A Brief History of Math by Fink - 1900- 29 pgs

.7 . e an ABRIEF HISTORY OF MATHEMATICS AN AUTHORIZED TRANSLATION OF DR. KARL FINK'S GESCHICHTE DER ELEMENTAR-MATHEMATIK BY d3 @ .WOOSTER WOODRUFF BEMAN Vi3sav g PROFESSOROFMATHEMATICS INTHEUNIVERSITY OFMICHIGAN, x iy . AND .uégaz PAVIDEUGENE SMITH Aa2= | 7 Pat PROPERTYOF x +THIRD,REVISEDEDITION 334a6 CARNEGTETECHNICALSCHOOLS 4tee ®:os ay2% ! {ewtcaco |. . THEoprn| COURT PUBLISHING COMPANY , vi HISTORY OFMATHEMATICS. pols. areconsidered inordernumber-systems andnumber-syo? gar e arithmetic, algebra, geometry andtrigonometry, allowing, @520°aspossible, within thenarrow confines ofasingle branch o£ * elements,arapidandsureorientation. AgainstsuchaprocedaF*theobjection mayberaised thatinthiswaythegeneral surveY 7 theculture history ofacertain epoch willsuffer. OntheotP©™ hand, inahistory ofelementary mathematics, especially one CO" fined within such modest bounds, anexhaustive description whole periods withalltheircorrelations ofpastandfuture cannot well bepresented. Itisnotthepurpose ofthisworktosetforththeinterestir= historical development ofmechanics andastronomy. Althougb #* cannot bedenied that bythisseparation ofrelated branches theé©@ ~ iswanting acertain definitiveness tothework, yetthehope 23 beexpressed that thislackwillnotbefelttookeenly, The ele- mentary parts ofmathematics haveonlyfewpoints ofcontact with thesebranches,andourendeavoristopresentinbriefcomp25% r) onlythatwhichismostessential. Further, intheinterest ofapresentation ascondensed asPOS sible, thebiographical notices which often lend great attraction toamore extended treatment ofasubject must berelegated tothe appendix andthere treated butbriefly. ‘Thework haditsinception incertain suggestions which the author received atthesemi-monthly meetings ofamathematical club inTibingen, founded andconducted byProf. Dr.A.Brill, forwhich suitable thanks ought heretobeexpressed. Acknowl- edgment isespecially duetothepresident oftheclubwhose im- terpretations have been decisive forcertain parts ofthepresent work. These meetings furnished theauthor thedesired oppor- tunity, through the lectures connected with themost diverse branches ofthescience and through thediscussions which often followed, with references torecent literature, topenetrate into those circles ofthought which to-day dominate thehigher branches e ofmathematics. Thewriter wasthusledtocomplete hisstudies PREFACE. vii ® bygoingintotherecenthistoryofthescience.Theresultsof such investigations arehere presented with perhaps greater full- ness than seems necessary forthemain purpose ofthebook or justified byitstitle. But indefault ofsuch adigest, afirst experi- ment may layclaim toafriendly judgment, inspite ofthe con- tinually increasing subdivisions ofthe science; nor will such an attempt bethought inappropriate, inasmuch asitdoes not seem possible todraw asharp line ofdemarcation between theelemen- taryandhigher mathematics. Forontheonehand certain prob- lems ofelementary mathematics have from time totime furnished theoccasion forthedevelopment ofhigher branches, andonthe other from theacquisitions ofthese new branches aclear light has fallen upon theelementary parts. Accordingly itmay begratify- ingtomany astudent and teacher tofind here atleast that which isfundamental. . r) ‘Theexceedingly richliterature, especially inGerman, atthe disposal oftheauthor isreferred tointhefootnotes. Hehasmade free use ofthe excellent Jahrbuch aber die Fortschritte der Mathematik, which with clear and systematic arrangement enu- merates and discusses the most recent mathematical literature. K, Fink. Tvamezn, June, 1890, Vii>Wake ° | . ix 6 CONTENTS. ox ‘Translators’Preface, 2ss+speeeelf Author's Preface, 4.545ssnes L_NUMBER-SYSTEMS ANDNUMBER-SYMBOLS. 6 UL_ARITHMETIC. A.General Survey. oss) ssss 8 ‘_Rirst Period. TheArithmetic oftheOtdestNations totheTimeoftheArabs," . 1,TheArithmetic ofWhole Numbers. ......42.TheArithmotio ofFractions .......-+3F8.Applied Arithmetic... .1.1.+ss+3g .Second Perlod. FromtheEighthtotheFourteenthCen tury, 2.‘TheArithmetic ofWholeNumbers... ....36 2.The ArithmeticofFractions...» ++++49e g.AppliedArithmetic... ..++. at‘D._Third Period. FromtheFifteenth totheNineteenth Cen. tory.1,TheArithmeticofWhole Numbers...5...4 2.TheArithmeticofFractions. ....++»++49 3.AppliedArithmetic... ..+.++e+>Sf TIL_ALGEBRA, A.General Survey... 11seesBE B.EFirst Period. FromtheEarliest TimestotheArabs.