931. He died on May 6, 1951, in Paris.

Cartan’s mathematical achievements havehad a great impact on the subsequent develop-ment of many areas of modern mathematics.After 1930 the younger generation of mathe-maticians became increasingly appreciative ofhis work, which they eagerly digested and sup-plemented. Now, the unified disciplines of Liegroups, differential systems, and differentialgeometry constitute a powerful tool and a cen-tral place in current mathematical research.

Further ReadingChern, S., and C. Chevalley. “Elie Cartan,” American

Mathematical Society. Bulletin. New Series 58 (1952).

� Cauchy, Augustin-Louis(1789–1857)FrenchComplex Analysis, Algebra,Differential Equations

Cauchy’s profundity can be compared with thatof CARL FRIEDRICH GAUSS, as to the quantity,quality, and variety of mathematical material

considered. He made outstanding contributionsto real analysis and calculus, complex functiontheory, differential equations, and algebra, as wellas elasticity theory and celestial mechanics. Hisstrange personality, alternately described aschildishly naive and flamboyantly melodramatic,together with his profuse literary style combineto form a singular character in the history ofmathematics. Indeed, the name of Cauchy is at-tached to more theorems and mathematical con-cepts than that of any other mathematician.

Augustin-Louis Cauchy was born on August21, 1789, in Paris to Louis-François Cauchy, apowerful administrative official, and Marie-Madeleine Desestre. The couple had wed in 1787and had four sons and two daughters. His father,who was an expert in classics, first educated

Augustin-Louis Cauchy made great advances incomplex analysis, infinite series, calculus, algebra,and differential equations and developed an integralformula for the calculation of residues for complexintegrals. (Courtesy of the Library of Congress)

Cauchy, Augustin-Louis 61

Augustin Cauchy, the eldest child. Later he metseveral leading scientists, such as PIERRE-SIMON

LAPLACE. Cauchy next attended the ÉcoleCentral du Panthéon, and he was admitted tothe École Polytechnique at the age of 16. A fewyears later he left school to become an engineer,and in 1810 he worked at the harbor ofCherbourg, where Napoleon was building up hisnaval operations against England. By 1813Cauchy had returned to Paris.

Meanwhile, in 1811 Cauchy had solved ageometrical problem posed by JOSEPH-LOUIS LA-GRANGE: the determination of the angles of aconvex polyhedron from its faces. In 1812 hecracked a problem of PIERRE DE FERMAT—whether every number is the sum of n ngonalnumbers. His 1814 treatise on definite integralswas submitted to the French Academy, and thisessay would later become the foundation of thetheory of complex functions. Two years laterCauchy won a prize contest of the FrenchAcademy on the topic of wave propagation onthe surface of a liquid. By 1819 he had inventedthe method of characteristics used to solve par-tial differential equations, and in 1822 he laidthe basis for elasticity theory.

These represent a small sample of Cauchy’sextensive writings. He had obtained the positionof adjoint professor of the École Polytechniquein 1815, and the next year was promoted to fullprofessor and was appointed as a member of theAcadémie des Sciences; before 1830 he heldchairs at both the Faculté des Sciences and theCollège de France. In the meantime he wrotemany notable textbooks, which were remarkablefor their precision.

In 1818 Cauchy married Aloïse de Bure, bywhom he had two daughters; the family settledin nearby Sceaux. He was a devout Catholic, de-voted to several charities throughout his life andhelping to found the Institut Catholique.Cauchy served on a committee to promote sab-bath observance, and apparently spent his entiresalary on the poor of Sceaux. His personality has

been described as bigoted, self-centered, and fa-natical; others paint him as merely childish. Forexample, Cauchy wrote a defense of the Jesuits,contending that they were hated because oftheir virtue. His treatment of the memoirs ofNIELS HENRIK ABEL and EVARISTE GALOIS hasbeen cited as proof of his selfish egotism, thoughin general he recognized other persons’ workand was careful in his referencing. Before pre-senting another’s paper to the academy, Cauchywould often generalize and improve the author’sresults; it seems that his obsession with math-ematics transgressed the boundaries of propri-ety, driving him to publish an idea as soon asit was developed. And Cauchy was prodigious:He produced at least seven books and morethan 800 papers!

In the July revolution of 1830, the Bourbonmonarchy was replaced by Louis-Philippe. Aroyalist, Cauchy refused to swear allegiance tothe new king. As a result, he lost his chairs andwent into self-imposed exile to Fribourg,Sardinia, and finally Prague, where he tutoredthe Bourbon crown prince and was later re-warded by being made a baron. In 1834 his wifeand daughters joined him in Prague, but Cauchyreturned to Paris in 1838, resuming his mathe-matical activity at the academy. Various friendsattempted to procure a position for Cauchy, buthis steadfast refusal to make the oath of alle-giance rendered these efforts abortive. After theFebruary Revolution of 1848, the republicans re-sumed power and Cauchy was permitted to takethe chair at the Sorbonne. He continued pub-lishing at an enormous rate until his death onMay 22, 1857.

Cauchy had written a masterful calculus textin 1821, notable for its rigor and excellent style.This approach to mathematics was characteristicof him: He rejected the “generality of algebra,”which was an illogical argument for treating in-finitesimal quantities the same as finite ones.Cauchy distinguished between a convergent anddivergent series (and refused to treat the latter

62 Cauchy, Augustin-Louis

type), laying specific conditions for convergence,such as the so-called Cauchy property for con-vergence of a sequence, as well as the root, ra-tio, and integral tests. Cauchy defined upper andlower limits for nonconvergent sequences, es-tablished the limit and series representations forthe transcendental number e, and was the firstto use the limit notation. He derived variousrules for the manipulation of convergent series,and computed radii of convergence for powerseries, cautioning against rash use of Taylor’s ap-proximation. Cauchy proved a remainder theo-rem for series, invented the modern concept ofcontinuity, and obtained a version of the in-termediate value theorem—later proved byBERNHARD BOLZANO. Cauchy stressed the limitdefinition of both the derivative and the defi-nite integral, as well as discussing indefinite andsingular integrals. He made extensive use of theFourier transform (discovered prior to JEAN-BAP-TISTE-JOSEPH FOURIER) in differential equations,invented the so-called Jacobian (a special de-terminant), and gave a proof of the fundamen-tal theorem of algebra.

In statistics, Cauchy treated regression the-ory using absolute errors, in contrast to Gauss’stheory of least squares; one result of this inves-tigation is the creation of the so-called Cauchydistribution. In algebra, Cauchy researched theinverse of a matrix, provided theorems on de-terminants, and investigated orthogonal trans-formations. He contrasted the geometric andalgebraic construction of the complex number.He also laid the fundamentals of group theory,including the concepts of group, subgroup, con-jugation, and order, and proved Cauchy’s the-orem for finite groups. He also unsuccessfullyattempted the proof of Fermat’s last theorem,which was proved only in 1994.

Cauchy’s methods in the theory of differen-tial equations include use of the Fourier trans-form and the method of characteristics. Cauchystressed that not all such equations had solutions,and uniqueness could be stipulated only under

important initial and boundary conditions; awell-specified partial differential equation withinitial and boundary data is called the Cauchyproblem. He also founded elasticity theory, gen-eralizing it from the one-dimensional examplesconsidered by 18th-century mathematicians;this was one of his most elegant and praisewor-thy contributions to science. Cauchy also wroteon the topic of celestial mechanics, solving theKepler equation and researching the perturba-tive function.

The theory of complex functions is, perhaps,most heavily indebted to Cauchy. He first justi-fied limit and algebraic operations on complexnumbers, and later developed Cauchy’s integralformula and the calculus of residues. These toolshave a remarkable breadth of application. It isinteresting that Cauchy failed to deduceLiouville’s theorem (that bounded regular func-tions must be constant), but he had no global,connecting perspective on the new science.However, his numerous contributions and in-sights greatly advanced the field of complexanalysis.

Further ReadingBelhoste, B. Augustin-Louis Cauchy: A Biography.

New York: Springer-Verlag, 1991.Dubbey, J. “Cauchy’s Contribution to the Establishment

of the Calculus,” Annals of Science 22 (1966):61–67.

Fisher, G. “Cauchy’s Variables and Orders of theInfinitely Small,” British Journal for the Philosophyof Science 30, no. 3 (1979): 261–265.

Grabiner, J. The Origins of Cauchy’s Rigorous Calculus:Cambridge, Mass.: MIT Press, 1981.

———. “Who Gave You the Epsilon? Cauchy andthe Origins of Rigorous Calculus,” The AmericanMathematical Monthly 90, no. 3 (1983): 185–194.

Grattan-Guinness, I. “Bolzano, Cauchy and the ‘NewAnalysis’ of the Early Nineteenth Century,”Archive for History of Exact Sciences 6 (1970):372–400.

Cavalieri, Bonaventura 63

� Cavalieri, Bonaventura(ca. 1598–1647)ItalianGeometry, Calculus

Before SIR ISAAC NEWTON and GOTTFRIED WIL-HELM VON LEIBNIZ systematically developed theintegral calculus, a few other mathematicians la-bored as predecessors, drawing on the ideashinted at by ARCHIMEDES OF SYRACUSE. The con-cept of indivisibles—those quantities so smallthat they cannot be divided in half—had begunto take hold, and Cavalieri was one of the firstproponents; his work on integration would laterinspire BLAISE PASCAL, Newton, and Leibniz.

The exact birth date of BonaventuraCavalieri is unknown, and nothing is known ofhis family. He was born in Milan, Italy, and headopted the name Bonaventura upon enteringthe Jesuit religious order as a boy, and he re-mained a monastic throughout his life. In 1616he was transferred to the monastery at Pisa,where he met Castelli, a Benedictine monk andstudent of GALILEO GALILEI. At the time, Castelliwas lecturer in mathematics at Pisa, and he tookCavalieri as his student. The boy quickly mas-tered EUCLID OF ALEXANDRIA, Archimedes, andAPOLLONIUS OF PERGA, and demonstrated a re-markable talent for geometry, sometimes actingas substitute for Castelli. Later, Cavalieri was in-troduced to Galileo, with whom he exchangedmany letters over the years.

From 1620 to 1623 Cavalieri taught theol-ogy at Milan, having been ordained deacon toCardinal Borromeo. During this period he de-veloped his first ideas on the method of indivis-ibles: One views a plane surface as the union ofinfinitely many parallel lines (the indivisibles),so that area is computed from the sum of all theirlengths. In the same fashion, a solid figure wascomposed of infinitely many stacked surfaces, sothat volume could be computed by summing allthe areas. His next assignment was at Lodi, wherehe stayed three years, and in 1626 he became

prior of the monastery in Parma; he sought a lec-tureship at Parma, but without success. Fallingill in 1626 with gout, which plagued himthroughout his life, Cavalieri recovered in Milanand soon announced to Galileo the completionof his Geometria Indivisibilibus Continuorum NovaQuadam Ratione Promota (A certain method forthe development of a new geometry of contin-uous indivisibles). Through the latter’s assis-tance, Cavalieri obtained in 1629 the first chairof mathematics at Bologna, which he held un-til his death on November 30, 1647.

It had become apparent to Cavalieri thatArchimedes was aware of a method for calcu-lating areas and volumes that he was unwillingto disclose—either out of competitive secrecy ora desire to avoid derision from his conservativecolleagues. Cavalieri developed a rational systemof the so-called indivisibles and attempted to es-tablish the validity of this approach. From hisprinciples, Cavalieri deduced several of the ba-sic theorems of integral calculus, but without theformalism of the integral itself. His method ofcalculation, which involves the concept of con-gruence under translation, is shown to be validfor parallelograms and plane figures lying be-tween two parallel lines (and conglomerationsof such).

His contemporaries largely rejected Cavalieri’smethodology, unaware that Archimedes himselfhad utilized similar techniques. Cavalieri ob-tained some basic formulas, such as the powerrule for integration of a polynomial, in 1639,though it had been discovered three years ear-lier by PIERRE DE FERMAT and Gilles de Roberval.He also discovered the volume of solids obtainedby rotating about an axis.

Also in Geometria there is an early formula-tion of the mean value theorem, which statesthat between any two points on a curve can befound a tangent line parallel to the chord con-necting the two points. Cavalieri also investigatedlogarithms, which had recently been inventedby JOHN NAPIER, as well as trigonometry with

64 Chebyshev, Pafnuty Lvovich

applications to astronomy. His Centuria di variiproblemi (A century of various problems) of 1639treated the definition of cylindrical and conicalsurfaces, and also gave formulas for the volume ofa barrel and the capacity of a vault. Among hisother contributions to science are a theory of con-ics applied to optics and acoustics, the idea for thereflecting telescope (apparently prior to Newton),determination of the focal length of a lens, andexplanations of Archimedes’s burning glass.

Cavalieri’s work was a first modern step to-ward calculus, and he should be seen as an es-sential link in the chain between Archimedesand the great 17th-century mathematicianswho developed calculus—Pascal, Leibniz, andNewton, along with JOHN WALLIS and ISAAC

BARROW.

Further ReadingAnderson, K. “Cavalieri’s Method of Indivisibles,”

Archive for History of Exact Sciences 31, no. 4 (1985):291–367.

Ariotti, P. “Bonaventura Cavalieri, Marin Mersenne,and the Reflecting Telescope,” Isis 66 (1975):303–321.

� Chebyshev, Pafnuty Lvovich(1821–1894)RussianProbability

Russian mathematics in the early 19th centurywas relatively inactive, especially in comparisonto the abundant researches carried out by LEON-HARD EULER in St. Petersburg. The primary im-pact of Chebyshev was to revive the pursuit ofmathematical knowledge in Russia; the influ-ence of the Petersburg mathematical school,founded by Chebyshev, extended beyond thattown to Moscow and much of Europe as well.

Pafnuty Chebyshev was born to an aristo-cratic family on May 16, 1821, on the family es-tate in Okatovo, Russia. His father, Lev Pavlovich

Chebyshev, was a retired army officer who hadfought against Napoleon; his mother, AgrafenaIvanovna Pozniakova Chebysheva, bore ninechildren. The family moved to Moscow in 1832,and Chebyshev received his early educationfrom a famous tutor. In 1837 he enrolled in themathematics and physics department of MoscowUniversity, where he came under the influenceof Brashman, who encouraged his students topursue the most important problems of scienceand technology.

At this time Chebyshev wrote his first pa-per, which generalizes the Newton-Raphsonmethod for finding the zeroes of a differentiablefunction; for this work he earned a silver medalin a departmental contest. In 1841 Chebyshevgraduated with his bachelor’s degree, and twoyears later he passed his master’s examinations;meanwhile he published papers on the theory ofmultiple integrals and the convergence of Taylorseries. His master’s thesis, which he defended in1846, was “An Essay on an Elementary Analysisof the Theory of Probability,” and treated SIMÉON

DENIS POISSON’s law of large numbers. This earlywork characterizes Chebyshev’s general ap-proach: the precise calculation of bounds for alimit demonstrated with elementary techniques.

Chebyshev took an assistant professorship atSt. Petersburg, where he worked toward com-pletion of his doctoral thesis. His lectures atPetersburg University ranged from higher alge-bra to integral calculus and the theory of num-bers; his teaching focused on particular topics ofinterest, and the special techniques that wererelevant. While editing an edition of Euler’sworks, Chebyshev became increasingly inter-ested in the theory of numbers. His monographTheory of Congruences earned him his doctoratein mathematics in 1849 and a prize from theAcademy of Sciences. One of the most impor-tant topics considered was the distribution ofprime numbers; Chebyshev made substantialprogress in finding precise asymptotics for theprime number function (given any integer, this

Chebyshev, Pafnuty Lvovich 65

function reports how many prime numbers areless than the given integer).

As a result of this work, which becamequite well known in the scientific community,Chebyshev was promoted to extraordinary pro-fessor in 1850, and 10 years later attained fullprofessor. During this decade Chebyshev inves-tigated the theory of mechanisms, which led tohis study of the best approximation of a givenfunction. This topic involved the uniform ap-proximation of a function by a polynomial. Hehad long possessed an interest in mechanics,which received additional stimulation from vis-iting several factories throughout Europe, andthrough 1856 Chebyshev delivered numerouslectures in practical mechanics. He also con-tinued his research of the integration of alge-braic functions, extending the achievements ofNIELS HENRIK ABEL in the realm of elliptic inte-grals, and lectured on elliptic functions at theuniversity.

For his work in practical mechanics andmathematics, Chebyshev was elected to thePetersburg Academy of Sciences in 1853, ob-taining the chair of applied mathematics. In1856 he became associated with the ArtilleryCommittee, which was concerned with devel-oping Russian ballistic technology; Chebyshevintroduced mathematical solutions for artilleryproblems, and in 1867 he gave a formula for thetrajectory of spherical missiles. Meanwhile, heactively contributed to the Ministry of Education,endeavoring to improve the teaching of sciencein secondary schools.

Chebyshev is recognized for his foundationof the Petersburg mathematical school, whichconsisted of an informal collection of his nu-merous pupils, including ANDREI MARKOV. Theagenda of this community was directed towardthe solution of pure and applied mathematicalproblems that arose from practical applications.Chebyshev was a proponent of finding practicalsolutions to problems, and he often tried to de-vise algorithms that would provide a numerical

answer. Toward this end he advanced the use ofinequalities, which allowed the construction ofbounds and approximations simpler than an ex-act solution. The school pursued many branchesof mathematics, but was connected in its devel-opment of thought from Euler, whose works hadexercised a great influence upon Chebyshev.

From the 1850s a dominant aspect ofChebyshev’s work was the theory of best ap-proximation of functions, which led to researchof orthogonal polynomials, limiting values of in-tegrals, the theory of moments, interpolation,and methods of approximating quadratures; healso improved the methodology of continuedfractions. Starting in the 1860s Chebyshev wroteseveral articles about technological inventions,and returned to the theory of probability. Twoimportant works include his 1867 paper “OnMean Values,” in which he generalizes the lawof large numbers by implementing the so-calledChebyshev inequality, and an 1887 work, “OnTwo Theorems Concerning Probability,” thatgeneralized the central limit theorem. This lat-ter achievement effectively extended the normaldistribution’s range of applicability in the fieldof statistics.

Many other problems of applied mathemat-ics piqued his interest; Chebyshev investigatedthe problem of binding a surface with a cloth,giving rise to the “Chebyshev net,” and he con-sidered the equilibrium of a rotating liquid mass.Chebyshev built a calculating machine in thelate 1870s that performed multiplication anddivision.

In 1882 he retired from Petersburg University,but he kept his academic salon active, supplyingthe new generation of intellectuals with ideasand encouragement. He continued working un-til his death on December 8, 1894. Chebyshevhad been elected to numerous academic soci-eties and had greatly influenced the develop-ment of Russian mathematics. The Petersburgschool developed whole new areas of study, butwas mainly characterized by Chebyshev’s vision

66 Ch’in Chiu-Shao

of the unity of theory and practice. His numer-ous contributions to diverse areas of mathemat-ics, as well as his industrious mentorship, haveserved to propagate Chebyshev’s ideas through-out modern mathematics.

Further ReadingButzer, P., and F. Jongmans. “P. L. Chebyshev

(1821–1894): A Guide to His Life and Work,”Journal of Approximation Theory 96, no. 1 (1999):111–138.

———. “P. L. Chebyshev (1821–1894) and HisContacts with Western European Scientists,”Historia Mathematica 16 (1989): 46–68.

Stander, D. “Makers of Modern Mathematics: PafnutyLiwowich Chebyshev,” Bulletin of the Institute ofMathematics and Its Application 26, nos. 1–2(1990): 18–19.

� Ch’in Chiu-Shao(ca. 1202–ca. 1261)ChineseAlgebra

Mathematics in 13th-century China was mainlyconcerned with solutions of algebraic equationsand determination of areas of certain shapes, withimportant applications to finance, commerce,agriculture, and astronomy. Ch’in Chiu-Shaoworked on problems of this type, leaving behindsome general methods for their solution. He wasalso an important influence on later Chinesemathematicians, such as CHU SHIH-CHIEH.

Ch’in Chiu-Shao was born around 1202 inthe Szechuan province of China; his father wasa civil servant, and Ch’in would follow his fa-ther’s vocation. He was an ill-disciplined youth,and later became a vindictive, unprincipledadult. During one of his father’s banquets, Ch’inrecklessly hurled a stone into the midst of theassembled guests in a display of his archery skills;later in life, he would become known for poi-soning his adversaries.

In 1219 Ch’in joined the army as the cap-tain of a squad of volunteers, who helped sup-press a local uprising. In 1224 and 1225 Ch’infollowed his father to the capital city ofHangchow. There he studied astronomy underthe tutelage of the official astronomers. He soonleft the capital when his father was transferredto another position, and in 1233 it is known thatCh’in served as a sheriff.

In 1236 the Mongols invaded Szechuan, andCh’in fled east, becoming a vice-administratorin Hupeh Province, and later governor of Ho-hsien in Anhwei Province. His next post was inNanking in 1244, which he held briefly, and fi-nally he came to Wu-hsing, where he wrote hisMathematical Treatise in 1247. According to hisown account, Ch’in learned his mathematicsfrom an unnamed mathematician.

Ch’in’s manuscript, his only known mathe-matical writing, consisted of nine parts, each ofwhich had two chapters. They deal with inde-terminate analysis, astronomical calculations,land measurement, surveying via triangulation,land tax, money, structural works, military mat-ters, and barter, respectively. Ch’in representedthe height of Chinese attainment in the arenaof indeterminate analysis, which had first ap-peared in the fourth century. One type of prob-lem involved finding a number with variousgiven remainders for given divisors; these typesof problems now fall generally under the domainof the so-called Chinese remainder theorem.One could apply these results to calendar calcu-lations and military logistics, among otherthings.

Ch’in’s method for such remainder problemswas general; he gave a formula for solving suchquestions that was not discovered in Europe un-til the 16th century. This technique was appli-cable when the various divisors were relativelyprime (when they themselves had no commonfactors); but Ch’in also extended his method tothe more general situation when the divisorswere not relatively prime. This technique came

Chu Shih-Chieh 67

to be known as “the Great Extension methodof searching for unity.” Of course, Ch’in did notuse the modern notations of modular arith-metic, but introduced many technical terms ofhis own, such as celestial monads and operationnumbers.

In solving algebraic equations, Ch’in used acounting board with rods arranged in certain for-mations to represent numbers and unknownquantities. In this way, he would calculate solu-tions to various equations, with degree as highas 10. His method, identical to that discoveredby Paolo Ruffini in 1805, was labeled the “har-moniously alternating evolution”; it seems, how-ever, that Ch’in was not the inventor of thistechnique, as his contemporaries were also fa-miliar with it.

It is interesting that Ch’in’s book gives var-ious values for pi, such as 22/7 and square rootof 10, as well as the old value of 3. He gives theareas of several geometric shapes, such as trian-gles (without the use of trigonometry), circulararcs, and quadrangles. He treats several simulta-neous linear equations in several variables, andCh’in also discusses the sum of certain series ofnumbers. Finally, he deals with problems in-volving finite differences, as these were of in-terest to calendar makers.

After this work Ch’in returned to the civilservice in 1254, and he was appointed as gover-nor of Hainan in 1258. He was dismissed threemonths later on charges of corruption, and hereturned home with an immense fortuneamassed from his acceptance of numerous bribes.He later became an assistant to his good friendWu Ch’ien (of whom it is related that Ch’incheated him out of a plot of land), and followedhim to the provinces of Chekiang andKwangtung. Shortly after receiving an appoint-ment in Mei-hsien, Ch’in died. The year of hisdeath is estimated to be 1261.

Ch’in was known as an expert poet, archer,fencer, equestrian, and musician, as well as beinga foremost mathematician of his age and country.

The stories about him paint a disreputable pic-ture; it is related that he punished a female mem-ber of his household by starvation. However, hismathematical talent is undisputed, and his mas-tery of indeterminate analysis reserves him aspace in mathematical history.

Further ReadingMartzloff, J. A History of Chinese Mathematics. Berlin

and Heidelberg: Springer-Verlag, 1997.Mikami, Y. The Development of Mathematics in China

and Japan. New York: Chelsea PublishingCompany, 1974.

Shen, K. “Historical Development of the ChineseRemainder Theorem,” Archive for History ofExact Sciences 38, no. 4 (1988): 285–305.

� Chu Shih-Chieh (Zhu Shijie)(ca. 1279–ca. 1303)ChineseAlgebra

Much of Chinese mathematics focused on prob-lems of algebra and the summation of series. ChuShih-Chieh represented a significant advance inthe knowledge of these areas, adding to the workof great 13th-century mathematicians such asCH’IN CHIU SHAO. He was certainly the greatestmathematician of his time and country.

Little is known of the personal life of ChuShih-Chieh, but he was born sometime prior to1280 and died after the publication of his sec-ond book in 1303. In the preface to this work,called the Precious Mirror of the Four Elements,the author claims to have spent 20 years jour-neying around China as a renowned mathe-matician. Afterward, he visited the city ofKuang-ling, where he attracted numerous pupils.It seems that Chu Shih-Chieh flourished in thelatter part of the 13th century after the reunifi-cation of China through the Mongol conquest.

His first work, Introduction to MathematicalStudies of 1299, was a textbook for beginners.

68 Chu Shih-Chieh

The Four Elements contains the method of thatname, evidently invented by Chu. This was the“method of the celestial element” extended tofour variables; this prior method was apparentlywell known in China, though no record of it cur-rently exists. Chu’s method represents the fourunknown quantities graphically, and he solvedhigh-degree equations by using a transformationmethod. Chu does not describe this, but he wasable to solve complicated quartic (fourth degree)equations. When exact solutions of such equa-tions were not possible, Chu used an approxi-mation. It is interesting that in finding squareroots, Chu used a substitution technique knownto Ch’in Chiu-Shao that is similar to modernmethods.

In the same book, Chu has a drawing ofPascal’s triangle, containing the coefficients ofthe expansion of a binomial. He gives an expla-nation of its use, and refers to this diagram asthe “old method” (it was already known to 12th-century Chinese mathematicians). Considerableinterest focused on the calculation of series, suchas the sum of n consecutive integers. Chu’s workrepresents an advance over previous knowledge

of such sums, and he applied his results to cross-sections of pyramids and cones (for instance, thedetermination of how many balls are in a py-ramidal stack of a given height). After Chu,Chinese mathematicians made little additionalprogress in the study of such higher series.

Chu also examined finite differences, whichwere important to Chinese astronomers in theirformulas for celestial motion. Finite differencemethods were known since the seventh centuryin China, but Chu applied them to several wordproblems.

Chu Shih-Chieh helped to advance Chinesemathematics through his techniques for solvingalgebraic equations and his computation of se-ries. His works remained hidden for centuries,but when rediscovered they stimulated addi-tional research in the 18th century.

Further ReadingMartzloff, J. A History of Chinese Mathematics. Berlin

and Heidelberg: Springer-Verlag, 1997.Mikami, Y. The Development of Mathematics in China

and Japan. New York: Chelsea PublishingCompany, 1974.

D

69

� Dedekind, Richard(1831–1916)GermanAnalysis, Logic

Much of the mathematical work of the later 19thcentury was concerned with establishing rigorousfoundations for previous mathematical topics,such as the concepts of function, infinity, andnumber. Dedekind worked in this latter area, be-ing concerned with the definition of the realnumber and the concept of continuity.

Richard Dedekind was born in Brunswick,Germany, on October 6, 1831. His father, JuliusDedekind, was a professor of law at the CollegiumCarolinum of Brunswick, and his mother,Caroline Emperius, was the daughter of anotherprofessor at the same institution. RichardDedekind was the youngest of four children inthis intellectual family; he lived with his secondsister for most of his life. As a youth, Dedekindattended the local Gymnasium, where he even-tually switched his focus from physics to mathe-matics, claiming that physics was too disordered.At the Collegium Carolinum, which CARL

FRIEDRICH GAUSS had also attended, Dedekindmastered analytic geometry, differential and in-tegral calculus, and higher mechanics.

In 1850 he entered the University ofGöttingen, and he developed a close friendship

with BERNHARD RIEMANN, while attending lecturesby Moritz Stern, Wilhelm Weber, and CarlGauss. Only two years later Dedekind obtainedhis doctorate under Gauss with a thesis onEulerian integrals; he went on to Berlin to at-tend lectures by CARL JACOBI and GUSTAV PETER

LEJEUNE DIRICHLET, filling in the rest of his edu-cation. In 1854 he obtained a lectureship at theUniversity of Berlin, where he taught probabil-ity and geometry. Also at this time, Dedekindbecame friends with Dirichlet, who expanded hissocial and intellectual horizons. In 1858 Dedekindobtained an appointment at the Polytechnikumin Zurich, and four years later returned to hishometown, Brunswick, where he remained un-til his death.

Dedekind is well known among mathemati-cians for the so-called Dedekind cut, which wasan element in his construction of the real num-bers. He had already noticed the lack of a trulyrigorous foundation of arithmetic; he success-fully constructed a purely arithmetic definitionof continuity, and exactly formulated the notionof an irrational number. In this connection,Dedekind’s work builds on EUDOXUS OF CNIDUS’sancient theory of proportion as a foundation forthe real numbers, although the two versions arenot exactly identical; Dedekind established thefact that Euclidean postulates alone, devoid of aprinciple of continuity, could not establish a

70 Democritus of Abdera

complete theory of real numbers. His conceptshave had enduring significance for the field ofmathematical analysis, especially through his useof order to understand the real numbers.

He published these ideas in an 1872 manu-script called Stetigkeit und Irrationale Zahlen(Continuity and irrational numbers), which es-tablished Dedekind as a leading researcher in thefoundations of mathematics, along with GEORG

CANTOR and BERNHARD BOLZANO. His 1888 bookon numbers—Was Sind und was Sollen die Zahlen(What Numbers Are and Should Be)—delineatedthe logical theory of number, treating such top-ics as the continuity of space, the essence of arith-metic, and the role of numbers in geometry. Oneimportant discovery was the definition of in-finiteness of a set through mappings, which wasvital to Cantor’s later research into set theory.

There are many similarities between Gaussand Dedekind, including their personalities:Like Gauss, Dedekind was an intense, disci-plined worker who enjoyed a frugal lifestyle. Hewas a deep thinker who preferred mathematicalnotions to useful notations. Due to their closekinship, and the fact that Dedekind understoodGauss’s work better than anyone else, he editedseveral of Gauss’s unpublished manuscripts, andwas able to comment cogently upon theseworks. This project led Dedekind toward ex-amination of the complex numbers, and he gavethe general definition of an algebraic ideal andestablished several classical results. This workin algebra, for which Dedekind is most famous,led to many fruitful developments by later math-ematicians, such as EMMY NOETHER and DAVID

HILBERT.Dedekind was active at the Polytechnikum

of Brunswick, of which he assumed the direc-torship from 1872 to 1875. He received manyhonorary doctorates during his life and had awide number of correspondents. In 1894 he be-came an emeritus professor, and after his deathon February 12, 1916, mathematicians in manycountries mourned him.

Dedekind’s contribution to mathematicsmight be measured through the quantity of ideasnamed after him—about a dozen. His contribu-tions to the foundations of number allowed realanalysis to progress, developing a deeper knowl-edge of the real numbers and the concept of con-tinuity; his theorems in algebraic ideals havestimulated much additional activity in the 20thcentury.

Further ReadingBell, E. Men of Mathematics. New York: Simon and

Schuster, 1965.Edwards, H. “Dedekind’s Invention of Ideals,” Bulletin

of the London Mathematical Society 15, no. 1(1983): 8–17.

Ferreirós, J. “On the Relations between Georg Cantorand Richard Dedekind,” Historia Mathematica 20(1993): 343–363.

Gillies, D. Frege, Dedekind, and Peano on the Foundationsof Arithmetic. Assen, Netherlands: Van Gorcum,1982.

� Democritus of Abdera(ca. 460 B.C.E.–ca. 404 B.C.E.)GreekGeometry

Democritus is numbered among the very earlyGreek mathematicians who influenced the laterdevelopment of geometry. Although his mathe-matical works have not survived, it is clear thatDemocritus possessed an extensive interest inconics and other aspects of solid geometry. Thegreat Greek geometers APOLLONIUS OF PERGA

and ARCHIMEDES OF SYRACUSE came much later,but even they studied some of the problems in-vestigated by Democritus.

Information on Democritus’s life is distortedby several unverifiable accounts. One chronol-ogy places his birth after 500 B.C.E. and his deathabout 404 B.C.E., and represents him as theteacher of Protagoras of Abdera; another version

Democritus of Abdera 71

frames his life much later, depicting him as acontemporary of Socrates, being born around460 B.C.E. and dying in about 404 B.C.E. Mostscholars accept the latter dates.

Democritus was born in Abdera, Thrace, andwas fairly wealthy. He traveled widely, visitingAthens, and was known as the “laughing philoso-pher” since he found the follies of humankind

amusing. Although more than 60 of his workswere contained in the library at Alexandria,none of these have survived intact, so knowl-edge of his writings comes completely from quo-tations and commentaries. Democritus’s pupilspropagated his doctrine, and one of them was ateacher of Epicurus. Hence it is thought thatEpicureanism represents an elaboration ofDemocritus’s physical theories.

The theory of atoms did not originate withDemocritus—its roots have been traced back tothe second millennium B.C.E.—but Democritusdeveloped the idea extensively. Atoms were in-divisible objects, the basic building blocks ofboth matter and soul; perceived differences inobjects arise from the arrangement and positionof atoms within the ambient void. Democritus’sEarth was flat and long, with earthquakes causedby fluctuations in the quantity of water in un-derground cavities.

Democritus made use of the sphere in hisnatural philosophy, as both fire and soul consistof spherical atoms. He viewed the sphere as pureangle, by which he meant that it was uniformlybent. It is thought that Democritus conceived ofthe sphere as a polyhedron with minutely smallfaces, and therefore believed that a tangentmakes contact with a circle over a distancerather than in a single point.

He discusses the question of whether thetwo contiguous surfaces produced when a coneis sliced horizontally are equal. If equal, it mightseem that a cone is more like a cylinder; if un-equal, the cone must have minute steps. Thesequestions strike to the heart of divisibility andthe concept of the continuum, which wouldbecome so intriguing for later Greeks andmodern European mathematicians. Democri-tus’s own opinion on the dilemma is unclear.Archimedes also records that Democritus exam-ined the ratio of sizes between cylinders, pyra-mids, and prisms of the same base and height,but we are unaware of the extent of Democritus’sknowledge.

Democritus developed early ideas about thecontinuum. (Courtesy of the National Library ofMedicine)

72 De Morgan, Augustus

Democritus also explored the field of biol-ogy, and he derived a doctrine of the evolutionof culture that represents the development fromlesser to greater civilizations; this was contraryto the prevailing view that humankind hadfallen from an original golden age.

Democritus commanded a significant swayover later thought in natural philosophy, and hisatomism later influenced Epicureanism. It is dif-ficult to gauge the extent of his mathematicalaccomplishments, but it is apparent that his re-searches and questions prompted the inquiries offollowing mathematicians.

Further ReadingBrumbaugh, R. The Philosophers of Greece. New York:

Crowell, 1964.Guthrie, W. A History of Greek Philosophy.

Cambridge: Cambridge University Press, 1981.Hahm, D. “Chrysippus’ Solution to the Democritean

Dilemma of the Cone,” Isis 63, no. 217 (1972):205–220.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Laertius, D. Lives, Teachings, and Sayings of FamousPhilosophers. Cambridge, Mass.: Harvard UniversityPress, 1959.

Russell, B. History of Western Philosophy and ItsConnection with Political and Social Circumstancesfrom the Ancient Times to the Present Day. London:Allen and Unwin, 1961.

� De Morgan, Augustus(1806–1871)EnglishLogic

Aristotelian logic, for centuries the dominantmode of logical argument, was unable to evalu-ate logical statements which involved quantity.This inadequacy pointed to a general need toplace logic on a mathematical foundation. Thiseffort, which took place in the 19th and 20th

centuries, was greatly assisted by Augustus DeMorgan.

Augustus De Morgan was born in India onJune 27, 1806. His father was an officer in theIndian colonial army, and his mother was thedaughter of a pupil of ABRAHAM DE MOIVRE; thisconnection may account for De Morgan’s latertalent in mathematics. While he was still quiteyoung, the family relocated to England, eventu-ally settling in Taunton. De Morgan attendedseveral private schools, gaining facility in Latin,Greek, and Hebrew, as well as a fervent interestin mathematics by age 14.

In 1823 he entered Trinity College. He latergraduated fourth in his class. Upon the strongrecommendation of his tutors, De Morgan ob-tained the chair of mathematics at UniversityCollege in 1828. His life was characterized bystrong convictions about Christianity; he wishedto avoid even the appearance of hypocrisy, andas a result refused a fellowship at Cambridgethat required his ordination. The college coun-cil dismissed a fellow professor in 1831 withoutgrounds, and De Morgan immediately resignedon principle. He resumed the post in 1836 uponthe death of his successor. Meanwhile, DeMorgan became a fellow of the AstronomicalSociety in 1828, and later helped found theLondon Mathematical Society. In 1837 he mar-ried Sophia Elizabeth Frend.

He was interested in all branches of knowl-edge, and published prolifically—more than 850articles and numerous textbooks on arithmetic,algebra, trigonometry, calculus, complex num-bers, probability, and logic. These texts areknown for their logical presentation and preci-sion—as a teacher he endeavored to demon-strate principles rather than techniques. DeMorgan’s interests extended to the history ofmathematics: he wrote a biography of SIR ISAAC

NEWTON and composed one of the first signifi-cant works in the field of scientific bibliography.

De Morgan’s mathematical contributionslay mainly in the fields of logic and analysis. In

Desargues, Girard 73

an 1838 article he invented the term mathemat-ical induction as a technique of proof, formalizinga method that had long been used by mathe-maticians. His Differential and Integral Calculus(1842) gives a precise analytical formulation ofAUGUSTIN-LOUIS CAUCHY’s intuitive concept ofa limit and a discussion of convergence rules forinfinite series. This includes De Morgan’s rule,a test for convergence involving the limit of thesequence of a series, which is useful when sim-pler tests are not informative. Elsewhere DeMorgan describes a system that he created as“double algebra,” which assisted in the geomet-rical interpretation of complex numbers, andsuggested the idea of quaternions to SIR WILLIAM

ROWAN HAMILTON.Most important are his researches into logic.

De Morgan was among the first to perceive theneed for a better system than the Aristotelian,which could not handle statements involvingquantity. Following the ideas of George Bentham,De Morgan invented algebraic notations for ex-pressing logical statements, and his work laterinspired GEORGE BOOLE’s analytical formulation.Some of the basic rules of Boolean algebra aretherefore called the De Morgan formulas. DeMorgan was the first person to present a logicalcalculus of relations; he used an algebraic nota-tion to express relations between objects, andwent on to consider the composition of relationsand the inverse of a relation.

De Morgan’s other accomplishments includethe promotion of a decimal monetary system, analmanac covering 4,000 years of lunar dates, anda probability text giving applications to life in-surance. He died in London on March 18, 1871.

De Morgan is very important in the historyof mathematics, as he substantially developedthe algebraic foundation of logic (this wouldbe further worked out by subsequent mathe-maticians such as Boole), and thus was an earlyadvocate of the view that logic should be con-sidered as a branch of mathematics, not a sep-arate field of knowledge.

Further ReadingHalsted, G. “Biography: De Morgan,” The American

Mathematical Monthly 4 (1897): 1–5.Rice, A. “Augustus De Morgan: Historian of Science,”

History of Science 34 (1996): 201–240.———.“Augustus De Morgan (1806–1871),” The

Mathematical Intelligencer 18, no. 3 (1996):40–43.

Richards, J. “Augustus De Morgan, the History ofMathematics, and the Foundations of Algebra,”Isis 78, no. 291 (1987): 7–30.

� Desargues, Girard(1591–1661)FrenchGeometry

In the late 16th century, mathematics was on theeve of a dramatic rebirth. It would emerge fromthe darkness of the medieval age, and the prin-cipal mathematicians of this time sought to re-cast and develop classical Greek mathematics.Concurrent with the powerful algebraic approachto geometry developed by RENÉ DESCARTES,Desargues advocated a unified graphical per-spective. His ideas influenced such great thinkersas BLAISE PASCAL and SIR ISAAC NEWTON, eventhough they were inevitably overshadowed bythe power of the Cartesian system.

Girard Desargues was born into a family ofnine children on February 21, 1591, in Lyons.His father was an affluent tithe collector, andGirard’s early studies took place in Lyons.However, little is known of his early years andeducation, and his first recorded scientific ac-tivity took place in 1626, when he proposed thatthe state construct machines to raise the waterlevel of the Seine.

Around 1630 Desargues became friendlywith several Paris mathematicians, such asMarin Mersenne and Gilles de Roberval; he reg-ularly attended the meetings of the AcadémieParisienne, in which the young Blaise Pascal

74 Desargues, Girard

would later participate. In 1636 Desargues pub-lished a work describing his so-called universalmethod of perspective. There he outlined a vastresearch program: to unify the various graphicaltechniques (such as those used by architects anddraftsmen) via his “universal methods,” whilealso incorporating projective geometry into thebody of mathematical knowledge through therigorous study of perspective. Projective geome-try had few followers until the 19th century, andDesargues’s work was greatly overlooked (andeven ridiculed) during his own lifetime; its sig-nificance would be realized centuries later.

A competing work by Jean de Beaugrandevoked intellectual discussions on the topics ofcenter of gravity, optics, and tangents, as well asother geometrical issues. Desargues was an avidparticipant in these disputations, distinguishinghimself by his intent to understand the mostgeneral aspect of given problems. Although healienated Beaugrand, he gained the esteem ofDescartes and Mersenne. In contrast toDescartes’s program to give geometry an alge-braic foundation, Desargues wished to extendthe influence of geometrical methods to graph-ical techniques via mechanics. His Brouillonprojet d’une atteinte aux événements des recontresdu cône avec un plan (Rough draft for an essayon the results of taking plane sections of a cone)of 1639 gave a description of conic sections fromthe perspective of projective geometry; thiswork was not popular, perhaps due to its diver-gence from the Cartesian approach. However,Blaise Pascal was able to appreciate the worthof Desargues’s Brouillon project, and Pascal wrotehis Essay pour les coniques (Essay on conic sec-tions) as a tribute. Although this work influ-enced other great mathematicians, such asNewton, its full impact was not realized untilthe 19th century, when geometers acquired a re-newed interest in projective geometry. This lackof popularity was surely the result of the in-creasing interest in the problems of infinitesi-mal calculus.

Beaugrand criticized Desargues’s work vehe-mently, seemingly for personal rather than pro-fessional reasons. In 1640 Desargues publishedan essay concerning stonecutting, applying histechniques to practical graphing methods.However, his useful innovations were contraryto the established practice of the powerful tradeguilds, and thus Desargues attracted the ani-mosity of veteran artisans.

In 1641 Desargues became interested in theproblem of determining various circular sectionsof cones having a conic base. His general solu-tion relied solely on pure geometry. Roberval,Descartes, and Pascal became interested in thisproblem, and Desargues later generalized histechnique.

Subsequently Desargues became involved ina dispute over the priority of his methods. Hehad worked to spread his graphical techniquesamong stonecutters, but in 1642 an anonymouspublication plagiarized and distorted his work.The resulting debate injured Desargues’s credi-bility and confidence. Desargues appointed thetask of dissemination of his work to his discipleAbraham Bosse, an engraver, who later publishedtwo treatises discussing Desargues’s method. Anew dispute arose in 1644 with a stonecutternamed Curabelle, with the apparent result thatDesargues decided to forgo further publishing.However, Bosse continued to compose treatiseson his master’s method of perspective.

After 1644 Desargues’s scientific activity de-clined. He started a new career as architect, fash-ioning some spectacular structures of a delicatefigure. For example, Desargues used his mathe-matical knowledge to constuct curved staircasesthat were both functional and elegant. He im-plemented his techniques directly in practicalapplications, and in this way silenced the accu-sation of his enemies that his methods werepurely theoretical. His engineering feats were ex-tensive, including a system for raising water,which was installed near Paris. He died in Parisin October 1661.

Descartes, René 75

Desargues must be credited as being one ofthe founders of projective geometry—the study ofspaces of straight lines meeting infinitely far awayfrom the observer. He also contributed to theknowledge of conics, introduced projective trans-formations, and furthered the development ofgraphical techniques useful in drawings. Sadly, hiswork was savagely contested in his own lifetime,and even among supportive mathematicians itreceived little circulation due to his cumbersomestyle of writing. His labors bore fruit only cen-turies after his death, when his papers were re-discovered by grateful 19th-century geometers.

Further ReadingField, J. The Invention of Infinity: Mathematics and Art

in the Renaissance. New York: Oxford UniversityPress, 1997.

———. “Linear Perspective and the ProjectiveGeometry of Girard Desargues,” Istituto e Museodi Storia della Scienza Firenze. Nuncivs. Annali diStoria della Scienza 2, no. 2 (1987): 3–40.

———, and Gray, J. The Geometrical Work of GirardDesargues. New York: Springer-Verlag, 1987.

� Descartes, René(1596–1650)FrenchGeometry, Algebra

The 17th century was a time of heightened ac-tivity in science and mathematics, and RenéDescartes is one of the men who substantially af-fected the development of scientific knowledgeand philosophy. His idealistic system attemptedto explain all of human knowledge from a fewbasic principles. Although his program wasoverly optimistic, Descartes’s influence was enor-mous, especially in mathematics. Perhaps hisgreatest contribution was his placement ofgeometry within the domain of algebra, allow-ing mathematicians to study curves and figuresthrough analysis of algebraic equations.

René Descartes was born in La Haye,France, on March 31, 1596, to an aristocraticfamily. His father was a member of the parlia-ment of Brittany, and his mother was a wealthynoblewoman. Descartes later inherited an estatein Poitou from her, which granted him financialindependence and the leisure to pursue scien-tific studies. Descartes was educated by Jesuits,and became familiar with modern developmentsin mathematics and physics, including the re-cent researches of GALILEO GALILEI, as well asphilosophy and classical literature.

He graduated with a law degree from theUniversity of Poitiers and became a volunteer inthe army of Prince Maurice of Nassau. On thenight of November 10, 1619, Descartes reached

René Descartes, an outstanding French philosopher,introduced the idea of analytic geometry; namely, thatgeometric objects could be represented by equationsthat could be graphed on coordinate axes. (Courtesyof the National Library of Medicine)

76 Descartes, René

two conclusions after a day of solitary thought:that a program of true knowledge must be car-ried out by himself, and that methodical doubtof current philosophical knowledge was the rightway to begin such a task. He would look for self-evident principles as a starting point, from whichone could deduce each of the sciences.

As a result of this epiphany, Descartes’s laterwork was characterized by intensity, confidence,and a commitment to working alone. Later,Descartes would increasingly realize the impor-tance of experimentation and empirical obser-vation to the attainment of true knowledge.However, he did not immediately embark on thisintellectual quest, but continued his travels inEurope until 1628. In this year, after a success-ful public debate on the topic of how to distin-guish certain and probable knowledge, Descarteswithdrew to a solitary life of scientific work inthe Netherlands.

Descartes contributed many radical ideas toscience. He treated animals and humans as me-chanical objects and viewed the laws of motionas the ultimate laws of nature. In his opinion,science should not only demonstrate informa-tion, but it should explain as well. Some of hiswork on cosmology was merely qualitative, andhis work suffered from a lack of empirical veri-fication. However, Descartes spent an immenseamount of time in experimentation in anatomy,chemistry, and optics.

As a mathematician, Descartes greatly ad-vanced the discipline of algebra and laid thefoundation for analytic geometry. Up to histime, few of the modern algebraic notationswere employed. Descartes introduced the nowfamiliar alphabetic symbols x, y, and z for un-known quantities, as well as superscripts for thepowers of a variable. For example, x2 was alwaysinterpreted as the area of a square on the linesegment of length x; but Descartes abstractedthe meaning of this symbol, so that one couldmanipulate x2 without reference to any geo-metric construction.

Descartes’s main goal in his Géométrie(Geometry), his masterpiece of 1637, was to ap-ply algebra to geometry, providing a convenientnotation for analyzing figures. He defined the sixbasic algebraic operations (addition, multiplica-tion, raising to a power, and their inverses of sub-traction, division, and taking a root), and definedan algebra of lines that extended the initial no-tions of the Greeks. But his most important ideawas the graph of a function: Given a function,such as a polynomial f(x), one could draw thecorrespondence between y and x via the equa-tion y = f(x) by using coordinate axes.

In another part of Géométrie, Descartes de-scribes how to construct a normal line at any pointof a curve by constructing an appropriate coordi-nate system and inscribing a circle that contactsthe curve at one point. His method is similar toPIERRE DE FERMAT’s method for finding extrema ofa curve, and constitutes one of the first steps inthe development of differential calculus. He alsogives a purely algebraic theory of equations andstates the fundamental theorem of algebra.

Although this work was an influential con-tribution to mathematics, it did not representthe total of Descartes’s knowledge: His insistenceon clear deduction from intuitive principles prevented him from stating or accepting morequestionable ideas, such as the concept of theinfinitesimal. After he completed the Géométrie,Descartes’s mathematical studies were for themost part complete, and he spent the rest of histime on philosophy.

Descartes remained in the Netherlands un-til 1649, when he accepted a position as courtphilosopher to Christina, the queen of Sweden.Queen Christina interrupted his lifelong habitof sleeping through most of the morning, andDescartes died on February 11, 1650, from ex-posure to the cold morning air.

Descartes was an excellent scientist. Heforms an interesting contrast with SIR ISAAC NEW-TON, who emphasized the vital role of experi-mentation and observation. Descartes, however,

Diophantus of Alexandria 77

was more concerned with carefully reasoned de-duction from a few basic principles, and he wasoptimistic that induction (that is, knowledge ob-tained through experimentation) would eventu-ally be unnecessary. In practice, Descartes foundthis goal impossible to obtain, and he was forcedto experiment in order to attain the knowledgeof natural phenomena that he desired. His ra-tionalistic approach to science and philosophyis an enduring legacy for modern man. For math-ematics, his development of algebraic methodsfor geometry revolutionized the study of curvesand figures, giving a tremendous forward push tothese disciplines.

Further ReadingClarke, D. Descartes’ Philosophy of Science. University

Park: Pennsylvania State University Press, 1982.Gaukroger, S. Descartes: Philosophy, Mathematics, and

Physics. Sussex, U.K.: Harvester Press, 1980.Haldane, E. Descartes: His Life and Times. London: J.

Murray, 1905.Pearl, L. Descartes. Boston: Twayne Publishers, 1977.Scott, J. The Scientific Work of René Descartes. New

York: Garland, 1987.Shea, W. The Magic of Numbers and Motion: The

Scientific Career of René Descartes. Canton, Mass.:Science History Publications, 1991.

Sorell, T. Descartes. New York: Oxford UniversityPress, 1987.

� Diophantus of Alexandria(ca. 200–284)GreekNumber Theory, Algebra

By the third century, Greek mathematics was al-ready in decline. Diophantus represents a lonepinnacle of thought and activity (excepting thelater ARCHIMEDES OF SYRACUSE), and is thefounder of the field of indeterminate analysis(the solving of algebraic equations with morethan one unknown). The number theory of

16th- and 17th-century Europe can trace muchof its genesis to Diophantus’s labors.

As with many ancient mathematicians,Diophantus’s life is a mystery. We know that helived in the third century in Alexandria, and heis thought to have married at age 33 and died atage 84. His four surviving writings are Moriastica,Porismata (Porisms), Arithmetica (Arithmetic),and a fragment of On Polygonal Numbers. The firstis concerned with computations involving frac-tions, and the second is a collection of proposi-tions cited in the Arithmetica. The fragment of OnPolygonal Numbers uses geometric proofs for cer-tain results on polygonal numbers that involve thenumber of vertices and sides of a given polygon.

The Arithmetica deals with practical, com-putational arithmetic; it is a collection of prob-lems that Diophantus solves in a great variety ofways. Through his solutions he demonstrated hisvirtuosity as a mathematician, but presented nogeneral technique or theory. Using certain tricks,Diophantus would commonly reduce the degreeof the equation and the number of unknowns(as many as 10) to obtain a solution.

Six books of the Arithmetica have survived,in which Diophantus introduces the techniquesof algebra and treats many specific problems. Hisnotation differs substantially from the modernsymbolism, and this must have hampered the de-velopment of general techniques for equationswith multiple unknowns. There are 189 prob-lems, ordered from simplest to most difficult,treating various determinate and indeterminateequations. Among the great diversity of mate-rial, Diophantus considers multiple equations ofpolynomials as well as the decomposition ofnumbers into parts (such as sums of squares).

In solving such problems, Diophantus isaware that multiple solutions are possible, al-though he discounts negative numbers, as thesedid not yet exist. Rational number solutionswere acceptable, but irrationals such as squareroots were excluded. For determinate linear andquadratic equations, Diophantus used balancing

78 Dirichlet, Gustav Peter Lejeune

and completion. He would typically reduce thenumber of unknowns and the degree through themethod of “false position” and by making sub-stitutions; and he made approximations utilizingalgebraic inequalities when direct methods werenot possible. It is noteworthy that Diophantus’ssolutions are remarkably free from error.

Through his writings, Diophantus demon-strates himself to be a first-rate mathematicianand ingenious problem-solver. He probably drewupon current Greek knowledge of procedures forsolving linear and quadratic problems, thoughDiophantus made great advances over his pred-ecessors. His work influenced later Arabic math-ematicians, who absorbed the corpus of Greekmathematical thought. When European mathe-maticians such as FRANÇOIS VIÈTE and PIERRE DE

FERMAT rediscovered him in the 17th century,the so-called Diophantine problems became acornerstone of number theory.

Further ReadingHeath, T. Diophantus of Alexandria: A Study in the

History of Greek Algebra. New York: DoverPublications, 1964.

———. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Klein, J. Greek Mathematical Thought and the Origin ofAlgebra. Cambridge, Mass.: MIT Press, 1968.

Sesiano, J. Books IV to VII of Diophantus’ ‘Arithmetica’in the Arabic Translation Attributed to Qusta ibnLuqa. New York: Springer-Verlag, 1982.

Swift, J. “Diophantus of Alexandria,” The AmericanMathematical Monthly 63 (1956): 163–170.

� Dirichlet, Gustav Peter Lejeune(1805–1859)GermanNumber Theory, Analysis, DifferentialEquations

In the beginning of the 19th century inGermany, CARL FRIEDRICH GAUSS had already

made outstanding discoveries in the theory ofnumbers. Of his many successors, Dirichlet ismemorable as a mathematician of great abilitywho significantly extended the knowledge ofnumber theory.

Gustav Dirichlet was born in Düren in1805. His father was a postmaster, and Gustavwas educated at a public school. Before age 12,he expressed a zealous interest in mathematics,even spending his spare money on mathemati-cal books. In 1817 he entered the Gymnasiumin Bonn, and he progressed rapidly in his stud-ies. Two years later Dirichlet was transferred toa Jesuit college in Cologne, where he com-pleted his major examinations by age 16.Although his parents desired Dirichlet to studylaw, he instead chose to follow his passion—mathematics—and traveled to Paris in 1822 topursue further studies with the great Frenchmathematicians.

In Paris, after surviving a bout of smallpox,Dirichlet attended the lectures at the Collège deFrance, and in 1823 secured an attractive posi-tion as tutor to the children of a famous Frenchgeneral. Through this situation, Dirichlet wasintroduced to French intellectual life. Amonghis mathematical acquaintances, he was espe-cially drawn to JEAN-BAPTISTE-JOSEPH FOURIER,who would continue to exert a significant influ-ence on Dirichlet’s later work.

His first interest was number theory, and thiscontinued to be Dirichlet’s major area of con-tribution to mathematics. In 1825 he presentedhis first paper on so-called Diophantine equa-tions of degree five (equations in more than twovariables involving fifth powers, and requiringinteger solutions). The results of this work in al-gebraic number theory led to additional progresson Fermat’s last theorem, a famous unsolvedconjecture (proved in 1994 by Andrew Wiles)in number theory.

In 1825 Dirichlet’s employer died, and hereturned to Germany. Although he did not havea doctoral degree, he obtained a position at the

Dirichlet, Gustav Peter Lejeune 79

University of Breslau; Dirichlet subsequentlyearned his spurs with a thesis concerning theprime divisors of certain polynomials. He alsocontributed the law of biquadratic reciprocity,building on Gauss’s own pioneering work. Theatmosphere in Breslau was not conducive to re-search, and so Dirichlet moved to Berlin in 1828,becoming a lecturer at the military academythere. In 1831 he married Rebecca Mendelsohn-Bartholdy, and also became a member of theBerlin Academy of Sciences.

Dirichlet spent almost three decades as aprofessor in Berlin, and during this time he ex-erted a profound influence on the developmentof German mathematics. He trained many stu-dents and continued to author papers of thehighest quality and relevance. Dirichlet was shyand retiring, expressing a modest reluctance tomake public appearances. During these years,Dirichlet communicated with KARL GUSTAV

JACOB JACOBI, another great German mathe-matician.

Dirichlet’s early work in number theorydelved into topics that Gauss had sketched in hisDisquisitiones Arithmeticae (Arithmetic Investi-gations) of 1801—topics such as quadratic forms,the quadratic and biquadratic laws of reciproc-ity, and the number theory of Gaussian integers.At an 1837 meeting of the Academy of Sciences,Dirichlet presented his first work on analyticnumber theory, in which he established that anyarithmetical progression must contain an infi-nite number of primes. Some follow-up papersworked out the convergence of so-calledDirichlet series, as well as determining the classnumber for quadratic forms. In this literature onefirst encounters Dirichlet’s “pigeon-hole princi-ple,” used in many mathematical proofs, whichstates that if more than n objects are placed inn holes, at least one hole must have more thanone object contained inside it. Dirichlet wassearching for a general algebraic number theorythat would be valid for fields of arbitrary degree.His techniques included a generalization of

quadratic forms, and he founded the algebraictheory of units.

Meanwhile, Dirichlet was researching math-ematical analysis, and was especially interestedin Fourier series. These power series could ap-proximate both continuous and discontinuousfunctions, and had been utilized by DANIEL

BERNOULLI and LEONHARD EULER to model thevibrations of strings. Dirichlet’s method ofproving convergence of a trigonometric seriesdiffered from AUGUSTIN-LOUIS CAUCHY’s priorapproach, and his approach later becamestandard. In an 1837 paper Dirichlet formu-lates the modern notion of a function as aspecial correspondence between a pair of variables.

Dirichlet is also known for his contributionsto mechanics, where he develops a method toevaluate integrals via a discontinuity factor. TheLaplace partial differential equation with a con-straint on the boundary values of the solution isnow known as the Dirichlet problem; an 1850paper by Dirichlet deals with this equation,which has applications to heat, magnetism, andelectricity. An 1852 paper treats hydrodynam-ics, giving the first exact integration for the hy-drodynamic equations.

After Gauss’s death in 1855, the Universityof Göttingen solicited Dirichlet as a replacementfor the deceased prince of mathematics, andDirichlet spent his remaining few years in thesuperior research environment of Göttingen. Hecontinued his work in mechanics until he suf-fered a heart attack, from which he later died onMay 5, 1859.

Dirichlet should be viewed as a successorto Gauss in his work in number theory.However, his accomplishments in analysis,mechanics, and differential equations are alsoquite noteworthy. Through his extensive tute-lage, Dirichlet also passed on his passion andknowledge for mathematics to a new generationof pupils, who included RICHARD DEDEKIND andLEOPOLD KRONECKER.

80 Dirichlet, Gustav Peter Lejeune

Further ReadingButzer, P. “Dirichlet and His Role in the Founding of

Mathematical Physics,” Archives Internationalesd’Histoire des Sciences 37, no. 118 (1987): 49–82.

Fischer, H. “Dirichlet’s Contribution to MathematicalProbability Theory,” Historia Mathematica 21(1994): 39–63.

Koch, H. “Gustav Peter Lejeune Dirichlet,” inMathematics in Berlin. Boston: Birkhäuser Verlag,1998.

Rowe, D. “Gauss, Dirichlet and the Law of BiquadraticReciprocity,” The Mathematical Intelligencer 10(1988): 13–26.

E

81

� Eratosthenes of Cyrene(ca. 276 B.C.E.–ca. 195 B.C.E.)GreekGeometry, Number Theory

In ancient Greece it was well known that theEarth was a sphere, and Eratosthenes first cal-culated its circumference to a surprising degreeof accuracy. Although principally known as a ge-ographer, he was actually one of the foremostmathematicians of his time, having accom-plished important work in geometry and num-ber theory.

The most likely account of Eratosthenes’ lifeinforms us that he was born around 276 B.C.E. toAglaos in the town of Cyrene. In his early man-hood he studied in Athens, where he became as-sociated with the members of the Academy.Around 235 B.C.E., he was invited to Alexandriaby King Ptolemy III to serve in the library there,and five years later he became its head. He be-came tutor to the king’s son, and remained infavor with the royal family until his death in 195B.C.E.

Eratosthenes was well known among hiscontemporaries as being an extraordinaryscholar, having produced works in mathemat-ics, geography, philosophy, literary criticism,grammar, and poetry. Laboring in so manyfields, he was foremost in none, and therefore

was known as “Beta” (the second letter of theGreek alphabet). However, his great accom-plishments in mathematics clearly demonstratethat he was no second-rate mathematician.Indeed, even ARCHIMEDES OF SYRACUSE dedi-cated a treatise to Eratosthenes.

Eratosthenes’ Geography long remained anauthority, and he was the first to give the fielda mathematical foundation. He considered theentire terrestrial globe, which was divided intozones, and he attempted to estimate distancesalong vaguely defined parallels. His new worldmap replaced the traditional Ionian depiction ofthe known world; he had much information athis disposal, as the library in Alexandria was thegreatest repository of ancient records.

Eratosthenes’ famous estimate of the Earth’scircumference was based on some simple princi-ples of similar triangles; due to errors in certainestimates, he arrived at 250,000 stades (about29,000 miles), which is fairly close to the truevalue of about 25,000 miles. His depictions ofother geographical distances also suffer from in-accuracies, largely due to errors in the recordedestimates of travelers. Eratosthenes divided theglobe into five latitudinal zones: two frigid, twotemperate, and one torrid (tropical). He calcu-lated the radius of each latitude that separatesthe various zones, using his computation of theEarth’s circumference.

82 Euclid of Alexandria

Besides his geographical calculations,Eratosthenes contributed to mathematics in hiswork Platonicus (Platonic), which treats pro-portion, progression, and musical scales. Hegives his solution to the famous problem of dou-bling the cube, which involves constructing alarge cube from a given cube that has exactlytwice the volume of the original. Eratosthenesreduced the mathematical problem to a geo-metrical one of finding two mean proportion-als in continuous proportion between twostraight lines. He even described a mechanicalapparatus for finding these so-called mean proportionals.

He is also famous for his method of find-ing prime numbers, known as the sieve ofEratosthenes. This involves writing out all oddintegers and successively crossing out multiplesof three, five, seven, and so forth, so that the re-maining numbers must be the primes. Althoughthis method is slow, it is guaranteed to locate allthe primes less than a given threshold.

Eratosthenes is ranked among the greatGreek mathematicians for his many contribu-tions. He was the first to accurately estimatethe circumference of the Earth, using onlymathematical principles and a few measure-ments. The doubling of the cube was an outstanding problem that had long defied so-lution, and Eratosthenes’ genius is evident inhis solution.

Further ReadingFowler, D. The Mathematics of Plato’s Academy: A

New Reconstruction. Oxford: Clarendon Press,1999.

Goldstein, B. “Eratosthenes on the ‘Measurement’ ofthe Earth,” Historia Mathematica 11, no. 4(1984): 411–416.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Rawlins, D. “Eratosthenes’ Geodest Unraveled: WasThere a High-Accuracy Hellenistic Astronomy,”Isis 73 (1982): 259–265.

� Euclid of Alexandria(ca. 325 B.C.E.–ca. 265 B.C.E.)GreekGeometry

The mathematical activity of ancient Greecewas not so organized as in modern times, as therewas little uniformity of notation or effort. Euclidof Alexandria is credited for organizing the vastmaterial of known but incoherent theorems andcollecting them in the single work known as theElements. Although he made some contributionsof his own, Euclid is mainly distinguished forgathering the great quantity of geometric infor-mation.

We know only that Euclid lived in Alexandriaduring the early part of that city’s life, being ac-tive in the years between 325 B.C.E. and 265B.C.E. He worked after the time of Plato’s disci-ples, but before the rise of ARCHIMEDES OF SYRA-CUSE. He dwelt in Alexandria during the reignof the first Ptolemy, and founded a thrivingschool of mathematics in that city.

Euclid may have been a Platonist, as he wasfriendly with Plato’s associates EUDOXUS OF

CNIDUS and Theaetetus. When asked by a stu-dent about the use of geometry, Euclid respondedby giving him three obols (the coins of thattime), since “he must need make gain out ofwhat he learns.” Euclid is described as a fair man,not given to vaunting. It is probable that he at-tended the Academy at Athens as a young man,where he would have studied mathematics, andwas later invited to Alexandria during the foun-dation of its famous library.

Besides the Elements, Euclid wrote some mi-nor mathematical works: Data, On Divisions ofFigures, Porisms, Surface Loci, and Book ofFallacies. These treat a variety of topics, such asmagnitudes, analysis, divisions of circles, conicsections, and locus theorems. The Elements,written in 13 books, constitute a work that hasgreatly influenced human thought for more thantwo millennia. The work is mainly concerned

Euclid of Alexandria 83

with certain geometrical problems, and it setsout to solve them in a series of propositions withproofs. Euclid’s method always proceeds fromwhat is known to the unknown through care-fully reasoned steps. The various propositions areplaced in a majestic order, so that previous re-sults are used in subsequent proofs, in a sort ofmathematical progression. The books also con-tain various essential definitions as well as certainpostulates and axioms, which were assumptionsto be taken for granted. For example, he takes theexistence of points, lines, and circles for granted(as well as the ability to construct them), and fromhere shows how many other figures can be drawn.

It is interesting to note that from his firstthree postulates one can deduce that the spaceof Euclid’s geometry is infinite and continuous,embracing both the infinitely large and the in-finitely small. The fifth postulate has drawnmuch attention, especially in the last two cen-turies, and essentially states that parallel linesnever meet. Many mathematicians subsequentlyattempted to deduce this fifth postulate from theother four, thus making it redundant; their fail-ure (see JÁNOS BOLYAI and CARL FRIEDRICH

GAUSS) has resulted in the so-called non-Euclidean geometries.

Book I treats the geometry of points, lines,triangles, squares, and parallelograms; this in-cludes proposition 47, more well known as thePythagorean theorem. Book II develops thetransformation of areas, and Book III deals withthe intersections of circles. Book IV is concernedwith the inscribing of rectilinear figures in cir-cles, with applications to astronomy. Next,Euclid develops a general theory of proportion,which has been praised by mathematicians forits elegance and accuracy. His definition of pro-portion in Book V has never been superseded asa formulation of the concept of proportion, andis recognized as a wonderful contribution tomathematics.

Next, Book VI treats the general theory ofproportion applied to similar figures, illustrating

the importance of the previously discussed defi-nition of proportion. The next three books dealwith arithmetic; these present more of an out-moded approach to the study of numbers, whichEuclid probably included out of deference to tra-ditional doctrines. The topics include primenumbers and least common multiples, as well asgeometric progressions of numbers. Book Xtreats irrational magnitudes, as Euclid sets outthe “method of exhaustion” used to calculate ar-eas and volumes. His final three books treat solidgeometry, such as parallelepipeds, pyramids,cones, and spheres. Euclid ends with considera-tion of the five Platonic figures, called the pyra-mid, cube, octahedron, dodecahedron, andicosahedron, which had been objects of ferventstudy for the Greeks. Not only does he deter-mine their angles, but also he determines thatthese are the only possible regular solids.

Of this huge body of work, Euclid is mainlycredited with organizing the known material andproviding simplified proofs in some cases. Thetheory of proportion and method of exhaustionis attributed to Eudoxus, and the knowledge ofirrational quantities is due to Theaetetus.However, Euclid was the inventor of the parallelpostulate, giving an insightful definition to theconcept of a parallel line. The Elements intro-duced new standards of rigor into mathematicalthinking, and also moved mathematics furtherinto the arena of geometry. Euclid’s influence, forboth these reasons, was profound and long last-ing; numerous later mathematicians, such as SIR

ISAAC NEWTON, would also cast their mathe-matical ideas in the rigorous geometrical moldthat Euclid set forth so long ago.

Further ReadingArchibald, R. “The First Translation of Euclid’s

Elements into English and Its Source,” TheAmerican Mathematical Monthly 57 (1950):443–452.

Berggren, J., and Thomas, R. Euclid’s “Phaenomena”:A Translation and Study of a Hellenistic Treatise in

84 Eudoxus of Cnidus

Spherical Astronomy. New York: GarlandPublishing, 1996.

Busard, H. The Latin Translation of the Arabic Versionof Euclid’s “Elements” Commonly Ascribed toGerard of Cremona. Leiden, the Netherlands:Brill, 1984.

Fraser, P. Ptolemaic Alexandria. Oxford: ClarendonPress, 1972.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

———. The Thirteen Books of Euclid’s Elements. NewYork: Dover Publications, 1956.

Knorr, W. “What Euclid Meant: On the Use ofEvidence in Studying Ancient Mathematics,” inScience and Philosophy in Classical Greece. NewYork: Garland, 1991.

� Eudoxus of Cnidus(408 B.C.E.–355 B.C.E.)GreekAnalysis, Geometry

Our modern theory of real numbers is essentialto the solution of algebraic equations and all ofmathematical analysis; and yet, for many of theGreeks, there was no concept of an irrationalnumber—it did not exist. Eudoxus, who cre-ated a rigorous mathematical foundation forreal numbers through his theory of proportions,removed this conceptual block. As a result,Greek mathematics was able to continue advancing.

Eudoxus was born in Cnidus in 408 B.C.E.,the son of Aischines. While still a young man,he studied geometry with Archytas of Tarentum,and his philosophical inquiries may well havebeen inspired by Plato, whose lectures he at-tended while studying in Athens. After return-ing to his native city, Eudoxus went on a trip toEgypt, spending some of his time with the priestsof Heliopolis. He there composed his eight-yearcalendric cycle, which probably included the ris-ings and settings of constellations. After a year

in Egypt, he settled in Cyzicus and founded aschool (probably dealing with mathematics andphilosophy), and later made a second visit toAthens. It seems that he had some additionalinteraction with Plato at this time, althoughPlato did not exert much influence uponEudoxus’s philosophy. He returned to Cnidus,where he lectured, wrote textbooks, and pro-vided laws for the citizens.

Eudoxus’s mathematical thinking lies be-hind much of the material of Books V, VI, andVII of EUCLID OF ALEXANDRIA’s Elements. Sincenone of Eudoxus’s written works are extant, wecan rely only on Euclid’s account. Eudoxus re-searched mathematical proportion, for the firsttime giving a sensible, rigorous definition of theconcept (which is still in use today). He also in-vestigated the method of exhaustion (a protocal-culus idea, used to compute areas and volumes),and was interested in the axiomatic developmentof mathematics (this approach greatly influ-enced Euclid, who carefully laid out various pos-tulates and axioms of geometry in the Elements).Eudoxus may have been the first to approachmathematics in this systematic fashion.

Before Eudoxus’s theory of proportion,Greek mathematics was immobilized by the ir-rational numbers—the Pythagoreans had al-ready discovered square roots, but to their wayof thinking these quantities did not really exist.Only rational numbers (ratios of integers) ex-isted for these earlier Greeks. In order to makeprogress in number theory and the solutions ofequations (and also geometry), it was necessaryto include irrational numbers; Eudoxus’s theoryof proportions gave a rigorous definition of thereal numbers, which in particular showed the ex-istence of irrational quantities. It is interestingthat modern definitions of the real numbers,such as those propounded by RICHARD DEDEKIND

and KARL WEIERSTRASS, are virtually identicalwith Eudoxus’s ancient formulation.

Eudoxus worked on the old “Delian prob-lem” of duplicating the cube (see ERATOSTHENES

Euler, Leonhard 85

OF CYRENE). According to Archimedes, Eudoxusproved that the volume of a pyramid is one-thirdthe volume of the prism containing it, with sim-ilar results for the cone. Though DEMOCRITUS OF

ABDERA already knew these facts, Eudoxus wasthe first to prove them. It seems that he alsodiscovered formulas for the area and volume ofcircles and spheres, respectively. These proposi-tions are given in Book XII of the Elements,which reflects much of Eudoxus’s work in thisarena.

Another important aspect of his work wasthe application of spherical geometry to astron-omy. Eudoxus, in his work On Speeds, expoundsa geocentric astronomical system involving ro-tating spheres. Although the model was highlyidealized, having a poor fit to the known ob-servational data, Aristotle took the idea quiteliterally and popularized it through his ownwork. Eudoxus had his own observatory andcarefully watched the heavens as part of his ownstudies; he published his results in the Enoptronand the Phaenomena, which were well-used ref-erences for two centuries. Eudoxus was alsoknown as a great geographer, and his Tour of theEarth gave a systematic description of the knownworld, including political, historical, and ethno-graphical information.

Eudoxus was certainly one of the greatest in-tellectuals of his time, though his work is knowntoday only through secondhand accounts. Hiscontribution to mathematics through the for-mulation of the real number system cannot beoveremphasized; this work allowed the furtherdevelopment of Greek mathematics throughsuch persons as Archimedes and Eratosthenes.

Further ReadingHeath, T. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.———. The Thirteen Books of Euclid’s Elements. New

York: Dover Publications, 1956.Huxley, G. “Eudoxian Topics,” Greek, Roman and

Byzantine Studies 4 (1963): 83–96.

Stein, H. “Eudoxos and Dedekind: On the AncientGreek Theory of Ratios and Its Relation toModern Mathematics,” Synthese 84, no. 2(1990): 163–211.

� Euler, Leonhard(1707–1783)SwissNumber Theory, Mechanics, Analysis

The 18th century witnessed the developmentof various 17th-century mathematical ideas—calculus was one important example. LeonhardEuler was exceptional among his peers for notonly the breadth and profusion of his work butalso his great originality; he largely foundednumber theory, defined the modern concept of

Leonhard Euler was a great Swiss mathematician whoworked principally in number theory and differentialequations. For his profundity the 18th century wasoften known as the “Era of Euler.” (Courtesy of theLibrary of Congress)

86 Euler, Leonhard

a function, and formulated a general theory forthe calculus of variations. His renown and vir-tuosity were such that the 18th century wassometimes referred to as the “Era of Euler.”

Leonhard Euler was born in Basel on April15, 1707. His father, Paul Euler, was a Protestantminister, and his mother, Margarete Brucker, wasa minister’s daughter; this religious backgroundremained with Euler throughout his life. Euler’sfather, who was interested in mathematics him-self, having attended JAKOB BERNOULLI’s lecturesat the University of Basel, educated his son in hisearly years. Because his Gymnasium did not teachmathematics, Euler studied privately with an am-ateur mathematician, and he displayed remark-able talent for one his age. In 1720 he enteredthe University of Basel and soon came under theguidance of JOHANN BERNOULLI. In 1722 he re-ceived his bachelor of arts degree, and a yearlater his master’s in philosophy; at age 16, hejoined the theology department.

However, Euler’s strength lay in mathemat-ics, and he soon gave up the ambition to be aminister. About this time he began his own re-search in mathematics and published an articleon algebraic reciprocal trajectories. There werefew opportunities in Switzerland for youngmathematicians, so Euler accepted an offer tojoin the new St. Petersburg Academy of Sciencesin 1727. His official appointment was as an ad-junct of physiology, although he was permittedto work in mathematics. Euler became professorof physics in 1731 and professor of mathematicsin 1733; the atmosphere at the young academywas stimulating for Euler, who interacted withJakob Hermann, DANIEL BERNOULLI, and CHRIS-TIAN GOLDBACH.

Euler’s life was marked by his remarkable dili-gence and activity. His mathematical researchwas reported in the sessions of the academy;meanwhile, he was involved in the training ofRussian scientists, as well as the study of Russianterritory (Euler assisted in the construction ofgeographical maps) and development of new

technology (Euler studied problems in ship-building and navigation). But his contributionsto mathematics were prolific—Euler preparedmore than 80 works in his first 14 years at St.Petersburg.

Many of his best ideas were formulated inhis youth, even while at Basel, and were fleshedout much later. Due to his voluminous corre-spondence with other scientists, Euler’s discov-eries were often made public before they wereeven published; this brought him no smallamount of fame. In 1733 he married KatharinaGsell, and he soon had two sons. His quiet lifewas marred only by the blinding of his right eyein 1738; according to Euler, this was due to over-strain from his cartographic work. However, in1740 the political situation in Russia became un-stable, and Euler accepted an offer to work atthe Berlin Society of the Sciences.

Euler stayed in Berlin for 25 years, duringwhich time he was blessed with many additionalchildren. During this period, he worked in boththe Berlin and St. Petersburg academies. He wasdirector of the Berlin Society of the Sciences,which he largely transformed. Besides the nu-merous administrative duties, he dealt with sev-eral practical problems, such as correcting thelevel of the Finow Canal. He consulted with thegovernment on problems of insurance, pensions,and hydraulics, and even organized a few statelotteries. Meanwhile, Euler received a pensionfrom the St. Petersburg Academy, and in returnhe edited the academy’s journal, apprised it ofnew scientific ideas, and oversaw competitions.Euler received 12 prizes from the Paris Académiedes Sciences from 1738 to 1772.

The Berlin period was fruitful, as Euler pro-duced more than 380 works—some of which werelengthy books—on topics such as the calculus ofvariations, the calculation of orbits, ballistics,analysis, lunar motion, and differential calculus.His famous Lettres à une princesse d’Allemagne surdivers sujets de physique et de philosophie (Lettersto a German princess on diverse topics of physics

Euler, Leonhard 87

and philosophy) were written in a popular man-ner and became a great success in Europe. Eulerparticipated in many academic debates on suchtopics as the religion of pure reason espoused byGOTTFRIED WILHELM VON LEIBNIZ, and the prin-ciple of least action.

After 1759, Euler’s relationship with KingFrederick of Prussia deteriorated, and he even-tually returned to St. Petersburg in 1766. Soonafter his return, a brief illness left him completelyblind; this hampered his ability to do research,but with the help of assistants he was able to dic-tate his thoughts, and so continue his work. Theonly change seems to be that his articles becamemore concise, and half of his total works were pro-duced after 1765. His memory (he could reciteVirgil’s Aeneid verbatim) remained flawless, andhe continued to have original ideas. Euler’s ac-tivity in the academy was unabated when he diedon September 18, 1783, of a brain hemorrhage.

Euler was one of the most important math-ematicians since SIR ISAAC NEWTON. He wasdeeply interested in applications, but would de-velop the pertinent mathematics to deep levelsof abstraction and generality. His foremost sub-ject was analysis, contributing to the calculus ofvariations, the theory of differential equations,functions of a complex variable, and the theoryof special functions. Many modern conventionsand notations are due to him, such as the sym-bol f(x) for the value of a function and i for thesquare root of –1.

In number theory Euler was concerned withthe theory of divisibility, introducing the so-called Euler’s function, which tallies the quan-tity of divisors of a given integer. These studiesled him to the discovery of the law of quadraticreciprocity, whose complete proof was laterestablished by CARL FRIEDRICH GAUSS. Eulerinvestigated decompositions of prime numbersas linear combinations of squares, and workedon Diophantine analysis via continued fractions.His methods were algebraic, but Euler was thefirst to introduce analytic methods into number

theory—in particular, he deduced a famous iden-tity relating sums of reciprocal squares to a prod-uct of prime numbers, which was a first step inthe study of the Riemann zeta-function. Eulerstudied mathematical constants, such as e andpi, as well as Euler’s constant (which comes upin the study of the divergent harmonic series).

Euler stated the theorem that an algebraicpolynomial of degree n had n roots of the forma1bi, which is now known as the fundamentaltheorem of algebra. His 1751 proof had someomissions, which were later corrected by Gauss.Euler also attempted to derive an exact formulafor the roots of the fifth degree polynomial, andhis failures led him to approximation methodsof numerical analysis.

