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I 

1 


A  SHORT  ACCOUNT 


fflSTOBY  OF  MATHEMATICS 


^eem. 


I 


A  SHORT  ACCOUNT"/' 


HISTORY   OF    MATHEMATICS 


: 


W.  W.   ROUSE  BALL, 


TBIRO   KPITIOS. 


lontian 


MACMILLAN    AND  CO.,   Livitrd 

NBW  YORK:  THK  MArMlLLAN  OOHPAHT 
1901 


[iUfI 


H 1  ^  :l 


Reproduced  by 

DUOPAGE  PROCESS 
in  the 
U.S.  of  America 


Micro  Phoco  Divisioa 
Bell  ft  Howell  Conraii7 
aerelaod  12,  Ohio 


OP  »  1392 


IMS. 


Third  Kdittm  1901. 


fc 


I 


V 


/' 


PREFACE. 

Thb  subject-matter  of  this  book  is  a  historical  sum- 
mary of  the  development  of  mathematics,  illustrated  by 
the  livte  and  discoveries  of  those  to  whom  the  progress  of 
the  science  is  mainly  due.  It  may  serve  as  an  introduction 
to  more  elaborate  works  on  the  subject,  but  primarily  it 
is  intended  to  give  a  short  and  popular  account  of  those 
leading  &cts  in  the  history  of  mathematics  which  many 
who  are  unwilling,  or  have  not  the  time,  to  study  it 
'systematically  may  yet  desire  to  know. 

The  first  edition  was  substantially  a  transcript  of  some 
lecttires  which  I  delivered  in  the  year  1888  with  the 
object  of  giving  a  sketch  of  the  history,  previous  to  the 
nineteenth  century,  that  should  be  intelligible  to  any  one 
acquainted  with  the  elements  of  mathematics.  In  the 
second  edition  I  introduced  a  good  deal  of  additional 
matter  and  re-arranged  parts  of  it.  The  present  edition 
has  been  again  revised  but  not  materially  altered. 

The  scheme  of  arrangement  will  be  gathered  from  the 
table  of  contents  at  the  end  of  this  preface.  Shortly  it  is 
as  follows.  The  first  chapter  contains  a  brief  statement 
of  whki  is  known  oonoeming  the  mathematics  of  the 
Egyptians  and  Phoenicians:  this  is  introductory  to  the 


n\? 


VI 

history  of  mathetnatioi  under  Greek  influeiioe.  The 
flubeequent  history  is  divided  into  three  periods:  fint» 
that  under  Ureek  influence^  chapters  il  to  vn;  second, 
that  of  the  middle  ages  and  renaissance,  chapters  vm  to 
XIII ;  and  lastly  that  of  modern  times,  chapters  xiv  to 

In  discussing  the  mathematics  of  these  periods  I  have 
confined  myself  to  giving  the  leading  events  in  the  history, 
and  frequently  have  passed  in  silence  over  men  or  works 
whose  iuflueuce  was  comparatively  unimportant;  doubtless 
an  exaggerated  view  of  the  discoveries  of  those  mathe- 
maticians who  are  mentioned  may  be  caused  by  the  non- 
allusiou  to  minor  writers  who  preceded  and  prepared  the 
way  for  them,  but  in  all  historical  sketches  this  is  to  some 
eiteut  inevitable,  and  I  have  done  my  best  to  guard 
against  it  by  interpolating  remarks  on  the  progress  of  the 
science  at  different  timea  Perhaps  also  I  should  here 
state  that  generally  I  have  not  referred  to  the  results 
obtained  by  practical  astronomers  and  physicists  unless 
there  was  some  matheumtical  interest  in  them.  In  quot- 
ing results  I  have  commonly  made  use  of  modem  notation; 
the  reader  must  therefore  recollect  that,  while  the  matter 
is  the  same  as  that  of  any  writer  to  whom  allusion  is 
matle,  his  proof  is  sometimes  translated  into  a  more 
convenient  and  familiar  language. 

The  greater  ptui  of  my  account  is  a  compilation 
from  existing  histories  or  memoirs,  as  indeed  must  be 
necessarily  the  case  where  the  works  discussed  are  so 
numerous  and  cover  so  much  ground;  when  authorities 
disagree  I  have  generally  stated  only  that  view  which 


i 


I: 


PREFACE.  Til 

seems  to  me  to  be  the  most  probable,  but  if  the  question 
be  one  of  importance  I  believe  that  I  have  always  indi- 
cated that  there  is  a  difference  of  opinion  about  it 

I  think  that  it  is  undesirable  to  overload  a  popular 
■ooonnt  with  a  mass  of  detailed  references  or  the 
authority  for  every  particular  fact  mentioned.  For  the 
history  previous  to  1758, 1  need  only  refer,  once  for  all, 
to  the  closely  printed  pages  of  M.  Cantor's  monumental 
Vorlemingen  fiber  die  Geschichte  der  MathemaJtik  (hereafter 
alladed  to  as  Cantor)  which  may  be  re^i^arded  as  the 
standard  treatise  on  the  subject,  but  generally  I  have 
also  added,  in  footnotes  to  the  text,  references  to  the  other 
leading  authorities  on  which  I  have  based  my  account  or 
with  which  I  am  acquainted :  for  the  period  subsequent 
to  1758,  it  is  necessary  to  consult  memoirs  or  monographa 
I  hope  that  the  footnotes  will  supply  the  means  of  study- 
ing in  detail  the  history  of  mathematics  at  any  specified 
period  should  the  reader  desire  to  do  so. 

My  thanks  are  due  to  various  firiends  and  oorre- 
qwodents  who  have  called  my  attention  to  points  in  the 
previoas  editions.  I  shall  be  grateful  for  notices  of  addi- 
tioDS  or  ooirectiona  which  may  occur  to  any  of  my  readers. 

•  a 

W.  W.  BOUSE  BALL. 

Tmnrr  CollboIi  CAMBanKn, 
mriiry,  1901. 


i 


IX 


TABLE  OF  CONTENTS. 


rAOB 

Pi«fae« ▼ 

Table  of  eontciiU .      U 

Chaptbr  I.     Egtptiav  axd  Phoenician  Matiikmaticb. 

The  hisUwy  of  mathematieii  bejnns  with  thiit  of  the  Ionian  Oreeki 
Greek  indebtodness  to  Efi^iHmni  and  Phoenicians   . 
Kaowledse  of  the  ecienee  of  nnmlien  pomemcd  bj  the  Phoenicians 
Kaowledse  of  the  eeienoe  of  numbers  posspssiMl  bj  tlie  Egyptians 
Knowledge  of  the  science  of  geometiy  posKCftsed  br  thr  Efcyptians 
Hole  OB  ignoraaee  of  mathematics  shewn  bj  the  Chinese 

:f  tot  9efMl.    iCIattnnaticf  unlrcr  CBmk  Influencf. 

ThU  perioi  befin$  with  the  teaekimff  nf  Thalet,  eire.  600  •.€.,  emf  emh 
miik  the  cnptmrt  cf  AUxandrin  btf  the  MohnmwteHatu  in  or  nhimt  641  A.ix. 

The  chmneteriitie  fmtmrt  of  thit  period  i$  the  devtiommemi  of  9eometrw, 

J*.' 

•  '  Chaptbb  II.    The  Ioxian  and  Pttiiaoorban  Schoolb. 

CiBO.  600  B.C— 400  &c. 

AstboritiM      .  .       .  .18 

TAe  TMlflm  Sr  Aoel 14 

64a-M0B.e. 14 

HiB  ijeooieiiWM  diseoreiies        ••••.•      15 

Hb  MifOBoiiiiGal  teaehing 17 

%  $11-446  B.C    Ifameraoa.    MaadiTatofl .  18 

;/     a  6 


TABLE  OF  OONTENTa. 


The  Pyihagoneam  Sekooi 

FmuooBAs,  569~iS00  MX 

Tbfl  I^thagonaB  U>a«hing . 

Tbfl  Pytlugorean  geooMiiy 

The  Pytha|{or«aii  tbeoiy  of  aainlMn  • 
EpiduunuiM.    Hipiiftsat.    PhiloUos.    Aichipptts. 
Amcu¥tai»,  cire.  400 11.C.    ••••.• 

HU  lolutiou  of  the  dopUcation  of  a  eubo    • 
Theodorm.    Timaeua.    Bryio 
Other  Greek  JIatkematieal  SekooU  in  tke/fth  eeatiify 
O^nopides  of  Chios.    Zeno  of  Elea  •        .        •       • 
l>ianoctitus  of  Abdeia      •  •       • 


CuAPTEB  III.  The  Scuoolb  of  Athbxs 

CiBC.  420— 300ii.a 


it 
it 


Lriii 


•  • 


•     ^« 


81 


•  • 


S4 


AMD  OtUCUB. 


Anthoriiies 

Mathematical  teachers  at  Athens  prior  to  4!i0  b.g;    • 

Auaugoraa.    The  Sophists.    Hippias  (The  nnadimlrU)   • 
Antipbo      •        •        •        •        •        •        *|*        *        * 

Th«  three  problems  in  which  these  sohools  were  speciall|y  inlsiitled 

Hii'i'uciuTKs  of  Chios,  cire.  420  b.  c I     • 

Letters  used  to  describe  geometrical  diai^rame  • 
Introduction  in  geometry  of  the  method  of  lidnetkNi 
The  quadrature  of  certain  lanes .       •        •  !     • 
The  Delian  problem  of  the  duplication  of  the  euha    • 

PUto,  429—348  b.c |     .        .        . 

Introduction  in  geometiy  of  the  method  of  acnaljsis  • 
Theorem  on  the  duplication  of  the  cube     •  !     • 

Eoiioicii,  40rt— 3(»5  S.C. j     • 

Theorems  on  the  golden  section 

lutroductiou  of  the  method  of  eihaui*tions         .        •        • 

Pupils  of  Plato  aud  Eudoxus ,     .        •        • 

IfexiBCUMi'ii,  circ.  31O11.C •        • 

Di8cuK«ion  of  the  conic  sections 

His  two  solutions  of  the  duplication  of  the  enba        •       • 

Aristaeun.    Theaetetus •        • 

Aristotle,  384--S22  U.C 

Questions  on  mechanics.    Letters  used  to  indieata  magnitidM 


87 


40 
41 
41 
48 
44 
45 
46 
46 
46 
47 
48 
49 
49 
49 
50 
50 
81 


I 
t 


V 


) 

^ 


TABLE  OP  OOMTKNTB. 


Zl 


\ 

t 
\ 


IV.    Thb  First  Albxaitdiuan  School. 
CiRC.  300--dOBx. 


ri0B 

ABthoriiics               52 

FoQodatkm  of  Alexandri* 53 

The  Ukird  cenimrp  before  CkrUt 54 

BocuD^  eire.  330—275  b.c.                       54 

Euclid's  ^IrM^nffl «        .55 

The  KlewunU  as  a  text-book  of  geometry   •  ^6- 

The  EtewkentB  as  a  text-book  of  the  theory  of  oambera  ^     .  59 

Euclid's  other  works .,       .  63 

Aiiitarahiis,  cire.  310— 250 1I.C .y^ 

Method  of  determining  the  distance  of  the  son  .       •  05 

OoooB.    Dositheaa.    Zenxippns.    Nieoteles  ••       .             ...  06 

ABomiBDBS,  287— 212  B.C.                                                       .  ■     .  66 

His  works  on  plane  geometry                     .       •        .        •  69 

His  works  on  geometry  of  three  dimensions       •        •        •  78 

His  two  papers  on  arithmetic,  and  the  **  cattle  piobleai  **  •  74 

His  works  on  the  statics'of  solids  and  fluids      ...  76 

His  astronomy 78 

The  principles  of  geometry  assumed  by  Archimedes  •  79 

Apouxncirs,  eire.  260— 200  BX.                79 

His  conic  sections 80 

His  other  works 82 

His  solntion  of  the  daplication  of  the  cnbe        •       .       •  84 

^  V            Contrast  between  his  geometry  and  that  of  Archimedes    .  85 

Enatoethenes,  275—194  m.c 85 

\        The  Sicfe  of  Eratosthenes .                                       ...  86 

7%r  WofMi  etntmrff  hefon  Ckriet .86 

HypMes  (Eoelid,  book  ut).    Nicomedes;  theeoncfaoid        .       .87 

Dioe^i;  thecissoid.    Persens.    Zenodoros     .                       •  ■     .  88 

HoffABonrnfCirQ.  130a.c. 88 

Foundation  of  scientific  astronomy  and  of  trlgoDometiy    •  89 

Hbbo  of  Alexandria,  dre.  125  B.C 91 

Fooadatioa  of  scientifie  engineering  and  of  land-fomyiBB  91 
Aim  of  a  trianfle  determiMd  in  terms  of  its  sidea     • 

Featnw  of  Bcfo's  work  

62 


XII 


TABLE  OF  00NTKN1K 


Tk€  fini  ceniu^  kefoM  CkrUi fl 

Theodosins .94 

Dionjmodonu  •• .95 

Emdof  ike  Fini  AUxmuiriam  Sekoa                                                .95 
Egypt  eoiuiiitntfid  a  Banuun  piofiiiot 95 


CuAFTBft  V.    Tub  Skooxo  AutZAXoiuAV  Scaook 

SOac — 641  A.1II. 


▲uthoritieB 

Tkt/ini  centurfi  afier  CkrUt 

Seraniit.    llendau* 


>  •  •  •  4  •  •  •  Vf 

...     .  n 

.       .       .       ^       .       .       .  OT 

Introduction  of  the  arithmfltie  ettmal  in  flMditfil  Kowf  96 

The  sseomd  century  q/Ur  CkriMi        •••••••  96 

Tbeon  of  Smyrna.    ThymaridM      •••...  96 

PiOLBMY,  died  in  168 99 

Thi  Atmagesi 99 


I 


100 
101 
lOS 
109 
109 
104 
105 
106 


Ptolemy's  astronom/  • 
Ptolemy's  geometiy    • 
The  third  century  after  ChrUt  . 
Pappus,  cire.  280  •        • 

The  2iv«>«rfYi|,  a  synopsis  of  Qntk 

The /ourth  century  after  ChrUt 

Metrodorus.    Elementary  proltiemt  in  aritlmeiie  and  algsttm 
Three  stages  in  the  development  of  algebra     •        «        •        • 

DiOFBAXTus,  cire.  S*iO(?) 107 

Introduction  of  syncopated  algebra  in  his  ArithmHit        •    108 
The  notation,  methods,  and  subjeel-mattir  of  the  woik    •    108 

His  Poritms •    118 

Subsequent  neglect  of  his  disooveries 114 

lamUichus.    Th^on  of  Alexandria  .  •        •       .114 

HypatU 115 

Hostility  of  the  Eastern  Chureh  to  Greek  scienee    .        •       •       •    115 

The  Athenian  School  {in  the  fifth  eemtmry) 115 

Produs,  413— 485.    Damascius  lEuclid,  book  zr).    Entoein.       .    116 


i 


•  •• 


*• 


r 


i 


TABLE  OF  CQNT15NTA.  XIU 

rAoi 

Rm^n  MttttkewtaiieM .       ....••..•  117 

Nature  iDd  eiteni  of  the  matheniAiics  read  at  Borne                      •  117 

OoBtraet  between  the  eonditioni  for  itiidj  at  Boiiie  and  at  Akiandiia  118 

Kml  of  tke  Second  AUxandriaH  School 119 

The  capture  of  Aleiaadria,  and  end  of  the  Akxaiidriao  Schoola     •  119 


CBAmR  VL    Ths  Btzantixb  School.    641 — 1453. 

Preeerratkni  of  works  of  the  great  Greek  mathematieiami  .  190 

Hero  of  Conetaatiiiople.    Peellos.    Planndea.    Bartoam.    Argyms  131 

I  MiehohM  Bhabdae  of  Smjma.    Pachjnneren 199 

Moediopolos  (Uagie  Squares) 199 

Caftnre  of  Coostaotiiiople,  and  disperse!  of  Greek  mathematicians  194 


Chapter  VII.    8t9tbm9  or  Numrratioh  and  Primitivb 

Arithmetic. 

Anthorities 195 

Methods  of  eoanting  and  indieating  nambers  among  primitifv  raees  195 

t  \            Use  of  the  abaeos  or  swan-pan  for  practical  calculation  •               .  197 

Methods  of  representing  nombers  In  writing 190 

The  Boman  and  Attic  symbols  for  numbers     •                               •  181 

The  Aleiandrlao  (oc  later  Greek)  symbols  for  nombers   .               •  ISl 

Greek  aiithmetie 199 

Aieptfaii  of  the  Arabie  system  of  notation  among  ciTilised  raees     .  199 


jdv  TABU  or  OONTBmB. 


Sbmiife  yeiM.    mat&ematicf  if  tftt  mObb  ft|ni 

aKb  if  Hk  Iftmatottiicf. 

Tkii  pnicd  be§itu  €bau$  the  iiMih  emlHry,  mud  mm^  W  mM  !•  Mtf 
irffA  Mr  inrentUm  uf  analytical  §eomelry  and  of  tkg  i^flaiin^wuU  Mlralnf. 
Tki  ckarueUrUlic /iaiure  of  Ikis  period  i»  ike  ereatUm  «f  devthprntaj  1/ 
wmodem  aritkioiUet  algehra^  and  tri$oaometry. 


VIII.    Tui  R18K  OP  Lkaexinq  IV  Wistsev  Bubope. 
CiMX  GOO— 120a 


AathorillM  m 

Edaeaiiom  im  tkg  tijrlA,  Mrmf  A,  mmd  ti$kik  etatmriea  .  117 

Tbfl  Mooaitio  Seboob 187 

Boeihius,  die.  476—596 188 

IfedMval  toxt*books  in  geomeUy  and  tiiUiiiieCio  •  189 

CMniodonu.  490-660  189 

Iniaorui  of  H«vUle,  670-636 140 

The  Caiktdral  and  Comvtniual  SekooU     ......  140 

The  Sdiook  of  CharlM  the  Oraai 140 

Alcuin,  785— 804 140 

EducaUon  iu  the  ninth  and  tenth  eentnriet     .        •        •        •        «  148 

Oerhert  (Sjlfeiter  II),  died  ui  1008 148 

Bernelinut 146 

The  Karl^  Mtdieral  Univeniiiet 145 

Riee  durinK  tlie  twelfth  eentuiy  of  the  enrliert  VBlTanitlea  .  |46 

Development  of  the  medieval  oni? enitiet         .        •        •        •        •  147 

Oatlino  of  the  conrao  of  utadiet  in  a  medietal  imifwndlgr  .  147 


CUAPTER   IX.      TUE   MaTUEMATICS  OP  THE  A 

Anthoritlea 160 

Extent  0/ wuiikewiatici  obtained /ram  Ortek  mmrt€$  .  •  160 

The  College  of  Seribee 161 


TABLE  OP  CONTENTS.  IV 

PAOB 

Bxieui  €f  mathewiafic$  obinimeHfrom  ike  {Ar^an)  iiindooii                .  152 

Abta-Bkata,  eire.  &30       .       ^ 153 

f                              Hie  chmpten  on  algebts  St  Irigonooielry  of  hit  A  rfakkatkfffm  153 

^               BumJUOvrTK,  eire.  640 154 

.  The  ehaptera  on  algpbn  and  ijeomelrj  of  hit  SiHtlh&mim    •  154 

Bbamuba,  eire.  1140                        156 

The  LiUtrati  or  arithmetic;  decimal  nnraeration  Qped       .  157 
The  Bija  Ganiin  or  algebra                                                  .159 

Tkt  devttofmemt  of  mnthematicM  in  Arabin 161 

AuuBDun  or  AL-KnwiMmi,  eire.  930 162 

B\»  At'^ehr  we' I  mmhnhaU 163 

Hie  eolation  of  a  quadratic  equation 163 

Introdoetion  of  Arabic  or  Indian  ejstem  of  nvmeration    .  164 

TAarr  imc  KoamA,  836—901 ;  eolntioo  of  a  enbie  equation               •  165 

Alkajani.    Alkarki.    DcTelopnent  of  algebra        ....  165 

Albalegnl.    Albuijanior  Abul-Wafk.    DeTelopment  of  trigonometry  167 

'Alhaiea.    Abd-al-gehl.    DerdopDcnt  of  geometry .                       •  168 

Chanelcriiitiee  of  the  Arabian  nehool 169 


OHAPTim  X.    IifTRonucTiox  OP  Arabian  Works  irto  Europe. 

CiRC  1150—1450. 

The  etertnth  tent unf 171 

Ocber  iVn  AphU.    Anaehel .171 

The  twelfth  century 171 

AMhaidofBath 171 

Ben-Bsra.    Genud 172 

JoliB  Hlspalensit 173 

ne  lAfrfffRf  A  eealarry 178 

Lborardo  or  Pnu,  eire.  1175— 1230 .173 

The  LI6fr  il6acl,1202  .173 

The  iBtroduction  of  the  Arable  nmnerale  into  eommeree  •    174 
The  introdoetion  of  the  Arahie  Bttmerale  into^eeienee       .    175 

The  mathematieal  tournament 175 

fMtriek  n.,  1194-1250  176 


ZVI  TABLE  OF  OOMTEim. 

WAom 

JouMUiui,  oira.  IttO •       •    in 

Hk  De  NmuurU  Dmih  |  ^yaflopOdl  ftlfebim  .ITS 

H<4jwood 180 

BooBB  Bicox,  1314^1394 181 

Campaniis        .       •       • 184 

The  fourtetuik  cetUurff  •        •        •    184 

BndwardiiM     .' 184 

Orecmiui ••••••    185 

Tha  refonn  of  Um  anif enity  ennieiilBai  ••••••    18ft 

Thejifieentkcemtmr^ .186 

Belilomandi -•       •       •    187 


ClIAPTKIt    XI.      TUK    DbTBLOPMSMT    of    AUTBHBTia 

CiBa  1300—1637. 

AutboritiM '    •        •  188 

The  Boethiaa  uriUinieUo  •                        188 

AlKorUm  or  modern  arithmeUe •        •  189 

The  Arable  (or  Iiuluin)  sjmbole :  histoiy  of 180 

Introduction  into  Europe  by  ecienee,  oommerat*  iDd  tnlendnn       .  198 

Iniproveuiente  introduced  in  al|{oriitic  arithmetie    •        •        •        •  194 

(i)    SiuipUtication  of  the  fundamental  iMrooeeera                       •  194 

(ii)    Introduction  of  eigne  for  addition  and  eubiraction, eira.  1480  801 

(iii)   Invention  of  lo|{aritbmB,  IGI4 801 

(iv)   UHe  uf  deoimab,  1019 808 


CiiAPTEK  XII.     The  AIatbkmatics  of  tub  RbVAI88AMOIC 

Ciua  1450—1637. 

Anthoritiee 805 

Effect  of  invention  of  printing.    The  renaieaanee    ....  805 

The  develvpmeHt  vj  BjfHCopaUd  algebiu  and  triifOHomgirp  ,  .  807 

Ukoiomoktamdm,  14»G— 1470 907 

His  De  TriaHifulU  (printed  in  1490) 808 

Pnrhach,  1423— 1461.    Cnaa,  1401— 1404 811 

Chuquct,  ciro.  1484 818 

Introduction  end  origin  of  ejrmboU  4-  and  > 818 


I 


t 


TABLE  OF  OOKTENTBL  XVll 

rxam 

IMoli  or  LueM  di  Bvigo,  cire.  1500      ....••    315 
His  ariUimeiio  and  geometiy,  1494    .       •       •       •       •    315 

Leonardo  da  Tinei,  1453—15111 318 

DOrer.  1471- 153*^.    Copemicai,  1473—1543 319 

Reeord,  1510—1588;  introdoetion  of  sjmbol  for  cqoalilj  .    330 

Bodolff,  rire.  1535.    RieM,  1489— 1559 331 

Bnnh.  1480—1507 331 

His  Aritkmetiea  Integra,  1544    .  .  '      •        .        .     333 

Tabtaoua,  1500-1559 333 

His  solation  of  a  eobio  equation,  1535       •        •        .        .    .334 

His  arithmelie,  1550— 1560 835 

CAanui,  1501— 1576  337 

His  An  Jffif  R«,  1545 ;  the  third  work  printed  on  algebra .    339 

His  solation  of  a  cubic  eqnation         .  .     331 

Ferrari,  1533 — 1563;  solution  of  a  biqnadratie  eqnation  .    331 

Bbelieos,  1514—1576 333 

llanroljens.    Borrel.    Xjlander.    Commandino    ....    333 

Metier.    Bomanns.    Pitixcus 333 

Bamns,  1515—1573.    Bombelli,  cire.  1570  .334 

The  deretopwient  of  fjftnbolic  algebra 335 

Vnrri,  1540—1608 .335 

Introdoetion  of  symbolic  algebra,  lo91      ....    337 

Tieta*s  otherworks 339 

Otrard,  1590—1683 ;  development  of  trigonometrj  and  algebra  341 

NArm,  1560—1617 ;  introduction  of  logarithms,  1614    .  .843 

Briggt,  1556—1681 ;  ealeulations  of  tables  of  logarithms  .    348 

HjumoT,  1560—1631 ;  deTelopnient  of  analysis  in  algebra  .    844 

Oi«htfed,  1574^1660 .845 

The  origin  of  ike  mart  eownaaa  9gmboti  in  atgehra    ....    846 


CBAPnm  XIII.    TiiR  Cjmb  of  thr  ReiiAiaiAifCB. 

Cite.  1.586—1637. 

AailMtltica 851 

Tki  4ev€lo/memt  of  mechanieM  and  erperiwuntat  wtetkoda  .  858 

SrsviinH  1M8— 1608 858 

It  of  the  modem  traalmeBl  of  statiea,  1686  .  858 


ZTUl  TABU  OF  OONtKNTB. 


OiULM^lMi— IMS iM 

ComneiioeiiMBlof  tlMMlMMtof  ^yattate                      •  SM 

OalUeo'i  Atlroiioiiij SM 

ftMieii  Bmob,  1561— ie96     .'      .                                               .  iM 

OnldiniM,  1577—1613.   Wright,  1560-1615 ;  MMlnMllMi  of  wmr^  MO 

SneU,  1591— 16S6 Ml 

JUvirmi  of  interui  in  pure  §eamitr]f •  MS 

Kbplrb,  1571—1680 .MS 

Hii  Paratipomemtt,  1601 ;  priadpla  oT  eontiBvilj               »  MS 

Hm  Stereometric,  1615;  ait  of  inftnitarimih    •        .        .  S68 

Kepler's  Uws  of  pUoeUiy  motion,  160S  wad  1610     .        .  S64 

DeMTgnes,  1599- 166S    .        .' S64 

Hit  BruuilloH  project ;  vaa  of  projaetifo  geottij              .  S65 

Jiiatkematieal  kuoirledge  at  the  close  of  lAe  rtimismmeg    .                 •  M6 


IRftMi  9erfoti.    mobmi  inatbematicf. 


TAIf  jieniMi  frrpiiie  with  the  inrtuiiom  tf  oiMlyliWU  fii«ilrf  imhI  tA« 
it^miletitml  caieulmi.  The  tmthemmtieg  U  far  wMrt  eompkg  tktm  tkmi 
pndmeed  in  either  of  the  yrteediuf  periodt:  hmi  it  wm^  W  pmnrnil^  di* 
§eribed  m*  charaeterized  6y  the  devetopment  o/ono^f' **,  and  iU  mppHtmtiom 
to  the  phenomena  of  nature. 


CuAmB  XIV.    Thb  Hi8tc?t  of  Modbut  Mathbiiaticsb. 

TraUmeni  of  iho  tobjoot S7I 

Invention  of  analjtinl  goometiy  and  the  BMthod  of  indiTieiblet     .  S7S 

Invention  of  the  calcnlna STS 

Development  of  meehenict S74 

Application  of  mathematics  to  physics S74 

Becent  development  of  pate  mathematics S75 


TABLE  OF  CONTENTS. 


XIX 


3 


OHAPncE  XV.    History  or  Mathexatigb  rBOM  Dbscaktes 

TO  HuTGBXs.    Cina  1635 — 1675. 

MOB 

Aatborilics S77 

DncABTcs,  1596—1690  .  ......    278 

His  TiewB  on  philosopbj 281 

His  InTeDtion  of  analjtical  gtmnetiy,  16S7  .281 

His  algebra,  optics,  and  theory  of  Tortiees  •    285 

Oatausbi,  1598--1647 .        .    287 

The  method  of  indiTitibles .288 

PakaU  1623— 1662 290 

His  geometrical  conies 293 

The  arithmetical  triangle .293 

Foundation  of  the  theorj  of  probabilities,  1654         •  294 

His  diwassion  of  the  cycloid  .        .        .    296 

Walus,  1616-1703 .297 

The  Arithmetiea  l^finitonm,  1656 298 

Law  of  indices  in  algebra •        .    299 

Use  of  series  in  qoadratores 299 

Earliest  rectification  of  enrres,  1657 301 

Wallis's  algebra  302 

FnxAT,  1601— 1665 .        .302 

His  investigations  on  the  UiPory  of  nnmbers  .        •    303 

His  nw  in  geometry  of  analysis  and  of  infinltesiRialfl  308 

Foundation  of  the  theory  of  probabilities,  1654         •        .    309 

HmtoBiB,  1629—1695  311 

The  Horofoffium  Oicilhtoritim^  1673 312 

The  undolatoiy  theory  of  light 313 

Other  wutikewtaiieian$  of  ihii  iitme 315 

Baehet.    Mersenne;  theorem  on  primes  and  perfect  nnmbers  315 

BoberraL    YaaSchooten.    Saint-Vincent 317 

Tonieelli.    Hndde.    Fr^nicle.    DeLalonb^        .        •        •        •    318 
Itooator.    Barrow;  the  differential  triangle  .        .        .        •        •    319 

Bimuieker;  eontinned  fnctiona 822 

Janet  Oregoiy ;  dUtlDotlon  between  eontergenl  and  divergent  eeriee    823 

Or  Chriatopher  Wien 324 

Hmke.    CoDlna.    Pell   . 825 

Bfana.    TiviMiL    IwMmiuMmm    •                                             .826 
Dak  ran.    Bocner.    BoOe 827 


zx 


TABLE  OF  OOHTENTB. 


Chaptbe  XVI.    Thi  Lifi  avd  Wobks  or  Niwiov. 


ptoiMior, 


AathoriUes 

N«irloo*s  teliool  aod  nndargrmiliuiU  life 

luf etligatioiM  in  164i5^1(iliC  on  lloiiom,  opCiei,  and  cmfttnlkNi 

His  Tiewt  on  gmviution,  16C0  . 
BeMarahes  in  1667—1666.    Eleeled  LneaalAn  pffoiBtM|r»  1669 
Optiod  loetorM  and  di«oof eriM,  166»— 1671 
Emittion  tbeorj  of  light,  1675 
Tha  LeibniU  Letters,  1676      .... 
Diseoreries  on  gniTitstion,  1679 
Diseof  eries  sud  leettues  on  slgebrm,  1678 — 1688 
Disco"*  ^ries  and  lectures  on  gravitation,  1684  • 
The  l-rincipia,  1685—1686      .... 

The  subject-iustter  of  the  PrtHtipia  . 

Publication  of  the  PriHcipia      .        • 
Investigations  snd  work  from  1686  to  1696 
Appointment  st  tlie  Mint,  and  removal  to  London,  16S|8 
Publication  of  the  Optie*^  1704        .... 

Appendix  on  dasiufication  of  euUe  enrvca . 

Appendix  on  quadrature  bj  means  of  ii 

Appendix  on  method  of  fluxions 
The  invention  of  fluxions  and  the  iufinitesimal  eaknlali . 
Newton's  death.  1727       .... 
List  of  his  works 

Newton's  character 

Newton's  discoveries        .... 


