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MATHEMATICS

BY

PLORIANCAJORI

OFLOUISIANA;NOWPROCESSORorPHYSICS

INCOLORADOCOLLEGE

GLAISHMR

gatfcMACMILLANANDCO.

ANBLONDON

1894

AllriffMsreserved

COPYRIGHT,189,1,

BYMAOM1LLANAND00,

ITortooobprt;

J.S,Gushing&Co,-Berwick&Smith,

Boston,Mass,,U.S.A.

PREFACE.

v viPKEFACE.

FLORIANCAJOBL

COLORADOCOLLEGE,December,1893.

TABLEOFCONTENTS.

PAGE

INTRODUCTION1

,ANTIQUITY5

THEBABYLONIANS5

THEEGYPTIANS9

THEGREEKS16

GreekGeometry16

TheIonicSchool17

TheSchoolofPythagoras19

TheSophistSchool23

ThePlatonicSchool29

TheFirstAlexandrianSchool34

TheSecondAlexandrianSchool54

GreekArithmetic63

TUBROMANS77

^MIDDLEAGES84

THEHINDOOS84

THEARABS100

EtJBOPEDURINGTHEMIDDLEAOES117

IntroductionofRomanMathematics117

TranslationofArabicManuscripts124

TheFirstAwakeninganditsSequel128

MODERNEUROPE138

THERENAISSANCE:....189

VIETATODJCSOARTES^

DBSGARTESTONEWTON183

NBWTGNTOEULBK199

vii

ViiiTABLEOFCONTENTS.

PAGE

EULER,LAGRANGE,ANDLAPLACE246

TheOriginofModernGeometry285

KECENTTIMES291

SYNTHETICGEOMETRY293

ANALYTICGEOMETRY307

ALGEBRA315

ANALYSIS331

THEORYOPFUNCTIONS347

THEORYOFNUMBERS362

APPLIEDMATHEMATICS373

INDEX405

BOOKSOFREFEKENCE.

hasbeenmade. torischenJForschung.Erlangen,1876.

Washington,1890.

Leipzig.BelI.,1880;Bd.II.,1892.

P.J.K.STUASSMAIER.Freiburg,1889.

des.Leipzig,1870. 1884.
undMittelalter.Leipzig,1874. 1889.

ItomanBiographyandMythology.

zig,1807.

KopQnlaagen,1886.

ix

XBOOKSOFREFERENCE.

siques.TomeI.-XII.Paris,1883-1888.

LESPIE.

tenJahrhunderten.Tubingen,1884.

VoUcer.Halle,1863.

22.PEACOCK,GEORGE.Article

"Aritlimetic,

1inTheEncyclopedia,of

PureMathematics.London,1847.

23.HERSCHEL,J.!F.W.Article

*Mathematics,"inEdinburghJfflncy- dopcedia.

Zurich,1873-75.

Bruxelles,1866.

tinuedinthe8tlxeditionbySIKJOHNLESLIE. tothePresentTime. burgh,1834.

JournalofMathematics,Vol.XL,1879.

EnglishbyW.A.,London,1744.

Mechanik.Leipzig,1887.

London,1888,2ndedition,189S,

PhilosophicalMagazine,November,1852.

BOOKSOFREFERENCE,xi

Stockholm.

Miinclien,1877.

niz.Halle,1848.

40.GERIIARBT,K.I.

FeTbruar,1891.

cum,"intliePennyCyclopaedia,

London,1865.

Cambridge,1886.

44.TOBHUNTKR,I.

BaHol,1884,

zig,1850.

Leipzig,1885.

theN.Y.MathematicalSociety,I.3.

51.AUAOO,1).F.J.

"EulogyonLaplace.

7TranslatedbyB.POWELL,

SmitlisonianllPfiort,1874,

C.A.ALEXANDISR,SmithsonianIteport,1867.

1871.
zig,1884. genvonFitmSOHUTTB.Leipzig,1888.

XllBOOKSOFREFERENCE.

1883.

Association,1878.

58.GIBBS,J.WILLARD.

1890.

Nordlingen,1878.

trischeForschungen.Erlangen,1872.

Cambridge,1893.

Archiv,48:2,1868.

undPhysik,26:5,1881.

66.BRONICE,AD.JuliusPlucker.Bonn,1871.

1882.
zig,1873.

Krwnmungsmasses.Tubingen,1881.

NewYork,1890.

Werke.Leipzig,1878.

72.ZAHN,W.v.

MathematischeAnnalen,VII.4,1874.

1883.
1889.
ment,1892.

BOOKSOFREFERENCE.xiii

1869.

II.,Paris,1868.