Although many mathematicians had stud-ied infinite series, Euler was unusually successfulin their calculation, deriving simple formulas forsums of reciprocal even powers of integers.Through these studies, Euler studied specialfunctions (such as the Bessel functions) and dis-covered Euler’s constant for the approximationof the harmonic series. He made great use ofpower series, and introduced trigonometric se-ries before JEAN-BAPTISTE-JOSEPH FOURIER as ananalytic tool. Euler believed that divergent se-ries could be useful, and this effort would cometo fruition much later, in the 20th century.

Euler presented the idea that mathematicalanalysis is the study of functions; to this end, hemore clearly defined the concept of a function,which closely approximates the modern notion.Through consideration of the logarithm of nega-tive numbers, Euler came to an understanding ofthe exponentiation of imaginary numbers, deriv-ing many crucial elementary facts. He advancedthe knowledge of complex numbers, discoveringthe differential equations that relate the real andimaginary parts of an analytic function. Euler ap-plied his techniques to the computation of realintegrals.

He also made numerous discoveries in differ-ential and integral calculus, deriving substitution

88 Euler, Leonhard

rules, validating the interchange of partial deriv-atives, and founding the concept of multiple in-tegrals. As a result of the many special cases andtechniques of integration he employed, the betaand gamma functions were discovered, which areuseful in physics. Euler made great contributionsto the field of differential equations, includingthe method of variation of constants as well asthe use of characteristic curves. Some of the ap-plications of this work include vibrating stringproblems, hydrodynamics, and the motion of airin pipes.

His studies in the calculus of variations ledhim to the Eulerian differential equation, andhis exposition of the subject became classical.Euler was the first to formulate the principalproblems of this subject and the main methodsof solution. In geometry, Euler investigatedspherical trigonometry and founded a theory oflines on a surface—one of the initial steps towardthe modern subject of differential geometry. Heanalyzed the curvature of a surface in terms ofthe curvature of embedded principal curves, andintroduced Gaussian coordinates, which were ex-tensively used in the 19th century.

Euler was also the first author in topology,solving the famous riddle of seven bridges ofKönigsberg; he studied polyhedra, deriving whatlater became known as the Euler characteris-tic—a formula relating the number of edges,faces, and vertices.

Besides these contributions to pure mathe-matics, Euler labored in mechanics, astronomy,and optics. Euler systematized mechanics, intro-ducing analytical methods that greatly simplifiedthe subject. He studied celestial mechanics andelasticity, deriving the famous Euler bucklingformula, used to determine the strength ofcolumns. In fluid mechanics, he studied equilib-rium positions and presented three classic worksdiscussing the motion of incompressible fluids;

Euler also improved the design of the hydraulicturbine.

In astronomy, Euler was interested in the de-termination of orbits of comets and planets, thetheory of refraction, and the physical nature ofcomets. He presented an extensive lunar theory,which enabled a more precise calculation of aship’s longitudinal position at sea. Euler assistedphysics by mathematizing it (that is, by intro-ducing many techniques of analysis to better un-derstand certain problems). Indeed, he is creditedas founding mathematical physics. He studied op-tics as well, constructing a nonparticle theory oflight that viewed illumination as the product ofcertain oscillations in the ambient ether.

Euler was a humble man, but also one of thegreatest scientists and mathematicians of alltime, and especially the 18th century; he wasrecognized by his peers as an outstanding genius.His mathematical research has stimulated anenormous amount of subsequent activity, andmany of his ideas were ahead of their time.

Further ReadingAyoub, R. “Euler and the Zeta Function,” The

American Mathematical Monthly 81 (1974):1067–1086.

Deakin, M. “Euler’s Invention of IntegralTransforms,” Archive for History of Exact Sciences33, no. 4 (1985): 307–319.

Dunham, W. Euler: The Master of Us All.Washington, D.C.: The MathematicalAssociation of America, 1999.

Edwards, H. “Euler and Quadratic Reciprocity,”Mathematics Magazine 56, no. 5 (1983): 285–291.

Ewing, J. “Leonhard Euler,” The MathematicalIntelligencer 5, no. 3 (1983): 5–6.

Finkel, B. “Biography. Leonard Euler,” The AmericanMathematical Monthly 4 (1897): 297–302.

Weil, A. “Euler,” The American Mathematical Monthly91, no. 9 (1984): 537–542.

F

89

� Fatou, Pierre-Joseph-Louis(1878–1929)FrenchAnalysis

The theory of integration was an intensely re-searched topic in the late 19th century, andPierre Fatou contributed to certain basic aspectsof HENRI-LOUIS LEBESGUE’s theory of integration.His results, later developed by Caratheodory andFRIGYES RIESZ, were fundamental in the youngtheory of integration.

Fatou was born in Lorient, France, on Febru-ary 28, 1878. He attended the École NormaleSupérieure from 1898 to 1901, where he studiedmathematics. Since there were few positions formathematicians, Fatou accepted a position atthe Paris observatory, where he continued towork until his death on August 10, 1929. Whileworking there, he obtained his doctorate in1907 and was appointed titular astronomer in1928.

One of the applied mathematical problemsthat fascinated Fatou was the calculation of theperturbation of a planet due to the movementsof a second planet. CARL FRIEDRICH GAUSS hadformulated some intuitive results, but Fatou wasable to give these a rigorous mathematicalfoundation through the study of differential

equations. Fatou also investigated the move-ment of a planet through a resistant medium,which was based on the supposition that stel-lar atmospheres were more extensive in thepast.

In pure mathematics, Fatou contributed tothe theory of Taylor series, and was particularlyinterested in the circle of convergence. His re-searches in this direction led him to formulatethe so-called Fatou’s lemma, which gives a suf-ficient condition for convergence of a series offunctions. He formulated his results under theumbrella of Lebesgue integration theory, and histheorem is also a key result toward determiningthe validity of swapping limit and integral. Fatoualso studied rational functions of a complex vari-able and determined how the coefficients of aTaylor series affect the singularities of the re-sulting function.

Lebesgue integration theory later became anextremely important part of real analysis, andFatou’s researches in this field have had an en-during significance.

Further ReadingAlexander, D. A History of Complex Dynamics: From

Schröder to Fatou and Julia. Braunschweig/Wiesbaden: Friedrich Vieweg and Sohn, 1994.

90 Fermat, Pierre de

� Fermat, Pierre de(1601–1665)FrenchNumber Theory, Calculus, Probability

Pierre de Fermat is known as one of the great-est mathematicians of the 16th century, havingmade contributions to the foundations of calcu-lus, probability, and number theory. In the last-named subject he is particularly renowned, forhis research into divisibility and the propertiesof prime numbers would later fuel much 19th-and 20th-century investigation.

Pierre de Fermat was born in Beaumont-Lomagne, France, on August 20, 1601. His father,Dominique Fermat, was a prosperous merchant,while his mother, Claire de Long, was a noble-woman. As a result of his mother’s pedigree,

Fermat enjoyed a high social status and laterchose the profession of law. He received a clas-sical secondary education, and probably studiedat the University of Toulouse. In any event, hecertainly lived in Bordeaux in the late 1620s,and at this time he first began his mathematicalinvestigations.

Fermat received the degree of bachelor ofcivil laws from the University of Orleans in1631, and embarked on his legal career in thelocal parlement. The same year, Fermat marriedhis cousin Louise de Long, by whom he had fivechildren. It seems that Fermat enjoyed financialprosperity, and he was allowed the privilege, asa member of the aristocracy, to append “de” tohis last name. However, his performance in hisoffice was unsatisfactory, and Fermat advancedonly through the death of his professional col-leagues. In 1642 he ascended to the highestcouncils of the parlement, later serving as presi-dent of the Chambre de l’Édit, which had juris-diction over legal suits between Huguenots andCatholics. Fermat was a devoted Catholicthroughout his life.

Fermat enjoyed some fame as a mathemati-cian during his own life, though his unwilling-ness to publish kept him from the renown hemight have obtained. He was also reputed as aclassical scholar, being fluent in several lan-guages. He enjoyed good health, surviving an at-tack of plague in 1652, and died in Castres onJanuary 12, 1665.

Fermat’s development as a mathematicianmay have commenced during his Bordeauxperiod, at which time he became familiar withthe works of FRANÇOIS VIÈTE. From Viète Fermatacquired the new symbolic algebra, as well as theconception of algebra as a tool useful for geo-metric problems. Fermat sought to build onViète’s concepts, including the ability to solveand construct determinate equations; his methodoften involved reducing a given problem to aknown class of problems (much like an inversetype of induction). At first, Fermat heavily

Pierre de Fermat formulated the famous “Fermat’s LastTheorem,” which remained unproved for 300 years.He also contributed to number theory and probability.(University of Rochester, courtesy of AIP Emilio SegrèVisual Archives)

Fermat, Pierre de 91

relied on the ancient Greeks for ideas on math-ematical analysis, but he often generalized theoriginal problems considered, using reductionanalysis and his natural genius to arrive at gen-eral solutions.

By spring 1636, Fermat had already com-pleted his Ad locos planos et solidos isagoge(Introduction to planes and solids), a work set-ting forth an analytic geometry that was ex-tremely similar to RENÉ DESCARTES’s Géométrie(Geometry) of 1637. Although these works werevirtually identical in their use of algebraic equa-tions to describe geometric curves, the issue ofpriority is unresolved, as each mathematicianwas working independently. Fermat started fromthe works of PAPPUS OF ALEXANDRIA and APOL-LONIUS OF PERGA, realizing that the loci ofpoints discussed by the latter could all be de-scribed by algebraic equations in two unknowns.He then employed a single axis with origin andmoving ordinate (similar to Descartes’s graphi-cal method, which did not involve coordinates)to describe a given curve. Fermat then consid-ered the general second-degree equation by re-ducing it into seven irreducible forms (or specialcases), which included lines, hyperbolas, ellipses,parabolas, and circles. Fermat’s presentation dif-fered substantially from that of Descartes, whopassed over the issue of construction and fo-cused on an advanced theory of equations.Pursuing the implications of his research after1636, Fermat demonstrated the graphical solu-tion of determinate algebraic equations. In1643 he tried to extend his methods to solidsof revolution (those solids obtained by revolv-ing a curve about a fixed axis). This latter effortwas not successful, as Fermat did not yet havethe tools of a three-dimensional coordinate sys-tem, although he laid the correct algebraic foun-dation for such a system of solid geometry.Fermat established the connection between di-mension and the number of unknowns, an im-portant conceptual contribution to 17th centurymathematics.

Fermat also developed a method of com-puting maxima and minima of curves, which es-sentially involved a calculation of the derivativeof a polynomial. However, Fermat did not useany infinitesimals in his method, and thus hiswork was peripheral to the foundation of calcu-lus. Using his technique, Fermat could determinethe centers of gravity for geometrical figures, aswell as the formation of tangent lines to a givencurve. This work became a pivotal point in a1638 debate with Descartes, who criticizedFermat’s work because it rivaled his own math-ematics set forth in Géométrie. Although theyeventually made peace when Descartes admittedthat his critique of Fermat’s work was invalid,the two men remained at strife; the reputationof Fermat, who adamantly refused to publish hiswork, suffered as a result.

The quadrature of curves (that is, the com-putation of the area under a curve by means ofapproximating it by rectangles) was also studiedby Fermat, who expanded on ARCHIMEDES OF

SYRACUSE’s labors on the spiral. Fermat was ableto approximate a given area with arbitrary accu-racy (through the number of rectangles chosen),and thus calculate the area beneath certain sim-ple polynomials. At first his style was geometri-cal, relying on carefully drawn figures, but laterhe adopted a more algebraic approach. His var-ious results on quadrature eventually circulatedin 1679, by which time they were obsolete, inview of the more comprehensive work of SIR

ISAAC NEWTON and GOTTFRIED WILHELM VON

LEIBNIZ. It seems that Fermat did not realize thatthe method of tangents and quadrature were in-verse to one another, and this work exerted lit-tle influence on later mathematics.

Fermat is best known for his work in numbertheory, which was largely neglected by his 17th-century colleagues. His labors went completelyunappreciated until LEONHARD EULER revivedinterest in number; finally, in the 19th centuryCARL FRIEDRICH GAUSS and others proved manyof the important results and established number

92 Ferrari, Ludovico

theory as a modern field of mathematical in-quiry. Fermat was interested in integer solutionsof algebraic equations, and his early researchcentered on divisibility and the study of theprime numbers. His methods are not known, be-cause most of his results were written in lettersto friends or in the margins of other books; ap-parently, Fermat used the sieve of ERATOSTHENES

OF CYRENE as a criterion for primeness. He de-rived several important theorems (withoutproof), investigating the decomposition ofprimes as sums of squares. In this connection,Fermat was interested in integer solutions to xn 1 yn 5 zn where n is at least two. The fact(proved only recently, by Andrew Wiles in1994) that there are no solutions for n largerthan two is known as Fermat’s last theorem; hejotted this conjecture in the margin of one of hisbooks.

One technique that Fermat applied repeat-edly was the method of infinite descent: Hewould argue by contradiction, constructing aninfinite sequence of decreasing (positive) inte-gers, which could not exist. The main impor-tance of Fermat’s work in number theory is thestimulus that it gave to research in the late 18thand 19th centuries.

Fermat also contributed to the study of op-tics (over which subject he also debated withDescartes, objecting to his a priori reasoning),and is credited, along with BLAISE PASCAL, as be-ing the founder of the theory of probability.Through a series of letters written during 1654,these two mathematicians corresponded on a va-riety of probability questions, such as how tofairly divide the stakes of an interrupted game.Though their methods differed somewhat (Fermatmade direct calculations rather than derivinggeneral formulas), they both used the concept of“expected winnings,” defined through the math-ematical expectation.

The later years of Fermat’s life saw little in-teraction with other mathematicians, as heincreasingly devoted his spare time to number

theory. Although his work, especially his effortsin number theory, deserved the acknowledgmentof his colleagues, Fermat fell into increasing ob-scurity due to his reluctance to publish. Afterthe 17th century he was completely forgotten,until rediscovered by Euler and others in the19th century, when the renewed interest in num-ber theory drew inspiration from his intellect.

Further ReadingEdwards, H. Fermat’s Last Theorem, a Genetic

Introduction to Algebraic Number Theory. NewYork: Springer-Verlag, 1977.

Indorato, L., and P. Nastasi. “The 1740 Resolution ofthe Fermat-Descartes Controversy,” HistoriaMathematica 16, no. 2 (1989): 137–148.

Singh, S. Fermat’s Last Theorem: The Story of a RiddleThat Confounded the World’s Greatest Minds for358 Years. London: Fourth Estate, 1998.

� Ferrari, Ludovico(1522–1565)ItalianAlgebra

One of the hot mathematical problems of the16th century was the solution of the cubic (orthird degree) polynomial; mathematicianssearched for a general formula, in terms of thecoefficients, that would instantly produce theanswer. Some of the famous characters in thisintellectual quest were GIROLAMO CARDANO,NICCOLO TARTAGLIA, SCIPIONE DEL FERRO, andLudovico Ferrari. Ferrari probably made the mostsubstantial contribution of that time towardsolving the cubic and quartic equations.

Ludovico Ferrari was born in Bologna, Italy,on February 2, 1522. His father, AlessandroFerrari, was a refugee from Milan, and when hedied Ludovico went to live with his uncle. In1536 he traveled to Milan to serve in the house-hold of Girolamo Cardano, where he could re-ceive an education. Ferrari was very intelligent,

Ferro, Scipione del 93

and he quickly absorbed Latin, Greek, andmathematics.

In 1540 Ferrari was appointed as a publiclecturer in Milan, and soon defeated a compet-ing mathematician in a public disputation. Hecollaborated with Cardano on his research intothe cubic and quartic equations, and the resultswere published in the Artis magnae sive de reg-ulis algebraicis liber unus (also known as Arsmagna [Great art]) in 1545. Because Cardanohad published information that Tartaglia had re-vealed to him under an oath of secrecy, Tartagliabecame enraged at the betrayal, and as a resulta feud erupted. Cardano claimed that Tartaglia’ssecret method for solving a certain cubic was al-ready known to Scipione del Ferro, and hencewas essentially public knowledge. Tartagliamaintained that Cardano had stolen his re-search. Ferrari vigorously defended his mentor,and refuted Tartaglia’s public defense through aseries of letters (which were rife with personalattacks), and eventually through a public dispu-tation in 1548. This debate, over which the gov-ernor of Milan presided, seems to have been wonby Ferrari, who was probably a superior mathe-matician to Tartaglia.

Ferrari’s method for solving a certain typeof quartic (or fourth degree polynomial) in-volved introducing a second variable, whichhad to satisfy certain constraints. This secondvariable’s constraints led to a cubic equation.With this ingenious method, Ferrari could firstsolve the derived cubic and then go back to solvethe original quartic.

As can be surmised from his letters, Ferrari’scharacter was both loyal and pugnacious. Hishandling of Tartaglia was belligerent, but thismay have been due to his personal bond withhis former master Cardano. As a result ofFerrari’s success in the debate with Tartaglia, hereceived many offers of employment, and heagreed to become the tutor to the son of the car-dinal of Mantua. He carried out a survey of theprovince of Milan, and after eight years he re-

tired to Bologna due to ill health. In 1564 heobtained a post at the University of Bologna,which he held until his death on October 5,1565, in Bologna, Italy.

Of the mathematicians of that time, Ferrariwas probably the best at solving polynomialequations. His work would constitute some ofthe early steps in the grand effort to solve fourthand fifth degree polynomials; these problemswould be definitively answered by NIELS HENRIK

ABEL and EVARISTE GALOIS in the 19th century.

� Ferro, Scipione del(1465–1526)ItalianAlgebra

In mid-16th-century Italy, there was much inter-est and debate over the solution of the cubic, orthird degree, polynomial. Before the publicationin about 1545 of GIROLAMO CARDANO’s contro-versial book Artis magnae sive de regulis algebraicisliber unus (also known as Ars magna [Great Art]),Scipione del Ferro had already discovered thegeneral solution of certain cubic equations somedecades earlier.

Scipione del Ferro was born on February 6,1465, in Bologna, Italy. His father was FlorianaFerro, a papermaker, and his mother was namedFilippa. Of his early education nothing is known,but del Ferro became a lecturer in arithmetic andgeometry at the University of Bologna, where heremained from 1496 to 1526. Del Ferro was ac-tive as a businessman in the years 1517 to 1523.He died on November 5, 1526, in Bologna.

He left no writings behind him, but othersources declare him to be a great algebraist andarithmetician. Of all his accomplishments, themost renowned is his solution of the cubic of theform x3 1 ax 5 b. The discovery of a formula forthe solutions of such an equation had eludedmathematicians since antiquity; the Greeks werefascinated by this problem. In the 15th century,

94 Fibonacci, Leonardo

despite much effort after the time of LEONARDO

FIBONACCI, it was thought to be impossible. DelFerro’s path to discovery can only be surmised,but he probably relied on Fibonacci’s Liber ab-baci (Book of the abacus). It seems that hisachievement took place within the first twodecades of the 16th century, but del Ferro didnot give any printed account. Rather, he passedhis technique on orally to certain individuals.

One of these heirs, Antonio Maria Fior,became involved in a dispute in 1535 with NIC-COLO TARTAGLIA, in which Fior used hisknowledge of the solution of the cubic to ad-vantage. The latter was then prompted to makehis own discovery of the solution to the cubic,although unbeknownst to Tartaglia, del Ferrohad priority. In 1542 Annibale dalla Nave—thesuccessor of del Ferro—revealed the method toCardano, who subsequently included it in his1545 Ars magna.

Del Ferro also contributed to the knowledgeof algebra through the study of fractions with ir-rational denominators. The problem of ration-alizing the denominator goes back to EUCLID OF

ALEXANDRIA, who considered square roots, but delFerro was the first to consider more complicatedirrational numbers, such as those involving cuberoots. From the testimony of LUDOVICO FERRARI,it is known that del Ferro also investigated thegeometry of the compass, although it is unclearwhat contributions he made.

Del Ferro was certainly one of the most ac-complished algebraists of his time, although allmodern knowledge of him is secondhand. Hissolution of the cubic was the first successful stepin a long progression of mathematical interest;interest in related questions would eventuallygive rise to the elegant Galois theory of numberfields.

Further ReadingCrossley, J. The Emergence of Number. Victoria,

Australia: Upside Down A Book Company,1980.

Rose, P. The Italian Renaissance of Mathematics: Studieson Humanists and Mathematicians from Petrarch toGalileo. Geneva: Droz, 1975.

� Fibonacci, Leonardo (Leonardo of Pisa)(ca. 1170–ca. 1240)ItalianArithmetic

After the blaze of light in Greece during the clas-sical era, a great intellectual and cultural dark-ness consumed Europe; Leonardo Fibonaccirekindled that light during the first stirrings ofthe Italian Renaissance, and that illuminationwas destined to wax greater into the brilliant ra-diance of current mathematical achievement.Certainly, others made important contributionsin earlier centuries in other parts of the world,but Fibonacci was the first great mathematicianof the Christian West. As a Renaissance char-acter, he revived interest in classical literatureand values and, in particular, renewed the ap-preciation for mathematical knowledge.

Leonardo Fibonacci was a member of theBonacci family, born in Pisa, Italy, to GuglielmoBonacci. We only have an estimate of the dateof his birth, 1170, since there are few records ofthe details of his life—one of the main sources ishis own book, Liber abbaci (Book of the abacus).He was nicknamed Bigollo, a term that desig-nates a loafer, which may have been an epithethurled by those who thought lightly of the valueof mathematical work. His father was an officialin the Republic of Pisa, and in 1192 he receiveda commission to direct a Pisan trading colony inAlgeria. The young Fibonacci accompanied hisfather, who hoped to school his son in the artsof calculation so that he could one day becomea merchant. Fibonacci far exceeded his father’sexpectations.

The instruction in Africa from an Arab mas-ter was quite good, probably much better than

Fibonacci, Leonardo 95

in Europe, and Fibonacci encountered the “new”Indian numerals. These numbers were symboli-cally quite similar to modern digits, and consistedof 10 distinct numerals, which could describe anyquantity merely through their proper arrange-ment (the same as the modern numeric system).At the time in Europe, most merchants still usedRoman numerals, for which the calculations ofaddition and multiplication are far more diffi-cult. Fibonacci quickly mastered this superiornumber system. In the ensuing years, he traveledwidely, including Egypt, Greece, Sicily, Syria, andProvence, in pursuit of his mercantile vocation,and in all the cities he would communicate withlocal scholars on their methods of calculation.Through these disputations, Fibonnaci came tosee that these other learned men, who did notunderstand the Indian system, were at a greatmathematical disadvantage, and were often inerror.

These experiences were crucial for Fibonacci’sdevelopment. In 1200 he returned to his nativecity and worked for the next 25 years on calcu-lation with Indian numerals. Due to his back-ground in business, he was driven by practicalconcerns, and thus his investigations were mo-tivated by a desire to apply them to commercialmatters; however, he also did considerable the-oretical work in algebra and geometry.

In 1202 Fibonacci’s Liber abbaci was com-pleted; like the abacus it was named after, thiswork focused on computation. New materialwas added in a second version in 1228. The firstsection dealt with Roman numerals and fingercalculations, then Indian numerals were intro-duced, along with the fraction bar. The nextportion was mainly relevant to merchants, re-sembling an almanac—there was informationon the price of goods, the calculation of inter-est and wages, measurement of quantities, andexchange of currencies. The third section con-tained puzzles and mathematical riddles, andrules for the summation of series (for instance,there was a formula for the sum of a geometric

series). One famous problem is stated as follows:Given a pair of rabbits, which take a month tomature and then produce a pair of offspring everymonth, how does the population increase?Assuming that the offspring mature in the samefashion as their parents, the monthly populationfollows the sequence 1, 1, 2, 3, 5, 8, . . . Thesenumbers, now known as the Fibonacci sequence,are one of the first examples of recursion, sinceeach term is equal to the sum of the two pre-ceding terms. Recursion, the concept that athing’s definition depends on itself (or at leastits past self), is a powerful concept in modernmathematics, computer science, and philosophy.

Even more important for the history of math-ematics is Fibonacci’s introduction of negativenumbers. Prior to this time, merchants had a con-cept of subtraction, as a way of keeping track oftheir inventory. But since it was impossible tohave negative inventory, the concept of a nega-tive number was nonsensical to them. For ex-ample, they would say that the equation (thoughthey would not write it this way) x 1 2 5 1 hasno solution. However, Fibonacci used negativenumbers, thought of as debits or debts, in orderto solve equations, and it seems that he was thefirst to do so. Others who came after him wouldformalize the notion of a negative number andconstruct the integers. It is interesting to notethat some mathematical concepts now taken forgranted, like negative numbers, were once verymysterious, and it required genius and creativityto arrive at the new idea. Certainly, Fibonacci’speers caught on slowly.

The fourth section of this book dealt with thecalculation of radicals, using formulas of arith-metic from EUCLID OF ALEXANDRIA’s Elements, andcontained examples of the ancient method ofapproximation. For example, to approximate pi,the ancients would find two fractions, one a bitsmaller (such as 223/71) and the other a bitlarger (such as 220/70) than pi, which could beeasily calculated. In general, the Liber abbaci isremarkable for the richness of its examples and

96 Fisher, Sir Ronald Aylmer

the rigor of proof. Fibonacci was a master of hisart, and he would present several different meth-ods of solution, including algebraic and geomet-ric approaches.

Fibonacci also wrote Practica Geometriae(Practice of geometry) in 1220 or 1221, whichfocuses on geometry. Appealing to Euclid’sElements, he solves square and cube root prob-lems, and gives various calculations of segmentsand surfaces of plane figures. An approximationof pi is given by inscribing a 96-gon. There arealso some practical directions for the field sur-veyor; for example, he gives directions for theuse of the “archipendulum,” a geodetic instru-ment used to find horizontal projections ofstraight lines lying on an inclined hill.

Fibonacci also made great progress in inde-terminate analysis, the study of several equationsin several unknowns. In Flos (Flower) (1225)and Liber Quadratorum (Book of square num-bers) (1225) he demonstrates his facility withnumber theory, posing and solving various an-cient problems in indeterminate analysis. In1225 he was presented to Emperor Frederick II,and his latter writings were in response to ques-tions posed by the imperial philosopher Theo-dorus. The last record of Fibonacci is in 1240,when he was awarded an annual salary by his cityfor his advice about accounting practices.

Certainly Fibonacci played a pivotal role inthe rebirth of mathematics in western Europe.His systematic presentation of new and ancientknowledge, moving fluidly from easier to harderproblems, assisted the dissemination of mathe-matical ideas. Most important, through Fibonaccia new concept of number emerged in the West.His endorsement of the Indian numeral was cru-cial to advance the science of calculation, but hewas the first to recognize negative quantities, andzero as well, as genuine numbers. In addition, hisuse of a symbol or letter as an abbreviated repre-sentation of a generic number was an importantstep toward modern algebra, which is abstract andtotally symbolic. Fibonacci was familiar with

Arabic texts, which had preserved the flower ofGreek mathematical endeavor; in transmittingand systematizing this material, Fibonacci re-vived interest in the classics. In the ensuingyears, European mathematicians would makewondrous advances from the Greek foundation.

Further ReadingChong, P. “The Life and Work of Leonardo of Pisa,”

Menemui Matematik. Discovering Mathematics 4,no. 2 (1982): 60–66.

Gies, J., and F. Gies. Leonard of Pisa and the NewMathematics of the Middle Ages. New York:Crowell, 1969.

Hooper, A. Makers of Mathematics. New York:Random House, 1958.

� Fisher, Sir Ronald Aylmer(1890–1962)BritishStatistics

Ronald Aylmer Fisher was born into a large fam-ily of ordinary origins; before his death he wouldbe bestowed with numerous honors, and wouldreceive knighthood. This talented man, whommany would describe as a genius, made severalimportant contributions to the theory of statis-tics. Indeed, the theories of hypothesis testing,analysis of variance, experimental design, andestimation are much indebted to his labors.

Twin sons were born on February 17, 1890,in London, England, to Fisher’s father, a Britishauctioneer. One of them soon died, and Ronaldwas the survivor. In the following years he de-veloped his mathematical gifts, directing all hisenergy in this direction. He studied at Cambridgefrom 1909 to 1912, concentrating on mathe-matics and theoretical physics. After graduationFisher took up diverse employments unrelated tomathematics, and in 1917 he married RuthEileen Guiness, who bore him eight children.Fisher was eccentric, and with his eloquent

Fisher, Sir Ronald Aylmer 97

tongue made a strong impression on most peo-ple he encountered. However, despite being ami-able toward his own following, he could be quiteabrasive toward his rivals. In fact, Fisher createdmany enemies through his unveiled hostility to-ward dissenters, expressed through scathing per-sonal remarks.

Fisher started his mathematical career whenhe joined Rothamsted Experimental Station in1919, with the task of sorting and analyzing nu-merous field data. Through his time atRothamsted, Fisher would establish himself as aleading statistician; later he would become a pro-fessor of eugenics at University College and a

professor of genetics at Cambridge. His first con-tribution to mathematics was the discovery ofthe sampling distributions of various statistics,such as the correlation coefficient. The samplingdistribution is important, since it tells the prac-titioner how unlikely an extreme value of a sta-tistic, computed on a data set, would be. Fisherdeveloped WILLIAM GOSSET’s work on t statistics,constructing a complete theory of hypothesistesting for small samples. Hypothesis testing isnow a huge component of statistical theory, andis basically a mathematical formulation of themethods by which Western civilization has pur-sued empirical science.

Subsequently, Fisher developed an exten-sion of the t test to multiple groups called theANOVA (analysis of variance)—now one of themost widely used statistical procedures by med-ical and industrial researchers. Given severalgroups of data, each separated by a factor (suchas males and females or blood type), theANOVA procedure assesses whether there areany significant differences between the groups.Fisher further pointed out that the considerationof several factors simultaneously (so, for exam-ple, one might vary gender and blood type con-currently) was not only possible but was also cru-cial for determining whether factors affect eachother. These contributions to experimental de-sign are viewed by many as the most importantof all that Fisher has done—and indeed, theseideas and procedures are very widely usedthroughout science.

In terms of theory, Fisher desired to placestatistics on a firm mathematical foundation. Inthe paper “On the Mathematical Foundation ofTheoretical Statistics” (1922) he developed asensible theory of estimation. The central issuewas the following: Given a collection of numer-ical data, how can one summarize—with onenumber or statistic—a particular feature of themeasurements in the “best” possible way? Fisherformulated the following criteria for a “good” sta-tistic: it should be consistent (increased accuracy

Sir Ronald Aylmer Fisher laid the mathematicalfoundations for modern statistics. (Courtesy of theLibrary of Congress)

98 Fourier, Jean-Baptiste-Joseph

with larger sample size), efficient (precise), andsufficient (all relevant information has been uti-lized). He made the concept of “information”quantitative, which gave a method for measur-ing the amount of order within data.

Besides making the theory of estimationmore rigorous and structured, Fisher developedthe concept of standard error mathematically.Typically, estimates would be accompanied by a“plus or minus” band of values, which wouldform an interval around the estimate. Fisherwould associate a probability to each interval; ifthe interval was widened, the probability wouldincrease toward one. The probability was thelikelihood of capturing the desired parameterwithin the interval. In this way, one could rig-orously quantify “how good” an interval was.Because this procedure would involve the prob-able location of a parameter—which was sup-posed to be fixed and unknown—a controversyarose between Fisher’s school of thought and hisopponents. Debate still continues today, be-tween Bayesian statisticians, who view unknownparameters as being themselves random, andFrequentists, who hold to the orthodox view.

Ronald Fisher also labored extensively in thefield of genetics, and was particularly interestedin the theory of natural selection. He formulateda “fundamental theorem of natural selection,”which stated, “The rate of increase in fitness ofany organism at any time is equal to its geneticvariance in fitness at that time.” His work, typ-ically, was a blend of theory and practice, andhe conducted some breeding experiments in hisown home.

Later in life, Fisher was bestowed with manyhonors: In 1929 he was elected to fellowship ofthe Royal Society, and in 1952 he receivedknighthood. In 1959 he retired from his positionat Cambridge and moved to Australia. There hespent his last three years working on mathemat-ical statistics at the Commonwealth Scientificand Industrial Research Organization. He diedin Adelaide, Australia, on July 29, 1962.

Fisher is famous among modern statisticiansfor his work on the mathematical foundations ofstatistics. He is also well known for his researchinto genetics. Many statistical objects, such asthe Fisher Information Criterion, can be attrib-uted to his genius, and the modern field of math-ematical statistics is founded on much of hiswork.

Further ReadingBox, J. R. A. Fisher, the Life of a Scientist. New York:

John Wiley, 1978.Gower, J. “Ronald Aylmer Fisher 1890–1962,”

Mathematical Spectrum 23 (1990–91): 76–86.

� Fourier, Jean-Baptiste-Joseph(1768–1830)FrenchAnalysis

Jean-Baptiste-Joseph Fourier was the son of a tai-lor. He was born on March 21, 1768, in the townof Auxerre, France. By age nine he had lost bothhis parents, Joseph and Edmée. The archbishopplaced him in the local military school, wherehe developed a strong inclination toward math-ematics. Before the end of his life, Fourier wouldgo on to found the theory of trigonometric se-ries and make great advances in the under-standing of the dynamics of heat.

Fourier was born in a tranquil period ofFrance’s history, which was soon to erupt intothe chaos of the French Revolution. Initially, theyoung man wished to join the artillery or engi-neers, but was instead sent to a Benedectineschool in St.-Benoît-sur-Loire. When the revo-lution began in 1789, he returned to Auxerre asa teacher in his former military school. He be-came prominent in local affairs, and defied thegovernment through his brave defense of thevictims of the Terror. In 1794 he was arrested,but was released after Robespierre’s executionand briefly attended the École Normale. Though

Fourier, Jean-Baptiste-Joseph 99

this school only existed for a year, it seems thatFourier made a strong impression on the faculty,and he was appointed as an assistant lecturer in1795 at the École Polytechnique. There he fellafoul of the reaction to the previous regime(which he had actually struggled against) andwas imprisoned, but his colleagues managed toprocure his release. In 1798 he was chosen to ac-company Napoleon on his Egyptian campaign,where he became secretary of the Institut d’Égypteand conducted various diplomatic missions.Despite these duties, Fourier found time to pur-sue his mathematical interests.

In 1801 Fourier returned to France, but hisdesire to return to his post at the ÉcolePolytechnique was not realized—Napoleon, hav-ing ascertained Fourier’s administrative talent,

appointed him as prefect of the department ofIsére. He was successful in this appointment, andwas made a baron in 1808 by Napoleon. At thistime he wrote the historical preface of theDescription de l’Égypte, completed in 1809, whichwas a record of the work of the Institut d’Égypte.When Napoleon was defeated in 1814, the partyescorting him to Elba planned to pass throughGrenoble, where Fourier was installed as prefect.Fourier negotiated a detour for the group inorder to save Napoleon from an embarrassingencounter. On Napoleon’s return from Elba in1815, Fourier fulfilled his duties as prefect by or-ganizing a token resistance at Lyons. Later, thetwo friends met up in Bourgoin, and Napoleonreestablished his trust in the mathematician bymaking him a count and the prefect of theRhône.

However, the new regime was brutal, andbefore the end of Napoleon’s brief restoration,Fourier had resigned his commission, and hecame to Paris to pursue his research without dis-traction. Things were difficult for Fourier, sincehe was unemployed with a poor political repu-tation. Soon an old friend secured him a posi-tion as director of the Bureau of Statistics in thedepartment of the Seine, which provided for hisneeds while leaving sufficient time for him toprogress in his mathematical studies.

Fourier’s main achievements lie in the area ofheat diffusion. Much of the work was completedduring his tenure at Grenoble, though his inter-ests in heat go back to his sojourn in Egypt. In1807 he presented a lengthy paper on heat diffu-sion in special continuous bodies to the Academy;the content was based on the diffusion (or heat)equation in three variables. Due to the use of so-called Fourier series in the paper, one of the re-viewers, JOSEPH-LOUIS LAGRANGE, prevented thework’s publication—Lagrange felt that trigono-metric series were of little use. In 1810 a revisedversion of the paper was submitted in competi-tion for a proffered prize, and the update con-tained new material on heat diffusion in infinite

Joseph Fourier studied the theory of heat anddeveloped trigonometric series as representations offunctions. He invented harmonic analysis.(J. Boilly Del., Geille Scup., Deutsches MuseumMünchen, courtesy AIP Emilio Segrè Visual Archives)

100 Fourier, Jean-Baptiste-Joseph

bodies. The latter sections of the paper dealtwith the physical aspects of heat, such as radia-tion, which would occupy Fourier increasinglyin later years. This excellent work won the prize,and later it was expanded into the book Théorieanalytique de la chaleur (Analytic theory of heat).

The importance of Fourier’s contributionscan be seen in two aspects: first, the formulationof the physical problem as a boundary-valueproblem in the theory of linear partial differen-tial equations; and second, the powerful mathe-matical tools for the solution of these problems.These tools would become vastly influential tothe development of later mathematics, leavingnumerous descendants behind.

Early notions on the mechanics of heat in-volved the notion that some type of shuttletransferred heat between discrete particles.Eventually, Fourier was able to discover a differ-ential equation that smoothly described the dy-namics of heat—this was the so-called diffusionequation. The domain for this equation was a“semi-infinite” strip—essentially the positivepart of the x-axis—that was uniformly hot at oneend and uniformly cold on the sides. This con-figuration of the problem was both simple andphysically meaningful. In this context, Fourierconstructed a series solution to the problem thatinvolved trigonometric terms. He was aware ofthe convergence difficulties involved with thistype of approach, and he handled these issuesquite effectively. The surprising thing about hiswork was that he demonstrated that for manygeneric functions one could construct trigono-metric series that were identical with the func-tion on an interval. For nonperiodic functions,it seems strange that one can express them as asum of sines and cosines.

Fourier then generalized his solutions tothree dimensions and to other configurations,such as a cylinder. Some of his last creative workcame in 1817 and 1818, wherein he developeda relation between integral-transform solutionsand operational calculus. The so-called Fourier

transform of a function, so useful for the solu-tion of differential equations, came out of theselabors.

In many ways Fourier was very practical inhis approach to mathematics: Every statementhad to possess physical meaning, and he wasguided in his investigations by his excellentphysical intuition. He developed a coherentpathway through a jumble of ad hoc techniquesfor solving differential equations, and a gift forinterpreting the asymptotic properties of his so-lutions for physical meaning. When possible, hewould test his results against the outcome of ex-perimentation. His mathematical legacy is enor-mous, with such giants as BERNHARD RIEMANN,GEORG CANTOR, and HENRI-LÉON LEBESGUE fol-lowing his work in mathematical analysis.

In 1817 Fourier was elected to the reconsti-tuted Académie des Sciences after some politi-cal troubles. Gradually he advanced in his career,despite the enmity of SIMÉON-DENIS POISSON andthe opposition of royalists. His later honors in-clude election to the Académie Française in1827 and election as a foreign member of theRoyal Society. Throughout his life, he won thesupport of many friends through his unselfishsupport and encouragement, and he aided manymathematicians and scientists in his later years.

While in Egypt, he had developed some ill-ness, possibly myxedema, so that increasingly hewas confined to his own heated quarters. OnMay 4, 1830, he had an attack while descend-ing some stairs in his Paris home. He died 12days later. He is certainly one of the greatest ofall mathematicians, as Fourier analysis is anenormously successful method in engineeringand statistics; its applications to differentialequations, called harmonic analysis, is a lovelyand thriving branch of mathematics.

Further ReadingArago, F. “Joseph Fourier,” in Biographies of

Distinguished Scientific Men. Boston: Ticknor andFields, 1859.

Fréchet, René-Maurice 101

Bose, A. “Fourier, His Life and Work,” Bulletin of theCalcutta Mathematical Society 7 (1915–16):33–48.

———. “Fourier Series and Its Influence on Some ofthe Developments of Mathematical Analysis,”Bulletin of the Calcutta Mathematical Society 9(1917–18): 71–84.

Grattan-Guinness, I. Joseph Fourier, 1768–1830: ASurvey of His Life and Work. Cambridge, Mass.:MIT Press, 1972.

Herivel, J. Joseph Fourier: The Man and the Physicist.Oxford: Clarendon Press, 1975.

———.“The Influence of Fourier on BritishMathematics,” Centaurus 17, no. 1 (1972):40–57.

� Fréchet, René-Maurice(1878–1973)FrenchAnalysis

Functional analysis arose in the 20th centurythrough the genius of several mathematicians.René-Maurice Fréchet was one of these impor-tant individuals. Fréchet was the first to presenta notion of topology (the study of continuousfunctions and their effects on high-dimensionalsurfaces) for the new function spaces, and his ab-stract ideas became fundamental to the later de-velopment of this field.

Fréchet was born on September 10, 1878, inMaligny, France. He was the fourth of six chil-dren, and his parents were middle-classProtestants. His father, Jacques Fréchet, was thedirector of an orphanage, but later became aschoolteacher when the family moved to Paris.Fréchet’s mother, Zoé, ran a boardinghouse forforeigners. At the Lycée Buffon in Paris, Fréchetlearned mathematics from Jacques Hadamard,who recognized the youngster’s budding talent.

Fréchet entered the École Normale Supér-ieure in 1900, and graduated three years later.During this time he made the acquaintance of

ÉMILE BOREL, and this developed into a lifelongfriendship. Fréchet continued his studies underthe tutelage of Hadamard, and completed his dis-sertation in 1906 on the topic of functional cal-culus. The study of functionals (or numericallyvalued functions of ordinary functions) was a newtopic, and Fréchet introduced topological no-tions into the space of functionals in severalnovel ways. In particular, Fréchet was able to de-fine the notions of continuity and limit for thesefunctionals. Generalizing the work of GEORG

CANTOR to function spaces, Fréchet was able todefine the notions of compactness, separability,and completeness that are encountered in pointspaces. These early ideas later became pivotalelements of the modern theory of functionalanalysis. Today, functional analysis is used in awide variety of engineering and statistical appli-cations, and is largely responsible for the rapidadvance of many technologies (for instance, me-diating the effect of wind turbulence on aircraftflight) of the present day.

Fréchet eventually obtained a professorshipat the University of Poitiers in 1910. DuringWorld War I, he served as an interpreter to theBritish army. After the war he headed theInstitute of Mathematics at the University ofStrasbourg in 1919. He married Suzanne Carrivein 1908, and they had four children.

Meanwhile, Fréchet continued his researchinto functional analysis, establishing the repre-sentation theorem for continuous linear func-tionals in 1907. FRIGYES RIESZ independentlydiscovered this important result. Fréchet gen-eralized the notion of derivative from ordinarycalculus to operators in function spaces and alsoextended the integration ideas of Johann Radonand HENRI-LOUIS LEBESGUE to spaces without atopology. He also developed some of the first ab-stract topological spaces from certain set axioms.These results are now classical in the field offunctional analysis and point-set topology.

Fréchet moved to the University of Paris in1928, where he taught mathematics until his

102 Fredholm, Ivar

1949 retirement. During this period he focusedon probability and statistics, using functionalanalysis as a tool to solve several concrete prob-lems in probability and statistics. However, thiswork was not equal in originality to his first con-tributions to the topology of function spaces. Hedied in Paris on June 4, 1973.

Fréchet’s work received accolades from hiscolleagues in America, and he received severalhonors during his lifetime, being made a fellowof the Royal Society of Edinburgh and anHonorary Fellow of the Edinburgh MathematicalSociety. Hadamard, in a 1934 report to theAcademy of Sciences, announced that Fréchet’swork in functional analysis, in terms of abstrac-tion and generality, could be compared with thepioneering labors of EVARISTE GALOIS in alge-braic field theory. Certainly, Fréchet’s resultsgave a firm topological foundation to functionalanalysis, which has proved to be one of the mostuseful tools of modern mathematics, with man-ifold applications to statistics and engineering.

Further ReadingKendall, D. “Obituary: Maurice Fréchet, 1878–1973,”

Journal of the Royal Statistical Society. Series A.Statistics in Society 140, no. 4 (1977): 566.

Taylor, A. “A Study of Maurice Fréchet I,” Archivefor History of Exact Sciences 27 (1982): 233–295.

———. “A Study of Maurice Fréchet II,” Archive forHistory of Exact Sciences 34 (1985): 279–380.

———. “A Study of Maurice Fréchet III,” Archive forHistory of Exact Sciences 37 (1987): 25–76.

� Fredholm, Ivar(1866–1927)SwedishAnalysis

The theory of integral equations, which is con-cerned with finding an unknown function thatsatisfies an equation involving its integral, haswide applications to mathematical physics, and

has become a field of interest in its own right.Ivar Fredholm made a few significant contribu-tions to this field, and his work has attained clas-sical importance to this branch of mathematics.

Ivar Fredholm was born in Stockholm,Sweden, on April 7, 1866, the son of an upper-class family. His father was a wealthy merchant,and his mother came from the elite of Sweden;as a result, Fredholm received the best educa-tion. He displayed his brilliance at an early age,passing his baccalaureate in 1885. A year at thePolytechnic Institute of Sweden fostered an en-during interest in applied mathematics and prac-tical mechanics. He enrolled in the Universityof Uppsala the next year, obtained his bachelorof science degree in 1888 and his doctorate in1898. Fredholm also took classes at theUniversity of Stockholm, studying under therenowned Magnus Mittag-Leffler, and he re-ceived an appointment there in 1898; in 1906he became a professor of mathematical physics.He died on August 17, 1927, in Stockholm.

Fredholm’s thesis treated a topic in the the-ory of partial differential equations, which hadapplications to the study of deformations of ob-jects subjected to interior or exterior forces. Adecade later, Fredholm completely generalizedhis work to solving the general elliptic (a par-ticular type of differential equation) partial dif-ferential equation.

Fredholm acquired fame for his solution ofthe so-called Fredholm integral equation, whichhas wide applications in physics—for example,these equations arise in the study of the vibrat-ing membrane. Many mathematicians, such asNIELS HENRIK ABEL and VITO VOLTERRA, had al-ready attacked this problem with only partialsuccess. Fredholm presented a complete solutionin 1903 after recognizing a fundamental anal-ogy between the Fredholm equation and anequation of matrices; this observation wouldlater become a cornerstone in functional analy-sis—namely, that integration against a kernelfunction was a linear operator similar to matrix

Frege, Friedrich Ludwig Gottlob 103

multiplication. By examining the analogy of thematrix determinant for the integral operator,Fredholm was able to formulate precise condi-tions under which the equation was even solv-able, and to give a formula for the solution whenit existed.

This work certainly constitutes a milestonein functional analysis and the theory of integralequations. When these advances were commu-nicated to DAVID HILBERT, he was inspired to fur-ther the study of these equations; soon afterward,Hilbert invented a theory of eigenvalues for in-tegral operators, in analogy with matrices, andconstructed Hilbert spaces. Thus, Fredholm in-spired the construction of some of the most valu-able tools of functional analysis.

Further ReadingGarding, L. Mathematics and Mathematicians: Mathe-

matics in Sweden before 1950. Providence, R.I.:American Mathematical Society, 1998.

� Frege, Friedrich Ludwig Gottlob(1848–1925)GermanLogic

Gottlob Frege performed substantial work onmathematical logic in the 19th century; indeed,he is viewed by many as the father of modernmathematical logic. The language that he cre-ated in order to rigorously analyze arithmeticwould later develop into the syntax and nota-tion of modern proof theory.

Gottlob Frege was born on November 8,1848, in Wismar, Germany, to Alexander Fregeand Auguste Bialloblotzky. His father was theprincipal of a girls’ high school in Wismar, andGottlob attended the Gymnasium there. From1869 to 1871 he was a student at Jena, and af-ter this period matriculated at the University ofGöttingen, where he took courses in mathe-matics, physics, chemistry, and philosophy. Two

years later he had earned his doctorate in phi-losophy with the thesis Über eine geometrischeDarstellung der imaginaren Gebilde in der Ebene(Over a geometrical representation of imaginarythings in the plane). His 1874 dissertation wasconcerned with certain groups of functions, andFrege’s ambition was to give a definition of quan-tity that would greatly extend the applicabilityof the resulting arithmetic. At about this timehe began work on the project of providing a rig-orous foundation for arithmetic. Frege wished todefine number and quantity in a satisfactorymanner, and he turned to logic as an appropri-ate vehicle.

At this period of history there was little inthe way of a coherent treatment of mathemati-cal logic. Since Frege wanted to be precise in hisdevelopment of the theory of numbers, he de-cided to construct a language of logic in whichto formulate his ideas. The tools for analyzingmathematical proofs were published in Begriff-schrift (Concept script) in 1879, and some of theideas from his Jena dissertation entered into hisconcept of quantity. In the same year, he was ap-pointed extraordinary professor in Jena, and wasmade an honorary professor in 1896. His dili-gent work toward the logical construction ofarithmetic over the years resulted in his two-vol-ume Grundgesetze der Arithmetik (Basic laws ofarithmetic) (1893–1903). In 1902 BERTRAND

RUSSELL pointed out a contradiction in Frege’ssystem of arithmetic; this comment proved to bedisastrous, as Frege could find no way of reme-dying the problem. Indeed, as later work by KURT

FRIEDRICH GÖDEL would demonstrate, any effortsto construct complete and consistent numbertheories were doomed to failure.

The Begriffschrift should be viewed as a for-mal language as a vehicle for pure thought. Thislanguage consisted of various symbols (such asletters) that could be combined together ac-cording to certain rules (the grammar) to formstatements. As with arithmetic, after whichFrege’s language was modeled, one could then

104 Frege, Friedrich Ludwig Gottlob

perform calculations whose result would be a log-ical calculation rather than a numeric quantity.The idea of a logical calculus goes back at leastto GOTTFRIED WILHELM VON LEIBNIZ, who sup-posed that one day all philosophical debatecould be reduced to logical calculations. Frege’scalculus could be used to formalize the notion ofa mathematical proof, so that one could, essen-tially, compute the conclusion.

The basic components of Frege’s calculus arean assertion symbol (represented by a verticalstroke), a conditional symbol (for instance, Aimplies B), and a deduction rule, which statesthe following: If we assert A, and A implies B,then we may assert B. Frege also developed no-tation for negation, and demonstrated that log-ical and and or could be expressed in terms ofthe conditional and negation symbols. On topof these basic notions he added a theory of quan-tity, rigorously defining such notions as for alland there exists.

There is a school of mathematics calledformalism, whose adherents believe that thereis no true or inherent meaning to mathemat-ics, but that mathematics is purely a formal lan-guage with which other ideas may be expressed,and mathematical truth can be arrived at onlyby playing according to the rules of the game.Frege was not a formalist and was not inter-ested in applying his system to questions per-taining to a formalist agenda. Ironically, his workwas quite suitable as a foundation for formallogic.

Frege’s work Grundlagen der Arithmetik(Foundations of arithmetic) (1884) defines thenotion of number and relies upon the languageintroduced in Begriffschrift. Here he gives a criti-cism to the previous theories of number, pointingout their inadequacies; he argues that equality ofnumber is an essential component to the notionof number. The Grundgesetze incorporates andrefines his previous work, including improve-ments based on several papers. Many of these

ideas had great influence on subsequent philo-sophical discussion, in particular influencing thephilosophy of Wittgenstein.

After 1903 Frege’s powers of thought werein decline; he seemed unable to keep up withan increasingly modern and alien mathemati-cal culture. In this latter period, he spent hisenergy reacting against various new develop-ments in mathematics, and especially came intoconflict with DAVID HILBERT and his program forthe axiomatization of mathematics. In 1917Frege retired, and after this produced LogischeUntersuchungen (Logical investigations) as anextension of previous work. He died in BadKleinen, Germany, on July 26, 1925.

Frege is principally remembered for his workon mathematical logic, which led to modernproof theory. Other great logicians such asRussell and Gödel continued his work. AlthoughFrege’s effort to construct a complete, consis-tent number theory was doomed to failure, theideas that he formulated in the course of his re-search greatly influenced later generations ofmathematicians.

Further ReadingAngelelli, I. Studies on Gottlob Frege and Traditional

Philosophy. Dordrecht, the Netherlands: KluwerAcademic Publishers Group, 1967.

Dummett, M. Frege: Philosophy of Language.Cambridge, Mass.: Harvard University Press,1981.

———. Frege: Philosophy of Mathematics. London:Ducksworth, 1991.

———. The Interpretation of Frege’s Philosophy.Cambridge, Mass.: Harvard University Press,1981.

Kenny, A. Frege. London: Penguin Books, 1995.Weiner, J. Frege. New York: Oxford University Press,

1999.Wright, C. Frege’s Conception of Numbers as Objects.

Aberdeen, U.K.: Aberdeen University Press,1983.

Fubini, Guido 105

� Fubini, Guido(1879–1943)ItalianAnalysis, Geometry

Guido Fubini was one of Italy’s most productivemathematicians, opening up new areas of re-search in several areas of analysis, geometry, andmathematical physics. His unfailing intuition,together with his mastery of the techniques ofcalculation, made him a formidable mathemati-cian. His accomplishments earned him the royalprize of Lincei in 1919, and he was a member ofseveral Italian scientific academies.

On January 19, 1879, in Venice, GuidoFubini was born to Lazarro Fubini and ZoraideTorre. Lazarro Fubini taught mathematics inVenice, and the son followed in his father’s foot-steps. He quickly completed his secondary stud-ies in Venice, and entered the Scuola NormaleSuperiore di Pisa at age 17. A few years later, in1900, Fubini defended a thesis on Clifford’s par-allelism in elliptic spaces, a topic in differentialgeometry. This subject studies smooth surfacesand solids (and their higher-dimensionalanalogs), and Fubini’s contribution came tohave great importance for this area of mathe-matics after the thesis was included in a 1902edition of Luigi Bianchi’s treatise on differentialgeometry.

Fubini had already made great progress forone so young, and he remained at Pisa for an-other year to obtain the diploma, a higher degreethat would allow him to teach at the universitylevel. The memoir that he wrote for his diplomainvestigates the theory of harmonic functions inspaces of constant curvature—a topic quite dif-ferent from that explored in his doctoral thesis.

It is said that Guido Fubini was a man ofgreat cultivation and kindness; his affability andwit made him a pleasant companion. As ateacher, Fubini was possessed of great talents,and over many years as a professor he was able

to influence many young mathematicians. Hewas a small man, but had a vigorous voice. Familywas quite important to Fubini, and he took a se-rious interest in his sons’ engineering studies.Luigi Bianchi, one of his teachers from Pisa, wasa role model for him, and Fubini was grateful forBianchi’s assistance and guidance.

In 1901 Fubini was placed in charge of acourse at the University of Catania, and was soonnominated to the position of full professor. FromCatania he went to the University of Genoa, andin 1908 again transferred, to the Politecno inTurin. At the Politecno and University of Turin,Fubini taught mathematical analysis.

In the subject of analysis Fubini did much ofhis best work, focusing on linear differentialequations, partial differential equations, analyticfunctions of several complex variables, and mo-notonic functions. Within the calculus of vari-ations, he studied various problems involvingintegration, such as nonlinear integral equa-tions with asymmetric kernels. He also exam-ined discontinuous groups, in particular studyingmotion on Riemannian surfaces. For non-Eucli-dean spaces, he introduced sliding parameters,which made possible a transposition of results from ordinary differential geometry to ellipticalgeometry.

Differential projective geometry was thebranch of mathematics to which Fubini madehis most significant contributions. He developedgeneral procedures, which still bear his name.Projective geometry is the study of spaces oflines, and arose out of medieval artistic studiesof perspective. To this difficult subject, Fubinibrought the tools of differential calculus andgroup theory.

Fubini also made contributions to mathe-matical physics. During World War I he madetheoretical studies on the accuracy of artilleryfire, and later examined problems in acousticsand electricity. The mathematical aspects of en-gineering interested him, and a posthumous

106 Fubini, Guido

work on engineering mathematics appeared in1954.

Fubini continued for many years at Turin,and in 1928, upon the death of Bianchi, he be-came coeditor of the Annali di matematica puraed applicata (Annals of pure and applied mathe-matics). He held this position for 10 years, butin 1938 faced forced retirement due to new raciallaws instituted by the Fascist government. Hewas concerned for the future of his two sons, andthe following year he received and accepted anoffer from the Institute for Advanced Study inPrinceton, New Jersey. In the United States hewas made welcome, and his voluntary exileproved to be wise due to the unfolding events inEurope.

At this point his health was poor, but hecontinued to teach at New York University un-til he died of a heart ailment on June 6, 1943,in New York City. Fubini left a great legacy inmathematics: He had stimulated many branches,opening up new areas of inquiry and providinginnovative techniques. His textbooks have beenwidely employed for courses in analysis, with col-lections of problems used by many generationsof students. Indeed, his labors were varied, yetwith great influence on the future of mathe-matics.

Further ReadingPapini, P. “Guido Fubini 1879–1943,” European

Mathematical Society Newsletter 9 (1993) 10.

G

107

� Galilei, Galileo(1564–1642)ItalianMechanics, Geometry

Galileo Galilei is one of the best-known namesin the history of science. This man lived in atime when speculative philosophy was graduallysupplanted by mathematics and experimentalevidence, and indeed he contributed, perhapsmore than any of his contemporaries, to this par-adigm shift. Galileo’s research into mathematics,mechanics, physics, and astronomy completelyaltered the way that people pursued knowledgeof the natural world, and started an avalancheof scientific inquiry throughout Europe.

Galileo was born on February 15, 1564, inPisa, Italy. His father, Vincenzio Galilei, was amusician and member of an old patrician fam-ily. Vincenzio married Giulia Ammannati ofPescia in 1562, and Galileo was born two yearslater. He would be one of seven children. He wasfirst tutored in Pisa, but the family returned toFlorence in 1575. He studied at the monasteryof Santa Maria at Vallombrosa until 1581, whenhe was enrolled at the University of Pisa as amedical student. Galileo had little interest inmedicine, but preferred mathematics, in whichhe progressed rapidly despite his father’s disap-proval. In 1585 he left school without a degree

and pursued the study of EUCLID OF ALEXANDRIA

and ARCHIMEDES OF SYRACUSE privately.During the next four years, Galileo gave pri-

vate mathematics lessons in Florence, whilecomposing some minor works on mechanics andgeometry. It was at this time that Galileo’s fa-ther became engaged in a musical controversy.Vincenzio Galilei resolved the dispute throughexperimental investigations, and this approachproved to have a great influence on his son.Galileo would mature into a great experimen-talist, testing mathematical theories with phys-ical evidence.

In 1589 Galileo obtained the mathematicschair at Pisa, where he performed some of hisfirst experiments on falling bodies. At about thistime, Galileo embarked on a lifelong campaignto discredit Aristotelian physics, the official viewof the world espoused by the Roman CatholicChurch, which, among other things, declaredthat the denser objects fall faster. Galileo infu-riated many of his fellow professors by publiclydemonstrating that bodies of different weight fallat the same speed—by dropping such objects outof the Leaning Tower of Pisa. His treatise onthese topics was De motu (On motion), and itrelied on some ideas from Archimedes.

His father died in 1591, creating an uncer-tain financial situation for Galileo. Due to theanimosity that he had aroused, his position at

108 Galilei, Galileo

Pisa was not renewed; however, his friends as-sisted him in obtaining an appointment atPadua, where the community was less conserva-tive. He lectured on Euclid, CLAUDIUS PTOLEMY,and mechanics, but did not become interestedin astronomy until much later. In 1597 Galileoexpressed his sympathies to the Copernican sys-tem to Johannes Kepler, but he did not publiclyadvance the anti-Aristotelian astronomy at thispoint. In the same year he manufactured an in-strument (the proportional compass) for sale,over which a controversy later arose in whichGalileo obtained the priority of its invention.

While in Padua, Galileo took a mistressnamed Marina Gamba, who later bore him twodaughters and a son. His eldest daughter,Virginia, would be a great solace to him in later

years of strife and conflict. In 1602 he becameinterested in the motions of pendulums and theacceleration of falling bodies, and he derived thesquare law for free fall correctly in 1604, thoughwith an incorrect assumption. In the same year,a supernova led to dispute over the Aristoteliannotion of the incorruptibility of the heavens,and Galileo delivered several public lectures onthis topic. He was soon to become increasinglyinterested in the study of the skies.

In 1609 Galileo learned of the invention ofa telescope by Hans Lipperhey, a Dutch lensgrinder, and the Paduan professor set about con-structing his own version, which was eventually30 times more powerful than the original. Thisdevice, so useful for navigation, won him a life-time appointment at Padua, and he set about us-ing it to view the heavens. He soon discoveredthat the Moon had mountains and that theMilky Way consisted of many separate stars.Galileo published many additional discoveries inSidereus nuncios (The siderial messenger) (1610).His resulting fame gained him the post of math-ematician and philosopher to the grand duke ofTuscany, where he could focus on his researchwithout having to teach.

The book created a furor in Europe, andmany claimed that it was a fraud, though Keplerendorsed it. In the satellites of Jupiter, Galileo nowsaw decisive evidence against the Aristotelianconception that all heavenly bodies revolvedaround the Earth. In 1611 he journeyed toRome, where he was honored by the Jesuits ofthe Roman College and admitted to the LinceanAcademy.

After this time, Galileo turned back tophysics and became embroiled in more con-troversies in Florence. The dispute concernedthe behavior of bodies floating in water, andGalileo supported the theories of Archimedesagainst those of Aristotle; he was able, usingthe concepts of moment and velocity, to ex-tend Archimedes’s ideas beyond hydrostaticsituations.

Galileo Galilei studied mechanics and derived thesquare law for free fall. He also advanced theseparation of science and theology. (Courtesy of theLibrary of Congress)

Galilei, Galileo 109

In 1613 Galileo published Letters onSunspots, which spoke out decisively for theCopernican system for the first time in print.Certain Catholics did not favorably regard thisdocument, and the opposition grew over the nextfew years. In Galileo’s opinion, theology shouldnot interfere with purely scientific questions—those that could be resolved experimentally; thisopinion later came before the Inquisition, and in1615 Galileo went to Rome to fight the sup-pression of Copernicanism. Pope Paul V, annoyedby questions of theological authority, appointeda commission to determine the Earth’s motion:in 1616 the commission ruled against theCopernican system, and Galileo was prohibitedfrom advocating that view.

Returning to Florence, Galileo turned to theproblem of determining longitudes at sea. Healso took up mechanics again, correctly defininguniform acceleration and putting forth many ofhis kinematic principles. But Galileo had a feistypersonality, and he was soon drawn into a newcontroversy regarding the motion of three cometsthat appeared in 1618. In a highly celebratedpolemic of science, Il saggiatore (The assayer),Galileo set forth a general scientific approach tothe investigation of celestial phenomena with-out direct reference to the Copernican system.In this essay, Galileo repudiates any authoritythat contradicts direct investigation, and thusputs forth empirical science as a sole foundationof knowledge of the universe. This work waspublished in 1623 and dedicated to Pope UrbanVIII on his installation. Galileo secured his oldfriend’s permission to write a book that wouldimpartially discuss the Copernican and Ptolemaicsystems, called Dialogue Concerning the Two ChiefWorld Systems.

This work, which occupied Galileo for thenext six years, consisted of a dialogue betweentwo advocates—for the Copernican andPtolemaic systems, respectively—attempting towin a layman over to their side. Galileo remainsofficially uncommitted, except in the preface;

the important concepts include the relativityand conservation of motion. Sunspots andocean tides were presented as pro-Copernicanarguments, as they could not be explained with-out terrestrial motion. The book was printed inFlorence in 1632, and soon its author was com-manded to come before the Inquisition inRome.

The pope, though once a friend of Galileo,had become convinced by Galileo’s enemies thatthe Aristotelian perspective was deliberatelymade to look foolish by the author. The trial wasprosecuted with vindictiveness, and Galileo wassentenced to life imprisonment after abjuringthe Copernican heresy. Under house arrest, hespent his remaining years completing his unfin-ished work on mechanics. By 1638 Discoursesand Mathematical Demonstrations Concerning TwoNew Sciences had appeared in France (he couldnot publish in Italy, as his works were banned).The content deals with the engineering scienceof materials and the mathematical science ofkinematics, and underlies much of modernphysics. Both the pendulum and the inclinedplane play a large role in Two New Sciences,and Galileo deduces the parabolic motion oftrajectories.

In the last four years of his life Galileo wasblind, and before his death he was denied therequest to attend Easter services or consult doc-tors. Finally, on January 8, 1642, in Arcetri, Italy,he passed away. He was certainly one of thegreatest scientists of all time, and an able math-ematician as well. Not only did he make greatcontributions to science, but also advanced anew epistemology—that knowledge of the nat-ural world (including mathematical knowledge)should be acquired through reason and experi-ment.

Further ReadingCampanella, T. A Defense of Galileo, the Mathematician

from Florence. Notre Dame, Ind.: University ofNotre Dame Press, 1994.

110 Galois, Evariste

Drake, S. Galileo at Work: His Scientific Biography.Chicago: University of Chicago Press, 1978.

———. Galileo Studies: Personality, Tradition, andRevolution. Ann Arbor: University of MichiganPress, 1970.

———. Galileo. New York: Hill and Wang, 1980.———. Galileo: Pioneer Scientist. Toronto, Ont.:

University of Toronto Press, 1990.Fantoli, A. Galileo: for Copernicanism and for the

Church. Vatican City: Vatican ObservatoryPublications, 1994.

Geymonat, L. Galileo Galilei. Turin: Piccola BibliotecaEinaudi, 1969.

Redondi, P. Galileo Heretic. Princeton, N.J.: PrincetonUniversity Press, 1987.

Ronan, C. Galileo. New York: Putnam, 1974.Shea, W. Galileo’s Intellectual Revolution: Middle

Period, 1610–1632. New York: Science HistoryPublications, 1977.

� Galois, Evariste(1811–1832)FrenchAlgebra

The French mathematician Evariste Galois leda meteoric life, notable for its short duration andmathematical profundity. Aside from his in-volvement in politics as a demagogue, Galois inhis work focused on the solution of algebraicequations and the foundation of modern grouptheory. It seems that he condensed a lifespan ofdeep mathematical work into just a few years oftroubled existence, accomplishing more in hisbrief time than many achieve through decadesof labor. His writings, though despised and un-recognized in his own lifetime, later proved tobe the fruit of a great genius, and his work in thetheory of groups and fields is now a pillar of themodern discipline of algebra. Even his writings onmathematical epistemology demonstrated re-markable insight, outlining an uncannily accurateprediction of the course of modern mathematics.

Evariste Galois was born on October 25,1811, in Bourg-la-Reine, France. He was the sonof Nicolas-Gabriel Galois, a liberal town mayor,and Adelaïde-Marie Demante, an eccentricwoman who determined to raise her son in theprinciples of stoic morality and austere religion.However, the young Galois had a happy child-hood and a good education. In October 1823he continued his studies at College Louis-le-Grand in Paris. His discipline problems there,perhaps in response to a royalist establishment,would prove to be an enduring characteristic ofhis personality.

In 1827 Galois entered his first mathemat-ics courses, and was so deeply impressed that hebegan reading the original works on his own ini-tiative. When he continued into more advancedinstruction, he was at the same time pursuing hisown personal investigations. Through 1828 hestudied recent literature on the theory of equa-tions, number theory, and the theory of ellipticfunctions (these are a class of functions, relatedto the cubic, which have many interesting prop-erties), and in March 1829 he started to publishhis results.

For centuries, mathematicians had been do-ing work on quadratic, cubic, and quartic equa-tions, attempting to provide general formulas forthe solutions. But an outstanding problem thatremained impregnable to attack was the solutionof the quintic, or fifth degree, polynomial withinteger coefficients. In 1828 Galois believed thathe had solved this problem, unaware that NIELS

HENRIK ABEL had previously shown that this wasimpossible; however, he soon realized his error,and instead set forth to investigate solubilitymore generally. The methods he used involvedgroup theory, which at the time was not very ad-vanced; in the course of his labor, Galois wouldprecisely formulate this theory, and introducevital, original ideas to it. A mathematical groupis a set with certain prescribed arithmetic opera-tions—examples include the integers under addi-tion, or the rotational and reflective symmetries

Galois, Evariste 111

of a polygon. The so-called Galois theory stud-ies the various groups that are related to a par-ticular equation. In May 1829, he communi-cated his results to the Academy of Sciences viaAUGUSTIN-LOUIS CAUCHY.

However, what seemed to be a promising ca-reer for the young Galois would evaporatethrough a series of frustrating circumstances,some of which were due to his volatile charac-ter. In July 1829 his father committed suicide,despondent over persecution for his liberalviews; Evariste, who was of the same liberal po-litical disposition (liberalism at this timemeant antimonarchical republicanism) seemeddoomed to suffer his father’s fate. Shortly after-ward, in February 1830, Galois prepared a mem-oir containing his new results for the Academyof Sciences, hoping to win that year’s grandprize. However, JEAN-BAPTISTE-JOSEPH FOURIER

lost the manuscript, and Galois was ejected fromthe competition. Suspecting this to be additionalpersecution, Galois became increasingly alien-ated from the scientific community. This mem-oir commemorated the death of Abel, anothermathematician who had done work in the samearea, and the work established that Galois hadmade significant progress beyond Abel, althoughthere remained some obstacles to obtaining ageneral answer to the question of the quintic’ssolvability.

The next major event was the July revolu-tion of 1830, in which Galois was politically in-volved. In December of that year, Galois wasexpelled from the École Normale Supérieure forwriting an antiauthoritarian letter, and he sub-sequently devoted himself to writing politicalpropaganda, even participating in the Paris ri-ots. In May he was arrested and imprisoned formaking a “regicide toast,” but was released amonth later.

Meanwhile, Galois published some work onanalysis, and presented an updated version of theprevious memoir, in which he demonstrates howthe former difficulties were overcome. The editor

of the journal, SIMÉON-DENIS POISSON, disparagedthe work and returned it to Galois. As a result,Galois’s frustration grew such that he came todespise his fellow scientists.

In July 1831 Galois was again arrested, andhe continued his mathematics while incarcer-ated. Later, in March 1832, he was transferredto a nursing home, where he wrote some essayson science and philosophy. The circumstancesof his death are cluttered, and there are com-peting theories as to the cause; however, it seemsthat he had some premonition of it. After a loveaffair that ended unhappily, he set about writingall his main results down. On May 30, 1832, hewas mortally wounded in a duel, and he died inParis on May 31.

Despite his contemporaries’ inability to rec-ognize the value of his work, it has since provedto be of profound influence in the creation ofmodern algebra. Not only did Galois theory pro-vide a complete answer to the question of thequintic, but it could be applied to many otherproblems as well; for example, one can prove theimpossibility of trisecting an angle with ruler andcompass. Perhaps Galois went unnoticed in hisday due to the profundity of his thought, beingtoo far beyond his peers to be properly appreci-ated; or it may have been due to his irascibletemperament and inability to get along with hisfellows. Fifteen years after his death, his cele-brated memoir was finally published, and math-ematicians, by this time more receptive, came togreatly appreciate his influence and thought. Hisintuition about the direction of future mathe-matics was remarkable, sketching in a few sen-tences the main highways of modern research,and his importance for modern algebra cannotbe overstated.

Further ReadingNg, L. “Evariste Galois,” Mathematical Medley 22, no.

1 (1995): 32–33.Ritagelli, L. Evariste Galois (1811–1832). Boston:

Springer-Verlag, 1996.

112 Gauss, Carl Friedrich

� Gauss, Carl Friedrich(1777–1855)GermanStatistics, Geometry, Analysis,Complex Analysis, Number Theory,Probability, Algebra

Known as the “prince of mathematicians,” CarlGauss is often ranked with SIR ISAAC NEWTON

and ARCHIMEDES OF SYRACUSE as the foremost ofthinkers; certainly, among his contemporaries hehad no rivals, as even they acknowledged.Conservative, cold, introspective, brilliant, pro-lific, tragic, and ambitious—Gauss’s life repre-sents that of the ideal or archetypal mathemati-cian in many respects. His work extendedthrough pure mathematics, including arithmeticand number theory, geometry, algebra, andanalysis, to applied mathematics—probabilityand statistics, mechanics, and physics—to thesciences of astronomy, geodesy, magnetism, anddioptrics, to industrial labors in actuarial scienceand financial securities. Gauss was an active fieldresearcher, empiricist, data analyst and statisti-cian, theorist, and inventor, with more than 300publications and more than 400 original ideasthroughout a long lifetime of intense and sus-tained effort. His genius flourished in a time oflittle mathematical activity in Germany, and isthe more remarkable for his solitary and reclu-sive style.

Carl Friedrich Gauss was born on April 30,1777, in Brunswick, Germany, to lower-class par-ents. Gauss’s mother was highly intelligent butsemiliterate, and was a devoted supporter of herson throughout her long life. His father workedvarious professions in an attempt to extricate hisfamily from poverty; of a practical bent, he neverappreciated his son’s extraordinary gifts, whichwere manifested at a young age. Before he couldtalk, Carl had learned to calculate, and at agethree he had corrected mistakes in his father’swage calculations! In his eighth year, while inhis first arithmetic class, Gauss found a formula

for the sum of the first n consecutive numbers.His teacher, suitably impressed, supplied the boywith literature to encourage his intellectual de-velopment.

In 1788, at age 11, the prodigy entered theGymnasium, where he made rapid progress in allhis studies, especially classics and mathematics.Through the benevolence of his teachers, theduke of Brunswick appointed Gauss a stipend,effectively making him independent; he was 16at the time. In 1792 he entered the CollegiumCarolinum, already possessed of a thorough sci-entific education. His extensive calculations andempirical investigations had led him to deepfamiliarity with numbers and their properties; hehad already independently discovered Bode’s law

Carl Gauss, greatest of mathematicians, dominatedalgebra, geometry, complex analysis, number theory,and statistics and invented the principle of leastsquares in regression. (Courtesy of AIP Emilio SegrèVisual Archives, Brittle Book Collection)

Gauss, Carl Friedrich 113

of planetary motion and the binomial theoremfor rational exponents.

While at the Collegium, Gauss continuedhis investigations in empirical arithmetic andformulated the principle of least squares used instatistics. In 1795 he entered the University ofGöttingen, and by this time he had rediscoveredthe law of quadratic reciprocity, related the arith-metic-geometric mean to infinite series expan-sions, conjectured the prime number theorem,and found some early results in non-Euclideangeometry. Gauss read Newton, but most mathe-matical classics were unavailable; as a result, henearly became a philologist. However, in 1796he made the significant discovery that the reg-ular 17-gon could be constructed by ruler andcompass, an outstanding problem that had beenunsolved for 2,000 years. This success encour-aged him to pursue mathematics.

His fate as a mathematician was set, and theyears until 1800 were marked by a remarkableprofusion of ideas. In style, Gauss adopted therigor of Greek geometry, although he thought al-gebraically and numerically. He pursued intenseempirical researches, followed by the construc-tion of rigorously laid theories. This approach toscience ensured that there was a close connec-tion between theory and practice.

In 1798, finished with the university, Gaussreturned to Brunswick, where he lived alone andworked assiduously. The next year he presentedthe proof of the fundamental theorem of alge-bra, which states that any degree n polynomialhas exactly n roots in the complex numbers; withthis result, the first of four proofs he would writefor this theorem, he earned his doctorate fromthe University of Helmstedt. The year 1801 sig-naled two great achievements for Gauss: theDisquisitiones arithmeticae (Arithmetical investi-gations) and the calculation of the orbit of thenewly discovered planet Ceres. The former was asystematic summary of previous work in numbertheory, in which he solved most of the difficultoutstanding questions and formulated concepts

that would influence future research for two cen-turies. He introduced the concept of modularcongruence, proved the law of quadratic reci-procity, developed the theory of quadratic forms,and analyzed the cyclotomic equation. Thisbook won Gauss fame and recognition amongmathematicians as their “prince,” but his austerestyle ensured that his readership was small. Asfor Ceres, it was a new planet that had been ob-served by Giuseppe Piazzi and subsequently waslost. Gauss, equipped with his computational tal-ents, took on the task of locating the truant ce-lestial body. With a more accurate orbit theory,which used an elliptical rather than circular or-bit, and his least squares numerical methods, hewas able to predict Ceres’ location. Because hedid not disclose his methods, the feat seemed su-perhuman, and established Gauss as a first-classscientific genius.

During the next decade, Gauss exploited thescientific ideas of the previous 10 years. He tran-sitioned from pure mathematician to astronomerand physical scientist. Although he was treatedwell by the duke of Brunswick, who still sup-ported him with a stipend, Gauss decided on as-tronomy as a stable career in which he couldpursue research without the burden of teaching;in 1807 he accepted the directorship of theGöttingen observatory. He made some contactsamong other scientists that sprouted into col-laborations, but had little interaction with othermathematicians—he exchanged a few letterswith SOPHIE GERMAIN and later had GUSTAV PETER

LEJEUNE DIRICHLET and BERNHARD RIEMANN asstudents, but he did not work closely with anyof these persons. This seems to be due to in-grained introspection, a consequence of his un-derappreciated childhood talents, and a drivingambition that made him unwilling to share dis-covery with others. Much of Gauss’s work wentunpublished, ostensibly because he thought itunworthy of dissemination; the real reason seemsto be his possessive secretiveness that fostered areluctance to reveal his methods.

114 Gauss, Carl Friedrich

In this time period, Gauss’s political viewswere fixed: A staunch conservative, he was dis-concerted by the chaos of revolution and wasskeptical of democracy. In philosophy he was anempiricist, rejecting the idealism of ImmanuelKant and Georg Hegel. He also experiencedsome personal happiness in this time; in 1805he married Johanna Osthoff, by whom he begota daughter and son. But in 1809 she died inchildbirth, and Gauss was plunged into loneli-ness. Though he soon remarried, to MinnaWaldeck, this marriage was less happy, as she wasoften sick. Gauss dominated his daughters andquarreled with his sons, who left Germany forthe United States.

In his early years at Göttingen, Gauss hadanother surge of mathematical ideas on hyper-geometric functions, the approximation of in-tegration, and the analysis of the efficiency ofstatistical estimators. His astronomical dutiesdevoured much of his time, but he continuedwith mathematical investigations in his sparemoments. At this time he developed many ofthe notions of non-Euclidean geometry, workedout from his early years in Göttingen as a stu-dent. However, his conservatism made him re-luctant to accept the truth of his discoveries, andhe was unwilling to face the public ridicule at-tendant on such novel mathematics. This led tolater arguments over priority with JÁNOS BOLYAI,who independently developed non-Euclideangeometry despite Gauss’s negative influence.

Gauss’s endeavors in science were also con-siderable, but we shall pass over them briefly, andfocus on their mathematical aspects. In 1817Gauss became interested in geodesy, the meas-urement of the Earth. He completed, after manyadministrative obstacles, the triangulation ofHannover 30 years later. As a result of his ar-duous fieldwork, he invented the heliotrope, adevice that could act as a beacon even in the day-time by reflecting sunlight. His work in geodesyinspired the early mathematics of potential the-ory, and the mapping of one surface to another,

an important concept in differential geometry. Hewas also stimulated to continue his research inmathematical statistics, and his Disquisitionesgenerales circa superficies curves (General investi-gations of curved surfaces) in 1828 would fuelmore than a century of activity in differentialgeometry. By 1825 Gauss had new results on bi-quadratic reciprocity and was working on non-Euclidean geometry and elliptic functions.Slowing down due to age, Gauss turned towardphysics and magnetism for fresh inspiration. In1829 he stated the law of least constraint, andin 1830 he contributed to the topic of capillar-ity and the calculus of variations. The year1830–31 was quite difficult, as Gauss was afflicted by a heart condition and his wife soondied from tuberculosis. At this time Gauss be-gan collaboration with Wilhelm Weber in mag-netism, and they invented the first telegraph in1834. Gauss’s 1839 work based on worldwidemagnetic observatory data expressed the mag-netic potential on the Earth’s surface by an in-finite series of spherical functions. His fruitfulcollaboration with Weber had already endedwith the latter’s exile for political reasons. In1840 Gauss gave a systematic treatment of po-tential theory as a mathematical topic, and in1841 he analyzed the path of light through a sys-tem of lenses.