%.Genoral Arithmetic .0.2.ss ws 3 Bgyptian Symbolism 63.Greek Arithmetio6%;Symbolism 64;Theory ofNumbers 66;Series67;theIrrational 68;Neg 1 |xHISTORYOFMATHEMATICS. e PAGEative Numbers 70;Archimedes's NotationforLargeNumbers7x.Roman Arithmetic 71,Hindu Arithmetic 72;Symbolism 92;Negative Numbers 72;Involution andEvolution 73;Per- mutations and Combinations 74;Series 74. Chinese Arith- metic7s. Arab Arithmetic74;“Algorism”75;RadicalSigns 76;Theory ofNumbers 76;Series 76. 2.Algebra. 26ee ee ee ee TT ‘The Egyptians 77.The Greeks; Form oftheEquation 773 Equations oftheFirst Degree 78;Equations oftheSecond Degree (Application ofAreas) 79;Equations oftheThird De- gree 81;Indeterminate Equations (Cattle Problem ofArchi- medes; Methods ofSolution ofDiophantus) 83. Hindu Al- gebra 84. Chinese Algebra 87. Arab Algebra 88, C. Second Period. To the Middle ofthe Seventeenth Cen- tury. x.General Arithmetic. ©2. 2... ee 8S Symbolism oftheItalians and theGerman Cossists 95; Irra~ tional and Negative Numbers 99;Imaginary Quantities rox5 Powers20a;Series103;Stifel’sDuplicationoftheCube104} e@Magic Squares 105. 2Algebras 22ee 05 Representation ofEquations x07; Equations ofthe First and Second Degrees 108; Complete Solution ofEquations ofthe ThirdandFourthDegreesbytheItalians112;Workofthe |German Cossists 113; Beginnings ofaGeneral Theory ofAl- gebraic Equations 115. D. Third Period. From the Middle ofthe Seventeenth Cen- tury tothe Present Time. Symbolism 117; Pascal's Arithmetic Triangle 118; Irrational Numbers 139; Complex Numbers 123; Grassmann’s Aus- deknungelehre x27; Quaternions 129; Calculus ofLogic 131; Continued Fractions 131; Theory ofNumbers 133; Tables of 5 Primes 142; Symmetric Functions x42; Elimination 143;The- oryofInvariants andCovariants 145; Theory ofProbabilities x48; Method ofLeast Squares 149; Theory ofCombinations | 330;Infinite Series (Convergence andDivergence) 252;Solu- tion ofAlgebraic Equations 155; theCyclotomic Equation x60; Investigations ofAbel andGalois 63; Theory ofSubsti- tutions 164;theEquation oftheFifth Degree 165;Approxi-mationofRealRoots166;Determinants 107;Differentialand eIntegral Calculus 168; Differential Equations 174; Calculus ofVariations 178; Elliptic Functions :80; Alelian Functions 186; More Rigorous Tendency ofAnalysis 189, “yt Iv. GEOMETRY. Pacr e A.GeneralSurvey2221.1eeee190B.First Period. Egyptiansand Babylonians .....192 C.Second Period. TheGreeks. ..2... 1... 193 ‘The Geometry ofThales andPythagoras 194; Application of theQuadratrix totheQuadrature oftheCircleandtheTrisee-tionofanAngle196;theElements ofBuclid198;Archimedesand hisSuccessors 199; theTheory ofConic Sections 203; theDuplication oftheCube, theTrisection ofanAngle and the Quadrature oftheCircle 209; Plane, Solid, and Linear Loci 209; Surfaces ofthe Second Order 2x2; the Stereo- graphic Projection ofHipparchus 213. D.Third Period. Romans, Hindus, Chinese, Arabs. ..214 E.Fourth Period. From Gerbert toDescartes. ....218 Gerbert andLeonardo 218;Widmann andStifel220;Vietaand Kepler 222; Solution ofProblems with butOne Opening oftheCompasses 225; Methods ofProjection 226. F.Fifth Period. From Descartes tothePresent ....228 Descartes’s Analytic Geometry 230; Cavalieri's Method ofIn- . divisibles234;Pascal'sGeometricWorks237;Newton'sIn- e@vestigations 239; Cramer's Paradox 240; Pascal’s Limagon and other Curves 241; Analytic Geometry ofThree Dimen- sions 242; Minor Investigations243;Introduction ofProjec- tiveGeometry 246;M&bius's Barycentrischer CaleRi250;Bel-lavitis's Equipollences 250; Plticker’s Investigations 251; Steiner's Developments 256; Malfatti's Problem 256; Von Staudt’s Geometric derLage258;Descriptive Geometry 259; Form-theory and Deficiency ofanAlgebraic Curve 261; Gauche Curves 263; Enumerative Geometry 264; Conformal Representation 266; Differential Geometry (Theory ofCurva- | tureofSurfaces) 267;Non-Euclidean Geometry 270;Psendo-Spheres 273; Geometry of#Dimensions 275; Geometria and Analysis Situs 275; Contact-transformations 276; Geometric ‘Theory ofProbability 276; Geometric Models 277; theMath- ematics ofTo-day 279. V.TRIGONOMETRY. | A.General Survey, 2626s2eeee 88E B.First Period. From theMost Ancient Times totheArabs 282 | ‘TheEgyptians 282,‘TheGreeks 82,TheHindus8.The Arabs 285. . — xii HISTORY OFMATHEMATICS, CSecond Period. From theMiddle Ages totheMiddle of theSeventeenth Century. ......++28 Vieta and Regiomontanus: 987; Trigonometric Tables 289; Logarithms 290, D.Third Period From theMiddle oftheSeventeenth Cen- turytothe Present. ©..2... 1e+20 Biographical Notes... .1... 