Meihen.Gottingen,1879.

Heilien,MineJBntgegnung.Tubingen.

Poincare.Paris,1886.

actionscientifique.Paris,1885.

Jacobi.1852.

schichte.Hallea/S.,1876. tonDarboux.Paris,1884.

Berlin,1860.

90.GLAISUISH,J.W.L.

Geburtstag.Konigsherg,1884.

1890.
ressofPhysics,"Nature,26:17,1882.

MatematicodiPalermo,V.,1891.

xivBOOKSO:FBEFBBBNCE,

Dynamics.1857.

AHISTORYOFMATHEMATICS.

INTEODUCTION.

1

2AHISTORYOFMATHEMATICS.

<>

INTBODUCTION.3

thefirmamentofheaven. the "Arabicnotation " ;theywillmarvelatthethousands him.Whenthevalueofmathematicaltrainingis iacademyofPlato,thephilosopher "Letnoonewhois

4AHISTORYOFMATHEMATICS.

oneinwhichsteadyprogressismade.2

ANTIQUITY,

THEBABYLONIANS,

theIcharacters"^and y>*.signified10and100respec theeaf5

6AHISTOKYOFMATHEMATICS.

itsleft.Thus,^^f >*"denoted,not20times100,but amillion.3

1.4=8s

,1.21=92 ,1.40=102 ,2.1=11*,etc.This"reinLfta makes1.4=60+4,1.21=60+21,2.1=2.60+1.Thei catestheacquaintanceoftheBabylonianswith

THEBABYLONIANS.7

hismind,tosupplytheword "sixtieths."TheGreekgeom

8AHISTOKYOFMATHEMATICS.

Babylonians.

THEEGYPTIANS.9

Oppert:

"

THEEGYPTIANS.

uncivilisedstateofsociety. "Menes,thefirstking,changes

10AHISTOBYOFMATHEMATICS.

sciences.PlatoinPho&drussays "AttheEgyptiancity 15 writerstohaveoriginatedinEgypt.

5InHerodotuswofind

this(II.c.109) arithmeticandgeometry.ItwaswrittenbyAtaw

THEGREEKS.17

underthenameofEudemianSummary.

TheIonicSchool

staffwas0(jualtoitsownlength.

18AHISTORYOFMATHEMATICS.

angulartrianglestobeproportional.

8TheEgyptiansmust

thouseestnotwhatisatthyfeet? "

THEGREEKS.19

thesolutionofthefollowingproblems :Fromapointwithout, structions.

TJieSchoolofPythagoras.

20AHISTORYOFMATHEMATICS.

tothegreatfounderofthesect. rasfledtoTarentumandthencetoMetapontum, murdered. icipal iencc.

THEGrKEEKS.21

ofarithmeticalexpression. ficebythatofanoxmadeofflour " !Theproofofthelaw conjecture.

AHISTORYOFMATHEMATICS.

divulged

1Pythagoreans,andwascalledbythemHealth.

headofarithmetic. ~fdefectandexcess,asinEuclid,VI.28,29. t discoveredbythisschool,,

THEGREEKS.23

wasadvancedthroughhim. andhadawarmfriendinArchytas.

TheSophistSchool

24
*AHISTORYOFMATHEMATICS. atAthens, famousproblems: (1)Totrisectanarcoranangle. (2)To isdoublethatofagivencube.

THEGREEKS.25

Onthisaccountitiscalledthequadratrix.

:2a,sincea?2=ayandy

2=2axandce*=a2/,we

havea;4=2cfxanda?3=2a8 .Buthefailedtofindthetwo thisresulttothesquaringofthecircle. contributedmuchtothegeometryofthecircle.

26AHISTORYOFMATHEMATICS.

" importantstep.

THEGREEKS*27

AHISTOEYOFMATHEMATICS.

:d2=C:cisnottrue,supposethatD2 :$=:c.Ifd<c, nearertoitinareathandoesc f .IfPbethecorresponding polygoninC,thenP:p=D2 ;d2=G:c,andP:O=p:c. Since j>>c f ,wehaveP>C,whichisabsurd.Nextthey falsityofthesupposition,thatcf >c.Sinceccanbeneither

THEGREEKS.29*

andabilityoftheEgyptians.

TJiePlatonicSchool.

"Hegeom-

30AHISTOBYOFMATHEMATICS.

philosophy.

1Platoobservedthatgeometrytrainedthemind

EudemianSummarysays,

"Hefilledhiswritingswithmathe

7butthisisa

Platoobjectedtocallingapointa

"geometricalfiction."He ibleline,"andalineasquotesdbs_dbs47.pdfusesText_47
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