From the early 1840s Gauss’s productivitygradually decreased. He had more liking forteaching, and Dedekind and Riemann wereamong his most gifted students. Working in ac-tuarial science, he collected many statistics fromperiodicals; this data aided him in his financialspeculations, which made him quite wealthy. Hishealth gradually failed, until he died in his sleepon February 23, 1855, in Göttingen.

Gauss was one of the greatest mathemati-cians of all time. Later mathematicians, unawarethat Gauss had already gone before them, repli-cated many of his discoveries. His name is asso-ciated with many diverse areas of mathematics,and his impact cannot be overestimated.

Germain, Sophie 115

Further ReadingBühler, W. Gauss: A Biographical Study. New York:

Springer-Verlag, 1981.Coxeter, H. “Gauss as a Geometer,” Historia Mathematica

4, no. 4 (1977): 379–396.Dunnington, G. Carl Friedrich Gauss, Titan of Science.

New York: Hafner, 1955.Forbes, E. “The Astronomical Work of Carl Friedrich

Gauss (1777–1855),” Historia Mathematica 5, no.2 (1978): 167–181.

———. “Gauss and the Discovery of Ceres,” Journalfor the History of Astronomy 2, no. 3 (1971):195–199.

Hall, T. Carl Friedrich Gauss: A Biography. Cambridge,Mass.: MIT Press, 1970.

Plackett, R. “The Influence of Laplace and Gauss inBritain,” Bulletin de l’Institut International deStatistique 53, no. 1 (1989): 163–176.

Sheynin, O. “C. F. Gauss and the Theory of Errors,”Archive for History of Exact Sciences 20, no. 1(1979): 21–72.

Sprott, D. “Gauss’s Contributions to Statistics,”Historia Mathematica 5, no. 2 (1978): 183–203.

Stigler, S. “Gauss and the Invention of Least Squares,”The Annals of Statistics 9, no. 3 (1981): 465–474.

� Germain, Sophie(1776–1831)FrenchNumber Theory, Geometry

Sophie Germain is known as one of France’sgreatest mathematicians. She made importantcontributions to number theory, partial differ-ential equations, and differential geometry.Germain was able to accomplish much despitea lack of formal education and her parents’ op-position.

Born the daughter of Ambroise-FrançoisGermain and Marie-Madeleine Gruguelu onApril 1, 1776, in Paris, Sophie Germain lived inan affluent household during turbulent times.Her father was a deputy to the States-General,

and was by profession a merchant; later he be-came a director of the Bank of France. Underthis comfortable situation, Germain grew upwith her father’s extensive library at her disposal.At a time when women did not regularly receiveeducations, Germain supplied the lack herself byreading at home. At age 13, she read an accountof the death of ARCHIMEDES OF SYRACUSE by acareless soldier, and the Sicilian mathematicianbecame a heroic symbol for her. At this youngage, she decided to be a mathematician. Althoughher parents were opposed to this direction of herenergies, she first mastered Latin and Greek, andthen started to read SIR ISAAC NEWTON andEUCLID OF ALEXANDRIA.

Eventually, the library at home became in-sufficient for Germain’s intellectual needs, andat age 18 she sought a better situation. She wasable to obtain lecture notes of courses taught atthe École Polytechnique, and was particularlyinterested in JOSEPH-LOUIS LAGRANGE’s lectureson analysis. Although not registered, Germainpretended to be a student, taking the pseudonymLe Blanc, and submitted a term paper on analy-sis to Lagrange. Lagrange was duly impressed byits originality, and he sought out its author. Ondiscovering that the writer was actually Germain,Lagrange became her sponsor and mathematicaladviser.

Germain obtained higher education purelythrough correspondence with the great scholarsof Europe; by this means she became well versedin mathematics, literature, biology, and philoso-phy. She became interested in certain problemsof number theory after reading ADRIEN-MARIE

LEGENDRE’s 1798 Théorie des nombres (Theory ofnumbers), and a voluminous correspondence be-tween the two soon arose. In the course of thesecommunications they collaborated on mathe-matical results, and some of Sophie’s discoverieswere included in the second edition of theThéorie.

Also at this time she read CARL FRIEDRICH

GAUSS’s Disquisitiones arithmeticae (Arithmetical

116 Gibbs, Josiah Willard

investigations), and entered into a correspon-dence with him under the pseudonym of LeBlanc. In 1807, when French troops were occu-pying Hannover, she feared for Gauss’s safety inGöttingen. Hoping there would be no repeat ofArchimedes’s death in the person of Gauss, shecommunicated to the French commander there,who was a friend of her family. In this fashion,Gauss came to know her true identity.

Among her work on number theory,Germain worked on the famous problem calledFermat’s last theorem, which was solved byAndrew Wiles in 1994. The theorem is a con-jecture by PIERRE DE FERMAT, which states thatthere are no integer solutions x, y, z to the equa-tion xn 1 yn 5 zn if n is an integer greater thantwo. Germain was able to show that no positiveinteger solutions exist if x, y, and z are rela-tively prime (have no common divisors) to oneanother and to n, where n is any prime lessthan 100.

Germain was interested in mathematicsother than number theory; indeed, she madecontributions to applied mathematics and phi-losophy. In 1808 the German physicist ErnstChladni visited Paris and conducted experi-ments in acoustics and elasticity. He would takea horizontal plate of metal or glass, sprinkle sanduniformly on top of it, and then cause vibrationsin the plate by rubbing the edge with a violinbow. The resulting oscillations would move theparticles of sand into certain stable clusters,called Chladni figures. In 1811 the Académie desSciences offered a prize for the best explanationof the phenomenon; the challenge was to for-mulate a mathematical theory of elastic surfacesthat would agree with the Chladni figures.

Germain attempted to solve the problem,and after a series of revisions and subsequentcontests, she won the prize in 1816 with a pa-per bearing her own name. Her work treated thevibrations of curved and plane elastic surfaces ingenerality. In 1821 she produced an enhancedversion of her prize paper, in which she stated

that the law for the general vibrating elastic sur-face is given by a fourth order partial differen-tial equation. One of the concepts playing a rolein this work was the notion of mean curvature,which was an average of the principal curva-tures, that is the curvatures of a surface in twoperpendicular directions.

In later work Germain expanded on thephysics of vibrating curved elastic surfaces, con-sidering the effect of variable thickness. She alsocontributed to philosophy, developing the con-cept of unity of thought—that science and thehumanities would always be unified with respectto their motivation, methodology, and culturalimportance. She died on June 27, 1831, in Paris.

Germain’s work has not received a great fol-lowing, and this may be partly due to her gen-der. Her work on number theory and differentialequations was of the highest quality, and shecontributed to the development of differentialgeometry through her notion of mean curvature.

Further ReadingBucciarelli, L. Sophie Germain: An Essay in the History

of the Theory of Elasticity. Boston: Kluwer, 1980.Dalmedico, A. “Sophie Germain,” Scientific American

265 (1991): 117–122.Grinstein, L., and P. Campbell. Women of Mathematics.

New York: Greenwood Press, 1987.

� Gibbs, Josiah Willard(1839–1903)AmericanMechanics, Geometry

In the 19th century physicists were tremen-dously interested in several topics of thermody-namics (the theory of heat); even the basicaxioms of the discipline were not universallyagreed upon. Josiah Gibbs made importantmathematical and physical contributions to thisproject, laying the foundations for the moderntheory of statistical mechanics.

Gibbs, Josiah Willard 117

Josiah Gibbs was born in New Haven,Connecticut, on February 11, 1839, to an afflu-ent family. His father, of the same name, was awell-known professor of sacred literature at Yale,and Gibbs’s mother, Mary Anna Van CleveGibbs, raised four daughters in addition toJosiah. The boy grew up in New Haven and at-tended Yale, winning several prizes in Latin andmathematics and graduating in 1858. Gibbs con-tinued his studies in the graduate engineeringschool, obtaining his Ph.D. in 1863, two yearsafter his father’s death.

He spent the next three years as a Latin tu-tor, and after the death of his mother and twosisters, Gibbs traveled to Europe to further hisstudies. He visited the universities of Paris,Berlin, and Heidelberg, remaining a year in eachcity, and greatly extended his knowledge of both

mathematics and physics. This study abroad laidthe foundation for his subsequent accomplish-ments in theoretical physics.

Gibbs returned to America in 1869, and wasable to live, along with his two sisters, off his in-heritance, residing in his childhood home closeto Yale. Two years later he was appointed pro-fessor of mathematical physics at Yale, a positionhe held without salary for nine years. At thistime Gibbs wrote his memoirs on thermody-namics, which undoubtably constitute his great-est contribution to science and mathematics.

His first published paper, “GraphicalMethods in the Thermodynamics of Fluids,”displayed an admirable mastery of thermody-namics. Gibbs assumed that entropy—the ten-dency for heat to dissipate—was as importantas energy, temperature, pressure, and volumetoward understanding the properties of heatflow, and he formulated a differential equationthat relates these quantities elegantly. A sec-ond paper extended his results to three dimen-sions; characteristic of Gibbs’s talent was hisemphasis on a geometrical approach (as op-posed to the algebraic approach). Through hisanalysis, Gibbs was able to demonstrate howvarious phases (solid, liquid, gas) of a substancecould coexist.

Although these initial articles had a narrowreadership in their respective journals, Gibbssent copies of his work to various leadingEuropean physicists, including James Maxwell,and in this way gained wider recognition. Soonafterward, Gibbs completed his memoir On theEquilibrium of Heterogeneous Substances, whichgeneralized his previous work and greatly extendedthe domain of thermodynamics to chemical,elastic, electromagnetic, and electrochemicalphenomena. Generally, Gibbs stressed the im-portance of characterizing equilibrium as thestate where entropy is maximized. This is equiv-alent to the minimum energy principle of me-chanics. In this way, Gibbs’s thought greatlyimpacted chemistry.

Josiah Willard Gibbs studied the theory of heat andpromoted the spread of vector analysis. (Courtesy ofAIP Emilio Segrè Visual Archives)

118 Gödel, Kurt Friedrich

Gibbs’s work reached continental scientistssuch as Max Planck, where it eventually exerteda substantial influence. Meanwhile, Gibbs wasturning toward optics and the electromagnetictheory of light; he defended the electromagnetictheory of light against purely mechanical theo-ries based on elastic ethers. In the realm of puremathematics, Gibbs rejected the use of quater-nions in the study of theoretical physics, and sohe developed his own theory of vector analysis.A textbook based on his lectures was publishedon this topic in 1901.

In 1902 Gibbs produced a book on statisti-cal mechanics that takes a statistical approachto physical systems by representing the coordi-nates and momenta of particles with probabilitydistributions. The main theme of his work wasthe analogy between the average behavior ofsuch statistical mechanical systems and the be-havior produced by the laws of thermodynamics.Thus, his theory of statistical mechanics laid amathematical foundation for the physics of heatflow. Although incomplete, Gibbs’s work was auseful contribution to rational mechanics.Indeed, Gibbs’s labors greatly advanced physics,and later came to dominate the whole field ofthermodynamics.

Gibbs never married, and continued teach-ing at Yale until his death on April 28, 1903.Although he had made some contributions toengineering, such as the design of a new gover-nor for steam engines, it is for his insightfulanalysis of thermodynamics—the importance ofentropy, the relationship of entropy to equilib-rium, and his statistical mechanical formulationof the field—that Gibbs is known.

Further ReadingBumstead, H. “Josiah Willard Gibbs,” American

Journal of Science 4, no. 16 (September 1903).Crowther, J. Famous American Men of Science.

Freeport, N.Y.: Books for Libraries Press, 1969.Rukeyser, M. Willard Gibbs. Garden City, N.Y.:

Doubleday, Doran and Company, 1942.

Seeger, R. J. Willard Gibbs, American MathematicalPhysicist par Excellence. Oxford: Pergamon Press,1974.

Smith, P. “Josiah Willard Gibbs,” American Mathe-matical Society. Bulletin. New Series 10 (1903).

Wheeler, L. Josiah Willard Gibbs, the History of a GreatMind. New Haven, Conn.: Yale University Press,1962.

� Gödel, Kurt Friedrich(1906–1978)AustrianLogic

In the beginning of the 20th century, threeschools of thought held sway over mathematicallogic: formalism, intuitionism, and logicism.Formalism taught that mathematics was prima-rily a syntax into which meaning is introduced,intuitionism stressed the role of intuition overpure reason, and logicism viewed mathematicsas part of logic. Kurt Gödel established a newmode of thought—namely, that mathematicallogic was a branch of mathematics, that had onlyindirect ramifications on philosophy. His theo-rems, especially his incompleteness theorem,have earned him considerable fame as a first-ratemathematician, since his work is extremely rel-evant to epistemological questions (questions re-lating to the foundations of knowledge).

Kurt Gödel was born on April 28, 1906, inBrno, the Czech Republic, which at the time waspart of the Austrian Empire. Rudolf Gödel, hisfather, was a weaver who eventually attained asignificant amount of property. MarianneHandschuh, his mother, had a liberal education,and the household in which Gödel and his olderbrother Rudolf grew up was upper class. Gödelhad a happy childhood, and was called “Mr.Why” by his family, due to his numerous ques-tions. He was baptized as a Lutheran, and re-mained a theist (a believer in a personal God)throughout his life.

Gödel, Kurt Friedrich 119

came interested in philosophy after 1920, andthe famous philosopher Immanuel Kant was in-fluential throughout Gödel’s life. When he grad-uated in 1924, Gödel had already masteredmuch of university mathematics, and thus hewas very well prepared to enter the Universityof Vienna. Initially he considered taking a de-gree in physics, but after some classes on num-ber theory Gödel switched to mathematics. From1926 to 1928 he was involved in the ViennaCircle, a group of logical positivists interested inepistemology. Gradually, Gödel fell away fromthese philosophers due to his own Platonic po-sition. Platonism, as applied to the philosophyof mathematics, espouses a belief in the true ab-stract reality of mathematical objects (such asnumbers), which attain concrete particular real-izations in the world.

In 1929 Gödel’s father died, and in thesame year Gödel completed his dissertation.He received his doctorate in mathematics in1930. This paper provided the completenesstheorem for first-order logic, which showedthat every valid formula in first-order logic wasprovable. The term completeness refers to theissue of whether every true mathematical the-orem has a proof; thus, incomplete systems aresomewhat mystical, in that they contain truestatements that cannot be established throughreason and logic alone. Later in 1930 Gödelannounced his famous incompleteness theo-rem: There are true propositions of numbertheory for which no proof exists. This resulthad enormous ramifications in mathematics,since it effectively destroyed the efforts ofmathematicians to construct a logical calculusthat would prove all true statements; it also in-fluenced philosophy and epistemology. Thephilosophical version of the theorem says thatin any system of thought, one cannot producea proof for every true statement, as long as oneis restricted to that system.

In the following years Gödel published nu-merous articles on logic and worked as a lecturer

Kurt Gödel was a great logician famous for theincompleteness theory, which is concerned with thefoundations of mathematics. (Photograph by RichardArens, courtesy of AIP Emilio Segrè Visual Archives)

Gödel advanced rapidly through school, ex-celling in mathematics, languages, and religionat a German high school in Brno. He also be-

120 Gödel, Kurt Friedrich

at the University of Vienna. While extreme shy-ness made him a poor public speaker, the contentof his lectures included the most recent researchin the foundations of mathematics. In 1933 hevisited the Institute for Advanced Study atPrinceton (a mathematical think tank affiliatedwith Princeton University), where he wouldspend increasing amounts of time as the politi-cal situation in Europe deteriorated. Gödel alsosuffered from mental depression, and he stayedat a sanatorium in Europe in 1934 after a nerv-ous breakdown. In 1935 he returned to Americaand continued his important new work in settheory, obtaining a significant breakthroughconcerning the axiom of choice. Soon afterwardhe resigned, suffering from overwork and de-pression, and returned to Austria. His work fromthis time period showed that the axiom ofchoice and the continuum hypothesis, two im-portant postulates of set theory, were relativelyconsistent (consistency means that a given pos-tulate does not contradict the other axioms ofthe system).

In 1938 Gödel married Adele PorkertNimbersky, a nightclub dancer. They were soonforced to flee back to the United States due tothe Nazi persecution in Austria—Gödel’s asso-ciation with Jews and liberals made him a tar-get for discrimination. He was prevented fromcontinuing his lectureship at Vienna, and waseven attacked by right-wing students. As a re-sult, Gödel and his wife moved back toPrinceton in 1940, escaping Austria to the eastvia the Trans-Siberian Railway.

At Princeton the introverted Gödel had aquiet social life; however, he did develop someclose friendships with his colleagues, includingAlbert Einstein. Out of this relationship, Gödelbecame increasingly interested in the theoryof relativity—later, after 1947, he contributedto cosmology by presenting mathematical mod-els in which time travel was logically possible.In 1943 Gödel turned increasingly towardphilosophical research, where he expressed his

Platonist views and criticized BERTRAND RUSSELL’slogicism.

In the later portion of his life, Gödel re-ceived numerous honors and awards, such asthe Einstein Award in 1951 and the NationalMedal of Science in 1974. It is interesting thathe steadfastly refused to receive any honorsfrom the Austrian academic institutions be-cause of their previous treatment of him. In1953 he became a full professor at the institute,continued his work on logic and cosmology,and in 1976 retired as an emeritus professor. Hedied on January 14, 1978, in Princeton, aftersuffering from depression, paranoia, and mal-nutrition—believing that his food was beingpoisoned, Gödel refused to eat and starved todeath.

Kurt Gödel made extraordinary discoveriesin mathematical logic and set theory. His workin cosmology and philosophy is also noteworthy.Gödel essentially established the framework formodern investigations. Since he showed thatnumber theory was incomplete, the project ofDAVID HILBERT and previous logicians to mecha-nize the proof making of mathematics becameimpractical. Instead, logicians began to focus onthe completeness and consistency questions ofvarious types of logical systems. This paradigmshift was due to Gödel’s epochal incompletenesstheorem. His results on the axiom of choice andcontinuum hypothesis emphasized the relativenature of any answer to these questions; herealso, a rich new field of set theoretical researchwas spawned by Gödel’s initial discoveries. In abroader sense, Gödel’s ideas have influencedcountless philosophers and computer scientists,with ramifications in epistemology and artificialintelligence.

Further ReadingDawson, J. “Kurt Gödel in Sharper Focus,” The

Mathematical Intelligencer 6, no. 4 (1984): 9–17.Hofstadter, D. Gödel, Escher, Bach: An Eternal Golden

Braid. New York: Basic Books, 1999.

Goldbach, Christian 121

Wang, H. “Kurt Gödel’s Intellectual Development,”The Mathematical Intelligencer 1, no. 3 (1978):182–185.

———. Reflections on Kurt Gödel. Cambridge, Mass.:MIT Press, 1987.

———. “Some Facts about Kurt Gödel,” Journal ofSymbolic Logic 46, no. 3 (1981): 653–659.

� Goldbach, Christian(1690–1764)GermanNumber Theory, Analysis

Christian Goldbach was an amateur mathemati-cian, possessing no formal training. Nevertheless,he corresponded with many scientists and math-ematicians around Europe, and was one of thefew persons who understood the works of PIERRE

DE FERMAT and LEONHARD EULER. His contribu-tions to mathematics were sporadic with flashesof brilliance. There were also surprising gaps inhis knowledge. However, through his mathe-matical communications, he was able to partic-ipate in the mathematical inquiries of his time and stimulate others toward fundamentalresults.

Christian Goldbach was born in Königsberg,Prussia, on March 18, 1690. His father was aminister, and Goldbach received a good edu-cation, studying mathematics and medicine atthe University of Königsberg. Around 1710 hebegan traveling about Europe, and made the ac-quaintance of several leading mathematicians,such as GOTTFRIED WILHELM VON LEIBNIZ, ABRAHAM

DE MOIVRE, and DANIEL BERNOULLI. Sometime af-ter 1725 Goldbach received a position as pro-fessor of mathematics as the Imperial Academyof Russia.

Goldbach was a skilled politician, and headvanced quickly in political circles to the detri-ment of his mathematical research. In 1728 hemoved to Moscow to become tutor to the king’sson Peter II; he returned to St. Petersburg in

1732 and quickly rose to a powerful position inthe Imperial Academy. In 1737 he had admin-istration of the academy, but was simultaneouslyrising in government circles. In 1742 he severedhis ties with the academy, and eventually roseto the rank of privy councilor in 1760, oversee-ing the education of the royal family.

Goldbach’s knowledge of advanced mathe-matics was acquired informally through discus-sions with mathematicians rather than throughconsistent reading. He became intrigued with in-finite series in 1712 after meeting NikolausBernoulli, and this date probably marks the be-ginning of his own research into that subject. Ofhis various papers, some of which repeat mate-rial already published by others, two show gen-uine originality: One treats the manipulation ofinfinite series, and the other concerns a theoryof equations. Goldbach developed a method fortransforming one series into another by addingand subtracting certain terms successively. Thesenew terms were allowed to be divergent, so longas the end result was convergent. Second,Goldbach applies some results from number the-ory to test whether a given algebraic equationhas a rational root. This method developed froma correspondence with Leonhard Euler, withwhom Goldbach began communicating in 1729.

Besides these original contributions tomathematics, Goldbach kept abreast of currentdevelopments and entered into the dialogue ofmathematicians regarding new results. For exam-ple, Goldbach communicated one of Fermat’sconjectures on prime numbers to Euler, who wasable to construct a counterexample. He is alsofamous for the Goldbach conjecture that everyeven integer could be expressed as the sum oftwo prime numbers. This conjecture remainsunproved today.

Goldbach died on November 20, 1764, inMoscow. Although Goldbach undoubtably pos-sessed considerable mathematical talent, thiswas not developed due to his success in civic af-fairs. However, Goldbach was able to stimulate

122 Gosset, William

research into mathematical ideas in his owntime, and also in the modern era through hismysterious conjecture.

Further ReadingYuan, W. Goldbach Conjecture. Singapore: World

Scientific, 1984.

� Gosset, William(1876–1937)BritishStatistics

William Gosset, known informally as “Student,”his common pseudonym, was an important figurein the development of mathematical statistics.Because he was not a professor at a university, hisperspective on data analysis was practical, and hisresearch addressed tangible, real-world statisti-cal problems. Perhaps Gosset’s most vital con-tribution to statistics was the realization that the“sampling” distribution is critical for inference.His work has had enduring relevance and influ-ence on modern statistical procedures com-monly used in science and medicine today.

William Sealy Gosset was born on June 13,1876, in Canterbury, England, the eldest son ofColonel Frederic Gosset and Agnes Sealy. As ayoung man he studied mathematics and chem-istry at Winchester College and New College atOxford, where in 1899 he obtained a degree inthe natural sciences. Soon afterward he joinedthe brewing company Arthur Guinness & Sons,which was located in Dublin, being employed asa chemist. In 1906 Gosset married MarjorySurtees, and later had two children.

In the course of his work on quality control,it became necessary to analyze the brewingprocess statistically. Eventually, in 1906 Gossetcame to University College, London, to work un-der the statistician Karl Pearson. Previous workin statistics, much of it due to the efforts of CARL

FRIEDRICH GAUSS, emphasized the importance of

large samples and asymptotic distributions. Theterm sample refers to the data set, usually a col-lection of numbers, that represents repeatedmeasurements of a phenomenon. The older workin statistics used approximations that reliedupon large samples (roughly speaking, 30 ormore data points), whereas the data available toGosset came in a much smaller size. Since thepresent theory was inadequate to handle this sit-uation, Gosset was forced to develop new meth-ods applicable to small samples.

Over the next several years, Gosset con-tributed to statistical theory under the pseudo-nym “Student,” and corresponded with a varietyof statisticians, including SIR RONALD AYLMER

FISHER, Jerzy Neyman, and Karl Pearson. Gosset’smost famous paper, entitled The Probable Errorof a Mean, analyzed the ubiquitous sample meanstatistic from a small sample perspective and de-rived its distributional properties without relyingupon a large sample approximation. The distri-bution of the scale-normalized sample mean sta-tistic that Gosset discovered came to be knownas Student’s t distribution, and there is a corre-sponding statistical test, which is called Student’st test. This work, accomplished through a com-bination of mathematical analysis and MonteCarlo simulation, was later found to be optimalin statistical testing theory, and its importanceis demonstrated through its continued use today,along with its descendant, the ANOVA (analysis of variance) procedure.

Gosset stayed with Guinness, obtaining anassistant in 1922, and gradually built up a mod-est statistics department there until 1934. In1935 he moved to London to head the newGuinness brewery there. He continued his sta-tistical work until his death on October 16,1937, in Beaconsfield, England.

Gosset’s contribution to mathematics wassomewhat untraditional, for at that time most ofthe important research was taking place in thevarious universities. However, in the field of sta-tistics more than other branches of mathematics,

Grassmann, Hermann Günter 123

it is crucial for investigations to be motivated bypractical problems; thus, it seems that Gossetwas well situated. His emphasis on and devel-opment of small sample theory was pivotal forthe evolution of mathematical statistics; mostimportant, perhaps, was the attention he drewtoward the sampling distribution of a statistic.

Further ReadingPearson, E. Student—A Statistical Biography of William

Sealey Gosset. Oxford, U.K.: Oxford UniversityPress, 1990.

———. “Student as a Statistician,” Biometrika 30(1939): 210–250.

———. “Some Reflections on Continuity in theDevelopment of Mathematical Statistics1840–1894,” Biometrika 54 (1967): 341–355.

� Grassmann, Hermann Günter(1809–1877)GermanGeometry, Algebra

Hermann Grassmann made substantial contri-butions to algebra and geometry during the 19thcentury. His ideas were so advanced that manyof his colleagues failed to recognize their merit,but later generations quickly gravitated towardGrassmann’s highly abstract and beautiful work.

Hermann Günter Grassmann was born onApril 15, 1809, in Stettin, a town in Prussia,though now it lies in Poland. Grassmann taughtat the high school in Stettin for most of his life;he began teaching in 1831 and continued untilhis death, except for a brief period (1834–36) atBerlin. While teaching, he was able to devotesome of his time to personal research into alge-bra and geometry.

Grassmann is well known for his develop-ment of vector calculus, but his most importantwork was his 1844 Die lineale Ausdehnungslehre(The theory of linear extension). This book de-veloped an abstract algebra—a set with certain

rules of arithmetic operations that define howthe symbols in the set interact—in which thesymbols were geometric objects such as points,lines, and planes. His algebra gave certain rulesfor the interactions of these things. Grassmannalso studied the subspaces of a given geometricspace and developed a certain type of algebraicmanifold (a high-dimensional surface given asthe solution of an algebraic equation) that waslater called the Grassmannian.

Grassmann also invented the concept of anexterior algebra—another algebra with a specialproduct called the exterior product. This ab-stract structure was related to the quaternions ofSIR WILLIAM ROWAN HAMILTON, and was later de-veloped by William Clifford into a tool that hasbeen quite useful in quantum mechanics. Theexterior algebra is an important object of studyin modern differential geometry. Grassmann’sideas were quite advanced for his time, and theywere accepted slowly; this led to frustration forGrassmann, who in his later years turned awayfrom mathematics to the study of Sanskrit. (HisSanskrit dictionary is still used today.) Besideshis mathematical work, Grassmann also con-tributed to the literature of acoustics, electricity,and botany.

Grassmann died on September 26, 1877,in Stettin. Although disappointed by the un-deracceptance of his brilliant ideas, Grassmannachieved fame later on. At the end of the 19thcentury, more geometers began to discover hiswork; ÉLIE CARTAN was inspired to study dif-ferential forms (an example of the exteriorproduct), which are important to differentialgeometry. Today, Grassmann is viewed as anearly contributor to the budding field of alge-braic geometry.

Further ReadingFearnley-Sander, D. “Hermann Grassmann and the

Creation of Linear Algebra,” The AmericanMathematical Monthly 86, no. 10 (1979):809–817.

124 Green, George

———. “Hermann Grassmann and the Prehistory ofUniversal Algebra,” The American MathematicalMonthly 89, no. 3 (1982): 161–166.

Heath, A. “Hermann Grassmann: The Neglect of HisWork,” Monist 27 (1917): 1–56.

� Green, George(1793–1841)BritishCalculus, Differential Equations

One of the most remarkable scientific theoriesof the 19th century—the mathematical theoryof electricity—was largely developed by a self-taught miller, George Green. His contributionswere outstanding, but Green little realized theirimportance, and much of his life is shrouded inobscurity.

George Green was born in July 1793 inSneinton, England. His exact birthday is notknown, but he was baptized on July 14, 1793.He was named after his father, a prosperousbaker. Green’s mother was Sarah Butler, whohelped Green’s father to buy his own bakery inNottingham. Green had one sister, Ann, whowas two years younger.

Green received very little education. Whenhe was nine years old, he was sent to RobertGoodacre’s school for two years, where he firstbecame interested in mathematics. It is unknownhow Green became interested in mathematics,but he applied himself to the subject vigorously,until he surpassed Goodacre. Throughout his life,Green continued to study mathematics in hisspare moments.

In 1802 Green left school to work in his fa-ther’s business, which became quite successfulover the years. Green’s father bought a piece ofland in 1807, and there built a mill; this was des-tined to become Green’s private study. In 1817a house was built beside the mill, and Green’sfamily moved there. Green continued his own

mathematical investigations in his spare mo-ments away from work.

Green had little access to current mathe-matics and could not stay abreast of the Frenchdevelopments. However, he nevertheless dis-played a familiarity with contemporary mathe-matics in his later works, and it is hypothesizedthat the Cambridge graduate John Toplis tutoredhim. In any event, Green acquired great famil-iarity with British and French mathematics, andhe mastered calculus and differential equations.Meanwhile, Green had several children by JaneSmith, the daughter of the mill’s manager; hemet Smith sometime before 1823. In this yearhe also gained access to the transactions of theRoyal Society, which increased his resources.

The next years were difficult: His parentsdied, and two daughters were born. Despite thenumerous disturbances, Green managed to studymathematics in the top floor of the mill, and in1828 published An Essay on the Application ofMathematical Analysis to the Theories of Electricityand Magnetism. This was certainly one of the mostimportant mathematical works ever, since it gavea mathematical theory for electricity—his workintroduced the potential function, demonstratedthe relationship between surface area integralsand volume integrals (known as Green’s formula),and defined the important “Green’s function”that is ubiquitous in the theory of differentialequations. This monumental work was little ap-preciated by its provincial audience, but onereader named Sir Edward Bromhead encouragedGreen to pursue his mathematical gifts.

As a result of Bromhead’s friendship, Greenwas introduced to several mathematicians, in-cluding CHARLES BABBAGE. From 1830 Greenpublished three additional papers of great worth,treating the topics of electricity and hydrody-namics. Upon Bromhead’s advice, Green left hismill in 1833 to enroll at Cambridge as an under-graduate. There Green excelled at mathematics,but had difficulty in his other subjects due to his

Gregory, James 125

poor education. Nevertheless he graduated fourthin his class in 1837, and afterward stayed on atCambridge, conducting his own research. He ob-tained a Perse fellowship in 1839 despite the re-quirement necessitating bachelorhood, sinceGreen was legally unmarried. However, he hadseven children by Jane Smith.

Green produced papers on hydrodynamics,optics, and the refraction of sound, such asMathematical Investigations Concerning the Lawsof the Equilibrium of Fluids Analagous to theElectric Fluid (1833) and On the Determination ofthe Interior and Exterior Attractions of Ellipsoids ofVarying Densities (1835). In 1840 he fell ill, andreturned to Nottingham to recuperate. Greenmade out his will (his estate went to his family)and died on May 31, 1841, in Nottingham.

The importance of Green’s work was hardlyrealized (neither by him nor by his contempo-raries) during his own lifetime. A few years later,in 1850, Green’s works were republished byWilliam Thomson, an event of great interest toJOSEPH LIOUVILLE. Through James Clerk Maxwell(a great scientist who worked on electricity, mag-netism, and heat) and others, Green’s theory ofelectricity would be fully developed by the turnof the 20th century. Thus, Green’s work becamegreatly influential to science. But Green is bestknown today for Green’s theorem and Green’sfunction, which are both highly significant inthe theory of differential equations.

Further ReadingCannell, D. George Green: Mathematician and Physicist

1793–1841: The Background to His Life and Work.London: Athlone Press, 1993.

Cannell, D., and N. Lord. “George Green, Mathe-matician and Physicist 1793–1841,” MathematicalGazette 66 (1993): 26–51.

Edge, D. “The Omission of George Green from‘British Mathematics (1800–1830),’” Bulletin ofthe Institute of Mathematics and Its Applications 16,no. 2–3 (1980): 37.

� Gregory, James(1638–1675)BritishCalculus, Geometry, Algebra

James Gregory, one of the greatest mathemati-cians of the 17th century, was also one of theleast known, largely due to his own isolation.Indeed, his work on the foundations of calcu-lus, conducted independently of and contem-poraneously with SIR ISAAC NEWTON, were notrecognized until centuries after his death; andin many other areas of mathematics, such asnumber theory, algebraic equations, integration,and differential equations, Gregory significantly

James Gregory, a little-known mathematician whomade many early discoveries in calculus (Courtesy ofthe Library of Congress)

126 Gregory, James

anticipated the discoveries of later mathemati-cians.

James Gregory was born in November 1638in Drumoak, Scotland, to John Gregory, a clericlearned in theology, and Janet Anderson, whosebrother was a pupil of FRANÇOIS VIÈTE. Gregorywas the youngest child of his parents, having twoolder brothers, Alexander and David. Hismother, who was intellectually gifted, educatedGregory. After his father’s death in 1651, hisbrother David oversaw his mathematical in-struction. This involved reading EUCLID OF

ALEXANDRIA’s Elements, which James found quiteeasy.

Gregory pursued higher education atMarischal College in Aberdeen. His health waspoor, and he suffered from quartan fever whilestill a youth. While in school, Gregory becameinterested in telescopes and completed his firstbook, Optica Promota (The advance of optics).This work described a reflecting telescope—anovel invention that increased the telescope’spower—and the mathematical equations of re-flection. The book covers mathematical astron-omy through several postulates and theorems. In1663 Gregory traveled to London, and was ableto get his book published; his telescope was con-structed by Robert Hooke some 10 years later.

In 1664 Gregory traveled to Italy to con-tinue his mathematical studies. He spent muchof his time at the University of Padua with themathematician Stefano Angeli; there Gregoryapplied infinite series to the calculation of theareas of the circle and the hyperbola. Gregoryalso learned the method of tangents (the pred-ecessor of the theory of differentiation), andmade such discoveries in early calculus to earnhim the right of discovery. Indeed, his early workon calculus mirrored the labors of Newton, al-though neither mathematician was aware of theother’s work. Certainly, Gregory received nocredit for his discoveries during his own lifetime,since he was reclusive and unwilling to dissem-inate his knowledge. For example, in Vera circuli

et hyperbolae quadratura (The true squaring of thecircle and the hyperbola) (1667), Gregory laysthe foundations for infinitesimal geometry. Thethesis of this document was that the numbers piand e are transcendental—a concept that fewmathematicians of the time even grasped.Although flawed, Gregory’s arguments includedan amazing breadth of ideas, such as conver-gence, algebraic functions, and iterations.

His Geometriae pars universalis (The univer-sal part of geometry) (1668), a first attempt at acalculus textbook, clearly showed that differen-tiation and integration are inverse operations.After concluding his work at Padua, Gregory re-turned to London in 1668 and became involvedin a dispute with CHRISTIAAN HUYGENS. The lat-ter had received Gregory’s Geometriae pars uni-versalis for review, and accused Gregory of havingstolen some of his own ideas. In retrospect, thiswas impossible, but it nevertheless soured rela-tions between the two mathematicians and fos-tered Gregory’s reluctance to publish.

During the summer of this year, Gregory ob-tained the infinite series expansions for thetrigonometric functions and found the anti-derivative of the secant function, solving a prob-lem in navigation. At meetings of the RoyalSociety—to which he was elected a fellow in1668—Gregory discussed scientific topics suchas astronomy, gravitation, and mechanics.Through the intervention of another fellow,Charles II was persuaded to create the Regiuschair at St. Andrews University for Gregory, inorder to facilitate his continuing mathematicalresearch.

Gregory took up the post, and in 1669 wasmarried to Mary Jamesome; they had two daugh-ters. Gregory’s teaching was not well received,partly because the students were inadequatelyprepared. The educational style of St. Andrewsfocused on the classics and placed little empha-sis on current developments in science andmathematics. Later, the faculty rebelled againsthim, claiming that his mathematical teachings

Gregory, James 127

had a negative impact on their own courses; hissalary was even withheld.

Meanwhile, Gregory continued his own re-search. On learning of ISAAC BARROW’s work, hewas able to extend it and greatly develop thefoundations of calculus—for example, he dis-covered Taylor’s theorem long before BrookTaylor did. When Gregory became aware of thework Newton was doing on calculus, he deferredhis own publications in order to avoid anothernasty dispute. He also discovered the refractionof light, but did not pursue further research onthis topic, again in deference to Newton.Gregory was also involved in astronomy, and ac-tually took up a collection personally in order tobuild an observatory at St. Andrews in 1673.Relations with the St. Andrews faculty had be-come so odious that he left for the University ofEdinburgh in 1674, taking up the chair of math-ematics there.

Gregory held his post for only one year be-fore dying of a stroke in October 1675 inEdinburgh. He was only 36 years old. His lastyear was especially productive, as Gregory made

inroads into Diophantine equations. Beforemany other mathematicians, he began to realizethat the quintic equation did not admit rationalsolutions; NIELS HENRIK ABEL would prove thismore than a century later. Gregory is a fascinat-ing character, since his tremendous contribu-tions to science and mathematics had gonelargely unnoticed until the early 20th centurywhen his true contributions were revealed.Besides being a codiscoverer of calculus and in-finitesimal geometry, Gregory developed tests forconvergence (such as AUGUSTIN-LOUIS CAUCHY’sratio test), defined the Riemann integral, devel-oped a theory of differential equations that al-lowed for singularities, and attempted to establishthe transcendence of pi. Thus, Gregory was farahead of most of his contemporaries; however,his influence was negligible due to the obscurityof his life and his reluctance to publish.

Further ReadingDehn, M., and E. Hellinger. “Certain Mathematical

Achievements of James Gregory,” The AmericanMathematical Monthly 50 (1943): 149–163.

H

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� Hamilton, Sir William Rowan(1805–1865)IrishComplex Analysis, Algebra, Mechanics

In the 19th century the concept of the complexnumber was further developed, and it was foundto be a useful tool for understanding other branch-es of mathematics. A similar type of number rel-evant to quantum mechanics is the so-calledquaternion, which was discovered by the prodigySir William Rowan Hamilton. Although hemade contributions to optics, mechanics, andgeneral algebra, he is most famous for the quater-nions and his acceptance of noncommutativityin algebraic systems.

William Rowan Hamilton was born inDublin, Ireland, on August 4, 1805, and from anearly age he displayed a remarkable genius forlanguages. Hamilton was educated by his uncle,and the boy had mastered Greek, Hebrew, andLatin by age five. He also possessed mathemat-ical talent, being capable of rapid calculation;around 1820 Hamilton read through SIR ISAAC

NEWTON’s works and began his own astronomi-cal observations through his telescope.

Around 1822 Hamilton’s interest in math-ematics reached a new phase, and he began hisown research. He investigated the properties ofcurves and surfaces, which eventually led to his

Theory of Systems of Rays of 1827. Meanwhile,he had entered Trinity College, having achievedthe top score on his entrance examinations; hecontinued to win special honors in classics andscience throughout his college career.

Upon graduation in 1827, Hamilton was ap-pointed astronomer royal at Dunsink Observatory,and also became a professor of astronomy atTrinity. However, he did little actual observa-tion, focusing his energy on pure mathematics.His first work on optics introduced his notion ofthe “characteristic function,” which describedthe mathematical portion of optics completely.His theory was independent of whether oneviewed light as particle or wave, and thusHamilton largely freed himself from this con-tentious debate.

Next Hamilton applied his characteristicfunction to celestial mechanics in his 1833 pa-per On a General Method of Expressing the Pathsof Light and of the Planets by the Coefficients of aCharacteristic Function. Later in 1834 he appliedthe same principles to dynamics in On aGeneral Method in Dynamics. These papers werehard to read, owing to Hamilton’s terse style,and thus they did not enjoy much popularity.He introduced the so-called Hamilton equationfor the energy of a conservative system, whichis ubiquitous in modern mechanics. Althoughthe Hamiltonian formulation of mechanics had

Hamilton, Sir William Rowan 129

little practical advantage over the prior Lagrangianapproach, it did carry over easily into the quan-tum mechanical situation. Hamilton’s theoryalso maintained a close affinity between opticsand mechanics.

His research led him to the discovery of gen-eral methods in the calculus of variations. Moreimportant, Hamilton discovered the quaternionsin 1843. He had earlier become interested in ageometric explanation of the complex numbers,and in 1835 he presented a Preliminary andElementary Essay on Algebra as the Science of PureTime, which describes the complex numbers aspoints in the plane with a special rule for additionand multiplication. The intriguing title reveals

Hamilton’s Kantian philosophy—geometry wasthe science of pure space, and algebra was thescience of pure time. From this standpoint,Hamilton attempted to construct an analogousalgebra for three-dimensional space. This wassubsequently proved to be impossible, and Hamil-ton eventually formulated his new algebra infour dimensions. The resulting quadruples werecalled quaternions; they could be added, multi-plied, and divided into each other according tocertain rules. The resulting algebra was a first ex-ample of a noncommutative system; part ofHamilton’s genius lay in recognizing that therewas nothing innately illogical about this situa-tion. More than a century later, quaternions andother noncommutative algebras exert a profoundinfluence on quantum mechanics and quantumcomputers.

The story is related that Hamilton’s suddendiscovery occurred on a bridge across the RoyalCanal in Dublin, on October 16, 1843; hescratched the relevant formula into the stonesof the bridge. The quaternions were not imme-diately accepted, as the vector analysis of JOSIAH

WILLARD GIBBS was more popular; Hamilton’sdifficult treatise, the Elements of Quaternions(published in 1867 after his death), was inac-cessible to most of his contemporaries.

Hamilton received many honors during hislife, and served as the president of the Royal IrishAcademy from 1837 to 1845. The lord lieu-tenant of Ireland knighted him in 1835 for hiswork as a scientist. Hamilton was thwarted inlove as a young man, but in 1833 he marriedHelen Bayly, by whom he had two sons and adaughter. Hamilton was an avid poet, but hisfriend William Wordsworth encouraged him toconcentrate on mathematics, where his true giftslay. He was full of energy and had many literaryfriends, including Ellen de Vere and SamuelColeridge. After the death of his first love,Catherine Disney, in 1853, Hamilton fell intoalcoholism; he died in Dublin on September 2,1865.

Sir William Rowan Hamilton worked in the field ofalgebra, and developed a number system for four-dimensional spaces, known as quaternions, which isused in physics. (Courtesy of the Library of Congress)

130 Hardy, Godfrey Harold

Given the early genius of his childhood,Hamilton may have expected more of himselfmathematically. However, he left his mark onthe field of mechanics through his use of theprinciple of least action, the derivation of thecharacteristic function, and the Hamiltonianequation for the energy of a system. He also pro-foundly influenced the development of abstractalgebra through his discovery of the quaternions.The applications of these strange numbers arestill being explored today.

Further ReadingArunachalam, P. “W. R. Hamilton and His Quater-

nions,” Mathematical Education 6, no. 4 (1990):261–266.

Graves, R. Life of Sir William Rowan Hamilton. NewYork: Arno Press, 1975.

Hankins, L. Sir William Rowan Hamilton. Baltimore,Md.: Johns Hopkins University Press, 1980.

MacDuffee, C. “Algebra’s Debt to Hamilton.” ScriptaMathematica 10 (1944): 25–35.

Synge, J. “The Life and Early Work of Sir WilliamRowan Hamilton,” Scripta Mathematica 10(1944): 13–24.

� Hardy, Godfrey Harold(1877–1947)BritishNumber Theory, Analysis

One of the most famous mathematical collabo-rations of the 20th century was between GodfreyHardy and John Littlewood. Their research intomodern number theory, together with the geniusof SRINIVASA RAMANUJAN, greatly advanced thefield. During his lifetime, Hardy was recognizedas the leading British mathematician in puremathematics.

Godfrey Harold Hardy was born in Cranleigh,England, on February 7, 1877. His parents wereboth intellectuals with an interest in mathe-matics; his father, Isaac Hardy, was a master at

Cranleigh School. Thus, the young Hardy andhis two sisters received an excellent education,in which they were encouraged to ask ques-tions.

From a young age, Hardy was good withnumbers; at age 13 he attended WinchesterCollege on a scholarship, and in 1896 came toTrinity College in Cambridge. Hardy was suc-cessful at several competitions involving therapid calculation of problems, earning a Smith’sPrize in 1901. He developed an affinity for math-ematical analysis after reading CAMILLE JORDAN’sCours d’analyse (Course in analysis) at the in-stigation of one of his professors. In the next 10years Hardy attacked a series of research prob-lems concerning the convergence of infinite se-ries and integrals. His 1908 text A Course onPure Mathematics became a classic, effectivelytransforming undergraduate mathematics educa-tion through its clear exposition of number,function, and limit.

Hardy was a lecturer at Trinity College un-til 1919, when he became a professor of geome-try at Oxford. During this time, he had begunhis fruitful collaboration with Littlewood, whichresulted in about 100 joint papers on Diophantineapproximation, the additive and multiplicativetheory of numbers, the Riemann zeta function,and infinite series.

In 1913 Hardy received a manuscript fromthe amateur Indian mathematician SrinivasaRamanujan. He quickly discerned the genius ofthe author, who had been able to obtain ad-vanced mathematical results in number theorywithout formal training, and arranged forRamanujan to come to England in 1914. For thenext three years, the two worked intensively ona series of mathematical results—Hardy’s expe-rience with analysis and Ramanujan’s intuitivegenius with numbers led to many great achieve-ments, including an asymptotic formula for thenumber of partitions of a given integer. Sadly,Ramanujan fell ill in 1917, and he returned toIndia two years later, dying in 1920.

al-Haytham, Abu Ali 131

Hardy founded a flourishing school of re-search at Oxford, and returned to Cambridge in1931 as a professor of pure mathematics. Henever married, but dwelt with his devoted sister.Hardy’s main passion was mathematics, althoughhe was also interested in cricket and was a con-firmed atheist. His mastery of the English lan-guage increased the popularity of his literature,and he was known to be lively and enthusiastic.He died in Cambridge on December 1, 1947.

Hardy’s research into analytic number the-ory advanced the field. His work with Ramanu-jan and Littlewood led to fundamental results inthe study of number, and he stimulated and en-couraged many younger mathematicians duringhis life.

Further ReadingChan, L. “Godfrey Harold Hardy (1877–1947)—the

Man and the Mathematician,” Menemui Mathe-matik. Discovering Mathematics 1, no. 3 (1979):1–13.

Fletcher, C. “G. H. Hardy—Applied Mathematician,”Bulletin of the Institute of Mathematics and ItsApplications 16, nos. 2–3 (1980): 61–67.

Milne, E. “Obituary: Godfrey Harold Hardy,” RoyalAstronomical Society. Monthly Notices 108 (1948):44–46.

Newman, M. “Godfrey Harold Hardy, 1877–1947,”The Mathematical Gazette 32 (1948): 50–51.

Snow, C. “Foreword,” in A Mathematician’s Apologyby G. H. Hardy. Cambridge, U.K.: CambridgeUniversity Press, 1967.

� al-Haytham, Abu Ali (Alhazen)(ca. 965–ca. 1040)ArabianGeometry

The Arabic mathematician and natural philoso-pher Abu Ali al-Haytham, also known asAlhazen, played an important role in preservingand transmitting the classical knowledge of the

Greeks. He made numerous contributions tooptics and astronomy, investigating light, vi-sion, refraction, sundials, and the height of thestars. In mathematics, he is known for his treat-ment of “Alhazen’s problem,” where he buildson the knowledge of his Greek and Arabicpredecessors.

Little is known of al-Haytham’s early life,and many of the accounts of his middle years areconflicting. Apparently, he left Iraq during thereign of the Egyptian caliph al-Hakim, who hadfounded a famous library in Cairo. Al-Haytham,already a famous mathematician, had made theclaim that he could regulate the flow of waterson the Nile through certain constructions; theEgyptian caliph then invited him to Egypt tocarry out his boast. Al-Haytham had based hisclaim on the assumption that the upper Nile en-tered Egypt through high ground, and he soondiscovered that his project would be impossibledue to the unexpectedly different terrain.Ashamed and afraid of reprisal, he confessed hisfailure to the caliph, who put him in charge ofa government office. There, al-Haytham pre-tended to be insane, fearing the anger of thecapricious caliph, and was confined in his houseuntil al-Hakim’s death. Then al-Haytham re-vealed his sanity, and he spent the rest of his lifewriting scientific texts and teaching students.

Another account relates that al-Haythamfirst occupied the office of minister at Basra, butin order to devote himself purely to the pursuitof science and learning, he feigned madness toescape his official duties. Afterward he journeyedto Egypt, where he spent his life at the AzharMosque, making copies of EUCLID OF ALEXAN-DRIA’s Elements about once a year. He diedaround 1040 in Cairo.

In his autobiography al-Haytham reflectedon his doubts regarding various religious sectsand became convinced that there could be onlyone truth. He turned toward the philosophicalsciences of mathematics, physics, and meta-physics as subjects in which truth could be more

132 Heaviside, Oliver

easily obtained by rational inquiry after the man-ner of Aristotle. Al-Haytham wrote on manysubjects, including logic, ethics, politics, poetry,music, and theology. He achieved fame in math-ematics for his treatment of “Alhazen’s prob-lem,” which concerns the reflection of light froma surface. If one takes any two points on the re-flecting surface, be it flat or curved, the problemis to find a third position on the surface wherelight from one point will be reflected to theother. CLAUDIUS PTOLEMY had shown that thereexists a unique point for concave spherical mir-rors. Al-Haytham set out to solve the problemfor all spherical, cylindrical, and conical surfaces,whether convex or concave. Although he wasnot always successful, he did demonstrate hisgreat facility with higher Greek mathematics.His general solution is based on six geometricallemmas, or steps; he would apply these lemmasin succession to various kinds of surfaces. Thesesolutions are included in his Optics.

About 20 other writings of al-Haytham dealentirely with mathematics. Some of them dealwith the solution of difficulties arising from cer-tain parts of Euclid’s Elements. His Solution of theDifficulties in Euclid’s Elements attempts to treatmost of the problems arising from Euclid, givingalternative constructions in certain cases and re-placing indirect proofs with direct proofs. Itseems that al-Haytham intended this, along withanother work, to form a commentary on Euclid.In the axioms of Euclid, he attempts to replacethe troublesome fifth postulate—which statesthat parallel lines never intersect—with a postu-late involving equidistance. There had beenmany Islamic attempts to prove the fifth postu-late, and al-Haytham was able to deduce the par-allel postulate from his postulate on equidistance,though he used the concept of motion in hisproof, which is somewhat foreign to Greekgeometry.

Al-Haytham also composed two works onthe quadrature of lunes—crescent-shaped fig-ures. These contain various propositions on the

geometry of lunes, and the subject is related tothe topic of squaring the circle (the construc-tion of a square with area equal to that of a spec-ified circle). In another tract al-Haythamdemonstrates the possibility of squaring the cir-cle, without providing an explicit construction.In On Analysis he discusses the principles ofanalysis (breaking down) and synthesis (puttingtogether) used in the discovery and proof ofmathematical theorems and constructions. Heillustrates these principles through applyingthem to arithmetic, geometry, astronomy, andmusic, which at that time were considered to bethe four mathematical disciplines. He stressesthe role of “scientific intuition,” when a certainproperty is yet to be proved, and can be conjec-tured only from evidence. Such conjectures, ledby the intuition, must be made before the processof analysis and synthesis can be made. Thisseems to be related to more modern notions ofscientific investigation.

Further ReadingHogendijk, J. Ibn al-Haytham’s “Completion of the

Conics.” New York: Springer-Verlag, 1985.Langermann, Y. Ibn al-Haytham’s “On the Configuration

of the World.” New York: Garland, 1990.Rashed, R. The Development of Arabic Mathematics:

Between Arithmetic and Algebra. Boston: KluwerAcademic Press, 1994.

Sabra, A. The Optics of ibn al-Haytham. London:Warburg Institute, University of London, 1989.

� Heaviside, Oliver(1850–1925)BritishMechanics

Oliver Heaviside is known for his use of math-ematical techniques and ideas in the fields ofelectrical engineering and physics, and in thissense anticipated the use of Laplace and Fouriertransforms, which became ubiquitous later on.

Hermite, Charles 133

In his own day he acquired some fame due to hisinnovations in electrical engineering, and he in-fluenced the development of mathematicalphysics.

Heaviside was the youngest of four sons bornto Thomas Heaviside, an artist, and his wifeRachel Elizabeth West. He was born on May 18,1850, in London. He was entirely self-taught andacquired a level of understanding that eventu-ally earned him an honorary doctorate from theUniversity of Göttingen. While a teenager,Heaviside engaged in electrical experimenta-tion, and he published his first technical articleat age 22. From 1870 to 1874 he was a telegraphoperator at Newcastle-on-Tyne. After this timehe lived privately (with narrow means) throughthe assistance of his brother.

Starting in 1873 Heaviside published sev-eral papers on electrical engineering that madetelegraphy practical, despite the opposition ofseveral powerful engineers who disagreed withHeaviside’s correct theories. Heaviside claimedthat additional coils added to a long-distance ca-ble would improve performance, and this waslater shown to be correct. The slow acceptanceof Heaviside’s ideas was due not only to his lackof reputation and credentials, but also to his uti-lization of sophisticated mathematical tools toformulate and express his theories.

Heaviside saw the benefit of operational cal-culus to the investigation of transients, foresee-ing the use of Laplace and Fourier transforms inelectrical engineering. He also developed a vec-tor notation for performing calculations inthree-dimensional system, which was along thelines of JOSIAH WILLARD GIBBS’s system and incontrast to SIR WILLIAM ROWAN HAMILTON’squaternionic formulation. Heaviside was thefirst to write the “telegrapher’s equation,” whichis a differential equation involving voltage andresistance. This equation also depends on con-stants representing capacitance and inductance,terms that Heaviside invented. This equationhas numerous applications to dynamical systems.

Besides these mathematical contributions,Heaviside introduced a new system of electro-magnetic units, correctly predicted the existenceof a reflecting ionized region surrounding theEarth, and proposed a theory of motion for anelectric charge. His fame spread during his life,resulting in his election to the Royal Society in1891. Although he was extremely generous toothers who needed his scientific help, his finan-cial situation inhibited his continued research;various professional societies and his friends latersupported him. He died in his seaside cottage inPaignton on February 3, 1925.

Heaviside introduced useful mathematicsinto physics and electrical engineering. Theformer subject was, of course, already mathe-matically oriented, but neither took full ad-vantage of current mathematical techniques. Inthis way, Heaviside set a pattern for 20th-cen-tury engineering, as mathematical ideas havebeen increasingly instrumental for designingnew technologies.

Further ReadingNahin, P. Oliver Heaviside, Sage in Solitude: The Life,

Work, and Times of an Electrical Genius of theVictorian Age. New York: IEEE Press, 1988.

Yavetz, I. From Obscurity to Enigma: The Work ofOliver Heaviside, 1872–1889. Boston: BirkäuserVerlag, 1995.

� Hermite, Charles(1822–1901)FrenchAlgebra, Analysis

After the deaths of CARL FRIEDRICH GAUSS, AU-GUSTIN-LOUIS CAUCHY, CARL JACOBI, and GUSTAV

PETER LEJEUNE DIRICHLET in the 1850s, Europewas deprived of its best mathematicians. Inthe areas of arithmetic and analysis, CharlesHermite became the sole successor to thesegiants, retaining his position of glory for many

134 Hermite, Charles

years. Not only was he exceedingly influentialin his own time, but Hermite also laid impor-tant groundwork for 20th-century research aswell.

Charles Hermite was born in Dieuze,France, on December 24, 1822, the sixth childof seven. His mother was Madeleine Lallemandand his father was Ferdinand Hermite, an artistand engineer. In 1829 the family moved toNancy, where Charles attended the Collège deNancy. He continued his studies in Paris at theCollège Henri IV and the Collège Louis-le-Grand, but his performance was not spectacular.Hermite focused on reading the works of LEON-HARD EULER, Gauss, and JOSEPH-LOUIS LAGRANGE

instead of preparing for his examinations.From 1840 to 1841, while at Louis-le-

Grand, Hermite published his first two papers,wherein he attempted to prove the impossibil-ity of solving the quintic by radicals—he was un-aware of NIELS HENRIK ABEL’s results. Despitescoring poorly on his entrance examinations,Hermite was admitted to the École Polytechniquein 1842. He was refused further study due tolameness in his right foot that forced him to usea cane; he was allowed to resume his studies dueto the intervention of certain influential per-sons. At about this time Hermite entered the so-cial circle of Joseph Bertrand—he would latermarry Bertrand’s sister.

Hermite began serious work on Jacobi’s fa-mous inversion problem for hyperelliptic inte-grals, and in 1843 he succeeded in generalizingAbel’s work on elliptic functions to hyperellip-tic functions. He communicated his result toJacobi, which initiated a six-year correspon-dence and gained him renown in the mathe-matical community.

Hermite became an admissions examiner atthe École Polytechnique in 1848, and he acquireda more permanent position in 1862, eventuallybecoming professor of analysis in 1869. Duringthese years he was enormously productive, andcorresponded heavily with many mathematicians.

From 1843 to 1847 he focused on elliptic func-tion. One of the most intriguing problems of thetime was the inversion of integrals of algebraicfunctions, and Hermite made progress on thisquestion by introducing theta functions.

Next, in 1847 Hermite turned to numbertheory, generalizing some of Gauss’s results onquadratic forms. From here he further extendedhis results to algebraic numbers (which includedsquare roots), deriving some of their fundamen-tal properties. In 1854 he studied the theory ofinvariants, discovering the reciprocity law,which gave a correspondence between binaryforms. Hermite then applied the theory of in-variants to abelian functions in 1855, and his re-sults became a foundation for CAMILLE JORDAN’stheory of “abelian groups.” From 1858 to 1864he researched the quintic equation and classnumber relations, and in 1873 he turned to theapproximation of functions. It is noteworthythat Hermite proved the transcendence of thenumber e in 1873; his methods would later beextended to establish the transcendence of pi.His work covered Legendre functions, series forelliptic integrals, continued fractions, Besselfunctions, Laplace integrals, and special differ-ential equations.

Hermite was known as a cheerful man whowas unselfish in sharing his discoveries with oth-ers. He fell ill with smallpox in 1856, and laterbecame a devout Catholic under Cauchy’s in-fluence. During his life he received many hon-ors, being awarded membership at severallearned societies. He died in Paris on January 14,1901.

Hermite is chiefly remembered as an alge-braist and analyst. He invented the so-calledHermitian forms, which are a complex general-ization of quadratic forms, as well as theHermitian polynomials, which are of use in theapproximation of functions. His work has beenabsorbed into more general structures, and inthis sense his thought lives on in the more ab-stract mathematics of the 20th century.

Hilbert, David 135

Further ReadingStander, D. “Makers of Modern Mathematics: Charles

Hermite,” Bulletin of the Institute of Mathematicsand Its Applications 24, no. 7–8 (1988): 120–121.

� Hilbert, David(1862–1943)GermanLogic, Algebra, Analysis, Geometry

David Hilbert is probably best known for his listof 23 outstanding mathematical problems, whichguided much of 20th-century research. However,much more important were his contributions tothe theory of algebraic invariants, algebraicnumber theory, and the foundations of mathe-matics. Few mathematicians have had so pro-found an impact on subsequent research asHilbert. His foresight into the future of mathe-matics was uncannily accurate, even borderingon the prophetic.

Hilbert’s family consisted of German Pro-testants living in East Prussia. His father, OttoHilbert, was a judge in Königsberg, where DavidHilbert was born on January 23, 1862. It is saidthat he inherited his mathematical talents fromhis mother. The young Hilbert attended theFriedrichskolleg in Königsberg beginning in1870, and he studied at the University ofKönigsberg from 1880 to 1884, obtaining hisPh.D. in 1885. After some traveling, Hilbert ob-tained a position at the University of Königsbergin 1892, and in the same year he married KätheJerosch. In 1895 he was appointed to a chair atthe University of Göttingen, where he remaineduntil his retirement in 1930.

Hilbert’s first research, in the period up to1893, was on algebraic forms. He studied thetheory of algebraic invariants, approaching thesubject from a revolutionary symbolic standpointand dispensing with the algorithmic methods ofthe past. It is noteworthy that the modern ap-proach to algebra follows Hilbert’s abstract path.

He was the first to propose these new techniques,which later became classical. Mention should bemade of his famous Nullstellensatz (zero positionprinciple), which gives a condition for a poly-nomial to be included in a special set of func-tions called an ideal.

From 1894 to 1899 Hilbert turned to num-ber theory, taking an algebraic perspective to thesubject. His Der Zahlbericht (Commentary onnumbers) of 1897 was a summary of all current

David Hilbert was a versatile mathematician whocontributed to logic, functional analysis, and algebra.Hilbert spaces were a powerful generalization of flatEuclidean space. (Aufnahme von A. Schmidt,Göttingen, courtesy of AIP Emilio Segrè VisualArchives)

136 Hilbert, David

knowledge, organized from a contemporaneousviewpoint. It proved to be a crucial guidebookfor the next half-century of research into alge-braic number theory. Hilbert’s contributions toalgebraic number theory were so profound andextensive, it almost seems that he began thetopic. His work focused on the reciprocity lawand the notion of the class field. Although hisbrilliant work provided stimulating ideas fordecades to come, Hilbert moved on to the foun-dations of geometry, leaving the further detailsto his students and successors.

From 1899 to 1903 Hilbert emphasized theaxiomatic character of geometry. His Grundlagender Geometrie (Foundations of geometry) (1899)attempted to establish the consistency of geo-metric axioms and to determine which theoremswere independent of certain axioms. For exam-ple, non-Euclidean geometry had already beeninvented, and was a geometrical system that wasindependent of EUCLID OF ALEXANDRIA’s parallelpostulate. Hilbert brought algebra to bear ongeometry to obtain results concerning consis-tency and independence. His work in this areawould lead to future ideas of field and topologi-cal space.

Although the Dirichlet principle for solvingthe boundary value problem in the theory of par-tial differential equations had been discreditedby KARL WEIERSTRASS, Hilbert succeeded in giv-ing a rigorous proof. His technique of diagonal-ization later became classic in abstract analysis.From 1904 to 1909 Hilbert labored on the cal-culus of variations and integral equations.Hilbert focused on homogeneous equations, ar-riving at the notion of a function space with aninner product (later these would be calledHilbert spaces, and are of great practical use infunctional analysis and statistics), and definedthe spectrum of an operator. The term spectrumwas prophetic, as physicists some two decadeslater would connect the spectra of operators tooptical spectra. FRIGYES RIESZ and John Von

Neumann would later follow up Hilbert’s firstclumsy steps in the new field of operator theory.

Next, Hilbert turned to mathematicalphysics, feeling that the subject was too impor-tant to be left to the physicists. He obtained re-sults on kinetic gas theory and relativity, but thiswork was not as influential as his previous suc-cess in algebra and analysis. After 1918 Hilbertwas heavily involved in the foundations ofmathematics. He was eager to demonstratethe consistency of number theory in particular.Although some of his concepts, such as thetransfinite function, were quite brilliant, ulti-mately the program was doomed to failure.Essentially, Hilbert was obsessed with provingthat mathematical proof was valid—thus he wasengaged in “metamathematics” (that is, thestudy of mathematical thought processes, in-cluding the structure of proofs). KURT FRIEDRICH

GÖDEL dealt a death blow to the prospect of es-tablishing the consistency of number theory in1931. From this time is dated the genesis of themodern studies in the foundations of mathe-matics.

Hilbert fell ill from anemia in 1925, and hemade only a partial recovery due to new treat-ments. His list of 23 problems, set as tasks for the20th century and delivered at a 1900 address atthe International Congress of Mathematicians,has proved to be of enduring importance. It treatsa wide breadth of problems, including the cardi-nality of the continuum, the consistency ofarithmetic, the axiomatization of physics, theRiemann hypothesis, and the study of generalboundary value problems. Some of these havebeen completely or partially solved, others areunsolved, and still others have been shown tobe unsolvable (or dependent on a choice of ax-iom). In any event, they have provided an enor-mous stimulus for mathematicians. The topicsenumerated in the 23 problems have for themost part been major areas of research into puremathematics.

Hipparchus of Rhodes 137

Hilbert died on February 14, 1943, inGöttingen. He was known for his intense per-sonality and mathematical boldness. HERMANN

MINKOWSKI was both a close friend and a signif-icant influence, as they studied together inKönigsberg. One of his most important mentorswas LEOPOLD KRONECKER. Hilbert had many fa-mous students, including HERMANN WEYL. Hisimpact on 20th-century mathematics has beentremendous.

Further ReadingBernays, P. “David Hilbert,” Encyclopedia of Philosophy

III. New York: Macmillan, 1967.Lewis, D. “David Hilbert and the Theory of Algebraic

Invariants,” Irish Mathematical Society Bulletin 33(1994): 42–54.

Reid, C. Hilbert-Courant. New York: Springer-Verlag,1986.

Rowe, D. “David Hilbert on Poincaré, Klein, and theWorld of Mathematics,” The MathematicalIntelligencer 8, no. 1 (1986): 75–77.

Weyl, H. “David Hilbert and His MathematicalWork,” American Mathematical Society. Bulletin.New Series 50 (1944): 612–654.

� Hipparchus of Rhodes(ca. 190 B.C.E.–ca. 120 B.C.E.)GreekTrigonometry

Hipparchus of Rhodes was one of the greatestastronomers of antiquity, and was also wellknown for the mathematical techniques that heimported into the study of the stars. Many ofthe astronomical calculations were made acces-sible through the trigonometric formulas ofHipparchus; he may also have been the first touse spherical trigonometry and stereographicprojection.

Hipparchus of Rhodes was not born inRhodes, though he spent much of his later career

at that island. He was born in Nicaea, in thenorthwestern portion of Turkey, sometime in thefirst quarter of the second century B.C.E. He wasmost active between 147 B.C.E. and 127 B.C.E.,which are known to be the dates of his first andlast astronomical observations.

Little is known of Hipparchus’s early life, butit is probable that he began his scientific careerin Nicaea and moved to Rhodes before 141 B.C.E.He was quite famous and respected during hislifetime, but later fell into obscurity due to thesmall circulation of his published works.CLAUDIUS PTOLEMY wrote the Almagest, whichdrew heavily on the previous writings ofHipparchus, and it is from this source that schol-ars’ scanty knowledge of Hipparchus is drawn.

In order to calculate the position of heav-enly bodies, it was necessary for the Greeks tosolve certain trigonometric problems. Hipparchuswrote a work (its name is unknown and its ex-istence is known only indirectly) on the chordsof a circle, and produced a chord table, whichwas basically an early table of sines. He con-structed the table by computing the values di-rectly at intervals of 71⁄2 degrees, and then bylinearly interpolating between these values.Although Indian astronomers later used thischord table, Ptolemy’s superior chord table su-perseded it. Nevertheless, Hipparchus was thefirst to construct such a table, and this tool al-lowed him to solve many trigonometric prob-lems relevant to astronomy. In this sense,Hipparchus can be regarded as the founder oftrigonometry; he also transformed astronomyinto a quantitative science.

Many of the astronomical calculations per-formed by Hipparchus involved spherical trigono-metry (the measurement of triangles located ona sphere), but many scholars think that he wasable to solve these problems without an explicitknowledge of spherical trigonometry. There isalso evidence that Hipparchus used stereo-graphic projection, which is a method of mapping

138 Hippocrates of Chios

the sphere (minus the North Pole) onto a planethrough the equator, thus providing a way oftranslating coordinates.

Besides these mathematical accomplish-ments, Hipparchus was most famous for hisnumerous contributions to astronomy. It is in-teresting that he drew upon Babylonian lunardata for some of his theories; he was able to cal-culate the size of the Moon and Sun (the lattercalculation was inaccurate). He composed workson geography and astrology, as well as a book onoptics.

Hipparchus’s main achievement was thetransformation of astronomy from a qualitativeto a quantitative science; he successfully em-ployed mathematical formulas to calculate dis-tances and angles of various celestial bodies. Hisown observational data, along with the trans-mission of Babylonian astronomical data, wereboth important to the development of astron-omy. Hipparchus was known to be open-minded,critical, and empirical—he tested scientific the-ories against observations. His own writings werehighly specialized, but it seems that Hipparchushad conceived of a complete astronomical sys-tem. Ptolemy fleshed this out. Mathematically,Hipparchus deserves fame for founding the dis-cipline of trigonometry and constructing the firsttrigonometric table.

Further ReadingDicks, D. The Geographical Fragments of Hipparchus.

London: University of London, Athlone Press,1960.

Dreyer, J. A History of Astronomy from Thales to Kepler.New York: Dover Publications, 1953.

Grasshoff, G. The History of Ptolemy’s Star Catalogue.New York: Springer-Verlag, 1990.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Neugebauer, O. A History of Ancient MathematicalAstronomy. New York: Springer-Verlag, 1975.

Sarton, G. A History of Science. Cambridge, Mass.:Harvard University Press, 1959.

� Hippocrates of Chios(ca. 470 B.C.E.–ca. 410 B.C.E.)GreekGeometry, Analysis

Hippocrates of Chios was an early Greek math-ematician who made contributions toward theproblem of duplicating the cube, the quadratureof lunes, and the elements of geometry. In fact,he was the first to compose an Elements ofGeometry, although Hippocrates’ version did notachieve the fame and renown of EUCLID OF

ALEXANDRIA’s work of the same name.As with most persons of antiquity, the spe-

cific dates of his birth and death are unknown,although scholars have been able to pinpoint hisactivity to the latter portion of the fifth centuryB.C.E. Not to be confused with the physicianHippocrates of Cos, this Hippocrates was bornon the island of Chios in the Aegean Sea.Reportedly, Hippocrates of Chios was a mer-chant whose wealth was ruined by pirates. Hethen pursued his enemies to Athens, where hespent much of his remaining life.

In Athens Hippocrates attended lectures,becoming proficient in geometry and analysis. Itis probable that Hippocrates already possessedsome mathematical education, since there was aflourishing school in Chios; it is possible thatearly in his life Hippocrates came under the in-fluence of the Pythagoreans, who were based innearby Samos. Three mathematical problems—the duplication of the cube, the squaring of thecircle, and the trisection of an angle—held theattention of the Athenians, and Hippocratesstudied the first two.

The method of analysis refers to the processof reducing a problem into smaller parts that maybe easier to solve. Hippocrates applied themethod of analysis to the duplication of thecube. This famous problem amounted to findinga geometric construction for the cube root of 2;Hippocrates reduced this to the task of findingtwo mean proportionals x and y between the

Hopf, Heinz 139

numbers a and 2a, where a is the given side ofthe cube. In other words, one needed to solvethe ratio a/x 5 y/2a. ERATOSTHENES OF CYRENE

later completed the solution by demonstratinghow to solve this ratio geometrically.

Lunes are shapes similar to a crescent moon.Greek mathematicians attempted to find thearea within a given lune by inscribing and cir-cumscribing various shapes with known areas—a method known as quadrature. Hippocrates wassuccessful in squaring the lune (that is, he wasable to construct a rectangular figure with areaequal to that contained within certain givenlunes), although he was unable to square the cir-cle (it is impossible, since the number pi cannotbe expressed in terms of algebraic numbers).Some ancient commentators mistakenly attrib-uted the latter claim—that he was able to squarethe circle—to Hippocrates, but Hippocrates wasa talented mathematician, and it is unlikely thathe could have made such a fallacy. He also gavethe first known example of the Greek construc-tion of “verging,” which is used in establishingthe quadrature of a lune.

Hippocrates was aware of several propertiesof triangles, such as the Pythagorean theorem andvarious relationships between the angles. It isprobable that Hippocrates’ Elements of Geometrycontained such knowledge, covering much of thefirst two books of Euclid’s Elements. Hippocrateswas also familiar with the geometry of the circle.For example, he knew how to circumscribe a cir-cle about a given triangle, and was familiar withthe angles within the circle. It is conjectured thatHippocrates’ Elements contained some solidgeometry as well, although he was unaware of thetheory of proportion encountered in Book V ofEuclid’s Elements.

Hippocrates was probably the first to artic-ulate the theorem relating a circle’s radius to itscircumference, although this was not proved un-til later. Besides these mathematical works, hecontributed to astronomy through his specula-tions on the nature of the comet.

The work of Hippocrates has been lost due tothe ravages of time, and his accomplishments havebeen pieced together from secondary sources.From these it is clear that Hippocrates was notonly a foremost intellectual of his time and one ofthe greatest mathematicians of Athens, helping tomake that city a center of learning, but he wasalso a contributor to Greek mathematics whoseenduring work stimulated his successors.

Further ReadingAaboe, A. Episodes from the Early History of

Mathematics. Washington, D.C.: MathematicalAssociation of America, 1964.

Amir-Moéz, A., and J. Hamilton. “Hippocrates,”Journal of Recreational Mathematics 7, no. 2(1974): 105–107.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Hughes, B. “Hippocrates and Archytas Double theCube: A Heuristic Interpretation,” The CollegeMathematics Journal 20, no. 1 (1989): 42–48.

� Hopf, Heinz(1894–1971)GermanTopology

Topology, the study of the form of mathemati-cal sets, became an increasingly popular disci-pline in the 20th century. Powerful new toolssuch as homology and algebraic ring theory werebrought to bear on important problems, andmuch progress was made in the project of topo-logical classification. Heinz Hopf was a keyplayer in these developments.

Heinz Hopf was born in Breslau, Germany,on November 19, 1894. He studied mathemat-ics at the university in his hometown, but WorldWar I interrupted his career. After a long periodof service, Hopf was able to attend a set theorycourse at the University of Breslau in 1917; hebecame fascinated with the topological work of

140 Huygens, Christiaan

LUITZEN EGBERTUS JAN BROUWER and pursued fur-ther studies in Berlin. In 1925, Hopf received aPh.D. in mathematics. Later he received a fel-lowship to study at Princeton University in 1927.In 1931 Hopf was appointed full professor ofmathematics at the Eidgenössische TechnischeHochschule in Zurich, Switzerland.

Although Hopf published only a small num-ber of papers, such as Fundamentalgruppe undZweite Bettische Gruppe (Fundamental groupsand second Betti groups), he was tremendouslyinfluential in the field of topology, due to thebreadth and originality of his ideas. He built onBrouwer’s methods, developing the concepts ofmapping degree and homotopy class—thesewere mathematical tools that brought algebraicgroup theory into play to assist in the classifica-tion of continuous manifolds. Hopf had an ex-cellent geometric intuition, but his argumentsbecame increasingly algebraic under the influ-ence of EMMY NOETHER.

Some of Hopf’s most important work fo-cused on mappings of high-dimensional spheres,vector fields, and fixed point theorems. He de-fined the inverse homomorphism, which becamea powerful tool in the study of manifolds. In 1931Hopf identified an infinite number of relation-ships between the three-dimensional sphere andthe two-dimensional sphere, and made conjec-tures about the so-called Hopf fiber map. AfterWorld War II, Hopf’s early research in these ar-eas developed into a thriving field of study.

In the area of vector fields, Hopf studied bi-linear forms, almost complex manifolds, and thehomology of group manifolds. His main tool inthese researches was homology, which refers toclasses of embeddings of simple objects (like tri-angles and tetrahedrons) into a given manifold.Other work by Hopf in the 1940s led to the de-velopment of cohomological algebra, a vigorousnew field of pure mathematics.

Besides all this excellent work in algebraictopology, Hopf also explored differential geometry.For example, he investigated complete surfaces,

congruency of convex surfaces, and isometry; healso wrote about the tangent of a closed planecurve. Heinz Hopf was energetic and cheerful,and he gave clear, stimulating lectures. With hiswife, Anja Hopf—whom he married in October1928—he was a gracious host to colleagues andpolitical exiles. During World War II, he pro-vided refuge for his German friends inSwitzerland, where they would be safe from Nazipersecution. He died on June 3, 1971, inZollikon, Switzerland. He earned several hon-orary doctorates from such institutions asPrinceton, Freiburg, and the Sorbonne, and waspresident of the International MathematicalUnion from 1955 to 1958.

Heinz Hopf was enormously influential in thearea of topology: He developed homology into oneof the most widely used tools of algebraic topol-ogy, and obtained several important results, espe-cially in the study of high-dimensional spheres.The origins of cohomology can be traced back toHopf, as can the concept of homotopy groups.

Further ReadingBehnke, H., and F. Hirzebruch. “In Memoriam Heinz

Hopf,” Mathematische Annalen 196 (1972): 1–7.Cartan, H. “Heinz Hopf (1894–1971),” International

Mathematical Union (1972): 1–7.Frei, G., and U. Stammbach. “Heinz Hopf,” in History

of Topology. Amsterdam: Elsevier Science B.V.,1999.

� Huygens, Christiaan (ChristianHuyghens)(1629–1695)DutchMechanics, Geometry, Analysis,Probability

Between the time of RENÉ DESCARTES and SIR

ISAAC NEWTON, it is said that Christiaan Huygens(sometimes referred to as Christian Huyghens)was Europe’s greatest mathematician. He made

Huygens, Christiaan 141

substantial contributions to mechanics, astron-omy, the measurement of time, the theory oflight, and geometry. His work demonstrated theefficacy of a mathematical approach to the studyof nature, and Huygens developed many sophis-ticated mathematical tools.

Christiaan Huygens was born in The Hague,the Netherlands, on April 14, 1629. His familywas prominent, having a long history of diplo-matic service to the royal house. Christiaan’s fa-ther, Constantijn Huygens, educated both hissons personally, covering music, ancient lan-guages, mathematics, and mechanics. ChristiaanHuygens displayed his considerable intellectualtalents at a young age, and he had a gift for ap-plying theory to actual constructions—at age 13he constructed a lathe.

In 1645 Huygens attended the University ofLeiden, where he studied law and mathematics.During his two years there, he became familiarwith the recent works of FRANÇOIS VIÈTE, PIERRE

DE FERMAT, and Descartes. Huygens began to re-search the mechanics of falling bodies, and be-gan a correspondence with Mersenne. Aftercompleting his university studies, he matricu-lated at the College of Orange from 1647 to1649, where he pursued law. However, Huygensdid not pursue a career in diplomacy, but ratherchose to be a scientist.

Huygens lived at home until 1666, receiv-ing financial support from his father that enabledhim to focus on his scientific research. He firstinvestigated mathematics, considering quadra-tures of curves as well as algebraic problems.Huygens’s mathematical contributions are im-portant, in that he improved on existing methods,and was successful in his application of these tonatural phenomena. He also developed the newtheory of evolutes and was one of the foundersof the theory of probability.

In 1651 Huygens produced a manuscriptthat refuted Gregory of St. Vincent’s quadratureof the circle. In the same work, he derived aconnection between the quadrature and the

center of gravity for circles, ellipses, and hyper-bolas. His next publication, in 1654, approxi-mates the center of gravity of any arc of a circleand thus obtains an approximate quadrature. Asimilar technique, developed more than a decadelater, produced a quick method for calculatinglogarithms.

Upon learning of BLAISE PASCAL’s work inprobability, Huygens began studying gamblingproblems in 1656, such as the fair division ofstakes for an interrupted game. He invented theconcept of the mathematical expectation, whichrepresents the long-term winnings of a game ofchance. This idea, expressed by Huygens in aprimitive form, is now of central importance tothe modern theory of probability.

In 1657 Huygens related the arc length ofthe parabola to the quadrature of the hyperbola,and he used this property to find the surface area

Christiaan Huygens contributed to mechanics,geometry, and probability, and invented the pendulumclock. (Courtesy of the Library of Congress)

142 Huygens, Christiaan

of a paraboloid of revolution. A year later, hediscovered a vital theorem of modern calculus—that the calculation of the surface area of a sur-face of revolution could be reduced to find thequadrature of the normal curve.

