School of mathematics

After innumerablefail ures to solvethe problem at a time, even, wheninvestigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the

BY

FOHMEBLYPROFESSOR 03T APPLIED MATHEMATICSINTHE TULANE UNIVERSITY

OF LOUISIANA; NOWPROCESSOR or PHYSICS

IN COLORADO COLLEGE

*

AN increased interest in the history of the exact sciencesmanifested in recent years by teachers everywhere, and the

class-rooms and seminaries of our leading universities, cause

will be found acceptable to teachers and students

The pages treating necessarily in a very condensed

form of the progress made during the present century,

much time in the effort to render them accurate and

reasonably complete Many valuable suggestions and criticisms on the chapter on "B/ecent Times" have been made

by ,I)r. E W. Davis, of the University of Nebraska The

Dr J E Davies and Professor C.A Van Velzer,both of theUniversity of Wisconsin; to Dr G-. B Halsted, of theUniversity of Texas; ProfessorL M HosMns,of the Leland

StanfordJr University; and ProfessorGr.D Olds, ofAmherst

1 am specially indebted to Professor3T. H. Loud, ofColorado

the gentlemen abovenamed, aswell as toDr Carlo Veneziani

vi PKEFACE.

of Salt Lake City, who read the first part of my work in.

manuscript, I desire to express my hearty thanks But inacknowledging their kindness, I trust that I shall not seem

to lay upon them any share in the responsibility for errors

which I may have introduced in subsequent revision of the

FLORIAN CAJOBL

COLORADO COLLEGE, December, 1893.

TheSchoolofPythagoras 19

IntroductionofRomanMathematics 117TranslationofArabic Manuscripts 124

Viii TABLE OF

PAGE

BOOKS OF REFEKENCE.

The followingbooks,pamphlets,andarticles havebeen used

inthe preparation of this history Referenceto anyofthem

ismadeinthetextbygivingtherespectivenumber. Histories

marked with astararethe only ones of which extensive usehasbeen made.

1. GUNTHER, S Ziele tmd Hesultate der neueren torischen JForschung Erlangen, 1876.

Washington, 1890.

3. *CANToit, MORITZ Vorlesungen uber Gfeschichte der MathematiJc

4. EPPING,J. Astronomisches aus Babylon Unter Mitwirlcungvon

7. *HANKBL, HERMANN ZurGfeschichte derMathematiJcim Alterthum

8. *ALLMAN, G.J G-reek G-eometryfrom Thalesto JEuclid. Dublin,

11. WmcwELL,WILLIAM History oftheInductiveSciences.

KopQnlaagen, 1886.

X BOOKS OF REFERENCE.

13. *CHASLES,M G-eschichteder Geometric Aus dem Franzosischen

14.MARIE, MAXIMILIEN Histoire des Sciences MatheniatiquesetPhy

15. COMTE, A Philosophy of Mathematics, translated by W.M LESPIE

GIL-16. HANKEL, HERMANN DieISntwickelung derMathematikin den

letz-ten Jahrhunderten Tubingen, 1884.

17. GUNTHER, SIEGMUND und WiNBELBAND,W. GesckicJitederantiJcen

NaturwissenschaftundPhilosophic. Nordlingen,1888.

18. ARNETH,A GeschichtederreinenMathematik Stuttgart,1852

19. CANTOR, MOIUTZ Mathematische Beitrage zum Kulturleben derVoUcer Halle, 1863.

20. MATTIIIESSEN, LTIDWIG Grundzilge der Antiken und ModernenAlgebraderLitteralen GUichungen Leipzig, 1878.

21. OURTMANNund MULLER Fort$chrittederMathematik

22. PEACOCK, GEORGE Article "

PureMathematics London,1847.

26. PLAYFAIR, JOHN Article u

ProgressoftheMathematicalandPhys

ical Sciences," in Encyclopedia Britannica, 7th editi6n, con

tinuedinthe8tlx editionby SIK JOHN LESLIE

27. BEMORGAN,A ArithmeticalBooks fromtheInvention of Printing

to thePresent Time

28. NAPIER, MARK Memoirs ofJohn Napier of Merchiston Edin

burgh, 1834.

29. HALSTEB, G B "Note on the First English Euclid,"AmericanJournalof Mathematics, Vol XL, 1879.

30. MADAME PERIER The Life of Mr Paschal Translated into

EnglishbyW.A.,London, 1744.

31. MONTUCLA, J F Histoire des Mathematiques Paps, 1802.

32. BtiHRiNG- E Kritische Geschichte derallgemeimn Principien der

Mechanik Leipzig, 1887.

33. BREWSTER,D The Memoirsof Nc.wton. Edinburgh, 1860.

^81. BALL,W W.R A Short Account oftheHistory of Mathematics

London,1888,2ndedition, 189S,

35. DE MORGAN,A "On the Early History of lEfiEitesixualB," inthe

Leibniz in London," in jSitzirngsberichte der

Koniglich PreussischenAcademicderWissenschaftenzuBerlin,FeTbruar, 1891.

41. DBMoR(UK,A Articles "Muxions"anduCommercimn

42. *TODUUNTEK,I. AHistory oftheMathematical Theory of Probabil

ityfromthe Timeof Pascal to thatof Laplace Cambridge andLondon, 1865.

43 *Toi>iniNTBK, I. AHistory ofthe TheoryofElasticityandofthe

Strength ofMaterials. EditedandcompletedbyKARLPEARSON

Cambridge,1886.

44. TOBHUNTKR, I "

NoteontlieHistoryofCertain FormulasinSpher

BaHol, 1884,

46. RKIFF, R Gfeschichte der Unendlichen Heihen Tubingen,1889.

47. WALTKRSIIAUSKN, W. SAHTOUIXIS. Gauss, mmQ-ed&chniss. Leip

52* BEAUMONT,M I^LIK DB "Memoir of Legendre." Translatedby

C.A ALEXANDISR, SmithsonianIteport, 1867.

58. AUAOO, I) F. X Joseph Fourier." Smithsonian Eeport,

58. GIBBS, J. WILLARD "

Multiple Algebra," Proceedings of theAmericanAssociation fortheAdvancementofScience, 1886.