1. ee +89 Index ee . [ ad GENERAL SURVEY. Te beginnings ofthedevelopment ofmathemat-ical truths date back tothe earliest civilizations ofwhich any literary remains have come down tous, namely theEgyptian and the Babylonian. Onthe onehand, brought about bythedemands ofpractical life, onthe other springing from thereal scientific spirit ofseparate groups ofmen, especially ofthe. priestly caste, arithmetic and geometric notions came into being. Rarely, however, was this knowledge transmitted through writing, sothat ofthe Babylo- e niancivilization wepossess onlyafewtraces. From the ancient Egyptian, however, wehave atleast one manual, that ofAhmes, which inallprobability ap- peared nearly twothousand years before Christ. ‘ The real development ofmathematical knowledge, obviously stimulated byEgyptian and Babylonian in- fluences, begins inGreece. This development shows itself predominantly intherealm ofgeometry, and enters uponitsfirstclassic period, aperiod ofno great duration, during theeraofEuclid, Archimedes, Eratosthenes, and Apollonius. Subsequently itin- clines more toward the arithmetic side; but itsoon becomes socompletely engulfed bytheheavy waves @ 2 HISTORY OFMATHEMATICS. . e ofstormyperiodsthatonlyafterlongcenturies and inaforeign soil,outofGreek works which hades- caped thegeneral destruction, could aseed, newand fullofpromise, take root. ° One would naturally expect tofindtheRomans entering with eagerness upon therich intellectual inheritance which came tothem from theconquered Greeks, and tofind their sons, who sowillingly re- sorted toHellenic masters, showing anenthusiasm for Greek mathematics. Ofthis, however, wehave scarcely anyevidence. The Romans understood very well thepractical value tothestatesman ofGreek geometry and surveying—a thing which shows itself e@ alsointhelaterGreekschools—but norealmathe-maticaladvanceistobefoundanywhereinRonan} history. Indeed, theRomans often had somistaken anidea ofGreek learning that notinfrequently they handed itdown tolater generations inaform entirely distorted. More important forthe further development of mathematics aretherelations oftheGreek teachings tothe investigations ofthe Hindus and theArabs. The Hindus distinguished themselves byapronounced talent fornumerical calculation. What especially dis- tinguishes them istheir susceptibility totheinfluence _ofWestern science, theBabylonian andespecially theGreek,sothattheyincorporated intotheirown t) system what theyreceived fromoutside sources and . then worked outindependent results. . , GENERAL survey, 3 . TheArabs,however, inSeneraldonotshowthis @sameindependence ofapprehension andofjudgment.Theirchiefmerit,nonethelessarealonehowever,liesintheuntiring industry whichtheyshowedintranslating intotheirownlanguage theliterarytreas-uresoftheHindus,PersiansandGreeks.ThecourtsoftheMohammedan Princesfromtheninthtothethirteenth centuries weretheseatsofaremarkable .Scientific activity,andtothis.circumstance alonedoWeoweitthatafteraPeriodoflonganddensedark-nessWestern Europe wasinacomparatively shorttimeopeneduptothemathematical Sciences,Thelearning ofthecloisters intheearlierpart oftheMiddleAgeswasnotbynatureadaptedto r)enterseriously intomatters mathematical ortosearchfortrustworthy sourcesofsuchknowledge. ItwastheItalianmerchants whosePractical turnandeasyadaptability firstfound,intheircommercial relations ,withMohammedan WestAfricaandSouthernSpain,abundant useforthecommon calculations ofarith. metic.Norwasitlongafterthattheredeveloped | among themarealspiritofdiscovery, andthefirst greattriumph ofthenewlyrevived science wasthesolution ofthecubicequation byT,lia.Itshouldbesidhowever,hattheial chorezealouslytoextendAAgbymeansoftranslations inkswenn oeInthefifteenth Conti:ArEGRPROMSOLBest, yo bachandRegi nts,GeraneBRookPosition... . 4 HISTORY OFMATHEMATICS, e inthegreat rivalry fortheadvancement ofmathemat ics. From that time until the middle oftheseven teenth century theGermanmathematicianswer chieflycalculators,thatisteachersintheperhinin schools (Rechenschulen), Others, however, wereale braists, and thefact isdeserving ofemphasis that there were intellects striving toreach still lnftier heights. Among themKepler stands forthpreetn nent, butwith him areassociated Stifel, Rucolff, and Biirgi. Certain isitthat atthis time and onGrr man soil elementary arithmetic and common aljebea, vitally influenced bythe Italian school, attained a standing very conducive tosubsequent progress @ Themodern periodinthehistory ofmathematics 5. begins about themiddle oftheseventeenth century, {b Descartes projects thefoundation theory oftheana lytic geometry. Leibnitz and Newton appear asthe discoverers ofthe differential calculus. ‘The time ha» now come when geometry, ascience only rarely, and even then but imperfectly, appreciated after itsban ishment from Greece, enters along with analysis upon aperiod ofprosperous advance, and takes fulladvan tage ofthislatter sister science inattaining itsresultn, Thus there were periods inwhich geometry wan able through itsbrilliant discoveries tocast analysis, ten | porarily atleast, intotheshade. The unprecedented activity ofthe great Gausy e divides themodern periodintotwoparti:before oe Gauss—the establishment ofthe methods ofthedif : GENERAL SURVEY. 