881 


885 


889 

840 
848 

848 
845 

847 
848 
849 
849 
850 
859 
858 
856 
857 
857 
858 
860 


Chapter  XVH.     Lbibniti  avd  the  MAroBiiATioiAMi 

or  THR   FIRST   ttALP  OP  TUB    ElGllTBBlfTH  CSMTURV. 


Authorities 

IMbHitz  and  the  iiermmllU 

LxiBNiTZ,  1646—1716     . 


His  system  of  pliilosophj,  and  servioes  to  litoimUu* 

*  The  controversy  as  to  the  origin  of  the  ealenlna 

*  His  memoirs  on  the  inflnitesimal  calcnlns 
His  papers  on  various  mechanieal  problema    • 
Characteristics  of  his  work  .        . 


868 
868 
865 

866 
879 
878 
878 


1 


f 


TABLE  OF  OONTENT8.  XXI 

PAOB 

'  Jambs  Debxouilli,  1654 — 1705 876 

JoBM  DEBiffOUiLU,  1667—1748 377 

The  yoanger  BemouiUiH 879 

The  ^erelnpiRfmt  of  antilif»i$  on  the  coMtimfni 379 

L'HospiUl,  1661—1701.    TarignoD,  1654—1722     ....  380 

De  MontmoH.    Nicole.    Parent.    Banrin.    De  Gna  .  381 

;  Cnuner,  1704— 1752.    Biceali,  1676— 1754.    Fagnaiio,  1662— 1766  3K2 

Clubavt,  1713—1765  . 383 

•  D^Albmbebt,  1717—1783 385 

Bolotion  of  a  partial  differential  equation  of  the  eeeond  order  387 

Daniel  Bemoalli,  1700-1782 .388 

The  EmgliMh  WMthcwMiicitina  of  the  tighUtnih  eentmrg  .  389 

DaTid  Orei^ny,  1661— 1708.    Halley.  1656— 1742  ....  390 

DmoB,  167^—1715 391 

BnooB  Tatlob,  168.>— 1731 .   .  391 

Taylor*^  theorem 892 

\  Tajlor*t  ph jBieal  icsearches 392 

j  Oolea,  1682— 1716 393 

I  I>MMiTie,  1667— 1754 394 

I;  * llMa.AUBiir,  1698— 1746 895 

Hit  geometrical  disooTcrieii  396 

The  Trtati$€  of  Flmjcidn$,  and  propositions  on  attractions .  897 

8l«WBii»  1717-1785.    Thomas  Simpson,  1710—1761     .  .899 


4 


Chapter  XVIII.    Laoranuk,  Laplace,  akd  tubir 
€k>irrEMPORARiE8.    CiRc.  1740 — 1830. 


CShafBoteristics  of  the  mathematics  of  the  period     ....  401 

Tke  devehfwteni  of&nalgn$  and  meehomiet 402 

BoLBB,  1707— 1783 402 

The  JnlrMiectfo  la  AMlyrin  Imjimitonm^  1748  .        •        •  408 

•  The  Intf ^iNf lone*  Cale«IJZ>(jfirrriit/ali#,  1755     .        .    •    .  406 

•The  InffJlNfioiKf  C^Umli  Inteffrolh,  1768—1770  .  4C3 

The  Jnleffaa^  tvr  il(^m,  1770  ...  406 

Bnkff'i  works  oo  aveohanies  and  astroDony  •  408 

itai  171»-1777     .       •       •       • 410 


J 


5 
t 


xxn  TABLE  OP  OONTKNTB. 

Bteoaft,  1780-1788.  Tnmbligr,  1749-1811.  AibcvMl,  17iO-ia08    4U 

loamAMOi,  1786— 18U 411 

Memoin  on  TarioQS  nibjceta  •    418 

TIm  if^cR^gM  ciialyf  igM,  1788  .417 

Thb  Thione  BJod  C^kMt  d€t /oiKtimii.  Vm.  19H     •       •    480 
The  XUoiution  dt$  /gualioiM  miM^rlf  tiei,  1788  ,        •        •    481 

ChAnetariitifit  of  Lagrange't  work 481 

Latlaci,  1749—1887 •        .488 

Memoin  on  ftstronomjr  and  attrmoUont,  1778—1784 .        •    488     * 
Uie  of  spherical  harmoniea  and  tho  potontial    .  .    488 

If  emoin  on  problems  in  attronomy,  1784 — 1788  •    484 

The  M^caHiqueeiU$U%ndExpo$UUm  dm  9^$tbtigduwmdt    486 

The  NeMar  HjrpoUieua 485 

The  Thiorie  analytiqu^  tU$  probabUitiM,  1818    ...    487 
Other  reaearches  in  pure  mathematici  and  in  pfajriiei       .    488 

CharacteriBties  of  Laplaoe*i  work 489 

Character  of  Laplace 480 

LnuomBB.  1758—1833 481 

Uifl  memoirs  on  attractions 488 

The  Th^orie  de$  nombn*,  1798 438 

Law  of  qnadratie  reciprocity 488 

The  Cd/cu/ inf/praf  and  the  l-'oNclioiM  r//i>lif  lief      .        .434 

Pfaff,  17C5— 1825 435 

TAe  creation  of  modern  peometry  .        '      ;  •        •        •    ^^ 

Monge,  1748— 1818 485 

Lazare  Camot,  1753—1883.    Poncelet,  1788—1867         ...    438 
The  dtvelopment  of  mathematical  phyeice  .        .     i   .        •        .     439 

Cayendish,  1731—1810 439 

Bumford,  1753—1815.    Yonng,  1773—1889    .        .     |   .        .        .    440 
Dalton,  17CC— 1844         .  .        .        .     |   .        .        .441 

FouaiEB,  17C8— 1830 j   ...    448 

Sadi  Camot;  foandation  of  thermodynaniiea  .        .     '  .  «    448 

PoissoH,  1781— 1H40 448 

Ampere,  1775— 1836.    Fresnel.  1786-1887.    Bioi.  17^4— 1868      .446  j 

Arago,  1786-1853 447 

The  introduction  of  analyti$  into  England        .        .     i    .        .        .    449 

Ivory,  1765—1848 !   ...     449 

The  Cambridge  Analytical  School 449 

Woodboose,  1773—1887 .450 

Peacock,  1791— 185a    Babbage,  1798—1871 451 

bir  John  Uemchel,  1798—1871 458 


I  • 


I 


TABLE  or  CONTENTH.  ZXIU 


•  ■  • 

1}  Chaptbr  XIX.    Mathematics  op  the  Nikbtbehth  Obiturt. 

FAOB 

Graatkm  of  new  branebet  of  matheiDAtics                                       •  454 

Diffieolljr  ia  disciMsiiig  tbe  Dwtbcmaties  of  Ibis  eefitmy .                .  454 

Acoonni  of  eootemporaiy  work  not  intended  to  be  eibaaslife .        •  455 

Antboriiiet 455 

•  Gaom»  lTn--1855 457 

LiTestigAtions  in  Mtrononj,  eleelricity,  Ae.      •                •  458 
Thm  Di$quMtione$  Aritkmetic^e,  1901                               .461 

His  otber  diseoferiee 463 

CompAriflon  of  Lftgnngei,  IiAplftoe»  and  OaoM  .               •  463 

Dirichlel,  1805— 1859                                      464 

Development  of  the  Theory  of  Nwmben 464 

Eiwnitcin,  182S— 1852 ...  464 

Henry  Smitb,  1825—1883 465 

Kolc9  on  otber  writers  on  tbe  Tbeory  of  Nombers  .                .        .  468 

Development  of  the  Theory  of  Fmnetiom  of  Mtultiple  Periodieitp       .  469 

Abbl^  1802—1829.    Abel's  Tbeorem 469 

Jaoom,  1804— 1851 471 

BuBMissf,  1826—1866 472 

Notes  on  otber  writers  on  Elliptic  and  Abelian  FonctionB               .  473 

WEnssnAss,  1815— 1897 474 

Notes  on  recent  writers  on  Elliptic  and  Abelian  Functions     •       .  475 

Tho  Theory  of  Fmnetiom 475 

DevehpmaU  of  Hiffher  Alsehfn 476 

Cavcbt,  1759—1857 .476 

Argandy  bom  1825;  geometrical  interpretation  of  complei  nnmben  479 

Sim  Wnuui  Hajultoh,  1805—1865;  introdoction  of  qoatemiona  •  479 

GBASSMAsm,  1809—1877;  bit  non-commntatire  algebta,  1884  •       •  481 

Bode,  1815—1864  . 481 

Galois,  1811—1832;  tbeoiy  of  disoontinoom  snbsUtation  ginnpa   .  482 

Do  Moffgan,  1806—1871 482 

Gailbt,  1821— 1895 .  483 

BnTianB,  1814—1897 ...  484 

IiiB»  1842— 1889;  theoiy  of  eontinnoQssQbstitationBioQps    .        .  484 

Notes  on  other  writers  on  Higber  Algebra       .....  486 

DeoeHopmund  of  Anolytitol  Ocomttry 487 

IMw  om  aoaw  ises»l  writers  om  Anajytieai  Oewnetiy    ...  486 


XZIV 


TAUJS  OP  00NTKMT8. 


Ammiggi*.    Mmum  of  aome  veoeni  wiitan  on  Aaallywi  •    489 

Dewtlopmsmi  uf  Syntketit  Oeumelrg 489 

Btetncr,  1796—1068.    Von  BUiidl.  1796—1967       ....    496 

Othiir  wrilera  oo  modern  Sjntlietie  Ooometiy .        •        •        •        •    491 

DevtlopmeM  of  maH-Euclidtam  Oeowieirjf .        •        •        •  •    491 

Dtrelopmeut  of  the  Theory  of  Meehamict^  trtaUd  QtmfklaM§  •        •    494 
ClUIoid,  1845—1879        .        .  .        .        .        .        .        .496 

Development  of  Theoretical  Meehamies,  treated  Amtlgticmttg  •     496 

Onsen,  1793—1841 497 

Noteii  ou  other  writers  on  Mechenies  .        .  ■      •        •    496 

Derelopment  of  Theoretical  JttruHomg 498 

Ikhisel,  1784—1846 •        .    496 

LeTenrier,  1811—1877 .        .499 

AdauiM.  1819—18112 .666 

Not€fl  on  otbcr  writers  on  Theoretical  Aslionon^  .        .        •        •    Ml 
Development  of  Mathematical  Phytic t 509 


IxtHLX  606 

PJUUM  NOTICBS •  •  •  •  • 


AbKKMKUM.  A  leleranee  to  Plot  O.  hoiWu  L$  ScUmu  MmUt  melt 
Ataica  Orecia,  Modena,  1896—1900,  ■hoold  baf«  been  tdded  to  the 
lootnotee  on  pp.  IS,  86,  and  59;  and  a  referenoe  to  hit  IMe  hmupuack* 
liehtUn  Theorieen  der  Geowutrie,  Leipiig,  1888,  ihottld  have  heen  added 
to  the  works  mentioneil  on  p.  456. 


f 


r 


ERRATA  ET  ADDENDA. 


Pli0e  17,  liiM  4  from  end.    AJUr  %al  imert  or  poesiUj  September  90, 

€091.6 
Pli0e  74,  line  6.    Add  Inii  references  to  il  are  giten  bw  Pappoe. 
Fife  79,  note  *,  line  4.    After  by  insert  J.  L.  Heiberg,  Leiptig,  1990, 

1993,  and  another  bj 
PUge  91,  note  *,  line  8.  .  Afier  m  4mM  and  Loria,  book  m,  chapter  ▼• 

pp.  107—138. 
PUge  97»  line  19.    Heiberg  thinks  that  Serenas  lived  at  Antinoe  and 

not  at  Antissa. 
PUge  103,  line  3  from  end,  and  Index.    For  Oltaiano  read  A.  GUMrdano. 
PUge  150,  note  *,  line  9.    After  1877  lufd  and  1^  H.  Sater,  JHe  Matke- 

wiatiker  und  Attnmtfmen  der  Araher  tuid  ihre  Werke,  ZeiUckrift  ptr 

Maihewuitik  wid  Physik^  Ahhafudlungen  Mur  OeeehiekU  der  Mmike* 

auitjft,  Leipsig,  vol.  1L7/1900. 
PUge  918,  note  *.    Dele  reference  to  Woepeke's  memoir. 
Pftge  934,  lines  18,  35;   Page  339,  line  10;   and  Index,  page  511. 

For  Fiori  read  Fiore. 
PUge  334,  lines  19,  33,  and  Index,  page  518.    For  Feneo  read  Ferro. 
FwjgB  341,  line  6  from  end.    After  rooto  ineert  the  theorem  known  as 

Newton's  mle  for  finding  the  sam  of  the  like  powers  of  the  roote  of 

an  equation ; 
PUge  348,  lines  8,  4, 5.    The  eolon  (or  ^jmbol|)  need  te  denote  a  ratio 

oeears  on  the  last  two  pages  of  Ooghtred's  iammei  Sinmwn,  1657. 
Pfegc  363,  note  \  line  3.    For  1871  read  1877. 
Pfege  388,  line  3  from  end.    For  algebraical  read  geometrical. 
PUge  386,  line  i.    After  coeiBciente  ineeri  It  maj  be  also  mentioned  that 

he  enunciated  the  theorem,  commonlj  attriboted  to  Eoler,  on  the 

relation  between  tfab  -nomber  of  Amcs,  edges,  and  angles  of  a 

poljhedroQ. 
Pfege  801,  line  8.    The  logarithmic  spiral  had  been  rsetifled  hj  TorriosDi 

shortly  before  Neil's  aiscorery. 
PUge  818,  note  *.    Add  See  also  a  memoir  1^  O.  Loria,  BMiaikam 

sMf AesMfiM,  series  8,  toI.  i,  pp.  75—89,  Leipsig,  1900. 
Page  885,  note  *,  line  1.    After  Cordoroet  add  Bertfand 
Page  403,  note  *,  line  8.    After  1843  ineert  See  also  Index  Opermm  Ewteri 

by  J.  O.  Hagen,  Berlin,  1896. 
PMige  463,  line  5.    After  have  insert  his  theorems  oo  the  onrvatore  of 

sorfrMDes,  and 
P)H{e  479,  linee  15, 16.   ArgandwasbomJa^l8^  1768anddiedAngosll8, 

1838. 
PH{e  486,  note  f.    The  third  Tolome  has  not  been  issoed. 
PUge  488,  note  f.    8ehabert*s  Leetorss  wers  pablisbed  al  Leipsig,  1879; 

Lindemann  edited  Clebeeh*s  leetoree ; 
Page  489,  line  U.    4/l<r  Fochs  Ifwerf  (1888— 1903). 
Page  489,  Use  17.    /or  1858  rfod  1850. 
ff^  491,  Bote  •,!&■•  8.    4ftar  fai  fnwrf  the  wmkM  by  W.  lagsl 


•     i    ; 


r 


CHAPTER   I. 


EorrnAN  and  phormdak  mathematics. 


Tub  history  of  tnnthematics  cannut  with  ccrlainty  ha 
traced  hiu^k  to  any  arhool  or  prrioil  Ixrfiire  that  of  the  Ionian 
Greeks.  The  llu)>^^ue^l  hintory  limy  Ijo  tJivide^l  into  three 
perioilis  the  (liHtiiictions  l)f>twwn  which  npc  tolerably  well 
mnrhed.  Tlie  tint,  period  is  t)uit  of  the  hislory  of  innthemnticv 
nnder  Greek  influence,  thin  it  dixcuxseil  in  chapters  li  (o  vir : 
the  scKvnd  'm  thnt  nf  the  mntlicninticN  of  the  miihllp  agi^  and 
the  rrnaissnnce,  thin  is  discuKE<r<l  in  chR[iti-n  vlil  to  XIM  :  the 
third  is  that  of  modem  mathemntio,  and  this  is  rliHciiHtrd 
in  chaptcnt  xiv  to  iix. 

Although  thfi  history  of  mathematicfi  commences  with  that 
of  the  Ionian  eclionls,  there  is  no  duaU  that  those  Greeks  who 
Snt  paid  attention  to  the  subject  were  lar^ly  indebtixl  to  the 
previoua  investigations  of  the  Egyptians  and  Phoenicians. 
Onr  knowledge  of  the  muthematical  attainment* of  Ihrwe  race* 
ia  imperfect  and  partly  cunjectumi,  Imt,  such  ns  it  is,  it  in 
here  briefly  summarized.  Tlio  delinite  hintory  begins  with 
tbe  next  chapter. 

On  the  Rabject  of  pn^historic  matliemalic",  we  mnyubservfl  * 
in  the  first  place  that,  titovgh  all  early  races  which  have  left 
rcGonla  behind  them  knew  something  of  nnmeration  and 
^  aad  thoBgh  the  majority  were  also  acquainted  with 
I  flf  laad'mrTvjing,  yet  the  mlea  which  they 
1 


J 


2  EOTFTIAN  AND  PHOKNICniN  MATHEMATICB. 

pdaaeiaed  wero  in  general  founded  onlj  on  tlie  retolU  of 
observation  and  experiment,  and  were  neither  deduced  bom 
nor  did  they  form  part  of  any  science.  The  fact  then  that 
various  nations  in  tlie  vicinity  of  Greece  had  reached  a  high 
state  of  civilization  does  not  justify  us  in  assuming  that  they 
had  studied  niatheuiatica. 

The  only  races  with  whom  the  Greeks  of  Asia  Minor 
(amongHt  wlioiii  our  liintory  bi*gins)  were  likely  to  have  come 
into  frequent  contact  were  those  inhabiting  the  eastern  littoral 
cif  tlie  Mediterranean  :  and  Greek  tradition  uniformly  assigned 
the  special  development  of  geometry  to  the  Egyptians,  and 
tliat  of  the  science  of  nunibi*rs  either  to  tlie  Egyptians  or  to 
the  Phoenicians.     I  discuss  these  subjects  separately. 

First,  as  to  the  science  of  uuMibers,  So  far  as  the  acquire- 
ments of  the  PliueniciAiiH  on  this  subject  are  concerned  it  it 
impossible  to  speak  with  certainty.  The  magnitude  of  the 
coi>;iiiercittl  tninMiictioiia  uf  Tyre  and  Sidoii  necessitated  a  con- 
siderable development  of  arithmetic,  to  which  it  is  probable 
the  name  of  science  might  lie  pmperly  applied.  A  liubylonian 
table  of  the  nunieriiral  value  of  the  squares  of  a  series  of 
consecutive  integers  has  been  found,  and  this  would  seem  to 
indicate  that  properties  of  numbers  were  studied.     According 

\to  Strabo  the  Tyriaiis  paid  particular  attention  to  the  sciences 
of  numbers,  navigation,  and  astronomy;  they  had  we  know 
considerable  commerce  with  their  neighbours  and  kinsmen  the 

'  Chaldiieans ;  and  lliickh  says  that  they  regularly  supplied  the 
weights  and  measures  used  in  lUbylon.  Now  the  Chaldaeans 
had  certainly  |taid  some  attention  to  arithmetic  and  geometry, 
OS  is  shewn  by  their  astrfmuinicol  calculations ;  and,  whatever 
was  the  extent  of  their  attainments  in  arithmetic,  it  is  almost 
certain  that  the  Phoenicians  were  equally  proticient,  while  it 
is  likely  that  the  knowledge  of  the  latter,  such  as  it  was,  was 

I  communicated  to  the  Greeks.  On  the  whole  it  seems  probable 
that  the  early  Greeks  were  largely  indebted  to  the  Phoenicians 
for  their  knowledge  of  pnuftical  arithmetic  or  the  art  of  calca- 
lation,  and  perhaps  also  learnt  from  them  a  few  properties  of 


I 


t 


EARLY  EGTFTIAN  ARITHMCTIC.  3 

nnmbera.  It  maj  be  worthy  of  note  that  Pythagoras  was  a 
Phoenician ;  and  according  to  Herodotus,  but  this  is  more 
doubtful,  Thales  was  also  of  that  race. 

I  may  mention  that  the  almost  universal  use  of  the  abacus 
or  Bwanpan  rendered  it  easy  for  the  ancients  to  add  and 
subtract  without  any  knowledge  of  theoretical  arithmetic 
These  instruments  will  lie  def^cribed  later  in  chapter  vii ;  it 
will  be  Hufficient  here  to  Hay  that  they  afford  a  concrete  way 
of  representing  a  nunili«*r  in  the  decimal  scale,  and  enable  the 
results  of  addition  and  subtraction  to  Ije  obtained  by  a  merely 
mechanical  process.  This,  coupled  with  a  means  of  represent- 
ing the  result  in  writing,  was  all  that  was  required  for  practical 
pucposes. 
I     .  We  are  able  to  speak  i^ith  more  certainty  on  the  arithmetic 

of  the  Egyptians.  Aljont  thirty  years  ago  a  hieratic  papyrus*, 
forming  part  of  the  Rhind  colliH^tion  in  the  British  Museum, 
•  was  deciphered,  which  Ims  thrown  cor.siderable  light  on  their 
mathematical  attainments.  The  msnuscript  was  written  by  a 
priest  named  Ahmes  at  a  date,  according  to  Egyptologists, 
considerably  more  than  a  thousand  years  liefore  Christ,  and  it 
is  believed  to  be  itself  a  copy,  with  emendations,  f>f  a  treatise 
HKHne  than  a  thousand  years  older.  The  work  is  called  **  direc- 
tions for  knowing  all  dark  thingn,''  and  consists  of  a  collection 
of  problems  in  arithmetic  and  geometry ;  the  answers  are 
given,  but  in  general  not  the  processes  by  which  they  are 
obtained.  It  appears  to  lie  a  summary  of  rules  and  questioi^s 
familiar  to  the  priests. 

The  first  part  deals  with  the  reduction  of  fractions  of  the 
form  2/(2n-i- 1)  to  a  sum  of  fractions  whose  numerators  are 
each  unity :  for  example,  Ahmes  states  that  ,',  is  the  sum  of 


i 


I 


*  See  Eim  wmtkemati»€he»  itnnihneh  der  aften  Aegyplrr  by  A.  Eisen- 
lohr,  seeond  edition,  Lei|»«ii;,  1891 ;  see  sIko  dntor,  clisp.  i ;  and 
A  Short  auioqf  tf  Orttk  JUmtkemttticM^  by  J,  Oow,  Cainbridi:«>,  1R84.  arts. 
IS— 14.  Benides  three  aathoritice  the  i«|ijnifl  has  been  diseasved  ia 
by  h.  Bodel,  A.  Favaio,  V.  Bobynin,  and  E.  Wcyr. 

1—2 


Vpn 


J 


4  EOTFTIAir  AND  PHOENICIAN  MATHDUTICBL 

In  all  the  ezamplet  n  it  low  than  50.  Pinobablj  ha  had  no 
rule  for  forming  the  component  fractions,  and  the  antwert 
given  reprefient  the  accumulated  experiences  of  previous 
writers :  in  one  solitary  case  however  he  has  indicated  his 
method,  for,  after  having  asserted  tliat  |  is  the  sum  of  | 
and  I,  he  adds  that  therefore  two- thirds  of  one-fifth  is  equal 
to  the  sum  of  a  half  of  a  fifth  and  a  sixth  of  a  fifth,  that  is^ 

That  so  much  attention  should  have  been  paid  to  fractions 
may  be  explained  by  the  fact  that  in  early  times  their  treat- 
ment preiiented  great  difficulty.  The  Egyptians  and  Greeks 
simplified  the  problem  by  reducing  a  fraction  to  the  sum  of 
several  fractious,  in  each  of  which  the  numerator  was  unity,  so 
that  they  had  to  consider  only  the  various  denominators :  the 
sole  exceptions  to  this  rule  bein<(  the  fractions  |  and  ] .  This 
mained  the  Ureek  practice  until  the  sixth  century  of  our 
era.  The  HomuiiH,  on  the  other  hand,  generally  kept  tlie 
.denominator  constant  and  ei|ual  to  twelve,  expressing  the 
fraction  (approximately)  as  ho  umny  twelfths.  The  Babylonians 
did  the  minie  in  aHtroaomy,  except  that  they  used  sixty  as  the 
constant  denominator;  and  fix)m  them  through  the  Greeks 
the  mftdern  division  of  a  degree  intii  sixty  equal  parts  is 
derived.  Thus  in  one  way  or  the  other!  the  difficulty  of 
having  to  consider  changes  in  both  nunjerator  and  deno- 
minator was  evaded.  ] 

After  considering  fractions  Ahmes  {Proceeds  to  some 
examples  of ,  the  fundamental  processes  of  arithmetic.  In 
ultiplication  he  seems  to  have  relied  on  repeated  additions. 
Thus  in  one  imnierical  example,  where  he  reiquires  to  multiply 
a  certain  numlier,  say  a,  by  13,  he  tirsi  multiplies  by  2 
and  gets  2<i,  then  lie  doubles  the  result  i^nd  gets  4a,  then 
he  again  doubles  the  result  and  gets  8a,  apd  lastly  he  adds  ( 

together  a,  4a,  and  Sa.  Probably  division  vjras  alito  performed 
by  repeated  subtractions,  but  as  he  rarely  explains  the  process 
by  which  he  arrived  at  a  result  this  is  nc|t  certain.  After 
these  examples  Almies  goes  on  to  the  solution  of  some  simple 


yAn 


EARLY  EOTPTIAN   MATHEMATICS.  5  ^ 

numerical  equationn.  For  example,  he  says  "heap,  its  se^-enth, 
its  whole,  it  makes  nineteen,"  by  which  he  means  that  the 
olject  is  to  find  a  number  such  that  the  sum  of  it  and  one- 
seventh  of  it  shall  lie  together  equal  to  19;  and  he  gives  as 
the  answer  16  +  |  -•-  ^t  which  is  correct. 

The  arithmetical  part  of  the  papyrus  imiicates  that  he  had 

yfeome  idea  of  algebraic  symbols.     The  unknown  quantity  is 

I  V always  represented   by   the  symbol    which   means  a   heap; 

addition  is  represented  by  a  pair  of  legs  waikitig  fori^ards, 

subtraction  by  a  pair  of  legs  walking  liack wards  or  by  a  flight 

of  arrows;  and  ec|uality  by  the  sign  ^. 

The  latter  part  of  the  lMM>k  contains  various  geometrical 
problems  to  which  I  allude  later.  He  concludes  the  work 
with  some  arithmetioo-algebraical  quefttitms,  two  of  which  deal 
with  arithmetical  prngressions  and  Kceni  to  indicate  that  he 
knew  how  to  sum  such  series. 

Hecond,  as  to  the  science  i »f  tj^om^trff.  Geometry  is  suppoHed 
to  have  had  itx  origin  in  land-survejring;  but  while  it  isdiflicult 
to  say  when  the  study  of  nunilM*rs  ami  calculation— some 
knowledge  of  which  is  e5isential  in  any  civiliznl  state — liccame 
a  science,  it  is  comparatively  eaMy  to  distingiiinh  lietween  the 
abstract  reasonings  of  geometry  and  the  practical  rules  of  the 
land-surveyor.  Some  methods  of  land-surveying  must  have 
been  practised  from  veiy  early  times,  but  the  universal 
tradition  of  antiquity  asserted  that  the  origin  of  geometiy 
was  to  be  sought  in  Egypt.  That  it  was  not  indigenous  to 
Greece  and  that  it  aniee  fn>m  the  necessity  of  surveying  is 
rendered  the  more  proliable  by  the  derivatitm  of  the  word 
from  y7  the  earth  ami  /act/km  1  mennnre.  Now  the  Greek 
igoometricians,  as  far  as  we  can  judge  by  their  extant  works, 
always  dealt  with  the  science  as  an  abstract  one :  they  sought 
for  theorems  which  should  be  kbsolutely  true,  and  (at  any 
rate  in  historical  times)  would  have  argued  that  to  measure 
quantities  in  terms  of  a  unit  whicn  might  have  been  inctmi- 
menattrable  with  some  of  the  magnitudes  considered  would 
haira  made  their  results  mere  approzim4*.^ioiM  to  the  truth. 


\ 


6  ■QTPTIAK  AND  PHOKNIOIAIT  MATHDUTIOHL 


The  DMiie  does  not  tliarefoi^  refer  to  their  ^nictioe.  li  k  «ot 
however  nnlikelj  that  it  indicstes  the  uae  which  wee  made  of 
geometry  among  the  Egyptiaiw  from  whom  the  Greeks  teamed 
it  Tliis  also  agrees  with  the  Greek  traditions^  which  in 
themselves  appear  probable;  for  Herodotus  states  that  the 
periodical  iiiundatious  of  the  Nile  (which  swept  away  the 
laiid-oiarks  in  the  valley  of  the  river,  and  by  altering  its 
oourae  increased  or  decreased  the  taxable  value  of  the  adjoin- 
ing lands)  rendered  a  tolerably  accurate  syatem  of  surveying 
incliKpciisable,  and  thus  led  to  a  systematic  study  of  the 
subject  by  the  prients. 

We  have  no  reason  to  think  that  any  special  aUention 
was  paid  to  geometry  by  the  Phoenicians,  or  other  neighbours 
of  the  Egj'ptians.  A  small  piece  of  evidence  which  tends  to 
shew  that  the  Jews  had  not  paid  much  attention  to  it  is 
to  be  found  in  the  mistake  mode  in  their  sacred  books*, 
where  it  is  stated  that  the  circumference  of  a  circli«  is  three 
times  its  diameter:  the  Babylonians f  also  reckoned  that  w 
was  equal  to  3. 

Assuming  then  that  a  knowledge  of  geometry  was  first 
derived  by  the  Greeks  from  Egypt,  we  must  next  discuss  the 
range  and  nature  of  Egyptian  geometry  |.  That  some  geo- 
metrical results  were  known  at  a  date  anterior  to  Ahmes's 
work  seems  clear  if  we  admit  (as  we  have  reason  to  do)  that, 
centuries  liefoi^K  it  was  written,  the  following  method  of 
obUiining  a  right  angle  was  used  in  laying  out  the  ground- 
plan  of  certiiin  buildings.  The  Kgyptians  were  very  particular 
about  the  exact  orientation  of  their  temples;  and  they  hod 
therefore  to  obtain  with  accuracy  a  north  and  south  line,  and 
also  an  east  and  west  line.  By  observing  the  points  on  the 
horizon  where  a  star  rose  and  set,  and  taking  a  plane  midway 
between  them,  they  could  obtain  a  north  and  south  line.     To  (' 

*  L  Kiiigi»t  chap,  vii,  versv  23,  sud  II.  Chronicle!*,  chap,  iv,  Yene  2. 
t  See  J.  Oppert,  Jtwruul  ABtaiiqur,  August,  187*1,  sod  October,  1871. 
X  See  EiMulohr;  Cantor,  chap,   u;  Oow,   srt*.  75,  76;  and  Dk 
Qeomttr'u  lUr  mlten  Aeyypter  by  £.  Weyr,  Vieoiis,  1884. 