59. FINK, KARL. Geschichte der Elenientar-Mathematik Tubingen,1890.

60. WITTSTEIN, ARMIN Zur Qeschichte des Malfatttfschen Problems

Nordlingen,1878.

61. KLEIN,FELIX Vergleichende Betrachtimgen uber neuere

62. FORSYTH, A R Theory of Functions of a Complex Variable

Cambridge, 1893.

63. GRAHAM,R H Geometry ofPosition. London,1891.

64. SCHMIDT, FRANZ "Aus dem Leben zweier ungarischer matikerJohannund Wolfgang Bolyaivon Bolya." Grrunertfs

Mathe-Archiv,48:2,1868.

65. FAVARO, ANTON JustusBellavitis," Zeitschriftfur Mathematik

undPhysik, 26: 5, 1881.

66. BRONICE,AD. Julius Plucker Bonn,1871.

67. BAUER, GUSTAV Gfedachnissrede auf Otto Hesse Miinchen,

1882.

68. ALFREDCLEBSCH VersucheinerDarlegungund Wunligungseiner

EinigeWortezumAndenkonanHermannIlankol,"

Mathematische Annalen, VII.4, 1874.

73. MUIR, THOMAS ^1 Treatiseon Determinants* 1882.

74. SALMON, GEORGE "Arthur Cayley," Nature, 28:21, September,1883.

75. CAYLEY, A "JamesJoseph Sylvester," Nature, 39:10, January,1889.

76. BURKHARDT, HEiNRicii. Die AnfUngo der Gruppontliooiie und

Paolo Ikiffim," Zeitschrift der MathemaUk undPhysik, Supple

BOOKS OF REFERENCE. xiii

77. SYLVESTER, J J. Inaugural Presidential Address to the Mathe

maticalandPhysical Section ofthe BritishAssociationat Exeter 1869.

78. YALSON,C. A La Vieet les travauxduBaron Cauchy. TomeI.,

II., Paris, 1868.

79. SACHSE,ARNOLD Versuch einer Qeschichte der Darstellung

Meihen Gottingen, 1879.

80. BOIS-KEYMOND, PAUL DU Zur G-eschichte der Trigonometrischen

81. POINCARE, HENRI Notice sur les Travaux Scientifiques de Henri

Poincare Paris, 1886.

82. BJERKNES,C. A Niels-HenriTc Abel, Tableau de sa vie et deson

action scientifique Paris, 1885.

83. TUCKER,R "CarlFriedrichGauss,"Nature, April, 1877.

84. DIRICHLET, LEJEUNE Gfedachnissrede auf Carl Gf-iistav Jacob

Jacobi 1852.

85. ENNEPER, ALFRED JUlliptische JFunktionen Theorie und

86. HENRICI,O "Theory ofFunctions,"Nature, 43: 14and15, 1891.

87. DARBOUX, GASTON Noticestir lesTravauxScientijlquesdeM.tonDarboux Paris, 1884.

Gas-88. KUMMER,E.E GfedachnissredeaufG-ustav PeterLejeune-Dirichlet.

Berlin, 1860.

89. SMITH, H.J. STEPHEN "On the Present State and Prospects ofSomeBranchesofPureMathematics,"Proceedings oftheLondonMathematicalSociety, Vol.VIII,Nos 104, 105, 1876.

90. GLAISUISH,J.W.L "

Henry JohnStephenSmith, MonthlyNotices

oftheEoyalAstronomical Society,XLIV.,4, 1884.

91. BesselalsBremerIfandlungslehrling Bremen,1890.

92. FRANTZ,J*. Festrede aus Veranlassung vonHesseVs hundertjahrigemGeburtstag Konigsherg, 1884.

93. DZIOBEK, 0 Mathematical Theories of Planetary Motions.TranslatedintoEnglishby M."W". Harringtonand"W J.Hussey

94. HERMITB, Cn "Discoursprononc6devantlepresident dela

xiv BOOKS BEFBBBNCE,

08) Bdci-iER, MAXIME uABit of MathematicalHistory,"Bulletin of

the 2V" T. Math./SV>c., Vol II., No 5.

99. CAY:LEY, ARTHUR Report on the Recent Progress of Theoretical

Dynamics 1857.

100. GLAZEBROOK,U.T. Report on Optical Theories 1885.

101. ROSENBERGER,If. GeschichtetierPhysik Braunschweig,1887-1890

A HISTORY OF MATHEMATICS.

INTEODUCTION.

THE contemplation of the various steps by which mankind

has come into possession of the vast stock of mathematical

knowledge can hardly failto interest the mathematician He

takes pride inthe fact that his science, more than any other,

jnatheBiati.es has proved to be useless The chemist smiles

at the childish, efforts of alchemists, but the mathematician

finds the geometry of the Greeks and the arithmetic of the

Hindoos as useful and admirable as any research of to-day.

He is pleased to notice that though, in course of its development, mathematics has had periods of slow growth, yet inthemainit hasbeen pre-eminentlya progressive science

inayalso teach ushow toincreaseour store Says De Morgan,

*Theearly history of the mind ofmen with regard to mathe

matics leads us to point out our own errors; and in this

*aspect it is well to pay attention to the history of mathematics." Itwarnsus againsthasty conclusions; itpoints outthe importance of a good notation upon the progress of thescience; it discourages excessive specialisationon the part of