5 e ferential andintegral calculus andofanalytic geom- etry aswell asmore restricted preparations forlater advance; withGauss andafterhim—the magnificent AS: development ofmodern mathematics with itsspecial regions ofgrandeur anddepth previously undreamed So of.Themathematicians ofthenineteenth century \e@ aredevoting themselves tothetheory ofnumbers, modern algebra, thetheory offunctions andprojec- tive geometry, and inobedience totheimpulse of human knowledge areendeavoring(to carrytheirlight into remote realms which tillnow have remained in darkness.) e | e I.NUMBER-SYSTEMS AND NUMBER-~ SYMBOLS. ANinexhaustible profusion ofexternal influencesupon thehuman mind hasfound itslegitimate expression intheformation ofspeech and writing innumbers and number-symbols. Itistrue that a counting ofacertain kind isfound among peoples of alowgrade ofcivilization and even among thelower | animals. ‘‘Even ducks cancount their young.”* But wherethenatureandthecondition oftheobjects @ have been ofnoconsequence intheformation ofthe number itself, there human counting hasfirst begun. | Theoldest counting waseven initsorigin apro- cess ofreckoning, anadjoining, possibly also inspecial elementary cases amultiplication, performed upon | theobjects counted orupon other objects easily em- ployed, such aspebbles, shells, fingers. Hence arose “number-names. The most common ofthese undoubt- edly belong totheprimitive domain oflanguage; with theadvancing development oflanguage their aggre- gate was gradually enlarged, thelegitimate combina- ‘*Hankel, Zur Geschichte derMathematik imAlterium und Mittdlalter, 1874, p.7.Hereafter referred toasHankel, Tylor's Primitive Culture also e hasavaluablechapteruponcounting. e NUMBER-SYSTEMS ANDNUMBER-SYMBOLS. 7 tion ofsingle terms permitting andfavoring thecrea- tion ofnewnumbers. Hence arose number-systems. Theexplanation ofthefactthat10isalmost every- where found asthebase ofthesystem ofcounting is seen inthecommon useofthefingers inelementary calculations. Inallancient civilizations finger-reckon- ingwas known and even to-day itiscarried ontoa remarkable extent among many savage peoples. Cer- tain South African races usethree persons fornum- : bers which runabove 100, thefirst counting theunits onhisfingers,thesecondthetens,andthethirdthe C hundreds. (They always begin withthelittlefinger of | thelefthandandcounttothelittlefingeroftheright.) The first counts continuously, theothers raising a finger every time atenorahundred isreached.* Some languages contain words belonging funda- mentally tothescale of5or20without these systems | having been completely elaborated; only incertain places dothey burst the bounds ofthedecimal sys- tem. Inother cases, answering tospecial needs, 12 a and60appear asbases. TheNewZealanders have ~~“ ascale of11,their language possessing words forthe first few powers of11,and consequently 12isrepre- sentedas11and1,13as11and2,22astwo11's, eand soon.t + *€antor, M., Vorlesungen iber Geschichte derMathematik. Vol. 1,1880; anded.,1894, p.6,Hereafter referredtoasCantor,Conant,L.L.,TheNum- r) 8 HISTORYOFMATHEMATICS. Intheverbal formation ofanumber-system addi- tionandmultiplication stand outprominently asdefin- itiveoperations forthecomposition ofnumbers ;very rarely does subtraction come intouseandstillmore rarely division. Forexample, 18iscalled inLatin 10-48 (decem etocto), inGreek 8-++10 (éxrw-xal-Sexa) , inFrench 108(dix-huif), inGerman 810(achi-zehzz), inLatin also 20—2 (duo-de-viginti), inLower Breton 3-6 (ért-ome'h), inWelsh 2-9(dew-naw), inAztec 1548 (caxtulli-om-ey), while 50iscalled intheBasque half-hundred, inDanish two-and-a-half times twenty. * Inspite ofthegreatest diversity offorms, thewritten representation ofnumbers, whennotconfined tothe ®mere rudiments, shows ageneral law according to which thehigher order precedes the lower inthe di- rection ofthewriting. Thus inafour-figure number the thousands arewritten bythePhoenicians atthe right, bytheChinese above, theformer writing from right toleft, ‘the latter from above downward. A striking exception tothis law isseen inthe sub- tractive principle ofthe Romans inIV, IX, XL, etc., where the smaller number iswritten before the larger. Among theEgyptians wehave numbers running from right toleftinthehieratic writing, with varying direction inthehieroglyphics. Inthelatter thenum- e berswereeitherwritten outinwordsorrepresentedbysymbols foreach unit, repeated asoften asneces- *Hankel, p.22. +Hankel, p.32. e NUMBER-SYSTEMS ANDNUMBER-SYMBOLS. 