I  i 


\ 
I" 


i 


/ 

r 


^ 


^ 


EARLY   GOrrriAN  GEOMETRY.  7 

get  an  east  And  west  line,  which  had  to  be  drawn  at  right 
angles  to  thi-s  certain  profesKional  "ropefaHteners'*  were 
employed.  The^e  men  used  a  n)[)e  .-I  /iCD  divided  l>y  knots  or 
marks  at  B  and  C  so  that  the  IcngthK  A/{^  //(7,  CD  were  in  the 
ratio  3:4:5.  The  len^i^h  HC  was  placed  along  the  north  and 
south  line,  and  pegs  P  and  Q  insert ed  at  the  knots  /?  and  C. 
The  piece  BJ  (keeping  it  stretched  all  the  time)  was  then 
rotated  round  the  peg  /*,  and  similarly  the  piece  CD  was 
rotated  round  the  peg  Q,  until  the  ends  A  and  D  coincided ; 
the  point  thus  indicate<l  was  marked  by  a  peg  Ji,  The  result 
was  to  form  a  triangle  /V^ A'  whose  siiles  /i7\  PQ^  QK  v;ere  in 
the  ratio  3:4:5.  The  angle  of  the  triangle  at  P  would  then 
be  a  right  angle,  and  the  line  /Vi'  ii^-ould  give  an  east  nnd  west 
line.  A  similar  metluid  is  ctmstantly  u.scd  at  the  present  time 
by  practical  engineers  f«>r  measuring  a  right  angle.  Tlie 
property  employed  can  be  do<lure<l  as  a  particular  case  of  Eua 
I,  48 :  and  there  is  n*as<m  to  think  that  the  Egyptians  were 
acquainted  with  the  results  of  this  pro|Kisiti«m  and  of  Euc. 
I,  47,  for  triangles  whose  sides  are  in  the  ratio  mentioned 
above.  They  must  also,  there  is  little  doubt,  have  known  that 
the  latter  proposition  was  true  for  an  isosct*les  riglit-angl<^ 
triangle,  as  this  is  obvious  if  a  floor  lie  paved  with  tiles  of  that 
shape.  But  though  these  are  interesting  facts  in  the  history 
of  the  Egyptian  arts  we  must  not  press  them  too  far  as 
shewing  that  geometry  was  then  studied  as  a  science.  Our 
real  knowledge  of  tlia-xiaturc  of  Egyptian  geometry  depends 
jnainly  ontheHhind  |Mipyrus.  '^ 

XHnies  commeu^es^tliat  part  of  his  papyrus  which  deals 
with  geometry  by  giving  some  numerical  instances  of  the 
oontents  of  bama  Unluckily  we  do  not  know  what  was  the 
usual  shape  of  an  Egyptian  bam,  but  where  it  is  defined  by 
three  linear  measurements^  say  ci,  6,  and  c,  the  answer  is 
always  given  as  if  he  had  formed  the  expression  a  x  A  x  (c-f  |c). 
He  next  proceeds  to  lind  the  areas  of  certain  rectilineal  figures ; 
if  the  text  be  oorrectly  interpreted,  some  of  these  results  are 
wrong.    He  then  goes  on  to  find  the  area  of  a  circular  field  of 


8  EOTFTIAM  AND  PHOENICIAN  MATHHUTICB. 

diameter  13 — no  iiiiit  of  length  being  mentioned— >  and  gives 
the  result  aa  (<l  -  IJ)',  where  d  it  the  diameter  of  the  eirele : 
thiB  it  equivalent  to  taking  3*1604  at  the  value  of  v,  the 
actual  value  being  very  approximately  3*1416.    liutly  Ahmet  j 

givet  tome  problemt  on  pyramidt.  These  long  proved  incapable 
of  interpretation,  but  Cantor  and  Eitenlohr  have  sliewn  that 
Ahmet  was  attempting  to  find,  by  means  of  data  obtained 
fr»m  the  measurement  of  the  external  dimentiont  of  a 
building,  tlie  ratio  of  certain  other  dimentiont  which  could 
not  be  dinnstly '  meaM^red :  hit  process  is  equivalent  to  de- 
teniiining  the  trigonometrical  ratios  of  certain  angles.  The 
data  and  the  results  given  agree  closely  with  tlie  dimensions 
of>M>uie  of  tlie  existing  pyramids.  I 

^/   It  is  noticeable  that  all  the  specimens  of  Egyptian  geo-  ^ 

metry  which  we  possess  deal  only  with  particular  numerical 
problems  and  nut  with  general  theorems ;  and  even  if  a  result 
be  stated  as  universally  true,  it  was  proliably  proved  to  be 
so  only  by  a  wide  induction.  We  slmll  see  later  that  Greek 
geometry  was  from  its  coninieiiceiiient  deductive.  There  are 
re;(Sons  fur  thinking  tliat  Egyptian  geometry  and  arithmetic 
mode  little  or  no  progress  subsequent  to  the  date  of  Ahmes's 
work :  and  though  fur  nearly  two  hundred  years  after  the 
time  of  Thales  Egypt  was  recognized  by  the  Greeks  as  an 
impi»rtant  school  of  mathematics,  it  would  seem  tlmt,  almost 
from  the  foundation  of  the  Ionian  school,  the  Greeks  out- 
stripped their  funiier  teachers. 

It  may  lie  added  that  AhiiieH's  book  gives  us  much  tlmt 
idea  of  Egyptian  niathi'nmtics  which  we  should  have  gathered 
from  statements  aUmt  it  by  various  Grm*k  and  Latin  authors, 
who  lived  centuries  later.  Previous  to  its  translation  it  was 
comnumly  thought  that  these  statements  exaggerated  the 
acquirements    of    the    Egyptians,   and    its    discovery    must  ) 

increase  the  weight  to  be  attached  to  the  testimony  of  these 

2horities. 
We  know  nothing  of  the  applied  matliematica  (if  thero 
re  any)  of  the  Egyptians  or  Phoeniciant.  The  attronomical 


EABLT  CUINESK  MATHEMATIOl 


9 


J 


I 


f 


I  < 
h 
$' 


k 


Altaininefits  cvf  the  E^ptians  and  Chaldaeans  were  no  doubt 
oonskimihle,  though  they  wore  chiefly  the  malts  of  obwr- 
▼ation :  tlie  Phoenicians  are  said  to  have  confined  tlieniBelves 
to  studying  what  wan  re(|uired  for  navigation.  Astronomy 
however  lies  outsiiie  the  range  of  this  Imok. 

«^  do  not  like  to  conclude  the  clia]>ter  without  a  brief 
mention  M  tlie  Chinese,  since  at  one  time  it  was  asserted  tliat 
they  were  familiar  with  the  sciences  of  arithmetic,  geometry, 
mechanics,  optics,  •  navigation,  and  astronomy  nearly  three 
thousand  years  ago,  and  a  few  writers  were  inclined  to 
suspect  (for  no  evidence  was  forthcoming)  that  some  know- 
ledge of  this  learning  had  filtered  across  Asia  to  th«(  West.  It 
is  true  that  at  a  very  ^rly  period  the  Chinese  were  acquainted 
with  several  geometrical  or  rather  architectural  implements, 
such  as  the  rule,  square,  compasses,  and  level ;  with  a  few 
mechanical  machines,  such  as  the  wheel  and  axle;  that  they 
knew  of  the  characteristic  property  of  the  magnetic  needle; 
and  were  aware  that  astnmomical  events  occurred  in  cycles. 
But  the  careful  investigations  of  L.  A.  Sedillot^  have  shewn 
that  the  Chinese  mmle  no  serious  attempt  to  classify  or  extend 
the  few  rules  of  arithmetic  or  geometry  with  which  they  were 
acquainted,  or  to  explain  the  causes  of  the  phenomena  which 
they  obsen'ed. 

The  idea  that  tlie  Chinese  had  made  considerable  progress 
in  theoretical  mathematics  seems  to  have  lieen  due  to  a 
misapprehension  of  the  Jesuit  missionaries  who  went  to  China 
in  the  sixteenth  centifry.  In  the  first  place  they  fsiled  to 
distinguish  between  the  original  science  of  the  Chinese  and 
the  views  which'  they  found  prevalent  on  their  arrival ;  the 
latter  being  founded  cm  the  work  and  teaching  of  Arab  or 
Hindoo  misiitmarieii  who  had  come  to  China  in  the  oourse  of 
the  thirteenth  century  or  later,  and  while  there  introduced  a 
knowledge  of  spherical  trigonometry.     In  the  second  plaoe^ 

*  See  Doocoiapsgni's  ItwUetino  di  hihHo$rafiii  t  di  $lorin  Mte  tcietue 
wmUmmtielu  €jMtke  Une  May,  1868,  vol.  i,  pp.  161—108.    On  Chinest 


10  lOaTFTIAir  AHO  PHOBNICIAH  IIATHSlUTIOiL 

finding  thftt  one  of  the  luoiit  imporUuit  governmeiii  depMi- 
menU  wrns  known  ms  the  Board  of  MAtheniatiei»  they  mppoeed 
that  ite  function  wm  to  promote  and  superintend  wwtheinaUcal 
studies  in  the  empire.  Its  duties  were  really  confined  to  the 
annual  preparation  of  an  almanack,  the  dates  and  predictaoos 
in  which  regulated  Qiany  afiiiirs  both  in  public  and  domestic 
life.  All  extant  specimens  of  these  almanacks  are'd^ective 
and  very  inaccurate. 

The  only  geometrical  theorem  with  whidi  we  can  be 
certain  that  the  ancient  OhineMO  were  acquainted  is  that  in 
certain  casntH  (niiuiely  when  the  ratio  of  the  sides  is  3  :  4  :  5 
or  1  :  1  :  ^/2)  Uie  area  of  the  square  described  on  the  hypo- 
tenu«e  of  a  right-angled  triangle  is  equal  to  the  sum  of  the 
areas  of  the  wiuares  deticrilieil  on  the  sides.  It  is  barely 
pusnible  tliat  a  few  geumetrical  tlieorenis  which  can  be 
demonstrated  in  tlie  quiuii-experiniental  way  of  super-position 
were  also  known  to  them.  Their  arithmetic  was  decimal 
in  notation,  but  their  knowledge  seems  to  have  been  con- 
fined to  the  art  of  calculation  by  means  of  the  swan-pan, 
and  the  power  of  expressing  the  results  in  writing.  Our 
acquaintance  with  the  early  attainments  of  the  Chinese,  slight 
though  it  is,  is  more  complete  than  in  tlie  case  of  most  of 
their  conteiiqioraries.  It  is  thus  specially  instructive,  and 
serves  to  illustrate  the  fact  that  a  nation  may  possess  con- 
siderable skill  in  the  applied  arts  while  they  are  ignorant  of 
the  sciences  on  which  those  arts  are  founded. 

From  the  f«»regoing  summary  it  will  be  seen  that  our 
knowledge  of  the  mathematical  attainments  of  those  who 
preceded  the  Greeks  is  very  limited ;  but  we  may  n-asonably 
infer  that  from  one  source  or  another  the  early  Greeks  learned 
the  use  of  the  aliacus  for  practical  calculations,  symbols  for 
recording  the  results,  and  as  much  mathematics  as  is  contained 
or  implied  in  the  lihiml  papyrus.  It  is  probable  tliat  this 
sums  up  their  indebtedness  to  other  races.  In  the  next  six 
chapters  I  shall  trace  the  development  of  mathematics  under 
Greek  influence. 


I 
I 


11 


. 


FIRST  PERIOD. 

iilatlKmatics  untei  CSmk  influence. 

nin  imriml  b^ins  with  th^  t^rhin^  of  Thnfeg,  nrr,  fiOO  H.C, 
nnd  en*h  trith  Ih^  cnpittr^  of  Afe.mwinn  h»f  thf.  MohmntitedoHn 
in  or  abotfi  OH  a.i».  7%^  chnmrfrriitfir  J^atiity!  of  lh%n  ftrrumi 
is  the  tktrffppment  of  geometry. 

It  will  be  romrmbrrcd  thnt  I  r«iinm(*nc«Hl  tho  Inst  cliapirr 
bj  mjing  that  the  hiNtorr  of  mathoiimtics  mi^ht  lie  dividc-d 
into  tliree  period^  namrly,  thnt  of  iiiatlieinAticH  ttnfK*r  Greok 
inflaetice,  that  of  the  matheiiiatica  of  tho  middle,  a^^oR  and  of 
the  renaissance,  and  lastly  that  of  modern  mathematics.  The 
next  four  chapters  (rhapters  ii,  ill,  iv,  and  v)  deal  with  the 
history  of  mathemntics  under  Greek  influence :  to  these  it  will 
be  oonvenient  to  aiM.one  (chapter  vi)  on  the  Byzantine  scIhioI, 
rince  through  it  the  results  of  Gn*ek  matliematics  were  trans- 
mitted to  western  Europe ;  and  another  (cliapter  vii)  on  the 
systems  of  numeration  which  were  ultimately  displaced  hy 
tiie  system  introduced  by  the  Araba.  I  shf»uld  add  that  many 
oC  the  dates  mentioned  in  these  chapters  are  not  known  with 
eertftinty  and  must  be  regarded  as  only  approximately  correct* 


i  ■ 


'\ 


13 


CHAPTER  It. 

THE  IONIAN   AND  PYTHAGOREAN  8CHOOL8' 

CIRC.  600  B.a — 400  B.c. 


V 


With  the  foundation  of  the  Icmian  unci  Pythagorean 
■chools  we  emerge  from  the  region  of  antiquarian  research  and 
eonjecture  into  the  light  of  higtonr.  The  materiaU  at  our  dis- 
posal for  estimating  the  knowledge  of  the  philosophers  of  these 
■chools  previous  to  about  the  jear  430  B.C.  are  however  very 
scanty.  Not  only  have  all  but  fraginenU  of  the  different 
mathematical  treatises  then  written  Ijeen  lost,  but  we  possess 
no  copies  of  the  histories  of  mathematics  written  about 
325  &a  by  Eudemus  (who  was  a  pupil  of  Aristotle)  and 
Theophrastus  respecti%'ely.  Luckily  Proclus,  who  about 
450  A.D.  wrote  a  commentary  on  Euclid's  EfemenU^  was 
familiar  with  the  history  of  Eudemus  and  gives  a  summary  of 
thai  pari  of  it  which  dealt  with  geometry.  We  have  also  a 
fmgmeni  of  the  General  Vuffo  of  .Vaihemaiicti  written  by 
Geminus  about  50  ac.,  in  which  the  methods  of  proof  used  by 
the  earlj  Greek  geometricians  are  compared  with  those  current 
ai  a  later  date.     In  addition  to  thiese  general  statements  we 

*  The  hisloiy  of  tbeM  pcbooU  is  diacusMd  by  Csotor,  cfasps.  ▼ — vm ; 
by  O.  J.  Allmsa  in  his  Oretk  Geomeirf  from  TlmU$  to  Kmelid,  Dnblio, 
I8a9 1  by  J.  Oow,  io  his  Greek  Iimihemniic§,  Combridgs,  1884 ;  by  C.  A. 
Bwtschntider  ia  his  Die  Oeometrie  mud  die  Geometer  tor  Emkleide$f 
Li^pB%.  1870;  and  partially  by  H.  Hanksl  ia  his  potthanMias  GeerMeUe 
4gr  Mmikewmtik,  Ltipsif,  1874. 


14  THE  lONIAH  AVO  PTTHAOOBIAII  flCHOOUk 

have  biogimphiet  of  a  few  of  the  lending  mathemAtieiMM^  and  ' 
some  acattered  notes  in  variooa  writers  in  which  aUnsions  are 
made  to  the  lives  and  works  of  others.    The  original  anthort* 
ties  are  criticised  and  discussed  at  length  in   the   works 
mentioned  in  the  footnote  to  the  heading  of  the  chapter. 

The  Ionian  School, 


1 


I 


y  Thalea^.  T\w  founder  of  the  earliest  Greek  school  of 
niathouiatioi  and  philosophy  was  Thaie^^  one  of  the  seven  sages 
of  Greece,  who  was  horn  ahout  640  &c.  at  Miletus  and  died  in 
the  same  town  aUiut  550  aa  The  materials  for  an  account  of 
his  life  consist  of  little  more  than  a  few  anecdotes  whidi  have 
been  handed  down  hy  tradition. 

During  the  eariy  part  of  his  life,  Thales  was  engaged  partly 
in  commerce  and  partly  in  public  afiairs ;  and  to  judge  by  two 
stories  that  have  been  preserved,  he  was  then  as  distinguished 
for  shrewdness  in  buniness  and  readiness  in  resource  ss  he  was 
subsequently  celebrated  in  science.  It  is  said  that,  once  when 
transporting  some  salt  which  was  loaded  on  mules,  one  of  the 
animals  slipping  in  a  stream  got  its  load  wet  and  so  caused 
some  of  the  salt  to  be  dinsolved,  finding  its  burden  thus 
lightened  it  rolled  over  at  the  next  ford  to  which  it  came ;  to 
break  it  of  this  trick  Thales  loaded  it  with  rags  and  sponges 
which,  by  absorbing  the  water,  made  tlie  load  heavier  and  soon 
effectually  cured  it  of  its  troublesome  habit.  At  another  time^ 
according  to  Aristotle,  when  there  was  a  prospect  of  an 
unusually  abundant  crop  of  olives  Thales  got  possession  of  all 
the  olive-preHst-s  of  the  district ;  and,  having  thus  "cornered" 
them,  he  was  able  to  make  his  own  terms  for  lending  tliem  out^ 
and  thus  realized  a  large  sum.  These  tales  may  lie  apocryphal, 
but  it  is  certain  tliat  he  must  have  liad  considerable  reputation  i 

as  a  man  of  affairs  and  as  a  good  engineer  since  he  was  em- 
ployed to  construct  an  embankment  so  as  to  divert  the  river 
Ualys  in  such  a  way  as  to  permit  of  the  construction  of  a  ford. 

*  See  Csutor,  chsp.  v ;  Alhnsn,  chap.  i. 


tHALES. 


15 


PkobaUj  it  was  lu  »  merchant  that  Thales  first  went  to 
'Egypt,  hut  during  his  leisui^  there  he  studied  astronomy  and 
geometry.  He  was  middle-aged  when  he  returned  to  Miletus ; 
he  seems  then  to  have  abandoned  business  and  public  life, 
and  to  have  devoted  himself  to  the  study  of  philosophy  and 
science— subjects  which  in  the  Ionian,  Pythagoronn,  and 
perhaps  also  the  Athenian  schools,  were  closely  connected : 
his  views  on  philosophy  do  not  here  concern  us.  He  continued 
to  live  at  Miletus  till  his  death  circ  550  ac, 

Wn  ounnot  form  nny  oxAot  idM  nn  to  how  Xhg[w  pmsentiHl 
Mm  gMNBetrienl  tMching t  w«  Infer  hownvpr  (rotnTReluii  tliiit 

it  consisted  of  a  numlier  of  isolated  propOf»itions  whi<^h  were 
not  arr&nged  in  a  logical  sequence,  but  that  the  proofs  were 
deductive,  so  that  the  theorems  were  not  a  mere  statement 
of  an  induction  from  a  large  numlier  of  special  instances, 
as  probably  was  the  case  with  tlie  Egyptian  gcHNnetricians. 
The  deductive  character  which  he  thus  gave  to  the  science 
is  his  chief  claim  to  distinction. 

The  following  comprise  the  chief  propositions  that  can 
DOW  with  reasonable  probability  lie  attributed  to  him. 

(i)  The  angles  at  the  Imse  of  an  isosceles  triangle  are 
et|iial  (fiaSr%'  5).  Proclus  seems  to  imply  that  this  was 
proved  by  taking  another  exactly  equal  isosceles  triangle, 
taming  it  over,  and  then  superposing  it  on  the  first ;  a  sort 
of  experimental  demonstration. 

(ii)  If  two  straight  lines  cut  one  another,  the  vertically 
opposite^  angles  are  equal  (Euc.  i,  15).  Thales  may  have 
regarded  this  as  obvious,  for  Proclus  adds  that  Euclid  was  the 
first  to  give  a  strict  proof  of  it. 

(iii)^  A  triangle  is  determined  if  its  base  and  base  angles 
be  given  (^  Euc  i,  26).  Apparently  this  was  applied  to  find 
the  distance  of  a  ship  at  sea ;  the  base  being  a  tower,  and  the 
base  angles  being  obtained  by  observation. 

(iT)^Tbende8  of  equiangular  triangles  are  proportionals 
(Boo.  TiT^  or  perfaapa  rather  Euc  ti,  2).  This  is  said  to 
httvo  beoi  Med  by  Thalea  when  in  Egypt  to  find  the  height  of 


16 


THE  IONIAN  AND  PTTHAOOREAV  HSHOOIA 


m  pjnuDid.  In  m  dialogue  given  bj  PlaUivfa,  the 
addrening  Thalet  sajt  **  placing  your  etielf  at  the  end  of 
the  shadow  of  the  p/ramid,  you  made  hj  the  ran't  njt  two 
triangleiii  and  so  proved  that  the  [height  of  tne]  pjimmid  waa 
to  the  [length  of  the]  stick  as  the  shadow  of  the  pyramid  to 
the  shaclow  of  the  stick."  Tlie  king  Amasisi  who  was  present^ 
is  said  to  have  been  aniased  at  this  application  of  abstract 
science,  and  the  Egyptians  seem  to  have  been  previously  unao> 
quainted  with  the  theorem. 

(v)  A  circle  is  bisected  by  any  diameter.  I  This  may  have 
been  enunciated  by  Thales,  but  it  must  have  been  recognised 
as  an  obvious  fact  from  the  earliest  times. 

(vi)  The  angle  subtended  by  a  diameter  of  a  circle  at  any 
point  in  the  circuuiference  is  a  right  angle  (Euc.  ill,  31). 
This  appears  to  have  been  regarded  as  the  most  remarkable 
of  the  geometrical  achievements  of  Thales,  and  it  is  stated 
that  on  inscribing  a  right-angled  triangle  in  a  circle  he  sacri- 
ficed an  ox  to  the  iiii mortal  guds.  It  is  sup|i06ed  that  he 
proved  the  proposition  by  joining  the  centre  of  the  circle  to 
tlie  apex  of  the  right  angle,  thus  nplitting  the  triangle  into  two 
imnsceles  triangles,  and  then  applied  the  proposition  (i)  above : 
if  this  be  the  correct  account  of  his  pruof,  he  must  have  been 
aware  that  the  sum  of  the  angles  of  a  right-angled  triangle 
is  equal  to  two  right  angles. 

It  has  been  ingeniously  suggested  that  the  shape  of  the 
tiles  used  in  paving  doors  may  have  afforded  an  experimental 
demonstration  of  the  latter  result,  namely,  that  the  sum  of 
the  angles  of  a  triangle  is  e<|ual  to  two  right  angles^  We 
know  from  Eudenms  that  the  first  geometers  proved  the 
general  property  separately  for  tlin*e  s|)ecies  of  triangles,  and 
it  is  not  unlikely  that  they  proved  it  thus.  The  area  about  a 
point  can  be  filled  by  the  angles  of  six  equilateral  triangles  or 
tiles,  hence  the  proposition  is  true  for  an  equilateral  triangle. 
Again  any  two  equal  right-angled  ti-iangles  can  lie  placed  in 
juxtaposition  so  as  to  fonu  a  rectangle,  the  sum  of  whose 
angles  is  four  right  angles ;  hence  the  proposition  is  true  for  a 


.> 


1 


A." 


THALES. 


^^eighi-Migled  triangle :  mkI  it  will  be  noticed  tliat  tiles  of  n 
shape  woaM  give  an  ocular  demonstration  of  thi^  case. 
iM  appear  that  this  proof  was  given  at  fint  onlj  in  i 
of  ieoeoelet  right-angled  triangles,  hat  probahl/  it  i 
Ltended  Uler  so  as  to  cover  anj  right*angl<*d  triangle.     Las 
ly  triangle  can  bo  split  into  the  sum  of  two  right-ang 
bj  drawing  a  perpendicular  from  the  biggest  an 
the  opposite  side,  and  therefore  again  the  proposition 
The  first  of  these  proofs  is  evidently  included  in  i 
bat  there  is  nothing  improbable  in  the  suggestion  t1 
eariy  Greek  geometers  conti'.ued  to  teach  the  first  prop 
'^km  in  the  form  above  given. 

Tliales  wrote  an  astronomy,  and  among  his  contemporai 
"was  more  famous  as  an  aHtnmonier  than  as  a  geometrician, 
is  said  that  one  night,  when  walking  out,  he  was  looking 
hitently  at  the  stars  that  he  tumbled  into  a  ditch,  on  which 
okl  woman  exclaimed  *'  How  can  you  tell  what  is  going  on 
the  sky  when  y<ni  can*t  see  what  is  lying  at  your  own  feet 
— aa  anecdote  which  wax  often  quoted  to  illustrate  the  i 
practical  character  of  philosophers. 

Without  going  into  astronomical  details  it  may  be  mentioi 
that  he  taught  that  a  year  crTitained  about  3G5  days,  and  i 
(as  is  said  to  have  been  previously  reckoned)  twelve  montlu 
thirty  days  each.  It  is  said  that  his  predeceftsors  occasions 
intercalated  a  month  to  keep  the  seasons  in  tlieir  customi 
plaoesi  and  if  so  they  must  have  realized  that  the  year  contai 
en  the  average,  more  than  360  days.  According  to  so 
leoent  critics  he  believed  the  earth  to  be  a  disc,  but  it  sec 
to  be  more  probable  that  he  was  aware  that  it  was  spheric 
He  exphuned  the  causes  of  the  ecli|ises  both  of  the  sun  a 
■Mon,  and  it  is  well  known  that  he  predicted  a  solar  ecli 
which  took  ph^e  at  or  about  the  time  he  foretcM :  the  act 
dale  was  Hay  28^  585  B.C.  But  though  this  prophecy  a 
its  fvlfibneni  gave  extraordinaiy  prestige  to  his  teaching,  a 
ssesred  hisn  the  name  of  one  oi  the  seven  sages  of  Greece 
is  aMSl  UUtf  that  be  only  made  ase  of  one  of  the  Egyptian 

a  a 


18  THS  lONUM  AVO  PTTHAOOBSAV  8CB00U1 

i 

ChaMaaan  reg[istera  frhieh  aUlad  that  aolsr  adipiM 
intenrmb  of  18  yean  11  dajib 

AmcMig  the  pupils  of  Thalet  were  All>¥linMullir,  1I»- 
meronik  and  Mandryatiu.  Of  the  two  mentioDed  fast  wo 
koow  next  to  nothing.  Anaxiwumder  la  hotter  known;  ho 
was  bom  in  611  B^a  and  died  in  545  E.a,  and  ■occeeded  ^ 

Thales  as  head  of  the  school  at  Miletus.  Aocording  to  Suidas 
he  wrote  a  treatise  on  geometry  in  which  tradition  sajs  he 
paid  particular  attention  to  the  properties  of  spheresi  and 
dwelt  at  length  on  the  philosophies)  ideas  involi-ed  in  the 
conception  of  infinity  in  space  and  time.  He  constmcted 
terrestrial  and  celestial  globes. 

Anaximander  is  alleged  to  have  introduced  the  use  of  the 
siffle  or  gtwtt^n  into  Greece.  ThiS|  in  principle,  consisted  only 
of  a  stick  stuck  upright  in  a  horisontal  piece  of  ground.  It 
was  originally  used  as  a  sun-dial,  in  which  case  it  was  plaoed 
at  tlie  centre  of  three  concentric  circles  so  that  every  two 
hours  the  end  of  its  shadow  passed  from  one  circle  to  another. 
8uch  sun-dials  have  been  found  at  Pompeii  and  Tusculum.  It 
IB  said  that  he  employed  these  styles  to  determine  his  meridian 
(presumably  by  marking  the  lines  of  shadow  cast  by  the  stylo 
at  sunrise  and  sunset  on  the  same  day,  and  taking  the  plane 
bisecting  the  angle  so  formed) ;  and  thence,  by  observing  the 
time  of  year  when  the  noon -altitude  of  the  sun  was  greatest 
and  least,  he  got  the  solstices ;  thence,  by  taking  half  the  sum 
of  tlie  noon-altitudes  of  the  sun  at  the  two  solstices,  he  found 
the  inclination  of  the  equator  to  the  horizon  (which  determined 
the  latitude  of  the  place),  and,  by  taking  half  their  difference, 
he  found  the  inclination  of  the  ecliptic  to  the  equator.  There 
seems  guod  reason  to  think  that  he  did  actually  determine  the 
latitude  of  Sparta,  but  it  is  more  doubtful  whether  he  really 
made  the  rest  of  these  astronomical  deductions. 

We  need  not  here  concern  oursch'es  further  with  the 
successors  of  Thales.  The  school  he  established  continued  to 
flourish  till  about  400  B.C.,  but,  as  time  went  on,  its  members 
occupied  themselves  more  and  more  with  philosophy  and  less 


PTTHAOORAS. 


19 


II 


mathematics.  We  know  very  little  of  the  mathematicians 
comprised  in  it^  but  they  would  seem  to  have  devoted  most  of 
their  attention  to  astronomy.  Iliey  exercised  hot  slight  in- 
fluence on  the  farther  advance  of  Greek  mathematics,  which 
was  made  almost  entirely  under  the  influence  of  the  Pythago- 
reanSi  who  not  onljr  immensely  fleveloped  the  science  of 
geometry  but  created  a  science  of  numbers.  If  Thales  was  the 
first  to  direct  general  attention  to  geometry,  it  was  I^hagoras, 
says  Proclus,  f|Uoting  from  Eudemus,  who  '*  changed  the  study 
of  geometry  into  the  form  of  a  liberal  education,  for  he  ex- 
amined its  principles  to  the  bottom  and  investigated  its  theo- 
rems in  an... intellectual  manner":  and  it  is  accordingly  to 
^rthagoras  that  we  must  now  direct  attention.  * 


The  Pythagorean  School, 

^Pythagoras  ^.  Pythagoras  was  bom  at  Samos  about 
569  ac.,  perhaps  of  Tynan  parents,  and  died  in  500  B.a  He 
was  thus  a  contemporary  of  Thales.  The  details  of  his  life  are 
somewhat  doubtful,  but  the  following  account  is  I  think  sub- 
stantially correct.  He  studied  lirst  under  Plierecydes  of  Syros, 
and  then  under  Anaximshder ;  by  the  latter  he  was  recom- 
mended to  go  to  Thrives,  and  there  or  at  Memphis  he  spent 
some  years.  After  leaving  Egypt  he  travelled  in  Asia  Minor, 
and  then  settled  at  Samoe,  where  he  gave  lectures  but  without 
much  success.  About  529  b.c.  he  migrated  to  Sicily  with  his 
mother,  and  with  a  single  disciple  who  seems  to  have  been  the 
sole  fruit  of  his  labours  at  Samos.  Thence  he  went  to  Tarcntum, 
but  very  shortly  moved  to  Croton,  a  Dorian  colony  in  the  south 
of  Italy.  Here  th^  schools  that  he  opened  were  crowded  with 
ao  eotfiusiastio  audience;  citizens  of  all  ranks,  especially  those 

*  8cs  Cantor,  ehapi.  vi,  vn;  AUmsn,  chap,  n ;  Haukel,  pp.  9S— 111; 
Hoefer,  BiBioirt  det  wuihitMiiqw^  Pftriw,  Diird  editioD,  1886,  pp. 
87 — 180 ;  sad  vsrioiw  papers  bj  8.  P.  TAoneiy.  For  sa  soeooot  of  Fytha- 
forss's  lifi^  smlodjing  the  ^Ihsgonaa  trsditkms,  ses  the  hlognifhj  bj 
IssiMIAm,  8l  wiiiA  Ihsir  srs  Iwe  or  Ibiss  Bagiidi  trsnslstions. 


so  TUK  lOXUV  AMD  PTtHAOOBIAK  flOOOOIA 


I 


of  the  upper  cUiiaet,  attended,  and  tfeii  the  women  bitilw*kw 
which  forbade  their  going  to  publio  meetings  aiid  ilodced  to  I 

hear  him,  Amongiit  his  moat  attentive  aoditors  was  Theano^ 
the  young  and  beautiful  daughter  of  his  host  Milo,  whoa^  in 
spite  of  the  dinparity  of  their  ages,  he  married :  she  wrote  a 
biography  of  her  hunband  but  unfortunately  it  is  lost. 

Pytliagoras  was  really  a  philosopher  and  moralist  of  a 
religious  and  somewhat  ascetic  type,  Imt  his  philosophical 
and  ethical  teaching  were  preceded  by  and  founded  on  a 
study  of  matheuiatics.  He  divided  tliose  who  attended  his 
lectures  into  two  classes,  the  Ivtieiten  and  the  madkenuUieiatu. 
In  general,  a  listener  after  pausing  three  years  ai«  such  could 
be  initiated  into  the  second  class,  to  whom  alone  were  con- 
fided the  chief  discoveries  of  the  school.  Following  the 
modern  usage  I  confine  the  use  of  the  word  Pythagoreans 
to  the  latter  class. 