2 A HISTORY OF MATHEMATICS.

investigators, by showing how apparently distinct brandieshave been found to possess unexpected connecting links; it

lems which were, perhaps, solved long since; it discourages

whichhas led othermathematiciansto failure; itteachesthat

fortificationscan be takeninotherways than bydirect attack,thatwhenrepulsedfrom a direct assault it is well to reconnoitre andoccupythesurrounding ground andto discoverthesecretpaths by which the apparently unconquerable position

be emphasisedby citing a case in which ithas been violated

(Anuntold amount of intellectual energy has been expended

onthe quadrature ofthecircle,yetnoconquest hasbeen made

bydirectassault The circle-squarers have existed incrowds

ever since the period ofArchimedes After innumerablefail

ures to solvethe problem at a time, even, wheninvestigators

possessed that most powerful tool, the differential calculus,

persons versed in mathematics dropped the subject, wMlo

thosewho still persisted were completely ignorant of its

Ms-toryand generallymisunderstood the conditions of the problem.^ "Our problem," says De Morgan, "is to square the

circle with the old allowance of means: Euclids postulates

and nothing more. Wecannotrememberan instance tyx

a questionto be solved by adefinite method was tried by\$k6best heads, and answered at last, by that method, after thou

sandsof complete failures." But progresswas made on this

problem byapproaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that

ratioofthe circumference ofacircleto itsdiametot is

this ratio is also transcendental and that the quadrature <>

the by means of the ruler and compass

keen-minded mathematicians had long suspected; namely, thatthegreat army of circle-squarers

have, for two thousand years,been assaulting a fortification which is as indestructible asthefirmament ofheaven

Another reason for the desirability of historical study isthe value of historical knowledge to the teacher of mathe

bo greatly increasedif the solution of problems and the cold

logic of geometrical demonstrations arc interspersed with

historicalremarks and anecdotes A class in arithmetic will

be pleasedtohear about the Hindoos and their invention ofthe "

Arabic notation"

; they will marvel at the thousands

of years which elapsed before people had even thought ofintroducing into the numeral notation that Colunibus-eggthe zeroj they will find it astounding that it should have

taken solong to invent a notation which they themselves can

bisect a given angle, surprise them by telling of the many

futile attemptswhich havebeenmadeto solve, by elementarygeometry, the apparently very simple problem of the trisec-

whose area is double the area of a given square, tell them

about theduplication of the cube how the wrath of ^Apollocouldbe appeased only by the construction of a cubical altar

double the givenaltar, and how mathematicians long wrestledwiththis problem After the classhave exhaustedtheir ener

giesonthetheorem oftherighttriangle, tell them something

aboutits discoverer howPythagoras, jubilant overhis greataccomplishment, sacrificed a hecatomb to the Muses who in-him When the value of mathematical training is

in question, quotethe inscription overthe entrance into

i academyof Plato, the philosopher:

"

Let no one who is

4 A HISTORY OF MATHEMATICS.

unacquaintedwith geometryenterhere." Studentsin analyt^

ical geometry shouldknow something of Descartes, and, after

taking up the differential and integral calculus, they shouldbecome familiar with the parts that Kewton, Leibniz, and

Lagrange played in creating that science. In his historicaltalk it is possible for the teacher to make it plain to thestudentthat mathematics is not a dead science, but a livingoneinwhichsteadyprogress ismade.2

Thehistory ofmathematicsis important also as avaluablecontribution to the history of civilisation Human progress

is closely identified with scientific thought Mathematical and physical researches are a reliable record of intellectual

windows through which the philosophic eye looks into pastages andtracesthe lineofintellectual development

THE BABYLONIANS, THEfertilevalley of the Euphrates and Tigris was one ofthe primeval seats of human society. Authentic history ofthe peoples inhabitingthis region begins onlywith the foun

dation, in Chaldaoa and Babylonia, of a united kingdom out

thrown ontheir historybythe discovery of the artof readingthecuneiformorwedge-shaped system ofwriting.

In the studyof Babylonian mathematics webeginwith the

notation ofnumbers, Avertical wedge If stood for 1, while

theI characters" ^ and

y>*. signified 10 and 100 respec

tively G-rotefend believes the character for 10 originally tobeen the picture of two hands, as held in prayer, the

palniisbeing pressedtogether, the fingers close to each other,btiTOhethumbs thrust out, In the Babylonian notation two

ptincjiiples were employed the

^ditive) and multiplica

tive.

|i Numbers below 100 were

expressed by symbols whose

respt-Mctive values had to be added

^ Thus, y stood for 2,

for30

J Herethe

of higher orderappear alwaystothe left of those of

Iorder In writing the hundreds, on the other hand, a

Irsymbolwas placedtothe left of the 100,and was, in

the

6 A HISTOKY OF MATHEMATICS.

10 times 100, or 1000 But this symbol for 1000 was itselftakenfora newunit, whichcould take smaller coefficients to

its left. Thus, ^ ^ f>*"

denoted, not 20 times 100, but

10 times 1000 Of thelargest numbers written incuneiformsymbols, whichhavehithertobeenfound, none go as high as

amillion.3

If, as is believed by most specialists, the early Sumerianswerethe inventors of the cuneiform writing, then they were,

inallprobability, alsofamiliarwith the notation of numbers

Mostsurprising, in this connection, is the fact that Sumerian

inscriptions disclose the use, not only of the above decimalsystem, biit also of a sexagesimal one The latter was used

chiefly in constructingtables forweights andmeasures Itis

full of historical interest Its consequential development,both for integers and fractions, reveals a high degree ofmathematical insight. We possess two Babylonian tablets

whichexhibit itsuse One ofthem, probablywritten between

2300 and 1600 B.C., containsa table of square numbers up to

601 The numbers 1, 4, 9, 16,,25, 36, 49, are given asv thesquares of the firstsevenintegers respectively. Wehave next

(=80), which is a geometrical

the series becomes an arithmetical progression, the

only exhibitsthe useof the sexagesimal system, but

cates the acquaintance of the Babylonians with

THE BABYLONIANS. 7

Not tobe overlookedisthe factthat in the sexagesimal nota-.tion of integers the

Thus, in 1.4 (=64), the 1 is made to stand for 60,the unit

of the secondorder, byvirtue of its position with respect tothe4. Theintroduction of this principle at so early a date

is the more remarkable, because in the decimal notation

it

wasnot introducedtill about the fifth or sixth centuryafterChrist The principle of position, in its general and syste

matic application, requires a symbol for zero We ask, Did

the Babylonians possess one? Had they already taken the

units? Neither of the above tables answers this question,

for they happen to contain no number in which there was

occasion touse azero The sexagesimal systemwasusedalso

in fractions Thus, in the Babylonian

inscriptions, | and |

are designated by 30 and 20, the reader being expected, in

his mind, to supply the word "

sixtieths." The Greek geom

eter Hypsicles and the Alexandrian astronomer Ptolemaeus

borrowed the sexagesimal notation of fractions from theBabylonians andintroduced it into Greece From that timesexagesimal fractions held almost full sway in astronomical

and mathematical calculations until the sixteenth century,

when theyfinallyyielded theirplace tothe decimal fractions

It may beasked, What led to the invention of the sexagesi

mal system? Why was it that 60 parts were selected? To

decimal system,because it represents the number of fingers.