His theory of evolutes, which concerns thegeometry of cords dangling from a convex sur-face, was developed in 1659 as a component ofhis research into pendulum clocks. His methodof evolutes essentially determines the radius ofcurvature of a given algebraic curve. Huygensalso studied logarithms, starting in 1661, and inthis connection introduced the natural expo-nential function.

Huygens also contributed to areas of science.He completed a manuscript on hydrostatics in1650, in which he derived the law of ARCHIMEDES

OF SYRACUSE from a basic axiom. In 1652 he for-mulated the rules of elastic collision and com-menced his study of optics. Later in 1655, togetherwith his brother, he turned to lens grinding andthe construction of telescopes and microscopes.He built some of the best telescopes of his time,and was able to detect the rings of Saturn.Huygens also observed bacteria and other mi-croscopic objects.

In 1656 Huygens invented the pendulumclock as a tool for the measurement of time. Ithad become increasingly important to accuratelymeasure time, since this technology was necessaryto astronomy and navigation; Huygens’s inven-tion had much success. In his theoretical investi-gation of pendulum swing, Huygens discoveredthat the period could be made independent ofthe amplitude if the pendulum’s path was acycloid. He then constructed the pendulumclock such that the bob of the pendulum wouldbe induced to have a cycloidal path. This so-called tautochronism of the cycloid is one ofHuygens’s most famous discoveries.

Next, Huygens began studying centrifugalforce and the center of oscillation in 1659, ob-taining several fundamental results. He rigorouslyderived the laws of descent along inclined planes

and curves and derived the value for the accel-eration due to gravity on the Earth, which isabout 9.8 meters per second squared. Huygensturned to fall through resisting media (such asair) in 1668, and conceived of the resistance (orfriction) as proportional to the object’s velocity.Huygens also investigated the wave theory oflight; he explained reflection and refraction in1676 by his conception of light as a series of fast-moving shock waves.

It is interesting that Huygens did not acceptthe Newtonian concept of force, and he was ableto circumvent it completely. He was also criti-cal of GOTTFRIED WILHELM VON LEIBNIZ’s conceptof force, although he agreed with the principleof conservation in mechanical systems. In hisnatural philosophy, he agreed with Descartes,attempting to arrive at a mechanistic explana-tion of the world. One of his most popular worksspeculated on the existence of intelligent life onother planets, which Huygens thought washighly probable.

During the period 1650–66, Huygens metmany French scientists and mathematicians, andhe visited Paris several times. In 1666 Huygensaccepted membership in the newly foundedAcadémie Royale des Sciences and moved toParis, where he remained until 1681. He was themost prominent member of the academy, and re-ceived a generous stipend. He spent this timedeveloping a scientific program for the study ofnature, observing the heavens, and expoundinghis theories of gravity and light.

Huygens suffered from ill health, and sev-eral times was forced to return to the Hague. In1681 he again left due to illness and, due to po-litical and religious tensions, was not invited tocome back to France. Huygens had never mar-ried, but was able to live off the family estate. Inthe last decade of his life he returned to math-ematics, having become convinced of the fruit-fulness of Leibniz’s differential calculus. However,Huygens’s mathematical conservatism led him toemploy his old geometrical methods, and this

Huygens, Christiaan 143

somewhat inhibited his progress and under-standing of calculus. Nevertheless, Huygens wasable to solve several mathematical problemsposed publicly, such as Leibniz’s isochrone, thetractrix, and the catenary.

Huygens finally succumbed to his ill consti-tution, and he died on July 8, 1695. He was themost preeminent scientist and mathematician ofhis time (at least before Newton and Leibniz be-came active), and he made brilliant contribu-tions to diverse areas of science. However,Huygens’s reluctance to publish theories thatwere insufficiently developed limited his influ-ence in the 18th century; neither did he haveany pupils to carry on his thought. His work inmechanics opened up new frontiers of research,but his mathematical work mostly extendedolder techniques rather than opening up new

vistas for exploration. Nevertheless, Huygenswas a master at applying mathematical methodsto scientific problems, as his work in the meas-urement of time eminently demonstrates.

Further ReadingBarth, M. “Huygens at Work: Annotations in His

Rediscovered Personal Copy of Hooke’s ‘Micro-graphia,’” Annals of Science 52, no. 6 (1995):601–613.

Bell, A. Christian Huygnes [sic] and the Development ofScience in the Seventeenth Century. London: E.Arnold, 1950.

Dijksterhuis, E. “Christiaan Huygens,” Centaurus 2(1953): 265–282.

Yoder, J. Unrolling Time: Christiaan Huygens and theMathematization of Nature. New York: CambridgeUniversity Press, 1988.

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� Ibrahim ibn Sinan (Ibrahim Ibn Sinan Ibn Thabit Ibn Qurra)(908–946)ArabianGeometry

The Arabs inherited the works of famous Greekmathematicians, such as EUCLID OF ALEXANDRIA,APOLLONIUS OF PERGA, and ARCHIMEDES OF SYRA-CUSE. Once they had mastered the ideas therein,several of them were able to extend the methods.Ibrahim ibn Sinan was one Arab who employedgreat originality in his study of mathematics, andhe represents a high point in Arabian scientificknowledge.

Ibrahim ibn Sinan was born in 908, proba-bly in Baghdad, into a family of famous schol-ars. His father, Sinan ibn Thabit, was a physi-cian, astronomer, and mathematician. Ibrahimled a brief life, dying at age 38 in Baghdad, buthe accomplished a significant amount of scientificactivity. Besides his labor in mathematics, Ibrahimexamined the apparent motions of the Sun, stud-ied the optics of shadows, and investigated astro-nomical instruments such as the astrolabe.

In mathematics proper, Ibrahim’s writtenworks cover the tangents of circles and geome-try in general. His quadrature of the parabola(determination of the area enclosed by a givenparabola) involves an expansion of Archimedes’

method. Ibrahim’s grandfather Thabit ibn Qurrahad already generalized Archimedes’ technique,which was equivalent to summing definite inte-grals, but his exposition was quite lengthy. Incontrast, Ibrahim’s analysis is simple and elegant.He decomposes the area of the parabola into anapproximating collection of inscribed triangles,and he proves an elementary relation betweenthe areas of the inscribed polygons. As a result,the desired area is four-thirds of the first in-scribed triangle. Ibrahim’s genius is evident inhis graceful solution to this problem.

He also sought to revive classic geometry,which had been neglected by his contempo-raries. Ibrahim desired to both provide a practi-cal method for solving geometrical problems andto categorize problems according to their diffi-culty and method. Following the epistemologyof the ancient Greeks, Ibrahim argued for thedual importance of synthesis and analysis. Thework of Ibrahim ibn Sinan exerted a deep in-fluence on the mathematical philosophy of sub-sequent Arabic mathematicians.

Further Readingal’Daffa, A. The Muslim Contribution to Mathematics.

London; Prometheus Books, 1977.Rashed, R. The Development of Arabic Mathematics:

Between Arithmetic and Algebra. Boston: KluwerAcademic, 1994.

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� Jacobi, Carl (Carl Gustav Jacob Jacobi)(1804–1851)GermanAnalysis, Number Theory,Mathematical Physics

In early 19th-century Germany, Carl Jacobiranked among the foremost mathematical suc-cessors of CARL FRIEDRICH GAUSS. Jacobi distin-guished himself by his numerous contributionsto analysis, especially in the area of elliptic in-tegrals; for the diversity of his activity and thebreadth of his intellect, he has been comparedto LEONHARD EULER.

Born on December 10, 1804, in Potsdam,Germany, Jacobi was the second son of SimonJacobi, a wealthy Jewish banker, and he receivedan excellent education from his uncle. Jacobihad an older brother, Moritz, who became aphysicist in St. Petersburg, and a younger brotherand sister. Jacobi was intellectually advanced asa young boy, and he entered the high school inPotsdam in 1816. He was soon promoted to thehighest class despite his youth; when he gradu-ated in 1821, Jacobi had mastered Greek, Latin,and history and possessed an extensive knowl-edge of mathematics, already having attemptedthe solution of the quintic equation.

Jacobi went on to the University of Berlin,where he concentrated on mathematics. Work-ing privately, he soon mastered the works ofEuler, JOSEPH-LOUIS LAGRANGE, and severalother leading mathematicians. In 1824 he passedhis preliminary exams, and soon submitted aPh.D. thesis. Having converted to Christianity,he was allowed to commence his academic ca-reer at the University of Berlin at the young ageof 20.

Jacobi’s lectures were stimulating, since hedescribed his current research to his audience.His first lecture in 1825 dealt with the analytictheory of curves and surfaces. This was Jacobi’smost prolific period, and he established contactwith fellow mathematicians such as Gauss,ADRIEN-MARIE LEGENDRE, and NIELS HENDRIK

ABEL. Much of Jacobi’s research built upon anddeveloped Gauss’s explorations. Legendre wasthe first to study elliptic integrals in a system-atic fashion, and both Abel and Jacobi becamehis intellectual heirs, competing in their inves-tigations of transcendental functions.

Jacobi relocated to the University ofKönigsberg in 1826, as there were more oppor-tunities for advancement there. Through in-teractions with FRIEDRICH WILHELM BESSEL,Jacobi became increasingly interested in ap-plied problems. Jacobi’s publications enjoyed a

146 Jacobi, Carl

wide popularity, and he soon advanced to asso-ciate professor in 1827 and full professor in 1832.During his 18 years at Königsberg, Jacobi pro-duced amazing results on the theory of ellipticfunctions, analysis, number theory, geometry,and mechanics. Many of his papers were pub-lished in Crelle’s Journal for Pure and AppliedMathematics, and Jacobi was partly responsiblefor its rise to international repute. Although heenergetically pursued his research, Jacobi alsolectured about 10 hours a week, often discussingthe most recent advances in knowledge. Jacobideveloped the research seminar—essentially acollection of advanced students—and also en-couraged the research-oriented approach to uni-versity teaching.

Jacobi married Marie Schwinck in 1831, andhad five sons and three daughters with her. Hetraveled to Paris in 1829 in order to meet theleading French mathematicians, and visitedLegendre, JEAN-BAPTISTE-JOSEPH FOURIER, andSIMÉON-DENIS POISSON. Later, he attended amathematical conference in Britain in 1842. In1843 Jacobi became quite ill with diabetes, andhe traveled through Italy in the hope that themilder climate would improve his health; on hisreturn, Jacobi moved back to Berlin, and he oc-casionally lectured at the University of Berlin.

Up to this time, Jacobi’s research was mainlyconcerned with elliptic functions. A summary ofhis initial results was published in Fundamentanova theoriae functionum ellipticarum (New foun-dations of the theory of elliptic functions) in1829; herein, Jacobi discussed the transforma-tion and representation of elliptic functions, re-vealing many of the most important properties.One of Jacobi’s important ideas (which was in-dependently developed by Abel and Gauss) wasthe inversion of an elliptic integral, and this ledto several important formulas. Along with Abel,Jacobi also introduced imaginary numbers intothe theory of elliptic functions and discoveredtheir double periodicity. Throughout his com-petition with Abel, Jacobi remained generous

and good-willed, advocating the term abelian forcertain results on transcendental functions.Later in the same work, Jacobi expressed theseintegrals as infinite products, and was able to ap-ply his results to number theory—for example,he was able to prove that any integer can be rep-resented as the sum of at most four squares,which had been previously conjectured by PIERRE

DE FERMAT.This work continued through the 1830s,

with additional results on the theta function. Innumber theory, Jacobi studied the theory ofresidues, quadratic forms, and the representa-tions of integers as sums of squares and cubes.Jacobi also contributed to partial differentialequations (into which field he introduced ellip-tic functions), mathematical physics (Jacobistudied the configurations of rotating liquidmasses), and the theory of determinants. Jacobigave a systematic presentation of determinantsin 1841, and he introduced the “Jacobian,” thedeterminant used in change of variable calcula-tions in integral calculus. Besides these labors inmathematics, Jacobi lectured on the history ofmathematics, and even started the immenseproject of producing a volume of Euler’s com-plete works.

Jacobi took an interest in Euler as a kindredspirit, since their view of mathematics was sim-ilar. Jacobi, like Euler, was a good calculator, andenjoyed an algorithmic perspective on solvingproblems; Jacobi was versatile in many areas ofmathematics, and he wrote prolifically.

He committed some political blunders in1848, alienating himself from the Prussianmonarchy. As a result his salary was cut, and hewas forced to sell his house in Berlin. In 1849he received an offer from Vienna, and Prussia re-stored his salary—evidently, they were loath tolose such an eminent mathematician. In 1851Jacobi contracted influenza, followed by small-pox, which proved fatal. He died on February18, 1851, in Berlin. His good friend GUSTAVE-PETER-LEJEUNE DIRICHLET gave a eulogy in 1852,

Jordan, Camille 147

describing Jacobi as the greatest mathematicianin the Berlin Academy since Lagrange.

Jacobi’s work spanned several fields, but hiswork on elliptic functions and integrals is the mostsignificant. In his own time he was recognized,along with Dirichlet, as one of the greatestGerman mathematicians. Yet even after his death,his work continued to prove influential; he left be-hind a school of mathematicians and an impres-sive body of mathematical ideas.

Further ReadingHawkins, T. “Jacobi and the Birth of Lie’s Theory of

Groups,” Archive for History of Exact Sciences 42,no. 3 (1991): 187–278.

Pieper, H. “Carl Gustav Jacob Jacobi,” in Mathematicsin Berlin. Berlin: Birkhäuser Verlag, 1998.

� Jordan, Camille(1838–1921)FrenchAlgebra, Analysis, Topology

In the latter portion of the 19th century, con-siderable mathematical activity focused on thedevelopment of group theory, flowering from thebrilliant research of EVARISTE GALOIS. Amongthe many talented algebraists, Camille Jordanwas notable for his virtuosity and leadership ofthe subject. He also made substantial contribu-tions to other areas of mathematics, such as com-binatorial topology and analysis.

Camille Jordan was born on January 5, 1838,in Lyons, France. His father was an engineer, andmany members of his family were well known(his cousin Alexis Jordan was a famous botanist).Camille Jordan was successful in mathematicsfrom a young age, and he entered the ÉcolePolytechnique at age 17. His nominal career asan engineer lasted until 1885, during which timehe wrote more than 100 papers.

Jordan was an excellent all-around math-ematician, publishing noteworthy papers in all

fields of study. He investigated the symmetriesof polyhedrons from a purely combinatorialviewpoint, which was original. Jordan’s con-ception of rigor in analysis exceeded that of hispeers, and his text Cours d’analyse (Course inanalysis) became a popular classic. Jordan wasactive in the first developments of measuretheory, constructing the notion of exteriormeasure, inventing the concept of a functionof bounded variation, and proving that anysuch function could be decomposed as the dif-ference of two increasing functions. This pro-found result paved the way for later advancedresults concerning the decomposition of posi-tive and signed measures. In topology, he real-ized that it was possible to decompose a planeinto two regions via a simple closed curve (thisintuitive concept was already known to many,but Jordan was the first to suggest some waysto prove it).

Jordan’s chief mathematical fame is derivedfrom his talent as an algebraist. He was the firstto systematically develop finite group theory,with the applications of Galois theory in mind.One of his most famous results was part of theJordan-Hölder theorem concerned with the in-variance of certain compositions of groups.Jordan researched the general linear group (thegroup of invertible matrices), and applied his re-sults to geometric problems.

Jordan was interested in the “solvability” offinite groups (the characterization of the finitegroups by certain numerical invariants). He setup a recursive machinery in an attempt toresolve this problem, and in the process discov-ered many new algebraic techniques and con-cepts, such as the orthogonal groups over afield of characteristic two. In 1870 Jordan pro-duced his Traité des substitutions et des équationsalgébriques (Treatise on substitutions and alge-braic equations), a valuable summary of all hisprevious results on permutation groups. Thiswork heavily influenced algebraic research forthe next three decades.

148 Jordan, Camille

As a result of his outstanding and innova-tive work, Jordan’s fame spread, attracting nu-merous foreign students, including FELIX KLEIN

and SOPHUS LIE. From 1873 to 1912 he taught(while still working as an engineer) at the ÉcolePolytechnique and the Collège de France. Jordan’smost profound results are his “finiteness theo-rems,” which provide bounds on the number ofsubgroups of a given mathematical group. Theseresults and others have now become classical inthe study of abstract algebra.

Jordan died on January 22, 1921, in Paris.He was honored during his lifetime, having beenelected to the Academy of Sciences in 1881. Hewas the undisputed master of group theory dur-ing the time he was active, and his work greatlyinfluenced the subsequent evolution of the mod-ern study of algebra.

Further ReadingJohnson, D. “The Correspondence of Camille Jordan,”

Historia Mathematica 4 (1977): 210.

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� al-Karaji, Abu (al-Karaji, Abu Bakr Ibn Muhammad Ibn Al-Husayn; al-Karkhi)(ca. 953–ca. 1029)ArabianAlgebra

Al-Karaji made great contributions to algebra byfirst treating numbers independently of geome-try. In this respect he differed from the Greeksand from his Arabic predecessors. He was thenable to develop many of the basic algebraic prop-erties of rational and irrational numbers, andtherefore represents an important step in theevolution of algebraic calculus. Al-Karaji isknown as the first author of the algebra of poly-nomials.

There is much dispute among scholars aboutthe spelling of this man’s name. From earliertranslations he was known as al-Karkhi, but thiswas later disputed, and the name al-Karaji wasput forth. The latter name has stuck. The con-troversy is of some pertinence, since the nameal-Karkhi would indicate Karkh, a suburb ofBaghdad, whereas al-Karaji is indicative of anIranian city. In any event, al-Karaji dwelled inBaghdad, where he produced most of his math-ematical work, and his books were written fromthe end of the 10th century through the begin-ning of the 11th century. Some scholars believe

he was born on April 13, 953. After this period,he apparently departed for the “mountain coun-tries” in order to write works on engineering.

His treatise on algebra, called al-Fakhri fi’-ljabr wa’l-mugabala (Glorious on algebra), offersthe first theory of algebraic calculus developedby the Arabs. Al-Karaji built upon the tech-niques of previous Arab mathematicians, such asABU AL-KHWARIZMI, but his approach was com-pletely new. He sought to separate algebraic op-erations from the geometric representation giventhem by the Greeks. The Arithmetica of DIO-PHANTUS OF ALEXANDRIA influenced al-Karajiand played a part in al-Karaji’s arithmetizationof algebra.

In the al-Fakhri, al-Karaji first studies thearithmetic of exponents—multiplication and di-vision of monomials translates into addition andsubtraction of their exponents. His successorswere able to apply these rules toward the ex-traction of square roots. Al-Karaji took a boldstep in producing algebraic rules for real num-bers, independent of any geometrical interpre-tation. For one thing, algebraic operations (suchas addition and multiplication) and their basicrules (such as associativity and commutativity)were known to be true for rational numbers, butno theory had been developed for irrationalnumbers (such as square roots). Al-Karaji de-fined the notion of irrational numbers from Book

150 al-Khwarizmi, Abu

X of EUCLID OF ALEXANDRIA’s Elements. For Euclidthis theory of incommensurability applied onlyto geometrical quantities, not to numbers. Thus,al-Karaji extended this concept of irrationalityto numbers in a leap of faith, and extended thealgebraic operations to this class. Modern math-ematicians would later rigorously develop analgebra of real numbers that was purely arith-metical.

One consequence of this conceptual leapwas that Euclid’s Elements would no longer beconsidered as a purely geometrical book. Al-Karaji went on to develop the calculus of radi-cals, deriving rules that allowed the calculationof simple expressions involving square roots. Ina similar vein, al-Karaji gave formulas for the ex-pansion of binomials. In his proof of the so-called binomial theorem, the beginnings ofmathematical induction can be seen. Al-Karajialso derived formulas for the sum of consecutiveintegers and consecutive squares.

Al-Karaji was interested in applying thesemethods to the solution of polynomial equa-tions. He considered linear, quadratic, and cer-tain special higher-degree equations—in thisarea the influence of Diophantus on al-Karajiis evident. In the area of indeterminate analy-sis, al-Karaji was able to clarify and extendDiophantus’s work, and considered problemsinvolving three nonlinear equations in three un-knowns. Diophantus was known for his ingenu-ity in deriving special tricks for individualproblems. In contrast, al-Karaji strove to developgeneral methods that could handle even morecases.

Little is known of the conclusion of Al-Karaji’s life, but some scholars believe that hedied in 1029. Al-Karaji produced a fresh per-spective on algebra. Under his guidance, algebrawas made independent of geometry and moreclosely linked to analysis. This attitude divergedsignificantly from Greek thought and becamenormative for later Arab mathematicians. Histransformation of algebra later had an impact on

Europe through LEONARDO FIBONACCI, who im-ported Arabic ideas and methods to Italy in the12th century.

Further ReadingCrossley, J. The Emergence of Number. Victoria,

Australia: Upside Down A Book Company,1980.

Neugebauer, O. The Exact Sciences in Antiquity.Providence, R.I.: Brown University Press, 1957.

Parshall, K. “The Art of Algebra from al-Khwarizmito Viète: A Study in the Natural Selection ofIdeas,” History of Science 26, no. 72, 2 (1988):129–164.

Rashed, R. The Development of Arabic Mathematics:Between Arithmetic and Algebra. Boston: KluwerAcademic, 1994.

� al-Khwarizmi, Abu (Abu Ja’far Muhammad Ibn Musa al-Khwarizmi)(ca. 780–850)ArabianAlgebra

Al-Khwarizmi is an important Arab in the his-tory of Middle Eastern mathematics, since heplayed a role in the transmission of Hindu knowl-edge to Arabia, from whence it made its way toEurope. His development of algebra, even thoughrudimentary, was an important foundation forlater mathematicians such as ABU AL-KARAJI.

The life of al-Khwarizmi is fairly obscure,but he was probably born sometime before 800in Qutrubbull, a district between the Tigris andEuphrates Rivers near Baghdad. Under the reignof the Caliph al-Mamun from 813 to 833, al-Khwarizmi became a member of the House ofWisdom, an academy of scientists in Baghdad.Al-Khwarizmi wrote books that treated astron-omy, algebra, Hindu numerals, the Jewish cal-endar, geography, and history.

His Algebra was an elementary work, de-signed to provide practical help with common

Klein, Felix 151

calculations used in commerce. The first part isconcerned with the solution of actual algebraicequations, while the second and third sectionstreat measurement and applications. Al-Khwari-zmi gives six basic types of equations that in-clude linear and quadratic equations in onevariable. At this stage there is no notion of zeroor negative number, and a substantial portion ofthe techniques is concerned with removing neg-ative quantities. In fact, the word algebra comesfrom al-jabr, which means “restoration.” Thisrefers to the operation of adding a positive quan-tity to both sides of an equation to remove anegative quantity. A similar operation calledbalancing is also used. The full name of the bookis The Compendious Book on Calculation byCompletion and Balancing. Besides these basicrules, the author provided information on howto find the area of various plane figures, such astriangles and circles, as well as the volume ofsolids such as the cone and pyramid.

Al-Khwarizmi’s Algebra is believed to be thefirst Arabic work on the topic of algebra. Thereis some dispute among scholars over whether hederived his information from Greek or Hindusources. His use of diagrams indicates that hemay have been familiar with EUCLID OF ALEXAN-DRIA’s Elements.

Al-Khwarizmi’s treatise on Hindu numeralsis also quite important for the history of math-ematics, since it is one of the earliest works toexpound the superior numerical system of theHindus. This is essentially the modern system,which involves 10 numerical symbols in a place-value system. They are mistakenly referred to as“Arabic numerals,” since they came to Europeansvia the Arabs. It is likely that the Hindu nu-meric system had already been introduced to theArabs, but al-Khwarizmi was the first to presenta systematic exposition.

Besides these mathematical labors, al-Khwarizmi composed a work on astronomy thatderived from the knowledge of the Hindus. HisGeography was an improvement over Ptolemy’s,

as it included the greater knowledge of theArabs.

Al-Khwarizmi died sometime in the ninthcentury, perhaps around 850. Al-Khwarizmi’sAlgebra was used widely in both Arabia and Europeafter the 12th century. More important, perhaps,is the impact of his treatise on Hindu numerals,which facilitated the explosion of European math-ematics following the 12th century.

Further ReadingCrossley, J. The Emergence of Number. Victoria,

Australia: Upside Down A Book Company, 1980.Hogendijk, J. “Al-Khwarizmi’s Table of the ‘Sine of

the Hours’ and the Underlying Sine Table,”Historia Scientiarum. Second Series. InternationalJournal of the History of Science Society of Japan42 (1991): 1–12.

Kennedy, A., and W. Ukashah. “Al-Khwarizmi’sPlanetary Latitude Tables,” Centaurus 14 (1969):86–96.

Kennedy, E. “Al-Khwarizmi on the Jewish Calendar,”Scripta Mathematica 27 (1964): 55–59.

� Klein, Felix (Christian Felix Klein)(1849–1925)GermanAlgebra, Analysis, Geometry

Felix Klein was an important character in thedevelopment of 19th century mathematics: headvanced current knowledge of group theory andRiemann surfaces, contributed to hyperbolic andprojective geometry, and developed the theoryof automorphic functions. His work gave direc-tion and impetus to the next generation ofmathematicians in Europe.

Christian Felix Klein was born inDüsseldorf, Germany, on April 25, 1849. Aftergraduating from the local high school, he beganstudying mathematics and physics at theUniversity of Bonn, obtaining his doctorate in1868. Initially he wanted to be a physicist, but

152 Klein, Felix

under the influence of his teacher Julius Plücker,he switched to mathematics. His dissertationdealt with topics in line geometry (also knownas projective geometry).

Next, Klein pursued further education inGöttingen, Berlin, and Paris, with a brief inter-ruption due to the Franco-Prussian War. In 1871he obtained a lectureship at Göttingen, and thefollowing year he became a professor at theUniversity of Erlangen. During this time, Kleinstudied certain special surfaces and curves and

provided important models for the new hyper-bolic and elliptic geometries previously discov-ered by NIKOLAI LOBACHEVSKY and JÁNOS BOLYAI.His examples were based on projective geometry,and he was the first to give such constructions;one of his important papers from this time wasOn the So-Called Non-Euclidean Geometry, whichput non-Euclidean geometry on a solid footing.Next, Klein identified the groups that are natu-rally associated with various types of geometries.Later he connected these theoretical results tophysics through the theory of relativity.

Klein held professorships at Munich,Leipzig, and finally Göttingen in 1886. In 1875he married Anne Hegel, and they had one sonand three daughters. Besides his early work inprojective geometry and group theory, Kleinmade contributions to function theory—he con-sidered this to be his most important work. Kleinwas able to successfully relate Riemann surfaces,a class of surfaces encountered in complex analy-sis, to number theory, algebra, differential equa-tions, and the theory of automorphic functions.Klein’s excellent spatial intuition enabled himto trace remarkable relationships. He divergedfrom the school of thought led by KARL WEIER-STRASS, which took an arithmetic approach. In1913, HERMANN WEYL gave a rigorous foundationto many of Klein’s important ideas.

Klein was also interested in the solution ofthe quintic, since the theory that had developedinvolved algebra, group theory, geometry, differ-ential equations, and function theory. He deriveda complete theory for the quintic through stere-ographic projection of the group symmetries ofthe icosahedron, and consequently discovered theelliptic modular functions. After studying theirproperties, he went on to investigate automorphicfunctions and algebraic function fields. All of thisresearch stimulated further exploration by hisown students, and the ideas have continuing rel-evance for many areas of modern mathematics.

Although Klein did most of his work in puremathematics, he was very concerned with

Felix Klein, a geometer and algebraist who madeconnections between mathematical groups and certainclasses of surfaces (Aufnahme von Fr. Struckmeyer,Göttingen, courtesy of AIP Emilio Segrè VisualArchives, Landé Collection)

Kovalevskaya, Sonya 153

applications. In the 1890s he turned to physicsand engineering. He even helped to found theGöttingen Institute for Aeronautical andHydrodynamical Research, and attempted to en-courage engineers to a greater appreciation ofmathematics. He retired in 1913 due to poorhealth, and he died on June 22, 1925, inGöttingen.

Klein was a versatile mathematician, con-tributing to several branches of mathematics,and he helped to establish Göttingen as a cen-ter of mathematical activity in Germany.Through his many pupils and his extensive re-search, Klein exerted a great influence on thedevelopment of mathematics from the 19th tothe 20th century.

Further ReadingHalsted, G. “Biography. Professor Felix Klein,” The

American Mathematical Monthly 1 (1894): 416–420.

Rowe, D. “The Early Geometrical Works of SophusLie and Felix Klein,” in The History of ModernMathematics. Boston: Academic Press, 1989.

———. “Klein, Hilbert, and the GöttingenMathematical Tradition,” Osiris 2, no. 5 (1989):186–213.

———. “The Philosophical Views of Klein andHilbert, The Intersection of History andMathematics,” Science Networks. Historical Studies15 (1994): 187–202.

Yaglom, I. Felix Klein and Sophus Lie: Evolution of theIdea of Symmetry in the Nineteenth Century.Boston: Birkhäuser, 1988.

� Kovalevskaya, Sonya (Sofia Kovalevskaya, Sofya Kovalevskaya)(1850–1891)RussianDifferential Equations, Analysis

In the opinion of some historians, Sonya Kovale-vskaya was the greatest female mathematician

prior to the 20th century. She made outstand-ing contributions to the theory of partial differ-ential equations and also advanced the study ofelliptic functions.

Born on January 15, 1850, in Moscow, shewas the daughter of Vasily Korvin-Kukovsky, aRussian nobleman and officer, and YelizavetaShubert, also an aristocrat. Sonya was educatedby an English governess, and participated in asophisticated social circle after the family movedto St. Petersburg. Around age 14 she became in-terested in mathematics, apparently stimulatedby the wallpaper of her father’s country estate,which consisted of lithographs of his notes ondifferential and integral calculus. Sonya showedgreat potential while taking an 1867 course atthe naval academy of St. Petersburg.

Sonya and her sister Anyuta subscribed tothe radical ideology of the late 19th century, andboth were unwilling to accept the traditionallifestyle that Russian society advocated. ThereforeSonya contracted a marriage with VladimirKovalevsky, a young paleontologist, which madefeasible her desire to study mathematics at a for-eign university. In 1869 the couple moved toHeidelberg, and later, in 1871, Sonya came toBerlin, where she studied under KARL WEIER-STRASS. Since she was a woman, she was not al-lowed to attend lectures; instead, she receivedprivate instruction from Weierstrass. By 1874Kovalevskaya had already completed three re-search papers on partial differential equationsand abelian integrals. As a result of this work,she qualified for a doctorate at the University ofGöttingen.

Despite Kovalevskaya’s impressive mathe-matical talent, she was unable to obtain an aca-demic position in Europe, and so she returned toRussia to live with her husband. They had onedaughter, born in 1878. The couple worked oddjobs for several years, but separated in 1881.During this time Kovalevskaya’s husband becameinvolved with a disreputable company, resultingin his disgrace and suicide in 1883. Appealing to

154 Kronecker, Leopold

Weierstrass for assistance, Kovalevskaya ob-tained an appointment at the University ofStockholm. In Sweden Kovalevskaya continuedher research into differential equations, which isdiscussed below. In 1889 she was elected to theRussian Academy of Sciences; at the height ofher career she fell ill from pneumonia, and diedon February 10, 1891, in Stockholm.

Kovalevskaya is most famous for her work inpartial differential equations. She expanded onthe work of AUGUSTIN-LOUIS CAUCHY and formu-lated the existence and uniqueness of solutions ina precise and general manner, introducing the im-portant boundary and initial conditions to theproblem. The resulting Cauchy-Kovalevskayatheorem gave necessary and sufficient conditionsfor a solution of a given partial differential equa-tion to exist.

Kovalevskaya also contributed to the im-portant field of abelian integrals, explaining howto express some of these integrals in terms of sim-pler ones. She won a prize for her memoir Onthe Rotation of a Solid Body about a Fixed Point(1888), which generalized the previous work ofLEONHARD EULER and JOSEPH-LOUIS LAGRANGE.She also studied the motion of Saturn’s rings,which earned her the epithet “Muse of theHeavens.” Parallel to her career as a mathe-matician, Kovalevskaya also wrote several worksof literature that were favorably received.

In a period when it was exceedingly difficultfor women to enter academics, Sonya Kovalevs-kaya was rare in her ability to enter the field ofmathematics and make significant discoverieswith far-reaching impact. However, independentof her gender, she certainly ranks as one of themost talented and influential mathematicians ofthe 19th century. Her work on partial differentialequations has become emblematic of modern ap-proaches to the subject—namely, to focus onquestions of existence and uniqueness of solutionsthrough specification of certain boundary condi-tions. In this way, Kovalevskaya’s work had guidedthe development of the theory of differential

equations, which has numerous applications toscience and engineering today.

Further ReadingCooke, R. The Mathematics of Sonya Kovalevskaya.

New York: Springer-Verlag, 1984.Grinstein, L., and P. Campbell. Women of Mathematics.

New York: Greenwood Press, 1987.Kennedy, D. Little Sparrow: A Portrait of Sophia

Kovalevsky. Athens: Ohio University Press,1983.

Koblitz, A. A Convergence of Lives: Sofia Kovalevskaia:Scientist, Writer, Revolutionary. Boston: Birkhäuser,1983.

� Kronecker, Leopold(1823–1891)GermanNumber Theory, Algebra

Leopold Kronecker was an eminent Germanmathematician in the late 19th century who wasknown for his ability to unite separate areas ofmathematics. However, his puritanical outlook,which struck out against the current trends inanalysis, tended to inhibit the growth of newmathematics.

Leopold Kronecker was born on December7, 1823, in Liegnitz, Germany, to Isidor Kroneckerand Johanna Prausnitzer. They were a wealthyJewish family, and their son received private tu-toring at home. At the high school, Kroneckerwas taught by ERNST KUMMER, who encouragedthe boy’s natural mathematical talent.

In 1841 Kronecker went on to the Universityof Berlin, where he attended mathematics lec-tures by GUSTAV PETER LEJEUNE DIRICHLET. He wasinitially interested in philology and philosophy,and later studied astronomy at the University ofBonn. However, he focused his energies onmathematics and completed his doctorate in1845 with a dissertation on complex numbers.Dirichlet, who was one of Kronecker’s examiners

Kronecker, Leopold 155

and remained a lifelong friend, was impressed byKronecker’s penetration and knowledge ofmathematics.

Kronecker returned to his hometown ofLiegnitz on family business. During this time hepursued mathematics in his spare time as an am-ateur, and continued his correspondences withleading mathematicians while managing hisfamily affairs. He married his cousin FannyPrausnitzer in 1848, and his financial situationimproved so much over the years that he wasable to return to Berlin as a private scholar in1855. The following year, KARL WEIERSTRASS

came to Berlin and became friends withKronecker and Kummer.

At about this time, Kronecker’s mathemat-ical productivity increased greatly. He wroteabout number theory, elliptic functions, and al-gebra. He also related the various branches ofmathematics to one another. As a result of hiswork, he was elected to the Berlin Academy in1861. Exercising his right as a member,Kronecker gave a series of lectures at theUniversity of Berlin on the topics of algebraicequations, number theory, determinants, andmultiple integrals. Kronecker did not attractmany students, but his ideas were neverthelessquite influential within the academy.

During the 1870s Kronecker’s relationshipwith Weierstrass gradually disintegrated. Thiswas primarily due to a divergence in their ap-proach to analysis. Weierstrass emphasized theimportance of irrational numbers and more mod-ern methods, while Kronecker believed thatmost of mathematics—including algebra andanalysis—should be studied under the categoryof arithmetic. In particular, he dispensed with ir-rational numbers completely; he uttered thewell-known saying “God Himself made thewhole numbers—everything else is the work ofmen.” These views, which now seem antiquatedand preposterous, rubbed shoulders with the tideof new ideas in analysis. Eventually, Weierstrassand Kronecker ceased all communication.

Kronecker’s orthodoxy prevented him from ap-preciating the value of GEORG CANTOR’s new settheoretic results on infinity. Because he was in-fluential, he actually prohibited the develop-ment of new ideas.

Nevertheless, Kronecker was able to ad-vance mathematics through his talent at unify-ing and connecting the different branches ofarithmetic, analysis, and algebra. His theoremson boundary formulas, cyclotomic theory, andthe convergence of infinite series are particularlynoteworthy. His paper “Über den Zahlbegriff”(About the concept of number) of 1887 out-lined his program to study only mathematicalobjects that could be constructed in a finitenumber of steps. Out of his dissertation camethe theory of units for an algebraic number field,which would later become an important topicin modern algebra. As a mathematician,Kronecker stressed the utility of the algorithmas a means of calculation rather than as a wor-thy idea in itself.

Kronecker continued at Berlin, while stillfeuding with Weierstrass. In 1891 Kronecker’swife died, and Kronecker himself died soon af-terward, on December 29, 1891, in Berlin.Although of Jewish heritage, he converted toChristianity in the last year of his life.

Kronecker represents an orthodoxy of 19th-century thought, which resisted the new waveof ideas ushered in by younger mathematicians,like Cantor. The very movement he sought tocombat later became the mainstream of modernmathematics, and thus Kronecker seems, in ret-rospect, to be merely an obstacle to progress. Theold school of mathematics still clung to a moreintuitive conception of mathematics, which theincreasingly abstract and formalistic mathematicsof the late 19th century ignored. On the positiveside, Kronecker was successful in his attempts tounify the different branches of mathematics. Onecan also view Kronecker’s stress on finitely con-structed mathematics as anticipatory of thelater 20th-century movement of intuitionism,

156 Kummer, Ernst

spearheaded by LUITZEN EGBERTUS VAN BROUWER

and HENRI-JULES POINCARÉ.

Further ReadingEdwards, H. “An Appreciation of Kronecker,” The

Mathematical Intelligencer 9 (1987): 28–35.––––––. “Kronecker’s Place in History,” in History and

Philosophy of Modern Mathematics. Minneapolis:University of Minnesota Press, 1988.

———.“On the Kronecker Nachlass,” HistoriaMathematica 5, no. 4 (1978): 419–426.

� Kummer, Ernst (Ernst Eduard Kummer)(1810–1893)GermanAlgebra

Ernst Kummer was one of the great creativemathematicians of the 19th century, contribut-ing to function theory, algebra, and geometry.Several mathematical techniques and ideas areattributed to him, and his efforts helped to ad-vance modern mathematics.

Ernst Kummer was born on January 29, 1810,in Sorau, Germany, to Carl Gotthelf Kummer, aphysician who died in 1813, and FrederikeSophie Rothe. Kummer entered the Sorau highschool in 1819, and pursued Protestant theologyat the University of Halle in 1828. However, hesoon turned to mathematics, at first as a prepa-ration for philosophy. In 1831 he received hisdoctorate, and he taught mathematics andphysics at the Gymnasium in Liegnitz from 1832to 1842. During this time, LEOPOLD KRONECKER

was one of his students, and Kummer was able toencourage his natural talent.

His research at this time focused on the hy-pergeometric series introduced by CARL FRIEDRICH

GAUSS. Kummer probed deeper than anyone else,obtaining several remarkable discoveries. Thefailed attempts to prove Fermat’s last theorem ledKummer to study the factorization of integers and

develop the theory of ideals. He also discoveredthe Kummer surface, a four-dimensional manifoldwith 16 conical double points and 16 singulartangent planes. A gifted teacher, he succeeded ininspiring several students to carry out independ-ent investigations. He had previously sent someof his work on function theory to CARL JACOBI,who helped procure a professorship for Kummerat the University of Breslau in 1842. In 1840Kummer married Ottilie Mendelssohn, a cousinof GUSTAV PETER LEJEUNE DIRICHLET’s wife. Heheld his position at Breslau until 1855, andthere he did his important work on numbertheory and algebra. Kummer introduced idealnumbers and ideal prime factors in order toprove a great theorem of PIERRE DE FERMAT. Inlater years, Kronecker and RICHARD DEDEKIND

further developed his initial results.In 1855 Dirichlet left the University of

Berlin to succeed Gauss at Göttingen, andKummer was named as Dirichlet’s replacement.By 1856 both KARL WEIERSTRASS and Kroneckerhad also come to Berlin, ushering in a period ofmathematical productivity at the university.Kummer and Weierstrass constructed the firstGerman seminar in pure mathematics in 1861,which attracted many young students. Kummer’slectures, which covered topics such as analyticgeometry, mechanics, and number theory, wereheavily attended due to his excellent exposition.

Kummer was blessed with an immenseamount of energy. He simultaneously taught at theKriegsschule from 1855 to 1874, was secretary ofthe mathematical section of the Berlin Academyfrom 1863 to 1878, and served several times asdean and rector of the University of Berlin. Duringthis last phase of his career, Kummer focused ongeometry, with applications to ray systems and bal-listics. His study of ray systems followed the workof SIR WILLIAM ROWAN HAMILTON, althoughKummer took an algebraic perspective. In thecourse of this research he discovered the so-calledKummer surface. Numerous mathematical con-cepts have been named after him.

Kummer, Ernst 157

When Kronecker and Weierstrass partedways in the 1870s, Kummer may also have beenestranged from Weierstrass. Certainly, Kummerwas politically and mathematically conservative,shying away from many of the new developments.For example, Kummer rejected non-Euclideangeometry as pointless. He also considered math-ematics a pure science, and believed that theappeal of mathematics lay in its dearth of ap-plications. It is noteworthy that this has proba-bly been the view of mathematicians for most ofhistory, and only in the modern era has the opin-ion emerged that mathematics is valuable onlyif it can contribute to technology and societalimprovement.

In 1882 Kummer retired from his position,claiming that his memory had been weakening.He died on May 14, 1893, in Berlin. Both Gaussand Dirichlet exerted a great influence overKummer’s development as a mathematician, and

he had a great respect for them both. Despite hisconservatism, Kummer was able to affect theevolution of mathematics through his numerouspupils and his raw creativity. His work in alge-bra on the arithmetization of mathematics wasperhaps the most important.

Further ReadingCalinger, R. “The Mathematics Seminar at the

University of Berlin: Origins, Founding, and theKummer-Weierstrass Years,” in Vita Mathematica.Washington, D.C.: Mathematical Association ofAmerica, 1996.

Edwards, H. “Mathematical Ideas, Ideals, andIdeology,” The Mathematical Intelligencer 14 (1992):6–18.

Ribenboim, P. “Kummer’s Ideas on Fermat’s LastTheorem,” L’Enseignement Mathématique. RevueInternationale. Deuxième Série (2) 29, nos. 1–2(1983): 165–177.

158

L� Lagrange, Joseph-Louis (Giuseppe

Lodovico Lagrangia)(1736–1813)ItalianMechanics, Analysis, Algebra,Differential Equations

Joseph Lagrange has been described as the lastgreat mathematician of the 18th century. Hismathematical ideas were highly original and in-fluential, paving the way for the more abstractstudies of the 19th century. Perhaps his mostimportant contribution lies in his mechanisticformulation of the universe, giving exact math-ematical formulas for the laws governing mo-tion and mechanics.

Joseph-Louis Lagrange was born on January25, 1736, in Turin, Italy. His name at birth wasGiuseppe Lodovico Lagrangia, but later in lifehe adopted the French formulation Joseph-Louis Lagrange. Lagrange’s father was GiuseppeFrancesco Lodovico Lagrangia, and his motherwas Teresa Grosso. His family was mostly ofFrench descent, although Lagrange’s mother wasthe only daughter of a Turin physician. Lagrangewas the eldest of 11 children, most of whom diedduring childhood. Lagrange’s father held thepost of treasurer of the Office of Public Worksand Fortifications at Turin. Despite this presti-gious position, the family lived modestly.

Lagrange was originally intended for a ca-reer in law, but once he began studying physics,Lagrange recognized his own talent for the math-ematical sciences. At first he developed an inter-est in geometry, but by age 17 was turning towardanalysis. His first paper (1754) developed a for-mal calculus, which he later realized was alreadyknown to GOTTFRIED WILHELM VON LEIBNIZ. Subse-quently, he began work on the problem of thetautochrone and began the development of hiscalculus of variations. This was essentially an ap-plication of the ideas of calculus to bundles offunctions, rather than a single function.

In 1755 Lagrange sent his early results onthis new calculus of variations to LEONHARD

EULER. Lagrange developed this original and highlyuseful piece of mathematics when he was only19. At the end of his life, he considered it to behis most important contribution. Euler expressedhis interest in the novel method for solving op-timization problems and, as a result of his grow-ing renown, Lagrange was named professor at theRoyal Artillery School in Turin in 1755. Thisposition was poorly paid, and Lagrange felt un-derappreciated by his fellow citizens, which ledhim to later leave Italy.

The next year Lagrange applied his methodto mechanics. He was able to describe the tra-jectory of an object subject to certain forces asthe solution to an optimization problem in the

Lagrange, Joseph-Louis 159

the trajectory that minimizes energy—the founda-tion of dynamics. Many French mathematicians,including JEAN D’ALEMBERT and PIERRE-SIMON

LAPLACE, recognized the excellent quality of hiswork.

In 1763 Lagrange was invited to Paris, wherehe was heartily received by the mathematicalcommunity there. D’Alembert attempted to se-cure Lagrange a superior position in Turin, butthe promises of the royal court failed to materi-alize. As a result, Lagrange accepted an offer tofill Euler’s vacant position in Berlin in 1766,which initiated the second scientific period ofLagrange’s life.

Lagrange made friends with Johann Lambertand JOHANN BERNOULLI, and was named directorof the Berlin Academy of Sciences. He had noteaching duties, which allowed him to focus onhis mathematical research. Lagrange wed hiscousin, Vittoria Conti, in 1767, and they weremarried—though childless—for 16 years, untilVittoria’s health declined and she died in 1783after a protracted illness.

While in Berlin Lagrange enjoyed contin-ued participation and success in the Paris com-petitions, making outstanding contributions tothe three-body problem. Besides these publiccontests, Lagrange developed his own personalwork on celestial mechanics, publishing severalimportant papers from 1782 onward. Meanwhile,he had already begun to investigate certain prob-lems in algebra, completely solving a celebratedindeterminate equation posed by PIERRE DE FER-MAT in 1768. Building on Euler’s previous work,Lagrange proved that every integer can be ex-pressed as the sum of, at most, four perfect squares(1770); he characterized prime numbers througha divisibility criterion, and further developed thetheory of quadratic forms (1775), opening av-enues of future research for CARL FRIEDRICH

GAUSS and ADRIEN-MARIE LEGENDRE. He gave anexposition of the method of infinite descent, in-spired by Fermat, and utilized the method of con-tinued fractions.

calculus of variations. This elegant mathemati-cal formulation of mechanics would revolution-ize the study of dynamical systems.

Meanwhile, the Royal Academy of Sciencesat Turin was founded, to which Lagrange madenumerous fundamental contributions over thenext decade. His works from the time period upto around 1770 include material on the calculusof variations, differential equations, the calculusof probabilities, celestial mechanics, and fluidmotion. He developed the technique of integra-tion by parts, so familiar to calculus students, andwon several prizes offered by the Academy ofSciences in Paris, for his outstanding work onthe motions of the Moon and other celestialbodies. Lagrange’s system of mechanics made theprinciple of least action—that a particle chooses

Joseph-Louis Lagrange developed a mechanisticmathematical formulation of the universe known asLagrangian mechanics. (Courtesy of AIP Emilio SegrèVisual Archives, E. Scott Bar Collection)

160 Laplace, Pierre-Simon

He made one particularly important contri-bution to analysis in 1770, when he gave a se-ries expansion involving the roots of a givenequation, which had useful scientific applica-tions. Lagrange’s formula proved to be of endur-ing interest to mathematicians, as most of thegreat analysts of the 19th century, including AU-GUSTIN-LOUIS CAUCHY, studied the consequencesof this idea. This work, in conjunction with thatof Alexandre Vandermonde, reveals the conceptof the permutation group, which would later bedeveloped precisely by Galois.

Lagrange also contributed to fluid mechan-ics in the 1780s, imaginary roots of algebraicequations in the 1770s, and infinitesimal analy-sis from 1768 to 1787. His work on the integra-tion of differential equations, expanding on theideas of Euler, represents an early step in the the-ory of elliptic functions, which would attractmuch interest in the 19th century. His work onpartial differential equations must also be men-tioned, as he brought integration powerfully tobear on several problems. His work in probabil-ity is of lesser significance.

Lagrange’s considerable contributions to me-chanics were scattered among several publications,and he summarized them in a 1788 treatise.Around this time, Lagrange had settled in Paris.Although Turin had attempted to lure Lagrangeto return to his native city, he was not eager toleave Berlin until the death of his wife in 1783.But the French mathematicians, who aggres-sively solicited his presence, were successful inattracting Lagrange. In 1787 he became a pen-sioned member of the Academy of Sciences,where he weathered the chaotic political turmoilof the succeeding decades.

In 1792 Lagrange married Renée-Françoise-Adélaïde Le Monnier, with whom he also had ahappy marriage. During the beginning of theParisian phase of his career, Lagrange’s activityabated somewhat. He was active in the 1790Constituent Assembly on the standardization ofweights and measures, and later taught analysis

at the newly founded École Polytechnique until1799. After Napoleon rose to power Lagrangewas appointed a grand officer of the Legion ofHonor and in 1808 a count of the Empire. Hedied on the morning of April 11, 1813,in Paris. Universities throughout Europe ob-served his death, and Laplace gave his funeraloration.

Lagrange made extensive contributions tomany areas of mathematics; these labors oftenopened up new areas of inquiry (such as ellipticfunctions, quadratic forms, and the calculus ofvariations). Most significant was his formulationof mechanics, sometimes called Lagrangianmechanics, which essentially mechanized theunderstanding of the physical universe. Thisproved to be a powerful and influential mode ofdescribing the known world, and continues toaffect mathematical inquiry today.

Further ReadingBaltus, C. “Continued Fractions and the Pell Equations:

The Work of Euler and Lagrange,” Communicationsin the Analytic Theory of Continued Fractions 3(1994): 4–31.

Fraser, C. “Isoperimetric Problems in the VariationalCalculus of Euler and Lagrange,” Historia Mathe-matica 19, no. 1 (1992): 4–23.

Koetsier, T. “Joseph Louis Lagrange (1736–1813): HisLife, His Work and His Personality,” Nieuw Archiefvoor Wiskunde 4, no. 4 (3) (1986): 191–205.

Sarton, G. “Lagrange’s Personality (1736–1813),”Proceedings of the American Philosophical Society88 (1944): 457–496.

� Laplace, Pierre-Simon(1749–1827)FrenchProbability, Mechanics, DifferentialEquations

Pierre-Simon Laplace was certainly one of themost eminent scientists and mathematicians of

Pierre-Simon Laplace studied the heat equation, wherethe Laplace operator appears, and developed manyfirst results of probability. (Courtesy of AIP EmilioSegrè Visual Archives)

Laplace, Pierre-Simon 161

the late 18th century in Paris, and may be re-garded as one of the principal founders of thetheory of probability. His scientific theories, re-markable for their modernity and sophistication,held sway for many years and profoundly influ-enced the later evolution of thought. Laplacewas remarkably talented as a mathematician,and he was a prominent intellectual in Franceduring his lifetime.

Pierre-Simon Laplace was born on March23, 1749, in Beaumont-en-Auge, in the provinceof Normandy, France, to Pierre Laplace, a pros-perous cider trader, and Marie-Anne Sochon,who came of a wealthy family of landowners.Although Laplace’s family was comfortable fi-nancially, they did not possess an intellectualpedigree. In his early years Laplace attended aBenedectine priory, his father intending him for

a career in the church. At age 16 Laplace en-rolled at Caen University, where he studiedtheology; however, some of his professors rec-ognized his exceptional mathematical talent,and stimulated him to pursue his innate gifts atParis.

So it was that at age 19, Laplace departedfrom Caen without a degree and arrived at Paris,with a letter of introduction to the eminentJEAN D’ALEMBERT. D’Alembert warmly acceptedLaplace, and would fill the role of a mentor tothe brilliant young mathematician. D’Alembertalso procured a position for Laplace as professorat the École Militaire. This position affordedLaplace little intellectual stimulus, but it en-abled him to remain in Paris, where he was ableto interact with the Parisian mathematical com-munity and produce his first papers. Theseworks, read before the Academy of Sciences in1770, built upon the work of JOSEPH-LOUIS LA-GRANGE on the extrema of curves and differenceequations.

Laplace was an ambitious young man whowas well aware of his own talent, and he madethis fact apparent to his inferior colleagues. Thisarrogance alienated many, while they were at thesame time forced to admit his brilliance. Laplacewas indignant at being passed over for electionto the Academy of Sciences due to his youth,but by 1773 he was made an adjoint of that in-stitution after having read 13 papers before thecommunity within three years. Laplace’s earlywork was of high quality on a variety of topics,including differential equations, difference equa-tions, integral calculus, mathematical astronomy,and probability. These latter two topics wouldform a recurring theme in Laplace’s lifetimework.

In the 1770s Laplace built up his reputationas a mathematician and scientist, and in the1780s he made his most important contribu-tions. Laplace demonstrated that respiration wasa form of combustion, studied the impact ofmoons upon their planets’ orbits, and formulated

162 Laplace, Pierre-Simon

the classical mathematical theory of heat. TheLaplace operator (referred to as the “Laplacian”by mathematicians) forms an important role inthe basic differential equation for heat. His workin astronomy at this time laid the groundworkfor his later seminal masterpiece on the struc-ture and dynamics of the solar system.

In 1784 Laplace became an examiner of theRoyal Artillery Corps, and served on various sci-entific committees. He used his expertise inprobability to compare mortality rates amonghospitals—one of the first exercises in survivalanalysis. In 1785 Laplace was promoted to a sen-ior position in the Academy of Sciences, andsoon afterward Lagrange joined the same insti-tution. The proximity of both these eminentscientists led to an explosion of scientific ac-tivity in Paris.

On May 15, 1788, Laplace married Marie-Charlotte de Courty de Romanges, who was 20years younger; they had two children. Laplacebecame involved in a committee to standardizeweights and measures in 1790 that advocatedthe metric system. By this time the FrenchRevolution had already commenced, and Laplaceposed as a republican and antimonarchist toavoid political persecution. He was somewhat ofan opportunist, strategically altering his politi-cal opinions throughout the Revolution in or-der to escape attack. In 1793 he fled the Reignof Terror, but was later consulted by the gov-ernment on the new French calendar; althoughthis calendar was incompatible with astronomi-cal data, Laplace refrained from criticism to pro-tect himself.

Laplace taught probability at the ÉcoleNormale, but his abstraction made his lecturesinaccessible to the students there. These lectureswere later published as a collection of essays onprobability in 1814, giving basic definitions aswell as a wealth of applications to mortalityrates, games of chance, natural philosophy,and judicial decisions. In 1795 the Academy of

Sciences, which had been shut down by the rev-olutionaries, was reopened, and Laplace becamea founding member of the Bureau des Longitudesand head of the Paris Observatory. However, inthe latter capacity Laplace proved to be too the-oretical, as he was more concerned with devel-oping his planetary theory for the dynamics ofthe solar system than with astronomical obser-vation. In 1796 Laplace presented his famousnebular hypothesis in his Exposition du systèmedu monde (The system of the world), which de-lineated the genesis of the solar system from aflattened, rotating cloud of cooling gas into itspresent form. In five books Laplace described themotion of celestial bodies, the tides of the sea,universal gravitation, the mechanical conceptsof force and momentum, and a history of the so-lar system. Much of his material is remarkablymodern in its description of the world, which isa testimony to the enduring legacy of his scien-tific thought.

This important work was followed up by theTraité du Mécanique Céleste (Treatise on celestialmechanics), which gave a more mathematicalaccount of his nebular hypothesis. Here Laplaceformulates and solves the differential equationsthat describe the motions of celestial bodies, andmore generally applies mechanics to astronomi-cal problems. Herein appears the Laplace equa-tion, which features the Laplacian, although thisdifferential equation was known previously.Characteristically, Laplace failed to give creditto his intellectual progenitors. However, it isclear that Laplace was heavily influenced byLagrange and ADRIEN-MARIE LEGENDRE.

Laplace received a variety of honors under theempire of Bonaparte, including the Legion ofHonor in 1805; he was chancellor of the senateand briefly served as minister of the interior. In1806 he became a count, and after the restorationbecame a marquis in 1817. In 1812 he publishedhis Théorie Analytique des Probabilités (Analytictheory of probability), which summarized his

contributions to probability. Therein Laplacedetailed Bayes’s theorem (see THOMAS BAYES),the concept of mathematical expectation, andthe principle of least squares (simultaneously in-vented by CARL FRIEDRICH GAUSS); these threeideas have had a momentous impact on scienceand statistics. He applied his techniques to awide variety of topics, such as life expectancyand legal matters. Although others (such asBLAISE PASCAL) had contributed to probabilitypreviously, Laplace gave a more systematictreatment, and clearly demonstrated its utility inpractical problems.

Laplace’s scientific ideas were profound. Hesought to reduce the study of physics to the in-teractions between individual molecules, actingat a distance. This singularly modern formulationwas revolutionary in its wide range of applications,including the study of pressure, density, refraction,and gravity. His work is also distinguished fromprevious molecular theories in its precise math-ematical formulation. In the first decades of the19th century, Laplace went on to apply his prin-ciples to a variety of scientific problems, such asthe velocity of sound, the shape of the Earth,and the theory of heat. He also founded theSociété d’Arcueil in 1805, in which SIMÉON DE-NIS POISSON was also active; this group activelyadvocated a prominent role of mathematics inscientific exploration. After 1812 this group’senergy waned, and Laplace’s ideas came underattack as newer paradigms were advanced. Forexample, Laplace held to the fluid theories ofheat and light, and these fell out of favor withthe advance of JEAN-BAPTISTE-JOSEPH FOURIER’sideas.

The wane of Laplace’s scientific hegemonywas also accompanied by increasing social isola-tion, as his colleagues became disgusted with hispolitical infidelity. In later life he supported therestoration of the Bourbons, and was forced toflee Paris during the return of Bonaparte. Hedied on March 5, 1827, in Paris, France. The

Academy of Sciences, in honor of his passing,canceled its meeting and left his position vacantfor several months.

Laplace contributed to the wave of scien-tific thought in Paris in the late 18th century.Although many of his ideas were soon discarded,others have endured to modern times; even hisoutdated theories were influential on the nextgeneration of scientists. More notable was hisadvocacy of a stronger role of mathematics inscientific enterprise, and his precise formulationof mathematical laws for scientific phenomena.His mathematical work has proved to be moreenduring; in particular, his efforts in differentialequations, probability, and mechanics have be-come classical in these disciplines. Of particularnote is the mathematical theory of heat and hisfundamental work on basic probability.

Further ReadingBell, E. Men of Mathematics. New York: Simon and

Schuster, 1965.Crosland, M. The Society of Arcueil: A View of French

Science at the Time of Napoleon I. Cambridge,Mass.: Harvard University Press, 1967.

Gillispie, C. Pierre-Simon Laplace, 1749–1827: A Lifein Exact Science. Princeton, N.J.: Princeton Univer-sity Press, 1997.

Stigler, S. “Laplace’s Early Work: Chronology andCitations,” Isis 69, no. 247 (1978): 234–254.

———. “Studies in the History of Probability andStatistics XXXII: Laplace, Fisher, and theDiscovery of the Concept of Sufficiency,”Biometrika 60 (1973): 439–445.

� Lebesgue, Henri-Léon(1875–1941)FrenchAnalysis

Henri Lebesgue played an important role in thedevelopment of integration theory as one of the

Lebesgue, Henri-Léon 163

164 Legendre, Adrien-Marie

most active branches of 20th-century mathe-matics. The so-called Lebesgue integral is nowclassic in integration theory and has been a cor-nerstone for continuing research as well as anaid in applications.

Henri Lebesgue was born on June 28, 1875,in Beauvais, France. As a young man, Lebesguestudied at the École Normale Supérieure from1894 to 1897. After graduation, Lebesgue spentthe next two years working in his school’s library,where he became acquainted with the work ofRENÉ-LOUIS BAIRE on discontinuous functions.Lebesgue taught at the Lycée Centrale in Nancyfrom 1899 to 1902 while completing his doctoralthesis for the Sorbonne. He set out to developa more general notion of integration that wouldallow for the discontinuous functions discoveredby Baire.

In the process of working on his thesis,Lebesgue became acquainted with the work ofCAMILLE JORDAN and ÉMILE BOREL on measura-bility and integration; since the time of BERN-HARD RIEMANN’s integral, mathematicians hadgradually introduced concepts from measure the-ory. Lebesgue’s early work expanded Borel’s ef-forts, and he successfully constructed a definitionof integration that was more general thanRiemann’s. More significantly, Lebesgue wasable to apply his integral to several problems ofanalysis, including the validity of term-by-termintegration of an infinite series. Also, the fun-damental theorem of calculus, which relates in-tegration and differentiation, was not able tohandle nonintegrable functions with boundedderivative. Lebesgue’s new integral resolved thisdifficulty. He also worked on curve rectification(computing the length of a curve).

After submitting his thesis, Lebesgue wasgiven a position at the University of Rennes,which lasted until 1906. He was at the Universityof Poitiers from 1906 to 1910, and then was ap-pointed as a professor at the Sorbonne from 1910to 1919. Lebesgue continued his research on thestructure of continuous functions and integration;

he took the first steps toward a theory of doubleintegrals, which was later completed by GUIDO

FUBINI. A wave of research followed in the wakeof Lebesgue’s work, and this effort includedsuch mathematicians as PIERRE FATOU. ThroughFRIGYES RIESZ, the Lebesgue integral became aninvaluable tool in the theory of integral equa-tions and function spaces. His other researchincluded the structure of sets and functions,the calculus of variations, and the theory of dimension.

Lebesgue received many honors, such as thePrix Santour in 1917 and election to the FrenchAcademy of Sciences in 1922. During the lasttwo decades of his life, his interests shifted in-creasingly to pedagogical issues and elementarygeometry. He died on July 26, 1941, in Paris.Besides his own pivotal contributions, Lebesguewas an active proponent of abstract theories ofmeasure, and thus is largely responsible for theimportant position of measure theory in modernmathematics. His essential improvement on thebasic Riemann integral resolved many out-standing mathematical questions and opened upa new vista for future exploration.

Further ReadingBurkill, J. “Henri Lebesgue,” Journal of the London

Mathematical Society 19 (1944): 56–64.———.“Obituary: Henri Lebesgue. 1875–1941,”

Obituary Notices of Fellows of the Royal Society ofLondon 4 (1944): 483–490.

May, K. “Biographical Sketch of Henri Lebesgue,” inMeasure and the Integral by Henri Lebesgue. SanFrancisco: Holden-Day, 1966.

� Legendre, Adrien-Marie(1752–1833)FrenchNumber Theory, Analysis, Geometry

Legendre was an important figure in the transi-tion from 18th- to 19th-century mathematics.

Legendre, Adrien-Marie 165

His contributions to number theory and analy-sis raised many significant questions that futuremathematicians would pursue. He was also oneof the founders of the central theory of ellipticfunctions.

Adrien-Marie Legendre was born onSeptember 18, 1752, in Paris, France. His fam-ily was affluent, and Legendre received an ex-cellent scientific education at the schools ofParis. In 1770 Legendre defended his theses onmathematics and physics at the Collège Mazarin.

Legendre possessed a modest fortune, whichgave him the liberty to pursue mathematicalresearch in his leisure time. Nevertheless, hetaught mathematics at the École Militaire inParis from 1775 to 1780. Legendre won a 1782prize from the Berlin Academy, with a paperconcerning the trajectory of cannonballs andbombs, taking air friction into account. Over thenext few years he increased his scientific output,attempting to gain more renown among theFrench scientists; he studied the mutual attrac-tions of planetary bodies, indeterminate equa-tions of the second degree, continued fractions,probability, and the rotation of accelerating bod-ies. Throughout his life, Legendre’s favorite areasof research were celestial mechanics, numbertheory, and elliptic functions.

In 1786 Legendre published Traité des func-tions elliptiques (Treatise on elliptic functions)that outlined methods for discriminating be-tween maxima and minima in the calculus ofvariations, and the so-called Legendre condi-tions gave rise to an extensive literature. He alsostudied integration by means of elliptic arcs,which was really a first step in the theory of el-liptic functions. Around this time Legendre waspromoted in the Academy of Sciences, andcontributed to some geodetic problems, bring-ing his expertise with spherical trigonometry tobear.

Legendre next studied partial differentialequations, expressing the so-called Legendretransformation. He self-published his 1792 work

on elliptic transcendentals, since the Frenchgovernment suppressed the academies. This wasa strenuous time for Legendre. He married ayoung girl, Marguerite Couhin, while theFrench Revolution destroyed his personal fortune.His young wife was able to give him emotionalstability while he continued writing new scien-tific works.

In 1794 Legendre received a new post con-cerned with weights and measures. In the mean-time, he published his Elements of Geometry,which would dominate elementary instructionin geometry for the next century. During thenext decade he directed the calculation of new,highly accurate trigonometric tables; they werebased on the newer mathematical techniques ofthe calculus of variations.

Legendre published his Essay on the Theoryof Numbers in 1798, which expanded on his pre-vious work of 1785, with material on indetermi-nate equations, the law of reciprocity of quadraticresidues, the decomposition of numbers into threesquares, and arithmetical progressions. His 1806work on the orbits of comets gave the first pub-lic exposition of the method of least squares.However, Legendre was infuriated to learn thatCARL FRIEDRICH GAUSS had been using the methodin private since 1795.

In the succeeding decades Legendre ex-panded on the theory of elliptic functions, inde-terminate equations, and spherical trigonometry.His work in number theory was noteworthy forthe law of quadratic reciprocity. Legendre gavean imperfect demonstration of this law in1785, and Gauss proved it rigorously in 1801.Legendre contributed to the knowledge ofFermat’s last theorem, establishing the result ina special case, and was a precursor of analyticnumber theory—he studied the distributionof prime numbers, stating their asymptotics in1798. His best achievements lie in the theoryof elliptic functions; expanding on the work ofLEONHARD EULER and JOSEPH-LOUIS LAGRANGE,Legendre essentially founded this theory in

166 Leibniz, Gottfried Wilhelm von

1786 by expressing elliptic integrals in terms ofcertain more basic types called transcendentals.A master calculator, Legendre developed ex-tensive tables for the values of these ellipticfunctions. NIELS HENRIK ABEL and KARL JACOBI

substantially developed his early work in the fol-lowing years. Legendre succeeded PIERRE-SIMON

LAPLACE in 1799 as the examiner of mathemat-ics at the school of artillery, and resigned in1815, succeeding Lagrange at the Bureau deLongitudes in 1813. He received several hon-ors, including membership in the Legion ofHonor. On January 9, 1833, he died in Paris af-ter a painful illness.

Legendre’s approach to mathematics wastypical of the 18th century. Many of his argu-ments lacked rigor, and he was highly skepticalof innovations such as non-Euclidean geometry.He was, in many respects, a disciple of Euler andLagrange, whose view on mathematics influ-enced him greatly. But Legendre’s contributionsto number theory and elliptic functions led towhole new arenas of inquiry, and it is here thathis impact was so pronounced.

Further ReadingGrattan-Guinness, I. The Development of the Foundations

of Mathematical Analysis from Euler to Riemann.Cambridge, Mass.: MIT Press, 1970.

Hellman, C. “Legendre and the French Reformof Weights and Measures,” Osiris 1 (1936):314–340.

Pintz, J. “On Legendre’s Prime Number Formula,” TheAmerican Mathematical Monthly 87, no. 9 (1980):733–735.

� Leibniz, Gottfried Wilhelm von(1646–1716)GermanCalculus, Logic, Differential Equations

One of the most hotly debated issues in the his-tory of mathematics was the question of priority

in the discovery of infinitesimal calculus. SIR

ISAAC NEWTON and Gottfried Leibniz had eachmade remarkable discoveries in differentialcalculus, and each man’s followers fostered anugly argument over who should be creditedwith the original discovery. Whatever the truthmay be, there is no doubt that Leibniz was oneof the greatest mathematicians of his time,which is apparent not only from the breadthand depth of his original ideas, but also fromhis ability to organize others’ thoughts moreefficiently.

Gottfried Wilhelm von Leibniz was born onJuly 1, 1646, in Leipzig, Germany, to FriedrichLeibniz, a professor at the University of Leipzig,and Katherina Schmuck. The family was ofSlavonic descent, but had been dwelling in

Gottfried Leibniz, recognized as cofounder ofcalculus, formulated the notion of limit, which iscrucial to real analysis. (Joh. Gottfr. Auerbach ad viv.Delin Viennae 1714. Joh. Elias Haid sc. 1781. Aug.Vind., AIP Emilio Segrè Visual Archives)

Leibniz, Gottfried Wilhelm von 167

Germany for several generations. Leibniz was aprecocious student, and his teachers initially at-tempted to restrain his curious nature. After hisfather died in 1652, he was allowed access to hisfather’s library. In this manner Leibniz educatedhimself, so that when he entered the Universityof Leipzig at age 15, he had already mastered theclassics. His voracious appetite for reading re-mained with him throughout his life, andLeibniz was able to digest a great variety ofscholarly subjects.

Leipzig held to the unscientific Aristoteliantradition, first learning Euclidean geometry atthe University of Jena, where he briefly studiedafter 1663. He completed his doctorate atAltdorf in 1666, and soon entered the serviceof a nobleman of the Holy Roman Empire.Leibniz initiated a correspondence with manyscientific societies, and he began work on acalculating machine that was finally completedin 1674. In 1671 he journeyed to Paris on adiplomatic mission designed to forestall theFrench monarch’s invasion of the Rhineland.This project was unsuccessful, but while in ParisLeibniz developed a lifelong friendship withCHRISTIAAN HUYGENS.

During these years, Leibniz expanded onhis earlier instruction in mathematics, devel-oping calculation rules for finite differences.Continued peace negotiations led him toLondon in 1673, where he was admitted to theRoyal Society and became familiar with theworks of ISAAC BARROW. At this time Leibniz re-ceived hints of Newton’s work on infinitesimalcalculus, and he soon developed his own com-putational techniques and notation. By 1674Leibniz effected the arithmetical quadrature ofthe circle.

Leibniz’s previous patron had died, and in1676 he took on a new position in Hannover,acting as librarian and engineer. A few years laterhe became a court councilor and was busily en-gaged in genealogical research for the duke.Meanwhile, Leibniz had started researching

algebra, and had obtained several important re-sults by 1675, such as the determination of sym-metric functions and an algorithm for the solutionof higher-degree algebraic equations. He conjec-tured that the sum of complex conjugate num-bers is always a real number. ABRAHAM DE

MOIVRE later proved this result. Leibniz also in-vestigated progressions of primes and arithmeticseries, such as the sum of reciprocal squares. Helearned of the transcendence of the logarithmicand trigonometric functions and their basicproperties, and investigated some problems inprobability.

But his greatest discovery came late in 1675,when he introduced the notion of the limit toinfinitesimal calculus. This method, and his cor-responding notation, facilitated the furtherspread and understanding of the new mathe-matics. Newton disparaged his work, as it did notsolve any new problems; but the strength ofLeibniz’s system was its clarity and abstraction ofthe general principles of calculus. Leibniz didproceed to solve several important differentialequations with his techniques. Many of his dis-coveries of this time were written only as notesand ideas in letters, and were not systematicallydeveloped or published until 1682. In the nextfew years he presented a few papers to the pub-lic that treated arithmetical quadrature, the lawof refraction, algebraic integrations, and differ-ential calculus.

In 1687 Leibniz traveled around Germanyto continue his genealogical research. He alsovisited Italy, and finally completed his project in1690—his efforts helped raise the duchy ofHannover to electoral status in 1692. Leibniz at-tracted attention from the scientific communitythrough his attack on Cartesian dynamics in1686. Out of this controversy, several dynami-cal questions were posed and solved by Leibniz,Huygens, and JAKOB BERNOULLI, including thefamous problems of the catenary (1691) andthe brachistochrone (1697). Characteristically,Leibniz disclosed only his results and not his

168 Leonardo da Vinci

methods. In fact, his papers were often writtenhastily. Despite some errors, his work was note-worthy for the originality of his ideas, some ofwhich were precursors to EVARISTE GALOIS’s workon the solubility of equations. Leibniz definedthe center of curvature, developed the methodof undetermined coefficients in the theory ofdifferential equations, and constructed thepower series for exponential and trigonometricfunctions.

In the later years of the 17th century, muchof Leibniz’s time was taken up in the controversywith Newton over the discovery of calculus. Thefollowers of Newton contended that Leibniz hadplagiarized his ideas directly from Newton andBarrow. Leibniz defended himself in 1700, stress-ing that he had already published his materialon differential calculus in 1684. The ugly pub-lic debate raged back and forth, egged on by na-tionalistic considerations, until the RoyalSociety held a biased investigation, which ruledin favor of Newton in 1712. This verdict was ac-cepted without question for about 140 years.Now it is thought that Leibniz developed hismethods independently of Newton.

Leibniz journeyed to Berlin in 1700 andfounded the Berlin Academy, becoming presi-dent for life. He labored to make certain politi-cal and religious reforms and was appointed acouncilor to Russia in 1712. He spent the lastfew years of his life attempting to complete thehistory of the house of Brunswick while plaguedby gout. He died on November 14, 1716. Besideshis remarkable contributions to mathematics,Leibniz researched physics, logic, and philoso-phy. He wrote on topics as diverse as religiousdogma (theodicy) and planetary motion, anddeveloped a logical calculus that would allow forthe certainty of deductions through an algebraicsystem. In this aspect, Leibniz was a forefatherof numerous other formal logicians such asGEORGE BOOLE and FRIEDRICH LUDWIG GOTTLOB

FREGE.

His greatest talent as a mathematician washis ability to penetrate the thoughts of other sci-entists and present them in a coherent fashionsuitable for computation. The notation that hedeveloped for differential calculus is the quintes-sential example of this power—he assiduouslyperceived that the notion of limit was crucial tothe study of infinitesimal calculus. The details, forLeibniz, were not so important as the underlyingabstract concepts. His legacy in mathematicscontinues to this day.

Further ReadingBroad, C., and C. Lewy. Leibniz: An Introduction.

London: Cambridge University Press, 1975.Hostler, J. Leibniz’s Moral Philosophy. London:

Duckworth, 1975.Ishiguro, H. Leibniz’s Philosophy of Logic and Langu-

age. New York: Cambridge University Press,1990.

Merz, J. Leibniz. Edinburgh: Blackwood, 1884.Rescher, N. Leibniz: An Introduction to His Philosophy.

Oxford: Blackwell, 1979.Ross, G. Leibniz. New York: Oxford University Press,

1984.Rutherford, D. Leibniz and the Rational Order of

Nature. New York: Cambridge University Press,1995.

� Leonardo da Vinci(1452–1519)ItalianGeometry

Leonardo da Vinci is one of the most famouspersons of the medieval period. An artist, engi-neer, and scientist, he was both diverse andprophetic. He made substantial contributions toart, anatomy, technology, mechanics, geology,and mathematics.

Leonardo da Vinci was born on April 15,1452, in Empolia, Italy. He was the illegitimate

son of Piero da Vinci, a Florentine citizen. Hismother was a peasant girl named Caterina.Leonardo’s father was soon married to a re-spectable Italian woman, Albiera di GiovanniAmadori. Leonardo received a rudimentary ed-ucation, and he displayed his talents for musicand art at a young age. In 1467 he was appren-ticed to Andrea del Verrocchio, with whom hestudied painting, sculpture, and mechanics.

Leonardo completed some of his early paint-ings during this time, including Baptism of Christ.In 1482 he departed in order to work for the dukeof Milan; by this time he was already accom-plished in architecture, painting, and sculpture,as well as military engineering. He remained inMilan until 1499, during which time he became

more interested in physics and mechanics andin the properties of light. He also augmented hismeager education in mathematics, studyingLatin and geometry concurrently.

Leonardo formulated his theory of the su-premacy of painting on carefully laid mathe-matical principles of proportion and perspective.His interest in proportion led him to conductfurther research in physics and mathematics.Some of his early work in mathematics was quiteerroneous, as he did not have an adequate graspof arithmetical computation—one example ishis claim that the fraction 2/2 is the square rootof 2, since he falsely asserts that 2/2 times 2/2 is4/2.

His other projects during the time in Milaninclude the physics of light, the physics of vi-sion, and the problem of mechanical flight. Hecollaborated with the mathematician Paciolion the Divina proportione (Divine proportion).It is probable that Leonardo had read EUCLID

OF ALEXANDRIA’s Elements before making thedrawings in this book. Leonardo’s notebookscontain proofs of various propositions in theElements, and it is likely that his friend Pacioliencouraged and assisted him in his study ofEuclid.

Leonardo departed for Venice after theFrench captured the duke of Milan, and later hereturned to Florence. He briefly served underCesare Borgia as a military engineer, and latercompleted his famous Mona Lisa. From 1500 to1506 he pursued research into human anatomy,and more of his time was occupied with mathe-matics and mechanics. After completing hisstudy of Euclid—Leonardo was especially inter-ested in the treatment of proportion in Book Xof the Elements—he began his own research onequiparation. He was mainly interested in thesquaring of curvilinear surfaces (transformingthese curved regions into squares with the samearea), although his method of proof was oftenmechanical rather than strictly geometrical.

Leonardo da Vinci held mathematics to be the key toall the other sciences and studied curvilinear surfaces.(Courtesy of the Library of Congress)

Leonardo da Vinci 169

170 Lévy, Paul-Pierre

Leonardo proposed several methods for squaringthe circle; he was familiar with ARCHIMEDES OF

SYRACUSE’s method, but rejected the latter’s ap-proximation of pi by 22/7. He attempted to im-prove the approximation by inscribing a 96-sided polygon in the circle.

Encouraged by his supposed discovery ofthe quadrature of the circle on November 30,1504, he pursued similar research on doublingsquares and quadrupling circles. He also be-came interested in the doubling of the cube(which had already been solved by ERATOS-THENES OF CYRENE centuries ago), dissatisfied bya recent solution given by Valla. Eventually,Leonardo conceived of a solution that elimi-nated the need for a mechanical apparatus, andhe was thereby able to obtain extremely accu-rate approximations for the cube root of two.However, he was unable to provide a rigorousproof for his method.

Many of his mathematical writings are in-cluded in the Codex Atlanticus (Atlantic codex).Leonardo continued to investigate the propertiesof curvilinear surfaces, such as the portions leftbetween a circle and an inscribed square or hexa-gon. He also explored the possibility of humanflight by studying the anatomy of birds, as well asthe movement of water. In 1506 he returned toMilan, where he served under the French gover-nor. In this latter period of his life, he producedsome of his best anatomical drawings, and his sci-entific efforts spread to hydrology, geology, mete-orology, biology, and human physiology. In allthese areas, he felt that mathematics held the keysof knowledge, and he attempted to formulate geo-metrical laws for these disciplines. The Frenchwere expelled in 1513, and Leonardo departed forRome, hoping to find work with Pope Leo X; thisfailed to materialize, and he returned to the serv-ice of France in 1516, working under Francis I.He suffered a stroke in Amboise, and died on May 2, 1519.

Leonardo’s approach to the study of naturecannot be called scientific in the modern sense.

He did believe in the importance of empiricalinvestigation, but many of his ideas were purelyspeculative, without solid reasoning behindthem. Of course, many of his concepts werebrilliant contributions as well. In mathemat-ics, he seems to have been an amateur. He cer-tainly made some worthwhile discoveries, andhe deeply respected the role of mathematics inthe investigation of nature. But many of hisworks were deeply flawed, and his approach toproofs was more typical of his identity as anartist. In addition, his mathematical workshave not influenced the subsequent progress ofmathematical thought. His geometrical re-search on curvilinear areas did develop an as-pect of Euclid’s work, but his writings were notwell known in his own time, and thus did notexert an influence on other mathematicalthinkers.

Further ReadingClark, K. Leonardo da Vinci. London: Viking, 1975.Kemp, M. Leonardo da Vinci, the Marvelous Works of

Nature and Man. London: J.M. Dent, 1981.McLanathan, R. Images of the Universe; Leonardo da

Vinci: The Artist as Scientist. Garden City, N.Y.:Doubleday, 1966.

Moody, D., and M. Clagett. The Medieval Science ofWeights. Madison: University of Wisconsin Press,1960.

Zubov, V. Leonardo da Vinci. Cambridge, Mass.: HarvardUniversity Press, 1968.