9 sary. Inoneofthetombs near thepyramids ofGizeh havebeenfoundhieroglyphic numerals inwhich1is represented byavertical line, 10byakind ofhorse- shoe, 100byashort spiral, 10000byapointing finger, 100000 byafrog, 1000 000 byaman intheattitude’ ofastonishment. Inthehieratic symbols thefigure fortheunit ofhigher order stands totheright ofthe one oflower order inaccordance with the law ofse- quence already mentioned. The repetition ofsym- bolsforaunit ofanyparticular order does notobtain, because there arespecial characters forallnine units, allthetens, allthe hundreds, and allthe thousands.* Wegivebelowafewcharacteristic specimens ofthe ®hieratic, symbols : ruw -TaAAA + |1 3 8 4 5 10 20 30 40 TheBabylonian cuneiform inscriptionst proceed |from lefttoright, which must belooked upon asex- ceptional inaSemitic language. Inaccordance with thelawofsequence theunits ofhigher order stand on theleftofthose oflower order. The symbols used inwriting arechiefly thehorizontal wedge >,thever- tical wedge Y,and thecombination ofthe twoatan angle¢. Thesymbols were written beside oneanother, or,forease ofreading and tosave space, over one another. Thesymbols for1,4,10,100,14,400,re- @spectively, areasfollows: ’ ; “Cantor, 1.,PP.4344+ tCantor, I,pp.77,78. > e10 HISTORY OF MATHEMATICS. RAAaD AAA Yue evVY 1 4 10 100 4 400 -‘ For numbers exceeding 100 there was also, Besides the mere juxtaposition, amultiplicative principle ; thesymbol representing thenumber ofhundreds was placed attheleftofthesymbol forhundreds asinthe case of400 already shown. The Babylonians probably had nosymbol forzero.* The sexagesimal system . (i.e.,with thebase 60), which played such apart in thewritings oftheBabylonian scholars (astronomers and mathematicians), will bementioned later. ThePheenicians, whosetwenty-two letterswere ederived from thehieratic characters oftheEgyptians, either wrote thenumbers outinwords orused special numerical symbols—for the units vertical marks, for | thetens horizontal.t Somewhat later theSyrians used | thetwenty-two letters oftheiralphabet torepresent ~ : thenumbers 1,2,..9,10,20,.. .90,100, ...400; 500was 400+ 100, etc. The thousands were repre sented bythe symbols forunits with asubscript comma atthe right.{ The Hebrew notation follows the same plan. The oldest Greek numerals (aside from thewritten words) were, ingeneral, theinitial letters ofthefunda- mental numbers. Ifor1,Iffor5(wére), Afor10 @ (Sea),§andthesewererepeated asoftenasnecessary. *Cantor,L, p.8% tCantor, L,p.113. Cantor. I.,pp.113-114. §Cantor, I.,p.tro, . . NUMBER-SYSTEMS AND NUMBER-SYMBOLS. Ir e@ Thesenumerals aredescribed bytheByzantine gram- marian Herodianus (A.D.200) andhence arespoken ofasHerodianic numbers. Shortly after 500 B.C. | twonewsystems appeared. One used the24letters ofthe Ionic alphabet intheir natural order forthe numbers from 1to24. The other arranged these letters apparently atrandom butactually inanorder fixed arbitrarily; thus, a=1, B=2,...., s=10, c= 20,...., p==100, «==200, etc. Here too there is nospecial symbol forthe zero. The Roman numerals* were probably inherited fromtheEtruscans. Thenoteworthy peculiarities | are the lack ofthe zero, the subtractive principle e wherebythevalueofasymbolwasdiminisled byplacing before itone oflower order (IV=4, IX=9, XL=40, XC=90), even incases where thelanguage itself did not signify such asubtraction; and finally themultiplicative effect ofabar over the numerals (XXX=80 000, T=100000). Also forcertain frac- tions there were special symbols and names. Accord- ingtoMommsen theRoman number-symbols I,Vv, Xrepresent the finger, thehand, and the double . hand. Zangemeister proceeds from thestandpoint that decem isrelated todecussare which means a :perpendicular oroblique crossing, andargues that every straight orcurved linedrawn across thesymbol ofanumber inthe decimal system multiplies that @ number byten.Infact,thereareonmonuments *Cantor, I.,p.486. e@ 12 HISTORY OFMATHEMATICS. representations of1,10,and1000, aswellasof5and 500, toprove his-assertion.