But nothing of the human body could have suggested 60.Cantoroffers the following theory: At first the Babyloniansreckoned the year at 360 days. This led to the division ofthe circle into360 degrees, each degree representing the daily

amount of the supposed yearly revolution of the sun aroundthe earth Now

8 A HISTOKY OF MATHEMATICS.

factthatthe radius can be applied to its cir%umference as achord6 times, and that each of thesechords subtends an arc

measuring exactly 60 degrees. Fixing their attention upon

these degrees, the division into 60 parts may have suggested

itself tp them Thus, when greater precision necessitated asubdivision of the degree, itwas partitioned into60minutes

In this way the sexagesimal notation may have originated.

into minutes and seconds on the scale of 60, is due to theBabylonians

Itappears thatthe peoplein the Tigro-Exiphrates basinhad

madeverycreditableadvance inarithmetic Theirknowledge

of arithmeticaland geometrical progressions has alreadybeenalludedto. lamblichus attributes to them also a knowledge

of proportion,and eventheinvention of the so-called musicalproportion Though we possess-no conclusive proof, we havenevertheless reason to believe that in practical calculationthey used the abacus Amongthe races of middle Asia, even

asfaras China, the abacusis as old as fable Now, Babylon,

wasonce agreatcommercialcentre, the -metropolis ofmany

course, no trace "As arule, inthe Oriental mind the intuitive powerseclipsethe severely rational andlogical."

The astronomy of the Babylonians has attracted much

attention They worshipped the heavenly bodies from tie

earliest historic times, When the after

THE EGYPTIANS. 9

the battle of Arbela (331 B.C.), took possession of Babylon,Callisthenes found there onburned brick astronomical recordsreachingback as far as2234B.C. Porphyrius says that these

mer, possessed aBabylonian record of eclipses going back to

747B.C. EecentlyEpping and Strassmaier4

threwconsidera

ble light on Babylonianchronology and astronomy byexplaining two calendars of the years 123 B.C. and 111 B.C., taken

from cuneiform tablets coining, presumably, from an old

account of the Babylonian calculation of the new and full

moon, and have identified by calculations the Babylonian

names of the planets, and of the twelve zodiacal signs and

twenty-eight normal stars which correspond to some extentwiththetwenty-eightnaksJiatras oftheHindoos We append

part of an Assyrian astronomical report, as translated by Oppert:

"Tothe King,mylord, thyfaithful servant, Mar-Istar."

"

Ihadalready predictedtomymasterthe King, Ierrednot."

THE EGYPTIANS.

Though there is great difference of opinion regarding theantiquity ofEgyptiancivilisation, yet allauthorities agree inthe statement that, however far back they go, they find no

uncivilisedstate of society. "

Menes, the first king, changesthecourse oftheWile, makes agreat reservoir, andbuilds thetemple of Phthah at Memphis." The Egyptians built the

10 A HISTOBY OF MATHEMATICS.

mathematics atleast of practical mathematics

All Greek writers are unanimous in ascribing, withoutenvy, to Egypt the priority of inventionin the mathematical

sciences Plato in Pho&drus says:

"

At the Egyptian city

of Naucratis there was a famous old god whoso name was

Theuth; the bird which is called the Ibis was sacred tohim, and he was the inventor ofmanyarts, suchas arithmetic

anddice, buthis great discoverywas the use ofletters/15

because there the priestly class had the leisure needful forthestudyofit. Geometry,inparticular,is saidby Herodotus,

writers to have originated in Egypt.5 In Herodotus wo find

this (II. c. 109):

"

They said also that this king [Sesostjris]

dividedthe landamongall Egyptians so as to giveeach 0110a

quadrangle of equal size and to draw fromeachMs revenues,

byimposing a tax to be levied yearly But every one from whose part the river tore away anything,had to go to hh^i

who had to measure out by how much th0land lxad>become

smaller,,in order that the owner might pay on what was left,

in/ proportion to the entire tax imposed, Iti this wny/ifcappears to me, geometry originated, which passed thence to

regarding Egyptian mathematics, or from indulging in wild

A hieratic papyrus, included in the Rhine! collection of tha

found to be a mathematical manual containing problems in

THE GREEKS 17

left behind no written records of their discoveries A full

jdstory ofGreek geometry and astronomy during this period,

written by Eudenus,a pupil of Aristotle,has been lost. It

was well known to Proclus, who, in his commentaries on

Euclid, givesabriefaccount of it. This abstract constitutesourmostreliable information We shall quote it

frequentlyunder thenameofEudemian Summary.

ToThales of Miletus (640-546B.C.),one ofthe "sevenwise

having introduced thestudy ofgeometryinto Greece During

middle life he engaged in commercial pursuits, which took

himto Egypt Heissaidtohave resided there, and to have

studied thephysical sciences and mathematics withthe Egyptian priests Plutarchdeclares thatThales soon excelled his

ofthe pyramids fromtheir shadows According to Plutarch,

thiswas dono by considering that the shadow castbyavertical staff ofknown lengthbearsthe same ratio to the shadow

ofthepyramidas the height of the staff bears to the height

of the pyramid This solution presupposes a knowledge ofproportion? and the Ahmes papyrus actually shows that therudiments of proportion were known to the Egyptians. Ac

cordingto Diogenes Laertius, the pyramids were measured by

Thales in. a different way; viz. by finding the length of the

shadowofthepyramidatthemoment when the shadow of a

The JSud&mian Summary ascribes to Thales the invention

ofthetheorems onthe equality ofvertical angles,the equality

af the angles at the base of an isosceles triangle,the bisec

18 A HISTORY OF MATHEMATICS.

triangleshaving aside and the two adjacent angles equal respectively The lasttheorem he applied to the measurement

theorem that all angles inscribed in a semicircle are right

inferred that he knew the sum of the three angles of a triangle tobe equalto two right angles, and the sides of equiangular triangles to be proportional.8