� Lévy, Paul-Pierre(1886–1971)FrenchProbability, Analysis, DifferentialEquations

In the early 20th century, the field of probabil-ity lacked unity and cohesion. Paul Lévy madefundamental contributions to this area, formingit into one of the major divisions of modern

Li Chih 171

mathematics. He also developed the theory ofpartial differential equations and functio-nal analysis, propelling these fields of thoughtforward.

Paul Lévy was born on September 15, 1886,in Paris, France. He belonged to a family ofmathematicians, including his father and grand-father. His father, Lucien Lévy, was an examinerat the École Polytechnique. Paul Lévy was anoutstanding student, attending the Lycée SaintLouis in Paris, where he won prizes in mathe-matics and science. In his entrance examina-tions to college, he scored first place for theÉcole Normale Supérieur and second place forthe École Polytechnique.

He chose to attend the latter institutionand began publishing papers while still an undergraduate. His first paper (1905) studiedsemiconvergent series. Lévy later graduated infirst place and spent a year in the military be-fore joining the École des Mines in 1907. Whilethere, Lévy also attended lectures by CharlesPicard and Jacques Hadamard. The latter greatlyinfluenced Lévy’s research, encouraging him topursue functional analysis.

In 1910 Lévy started to research functionalanalysis, and Picard, JULES-HENRI POINCARÉ, andHadamard examined his thesis the followingyear. He obtained his doctorate in 1912. Lévybecame professor at the École des Mines in 1913and, in 1920, became professor of analysis at theÉcole Polytechnique. During World War I Lévyserved in the French military and worked onmathematical ballistics problems to aid the wareffort.

His work on functional analysis extendedthe calculus of variations to function spaces, andfollowed the same lines of thought as those ofVITO VOLTERRA. But his greatest work lay inprobability, where he labored extensively formany years. Lévy borrowed many techniquesfrom analysis to attack probability problems, andalso initiated the development of probabilityinto a mature field of mathematics, capable of

being applied to other disciplines. In particular,he worked on limit laws, the theory of martin-gales, and the properties of Brownian motion.The latter two areas form two large branches ofthe theory of stochastic processes, which arewidely used in engineering, statistics, and thesciences to model and solve a variety of practi-cal problems.

Beyond these advances in probability, Lévyalso studied the theory of partial differentialequations and geometry. In the former area, Lévyextended Laplace transforms and generalized thenotion of functional derivatives. He producedseveral texts that have been widely used by stu-dents of mathematics. Lévy died on December15, 1971, in Paris, France.

Lévy made important contributions toprobability and functional analysis, which havebeen two of the most important areas of math-ematics for modeling real scientific problems inthe 20th century. He was a deep thinker withan appreciation for mathematical beauty andutility.

Further ReadingDoob, J. “Obituary: Paul Lévy,” Journal of Applied

Probability 9 (1972): 870–872.Kendall, D. “Obituary: Paul Lévy,” Journal of the Royal

Statistical Society. Series A. Statistics in Society 137(1974): 259–260.

� Li Chih (Li Yeh)(1192–1279)ChineseAlgebra

Li Chih, also known as Li Yeh, was one of thegreatest Chinese mathematicians. He is knownfor developing methods to solve algebraic equa-tions. The method of the celestial element, whichLi Chih helped to propagate, has exerted an en-during influence on Chinese and Japanese math-ematics up to present times.

172 Li Chih

Li Chih was born in 1192 in Beijing, China.His father was Li Yü, an official in the northerngovernment. Li Chih received his education inhis father’s native province of Hopeh. In 1230he passed his civil service examination and wasappointed as a registrar in Shensi Province.Before he started this post, he became the gov-ernor of Chün-chou in Honan Province. In 1232the Mongols captured this city, and Li Chih fledfor safety.

By 1234 the kingdom of the Jurchen had fallento the Mongol invaders, and Li Chih devotedhis energies to private studies in mathematics.During this time, he frequently lived in poverty,but he managed to complete his most importantwork, Sea Mirror of the Circle Measurements,which is discussed below. By 1251 his circum-stances had improved, and he settled in Feng-lung, in Hopeh Province. Although he still livedin isolation, he made friends with two other fa-mous men, Chang Te-hui and Yuan Yü. The triobecame known as “the Three Friends of LungMountain.”

In 1257 Kublai Khan consulted Li Chih oncertain matters of government and the rea-sons for earthquakes. Shortly, Li Chih completeda second mathematical work: New Steps inComputation in 1259. When Kublai Khan as-cended the throne in 1260, he offered Li Chiha government post, which he refused due to hisage. In 1264 the Mongolian emperor created theHan-lin Academy, designed to record the offi-cial history of the Liao and Jurchen kingdoms,and Li Chih was recruited to join this body. Hesoon resigned, again claiming old age, and re-treated to Feng-lung, where he attracted a groupof pupils.

Li Chih later changed his name to Li Yeh, inorder to avoid having the same name as the thirdTang emperor. Before his death in 1279, he in-structed his son to burn all his writings except SeaMirror of the Circle Measurements, as this aloneseemed—to him—to be of benefit to posterity.

New Steps in Computation also survived confla-gration, as did some other nonscientific works.

Sea Mirror of the Circle Measurements wascompleted in 1248. Li Chih therein introducedan algebraic process known as the “method ofcelestial elements” and the “coefficient arraymethod.” This method exerted much influencethrough China and Japan, even though it wasnot well understood. Li Chih was not themethod’s inventor, having learned it from themathematician P’eng Che, of whom little isknown. The method is used to solve algebraicequations of high degree by arranging the coefficients in vertical columns, ascending withthe power of the unknown. The absolute term,which refers to the additive constant in theequation, occupies a middle place, and recipro-cal powers of the unknown descend beneath it.It is interesting that Li Chih was able to denoteboth negative quantities and zero.

Li Chih would display equations of up tothe sixth degree, but he did not describe theprocess used to solve them. Scholars havetherefore concluded that the method of solu-tion was common knowledge in China at thattime. He was largely responsible for standard-izing the terminology used to describe the var-ious coefficients. The term celestial elementrefers to the unknown variable. In the book,Li Chih considers some 170 problems con-cerned with circles inscribed in or circum-scribed around triangles, and then he appliesthe solutions to a variety of word problems,such as finding the diameter of a circular citywall given certain conditions.

The New Steps in Computation is a muchsimpler text. Perhaps Li Chih realized that manypeople had trouble understanding his first book.It contains 64 problems, some of which treat thequadratic equation, and some that deal with re-lationships between circles and squares. Towardthe end, he provides three values for pi: the oldvalue of 3, as well as 3.14 and 22/7.

Lie, Sophus 173

Li Chih was significant in his dissemina-tion of the method of the celestial element,and was certainly one of the most talentedmathematicians of his time. Although CH’INCHIU-SHAO was a contemporary, they wereprobably ignorant of one another due to their dis-tant geographical locations. Essentially, scholarscategorize Li Chih as an algebraist, since hewas more concerned with the algebraic formu-lation of equations than with the exposition oftheir solution.

Further ReadingMartzloff, J. A History of Chinese Mathematics.

Berlin/Heidelberg: Springer-Verlag, 1997.

� Lie, Sophus (Marius Sophus Lie)(1842–1899)NorwegianGeometry, Differential Equations,Algebra

One of the most popular and elegant branchesof mathematics in the 20th century has been thetheory of Lie groups. This discipline combinesideas from algebra, geometry, and analysis, andis relevant to theoretical physics. Sophus Lie firstdiscovered these objects, and thus founded afruitful arena for future research.

Marius Sophus Lie, commonly known asSophus Lie, was born on December 17, 1842, inNordfjordeide, Norway. He was the sixth andyoungest child of Johann Lie, a Lutheran pastor.He attended a local school, and from 1857 to1859 studied at Nissen’s Private Latin School inOslo. From 1859 to 1865 he continued his educa-tion at Christiania University in Oslo. Originallyhe showed little interest in mathematics, and hefocused on the sciences. After his examinationin 1865, Lie gave private lessons, and he becameinterested in astronomy.

Lie’s life acquired new direction after hisdiscovery in 1868 of some geometrical papers bythe mathematicians JEAN-VICTOR PONCELET andJulius Plücker. The idea that space could bemade up of lines instead of points had a pro-found impact on Lie’s conception of geometry.He obtained a scholarship abroad, living inBerlin in winter 1869, where he became ac-quainted with FELIX KLEIN. Both men’s scientificendeavors benefited greatly from the friend-ship that ensued. Klein was an algebraist in-trigued by particular problems, whereas Lie wasa geometer and analyst interested in generaliz-ing concepts.

They spent summer 1870 in Paris, wherethey came in contact with CAMILLE JORDAN andGASPARD MONGE as well as other French math-ematicians. Lie discovered his famous contacttransformation, which was an important initialgeometrical discovery—it was a first step towardhis later development of the theory of Lie groups.The Franco-Prussian War erupted in the sameyear, and Lie was arrested as a spy while hikingthrough the countryside. He was soon freed andmanged to escape France before the blockade ofParis. In 1871 he returned to Oslo, where hetaught at Nissen’s Private Latin School. He ob-tained his doctorate in 1872.

At this time Lie developed the integrationtheory of partial differential equations, whichis still taught as a classical method in mathe-matics texts. His initial work on differentialgeometry later led to his important work ontransformation groups and differential equa-tions. The transformation group, later known asthe Lie group, brought algebraic tools to bearon geometric and analytic problems, and in par-ticular resulted in his powerful approach to par-tial differential equations. Although these ideaswere not initially accepted—largely due to thecumbersome style of presenting analytic ideas thatwas fashionable at the time—their significance tomodern mathematics cannot be overestimated.

174 Liouville, Joseph

He completed his work on Lie groups in the1870s, but their publication took several decadesof effort.

In 1872 a chair in mathematics was createdfor Lie at Christiania University. Besides theabove-mentioned research on contact transfor-mations, he was busy editing the collected worksof NIELS HENRIK ABEL. Lie married Anna Birchin 1874, and together they raised two sons andone daughter.

In Oslo Lie was isolated from other math-ematicians; he had no pupils, and only twomathematicians—Klein and Emile Picard—paid attention to his work. Friedrich Engel as-sisted Lie in the publication of a lengthy texton transformation groups, which appeared inthree parts between 1888 and 1893. His paral-lel work on contact transformation and partialdifferential equations with Felix Hausdorff wasnot completed. In 1886 Lie came to Leipzig, suc-ceeding Klein, and his collaborative situationimproved.

Lie’s health had been excellent, and he wasdescribed as an openhearted man of huge stature.However, in 1889 he was struck with mental ill-ness. When he resumed work in 1890 his char-acter had changed greatly—now he was paranoidand belligerent. Eventually he returned toChristiania University with the lure of a specialchair in 1898. He died a year later on February18, 1899, in Oslo from anemia.

Lie’s work revolutionized the study of geom-etry and differential equations, since group theo-retic and algebraic techniques could now solveproblems. The study of Lie groups eventuallybecame a discipline of its own. The apprecia-tion for Lie’s work grew gradually. InitiallyEngel and Issai Schur further developed hisideas, and later Picard, Killing, ÉLIE CARTAN,and HERMANN WEYL continued Lie’s theoreticalwork in the 20th century. In the early 20th cen-tury Lie algebras were discovered, and Lie’s orig-inal work has been generalized in many ways. Onereason for the enduring popularity of his thought

is the application of Lie groups to quantumphysics.

Further ReadingAckerman, M., and R. Hermann. Sophus Lie’s 1880

Transformation Group Paper. Brookline, Mass.:Math Sci Press, 1975.

Fritzsche, B. “Sophus Lie: A Sketch of His Life andWork,” Journal of Lie Theory 9, no. 1 (1999):1–38.

Hermann, R. Lie Groups: History, Frontiers andApplications. Brookline, Mass.: Math Sci Press,1979.

Yaglom, I. Felix Klein and Sophus Lie: Evolution of theIdea of Symmetry in the Nineteenth Century. Boston:Birkhäuser, 1988.

� Liouville, Joseph(1809–1882)FrenchAnalysis, Mechanics, Geometry,Number Theory

Joseph Liouville played an important role in theadvancement and promotion of 19th-centurymathematics, accomplished through his ownpublications as well as his editing of the influ-ential Journal de Liouville (Liouville’s Journal).He principally researched analysis, geometry,and number theory, publishing a series of nu-merous short notes and papers. Through the ac-ademic positions that he held, Liouville was ableto mold the mathematical interests of the nextgeneration.

Joseph Liouville was born on March 24,1809, in St.-Omer, France. He was the secondson of Claude-Joseph Liouville, an army cap-tain, and Thérèse Balland, both of whom camefrom the province of Lorraine. He initially stud-ied in the towns of Commercy and Toul beforeattending the École Polytechnique in 1825. Hetransferred to the École des Ponts et Chausséesin 1827, where he began original research in

Liouville, Joseph 175

mathematics. During the next few years, Liouvillepresented several memoirs to the Academy ofSciences, treating heat, electricity, and mathe-matical analysis. These were favorably received,and enabled Liouville to obtain a teaching po-sition at the École Polytechnique in 1831, fol-lowing his graduation the previous year. Also in1831 he married his cousin Marie-LouiseBalland. They would have three daughters andone son.

Liouville remained in the teaching profes-sion for 50 years, and he was able to teach pureand applied mathematics at the leading institu-tions of Paris. In 1838 he occupied a chair of analy-sis and mechanics at the École Polytechnique,and in 1851 gained the chair of mathematics atthe Collège de France, remaining there until1879. Meanwhile, he taught classes at a moreelementary level, and dabbled in politics—hewas elected to the Constituent Assembly in1848, but left his political career after an 1849defeat. Liouville earned his doctorate on the ap-plications of Fourier series to mathematicalphysics in 1836, which allowed him to teach atthe university level. In 1857 he simultaneouslytaught mechanics at the Paris Faculty ofSciences.

While he was teaching, Liouville also par-ticipated in several societies, such as the Academyof Sciences and the Bureau des Longitudes, andmost importantly launched the Journal of Pureand Applied Mathematics (later known asLiouville’s Journal) in 1836. This forum, createdafter the demise of two influential journals, wascrucial for the dissemination of mid-19th-cen-tury mathematics. As the journal’s editor,Liouville was able to affect the development ofmathematics at that time; he remained the chiefeditor until 1874.

His initial research interests focused onmathematical analysis. Most important, Liouvilledealt with such topics as the classification of al-gebraic functions (he defined the Liouvilliannumber, an example of a transcendental number),

the theory of elliptic functions (building on thework of NIELS HENRIK ABEL and CARL JACOBI), anddifferential equations. Between 1832 and 1837he formulated a notion of fractional derivative;he also extended the knowledge of oscillations,and contributed the theories of electricity andheat. Liouville’s work was quite interdisciplinary,and he took an interest in the applications ofmathematical methods to problems in celestialmechanics.

Liouville also contributed to algebra, fur-nishing a new proof of the fundamental theo-rem of algebra as well as proving one ofCauchy’s theorems. More important, he publi-cized the works of EVARISTE GALOIS, exposingthem to a wider audience of mathematicians,and thus ushered in new techniques that wouldbecome classical in modern algebra and grouptheory. Liouville wrote numerous papers ongeometry, studying the calculus of variations,geodesic lines of ellipsoids, and properties ofpolygons. He introduced the new notion of to-tal curvature, and studied the deformations ofa surface.

In addition to these studies, Liouville laterbecame interested in number theory. From 1858to 1865 he published several theorems that allbelong to analytic number theory—some ofthe first publications in this new field of in-quiry. In his later years Liouville’s research fo-cused on more particular problems outside themainstream, and was of less interest to othermathematicians.

Liouville lived a quiet, studious life. He diedon September 8, 1882, in Paris. He was able toaffect the development of mathematics throughhis distinguished academic career. His promo-tion of the work of Galois, and more generallyhis encouragement of younger mathematiciansthrough his journal, changed the landscape ofmathematics. His own research contributionswere significant in their scope and maturity, ex-cepting his later work in number theory towardthe end of his life.

and was distinguished by the variety and depthof his research.

Lipschitz investigated number theory, Besselfunctions, Fourier series, differential equations,and mechanics. Most important was his work onhigh dimensional differential forms and their re-lationships to the calculus of variations andgeometry. In this area, Lipschitz developed theearly ideas of Riemann and was able to developa new area of mathematics that has proved tohave enduring interest and relevance in the 20thcentury.

Lipschitz also wrote a book, Foundation ofAnalysis, which gathered together various topicsof mathematical research in one work and wasthe first of its kind written in German. He for-mulated a continuity condition for functionscalled the Lipschitz condition that has provedto be important in function theory and approx-imation theory, and relates to questions of exis-tence and uniqueness of solutions of differentialequations.

Lipschitz’s work on differential forms wasdone in collaboration with the mathematicianElwin Christoffel. Lipschitz obtained manysignificant results concerning the curvature of Riemannian manifolds and submanifolds.His investigations were later continued byGregorio Ricci-Curbastro, and implementedby Albert Einstein in his theory of generalrelativity.

Lipschitz died on October 7, 1903, in Bonn.His principal contribution lies in the founda-tion of the theory of differential forms; thisbranch of mathematics is both elegant and use-ful for understanding high-dimensional geome-tries. His work on manifolds was indicative ofthe direction that geometrical inquiry wouldsoon take.

Further ReadingScharlau, W. “The Mathematical Correspondence of

Rudolf Lipschitz,” Historia Mathematica 13, no. 2(1986): 165–167.

176 Lipschitz, Rudolf

Further ReadingChrystal, G. “Joseph Liouville,” Proceedings of the

Royal Society of Edinburgh 14 (1888): 83–91.Lützen, J. Joseph Liouville, 1809–1882, Master of Pure

and Applied Mathematics. New York: Springer-Verlag, 1990.

Stander, D. “Makers of Modern Mathematics: JosephLiouville,” Bulletin of the Institute of Mathe-matics and Its Applications 24, nos. 3–4 (1988):59–60.

� Lipschitz, Rudolf (Rudolf Otto Sigismund Lipschitz)(1832–1903)GermanAnalysis, Geometry

Rudolf Lipschitz was an important analyst andgeometer in the latter half of the 19th centurywho advanced the knowledge of Riemannianmanifolds, differential forms, and continuousfunctions, thereby contributing to the founda-tions of much of 20th-century mathematics. Hisresearch interests were quite broad, but his laborsin geometry (building on BERNHARD RIEMANN’swork) are most noteworthy.

Rudolf Lipschitz was born on May 14, 1832,in Königsberg, Germany. His father was a land-holder, and Lipschitz received a good education.He began his study of mathematics at theUniversity of Königsberg at age 15, but laterwent to Berlin where he became GUSTAV PETER

LEJEUNE DIRICHLET’s pupil. He suffered a delay inhis studies due to illness, but was able to com-plete his doctorate at the University of Berlin in1853.

After several years teaching at localGymnasiums, Lipschitz obtained a position atthe University of Berlin in 1857, and in the sameyear he married Ida Pascha. He first became aprofessor at the University of Breslau in 1862,and then at the University of Bonn in 1864.Lipschitz was a member of several academies,

Maksimovich, was a clerk, and his mother wasnamed Praskovia Aleksandrovna Lobachevskaya.In 1800 Lobachevsky’s mother moved, togetherwith Lobachevsky and his two brothers, toKazan. There the three boys were enrolled in theGymnasium on scholarships. In 1807 Lobachevskyentered Kazan University, where he studiedmathematics and physics, obtaining his master’sdegree in 1812.

In 1814 Lobachevsky lectured on mathe-matics and mechanics as an adjunct, and becamea professor the same year; he was promoted in1822, and served a variety of positions at KazanUniversity, including dean of the department ofphysics and mathematics, librarian of the uni-versity, rector, and assistant trustee for the Kazandistrict. His first major work, written in 1823,was called Geometriya (Geometry), and its basicgeometrical studies led Lobachevsky to his laterresearches into non-Euclidean geometry. He re-ported his early discoveries in 1826, and pub-lished these ideas in 1829–30.

Lobachevsky initially attempted to provethe fifth postulate of EUCLID OF ALEXANDRIA, asmany before him (including CLAUDIUS PTOLEMY,Thabit ibn Qurra, ABU ALI AL-HAYTHAM, ADRIEN-MARIE LEGENDRE, and JOHN WALLIS) had tried andfailed to accomplish. He soon turned to the con-struction of a more general geometry that didnot require the fifth postulate, which states thatgiven a line and a distinct point, there exists aunique line through the point that is parallel tothe given line. The resulting geometry, whichhe named “imaginary geometry,” allowed for theconstruction of multiple distinct parallel linesthrough the given point. From here he was ableto deduce several interesting properties: mostimportant, the geometry was consistent (therewas no contradiction in its rules, however coun-terintuitive its characteristics). Curiously, thesum of angles in a triangle is less than 180 de-grees; Lobachevsky later attempted to deduce thegeometry of the universe by measuring the an-gles of a vast cosmic triangle spanned by distant

Lobachevsky, Nikolai 177

� Lobachevsky, Nikolai (Nikolai Ivanovich Lobachevsky)(1792–1856)RussianGeometry

A major paradigm shift in geometrical intuitiontook place in the 19th century, when CARL

FRIEDRICH GAUSS, JÁNOS BOLYAI, and Lobachevskyeach independently developed alternative geome-tries to flat space. Lobachevsky was the first topublish this discovery. His generalizations ofthe intuitive notion of space have since provedextremely relevant within mathematics (pavingthe way for the abstract definition and study ofgeometry) and physics, through the modeling ofgravity’s effect on the shape of the universe.

Nikolai Lobachevsky was born on Decem-ber 2, 1792, in Gorki, Russia. His father, Ivan

Nikolai Lobachevsky invented an alternative,consistent geometry known as hyperbolic geometry.(Library of Congress, courtesy of AIP Emilio SegrèVisual Archives)

178 Lovelace, Augusta Ada Byron

stars. He concluded that, within the margins ofmeasurement error, the angles summed to 180degrees, and hence the universe is Euclidean.

Lobachevsky produced several more paperson this subject, including the above stellarcomputation; he gave both an axiomatic anda constructive definition of his “pangeometry,”which later came to be known as hyperbolicgeometry. His ideas were not initially acceptedabroad, although he was promoted at Kazanand made into a nobleman in 1837. He mar-ried a wealthy aristocrat, the Lady VarvaraAleksivna Moisieva, in 1832, and they hadseven children.

Besides his important geometrical work,Lobachevsky contributed to algebra, infinite se-ries, and the theory of integration. However, thiswork was flavored by his geometrical ideas andwas related to his “imaginary geometry.” Gaussappreciated Lobachevsky’s efforts, which weresimilar to his own work on non-Euclidean geom-etry, and assisted his election to the GöttingenAcademy of Science after 1842.

Lobachevsky, despite his advantageous mar-riage, experienced financial difficulties in hislater years, due to the cost of his large family andthe maintenance of his estate. His eyes deterio-rated with age until he became totally blind. Hedied on February 24, 1856, in Kazan.

Recognition of Lobachevsky’s pioneeringwork came slowly. Many mathematicians, suchas Arthur Cayley, failed to comprehend its sig-nificance, and denigrated it. In the 1860s theworks of Bolyai and Lobachevsky gained in-creasing renown among the French, andEugenio Beltrami later gave a construction ofLobachevskian geometry on a closed circle of theplane. After 1870 KARL WEIERSTRASS and FELIX

KLEIN became interested in Lobachevsky’s work,and Klein eventually formulated the variousgeometries—elliptic, flat, and hyperbolic—interms of invariants of group transformations.Lobachevskian geometry was later shown to be

a special case of Cayley geometries. JULES-HENRI POINCARÉ, together with Klein, furtherbuilt on the ideas of BERNHARD RIEMANN andLobachevsky. In the 20th century non-Euclideangeometry was shown to be relevant to the gen-eral theory of relativity. It is intriguing that thespace of the universe was later demonstrated tohave variable curvature, with the warp andwoof of its fabric defined by gravitationalforces. This reality is modeled by Lobachevsky’sgeometry.

Further ReadingBonola, R. Non-Euclidean Geometry: A Critical and

Historical Study of Its Developments. New York:Dover Publications, 1955.

Halsted, G. “Biography. Lobachevsky,” The AmericanMathematical Monthly 2 (1895): 137.

Householder, A. “Dandelin, Lobachevskii, or Gräffe?”The American Mathematical Monthly 66 (1959):464–466.

Kagan, V. N. Lobachevsky and His Contribution toScience. Moscow: Foreign Languages PublishingHouse, 1957.

Vucinich, A. “Nicolai Ivanovich Lobachevskii: TheMan behind the First Non-Euclidean Geometry,”Isis 53 (1962): 465–481.

� Lovelace, Augusta Ada Byron(Ada Lovelace)(1815–1852)BritishLogic, Algebra

Computers, and the theory accompanying them,have been one of the most significant intellec-tual developments of the latter half of the 20thcentury. However, the history of the computergoes back several centuries to the first calculators,developed by BLAISE PASCAL and others; later, inthe early 19th century, the first primitive com-puter was developed by CHARLES BABBAGE, and

Lovelace, Augusta Ada Byron 179

the first work on computer science was writtenby Ada Lovelace.

Ada Lovelace was born on December 10,1815, in Piccadilly, England, with the nameAugusta Ada Byron. Her mother was AnneIsabelle Milbanke, and her father was GeorgeGordon, Lord Byron, the famous poet. Lovelace’sparents were separated shortly after her birth,and her manipulative mother kept the identityof Lovelace’s father a secret. Byron died inGreece when Lovelace was eight years old. LadyByron was determined that her daughter wouldnot become a poet like her father, and so trainedher in mathematics, a subject to which she hadpredilection herself.

Thus Lovelace’s early education, overseenby her mother, had an intense focus on mathe-matics. In fact, it seems that Lady Byron wasoverbearing and tyrannical in her efforts; how-ever, Lovelace’s native mathematical talent dideventually flower in the latter portion of herlife. Lovelace herself preferred geography toarithmetic, and only became interested in sci-ence and mathematics after meeting Babbage in 1833, when she learned about his differenceengine.

In 1835 Lovelace married William King,who was made the earl of Lovelace in 1838; thus,Ada Lovelace gained the name by which she iscommonly referred. The next few years werespent in childbearing (she had two boys and agirl), and Lovelace began an earnest study ofmathematics only in 1841. Her advanced stud-ies were guided by AUGUSTUS DE MORGAN.

In 1842 she published a translation of LuigiMenabrea’s description of Babbage’s analyticalengine, and she added much of her own materialto the original text. Lovelace’s notes entereddeeply into the abstract algebraic questions raisedby the engine, which essentially attempted tomap algebraic operations into the mechanical ac-tions of moving machine parts. She also de-scribed how the engine could be manipulated, in

essence presenting the first computer programever written. It is interesting that Lovelace wasvery optimistic about the capabilities of the en-gine, believing that every algebraic operationcould be mechanized and computed. Moderncomputer science theory has further exploredwhat types of mathematical operations can beperformed by a computer program (or at least bya Turing machine); much present effort is focusedon the design of quantum computers with the ca-pability of performing different operations (orperforming the classical ones faster). Lovelace,together with Babbage, was the progenitor of thisintellectual journey.

Although she published under a pseudonym,Lovelace enjoyed some recognition among herfriends. However, her troubled personal life in-terfered with further contributions. Lovelacebecame involved in several marital scandals, de-veloped addictions to gambling, alcohol, andopium, and suffered health problems. She wasafflicted with cancer, and its effects interferedwith her ability to concentrate. After severalyears of struggle, she died on November 27,1852, in London.

Lovelace might well have contributed moreto the foundations of computer science and ab-stract algebra if she had lived longer. Neverthe-less, her one published work presented the firstexplicit computer program (in a primitiveform), and she gave the mathematical justifica-tion for the analytical engine. Therefore, she isremembered for pioneering this branch of ab-stract algebra, which eventually would becomethe separate field of computer science in the20th century.

Further ReadingAngluin, D. “Lady Lovelace and the Analytical

Engine,” Association for Women in MathematicsNewsletter 6 (1976): 6–8.

Babbage, C. Passages from the Life of a Philosopher.London: Dawsons, 1968.

Morrison, P., and E. Morrison. Charles Babbage andHis Calculating Engines. New York: DoverPublications, 1961.

Stein, D. Ada: A Life and a Legacy. Cambridge, Mass.:MIT Press, 1985.

Toole, B. Ada, the Enchantress of Numbers: Prophet ofthe Computer Age, a Pathway to the 21st Century.Mill Valley, Calif.: Strawberry Press, 1998.

180 Lovelace, Augusta Ada Byron

Baum, J. The Calculating Passion of Ada Byron.Hamden, Conn.: Archon Books, 1986.

Elwin, M. Lord Byron’s Family: Annabelle, Ada, andAugusta, 1816–1824. London: J. Murray,1975.

Moore, D. Ada, Countess of Lovelace: Byron’s LegitimateDaughter. London: John Murray GeneralPublishing Division, 1977.

181

M� Maclaurin, Colin

(1698–1746)BritishCalculus, Geometry

England suffered a drought of mathematicians inthe latter portion of the 18th century, andMaclaurin represents the last great mathemati-cian of the Newtonian era. He contributed tothe defense of SIR ISAAC NEWTON’s calculus andmade some impressive discoveries of his own, in-cluding the series expansion of a function.

Colin Maclaurin was born in February 1698,in Kilmodan, Scotland. He was the youngest ofthree sons, and his father, John Maclaurin, wasthe local minister. Soon after Maclaurin was born,his father died, and his mother died when he wasnine. After the death of his parents, Maclaurinwas raised by his uncle. In 1709 Maclaurin en-tered the University of Glasgow, where he be-came interested in mathematics. He defendedhis thesis in 1715 On the Power of Gravity, ob-taining his master’s degree. In 1717 he was ap-pointed as professor of mathematics at MarischalCollege, although he was still quite young.

He visited London in 1719 and made theacquaintance of Newton, whose work made aprofound impression on him. By 1720 Maclaurinpublished his Geometrica organica (Organicgeometry), which contained many proofs of

Newton’s unproved results, as well as many ofMaclaurin’s own discoveries. His approach, likeNewton’s, was highly geometrical.

In 1722 Maclaurin left Scotland to serve astutor to the son of Lord Polwarth. They traveledin France, where Maclaurin continued his re-search, winning a prize from the FrenchAcademy of Sciences in 1724. In this year, hispupil died unexpectedly, and Maclaurin wasobliged to return to Scotland, where he obtainedthe chair of mathematics at the University ofEdinburgh through the intervention of Newton.There, Maclaurin lectured on EUCLID OF

ALEXANDRIA, spherical trigonometry, conic sec-tions, fortification, astronomy, and perspective.He was also one of the foremost expositors ofNewton’s calculus. In 1733 he married AnneStewart, with whom he raised seven children.

In 1742 Maclaurin published his Treatise onFluxions, which was a defense of Newton’s meth-ods and a response to the criticism of GeorgeBerkeley. Many scientists and mathematicianswere skeptical of infinitesimals, and Maclaurinundertook to provide the theory of fluxions witha rigorous logical foundation. It is noteworthythat Maclaurin rejected the advantageous no-tation of GOTTFRIED WILHELM VON LEIBNIZ in fa-vor of Newton’s clumsier nomenclature, due tohis loyalty and partisanship. As a result, theNewtonian style came to dominate thought in

England, and consequently impaired the com-putational and analytical abilities of subsequentmathematicians. In this sense, Maclaurin was tosome extent responsible for retarding mathe-matical progress in England.

The treatise contained the solutions of anumber of problems in geometry, statics, and in-finite series. It contained Maclaurin’s test forconvergence and, more important, the series ex-pansion of a differentiable function around theorigin. Although the Taylor series is more gen-eral, Maclaurin’s series was the first step in de-veloping an analytical tool of great utility.

Maclaurin also investigated bodies of at-traction, and he competed for a French prizewith his “On the Tides” in 1740, sharing theaward with LEONHARD EULER and DANIEL

BERNOULLI. He is also remarkable for being thefirst mathematician to correctly distinguish be-tween the maxima and minima of a function. Hewas a skilled experimentalist and inventor, andhe performed astronomical observations and ac-tuarial computations. In 1745 a Highland armymarched on Edinburgh, and Maclaurin vigor-ously organized the defense of the city. As a re-sult of his exertions, his health began to fail. Hedied on January 14, 1746, in Edinburgh.

Maclaurin was described as a benevolentand pious man, and in his controversies he waspolite to his adversaries. His defense ofNewtonian methods and notation partially ledto the English neglect of calculus, which de-prived England of good analysts for the next cen-tury. Nevertheless, he positively contributed tothe rigorous development of Newtonian calcu-lus, and he was easily one of the most talentedBritish mathematicians of his own day.

Further ReadingGiorello, G. “The ‘Fine Structure’ of Mathematical

Revolutions: Metaphysics, Legitimacy, and Rigour,The Case of the Calculus from Newton to Berkeleyand Maclaurin,” in Revolutions in Mathematics.New York: Oxford University Press, 1995.

Schlapp, R. “Colin Maclaurin: A Biographical Note,”Edinburgh Mathematical Notes 37 (1949): 1–6.

Turnbull, H. “Colin Maclaurin,” The AmericanMathematical Monthly 54 (1947).

� Madhava of Sangamagramma(ca. 1350–ca. 1425)IndianTrigonometry, Analysis

The great European mathematicians had manypredecessors within the Arabian, Indian, andChinese cultures, but the latter groups are nottypically acknowledged as the original discover-ers because their knowledge was not dissemi-nated as broadly. Madhava of Sangamagrammais an example of this, as he began the explo-ration of infinitesimal calculus and so-calledTaylor expansions two centuries before COLIN

MACLAURIN began his investigations. In thissense, Madhava must be recognized as the firstanalyst, even though his ideas did not blossomas fully in India.

None of the original works of Madhava re-main, and his life and mathematical contribu-tions must be reconstructed from the accountsof later Indian mathematicians. He was bornaround 1350 in Sangamagramma, in the state ofKerala, India. In about 1400 Madhava discov-ered the series expansion for various trigono-metric functions, such as the sine and cosine.These formulas are similar to the Taylor seriesthat were later discovered in Europe, and can beused to develop computable approximations tosines and cosines of angles.

Madhava came from a tradition of mathe-matics that emphasized finite procedures; thevery idea of an infinite sum of terms is a novelinnovation that completely departs from pre-ceding concepts of mathematics. Madhava ap-plied these series to trigonometry, developinghighly accurate tables for trigonometric values.In developing the infinite series for the arcsine

182 Madhava of Sangamagramma

function, Madhava was able to produce an ex-cellent approximation for pi, producing its valueto 11 decimal places. He also analyzed the re-mainder terms when the exact infinite series istruncated to a finite sum. Scholars believe thatMadhava used the method of continued frac-tions to derive these remainder terms.

Again, little is known of Madhava’s life, butit is thought that he died around 1425 in India.It is amazing that Madhava developed such tech-niques far before the Europeans did, given thatthey had the benefit of an intellectual progres-sion. In Europe the progress toward calculus canbe traced through several characters who eachparticipated in this mathematical process. InIndia the scholastic community was sparser, andthere was less of a concerted effort to producemathematics useful to science. Many historiansbelieve that Madhava’s discovery of infinite se-ries expansions is akin to a term-by-term inte-gration technique of calculus—a few hundredyears before the official discovery of calculus!

Further ReadingJoseph, G. The Crest of the Peacock: Non-European

Roots of Mathematics. London: Penguin Books,1990.

� Markov, Andrei(1856–1922)RussianProbability

The theory of probability, which had its foun-dation in the 17th-century studies of BLAISE PAS-CAL and PIERRE DE FERMAT, became one of themost important and influential mathematicalsubjects in the 20th century. The work of AndreiMarkov contributed some fundamental conceptsto the discipline of probability, and so-calledMarkov chains have been one of the mostwidely used probabilistic concepts in scienceand statistics.

Andrei Markov was born on June 14, 1856,in Ryazan, Russia. He graduated from SaintPetersburg University in 1878, and became aprofessor of mathematics there in 1886. His earlyresearch efforts focused on number theory andanalysis, dealing with such topics as continuedfractions and convergence of infinite series.

After 1900 Markov turned increasingly toprobability theory, wherein he would achieve hisgreatest work. Following in the footsteps of histeacher PAFNUTY LVOVICH CHEBYSHEV, Markovapplied his knowledge of continued fractions toprobability. He began the study of relationshipsbetween dependent random variables, whichwould become quite important to later work onstochastic processes. For example, Markov wasable to prove the central limit theorem, the mostimportant result of mathematical statistics, un-der more general assumptions on the depend-ence structure of the random variables beingsummed.

These results are of fundamental importanceto the study of time series, or chronologically or-dered data, where future values depend, in a sto-chastic manner, upon present and past data. Inparticular, Markov invented and studied Markovchains, which are essentially sequences of ran-dom variables where the probabilistic structure ofa future value only depends on its immediatepredecessor. This simple structure has sinceproved to be applicable to a variety of scientificproblems, while at the same time being mathe-matically tractable. The invention of Markovchains constitutes a first step in the study of sto-chastic processes, and so Markov is arguably thefounder of this important branch of probability.Later in the early 20th century, NORBERT WIENER

and Andrei Kolmogorov would generalizeMarkov’s early work on stochastic processes.

Markov died on July 20, 1922, in St.Petersburg, Russia. He represents an importantlink in the sequence of great Russian proba-bilists, including Chebyshev and Kolmogorov.Markov’s work is heavily cited in the theory of

Markov, Andrei 183

184 Menelaus of Alexandria

probability, and is now classical in its importanceand influence.

Further ReadingSheynin, O. “A. A. Markov’s Work on Probability,”

Archive for History of Exact Sciences 39 (1988):337–377.

� Menelaus of Alexandria(ca. 70–ca. 130)GreekGeometry

Menelaus of Alexandria, one of the later greatGreek mathematicians, is the founder of spher-ical trigonometry (the study of triangles definedon spheres). Both Earth and the heavens arespherical, so this subject is relevant to naviga-tion, geography, and the study of the calendar.By defining spherical triangles properly,Menelaus greatly advanced this subject, therebyalso advancing astronomy.

Only scraps of information are available onthe life of Menelaus, and only one of his workshas survived. Scholars believe that he was bornin approximately the year 70 in Alexandria,Egypt, and later spent much of his adult life inRome. Based on historical records, he made as-tronomical observation in Rome on January 14,98. Plutarch records a conversation betweenMenelaus and another man long after the year75 in Rome, concerning the reflection of light.These facts constitute the only evidence of hisactivity in Rome.

Menelaus wrote several books, includingSphaerica (The Book of Spherical Propositions), Onthe Knowledge of the Weights and Distribution ofDifferent Bodies, and The Book on the Triangle.Only the first of these has survived. He was thefirst to write down the definition of a sphericaltriangle as the figure enclosed by the intersectionof three great circles on a sphere (a great circleon a sphere is a circle of maximal diameter).

Menelaus proceeded in analogy with EUCLID OF

ALEXANDRIA’s treatment of plane geometry, andestablished many basic results. His success wasdue to his superior definition of a triangle, as pre-vious works utilized lesser circles. In fact, it is nowknown that great circles are geodesics, the equiv-alent of straight lines on a plane (they give theshortest path between two points). Hence, tri-angles should have sides determined by geodes-ics, and this is exactly how Menelaus proceeded.

In paralleling Euclid’s Elements, Menelausproved many propositions. It is interesting that herejected the reductio ad absurdum argument, whichinvolves an infinite chain of arguments leading toan absurdity. Menelaus instead used other tech-niques that he believed to be more rigorous, andhis treatment of the spherical trigonometry issomewhat more complete than that of Euclid forthe plane trigonometry.

The second part of Sphaerica gives the ap-plications of spherical trigonometry to astron-omy, and the third part presents Menelaus’stheorem, which was a generalization to spheri-cal trigonometry of a plane geometry result con-cerning the intersection of a line with the sidesof a triangle.

Menelaus’s Sphaerica comes to the modernreader through several Arab translators andcommentators and, unfortunately, their ac-counts of the book differ somewhat. Menelaus’sother works, listed above, were referenced byArabs such as Thabit ibn Qurra. Only fragmentsof the original remain. The Arab commentatorsalso mentioned Menelaus’s work on mechan-ics—apparently he studied the balances devisedby ARCHIMEDES OF SYRACUSE.

Scholars believe that Menelaus died around130. It seems that he was somewhat well knownas a mathematician in his own time, and thelater Arab mathematicians certainly referencedhim heavily. Menelaus’s most important contri-bution lies in his solid definition of spherical tri-angles, which allowed the field of astronomy toprogress further.

Minkowski, Hermann 185

Further ReadingHeath, T. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.Neugebauer, O. A History of Ancient Mathematical

Astronomy. New York: Springer-Verlag, 1975.

� Minkowski, Hermann(1864–1909)LithuanianGeometry, Analysis

Albert Einstein’s special theory of relativityposited space and time as a unified structure withits own geometry. The work of HermannMinkowski, building on the general geometricaltheories formulated by BERNHARD RIEMANN,formed the mathematical basis for this model ofthe universe.

Hermann Minkowski was born on June 22,1864, in Alexotas, part of the Russian Empire.Now the town is known as Kaunas, and is partof Lithuania. Minkowski pursued mathematicalstudies at the universities of Berlin andKönigsberg, receiving his doctorate from the lat-ter institution in 1885. Following graduation,Minkowski taught at several schools, includingBonn, Zurich, and Königsberg.

Minkowski accepted a chair at theUniversity of Göttingen in 1902, where hestayed for the remainder of his career. There helearned mathematical physics from DAVID

HILBERT, and filled in the rest of his scientific ed-ucation. His main contribution to mathematicsarrived through his realization that Einstein’swork in physics could be mathematically formu-lated as a non-Euclidean (that is, nonflat) spacethat could be completely described throughRiemann’s metric description of manifolds.Minkowski viewed time and space as a joint con-tinuum that could not be thought of as beingformally independent; the dependence of timeand space was developed through Einstein’s studyof special relativity, and Minkowski supplied

the appropriate geometrical construction thatillustrated this dependence. Minkowski’s four-di-mensional manifold was summarized by a four-dimensional space-time metric, later known asthe Lorentz metric. This space-time continuumis sometimes referred to as Minkowski space, inrecognition of his contributions to this field,which are summarized in his 1907 Space andTime.

In addition, Minkowski developed a four-di-mensional treatment of electrodynamics, ex-posited in his 1909 Zwei Abhandlungen über dieGrundgleichungen der Elektrodynamik (Two paperson the principal equations of electrodynamics).He is less well known for his work in pure math-ematics, to which he devoted most of his atten-tion. Minkowski investigated quadratic forms andcontinued fractions, and he discovered an im-portant inequality in analysis. He made originaldiscoveries on the geometry of numbers, which

Hermann Minkowski developed the geometricalfoundations of the special theory of relativity.(H.S. Lorentz, A. Einstein, H. Minkowski, DasRelatitätsprinzip, 1915, courtesy of AIP Emilio SegrèVisual Archives)

186 Möbius, August

led to the study of packing problems—the ques-tion of how many objects of a given shape can bepacked into a given space. Packing problems havebecome a significant area of research in the 20thcentury, due to their intuitive appeal and easilyrealizable applications.

Minkowski died on January 12, 1909, inGöttingen of a ruptured appendix. His primaryachievement lay in his foundation of the math-ematical study of packing problems, though heis more famous for his geometric contributionsto the theory of special relativity. These earlystudies of special relativity led to the promulga-tion of mathematical methods in the generaltheory of relativity, developed by AlbertEinstein.

Further ReadingHancock, H. Development of the Minkowski Geometry

of Numbers. New York: Dover Publications,1964.

Pyenson, L. “Hermann Minkowski and Einstein’sSpecial Theory of Relativity,” Archive for Historyof Exact Sciences 17, no. 1 (1977): 71–95.

� Möbius, August (August FerdinandMöbius)(1790–1868)GermanTopology, Geometry, Number Theory

August Möbius was an excellent mathematicianwho pioneered many ideas in topology, the studyof continuous maps acting on high-dimensionalsurfaces. This field of mathematics was studiedpiecemeal in the early 19th century, and indeedwould only receive systematic investigation byJULES-HENRI POINCARÉ, LUITZEN EGBERTUS JAN

BROUWER, and others in the early 20th century.Möbius’s research presented the first investiga-tions of orientation, one-sided surfaces, and ho-mogeneous coordinates.

August Möbius was born on November 17,1790, in Schulpforta, Germany. His father,Johann Heinrich Möbius, was a dance instruc-tor who died when Möbius was only three yearsold. He was raised by his mother, a descendantof Martin Luther, and was educated by her un-til he was 13. Möbius pursued further study atthe local college, and he matriculated at theUniversity of Leipzig in 1809.

At Leipzig Möbius followed his family’s pref-erence for him to study law, but after his firstyear abandoned this program to pursue mathe-matics, physics, and astronomy instead. ThereKarl Mollweide, an astronomer with mathemat-ical inclinations, influenced Möbius. In 1813 hetraveled to the University of Göttingen for grad-uate studies, and was taught by CARL FRIEDRICH

GAUSS himself. As a result of having this greatmentor, Möbius had a solid background in math-ematics and astronomy. In 1815 Möbius com-pleted his doctoral thesis, which concerned theoccultation of the fixed stars, and next com-menced his postdoctoral research. Although hiswork at this time was in the field of astronomy,it was highly mathematical in flavor.

Avoiding the possibility of being draftedinto the Prussian army, Möbius completed hissecond thesis on trigonometric equations, andhe was soon appointed as professor of astronomyat Leipzig in 1816. Möbius’s career advancementcame slowly, essentially due to his poor lectur-ing abilities, even though his mathematical workwas of high quality and originality.

Möbius worked quietly and steadily on a va-riety of mathematical projects, producing fin-ished works of great quality and completeness.Besides his papers on celestial mechanics and as-tronomical principles, Möbius wrote about pro-jective geometry, number theory, topology, andpolyhedra. His classic work on analytical geom-etry of 1827 introduced homogeneous coordi-nates (a way of describing projective surfaces)and the Möbius net (a certain configuration inprojective space). This research was foundational

Moivre, Abraham de 187

to more modern studies in projective geometry.The Möbius function and Möbius inversion for-mula are both significant in the study of primenumbers and factorization in number theory. Butin the budding field of topology Möbius demon-strated his creative genius, with innovative in-vestigations of one-sided surfaces and the topicof orientation (the determination of clockwiseversus counterclockwise directions upon a sur-face). In particular, he rediscovered the so-calledMöbius strip in 1858 (it had previously been ex-plored by Johann Listing). This object is essen-tially a twisted strip of paper that has only oneside.

In 1844 Möbius became a full professor atLeipzig. In the meantime he took on astronom-ical duties, overseeing the local observatory’s re-construction from 1818 to 1821. He married in1820, and had one daughter and two sons. Alsoin 1844 he interacted briefly with HERMANN

GUNTER GRASSMANN, whose work on topologyand algebraic geometry was quite similar to thatof Möbius. Möbius died on September 26, 1868,in Leipzig, Germany.

Möbius is perhaps most well known for theMöbius strip and the Möbius inversion formula,although his most important work was probablyin projective geometry. His work was distin-guished in its originality and cohesion, as wellas the depth of penetration into the material.

Further ReadingFauvel, J., R. Flood, and R. Wilson. Möbius and His

Band. Oxford: Oxford University Press, 1993.

� Moivre, Abraham de(1667–1754)FrenchProbability, Analysis, Statistics,Geometry

Abraham de Moivre was an influential Frenchmathematician who took some of the important

initial steps in probability and statistics. He wasa contemporary of SIR ISAAC NEWTON, and par-ticipated in the calculus priority debate. In ad-dition, he advanced analytic geometry andmade some elegant discoveries in complexanalysis.

Abraham de Moivre was born on May 26,1667, in Vitry, France, into a Protestant family,and later in life was persecuted for his religiousbeliefs. His early education was at a Protestantacademy at Sedan. In 1682 he studied logic atthe school of Saumur, and two years later cameto Paris to study mathematics at the Collège deHarcourt.

In 1685 the Edict of Nantes (a 1598 decreethat granted French Protestants the liberty toworship God as they pleased) was revoked, whichsignified a resumption of hostilities toward theHuguenots. De Moivre fled to England, where heunsuccessfully attempted to secure a position asa mathematics professor. Instead he became a pri-vate tutor—a profession he pursued until the endof his life. Meanwhile, de Moivre continued hisown private researches in the area of analyticgeometry, but made a more significant mark inthe field of probability. He studied basic gamesof chance, and from his work formulated the first,most basic version of the central limit theorem,easily the most important result of probabilityand statistics. The theorem states that sampleproportions (the proportion of times a certainevent is observed to occur within a determinednumber of repeated trials of measurements) areclose to the underlying probabilities that theyestimate, assuming that the amount of infor-mation is sufficiently large. For example, theproportion of heads observed in a sequence ofcoin tosses should approximate the true proba-bility of one-half. Moreover, the error in esti-mating a probability with a proportion can bequantified, having an approximately bell-shapeddistribution.

De Moivre’s work in probability was sum-marized in his 1718 book The Doctrine of Chance.

188 Monge, Gaspard

This work was well received by the scientificcommunity, and greatly advanced the knowledgeof probability and statistics. Generalizations ofhis first central limit theorem would later becomea keystone in the theory of statistical estima-tion—the central limit theorem would be usedto compute probabilities of statistics such as thesample mean. De Moivre first introduced theconcept of statistical independence, which hasbeen a crucial concept for statistical inferenceup to the present day. He explored his new con-cepts through several examples from dice games,but he also investigated mortality statistics andfounded actuarial science as a statistical subject.

His later Miscellanea Analytica (AnalyticalMiscellany) of 1730 contained the famousStirling formula, which gives an asymptotic ex-pression for n! 5 n (n21) (n22) . . . 1 for largeintegers n. This formula has been wrongly at-tributed to James Stirling, who generalized deMoivre’s original result. De Moivre used thisformula to derive the approximation of thebell-shaped distribution from the binomial dis-tribution.

De Moivre is also famous for his work incomplex analysis—he gives an expression for thehigher powers of certain trigonometric functions.In fact, an arbitrary complex number could beexpressed with trigonometric functions; thus, hewas able to connect trigonometry to analysis.

Despite his poverty and French origins, deMoivre was elected to the Royal Society in 1697,and in 1710 he was asked to adjudicate over aheated dispute between Newton and GOTTFRIED

WILHELM VON LEIBNIZ. Both men claimed to havebeen the original inventors of calculus, but dueto the lateness in their publishing and the dis-tance of their native countries (Newton wasBritish and Leibniz was German), there wassome confusion about which one had priority.De Moivre was already a friend of Newton, andhe was selected in order to prejudice the verdicttoward the English favorably; as expected, deMoivre ruled in favor of Newton.

De Moivre died in financial dearth onNovember 27, 1754, in London. Some said thathe predicted the date of his own death, havingobserved that his slumber was steadily length-ening by 15 minutes each night. He is an im-portant character in the history of mathematics,most especially for his pioneering work in prob-ability, statistics, and actuarial science. In theseareas he showed the most originality, but he wasan excellent all-around analyst, and his complexvariables formula is of classical importance to thesubject.

Further ReadingStigler, S. The History of Statistics: The Measurement

of Uncertainty before 1900. Cambridge, Mass.:Belknap Press of Harvard University Press, 1986.

Walker, H. “Abraham De Moivre,” Scripta Mathematica2 (1934): 316–333.

� Monge, Gaspard(1746–1818)FrenchGeometry

Gaspard Monge was an important mathemati-cian of the late 18th century who also played asignificant political role during the FrenchRevolution. He is considered to be the father ofdifferential geometry and was renowned for hiscreative intellect. Monge diverged from thestandard modes of mathematical thought andwas equally adept at theoretical and appliedproblems.

Gaspard Monge was born on May 9, 1746,in Beaune, France, to Jacques Monge, a mer-chant from southeastern France, and JeanneRousseaux, a native of the province of Burgundy.Raised in the same region, Monge attended theOratorian College, a school intended for youngnoblemen; here Monge received education inthe humanities, history, natural sciences, andmathematics. He first showed his brilliance at

Monge, Gaspard 189

this school, and in 1762 he continued his stud-ies at the Collège de la Trinité. A year later hewas placed in charge of a physics course, al-though he was only 17 at the time. By 1764 hewas finished with his education, and he returnedto Beaune to draw up a plan for the city.

His plan was recognized for its genius, andhe was appointed as a draftsman at the ÉcoleRoyale du Génie at Mézières in 1765. This postbrought him into contact with Charles Bossut,the professor of mathematics. Meanwhile,Monge was developing his own ideas aboutgeometry in private. The next year, he solved aproblem involving the construction of a fortifi-cation, and he utilized his geometrical ideas inthe solution. After this event, the faculty of theÉcole Royale du Génie recognized Monge’s abil-ities as a mathematician. In 1771 Monge readan important paper before the French Academyof Sciences. The work generalized certain resultsof CHRISTIAAN HUYGENS on space curves, and itwas favorably accepted by the academy.

In 1769 Monge replaced Bossut, who hadmoved away to Paris, and also received a posi-tion as an instructor in experimental physics. Hesought out the great Parisian mathematicians inan effort to advance his career, and throughMarie-Jean Condorcet’s assistance, he was ableto present to the Academy his research on thecalculus of variations, partial differential equa-tions, infinitesimal geometry, and combina-torics. During the next few years he continuedto contribute in the area of partial differentialequations, which he approached from a geo-metrical perspective. At this time his academicinterests expanded to include problems inphysics and chemistry.

In 1777 he married Catherine Huart, ownerof a forge, and researched metallurgy at the forge.Later he organized a chemistry laboratory at theÉcole Royale du Génie. In 1780 he held an ad-junct position at the Academy of Sciences, andeventually resigned his job at Mézières in 1784when he became the examiner of naval cadets.

During the next five years, he researched topicsin chemistry, the generation of curved surfaces,finite difference equations, partial differentialequations, and refraction, as well as a variety ofother scientific topics.

The French Revolution struck Paris in 1789,and Monge became deeply involved. He washighly sympathetic to the republican cause, al-though he became a staunch supporter ofBonaparte in the later years of his life. Mongewas involved in various societies that supportedthe Revolution, and when a republic was formedin 1792, Monge was appointed as minister of thenavy. His tenure was unsuccessful, largely due tothe fickle nature of the new republic, and he re-signed in 1793. He briefly returned to theAcademy of Sciences (until it was abolished),and played a prominent role in the founding ofthe École Polytechnique. During this time,Monge wrote papers on military topics, such asballistics and explosives, and gave courses inthese subjects. He trained future teachers, andhis lectures on geometry were later published inhis text Application de l’analyse à la géométrie(Application of analysis to geometry).

From 1796 to 1797 Monge was in Italy, over-seeing the plundering of Italian art by theFrench. While there he became acquainted withNapoleon Bonaparte, who exerted a tremendousinfluence on Monge through his superlativecharisma. After some time spent in Paris andRome, Monge accompanied Bonaparte on theill-fated Egyptian expedition. After the Frenchfleet was wiped out, Monge was appointed pres-ident of the Institut d’Egypte in Cairo in 1798.The mathematical division of the institute had12 members, which included Monge and JEAN-BAPTISTE-JOSEPH FOURIER.

In 1799 Monge returned to Paris withBonaparte, who soon held absolute power. Mongebecame director of the École Polytechnique, andafter the consulate was established, was ap-pointed a senator. Monge abandoned his repub-lican views when Bonaparte showered him with

line of curvature on a surface in a three-dimen-sional space. Besides this important theoreticalwork, he developed what came to be known asdescriptive geometry, which was essentially away of giving a graphical description of a solidobject. Modern mechanical drawing utilizesMonge’s method of orthographic projection. Hisfresh, nonstandard approach to geometry greatlystimulated the subject, and his impact on math-ematics has endured far longer than his politicaland pedagogical efforts.

Further ReadingBikerman, J. “Capillarity before Laplace: Clairaut,

Segner, Monge, Young,” Archive for History ofExact Sciences 18, no. 2 (1977/78): 103–122.

Coolidge, J. A History of Geometrical Methods. NewYork: Dover Publications, 1963.

190 Monge, Gaspard

honors—Monge was made into the Count ofPéluse in 1808. During this first decade of the19th century, Monge’s research activity in math-ematics tapered off as he focused more on ped-agogical concerns. Later, in 1809, his health de-clined. After the failure of Bonaparte’s Russianexpedition, Monge’s health collapsed, and he ul-timately fled before the emperor’s abdication in1814. Upon Bonaparte’s escape from Elba in1815, Monge rallied to his support, but afterWaterloo he fled the country. Monge returnedto France in 1816, but his life was difficult, ashis political enemies harassed him. He died inParis on July 28, 1818.

Monge is considered one of the principalfounders of differential geometry, through his pi-oneering work Application de l’analyse à lagéométrie. Here he introduces the concept of a

191

� Napier, John(1550–1617)BritishAnalysis

In a time of great mathematical ignorance,John Napier made an outstanding contributionthrough his discovery of the logarithm. Not onlydid this discovery provide an algorithm thatsimplified arithmetical computation, but it alsopresented a transcendental function that has fas-cinated mathematicians for centuries. Napierwas regarded as one of the most impressive in-tellects of his time, and his creative genius rankshim among the best mathematicians of all time.

John Napier was born in 1550, in MerchistonCastle of Edinburgh, Scotland. His father wasArchibald Napier, a Scottish aristocrat (he wasknighted in 1565), and his mother was JanetBothwell, the sister of the bishop of Orkney.John Napier received his early education athome, and began studying at St. Andrew’sUniversity in 1563. Soon afterward his motherdied. It was at St. Andrew’s that Napier becamepassionately intrigued by theology, which wouldremain an enduring interest throughout his life.Indeed, it is ironic that Napier regarded his prin-cipal contributions as theological, since theworld remembers him for his mathematicalwork.

Napier probably journeyed to Europe tocontinue his education, acquiring knowledge ofclassical literature and mathematics; no recordsexist, but it is likely that he studied at theUniversity of Paris. Napier returned to Scotlandby 1571 to attend his father’s remarriage. He alsowas married in 1573, and took up residence withhis wife in his family’s Gartness estate in 1574.Here Napier occupied himself with running hisestates, proving to be a talented inventor andinnovator in agricultural methods. A ferventProtestant, he was also active in the religiouscontroversies of the time. In 1593 he publisheda work that he considered to be his best: ThePlaine Discovery of the Whole Revelation of St.John. This work, written to combat the spreadof Roman Catholicism, gained Napier a reputa-tion on the Continent.

Napier did much of his work on logarithmswhile at Gartness. His purpose was to simplifymultiplications by transforming them into addi-tions. The result was Napier’s logarithm. Today,this logarithmic function has the property thatproducts are transformed into sums, and it is eas-ily one of the most important and useful math-ematical tools. Napier’s logarithm was slightly dif-ferent from the modern definition, since his wasmotivated by analogy with dynamics rather thanby pure algebra. Napier’s first public discourse onlogarithms appeared in Mirifici logarithmorum

N

192 Navier, Claude-Louis-Marie-Henri

canonis descriptio (Description of the marvelouscanon of logarithms) in 1614. One difficultywith his definition was that the logarithm of onewas not zero, which was desirable so that the log-arithm could be interpreted as the inverse of ex-ponentiation.

Henry Briggs, who read Napier’s Latin dis-course, communicated with him about makinglogarithms to have base 10; this would put a con-venient interpretation on logarithms, since ournumerical system involved 10 digits. If log x = y,then y is equal to the number of powers of 10needed to obtain x. The revised logarithm alsogave the log of one the value zero. Napier andBriggs worked in tandem on new logarithmictables through 1616, but the collaboration wasinterrupted by Napier’s death in 1617.

Besides his work on the logarithm, Napierwas also the inventor of a calculational aid

known as “Napier’s bones.” These bones wereactually ivory rods with numbers inscribed, andproducts could be read off by arranging the rodsin certain patterns. His great intellect and cre-ativity gave Napier the reputation among localsof being a warlock, especially since he used towalk about in his nightgown. The rumors gaverise to the belief that Napier was in league withthe devil.

Napier died on April 4, 1617, in Edinburgh,Scotland. His invention of the logarithm was agreat aid to later calculators and mathemati-cians, who were able to carry out multiplicationswith increased speed and accuracy. The loga-rithm later became an important building stonein the foundation of modern analysis, and hascontinued to be widely used by mathematicians.

Further ReadingLeybourn, W. The Art of Numbring by Speaking Rods,

Vulgarly Termed Nepeirs Bones [sic]. 1667 AnnArbor, Mich.: University Microfilms, 1968.

Smith, R. Teacher’s Guide to Napier’s Bones. Burlington,N.C.: Carolina Biological Supply Company, 1995.

� Navier, Claude-Louis-Marie-Henri(1785–1836)FrenchMechanics, Differential Equations

Claude Navier is most famous today for theNavier-Stokes equations, which describe the dy-namics of an incompressible fluid. He is respon-sible for introducing analytical techniques intocivil engineering, and the cross-fertilization be-tween mathematics and engineering that Navierinitiated benefited both disciplines.

Claude Navier was born on February 10,1785, in Dijon, France. He was raised amidst thefuror of the French Revolution. His father wasa lawyer and a member of the NationalAssembly. Navier’s father died when he was onlyeight years old, and his mother retired to the

John Napier, inventor of logarithms (Courtesy of theLibrary of Congress)

Navier, Claude-Louis-Marie-Henri 193

countryside, leaving Navier in Paris in the careof his great-uncle Emiland Gauthey. Gautheywas the most renowned civil engineer of thattime, and he may have exerted some influenceon the young Navier.

Navier entered the École Polytechnique in1802, barely making the entrance requirementsdue to his scholastic mediocrity. However, hemade tremendous progress in his first year, ris-ing to the top 10 of his class. One of Navier’steachers was JEAN-BAPTISTE-JOSEPH FOURIER, whohad a significant impact on Navier’s mathemat-ical development. The two men remained life-long friends. In 1804 Navier matriculated at theÉcole des Ponts et Chaussées; he graduated twoyears later. When Emiland Gauthey died soonafterward, the Corps des Ponts et Chausséesasked Navier to edit his great-uncle’s collectedworks. In doing so, Navier gained an apprecia-tion for civil engineering as an application ofmechanics, and he inserted many elements ofmathematical analysis into Gauthey’s writings.

During the next decade Navier became rec-ognized as a leading scholar on engineering sci-ence, and he took a position at the École des Pontset Chaussées in 1819. Navier placed a strong em-phasis on the mathematical and analytical foun-dation of engineering, and this was evident in histeaching style. He was named professor there in1830, and replaced AUGUSTIN-LOUIS CAUCHY atthe École Polytechnique in 1831.

Navier had special expertise in buildingbridges. Traditionally, bridges were built onempirical principles, but Navier developed amathematical theory for suspension bridges. Heattempted to test his ideas by building a sus-pension bridge over the Seine, but the munici-pal council countered his efforts and eventuallydismantled his partially completed bridge.

During his lifetime, Navier was recognizedas a leading civil engineer, but he is famous to-day for his pioneering mathematical work onfluid mechanics. Navier worked on appliedmathematical problems, such as elasticity, the

motion of fluids, and applications of Fourier se-ries to engineering questions. In 1821 he gavethe Navier-Stokes differential equations for themotion of incompressible fluids. Today his deri-vation is known to be incorrect, as he neglectedto consider the effect of shear forces; neverthe-less, his equations were, providentially, correct.

Navier was not especially active in politics,though he favored a socialist position, aligninghimself with social philosophers such as AugusteComte. Navier believed in the power of scienceand technology to solve societal problems. AsFranz Kafka later pointed out, the mechaniza-tion of society addresses materialistic concernswhile leaving humankind spiritually alienated;from today’s perspective, Navier’s positivist phi-losophy seems naive. Navier did oppose thepropagation of violence through the militarycomplex, and in particular withstood the war-mongering of Napoleon.

Navier received many honors in his life, in-cluding being elected to the Academy ofSciences in 1824. From 1830 onward, Navierworked as a government consultant on how sci-ence and technology could be used to improvethe country. He died on August 21, 1836, inParis, France. His most important contributionlies in the Navier-Stokes equations for fluid flow,which were heavily studied in physics and engi-neering, and applied in many technical arenas.Fluid mechanics is certainly one of the most dif-ficult branches of applied mathematics, and isstill not completely understood. Navier mustalso be remembered for his introduction of math-ematics and physics to civil engineering, whichresulted in a more modern, effective science.

Further ReadingAnderson, J. A History of Aerodynamics and Its Impact

on Flying Machines. Cambridge, U.K.: CambridgeUniversity Press, 1997.

Picon, A. “Navier and the Introduction of SuspensionBridges in France,” Construction History 4 (1988):21–34.

194 Newton, Sir Isaac

� Newton, Sir Isaac(1643–1727)BritishCalculus, Geometry, Mechanics,Differential Equations

Sir Isaac Newton may well have been the great-est scientist of Western civilization. He madeoutstanding contributions to optics, mechanics,gravitation, and astronomy, using his newly dis-covered “method of fluxions”—a geometric formof differential calculus—to support his originalconclusions. Newton not only unified manybranches of physics through the umbrella of hisdifferential calculus, which provided a quantita-tive tool of great power to explain physical phe-nomena, but he also made amazing discoveriesthat revolutionized the way scientists under-stood the natural world. His intellect was highlycreative, and his genius was able to see to theheart of a scientific problem to provide a noveland viable explanation.

Isaac Newton was born on January 4, 1643,in Woolsthorpe, England. His father, also namedIsaac Newton, came from a long line of wealthyfarmers. He died a few months before the birthof his son. Young Isaac’s mother, HannahAyscough, soon remarried, and this resulted inan unhappy childhood for the young boy—hewas essentially treated as an orphan. Newton wassent to live with his maternal grandparents, andapparently he disliked his grandfather. He wasalso embittered toward his mother, and laterthreatened to burn her together with her newhusband. Newton was of a volatile temper, proneto fits of capricious rage.

When Newton’s stepfather died in 1653, helived with his mother, half-brother, and two half-sisters for a time. At this time he began attend-ing the Free Grammar School in Grantham, butthe initial reports indicated that Newton was apoor, inattentive student. His mother inflicted ahiatus on Newton’s education, bringing himhome to manage her estate. This was anotherfailure, as Newton had no interest in businessaffairs, and he resumed his education in 1660.Having shown a bit more promise in his latteryears at Grantham, Newton was allowed to pur-sue further university education. Due to hismother’s interference, he was older and less pre-pared than most students at Trinity College ofCambridge in 1661.

Despite his mother’s extensive propertyand wealth, Newton became a sizar atCambridge—a servant of the other students.He pursued a degree in law and became famil-iar with the classical philosophy of Aristotleand Plato, as well as the more modern ideas ofRENÉ DESCARTES. Newton’s personal scientificjournal, Certain Philosophical Questions, revealsthe early formulation of his most profoundideas. Newton’s passion for scientific truth (inthose days, scientific inquiry was dogmaticallyrestricted to the development of Aristotle’sideas) played a role in his remarkable profun-dity as a thinker.

Sir Isaac Newton, cofounder of calculus and oneof the greatest scientists ever, studied optics, gravity,mechanics, and astronomy. (Courtesy of the Libraryof Congress)

Newton, Sir Isaac 195

Newton’s interest in mathematics developedlater; the story is recounted that in 1663 he pur-chased an astrology text that was incomprehen-sible to him. This experience prompted Newtonto pursue geometry, commencing with EUCLID OF

ALEXANDRIA’s Elements. Next he devoured thegeometrical works of Descartes and FRANÇOIS

VIÈTE, as well as JOHN WALLIS’s Algebra. In 1663ISAAC BARROW took up a professorship at Cambri-dge, and there exerted some influence over theyoung Newton. However, Newton’s genius didnot emerge at this time. This flowering of his in-tellect occurred after his 1665 graduation, whenhe returned home for the summer to escape aplague that shut Cambridge down. During thenext two years at Lincolnshire, Newton achievedtremendous scientific breakthroughs in mathe-matics, physics, optics, and astronomy.

Newton’s development of differential calcu-lus took place at this time, well before GOTTFRIED

WILHELM VON LEIBNIZ made similar, independentdiscoveries. Newton called his calculus the“method of fluxions,” and it was very geometric(whereas Leibnizian calculus is more algebraic);central to Newton’s technique was the realiza-tion of differentiation and integration as inverseprocesses. His method was able to solve manyclassical problems in a more elegant, unifiedfashion, while also being capable of solvingwholly new problems unapproachable by oldermethodologies. The result of his labors, DeMethodis Serierum et Fluxionum (On the meth-ods of series and fluxions), was not publisheduntil 1736, many years after his death.

Cambridge was reopened in 1667 after ces-sation of the plague, and Newton obtained a mi-nor fellowship, shortly afterward obtaining hismaster’s degree and the Lucasian chair in 1669,which had been freshly vacated by Barrow.Barrow perused much of Newton’s work, andhelped to disseminate the novel ideas. He alsoassisted Newton in obtaining his new position.In 1670 Newton turned to optics, advancing aparticle theory of light as well as the notion that

white light was actually composed of a spectrumof different colors. Both of these ideas wentcounter to existing beliefs on the nature of light,but they were still well received. Newton waselected to the Royal Society in 1672 after do-nating a reflecting telescope of his own inven-tion. Some controversy erupted over his ideas,and Newton did not handle the criticism well.Throughout his life, he experienced a tensionbetween his desire to publish and gain recogni-tion for his genius, and his loathing of the bick-ering and politics of the academic arena.

In 1678 Newton suffered a nervous break-down, probably the result of overwork combinedwith the stress of his scholastic debates. The fol-lowing year his mother died, and Newton re-moved himself even further from society. Hiswork in gravity and celestial mechanics hasgained him the most renown, and his early ideason these topics date back to 1666. Based on hisown law of centrifugal force and Kepler’s thirdlaw of planetary motion, Newton was able todeduce his inverse square law for the force ofgravity between two objects. This work waspublished in the Philosophiae naturalis principiamathematica (Mathematical principles of naturalphilosophy) in 1687, commonly known as thePrincipia. This work is thought by many to bethe greatest scientific treatise of all time: It pres-ents an analysis of centripetal forces, with ap-plications to projectiles and pendulums. Newtondemonstrated the inverse square law for gravita-tional force and formulated the general (anduniversal) principle of gravity as a fundamentalartifact of our universe. Centuries later, Einsteinwould describe the force of gravity as the fabricof space itself. Of course, the Principia was anenormous success, and Newton became the mostfamous scientist of his age.

In the latter portion of his life, Newton fellaway from scientific research and becameinvolved in government. When James I at-tempted to appoint underqualified Catholics touniversity professorships, Newton (who was a

196 Noether, Emmy

fervent Protestant) withstood him publicly.After the king’s deposition in 1689, Newton waselected to Parliament and recognized as an aca-demic hero. He suffered a second emotionalbreakdown in 1693, perhaps brought on bylaboratory chemicals, and officially retired fromresearch. He became warden of the mint in 1696and master of the mint in 1699, and in thiscapacity he worked diligently to prevent coun-terfeiting of the new coinage.

Scientists and mathematicians throughoutEurope recognized Newton as one of the great-est intellects, although the controversy withLeibniz that erupted over the issue of priority inthe invention of calculus detracted from his in-tellectual reign. Much of his energy was devotedto this protracted debate, carried on by the dis-ciples of both men, and certainly Newton’sferocious temperament was conspicuous in histreatment of his competitor. (The details of thisargument are given in the biography of Leibniz.)Newton held the presidency of the Royal Societyfrom 1703 until his death, and he also holds thedistinction of being the first Englishman to beknighted (in 1705 by Queen Anne) for scien-tific achievements. He died on March 21, 1727,in London, England.

It is hard to overestimate the importance ofNewton’s work, so great has its impact on sci-ence and mathematics been. It must be under-stood that his invention of calculus was born outof a long progression of intellectual endeavor bysuch figures as ARCHIMEDES OF SYRACUSE, BLAISE

PASCAL, and Wallis, among others. However,Newton’s voice rang like a clarion call througha babble of disorganized and incoherent voices;his calculus not only provided a general systemthat revealed prior methodologies to be varia-tions on a theme, but also was a practical andpowerful tool capable of tackling thorny scien-tific questions. His calculus placed the sciencesfurther under the shadow of mathematics andquantitative reasoning, and thus increased theprecision and rigor of physics and astronomy. For

his mathematics alone, Newton would havebeen considered one of the greatest minds of hu-man history.

Further ReadingBechler, Z. Newton’s Physics and the Conceptual

Structure of the Scientific Revolution. Boston:Kluwer Academic Publishers, 1991.

Brewster, D. Memoirs of the Life, Writings, and Discoveriesof Sir Isaac Newton. New York: Johnson ReprintCorp., 1965.

Chandrasekhar, S. Newton’s “Principia” for the CommonReader. New York: Oxford University Press, 1995.

Christianson, G. In the Presence of the Creator: IsaacNewton and His Times. New York: Free Press,1984.

Cohen, I. Introduction to Newton’s “Principia.”Cambridge, U.K.: Cambridge University Press,1978.

Gjertsen, D. The Newton Handbook. New York:Routledge & Kegan Paul, 1986.

Hall, A. Isaac Newton, Adventurer in Thought. Oxford:Blackwell, 1992.

Meli, D. Equivalence and Priority: Newton versus Leibniz;Including Leibniz’s Unpublished Manuscripts on the“Principia.” Oxford: Clarendon Press, 1993.

Turnbull, H. The Mathematical Discoveries of Newton.London: Blackie and Son Ltd., 1945.

Westfall, R. Never at Rest: A Biography of Isaac Newton.New York: Cambridge University Press, 1980.

———. The Life of Isaac Newton. New York:Cambridge University Press, 1993.

� Noether, Emmy (Amalie Emmy Noether)(1882–1935)GermanAlgebra

Emmy Noether was an exceptional mathemati-cian who was able to overcome gender andethnic obstacles to make outstanding contribu-tions to abstract algebra. She is best known for

Noether, Emmy 197

her early work on ring theory. Her results onideals in rings were instrumental to the later de-velopment of modern algebra.

Emmy Noether was born on March 23,1882, in Erlangen, a town in the Germanprovince of Bavaria, to Max Noether, a notablemathematician at the University of Erlangen,and Ida Kaufmann. Both of her parents were ofJewish descent, which would later be a source ofpersecution for Noether. Emmy Noether was theeldest of four children—her younger siblingswere all boys.

Noether studied at the Höhere TöchterSchule in Erlangen from 1889 until 1897, whereshe studied languages and arithmetic. She orig-inally intended to become a language teacher,and became certified in 1900 to teach Englishand French in Bavarian schools. However,Noether instead pursued the difficult path ofmathematics, and began attending lectures atthe University of Erlangen in 1900. Womenwere allowed to study only unofficially, andNoether had to obtain permission to attendclasses. She also studied at Göttingen underDAVID HILBERT and FELIX KLEIN.

In 1904 Noether was allowed to matricu-late at Erlangen, and she obtained her doctor-ate in 1907 under the direction of Paul Gordan.Her thesis constructed several algebraic invari-ants, which was a constructive approach toHilbert’s basis theorem of 1888. Unable toprogress further in an academic career due toher gender, Noether spent the next few years as-sisting her father in his research. She also turnedtoward Hilbert’s more abstract approach to al-gebra, and made many contributions of her own.Gradually she gained recognition from themathematical community through her publica-tions, and in 1915 Hilbert and Klein invited herto Göttingen as a lecturer. It is a testimony toher talent that Hilbert and Klein fought longand hard with the university administration togrant Noether a position, which was finally ob-tained in 1919.

Noether’s first work in Göttingen was a the-orem of theoretical physics—sometimes referredto as Noether’s theorem—which relates particlesymmetries to conservation principles. AlbertEinstein later praised this contribution to gen-eral relativity for its penetration and value. After1919 Noether shifted from invariant theory toideals, which are certain special subsets of rings,a generalization of Euclidean space viewed froman algebraic perspective. One of her most im-portant papers, published in 1921, gave a fun-damental decomposition for these ideals. Herwork on ring theory was of great significance tolater developments in modern algebra; in 1927Noether investigated noncommutative rings(rings in which the commutative law does nothold). These algebraic spaces have become veryimportant for theoretical physics, where the in-teractions between particles follow noncommu-tative laws.

For her outstanding work, Noether receivedmuch recognition; in 1932 she shared the AlfredAckermann-Teubner Memorial Prize for theAdvancement of Mathematical Knowledge withEmil Artin. However, her Jewish ethnicity madeher a target of Nazi prejudice in 1933, and shewas forced to flee to the United States, whereshe lectured at the Institute for Advanced Studyat Princeton.

Noether died on April 14, 1935, in BrynMawr, Pennsylvania. Her colleagues recognizedher as an exceptional mathematician who didmuch to advance the knowledge of algebra. Herfundamental results in the theory of rings andinvariants left an enduring legacy in abstractalgebra, and her success in the presence of dis-crimination and persecution testifies to her spir-ited determination and firm character.

Further ReadingByers, N. “The Life and Times of Emmy Noether;

Contributions of E. Noether to Particle Physics,”in History of Original Ideas and Basic Discoveriesin Particle Physics. New York: Plenum, 1996.

198 Noether, Emmy

———. “E. Noether’s Discovery of the DeepConnection between Symmetries and Conserva-tion Laws,” Israel Mathematical ConferenceProceedings 12 (1999).

Dick, A. Emmy Noether, 1882–1935. Boston:Birkhäuser, 1981.

Kimberling, C. “Emmy Noether and Her Influence,”in Emmy Noether: A Tribute to Her Life and Work.New York: Marcel Dekker, 1981.

Srinivasan, B., and J. Sally. Emmy Noether in BrynMawr. New York: Springer-Verlag, 1983.

199

as royal chaplain to the king in 1364. From 1370onward he resided in Paris and busied himselfadvising the king and translating several ofAristotle’s works into French. Oresme chal-lenged some of Aristotle’s revered notions,redefining time and space in terms closer to whatis accepted today.

Oresme contributed to mathematics throughthe invention of a type of coordinate geometrythat traced the relationship between a table ofpaired values (such as for a function’s inde-pendent and dependent variables) and a two-dimensional plot. This concept anticipatesDescartes’s more sophisticated analytic geome-try by three centuries; it is likely that Descarteswas familiar with Oresme’s widely read Tractatusde configurationibus qualitatum et motuum(Treatise on the configurations of qualities andmotions).

Oresme was also the first mathematician touse the fractional exponent (though in a differ-ent notation), and made primitive investigationsinto infinite series. He posited the questionwhether the ratio of periods of two heavenlybodies could be an irrational number; this fairlydeep question of irrational periodicities has beenexplored in the nonlinear dynamical studies ofthe 20th century. Besides these mathematicaldiscoveries, Oresme also did some initial think-ing (although he did not completely formulate

O� Oresme, Nicole (Nicole d’Oresme,

Nicholas Oresme)(1323–1382)FrenchGeometry

Nicole Oresme was an excellent all-aroundscholar who proposed several ideas, in a primi-tive form, centuries before the persons to whomthey are usually credited. In particular, he was apredecessor to RENÉ DESCARTES in his graphicaldepiction of functional relationships.

Oresme was born in 1323 in Allemagne,France. There is no information about his earlylife, but it is known that he was of Norman an-cestry. He attended the University of Paris in the1340s, studying the arts under the philosopherJean Buridan. This teacher encouraged Oresme’sinterests in natural philosophy and pushed himto question the ideas of Aristotle.

Oresme later earned a degree in theology atthe College of Navarre in 1348, and he becamea master in theology in 1355. This led to his ap-pointment as grand master of the College ofNavarre in 1356. During this time he befriendedthe future king Charles V. This friendship wasborn of similar intellectual interests, and con-tinued through both of their lives.

Oresme took on increasingly prestigious re-ligious positions, culminating in his appointment

200 Oresme, Nicole

Further ReadingClagett, M. Nicole Oresme and the Medieval Geometry

of Quantities and Motions. Madison: University ofWisconsin Press, 1968.

Coopland, G. Nicole Oresme and the Astrologers: AStudy of His “Livre de Divinacions.” Liverpool:The University Press, 1952.