* Ofespecial interest inelementary arithmetic isthe number-system oftheHindus, because itistothese : Aryans thatweundoubtedly owethevaluable position. system now inuse. Their oldest symbols for1to9 were merely abridged number-words, and theuseof letters asfigures issaid tohave been prevalent from thesecond century A.D.t Thezero isoflater origin ; itsintroduction isnotproven with certainty tillafter 400A.D. The writing ofnumbers was carried on, chieflyaccording totheposition-system, invarious @ways. One plan, which Aryabhatta records,’ repre- sented the numbers from 1to25bythetwenty-five consonants oftheSanskrit alphabet, and thesucceed- ing tens (30, 40....100) bythesemi-vowels and sibilants. Aseries ofvowels and diphthongs formed multipliers consisting ofpowers often, gameaning 38,gé300, gw30000, gaux3-10%.t Inthis there isno application ofthe position-system, although itap- pears intwo other methods ofwriting numbers in “use among the arithmeticians ofSouthern India. | Both ofthese plans aredistinguished bythefactthat *Sitsungsberichte derBerliner Akademie vom 10.November 1887, Words- worth, inhisFragments andSpecimens ofEarlyLatin, 1874,derives Cfor centunt,Mformille,andLforguinguaginta fromthreelettersoftheChal- @ cidian alphabet, corresponding to0,4,andx.Hesays: “The origin ofthis notation is,Ibelieve, quite uncertain, orrather purely arbitrary, though, of course, weobserve that theinitials ofmille andcentusm determined thefinal - shape taken bythesigns, which atfirstwere very different inform."" +SeeEncyclopedia Britannica, under“Numerals" Cantor, I.,p.566. @ NUMSER-SYSTEMS ANDNUMBER-SYMBOLS, 13 thesame number canbemade upinvarious ways. Rules ofcalculation were clothed insimple verse easy tohold inmind andtorecall. For theHindu mathe- maticians thiswasallthemore important since they sought toavoid written calculation asfaraspossible. One method ofrepresentation consisted inallowing thealphabet, ingroups of9symbols, todenote the numbers from 1to9repeatedly, while certain vowels represented thezeros. IfintheEnglish alphabet ac- cording tothis method wewere todenote thenum- bers from 1to9bythe consonants 4,¢,... sothat after two countings one finally hasz==2, and were to e denote zerobyeveryvowelorcombination ofvowels,thenumber 60502 might beindicated bysiren orheron, and might beintroduced bysome other words inthe text. Asecond method employed type-words and combined them according tothe law ofposition. Thus addhi (one ofthe4seas) =4, surya (the sun with its12houses)=12, agoin (the two sons ofthe sun)==2. The combination abdhisuryaguinas denoted the number 2124.* Peculiar totheSanskrit number-language arespe- . cialwords forthe multiplication ofvery large num- bers. Arduda signifies 100 millions, padna 10000 millions; from these are derived maharbuda=1000 millions, mehapadma=100000 millions. Specially- r) formed words forlargenumbers runupto102?and even further. This extraordinary extension ofthe *Cantor, I.,p.567. e x4 HISTORY OFMATHEMATICS. decimal system inSanskrit resembles anumber- amania tograsp theinfinitely great. Ofthis enc tobring theinfinite into therealm ofnumber-p tion and representation, traces arefound also : the Babylonians and Greeks. This appearanc finditsexplanation inmystic-religious concepti philosophic speculations. . The ancient Chinese number-symbols areco toacomparatively fewfundamental elements ar! : inaperfectly developed decimal system. He F combination takes place sometimes bymult i tion,sometimes byaddition. Thus san=8, che y chesandenotes 18,butsazche30*Later,asa @F offoreign influence, therearosetwonewkinds i tation whose figures showsome resemblance ancient Chinese symbols. Numbers formed them were not written from above downwa after theHindu fashion from lefttoright beg withthehighest order. Theonekindcompris merchants’ figures isnever printed butisfour inwritings ofabusiness character. Ordina: ordinal andcardinal numbers arearranged .lines one above another, with zeros when nec inthe form ofsmall circles. Inthis notation i. H=2K=4, 1=6,p=10, h=10000,| ux eandhence’ F)O©“Ri =20046. i . *Cantor, 1,p.630. , NUMBER-SYSTEMS AND NUMBER-SYMBOLS. 