The Egyptians must have made use of theabove theorems on the straight line, in

someof their constructions foundinthe Ahmes papyrus, but

it was left for the Greek philosopher to give these truths,

which others saw, but did not formulate into words, an

explicit, abstract expression, and to put into scientific languageand subject to proof that which others merely felt to

be true Thales may be said to have created the geometry

of lines, essentially abstractinits character, while the.Egyp

tians studiedonlythe geometryof surfaces andtherudiments

ofsolid geometry, empirical intheir character.

8

WithThalesbegins also the study of scientific astronomy

He acquiredgreat celebritybythe prediction ofasolar eclipse

in 585 B.C. Whether hepredictedthe dayof the occurrence,

or simply the year, is not known. It is told of him thatwhile contemplatingthestars during an evening walk, hefellinto aditch The goodoldwoman attending him exclaimed,

thouseestnot whatisatthyfeet? "

The two mostprominent pupils ofThales wereAnaximander(b. 611 B.C.) and Anaximenes (b. 570 B.C.). They studied

chiefly astronomy andphysical philosophy OfAnaxagoras,;a

THE GREEKS 19

school, we knowlittle, exceptthat,whilein prison, he passed

his time attempting to square the circle. This is the first

time, in the history of mathematics, thatwe find mention ofthefamous problem of the quadratureof the circle, thatrock

upon which so many reputations have been destroyed. It

turnsuponthedeterminationofthe exact value ofIT. Approx

Hebrews, andEgyptians. Butthe invention of a method tofindits exactvalue,isthe knotty problem which has engaged

downtoour own Anaxagoras did not ofer any solution of

it,undseems tohaveluckilyescapedparalogisms.

Aboutthetimeof Anaxagoras, but isolated from the Ionic

school,flourished (Enopides ofChios Proclusascribes to him

the solution ofthefollowingproblems: Froma point without,

to drawaperpendicular to a given line,and to drawan angle

on. a line equal to a given augle That a man could gain a

reputation by solving problems so elementary as these, eates that geometry was still in its infancy, and that theGreeks had not yet gotten far beyond the Egyptian con

indi-structions

comparedwithits growth in a later epoch of Greek history

A newimpetus toits progresswas givenbyPythagoras.

TJie School of Pythagoras.

Pyrthagoras (580 ?-500? B.C.) was oneof those figureswhich

impressed the imagination of succeeding /ffmes to such an

eitenlt that their real histories have become difficult to be,d&a|medthroughthe mythical hazethat envelopsthem The

20 A HISTORY OF MATHEMATICS.

the fame of Pherecydes to the island of Syros. He thenvisited the ancient Thales,who incitedhimtostndy inEgypt

He sojourned in Egypt many years, and may have,visitedBabylon On his return to Samos, he found it under the

tyranny of Polycrates Failing in an attempt to found aschoolthere, hequittedhome againand, following the current

of civilisation,removedto MagnaGrsecia in South Italy. He

settledatCroton,and foundedthefamous Pythagoreanschool

Thiswasnot merelyan academyfortheteachingofphilosophy,mathematics, and natural science, but it was a brotherhood,themembersofwhichwere unitedforlife. This brotherhood

l|adobservances"approachingmasonic peculiarity. Thejrwore

forbidden to divulge the discoveries and doctrines of their

as a body, and find it difficult to determine to whom each

themselves were inthe habit of referring everydiscovep"back

to the greatfounderof thesect.

This school grew rapidly and gained considerable political

ascendency But the mystic and secret obseivaaptcfe, intro

duced in imitation of Egyptian usages, and the a*|stooratictendencies of the school, caused it to becoiae ai* object, ofsuspicion The democratic partyinLowerItely revolted and

destroyedthe buildings of the Pythagoreanschool*

rasfledtoTarentum andthencetoMetapontum,

murdered

Pythagoras has left behind no mathematical tventtees, and

our sources of information are rather scanty Certain it is

that,in thePythagorean school,mathematicswasthe

study Pythagorasraisedmathematicstothetaakofa soArithmetic wascourtedby himas ferventlyft&*geo$tetYj

arithmeticisthe foundationofhis

icipaliencc

especiallyfond of those geometrical relations which admitted

of arithmetical expression

Like Egyptiangeometry, the geometryof thePythagoreans

ismuchconcernedwithareas ToPythagoras is ascribed theimportant theorem that the square on the hypotenuse of aright triangleisequaltothe sumof the squares on the other

,two sides/He hadprobablylearned from the Egyptians thetruth of the theorem in the special case when the sides are

jubilant overthis discovery thathe sacrificedahecatomb Its

authenticityis doubted, because thePythagoreans believed inthe transmigration of the soul and opposed, therefore, thesheddingofblood. Inthelater traditionsof the!N"eo-Pythago-"

reansthis objectionis removed by replacingthis bloodysacri

ficebythat of an ox made of flour"

,of three squares, given in Euclids Elements, I. 47, is due toEuclidhimself, andnotto the Pythagoreans What the Py

thagoreanmethodof proofwas has been a favourite topic for

ispossibletodivideup a planeinto figures of either kind.Fromtheequilateral triangleandthe squarearisethesolids,

namely the tetraedron, octaedron, icosaedron, and the cube.Thesesolidswere, in all probability, knownto the Egyptians,

A HISTORY OF MATHEMATICS.

ophy, they represent respectively the four elements of thephysical world; namely, fire, air, water, and earth Lateranotherregularsolidwas/ discovered, namely the dodecaedron,which, in absence of a/fifth element, was made to representthe universe itself. lamblichus states that Hippasus, a Py-

thagorean, perished in thesea,"becausehe boastedthathefirstdivulged "

the sphere withthetwelve pentagons." The

star-fshapedpentagram wasusedas asymbol of recognition bythe

1Pythagoreans, and wascalledby themHealth

Pythagorascalledthe sphere themostbeautiful ofallsolids,

and the circle the mostbeautifttl of all plane figures The

treatment of the subjects of proportion and of irrational

quantitiesby him andhis school will be taken up under the

headof arithmetic

AccordingtoEudemus,the Pythagoreans inventedthe prob-*

lernsconcerningthe application of areas, including the cases

~fdefectandexcess, as in Euclid, VI 28, 29.