The  Pythagoreans  formed  a  brotherhood  with  all  things  in 
common,  holding  the  same  philosophical  beliefs,  engaged  in  the 
same  pursuits,  and  bound  by  oath  not  to  reveal  the  teaching  or 
secrets  of  the  schooL  Their  food  was  simple ;  tlieir  discipline 
severe ;  and  their  mode  of  life  arranged  to  encourage  self-com- 
mand, temperance,  purity,  and  obedienca  They  rose  before 
the  sun,  and  began  by  recalling  the  events  of  the  preceding 
day,  next  they  made  a  plan  for  the  day  tlien  commencing, 
and  finally  on  retiring  to  rest  they  were  expected  to  compare 
their  performiiiices  with  this  filaiL 

One  of  the  symbols  which  they  used  for  purposes  of  recog- 
nition was  the  pentagram,  sometimes  also  called  the  triple 
triangle — ro  rpirXow  rpiymtto^.  The  pentagram  is  merely  a 
regular  re-entrant  pentagon;  it  was  considered  symbolical  of 
health,  and  probably  the  angles  were  denoted  by  the  letters  of 
the  word  vyuia,  the  diphthong  ci  being  replaced  by  a  0,  it  will  ' 

be  noticed  that  the  figure  consists  of  a  single  broken  line^,  a 
feature  to  which  a  certain  mystical  im|iortance  was  attached. 

*  Oa  ilk)  tiieui>  ut  vuch  figures,  vee  liijr  Matkematicul  ReenttUtut 
ami  ProMemi,  Loodon,  third  edition,  IBSId,  chi«pi  ti. 


f 


PTTHAOOBAS. 


t  wboiB  w9  owe  the  ducloanra  ol  this  lymbolt 
tdli  w  bow  «  eertain  Pfthnjir'WMUi,  when  trnvelltDg^  Ml  ill 
•I » iMdHkb  ina  where  be  had  pat  ap  for  tbe  ni^i;  ha  wu 


poor  end  laek,  hot  the  Undlonl,  who  was  «  kjndhnirUd  fiellow, 
Boned  him  cnrefolly  and  spared  no  trouble  or  expense  to  re- 
liere  hin  p«im.  Howerer,  in  spite  of  all  elTorte,  the  Btadent 
got  worae ;  feeling  that  he  was  djritif;  mid  unable  to  make  the 
landkml  kuj  pecuniary  recoinpeniv,  he  ankml  for  a  bcMrd  on 
which  he  inscrilied  thr.  peiitngrnm-Hlar;  thin  he  gave  to  his 
host,  begging  him  to  hang  it  up  oatoiHe  so  tluit  all  piuseni-by 
might  Bee  it,  and  aasuring  him  that  the  resnlt  would  recom- 
penre  him  for  his  charity.  Tlie  scholar  died  and  was  honoar- 
aUjr  baried,  and  the  Inanl  was  duly  exjKMed.  After  a 
eoosidenUe  lime  had  elapsed,  a  traTeller  one  day  riding  by 
■aw  tho  Hcied  symbol  i  dismounting,  lie  entered  the  inn,  and 
after  bMuing  tbe  suiiy,  handsomely  rtmunemted  the  landlord. 
Sndi  is  the  «aecdot«,  which  if  not  true  is  at  least  appueite. 

Tba  B^iority  of  those  who  attended  the  lectures  of  Pythft- 
goraa  were  only  "IJMteni'rs";  but  lim  pliilf.M.i))1iy  was  intended  lo 
colour  the  whole  life,  piitittcal  and  Hocinl,  of  all  his  follnwen. 
In  advocating  self-conlnil.  {.itvprnment  by  tlie  1«st  men  in  tha 
slate,  strict  olirdieuce  to  Ic^lly  constituteil  aothoritte^  and  an 
o  vtanwl  priaeiples  of  right  and  wrong,  be  rapmeoted 


2S     THS  lONIAM  AVO  Ff THAOOBIAM  BOHOOLa 


A  view  of  society  toUU j  opposed  to  thai  of  the 
IMurtjof  that  time,  and  thus  natorallj  moat  of  the  htotherfaood 
were  aristocrata.  It  had  affiliated  meiiiben  in  manj  of  the 
neighbooring  cities,  and  its  method  of  organisation  rad  striet 
discipline  gave  it  great  political  power;  but  like  all  seciet 
societicM  it  was  an  object  of  suspicion  to  those  who  did  not  \ 

belong  to  it.     For  a  short  time  the  Pythagoreans  triumphed,  j 

but  a  popular  revolt  in  501  b.c.  overturned  the  civil  govern*  « 

meat,  aiul  in  the  riots  that  accompanied  the  insurrection  the 
mob  burnt  the  liouMe  of  Milo  (where  tlie  students  lived)  and  ^ 

killed  many  of  the  mont  pruiiiineut  members  of  the  schooL 
Pytbogorazi  himself  escaped  to  Tareutum,  and  thence  fled  to 
Metapontum,  where  he  was  murdered  in  another  popular 
outbreak  in  500  ac. 

Tliough  tlie  Pythagoreans  as  a  political  society  were  thus 
rudely  broken  up  and  deprived  of  their  head,  they  seem  to 
have  re-established  tbsnuielves  at  once  as  a  philosophical  and 
mathematical  society,  having  Tarentum  as  their  head-quarters. 
They  continued  to  flourish  for  a  hundred  or  a  hundred  and 
fifty  years  after  the  death  of  their  founder,  but  to  the  end 
they  remained  a  secret  society,  and  we  are  therefore  ignorant 
of  the  details  of  their  history. 

Pythagoras  himself  did  not  allow  the  use  of  text-bookSi 
and  the  assumption  of  his  school  was  not  only  that  all  their 
knowledge  was  held  in  common  and  veiled  fruni  the  outside 
world,  but  that  the  glory  of  any  fresh  discovery  must  he 
referred  back  to  their  founder :  thus  Uippasus  (circ  470  ao.) 
is  said  to  have  been  drowned  for  violating  his  oath  by  publicly 
boasting  that  he  had  added  the  dodecahedron  to  the  number 
of  regular  sulids  enumerated  by  Pythagoras.  Gradually,  aa 
the  society  became  more  scattered,  the  rule  was  abandoned, 
and  treatises  containing  the  substance  of  their  teaching  and 
doctrines  were  written.  The  first  book  of  the  kind  was  com* 
piMed,  aliout  370  ac,  by  Philolaus,  and  we  are  told  tliat  Plato 
secureil  a  copy  of  it.  We  may  say  that  during  ti«e  early  pari 
of  the  fifth  century  before  Christ  the  Pythagoreans  were  con- 


< 


PTTIlAGORAa. 


23 


I 


I 


i  #: 


tideimbljr  in  advance  of  their  contetiiporarien,  but  hy  the  end 
of  that  time  their  more  prominent  di5;coverie8  and  doctrines 
had  become  known  to  the  outHide  world,  and  the  centre  of 
intellectnal  activity  wan  transferred  to  Athemi. 

Though  it  is  impossible  to  separate  precisely  the  discoveries 
of  Pythagoras  himself  from  those  of  his  school  of  a  later  date, 
we  know  from  Proclas  that  it ,  was  Pytbap^oras  who  gave 
geometry  that  rigorous  character  of  de<luction  which  it  still 
bears,  and  made  it  the  foundation  of  a  liberal  education  ;  and 
there  is  reason  to  believe  that  he  was  the  first  to  arrange  the 
leading  propositions  of  the  subject  in  a  logical  order.  It  was 
also^  according  to  Aristoxenus,  the  glory  of  his  school  that 
they  raised  arithmetic  above  the  needs  of  merchants.  It  was 
their  boast  that  they  sought  knowledge  and. not  wealth,  or 
in  the  hmguage  of  one  of  their  maxims,  "a  figure  and  a  step 
forwards,  not  a  figure  to  gain  three  oboli." 

Pythagoras  was  primarily  a  moral  reformer  and  philo- 
•opher,  bat  his  system  of  morality  and  philonophy  was  built 
on  a  mathematical  foundation.  In  geometry  he  himself  pro- 
bably knew  and  taught  the  sul>stance  of  what  is  contained 
in  the  first  two  books  of  Euclid,  and  was  acquainted  with 
a  few  other  isolated  theorems  including  some  elementary  prr>- 
poaitions  on  irrational  magnitudes  (while  his  successors  added 
•everal  of  the  propositions  in  the  sixth  and  eleventh  Ixioks  of 
Enclid);  but  it  is  thought  that  many  of  his  proofs  were  not 
ligoitNMi  and  in  particular  that  the  converse  of  a  theorem 
was  aoaaeliiiies  assumed  without  a  proof.  What  philosophical 
joetrinea  were  based  on  these  geometrical  results  is  now  only 
cC  eoq)eot«re.  In  the  theory  of  numbers  he  was 
vilh  lo«r  difllNvnt  kinds  of  proUems  which  dealt 
vilh  polygonal  nambenii  ratio  and  proportion, 
and  nambers  in  series ;  but  many  of 
aad  in  particular  the  questions  on 
invopotiioiif  weretieated  by  geometrical 


Mi  was 


tial  to  the  aocvrate 


I 

I 

r 


24  THE  lONUN  AVO  PTTHAflOBKAM 


definitioii  of  fom  Pythagoras  thoaghi  thai  it  waa  alao  lo 

some  extent  the  cause  of  form,  and  be  therelune  taoghl  that 

the  foundation  of  the  theory  of  the  onivene  waa  to  be  IoomI 

ill  the  science  of  numbers.     He  waa  atwmgthened  in  this  '** 

opinion  by  discovering  that  the  note  sounded  by  a  Tibratuig 

string  depended  (other  things  being  the  same)  only  oo  the' 

length  of  the  string,  and  in  particukr  that  the  lengths  whidi 

gave  a  note,  itM  fifth,  and  its  octave  were  in  the  ratio  6:4:9^ 

fonuing  terms  in  a  musical  progression.     This  may  have  been 

tlie  reason  why  music  occupied  so  prominent  a  position  in  the 

exercises  of  his  school.     He  also  believed  that  the  distancfs 

fnmi  the  earth  of  the  astrological  planets  were  in  musical 

progression,  and  that  the  heavenly  bodies  in  their  motion  « 

thruugh  space  gave  out  harmonious  sounds :  hence  the  phrase 

"the  harmony  of  the  spheres."     if,  as  has  been  suggested,  he 

was  acquainted  with  the  fundamental  facts  of  crystallography 

he  must  have  regarded  them  as  still  further  confirming  these 

views. 

Taking  the  science  of  numbers  as  the  foundation  of  his 
philosophy  he  weut  on  to  attribute  properties  to  numbers 
and  geometrical  figures :  for  example  the  cause  of  colour  waa 
the  number  five;  the  origin  of  fire  was  to  be  found  in  the 
pyramid ;  a  solid  body  was  analogous  to  the  tetrad,  which 
represented  matter  as  composed  of  the  four  primary  elements^ 
fire,  air,  earth,  and  water;  and  so  on.  The  tetrad  like  the 
pentagram  was  a  sacred  symbol,  and  the  initiate's  oath  ran 
rai  fUL  Toy  a^cr^p^  ^X^  vapa&^*^a  rrrpocrvv  voyoy  itmrntm 
^Mrc«iK 

Tlie  Pythagoreans  began  by  dividing  the  mathematioal 
subjtH^lH  with  which  they  dealt  into  four  divisions:  numben 
alMolute  or  arithmetic,  numbers  applied  or  music,  magnitudes 
at  rent  or  geometry,  and  magnitudes  in  motion  or  astroooaqf!. 
This  **  qumlrivium  "  was  long  considered  as  constituting  tlm 
nectnisary  and  sudicicnt  course  of  study  for  a  liberal  edneatioB. 
Even  in  the  cose  of  geometry  and  arithmetie  (wliieh  wp 
founded  on  inferences  unconsciously  made  and  comMoa  la 


1 

I 


PTTHAOOKEAN  GEOMETRY. 


25 


I 
1 


men)  the  Fythngorean  presentation  was  involved  with  philo- 
sophy; and  there  is  no  doubt  that  their  teaching  of  tlie  sciences 
of  astronomy,  mechanics,  and  music  (which  can  rest  safely 
only  on  the  results  of  conscious  oliservation  and  experiment) 
was  intermingled  with  roetaphyHics  even  more  closely.  With 
the  philosophical  views  of  Pythagoras  I  neefl  not  concern 
myself  further,  nor  should  I  have  slluded  to  them  were  it  not 
that  the  Pythagorean  tradition  of  the  connection  lx*tween 
them  and  mathematics  confirmed  the  unfortunate  tendency  of 
the  Greeks  to  found  the  study  of  nature  on  philosophical 
conjectures  and  not  on  experimental  observations.  Of  the 
Pythagorean  researches  on  the  applied  subjects  of  the  quad- 
rivium  we  know  little,  and  I  here  confine  myself  to  describing 
their  treatment  of  geometry  and  arithmetic. 

First,  as  to  their  geometry.  We  are  of  course  unable  to 
reproduce  the  whole  body  of  Pythagorean  teaching  on  this 
•abject,  but  we  gather  from  the  notes  of  Froclus  on  Euclid 
and  from  a  few  stray  remarks  in  other  writers  that  it 
included  the  following  propositions,  most  of  which  are  on 
the  geometry  of  areas. 

(i)  It  commenced  with  a  number  of  definitions,  which 
probNbly  were  rather  statements  connecting  mathematical 
ideas  with  philosophy  than  explanations  of  the  terms  used. 
One  has  been  preserved  in  the  definition  of  a  point  as  unity 
having  position. 

(ii)  The  sum  of  the  angles  of  a  triangle  was  shewn  to 
be  equal  to  two  riglit  angles  (Euc  i,  32) ;  and  in  the  proof, 
which  has  been  preserved,  the  results  of  the  prepositional  Euc 
I,  13  and  the  first  part  of  Kuc.  i,  29  are  quoted.  The  demon- 
•timtion  is  substantially  the  same  as  tluit  in  Euclid,  and  it 
is  most  likely  that  the  proofs  there  given  of  the  two  propo- 
ntions  kst  mentioned  are  also  due  to  Pythagoras  himself. 

(iii)  Pythagoras  certainly  proved  the  propeKies  of  right- 
angled  triangles  which  are  given  in  Euc  I,  47  and  i,  48.  We 
know  that  the  proofs  of  these  propositions  which  are  found 
ia  Bodid  were  ct  Euclid's  own  invention ;  and  a  good  deal  of 


S6  THE  lOKUN  AMD  PTTBAOOKKAII  I  COOOUl 

cnriodtj  bu  been  «xoii«d  to  disoorcr  vbat  ««•  Um  dtmtm- 
■tntitm  which  wu  origiully  offered  hj  ^rthago>*>  o*  *^  "'■^ 
of  thew  theorems.  Not  improlMbly  it  dmj  ^mts  been  one  ot 
the  two  following* 

(«)  Anj  anvmn  ABCD  can  be  split  up,  ea  In  Kno.  u,  4, 
into  two  iquuvs  SK  and  l)K  and  two  equfi  rectanglee  AK 
and  CA' :  that  ia,  it  is  et|aal  to  the  etjnan  on'  FK,  tho  ■qnara 


on  KK,  and  (oar  times  (he  triangle  A  KF.  Kit,  if  poiate  be 
taken.  (7  on  BC,  11  on  CH,  and  f  on  /JJ,  ap  that  AO,  Ci7, 
and  /)£  are  each  equal  to  AF,  it  can  b«  cadly  iliewn 
that  EFQH  is  a  square,  and  that  the  triangles  AEF,  BFG, 
VQU,  and  DUE  are  equal:  thus  the  square  ABCD  is  also 
equal  to  the  squw«  on  EF^aA  four  times  the,  triangle  AEE. 
Hence  the  square  on  EF  ia  equal  to  the  snm  of  the  squares  on 
FK  and  EK. 

(;8)  Let  ABC  bo  a  right-angled  triangle,  A  being  the  right 
angle.  Draw  A0  perpendicular  to  BC.  llie  'triangles  ABC 
and  DBA  aro  similar,  ! 

.-.  BC  :  AB  =  AB  :  BD. 

'  K  collceUon  of  over  Ihirij  prouf<  of  Euo.  t,  47  was  pablbdwd  in  Drr 
Pflkatoriirk*  LtknaU  bj  Job.  Jos.  Ign.  l~ 
Main*,  lUl. 


PTTHAGORKAN  GEOXETRT. 


27 


KmilMrlj 


BC  :  AC^AC  :  DC. 
AR^^AC^^  nC(HD  ^DC)^ BC\ 

A 


This  proof  requires  a  knowlocl^  of  the  rpsulU  of  Euc.  ii,  2, 
Ti,  4,  and  ti,  17,  with  all  of  which  Pytha;i;oras  was  acquainted. 

(iv)  Pythagoras  is  also  creditod  with  the  discovery  of  tlie 
theorems  Euc  i,  44  and  i,  45,  and  with  giving  a  solution  of 
the  problem  Euc  ii,  1 4.  It  is  said  that  on  the  discovery  of 
the  neccAsary  construction  for  the  problem  la^t  mentioned  he 
sacrificed  an  ox,  but  as  his  school  had  all  things  in  common 
the  liberality  was  less  striking  than  it  seems  at  first.  The 
Pythagoreans  of  a  later  date  were  aware  of  the  extension 
given  in  Euc  vi,  25,  and  Allnian  thinks  that  Pjrthagoras 
himself  was  acquainted  with  it,  but  this  must  lie  reganlcd  as 
doubtful.  It  will  be  noticed  that  Euc.  ti,  14  provides  a  geo- 
metrical solution  of  the  equation  ^  =  ab, 

(v)  Pythagoras  shewed  that  the  plane  about  a  |N>int  could 
be  completely  filled  liy  equilateral  triangles,  by  si|uare8,  or  by 
regular  hexagons — results  that  must  have  lieen  familiar  where- 
ever  tiles  of  these  shapeti  were  in  common  use. 

(▼i)  The  Pythagoreans  were  said  to  have  solved  the  quad- 
rature of  the  circle :  they  stated  that  the  circle  was  the  most 
beaatiful  of  all  plane  figures. 

(vii)  They  knew  that  there  were  five  regular  solids  inscri- 
bable  in  a  sphere^  which  was  itself,  they  said,  the  most  beautiful 
ydtolidi. 

(viH)    Firam  their  phraseology  in  the  science  of  numliers 

1  4naai  otfMr  oeoasioiial  remarks  it  would  seem  that  they 

mied  wHh  the  methods  used  in  the  second  and 

fciHili  and  knew  eomethiDg  of   irratioiiai 


THK  lOHIAK  AMD  WTHAOOEmr  KHQOIA 


nugni^ades.  la  putiooUr,  tber*  b  i 
Pythagonw  proved  Uiat  the  eide  Mid  tit*  diigwiBl  nl  ft  m^Mim 
wen  incoauuenttinble ;  mmI  UtAt  it  wu  tbii  diaooveij  whid 
led  the  Oreeka  to  liuiuh  the  oonotiptiooi  of  number  and  meft- 
aureiueut  frum  their  geumetry.  A  prouf  of  thii  propoutioo 
which  uaj  be  that  due  to  Pytliagonw  is  given  below*. 

Next,  as  to  their  tlieorj  of  numberst.  I  have  alreudj  re- 
marked tluU  in  tliia  the  Pytliagoreans  were  chiefly  ooaoemed 
with  (i)  polygonal  numbera,  (ii)  the  (actors  tMt  numben, 
(iii)  nuuibure  wkidt  form  a  proportion,  and  (iv)  n 


^thagons  commenced  his  theory  of  arithnietic  by  dividing 
all  numbers  into  even  or  odd  :  the  odd  numbers  bung  tenued 
gnomom.  An  odd  number  such  aa  2m  +  1  wu  re}[arded  as  the 
difference  of  two  square  nuuibem  (t»  +  1)*  and  n';  and  thn  sum 
of  the  gnomons  fruiu  I  to  Sn  + 1  was  stated  to  be  a  iiquan 
number,  viz.  (n  +  I)',  its  square  tttot  was  t«mted  a  Me.  Pro- 
ducts  of  two  numbers  were  called  plane,  and,  if  a  prodnct  had  no 


K 
O      L 


ozact  squsra  root,  it  was  temted  an  oUoug.    A  pradnetof  thi** 
numbers  was  calkd  a  »olid  uumber,  and,  if  tlie  three  n 
were  equal,  a  ettt.     All  this  baa  obvious  referenc 

•  8ei'p.  69. 

t  8m  Uh  sppvndii  liar  ParMm/li^mt p$tliagortt»»t  to  S.  P.  1 
La  teirmci  htlUnt,  Puis,  IBS?. 


PTTHAOORBAN  ARITHMEna 


29 


I 


and  the  opinion  is  confirmed  by  Aristotle's  remark  that  when 
m  gnomon  is  pat  round  a  square  the  figure  remains  a  square 
though  it  is  increased  in  dimensions.  Thus,  in  the  figure  on 
the  opposite  page,  in  which  n  is  taken  equal  to  5,  the  gnomon 
AKC  (containing  11  sniall  squares)  when  |iut  round  the 
square  AC  (containing  5'  small  squares)  makes  a  square  f/L 
(containing  6*  sniall  squares).  It  is  possihle  that  several  of 
the  numerical  theorems  due  to  Greek  writers  were  discovered 
and  proved  by  an  analogous  method :  the  almcus  can  be  used 
for  many  of  these  demonstrations. 

The  numbers  (2n'  +  2ii  +  1),  (2n'  +  2n),  and  (2ii  +  I)  pos- 
sessed special  importance  as  representing  the  hypotenuse  and 
two  sides  of  a  right-angled  triangle:  (*antor  thinks  that 
Pjrthagoras  knew  this  fact  before  discovering  the  geometrical 
proposition  Euc  i,  47.  A  more  general  expression  for  such 
numbers  is  (nf-^n^  2mn,  and  (m'-n*):  it  will  be  noticed 
thai  the  result  ol>tained  by  Pythagoras  can  be  deduced  from 
these  expressions  by  assuming  m  ^  n  -i- 1 ;  at  a  later  time 
Archytas  and  Plato  gave  rules  which  are  equivalent  to  taking 
II  s.  1 ;  Diophantus  knew  the  general  expressions. 

After  this  preliminary  discussion  the  Pythngoreans  pro- 
ceeded to  the  four  special  problems  already  alluded  to. 
Pfthagorss  was  himself  acquainted  with  triangular  numbers ; 
polygonal  numbers  of  a  higher  order  were  discussed  by  later 
members  of  the  school.  A  triangular  number  represents  the 
sum  of  a  number  of  counters  laid  in  rows  on  a  plane;  the 
bottom  row  containing  ti,  and  each  succeeding  row  one  less ; 
it  is  therefore  equal  to  the  sum  of  the  series 

»  +  (n  -  1)  +  (n  -  2)  -I- ...  +  2  +  1, 

that  is,  to  J  »  (n  -I-  1).  Thus  the  triangular  number  corre- 
q[MXiding  to  4  is  10.  This  is  the  explanation  uf  the  language 
of  Pythagoras  in  the  well-known  passage  in  Lucian  where  the 
ipenAiani  asks  Pythagoras  what  he  can  teach  him.  Pythagoras 
replieSi  **I  will  teach  you  bow  to  oouni."  Jferekami^  **I  know 
thai  afaready.*   /yAoyorai^  **  How  do  you  count  t"  Merekmmi, 


i 


so     THK  lONlAH  AMD  PTTHAOORIAV  BOHOOIA 


* » 


**  One.  two^  thrae.  Ibar— *'  Pf^tKagonu. «« Slop  I  what  yw  tote 
to  be  four  U  ten,  a  perfect  triangle,  and  oar  symboL* 

As  to  the  work  of  the  Pythagoreans  on  the  fftolora  «l 
nnniben  we  know  very  little:  they.  daaaiAed  nnmberi  by 
comparing  them  with  the  sum  of  their  integral  subdiTison 
or  factors,  calling  a  number  excessive^  perfect,  or  deleetive 
aocunling  as  it  watf  greater  than,  equal  to,  or  less  than  the 
sum  of  tlieso  subdiviMors. .  These  investigations  led  to  no 
useful  reHult. 

The  third  class  of  problems  which  they  considered  dealt 
with  numben  which  foruuHl  a  proportion ;  presumably  these 
were  diHcuHHed  with  the  aicl  of  gecimetry  as  is  done  in  the  fifth 
book  of  Euclid. 

lastly  the  Pythagoreans  were  concerned  with  series  of 
numbers  in  arithmetical,  geometrical,  harmonical,  and  musioal 
progn*HHious.  The  three  pn>gn*iiHif»iis  first-mentioned  are  well 
known;  four  intt^gers  are  said  to  lie  in  musical  progression 
when  they  are  iii  the  ratio  a  :  2(i^/(tt  -^b)  :  |(a  <i>6)  :  A,  for 
example,  G,  8,  9,  and  12  are  in  musical  progression. 

After  the  death  of  Pythagoras,  his  teaching  seems  to  have 
been  carried  on  by  Epicharmofl,  and  Hippasui;  and  sub- 
sei|uently  by  Philolaui,  ArchippoB,  and  Lyais.  About  a 
century  after  the  murder  of  Pythagoras  we  find  Archytas 
recogni24Hl  as  the  hemi  of  the  bcliool. 

Archytas^.  Arvhytnn^  circ.  10(1  ar.|  was  one  of  the  most 
inlluciitiiil  citizens  of  Tarentum,  untl  uas  nuide  governor  of 
the  city  no  letis  than  seven  times.  His  influence  among  his 
conteuiporarii*H  wan  very  gn*at,  and  he  used  it  with  Dionysius 
on  one  occiiHion  to  have  the  life  of  Plato.  He  was  noted 
for  the  attention  he  paid  to  the  comfort  and  education  of  his 

*  8tf«  Alluisii,  chs|>.  IV.  A  ciitslogue  of  tlM  works  of  Archytsa  is 
given  hy  Fiibrieiua  iu  hi«  liiUiothrca  tiraeca,  vol.  i,  p.  8S8:  most  of 
tlic  fraKiiii'uU  on  |)hilui«oplt>  were  |iubli»hi-<l  bj  Thtomw  Gsis  ia  his 
Opu4Cula  Mythulugia,  CstubriUj^c,  1070;  and  bj  Tboiuss  Ti^lor  ss  aa 
sppeuJix  to  bi«  truusistioii  of  Isiublichus's  L{/t  of  Pythagoras^ 
Ittie.    8se  also  th«  isferenees  given  bj  Cantor,  voL  i,  p.  SOS. 


ARCHYTAS.  31 

slavea  and  of  children  in  the  city.  He  was  drowned  in  a 
shipwreck  near  Tarentuni,  and  his  body  washed  on  shore :  a 
fit  panishment,  in  the  eyes  of  the  more  rigid  Pythagoreans, 
for  his  haring  departed  fipm  the  linos  of  study  laid  down  by 
their  founder.  Several  of  the  lenders  of  the  Athenian  school 
were  among  his  pupils  and  friends,  and  it  is  lielieved  that 
much  of  their  work  was^duc  to  his  inspiration. 

Tlie  Pythagoreans  at  Orst  made  no  attempt  to  apply  their 
knowledge  to  mechanics,  but  Archytas  is  said  to  have  treated 
it  with  the  aid  of  geometry :  he  is  alleged  to  have  invented 
and  worked  out  tlie  theory  of  the  pulley,  and  is  credited  with 
the  construction  of  a  flying  bird  and  some  other  ingenious 
mechanical  toys.  He  introduced  various  mechanical  devices 
for  constructing  curves  and  solving  problems:  these  were 
oljected  to  by  Plato,  who  thought  that  they  destroyed  the 
value  of  geometry  as  an  intellectual  exercise,  and  later  Greek 
geometricians  confined  themselves  to  the  use  of  two  species 
of  instruments,  namely,  rulers  and  compasses.  Archytas  was 
also  interested  in  astronomy  ;  he  taught  that  the  earth  was  a 
sphere  rotating  round  its  axis  in  twenty-four  hours,  and  round 
which  the  heavenly  bodies  moved. 

Archytas  was  one  of  the  first  to  give  a  solution  of  the 
problem  to  duplicate  a  cube,  that  is,  to  find  the  side  of  a  cube 
whose  volume  is  double  that  of  a  given  cube.  This  was  one 
of  the  most  famous  problems  of  antiquity*.  The  construction 
given  by  Archytas  is  equivalent  to  the  following.  On  the 
diameter  OA  of  the  base  of  a  right  circular  cylinder  descrilie  a 
semicirele  whose  plane  is  perpendicular  to  the  base  of  the 
eylinder*  Let  the  plane  containing  this  semicirele  rotate 
RHUid  the  generator  through  0,  then  the  surface  traced  out  by 
the  semicirde  will  cut  the  cylinder  in  a  tortuous  eunre.  This 
curve  will  be  cut  by  a  right  oone  whose  axis  is  OA  and  semi- 
vertioal  angle  is  (say)  $0*  in  a  point  P,  such  that  the  jyrojec- 
two  of  OP  on  the  base  of  the  cylinder  will  be  to  the  radius  of 

•  8ss  btlow,  pp.  a^  48,  44. 


82  TUK  IONIAN  AND  J^TTBAOOHEAX  BOHOtOM. 


tlie  cqrlinder  in  tlie  r»tio  of  tlie  nde  of  tlia  feqaifed  o«bt  to 
that  of  the  given  cuba  The  proof  given  by  ArahjrUa  k  el 
coarae  geometrical*;  it  will  be  enonjgh  hero  lo  rainnrk  that 
in  the  oourae  of  it  he  shews  himself  acquainted  with  the 
rosults  of  the  propositions  Euc.  ill.  18,  ill,  35,  and  zi,  19. 
To  shew  analyticidly  that  th«  construction  is  oorrecti  take  OA 
as  the  axis  of  ^  and  the  generator  thnmgh  Oas  axis  of  s»  then, 
with  the  usual  notation  in  polar  coordinates,  and  if  «  be  the 
radius  of  the  cylinder,  we  have  for  the  equation  of  the  snilaoe 
described  by  the  seuicirolei  r = 2a  sin  ^ ;  f or  that  of  the  cylinder, 
rsin^  =  2acus^;  and  for  that  of  the  cone,  sin^cos^^i* 
These  three  surfaces  cut  in  a  point  such  that  sin'^s^,  and 
therefore,  if  p  be  the  projection  of  OP  on  the  base  of  the 
cylinder,  then  p*  =  (r  «in  6^}' ==  2a'.  Hence  the  volume  of  the 
cube  whose  side  is  p  is  twice  that  of  a  cube  whose  side  is  a. 
I  mention  the  problem  and  give  the  construction  used  fay 
Archytas  to  illustrate  how  considerable  was  the  knowledge  of 
the  Pytliagorean  scluiol  at  that  time. 

Theodoras.  Another  Pythagorean  of  about  the  same  date 
as  Archytas  was  TheodoruM  of  Cyrene  who  is  said  to  have 
proved  geometrically  tliat  the  numbers  ropresented  by  J\  ^5, 
s/6,  JTf  s!^t  s/l^»  sl^h  n/12,  V13,  ^14,  ^15  and  Jl7  aro 
incommensurable  with  unity.  Theaetetus  was  one  of  his 
pupils. 

Perhaps  Timaeos  of  Locri  and  Bryao  of  Heraclea  shoukl 
be  mentioned  as  other  distinguished  Pythagoreans  of  this 
time.  It  is  believed  that  Bryso  attempted  to  tind  the  area 
of  a  circle  by  inscribing  and  circumscribing  squares,  and 
finally  obtained  polygons  between  whose  areas  the  area  of  the 
cirole  lay ;  but  it  is  said  that  at  some  point  he  assumed  that 
the  area  of  the  circle  was  the  arithmetic  mean  between  an 
inscribed  and  a  circumscribed  polygon, 

*  It  is  printed  bj  Allmsn,  |ip.  Ill— lit. 