15 e@ Among theArabs,thoseskilfultransmitters of Oriental and Greek arithmetic tothe nations ofthe West, thecustom ofwriting out number-words con- tinued tillthebeginning ofthe eleventh century. Yetatacomparatively early period they had already formed abbreviations ofthenumber-words, theDivant figures. Intheeighth century theArabs became ac- quainted with theHindu number-system anditsfig- ures, including zero. From these figures there arose among theWestern Arabs, who intheir whole litera- ture presented adecided contrast totheir Eastern re- latives, theGubar numerals (dust-numerals) asvari- ants. These Gubar numerals, almost entirely forgotten e to-day among theArabs themselves, aretheancestors ofourmodern numerals,* which areimmediately de- rived from theapices oftheearly Middle Ages. These primitive Western formsusedintheabacus-calcula- | tions are found inthe West European MSS. ofthe eleventh and twelfth centuries and owe much oftheir prominence toGerbert, afterwards Pope Sylvester II. (consecrated 999 A.D.). Thearithmetic oftheWestern nations, cultivated : toaconsiderable extent inthecloister-schools from__ theninth century on,employed besides theabacus the Roman numerals, and consequently made nouseofa symbol forzero. InGermany uptotheyear 1500 the Roman symbols were called German numerals indis- rd tinction fromthesymbols—then seldom employed— *Hankel, p.255. 16 HISTORY OFMATHEMATICS. e ofArab-Hindu origin,whichincludedazero(Arabic as-sifr, Sanskrit sunya, the void). The latter were called ciphers (Ziffern)., From thefifteenth century on these Arab-Hindu numerals appear more frequently in Germany onmonuments and inchurches, butatthat time they had not become common property.* The oldest monument with Arabic figures (inKatharein near Troppau) issaid todate from 1007. Monuments ofthiskind arefound inPforzheim (1371), andinUlm (4388). Afrequent and free use ofthezero inthe thirteenth century isshown intables forthecalcula- tion ofthe tides atLondon and ofthe duration of e moonlight.t Intheyear1471thereappeared inCo- logne awork ofPetrarch with page-numbers inHindu figures atthe top. In1482 the first German arith- metic with similar page-numbering was published in Bamberg. Besides the ordinary forms ofnumerals everywhere used to-day, which appeared exclusively inanarithmetic of1489, thefollowing forms for4,5, 7were used inGermany atthe time ofthestruggle between the Roman and Hindu notations: RGA: The derivation ofthe modern numerals isillustrated aa bytheexamples belowwhich aretakeninsuccession Ps e fromtheSanskrit,theapices,theEasternArab,theva *Unger, DieMethodih derpraktischen Arithmetik, 1888, p.70, Hereatter referred toasUnger. +Gliniher, Geschichte desmathematischen Unterrichts imdeutechen Mittel alterdissumeJokr1525,1887,p.175.Hereafter reterred toasGiinther. , NUMBER-SYSTEMS AND NUMBER-SYMBOLS, 17 e Western ArabGubar numerals, thenumerals ofthe eleventh, thirteenth, and sixteenth centuries.* TU” 8 ° or A 3y?d GOV A Inthesixteenth century theHinduposition-arith- emetic and itsnotation first found complete introduc- tionamong allthecivilized peoples oftheWest. By this means was fulfilled one oftheindispensable con- ditions forthedevelopment ofcommon arithmetic in the schools and inthe service oftrade and commerce. *Cantor, table appended toVol. I,and Hankel, p.32s. | : » 8 e Il.ARITHMETIC. A. GENERAL SURVEY. Tesimplest number-words andelementary count )inghavealwaysbeenthecommon property of |thepeople. Quite otherwise isit,however, with the different methods ofcalculation which are derived from simple counting, andwith their application te complicated problems. Asthecenturies passed, that part ofordinary arithmetic which to-day every child knows, descended from theclosed circle ofpartienlar @ castes orsmaller communities tothecommon penple, soastoform animportant partof general culture, Among theancients theeducation oftheyouth had te doalmost wholly with bodily exercises. Only ariper age sought ahigher cultivation throngh intercourse with priests and philosophers, and this consisted in part inthe common knowledge oftoday: people learned toread, towrite, tocipher. Atthebeginning ofthefirst period inthehistoric development ofcommon arithmetic stand theExyy tians. To them the Greek writers ascribe the invert tion ofsurveying, ofastronomy, and ofarithmetica. ‘To etheir literature belongs also themost ancient book on . ARITHMETIC. 19 @ arithmetic, thatofAhmes, whichteaches operations. withwholenumbers andfractions. TheBabylonians ”’employed asexagesimal system intheir position-arith- metic, which latter must also have served thepur- poses ofareligious number-symbolism. The common arithmetic oftheGreeks, particularly inmost ancient times, wasmoderate inextent until bytheactivity of thescholars ofphilosophy there wasdeveloped areal mathematical science ofpredominantly geometric character. Inspite ofthis, skill incalculation was notesteemed lightly. Ofthis wehave evidence when Plato demands forhisidea]statethattheyouth should beinstructed inreading, writing, and arithmetic. Thearithmetic oftheRomanshadapurelyprac- @ tical turn; toitbelonged amass ofquite complicated problems arising from controversies regarding ques- tions ofinheritance, ofprivate property and ofreim- bursement