Theywere alsofamiliarwiththe construction of a polygon

iqualin area to agivenpolygon andsimilar to another given

)olygon This problem depends upon several important and

somewhat advanced theorems, and testifies to the fact that

t

jhePythagoreans madenomeanprogress in geometry.

Of the theorems generally ascribed to the Italian school,some cannot be attributed to Pythagoras himself,no* to his

earliestsuccessors Theprogressfrom empirical to reasoned

solutions must, of necessity, have been slow It is worth

noticing thatonthecircleno theorem of anyimportance *wa$

Though politics broke up the Pythagorean fraternity, yetthe school continued to exist at least two centuries longer*

Among the later Pythagoreans, Philolaus and Arckytas aw

THE GREEKS. 23

rean doctrines By him were first given to tlie world tlieteachings of the Italian school, which had been kept secretfor a whole century. The brilliant Archytas of Tarentum

(428-347B.C.), known as a great statesman and general, and

universallyadmired forhis virtues, wastheonlygreat geome

teramongtheGreeks when Platoopened his school

thelattersubject methodically. He alsofound a veryingeniousmechanical solutionto the problem of the duplication ofthe cube His solution involves clear notions on the generation of cones and cylinders. This problem reducesitself to

These mean proportionals were obtained by Archytas from

the section of a half-cylinder. The doctrine of proportion

was advancedthrough him

Thereis every reason tobelieve thatthelaterPythagoreansexercisedastrong influence onthe studyand development ofmathematics atAthens TheSophistsacquired geometryfrom Pythagorean sources. Plato bought the works of Philolaus,

and had awarmfriend in Archytas

After the defeat ofthePersiansunder Xerxes at^the battle

of Salamis, 480B.C., a league was formed among the Greeks

fcopreserve thefreedomof the now liberated Greek cities on

Athens soon became leader and dictator She caused the

Athens,and then spent the money of herallies for her own

tggrandisement Athens was also a great commercial centre

Phus she became the richest and most beautiful city of

an-All menial work was performed by slaves The

24 A HISTORY OF MATHEMATICS.

citizen of Athens was well-to-do and enjoyed a large amount

of leisure The government being purely democratic, everycitizen was a politician. To make his influence felt amonghis fellow-men he must,first of all, be educated Thus there

teachers were called Sophists, or "wise men." Unlike thePythagoreans, they acceptedpayfor their teaching. Although

rhetoric was the principal feature of their instruction, they

also taught geometry, astronomy, and philosophy. Athens

soonbecamethe headquarters of Grecian men of letters, and

of mathematicians in particular. The home of mathematics

among the. Greeks was first in the Ionian Islands, then in

Lower Italy, and during the time now under consideration,

at Athens,

\ The geometry of the circle, which had been entirelyneglectedbythePythagoreans,was takenup by the Sophists.Nearly all their discoveries were made in connection with

famousproblems:

(1) Totrisectanarc oran angle.

(2) To"

double the cube," i.e. tofind a cubewhose volume

is doublethat ofa givencube

(3) To "square the circle," i.e. to find a square or some

otherrectilinear figureexactly equalin areatoagivencircle*

These problems have probably been the subject of more

problems in geometry The trisection of an angle, on theotherhand, presented unexpected difficulties. A right iwaglehadbeendividedintothree equal parts bythe Pythagoreans,

THE GREEKS 25

fco wrestle with it was Hippias of Blis, a contemporary ofSocrates, and born about 460B.C. Like all the later geome

and compass only Prockts mentions aman, Hippias, presum

ablyHippiasof Elis, as the inventor ofatranscendentalcurve

whichserved todivide an angle not onlyinto three, but into

any numberofequal parts This same curve was used later

by Deinostratus and others for the quadrature of the circle.

Onthis accountit is calledthe quadratrix

ThePythagoreans had shown thatthe diagonalof a square

isthe side of another square having double the area of the

duplication of thecube, i.e. to findthe edge of a cube havingdouble thevolumeof a given cube Eratosthenes ascribes to

this problem a different origin The Delians were once suf

double acertain cubical altar. Thoughtless workmen simply

constructed acubewith edges twice as long,but this did notpacifythe gods. The error being discovered, Platowas consulted onthe matter He and his disciples searched eagerly

for asolutionto this "Delian Problem." Hippocrates of Chios(about 430B.C.), atalentedmathematician, but otherwise slow

and stupid, was the first to showthat the problem could bereduced to finding two mean proportionals between a given

line and another twice as long For, in the proportion a:a?

. But he failed to find the two

mean proportionals. His attempt to square the pircl& was

squaring akine, he committed anerror in attemptingtoapply

thisresultto thesquaring of the circle.

lujhis study of the quadrature and duplication-problems,

26 A HISTORY OF MATHEMATICS.

developed by Hippocrates This involved the theory of

Greeks only in numbers They never succeeded in unitingthe notions ofnumbers and magnitudes Theterm"number "

irrational numbers was not included underthis notion Not

even rational fractions were called numbers They used the

bers were conceived as discontinuous, while magnitudes were

distinct The chasm between them is exposed to full view

in the statement of Euclid that "incommensurable magni

tudes do not have the same ratio as numbers." In EuclidsElements we find the theory of proportion of magnitudesdevelopedand treated independent of that of numbers. The

nitudes (and to lengths in particular) was a difficult and

important step

Hippocrates added to his fame by writing a geometricaltext-book, called the Elements. This publication shows thatthe Pythagorean habit of secrecy was being abandoned;

secrecywas contraryto the spirit ofAthenian life.