Grant, E. “Nicole Oresme and the Commensurabilityor Incommensurability of Celestial Motions,”Archive for History of Exact Sciences 1 (1961).

———. “Nicole Oresme and His De ProportionibusProportionum,” Isis 51 (1960): 293–314.

a theory) on scientific problems, proposing thelaw of freefall, the rotation of the Earth, and thestructural theory of chemical compounds. Afterbeing appointed a bishop in 1377, Oresme diedon July 11, 1382, in Lisieux, France.

Oresme was an excellent scholar of the 14thcentury who made several innovative scientificand philosophic advances. In mathematics, heinvented coordinate geometry, which was a steptoward the full theory of analytic geometry pop-ularized in the 17th century. His widely readworks probably influenced later mathematicians.

201

� Pappus of Alexandria(ca. 290–ca. 350)GreekGeometry

Pappus of Alexandria is the last of the greatGreek mathematicians. He is principally knownfor considering certain geometrical questionsthat blossomed into the field of projective geom-etry. Virtually nothing is known of his personallife.

Pappus of Alexandria was born in approxi-mately 290 in Alexandria, Egypt. The ancientsources describing the dates of his activity are inconflict, but the consensus of scholars indicatesthat he was active from 284 to 305, during thereign of the emperor Diocletian. However, thismay also be too early, as it is now known thatPappus’s Almagest was written after 320. Hisdeath is conjectured to have occurred around350 in Alexandria.

Apparently Pappus lived in Alexandria all ofhis life. He had a family, for he dedicates one ofhis books to his son Hermodorus. Pappus also dis-cusses his philosopher friend Heirius, and it seemsthat Pappus headed a school in Alexandria.

Pappus’s major work on geometry is calledthe Mathematical Collection and is thought tohave been written around 340. A handbook ofgeometry designed to revive interest in the

classical works, this volume is divided into sev-eral books. Book I, on arithmetic, is lost, andBook II treats APOLLONIUS OF PERGA’s notationfor expressing large numbers. Book II discussesthe harmonic, geometric, and arithmetic meansand their accompanying constructions, as wellas some geometric paradoxes. In Book IV Pappustreats some special curves, such as the spiral andquadratrix. He divides geometric problems intoplane, solid, and linear problems. Book V de-scribes the construction of honeycombs by beesand the optimality of the circle for enclosingmaximal area with a minimal perimeter. He re-views the 13 semiregular solids of ARCHIMEDES

OF SYRACUSE and proves results relating sur-face area and volume for several types of solids.Book VI considers astronomy, reviewing theworks of EUCLID OF ALEXANDRIA, ERATOSTHENES

OF CYRENE, and Apollonius.Book VII contains the “Treasury of

Analysis,” in which Pappus sets out the methodof analysis and synthesis encountered in theclassic works of Euclid and others. He describesanalysis as a breakdown of a problem intosimpler, related problems; these are then syn-thesized into the final solution. This method ofthought was distinctively Greek, and was latermastered by the European mathematicians whostudied the classics. Pappus also lays out theso-called Pappus problem, which has greatly

P

202 Pascal, Blaise

influenced the evolution of geometry—RENÉ

DESCARTES and SIR ISAAC NEWTON later discussedthis topic of geometry. Book VIII treats me-chanics, which Pappus defines as the study ofmotion and force. The work as a whole demon-strates Pappus’s mastery of many mathematicalsciences, and his exposition is fairly good.

Besides the Mathematical Collection, Pappuswrote several commentaries of variable quality,including those on CLAUDIUS PTOLEMY’s Almagestand Euclid’s Elements. Pappus also wrote a workon geography, and he may well have writtenabout music and hydrostatics, but the primarysources have not survived.

Pappus was influential on later Europeanmathematicians, since he gave an insightfuloverview of all the older Greek mathematicalworks. After reading Pappus, a mathematiciancould track down the original sources of suchgreat mathematicians as Euclid and Archimedes;in this sense Pappus was quite influential. Hisown mathematical discoveries seem limited,though the Pappus problem can be viewed as thefoundation of projective geometry.

Further ReadingBulmer-Thomas, I. Selections Illustrating the History of

Greek Mathematics. Cambridge, Mass.: HarvardUniversity Press, 1980.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

Neugebauer, O. A History of Ancient MathematicalAstronomy. New York: Springer-Verlag, 1975.

Tahir, H. “Pappus and Mathematical Induction,” TheAustralian Mathematical Society Gazette 22, no. 4(1995): 166–167.

� Pascal, Blaise(1623–1662)FrenchGeometry, Probability, Mechanics,Calculus

Blaise Pascal is famous for his brilliant founda-tional work in probability, geometry, and hydro-statics, as well as for his insightful thoughts onphilosophy and religion. Pascal’s work in themathematics of gambling, along with that ofPIERRE DE FERMAT, laid the basis for modern the-ory of probability and statistics and sparked amovement in western Europe toward a “sto-chasticized” society. His labors in the arena ofhydrostatics were groundbreaking, providingmuch of the theory behind modern hydraulicstechnology, while his efforts in Christian apolo-getics are notable for their clarity of thought andinsight into human nature.

Pascal was born on June 19, 1623, inClermont (now Clermont-Ferrand) in theAuvergne region of France. Blaise was the third

Blaise Pascal, a geometer who formulated earlynotions of calculus, is also viewed as a founder ofthe theory of probability. (Courtesy of the Library ofCongress)

Pascal, Blaise 203

child of Étienne Pascal, a mathematician, whoeducated his only son himself. AntoinetteBegon, his mother, died when Pascal was onlythree years old. Interestingly, the young Pascalwas not permitted to study mathematics untilage 12, when he began reading EUCLID OF

ALEXANDRIA’s Elements. However, even beforethis time, the precocious child was investigatinggeometry on his own.

Blaise would accompany his father to meet-ings held in Paris by Marin Mersenne, a priestwho greatly promoted the spread and communi-cation of mathematics. In this context Pascalfurther developed his mathematical abilities,being influenced by the thought of GIRARD

DESARGUES. Pascal soon became Desargues’smain disciple in the study of geometry, and inJune 1639 he discovered the “mystic hexagram.”He had found that the opposite sides of a hexa-gon inscribed in a conic section form threepoints that are collinear.

In December 1639 the Pascal family movedto Rouen, where Étienne had an appointmentas tax collector. In 1640 Blaise published his firstwork, Essay on Conic Sections, an outline of atreatise on conics. Soon afterward, in 1642, hebegan an attempt to mechanize addition andsubtraction, in order to assist his father with hisaccounting calculations. By 1645 Pascal hadcompleted the construction of the first digitalcalculator (although Wilhelm Schickard haddesigned an earlier prototype in 1623, it was notmanufactured). The device, although unsuc-cessful financially due to the expense of con-struction, was quite similar to a mechanicalcalculator of the 1940s. After several experi-ments in atmospheric pressure, Pascal concludedthat as altitude increases the pressure of airdecreases, and that a vacuum exists above theatmosphere. Although true, these findings, pub-lished in 1647 as New Experiments ConcerningVacuums, were controversial in the scientificcommunity, and there was some debate overwho had priority in the discoveries, as several

scientists were researching hydrostatics. Hydro-statics is the study of fluids at rest and the pres-sures they exert, and Pascal’s 1654 Treatise on theEquilibrium of Liquids gave a rigorous account ofthe topic. This treatise clearly demonstrated theeffects of the weight of the air, as well as severallaws of hydrostatics, including Pascal’s law ofpressure. This principle, which states that fluidin a closed vessel transmits pressure undiminished(or, in other words, the fluid is incompressible),is the basis of the hydraulic press—essentially atype of lever. His treatment gave a synthesis ofprior knowledge and new work, and lucidly pre-sented the concept of pressure.

The young Pascal had been interested in re-ligion since 1646, and when his father died in1651, he became deeply contemplative aboutspiritual matters. His ideas would later be pub-lished in his philosophical work Pensées deMonsieur Pascal sur la religion et sur quelques autressujets (Monsieur Pascal’s thoughts on religionand some other subjects) of 1670. His work onprojective geometry, the mathematical study ofperspective, led to The Generation of ConicSections (1654). Conic section is the name for acurve obtained by slicing a cone by a plane atcertain angles. This great work dealt with theprojective generation of conics and their prop-erties, the definition of the mystic hexagram,and the projective theory of centers and diame-ters. In addition, his Treatise on the ArithmeticalTriangle appeared in the same year, dealing withthe so-called Pascal’s triangle, a triangle of num-bers, in which each entry is obtained by sum-ming the two entries above it. Although he didnot invent the arithmetical triangle, his workwas quite influential on the development of thegeneral binomial theorem.

In 1654 Pascal was working on some gam-bling problems with Fermat. The two main ques-tions they investigated were, the problem of thedice, calculating the probability of obtaining apair of sixes in a given number of throws; andthe problem of the stakes, concerned with how

204 Pascal, Blaise

to divide the pot fairly to the players if a gameof chance is interrupted. In formulating thesetypes of problems, Pascal became a foundingfather of the Western theory of probability. Inthe succeeding centuries, Western culture wouldbecome increasingly quantitative, embracing astochastic (i.e., relating to probability andchance) approach to phenomena; it became ap-parent, after these humble origins, that reliableinformation could be obtained from uncertainevents, as long as a large number of repeated tri-als could be performed and measured. Pascalworked on a calculus of probabilities, using in-ductive reasoning to find solutions. His work ingames seems to have affected Pascal’s view ofChristian apologetics (rhetorical or rationaldefenses of a belief), since Pensées includes thefamous “Pascal’s wager”:

If God does not exist, one will lose nothing bybelieving in him, while if he does exist, onewill lose everything by not believing.

Indeed, Pascal’s approach to probability fore-shadows the modern theory of decision, in whichchoice is intimately connected with the proba-bility of uncertain events; one can see Pascal’sapologetic method as a classic problem in deci-sion theory.

Pascal had long suffered from ill health(indigestion and constant headaches), beingsickly from his youth, and he had not long tolive. In 1654 he was drawn more deeply intoreligious concerns; on the night of November23, designated as his “night of fire,” he experi-enced a second conversion to Christianity; fromthis time, he would turn away from science andmathematics toward religion and epistemology(the study of knowledge and belief structures).In 1656 and 1657 he composed his Lettres provin-ciales (Provincial letters), a Jansenist polemicagainst the Jesuits. They were published anony-mously, and it is said that their rigor of thoughtand clarity of presentation dealt a wound to

Jesuitism from which it has never recovered. Hisdefense of Christianity to unbelievers, formu-lated in the Pensées, was written at this time.

Encouraged even by his Jansenist friends,Pascal did some final work in mathematics. In1657 he prepared the Elements of Geometry,which unfortunately was not completed. His lastwork in 1658 and 1659 was on the cycloid, acurve traced out by the path of a marked pointon the circumference of a rolling circle. In hisinvestigations of this curve, he developed the“theory of indivisibles,” which was a forerunnerof the integral calculus soon formulated bySIR ISAAC NEWTON and GOTTFRIED WILHELM VON

LEIBNIZ. Pascal considered such problems as cal-culating areas under curves, centers of gravity forsurfaces, and volumes beneath surfaces of revo-lution (the surface obtained by rotating a curveabout a fixed axis). Interestingly, it seems thatthis work developed over the course of severalpublic mathematical contests, in which Pascalposed calculus problems to the community.

In 1659 Pascal fell gravely ill and sought soli-tude, devoting himself to charitable works. His lastproject was the development of a public trans-portation project in Paris that involved horse-drawn carriages. He died at age 39 in great pain,on August 19, 1662, in Paris. For his contributionsto mathematics, as well as physics and religion,Pascal ranks as one of the greatest intellects of theWest. It seems that his soul was torn between prideover his intellectual abilities and accomplishmentsand self-denial of an austere Augustinian brand,but perhaps it was this very tension that producedsuch brilliant work. Though some mathematiciansmay exceed Pascal in terms of originality, profun-dity, or volume, Pascal’s systemization of much ofscience and mathematics must draw attention andadmiration.

Further ReadingAdamson, D. Blaise Pascal: Mathematician, Physicist,

and Thinker about God. New York: St. Martin’s,1995.

Peano, Giuseppe 205

Coleman, F. Neither Angel nor Beast: The Life and Workof Blaise Pascal. New York: Routledge and KeganPaul, 1986.

Edwards, A. Pascal’s Arithmetical Triangle. New York:Oxford University Press, 1987.

Hooper, A. Makers of Mathematics. New York:Random House, 1958.

Krailsheimer, A. Pascal. New York: Oxford UniversityPress, 1980.

Mesnard, J. Pascal. Paris: Hatier, 1951.Mortimer, E. Blaise Pascal: The Life and Work of a

Realist. New York: Harper, 1959.

� Peano, Giuseppe(1858–1932)ItalianLogic, Differential Equations

Giuseppe Peano was one of the most talentedmathematicians of the late 19th century; he wasconspicuous for his attention to rigor and detail.His work on mathematical logic and set theoryhas earned him fame, but he also contributed topedagogical projects that proved to be unimpor-tant. Peano’s creative genius gave birth to thefamous Peano curve, and he also constructed thePeano axioms.

Giuseppe Peano was born on August 27,1858, in Cuneo, Italy. His parents were farmers,and Peano traveled by foot each day to theschool in Cuneo. Peano’s uncle was a priest whorecognized the boy’s natural talents and took himto Turin in 1870 to prepare him for universitystudies. Peano started at the University of Turinin 1876, and there studied mathematics. Peanoreceived his doctorate in 1880.

Peano had a remarkable skill for detectingthe logical flaws in arguments. Upon graduationhe was appointed assistant to Angelo Genocchi,and Peano soon detected an error in the text-book for one of the courses. Peano largely taughtGenocchi’s classes, since the older professor wasill, and in 1884 published a text of the course

notes. Peano had also published several researchpapers after 1880, and became qualified to teachat the university level in 1884.

In 1886 Peano researched questions of exis-tence and uniqueness in the theory of differentialequations, and next developed a method forsolving such equations using successive approx-imations. He was also teaching at the MilitaryAcademy at this time, and was later appointedto Genocchi’s chair at Turin upon his death in1889. Peano meanwhile published GeometricalCalculus in 1888, which began with a chapteron mathematical logic, and developed HER-MANN GÜNTER GRASSMANN’s concept of a vec-tor space. Peano used a modern notation for thiswork, which built upon the ideas of CHARLES

PEIRCE and GEORGE BOOLE. In 1889 he publishedhis famous Peano axioms, which defined thenatural numbers in terms of sets, and he definedin a rigorous fashion such ideas as proof by in-duction. This was a significant contribution tothe foundations of mathematics, and it wouldbe exploited and developed by many of Peano’ssuccessors.

Peano is also famous for his “space-fillingcurves.” He defined a continuous mapping of theunit interval onto the unit square, in essenceconstructing a one-dimensional curve that filledup a two-dimensional space. This mapping doesnot have a continuous inverse, since that wouldbe tantamount to establishing that the line andthe plane have equal dimension. Nevertheless,many mathematicians were disturbed by thepathological result, which followed in the samespirit of the work of GEORG CANTOR.

Once appointed to his new post at theUniversity of Turin, Peano founded the journalRivista de matematica in 1891. As editor of thejournal, Peano was able to ensure that highstandards of rigor were maintained. In 1892 heembarked on a new project—the Formulariomatematico (Mathematical formulary), which wasto be a collection of definitions, theorems, andmethods of all mathematical subjects, that could

206 Pearson, Egon Sharpe

be used as a basic text for every mathematicscourse. This monumental effort was not com-pleted until 1908. It turned out to have littlepopularity, since this meticulous approach tomathematics did not facilitate learning. Peanowas considered a good teacher prior to his im-plementation of the Formulario; afterward,students and fellow faculty members complainedof the boring exactness of his method.

One of the high points of Peano’s careerwas the International Congress of Philosophyheld in Paris during 1900. Peano’s logical train-ing enabled him to shine among his less rigor-ous philosopher colleagues, as he was able towin all the philosophical arguments in whichhe became embroiled. His presence there madea great impression on the young BERTRAND

RUSSELL, who was excited by the power ofPeano’s notation and methodology. Peano alsoattended a similar congress of mathematicians,at which DAVID HILBERT stated his famous 23problems for the 20th century. Peano was in-trigued by Hilbert’s problem on the axioms ofarithmetic.

Peano’s last years were spent on a new proj-ect—the construction of a new language basedon French, Latin, English, and German. Theresultant “Latino sine flexione”—later calledInterlingua—has seen little use, and is irrelevantto the development of mathematics. Peano diedon April 20, 1932, in Turin, Italy. He was a bril-liant mathematician of great precision, settingstandards of rigor that were uncommon at thetime; his meticulousness seems more appropriatefor the present age of mathematics. Although hiswork on the Formulario and Latino sine flexionecan be seen as distractions, his contributions tomathematics are highly significant nonethe-less. Peano must be regarded as one of the earlyfounders of mathematical logic—his work on thePeano axioms was well known to his descen-dants. The Peano curve is also an important con-tribution to topology and the study of fractalgeometry.

Further ReadingGillies, D. Frege, Dedekind, and Peano on the

Foundations of Arithmetic. Assen, the Netherlands:Van Gorcum, 1982.

Kennedy, H. Peano: Life and Works of Giuseppe Peano.Dordrecht, the Netherlands: D. Reidel PublishingCompany, 1980.

———. “Peano’s Concept of Number,” HistoriaMathematica 1 (1974): 387–408.

� Pearson, Egon Sharpe(1895–1980)BritishStatistics

Egon Pearson is ranked among the greatestmathematicians of modern times due to hiswork, together with Jerzy Neyman, on hypoth-esis testing. Their work formulated the classicalstatistical decision theory familiar to mostscientists.

Egon Sharpe Pearson was born on August11, 1895, in Hampstead, England. His father wasthe famous statistician Karl Pearson, who firstinvented the correlation statistic to quantifylinear relationship between two statistical vari-ables. His mother was Maria Sharpe. Pearson wasthe middle child of the three children. His child-hood was somewhat sheltered, and he grew toadmire and revere the outstanding work of hisfather.

Pearson attended the Dragon School inOxford from 1907 to 1909, and later studied atWinchester College, from which he graduatedin 1914. At this time World War I erupted, butPearson did not serve in the military due to hispoor health—he had a heart murmur. Instead,he pursued university studies at Trinity College,although during his first year he was incapaci-tated by influenza. Determined to contribute tothe war effort, Pearson left Trinity to work forthe Admiralty; he resumed his undergraduatestudies after conclusion of hostilities, and earned

Pearson, Egon Sharpe 207

his diploma through a special test for veteransin 1920.

In the next phase of his life, Pearson tookup graduate-level research at Cambridge, origi-nally studying solar physics. In his astronomicalstudies he encountered a substantial amount ofstatistical theory, and in 1921 he joined his fa-ther’s Department of Applied Statistics atUniversity College of London as a lecturer.However, Pearson’s father ensured that his sondid little real lecturing; instead, his time wasspent attending talks and conducting research.He began to produce a significant quantity ofstatistical research papers, and in 1924 alsobecame an assistant editor of Biometrika, hisfather’s statistical journal.

Meanwhile, Pearson became emotionally in-volved in one of the most heated statistical con-troversies of the time. SIR RONALD AYLMER FISHER,who was also at the same department, promotedthe small-sample approach to statistical prob-lems. He emphasized the computation of exactdistributions, and more generally attempted toground statistical practice in rigorous mathemat-ics. Karl Pearson’s approach emphasized large-sample statistics and asymptotic theory instead;these differing viewpoints, coupled with Fisher’spugnacious personality, led to a savage public de-bate. Egon Pearson was caught between paternalfidelity and the realization that Fisher, antago-nistic as he was, seemed to be correct. After KarlPearson’s death years later, Fisher continued hisstatistical agenda in print, an ongoing source ofirritation for Egon Pearson.

In 1925 Pearson met with Neyman, anotheryoung statistician, and initiated a fruitful col-laboration with him. This work continued oninto 1927, resulting in excellent research intothe theory of hypothesis testing (that is, how totest scientific hypotheses with quantitative datain such a way as to minimize mistakes). Theirresearch in this area has now become a classicalsegment of basic statistical theory, althoughmuch of the current work is focused on the

Bayesian approach to inference and the resolu-tion of hypotheses. Pearson and Neyman con-tinued to collaborate, mostly by voluminouscorrespondence, over the next decade. Also atthis time, Pearson began working closely withWILLIAM GOSSET.

Pearson began lecturing in 1926. His fatherretired in 1933 from his position as the GaltonChair of Statistics, and University College decidedto split the department into two sections: Fisherbecame head of the Department of Eugenics, andPearson was head of the Department of AppliedStatistics. In 1934 Pearson married (he had twodaughters), and after his father’s death in 1936 hetook over managing and editing Biometrika. Inthis year Neyman also visited Pearson’s depart-ment, sparking further joint work. He was recog-nized for his exceptional labors through the 1935Weldon Prize.

Another great achievement of Pearson’s wasthe editing of his father’s substantial Tables forStatisticians and Biometricians, published in twovolumes in 1954 and 1972 with Hartley. Thesetables were easy to use, and became models forstatistical figures. In the advent of World WarII, Pearson shifted the thrust of his research totopics applicable to the war, such as the statis-tical analysis of the fragmentation of shells.Pearson was later recognized with a governmentaward for his service.

He was a quiet, introverted man who hadled a somewhat sheltered life. Pearson’s difficultrelations with Fisher were ameliorated in 1939,after Fisher moved away from London. Pearsonwas struck by personal tragedy in 1949, when hiswife, Eileen, died of pneumonia. He continuedhis prodigious contributions to the theory of sta-tistics, and retired from University College in1961. In 1966 he was belatedly elected to theRoyal Society, and he died on June 12, 1980, inMidhurst, England.

One of the greatest mathematicians, Pearsonwas an extremely renowned and noteworthystatistician who introduced many mathematical

208 Peirce, Charles

ideas into the practice of science and statistics.Most remarkable is his joint work with Neymanon hypothesis testing.

Further ReadingDavid, H. “Egon S. Pearson, 1895–1980,” The

American Statistician 35, no. 2 (1981): 94–95.Johnson, N., and S. Kotz. “Egon Sharpe Pearson,” in

Leading Personalities in Statistical Sciences. NewYork: John Wiley, 1997.

Moore, P. “A Tribute to Egon Sharpe Pearson,”Journal of the Royal Statistical Society. Series A 138,no. 2 (1975): 129–130.

Pearson, E. The Selected Papers of E. S. Pearson.Berkeley: University of California Press, 1966.

Reid, C. Neyman—from Life. New York: Springer-Verlag, 1982.

� Peirce, Charles(1839–1914)AmericanLogic, Topology

There were few American mathematicians ofnote before the 20th century, when Americanscame to dominate the mathematical scene. Oneexception is Charles Peirce, a notable mathe-matician who was active in the latter half of the19th century.

Charles Peirce was born on September 10,1839, in Cambridge, Massachusetts. His fatherwas the mathematician Benjamin Peirce, whosework he later extended, and his mother was SarahMills. Charles Peirce was educated at HarvardUniversity, and after graduation in 1859 becameactive on the Coast and Geodetic Survey.

After one year with the survey, Peirce en-tered the Lawrence Scientific School of HarvardUniversity, where he studied chemistry. He con-tinued his work for the survey as a computingaide to his father, who was also involved. CharlesPeirce contributed to the determination of theEarth’s ellipticity, and used the swinging of a

pendulum to measure the force of gravity.Despite his brilliant work for the survey, Peircewas in constant conflict with the administrators,who viewed his careful and meticulous prepara-tion of reports as procrastination. He eventuallyresigned in 1891, and thereafter had no steadyemployment or income.

Peirce is primarily known as a logician,though he was also interested in topological andgeometrical problems. In connection with hiswork on geodesy, he became interested in con-formal mappings, and he invented a quincuncialprojection that involved elliptic functions. LaterPeirce tackled the famous four color problem,which asks whether a political map can be col-ored with only four colors, such that no twoneighboring states have the same color. Peircewas intrigued by a wide selection of topologicalproblems, such as those found in the theory ofknots. He extended his father’s work on asso-ciative algebras, and made original contributionsto mathematical logic and set theory.

Peirce did not attain an academic career—there were no chairs in logic at that time in theUnited States—although he briefly taught coursesin logic at Johns Hopkins University from 1879to 1884. He supported himself through odd jobsand the generosity of his friends. Peirce was anearly member of the American MathematicalSociety, where he retained his membership despitebeing unable to pay his dues. He was admired forhis brilliance, although he often shrugged offmathematical details as unimportant.

Peirce identified logic with semiotics, thetheory of signs (symbols that signify some-thing). These ideas were summarized in his un-finished work A System of Logic, Considered asSemiotic. Peirce contributed to deductive logic,but his primary interest lay in induction, or inhis own words, abduction—the formation of ahypothesis in order to explain some strange ob-servation. Due to this unique emphasis in thetheory of logic, it is easy to see how Peirce wasinterested in science, broadly speaking, since

Poincaré, Jules-Henri 209

scientific investigations gave an application ofthe concept of abduction. He was also wellknown as a philosopher, contributing to thephilosophy of pragmatism, which he identifiedwith abduction.

Peirce was married twice: to Harriet MelusinaFay in 1862, who abandoned him in 1876, andto Juliette Pourtalai in 1883. He had no children,and spent the final years of his life on a farm inPennsylvania. In his last years, he fell seriously ill,and was also afflicted by poverty. Peirce died onApril 19, 1914, in Milford, Pennsylvania. He wasa famous logician, highly regarded by Americanmathematicians. Although he has not left an out-standing intellectual legacy, Peirce was one of thefirst notable American mathematicians; he flour-ished at a time when American mathematics wasin its infancy.

Further ReadingBrent, J. Charles Sanders Peirce: A Life. Bloomington:

Indiana University Press, 1998.Debrock, G., and M. Hulswit. Living Doubt: Essays

concerning the Epistemology of Charles SandersPeirce. Dordrecht, the Netherlands: KluwerAcademic Publishers, 1994.

Deledalle, G. Charles S. Peirce, 1839–1914: An Intel-lectual Biography. Amsterdam, the Netherlands:J. Benjamins Publishing Company, 1990.

Eisele, C., and R. Martin. Studies in the Scientific andMathematical Philosophy of Charles S. Peirce. NewYork: Mouton, 1979.

Kevelson, R. Charles S. Peirce’s Method of Methods.Amsterdam, the Netherlands: J. BenjaminsPublishing Company, 1987.

Roberts, D. The Existential Graphs of Charles S. Peirce.The Hague, the Netherlands: Mouton, 1973.

� Poincaré, Jules-Henri(1854–1912)FrenchTopology, Geometry, ComplexAnalysis, Differential Equations

Henri Poincaré has been described as the last ofthe great mathematicians adept in severalbranches of mathematics and science; however,a similar claim could be made about DAVID

HILBERT. Poincaré was a genius of the first rank,whose innovative contributions shaped (and insome cases, essentially founded) several areas ofmathematics, including algebraic geometry,algebraic topology, the theory of automorphicfunctions in complex analysis, and nonlineardynamics. His work continues to exert a pro-found influence on modern studies in topologyand geometry.

Jules-Henri Poincaré was born on April 29,1854, in Nancy, France, to Léon Poincaré, aprofessor of medicine at the University ofNancy, and Eugénie Launois. Henri Poincaré

Jules Henri Poincaré used algebraic methods to solvegeometrical problems and formulated the yetunproved Poincaré conjecture. (Courtesy of theLibrary of Congress)

210 Poincaré, Jules-Henri

was physically weak, suffering from nearsighted-ness and a lack of coordination; he was ill for atime from diphtheria. However, his intellectualgifts more than compensated for these deficien-cies. His mother taught him to write at a youngage, and Poincaré later became a powerfulauthor.

When Poincaré was still young, he startedat the local school in Nancy in 1862 (this schoolwas later renamed Lycée Henri Poincaré in hishonor). During the next 11 years, Poincaréproved to be the top student, excelling in all sub-jects, especially mathematics—he often wonfirst prize in competitions. He entered the ÉcolePolytechnique in 1873, and graduated two yearslater. Poincaré was far beyond his fellow studentsin most of the intellectual subjects; he also hada strong interest in music, especially the piano.He read widely in science, and thus obtained athorough knowledge of electricity, optics, andthermodynamics.

Next Poincaré pursued further studies at theÉcole des Mines, and briefly worked as a miningengineer while working on his doctorate at theUniversity of Paris. His mentor was CHARLES

HERMITE, and Poincaré completed a thesis on dif-ferential equations in 1879. From here, Poincaréwent through several appointments: a teacher ofanalysis at the University of Caen, a chair at theFaculty of Science in Paris in 1881, and the chairof mathematical physics and probability at theSorbonne in 1886. His lectures were disorgan-ized, but addressed new material each year;Poincaré seasoned his mathematical topics withapplications from optics, astronomy, electricity,and other cognate sciences.

Besides his scientific work, which includescontributions to celestial mechanics, fluid me-chanics, and the philosophy of science—he wasalso credited as a coinventor of the special the-ory of relativity along with Albert Einstein—Poincaré delved deeply into several of the ma-jor branches of pure mathematics. His thesiswork led to the definition of an automorphic

function, which is now a classical component ofthe theory of complex analysis (automorphismsalso play a substantial role in abstract algebra).These are complex functions whose values areinvariant under certain groups of transformationsof the domain space. Poincaré correspondedheavily with FELIX KLEIN regarding these new andintriguing functions, which had connections tonon-Euclidean geometry.

Poincaré’s Analysis Situs (Site analysis) of1895 was a systematic treatment of topology (thestudy of continuous mappings operating on high-dimensional surfaces), a fledgling subject in thelate 19th century. In this and in other papers overthe next decade, Poincaré developed the subjectof algebraic topology. Essentially, this subject usesalgebraic tools—such as groups and rings—to de-scribe and classify topological objects. For exam-ple, Poincaré’s homotopy group consisted ofequivalence classes of twisted circles embeddedon a manifold (a high-dimensional space); this af-forded a method of classifying manifolds. The fa-mous Poincaré conjecture, still unproved a centurylater, states that any three-dimensional manifoldwith homotopy group equal to that of a spheremust be topologically equivalent (that is, it canbe continuously deformed without tearing) intoa three-dimensional sphere. Poincaré conjecturedthis after proving it in the intuitive two-dimen-sional setting, and conjectured it for dimensionthree. It is intriguing that the conjecture has beenverified for higher dimensions, but a proof fordimension three has eluded a century of effort.Poincaré’s work dominated the scene of algebraictopology for the next four decades: his methods,his questions, and his results were all enormouslyinfluential.

Poincaré initiated the study of functions ofseveral complex variables through his 1883 workinvolving the Dirichlet principle. This difficultsubject is still being studied today. He laboredin the field of algebraic geometry, the study ofmanifolds given as the solution of algebraic equa-tions in several variables. In 1910 and 1911 he

Poisson, Siméon-Denis 211

developed powerful methods that allowed himto prove previously conjectured results concern-ing algebraic curves embedded in algebraic sur-faces. Poincaré studied number theory in 1901,examining Diophantine equations. He laterstated that an axiomatic approach to the foun-dations of arithmetic would be unable to furnisha rigorous proof of the consistency of numbertheory; his opinion was vindicated decades laterthrough the work of KURT GÖDEL.

Poincaré also studied optics, electricity,telegraphy, capillarity, elasticity, thermodynam-ics, potential theory, quantum theory, and thetheory of relativity and cosmology. In an 1889competition in Sweden, he developed new ideasin nonlinear dynamics concerning the three-body problem of celestial mechanics. Althoughhe won the prize, a perceived error in his man-uscript led to an extensive correspondence withthe mathematician Magnus Mittag-Leffler. Somedate the birth of chaos theory to this communi-cation. Besides his other work on fluid mechan-ics, Poincaré also wrote scientific articles aimedat a popular audience, and went a long way to-ward making mathematics and science of inter-est to the common people of France.

Poincaré also contributed to the philosophyof science, and he was a guiding influence in math-ematical logic, where he stressed the importanceof intuition over axiomatization. The thoughtprocess of Poincaré was the subject of a psycho-logical study by Toulouse, who described him asa true genius reliant on an amazing mathemati-cal intuition. Poincaré would leave problems fora time, letting his mind ruminate subconsciouslyover the issues; then, he would return to a projectin force, making sudden leaps of the intellect. Inthis way he was able to achieve a remarkable di-versity and profundity of mathematical material.Thus, logic alone was unfruitful, according toPoincaré, and was only useful as a tool for thecorrection of intuition. This mentality is quitesimilar to the philosophy of LUITZEN EGBERTUS

JAN BROUWER.

Poincaré was highly honored during his life-time, receiving many awards—he was elected tothe Academy of Sciences in 1887 and becamethe president in 1906. Due to the breadth of hisresearch, Poincaré was the only member of theacademy elected to all five sections—geometry,physics, geography, navigation, and mechanics.He died somewhat prematurely on July 17, 1912,in Paris, France. Although his contributions tomathematics were phenomenal, he did not havehis own school since he did not mentorstudents. Nevertheless, Poincaré’s ideas andmethods have proven to be enduringly influen-tial to modern mathematics—especially inalgebraic topology, complex analysis, and dif-ferential geometry.

Further ReadingBarrow-Green, J. Poincaré and the Three Body Problem.

Providence, R.I.: American MathematicalSociety, 1997.

Browder, F. The Mathematical Heritage of HenriPoincaré. Providence, R.I.: American Mathe-matical Society, 1983.

Miller, A. Imagery in Scientific Thought: Creating 20th-Century Physics. Cambridge, Mass.: MIT Press,1986.

———. Insights of Genius: Imagery and Creativity inScience and Art. New York: Copernicus, 1996.

� Poisson, Siméon-Denis(1781–1840)FrenchProbability, Differential Equations

Siméon-Denis Poisson was one of the greatFrench mathematicians active during the earlypart of the 19th century. He was recognized asone of the most brilliant young Frenchmen ofhis time, and his work in several branches ofmathematics has had an enduring influence; hisname is attached to numerous mathematicalobjects.

212 Poisson, Siméon-Denis

thereafter he earned first place in the entranceexams for the École Polytechnique. Poissonstarted there in 1798, and was finished by 1800.Although his background was not as thoroughas that of other young men, Poisson made ex-ceptional progress in his studies. He wrote hisfirst paper on finite differences at age 18,which attracted the attention of ADRIEN-MARIE

LEGENDRE; PIERRE-SIMON LAPLACE and JOSEPH-LOUIS LAGRANGE, who were among his teachers,were also duly impressed with his talent. In hisfinal year, Poisson substituted a paper of the the-ory of equations for his final examination andwas appointed to a position at the ÉcolePolytechnique. Attaining a post in Paris at sucha young age was unheard of.

Poisson gained a professorship there in 1802,and he spent all his energies on mathematicalresearch. Most of his work pertained to the the-ory of partial differential equations and themathematical aspects of various scientific prob-lems, such as the motion of the pendulum. Hewas averse to physical experiment and graphicaldrawings due to his lack of manual dexterity.Poisson’s success led to new appointments: Hewas made astronomer at the Bureau desLongitudes in 1808 and chair of mechanics atthe Faculté des Sciences in 1809.

In 1808 Poisson published Sur les inégalitésdes moyens mouvement des planets (On the in-equalities of planetary movements), which usedseries expansions to solve problems in celestialmechanics raised by Laplace and Lagrange. Thenext year he followed with two importantpapers that utilized Lagrange’s method of vari-ation. Poisson also published Treatise onMechanics in 1811, a lucid presentation of hiscourse notes.

Poisson also won the grand prize, set by theParis Institute in 1811, on the topic of how elec-trical fields are distributed over surfaces. Poissonwon the prize, and earned a place in the insti-tute as a result. He became increasingly busyover the next years, but never slowed the pace

Poisson was born on June 21, 1781, inPithiviers, France. His father was a formersoldier, who was embittered toward the nobilityfor discrimination he had suffered while in thearmy. At the time of Poisson’s birth, he workedas a government official. Poisson had manybrothers and sisters, some of whom died young;Poisson himself suffered from poor healththroughout his life, and was somewhat clumsy.Poisson’s father oversaw his early education,teaching him to read and write.

When the French Revolution struck in1789, Poisson’s father’s antiaristocratic outlookled to his appointment as president of the dis-trict of Pithiviers. He originally intendedPoisson to be a surgeon, but the boy’s physicalawkwardness made this idea impracticable.When Poisson enrolled in the École Central in1796, his exceptional aptitude for mathematicsbecame apparent to his teachers. Shortly

Siméon-Denis Poisson advanced the fields ofmechanics, algebra, and probability, and formulatedthe law of large numbers and the Poisson distribution.(Courtesy of the Library of Congress)

Pólya, George 213

of his investigations. Poisson would work ononly one mathematical problem at a time; hewould write new ideas in his wallet and put themaway for later development when he could de-vote his full attention. In 1815 he became anexaminer for the École Militaire. He marriedNancy de Bardi in 1817.

Poisson’s work focused on solving thornymathematical problems in developed fields, andthus he did not create any new branch ofresearch. However, his theorems and construc-tions propelled mathematics forward and as-sisted the general progress of knowledge. Someof Poisson’s important areas of work were elec-tricity, magnetism, elasticity, and heat. Again,his work was highly theoretical even though itconcerned scientific topics. Poisson’s work onthe velocity of sound was motivated by Laplace’sresearch, and GEORGE GREEN later drew inspira-tion from Poisson’s results on attractive forces.In 1837 Poisson produced an important bookon probability—Recherches sur la probabilité desjugements en matière criminelle et en matière civile(Research in the probability of criminal andcivil verdicts), in which he introduced thePoisson distribution—a tool that has been heav-ily used in probability and statistics. This dis-tribution models the likelihood of seeing re-peated rare events in a small window of time.He introduced the terminology for the law oflarge numbers, which is concerned with arith-metic means of independent random quantities.

Poisson died on April 25, 1840, in Sceaux,France. He was mainly influential in advancingdeveloped areas of mathematics. His name is at-tached to a wide variety of mathematical objects,indicating the scope of his research: the Poissonintegral, the Poisson differential equation of po-tential theory, the Poisson bracket (and alge-braic object), and the Poisson distribution. Hiswork was well known by his own teachers andalso by foreign mathematicians, but many of hiscountrymen failed to recognize Poisson’s merituntil after his death.

Further ReadingGeller, B., and Y. Bruk. “A Portrait of Poisson,”

Quantum (1991): 21–25.

� Pólya, George(1887–1985)HungarianProbability

George Pólya is one of the best-known figures ofthe 20th century to mathematicians, due to hispedagogical work on problem solving. His re-markably diverse mathematical work, whichachieved notable results in probability and com-binatorics, among other fields, merits him aplace among the top researchers of his time.

George Pólya was born on December 13,1887, in Budapest, Hungary, to Jakab Pólya andAnna Deutsch. Pólya’s parents were HungarianJews who had changed their last name to Pólyafrom Pollák for political reasons—Pólya soundedmore Magyar than Pollák. Pólya’s father hadoriginally been a lawyer, but he was more inter-ested in academics, and obtained a post at theUniversity of Budapest while George Pólya wasstill young. Pólya had an older brother, Jenö, twoolder sisters, Ilona, and Flóra, and a youngerbrother, Lásló.

Although Pólya’s parents were Jewish, thewhole family converted to Roman Catholicismbefore Pólya was born. His father died whenPólya was 10 years old, and the whole familylabored to assist with Pólya’s education. Pólyaperformed well in elementary school, but he wasindifferent to mathematics; he later claimed thathis mathematics teachers were terrible. He en-rolled in the University of Budapest in 1905,supported by his older brother, Jenö, who wasnow a surgeon. Pólya’s mother encouraged himto study law, but he found the subject boring andhe turned to languages, literature, and philoso-phy instead. His philosophy teachers informedPólya that he lacked an adequate background in

214 Pólya, George

mathematics and physics, so he studied thesesubjects next. He later attended the Universityof Vienna from 1910 to 1911, and upon his re-turn to Budapest was granted a doctor’s degreefor solving a problem in probability. He spentthe years 1912 and 1913 at the University ofGöttingen, pursuing further studies under suchmathematicians as FELIX KLEIN, DAVID HILBERT,and HERMANN WEYL.

Pólya’s experience in Germany greatly ad-vanced his development as a mathematician, buthe was forced to leave Göttingen after beinginvolved in lawlessness. On a train he becameinvolved in an altercation with a young man,and Pólya boxed his ears to further provoke him.The young man was a student at Göttingen, andhis father was a political official with the powerto bar Pólya from the campus. Later, Pólyaamended his feisty temperament, becoming apacifist and draft dodger by the start of WorldWar I.

Pólya did some more traveling, visiting themathematicians Émile Picard and JacquesHadamard in Paris. He later received an ap-pointment at the University of Zürich in 1914,where he collaborated with Adolf Hurwitz,whose work he found quite influential. Pólyaalso had Weyl and ERNST ZERMELO as colleagues,and his research was quite fruitful at this time.When World War I erupted, Pólya avoided mil-itary service in his native Hungary through aprevious soccer injury; later, he was drafted any-way but refused to serve, becoming a Swiss cit-izen instead. He married Stella Vera Weber, thedaughter of a physics professor, in 1918.

Pólya had met the mathematician GáborSzego in 1913 in Budapest, and soon after thewar contacted him with the idea of writing abook on mathematical problem solving.Although there are many such books now, their1925 Aufgaben und Lehrsätze aus der Analysis(Problems and theorems of analysis) was the firsttext of its kind. The authors classified problemsin analysis in a novel manner: They grouped the

material according to the method of solutionrather than the natural and historical develop-ment. This book was a great success, and helpedPólya to achieve some fame.

In 1920 Pólya was promoted to professor,and in 1924 obtained a fellowship to work withGODFREY HAROLD HARDY at Cambridge; they(along with Littlewood) started work on thebook Inequalities, later published in 1934. Pólyapublished more than 30 papers between 1926and 1928 on a wide range of mathematical top-ics, and he was promoted to full professor in1928 as a result. Pólya’s research touched onprobability, geometry, complex analysis, physics,and combinatorics. He also worked on numbertheory, astronomy, and many applied problems,such as the mathematics of voting. Some of hisresearch accomplishments include the study ofthe random walk, Fourier analysis applied toprobability, the central limit theorem, and geo-metric tilings. The random walk is a model ofmotion, where an object on a line moves eitherforward or backward with equal chances. Thiscan be generalized to higher-dimensionalspaces, giving the random walks on the planeand in space. Pólya showed that a randomwalker returns to his initial location only if thedimension of the random walk is at least three—one can get lost in space but not on a line orin a plane.

Pólya’s work on geometric configurations inthe plane was related to the various tilings of theplane—a partition of the plane into figures (suchas triangles or hexagons) that were invariantunder certain rotations and shifts. M. C. Escherlater used Pólya’s ideas to create his beautiful art-work. Pólya greatly added to the knowledge ofthis discipline, called crystallography, which hasmany applications in chemistry and art. In com-binatorics, Pólya’s greatest result was his enu-meration theorem, which provided a methodfor counting objects that share certain proper-ties; this later led to the new field of enumera-tive graph theory. In complex analysis, Pólya

Poncelet, Jean-Victor 215

contributed to potential theory and conformalmappings and explored the singularities of powerseries.

Pólya visited Princeton in 1933 on anotherfellowship, and while in the United States healso visited Stanford. By 1940 the political cli-mate in Europe led Pólya to emigrate, and heworked first at Brown University before settlingat Stanford. Around this time Pólya was get-ting his new book, How to Solve It, published,and it became an instant success among math-ematicians. Pólya stressed the idea of learningheuristic—the collection of methods and tech-niques that are used to solve classes of prob-lems. This was a landmark in the theory ofmathematics education, and over the yearsPólya followed it with similar books. One of hismain theses was that mathematics involvesthinking; it is a deeply intellectual subject, nota mechanical collection of methods and tech-niques. The mechanistic approach prevalent inU.S. secondary schools today differs greatlyfrom Pólya’s philosophy, and the consequencesof this departure are only beginning to beexperienced.

Pólya received many awards and honorsthroughout his life, including election to theNational Academy of Sciences and membershipin various mathematical societies. He retiredfrom Stanford in 1953, but he continued to re-search mathematics, being especially interestedin mathematics education. The last course hetaught was a combinatorics lecture at Stanfordin 1978, when he was more than 90 years old.He died on September 7, 1985, in Palo Alto,California.

Pólya was one of the most talented mathe-maticians of the 20th century, as his diverse andprofound research achievements can attest. Hiswork on mathematical learning and teachingwas profound, and he is perhaps the father of themodern studies in this area. His problem-solv-ing books are still classics, and his influence en-dures to the present day.

Further ReadingAlbers, D., and G. Alexanderson. Mathematical

People: Profiles and Interviews. Chicago:Contemporary Books, 1985.

Alexanderson, G. The Random Walks of George Pólya.Washington, D.C.: Mathematical Association ofAmerica, 2000.

———, and L. Lange. “Obituary: George Pólya,” TheBulletin of the London Mathematical Society 19,no. 6 (1987): 559–608.

� Poncelet, Jean-Victor(1788–1867)FrenchGeometry

Jean-Victor Poncelet was one of the founders ofmodern projective geometry, a subject that in-trigued many mathematicians in the 19th century.This subject, originally of interest to Renaissancepainters attempting to generate an aestheticallypleasing perspective, was picked up by modernmathematicians for its relevance to geometry.

Jean-Victor Poncelet was born on July 1,1788, in Metz, France. As a young man, hestudied under GASPARD MONGE at the ÉcolePolytechnique, where he learned analytic geom-etry and a wide range of mathematics. He wastrained as an engineer, and took part in thedisastrous 1812 march to Russia. After the de-struction of the French army by the Russianwinter, Poncelet was left for dead at the town ofKrasnoy, and was subsequently imprisoned bythe Russians. He spent the next two years study-ing projective geometry while in prison;Poncelet returned to France in 1814.

This period of study resulted in his Treatiseon the Projective Properties of Figures (1822). Inthis paper Poncelet developed fundamentalideas, such as polar lines of conics, the principleof duality, the cross-ratio, involution, and circu-lar points at infinity. For example, the point atinfinity for an unbounded plane enables one to

216 Ptolemy, Claudius

represent the plane as a sphere (minus its northpole); it can also be represented as a closedcircle—a model that has proved useful in hy-perbolic geometry. These original ideas becamevery influential in the evolution of projectivegeometry, and his work has also influenced thedevelopment of algebraic geometry more gener-ally. He also wrote a treatise on analytic geom-etry (Applications of Analysis and Geometry), butthis was not published until 1862.

Poncelet served as a military engineer inMetz from 1815 to 1825, and from 1825 to 1835was professor of mechanics there. He applied hisknowledge of mechanics to improve the effi-ciency of waterwheels and turbines. Ponceletproposed the first inward-flow turbine in 1826,and it was finally built in 1838. From 1838 to1848 he was professor at the Faculty of Sciences,and from 1848 to 1850 was commander of theÉcole Polytechnique, where he held the rank ofgeneral. He died on December 22, 1867, in Paris.

Poncelet’s innovative contributions to pro-jective geometry establish him as one of thefounders of the modern theory of the subject.His work certainly stimulated later mathemati-cians, such as JAKOB STEINER.

Further ReadingBos, H., et al. “Poncelet’s Closure Theorem,”

Expositìones Mathematicae. Internatìonal Journalfor Pure and Applied Mathematics 5, no. 4 (1987):289–364.

� Ptolemy, Claudius(ca. 85–ca. 165)GreekTrigonometry

The name of Ptolemy is associated with the geo-centric theory of the cosmos. But to some schol-ars, he is also perceived as an unethical scien-tist; he was accused by several mathematiciansof falsifying his data in order to fit the theory

that he proposed. Ptolemy developed a verycomplete system of mathematics, including orig-inal contributions to trigonometry that wereused in his cosmic system.

Ptolemy’s life is mostly constructed fromsecondary sources, so modern knowledge of himis scanty and unreliable. Ptolemy was born ap-proximately in the year 85, in Alexandria, Egypt.He may have been of Greek descent, but his firstname was Roman, and he may have been aRoman citizen as well.

Ptolemy made celestial observations be-tween the years 127 and 141 in Alexandria.Apparently he obtained some of his data fromTheon of Smyrna, who was likely his mentor aswell. Ptolemy’s early works were dedicated toSyrus, an individual of whom nothing else isknown—he may have been another of Ptolemy’s

Claudius Ptolemy developed an Earth-centered theoryof the universe that was supported mathematically.(Courtesy of the Library of Congress)

Ptolemy, Claudius 217

teachers. Even though Theon did not under-stand astronomy very well, Ptolemy had accessto the outstanding library of Alexandria, fromwhich he may well have gained his extensiveknowledge.

Ptolemy’s most important work was theAlmagest, a 13-book work that advanced hisgeocentric conception of the universe. In theAlmagest he presents a mathematical theory ofthe motion of Sun, Moon, and Earth that wasunsurpassed until Copernicus’s 1543 heliocen-tric theory. Thus this work, although incorrect,long remained the definitive masterpiece in as-tronomy, much as EUCLID OF ALEXANDRIA’sElements dominated geometry. This success waspartly due to Ptolemy’s masterful developmentof an extensive mathematical theory for hissystem.

In particular, Ptolemy developed trigono-metric functions similar to the familiar sine andcosine, and proved various trigonometric iden-tities to assist with computations. He also de-rived an accurate approximation to pi—namely3 + 17/120—and the square root of three. Usingthese tools, Ptolemy went on to describe themotions of the celestial bodies and the lengthsof seasons. In some of his observations TychoBrahe and SIR ISAAC NEWTON later discoveredgross inaccuracies; the latter accused Ptolemy offalsification of his data. However, scholars havesince found it more likely that Ptolemy was in-nocent of deliberate fraud, but merely made er-rors in judgment due to the lack of statisticalmethodology.

The motion of the Sun was circular, withthe Earth off-center; for the motion of the moonsPtolemy followed HIPPARCHUS OF RHODES. Fromhere he describes eclipses and the motions of thefive known planets. This latter model was a so-phisticated mathematical masterpiece to whichthere was no equal predecessor.

Besides the Almagest, Ptolemy wrote theHandy Tables—a compilation of expandedtrigonometric tables—and a popular scientific

account of his geocentric theory. He wrote aboutastrology—the application of the celestial theoryto people’s personal lives—and methods for con-structing a sundial. Ptolemy also gives an earlyaccount of the method of stereographic projec-tion, a commonly used projection of the sphereonto the plane used to make maps of the Earth.His efforts in geography were feeble, due to thedearth of reliable cartographic data during histimes. In Ptolemy’s studies of optics, he advancedthe concept that theory should be established byempirical evidence and experiment—a radicaldeparture from the Greek epistemology of natu-ral philosophy of the previous centuries.

Ptolemy died sometime around 165 inAlexandria. Although his theory of the motionof the heavens was distinctly incorrect, hismathematical system was exceptional in its de-tail and sophistication. His work on trigonome-try represented an early effort in this subject, andhis tables are the first of their kind in the an-cient world.

Further ReadingDreyer, J. A History of Astronomy from Thales to Kepler.

New York: Dover Publications, 1953.Gingerich, O. The Eye of Heaven: Ptolemy, Copernicus,

Kepler. New York: American Institute of Physics,1993.

Grasshoff, G. The History of Ptolemy’s Star Catalogue.New York: Springer-Verlag, 1990.

Heath, T. A History of Greek Mathematics. Oxford:Clarendon Press, 1921.

———. A Manual of Greek Mathematics. Oxford:Clarendon Press, 1931.

Newton, R. The Crime of Claudius Ptolemy. Baltimore:Johns Hopkins University Press, 1977.

Petersen, O. A Survey of the Almagest. Odense,Denmark: Odense Universitetsforlag, 1974.

Smith, A. “Ptolemy’s Theory of Visual Perception,”Transactions of the American Philosophical Society86, pt. 2 (1996).

Toomer, G. Ptolemy’s Almagest. London: Duckworth,1984.

218 Pythagoras of Samos

Pythagoras received the best education, beingtrained in poetry and music, and later in phi-losophy. He had two brothers, and the familytraveled extensively during Pythagoras’s youth,visiting Italy and Tyre.

Pythagoras was later taught by THALES OF

MILETUS and his pupil Anaximander; fromThales he gained an appreciation for geometry,and he traveled to Egypt in 535 B.C.E. to furtherhis studies. Before he left, the tyrant Polycratestook over Samos, and Pythagoras’s friendshipwith him facilitated his introduction intoEgyptian society, since Polycrates had an al-liance with Egypt. Pythagoras visited with thepriests there, but was admitted only to the tem-ple of Diospolis, where he was inducted into thereligious mysteries. It seems that many ofPythagoras’s later beliefs, as well as the rites ofthe cult he would later found, were drawn fromhis time among the Egyptian clerics—for in-stance, his vegetarianism and stress on ethicalpurity can be traced to his time in Egypt. Interms of mathematics, it is not likely that helearned much more there than Thales wouldhave taught him.

In 525 B.C.E., the Babylonian emperorCambyses invaded Egypt and sacked Heliopolisand Memphis; Pythagoras was taken as a pris-oner of war to Babylon. There he communedfurther with the magi and Chaldeans, learningfurther religious mysteries. More important, hestudied all the Babylonian mathematicalsciences, becoming an expert in arithmetic andthe mathematical theory of music (Pythagoraswas skilled at the lute). Somehow he obtainedhis freedom and returned to Samos in 520 B.C.E.

In his hometown, Pythagoras founded aschool called the semicircle, which taughtethics, philosophy, and mathematics. Pythagorashimself dwelled in a cave outside the city, wherehe researched the uses of mathematics.

In 518 B.C.E. Pythagoras left Samos for Italy,arriving at the town of Croton. His ostensible

� Pythagoras of Samos(ca. 569 B.C.E.–ca. 475 B.C.E.)GreekNumber Theory, Geometry

Pythagoras of Samos was one of the earliest Greekmathematicians, and is certainly one of the mostfamous of all time due to the well-knownPythagorean theorem. However, very little isknown of his life, and what details exist are re-constructed from several secondary sources; heleft no writings of his own behind him.

Pythagoras of Samos was born around 569B.C.E. on the island of Samos, Greece. His father,Mnesarchus, was a Phoenician merchant whoearned citizenship at Samos by delivering ashipment of grain during a time of famine.His mother was Pythais, a native of Samos.

Pythagoras was a philosopher who believed numbershad personalities. His cult formulated and proved thePythagorean theorem. (Courtesy of the NationalLibrary of Medicine)

Pythagoras of Samos 219

reason for departure was the rudeness withwhich his novel teaching methods were treated;however, it is more likely that he left in a de-sire to avoid public duty, being under continualpressure to execute diplomatic missions. InCroton, a town in southeastern Italy, Pythagorasfounded the cult that later came to be knownas the Pythagoreans. The inner circle, known asthe mathematikoi, held to principles of commu-nal property, vegetarianism, secrecy, and a strictethical code. They also investigated various as-pects of mathematics in a novel manner: Ratherthan merely formulating practical rules, they ex-plored mathematical objects—such as triangles,circles, and numbers—as concrete, real things.They abstracted the known mathematical rela-tionships into pure formulas and theorems, andthen claimed that these things were as real asthe material world. But their beliefs also tookon a religious aspect, since they held that real-ity was, at its foundation, constructed out ofnumbers.

The outer circle of the cult was known asthe akousmatics. They were allowed to own prop-erty and eat meat. This group produced a greatquantity of mathematical knowledge, the mostfamous of which is the Pythagorean theorem:The square on the hypotenuse of a right trian-gle is equal to the sum of the squares on the legs.For the Greeks, this would have been conceivedin terms of actual geometric squares rather thanthe second power of an abstract number. It is un-clear whether the proof of this result (it wasknown to the Egyptians and to the Babylonianscenturies earlier) was due to Pythagoras or hisfollowers. Other discoveries of the cult includethe irrational numbers, the fact that the sum ofthe angles of a triangle are 180 degrees, and thefive regular solids.

Pythagoras’s belief in the primacy of numberstemmed from his observations of mathematicalstructure in music and astronomy. It is interest-ing that today the modern culture is exceedingly

quantitative, even perceiving mathematical rela-tionships in human relationships (economics)and the human brain (cognitive science). So insome sense, the cult of Pythagoras endures tothe present era. However, Pythagoras’s viewswere a bit different—he held that each numberhad its own personality, complete with gender andcharacter traits.

Besides his mathematical research, Pythag-oras was a famous philosopher who emphasizedthe importance of the ethical life. He taught acosmology with the Earth at the center of theuniverse, and believed the soul to be a form ofnumber that moved through various reincarna-tion toward complete purity.

Despite Pythagoras’s aversion to politics, thecult was involved in Croton’s attack and defeatof the neighboring city Sybaris in 510 B.C.E. In508 B.C.E., an obnoxious nobleman by the nameof Cylon, incensed at his exclusion from themathematikoi, attacked the Pythagoreans. As aresult, Pythagoras fled the city, and some ac-counts say that he died in exile in Metapontium.Other sources say that he survived the persecu-tion to return to Croton, and that he died muchlater, around 475 B.C.E.

The cult itself flourished around 500 B.C.E.,spreading to other cities. It later became politi-cally oriented, and split into various factions;in 460 B.C.E. the members were again violentlypersecuted. Nevertheless, Pythagoras’s ideaspermeated much of the Greek mathematicalcommunity, and affected later thinkers such asEUCLID OF ALEXANDRIA. Pythagoras is most re-markable for his mathematical abstraction; hewas certainly one of the first thinkers in the worldto conceive of mathematics in this abstract form.

Further ReadingBrumbaugh, R. The Philosophers of Greece. New York:

Crowell, 1964.Heath, T. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.

220 Pythagoras of Samos

from the Ancient Times to the Present Day. London:Allen and Unwin, 1961.

Vogel, C. de. Pythagoras and Early Pythagorea-nism. Assen, the Netherlands: Van Gorcum,1966.

van der Waerden, B. Science Awakening. Leyden,the Netherlands: Noordhof International,1975.

Laertius, D. Lives, Teachings, and Sayings of FamousPhilosophers. Cambridge, Mass.: Harvard UniversityPress, 1959.

O’Meara, D. Pythagoras Revived: Mathematics andPhilosophy in Late Antiquity. New York: OxfordUniversity Press, 1989.

Russell, B. History of Western Philosophy and ItsConnection with Political and Social Circumstances

221

he spent his time jotting down results and com-putations in a little notebook. In 1909, at age22, he married, at the arrangement of his mother,a 9-year old girl. Shortly thereafter he secured ajob as a clerk, and in 1912 worked at the MadrasPort Trust. At this time, his first publication ap-peared, titled Some Properties of BernoulliNumbers (1911), a communication on series, in-finite products, and a geometric approximateconstruction of pi. In the Madras area, he wasincreasingly recognized for his brilliant work.

Ramanujan’s famous correspondence withthe British mathematician GODFREY HAROLD

HARDY, a specialist in analytic number theory,initiated the next phase of his life. In a letter toHardy, he outlined some of his principal results,and Hardy responded with enthusiasm. Throughthis credential, Ramanujan was able to obtain atwo-year fellowship at the University of Madras.In 1914 Ramanujan came to Trinity College inEngland at Hardy’s invitation, and in the nextfive years would produce 21 research papers ona variety of topics: approximations to pi, highlycomposite (that is, not prime) numbers, and theaverage number of prime divisors. Most impor-tant in terms of intellectual legacy, Ramanujanstudied partition of numbers into summands. Heproved many properties of this partition func-tion using elliptic function theory, and stimu-lated later work in this area. In addition, he

R� Ramanujan, Srinivasa Aiyangar

(1887–1920)IndianNumber Theory

The Indian mathematician Ramanujan led ashort life full of mathematics. From a highly dis-advantageous background, he was able to makesubstantial contributions to number theory. Hisfeverish preoccupation with mathematics, bor-dering on obsession, is remarkable for its inten-sity and devotion. He is remembered as one ofIndia’s greatest mathematical geniuses.

Srinivasa Aiyangar Ramanujan was born inErode, Madras Province, India, on December 22,1887. Although descended from the Brahmancaste, his family was quite poor, as his father wasa bookkeeper for a local cloth merchant. Heexcelled in his early education, and in 1900 hebegan his own investigations of mathematics.In 1903 he borrowed G.S. Carr’s Synopsis of PureMathematics, which contained thousands of the-orems. Ramanujan quickly devoured this book,and mathematics became his sole interest.

It is said of Ramanujan that he was quietand meditative, with a fondness for numericalcalculations and an unusual memory. In 1904 hewon a fellowship at Government College, butfailed to graduate due to his neglect of English.For a time he was without a definite occupation;

222 Regiomontanus, Johann Müller

labored in many other areas, such as combina-torics and function theory.

The remarkable thing about Ramanujan’sachievement is his lack of formal training. Atthe time of Hardy’s correspondence, there werelarge gaps in Ramanujan’s knowledge of mathe-matics, and his concept of proof was nebulous.His arguments were built from intuition andinduction, and lacked the characteristic rigor ofEuropean thought. Though his mastery ofcontinued fractions and elliptic integrals wasextensive, Ramanujan’s ignorance of other as-pects of mathematics was startling; some of histheorems about prime numbers were completelywrong. Nevertheless, his contributions to ellip-tic functions, continued fractions, and infiniteseries were profound.

He had struggled with ill health for manyyears, and in 1917 he fell ill again, perhaps withtuberculosis. In 1918 he was elected as a fellowto the Royal Society of London, the first Indianto receive that honor, and the accoladesseemed to improve his health. The followingyear he returned to India with the prospect of aprofessorship at the University of Madras.Unfortunately, his health worsened and he re-fused medical aid. Ramanujan continued hismathematical research until his last days, and hedied on April 26, 1920, in Chetput, India.

Mathematicians recognized Ramanujan asone of the greatest geniuses of all time. Giventhe lack of appropriate resources, the depth ofhis mathematical talent was truly exceptional.His most famous work addressed the topic ofthe partition of numbers, but his results onhypergeometric series have also fueled furtherresearch.

Further ReadingHardy, G. Ramanujan: Twelve Lectures on Subjects

Suggested by His Life and Work. New York: ChelseaPublishing Company, 1959.

Kanigel, R. The Man Who Knew Infinity: A Life of theGenius Ramanujan. New York: Scribner’s, 1991.

Ranganathan, S. Ramanujan: The Man and theMathematician. London: Asia Publishing House,1967.

� Regiomontanus, Johann Müller(1436–1476)GermanTrigonometry

The 15th century was a slow time for mathemat-ics in Europe, during which the knowledge oftrigonometry gradually broadened. Some of thisdevelopment was motivated by navigation and as-tronomy, two sciences that heavily utilizedtrigonometry. A century before GEORG RHETICUS’strigonometric tables were introduced, JohannRegiomontanus introduced trigonometric meth-ods that could be used to form accurate astro-nomical predictions.

Johann Müller Regiomontanus was born onJune 6, 1436, in Königsberg, Germany. His lastname is the Latin translation of his hometown,Königsberg, which means “king’s mountain.”Regiomontanus studied under Georg Peurbach,a professor of astronomy, at the University ofVienna. It is unknown when he finished his stud-ies, but in 1461 he was appointed to Peurbach’sposition after the death of Peurbach. In 1468Regiomontanus became the royal astronomer toKing Mathias Corvinus of Hungary.

Regiomontanus was an excellent scholarwho translated and published many documents.He advanced the knowledge of trigonometry inEurope by giving a systematic method for solvingtriangles by determining all the sides and anglesgiven some of them. This theory was developedin his De triangulis omnimodis (On triangles of allkinds) of 1464. He applied these mathematics toassist with the prediction of astronomical orbits,such as Halley’s comet. With financial supportfrom Corvinus, Regiomontanus built an obser-vatory in Nuremberg in 1471 with a workshopto produce instruments. The following year he

Rheticus, Georg 223

made highly accurate observations of a certaincomet, which 210 years later was verified to bethe same as Halley’s comet.

Regiomontanus was also interested in theMoon, observing several eclipses. He inventedthe idea of using lunar distances as a navigationalaid, although the full details of the method werenot worked out until later, when the position ofthe Moon could be measured with sufficient ac-curacy. He also worked on calendar reform, andin 1475 was summoned to Rome by the pope togive advice on this subject and to accept an ap-pointment as bishop of Regensburg. However,Regiomontanus died on July 9, 1476, in Rome be-fore he could take office; some accounts say hewas poisoned by political enemies, while othersclaim that he died from the plague.

Regiomontanus died before his time, but wasstill a significant figure in the history of mathe-matics. His work on trigonometry was certainlyan advance, especially in light of the dark in-tellectual times.

Further ReadingKren, C. “Planetary Latitudes, the Theorica, and

Regiomontanus,” Isis 68 (1977): 194–205.Zinner, E. Regiomontanus: His Life and Work.

Amsterdam: North-Holland, 1990.

� Rheticus, Georg (Georg Joachim von Lauchen Rheticus)(1514–1574)GermanTrigonometry

Georg Rheticus is known primarily for histrigonometric tables, being one of the firstEuropeans to produce such an item. He was anintense, busy man who advanced the cause ofmathematics in Europe at a time when there waslittle intellectual activity.

Rheticus was born on February 16, 1514, inFeldkirch, Austria. His father was named Georg

Iserin and his mother was Thomasina de Porris.Rheticus’s father was the town doctor, as well asa government official. When Rheticus was only14 years old, his father was tried and executedfor the crime of sorcery in 1528. As a conse-quence of the sentence, Rheticus was not al-lowed to use his father’s name; instead, he tookhis mother’s maiden name, (which means “of theleeks”) and translated it from Italian intoGerman, resulting in “von Lauchen.” He lateradded the name “Rheticus,” which referred tothe Roman province of Rhaetia in which he hadbeen born.

Rheticus’s father was succeeded by AchillesMasser as the town doctor, and this man sup-ported Rheticus’s further studies. Rheticuswent on to study at Zurich from 1528 to 1531after finishing his classical Latin education atFeldkirch. He entered the University ofWittenberg in 1533 and graduated with a mas-ter’s degree in 1536. At this point, Rheticusobtained an appointment to lecture at theUniversity of Wittenberg with the assistance ofPhilipp Melanchthon, a friend of Martin Lutherwho reorganized Germany’s educational system.Rheticus first taught mathematics and astron-omy at Wittenberg. In 1538 he traveled for ayear, visiting various other scholars, and in 1539journeyed to Frauenberg, where he was to spendthe next two years studying with Copernicus.This was a valuable friendship for Rheticus, whoeagerly devoured the astronomical and mathe-matical knowledge of the venerable Copernicus.In 1539 Rheticus also visited Danzig and pro-cured funding for the publication of Copernicus’sbook Narratio Prima (The first account of thebook on the revolution). This was essentially awork written by Rheticus, containing a mathe-matical presentation of Copernicus’s research.

Rheticus also obtained permission in 1541to publish another of Copernicus’s works (DeRevolutionibis) through an interesting gift tothe duke of Prussia—a map of Prussia and adevice that could determine the time of day

224 Riemann, Bernhard

(a prototype of the clock). The duke also re-quested that Rheticus return to his chair atWittenberg, and Rheticus was there electeddean of the faculty of the arts. In the same yearRheticus published the trigonometric tables ofDe Revolutionibis, supplemented with his owncalculations. This was the first such table of itskind in Europe, and the utility of this knowl-edge has helped to secure Rheticus’s positionin history.

Melanchthon again assisted Rheticus to at-tain a position at the University of Leipzig in1542 as a professor of higher mathematics. Afterthree years, Rheticus began another period oftravel, visiting GIROLAMO CARDANO in Italy aswell as Feldkirch. His physical health deterio-rated in 1547 at Lindau, and he also sufferedfrom mental problems. Once he recovered, hewent on to Zurich to study medicine. When hefinally returned to Leipzig in 1548, he was ap-pointed a member of the theological faculty.

Rheticus was very productive during theseyears, not merely in the area of mathematics, butmore widely; for instance, he produced a calen-dar for 1550 and 1551. He was soon involved ina scandal at Leipzig: he was accused of having ahomosexual affair with one of his students,and Rheticus chose to flee rather than defendhimself. As a result, his friends (such asMelanchthon) abandoned him, and he wassentenced to 101 years of exile. Meanwhile,Rheticus arrived at Prague, where he continuedhis medical studies. He used his knowledge ofmedicine mainly to treat patients rather thanconduct original research.

Rheticus was offered a post as mathematicsprofessor at the University of Vienna, but he in-stead moved to Krakow in 1554 where he set upa practice as a doctor. He remained in Krakow forthe next 20 years. While there, he continued hismathematical research, using his trigonometrictables to conduct further studies on astronomy andalchemy. He obtained funding for his projects fromEmperor Maximilian II, and employed six research

assistants. Rheticus’s most important work ontrigonometry, the Opus Palatinum de triangulis(The Palatine work on triangles), describes the useof the six main trigonometric functions: sine,cosine, tangent, cotangent, secant, and cosecant.Further trigonometric tables for these functionswere published in 1596, years after his death.

Rheticus died on December 4, 1574, inKassa, Hungary. Besides the mathematical workon trigonometric tables, he also completed abook on mapmaking as well as devising variousnavigational instruments, such as sea compasses.Rheticus was an intellectual of broad interestswho traveled widely to further his diverse schol-arly inclinations; his investigations were char-acterized by an uncommon vitality and energy.From the perspective of mathematics, he is animportant figure for his formation of trigono-metric tables, which were extremely useful forthe pursuit of astronomy and the other sciences.

Further ReadingArchibald, R. “The Canon Doctrinae Triangulorum

(1551) of Rheticus (1514–1576),” MathematicalTables and Other Aids to Computation 7 (1953):131.

———. “Rheticus, with Special Reference to HisOpus Palatinum,” Mathematical Tables and OtherAids to Computation 3 (1949): 552–561.

� Riemann, Bernhard (Georg FriedrichBernhard Rieman)(1826–1866)GermanComplex Analysis, DifferentialEquations, Geometry, Number Theory

Few mathematicians can compare to BernhardRiemann in terms of creativity and depth ofinsight. Not only did Riemann found the newdiscipline of Riemannian geometry that wouldbecome so important to the theory of generalrelativity a century later, but he significantly

Riemann, Bernhard 225

and in 1842 the boy joined the JohanneumGymnasium in Lüneburg. Riemann was a goodpupil, but did not yet show extraordinary talentin mathematics. Although his main studies wereclassics and theology, he became interested inmathematics after quickly devouring a numbertheory book by ADRIEN-MARIE LEGENDRE.

In 1846 Riemann enrolled at the Universityof Göttingen, where he pursued further study inmathematics. Although CARL FRIEDRICH GAUSS

was teaching there at the time, he did not rec-ognize Riemann’s talent, as did some of his otherteachers. The next year Riemann transferredto the University of Berlin, where he was ableto study under CARL JACOBI and GUSTAV PETER

LEJEUNE DIRICHLET; the latter was especially in-fluential on Riemann, who adopted Dirichlet’sintuitive, noncomputational approach to math-ematical ideas. Much of Riemann’s work lackedthe precise rigor common at that time—hefocused his energies on developing the correctconcepts and frameworks to understand mathe-matics. During this time, Riemann formulatedthe basic principles of his theory of complexvariables.

Riemann returned to Göttingen in 1849 fordoctoral work, and he submitted his thesis, con-ducted under Gauss’s supervision, in 1851. Thiswork introduces the geometrical objects thatcame to be known as Riemann surfaces. Riemannwas influenced by ideas from theoretical physicsand topology, and he brought these techniques tobear in his analysis of these surfaces, buildingupon AUGUSTIN-LOUIS CAUCHY’s more basic the-ory of complex variables. Some of his results wereproved using a variational technique known asDirichlet’s principle (Riemann attributed themethod to Dirichlet, although it had been devel-oped by Gauss and others previously). This the-sis was striking for its originality—even thesovereign Gauss was impressed.

For his postdoctoral work, Riemann beganinvestigating the representation of functions interms of a basis of trigonometric functions

advanced several other fields of mathematics, in-cluding complex analysis, the theory of ellipticfunctions, differential equations, the theory ofintegration, and topology. He is perhaps mostfamous for discovering the Riemann zeta func-tion, which is important to analytic numbertheory. Like those of many a genius, Riemann’sideas were so advanced that few were able to ac-cept them immediately; after his early death, theimpact of his research began to be appreciated.

Georg Friedrich Bernhard Riemann, com-monly known as Bernhard Riemann, was born onSeptember 17, 1826, in Breselenz, Germany. Hismother was Charlotte Ebell, and his father wasFriedrich Bernhard Riemann. Riemann main-tained a close relationship with his father, aLutheran minister, throughout his life. Riemannwas the second of six children. His father edu-cated him personally until he was 10 years old,

Bernhard Riemann, a founder of differential geometry,also contributed to complex analysis, topology, andthe theory of integration. (Courtesy of the Library ofCongress)

226 Riemann, Bernhard

(Fourier analysis); in the course of his research,he developed a rigorous theory of integration,constructing what later came to be known as theRiemann integral of a function. He was workingat Göttingen, and Gauss required him to give alecture on geometry for completion of his fel-lowship; Riemann’s lecture on geometry laterbecame very famous, as he laid down the basicprinciples and key ideas behind the theory ofdifferential geometry. This 1854 lecture devel-oped general concepts of space, dimension,straight lines, metrics, angles, and tangent placesfor curved surfaces. The result of this remarkablyoriginal exposition was the establishment ofdifferential geometry as a major field of mathe-matical inquiry (there were earlier works on dif-ferential geometry, but Riemann planted themajor ideas that would continue to guide thesubject throughout the next century), whichlater turned out to have a remarkable applica-tion to the general theory of relativity—AlbertEinstein, in the early 20th century, described theforce of gravity as essentially a curvature ofspace, and Riemann’s geometrical theory was theperfect mathematical basis for this importantnew branch of physics.

This lecture probed the fundamental con-cept of space to a remarkable depth, and fewscientists and mathematicians were able to ap-preciate the extraordinary genius of Riemann’spenetrating thought; perhaps Gauss alone wasable to truly grasp the significance of the newparadigm. Riemann next turned to the theory ofpartial differential equations, on which topic hegave a sparsely attended course. He obtained aprofessorship at Göttingen in 1857, the sameyear he published The Theory of AbelianFunctions. This work further investigates thetopological properties of Riemann surfaces, aswell as so-called inversion problems. Althoughother mathematicians—including KARL WEIER-STRASS—were working in this area, Riemann’swork was so far-reaching that he became a lead-ing thinker in this branch of mathematics.

Riemann again used the Dirichlet principle forhis results, and Weierstrass declared it to be in-valid for Riemann’s applications. The search foran alternative proof during the next severaldecades led to several other fruitful algebraicdevelopments; DAVID HILBERT eventually gavethe correct formulation and proof of Riemann’sresults around the turn of the century. As a resultof Weierstrass’s correct critique, many mathe-maticians abandoned the theories developed byRiemann, who maintained that they were true.

In 1858 Riemann was visited by ENRICO

BETTI, who imported Riemann’s topological ideasinto his own work. The next year Dirichlet died,and Riemann replaced him as the chair of math-ematics at Göttingen; Riemann was also electedto the Berlin Academy of Sciences through thestrong recommendations of ERNST EDUARD

KUMMER and Weierstrass. Riemann’s next area ofinquiry was number theory: he explored the zetafunction, already defined by LEONHARD EULER,by first extending it to the complex plane. Thiszeta function gives the sum of various infiniteseries and was already known to be related tothe set of prime numbers. Riemann’s workgreatly extended the knowledge of this function,as well as its applications; the famous Riemannhypothesis, which remains unsolved today, statesthat all the nontrivial roots of the zeta functionlie on the line in the complex plane defined bythe complex numbers whose real part is equal toone-half. This bizarre conjecture has been ex-tensively verified numerically, but a completeproof has escaped the concerted efforts of hun-dreds of mathematicians. The zeta function hasvarious applications to analytic number theory,such as estimating the number of primes lessthan a given integer.

Riemann suffered from poor health through-out his life. His weak constitution would laterimpede his research and take his life prematurely.Riemann married Elise Koch in 1862, but soonafterward he contracted a cold and then devel-oped tuberculosis. He spent much of his time

Riesz, Frigyes 227

over the next few years abroad in Italy, in thehope that the milder climate would soothe hisillness. Riemann returned to Göttingen in 1865,and his health declined rapidly thereafter; hetraveled to Italy in 1866 again for reasons ofhealth, but did not recover. He died on July 20,1866, in Selasca, Italy.

Riemann was easily one of the most influen-tial and creative mathematicians of the 19th cen-tury, and indeed of all history. He significantly af-fected geometry and complex analysis above all,essentially providing the framework throughwhich these subjects are studied today. And thedeep questions and issues that he addressed in thefield of geometry are extremely relevant to mod-ern conceptions of the physical universe. Hiswork in number theory has spurred an unparal-leled research effort—investigation of Riemann’szeta function must be one of the busiest arenas ofmathematical activity. Gauss would concur thatRiemann was certainly one of the greatest math-ematicians this world has seen.

Further ReadingKlein, F. Development of Mathematics in the 19th

Century. Brookline, Mass.: Math Sci Press, 1979.Monastyrsky, M. Rieman, Topology and Physics.

Boston: Birkhäuser, 1987.Weil, A. “Riemann, Betti and the Birth of Topology,”

Archive for History of Exact Sciences 20, no. 2(1979): 91–96.

� Riesz, Frigyes(1880–1956)HungarianAnalysis

Frigyes Riesz was one of the principal founders offunctional analysis in the early 20th century, ashe essentially invented operator theory and in-troduced many important concepts. His work hadimportant applications to physics—to quantummechanics in particular—and many of the rami-

fications of his theories were worked out over theensuing decades.

Frigyes Riesz was born on January 22, 1880,in Györ, Hungary. At the time of his birth, Györwas part of the Austro-Hungarian Empire. Riesz’sfather was Ignácz Riesz, a doctor; he had ayounger brother, Marcel, who also became afamous mathematician. After his preliminaryeducation, Riesz traveled to Budapest to study,and later journeyed to the Universities ofGöttingen and Zurich for further knowledge. Hereturned to Hungary to obtain his doctorate fromthe University of Budapest in 1902; his thesistopic was geometry.

After completing his doctorate, Riesz taughtat local schools before obtaining a university po-sition. In 1911 he finally earned a chair at theUniversity of Kolozsvár in Hungary. His mainwork was in functional analysis, and he built uponthe work of RENÉ-MAURICE FRÉCHET, building onhis idea of distance in a function space. Between1907 and 1909 Riesz developed some representa-tion theorems that expressed functionals in termsof integrals of other functions. Later he introducedthe notion of weak convergence for a sequence offunctions; this presented a convenient topologyfor the function spaces commonly used in physicsand engineering. Riesz also developed Lebesgueintegration theory, which facilitated the con-struction of orthonormal bases in Hilbert spaces.

In 1910 Riesz’s work marked the birth ofoperator theory; operators are the analogs of ma-trices in infinite-dimensional function spaces.The study and use of operators has continuedinto the present time, and they are certainly oneof the most effective mathematical tools of sta-tistics and engineering. In 1918 Riesz developedrigorous foundations for Banach spaces, whichwere axiomatically defined by STEFAN BANACH

two years later.In 1920 the territory of Hungary was se-

verely reduced as part of the aftermath of WorldWar I; as a result, Kolozsvár became located inRomania. The university that had been in

228 Russell, Bertrand

Kolozsvár was relocated to Szeged, and Rieszmoved with it. In 1922 Riesz founded theJános Bolyai Mathematical Institute and soonbecame editor of the Acta Scientiarum Mathe-maticarum, which became a renowned mathe-matical journal. Riesz later became the chair ofmathematics at the University of Budapest in1945.

Riesz’s research was important to the field offunctional analysis, but he also did work in er-godic theory (he proved the mean ergodic the-orem in 1938) and topology, the study of con-tinuously deformed surfaces. One of his mostimportant results, the Riesz-Fischer theorem of1907, is of great importance in Fourier analysis,which is used in engineering and physics. Besidesbeing an excellent researcher, Riesz was appre-ciated as a clear expositor of mathematics; hisstyle was lucid, with frequent reference to rele-vant applications. He received many honors andprizes throughout his life, including election tothe Hungarian Academy of Sciences.