ofinterest. The Romans used duodecimal fractions. Concerning themost ancient arithmetic of theHindus only conjectures can bemade ;onthecon- trary, theHindu elementary arithmetic after the in- troduction oftheposition-system isknown with toler- able accuracy from the works ofnative authors. The Hindu mathematicians laid the foundations for the ordinary arithmetic processes ofto-day. The influ- ence oftheir learning isperceptible inthe Chinese arithmetic which likewise depends onthedecimal sys- e tem;instillgreater measure, however, among the , 20 HISTORY OFMATHEMATICS. e Arabswhobesides theHindunumeral-reckoning also | employed acalculation bycolumns. opFr| Thetimefromtheeighthtothebeginning oftheeae"fifteenth century formsthesecond period. Thisisa noteworthy period oftransition, anepoch ofthetrans- | planting ofoldmethods intonewandfruitful soil, butalso one ofcombat between thewell-tried Hindu methods and theclumsy and detailed arithmetic ope- rations handed down from the Middle Ages. At first only incloisters and _cloister-schools could any arithmetic knowledge befound, and that derived from Roman sources. But finally there came new sugges tions from the Arabs, sothat from the eleventh tothe @ thirteenth centuries therewasopposed tothegroup of abacists, with their singular complementary methods, aschool ofalgorists aspartisans oftheHindu arith metic. Not until thefifteenth century, the period ofin vestigation oftheoriginal Greekwritings, ofthe yurapid development ofastronomy, ofthe rise ofthe vartsandofcommercial relations, doesthethirdpe-ag) riod inthehistory ofarithmetic begin. Asearly asthethirteenth century besides the cathedral and cloister-schools which provided fortheirownreligious andecclesiastical wants, there were, properly speak- ing,schools forarithmetic. Their foundation istobe ascribed totheneeds ofthebrisk trade ofGerman e townsWithItalianmerchants whowerelikewise skilled computers. Inthefifteenth andsixteenth centuries . ARITHMETIC. ar @ school affairs wereessentially advanced bythehuman- istic tendency andbythereformation. Latin schools, writing schools, German schools (inGermany) forboys : and even for girls, were established. Inthe Latin schools only theupper classes received instruction in atithmetic, inaweekly exercise: they studied thefour fundamental rules, thetheory offractions, and atmost therule ofthree, which may notseem s0very little when weconsider that frequently intheuniversities ofthat time arithmetic was not carried much further. Inthewriting schools andGerman boys’ schools the pupilslearnedsomething ofcalculation, numeration, |and notation, especially the difference between the_ German numerals (inRoman writing) andtheciphers .| e(aftertheHindufashion). Inthegirls’schools, which wywereintendedonlyforthehigherclassesofpeople,no.aes arithmetic was taught. Considerable attainments in computation could besecured only intheschools for arithmetic. The most celebrated ofthese institutions was located atNuremberg. Inthecommercial towns there were accountants’ guilds which provided forthe extension ofarithmetic knowledge. But real mathe- maticians and astronomers also labored together inde- veloping themethods ofarithmetic. Inspite ofthis assistance from men ofprominence, notheory ofarith- metic instruction had been established even aslate as inthe sixteenth century. What had been done be- r} forehadtobecopied. Inthebooksonarithmetic , a2 HISTORY OFMATHEMATICS. td werefoundonlyrulesandexamples, almostnever proofs ordeductions. The seventeenth century brought no essential change inthese conditions. Schools existed asbefore where theyhadnotbeen swallowed upbythehorrors oftheThirty-Years’ War. Thearithmeticians wrote their books onarithmetic, perhaps contrived calculat-~ ingmachines tomake thework easier fortheir pupils, orcomposed arithmetic conversations andpoems. A specimen ofthis isgiven inthefollowing extracts from Tobias Beutel’s Arithmetica, the seventh edition ofwhich appeared in1693.* . “Numerieren lehrt imRechen Zahlenschreiben undaussprechen."* e “In Summen bringen heisst addieren Dies muss dasWértlein Und vollfithren.* WieeineHandanunsdieandrewaschet rein Kann eine Species derandern Probe seyn.” “We aretaught innumeration Number writing’and expression," etc, etc. Commercial arithmetic wasimproved bythecultiva tion ofthestudy ofexchange anddiscount, andthe abbreviated method ofmultiplication. The form of instruction remained thesame,i.e.,thepupilreck- |oned according torules without anyattempt being made toexplain their nature. r) Theeighteenth century brought asitsfirstand Il ‘*Unger, p.124. .