The Sophist Antiphon, acontemporary of Hippocrates, introduced the process of exhaustion for the purpose of solvingthe problem of the quadrature Ho did himself credit by

remarking that byinscribing in a circle a square, and oa its

sides erecting isosceles triangles with their vertices itt thecircumference, and on the sides of these triangles erecting

new triangles, etc., one could obtain a succession of regularpolygons of 8, 16, 32, 64 sides, and so on, of "which eneh,

approaches nearer to the circle than the pxeviot^.o&f untilthe circle exhausted Thais obtained an

THE GREEKS* 27

polygonwhose sides coincide with the circumference Since

there also can be found a square equal to the last polygon

inscribed, and therefore equal to the circle itself. Brys0n

of Heraclea, a contemporary of Antiphon, advanced the prob

lem of the quadrature considerably by circumscribing polygons at the same timethat he inscribed polygons.- He erred,

however, in assuming that the area of a circlewas the arith

metical mean between circumscribed and inscribed polygons

Unlike Bryson and the rest of Greek geometers, Antiphon seems to have believed it possible, by continually doublingthe sides of an inscribed polygon, to obtain a polygon coin

that magnitudes are divisible ad infinitum. Aristotle alwayssupported the theory of tihe infinite divisibility, while Zeno,the Stoic, attempted to show its absurdity by proving that

if magnitudes are infinitely divisible, motion is impossible

Zeno argues that Achilles could not overtake a tortoise; forwhile he hastened to the place where the tortoise had been

when he started,the tortoise crept some distance ahead,and

while Achilles reached that second spot, the tortoise again

moved forward a little, and so on Thus the tortoise was

always in advance of Achilles Such arguments greatly con

founded Greek geometers. No wonder they were deterred

by such paradoxes from introducing the idea of infinity into

theirgeometry Itdidnotsuitthe rigour of theirproofs

brous but perfectly rigorous "method of exhaustion." In

determining the ratio of the areas between two curvilinearplane i|jp,|% s&y/two circles, geometers first inscribed or

and then bv

A HISTOEY OF MATHEMATICS.

the number of sides, nearly exhausted the spaces

between the polygons and circumferences IProm the theo

rem that similar polygons inscribed in circles are to eachothsr as the squares on their diameters, geometers may havedivined thetheorem attributed to Hippocrates of Chios thatthe circles, which differ but little from the last drawn poly

gons, mustbeto eachotheras the squares on their diameters

But in ordertoexcludeall vaguenessand possibilityof doubt,

later Greek geometers applied reasoning like that in Euclid,XII.2, as follows: Let and c, D and d be respectivelythe

circles and diameters in question Then if the proportion

, and P: O = p : c

Sincej> > c f

, we have P>C, which is absurd Next they

proved by this same method of reductio ad absurdum the

falsityof the supposition, thatcf

larger nor smaller than, c, it must be equal to it, QJE.D

Hankel refers this Method of Exhaustion back to Hippo

crates of Chios,but the reasons for assigning itto thisearlywriter, ratherthanto Eudoxus,seem insufficient.

Thoughprogress ingeometryatthis periodistraceable only

at Athens, yet Ionia, Sicily, Abdera in Thrace, and Gyrene

produced mathematicians who made creditable contributionB

to the science We can mention here only Bemociitus of

Abdera (about 460-370 B.C.), a pupil ofAnaxagoras, afriend

of Philolaus,- and an admirer of the Pythagoreans He

visitedEgypt and perhaps even Persia Ho was asuccessfulgeometer and wrote on incommensurable lines, on geometry,

on numbers, and on perspective. Hone of these works areextant, He used to boast that in the construction of plane

THE GREEKS 29*

the so-calledharpedonaptae ("rope-stretchers") ofEgypt By

this assertion he pays a flattering compliment to the skill

and ability of the Egyptians

TJie Platonic School

DuringthePeloponnesian War (431-404 B.C.) the progress

of geometry was checked After the war, Athens sank intothebackground as aminorpoliticalpower, but advancedmore and more to the front as the leader in philosophy,literature,

and science Plato was born at Athensin 429 B.C.,theyear

acquiredhis taste for mathematics Afterthe death of Soc

rates, Plato travelled extensively. In Cyrene he studiedmathematics under Theodoras He went to Egypt, then to

Lower Italy and Sicily, where he came in contact with thePythagoreans Archytas of Tarentum and Timaeus of Locri

became his intimate friends On his return toAthens^ about

anddevoted the remainder of hislife toteachingandwriting.

Platosphysical philosophy is partly based on that of thePythagoreans Like them, he sought in arithmetic and geometry the key to the universe When questioned about

theoccupation of the Deity, Plato answered that "

He

geom-etrises continually." Accordingly, a knowledge of geometry

is a necessary preparation for the study of philosophy. To

necessary it is forhigher speculation, Plato placedtheinscription over Ms porch, "Let no one who is unacquainted with

geometry enter here," Xenocrates, a successor of Plato asteacher intheAcademy, followedin his mastersfootsteps, by

admit a pupilwho had no mathematical training,

30 A HISTOBY OF MATHEMATICS.

with the remark, "Depart, for thou hast not the grip ofphilosophy

1

Plato observedthatgeometry trained the mind

Eudemian Summary says,

"

matical discoveries, and exhibited on every occasion the re

markable connection between mathematics andphilosophy."

the Platonic school producedsolarge anumber of mathemati

cians Plato did little real original work, but he made

in geometry It is true that the Sophist geometers of theprevious century were rigorous in their proofs, but as a rulethey did not reflect on the inward nature of their methods

Theyusedtheaxioms withoutgivingthemexplicitexpression,andthe geometrical concepts, suchasthe point, line, surface,

thagoreans called a point "unity in position/7

but this is astatementof a philosophical theory rather than a definition

Plato objectedto callingapoint a "

geometrical fiction." He

defined apoint as the "beginningof aline" or as "an indivis

lengthwithout breadth." He calledthepoint, line, surface, the boundaries of the line, surface,

true ofEuclidsaxioms Aristotle refers to Plato the axiom

that "equals subtractedfromequals leaveequals."