Riesz died on February 28, 1956, in Budapest.He principally contributed to functional analy-sis, where his ideas were foundational; the tech-niques and concepts that he developed continueto have an impact and influence on mathemat-ics, physics, engineering, and statistics.

Further ReadingBernkopf, M. “The Development of Functional

Spaces,” Archive for History of Exact Sciences 3(1966–67): 1–96.

Rogosinski, W. “Frederic Riesz,” Journal of the LondonMathematical Society 31, no. 4 (1956): 508–512.

� Russell, Bertrand (Bertrand Arthur William Russell)(1872–1970)BritishLogic

Bertrand Russell was one of the more colorfulmathematical personalities of the 20th century,and he is ranked among the most important logicians of the modern era. He believed in thepotential for all of mathematics to be reduced tologic, and exerted much effort to validate thisparadigm. Russell was also an active philosopherand social revolutionary, applying his logicalideas to science, ethics, and religion.

Bertrand Russell was born on May 18, 1872,in Ravenscroft, Wales. He was the grandson ofLord John Russell. His mother and father diedin 1874 and 1876, respectively, so his grandpar-ents raised him. This grandfather had twiceserved as prime minister under Queen Victoria,but he died in 1878 and Russell’s grandmothercontinued the boy’s education. He receivedprivate education at first, and later was in-structed at Trinity College, Cambridge, where heattained first marks in mathematics.

Russell became an academic, eventually be-ing elected to the Royal Society in 1908. Hespent his early years pursuing his program oflogicism, which believed that all of mathemat-ics could be reduced to logical statements. Inthis sense, Russell was a follower of FRIEDRICH

LUDWIG GOTTLOB FREGE, who held a similar phi-losophy. Russell’s 1910 work on the PrincipiaMathematica (The principles of mathematics),written together with Alfred Whitehead, estab-lished that mathematical proofs could be re-duced to logical proofs. The first volumes of thiswork dealt with set theory, arithmetic, andmeasure theory; a fourth volume, on geometry,was not completed. Part of this approach, in-spired by the ideas of Frege, was to express num-bers and other mathematical objects as sets ofclasses that share a common property. This am-bitious project lost steam in later years, proba-bly due to philosophical trends leading awayfrom logicism.

Prior to the Principia, Russell acquiredfame through construction of the so-called

Russell, Bertrand 229

Russell paradox. He formed the set (set A) ofall sets having the property that they are notmembers of themselves. Then one asks thequestion: Is A (viewed as an element) a mem-ber of the set A? This cannot be resolved aseither true or false, since either answer leads toa contradiction. This demonstrated the funda-mental problem with taking collections of setsand assuming that such a collection is itself aset; more generally, it pointed out the difficul-ties with self-reference in mathematics andphilosophy. This concept of self-referencewould later be utilized by KURT GÖDEL to pro-duce his incompleteness theorems.

Russell’s solution to the paradox was to de-velop his theory of types, mainly fleshed out inhis 1908 Mathematical Logic as Based on theTheory of Types. Therein Russell described ahierarchy of classes, for which the idea of set isspecially defined at each level. Other resolutionsof the paradox have resulted from weakening thepower of the basic axiom of comprehension for-mulated by GEORG CANTOR, which states thatone can always gather objects sharing a commonproperty into a set. The immediate consequenceof the paradox was to cast doubt on the logicprogram espoused by DAVID HILBERT, whichsought to rigorously establish the foundations ofmathematical logic and set theory. It seemedthat even the intuitive concept of a set was castin shadow.

Besides these important contributions tologic, Russell was also famous for his “analyticphilosophy,” which attempted to cast philo-sophical questions in the rigorous framework ofmathematical logic. Of course, this computa-tional approach to philosophy has a long his-tory, going back to RENÉ DESCARTES and othermathematicians.

Russell’s personal and public life both inter-fered with his career advancement. He wasconvicted of antiwar activity in 1916, and thisresulted in his dismissal from Trinity College.

Two years later he was again convicted andsentenced to a short prison term. During his in-carceration, he wrote his famous Introduction toMathematical Philosophy (1919). He stumbledthrough four marriages that were rife with extra-marital affairs, and was even fired from a teach-ing position at City College of New York in 1940after a judge ruled that he was morally unfit. Heran (but failed to be elected) for Parliament threetimes; Russell became Earl Russell in 1931 afterthe death of his brother. He opened an experi-mental school with his second wife in the late1920s. His antiwar sentiments gained better re-ception in the 1950s and 1960s, when he wasrecognized as a leader in the anti–nuclear prolif-eration movement. The Russell-Einstein mani-festo of 1955 called for the abandonment ofnuclear weapons. In 1957 Russell organized thePugwash Conference, a convention of scientistsagainst nuclear weapons, and he became presi-dent of the Campaign for Nuclear Disarmamentin 1958. In his 80s Russell was arrested again in1961 for nuclear protests.

After a full life of mathematics, philosophy,and public protest, Russell died on February 2,1970, in Penrhyndeudraeth, Wales. Russell wasrecognized for his extensive contributions toliterature and science, winning the Nobel Prizefor literature in 1950. He is best known for hisparadox and its subsequent resolution throughthe theory of types, but also through his laterinvestigations of logicism and the issue ofincompleteness studied by Gödel. Russell’sthought has been enormously influential onlogic, mathematics, and philosophy, as well asethics, religion, and social responsibility.

Further ReadingAyer, A. Bertrand Russell. Chicago: University of

Chicago Press, 1988.Clark, R. The Life of Bertrand Russell. New York:

Knopf, 1976.

230 Russell, Bertrand

Kuntz, P. Bertrand Russell. Boston: Twayne Publishers,1986.

Monk, R. Bertrand Russell: The Spirit of Solitude. NewYork: Free Press, 1996.

Watling, J. Bertrand Russell. Edinburgh: Oliver andBoyd, 1970.

———. Bertrand Russell and His World. New York:Thames and Hudson, 1981.

Hardy, G. Bertrand Russell and Trinity. Cambridge:University Press, 1942.

Klemke, E. Essays on Bertrand Russell. Urbana:University of Illinois Press, 1970.

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� Seki Takakazu Kowa(1642–1708)JapaneseAlgebra, Calculus

Seki Takakazu was a singular figure in the his-tory of mathematics: At a time when mathe-matical activity in Japan was quite limited, Sekimade amazing discoveries, rivaling those ofWestern mathematicians such as GOTTFRIED

WILHELM VON LEIBNIZ. His achievements areremarkable in light of the fact that Seki couldnot benefit from a mathematical culture andcolleagues with whom to exchange ideas.

Seki was born in March 1642 in Fujioka,Japan. His family was of the samurai caste, buta family of the nobility, known as SekiGorozayemon, adopted Seki. Afterward, he wasidentified by this adopted surname. Seki was achild prodigy in mathematics. A household ser-vant introduced him to the subject when hewas nine years old, and Seki taught himselffrom that time. As he became an adult, Sekibuilt up a library of Chinese and Japanesemathematical books, and was gradually recog-nized as an expert—he became known as the“Arithmetical Sage.” He attracted a body ofpupils and sparked an upsurge in mathematicalactivity in Japan.

Seki served as an examiner of accounts forthe lord of Koshu, and when his master waspromoted Seki became a shogunate samurai in1704. He was later advanced to master of cere-monies in the shogun’s household.

Seki’s mathematical work, building upon for-mer Chinese mathematicians, represented a con-siderable advance in knowledge. He publishedHatsubi Sampo in 1674, a work that treated andsolved algebraic equations. In his exposition, Sekishows himself to be a careful and thoroughteacher, accounting for his popularity with pupils.In 1683 Seki studied matrix determinants, whichwere not examined in the West until a decadelater, when Leibniz used them to solve certainproblems. The so-called Bernoulli numbers,named for JAKOB BERNOULLI, were investigated an-tecedently by Seki. He utilized the concept ofnegative numbers in solving equations, but hadno knowledge of complex numbers. Seki also re-searched magic squares, following the work ofYANG HUI, and used the Newton-Raphson methodfor solving algebraic equations, discovered inde-pendently of SIR ISAAC NEWTON. His work onDiophantine equations was also considerable.

Little else is known of Seki, except that hedied on October 24, 1708, in Tokyo, Japan. It isdifficult to determine the extent to which hisschool was familiar with calculus, but it seems that

232 Steiner, Jakob

Seki made some progress in this area. This is amaz-ing, since Japan did not have the historicaltradition that Europeans could claim—namely,the geometrical works of the earlier Greek andArab civilizations. Seki should be viewed in thelineage of Chinese mathematicians, even thoughhe was Japanese, since he thoroughly studied theprior mathematics of the mainland.

Further ReadingMikami, Y. The Development of Mathematics in China

and Japan. New York: Chelsea PublishingCompany, 1974.

Smith, D., and Y. Mikami. A History of JapaneseMathematics. Chicago: Open Court PublishingCompany, 1914.

Yosida, K. “A Brief Biography of Takakazu Seki(1642?–1708),” The Mathematical Intelligencer 3(1981): 121–122.

� Steiner, Jakob(1796–1863)SwissGeometry

Jakob Steiner made several important contribu-tions to projective geometry, the study of spacesmade up of lines rather than points. This sub-ject was historically of interest to artists, who inthe medieval age required knowledge of projec-tion in order to correctly render perspective; inthe 19th century the subject was pursued largelyfor its own beauty.

Jakob Steiner was born on March 18, 1796,in Utzenstorf, Switzerland. His farmer parentsdiscouraged him from pursuing education, andSteiner did not learn to read and write until age14. When he was 18, he went to school for thefirst time, and attended the Universities ofHeidelberg and Berlin. In light of this late start,his considerable achievement in abstract math-ematics is all the more remarkable. His parentsdid not endorse his pursuit of mathematics, and

Steiner was forced to support himself throughtutoring.

By 1824 Steiner had already studied a se-ries of geometric transformations that he for-mulated into the theory of inversive geometry,a type of non-Euclidean geometry. Steiner pub-lished many of his writings in Crelle’s Journal,the first periodical entirely devoted to mathe-matics. His work focused on projective geome-try, and he discovered the Steiner surface—asurface that receives considerable attentionfrom geometers—as well as the Steiner theo-rem, which describes the projective propertiesof conic sections. The Poncelet-Steiner theo-rem is another famous result, which states thatonly a circle and a straight edge are necessaryto complete the various Euclidean construc-tions. His colleagues’ recognition of his sub-stantial efforts in projective geometry resultedin an appointment to a chair at the Universityof Berlin in 1834.

Steiner was awarded an honorary doctoratefor his geometric discoveries from the Universityof Königsberg in 1832, and two years later hetook up a chair of mathematics at the Universityof Berlin, where he remained until his death. Hisvarious contributions were published after hisdeath, in 1881 in the Collected Works.

In terms of his mathematical philosophy,Steiner avoided computation and algebra, be-lieving that true insight came through geomet-rical understanding rather than mindlesscalculations. He suffered from rheumatism inhis later years, resulting in a characteristicpained facial expression. Steiner died onApril 1, 1863, in Bern, Switzerland.

Steiner’s work on projective geometryhelped to establish that discipline as a majorbranch of mathematical inquiry. He was influ-enced by JEAN-VICTOR PONCELET, whose work hefurther developed. The Steiner surface contin-ues to be an object of study today, and Steiner’sresults are now of classical importance in cur-rent studies.

Stokes, George Gabriel 233

� Stevin, Simon(1548–1620)BelgianTrigonometry, Mechanics, Algebra

The late 16th and early 17th centuries were anexciting time for Europe, as science and mathe-matics began to flourish during this period. SimonStevin was a Belgian engineer who made inno-vative contributions to a variety of different fieldsof knowledge, including mathematics. It is inter-esting that many of the notations and conceptshe introduced have become indispensable to themodern presentation of mathematics.

Simon Stevin was born in 1548 in Bruges,Belgium. Little is known of his early years. He didnot have a formal university education; he pur-sued higher learning only later in his life. Stevinfirst worked as a bookkeeper in Antwerp, and lateras a tax clerk in Bruges. Later he moved to Leiden,and began to study at the University of Leiden in1583. At some point after graduation, Stevin be-came a quartermaster in the Dutch army.

Stevin’s diverse scientific accomplishmentsare described in his 11 books. He essentiallyfounded the science of hydrostatics, discoveringthat the pressure exerted by water upon a sur-face principally depends on the height of the wa-ter and the surface area. He defended the helio-centric conception of the universe propoundedby Copernicus, and discovered (prior to GALILEO

GALILEI) that objects of diverse weight fell at thesame rate, thus arriving at the uniform acceler-ation due to gravity. He made numerous contri-butions to navigation, geography, mechanics,and the science of fortification.

Stevin was also an accomplished engineer,having constructed numerous windmills, locks,and ports. He was an advisor on the project ofbuilding military fortifications, and mastered theart of opening sluices in order to flood the low-lands before the advance of an invading army. Healso invented a 26-passenger carriage equippedwith sails for use along the seashore.

In terms of mathematical achievements,Stevin promulgated the use of the decimal systeminto European mathematics (it had been previ-ously discovered and used by Arabic mathemati-cians) through his exposition of decimal fractionsin his 1585 book The Tenth, and in his work onalgebra he introduced the modern symbols forplus, minus, and multiplication. His notion of realnumber, which includes the irrational numbers inaddition to the rationals, became widely acceptedand facilitated the progress of European mathe-matics beyond the knowledge of the Greeks. Inparticular, Stevin accepted and used negativenumbers, already advocated by LEONARDO FI-BONACCI and JOHN NAPIER, and other contempo-rary mathematicians took up his ideas. He formu-lated mathematical theorems that influenced thedevelopment of statics and the study of physicalforces. His 1586 Statics and Hydrostatics presentedthe theorem relating forces via a triangle, equiva-lent to the parallelogram diagram of forces.

Stevin died in 1620 in The Hague, theNetherlands. He is remembered for his contri-butions to algebra, trigonometry, and hydrostat-ics. His confidence in the decimal system as pos-sessing fundamental importance for continueddevelopments in mathematics proved to be wellfounded, as history has subsequently established.

Further ReadingStruik, D. The Land of Stevin and Huygens. Dordrecht,

the Netherlands: D. Reidel Publishing Company,1981.

� Stokes, George Gabriel(1819–1903)IrishMechanics

George Stokes made important contributions tothe mathematical theory of hydrodynamics,coderiving the famous Navier-Stokes equations.His work extends to optics, gravity, and the study

234 Stokes, George Gabriel

of the Sun; his mathematical work in the areaof vector calculus is familiar to modern under-graduates.

George Gabriel Stokes was born onAugust 13, 1819, in Skreen, Ireland. His fatherwas Gabriel Stokes, a Protestant minister, andhis mother was a minister’s daughter. Due to hisparents’ backgrounds, Stokes and his brothersreceived a very religious upbringing; he was theyoungest of six children, and his three olderbrothers all became clergy. His childhood washappy, full of physical and mental activity.Stokes learned Latin from his father at an earlyage, and in 1832 pursued further studies inDublin. During the next three years at Dublin,Stokes lived with his uncle and developedhis natural mathematical talents. His father diedduring this period, which greatly affected Stokes.

In 1835 Stokes entered Bristol College inEngland, and he won several mathematicalprizes with his native intelligence. His teachersencouraged Stokes to pursue a fellowship atTrinity College, but Stokes instead matriculatedat Pembroke College, Cambridge, in 1837.Upon entering, he had little formal knowledgeof differential calculus, although under the tute-lage of William Hopkins he quickly filled thegaps in his education; Hopkins stressed the im-portance of astronomy and optics. In 1841Stokes graduated first place in his class, and thecollege awarded him a fellowship. At this point,Stokes decided to work as a private tutor andconduct his own private mathematical research.

Stokes first began research into hydrody-namics, familiarizing himself with the work ofGEORGE GREEN. In 1842 he published a work onthe motion of incompressible fluids, which helater discovered was quite similar to the resultsof Jean Duhamel; however, Stokes’s formulationwas sufficiently original to merit public dissem-ination. His 1845 work on hydrodynamics re-discovered CLAUDE-LOUIS NAVIER’s equations, butStokes’s derivation was more rigorous. Part of thereason for this duplication of research was thelack of communication between British and con-tinental mathematicians. At this time Stokesalso contributed to the theory of light and thetheory of gravity.

Stokes became recognized as a leading math-ematician in Britain: He was appointed Lucasianprofessor of mathematics at Cambridge in 1849and elected to the Royal Society in 1851. To sup-plement his income, he also accepted a profes-sorship of physics at the Government School ofMines in London. Stokes next published an im-portant work treating the motion of a pendulumin a viscous fluid, and made significant contribu-tions to the theory of the diffraction of light;Stokes’s mathematical methods in this area be-came classical. In 1852 he explained and namedthe phenomenon of fluorescence, basing this onhis elastic theory of the ether.

George Stokes contributed to the mathematical theoryof hydrodynamics and derived the Stokes theorem ofvector calculus. (Courtesy of AIP Emilio Segrè VisualArchives, E. Scott Bar Collection)

Stokes, George Gabriel 235

In 1857 Stokes moved into administrativeand empirical work, leaving his more theoreti-cal studies behind. This was partly due to his1857 marriage to Mary Susanna Robinson, whoprovided him with a distraction from his intensespeculations. Stokes performed an importantfunction in the Royal Society, operating as sec-retary from 1854 to 1885, and presiding as pres-ident afterward until 1890. Stokes received theCopley Medal from the Royal Society in 1893and served as master of Pembroke College from1902 to 1903. All of this administrative work se-riously distracted him from his original research,but at that time it was not atypical for greatscientists to obtain financial support through avariety of positions, since there was no publicfunding for research.

Stokes died on February 1, 1903, inCambridge, England. He was a profound influenceon the subsequent generation of Cambridgescientists, such as James Maxwell, and formed animportant link with the previous French mathe-

maticians working on scientific problems, such asAUGUSTIN-LOUIS CAUCHY, SIMÉON-DENIS POISSON,Navier, JOSEPH-LOUIS LAGRANGE, PIERRE-SIMON

LAPLACE, and JEAN-BAPTISTE-JOSEPH FOURIER. Hismathematical work, which mainly focused on ap-plied physics problems, later became a standardelement of the modern calculus curriculum.

Further ReadingKinsella, A. “Sir George Gabriel Stokes: The Malahide

Connection,” Irish Mathematical Society Bulletin 35(1995): 59–62.

Larmor, J. Memoir and Scientific Correspondence of theLate Sir George Gabriel Stokes. Cambridge, U.K.:Cambridge University Press, 1907.

Tait, P. “Scientific Worthies V—George GabrielStokes,” Nature 12 (July 15, 1875): 201–202.

Wilson, D. Kelvin and Stokes: A Comparative Study inVictorian Physics. Bristol: A. Hilger, 1987.

Wood, A. “George Gabriel Stokes 1819–1903, anIrish Mathematical Physicist,” Irish MathematicalSociety Bulletin 35 (1995): 49–58.

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� Tartaglia, Niccolò (Niccolò Fontana Tartaglia)(1499–1557)ItalianAlgebra

Niccolò Tartaglia is one of the notable Italianmathematicians of the early 16th century, andhe figures as a prominent character in one of themost pugnacious mathematical feuds of history.He independently discovered a method for solv-ing a general cubic equation, and for thisachievement he is principally known.

Tartaglia was born in 1499 in Brescia, Italy(then the Republic of Venice). His last namemeans “stammerer,” which he received as a nick-name due to his slow and difficult speech.Tartaglia’s father was a mail rider. Little is knownof his early childhood. In 1512 French maraud-ers captured his hometown, and Tartaglia sufferedsevere sword wounds to his face; he survived theordeal only through his mother’s tender care, andhe forever afterward wore a beard to disguise thescars.

Tartaglia taught himself mathematics, havingno money for formal education, and supportedhimself as a private mathematics teacher inVenice and Verona. He increased his meager rep-utation by participating in several public mathe-matics debates, in which he was quite successful.

SCIPIONE DEL FERRO’s student Antonio Fior ac-quired his master’s secret of solving the cubicequation, and with this armory challengedTartaglia to a contest in 1535. Confident in his -superior mathematical ability, Tartaglia accepted,but was soon bewildered by Fior’s cubic equations.Fior made little progress on Tartaglia’s problemsdue to his ignorance of negative numbers, but ina flash of inspiration Tartaglia discovered the se-cret to solving the cubic on the last evening ofthe competition. After obtaining the key formula,he was easily able to solve all of Fior’s problems,demonstrating that he was clearly superior.

Tartaglia’s contemporary GIROLAMO CARDANO,already interested in the cubic equation himself,attempted to learn the method from Tartaglia,communicating with him in 1539. However,Tartaglia jealously guarded his knowledge, anddivulged his secrets only after repeated coercionfrom Cardano; he made Cardano swear an oathnot to reveal the secret formula. The relation-ship between the two men later deteriorated, asTartaglia became embittered over revealing hishidden knowledge. This hatred came to a headin 1545, when Cardano published the secret for-mula after learning that del Ferro was the pre-vious discoverer.

In the ensuing feud, which was carried out inprint and involved personal insults and childishbickering, Cardano’s assistant LUDOVICO FERRARI

Thales of Miletus 237

challenged Tartaglia to a debate. Tartaglia wishedto spar with Cardano, but he accepted Ferrari’schallenge in order to secure a lectureship at Brescia.On August 10, 1548, the debate took place, andTartaglia was defeated, despite his extensive debateexperience. As a result, he lost his lectureship atBrescia and returned home to Venice in shame.

Besides his work on the cubic, Tartaglia isalso known for his early work on ballistics andartillery fire, presenting the first known firingtables. He provided the first Italian translationof EUCLID OF ALEXANDRIA’s Elements in 1543,and published Latin editions of ARCHIMEDES OF

SYRACUSE’s works.Tartaglia died on December 13, 1557, in

Venice. Although he did not attain glory or pre-eminence in his lifetime, he is remembered todayfor his codiscovery of the formula for the rootsof the cubic equation (with rational coefficients),now known as the Cardan-Tartaglia formula.

Further ReadingDrake, S., and I. Drabkin. Mechanics in 16th-Century

Italy: Selections from Tartaglia, Benedetti, GuidoUbaldo, and Galileo. Madison: University ofWisconsin Press, 1969.

� Thales of Miletus(ca. 625 B.C.E.–547 B.C.E.)GreekArithmetic

Thales is credited with founding the study ofmathematics in Greece, though much of his his-tory is questionable due to the lack of reliablerecords. At a time when many in Greece weremore concerned with mere survival, Thales de-voted his energies to contemplating science and,in particular, mathematics. Perhaps he should beremembered best as introducing a logical struc-ture to mathematical investigations, and he isresponsible for the concept of proof. To theGreeks of the classical era, Thales was an ex-alted figure, ranking among the Seven Sages.

It is thought that Thales was born around625 B.C.E., in the city of Miletus. Even thoughthis town was located in what is today Turkey, atthe time it was inhabited by Grecian peoples, andit was a thriving economic center. His parentswere Examyes and Cleobuline, who were proba-bly members of a distinguished Milesian family. Itis recorded that he died in his 78th year.

Little is known of his youth and early years.Apparently Thales was fairly successful as a mer-chant, as he had the leisure to pursue naturalphilosophy and early science. The tale is toldthat he made his fortune by buying up all thesurrounding olive presses; in any event, he spentsome time in Egypt while trading. There was noformal mathematics in Egypt, but the Egyptiansdid possess certain rules of thumb about “earthmeasurements,” which consisted of practicalgeometrical knowledge useful for demarking thefrequently flooded Nile River area. What Thales

Niccolò Tartaglia solved the cubic equation. (Courtesyof the Library of Congress)

238 Tsu Ch’ung-Chih

brought from Egypt he called geometry, which isGreek for “earth measurement.” In the process,he transformed a loose bunch of facts and state-ments into a cohesive discipline. He introducedabstraction into the study of geometry, so thatone could make general statements, and he uti-lized deductive reasoning, what is known now asa mathematical proof, as a means to true knowl-edge. It is hard to appreciate the conceptual leapinvolved in this change of thinking. TheEgyptians had long known that the numbers 3,4, 5 form the sides of a right triangle, but theywere not concerned with why it was true. TheGreek mentality was different; their intellectualcuriosity propelled their mathematical culturefar beyond that of their predecessors, and Thaleswas the father of this movement.

Several other tales are told of Thales.Certainly he was interested in astronomy, andhe is credited with introducing that science toGreece from Babylonia, where he may well havetraveled. Reputedly he foretold an eclipse ofthe Sun using a Babylonian calendar of lunarmonths and computed the height of theEgyptian pyramids via the length of their shad-ows. Apparently he also wore a politician’s hat,persuading the surrounding states to form a con-federacy. The Greeks also claim him as theauthor of a work on navigation, The Nautical StarGuide, and he would calculate the distance fromshore to a ship out at sea via the angle-side-angletheorem on the congruency of triangles. LaterGreeks, such as Eudemus, credit Thales with var-ious theorems, such as the statement that thecircle is bisected by its diameter.

Thales is also known as the founder of nat-ural philosophy, which seeks to explain the phe-nomena of the world without reference to mys-terious mystical forces and spirits. He postulatedthat the world floated on water, the primordialelement, which gave an explanation for earth-quakes that did not depend upon Poseidon, theEarth-shaker. This approach represented a para-digm shift in thinking about reality, and would

serve as a foundation for centuries of Greek phi-losophy and science. Indeed, Thales is said to bethe first philosopher holding a belief in an in-nate structure to the world.

As for mathematics, it owes its current formand method to Thales. Many cultures, such asEgypt, India, and the Mayans, had pursued whatmight be called “mathematics,” but their investi-gations were limited to practical rules, and therewas no method for establishing the validity ofthose rules. Deductive reasoning—starting fromacceptable axioms and proceeding to conclusionsthrough carefully constructed logical steps—made mathematics into a credible science, andthis vital contribution is due to Thales.

Further ReadingAnglin, W., and J. Lambek. The Heritage of Thales.

New York: Springer, 1995.Brumbaugh, R. The Philosophers of Greece. New York:

Crowell, 1964.Guthrie, W. The Greek Philosophers: From Thales to

Aristotle. New York: Harper and Row, 1960.Heath, T. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.Hooper, A. Makers of Mathematics. New York:

Random House, 1958.Laertius, D. Lives, Teachings, and Sayings of Famous

Philosophers. Cambridge, Mass.: Harvard Univer-sity Press, 1959.

Russell, B. History of Western Philosophy and ItsConnection with Political and Social Circumstancesfrom the Ancient Times to the Present Day. London:Allen and Unwin, 1961.

� Tsu Ch’ung-Chih (Zhu Chongzhi)(ca. 429–ca. 500)ChineseGeometry

Also known as Zhu Chongzhi, Tsu Ch’ung-Chihwas an early Chinese mathematician whose

Tsu Ch’ung-Chih 239

main contribution was to improve the approxi-mation of pi. Little is known of his life, and hisworks have not survived to the present age.

Tsu Ch’ung-Chih was employed in the serv-ice of the emperor Hsiao-wu, who reigned from454 to 464, and it has been deduced that he wasborn around 429. Tsu Ch’ung-Chih first servedas an officer in the province of Kiangsu, and lateras a military officer in the capital city ofNanking. Apparently, he completed severalworks on mathematics and astronomy duringthis employment. When the emperor died in464, he left the government in order to focusentirely on science.

Several standard Chinese mathematicalworks were probably familiar to Tsu Ch’ung-Chih. He was particularly interested in findinga better approximation of pi, since his prede-cessors had given the values 3, 92/29, and142/35 in their attempts. The better value of22/7 was also known since the fourth century.Tsu Ch’ung-Chih attempted to produce a bet-ter calculation, and it is recorded by Chinesehistorians that he constructed a circle of vastdiameter and arrived at upper and lower boundsof 3.1415927 and 3.1415926, respectively. Itseems that he was the first (in the East) to for-mulate upper and lower bounds this way; it iscurious that, believing pi must be a ratio, heconcluded that the exact value should be355/133, but that 22/7 could be used as a con-venient approximation.

Unfortunately, nothing is known of TsuCh’ung-Chih’s methods, but the seventh-century Chinese historian Wei Cheng guessedthat he developed and solved a system of linearequations to arrive at the ratio 355/133. Anotherhistorian supposed that Tsu Ch’ung-Chih’smethod lay in computing the area of a many-sided polygon that was inscribed in the circle.Whatever the case, it seems that Tsu Ch’ung-Chih’s works were lost due to their advancednature—they were inaccessible to contemporarymathematicians, and hence were not preserved.

Tsu Ch’ung-Chih also produced works onastronomy and calendar reform. Although hisnew calendar year was more precise, it was notimplemented due to the opposition of TsuCh’ung-Chih’s political enemies. He is believedto have died around 500. Tsu Ch’ung-Chih isprincipally known for his improvement of thevalue of pi, and is one of the early great Chinesemathematicians.

Further ReadingChen, C. “A Comparative Study of Early Chinese and

Greek Work on the Concept of Limit,” in Scienceand Technology in Chinese Civilization. Singapore:World Scientific, 1987.

Dennis, D., V. Kreinovich, and S. Rump. “Intervalsand the Origins of Calculus,” Reliable Computing4, no. 2 (1998): 191–197.

Martzloff, J. A History of Chinese Mathematics.Berlin/Heidelberg: Springer-Verlag, 1997.

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� Venn, John(1834–1923)BritishProbability, Logic

John Venn contributed to both probability andlogic through his research, and was one of thefirst mathematicians to introduce the symbolismof logic into the study of probability. He is mostwell known for the Venn diagrams that are usefulin the study of logic.

John Venn was born on August 4, 1834, inHull, England. His family belonged to the evan-gelical wing of the Church of England, and Vennbecame a minister briefly. He attended the twoLondon schools of Highgate and Islington, andstudied at Cambridge from 1853 to 1857. He waselected a fellow of his college, and he retainedthis fellowship all his life.

Venn took holy orders in 1859 and workedfor a short time as a minister before returning toCambridge as a lecturer on moral philosophy. Heresigned his clerical orders in 1883, due to hisincreasing disagreement with Anglican dogma,although Venn remained a devout church mem-ber. In the same year he was also elected to theRoyal Society.

Venn wrote several texts on probability andlogic, and these were quite popular in the late19th and early 20th centuries. Venn’s Logic of

Chance offered criticism of AUGUSTUS DE MORGAN

and GEORGE BOOLE—he was especially critical ofBoole’s algebraic approach to logic. Venn alsoconstructed the empirical definition of probabil-ity, which states that the chance of an event oc-curring is defined to be the long-term limit of theratio of times it historically occurred. This defi-nition has many advantages over the more clas-sical approach, as it allows for events that are notequally likely. However, one drawback is that thenotion of such a limit is not well defined. Thisled to later work on laws of large numbers andthe modern (or axiomatic) formulation of proba-bility theory.

Venn’s works on logic also contain geomet-ric diagrams to represent logical situations—hewas not the first to use such diagrams, asGOTTFRIED WILHELM VON LEIBNIZ had previouslyused them systematically, and LEONHARD EULER

developed the notion further. Venn’s diagramswere therefore based on an existing historicaltradition of such geometric aids; nevertheless,Venn systematically developed these geometri-cal representations. These drawings have beenused extensively in elementary mathematics togive young students training in logic.

Venn died on April 4, 1923, in Cambridge.Besides his efforts on improving the foundationsof logic, his work on diagrammatic representa-tions of logical events and their applications to

Viète, François 241

probability is most noteworthy. His approach hasbecome fairly standard in elementary studies inprobability.

Further ReadingSalmon, W. “John Venn’s Logic of Chance,” in

Proceedings of the 1978 Pisa Conference on theHistory and Philosophy of Science. Dordrecht, theNetherlands: D. Reidel Publishing Company,1981.

� Viète, François(1540–1603)FrenchAlgebra, Geometry

François Viète, along with PIERRE DE FERMAT,RENÉ DESCARTES, and BLAISE PASCAL, was one ofthe principal founders of European mathematics.He is known as the “father of algebra” due to hisintroduction of so many important concepts andnotations that are still in use. However, his math-ematical work was not limited to algebra—healso contributed to geometry, trigonometry, andanalysis.

Viète was born in 1540 in Fontenay-le-Comte, a town in the province of Poitou, France.His father, Étienne Viète, was a lawyer inFontenay-le-Comte, and his mother wasMarguerite Dupont. Viète followed his father’sprofession, graduating with a law degree fromthe University of Poitiers in 1560. He pursued alegal career for four years before abandoning itto pursue science and mathematics. Viète be-came a tutor to a nobleman’s daughter in thetown of La Rochelle.

In the ensuing years, the French Wars ofReligion continued to rage between RomanCatholics and Protestants. Viète was a Huguenot,and so naturally sided with the Protestants. Laterin his life he became a victim of religious perse-cution. Before 1570, when he left La Rochelle forParis, Viète worked on various topics of mathe-

matics and science, and he published his firstmathematical work, the Canon Mathematicusseu ad triangular (Mathematical laws applied totriangles), in 1571. This book was designed toprovide introductory mathematical material per-tinent to astronomy; it included various trigono-metric tables, as well as techniques for studyingflat and spherical triangles. Herein Viète first givesa notation for decimal fractions that is a precur-sor of modern notations. Notation, especiallyat this immature stage in the history of mathe-matics, was tremendously important for theadvancement of knowledge, since it gave aconvenient and appropriate language to expresssubtle ideas. Arguably, good notation is still vitallyimportant for modern abstract mathematics. Asalient example of this point is the Arabic numeralsystem, which is essentially a notation that hasgreatly facilitated calculation and number theory;another example is the notations of algebraicequations (with exponents for powers of unknownquantities and letters to designate variables orconstants) largely introduced by Viète himself.

In 1572 King Charles IX authorized the mas-sacre of the Huguenots, but Viète escaped andwas appointed as a counselor to the governmentof Brittany in 1573. In the ensuing years ofpolitical unrest, Viète worked for Henry III and,after his assassination, for Henry IV. Viète wasfirst appointed as a royal counselor to Henry IIIin 1580 but, after the rise of Roman Catholicpower in Paris, was banished in 1584 for hisProtestant faith. Viète spent the next five yearsat Beauvoir-sur-Mer, devoting himself to math-ematical pursuits.

He focused his initial labors on astronomy,and Viète desired to publish a major book, theAd harmonicon coeleste (The celestial harmony),on astronomy. This was never completed, butfour manuscript versions have survived the rav-ages of time. These manuscripts show that Viètewas mainly concerned with geometry and theplanetary theories of Copernicus and CLAUDIUS

PTOLEMY.

242 Viète, François

In 1588 the Catholics forced Henry III to fleeParis, and he sent for Viète to accompany him inexile. Viète was made a member of the king’s par-liament in his government at Tours. A Catholicfriar murdered Henry III in 1589, and Viète en-tered the service of the heir, Henry IV. Henry IV,formerly a Protestant, relied heavily on the abili-ties of Viète, who eventually decoded the secrettransmissions of the king of Spain, who was plot-ting an invasion of France. It is interesting thatthe Spanish king Philip II, confident in his cipher,thought the French cognizance of his militaryplans was accomplished through black magic. Inthis case, it was mathematics rather than sorcery.

These events took place in 1590, and Viètemeanwhile gave lectures at Tours. His lec-tures concerned various supposed advances inmathematics—for example, there was a proofthat the circle could be squared—and Viètedemonstrated that these arguments were faulty.Perhaps it shows a weakness of character inViète that he converted to Roman Catholicismin 1593, following the lead of his liege, whoprobably converted for political reasons. As aresult, Viète returned to Paris.

Shortly thereafter, Viète entered a competi-tion with the Dutch mathematician Adriaanvon Roomen, who posed a problem involving adegree 45 equation. Viète solved this problemand posed a geometrical question of his own. Asa result of this interchange, a friendship arosebetween Roomen and Viète. Viète continued inthe king’s service until his dismissal in 1602. Hedied on December 13, 1603, in Paris, France.

Viète is considered to be the preeminentfounder of algebra. Of course, there are numer-ous Arabic mathematicians (not to mentionGreeks) who made pivotal contributions byshaping conceptions of what constitutes arith-metic (for instance, the introduction of zero andnegative numbers). However, Viète certainlyproduced the first complete algebraic systemwith a consistent notation. In Introduction to theAnalytic Art, published in Tours in 1591, Viète

used familiar alphabetic symbols to designatevariables and constants, using vowels for un-knowns and consonants for knowns. Descarteslater introduced the convention that letters fromthe end of the alphabet (such as x, y, and z)should designate unknowns, whereas lettersfrom the beginning of the alphabet (such as a,b, and c) should denote known quantities.Nevertheless, Viète made a convincing case forhis notational system; previous literature onalgebraic equations relied on inconvenientexpressions, and often equations were describedwith sentences rather than abstract symbols. Theuse of symbols facilitated computation.

Viète made little use of Arabic mathemat-ics, preferring the style of the Italian algebraistslike GIROLAMO CARDANO. He should have in-vestigated the Arabic writings more carefully, asmany of the ideas Viète introduced were alreadyknown to the Arabs. However, Viète made a su-perior algebraic framework generally available toEuropean mathematicians. He further developedthe theory of algebraic equations, although hestill attached a geometrical interpretation toquantities, much as the Greeks did. This inessence limited the types of equations that hecould examine (for example, homogeneousequations). The next level of algebraic abstrac-tion was ushered in by the next generation,including Descartes and Fermat. Nevertheless,Viète’s notation for algebraic equations wasadopted with minor adjustment by these succes-sors. One may measure his influence by notingthat the term coefficient for the known constantmultiplying an unknown variable is due to Viète.

Besides the strictly algebraic work, Viète alsoresearched analysis, geometry, and trigonometry.He produced early numerical methods for solvingalgebraic equations, gave a new decimal approxi-mation for pi (as well as an infinite product char-acterization of it), and presented geometrical meth-ods for doubling the cube and trisecting an angle.

Viète’s mathematical work is clearly part ofan intellectual movement from Arabia to Italy

Volterra, Vito 243

to France, and his ideas were dependent on var-ious contemporaries as well as predecessors suchas Cardano and LEONARDO FIBONACCI. But hisalgebraic system represents the next stage inmathematical thinking about algebra, as it pro-vided a foundation for future exploration andgeneralization. Even though he regarded himselfas an amateur (and indeed he lacked formaltraining in mathematics), Viète was able tomake intellectual contributions that would ef-fect a paradigm shift in mathematical circles.

Further ReadingCrossley, J. The Emergence of Number. Victoria,

Australia: Upside Down A Book Company,1980.

� Volterra, Vito(1860–1940)ItalianAnalysis

Vito Volterra helped to extend the ideas ofdifferential and integral calculus from sets tospaces of functions. His work on biology alsobrought mathematical concepts, such as partialdifferential equations, to bear on predator-preyrelationships. He is most famous for his work onintegral equations, producing the “integral equa-tions of Volterra type,” which were widely ap-plied to mechanical problems.

Vito Volterra was born on May 3, 1860, inAncona (a town in the Papal States of Italy) toa poor family. His father died when Volterra wasonly two years old, and his early education is un-known. He became interested in mathematicsafter reading ADRIEN-MARIE LEGENDRE’s Geometryat age 11, and two years later he began studyingthe three-body problem, an outstanding ques-tion in the theory of dynamical systems.

Volterra attended lectures in Florence, andlater matriculated at Pisa in 1878; there he stud-ied under the direction of Betti, and obtained

his doctorate in 1882 with a thesis treating hy-drodynamics. Betti died the next year, andVolterra succeeded him as professor of mathe-matics at the University of Pisa. He went on toserve at both Turin and Rome.

Volterra was the first mathematician to con-ceive of what later came to be known as the“functional,” a function of real-valued functions.An example of a functional (this terminology waslater introduced by Jacques Hadamard) is the op-eration of integration, which produces a realvalue for every input function. Volterra was ableto extend the integral methods of SIR WILLIAM

ROWAN HAMILTON and CARL JACOBI for differen-tial equations to other problems of mechanics,and he developed a wholly new functional cal-culus to perform the necessary computations.Hadamard, RENÉ-MAURICE FRÉCHET, and otherthinkers later developed this original idea.

From 1892 to 1894 Volterra moved on topartial differential equations, investigating theequation of the cylindrical wave. His mostfamous results were in the area of integral equa-tions, which relate the integrals of various un-known functions. After 1896 Volterra publishedseveral papers in this area; he studied what cameto be known as “integral equations of theVolterra type.” He was able to apply his func-tional analysis to these integral equations withconsiderable success.

Despite his age, Volterra joined the Italian airforce during World War I, assisting with the de-velopment of blimps into weapons of war.Afterward he returned to the University of Rome.He promoted scientific collaboration and laterturned to the predator-prey equations of biology,studying the logistic curve. In 1922 fascism spreadthrough Italy, and Volterra fought vehementlyagainst this tide of oppression as a member of theItalian parliament. In 1830 the Fascists gainedcontrol, and Volterra was forced to flee Italy. Hespent the rest of his life abroad in France andSpain. However, he returned to Italy before hisdeath on October 11, 1940, in Rome.

244 Volterra, Vito

Volterra was important as a founder of func-tional analysis, which has been one of the mostapplied branches of mathematics in the 20thcentury. Integral equations have been success-fully employed to solve many scientific prob-lems, and Volterra’s work greatly advanced theknowledge of these equations.

Further ReadingAllen, E. “The Scientific Work of Vito Volterra,” The

American Mathematical Monthly 48 (1941):516–519.

Fichera, G. “Vito Volterra and the Birth of FunctionalAnalysis,” in Development of Mathematics 1900–1950. Basel, Switzerland: Birkhäuser, 1994.

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� Wallis, John(1616–1703)BritishCalculus, Geometry, Algebra

John Wallis was the greatest English mathemati-cian of his time; in fact, he is the first significantBritish mathematician of the 17th century. Henot only stimulated the study of mathematics,making it an attractive subject for others to pur-sue, but directly influenced SIR ISAAC NEWTON

through his early discoveries in the area of dif-ferential calculus.

John Wallis was born on November 23,1616, in Ashford, England. His father, also calledJohn Wallis, was a widely respected minister inAshford. Wallis’s mother, Joanna Chapman, wasthe second wife of Wallis’s father, and Wallis wasthe third of her five children. Wallis’s father diedwhen Wallis was six years old.

Wallis’s early education was at Ashford, butwhen the plague struck his mother sent him toJames Movat’s grammar school in 1625. He firstdisplayed his potential as a scholar there, train-ing both his memory and his understanding.Throughout life, Wallis was able to achieve greatfeats of mental calculation, even taking thesquare roots of irrational numbers in his mind.Next Wallis attended Martin Holbeach’s schoolin Felsted from 1631 to 1632, where he mastered

Greek, Latin, and Hebrew. Although he learnedlogic there, he received no training in mathe-matics until his brother taught him the rules ofarithmetic during Christmas vacation. The sub-ject appealed to him as a diversion, but he didnot pursue mathematics formally.

Wallis next came to Emanual College,Cambridge, in 1632, where he studied ethics,metaphysics, geography, astronomy, and medi-cine. He later defended his teacher Glisson’s newtheory of the circulation of the blood in publicdebate. Wallis completed his bachelor’s degreein 1637 and his master’s degree in 1640. He wasthen ordained, and served as a chaplain at var-ious posts over the next few years.

Wallis’s career took a turn when he suc-cessfully deciphered an encoded Royalist mes-sage in only two hours. This made him popularwith the Parliamentarians, and Wallis continuedto provide them with cryptographical servicethroughout the Civil War. As a reward for hiswork, Wallis was awarded care of the church ofSt. Gabriel of Fenchurch Street in London in1643. His mother died that year, leaving Wallisa considerable inheritance.

Wallis briefly held a fellowship at Cambridgein 1644, but he was forced to forsake this whenhe married Susanna Glyde in 1645. In London hebegan to meet regularly with a group of scientistsinterested in discussing medicine, geometry,

246 Wallis, John

astronomy, and mechanics; this group laterevolved into the Royal Society. Through themeetings he encountered William Oughtred’sClavis Mathematica (The key to mathematics) in1647, which he devoured in a few weeks. Thiswork stimulated Wallis’s love for mathematics, andencouraged him to begin his own investigations.

Wallis first wrote the Treatise on AngularSections, and discovered methods for solvingequations of degree four. In 1649 Oliver Cromwellappointed him to the Savilian chair of geometryat Oxford; his opponents contended that he se-cured this position through politics, though itseems that the appointment was justified, basedon the exceptional service that Wallis provided.Wallis held the post for more than 50 years, un-til his death; he was also made keeper of the uni-versity archives in 1657. In 1648 Wallis publiclydisagreed with the motion to execute Charles I.As a result, Charles II rewarded Wallis when the

monarchy was restored: His appointment as theSavilian chair was continued, and he was alsomade royal chaplain.

Wallis’s primary mathematical contributionlies in his work on the foundations of calculus.He first studied the work of Johannes Kepler andRENÉ DESCARTES, and then extended their earlyresults. His Arithmetica Infinitorum (The arith-metic of infinitesimals) of 1657 establishes aninfinite product expansion for half of pi, whichWallis discovered in the course of computing acertain integral. Wallis discovered how to inte-grate functions of the form 1 – x2 that wereraised to an integer power, and extended hisrules to fractional powers via interpolation, re-lying on Kepler’s notions of continuity. His workin this area would later influence Newton, whocarried the basics of calculus to a much greaterextent.

Wallis’s Tract on Conic Sections of 1655 pre-sented parabolas and circles as sets of pointssatisfying abstract algebraic equations. This ap-proach, familiar to the modern reader, differs fromthe classical definition, which describes thesecurves as the intersection of tilted planes with acone (they are conic sections). Thus Wallis’s stylewas reminiscent of Descartes’s analytic geometry.Wallis’s 1685 Treatise of Algebra shows his ac-ceptance of negative and complex roots. HereinWallis solves many algebraic equations, andprovides a wealth of historical material. Herestored some of the ancient Greek texts,including works by ARISTARCHUS OF SAMOS andARCHIMEDES OF SYRACUSE.

Besides Wallis’s mathematical work, hewrote on a variety of other topics, including et-ymology, logic, and grammar. He became in-volved in a feisty dispute with the philosopherThomas Hobbes, who in 1655 claimed to havesquared the circle, which was tantamount to dis-covering a rational number whose square was pi.Wallis refuted this false claim publicly, and arather nasty dispute followed that ended onlywith Hobbes’s death.

John Wallis obtained early results in calculus.(Courtesy of the Library of Congress)

Weierstrass, Karl 247

Wallis slept badly, perhaps because his ac-tive mind could not easily find rest. He diedon October 28, 1703, in Oxford, England. Heis principally remembered for his work on thefoundations of calculus, which influenced latermathematicians such as Newton; however, hismathematical labors extended to geometry andalgebra as well. It is also notable that Walliswas the first great English mathematician;he had no predecessors or teachers, but in hiswake mathematics became a more popularsubject.

Further ReadingScott, J. “The Reverend John Wallis, F.R.S.,” Notes

and Records of the Royal Society of London 15(1960–61): 57–68.

Scriba, C. “The Autobiograhy of John Wallis, F.R.S.,”Notes and Records of the Royal Society of London25 (1970): 17–46.

———. “A Tentative Index of the Correspondenceof John Wallis, F.R.S.,” Notes and Records of theRoyal Society of London 22 (1967): 58–93.

———. “Wallis and Harriot,” Centaurus 10 (1965):248–257.

� Weierstrass, Karl (Karl TheodorWilhelm Weierstrass)(1815–1897)GermanAnalysis, Complex Analysis

Karl Weierstrass has been described as the fa-ther of modern analysis. Indeed, his exactingstandards of rigor have become embedded inthe modern discipline of analysis, and many ofthe methods and topics are due to him.Weierstrass also made fundamental contribu-tions to complex analysis and the theory ofelliptic functions.

Karl Theodor Wilhelm Weierstrass was bornon October 31, 1815, in Ostenfelde, Germany.His father, Wilhelm Weierstrass, was a highly

educated civil servant. Weierstrass’s mother wasnamed Theodora Vonderforst, and Weierstrasswas her eldest child of four. When Weierstrasswas eight years old, his father became a tax in-spector, which involved constant relocation. In1827 Weierstrass’s mother died.

The family settled down in 1829 whenWeierstrass’s father secured a more permanentposition in Paderborn, and Weierstrass attendedthe local high school. There he excelled atmathematics above all subjects, and developedan unusual facility and love for the discipline.He was already reading Crelle’s Journal in 1834when he entered a finance program at theUniversity of Bonn. The career of finance wasnot Weierstrass’s choice, but rather his father’s;in rebellion and vexation of spirit, Weierstrasswasted his college years in excessive drinkingand fencing. Although he was truant from mostof his classes, Weierstrass continued his private

Karl Weierstrass, father of modern real analysis, madethe subject rigorous and introduced new topics, suchas functions that are continuous but nondifferentiable.(Courtesy of AIP Emilio Segrè Visual Archives)

248 Weierstrass, Karl

study of mathematics, and decided that he woulddevote his life to this one branch of knowledge.As a result of his squandered time, he was un-able to pass his examinations (he never eventook the tests), and he instead enrolled at theacademy in Münster in 1839 in order to becomea secondary school teacher. His father was deeplydisappointed with his son’s change of direction,but it was a fortunate decision for the history ofmathematics.

In 1840 Weierstrass passed his exams withexcellent results, having proved a certain deri-vation of NIELS HENDRIK ABEL’s from a differen-tial equation; his examiner thought the proofworthy of publication. Weierstrass went on toteach at the high school in Münster, and hewrote three papers between 1841 and 1842 oncomplex variables. In these papers Weierstrassreformulated the concept of an analytic functionin terms of convergent power series, as opposedto the typical approach through differentiation.Meanwhile, Weierstrass taught a variety ofsubjects, such as history, geography, and evengymnastics, and was utterly bored. The workloadwas quite heavy, because he conducted researchinto theoretical mathematics in every sparemoment. This busyness may have caused hissubsequent health problems, which started in1850: he suffered from attacks of dizziness,followed by nausea.

Weierstrass worked in Brauensberg from1848, but after the 1854 publication of hisToward the Theory of Abelian Functions, whichwas widely acclaimed by mathematicians, he re-ceived several offers from prominent universi-ties. This paper sketched the representation ofabelian functions as convergent power series,and the University of Königsberg conferredan honorary doctorate on him in 1854. ERNST

EDUARD KUMMER attempted to procure a postfor Weierstrass at the University of Breslau, butthis attempt failed. Weierstrass remained assenior lecturer at Brauensberg until 1856, whenhe accepted his dream job at the University of

Berlin. In the meantime he published a follow-up to his 1854 paper, which gave the full detailsof his method of inversion of hyperellipticintegrals.

Weierstrass’s tenure at Berlin, together withKummer and LEOPOLD KRONECKER, made thatschool the mathematical mecca of Germany atthat time. Weierstrass’s well-attended lectures ofthe next few years give insight into the diversityand profundity of his mathematical research: In1856 he discussed the theory of elliptic functionsapplied to geometry and mechanics, in 1859 hetackled the foundations of analysis, and in 1860he lectured on integral calculus. His investigationsproduced a continuous function that was nowheredifferentiable; the existence of such a bizarrefunction shattered most analysts’ overreliance onintuition, since until that time mathematicianscould only conceive of nondifferentiability oc-curring at isolated points. Weierstrass’s 1863course founded the theory of real numbers—anarea that other mathematicians such as RICHARD

DEDEKIND and GEORG CANTOR would also workon. He proved that the complex numbers are theonly commutative algebraic extension of the realnumbers—a result that CARL FRIEDRICH GAUSS

previously stated but never proved.Weierstrass’s health problems continued,

and he experienced a total collapse in 1861; hetook the next year off to recover, but he wasnever the same. From that time, he had an as-sistant to write his lectures, and chronic chestpains replaced his dizziness.

Weierstrass organized his various lecturesinto four main courses: analytic functions, ellip-tic functions, abelian functions, and the calcu-lus of variations. The courses were fresh andstimulating, since much of the material was hisown innovative research. It is a testimony to thelegacy of his style that modern courses in analy-sis follow Weierstrass’s progression of topics,including the power series concept of a function,continuity and differentiability, and analyticcontinuation.

Weyl, Hermann 249

Weierstrass collaborated with Kummerand Kronecker profitably for many years, butlater he and Kronecker parted ways over theradical ideas of Cantor; Weierstrass was sup-portive of Cantor’s innovative ideas in settheory, but Kronecker could not accept thepathological constructions. Weierstrass hadmany excellent students, some of whom be-came famous mathematicians themselves, suchas Cantor, SOPHUS LIE, and FELIX KLEIN. He pri-vately instructed SOFIA KOVALEVSKAYA, whowas not allowed to formally enroll due to hergender. Weierstrass had great intellectual rap-port with this woman, whom he assisted infinding a suitable position.

Weierstrass was very concerned with math-ematical rigor. His high standards became im-pressed on the succeeding generation, andsparked intensive research into the foundationsof mathematics, such as the construction of thereal number system. Weierstrass’s studies of con-vergence led him to distinguish different types,thus sparking research into various topologies forfunction spaces. He studied the concept of uni-form convergence, which preserves continuity,and devised various tests for the convergence ofinfinite series and products. His approach topublishing was careful and methodical, so thathis publications were few but extremely deep andexact.

Weierstrass continued to teach until 1890.His last years were devoted to publishing the col-lected works of JAKOB STEINER and CARL JACOBI.He died of pneumonia on February 19, 1897, inBerlin, Germany. His contributions to mathe-matics, in particular to real and complex analy-sis, were extensive and far-reaching, earning himthe epithet of “father of modern analysis.” Hisinfluence was also extended through the largenumber of talented students whom he mentoredand who further developed his ideas in variousnew directions. From his humble beginnings asa high school teacher, Weierstrass accomplishedgreat things for the field of mathematics.

Further ReadingManning, K. “The Emergence of the Weierstrassian

Approach to Complex Analysis,” Archive forHistory of Exact Sciences 14, no. 4 (1975):297–383.

� Weyl, Hermann(1885–1955)GermanGeometry, Number Theory

Hermann Weyl, one of the great mathematiciansof the early 20th century, successfully developedthe ideas of others into rigorous theories. His pa-pers were remarkable for their originality anddepth of insight, and his work is quite influen-tial on present research.

Hermann Weyl was born on November 9,1885, in Elmshorn, Germany. As a boy he at-tended the Gymnasium at Altona, and enteredthe University of Göttingen at age 18. He re-mained there for several years, studying mathe-matics. After obtaining his degree, he became aprofessor at the University of Zurich in 1913.

Weyl had studied under DAVID HILBERT atGöttingen and was surely one of his most tal-ented pupils. Weyl’s first major work, in 1910,was on the spectral theory of differential equa-tions, which was an area that Hilbert was alsoinvestigating. In 1911 Weyl began studying thespectral theory of certain operators in so-calledHilbert spaces. His methods provided some geo-metric insight into these abstract spaces, andbecame important techniques in functionalanalysis.

In 1916 Weyl published a famous paper onanalytic number theory, treating the distributionof certain special sequences of numbers. Withcharacteristic ingenuity, Weyl gave a novel so-lution to the unsolved questions by making con-nections with integration theory. His techniqueshave remained relevant to the additive theoryof numbers.

250 Wiener, Norbert

After this work in number theory, Weylturned back to geometry (he had previously, in1913, given a rigorous foundation for the intuitivedefinition of a Riemannian manifold). In 1915 heattacked a problem concerned with certain de-formations of convex surfaces, and outlined amethod of proof that would eventually prove fruit-ful. Weyl was interrupted by World War I, but wasfreed from military duty in 1916. In Zurich heworked with Albert Einstein, and consequentlybecame interested in the general theory of rela-tivity. Weyl set out to provide a mathematicalfoundation for the physical ideas, discovering theconcept of a linear connection—ÉLIE CARTAN

further developed this important idea.In the 1920s Weyl became interested in Lie

groups, and his papers on this subject are prob-ably his most important and influential. Part ofthe genius of his approach was the use of topo-logical methods on algebraic objects such as Liegroups. SOPHUS LIE had introduced Lie groups asan interesting new field of mathematics, butWeyl greatly advanced this branch through hisnew methodology.

As a mathematician, Weyl believed in theimportance of abstract theories, and he believedthat they were capable of solving classical prob-lems when combined with careful, penetratingthought. He differed with the formalist Hilberton the philosophy of mathematical foundations,and instead accepted LUITZEN EGBERTUS JAN

BROUWER’s intuitionism. However, in manyother aspects he displayed Hilbert’s influence.In 1930 he succeeded Hilbert at Göttingen, butdecided to leave Nazi Germany in 1933, arriv-ing at Princeton’s Institute for Advanced Study.He remained in the United States until he re-tired in 1951. He divided the last years of hislife between Princeton and Zurich. He died onDecember 8, 1955.

Hermann Weyl made several significantcontributions to number theory, geometry, anddifferential equations. When he solved a diffi-cult problem, he often devised some wholly new

technique for the proof; these new methodsoften became standard tools, or sometimes ledto new areas of research. His work on the the-ory of Lie groups provided a foundation for lateradvances.

Further Readingvan Dalen, D. “Hermann Weyl’s Intuitionistic

Mathematics,” Bulletin of Symbolic Logic 1, no. 2(1995): 145–169.

Newman, M. “Hermann Weyl,” Journal of the LondonMathematical Society 33 (1958): 500–511.

Wheeler, J. “Hermann Weyl and the Unity ofKnowledge,” American Scientist 74, no. 4 (1986):366–375.

� Wiener, Norbert(1894–1964)AmericanProbability, Statistics

Norbert Wiener was one of the great Americanmathematicians of the 20th century. His ideaswere profound and rich, if poorly expressed, andrevolutionized the theory of communications aswell as harmonic analysis. Wiener is also fa-mous for founding the discipline of cybernet-ics, or the application of statistical ideas tocommunication.

Norbert Wiener was born on November 26,1894, in Columbia, Missouri. His father, LeoWiener, was a Russian Jew who had immigratedto the United States, and was a professor of mod-ern languages at the University of Missouri atthe time of his son’s birth. Wiener’s mother wasa German Jew originally named Bertha Kahn.He had one younger sister. Due to his father’sextensive intellectual interests (he publishedseveral books and was widely read in the sci-ences), Wiener received an excellent home ed-ucation that placed him far beyond boys his ownage. In fact, Wiener started high school at theage of nine and graduated in 1906.

Wiener, Norbert 251

It seems that Wiener’s father was largelyresponsible for the development of his child’sgenius. As a boy, Wiener was quite clumsy andsuffered from poor eyesight; when the doctor rec-ommended that Wiener cease reading for sixmonths, his father continued his mathematicaleducation. As a result, Wiener developed greatcapacities for memorization and mental calcula-tion at a young age. Wiener’s family had movedto Boston—Leo Wiener taught at Harvard—andthe boy attended Tufts College. He graduated in1909 with a degree in mathematics, and com-menced graduate school at Harvard at only14 years of age.

Originally Wiener studied zoology, but laterchanged to philosophy, earning his doctorate fromHarvard at age 18. Wiener then journeyed toEngland to continue his philosophical studies withBERTRAND RUSSELL, who told him that he neededto learn more mathematics. So Wiener studied un-der GODFREY HAROLD HARDY, and spent most of1914 at the University of Göttingen studyingdifferential equations under DAVID HILBERT. He

returned to the United States before the outbreakof World War I, and worked a number of odd jobs:He taught philosophy at Harvard, worked forGeneral Electric, and was also a staff writer forthe Encyclopedia Americana. At the end of the warhe obtained a position at the MassachusettsInstitute of Technology (MIT).

It was at MIT that Wiener first began study-ing Brownian motion, an important concept inprobability (it is a continuous time stochasticprocess used to model a variety of phenomena,from the motion of small particles to the evolu-tion of the stock market) and other topics ofprobability. He also investigated harmonicanalysis and its application to the statistical the-ory of time series. Much of the work that Wienerfound resulted from conversations with his en-gineering colleagues, who were eager to obtainmathematical assistance with their own engi-neering problems.

Wiener frequently traveled to France,Germany, and England in order to collaboratewith European mathematicians—he workedwith RENÉ-MAURICE FRÉCHET and PAUL LÉVY. Hemarried Margaret Engemann in 1926. He spent1931–32 in England working with Hardy, wherehe also met KURT GÖDEL.

Wiener’s genius certainly fulfilled many of thecommon stereotypes of mathematicians. His pa-pers were often difficult to read, with brilliant dis-coveries given inadequate proof; sometimes, hewould launch into great detail over trivial mat-ters. Despite his poor writing skills, Wiener’scontributions were outstanding. His 1921 work onBrownian motion set this important idea of par-ticle physics on a solid theoretical foundation; hisfurther research into the space of continuous one-dimensional curves led to the intuitively appeal-ing so-called Wiener measure, which facilitatedthe calculation of probabilities of the Brownianmotion paths. In 1923 he investigated the partialdifferential equation known as Dirichlet’s prob-lem, and this led to great advances in potentialtheory. From 1930 he labored in harmonic

Norbert Wiener researched statistics and harmonicanalysis and solved the signal extraction problem inthe theory of stationary time series. (MassachusettsInstitute of Technology Museum and HistoricalCollections, courtesy of AIP Emilio Segrè VisualArchives)

252 Wiener, Norbert

analysis, winning the Bôcher prize from theAmerican Mathematical Society in 1933. Wienerdelved into the various applications of the Fouriertransform—one large application was the so-called spectral analysis of time series. With thetools that Wiener developed, it was possible to fil-ter, forecast, and smooth data streams. His 1948Cybernetics: Or, Control and Communication in theAnimal and the Machine applied ideas from me-chanical systems to biology, such as feedback,stability, and filtering. Apparently this work wasa chaotic mess of poorly written text and brilliantflashes of insight.

Wiener died on March 18, 1964, inStockholm, Sweden. This child prodigy was a no-toriously poor lecturer, sloppy writer, and out-standing thinker. His most important work was inprobability theory and harmonic analysis, and his

influence is still felt today in such subjects as par-tial differential equations, stochastic processes,and the statistical analysis of time series.

Further ReadingHeims, S. John Von Neumann and Norbert Wiener:

From Mathematics to the Technologies of Life andDeath. Cambridge, Mass.: MIT Press, 1980.

Jerison, D., and D. Stroock. “Norbert Wiener,”Notices of the American Mathematical Society 42(1995): 430–438.

Mandrekar, V. “Mathematical Work of NorbertWiener,” Notices of the American MathematicalSociety 42 (1995): 664–669.

Wiener, N. Ex-Prodigy: My Childhood and Youth. NewYork: Simon and Schuster, 1953.

———. I Am a Mathematician. Garden City, N.Y.:Doubleday, 1956.

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� Yang Hui(ca. 1238–ca. 1298)ChineseArithmetic

Yang Hui was a Chinese official who developeda decimal system of numbers. The 10-digit num-ber system had already arrived in China throughIndia, but the use of decimals to represent frac-tions was previously unknown.

Little is known of Yang Hui’s life, but hisbirth is dated around 1238 in China (the loca-tion is unknown). A minor government offi-cial, Yang found time to write two books in1261 and 1275 that advanced the decimal rep-resentation of fractions, represented in themodern formulation. Apparently he relied onthe 11th-century work of Jia Xian, who deviseda method of calculating the roots of polynomi-als. Jia used the so-called Pascal’s triangle toextend the computation of square and cuberoots to higher-degree polynomials. Yang wasfamiliar with Jia’s work, and discusses hismethods in his own books.

Yang’s 1275 book, Alpha and Omega ofVariations on Multiplication and Division, providesan interesting document on mathematics edu-cation. Herein Yang emphasizes the importanceof true understanding over rote memorization,which would characterize modern approaches to

mathematics education in China and the East.Yang died sometime around 1298 in China.

Yang Hui did not contribute extensively tomathematics, but he is notable for his introduc-tion of decimals into Chinese mathematics, cen-turies before such an advance was made in Europe.

Further ReadingLam, L. A Critical Study of the “Yang Hui Suan fa”: A

Thirteenth-Century Chinese Mathematical Treatise.Singapore: Singapore University Press, 1977.

Needham, J. Science and Civilisation in China.Cambridge, U.K.: Cambridge University Press,1959.

� Yativrsabha(ca. 500–570)IndianNumber Theory

Mathematics in the sixth century in India wasdeveloped under the Jain culture; Yativrsabha,who was a Jain intellectual, developed currentmathematics and also tied his work to the oldertraditions.

Little is known of Yativrsabha’s life, but helived in the sixth century, which is known be-cause he refers to the termination of the Guptadynasty, which occurred in 551. He studied under

254 Yule, George Udny

Arya Manksu and Nagahastin, and he compiledseveral works that expounded Jain traditions.He is also dated by the fact that JinabhadraKsamasramana references Yativrsabha’s work in609; also, Yativrsabha refers to a work bySarvanandin in 458.

Yativrsabha’s main work, the Tiloyapannatti,contains a description of the universe and certainmathematical formulas that are typical of this erain India; his mathematics is representative of theprogress in Jain mathematical thought that haddeveloped from the older canonical works. Thebook describes various units for measuring dis-tance and time, notable for their massive scale. Infact, Yativrsabha’s system gives a first concept formeasuring infinite distances. In the Jaina cosmol-ogy, the universe was infinite in space and time;Yativrsabha’s work provided a method for meas-uring these increasingly larger quantities.

Yativrsabha’s mathematics are surely a prod-uct of his own Jain culture, and this comfort withinfinity facilitated the development of transfi-nite mathematics. Centuries before GEORGE

CANTOR, Yativrsabha first dabbled with the con-cept that there were varying degrees of infinity,and this hierarchy could be measured and stud-ied. However, he exerted little lasting influencein this direction since the study of infinity wouldnot be resumed until the 17th and 18th cen-turies in Europe.

Further ReadingJain, L. “Aryabhata I and Yativrsabha—a Study in

Kalpa and Meru,” Indian Journal of History ofScience 12, no. 2 (1977): 137–146.

� Yule, George Udny(1871–1951)BritishStatistics

George Yule was an important statistician of theearly 20th century who stimulated a generation

of students and contributed to a wide range oftopics. His work on correlation, regression, andtime series is most noteworthy.

George Udny Yule was born on February 18,1871, in Morham, Scotland. His father, whoshared the same name, was a colonial adminis-trator in India; he was later knighted for his serv-ices. Yule received a good education, and heattended Winchester College until 1887. Nexthe went on to study engineering at UniversityCollege, London, graduating in 1890. Yule spentthe next two years working in an engineeringlab, but this vocation failed to stimulate him,and he soon turned to experimental physics. In1892 Yule resided in Bonn, studying physicsunder the guidance of Heinrich Hertz, and pub-lished four papers on electricity.

Despite having a good start in physics, Yulewas dissatisfied, and in 1893 he returned toUniversity College as a demonstrator—a positionthat Karl Pearson had procured for him. WithPearson, Yule worked on the notion of correlationcoefficient and related this important statisticto linear regression. His 1895 work On theCorrelation of Total Pauperism with Proportion ofOutrelief addressed these topics, and its excellenceheralded Yule’s election to the Royal StatisticalSociety that same year. Yule’s subsequent workswere very influential in the social sciences, wherein the following decades his methods came to pre-dominate. He was advanced to assistant professorat University College in 1896, but three yearslater left for a better-paid position at the City andGuilds London Institute.

Yule continued to produce a large volumeof papers, and his annual Newmarch lectures atUniversity College became incorporated intohis widely read Introduction to the Theory ofStatistics (1911). The text was a great success,and in 1911 Yule was awarded the RoyalStatistical Society’s highest award, the GuyMedal in Gold. Yule was intimately connectedto the Royal Statistical Society, since he servedas secretary from 1907 to 1919 and president

Yule, George Udny 255

from 1924 to 1926. In 1912 he accepted a postat Cambridge and dwelled at St. John’s Collegefor most of his remaining life. Besides these ac-tivities, Yule served as a statistician in the armyduring World War I.

During the 1920s Yule was especially pro-ductive: He introduced the correlogram andmeasures of serial correlation to the statisticalstudy of time series. In addition, he promulgatedautoregressive time series models that are stillwidely used today. He retired from Cambridgein 1930, but he continued to make new editionsfor his enduringly popular Introduction to theTheory of Statistics. An aficionado of fast cars,Yule was determined to obtain a pilot’s licensefor his retirement. Unfortunately, he suffered aheart ailment in 1931 that incapacitated him.He died on June 26, 1951, in Cambridge,England.

Yule was a kind man who was very accessi-ble to students. Besides his contributions to theconcept of correlation and regression, Yule wasimportant for the impetus that he gave to math-ematical statistics through his textbook and hisinteractions with his colleagues.

Further ReadingJohnson, N., and S. Kotz. “George Udny Yule,” in

Leading Personalities in Statistical Sciences. NewYork: John Wiley, 1997.

Kendall, M. “George Udny Yule, 1871–1951,” Journalof the Royal Statistical Society 115 (1952): 156–161.

Stigler, S. The History of Statistics: The Measurementof Uncertainty before 1900. Cambridge, Mass.:Belknap Press of Harvard University Press, 1986.

Yates, F. “George Udny Yule,” Obituary Notices ofFellows of the Royal Society of London 8 (1952):309–323.

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� Zeno of Elea(ca. 490 B.C.E.–ca. 425 B.C.E.)GreekLogic

The classical Greek mathematicians shied awayfrom the study of infinity, both the infinitelylarge and the infinitely small (the infinitesimal).Infinitesimals are the cornerstone of calculus,and many Greeks, such as ARCHIMEDES OF

SYRACUSE, made the first faltering steps towarda full discovery of calculus. However, the ma-jority rejected the notion of infinitely divisiblequantities, such as a continuum, and this reac-tion was largely due to the paradoxes of Zeno.

Zeno of Elea was born in approximately490 B.C.E. in Elea, Italy. He was of Greek descentdespite his birth in Italy, and he is classed withthe Greek philosophers. There exists very littlereliable information on his life, but it is said thathis father was Telautagoras. Zeno eventuallystudied at the school of philosophy at Elea,where he met his master Parmenides. TheEleatic school, founded by Parmenides, taughtmonism—the concept that all is one. Thisphilosophy influenced Zeno to formulate vari-ous paradoxes that challenged the concepts ofinfinite divisibility.

Plato claims that Zeno and Parmenidestraveled to Athens in 450 B.C.E., where they

met the young Socrates and discussed philoso-phy with him. Before traveling to Athens, Zenohad already acquired some fame through thepublication of his book (which has not sur-vived) containing 40 paradoxes. These para-doxes form a deeply stimulating dissection ofthe concept of the continuum, thereby dis-turbing comfortable notions of such commonthings as motion, time, and space. One ofZeno’s assumptions is that of divisibility: If amagnitude can be divided in two, then it canbe divided forever. The work of RICHARD

DEDEKIND would later establish this continuumproperty for the real numbers. Zeno also as-sumed that any object of zero magnitude (hedid not express it this way, since the Greeks didnot have zero) does not exist.

In the paradox labeled “The Dichotomy,”Zeno states that in order to traverse a distance, itis first necessary to traverse half that distance; butto get halfway, it is first required to go a quarterof the way. Continuing this reasoning indefinitely,Zeno concludes that to begin is impossible, andthat therefore motion is impossible. This paradoxis typically resolved by summing the geometric se-ries of reciprocal power of two. In “The Arrow,”Zeno states that motion is impossible, because(assuming that the current “now” instance of timeis indivisible) if an arrow moves some distance inan indivisible instant of time, then it moved half

Zermelo, Ernst 257

that distance in half the time, thereby resultingin a division of the instant. This can be resolvedby allowing time to be a continuum—infinitelydivisible.

Zeno’s most famous paradox is that ofAchilles: It states that a race between the Greekhero Achilles and a tortoise is run, and the slowtortoise is given a head start. After some timehas elapsed, Achilles catches up half the inter-vening distance. But the tortoise has meanwhilemoved on; Achilles then runs half the remain-ing intervening distance, but again the tortoisehas advanced farther. Carrying this argument oninfinitely far, Zeno concludes that Achilles cannever catch up! This, too, can be resolved bysetting up an appropriate geometric series.However, the resolutions of these paradoxes relyon certain notions of infinity and properties ofthe continuum. The mathematical structurebehind these concepts was not developed untilmany centuries later. SIR ISAAC NEWTON,GOTTFRIED WILHELM VON LEIBNIZ, and BLAISE

PASCAL laid the modern foundations of calculus,along with a host of others discussed in thisbook. More advanced work on the continuum,as well as the basic properties of the real num-bers, was conducted in the late 19th century byGEORG CANTOR, FRIEDRICH LUDWIG GOTTLOB

FREGE, and BERTRAND RUSSELL, among others.Thus, Zeno’s influence was far-reaching, in thathe asked some very deep questions about time,space, and motion.

Zeno died sometime around 425 B.C.E., anda questionable source relates that he was exe-cuted after a failed attempt to remove a tyrantof Elea. Although he was a philosopher, Zeno’sideas sparked a mathematical revolution mil-lennia later, since his paradoxes pointed to theneed to provide a rigorous foundation to intu-itive concepts of space and time. His paradoxesconcerning motion demonstrated the difficul-ties of considering velocity as distance dividedby time, since this ratio appears to become zerodivided by zero when the elapsed time of travel

is reduced to zero; only with the discoveryof limits and infinitesimals in the discipline ofdifferential calculus was this conundrum re-solved. Besides providing a plethora of mentalobstacles for later intellectuals, Zeno alsoserved to inhibit the growth of Greek mathe-matics to encompass the infinite; thus, he wasa retarding influence classically, but millennialater became an impetus for mathematicaldevelopment.

Further ReadingBarnes, J. The Presocratic Philosophers. London:

Routledge and Kegan Paul, 1982.Grunbaum, A. Modern Science and Zeno’s Paradoxes.

London: Allen and Unwin, 1968.Guthrie, W. A History of Greek Philosophy. Cambridge,

U.K.: Cambridge University Press, 1981.Heath, T. A History of Greek Mathematics. Oxford:

Clarendon Press, 1921.Kirk, G., J. Raven, and M. Schofield. The Presocratic

Philosophers: A Critical History with a Selection ofTexts. Cambridge: Cambridge University Press,1983.

Laertius, D. Lives, Teachings, and Sayings of FamousPhilosophers. Cambridge, Mass.: Harvard Univer-sity Press, 1959.

Russell, B. The Principles of Mathematics. London:G. Allen and Unwin, 1964.

Salmon, W. Zeno’s Paradoxes. Indianapolis: Bobbs-Merrill, 1970.

Sorabji, R. Time, Creation and the Continuum. Ithaca,N.Y.: Cornell University Press, 1983.

� Zermelo, Ernst (Ernst Friedrich Ferdinand Zermelo)(1871–1953)GermanLogic

The revolution in mathematical logic and settheory that took place in the early 20th century

258 Zermelo, Ernst

had many important participants, includingErnst Zermelo. His axiomatic construction of settheory has been of great importance for the de-velopment of mathematics, since all of modernmathematics is now built upon set theoreticfoundations.

Ernst Zermelo was born on July 27, 1871,the son of Ferdinand Rudolf Theodor Zermelo,a college professor, and Maria Augusta ElisabethZieger. The young Zermelo received his sec-ondary education in Berlin, and he pursued thestudy of mathematics, philosophy, and physics atschools in Berlin, Halle, and Freiburg. His teach-ers included Georg Frobenius, Max Planck, andHerman Schwartz. In 1894 he obtained his doc-torate at the University of Berlin with a thesisstudying the calculus of variations. Even thoughZermelo would obtain fame through his re-searches into set theory, he maintained hisinterest and knowledge of the calculus of varia-tions throughout his life.

After working for several years on hydro-dynamics, Zermelo obtained a position at theUniversity of Göttingen in 1899, and he be-came titular professor there a year after his 1904proof of the well-ordering theorem. This result,which earned Zermelo instant recognitionamong contemporary mathematicians, statedthat any set could be well ordered (that is, onecan construct an ordering relation that allowsone to compare any two elements of the set,and determine which comes first). This sur-prising theorem says that any set looks like theset of real numbers (where the ordering is the“less than” symbol <).

Zermelo was interested in physics, and hada knack for finding applications of mathemat-ics to practical problems. For example, he an-alyzed the strength of chess competitors andstudied the fracture of a sugar cube. In 1900 hebegan lecturing on GEORG CANTOR’s set theory,which he had carefully digested; a few yearslater he produced his well-ordering theorem,

and in 1908 produced a second proof. In thesame year Zermelo set up an axiom system forCantor’s set theory that is commonly used to-day. The axioms carefully avoid Russell’s para-dox, but employ the controversial axiom ofchoice, which states that any disjoint union ofnonempty sets has a subset containing exactlyone element from each of the original sets. Theprovability of the famous continuum hypothe-sis is contingent on this axiom of choice. Somemathematicians dispense with it, while mostview it as intuitive.

In 1910 Zermelo accepted a professorship atZürich, but he retired six years later due to poorhealth. From 1916 to 1926 he lived in the BlackForest of Germany, regaining his health. He nextcame to the University of Freiburg im Breisgau.(He broke off connection with the school in1935 in protest against the Nazi regime. AfterWorld War II he was reinstated.) Meanwhile,the logicians Adolf Fraenkel and ThoralfSkolem had made certain criticisms of Zermelo’ssystem, pointing out the weakness of the seventhaxiom of infinity. In 1929 Zermelo responded tothis critique with an axiomatization of theproperty of definiteness, which is used to definesets through the common properties of theirelements.

Zermelo made a few additional contribu-tions to set theory, attempting to abolish prooftheory in 1935, but he had already accomplishedhis most important work. He died at theUniversity of Freiburg im Breisgau on May 21,1953. Although Zermelo’s proof of the well-ordering theorem was important, he is remem-bered principally for his axiomatic formulation ofset theory, which is still widely influential today.Besides providing a rigorous set theoretic foun-dation for most (or all) of modern mathematics,his work initiated pure research in mathematicalset theory. Today, many mathematicians are in-vestigating the consequences of various systemsof axioms, testing them for their strengths

Zermelo, Ernst 259

and weaknesses in terms of consistency andcompleteness.

Further Readingvan Heijenoort, J. From Frege to Gödel: A Source Book

of Mathematical Logic. Cambridge, Mass.: HarvardUniversity Press, 1967.

Moore, G. “Ernst Zermelo, A. E. Harward, and theAxiomatization of Set Theory,” Historia Mathe-matica 3, no. 2 (1976): 206–209.

Peckhaus, V. “Ernst Zermelo in Göttingen,” Historyand Philosophy of Logic 11, no. 1 (1990): 19–58.