I 


THE  SCHOOLS  OP  CHIOS   AND  ELBA. 


33 


(Hher  Greek  Matheinafiad  tSchaoU  in  the  fifih  century  B.C. 

It  wuald  be  a  miHtake  to  Kuppusc  that  Miletus  and  Tarentuin 
were  tlie  only  places  where,  in  the  fifth  century,  Greeks  were 
^ng>^g^  tn  laying  a  scientific  foundation  for  the  study  of 
uiathenMtics.  These  towns  represented  the  centres  of  chief 
activity,  but  there  were  few  cities  or  colonies  of  any  importance 
where  lectures  on  philosophy  and  geometry  were  not  given. 
Among  these  smaller  schools  I  may  mention  those  at  Cliios, 
Elea,  and  Thrace. 

The  liest  known  philosopher  of  the  Srhoof  of  Chio9  was 
CEnopides,  who  was  bom  alx>ut  5(^0  B.c.  and  died  alx>ut 
430  BLC  He  devoted  himself  chiefly  to  astronomy,  but  he  had 
studied  geometry  in  Eg}'pt,  and  is  c«Tdited  with  the  solution 
of  two  problems,  namely,  to  draw  a  straight  line  from  a 
given  external  point  |ierpendicular  to  a  given  straight  line 
(Eac  I,  12),  and  at  a  given  point  to  construct  an  angle  equal 
to  a  given  angle  (Euc.  t,  23). 

Another  important  centre  was  at  Elea  in  Italy.  This 
was  founded  in  Sicily  by  Xenophanes.  He  was  followed  by 
Pttnnenides,  Zeno,  and  MelissuB.  The  meml^ers  of  the 
EletUic  School  were  famous  for  the  difficulties  they  raised  in 
connection  with  questions  that  required  the  use  of  infinite 
aeries,  such  for  example  as  the  well-known  paradox  of  Achilles 
and  the  tortoise,  enunciated  by  Zerio^  one  of  their  most  promi- 
nent members.  Zeno  was  bom  in  495  b.c.,  and  was  executed 
at  Elea  in  435  &a  in  consequence  of  some  conspiracy  against 
the  state;  he  was  a  pupil  of  Parmenides,  with  whom  he  visited 
Athena,  circ  455-450  &a 

Zeno  argued  that  if  Achilles  ran  ten  times  as  fast  as  a 
tortoise^  yet  if  the  tortoise  had  (say)  1000  }'ards  start  it  could 
never  be  overtaken:  for,  when  Achilles  had  gone  the  1000 
yards,  the  tortoiae  would  still  be  100  yards  in  front  of  him ; 
fagr  the  time  he  had  covered  these  100  yards,  it  would  still  be 
10  yards  in  front  of  him ;  and  so  on  for  ever :  thus  Achillea 
would  get  neater  and  nearer  to  the  tortoiae  bat  never  overtake 

IL  3 


94  TUK  EUUmC  AND  ATOMUmO  BCBOOLII. 

it.  The  LdUcj  in  usiudlj  expUined  by  tlie  argiuneni  thai  tlia 
tune  raqoired  to  overtake  the  tortoiM  can*  be  divided  ialo 
an  infinite  number  ol  paria,  as  stated  in  the  question,  hot 
these  get  smaller  and  smaller  in  geometrical  progression,  and 
the  sum  of  them  all  is  a  finite  time :  after  the  lapse  of  that 
time  Achilles  would  be  in  front  of  the  tortoise. '  Probably 
Zeno  would  have  replied  that  this  argument  rests  on  the 
assumption   tliat  space   is  infinitely  divisible,  which  is  the  4 

question  under  diacuHsion ;  he  himself  asserted  that  magni- 
tudes were  not  infinitely  divisible. 

These  paradoxes  made  the  Greeks  look  with  suspicion  cm 
the  use  of  infiuitesiuials,  and  ultimately  led  to  the  invention 
of  the  method  of  exhaustions. 

The  Atomiulic  Schoui^  having  its  head-quarters  in  Thraoe, 
was  another  important  centre.  This  was  founded  by  Lea- 
oippos,  who  was  a  pupil  of  Zeno.  He  was  succeeded  by 
DemocritUB  and  Epioums.  Its  most  famous  matheuiatician 
was  Detnocriius,  lx>rn  at  Aljdera  in  460  B.C.  and  said  to  have 
died  in  370  ac\,  who,  besides  pliilosopliical  works,  wrote  on 
plane  and  solid  geometry,  incumiuensurable  lines,  perspective^ 
and  numbers.     These  works  are  all  lost. 

But  though  several  distinguished  individual  philosophen 
may  be  mentioned  who,  during  the  fifth  century,  lectured  at 
dilferent  cities,  they  mostly  seem  to  have  drawn  their  inspi- 
ration from  Tarentum,  and  towards  the  end  of  the  century  to 
have  looked  to  Athens  as  the  intellectual  capital  of  the  Greek 
world :  and  it  is  to  the  Athenian  schools  that  we  owe  the  next 
great  advance  in  mathematics. 


85 


CHAPTER    III. 

THE  SCHOOLS  OF  ATHENS  AND  CYZICUK* 

riRC.  420  RC.-300  RC. 


It  was  towardn  the  close  of  the  fifth  centary  before  Christ 
that  Athens  finit  became  the  chief  centi-e  of  mathematical 
ntudies.  Several  causes  coriHpired  to  bring  this  about.  During 
that  century  she  had  become,  partly  by  c<»mnieroey  partly  by 
approprinting  ft>r  her  own  purposes  the  contributions  of  her 
allieny  the  nitmt  wealthy  city  in  Greece ;  and  the  genius  of  her 
statesmen  had  made  her  the  centre  on  which  the  politics  of 
the  peninsula  turned.  Moreover  whatever  states  disputed  her 
claim  to  political  supremacy  her  intellectual  pre-eminence  was 
admitted  by  all  There  was  no  school  of  thought  which  had 
not  at  some  time  in  that  century  been  represented  at  Athens 
by  one  or  more  of  its  leading  thinkers ;  and  the  ideas  of  the 
new  science,  which  was  being  so  eagerly  studied  in  Asia  Minor 
and  Oraccia  Magna,  had  been  brought  before  the  Athenians 
OD  various  occasions. 

*  The  history  of  these  schools  is  diseotaed  al  lenjcth  in  O.  J.  Allman's 
Orttk  Gtimet9Tf  from  T9mU§  to  Emelk^  Piiblip,  1W9;  and  in  J.  Oow*« 
Oreek  Jinlkematie$^  Cambridge,  1884 ;  il  it  also  tn-aled  br  Cantor,  chaps, 
n,  I,  and  xi ;  bj  Hanhel,  pp.  111—156 ;  and  hj  C.  A.  Breischneidar  in 
Us  Die  OtomeiHe  iin4  dit  Gnmeter  ror  EukleidtB,  Ldpslir,  1870;  a 
CfilieAl  aeeooat  of  the  original  antboriiies  is  given  Vj  8.  P.  Tannery  in 
his  Qimtitrtt  Ofwefnr ,  Psrit»  1887,  and  ^Ihsr  papers. 


-\  . 


I 


S6  THK  SCHOOLS  OF  ATHINR  AMD  OTinOUa 

• 

AnaxagonuL  Amongtt  the  most  important  of  tlio  phib- 
iiophen  who  resided  at  AtheoH  and  prepared  the  way  for  the 
Athenian  school  I  may  mention  Anaxagorat  vf  Vtazom^mmB^ 
who  wan  almoHt  the  last  philosopher  of  the  Ionian  schopL  Ho 
was  born  in  500  B.C.  and  died  in  428  b.c  He  seems  to  have 
settled  at  Athens  about  440  ac,  and  there  taught  the  resulta 
of  the  Ionian  philosophy.  Like  all  meiabers  of  that  school 
he  was  much  interested  in  astronomy.  He  asserted  that  the 
sun  was  larger  than  the  Peloponnesus :  this  opinion,  together 
with  some  attempts  he  had  made  to  explain  various  physical 
phenomena  which  had  been  previously  supposed  to  be  due  to 
the  direct  action  of  the  gods  led  to  a  pruseciution  for  impiety^ 
and  he  was  convicted.  While  in  prison  he  is  said  to  have 
written  a  treatise  on  the  quadrature  of  the  circle. 

The  Sophists.  Tlie  sophists  can  hardly  be  considered  as 
belonging  to  the  Athenian  school,  any  more  than  Anaxagoras 
can;   but  like  him  they  immediately  preceded  and  prepared  '  i 

the  way  for  it,  so  that  it  is  desirable  to  devote  a  few  words  to  j 

them.  One  condition  for  success  in  public  life  at  Athens  was 
the  power  of  sfieaking  well,  and  as  the  wealth  and  power  of 
the  city  increostHl  a  considerable  number  of  ^  sophists  "  settled 
there  who  undertook  aimougst  other  things  t4  teach  the  art  of 
oratory.     Many  of  them  also  directed  the  gejieral  education  of  i 

their  pupils,  of  which  geometry  usually  formed  a  part.  We 
are  told  tliat  two  of  those  who  are  usually  termed  sophists 
made  a  special  study  of  geometry — these  were  Hippias  of  Elis 
and  Antipho,  and  one  made  a  special  studV  of  astronomy — 
this  was  Meton,  after  whom  the  metonic  cycle  is  named. 

Hippias.  /The  tirst  of  these  geometriciai^s,  JJippias  of  Elu 
(circ.  420  acA  is  described  as  an  expert  ari^imetician,  but  he 
is  best  knowa  to  us  through  his  invention  of  a  curve  called  the 
quadratrix,  py  means  of  which  an  angle  can  be  trisected,  or 
indeed  divided  in  any  given  ratia  If  the  radius  of  a  circle 
rotate  uniformly  i-ound  the  centre  0  from  the  position  OA 
through  a  right  angle  to  OB^  and  in  the  same  time  a  straight 
line  drawn  perpendicular  to  OB  move  uniformly  parallel  to 


1' 


THE  QUADRATRIX. 


S7 


itwif  from  tbe  ponition  OA  to  BC^  the  locus  of  their  inter- 
aeetMm  will  be  the  qoadrstrix. 


ft 


I; 


OH  and  MQ  .be  the  positioriA  of  these  lines  at  any 
time ;  and  let  them  cut  in  P^  a  point  on  the  curve.     Then 

angle  AOP  :  an^le  AOB^  OM  :  OB. 

Similarly,  if  O/T  be  another  position  of  the  radius, 

angle  AOT  i  angle  AOI(=  OSf  :  OB. 
:.  angle  ilO/'   :  anglr  ilO/*' =  O.V  :  OJ/'; 
.-.  angle  AOr  :  angle  P'OP-^  OM'  :  MM. 

Hence,  if  the  angle  AOP  lie  giv^n,  and  it  be  rpquired  to 
divide  it  in  any  given  ratio,  it  is  sufficient  to  divide  OM 
in  that  ratio  at  M\  and  draw  the  line  M'P' ;  thrn  OP'  will 
divide  AOP  in  the  required  ratio. 

If  OA  be  taken  as  the  initial  line,  OP  r,  the  angle 
AOP  =  0t  and  OA=a,  we  have  ^  :  |«'  =  rsin^  :  n,  and  the 
equati«»n  of  the  curve  is  «r  =  2a$  cosec  $. 

Hippias  devised  an  instrument  to  construct  the  curve 
mechanically;  but  constructions  which  invoh-ed  the  use  of 
any  mathematical  instruments  except  a  ruler  and  a  pair  of 
oompasses  were  objected  to  by  Plato^  and  rejected  by  most 
geometriciaDS  of  a  subsequent  date. 

AnilldlO.      The  second  sophist  whom  I  mentioned  waa 


38         THK  flcuooLA  OF  athhto  and  cnicoa 

Aniifko  (etre.  420  ac).  Ho  ia  one  of  tlie  very  hm  wrilon 
among  the  aneienU  who  attempted  to  find  tlio  ana  of  a  eiiele 
by  oonsidering  it  as  the  limit  of  an  inacribed  regular  polygon 
with  an  infinite  number  of  sides.  He  began  by  insoribing  an 
equiUteral  triangle  (or,  according  to  some  aooounts,  a  square) ; 
on  each  side  he  inscribed  in  the  smaller  segment  an  isosceles 
triangle,  and  so  on  oJ  infinitum.  This  method  of  attacking 
the  quadrature  problem  is  similar  to  that  described  tbove  as 
used  by  Bryao  of  Heraclea. 

No  doubt  there  were  other  cities  in  Greece  besides  Athens 
where  similar  and  equally  meritorious  work  was  being  done, 
though  the  record  of  it  has  now  been  lost ;  I  have  mentioned 
the  investigations  of  these  three  writers,  partly  in  ordiHr  to 
give  an  idea  of  the  kind  of  work  which  was  then  goinj^  on 
all  over  Greece,  but  chiefly  because  they  were  the  immediate 
predecesHoni  of  thuse  who  created  the  Athenian  scliool. 

The  hiiitory  of  the  Athenian  school  liegins  with  the  teaching 
of  Hippocrates  about  420  ac. ;  the  school  was  established  on 
a  permanent  basis  by  the  labours  of  Plato  and  Eudoxus ;  and, 
together  with  the  neigh liouring  school  of  Cyzicus,  continued 
to  extend  on  the  lines  laid  down  by  these  three  geometricians 
until  the  foundation  (about  300  b.c.)  of  the  university  at 
Alexandria  drew  thither  most  of  the  talent  of  Greece. 

Eudoxus,  who  was  among  the  most  distinguished  of  the 
Athenian  mathematicians,  is  also  reckoned  as  the  founder  of 
the  school  at  Cyzicus.  The  connection  between  this  school 
and  that  of  Athens  was  very  close,  and  it  is  now  impossible 
to  dineutanj^le  their  histories.  It  is  said  that  Hippocrates, 
Plato,  and  Tlieaetetus  lielonged  to  the  Athenian  school ;  while 
Eudoxus,  MenaechmuH,  and  Aristaeus  belonged  to  that  of 
Cyzicus.  There  was  always  a  constant  intercourse  between 
the  two  Hcluiols,  the  earliest  memliers  of  both  had  been  under 
the  intluence  either  of  Archytas  or  of  his  pupil  llieodorus  of 
Cyrene,  and  there  was  no  diflerence  in  their  treatment  of  the 
subject,  so  that  they  may  be  conveniently  treated  togetber. 

Before  discussing  the  work  of  the  geometricians  of  these 


I 


I 


THS  SCHOOLS  OF  ATHENS  AND  CTZICUS.  39 

•chools  in  detail  I  maj  note  that  they  were  especially  interested 
in  three  problems*:  namely,  (i)  the  duplication  of  a  cube, 
that  is,  the  determination  of  the  side  of  a  cube  whose  volume 
is  doable  that  of  a  given  cube ;  (ii)  the  trisect  ion  of  an  angle ; 
and  (iii)  the  squaring  of  a  circle,  that  is^  the  determination 
of  a  square  whose  area  is  equal  to  that  of  a  given  circle. 

Now  the  first  two  of  these  problems  (considered  analyti- 
cally) require  the  solution  of  a  cubic  e<|uation ;  and,  since  a  oon- 
ttruction  by  means  of  circles  (whose  equations  are  of  the  form 
af  -¥  ^  -¥  ax  4  bjf  -¥  e  =  0)  and  straight  lines  (whose  equations  are 
of  the  form  ax+fy^-y^O)  cannot  be  equivalent  to  the 
solution  of  a  cubic  equation,  the  pniblenis  are  insoluble  if  in 
our  constructions  we  restrict  ourselves  to  the  use  of  circles  and 
straight  lines,  that  is,  to  Euclidean  geometry.  If  the  use  of 
the  conic  sections  be  permitted,  both  of  these  questions  can 
be  solved  in  many  wajra.  The  third  problem  is  equivalent  to 
finding  a  rectangle  whose  sides  are  equal  respectively  to  the 
radius  and  to  the  semiperimeter  of  the  circle.  These  lines 
have  been  long  known  to  be  incommenHurable,  but  it  is  only 
recently  that  it  has  been  shewn  by  Lindeinann  that  their  ratio 
cannot  be  the  root  of  a  rational  algebraical  equation.  Hence 
this  problem  also  is  insoluble  by  Euclidean  geometry.  The 
Athenians  and  Cyzicians  were  thus  destined  to  fail  in  all  three 
problems,  but  the  attempts  to  solve  them  led  to  the  discovery 
of  many  new  theorems  and  processes. 

Besides  attacking  these  pn>blems  the  later  Platonic  school 
collected  all  the  geometrical  theorems  then  known  and  arranged 
them  systematically.  These  collections  comprised  the  bulk  of 
the  propositions  in  Euclid's  Efem^niti^  books  i — ix,  xt,  and  xii, 
together  with  tome  of  the  more  elementary  theorems  in  conic 
■eotions. 
;  Hippooimtes.     HippoeraUt  of  Chum  (who  must  be  care- 

fully distinguished  from  his  contemporary,  Hippocrates  of  Oos, 

*  On  thess  prolileiD«,  solulions  of  Iheni,  sod  the  aQtborities  for  their 
biiloty,  sss  mf  Mtttkewnitfemi  Herrmfimu  ^mti  Pnhkm,  London,  thiiil 
fdttliont  18M«  shap.  vni. 


w 

{ 


I 


40  THE  SCHOOLS  OP  ATHENS  AHD  CfmOOS. 

Uie  oelebfmtad  phjiidsn)  wtm  one  at  the  ymtcil  of  IIm  OiSik 
geometricUiia.  He  was  bom  abosl  470  aa  si  Chio^  ssd 
began  life  aa  a  merchant.  Tbe  acctmnta  difer  aa  to  whether 
he  waa  swindled  by  the  Athenian  custom-hoiiae  "^JM^W  who 
were  sUtioned  at  the  Chersoneae^  or  whether  one  of  hia 
vessels  was  captured  by  an  Athenian  pirate  near  BjamtiaBi ; 
but  at  any  rate  somewhere  about  430  B.C.  he  came  to  Athena 
to  try  to  recover  his  property  in  the  law  courta.  A  foreigner 
was  nut  likely  to  succeed  in  such  a  case^  and  the  Athenians 
seem  only  to  have  laughed  at  him  for  his  simplicity,  6rBt  in 
allowing  himself  to  be  cheated,  and  then  in  hoping  to  recover 
his  money.  While  prosecuting  his  cause  he  attended  the 
lectures  of  various  philosophers,  and  finally  (in  all  probability 
to  earn  a  livelihood)  opened  a  school  of  geometry  himsell  Hs 
seems  to  have  been  well  acquainted  with  the  Pythagorean 
philosophy,  though  there  is  no  sufficient  authority  for  the 
statement  that  he  was  ever  initiated  as  a  Pythagorean. 

He  wrote  the  first  elementary  text-book  of  geometry,  a 
text-book  on  which  probably  Euclid's  EhmentM  was  founded ; 
and  therefore  he  may  be  said  to  have  sketched  out  the  lines 
on  which  geometry  is  still  taught  in  English  schools.  It  is 
supposed  tliat  the  use  of  letters  in  diagrams  to  describe  a 
figure  was  made  by  him  or  introduced  about  his  time,  aa  he  } 

employs  expressions  such  as  ''the  point  <m  which  the  letter  [ 

A  stands  "and  ''the  line  on  which  AB  \a  marked."     Cantor  | 

liowover  thinks  that  the  Pytliagoreans  liad  previously  been 
accustomed  to  repreirent  the  five  vertices  of  the  pentagram- 
star  by  the  letters  v  y  i  ^  a*  ;  and  though  this  waa  a  single 
instance,  fierhaps  they  may  have  used  the  method  generally. 
The  Indian  geumeters  never  employed  letters  to  aid  them  in 
the  dcscripti<m  of  their  figures.  Hippocrates  also  denoted  the 
square  on  a  line  by  the  word  SiW/uc,  and  thus  gave  the 
t'*chnical  meaning  to  the  word  power  which  it  still  retains  in 
algebra:  there  is  rt^aMin  to  think  that  this  use  of  the  word 
was  derived  from  the  Pytliagoreans,  who  are  said  to  have 

*  8m  abovs,  p.  Sa 


I 


HIPPOCRATES. 


41 


iVBcialed  thm  ntnh  of  the  prtiposition  ms  Euc.  i.  47»  in  the 
thfti  ''the  total  power  of  tlie  nicies  of  a  right-angled 
1^  is  the  same  as  that  of  the  hypotheno!^.'* 

In  this  text-book  Hippocrates  intmluced  the  method  of 

**  ndndng "  one  theorem  to  another,  which  heing  proved,  the 

thing  proposed  necessarily  follows ;  of  wliich  plan  the  nrr/iir/ii 

mti  itbmtnium  is  a  paKicnlar  ca^.    No  doubt  the  principle  had 

been  nsed  occasionally  before,  bat  lie  drew  attention  to  it  as 

m  iegitiniate  mode  of  proof  which  was  capal>le  of  nanieroos 

triplications.    He  may  be  said  to  lia\*e  intniduced  the  geometry 

off  the  circle.     He  discovered  that  similar  segments  of  a  circle 

contain  ec|aal  angles :  that  the  angle  siibtende«l  by  the  chord 

off  a  circle  is  greater  than,  eqnal  to,  or  less  than  a  right  sngle 

mm  the  segment  of  the  cin*le  containing  it  is  less  than,  er|aal 

to^  or  greater  than  a  semicircle  (Euc  ill,  31) ;  and  prolmbly 

lieyeral  other  of  the  propositions  in.  the  third  book  of  Kuclid. 

It  is  most  likely  that  he  also  established  the  propositions  that 

[similar]  circles  are  to  one  another  as  tlie  squares  of  their 

diameters  (Euc  xii,  2),  and  that  similar  segments  are  as  the 

Wfoares  of  their  chords.     The  proof  gi%-en  in  Euclid  of  the  6nit 

of  these  theorems  is  believed  to  lie  due  to  Hip|Kicrates. 

The  most  celebrated  discoveries  of  Hippocrates  were  how- 
erer  in  connection  with  the  quadrature  of  the  circle  and  the 
duplication  ef  the  cube^  and  owing  to  his  influence  these 
problems  played  a  prominent  fiart  in  the  history  of  the 
Athenian  school. 

The  following  propositions  will  sufficiently  illustrate  the 
method  by  which  he  attacked  the  quadrature  problem. 

(a)  He  commenced  by  finding  the  area  of  a  lune  contained 
between  a  semicircle  and  a  quadrantal  arc  stsnding  on  the 
same  chord.  This  he  did  as  follows.  Let  if  /^C  be  an  isosceles 
right-angled  triangle  inscribed  in  the  semicircle  A  HOC  whose 
centre  b  O.  On  AB  snd  AC  as  diameters  describe  semicireles 
as  in  the  figure.    Then,  since  fay  Knc  i,  47, 

sq.  on  i9y*  =  sq.  on  ifC-fsq.  on  AB^ 


42 


THE  MCHOOLS  OP  ATUEN8  AND  0YS1CU& 


therefore,  by  Eoe.  iii,  2» 

area  }0  oa  BC^mnm  |0  on  AC-¥ 


|0  on  JA 


Take  away  the  common  parte 

/.  area  aABC  =  sum  of  araaa  of  lunee  AECD  and  AFBO. 

Henoe  the  area  of  the  lune  AECD  it  equal  to  half  that  of  the 
ABC. 


(fi)    He  next  inscribed  half  a  regular  hexagon  A  BCD  m  a 


eeniicirele  whoee  centre  was  O,  and  on  OA^  A  B^  BC^  and  CD 
as  diaiueteni  described  semicircles  of  which  those  on  OA  and 
AB  are  drawn  in  the  figure.  Then  il/>  is  double  any  of  the 
Unes  OA.AB,  BC  and  CD, 


HIPPOCRATEH.  48 

/.  iq.  on  AD^mun  of  sqs.  on  OA,  AB^  8C^  and  CD^ 
.'.  areA  }  0  ABCD^snm  of  areas  of  J  08on  OA^  AB^  BCjAtid  CD. 
Take  awaj  the  common  parts 

/.  area  trapezium  A  BCD  -  3  lune  A  EHF  -»•  1 0  on  OA^ 

If  therefore  the  area  of  this  latter  lune  lie  known,  so  is  that  of 
the  semicircle  described  on  OA  as  diameter.  According  to 
SimplicioSy  Hippocrates  assumed  that  the  area  of  this  lane 
was  the  same  as  the  area  of  the  lane  found  in  proposition  (a) ; 
if  this  be  so,  he  was  of  counte  mistaken,  as  in  this  case  he  is 
dealing  with  a  lane  contained  between  a  semicircle  and  a 
sextan tal  arc  standing  on  the  same  chord  ;  but  it  seems  more 
probable  {hat  Simplicias  misunderstood  Hippocrates. 

Hip|iocrates  also  enunciated  various  other  theorems  con- 
noted with  lunes  (which  have  been  collected  by  Bretschneider 
and  by  Alluian)  of  which  the  theorem  last  given  is  a  typical 
example.  I  believe  that  they  are  the  earliest  instances  in 
which  areas  bounded  by  curves  were  determined  by  geometry. 

Tlie  other  problem  to  which  Hippocrates  turned  his  atten- 
tion was  the  duplication  of  the  culie,  that  is,  the  rletermi nation 
of  the  side  of  a  cube  whose  volume  is  double  that  of  a  given 
cube. 

This  problem  was  known  in  ancient  times  as  the  Delian 
problem,  in  consequence  of  a  legend  that  the  Delians  had 
consulted  Plato  on  the  subject.  In  one  form  of  the  story, 
which  is  related  by  Philoponus,  it  is  a<^serted  that  the 
Athenians  in  \W  B.c,  when  suffering  from  the  plague  of 
eruptive  typhoid  fever,  consulted  the  oracle  at.  Delos  as  to 
how  they  could  stop  it.  Apollo  replied  that  they  must 
doable  the  sixe  of  his  altar  which  was  in  the  form  of  a  cabe. 
To  the  unlearned  suppliants  nothing  seemed  more  easy,  and 
a  new  altar  was  constructed  either  having  each  of  its  edges 
double  that  of  the  old  one^  (from  which  it  followed  that  the. 
volume  was  increased  eight^fbld)  or  by  phii*ing  a  similar  cubic 
ahar  next  to  the  oki  one.      Whereupon,  aoenrding  to  the 


V 


44  THE  HCHOOLft  OP  ATHSKll  AND  CYZ1CU& 

legend,  tbe  indigiuuit  god  made  tlie  peitiience  wone  tluui 
before,  and  informed  a  freeh  depatation  that  it  was  malen 
to  trifle  with  him,  as  hit  new  altar  miut  be  a  cube  and  havo 
a  volume  exactly  d<iulile  that  of  hiii  old  one.  HiMpecUng  m 
mystety  the  Atheniana  applied  to  Plato^  who  referred  iheoi 
to  the  geometriciauH,  and  eiipecially  to  Euclid,  who  had  made 
a  special  study  of  tlie  problem.  The  introduction  of  the  namei 
of  Plato  and  Euclid  is  an  obvious  anachronism.  Eratosthenes 
gives  a  somewliat  similar  account  of  its  origin,  but  with  king 
Minos  as  the  pnipounder  of  the  problem. 

Hippocrates  reducfd  the  problem  of  duplicating  the  cube 
to  that  of  finding  two  means  between  one  straight  line  (a) 
and  another  twice  as  long  (2if).  If  these  mcsans  be  x  and 
y,  we  have  a  :  x  -  x* :  y  -  y  :  2a,  from  which  it  follows  that 
x*  =  ^o*.  It  is  in  this  form  that  the  problem  is  usually  pre- 
sented now.  Hippocnites  did  not  succeed  in  finding  a  con- 
struction fur  these  means. 

Plato.  The  next  philusoplier  of  tlie  Athenian  school  who 
re<|uires  mention  here  was  Plato,  He  was  born  at  Athens  in 
4211  ac*.,  and  whs,  as  is  well  known,  a  pupil  lor  eight  years  of 
Bocrates ;  much  of  the  ti^iching  of  the  latter  is  infernnl  from 
Plat4/s  dialogues.  After  the  executitm  of  his  master  in  399  B.C. 
Plato  left  Athens,  an<l  being  posnesHed  of  considerable  wealth 
he  spent  some  years  in  travelling:  it  was  during  this  time 
that  he  studied  mathematics.  He  vinited  Egypt  with  Eudoxus, 
and  Strabo  says  that  in  his  time  the  apartments  they  occupied 
wX  Heliopolis  were  still  shewn.  Thence  Plato  went  to  Cyrene, 
where  he  studied  under  Theodorus.  Next  he  moved  to  Italy, 
whero  he  became  intimate  with  Arcliytas  the  then  head  of  the 
Pythagorean  scIkioI,  Eurytas  of  Metapontum,  and  Timaeus  of 
Locri.  He  returned  to  Athens  about  the  year  380  &c.,  and 
formed  a  school  of  students  in  a  suburban  gymnasium  called 
the  •<  Academy.*'     He  died  in  348  &c. 

Plato,  like  Pythagoras,  was  primarily  a  philosopher,  and 
perhaps  his  philosophy  should  be  regarded  as  founded  on  the 
Pythagorean  rather  than  on  the  Socratic  teaching.     At  any 


if 


* 


PLATO.  f  5 

.     •   •  • 
rmie  it,  like  thai  nf  the  PythagorcatiK,  wa>t  niloural  with. the 

idea  that  the  iiecrpt  of  the  universe  was  to  lie  found  in 
nmnlier  and  in  form  ;  hence,  as  Kudemus  kajii,  *<he  exhiluted 
on  every  occasion  the  remarkable  connection  lietween  mathe- 
matics and  pliilosophjr."  All  the  nuthoritien  agree  that,  unlike 
many  later  phihisophens  he  marie  a  study  of  geometry  or  . 
some  exact  science  an  indiRfienHalile  preliminary  to  that  of 
philosophy.  The  in^criptiim  over  th<*  entrance  to  his  school 
ran  "  Let  none  ignorant  ot  geometry  enter  my  door,**  and  on 
one  occNsion  an  applicant  who  knew  no  geometry  is  said  io 
have  been  refused  admiHsitm  as  a  student. 

Plato s  position  as  «ine  of  the  riiasters  of  the  Athenian 
mathematical  schcml  rests  not  mi  much  on  his  individual 
discoveries  ami  writings  as  on  the  extraordinary  influence 
he  exerted  on  his  contem|>oraries  and  successors.  Thus  the 
oljection  that  he  expressed  to  the  use  in  the  construction  of 
ciin'es  of  any  instruments  other  than  rulers  and  compasses 
was  at  once  accepted  as  a  canon  which  must  lie  observed  in 
such  problems.  It  is  proliably  due  to  Plato  that  subsequent 
geometricians  liegan  the  subject  with  a  can*fully  compiled  series 
of  definitions,  postulates,  and  axioms.  He  alsfi  systematized 
the  methods  which  could  lie  use<l  in  attacking  mathematical 
questions,  and  in  particular  directed  attention  to  the  %'alue  of 
analysis.  Tlie  analytical  method  of  proof  begins  by  assuming 
that  the  theorem  or  problem  is  wilved,  and  thence  deducing 
■ome  result :  if  the  result  be  false,  the  theorem  is  not  tnie  or 
the  problem  is  incapable  of  solution :  if  the  result  be  known  . 
to  be  true,  and  if  the  steps  lie  reversible,  we  get  (by  reversing 
them)  a  i^nthetic  proof ;  but  if  the  steps  be  not  reversible, 
no  conclusion  can  be  drawn.  Numerous  illustrations  of  the 
method  will  be  found  in  any  modem  text-book  on  geometry. 
H  the  claatiftcatioii  of  the  methods  of  legitimate  induction 
gireB  by  Hill  in  hb  work  on  logie  had  been  uikiverBally 
■ad  erveiy  new  diaoovery  in  science  had  been  justified 
■^  Um  ralei  there  Uud  down,  he  wonM,  I 
■4  n  poeilion  in  rBfarenca  to  modern 


.J 


4((  THE  SCHOOLS  OP  ATUKVS  AND  OTIICUa 

Mcienoe  somewhat  analogous  to  thai  which  Plato  oocspiMl  in 
r^ard  to  the  niathematicii  of  hw  tame. 