7 One of the greatestachievements of Platoand his school isthe invention ofanalysis as a method of proof. To be sure,

this method had been used unconsciously byHippocrates and

others; butPlato, like a true philosopher^ turned the instinc

tive logic into aconscious, legitimate method

The terms synthesis and analysis are used in mathematics

THE GREEKS 31

oldest definition of mathematical analysis as opposed to synthesisis thatgiven in Euclid, XIII.5,whichinallprobability

was framed by Eudoxus:

"

Analysis is the obtaining of thething sought by assuming it and so reasoning up to anadmitted truth; synthesis is the obtaining of the thingsought by reasoning up to the inference and proof of it."

The analytic method is not conclusive, unless all operationsinvolved in it are known to be reversible To remove all

a synthetic one, consisting of a reversion of all operations

occurring in the analysis. Thus the aim of analysiswas to

aid inthe discovery of synthetic proofs or solutions

; Plato is said to havesolved the problem ofthe duplication

of the cube Butthe solution isopentothevery same objec

tion which he made to the solutions by Archytas, Eudoxus,

and Menaeclmius He called their solutions not geometrical,

but mechanical, fortheyrequiredtheuse of otherinstrumentsthan the rulerand compass. He said that thereby

"

thegood

ofgeometry is set aside anddestroyed, for we again reduce it

to the worldofsense, instead of elevating and imbuingitwiththe eternal and incorporeal images of thought, even as it is

employed byGod, forwhichreasonHe alwaysis God." These

objections indicate either that the solution is wronglyattrib

uted to Plato or that he wished to show how easilynon-geo

metric solutions of that character can be found It is now

means oftherulerand compass only

Plato gave a healthful stimulus to thestudyof stereometry,

which untilhistime hadbeenentirely neglected. Thesphere

32 A HISTOBY OF MATHEMATICS.

epoch-making Menaechmus, an associate of Plato and pupil

of Eudoxus, invented the conic sections, which, in course ofonly a century, raised geometry to the loftiestheightwhich

obtuse-angled/ by planes at right angles to a side of thecones, andthusobtained the threesections which we nowcall

the parabola, ellipse, and hyperbola. Judging from the two

veryelegant solutions of the "Delian Problem" by means ofintersections of these curves,Mensechimis musthave succeededwellininvestigatingtheir properties

Another great geometer was Dinostratus, the brother of

Menaechmus andpupilofPlato Celebratedis hismechanicalsolution of thequadrature ofthecircle, by meansof the quad-

ratri of Hippias

Perhapsthemostbrilliantmathematician of thisperiodwas

Eudoxus He wasbornatCniclus about 408B.O.,studiedunder

Archytas, and later, for two months, under Plato He was imbued with a true spirit of scientificinquiry, and has beea

called the father ofscientificastronomical observation From

thefragmentarynotices of his astronomical researches,found

in later writers, Ideler and Schiaparolli succeeded in reconstructing thesystemofEudoxuswithitscelebrated representa

tion of planetarymotionsby "concentricspheres*" Eudoxus

hada schoolat Cyzicus,wentwithhispupils to Athens, visit

ing Plato, and then returned to Cyzicxis, where ho died 355

B.C. The fame of the academy of Plato is to a large extent

whom are Meneeclnnus, Dinostratus, Athensaus, and Helicon.DiogenesLaertius describesEudoxus asastronomer, physician,

as well The Eudemimi Summary

THE GREEKS 33

says that Eudoxus "

theorems, added to the three proportions three more, aixd

raised to a considerable quantitythe learning, begun byPlato,

onthesubject of thesection, to which he applied the analyt

and mean ratio The first five propositions in Euclid XIII

relate to lines cut bythis section, and are generally attributed

toEudoxus Eudoxus added much to the knowledge ofsolid

geometry He proved, says Archimedes, that a pyramid isexactly one-third ofaprism,and aconeone-third of acylinder,having equal base and altitude The proof that spheres are

to each other as the cubes of their radii is probably due tohim He made frequent and skilful use of the method ofexhaustion, of which he was in all probability the inventor

A scholiaston Euclid, thoughttobe Proclus, says further that

Eudoxus practicallyinvented thewhole ofEuclids fifthbook

Eudoxus also found two mean proportionals between two

given lines, but the method of solution is not known.

Plato has been called a maker of mathematicians Besides

natural gifts, to whom, no\loubt, Euclidwas greatlyindebted

inthe composition of the 10thbook;

8

treating of

incommensu-rables; Leodamas of Thasos; Feocleides and his pupil Leon,

who added much tothework of their predecessors, for Leonwrote an Elements carefully designed, both in number and

utility of its proofs; Theudius of Magnesia, who composed avery good book of Elements and generalised propositions,which had been confined to particular cases; Hermotimus of

and composed some on loci; and,finally,thenames of Amyclas

34 A HISTOBY OF MATHEMATICS,

A skilful mathematician of whose life and works we have

no details is Aristaelis,the elder, probablya senior contempo

His works contained probably a summary of the researches

of thePlatonicschool.8

Aristotle (384-322B.C.),the systematise! ofdeductivelogic,

though not a professed mathematician, promoted the science

of geometry by improving some of the most difficult definitions His Physics contains passages with suggestive hints

of the principle of virtual velocities About his time thereappeared a work called Mechanic, of which he is regarded

by some as the author Mechanics was totally neglected by

thePlatonicschool

The First Alexandrian School,

Inthe previous pages we have seen the birth of geometry

in Egypt, its transference to the Ionian Islands, thence toLower Italyand to Athens Wo have witnessed its growth

in Greece from feeble childhood to vigorous manhood, and

now weshall seeitreturn to the land of its birth and therederive newvigour.

During her declining years, immediately following theFeloponnesian War, Athens produced the greatest scientists

andAristotle In 338 B.C.,atthe battleof OUf&ronea, Athens was beaten,byPhilip of Macedon, and her power was broken

forever Soonafter, Alexander the Great,the son of Philip,