260

ENTRIES BY FIELD

ALGEBRA

Abel, Niels HenrikAryabhata IBrahmaguptaCardano, GirolamoCh’in Chiu-ShaoChu Shih-ChiehFerrari, LudovicoFerro, Scipione delGalois, EvaristeHermite, CharlesJordan, Camilleal-Karaji, Abual-Khwarizmi, AbuKlein, FelixKummer, ErnstLi ChihLiouville, JosephNoether, EmmySeki Takakazu KowaTartaglia, NiccolòViète, François

ANALYSIS

Agnesi, Maria GaetanaBabbage, CharlesBaire, René-LouisBanach, StefanBessel, Friedrich WilhelmBirkhoff, George DavidBolzano, Bernhard

Borel, ÉmileDedekind, RichardEudoxus of CnidusFatou, Pierre-Joseph-LouisFourier, Jean-Baptiste-JosephFréchet, René-MauriceFredholm, IvarFubini, GuidoJacobi, CarlLebesgue, Henri-LéonLiouville, JosephLipschitz, RudolfNapier, JohnRiesz, FrigyesVolterra, VitoWeierstrass, Karl

ARITHMETIC

Adelard of BathBacon, RogerFibonacci, LeonardoThales of MiletusYang Hui

CALCULUS

Barrow, IsaacCarnot, LazareGreen, GeorgeGregory, JamesLeibniz, Gottfried

Wilhelm von

Maclaurin, ColinNewton, Sir IsaacWallis, John

COMPLEX ANALYSIS

Cauchy, Augustin-LouisHamilton, Sir William RowanRiemann, Bernhard

DIFFERENTIAL EQUATIONS

Bernoulli, JakobBernoulli, JohannKovalevsky, Sonya

GEOMETRY

Apollonius of PergaArchimedes of SyracuseBolyai, JánosCartan, ÉlieCavalieri, BonaventuraDemocritus of AbderaDesargues, GirardDescartes, RenéEratosthenes of CyreneEuclid of AlexandriaGrassmann, Hermann Günteral-Haytham, Abu AliHippocrates of ChiosIbrahim, ibn SinanLeonardo da VinciLie, Sophus

Entries by Field 261

Lobachevsky, NikolaiMenelaus of AlexandriaMinkowski, HermannMonge, GaspardOresme, NicolePappus of AlexandriaPascal, BlaisePoncelet, Jean-VictorSteiner, JakobTsu Ch’ung ChiWeyl, Hermann

LOGIC

Boole, GeorgeBrouwer, Luitzen

Egbertus JanCantor, GeorgDe Morgan, AugustusFrege, Friedrich Ludwig

GottlobGödel, Kurt FriedrichHilbert, DavidLovelace, Augusta Ada ByronPeano, GiuseppePeirce, CharlesRussell, BertrandZeno of EleaZermelo, Ernst

MECHANICS

Alembert, Jean d’Bernoulli, Daniel

Galilei, GalileoGibbs, Josiah WillardHeaviside, OliverHuygens, ChristiaanLagrange, Joseph-LouisNavier, Claude-Louis-Marie-

HenriStokes, George Gabriel

NUMBER THEORY

Diophantus of AlexandriaDirichlet, Gustav Peter

LejeuneEuler, LeonhardFermat, Pierre deGermain, SophieGoldbach, ChristianHardy, Godfrey

HaroldKronecker, LeopoldLegendre, Adrien-MariePythagoras of SamosRamanujan, Srinivasa

AiyangarYativrsabha

PROBABILITY

Bayes, ThomasChebyshev, Pafnuty

LvovichLaplace, Pierre-SimonLévy, Paul-Pierre

Markov, AndreiMoivre, Abraham dePoisson, Siméon-DenisPólya, GeorgeVenn, JohnWiener, Norbert

STATISTICS

Fisher, Sir RonaldAylmer

Gauss, Carl FriedrichGosset, WilliamPearson, Egon SharpeYule, George Udny

TOPOLOGY

Betti, EnricoHopf, HeinzMöbius, AugustPoincaré, Jules-Henri

TRIGONOMETRY

Aristarchus of SamosBhaskara IIHipparchus of RhodesMadhava of

SangamagrammaPtolemy, ClaudiusRegiomontanus, Johann

MüllerRheticus, GeorgStevin, Simon

262

ENTRIES BY COUNTRY OF BIRTH

ARABIA

al-Haytham, Abu AliIbrahim, ibn Sinanal-Karaji, Abual-Khwarizmi, Abu

AUSTRIA

Gödel, Kurt Friedrich

BELGIUM

Stevin, Simon

CHINA

Ch’in Chiu-ShaoChu Shih-ChiehLi ChihTsu Ch’ung ChiYang Hui

CZECHOSLOVAKIA

Bolzano, Bernhard

FRANCE

Alembert, Jean d’Baire, René-LouisBorel, ÉmileCarnot, LazareCartan, ÉlieCauchy, Augustin-LouisDesargues, GirardDescartes, René

Fatou, Pierre-Joseph-LouisFermat, Pierre deFourier, Jean-Baptiste-JosephFréchet, René-MauriceGalois, EvaristeGermain, SophieHermite, CharlesJordan, CamilleLaplace, Pierre-SimonLebesgue, Henri-LéonLegendre, Adrien-MarieLévy, Paul-PierreLiouville, JosephMoivre, Abraham deMonge, GaspardNavier, Claude-Louis-Marie-

HenriOresme, NicolePascal, BlaisePoincaré, Jules-HenriPoisson, Siméon-DenisPoncelet, Jean-VictorViète, François

GERMANY

Bessel, Friedrich Wilhelm

Cantor, GeorgDedekind, RichardDirichlet, Gustav Peter

Lejeune

Frege, Friedrich LudwigGottlob

Gauss, Carl FriedrichGoldbach, ChristianGrassmann, Hermann GünterHilbert, DavidHopf, HeinzJacobi, CarlKlein, FelixKronecker, LeopoldKummer, ErnstLeibniz, Gottfried

Wilhelm vonLipschitz, RudolfMöbius, AugustNoether, EmmyRegiomontanus, Johann

MüllerRheticus, GeorgRiemann, BernhardWeierstrass, KarlWeyl, HermannZermelo, Ernst

GREAT BRITAIN

Adelard of BathBabbage, CharlesBacon, RogerBarrow, IsaacBayes, ThomasBoole, George

Entries by Country of Birth 263

Fisher, Sir RonaldAylmer

Gosset, WilliamGreen, GeorgeGregory, JamesHardy, Godfrey HaroldHeaviside, OliverLovelace, Augusta

Ada ByronMaclaurin, ColinNapier, JohnNewton, Sir IsaacPearson, Egon SharpeRussell, BertrandVenn, JohnWallis, JohnYule, George Udny

GREECE

Apollonius of PergaArchimedes of SyracuseAristarchus of SamosDemocritus of AbderaDiophantus of

AlexandriaEratosthenes of CyreneEuclid of AlexandriaEudoxus of CnidusHipparchus of RhodesHippocrates of ChiosMenelaus of AlexandriaPappus of AlexandriaPtolemy, ClaudiusPythagoras of SamosThales of MiletusZeno of Elea

HOLLAND/NETHERLANDS

Bernoulli, DanielBrouwer, Luitzen Egbertus JanHuygens, Christiaan

HUNGARY

Bolyai, JánosPólya, GeorgeRiesz, Frigyes

INDIA

Aryabhata IBhaskara IIBrahmaguptaDe Morgan, AugustusMadhava of

SangamagrammaRamanujan, Srinivasa

AiyangarYativrsabha

IRELAND

Hamilton, Sir William RowanStokes, George Gabriel

ITALY

Agnesi, Maria GaetanaBetti, EnricoCardano, GirolamoCavalieri, BonaventuraFerrari, LudovicoFerro, Scipione delFibonacci, LeonardoFubini, GuidoGalilei, GalileoLagrange, Joseph-Louis

Leonardo da VinciPeano, GiuseppeTartaglia, NiccolòVolterra, Vito

JAPAN

Seki Takakazu Kowa

NORWAY

Abel, Niels HenrikLie, Sophus

POLAND

Banach, Stefan

RUSSIA AND THE SOVIET UNION

Chebyshev, Pafnuty LvovKovalevskaya, SonyaLobachevsky, NikolaiMarkov, AndreiMinkowski, Hermann

SWITZERLAND

Bernoulli, JakobBernoulli, JohannEuler, LeonhardSteiner, Jakob

SWEDEN

Fredholm, Ivar

UNITED STATES

Birkhoff, George DavidGibbs, Josiah WillardPeirce, CharlesWiener, Norbert

264

ENTRIES BY COUNTRY OF MAJORSCIENTIFIC ACTIVITY

ARABIA

al-Haytham, Abu AliIbrahim ibn Sinanal-Karaji, Abual-Khwarizmi, Abu

BELGIUM

Stevin, Simon

CHINA

Ch’in Chiu-ShaoChu Shih-ChiehLi ChihTsu Ch’ung ChiYang Hui

CZECHOSLOVAKIA

Bolzano, Bernhard

EGYPT

Apollonius of PergaDiophantus of AlexandriaEratosthenes of CyreneEuclid of AlexandriaPappus of AlexandriaPtolemy, Claudius

FRANCE

Alembert, Jean d’

Baire, René-LouisBorel, ÉmileCarnot, LazareCartan, ÉlieCauchy, Augustin-LouisDesargues, GirardFatou, Pierre-Joseph-LouisFermat, Pierre deFourier, Jean-Baptiste-JosephFréchet, René-MauriceGalois, EvaristeGermain, SophieHermite, CharlesJordan, CamilleLagrange, Joseph-LouisLaplace, Pierre-SimonLebesgue, Henri-LéonLegendre, Adrien-MarieLévy, Paul-PierreLiouville, JosephMonge, GaspardNavier, Claude-Louis-Marie-

HenriOresme, NicolePascal, BlaisePoincaré, Jules-HenriPoisson, Siméon-DenisPoncelet, Jean-VictorViète, François

GERMANY

Bernoulli, JohannBessel, Friedrich

WilhelmCantor, GeorgDedekind, RichardDirichlet, Gustav Peter

LejeuneFrege, Friedrich Ludwig

GottlobGauss, Carl FriedrichGrassmann, Hermann

GünterHilbert, DavidJacobi, CarlKlein, FelixKronecker, LeopoldKummer, ErnstLeibniz, Gottfried

Wilhelm vonLie, SophusLipschitz, RudolfMinkowski, HermannMöbius, AugustNoether, EmmyRegiomontanus, Johann

MüllerRheticus, GeorgRiemann, Bernhard

Entries by Country of Major Scientific Activity 265

Weierstrass, KarlWeyl, HermannZermelo, Ernst

GREAT BRITAIN

Adelard of BathBabbage, CharlesBacon, RogerBarrow, IsaacBayes, ThomasDe Morgan, AugustusFisher, Sir Ronald

AylmerGreen, GeorgeGregory, JamesHardy, Godfrey HaroldHeaviside, OliverLovelace, Augusta

Ada ByronMaclaurin, ColinMoivre, Abraham deNapier, JohnNewton, Sir IsaacPearson, Egon SharpeRamanujan, Srinivasa

AiyangarRussell, BertrandStokes, George GabrielVenn, JohnWallis, JohnYule, George Udny

GREECE

Aristarchus of SamosDemocritus of AbderaEudoxus of CnidusHipparchus of RhodesHippocrates of Chios

Thales of MiletusZeno of Elea

HOLLAND/NETHERLANDS

Brouwer, Luitzen Egbertus Jan

Descartes, RenéHuygens, Christiaan

HUNGARY

Bolyai, JánosRiesz, Frigyes

INDIA

Aryabhata IBhaskara IIBrahmaguptaMadhava of SangamagrammaYativrsabha

IRELAND

Boole, GeorgeGosset, WilliamHamilton, Sir William

Rowan

ITALY

Agnesi, Maria GaetanaArchimedes of SyracuseBetti, EnricoCardano, GirolamoCavalieri, BonaventuraFerrari, LudovicoFerro, Scipione delFibonacci, LeonardoFubini, GuidoGalilei, GalileoLeonardo da Vinci

Menelaus of AlexandriaPeano, GiuseppePythagoras of SamosTartaglia, NiccolòVolterra, Vito

JAPAN

Seki Takakazu Kowa

NORWAY

Abel, Niels Henrik

RUSSIA AND THE SOVIET UNION

Banach, StefanBernoulli, DanielChebyshev, Pafnuty

LvovichEuler, LeonhardGoldbach, ChristianLobachevsky, NikolaiMarkov, Andrei

SWITZERLAND

Bernoulli, JakobHopf, HeinzPólya, GeorgeSteiner, Jakob

SWEDEN

Fredholm, IvarKovalevskaya, Sonya

UNITED STATES

Birkhoff, George DavidGibbs, Josiah WillardGödel, Kurt FriedrichPeirce, CharlesWiener, Norbert

266

ENTRIES BY YEAR OF BIRTH

699 B.C.E.–600 B.C.E.Thales of Miletus

599 B.C.E.–500 B.C.E.Pythagoras of Samos

499 B.C.E.–400 B.C.E.Democritus of AbderaEudoxus of CnidusHippocrates of ChiosZeno of Elea

399 B.C.E.–300 B.C.E.Aristarchus of Samos

299 B.C.E.–200 B.C.E.Apollonius of PergaArchimedes of SyracuseEratosthenes of CyreneEuclid of Alexandria

199 B.C.E.–100 B.C.E.Hipparchus of Rhodes

1–100Menelaus of AlexandriaPtolemy, Claudius

201–300Diophantus of AlexandriaPappus of Alexandria

401–500Aryabhata ITsu Ch’ung ChihYativrsabha

501–600Brahmagupta

701–800al-Khwarizmi, Abu

901–1000al-Haytham, Abu AliIbrahim ibn Sinanal-Karaji, Abu

1101–1200Adelard of BathBhaskara IIFibonacci, LeonardoLi Chih

1201–1300Bacon, RogerCh’in Chiu-ShaoChu Shih-ChiehYang Hui

1301–1400Madhava of SangamagrammaOresme, Nicole

1401–1500Ferro, Scipione delLeonardo da VinciRegiomontanus, Johann

MüllerTartaglia, Niccolò

1501–1550Cardano, GirolamoFerrari, LudovicoNapier, JohnRheticus, GeorgStevin, SimonViète, François

1551–1600Cavalieri,

BonaventuraDesargues, GirardDescartes, RenéGalilei, Galileo

1601–1620Fermat, Pierre deWallis, John

1621–1640Barrow, IsaacGregory, JamesHuygens, ChristiaanPascal, Blaise

Entries by Year of Birth 267

1641–1660Bernoulli, JakobLeibniz, Gottfried

Wilhelm vonNewton, Sir IsaacSeki Takakazu

Kowa

1661–1680Bernoulli, JohannMoivre, Abraham de

1681–1700Goldbach, ChristianMaclaurin, ColinNewton, Sir Isaac

1701–1720Agnesi, Maria

GaetanaAlembert, Jean d’Bayes, ThomasBernoulli, DanielEuler, Leonhard

1721–1740Lagrange, Joseph-Louis

1741–1760Carnot, LazareLaplace, Pierre-SimonLegendre, Adrien-MarieMonge, Gaspard

1761–1780Fourier, Jean-Baptiste-JosephGauss, Carl FriedrichGermain, Sophie

1781–1800Babbage, CharlesBessel, Friedrich WilhelmBolzano, BernhardCauchy, Augustin-Louis

Green, GeorgeLobachevsky, NikolaiMöbius, AugustNavier, Claude-Louis-Marie-

HenriPoisson, Siméon-DenisPoncelet, Jean-VictorSteiner, Jakob

1801–1810Abel, Niels HenrikBolyai, JánosDe Morgan,

AugustusDirichlet, Gustav Peter

LejeuneGrassmann, Hermann

GünterHamilton, Sir William

RowanJacobi, CarlKummer, ErnstLiouville, Joseph

1811–1820Boole, GeorgeGalois, EvaristeLovelace, Augusta

Ada ByronStokes, George GabrielWeierstrass, Karl

1821–1830Betti, EnricoChebyshev, Pafnuty

LvovichHermite, CharlesKronecker, LeopoldRiemann, Bernhard

1831–1840Dedekind, RichardGibbs, Josiah

Willard

Jordan, CamilleLipschitz, RudolfPeirce, CharlesVenn, John

1841–1850Cantor, GeorgFrege, Friedrich Ludwig

GottlobHeaviside, OliverKlein, FelixKovalevskaya, SonyaLie, Sophus

1851–1860Markov, AndreiPeano, GiuseppePoincaré, Jules-HenriVolterra, Vito

1861–1870Cartan, ÉlieFredholm, IvarHilbert, DavidMinkowski, Hermann

1871–1880Baire, René-LouisBorel, ÉmileFatou, Pierre-Joseph-

LouisFréchet, René-MauriceFubini, GuidoGosset, WilliamHardy, GodfreyHaroldLebesgue, Henri-LéonRiesz, FrigyesRussell, BertrandYule, George UdnyZermelo, Ernst

1881–1890Birkhoff, George David

268 A to Z of Mathematicians

Brouwer, Luitzen Egbertus Jan

Fisher, Sir Ronald Aylmer

Lévy, Paul-PierreNoether, Emmy

Pólya, GeorgeRamanujan, Srinivasa AiyangarWeyl, Hermann

1891–1900Banach, Stefan

Hopf, HeinzPearson, Egon SharpeWiener, Norbert

1901–1910Gödel, Kurt Friedrich

269

CHRONOLOGY

Gray lifelines indicate approximate dates

100 C.E.600 B.C.E. 500 400 300 200 100 0 700600 800200 300 400 500

925 950 975 1000 1025 1050 1075 1100 1125 1150 1175 1200 1225 1250 1275

I

Thales of Miletus

Pythagoras of Samos

Zeno of Elea

Hippocrates of Chios

Democritus of Abdera

Eudoxus of Cnidus

Aristarchus of Samos

Euclid of Alexandria

Archimedes of Syracuse

Apollonius of Perga

Eratosthenes of Cyrene

Hipparchus of Rhodes

Menelaus of Alexandria

Claudius PtolemyDiophantus of Alexandria

Pappus of Alexandria

Tsu Ch’ung ChiAryabhata I

Yativrsabha

Brahmagupta

Abu Al-Khwarizmi

Ibrahim ibn Sinan

Abu Al-Karaji

Abu Ali Al-Haytham

Bhaskara II

Adelard of Bath

Leonardo Fibonacci

Ch’in Chiu-Shao

Li Chih

270 A to Z of Mathematicians

Gray lifelines indicate approximate dates

i

Roger Bacon

Yang Hui

Chu Shih-Chieh

Nicole Oresme

Madhava of Sangamagramma

Johann Müller Regiomontanus

Scipione del Ferro

Leonardo da Vinci

Niccoló Tartaglia

Georg Rheticus

Girolamo Cardano

Ludovico Ferrari

François Viète

Simon StevinJohn Napier

Galileo Galilei

Girard DesarguesRené Descartes

Bonaventura Cavalieri

Pierre de Fermat

John Wallis

Blaise Pascal

Christiaan Huygens

Isaac Barrow

James Gregory

Seki Takakazu Kowa

Sir Isaac Newton

Jakob Bernoulli

Gottfried Wilhelm von Leibniz

Johann Bernoulli

Christian Goldbach

Abraham de Moivre

Colin Maclaurin

Daniel Bernoulli

Thomas BayesLeonhard Euler

Jean d’Alembert

Maria Gaetana AgnesiJoseph-Louis Lagrange

Gaspard Monge

Pierre-Simon Laplace

Adrien-Marie Legendre

Lazare Carnot

1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950

Chronology 271

Jean-Baptiste-Joseph Fourier

Sophie Germain

Carl Friedrich Gauss

Bernhard Bolzano

Siméon-Denis Poisson

Friedrich Wilhelm Bessel

Jean-Victor Poncelet

Claude-Louis-Marie-Henri Navier

Augustin-Louis Cauchy

Charles Babbage

August Möbius

Nikolai Lobachevsky

George Green

Jakob SteinerNiels Henrik Abel

János Bolyai

Carl JacobiGustav Peter Lejeune Dirichlet

Sir William Rowan Hamilton

Augustus De Morgan

Hermann Günter Grassmann

Joseph Liouville

Ernst Kummer

Evariste Galois

George Boole

Augusta Ada Byron Lovelace

Karl Weierstrass

Pafnuty Lvovich Chebyshev

George Gabriel Stokes

Charles Hermite

Leopold Kronecker

Enrico Betti

Bernhard Riemann

Richard DedekindRudolf Lipschitz

John Venn

Camille Jordan

Josiah Willard GibbsCharles Peirce

Sophus Lie

Georg Cantor

Friedrich Ludwig Gottlob Frege

Felix Klein

1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950

272 A to Z of Mathematicians

Oliver Heaviside

Sonya Kovalevskaya

Jules-Henri Poincaré

Andrei Markov

Giuseppe Peano

Vito Volterra

Hermann Minkowski

David Hilbert

Ivar Fredholm

Émile Borel

Élie Cartan

George Udny Yule

Ernst Zermelo

Bertrand RussellRené-Louis Baire

Henri-Léon Lebesgue

William GossetGodfrey Harold Hardy

Pierre-Joseph-Louis Fatou

René-Maurice Fréchet

Guido Fubini

Frigyes Riesz

Luitzen Egbertus Jan Brouwer

Emmy Noether

George Birkhoff

Hermann Weyl

Paul-Pierre Lévy

Srinivasa Aiyangar Ramanujan

George Pólya

Sir Ronald Aylmer Fisher

Heinz Hopf

Stefan Banach

Norbert Wiener

Egon Sharpe PearsonKurt Friedrich Gödel

1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950

273

BIBLIOGRAPHY

Aaboe, A. Episodes from the Early History ofMathematics. Washington, D.C.: Mathe-matical Association of America, 1964.

Ackerman, M., and R. Hermann. Sophus Lie’s1880 Transformation Group Paper. Brookline,Mass.: Math Sci Press, 1975.

Adamson, D. Blaise Pascal: Mathematician,Physicist, and Thinker about God. New York:St. Martin’s, 1995.

Albers, D., and G. Alexanderson. MathematicalPeople: Profiles and Interviews. Chicago:Contemporary Books, 1985.

Alexander, D. A History of Complex Dynamics:From Schröder to Fatou and Julia.Braunschweig/ Wiesbaden: Friedrich Viewegand Sohn, 1994.

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INTERNET RESOURCES

Eisenhower National Clearinghouse forMathematics and Science Education.Available online. URL: http://www.enc.org.Downloaded on June 2, 2003. As its nameimplies, this site is a clearinghouse for acomprehensive set of links to interestingsites in math and science.

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Fife, Earl, and Larry Husch. Math Archives.“History of Mathematics.” Available online.URL:http://archives.math.utk.edu/topics/

history.htm. Updated January 2002. Informa-tion on mathematics, mathematicians, andmathematical organizations.

Gangolli, Ramesh. Asian Contributions toMathematics. Available online. URL:http://www.pps.k12.or.us/depts-c/mc-me/be-as-ma.pdf. Downloaded on June 2, 2003. Asits name implies, this well-written onlinebook focuses on the history of mathematicsin Asia and its effect on the world historyof mathematics. It also includes informationon the work of Asian Americans, a welcomecontribution to the field.

291

INDEX

A

abduction 208–209Abel, Niels Henrik 1–3, 2,

145–146, 166and abstract algebra 36and quintic polynomial 56,

127successors of 61, 65, 102,

174, 175, 248Abelian functions 1–3, 134,

226, 248Abelian integrals 153–154,

248Abel’s theorem 2abstract algebra 123, 196–197Académie des Sciences. See

French Academy of SciencesAcadémie Française 7acoustics 64, 105, 116, 123Acta Scientiarum Mathematicarum

(journal) 228actuarial science 17, 28, 73, 114,

188Adelard of Bath 3–4Ad locos planos et solidos isagoge

(Introduction to planes andsolids) (Fermat) 91

Agnesi, Maria Gaetana 4–6, 5“witch of” (versiera) 5

akousmatics. See Pythagoreansalchemy 19

Alembert, Jean d’ (Jean Le Rondd’Alembert) 6, 6–8, 43, 159,161

algebra 5, 14, 110–111abstract 123, 196–197Boolean 45–46cohomological 140equations in 66–67, 77–78,

92–93, 231–232exterior 123foundations of 67–68,

241group theory in 110,

147–148ideals in 70idempotent 46noncommutativity in

128–129, 197notation in 241–242and number theory

135–136, 211ring theory in 197

Algebra (al-Khwarizmi)150–151

Algebra (Wallis) 195algebraic equations 1, 67, 76,

84, 91–92, 121, 149, 231–232,242

algebraic functions 65, 110, 165,175

algebraic geometry 36, 73, 75,90, 123, 187, 209–211, 216

algebraic topology 139–140,209–210

algorithms 155Alhazen. See al-Haytham,

Abu AliAlhazen’s problem 131–132Almagest (Pappus) 201Almagest (Ptolemy) 137, 202,

217Alpha and Omega of Variations on

Multiplication and Division(Yang) 253

American Mathematical Society40, 208, 252

Amsterdam, University of50

Analyse des infiniment petits(Analysis of the infinitelysmall) (L’Hôpital) 32

analysis 5, 36, 43–44, 46, 53,56, 60, 100

complex 188, 210–211,214–215, 225, 247

functional 21, 22–23,101–102, 171, 227–228,243–244, 249

harmonic 32, 99, 105,250–252

indeterminate 66, 77, 96of infinitesimals 11, 24,

32–33, 43, 58–59, 61,246, 256–257

Note: Page numbers in boldface indicate main topics. Page numbers in italic refer to illustrations.

analysis (continued)and number theory 79, 165,

175, 221, 225, 249and probability theory 171vector 118, 129, 133

Analysis situs (Site analysis)(Poincaré) 210

analytical engine. See computersanalytic geometry 5, 73–77, 91,

187, 199–200, 216Anaximander 218Angeli, Stefano 126angle, trisection of 111, 138,

242Annali di mathematica pura ed

applicata (Annals of pure andapplied mathematics)(journal) 106

ANOVA (analysis of variance)97, 122

Apollonius of Perga 8–9, 13,15, 63, 70, 91, 144, 201

Application de l’analyse à lagéometrie (Application ofanalysis to geometry) (Monge)189

Applications of Analysis andGeometry (Poncelet) 216

Arabic numerals. See numerals,Indian

Archimedes of Syracuse 10,10–12, 15, 70–71, 107, 112

Arabs on 144and geometry 15, 63, 82,

91, 184Greeks on 8–9, 201and hydrodynamics 28writings of 13, 81, 237, 246

Archytas of Tarentum 11, 84arcsine functions 182–183Arcueil, Société d’ 163Aristarchus of Samos 12–14,

246Aristotle 19, 85, 131, 167, 194,

199logic of 29, 72physics of 107–108

arithmetic 133, 155Arab 149Greek 77–78Indian 37, 49logical 44, 103–104medieval 4, 18–20notation in 201, 233

Arithmetica (Arithmetic)(Diophantus of Alexandria)77, 149

Arithmetica Infinitorum (Thearithmetic of infinitesimals)(Wallis) 246

Ars conjectandi (Art ofconjecturing) (JakobBernoulli) 30

Artin, Emil 197Artis magnae sive de regulis

algebraicis liber unus (ArsMagna; Book one of the greatart or concerning the rules ofalgebra) (Cardano) 56, 93

Aryabhata I 14–15, 37Aryabhatiya (Aryabhata I) 14Arya Manksu 254Association of German

Mathematicians 52asteroids 113astrolabe 4, 144Astrolabe (Adelard of Bath) 4astrology 19, 56, 217Astronomical Society (Great

Britain) 16, 72astronomy 48–49, 108

distances in 33, 35instruments of 4, 19, 64,

108, 114, 126, 142, 144mathematical 126, 161orbital calculation in 137

of asteroids 113of comets 34, 88, 165,

222–223of planets 9, 88, 89,

161, 168, 217spherical geometry in 85trigonometry in 184

atmospheric pressure 203

atomic theory 24, 71Aufgaben und Lehrsätze aus der

Analysis (Problems andtheorems of analysis) (Pólyaand Szego) 214

automorphic functions 152,209–210

B

Babbage, Charles 16–18, 17,124, 178–179

Babylonians 138Bacon, Roger 18, 18–20Baire, René-Louis 21, 47,

164ballistics

in 16th century 237in 17th century 109, 195in 18th century 5, 165,

189in 19th century 65, 156in 20th century 105, 171

Banach, Stefan 22–23Banach spaces 22, 227Banach-Steinhaus theorem 22Barrow, Isaac 23, 23–25, 29,

64, 127, 167students of 195

Basel, University of 30–31, 33,86

Bayes, Thomas 25–26, 163Bayesians 25, 98Bayes theorem 26, 163Beaugrand, Jean de 74Begriffschrift (Concept script)

(Frege) 103–104Beltrami, Eugenio 178Bentham, George 73Berkeley, George (bishop) 25,

181Berlin, University of

faculty of 69, 79, 145–146,154–155, 176, 232, 248

students of 140, 153, 154,176, 185, 225, 232, 258

292 A to Z of Mathematicians

Berlin Academy (Society) ofSciences 79, 86, 155, 156,159, 165, 168, 226

Bernoulli, Daniel 7, 25, 26–29,27, 35, 79, 86, 121, 182

Bernoulli, Jakob (Jacques) 7, 25,27, 29–31, 86, 167, 231

Bernoulli, Johann 7, 27, 30, 31,31–33, 35, 159

students of 86Bernoulli, Nikolaus 27, 121Bessel, Friedrich Wilhelm

33–35, 34, 145Bessel functions 35, 87, 176Betrachtungen über einige

Gegenstände derElementargeometrie (Views onsome articles of elementarygeometry) (Bolzano) 42–43

Betti, Enrico 35–37, 226, 243Betti numbers 36Betti’s theorem 36Bhaskara II 37–38, 49Bhillamalacarya. See

BrahmaguptaBianchi, Luigi 105–106Bijaganita (Root extraction)

(Bhaskara II) 37–38binary system 18, 46binomial equations 2, 150binomial theorem 2, 150, 203Biometrika (journal) 207Birkhoff, George David 38–40,

39Bôcher, Maxime 40Bologna, University of 4–5, 56,

63, 93Bolyai, Farkas 40Bolyai, János (Johann) 40–41,

43, 83, 177Bolzano, Bernhard (Bernardus

Placidus Johann NepomukBolzano) 42–44, 59, 62, 70

Bolzano function 44Bolzano-Weierstrass theorem

44Bonaventure 19

Bonn, University of 151, 154,176, 247

The Book of Spherical Propositions(Menelaus of Alexandria)184

Boole, George 44, 44–46, 73,168, 205, 240

Boolean algebra 45–46Boolean logic 44–46, 73Borel, Émile 21, 46–48, 101,

164Bosse, Abraham 74Bossut, Charles 189boundary formulas 155brachistochrone 30, 32, 167Bradley, James 34Brahe, Tycho 217Brahmagupta 37, 48–49Brahmasphutasiddhanta

(Brahmagupta) 49Breslau, University of 79, 139,

156, 176bridges, suspension 193Briggs, Henry 192Bromhead, Sir Edward 124Brouillon projet d’une atteinte aux

événements des rencontres ducône avec un plan (Rough draftfor an essay on the results oftaking plane sections of acone) (Desargues) 74

Brouwer, Luitzen Egbertus Jan49–51

intuitionism of 50–51,155–156, 211, 250

Brownian motion 171, 251Brown University 215Budapest, University of 213,

227–228Buridan, Jean 199

C

Caen, University of 210calculators 65, 178, 203

calculusdifferential 5, 23–24, 60–61,

86–87, 105, 125, 194–196discovery of 126–127, 246Fourier transform in

99–100infinitesimal 28, 32, 74,

166–168, 182integral 5, 63, 86–87, 146,

161logical 104, 168notation in 16–17,

167–168, 181of variations 248, 258

contributions to 114,164, 171, 176

foundations of 32, 158general theory for 86,

88vector 123, 205, 234

calendarancient Greeks and 184,

238Chinese and 67, 239French revolution 162Indians and 49medieval Western 20, 223,

22420th-century reform of 21

Cambridge Mathematical Journal45

Cambridge Universityfaculty of 16, 23–24, 97,

125, 128, 131, 195, 228,234, 240, 255

students of 16, 96, 124,128, 130, 194, 214, 221,228, 234, 240, 245

Canon Mathematica seu adtriangular (Mathematical lawsapplied to triangles) (Viète)241

Cantor, Georg 52–54, 53, 205,254, 255

functional analysis and100, 101

Index 293

Cantor, Georg (continued)real numbers and 42,

248–249set theory and 47, 70, 155,

229, 258Cardano, Girolamo 54–57, 55,

92, 224, 236–237, 242–243Cardan-Tartaglia formula 237Carnot, Lazare 57, 57–59Carr, George Shoobridge 221Cartan, Élie 59–60, 123, 174,

250cartography. See mapmakingCatania, University of 105catenary 31, 143, 167Cauchy, Augustin-Louis 2, 60,

60–62, 154, 235analysis and 79calculus and 58, 73, 154,

225contemporaries of 110–111,

193Cauchy distribution 62Cauchy-Kovalevskaya theorem

154Cauchy sequences 53Cavalieri, Bonaventura 12,

63–64Cayley, Arthur 178celestial elements, method of

171–172celestial mechanics

classical (Newtonian)60–62, 128, 158–160,165, 195, 212, 217

modern 39, 186, 210Ptolemaic 217

central limit theorem 187–188Centre National de la

Recherche Scientifique(National center for scientificresearch) (France) 48

centrifugal force 142, 195centripetal force 195Ceres 113Chang Te-hui 172chaos theory 211

Chebyshev, Pafnuty Lvovich64–66, 183

Chebyshev net 65chemistry 117, 189, 214Chicago, University of 39Ch’in Chiu-Shao 66–67, 68,

173Chinese remainder theorem 66Chladni, Ernst 116choice, axiom of 258Christiania University 173–174Christoffel, Elwin 176Chu Shih-Chieh (Zhu Shijie)

66, 67–68circle

relation of radius tocircumference in 138

squaring of 126, 132, 138,141, 170, 242, 246

City College of New York 229civil engineering 30, 192–193,

233Clavis Mathematica (Key to

mathematics) (Oughtred)246

Clifford, William 123clocks 142Codex Atlanticus (Atlantic

codex) (Leonardo) 170coefficient array method 172cohomological algebra 140Collected Works (Steiner) 232College de France (college of

University of Paris) 61, 78,148, 175

combinatorial topology 147combinatorics 222comets 29, 109

orbits of 34, 165, 222–223Commonwealth Scientific and

Industrial ResearchOrganization (Australia) 98

The Compendious Book onCalculation by Completion andBalancing (or Algebra) (al-Khwarizmi) 150–151

completeness theorem 119

complex analysis 188, 210–211,214–215, 225, 247

complex numbers 62, 70,128–129, 167, 248

complex variables 62, 225composite numbers 221computers 17–18, 120, 178–179Condorcet, Marie-Jean 189conics 8–9, 74–75, 215, 232,

246Conics (Apollonius of Perga) 8consistency

in geometry 136, 177in number theory 103, 136

continued fractions 14–15, 65,159, 165, 183, 185, 222

continuum 54, 69, 71, 120,256–257, 258

convergent series 61–62coordinate geometry 5, 199–200Copernican theory 13,

108–109, 233, 241Copernicus, Nicholas 13, 223correlation coefficient 97,

254–255cosmology 120Coulomb’s law 29Cours d’analyse (Course in

analysis) (Jordan) 147Crelle, August 2–3Crelle’s Journal for Pure and

Applied Mathematics 2, 146,232, 247

cryptography 16, 242, 245crystallography 214cube, doubling of (Delian

problem) 82, 84–85,138–139, 170, 242

cube roots 94cubic equations 55–56, 92–93,

236curves

algebraic 30, 91, 211cubic 5cycloid in 33, 142, 158in differential geometry

114, 152

294 A to Z of Mathematicians

probability 28rectification of 164space-filling (Peano) 205study of 32, 164

cybernetics 250, 252Cybernetics: Or, Control and

Communication in the Animaland Machine (Wiener) 252

cycloid. See curves, cycloid incyclotomic theory 155

D

decimal system 18, 233, 253Dedekind, Richard 69–70, 79,

156contemporaries of 52–53and continuum 256real numbers and 42–43, 84

Dedekind cut 69deductive reasoning 238De eodem et diverso (On

sameness and diversity)(Adelard of Bath) 4

Delian problem. See cube,doubling of

De mensura sortis (Concerningthe measure of chance)(Daniel Bernoulli) 28

De Methodis Serierum etFluxionem (On the methods ofseries and fluxions) (Newton)195

Democritus of Abdera 70–72,71, 85

De Morgan, Augustus 45,72–73, 179, 240

De motu (On motion) (Galileo)107

De Revolutionibus (Onrevolution) (Copernicus)223–224

Der Zahlbericht (Commentary onNumbers) (Hilbert) 135–136

Desargues, Girard 9, 73–75,203

Descartes, René 24, 28, 75,75–77, 141, 194, 246

dynamics of 167and geometry 9, 73–74philosophy of 229predecessors of 199, 202

descriptive geometry 190De triangulis omnimodis (On

triangles of all kinds)(Regiomontanus) 222

Dialogue Concerning the TwoChief World Systems (Galileo)109

Die lineale Ausdehnungslehre(The theory of linearextension) (Grassmann) 123

difference engine. See computersdifference equations 39, 161Differential and Integral Calculus

(De Morgan) 73Differential and Integral Calculus

(Lacroix) 16differential calculus 5, 23–24,

60–61, 86–87, 105, 125,194–196

differential equations 124–125,161, 168, 205. See also partialdifferential equations

of Bernoulli 30boundary-value problems in

39early work on 5, 6, 160,

173linear 39, 105

differential forms 176differential geometry 115, 140,

173, 176, 211exterior differential forms in

59–60, 123foundations of 88,

188–190, 225mean curvature in 116projective 105surface mapping in 114

Dijon, University of 21Diophantine problems 37, 47,

77–78, 87, 127, 130, 211, 231

Diophantus of Alexandria77–78, 149–150

Dirichlet, Gustav Peter Lejeune69, 78–80, 113, 136,146–147, 154, 156, 176

students of 69, 154, 225Dirichlet principle 210, 225–226,

251Dirichlet problem 79Dirichlet series 79Discourses and Mathematical

Demonstrations ConcerningTwo New Sciences (Galileo)109

displacement 10Disquisitiones Arithmeticae

(Arithmetic Investigations)(Gauss) 79, 113, 115–116

Disquisitiones generales circasuperficies curves (GeneralInvestigations of curvedsurfaces) (Gauss) 114

divergent series 7, 47, 61, 87Divina proportione (Divine

proportion) (Pacioli) 169The Doctrine of Chance (Moivre)

187–188drawbridge problem 30Duhamel, Jean 234dynamics 128, 159

of heat 98–99nonlinear 39, 209, 211topological 39

E

e (transcendental number) 62,87, 126, 134

Earthcircumference of 15, 81–82diameter of 14, 20measurement of (geodesy)

35, 66, 114École des Mines (Paris) 171,

210École Militaire (Paris) 165

Index 295

École Normale Supérieure(Paris) 21, 47–48, 59, 89,111, 164

École Polytechnique (Paris) 47,115, 189, 216

faculty of 61, 99, 134,147–148, 160, 171,174–175, 193, 212

students of 61, 134,147–148, 171, 174–175,210, 212, 215

Edinburgh, University of 127,181

education in mathematics 16,19, 36, 65, 164, 189, 215, 253

eigenvalues 103Einstein, Albert 176, 195, 229

colleagues of 120, 229, 250mathematical bases for

theories of 185–186, 197,210, 226

elasticity 30, 36, 60–61, 88,116, 117, 213

electrical engineering 132–133electricity 36, 46, 105, 124–125,

212–213electrodynamics 185electromagnetism 117, 133electrostatics 28Elements (Euclid of Alexandria)

11, 40, 82–85, 95, 150–151,237

commentaries on 24, 36,43, 131–132, 202

in mathematical education36, 126, 169, 195, 203

translations of 4, 131, 237Elements of Geometry

(Hippocrates of Chios)138–139

Elements of Geometry (Legendre)165, 243

Elements of Quaternions(Hamilton) 129

elliptic functions 155, 247–248Abelian 1–3, 153, 226,

248

early work on 145–146,160, 165–166

and hyperelliptic functions134

in mathematical physics 36theory of 175, 221, 225,

247and transcendental

functions 36elliptic geometry 152elliptic integrals 2–3, 65, 145,

166, 222Encyclopédie (Encyclopedia)

(Diderot) 7Engel, Friedrich 174engineering 105–106, 109,

153civil 30, 192–193, 233electrical 132–133Fourier analysis in

132–133, 226, 228mechanical 30, 57military 169, 189

entropy 117–118enumeration theorem 214epicureanism 72epistemology 118–119equations

algebraic 1, 67, 76, 84, 91–92, 121, 149,231–232, 242

binomial 2, 150cubic 55–56, 92–93, 236difference 39, 161differential 124–125, 161,

168, 205. See alsoequations, partialdifferential

of Bernoulli 30boundary-value

problems in 39early work on 5, 6,

160, 173linear 39, 105

Diophantine. SeeDiophantine problems

indeterminate 159, 164

integral 102–103, 136, 164,243–244

partial differential 61, 105,115–115, 153–154, 171,173, 212, 226, 243

polynomial 55, 63, 65, 77,87, 110, 149–150, 253

quadratic 14, 49, 172quartic 56, 92–93quintic 1–2, 87, 93,

110–111, 127, 145, 152trigonometric 186

Eratosthenes of Cyrene 81–82,92, 139, 201

ergodic theory 39, 228Erlangen, University of 152,

197error, standard 98Escher, M.C. 214Essai d’une nouvelle théorie de

la résistance des fluides(d’Alembert) 7

Essai sur les machines en générale(Essay on general machines)(Carnot) 58

An Essay on the Application ofMathematical Analysis to theTheories of Electricity andMagnetism (Green) 124

Essay on the Theory of Numbers(Legendre) 165

Essay pour les coniques (Essay onconic sections) (Pascal) 74,203

Euclid of Alexandria 4, 13, 19,42, 63, 82–84, 107

Arabs and 144, 177contemporaries of 84fifth postulate of 40, 83,

177Greeks and 8–9, 10,

201–202, 219Euclidus elementorum libri XV

(Euclid’s first principles in 15books) 24

Eudoxus of Cnidus 69, 82–83,84–85

296 A to Z of Mathematicians

Euler, Leonhard 1, 35, 64, 85,85–88, 145–146, 154

and calculus of variations158, 165

contemporaries of 7, 27,32, 121, 182

and logic 240and power series 75, 226

Euler buckling formula 88evolutes, theory of 142Exercitationes mathematicae

(Daniel Bernoulli) 27–28exhaustion, method of 10–11,

83exponents 149Exposition du système du monde

(The system of the world)(Laplace) 162

exterior algebra 123extreme value theorem 43

F

al-Fakhri fi’ljabr wa’l-mughabala(Glorious on algebra) (al-Karaji) 149–150

Fatou, Pierre-Joseph-Louis 89,164

Fatou’s lemma 89Fermat, Pierre de 5, 9, 24, 61,

90, 90–92, 121, 141and algebra 78, 159and calculus 63, 146contemporaries of 203and geometry 9, 76last theorem of 62, 78, 90,

92, 116, 156, 165and probability 56, 183,

202–203Ferrari, Ludovico 56, 92–93,

94, 236–237Ferro, Scipione del 56, 92,

93–94, 236Fibonacci, Leonardo (Leonardo

of Pisa) 12, 94, 94–96, 150,233, 243

Fibonacci sequence 95fields

algebraic number 155vector 140

finiteness theorems 148Fior, Antonio Maria 94, 236Fisher, Sir Ronald Aylmer 25,

96–98, 97, 122, 206Fisher information criterion 98flight, study of 169–170fluid, flow in 193fluid mechanics 6–7, 29, 88,

160, 193, 210–211. See alsohydrodynamics

fluxions. See infinitesimalsforce 142, 195formalism 104, 118Formulario matematico

(Mathematical formulary)(Peano) 205–206

Foundation of Analysis(Lipschitz) 176

four color problem 208Fourier, Jean-Baptiste-Joseph

62, 78, 98–101, 99, 111, 163,235

contemporaries of 78, 146,189

students of 193Fourier analysis 22, 35,

132–133, 226, 228Fourier series 99, 176, 193Fourier transform 62, 99–100,

132–133, 252fractal geometry 54, 205–206fractions, continued 14–15, 65,

159, 165, 183, 185, 222Fraenkel, Adolf 258Fréchet, René-Maurice 22,

101–102, 227, 243, 251Fredholm, Ivar 22, 102–103Fredholm integral equation

102Frege, Friedrich Ludwig Gottlob

103–104, 168, 228, 257Freiburg im Breisgau, University

of 258

French Academy. See AcadémieFrançaise

French Academy of Sciences 6,163, 189

competitions and prizes of2–3, 29, 58, 61, 116, 159

members of 7, 21, 60, 100,142, 148, 160, 161–162,164, 189, 193, 211

Frequentists 25, 98Frobenius, Georg 258Fubini, Guido 105–106, 164functional analysis 21, 22–23,

101–102, 171, 227–228,243–244, 249

functions 17Abelian 1–3, 134, 226, 248algebraic 65, 110, 165, 175arcsine 182–183automorphic 152, 209–210Bessel 35, 87, 176Bolzano 44continuous. See topologyelliptic 155, 247–248

Abelian 1–3, 153, 226,248

early work on145–146, 160,165–166

and hyperellipticfunctions 134

in mathematicalphysics 36

theory of 175, 221,225, 247

and transcendentalfunctions 36

logarithmic 167–168, 191–192

series expansion of 181theory of 21, 43, 59, 87,

156, 158, 171, 176, 222trigonometric 32, 167, 217zeta. See Riemann zeta

functionfunction spaces 101, 164, 171,

243

Index 297

Fundamenta nova theoriaefunctionum ellipticarum (Newfoundations of the theory ofelliptic functions) (Jacobi)146

G

Galilei, Galileo 12, 63, 75,107–110, 108

Galois, Evariste 56, 61, 102,110–111, 147, 168, 175

Galois theory 35–37, 94,110–111

game theory 46, 48Gauss, Carl Friedrich 12, 34, 60,

69–70, 112, 112–115,145–146

and algebra 2–3, 55–56and astronomy 89and calculus 59, 159contemporaries of 34, 41,

115–116, 145, 226and geometry 40, 83, 177and number theory 78, 87,

248and principle of least

squares 165and statistics 122students of 69, 186, 225

Gauthey, Emiland 193general theory of relativity

59–60, 176, 178, 186, 224,226, 250

Generation of Conic Sections(Pascal) 203

genetics 97–98Genoa, University of 105Genocchi, Angelo 205geocentric (Ptolemaic) theory

9, 13, 108–109, 241geodesy. See Earth, measurement

ofgeography 81–82, 84–85, 184,

202Geography (Eratosthenes) 81

Geography (al-Khwarizmi) 151Geometriae pars universalis

(Gregory) 126Geometria Indivisibilibus

Continuorum Nova QuadamRatione Promota (A certainmethod for the developmentof a new geometry ofcontinuous indivisibles)(Cavalieri) 63

Geometria organica (Organicgeometry) (Maclaurin) 181

Geometrical Calculus (Peano)205

Géométrie (Geometry)(Descartes) 29, 76, 91

Geometriyya (Lobachevsky) 177geometry 24, 164, 238

algebraic 36, 73, 75, 90,123, 187, 209–211, 216

analytic 5, 73–77, 91, 187,199–200, 216

consistency in 136, 177coordinate 5, 199–200descriptive 190differential 115, 140, 173,

176, 211exterior differential

forms in 59–60, 123foundations of 88,

188–190, 225mean curvature in 116surface mapping in 114

elliptic 152fractal 54, 205–206hyperbolic 32, 152, 178,

216“imaginary” 178infinitesimal 126–127inversive 232non-Euclidean 83, 152,

177–178, 232consistency in 136,

177foundations of 40–41,

114opponents of 157, 166

projective 9, 73–75, 105,152, 186–187, 201,215–216, 232

Riemannian 224–225solid 91of universe 177–178

Geometry (Legendre). SeeElements of Geometry

Germain, Sophie 113, 115–116Gibbs, Josiah Willard 116–118,

117, 129, 133Glasgow, University of 181Gödel, Kurt Friedrich 50, 52,

103–104, 118–121, 119, 136,229, 251

Goldbach, Christian 86,121–122

Goldbach conjecture 121Gordan, Paul 197Gosset, William 97, 122–123,

207Göttingen, University of 34,

133, 156faculty of 79, 113, 135,

152, 185, 197, 226, 250,258

students of 69, 103, 113,152, 186, 214, 225, 249,251

graphical solutions 73–76, 91,117, 199

Grassmann, Hermann Günter123–124, 205

contemporaries of 187gravity 234

Galileo and 107–109, 233measurement of 142, 208Newtonian mechanics of

28, 29–30, 162–163, 195relativity theory and 178

Green, George 36, 124–125,213, 234

Green’s formula 124Green’s function 124Gregory, James 24, 125,

125–127

298 A to Z of Mathematicians

Gregory of St. Vincent 141Groningen, University of 32Grosseteste, Robert 19–20group theory 62, 110–111, 134,

147–148in algebra 110, 147–148continuous groups in 49finite 147permutation groups in 160

Grundlagen der Arithmetik(Foundations of Arithmetic)(Frege) 104

Guinness, Arthur & Sons122

H

Hadamard, Jacques 40,101–102, 171, 214, 243

Hahn, Hans 22Hahn-Banach theorem 22“hairy ball theorem” 50Halle, University of 52, 156Halley, Edmund 34. See also

cometsHamilton, Sir William Rowan

45, 73, 128–130, 129, 133,156, 243

Hamiltonian equation 128–130Handy Tables (Ptolemy) 217Hansteen, Christoffer 1, 3Hardy, Godfrey Harold

130–131, 214, 221, 251harmonic analysis 32, 99, 105,

250–252Harvard University 39, 208,

251Hatsubi Sampo (On algebraic

equations) (Seki) 231Hausdorff, Felix 174al-Haytham, Abu Ali (Alhazen)

19, 24, 131–132, 177heat. See thermodynamicsHeaviside, Oliver 132–133Heidelberg, University of 232Heine-Borel theorem 47

heliocentric theory 11, 12–14,84, 233. See also Copernicantheory

Helly, Eduard 22Helmstedt, University of 113Heraclides of Pontus 13Hermann, Jakob 86Hermite, Charles 133–135,

210Herschel, John 16Hertz, Heinrich 254Hilbert, David 135, 135–137,

209, 226and complex numbers 70and consistency in

arithmetic 50, 104, 120,229

contemporaries of 185, 206and functional analysis 22,

103students of 185, 197, 214,

249, 251Hilbert spaces 35, 249Hindu numerals. See numerals,

IndianHipparchus of Rhodes 9, 13,

137–138, 217Hippocrates of Chios 138–139Hobbes, Thomas 246Holmboe, Bernt 1homeomorphisms 53homology 139–140homotopy groups 50, 140,

210Hooke, Robert 126Hopf, Heinz 139–140Hopf fiber map 140Hôpital, Guillaume de L’ 32Hopkins, William 234How to Solve It (Pólya) 215Hudde, Johann 24Hungarian Academy of Sciences

228Hurwitz, Adolf 214Huygens, Christiaan (Christian

Huyghens) 24, 28, 30, 126,140–143, 141, 167, 189

Hydraulica (Hydraulics) (JohannBernoulli) 27

hydraulics 27, 86, 88Hydrodynamica (Daniel

Bernoulli) 27–28hydrodynamics 7, 26–27, 79,

125, 192, 216, 233–234, 258hydrostatics 10–12, 108, 142,

202–203, 233hyperbolic geometry 32, 152,

178, 216hyperelliptic integrals 248hypergeometric series 156, 222hyperspace 36hypothesis testing 97, 206–207,

208

I

Ibrahim ibn Sinan (Ibrahim IbnSinan Ibn Thabit Ibn Qurra)144

ideals, theory of 70, 156, 197idempotent algebra 46Il Saggiatore (The assayer)

(Galileo) 109“imaginary” geometry 178imaginary numbers 41, 59incompleteness theorem

118–120, 229indeterminate analysis 66, 77, 96indeterminate equations 159,

164Inequalities (Pólya et al.) 214infinite series 5, 14, 26, 52,

60–61, 178, 199convergence in 47, 73, 121,

155, 249sums in 2, 87, 183

infinitesimalsanalysis of 11, 24, 32–33,

43, 58–59, 61, 246,256–257

calculus of 28, 32, 74,166–168, 182

geometry of 126–127

Index 299

infinity 11, 24, 37, 43, 47,52–54, 155, 215–216, 254,256–257, 258

Institut de France 34Institute for Advanced Study

(Princeton, New Jersey) 106,120, 197, 249

Institut Henri Poincaré (Paris)48

Institute of Technology (Lvov)22

Instituzioni analitiche ad uso dello gioventù italiana(Analytical methods for theuse of young italians)(Agnesi) 5–6

integral calculus 5, 63, 86–87,146, 161

integral equations 102–103,136, 164, 243–244

integralsAbelian 153–154, 248double 164elliptic 2–3, 65, 145, 166,

222hyperelliptic 248Lebesgue 164Riemann 164

integrationby parts 159theory of 32, 89, 163–164,

177, 225, 227, 249intermediate value theorem 43,

62International Mathematical

Union 140Introduction to Mathematical

Philosophy (Russell) 229Introduction to Mathematical

Studies (Chu Shih-Chieh)67–68

Introduction to the Analytic Art(Viète) 241

Introduction to the Doctrine ofFluxions (Bayes) 25–26

Introduction to the Theory ofStatistics (Yule) 254–255

intuition, scientific 132, 155, 211intuitionism 50–51, 118, 225,

250inverse square law of

gravitational force 195inversive geometry 232Investigation of the Laws of

Thought (Boole) 46irrational numbers 9, 69, 84, 94,

149–150, 155, 219, 233isochrone 143isoperimetric problem 32–33

J

Jablonow Society 41Jacobi, Carl (Carl Gustav Jakob

Jacobi) 3, 69, 78, 145–147,156, 165, 175, 243, 249

contemporaries of 3, 79, 134students of 69, 225

Jacobian 146Jacquard, Joseph-Marie 17Jains 254Jansenism 204Jena, University of 103, 167Jia Xian 253Johns Hopkins University 208Jordan, Camille 43, 134,

147–148, 164, 173Jordan curve theorem 43Jordan-Hölder theorem 147Journal de Liouville (Liouville’s

journal) 174Journal of Pure and Applied

Mathematics. See Journal deLiouville

Jupiter 108

K

al-Karaji, Abu 149–150Karanakutuhala (Calculation of

astronomical wonders) (Bhaskara II) 38

Kazan University 177Kepler, Johannes 12, 24, 108,

195, 246Khandakhadyaka (Brahmagupta)

49al-Khwarizmi, Abu (Abu Ja’far

Muhammad Ibn Musa al-Khwarizmi) 149, 150–151

Killing, William 59, 174kinematics 109Klein, Felix (Christian Felix

Klein) 148, 151–153, 152,173–174, 178

contemporaries of 173,208, 210

students of 197, 214Kolmogorov, Andrei 183Kolosvár, University of 227Königsberg, riddle of seven

bridges of 88Königsberg, University of 121,

135, 145, 176, 185, 232, 248Kovalevskaya, Sonya (Sofia

Kovalevskaya, SofyaKovalevskaya) 153–154

Kronecker, Leopold 79, 136,154–156, 248–249

students of 53, 137Ksamasramana, Jinabhadra 254Kummer, Ernst Eduard

154–155, 156–157contemporaries of 155,

226, 248–249students of 154, 156

Kummer surface 156

L

Lacroix, Sylvestre François 16Lagrange, Joseph-Louis

(Giuseppe LodovicoLagrangia) 145, 154,158–160, 159, 161, 235

and calculus of variations 6and elliptic functions 165and geometry 61

300 A to Z of Mathematicians

and number theory 43students of 115, 212and trigonometric series 99

Lambert, Johann 159Laplace, Pierre-Simon 36, 61,

132–133, 160–163, 161, 166,235

contemporaries of 61and dynamics 159and probability 26students of 212

Laplacian (Laplace equation,operator) 162, 171

large numberslaw of 30, 38, 64–65, 213,

240multiplication of 49

least squares principle 113, 163,165

Lebesgue, Henri-Léon 21, 47,89, 100–101, 163–164, 227

Lebesgue integrals 164Lectiones geometricae

(Geometrical lectures)(Barrow) 24

Lectiones mathematicae(Mathematical lectures) (Barrow) 24

Lectiones opticae (Lectures onoptics) (Barrow) 24

Legendre, Adrien-Marie 2, 159,162, 164–166, 177, 225

contemporaries of 3, 115,145–146

students of 212Legendre transformation 164Leibniz, Gottfried Wilhelm von

28, 121, 142–143, 166,166–168, 231

and astronomy 35and calculus 10, 30, 31, 43,

63–64, 142, 181, 195, 257contemporaries of 32, 121,

142, 167and logic 45, 104, 240and Newton 188, 195–196notation of 16

predecessors of 63, 204and reason 87

Leiden, University of 141, 233Leipzig, University of 152,

166–167, 174, 186–187, 224Lemma of Archimedes 11Leonardo da Vinci 55,

168–170, 169Leonardo of Pisa. See Fibonacci,

LeonardoLetters on Sunspots (Galileo)

109Lettres à une princesse

d’Allemagne sur divers sujets dephysique et de philosophie(Letters to a German princesson diverse topics of physicsand philosophy) (Euler)86–87

lever 10Levi-Civita, Tullio 40Lévy, Paul-Pierre 170–171, 251Liber abbaci (Book of the abacus)

(Fibonacci) 94–96Liber de ludo aleae (Book on

games of chance) (Cardano)56

Li Chih (Li Yeh) 171–173Lie, Sophus (Marius Sophus Lie)

59, 148, 173–174, 250Lie groups 59, 173, 250light. See opticsLilavati (The beautiful)

(Bhaskara II) 37Lille, University of 47limit, concept of 21, 43, 62, 73,

167, 171linear regression 254linear spaces 22Liouville, Joseph 36, 62, 125,

174–176Liouville’s journal. See Journal de

LiouvilleLiouville’s theorem 62Liouvillian number 175Lipschitz, Rudolf (Rudolf Otto

Sigismund Lipschitz) 176

Lipschitz condition 176Listing, Johann 187Littlewood, John 130, 214Li Yeh. See Li ChihLobachevsky, Nikolai (Nikolai

Ivanovich Lobachevsky) 41,43, 177, 177–178

logarithmic functions 167–168,191–192

logarithms 17, 30, 63, 141–142,167, 191–192

logicAristotelian 29, 72Boolean 44–46, 73deductive 43, 208mathematical 72–73,

103–104, 118–120, 168,205–206, 208, 228–229,257–258

intuition in 50–51,211

notation in 73, 205and probability theory 46proof theory in 103–104,

136, 205, 228, 237, 258logical calculus 104, 168Logic of Chance (Venn) 240London Mathematical Society

72Lorenz metric 185Lovelace, Augusta Ada Byron

(Ada Lovelace) 178–180lunes 132, 138–139Lvov, University of 22

M

Maclaurin, Colin 181–182Madhava of Sangamagramma

182–183Madras, University of 221–222magnetism 28, 114, 213manifolds 50, 59–60, 123, 140,

173, 210Riemannian 176, 250

mapmaking 20, 86, 223–224

Index 301

mappings, topological 50, 53,140, 186–187, 210

Markov, Andrei 65, 183–184Markov chains 183martingales, theory of 171Massachusetts Institute of

Technology (MIT) 251Mathematical Analysis of Logic

(Boole) 45Mathematical Collection (Pappus)

201mathematical expectation 92,

141, 163mathematical induction 73mathematical logic 72–73,

103–104, 118–120, 168,205–206, 208, 228–229,257–258

intuition in 50–51, 211Mathematical Logic as Based on

the Theory of Types (Russell)229

mathematical physics 88, 102,105–106, 133, 136

Mathematical Society (Germany) 52

mathematical statistics 47, 122Mathematical Treatise (Ch’in

Chiu-Shao) 66mathematics, logical

foundations of 50

mathematikoi. See PythagoreansMaxwell, James Clerk 117, 125,

235measure theory 22, 46–47, 164mechanical engineering 30, 57mechanics

celestialclassical (Newtonian)

60–62, 128, 158–160,165, 195, 212, 217

modern 39, 186, 210Ptolemaic 217

fluid 6–7, 29, 88, 160, 193,210–211. See alsohydrodynamics

quantum 39, 59, 123, 129,227

rational 7, 26, 28statistical 118

Menabrea, Luigi 179Menelaus of Alexandria

184–185Mersenne, Marin 73–74, 141,

203metric system 160, 162military engineering 169,

189Minkowski, Hermann 137, 185,

185–186Minkowski space 185Mirifici logarithmorumcanonis

descriptio (Description of themarvelous canon oflogarithms) (Napier)191–192

Miscellanea Analytica (Analyticalmiscellany) (Moivre) 188

Mittag-Leffler, Magnus 102,211

Möbius, August (AugustFerdinand Möbius) 186–187

Möbius inversion formula187

Möbius net 186Möbius strip 187Moivre, Abraham de 25, 72,

121, 167, 187–188Mollweide, Karl 186Monge, Gaspard 57, 173,

188–190, 215Monte Carlo simulation 122Moon 108

distance of 9, 13–14motions of 159, 217, 223size of 138

Moore, Eliakim Hastings 40motion, laws of 7, 58, 158–160Munich, University of 152music, mathematical theory of

218mystic hexagram 203mysticism, in mathematics 51

N

Nagahastin 254Napier, John 63, 191–192,

192, 233Napier’s bones 192Napier’s logarithm 191Narratio Prima (The first

account of the book on therevolution) (Copernicus) 223

Nash, John 48National Academy of Sciences

(United States) 215Nave, Annibale dalla 94Navier, Claude-Louis-Marie-

Henri 192–193, 234–235Navier-Stokes equations

192–193, 233–234navigation 86, 88, 109, 126, 184

instruments for 108, 224works on 33, 238

negative numbers 37, 49, 59,95, 233

Neumann, John Von 46, 48,136

New Experiments ConcerningVacuums (Pascal) 203

New Steps in Computation (LiChih) 172

Newton, Sir Isaac 12, 56, 72,112, 194, 194–196, 231

and calculus 10, 23–24, 25,32, 63–64, 181, 195–196,257

contemporaries of 24, 187and geometry 9, 73, 83, 195and infinitesimals 43, 246and Leibniz 188, 195–196and mechanics 6–7methodology of 76–77notation of 16and optics 195predecessors of 23, 63, 202,

204Newton-Raphson method 49,

64New York University 106

302 A to Z of Mathematicians

Neyman, Jerzy 122, 206–208Nobel prize for literature 229Noether, Emily (Amalie Emmy

Noether) 70, 140, 196–198Noether, Max 197Noether’s theorem 197noncommutativity 128–129, 197non-Euclidean geometry 83,

152, 177–178, 232consistency in 136, 177foundations of 40–41, 114opponents of 157, 166

notationin algebra 241–242in arithmetic 201, 233in calculus 16–17,

167–168, 181in logic 73, 205

Nullstellensatz (zero positionprinciple) 135

numbersBetti 36complex 62, 70, 128–129,

167, 248composite 221imaginary 41, 59irrational 9, 69, 84, 94,

149–150, 155, 219, 233large

law of 30, 38, 64–65,213, 240

multiplication of 49negative 37, 49, 59, 95,

233partition of, into summands

221–222prime 64–65, 82, 90, 92,

112, 121, 159, 221–222rational 42, 149real 42–43, 52–54, 69, 84,

248–249, 257transcendental 62, 87, 126,

134, 191number theory 43, 52, 85, 155,

165, 176, 186algebraic 78–79, 87, 91,

135–136, 211

analytic 79, 87, 115–116,165, 175, 225, 249

consistency in 103, 136Greek and medieval 77–78,

96modern 130–131, 221–222

numeralsIndian (Arabic, Hindu) 4,

14, 95, 151, 241Roman 14, 95

O

Olbers, Heinrich 34On Analysis (al-Haytham) 132On Floating Bodies (Archimedes

of Syracuse) 12On Speeds (Eudoxus of Cnidus)

85On Spheremaking (Archimedes of

Syracuse) 11On the Correlation of Total

Pauperism with Proportion ofOutrelief (Yule) 254

On the Equilibrium ofHeterogeneous substances(Gibbs) 117

On the Equilibrium of Planes(Archimedes of Syracuse) 12

On the Measurement of the Circle(Archimedes of Syracuse) 11

On the Rotation of a Solid Bodyabout a Fixed Point(Kovalevskaya)154

On the Sizes and Distances of theSun and the Moon (Aristarchusof Samos) 13–14

operational research 17operator theory 44, 227Optica Promota (The advance of

optics) (Gregory) 126optics 92, 125, 126, 128, 169

lenses and telescopes in 19,64, 108, 114, 126, 142,195

Newtonian 195theories of diffraction in 234theories of light in 24, 88,

118, 128, 142, 163, 195theories of reflection in 19,

24, 131, 184theories of refraction in

19,24Optics (al-Haytham) 132Opus maius (Great work)

(Bacon) 19Opus minus (Smaller work)

(Bacon) 19Opus Palatinum de triangulis

(Palatine work on triangles)(Rheticus) 224

Opus tertius (Third work)(Bacon) 19

Oresme, Nicole 199–200orthographic projection 190Oughtred, William 246Oxford University 19, 122, 130,

246

P

Pacioli, Luca 169packing problems 186Padua, University of 108, 126Pappus of Alexandria 9, 12, 91,

201–202Pappus problem 201–202paradoxes 256–257Paris, University of 19, 48, 101,

191, 210Parmenides 256partial differential equations 61,

105, 115–116, 153–154, 171,173, 212, 226, 243

Pascal, Blaise 24, 73, 178, 202,202–205

and calculus 10, 63–64, 257contemporaries of 92and geometry 9, 73and probability 56, 163,

183, 203–204

Index 303

Pascal’s triangle 68, 253Pascal’s wager 204Pavia, University of 56Peacock, George 16Peano, Giuseppe 205–206Peano axioms 205Peano curve 205Pearson, Egon Sharpe 206–208Pearson, Karl 122, 206, 254Peirce, Benjamin 208Peirce, Charles 205, 208–209pendulum, motion of 108–109,

142, 195, 212, 234P’eng Che 172Pensées de Monsieur Pascal sur la

religion et sur quelques autressujets (Monsieur Pascal’sthoughts on religion and someother subjects) (Pascal) 203

permutation groups 160perspective 169Peurbach, Georg 222Philolaus 13Philosophiae naturalis principia

mathematica (Mathematicalprinciples of naturalphilosophy, or Principia)(Newton) 45, 195

physicsAristotelian 107–108Fourier analysis in 226, 228mathematical 88, 102,

105–106, 133, 136quantum 174, 227

pi 87, 221as transcendental number

126–127, 134value of 95–96, 170, 242

among Chinese 67,172, 239

among Greeks 12, 217among Indians 14, 183

Piazzi, Giuseppe 113Picard, Charles 171Picard, Émile 174, 214Picard’s theorem 47“pigeon-hole principle” 79

Pisa, University of 36, 107, 243Planck, Max 118, 258plane, geometric configurations

of 214plane translation theorem 50planets

life on other 142motion of 9, 15, 33, 88, 89,

113, 137, 161, 164, 168,217

Plato 82, 84, 256Platonicus (Platonic)

(Eratosthenes) 82Platonism 4, 54, 82–84,

119–120Plücker, Julius 152, 173Plutarch 10, 184Poincaré, Jules-Henri 36, 39,

171, 178, 209, 209–211celestial mechanics and

39intuitionism and 155–156

Poincaré conjecture 210point-set topology 53–54, 101Poisson, Siméon-Denis 64, 100,

111, 146, 163, 211–213, 212,235

Poisson distribution 213Poitiers, University of 101, 164,

241polar lines 215Pólya, George 213–215polynomial equations 55, 63,

65, 77, 87, 110, 149–150, 253Poncelet, Jean-Victor 173,

215–216, 232Poncelet-Steiner theorem 232power series 2, 62, 87, 248Practica arithmeticae et

mensurandi singularis (Practiceof mathematics and individualmeasurements) (Cardano)55

Practica Geometriae (Practice ofGeometry) (Fibonacci) 96

pragmatism 209Prague, University of 42

Precious Mirror of the FourElements (Chu Shih-Chieh)67–68

prime numbers 64–65, 82, 90,92, 112, 121, 159, 221–222

Princeton University 140Principia. See Philosophiae

naturalis principia mathematicaPrincipia Mathematica (Principles

of mathematics)(Russell) 228probability theory 29–31, 251

analysis and 171foundations of 25–27, 56,

92, 161–163, 170–171,187,240

logic and 46mathematical expectation

in 92, 141, 163measure theory and 46–47

progressions, arithmetical 165projection

in geometry 9, 73–75, 105,152, 186–187, 201215–216, 232

orthographic 190stereographic 137–138,

152, 217projective geometry 9, 73–75,

105, 152, 186–187, 201,215–216, 232

proof theory 103–104, 136, 205,228, 237, 258

proportion, theory of 83–84,169

Propositiones philosophicae(Propositions of philosophy)(Agnesi) 5

Protagoras of Abdera 70Prussian Academy 7Ptolemaic (geocentric) theory

9, 13, 108–109, 241Ptolemy, Claudius 13, 132, 202,

216–217pulleys 10Pythagoras of Samos 218,

218–220

304 A to Z of Mathematicians

Pythagoreans 13, 84, 138, 219Pythagorean theorem 139,

218–219

Q

quadratic equations 14, 49, 172quadratic reciprocity, law of 165quadrature of circle. See circle,

squaring ofQuaestiones naturales (Natural

Questions) (Adelard of Bath)4

quantum mechanics 39, 59,123, 129, 227

quantum physics 174, 227quartic equations 56, 92–93quaternions 72, 118, 123,

128–130, 133quintic equations 1–2, 87, 93,

110–111, 127, 145, 152

R

Radon, Johann 101Radon-Nikodym theorem 22Ramanujan, Srinivasa Aiyangar

130, 221–222random walk 214rational mechanics 7, 26, 28rational numbers 42, 149ray systems 128, 156real numbers 42–43, 52–54, 69,

84, 248–249, 257Recherches sur la précession des

équinoxes et sur la nutation del’axe de la terre (Research onthe precession of the equinoxesand on the nodding of theearth’s axis) (Alembert) 7

Recherches sur la probabilité desjugements en matière criminelleet en matière civile (Research inthe probability of criminal andcivil verdicts) (Poisson) 213

Réflexions sur la métaphysique ducalcul infinitesimal (Reflectionson the metaphysics ofinfinitesimul calculus)(Carnot) 58–59

Regiomontanus, Johann Müller222–223

regression, linear 254Regule Abaci (By rule of the

Abacus) (Adelard of Bath) 4Rein analytischer Beweis (Pure

analytic proof) (Bolzano) 43relativity 39, 152, 211

general theory of 59–60,176, 178, 186, 224, 226,250

special theory of 185–186,210

Rennes, University of 164Revista de matematica (journal)

205Rheticus, Georg (Georg

Joachim, von LauchenRheticus) 223–224

Ricci-Curbastro, Gregorio 176Riemann, Bernhard (Georg

Friedrich Bernhard Rieman)100, 113, 178, 224–227, 225

contemporaries of 36, 69differential geometry and

60measure theory and 164

Riemannian manifolds 176, 250Riemann integrals 164Riemann surfaces 105, 152Riemann zeta function 87,

225–227Riesz, Frigyes 89, 101, 136, 164,

227–228Riesz-Fisher theorem 228ring theory 139–140, 197Rivista de matematica (journal)

205Roberval, Gilles de 63, 73–74Roman numerals. See numerals,

RomanRome, University of 243

Roomen, Adriaan van 242roots

extraction of 68, 94, 149Newton-Raphson

computation of 49, 64Royal Academy of Sciences at

Turin 159Royal Irish Academy 129Royal Society (Great Britain)

246competitions and prizes of

235fellows of

17th century 24, 126,195–196, 246

18th century 2619th century 16, 100,

133, 234, 24020th century 98, 207,

222, 228in Newton-Leibniz

controversy 168, 188Royal Society of Edinburgh 102Royal Statistical Society (Great

Britain) 254Ruffini, Paolo 36, 67Russell, Bertrand 50, 103–104,

120, 206, 228–230, 251, 257Russell paradox 229

S

St. Andrews University 126, 191St. Petersburg Academy of

Sciences 27, 86St. Petersburg University 183The Sandreckoner (Archimedes

of Syracuse) 12Saturn, rings of 154Schickard, Wilhelm 203Schooten, Frans van 24Schur, Issai 174Schwartz, Herman 258science

ethics in 216philosophy of 210–211

Index 305

Sea Mirror of the CircleMeasurements (Li Chih)172

Seki Takakazu Kowa 231–232semiconvergent series 171semiotics 208sensationalism 7series

convergent 61–62Dirichlet 79divergent 7, 47, 61, 87Fourier 99, 176, 193hypergeometric 156, 222infinite 5, 14, 26, 52,

60–61, 178, 199convergence in 47, 73,

121, 155, 249sums in 2, 87, 183

power 2, 62, 87, 248semiconvergent 171summing of 49Taylor 64, 89, 182time 183, 255trigonometric 28, 79, 87,

98–99set theory 52, 164, 205, 208

axiomatic 22, 120, 258and continuum 54, 120,

258and denumerable sets 47,

53and infinity 43, 54, 70and law of excluded middle

50and measure theory 47

Siddhantasiromani (Bhaskara II)38

Sidereus nuncios (The siderialmessenger) (Galileo) 108

“sieve” of Eratosthenes 82, 92sines 14–15, 38, 49, 137, 182Skolem, Thoralf 258small sample theory 122–123,

206–207Socrates 256solar system 162solid geometry 91

Solution of the Difficulties inEuclid’s Elements (al-Haytham) 132

Some Propertires of BernoulliNumbers (Ramanujan) 221

Sorbonne (college of Universityof Paris) 48, 61, 164, 210

soundrefraction of 125velocity of 163, 213

Space and Time (Minkowski)185

spacesfunction 101, 164, 243linear 22vector 123, 205

special theory of relativity185–186, 210

spectrum 136Sphaerica. See Book of Spherical

Propositionsspinors 59spirals 30Stanford University 215statics 11, 233Statics and Hydrostatics (Stevin)

233statistical independence concept

188statistical mechanics 118Statistical Society of London 16statistics

actuarial science and 17,28, 73, 188

correlation coefficient in97, 254–255

hypothesis testing in 97,206–7, 208

least squares principle in113, 163

linear regression in 254mathematical 47, 122small sample theory in

122–123, 206–207Steiner, Jakob 216, 232, 249Steiner surface 232Steiner theorem 232

Steinhaus, Hugo 22stereographic projection

137–138, 152, 217Stern, Moritz 69Stetigkeit und Irrationale Zahlen

(Dedekind) 70Stevin, Simon 12, 233Stirling, James 188Stirling formula 188stochastic processes 171, 183,

251–252Stockholm, University of 102,

154Stokes, George Gabriel

233–235, 234Strasbourg, University of 101Student’s t distribution 122–123Studia mathematica

(Mathematical studies)(journal) 22

summands, partition of numbersinto 221

Sun 109, 144, 217distance of 13–14size of 138

surfaces, study of 152, 156, 212Sur les inégalités des moyen

mouvements des planetes (Onthe inequalities of planetarymovements) (Poisson) 212

Sweden, University of 1–2Synopsis of Pure Mathematics

(Carr) 221A System of Logic, Considered as

Semiotic (Peirce) 208Szego, Gábor 214

T

Tables for Statisticians andDiometricians (Pearson) 207

Tartaglia, Niccolò 55, 92–94,236–237, 237

tautochronism of cycloid 142,158

Taylor, Brook 33, 127

306 A to Z of Mathematicians

Index 307

Taylor series (Taylor expansions)64, 89, 182

Taylor’s theorem 127telegraphy 133, 211telescopes 108, 126, 142, 195The Tenth (Stevin) 233Thabit ibn Qurra 144, 177, 184Thales of Miletus 218,

237–238Theaetetus 82–83theodicy 168Theodorus Physicus 96Theon of Alexandria 12Theon of Smyrna 216–217theorems. See specific theoremsThéorie analytique de la chaleur

(Analytic theory of heat)(Fourier) 100

Théorie Analytique des Probabilités(Analytic theory ofprobability) (Laplace)162–163

Théorie de la manoeuvre desvaisseaux (Theory of themaneuver of vessels) (JohannBernoulli) 33

Théorie des nombres (Theory ofnumbers) (Legendre) 115

The Theory of Abelian Functions(Riemann) 226

Theory of Congruences(Chebyshev) 64

Theory of Series (JakobBernoulli) 30

Theory of Systems of Rays(Hamilton) 128

thermodynamics 36, 99,116–118, 162, 211, 213

Thomson, William 125three-body problem 159, 211,

243Tiloyapannatti (Yativrsabha) 254time series 183, 255Toplis, John 124topology 88, 206, 249

algebraic 139–140,209–210

combinatorial 147homology in 139–140of linear spaces 22, 101mappings in 50, 53, 140,

186–187, 210point-set 53–54, 101

Torricelli, Evangelista 24Toulouse, University of 89Tour of the Earth (Eudoxus of

Cnidus) 85Toward the Theory of Abelian

Functions (Weierstrass) 248Tractatus de configurationibus

qualitatem et motuum (Treatiseon the configurations ofqualities and motions)(Oresme) 199

Tract on Conic Sections (Wallis)246

tractrix 143Traité de dynamique (Treatise on

Dynamics) (d’Alembert) 7Traité de l’équilibre de du

mouvement des fluides (Treatiseon the equilibrium andmovement of fluids)(Alembert) 7

Traité des functions elliptiques(Treatise on elliptic functions)(Legendre) 164

Traité des substitutions et deséquations algébriques (Treatiseon substitutions and algebraicequations) (Jordan) 147

Traité du Mécanique Céleste(Treatise on celestialmechanics) (Laplace) 162

transcendental numbers 62, 87,126, 134, 191

transfinite concepts 21, 254Treatise of Algebra (Wallis) 246Treatise on Angular Sections

(Wallis) 246Treatise on Fluxions (Maclaurin)

181Treatise on Mechanics (Poisson)

212

Treatise on the ArithmeticalTriangle (Pascal) 203

Treatise on the Equilibrium ofLiquids (Pascal) 203

Treatise on the ProjectiveProperties of Figures (Poncelet)215

triangles 13, 29, 42–43, 67, 83,139, 184, 219, 222, 241

Pascal’s 68, 253trigonometric equations 186trigonometric functions 32, 167,

217trigonometric series 28, 79, 87,

98–99trigonometry

plane 184spherical 38, 41, 87, 137,

184tables in 137, 165,

182–183, 217, 222,223–224

trisection, of angle 111, 138,242

Tsu Ch’ung-Chih (ZhuChongzhi) 238–239

turbines 216Turin, University of 5, 105, 205Turing machine 179

U

Ukrainian Academy of Sciences23

universe, geometry of 177–178University College, London

122, 207, 254Uppsala, University of 102

V

Valla, Lorenzo 170Vandermonde, Alexandre 160variables, complex. See complex

variables

308 A to Z of Mathematicians

variance, analysis of (ANOVA)97, 122

variations, calculus of 248, 258contributions to 114, 164,

171, 176foundations of 32, 158general theory for 86, 88

vector analysis 118, 129, 133vector calculus 123, 205, 234vector fields 140vector spaces 123, 205Venn, John 240–241Venn diagrams 240–241Vera circuli et hyperbolae

quadratura (The true squaringof the circle and thehyperbola) (Gregory)126

Verrocchio, Andrea del 169versiera (bell-shaped cubic curve;

witch of Agnesi) 5vibration 28, 79, 102, 116Vienna, University of 119–120,

222, 224Viète, François 78, 90, 126, 141,

195, 241–243Vitruvius (Marcus Vitruvius

Pollio) 13Volterra, Vito 22, 102,

243–244

W

Wallis, John 24, 29, 64, 177,245–247, 246

Was Sind und was Sollen dieZahlen (What Numbers Areand Should Be) (Dedekind)70

Weber, Wilhelm 69, 114Weierstrass, Karl (Karl Theodor

Wilhelm Weierstrass) 136,178, 247, 247–249

contemporaries of 155, 226and elliptic functions 36and real numbers 84students of 52, 152, 153,

173, 249and theory of functions 43

weights and measures 160, 162,165

well-ordering theorem 258Weyl, Hermann 137, 152, 174,

214, 249–250Whitehead, Alfred North 228Whittaker, Sir Edmund 40Wiener, Norbert 22, 183,

250–252, 251Wiener measure 251Wiles, Andrew 78, 92, 116Wisconsin, University of 39“witch of Agnesi” (versiera; bell-

shaped cubic curve) 5Wittenberg, University of

223–224women mathematicians 4–6,

115–116, 153–154, 178–180,196–198

Wren, Christopher 24

Y

Yale University 117Yang Hui 231, 253Yativrsabha 253–254Yuan Yü 172Yule, George Udny 254–255

Z

Zeno, paradoxes of 256–257Zeno of Elea 256–257Zermelo, Ernst (Ernst Friedrich

Ferdinand Zermelo) 214,257–259

zero, properties of 14, 38, 49zero position principle

(Nullstellensatz) 135Zhu Chongzhi. See Tsu Ch’ung-

ChihZhu Shijie. See Chu Shih-ChiehZürich, University of 249, 258Zwei Abhandlungen über die

Grundgleichungen derElektrodynamik (Two papers onthe principal equations ofelectrodynamics) (Minkowski)185