The  following  is  tlie  only  extant  theoieni  traditaonally 
attributed  to  PUta  li  CAB  and  DAB  be  two  right-angled 
triangles,  having  one  side»  AB^  common,  their  other  sides, 
AD  and  iJt\  parallel,  and  their  hjrpothenuses,  AC  and  BD^ 
at  right  angles,  then,  if  these  hypothenuses  cut  in  ^,  we  have 
/V  :  PH'-^PB  :  PA  =r  /M  :  />/>.  This  theorem  was  used  in 
duplicating  the  cube,  for,  if  such  triangles  can  be  constructed 
having  PD  ^  'IPC^  the  problem  will  be  solved.  It  is  easy  to 
make  an  iiiHtrumeut  by  which  the  triangles  can  be  con- 
structed. 

EudoxuB*.  Of  £uJiA4rus,  the  third  great  mathematician 
of  the  Athenian  school  and  the  founder  of  that  at  Cysicus,  we 
know  vei7  little.  He  was  born  in  Cnidus  in  40H  &c.  Like 
Plato,  he  went  to  Taivntum  and  studied  under  Archytas  the 
tlien  head  of  the  Pytlmgoreans.  Subtiequently  he  travelled 
with  PUto  to  Egypt,  and  then  settled  at  Cyzicus  where  he 
founded  the  school  of  that  name.  Finally  he  and  his  pupils 
moved  to  Athens.  There  he  seems  to  liave  taken  some  part 
in  public  affairs,  and  to  have  practised  medicine;  but  the 
hostility  of  Plato  and  hui  own  unpopularity  as  a  foreigner 
made  his  position  uncomfortable,  and  he  returned  to  Cyzicus 
or  Cnidus  shortl}'  before  his  death.  He  died  while  on  a  journey 
to  Egypt  in  355  &c. 

His  uiatheinatical  work  seems  to  have  been  of  a  high  order 
of  excellence.  He  discovered  most  of  what  we  now  know  as 
the  fifth  book  of  Euclid,  and  proved  it  in  much  the  same 
form  as  that  in  a-hich  it  is  there  given.  i 

He  discovered  some  theorems  on  what  was  called  ''the  I 

golden  section."     The  problem   to 

cut  a  line  AB  in  tho  goUeu  section,    A i^H  B 

tliat  is,  to  divide  it,  say  at  //,  in  I 

*  The  works  of  Eudoius  were  dibcumed  in  oonsiilersbls  detsil  by 
H.  Kiinssberg  of  DinkeUbiihl  in  18SS  and  1890;  see  sl^  the 
mentioned  above  in  the  footnote  on  p.  S5. 


( 


EUDOXUS.  47 

exiraue  mod meui  ratio(ihat  is,  so  tliat  AB :  AH^Aii  :  HB)  in 
■olved  in  Euc  ii,  11»  and  prohaMy  was  known  to  the  Pythago- 
reans at  an  early  date.  If  we  denote  AB  hy  l^  Ali  hy  Oy  and 
HB  by  by  the  theorems  that  Eudoxus  proved  are  equivalent 
to  the  following  algebraical  identities,  (i)  (a  -i-  \fY  =  ^(\^- 
(Eac.  llllf  1.)  (ii)  Conversely,  if  (i)  be  tme,  and  Aii  he 
taken  equal  to  a^  then  AB  will  lie  divided  at  //  in  a  golden 
■ecUon.  (Euc.  xiii,  2.)  (iii)  (6  -»•  ! a>*  =  5 (!fr)'.  (Euc.  xiii,  3.) 
{ir)  P-^li^^da^  (Euc.  XIII,  4.)  (v) /  +  a  :/  =  /:  a,  which  gives 
another  golden  section.  (Kuc.  xiii,  5.)  These  propositions 
were  subsequently  put  by  Euclid  at  the  commcncenient  of  his 
thirteenth  book,  Imt  they  might  have  lieen  equally  well  placed 
towards  the  end  of  the  second  book.  All  of  them  are  obvious 
algebraically,  since  i^a-^b  and  a' -  bi. 

Eudoxus  further  es^bliKhed  the  "method  of  exhaustions  '* ; 
which  depends  on  the  proposition  that  '*  if  from  the  greater 
of  two  unequal  magnitudes  there  be  taken  more  tlian  its  half, 
and  from  the  remainder  more  than  its  half,  and  so  on,  there 
will  at  length  n*main  a  magnitude  less  than  the  least  of  the 
proposed  magnitudes."  This  proposition  was  placed  by  Euclid 
as  tlie  first  proposition  of  the  tenth  book  of  his  EUmetiiBy 
but  in  most  modem  scIhioI  editions  it  is  printed  at  the 
beginning  of  the  twelfth  book.  By  the  aid  of  this  theorem 
the  ancient  geometers  were  able  to  avoid  the  use  of  infini- 
tesimals :  the  method  is  rigorous,  but  awkward  of  application. 
A  good  illustration  of  its  use  is  to  be  found  in  the  demon- 
stration of  Euc.  xil,  2,  namely,  that  the  square  of  the  radius 
of  one  circle  is  to  the  square  of  the  radius  of  another  circle 
as  the  area  of  the  first  circle  is  to  an  area  which  is  neither 
less  nor  greater  than  tlie  area  of  the  second  circle,  and  which 
therefore  must  lie  exactly  e«|ual  to  it:  the  proof  given  by 
Euclid  is  (as  was  usual)  completed  by  a  rednciio  ad  abturdum, 
Kadmus  applied  the  principle  to  shew  that  the  volume  of  a 
KfTHBid  (or  a  cone)  is  one-third  that  ct  the  prism  (or  cylinder) 

OMnMU  bMe.nnd  of  the  same  altitude  (Euc.  xii,  7  and  10). 
id  thai  he  proved  that  the  Tolumet  of  two  spheres 


48 


THB  MCHOOLB  OP  ATBBN8  AND  ICTUCO& 


were  to  one  anoilier  an  (be  ottbes  of  their  rfuiii ;  aome  wrileffs 
attribale  the  prupoeition  Eiic.  xii»  2  to  him,!  '^^  >>^  ^  Hippo- 
crAtee. 

EudoxuM  alau  considered  certain  curvee  other  than  the 
circle^  bat  there  seeiuii  to  be  no  authority  for  the  etatement^ 
which  is  found  in  some  old  books,  that  he  studied  the  proper- 
ties  of  the  oonic  sections.  He  discussHl  some  of  the  plane 
sections  of  the  anchor  ring,  that  is,  of  the  ik>lid  generated  by 
tlie  revolution  of  a  circle  round  a  straight  line  lying  in  its 
plane ;  but  he  assumed  that  the  line  did  not  cut  the  circle. 
A  section  by  a  plane  through  this  line  consists  of  two  circles ; 
if  the  plane  be  moved  parallel  to  itself  the  sections  are 
lemniscates;  when  the  plane  first  touches  the  surface  the 
section  is  a  ''figure  of  eight,"  generally  called  Bernoulli's 
lemniscate,  whose  equation  is  r'  =  rf*cos2^.  All  this  is  ex- 
plained at  length  in  books  on  solid  geometry.  Eudoxus 
applied  these  curves  to  explain  the  apparent  progressive  and 
retrograde  motions  of  the  planets,  but  we  do  not  know  the 
method  he  used. 

Eudoxus  constructed  an  orrery,  and  wrote  a  treatise  on 
practical  astronomy,  in  which  he  adopted  a  hypothesis  pre- 
viously propounded  by  Philolaus,  and  supposed  a  number  of 
moving  spheres  to  which  the  sun,  moon,  and  stars  were 
attached,  and  which  by  their  rotation  produced  the  effects 
obser\'ed.  In  all  he  required  twenty-seven  spheres.  As 
observations  became  more  accurate,  sulisequent  sstronomers 
who  accepted  the  theory  had  continually  to  introduce  fresh 
spheres  to  make  the  theory  agree  with  the  facts.  The  work 
of  Aratus  on  astronomy,  wliich  was  written  about  300  &c. 
and  is  still  extant,  is  founded  on  that  of  Eudoxus.    • 

Plato  and  Eudoxus  were  contemporaries.  Among  Plato's 
pupils  were  the  mathematicians  Leodamas,  Neocleidea, 
Amydas,  and  to  their  scliool  also  lielonged  Leon,  Theudiiui 
(both  of  whom  wrote  text-books  «m  plane  geometr}*),  Cysicd- 
,naa»  Thasua,  HermotimuB,  Fhilippua,  and  Theaetetnn. 
Among  the  pupils  of  Eudoxus  are  reckoned  Menaeohmua, 


I 


MENAECUMUS.  49 

hit  brother  Dinottraius  (who  applied  the  quadratrix  to  the 
duplication  and  trisection  problems),  and  Aristaeoa. 

Menaachnma.  Of  the  above-mentioned  mathematicians 
Jienaeekmu$  requires  Rpecial  mention.  He  was  bom  about 
375  B.GL  and  died  about  325  &r.  He  was  a  pupil  of  Eudoxus, 
and  probably  succeeded  him  as  head  of  the  school  at  CH'zicus. 
Menaechmuif  acquired  great  reputation  as  a  teacher  of  geo- 
nietrjy  and  was  for  that  reason  appointed  one  of  the  tutors 
of  Alexander  the  Great.  In  answer  to  his  pupiPs  request  to 
make  his  proofs  shorter,  he  made  the  well-known  reply  that 
tbongh  in  the  country  there  are  private  and  even  royal  roads, 
yet  in  geometry  there  is  only  one  road  for  all. 

Menaechmus  was  the  first  to  discuss  the  conic  sections, 
which  were  long  called  the  Menaechmian  triads.  He  divided 
them  into  three  classes,  and  investigated  their  properties,  not 
by  taking  different  plane  sections  of  a  fixed  cone,  but  by 
keeping  his  plane  fixed  and  cutting  it  by  difTerent  cones.  He 
shewed  that  the  section  of  a  right  cone  by  a  plane  perpen- 
dicular to  a  generator  is  an  ellipse,  if  the  cone  be  acute- 
angled  ;  a  larabola,  if  it  be  right-angled ;  and  a  h3rperlK>la,  if 
it  be  obtuse-angled ;  and  he  ga%'e  a  mechanical  construction 
for  cnivea  of  each  class.  It  seems  almost  certain  that  he  was 
acquainted  with  the  fundamental  properties  of  these  cur%'es ; 
but  some  writers  think  that  he  failed  to  connect  them  with 
the  sections  of  the  cone  which  he  hail  discovered,  and  there 
is  no  doubt  that  he  regarded  the  latter  not  as  plane  loci  but 
as  curves  drawn  on  the  surface  of  a  cone. 

He  also  shewed  how  these  cun-es  could  be  used  in  either 
of  the  two  following  ways  to  give  a  solution  of  the  problem 
to  duplicate  a  cube.  In  the  first  of  these,  he  pointed  out  that 
two  parabolas  having  a  common  vertex,  axes  at  right  angles, 
and  radi  that  the  Utns  rectum  of  the  one  is  double  that  of 
the  other  will  intersect  in  another  point  whose  abscissa  (or 
oidiiiate)  will  give  a  aolntion :  for  (using  analysis)  if  the 
ol  IIm  paimboiaa  he  j^^2ax mnd  g^'^aj^f  theyinter- 
whoio  abadna  is  given  by  af^ttf.    It  is 

4 


50  THE  8CHOOLH  OP  ATHBNB  AMDj  CTHCUH. 


probable  that  this  method  wm  mggeeted  bj  the  fom  In  whkh 
Hippocrates  had  cast  the  problem :  naoielj,  to  Ihid  x  and  jf  eo 
that  a  zx-sxzy  =  y  :  2a,  whenee  we  have  oe'say  and  y  »Saae. 

The  second  solution  given  by  Menaechmus  wss  as  loliowi. 
Describe  a  parabola  of  latiis  rectum  /.  Next  deseribe  a  rect- 
angular hyperbola,  the  length  of  whose  real  axis  is  41*  and 
having  for  its  asymptot^  the  tangent  at  the  vertex  of  the 
parabola  and  the  axis  of  the  parabola.  Then  the  ordinate  and 
the  abscissa  of  the  point  of  intersection  of  these  curves  are 
the  mean  proportionals  between  /  and  2/.  This  is  at  once 
obviouH  by  anal^'sis.  The  curves  are  x' ^  iff  end  ary  =  2i\ 
These  cut  in  a  puint  determined  by  a^  -•  2/'  and  y*  *  4/*. 
Hence  1 1  x  --.  x  :  y  -  y  :  ^L    . 

AriataeuB  and  Theaetetua.  Of  the  other  niemben  %d 
these  schools,  ArUtaeus  and  TkettetetUM^  whose  works  are 
entirely  lost,  were  luatheuiaticians  of  repute.  We  know  that 
Aristaeus  wrote  on  the  five  regular  solids  aiid  on  conic 
sections,  and  that  Theaetetus  developed  the  theory  of  incom- 
mensurable magnitudes.  The  only  theorem  we  can  now  de- 
finitely ascribe  tu  the  lattt^r  is  that  given  by  Euclid  in  the 
ninth  proposition  of  the  tenth  book  of  the  EUmentM^  namely, 
that  the  squares  on  two  coiumensurable  right  lin«^  hftveone 
to  the  other  a  ratio  which  a  square  number  has  to  a  square 
number  (and  conversely) ;  but  the  squares  on  two  incom- 
mensurable right  lines  have  one  to  the  other  a  ratio  ahich 
cannot  be  expressed  as  that  of  a  square  number  to  a  square 
number  (ami  conversely).     This  theorem  includes  the  results  '  \ 

given  by  Theodorus*.  1 

The  contemporaries  or  auccessors  of  these  mathematicians 
wrote  some  fresh  text-books  on  the  elements  of  geometry  and 
the  conic  sections,  intniduced  problems  concerned  with  Unding  i 

loci,  and  efficiently  carried  out  the  work  commenced  by  Plato  | 

of  systematizing  the  knowledge  already  acquired. 

Aristotle.  An  account  of  the  Atlieuian  school  would  be 
incomplete  if  there  were  no  mention  of  ArisioUe^  who  was  bom 

*  Bee  sbove,  p.  82. 


ARISTllTLE. 


51 


At  Htapra  in  Mncodonw  in  384  K,r,  and  died  at  Chalcis  in 
Eiiboea  in  322  ii.c.  Aristotle  however,  deeply  interested 
ihoagh  he  was  in  natural  philosophj,  was  chiefly  concerned 
with  mathematics  and  mathematical  physics  as  supplying 
illustrations  of  correct  reasoning.  A  small  book  containing  a 
few  questions  on  mechanics  which  is  sometimes  attributed  to 
him  is  of  doubtful  authority ;  but,  though  in  all  probability  it 
is  due  to  another  writer,  it  is  interesting,  partly  as  shewing 
that  the  principles  of  mechanics  were  beginning  to  excite 
attention,  and  partly  as  containing  the  earliest  known  em- 
pkijroent  of  letters  to  indicate  magnitudes. 

The  most  instructive  ftarts  of  the  book  are  the  d}'naniical 
proof  of  the  parallelogram  of  forces  for  the  direction  of  the 
resultant,  and  the  statement  that  "  if  a  lie  a  force,  fi  the  mass 
to  which  it  is  applied,  y  the  distance  through  which  it  is 
moved,  and  fi  the  time  of  the  motion,  then  a  will  move  |/3 
through  2y  in  the  time  ^  or  through  y  in  the  time  }  fi " :  but 
the  author  goes  on  to  say  that  "  it  does  not  follow  that  |  a 
will  move  fi  through  Jy  in  the  time  ^  because  }a  may  not  lie 
able  to  move  P  at  all ;  foi  100  men  may  drag  a  ship  100 
yards,  but  it  does  not  follow  that  oiwi  man  can  drag  it  one 
jrard."  The  first  part  of  this  statement  is  correct  and  is 
equivalent  to  the  statement  that  an  impulse  is  proportional  to 
the  momentum  produced,  but  the  second  part  is  wrong. 

The  author  also  states  the  fact  that  what  is  gained  in 
power  is  lost  in  speed,  and  therefore  that  two  weights  which 
keep  a  [weightless]  lever  in  equilibrium  are  inversely  pro- 
portional to  the  arms  of  the  lever;  this,  he  says,  is  the 
explanation  why  it  is  easier  to  extract  teeth  with  a  pair  of 
pinoeru  than  with  the  fingers. 

Among  other  questions  raised,  but  not  answered,  are  why 
a  prqjeetile  should  ever  stop^  and  why  carriages  with  Urge 
wbeeb  mn  easier  to  more  than  those  with  smalL  I  ought  to 
add  IW  Uin  iMok  eostaiiis  sobm  gross  blunders,  and  as  a 

TamelJTe  as  might  be  inferred  from 


62 


CHAPTER  IV. 


1 

-4 


THE  PIRST  ALBXANDRIAN  HCMOOL^ 

CIRC.  300  ac-SO  BLC 

Thb  Murliett  attempt  to  fomid  a  univenity,  mm  we  imdflnlMMl 
the  wurd,  was  made  at  Alexandria.  Richly  endowed,  impplied 
with  lecture  roomai  liUrariefl,  muaeumis  Uboratorieii  gaideii% 
and  all  the  plant  and  machinery  that  ingenuity  could  miggeMti 
it  became  at  once  tho  intellectual  metropolis  of  the  Greek  raoe^ 
and  remained  so  for  a  thouttand  yearr.  It  was  particularly 
fortunate  in  producing  within  the  first  century  of  its  existence 
three  of  the  greatest  mathematicians  of  antiquity — Euclid, 
Archimedes,  and  ApoUonius.  They  laid  down  the  lines  on 
which  mathematics  suUiequently  develo|ied,  and  treated  it  as 
a  subject  distinct  from  philosophy :  hence  the  foundation  of 
the  Alexandrian  Schools  is  rightly  taken  as  the  commencement 
of  a  new  era.     Thenceforward,  until  the  destruction  of  the 

*  The  hiitory  of  the  Alexsnclrisii  Scboolt  is  diiciuaed  by  Csntor, 
ehaps.  XII — xxin;  and  by  Gow  iu  his  UUtory  of  Greek  MaihemaiUt^ 
Cambridge,  1884.  Tbe  tubjt^k  of  Greek  algebra  it  treated  by  B.  H.  F. 
NeHselmaiiu  in  hit  Die-  Algebra  der  Grieckem,  Berlin,  1843;  tee  alto 
L.  Matlhiewen,  GruHdziige  dt'  antiktn  imii  wutderuen  Algebra  d^r 
liiieraleH  GUichungen,  Leipzig;,  1878.  Tbe  Greek  treatment  of  ths 
conie  tectioDi  formt  tbe  subject  of  Die  Lekrt  roa  den  KegeUekHiitem  is 
AlUrlum,  by  H.  O.  Zeuthen,  Copenhagen,  1886.  The  materUU  fdr  ths 
hUtoiy  of  tbete  tchoolt  bave  been  nubjeeted  to  a  tearehing  critidam 
by  S.  P.  Tannery,  and  muiti  of  hit  papers  are  collected  in  hit  Giomiirit 
Grecque,  Parin,  1887. 


4 


J 


THE   FIRSO*  ALBXANDRIAX  SCHOOL.  53 

citjT  by  the  Arab«  in  641  A.n.,  the  hintory  of  nmtheniatics 
cpntrt^  more  or  lem  round  that  of  Alexandria. 

The  city  and  univerKity  of  Alexandria  were  craited  under 
the  following  circuniHUinces.  Alexander  the  Great  had  as- 
cended the  thnme  of  Mace<1onia  in  336  ar.  at  the  early  a^  of 
twenty,  and  by  332  ac.  he  had  cohc|uerpd  or  Huhdued  Greece, 
Asia  Minor,  and  K;i^t.  Following  the  plan  he  adopted  wlien- 
e%*er  a  commanding  site  had  lieen  left  umiccupied,  he  founded  a 
new  city  on  the  ^[editer^anean  near  one  mouth  of  the  Nile ; 
and  he  himself  sketched  out  the  ground-plan,  and  arranged 
for  drafts  of  Greeks,  Egyptians,  and  Jews  to  lie  sent  to  occupy 
it.  Tlie  city  was  intended  to  be  the  most  magnificent  in  the 
world,  and,  the  lietter  to  secure  this,  its  erection  was  left  in 
the  hands  of  Dinocrates,  the  architect  of  the  temple  of  Diana 
at  Ephesus. 

After  Alexander's  death  in  323  B.C.  his  empire  w«s  di\i<led, 
and  Eg}'pt  fell  to  the  lot  of  Ptolemy,  who  chose  Alexandria 
as  the  capital  of  his  kingdom.  A  short  period  of  confusion 
followed,  but  as  soon  as  Ptolemy  was  settled  on  the  throne, 
say  about  306  B.C.,  he  determined  to  attract,  as  fsr  as  he  was 
able,  learned  men  of  all  sorts  to  his  new  city ;  and  he  at  once 
began  the  erection  of  the  university  buildings  on  a  piece  of 
ground  irom^xliately  adjoining  his  palace.  The  university  was 
ready  to  be  «ipeiied  somewhere  about  300  B.C.,  and  Ptolemy, 
who  withfd  to  sooure  for  its  stuff  the  most  eminent  pliilo- 
Miphen  of  the  time,  nntumlly  turtiMl  to  Athens  ko  find  them. 
The  great  library  which  was  the  central  feature  of  the  scheme 
was  placed  under  Demetrius  Phalereus,  a  distinguished 
Athenian ;  and  so  rapidly  did  it  grow  that  within  forty 
years  it  (together  with  the  Egyptisn  annexe)  possessed  about 
600,000  rolls.  The  mathematical  department  was  placed 
under  fiuclidy  who  was  thus  the  first,  as  he  was  one  of  the 
roost  lamoaa,  ^  ^^  mathematicians  of  the  Alexandrian 
sehooL 

It  happens  that  conteroponuieoiisly  with  the  fowidatioii 
of  this  sdiool  the  infomiatipii  on  whidi  oar  history  is  based 


M  THB  PIRST  AUUUNDRUN  liCHOOL. 

beoome*  more  ample  and  oertain.  Many  of  iho  works  of  the 
Alexandrian  muthematioiant  are  ttiU  extant ;  and  wo  bavo 
befudea  an  invaluable  treatise  by  Fsppusi  described  below,  in 
wliich  tbeir  bent  known  treatises  are  collated,  discuised,  and 
criticised.  It  curiously  turns  out  that  just  as  we  begin  to  be 
able  to  speak  with  certainty  on  the  subject-matter  wliicli  was 
taught,  we  find  that  our  information  as  to  the  personality 
of  the  teachers  becomes  uncertain ;  and  we  know  very  little 
of  the  lives  of  the  mathematicians  mentioned  in  this  and 
the  next  chapter,  even  the  dates  at  which  they  lived  being 
frequently  uncertain. 

The  third  century  be/are  ChrtMi, 

Euclid*.  This  century  produced  three  of  the  greatest 
mathematicians  of  antiquity,  namely  Kuclid,  Archimedes,  and 
ApoUonius.  The  earliest  of  these  was  EuelUI,  Of  hit  life  we 
know  next  to  nothing,  nave  that  he  was  of  Greek  descent, 
and  was  Ijom  about  330  n.c. ;  he  died  about  275  &c.  It  would 
appear  that  he  was  well  acc|uaiuted  with  the  Platonic  geometry, 
but  he  does  not  seem  to  have  read  Aristotle's  works;  and  these 
facts  are  supposed  to  strengthen  the  tradition  tliat  he  was 
educated  at  Athens.  Whatever  may  have  been  his  previous 
training  and  career,  he  proved  a  must  successful  teacher  when 
settled  at  Alexandria.  He  impressed  his  own  individuality  on 
the  teaching  of  the  new  university  to  such  an  extent  that  to 
his  Hucceiisors  and   almost  to  his  contemporaries  the  name 

*  litfrtidetf  C*ntur,  chaps.  &ii,  &111,  and  Gow,  pp.  7i-— 80,  195— SSI, 
mx  the  articlii  EucUuUs  by  A.  De  MitrKan  id  Smith*!  Dictionary  tf  Qngk 
and  UomuH  Itioyraithy,  Louclou,  1S49 ;  tlie  article  00  Irratiomal  Quamiitg 
by  A.  l>e  M(UK«n  in  the  Ptnuy  Cyclopaedia,  Londoo,  1839 ;  and  LiiUrmr* 
yr^'hickitii'hf  Studien  UUr  I'uklid,  by  J.  L.  lleibnTK.  Leipxig,  ISSS. 
Thts  latt'tit  compU'te  tuition  of  all  Euclid's  works  U  that  by  J.  L.  Heibsrg 
and  U.  Mt'UKt*.  Leipxig,  1883-1887.  Au  EugliMh  tran«latiou  of  the 
thirt«*en  boolui  of  the  EUmrtUt  was  published  by  J.  WilliamMm  ia 
S  vuluuien,  Oiford,  1781,  and  London,  1788,  but  the  notes  arc  not 
always  rrliable :  there  is  another  translation  by  Ituiac  Banow,  London 
and  Cambrtdics.  ISfiO. 


; 


i 


* 


RUCXID.  5.7 

Baclid  meant  (an  it  flora  to  uh)  the  liook  or  books  he  wrote, 
Mid  fiot  the  man  hinifielf.  Some  of  the  mediae^-al  writera 
went  so  far  an  to  denj  hiK  existence,  and  with  the  ingenuity 
of  philologintA  they  explained  that  the  tenn  was  only  a  corrop- 
tion  of  iNcAi  a  key,  and  &s  geometry.  The  former  word  wan 
presaniably  derived  from  kAccV  '  I  can  only  explain  the  mean- 
ing assigned  to  6cf  hy  the  conjecture  that  as  the  Pythagoreans 
said  thnt  the  nunilier  two  symliolisced  a  line  possibly  a  school- 
man may  have  tliought  that  it  could  lie  taken  as  indicative  of 
geometry. 

From  the  meagre  notices  of  Euclid  which  have  come  down 
to  us  we  find  that  the  sa^'ing  thnt  there  is  no  royal  mad  in 
geometry  was  attributed  to  Euclid  as  well  as  to  Menaechmus ; 
but  it  is  an  <*pigramniatic  remark  which  has  had  many  imi- 
tators. Euclid  is  also  said  to  have  insisted  that  knowledge 
was  worth  ac<|uiring  for  its  own  sake,  and  8toljaeus  (who  is  a 
somewhat  doubtful  authority)  tells  us  that,  when  a  lad  who 
had  just  begun  geometry  asked  **  What  do  I  gain  by  learning 
all  this  stuff*  f**,  Euclifl  made  his  slave  give  the  Isiy  some 
ooppersi  '*  since,**  said  he,  "  he  must  make  a  profit  out  of  what 
he  learns.**  According  to  tradition  he  was  noticeable  for  his 
gentleness  and  modesty. 

Euclid  was  tlie  author  of  several  works,  but  his  reputation 
rests  mainly  ou  his  Elenuint*.  This  treatise  contains  a  sys- 
tematic exposition  of  the  leading  propositions  of  elementary 
geometry  (exclusive  of  conic  sections)  and  of  the  theory  of 
numbers.  It  was  at  once  adopted  by  the  Greeks  as  the 
standard  text-book  on  the  elemeuts  of  pure  mathematics,  and 
it  is  probable  that  it  was  written  for  that  purpose  and  not  as 
a  philosophical  attempt  to  shew  tliat  the  results  of  georoetiy 
and  arithmetic  are  necessary  truths. 
)  The  modem  text^  is  founded  on  an  edition  prepared  by 

*  MoBlef  tbtBMdeni  lext-books  in  Englith  sre  founded  on  8iinson*s 

■saed  te  1758.    JMrrf  Sim»am,  who  wss  born  In  16S7  and  died 

"ilLinismiMMraf  ■sthfistics  at  the  aniversityof  Oksgow,  and 


56  THE  KIBST  ALKXANDftUK  8CB00L. 

Theon,  the  lather  of  HypAtia,  and  k  praetkally  m  tnuMfli^ 
of  Theon's  lectures  at  Alexandria  (eire.  380  aok).  Thera  is 
at  the  Vatican  a  copy  of  an  older  text,  and  we  have  bendee 
quotations  from  the  work  and  references  to  it  by  nunierona 
writers  of  various  dates.  From  these  sources  we  gather 
that  the  definitions,  axioms,  and  postulates  were  re-arranged 
and  slightly  altered  by  subsequent  editors,  but  that  the 
propositions  themselves  are  substantially  as  Euclid  wrote 
them. 

As  to  the  matUtr  of  the  work.  The  geometrical  part  is  to 
a  large  extent  a  compilation  from  the  works  of  previous 
writers.  Thus  the  substance  of  books  I  and  ii  is  probably 
due  to  Pythagoras ;  tliat  of  book  in  to  Hippocrates ;  that  of 
book  v  to  Eudoxus;  and  the  bulk  of  books  iv,  vi,  XI,  and  xil 
to  the  later  Pythagorean  or  Athenian  schools.  But  this 
material  was  rearrangtMl,  obvious  deductions  were  omitted 
(for  instance,  the  pruponition  that  the  perpendiculars  from  tlie 
angular  points  of  a  triungle  on  the  opposite  sides  meet  in  a 
point  was  cut  out),  and  in  sume  casen  new  proofs  substituted. 
The  part  concerned  with  the  theory  of  numbers  would  seem  to  { 

have  been  taken  from  the  works  of  Eudoxus  and  Pythagoras, 
except  tliat  |)ortion  (book  x)  which  deak  with  irrational 
nmgiiitudes.     This  latter  may  be  founded  on  the  lost  book  of  i 

TlieaetetUH ;  but  much  of  it  is  probably  original,  for  Proclus 
says  tliat  while  Eucliil  arranged  the  propotiitions  of  Eudoxus  f 

he  completed  many  of  tlioke  of  Theaetetus. 

The  way  in  aliich  the  propositions  are  proved,  consisting  i 

of  enunciation,  statement,  construction,  proof,  and  conclusion, 
is  due  to  Euclid :  so  nlso  is  the  synthetical  character  of  the 
work,  each  proof  being  written  out  us  a  logically  correct  train 
of  reasoning  but  without  any  clue  to  the  method  by  which  it 
was  obtained. 

The  defects  of  Euclid's  Element*  as  a  text-book  of  geometry 
have  been  often  stated  ;  the  most  prominent  are  these,  (i)  The 
definitions  and  axioms  contain  many  assumptions  which  ara 
not  obvious,  and   in   particular  the  so-called  axiom  about 


f 


EUCLID.  57 

pamllel  lines  is  not  self-evident^,  (ii)  No  explanation  is 
giren  as  to  the  reason  why  the  proofs  take  the  form  in  which 
thej  are  presented,  that  ih,  the  M^nthetical  proof  is  given  Uot 
not  the  analysis  by  which  it  was  obtained,  (iii)  There  is  ho 
attempt  made  to  generalize  the  results  arrived  at,  for  instance, 
the  idea  of  an  angle  is  never  extended  so  as  to  cover  the  case 
where  it  is  equal  to  or  greater  than  two  right  angles: — the 
second  half  of  the  thirty-third  propoMition  in  tlie  sixth  book,  as 
now  printed,  appears  to  lie  an  exception,  but  it  is  due  to  Theon 
and  not  to  Euclid,  (iv)  The  principle  of  superposition  as  a 
method  of  proof  might  \te  used  more  frequently  with  advantage, 
(v)  The  classification  is  imperfect.  And  (vi)  the  work  is  un- 
necessarily long  and  verbose. 

On  the  other  hand,  tlie  propositions  in  Euclid  are  arrangiHl 
so  as  to  form  a  chain  of  geometrical  reasoning,  proceeding 
from  certain  almost  obvious  assumptions  by  easy  steps  to 
results  of  cohsidt*rable  complexity.  The  demonstrations  are 
rigorous,  often  elegiint,  and  not  too  difiicult  for  a  beginner. 
Lastly,  nearly  all  the  elementary  metrical  (as  opposed  to  the 
graphical)  pro|»erties  of  space  are  investigated.  The  fact  that 
for  two  thousand  yeai-s  it  has  lieen  the  recognised  text-book 
on  the  subject  raises  further  a  strong  presumption  tliat  it  is 
not  unsuitable  for  the  purpose.  During  the  last  few  years 
some  determined  efforts  have  been  made  to  displace  it  in  our 
/  schools,  but  the  majority  of  teachers  still  appear  to  regard  it 
as  the  best  foundation  for  geometrical  teaching  that  has  been 
published :  the  liook  has  been  however  generally  abandoned 
on  the  continent.  To  those  arguments  in  its  favour  may  be 
added  the  fact  that  some  of  the  greatest  mathematicians  of 
modem  times,  such  as  Descartes,  Pascal,  Newton,  and 
Lagrange,  have  advocated  its  retention  as  a  text-book:  and 
Lagrange  said  that  he  who  did  not  study  geometry  in  Euclid 
would  be  as  one  who  should  learn  Latin  and  Gkvek  from 
modern  works .  written  in  those  tonguesi,     It  must  be  also 

*  It  «o«ld  SCOTS  flrosi  lbs  leseardMs  of 
UmI  it  is  ioeapabis  of  piooC. 


58  THK  HBST  ALBXAKOIUAV  SCHOOL.  I 

I 
remeoiberad  that  tliere  is  an  imnieiiM  advknUge  m  haTiDg  •  I 

•Ingle  text-book  in  universal  uiie  in  a  anbjlDet  like  geometrj.  I 

Tlie  unaattsfactory  condition  of  the  teaching  of  geometrical  I 

coiiicft  ill  ichooU  in  a  standard  illastratiuu  of  the  evils  lik«lj  to  I 

arise  from  using  dilferent  text-books  in  sud^  a  subject.     Some  1 

of  the  objections  urged  against  Euclid  do  niit  apply  to  certain  I 

of  the  recent  school  editions  of  his  UleuusuiMl  *  1 

I  do  not  think  that  all  tlie  cdijections  'above  stated  can  J 

fairly   be   urged   against   Euclid   himself.    !  He   published    a  a 

collection  of  problems  generally  known  ai|  the  Ac&i|am  or  I 

Daia.     This  contains  95  illustrations  of  the  kind  of  deductions  I 

which  frequently  have  to  be  made  in  uiialysis;  such  as  tliat,  if  I 

one  of  the  data  of  the  problem  under  consii Iteration  be  that  one  I 

angle  of  some  triangle  in  the  tig  are  is 'constant^  then  it  is  I 

legitimate  to  conclude  that  the  ratio  of  tlie  area  of  the  rectangle  I 

under  the  sides  oontHiuiiig  the  angle  to  the  area  of  the  triangle  I 

is  known  [prop.  G6].     Pappus  says  that  the  wofk  was  written  for  1 

those  **  who  wish  to  acquire  the  power  of  solving  problems."  I 

It  is  in  fact  a  graduated  series  of  exercises  in  geometrical  i 

analysis;  and  this  seems  a  sufficient  answer  to  the  second  i 

objection.  I  I 

Euclid  also  wrote  a  work  called   IIcpl  ^uupiatmv  or  De  1 

DivUionibuMf  known  to  us  only  through  an  Arabic  translation  I 

which  may  l)e  itself  imperfcH^t.     'Iliis  in  a  collection  of  36  \ 

problems  on  the  division  of  areas  into  parts  ^jrliich  bear  to  one  I 

another  a  given  ratio.     It  is  not  unlikely  tliat  this  was  only  j 

one  of  several  such  collections  of  examples — possibly  including 
the   Failncieti  and   the  PoriMiiu — but  even  by  itself  it  shews  { 

that  the  v«lue  of  exercises  and  riders  was  fully  recognized  by 
Euclid. 

I  may  here  add  a  suggestion  thrown  out  by  De  Morgan, 
who  is  perhaps  the  most  acute  of  all  the  modern  critics  of 
Euclid.  He  thinks  it  likely  that  the  ElemetUs  were  written 
towards  the  close  of  Euclid's  life,  and  tlist  their  present  form 
represents  tlie  tirst  draft  of  the  proposed  work,  which,  with  the 
exception  of  the  tenth  book,  Euclid  did  iu>t  live  to  revise.     If     * 


EUCLID.  59 

Ihk  o|niuon  be  correct^  it  is  prohable  that  Kaclid  woald  in 
bb  rerision  have  removed  the  fifth  objection. 

The  geometrical*  parts  of  the  EUmenfM  are  no  well  known 
that  I  need  do  no  more  than  allude  to  tliem.  The  first  fonr 
books  and  book  vi  deal  with  plane  y^eomotry ;  the  theory  of 
proportion  (of  any  ina^cnitudes)  is  di<tcuMsed  in  book  v ;  and 
bo^s  XI  and  xii  treat  of  solid  ^^eometry.  On  the  hypothesis 
thai  the  EUmetUt  are  the  first  draft  of  Euclid  s  proposed 
worky  it  is  possible  that  book  xiii  is  a  sort  of  appendix 
CDOtaining  some  additional  propositions  which  would  have 
been  put  ultimately  in  one  or  other  of  the  earlier  books. 
ThuSi  as  mentioned  above  the  first  live  propositions  which 
deal  with  a  line  cue  in  golden  section  might  be  added  to 
the  second  book.  The  next  seven  pro|Mmitions  are  concerned 
with  the  relations  between  certain  incommensurable  lines  in 
plane  figures  (such  as  the  radius  of  a  circle  snd  the  sides  of  an 
inscribed  regular  triangle,  pentagon,  hexagon,  and  decagon) 
which  are  treated  by  the  methods  of  the  tenth  bcMik  and  as  an 
ilKistration  <if  them.  The  five  regular  solids  are  discussed  in 
the  last  six  propositions.  Bretschneider  is  inclined  to  think 
that  the  thirteenth  liook  is  a  summary  of  part  of  the  lost  work 
of  Aristaeus :  but  the  illustrations  of  the  methods  of  the  tenth 
book  are  due  rocst  probably  to  Theaetetus. 

.Books. VII,  VIII,  IX,  and  x  of  the   KlementM  are  given  up 
/  to  the  theory  of  nuniliers.     The  mere  art  of  calculation   or 
Aoyurrunf  was  taught  to  Iwys  when  quite  young,  it  was  stig- 
matiied  by  Plato  as  childish,  and  never  recei%'ed  much  atten- 
tion from  Greek  mathematicians ;  nor  was  it  regarded  as  fbrm- 

*  Euclid  soppoaed  that  hi«  rcsdrrN  had  the  um  of  s  ruler  and  a  pair 
of  eonpamcs.  Lorento  Mnnciuroni  (who  was  horn  al  CanlagDeta  on 
May  14,  1750,  and  died  at  Parif  oa  July  BO.  1R00>  wt  hinmrlf  the  task  to 
obtain  by  neans  of  conntmetions  made  only  with  a  pair  of  eDinpaMics 
ths  laflM  results  as  Euelid  had  given.  Maselieroni*B  trvatias  on  the 
Kcometry  of  the  compass  which  was  pnbliiihed  at  Pavla  in  17115  \b 
80  earioQS  a  fovr  ie  font  that  it  is  worth  ehrooieling.  Hs  was  pro- 
ftrst  at  Bergamo  and  afterwards  at  Pavia,  and  left  oumeious  bIbot 


60 


THK  riRfrr  albxandkian  hchool. 


ing  fMuri  cif  m  ooorie  of  mAthematict.  We  do  not  knov  how  it 
was  taaglity  but  the  aliacutt  certainly  played  a  prominent  part 
in  it.  The  acienttlic  treatment  of  namhen  wan  called  ^a^/m- 
runjy  which  I  have  here  generally  trauMlated  as  the  science  of 
numbem.  It  had  special  rpferenoe  to  ration  proportion,  and  the 
theory  of  uuiubeRi.  It  is  with  this  alone  that  most  of  the 
extant  Greek  works  deal. 

In  disctissiug  Kiiclid's  arrangement  of  the  suhject^  we  most 
therefore  bear  in  mind  that  those  wlio  attended  his  lectures 
were  already  familiar  with  the  art  of  calculation.  The  system 
of  numeration  adopCtHl  by  the  (Greeks  is  described  later*,  but 
it  was  so  clumsy  that  it  rendered  the  scientific  treatment  of 
numbers  much  more  ditiicult  than  that  of  geometry ;  hence 
Euclid  ouniiiienced  his  matheuuitical  course  with  plane 
geometry.  At  the  same  time  it  must  be  observed  that  the 
results  of  the  second  book  though  geometrical  in  form  are 
capable  of  expression  in  algebraical  language,  and  the  fact 
that  numbers  could  be  repn*sented  by  lines  was  probably 
insisted  on  at  an  early  stage,  and  illustrated  by  concrete 
examples.  This  gra|iliical  method  of  using  lines  to  represent 
numbers  possesses  the  obvious  advantage  of  leading  to  proofs 
which  are  true  for  all  numliers,  rational  or  irrational  It  will 
be  noticed  that  among  other  propasitions  in  the  second  book 
we  get  geometrical  proofs  of  the  distributive  and  commutative 
laws,  of  rules  fi>r  multiplication,  and  finally  geometrical  solu- 
tions of  the  equations  a  (a  -  ^)  ^  .r*,  that  is,  x'-i-iix  — a'»0 
(£uc.  II,  11),  and  x*  -  m^  =  0  <  Euc.  ii,  1 4) :  the  solution  of  the 
first  of  these  ei|uatioiis  is  given  in  the  forui  ^/a*  +  (4a)*  "  J^'* 
The  solutions  of  the  equations  ax*- bx  -i-r.zO  and  ax*  +  &r-  e  =  0 
are  given  later  in  Euc.  vi,  28  and  vi,  29;  the  cases  when 
a  =B  1  can  be  deduced  from  the  identities  proved  in  Euc.  II, 
5  and  G,  but  it  is  doubtful  if  Euclid  recognized  this. 

The  results  of  the  fifth  bo4ik  in  which  the  theory  of  propor- 
tion is  considered  apply  to  any  magnitudes,  and  therefore  are 


] 


*  Hee  beluw,  chsp.  vii. 


EUCLin. 


01 


t 

t 


true  of  numbers  as  well  as  of  geoir  jtrical  magnitades.  In  the 
opinion  of  many  writers  this  is  the  most  satisfactory  way  of 
treating  the  theory  of  proportion  on  a  scientific  basis ;  and  it 
was  used  by  Enclid  as  the  foundation  on  which  he  built  the 
iheoiy  of  numbersw  The  theory  of  proportion  given  in  this 
book  is  believed  to  be  due  to  Eudoxus.  The  treatment  of  the 
same  subject  in  the  seventh  book  is  less  elegant,  and  is  sup- 
posett  to  be  a  reproduction  of  the  Pythagorean  teaching.  This 
double  discussion  of  proportion  is,  as  far  as  it  goes,  in  favour 
of  the  conjecture  that  Euclid  did  not  live  to  revise  the  work. 

In  books  VII,  viii,  and  ix  Euclid  discusses  the  tlieory  of 
rational  numbers.  He  commences  the  seventh  liook  with  some 
definitions  founded  on  the  Pythagorean  notation.  In  propo- 
sitions 1  to  3  he  shews  that  if,  in  the  usual  process  for  finding 
the  greatest  common  measure  of  two  numbers,  the  last  divisor 
be  unity,  the  numliers  must  be  prime ;  and  he  thence  deduces 
the  rule  for  finding  their  cs.c.M.  Propositions  4  to  22  include 
the  theory  of  fractions,  which  he  bases  on  the  theory  of  pro- 
portion ;  among  other  results  he  shews  that  ah  =  h<i  [prop.  16]. 
In  propositions  23  to  34  he  treats  of  prime  numliers,  giWng 
many  of  the  theorems  in  modem  text-books  on  algebra.  .  In 
propositions  35  to  41  he  discusses  the  least  common  multiple 
of  numbers,  and  some  miscellaneous  problems. 

The  eighth  book  is  chieOy  devoted  to  numbers  in  continued 
.  proportion,  that  is,  in  a  geometrical  progression  ;  and  the  cases 
I  where  one  or  more  is  a  product,  square,  or  cube  are  specially 
considered. 

In  the  ninth  book  Euclid  continues  the  discussion  of  geo- 
metrical progressions,  and  in  proposition  35  he  enunciates  the 
rule  for  the  summation  of  a  series  of  n  terms,  though  the 
proof  is  given  only  for  the  case  where  n  is  equal  to  4.  He 
also  develops  the  theoiy  of  primes,  shews  that  the  number  of 
primes  is  inBnite  [prop.  20],  and  discusses  the  properties  of 
odd  and  even  numbers.  He  concludes  by  shewing  that  a 
number  of  the  form  2*^*  (2*--lX  ^^^  ^*  1  is  a  primes  is  a 
**  perbei  **  number  [propi  36]. 


I 


tfS  THK  mm  AUXAVDUAN  8CHOOI.. 

Ill  the  tenth  book  Eadid  treete  e(  irimtioiial  nagnilwlee; 
and,  linoe  the  Greeks  poBseaaed  no  BymboUam  for  toid^  he  wee 
forced  to  adopt  a  geooietrical  reprMentation.  Propoiitioiie  1 
to  21  deal  generally  with  inoonimensurabld  magnitudes.  Hie 
rent  of  the  book,  namely,  propositions  22  to  117,  is  devoted  to 
the  discussion  of  every  possible  variety  of  lines  which  can  be 
represented  by  J(Ja  a  ^/6),  where  d  and  b  denote  oommensoiy 
able  lines.  There  are  twenty-tive  species  of  such  lines,  and 
that  Euclid  could  detect  and  classify  them  all  is  in  tlie  opinion 
of  so  competent  an  authority  as  Nesaelmann  the  most  striking 
illustration  of  his  geniuii.  It  seems  at  tirst  almost  impossible 
that  thin  could  have  been  done  without  the  aid  of  algebra,  bat 
it  is  tolerably  certain  that  it  was  actually  effected  by  abstract 
reasoning.  No  further  advance  in  the  theory  of  incom- 
mensurable magnitudes  was  made  until  the  subject  was  taken 
up  by  Leonardo  and  i*ardan  after  an  interval  of  more  than  a 
thousand  years. 

In  the  lost  proposition  of  the  tenth  book  [prop.  117]  the 
side  and  diagonal  of  a  square  are  proved  to  lie  incommensur- 
able. The  proof  is  so  short  and  easy  tliat  I  may  quote  it.  If 
possible  let  the  side  lie  to  the  diagonal  in  a  commensurable 
ratio,  namely,  that  of  the  two  integers  a  and  6.  Suppose  this 
ratio  reduci*d  to  its  lowest  terms  so  that  a  and  6  have  no 
common  divisor  other  than  unity,  that  is,  they  are  prime  to 
one  another.  Then  (by  Euc.  i,  47)  6*  =  2a* ;  therefore  6*  is  an 
even  nuiuber ;  therefore  6  is  an  even  number ;  hence,  since  a  is 
prime  to  6,  a  must  be  an  odd  number.  Again,  since  it  has 
been  shewn  that  b  is  an  even  number,  b  may  be  represented 
by  2/1 ;  therefore  (2ii)'«2a';  therefore  a*  =  2M';  therefore  a* 
is  an  even  nuinlier;  therefore  a  is  an  even  number.  Thus  the 
same  iiumlier  a  must  lie  both  odd  and  even,  which  is  absurd  ; 
therefore  the  side  and  diagonal  are  incommensurable.  Hankel 
believes  that  this  proof  was  due  to  P)'thagoras,  and  was 
inserted  on  account  of  its  historical  interest  lliis  proposition 
is  also  proved  in  another  way  in  Euc.  x,  9. 

In  addition  to  the  Elen^utu  and  the  two  oolleetions  of 


eucuD. 


69 


( 


i 


riders  above  mentknied  (which  are  extant)  Euclid  wrote  the 
Mlowing  books  on  geometry :  (i)  an  elementary  treatise  on 
eonie  9tcium$  in  fonr  Inrnks  ;  (ii)  a  book  on  cnrved  Murfhret 
(probably  chiefly  the  oon^  and  cylinder);  (iii)  a  collection  of 
gfnrneiticai  Jaffacie$y  which  werp  to  be  used  as  exercises  in  the 
detection  of  errors ;  and  (iv)  a  treatise  on  pori»m$  arranged  in 
three  books.  All  of  these  are  lo^t,  bat  the  work  on  porisros  was 
discassed  at  sach  length  by  Pappus,  that  some  writers  have 
thought  it  possible  to  restore  it.  In  particular  Chasles  in  1860 
published  what  purports  to  lie  a  reprorluGtion  of  it,  in  which 
will  lie  found  the  conceptions  of  cross  ratios  and  projection — 
in  fact  those  ideas  of  modem  geometry  which  Chasles  and  other 
writers  of  this  century  have  used  so  largely.  This  is  brilliant 
Mid  ingenious,  and  of  course  no  one  can  prove  that  it  is  not 
exactly  what  Euclid  wrote,  but  the  statements  of  Pappus  con- 
cerning this  book  have  come  to  us  only  in  a  mutilated  form, 
and  I>e  Morgan  frankly  says  that  he  found  them  unintelligible, 
an  opinion  in  which  most  «if  those  who  read  them  will,  I  think, 
concur. 

Euclid  published  two  liooks  on  optics,  namely  the  Optic$ 
and  the  Caiofttrica.  Of  these  the  former  is  extant.  A  work 
which  purports  to  lie  the  Utter  exists  in  the  form  of  an 
Arabic  translation,  but  there  is  some  doul>t  as  to  whether 
it  represents  the  original  work  written  by  Euclid ;  in  any 
ease,  the  text  is  corrupt.  The  Optictt  commences  with  the 
assumption  that  objects  are  seen  by  rays  emitted  from  the 
eye  in  straight  lines,  '*for  if  light  proceeded  from  the  object 
we  should  not,  as  we  often  do,  fail  to  perceive  a  needle  on  the 
floor."  It  contains  61  propositions  founded  on  12  assump- 
tions. The  Caioptrica  consists  of  31  propositions  dealing  with 
reflexions  in  plane,  convex,  and  concave  mirrors.  The  geometry 
of  both  books  is  Euclidean  in  form. 

Eadid  has  been  credited  with  an  ingenious  demonstration* 
of  the  principle  of  the  lever,  but  its  authenticity  is  doobtfuL 

«ItiBsivHi(flkoiS  the  Arable)  bjF.  Woepehelnths/Mnisliltisllfar, 
•tries  4,  voL  xvm,  October,  1851,  pp.  SS5— SSt. 


64 


THi  ram  alkxandrian  whool. 


He  alio  wrote  the  PAoMumena^  m  traaibe  on  fMselriod 
MtTODomy,  It  conUliM  referenon  to  Um  work  o(  A«loljeMi^ 
and  to  •ooie  book  on  ipherical  geometry  by  an  nnknovn 
writer.  Fappot  asserts  that  Euclid  also  oomposed  a  book  on 
the  elements  of  masio :  this  may  refer  to  the  Sedio  Cmm&mi$ 
which  is  by  Euclid,  and  deals  with  musical  intervals. 

To  thne  works  I  may  add  the  following  little  problem, 
which  occurs  in  the  Palatine  Anthology  and  is  attribnted  by 
tradition  to  Euclid.  (^A  mule  and  a  donkey  were  going  to 


market  laden  with  wheat.  Tlie  mule  said  'If  you  gave 
one  measure  I  iiliould  carry  twice  as  much  as  you,  bat  if 
I  gave  you  one  we  sliould  bear  equal  burdeniL*  T^  me^ 
learned  geometrician,  what  were  their  burdens.*/  It  is  im* 
possible  to  say  whether  the  question  is  due  fo  Euclid,  bat 
there  is  nothing  improbable  in  the  suggestion. 

It  will  be  noticed  that  Euclid  dealt  only  with  magni- 
tudes, and  did  not  concern  himself  with  their  numerical 
mesHures,  Imt  it  would  seem  from  the  works  of  Aristarchns 
and  Archimedes  that  this  was  not  the  case  with  all  the 
Orpek  mathematicians  of  tliat  time.  As  one  of  the  works 
of  the  former  in  extant  it  will  serve  as  another  illustration 
of  Greek  mathematics  of  this  period. 

ArifltarchuB.  Arisiarchnt  o/'Sawu^  bom  in  310  &a  and 
died  in  250  ii.c,  was  an  astronomer  rather  than  a  mathema- 
tician. He  asserted,  at  any  rate  as  a  working  hypothesis,  that 
the  sun  was  the  centre  of  the  universe,  and  that  the  earth 
revolved  round  the  sun.  This  view,  in  spite  of  the  simple 
explanation  it  afforded  of  various  phenomena,  was  generally 
rejected  by  his  contemporaries     But  his  propositionst  on  the 

*  Amioljft'tu  lived  at  PiUne  in  Acolis  and  ioariidMd  aboQl  SSO  sx. 
Hi«  two  worka  ou  astrooomy,  coataining  4S  j(wt>po«iiH>BS,  aic  the  oldsst 
eitaat  Greek  matlieiiuitical  treatiaes.  Thsj  aiist  ia  maauaeripl  at 
Oifonl.  A  LatiD  traaiaatioD  ba«  been  adiiea  by  F.  Hnltach,  Leipsia.  I88S. 

t  n«^  |uy#Mr  c«A  4w^€T^^Unm  UXim  ral  ZtX^r^f.  edited  by  E.  Xisss. 
8tralfttnd.  lH3e.  Latio  tran«latioos  were  isAOcd  by  F.  Comnisndino  im 
ISU  and  by  J.  Wallia  in  16h8  ;  and  a  Franch  traa^latka  was 
by  F.  d*Urban  ia  1810  and  Itttt. 


ARlSTARCHUa 


65 


nMamireineiit  of  the  suen  and  clistances  of  the  son  and  moon 
were  aocarate  in  principle,  and  his  results  were  generally  ac- 
cepted (for  example,  hy  Archimedes  in  his  i^afifunff,  mentioned 
below)  as  approximately  correct.  There  are  19  theorems,  of 
which  I  select  the  seventh  as  a  tjrpical  illustration,  because  it 
shews  the  way  in  which  the  Greeks  evaded  the  difficulty  of 
finding  the  numerical  value  of  surds. 

Aristarchus  observed  the  angular  distance  between  the 
moon  when  dichotomized  and  the  sun,  and  found  it  to  be 
twenty-nine  thirtieths  of  a  right  angle.  It  is  actually  about 
89*  2r,  but  of  course  his  instruments  were  of  the  roughest 
description.  He  then  proceeded  to  shew  that  the  distance  of 
the  ran  is  greater  tlian  eighteen  and  less  than  twenty  times 
the  distanoe  of  the  moon  in  the  following  manner. 

Lei  /?  be  the  sun,  E  the  earth,  and  M  the  moon.  Then 
when  the  noon  is  dichotomised,  that  is,  when  the  bright  part 
which  we  see  is  exactly  a  half-circle,  the  angle  between  M8 


mni  MS  km  rigjii  angle.  With  S  as  centre,  and  radii  £8 
and  SJf  diteribe  ciroki^  as  b  the  figure  abo?«.  Draw  SA 
parpmUcdar  to  Jft    Draw  EF  bbecti^g  the  angle  ABSf 


66  THE  riRST  ALBXANDEIAH  SCHOOL. 

mod£G  bbectiog  the  angle  A^*\  m  in  Um  flgsra.  IM  MM 
(fMrodiioed)  eut  AF  in  //.  The  angle  A£M  k  bj  IqrpotheM 
^^^th  of  a  right  angle.     Hence  we  have 

angle  A  EG  :  angle  AEH  ^  \  tt.  4,  i^wt,  i  »  l^  i% 

.\    AG  :  Aii[==tMnAEG  :  tan  AEH]>1&  :  2 (a). 

Again       FG'zAG'^EF':  EA'{E^e.  yi,  3)=:2 : 1  (Ene.  1, 47K 

/.  FG' :  AG*  >A9  :2b, 

:.  FG  xAG  >1  .b, 

:.  AF  .AG  >Uib, 

:.  AE  .AG  >12:6 (/B). 

Compounding  the  ratios  (a)  and  {fi),  we  have 

AE'.Aii>\%:\. 

But  the  triangles  EMS  and  EA  U  are  simikr, 

/.   AW  iEM>U.\. 

I  will  leave  the  second  half  of  the  proposition  to  anittse 
any  reader  who  may  care  to  prove  it :  the  analysts  is  straight- 
forward.  In  a  somewhat  similar  way  Aristarchns  ionnd  the 
ratio  of  the  radii  of  the  sun,  earth,  and  moon. 

We  know  very  little  of  Conon  and  T^^ftlWWfc  the 
immediate  sucoeHsors  of  Euclid  at  Alexandria,  or  of  their 
contemporaries  ZeuzippuB  and  NicotaleSi  who  most  likely 
also  lectured  there,  except  that  Arehimedes,  who  was  a  stu- 
dent at  Alexandria  prolmbly  shortly  after  Euclid's  death, 
had  a  high  opinion  «»f  their  ability  and  corresponded  with 
the  three  first  mentioned.  Their  work  and  reputation  has 
been  completely  overshadowed  by  tlmt  of  Arehimedes  whose 
nmrvellous  mathematical  powers  have  been  surpassed  only 
by  those  of  Newton. 

Archixnedea*.     Architnedes,  who  probably  was  related  to 

*  BcHidefi  Cantor,  chaps,  uv,  iv,  au«l  Gow,  pp.  Stil — 244,  see 
Quaettioue*  Archimedeae,  l*jr  J.  L.  Heiber^.  CopeDhsg«n,  1879  ;  snd  Marie, 
vol.  I,  pp.  81—134.  Tli«  UteMt  and  hv^i  edition  of  the  eitsnt  works  of 
Archimedes  is  tlist  by  J.  U  Ueiberg.  in  3  vols.,  Leipjti({,  1880-1881. 


1. 


ARCHIMEDES. 


67 


the  rojftl  family  at  Syrmenwe^  was  bom  there  in  287  &c.  and 
died  in  212  &c.  He  went  to  the  nniversity  of  Alexandria 
aad  attended  the  lectures  of  Gonon  hot,  as  soon  as  he  had 
finished  his  studies,  returned  to  Sicily  where  he  passed  the 
remainder  of  his  life  He  took  no  part  in  public  affairs, 
hot  his  mechanical  ingenuity  was  astonishing,  and,  on 
any  difficulties  which  oould  be  overcome  by  material 
means  arising,  his  advice  was  generally  asked  by  the  govern- 
ment. 

Archimedes,  like  Plato,  held  that  it  was  undesirable  for 
a  philosopher  to  seek  to  apply  the  results  of  science  to  any 
practical  use;  but,  whatever  might  have  lieen  his  view  of 
what  ought  to  be  the  case,  he  did  actually  introduce  a  Itivget 
number  of  new  inventions.  The  stories  of  the  detection  of 
the  fraudulent  goldsmith  and  of  the  use  of  burning-glasses 
to  destroy  the  ships  of  the  Roman  blockading  squadron  will 
recur  to  most  readen.  Perhaps  it  is  not  as  well  known  that 
Hiero,  who  had  built  a  ship  so  large  that  he  could  not  launch 
it  off  the  slips,  applied  to  Archimedes.  The  difficulty  was 
overcome  by  means  of  an  apparatus  of  cogwheels  worked 
by  an  endless  screw,  but  we  are  not  told  exactly  how  the 
machine  was  used.  It  is  said  that' it  was  on  this  occasion, 
in  adcnowledging  the  compliments  of  Hiero,  that  Archimedes 
made  the  well-known  remark  that  had  he  but  a  fixed  fulcrum 
be  could  move  the  earth. 

Most  mathematicians  are  aw«re  that  the  Archimedean 
ierew  was  another  of  his  inventions.  It  consists  of  a  tube, 
open  at  both  ends,  and  bent  into  the  form  of  a  spiral  like 
a  oork-eorew.  If  one  end  be  immened  in  water,  and  the  axis 
of  Um  instrument  {L€.  the  axis  of  the  cylinder  on  the  surfkoe 
oC  which  the  tube  lies)  be  inclined  to  the  vertical  at  a  suffi- 
eientfy  big  angle,  and  the  instrument  turned  round  it,  the 
water  will  flow  along  the  tube  and  out  at  the  other  end.  In 
order  that  it  may  work,  the  inclination  of  the  axis  of  the 
hutnuneat  to  the  vertical  must  be  greater  than  the  pitch  of 
Ihoaerew.    It  waa  need  in  ^gypi  to  drain  the  fiekb  alter  an 


08  THK  naar  AuocAVDEiAir  wbool. 

inondatioii  of  the  Nile;  And  was  abo  feequenUy  HV'^  ^ 
take  water  out  of  the  hold  of  a  ■hip. 

The  story  that  Archimedea  let  fire  to  the  Roomui  ahipa 
hy  means  of  burning-glasses  and  eoncaye  mirrors  is  not  men* 
tioned  till  some  centuries  after  his  death,  and  is  generally 
rejected :  but  it  is  not  so  incredible  as  is  oommonly  supposed. 
The  mirror  of  Archimedes  is  said  to  have  been  made  of  a 
hexMgon  surrounded  by  several  polygons,  each  of  24  sides; 
and  Buflbn^  in  1747  contriv^  with  the  aid  of  a  single  com- 
poftite  mirror  made  on  this  model  with  1 68 1  small  mirrors,  to 
set  fire  to  wood  at  a  distance  of  150  feet,  and  to  melt  lead 
at  a  distance  of  140  feet.     This  was  in  April  and  in  Pkris,  so 
in  a  Sicilian  summer  and  with  several  mirroi|s  the  deed  would 
be  possible,  and  if  the  ships  were  anchored  near  the  town 
would  not  be  ditKcult.     It  is  perhaps  worth ;  uieiiticming  that 
a  similar  device  is  said  to  have  lieen  used  in  the  defence  of 
Constantinople  in  514  A.p.,  and  is  alluded  to  by  writers  who 
either  were  present  at  the  siege  or  obtained  tlieir  information 
from  those  who  were  engagtKl  in  it     But  whatever  be  the 
truth  as  to  this  story,  there  is  no  doubt  that  Archimedes 
devised  the  catapults  which  kept  the  Romany  who  were  then 
besieging  Hyracuse,  at  bay  for  a  considerable  time.     These 
were  constructed  so  that  the  range  could  be  made  either  short 
or  long  at  pleasure,  and  so  that  they  could  be  discharged 
through  a  small  loophole  without  exposing  ihe  artillerymen 
to  the  fire  of  the  enemy.     So  effective  did  they  prove  that 
the  siege  was  turned  into  a  blockade,  and  three  years  elapsed 
before  the  town  was  taken  (212  &C.).  i 

Archimedes  was  killed  during  the  sack  off  the  city  which 
followed  its  capture,  in  spite  of  the  orders,  giv^n  by  the  consul 
Marcellus  who  was  in  command  of  the  Romanit,  that  his  house 
and  life  should  be  spared.  It  is  said  that  a'  soldier  entered 
his  study  while  he  was  regarding  a  geometrical  diagram 
drawn  in   sand  on  the  floor,  which  was  the  usual  way  of 

*  See  Mimoirti  de  Vaeadimu  royaU  det  teitmcek  for  1747,  Paris, 
1752.  pp.  62—101.  ! 


i 


ARCHIMEDES.  69 

drawing  figures  in  classical  times.  Archimedes  told  him  to 
get  off  the  diagram,  and  not  spoil  it.  The  soldier,  feeling 
insnlted  at  having  orders  given  to  him  and  ignomnt  of  who 
the  M  roan  was,  killed  him.  Acconling  to  another  and  more 
probable  aoconnt,  the  cupidity  of  the  troops  was  excited  by 
seeing  his  instruments,  constructed  of  polished  brass  which 
they  supposed  to  he  made  of  gold. 

The  Romans  erected  a  splendid  tomb  to  Archimedes  on 
which  was  engraved  (in  accordance  with  a  wish  he  had 
expressed)  the  figure  of  a  sphere  inscribed  in  a  cylinder,  in 
oommemoration  of  the  proof  he  had  given  that  the  volume 
of  a  sphere  was  equsl  to  twu-thinls  that  of  tfie  circumscribing 
right  cylinder,  *and  its  surface  to  four  times  the  area  of  a  great 
circle.  Cicero^  gi^M  a  charming  account  of  his  eflbrts  (which 
were  successful)  to  re-discover  the  tomb  in  75  B.C. 

It  is  difficult  to  explain  in  a  concise  form  the  works  or 
discoveries  of  Archimedes,  partly  because  he  wrote  on  nearly 
all  the  mathematical  subjects  then  known,  and  partly  because 
his  writings  are  contained  in  a  series  of  disconnected  mono- 
graphs. Thus,  while  Euclid  aimed  at  producing  sjntematic 
treatises  which  could  be  understood  by  all  students  who 
had  attained  a  certain  level  of  education,  Archimedes  wrote 
a  number  of  brilliant  essays  addressed  cliieOy  to  the  most 
educated  mathematicians  of  the  day.  The  work  for  which  he 
is  perhaps  now  best  known  is  his  treatment  of  the  mechanics 
of  solids  and  fluids ;  but  he  and  his  contemporaries  esteemed 
his  geometrical  discoveries  of  the  quadrature  of  a  parabolic 
area  and  of  a  spherical  surface,  and  his  rule  for  finding  the 
vuiuroe  of  a  sphere  as  more  remarkable;  while  at  a  some- 
what later  time  his  numerous  mechanical  inventions  excited 
nost  attention. 

(i)  On  pfane  geometry  the  extant  worics  of  Archimedes  are 
three  in  number,  namely,  (a)  the  Meature  t^the  Cirde^  (6)  the 
Qmadraiure  oftk^.  PamMoy  and  (e)  one  on  SpiraU, 

(a)    The  Meatmn  ^  the  Cireie  contains  three  propositionB. 
*  8ss  his  TJitruimmmm  DiafmimUmimtm^  v,  tlL 


70  THE  PIB8T  ALIXANDBUll  80HOQU 

In  tha  first  propotitioa  Arehimedet  provM  that  Am  mnm  k  IIm 
same  as  that  of  a  right-angled  triangle  wbose  sides  aro  ei|aal 
respectively  to  the  rsdius  a  and  the  circnmferenoe  of  the  eirale^ 
•'.••  the  area  is  equal  to  }a  (2«a).  In  the  seoond  prapositioii 
he  shews  tliat  «a*  :  (2a)' ss  U  :  14  very  nearly ;  and  nexti  in 
the  third  proposition,  that  w  is  less  than  Sf  and  greater  than 
3f  f.  These  theorems  are  of  coarse  proved  geometrically.  To 
deuionstrate  the  two  latter  propositions,  he  inscribes  in  and 
circumscribes  about  a  circle  r^ular  polygons  of  ninety  six 
sides,  calculates  their  perimeters,  and  then  assumes  the  eir> 
cumference  of  the  circle  to  lie  between  them :  this  leads  to 
the  result  6336/ 2017 1  <ir  <  14688/ 4673|,  from  which  hede- 
duces  the  limits  given  above.  It  would  seem  from  the  |»roof 
that  he  had  some  (at  present  unknown)  method  of  extracting 
the  square  roots  of  numbers  approximately.  The  table  which 
he  formed  of  the  numerical  values  of  the  chords  of  a  circle 
is  essentially  a  table  of  natural  sines,  and  may  have  sug- 
gested the  subsequent  work  on  these  lines  of  Hipparchus 
and  Ptolemy. 

(b)  The  QiuidraiHTfi  of  the  Parabola  contains  twenty- 
four  propuHitions.  Archimedes  liegins  this  work,  which  was 
sent  to  DuHitheus,  by  establishing  some  properties  of  conies 
[props.  1 — 5].  He  then  states  correctly  the  area  cut  off  from 
a  parabola  by  any  chord,  and  gives  a  proof  which  rests  on 
a  preliminary  mechr.nical  experiment  of  the  ratio  of  areas 
which  balance  when  suspended  from  the  arms  of  a  lever 
[props.  6 — 17];  and  laatly  he  gives  a  geometrical  demonstra- 
tion of  tliiM  result  [props.  18 — 24]^  The  latter  is  of  course 
based  on  the  method  of  exhaustions,  but  for  brevity  I  will,  in 
quoting  it,  use  the  method  of  limits.  i 

Let  the  area  of  the  parabola  (see  figure  on  next  page)  be 
bounded  by  the  chord  PQ,  Draw  YM  the  diameter  to  the 
chord  PQ^  then  (by  a  previous  proposition),  V  is  more  remote 
from  PQ  than  any  other  point  in  the  arc  PVQ.  Let  the 
area  of  the  triangle  PVQ  be  denoted  by  A.  In  the  segments 
bounded  by  VP  and  VQ  inscrilie  triangles  in  the  same  way  as 


ARCHIMEDBi 


71 


the  triMigle  PVQ  wiis  inncribed  in  the  given  segment.  Each  of 
these  trienglee  is  (bj  a  previous  proposition  of  his)  equal  to  \^ 
and  their  som  is  therefore  jA.  Simiiarlj  in  the  four  segments 
left  inscribe  triangles ;  their  sam  will  lie  ^V^.  Proceeding  in 
this  way  the  area  of  the  given  segment  is  shewn  to  be  equal 
to  the  limit  of 


n-^ 


when  ft  is  indefinitelj  large. 

Hie  problem  is  therefore  reduced  to  finding  the  sum 
of  9k  geometrical  series.  This  he  effects  as  follows.  Let 
A^  B^  C9  ;..|  /,  iT  be  a  series  of  magnitudes  such  that  each 


is  one-fourth  of  that  which  precedes  it.    Take  magnitudes 
6^  e,  ...»  k  equal  respectively  to  ^B^  jr,  ...,  J/T.    Then 

Hence  (i?  +  C+  ...  +ir)-i-(6  +  c+  ...  -••lr)  =  l(-4 +i? +  ...+/); 
bn^fajhjrpotbesis,(64.c4...-i->-Kib)=rJ(^4.C-i-...-fy)4.|ir; 


72  TBI  nBflT  AUXANINUAV  8CH0OU 

Henoa  tbe  ram  of  theae  magnitudai  axoeeds  Coor  tiowt  thm 
third  of  tho  burgest  of  them  by  one-thiid  of  tbe  ■ndliwt  of 
thorn. 

Rotuming  now  to  the  proUem  of  the  quadrmture  of  Iho 
parabohk  A  staiids  for  A,  aud  ultimately  K  is  imfefinitely 
■mall;  therefore  the  area  of  the  parabolic  aegmept  ia  iom^ 
thirds  that  of  the  triangle  PVQ,  or  two-thirds  that  of 
a  rectangle  whose  base  is  J^  and  altitude  tbe  distance  of 
r  from  PQ. 

Wliile  discussing  the  question  of  quadratures  it  may  be 
addetl  that  in  the  fifth  and  sixth  propositions  of  his  work 
on  conoids  and  spheroids  he  determined  the  area  of  an 
ellipse. 

(c)  The  work  on  SyiraU  contains  twenty-eight  proposi- 
tions on  Uie  properties  of  the  curve  now  known  as  the  spiral 
of  Archimedes.  It  was  sent  to  Dositheus  at  Alexandria  accom- 
panied by  a  letter,  from  which  it  appears  that  Archimedes  had 
previounly  sent  a  note  of  his  results  to  Conon,  who  had  died 
before  he  had  been  able  to  prove  tlieui.  The  spiral  is  defined 
by  saying  that  the  vectorial  angle  and  radius  vector  both 
increase  uniformly,  hence  its  equation  is  r  =  c^.  Archimedes 
finds  most  of  its  properties,  and  determines  the  area  inclosed 
between  the  curve  and  two  radii  vectores.  This  lie  does  (in 
effect)  b}'  saying,  in  the  language  of  the  infinitesimal  calculus, 
that  au  element  of  area  is  >  }  i'^dS  and  <  |(r  ••-  drYdO :  to  effect 
the  sum  of  the  elementary  areas  he  gives  two  lemmas  in  which 
he  sums  (geometrically)  the  series  a* -^  (2a)'  ••-  (3#i)' -r  ...  -i- (tia)* 
[prop.  10],  and  ii  •»-  ^u  <••  30  -»-  ...•»-  imi  [prop.  11]. 

(d)  In  addition  to  these  he  wrote  a  small  treatise  on  * 
geameiricai  tuethotU^  and  works  on  pandM  litteM^  irianylrit^  ike 
properties  of  riyhi-mntjled  triangles^  f/a/a,  the  Keptayon  iuseribed 

in  a  circle^  and  syBtems  o/cirtieg  iouchittg  one  another ;  possibly 
he  wrote  others  too.    Tliese  are  all  lost,  but  it  is  probable  that 


ARCHIMEDES. 


7d 


fragmenU  of  four  of  the  propositions  in  the  huit-mcntioned 
work  are  preserved  in  a  I^tin  translation  from  an  Arabic 
manuscript  entitled  Lemmas  of  Archimfntet. 

(ii)  On  ffeometry  of  three  dimenitiotui  the  extant  works 
of  Archimedes  are  two  in  number,  namely,  (a)  the  Sphere  and 
Cyiinder^  and  (6)  Ctrnouh  and  Spheroid tt. 

{a)  The  Sjthere  and  Cylinder  contains  sixty  propositions 
arranged  in  two  books.  Archimedes  sent  this  like  so  many 
of  his  works  to  Dositheus  at  Alexandria;  but  he  seems  to 
have  p1a}'ed  a  practical  joke  on  his  friends  there,  for  he  pur> 
posely  misstated  some  of  his  results  "to  deceive  those  vain 
geometricians  who  say  they  have  found  everything  but  never 
give  their  proofs,  and  sometimes  claim  that  they  have  dis- 
ooo-ered  what  is  impossible.**  He  regarded  this  work  as  his 
masterpiece.  It  is  too  long  for  me  to  give  an  analysis  of  its 
contents,  but  I  remark  in  passing  that  in  it  he  finds  expres- 
sions for  the  surface  and  volume  of  a  pyramid,  of  a  cone,  and 
of  a  sphere,  as  well  as  of 'the  figures  produced  by  the  revolution 
of  polygons  inscribed  in  a  circle  about  a  diameter  of  the  circle. 
There  are  several  other  propositions  on  areas  and  volumes  of 
which  perhaps  the  mmt  striking  is  the  tenth  proposition  of 
the  second  book,  namely  that  "of  all  spherical  sf*gments  whoFe 
surfaces  are  equal  the  hemisphere  has  the  greatest  volume.** 
In  the  second  proposition  of  the  second  book  he  enunciates 
the  remarkable  theorem  that  a  line  of  length  a  can  be  divided 
so  that  #1  — «  :  6  =  4a' :  9je*  (where  6  is  a  given  length),  only 
if  6  be  less  than  \a\  that  is  to  say,  the  cubic  equation 
3^-^031^  A- \tHf-0  can  have  a  real  and  positive  root  only  if 
a  be  greater  than  36.  This  proposition  was  required  to  com- 
plete his  solution  of  the  problem  to  divide  a  given  sphere 
by  a  plane  so  that  the  volumes  of  the  segments  should  be 
in  a  given  ratia  One  very  simple  cubic  equation  occurs  in 
the  Ariikmetie  of  Diophantus,  but  with  that  exceptiim  no  sudi 
equation  appears  again  in  the  history  of  European  mathemaiies 
for  more  than  a  thousand  years. 

(6)    Tlie  Conotdt  and  Spkermdn  oontains  forty  propositioiw 


74  THE  PIB8T  ALBXANDBUN  8CHOQU 


OQ  qiuulriot  of  rovolntioii  (aeiii  to  DodtbeiM  in  AlouuMlria) 
mostly  oonoerned  with  an  investigatioii  of  their  YolniiiM. 

(e)  AivhiDiedes  also  wrote  a  treaUae  oq  certain  jmt- 
rej^ular  polykedrans^  that  is,  solids  oontained  by  regular  bnt 
dissimilar  polygons.     This  is  lost. 

(iii)  On  ariikmeiie  Archimedes  wrote  two  papers.  One 
(addressed  to  Zeaxippas)  was  on  the  principles  of  numeration ; 
this  is  now  lost  The  other  (addressed  to  Gelon)  was  called 
^ofifun)^  (ike  Band'reckofi^)^  and  in  this  he  meets  an  objection 
which  had  been  urged  agaimit  his  fint  paper. 

Tlie  object  of  the  first  |iap«'r  had  been  to  suggest  a  con- 
venient system  by  which  numbers  of  any  magnitude  could 
be  represented ;  and  it  would  seem  that  some  philosophers  at 
Syracuse  had  doubted  whether  the  system  was  practicable. 
Archimedes  says  people  talk  of  the  sand  on  the  Sicilian 
shore  as  something  beyond  the  power  of  calculation,  but  he 
can  estimate  it,  and  further  he  will  illustrate  the  power  of 
his  method  by  Ending  a  superior  limit  to  the  number  of 
grainn  of  sand  which  would  fill  the  whole  universe,  i,e,  a 
sphere  whose  centre  ih  the  earth,  and  radius  the  distance 
of  the  sun.  He  begins  by  saying  that  in  ordinary  Greek 
nomenclature  it  was  only  passible  to  express  numbers  from 
1  up  to  10*:  these  are  expressed  in  what  he  says  he  may 
call  units  of  the  first  order.  If  10*  be  termed  a  unit  of  the 
second  order,  any  number  from  10*  to  10'*  can  be  expressed 
as  so  many  units  of  the  second  order  plus  so  many  units 
of  the  first  order.  If  10'*  be  a  unit  of  the  third  order 
any  numlier  up  to  10**  can  lie  then  expressed;  and  so  on. 
Assuming  that  10,000  grains  of  sand  occupy  a  bphere  whose 
radius  is  not  less  than  n^^th  of  a  finger  breadth,  and  that  the 
diameter  of  the  universe  is  not  greater  than  10'*  stadia,  he 
finds  that  the  number  of  grains  of  sand  requireo  to  fill  the 
universe  is  less  than  10^. 

Hrobably  this  system  of  numeration  was  suggested  merely 
as  a  scientific  curiosity.  The  Greek  system  of  numeration 
with  which  we  are  acquainted  had  been  only  recently  intro- 


ARCaiMEDES. 


76 


1 


diieed,  most  likelj  at  Alexandria,  and  was  sufficient  for  all  the 
purposes  for  which  the  Greeks  then  required  numbers;  and 
Archimedes  used  that  system  in  all  his  papers.  On  the  other 
hand  it  has  been  conjectured  that  Archimedes  and  ApoUonius 
had  some  symbolism  based  on  the  decimal  system  for  their 
own  investigations,  and  it  is  possible  that  it  was  the  one  here 
sketched  out.  The  units  suggested  by  Archimedes  form  a 
geometrical  progression,  having  10^  for  the  radix.  He  inci- 
dentally adds  that  it  will  lie  omvenient  to  remember  that  the 
product  6i  the  mth  and  itth  terms  of  a  geometrical  progression, 
wlioee  6rst  term  is  unity,  is  e(|ual  to  the  (m  -••  ii )th  term  of  the 
series,  i.^.  that  r^  xr^  -  r**". 

To  these  two  arithmetical  papers,  I  may  add  the  following 
celebrated  problem  which  he  sent  to  the  Alexandrian  mathe- 
maticiana  The  sun  had  a  herd  of  bulls  and  cows,  all  of 
which  were  either  white,  grey,  dun,  or  piebald :  the  number 
of  piebald  bulls  was  less  than  the  number  of  white  bulls  by 
5/6ths  of  the  number  of  grey  bulls,  it  was  less  than  the 
number  of  grey  bulls  by  9/20ths  of  the  numl)er  of  dun  bulls, 
and  it  was  lees  than  the  numlier  of  dun  bulls  by  \^fA2ndH 
of  the  number  of  white  bulls :  the  number  of  white  cows  was 
7/l2ths  of  the  number  of  grey  entile  (bulls  and  cows),  the 
number  of  grey  cows  was  9/20ths  of  the  number  of  dun 
cattle,  the  number  of  dun  cows  was  1  l/.'tOths  of  the  number  of 
piebald  cattle,  and  the  number  of  piebald  cows  was  13/42nds 
of  the  number  c^  white  cattle.  The  problem  was  to  find  the 
oomposition  of  the  herd.  •  The  pmblem  is  indeterminate,  but 
the  solution  in  lowest  integers  is 

white  bulls,    ......  10,306,482;  white  cows,   7,206,360; 

grey  bulls, 7,460,.514;  grey  cows, 4,8t»3,246; 

dun  bulbs 7,358,060;  dun  cows,  ....:....  3,515,820; 

piebakl  bulls, 4,149,387;  piebaki  cows, 5,439,213. 

In  the  classical  solution,  attributed  to  Archimedes,  these  num- 
bers are  multiplied  by  80. 

Nesiselmann  believes,  from  internal  evidence,  that  the  pro- 


76  THK  PiBinr  albxahdrian  Bomoou 

Mem  hM  been  fAlvely  atiribuled  to  Arohimedee.  It  eertsfadjr 
is  unlike  his  extant  work,  but  it  was  attribnted  to  bim  amoQg 
the  ancients,  and  is  generally  thooght  to  be  granine^  thongh 
possibly  it  has  cooie  down  to  as  in  a  modified  form.  It  is 
in  verse^  and  a  later  copyist- has  added  tlie  additional  con- 
ditions that  the  sum  of  the  white  and  grey  bulls  shall  be  a 
square  number,  and  the  sum  of  the  piebald  and  dun  bulls  a 
triangtilar  number. 

It  is  perhaps  worthy  of  note  that  in  tlie  enunciation  the 
fractions  are  represented  as  a  sum  of  fmctions  whose  numeim- 
tors  are  unity:  thuH  Archimedes  wrote  1/7  -t-  1/6  instead  of 
13/42,  in  the  same  way  as  Ahmes  would  Imve  done. 

(iv)  On  mechanics  the  extant  works  of  Archimedes  are 
two  in  number,  namely,  {a)  his  Mrchaniot^  and  (c)  his  Hydro- 
MtaiicM. 

(a)  The  Mechanics  is  a  work  on  statics  with  special  refer- 
ence to  the  et|uilibrium  of  plane  Ismiuas  and  to  properties  of 
their  ceutretf  of  gravity  ;  it  connists  of  twenty-five  propositions 
in  two  books.  In  the  first  part  of  iMok  i  most  of  the  ele- 
mentary properties  of  the  centre  of  gravity  are  proved  [props. 
1  —8] ;  and  in  the  remainder  of  book  i  [props.  9 — 15]  and  in 
book  II  the  centres  <if  gravity*  of  a  variety  of  plane  areas,  such 
as  parallelograms,  triangles,  trapeziums,  and  parabolic  areas, 
are  determined. 

(b)  Archimedes  also  wrote  a  treatise  on  levers  snd  perhaps 
on  all  the  mechanical  machines.  The  book  is  lost,  but  we 
know  from  Pappus  that  it  contained  a  discussion  of  how  a 
given  weight  could  be  moved  with  a  gi%*en  power.  It  was  in 
this  work  prolmbly  that  Archimedes  discussed  the  theory  of 
a  certain  compound  pulley  consisting  of  three  or  more  simple 
(Sulleys  which  he  had  invented  and  which  was  used  in  some 
public  works  in  Syracuse.  It  is  well  known*  that  he  boasted  \ 
that,,  if  he  hod  but  a  fixed  fulcrum,  he  could  move  the  whole  I 
earth  ;  and  a  commentator  of  later  date  says  that  he  added 

he  would  (k>  it  by  using  a  compound  pulley. 

*  See  above,  p.  67.  J 


ARCHI^IEDEa 


77 


', 


(c)     His  work  OfiJIoating  bodien  contains  nineteen  proposi- 
tions in  two  hooks,  and  was  the  first  attempt  to  apply  mathe- 
matical rmsoning  to  hydrostatics.    The  story  of  the  manner  in 
which  his  attentiim  was  directed  to  the  subject  is  told   by 
VitniTius.  '  Hiero,  the  kin^  of  Syracuse,  had  given  wime  gold 
to  a  goldsmitli  t^i  make  into  a  crown.    The  crown  was  delivered, 
made  up,  and  of  the  proper  weight,  but  it  was  suspected  that 
the  workman  had  appropriated  some  of  the  gold,  replacing  it 
hy  an  equal  weight  of  silver.     Archimedes  was  thereupon  con- 
sulted.    Shortly  afterwards,  when   in   the   public   baths,  he 
noticed  that  his  Ixidy  was  pressed  upwards  by  a  force  which 
increased  the  more  completely  he  was  immersefl  in  the  water. 
Recognizing  the  value  of  the  n)iser\'ation,  he  rushed  out,  just 
as  he  was,  and  ran  home  through  the  streets,  shouting  cvpif«ra, ' 
cvipifKa,  '*  I  have  found  it,  I  have  found  it.''     There  (to  follow 
a  later  account)  on  making  accurate  experiments  he  found 
that  when  equal  weights  of  gold  and  silver  were  weighed  in 
water  they  no  longer  appeared  equal :  each  seemed  lighter 
than  before  by  the  weight  of  the  water  it  displaced,  and  as  the 
rilTer  was  more  bulky  than   the   gold  its  weight  was   more 
diminished.     Hence,  if  on  a  lialance  he  weighed  the  crown 
against  an  equal  weight  of  gold  and  then  inmiersed  the  whole 
in  water,  the  gold  would  outweigh  the  crown  if  any  silver  had 
been  used  in  its  construction.     Tradition  says  that  the  gold- 
smith was  found  to  be  fraudulent 

Archimedes  began  the  work  by  proving  that  the  surface  of 
m  fluid  at  rest  is  spherical,  the  centre  of  the  sphere  being  at  the 
centre  of  the  earth.  He  then  proved  that  the  pressure  of  the 
fluid  on  a  body,  wholly  or  partially  immersed,  is  equal  to  the 
weight  of  the  fluid  displaced;  and  thence. found  the  position 
of  equilibrium  of  a  floating  body,  which  he  illustrated  by 
spherical  segments  and  paraboloids  of  revolution  floating  on  a 
fluid.  'Some  of  the  latter  problems  involve  geometrical  reason- 
ing of  great  complexity. 

The  following  is  a  bar  specimen  of  the  questions  considered. 
A  aolid  in  the  shape  of  m  paraboloid  of  revolution  of  height  k 


78 


THI  PIB8T  ALSXAVDEUlf  KHOOL. 


and  latiM  rocimn  4a  lloaU  in  wttler»  with  iU  vertex 
and  iU  liMe  whoUj  above  the  tarlace.  If  eqvilihruui  be 
poMiUe  when  the  axis  in  not  vertical^  then  the  deneatjr  of  the 
body  moat  be  lew  than  (A-da)'/A'  [book  ii,  prop.  A\  When 
it  11  recollected  that  Arehimedet  was  nnmi^g^^jn^  ^lli 
*  trigonometiy  or  analjrtical  geometry,  the  fact  that  he  ooold 
discover  and  prove  a  proposition  luich  a«  that  jnat  quoted  will 
serve  as  an  illustration  of  his  powers  of  analysis. 

As  an  illustration  of  the  induence  of  Archimedes  on  the 
history  of  mathematics  I  may  mention  that  the  science  of 
statics  rested  on  his  tlieory  of  the  lever  nntil  1586,  when 
Stevinus  published  his  treatise  on  statics;  and  no  distinct 
advance  was  made  in  the  theor}'  of  hydrostatics  until  Stevinus 
*in  the  same  work  investigated  the  laws  which  regulate  the 
pressure  of  fluids,  it  will  be  noticed  that  the  mechanical 
investigations  of  Archimedes  were  concerned  with  statics.  It 
may  be  added  that  though  the  Greeks  attacked  a  few  problems 
in  dynamicH,  they  did  it  with  but  indifferent  success :  some  of 
their  remairks  were  acute,  but  they  did  not  sufliciently  rraliae 
that  the  fundamental  facts  on  which  the  theory  must  be  bated 
can  be  established  only  by  carefully  devised  obseri'ations  and 
experiments,  it  was  not  until  the  time  of  Galileo  and  Newton 
that  this  was  done. 

(v)  We  know,  both  from  occasional  references  in  his  works 
and  from  remarks  b}'  other  writers,  that  Arehimedes  was  lately 
occupied  in  asirtrnomieai  observaiiutu.  He  wrote  a  book,  Utpi 
ir^ifKMro«ia«,  on  the  constructiou  of  a  celestial  sphere,  which  is 
lost ;  and  he  constructed  a  sphere  of  the  staraj  and  an  ortery. 
These,  after  the  capture  of  Syracuse,  were  taken  by  Maroellni 
to  Rome,  and  were  preserved  as  curiosities  for  at  least  two  or 
three  hundred  years. 

This  mere  catalogue  of  his  works  will  shew  how  wonderful 
Were  his  achieveiueuts ;  but  no  one  who  has  not  actually  read 
some  of  his  writings  can  form  a  just  appreciation  of  his  extra- 
ordinary ability.  Tliis  will  be  still  further  increased  if  we 
recollect  that  the  only  principles  used   by  Archimedes^  in 


ARCBIHEDES.       APOLLOMIUS.  79 

addition  to  tliu!<e  contained   in  Euclid'i  RIfmenU  •nd  Coiue 

irvlir)n»,  are  that  uf  nil  Vmc^  lihe  q 

ACB,  Al>B,  ...  connecting  two 

|>ointa  A  &nd  ft,  the  straight  line 

ia  the  short«t,  and  of  the  curved      _ 

lines,   the    inner    one   ADK  ia   A 

■borter  than  the  outer  one  ACS;  together  with  two  s 

atatementfl  (or  space  of  three  diniensionii. 

In  the  old  and  medievnl  world  Archimedea  wim  nckotied 
•a  the  first  of  mathemnticiitn!),  but  possiUj  the  best  tribute 
to  hia  fame  is  the  fact  that  those  writ^ra  who  hare  apoken 
most  highly  of  his  work  and  aliility  are  those  who  have  been 
themselves  the  moat  distinguished  men  of  their  own  generm- 
tion.  ' 

ApoUonitU*.  The  tliird  gn-at  inatheniatician  of  thia 
century  was  A/inlfoniiiii  "f  Pfrgn,  who  in  chiefly  celebrated  (or 
having  produced  a  systematic  treatise  on  the  conic  nctions 
which  not  only  include*!  all  that  was  previously  known  about 
them  but  immensely  extended  the  knowledge  of  theM  curvea. 
Thia  work  was  accepted  at  once  as  the  standard  t«xt-book  pn 
the  subject,  and  completely  superoeded  tlw  previous  treatises 
of  Menaechtnus,  AristHeus,  and  Eucli<l  which  until  that  time 
had  licen  in  general  use. 

We  know  very  little  of  Apollonius  himself.  He  waa  born 
about  260  b.c.  ami  died  about  200  na  He  studied  in 
Alexandria  for  many  years,  and  prohnbly  lectured  there ;  he 
is  representetl  by  Pappus  as  "  vain,  jealoua  of  the  reputatioB 
of  others,  and  ready  to  seize  every  opportunity  to  depreciate 
them."  It  is  curioos  that  while  we  know  next  to  nothing 
of  hia  life,  or  of  that  of  hia  contemporary  Eratostbeneai,  yet 
their   nicknames,  which  were   respectively  e/mton  and  beUt, 

*  In  addition  to  Zenthen'i  work  and  the  other  snIboHtias  mmtiooad 
in  the  (oolnote  on  p.  IH,  ate  LitlrrargtifliitliiUeln  SiaHtn  tber  Eaklld, 
by  J.  L.  HeibefK,  Leipiiff,  1883.  A  eollection  of  Iha  aitant  wsrki  of 
Apolkmias  waa  iMued  b;  B.  HaUej.  Oiroid,  ITM  and  ITMi  «  ediUni 
of  Um  eoMkM  was  pnUubed  by  T.  L.  Heath,  C*mMltt,  WH. 


80  THE  PffBflT  AUXANDEUir  SCHOOL. 

I 

have  oome  down  to  ua.     Dr  Oow  has  ingeiiioiuly  immMffiil 
that  the  lecture  roooMr  at  Alexandria  weni  numbend,  and 
that  they  always  aiied  the  rooms  numbered  1 5  and  3  i^eipee 
tively. 

Apullonitts  spent  some  years  at  Pergamum  in  Funphyliai 
where  a  university  hsd  been  recently  established  and  endowed 
in  imitation  of  that  at  Aleicandria.  There  he  met  Eudemus 
and  Attalus  to  whom  he  subsequently  sent  each  book  of  his 
conies  as  it  came  out  with  an  explanatory  note.  He  returned 
to  Alexandria,  and  lived  there  till  his  death,  which  was  nearly 
contemporaneous  with  that  of  Archimedeii. 

In  his  great  work  on  conic  neciionM  he  so  thoroughly 
investigated  the  properties  of  these  curves  that  he  left  but 
little  for  his  succesbors  to  add.  But  his  proofii  are  long  and 
involved,  and  I  tliiuk  most  readers  will  be  content  to  accept 
a  short  analysis  of  his  work,  and  the  assurance  that  his 
demonstrations  are  valid.  Dr  Zeutlien  believes  that  many  of 
the  properties  enunciated  were  obtained  in  the  first  instance 
by  the  use  of  coordinate  geometry,  and  that  the  demonstrations 
were  translated  subsequently  into  geometrical  form.  If  this 
lie  so,  we  must  suppose  tliat  the  classical  Writers  were  familiar 
with  some  branches  of  analytical  geometry — Dr  Zeuthen  says 
the  use  of  orthogonal  and  oblique  coordinates,  and  of  transfor- 
mations depending  on  abridged  notation — tliat  this  knowledge 
was  confined  to  a  limited  school,  and  was  finally  lost.  Tliis 
is  a  mere  conjecture  and  is  unsupported  by  any  direct  evidence, 
but  it  has  been  accepted  by  some  writers  as  affording  an  ex- 
planation of  the  extent  and  arrangement  of  the  work. 

The  treatise  contained  about  four  hundred  propositions 
and  was  divided  into  eight  books ;  we  have  the  Greek  text  of 
the  first  four  of  these,  and  we  also  possess  copies  of  the 
commentaries  by  Pappus  and  Eutocius  on  the  whole  work. 
In  the  ninth  century  an  Arabic  translation  was  made  of  the 
first  seven  books,  which  were  the  only  ones  then  extant; 
we  have  two  manuscripts  of  this  version.  The  eighth  book 
is  lost. 


APOLLONIUR. 


81 


In  the  letter  to  Eudemus  which  accoiDpanied  the  first  book 
ApoUonios  says  that  he  undertook  the  work  at  the  request  off 
Naucratesy  a  geometrician  who  had  been  staying  with  him 
at  Alexandria,  and,  though  he  had  given  some  of  his  friends  a 
rough  draft  of  it,  he  had  preferred  to  revise  it  carefully  before 
sending  it  to  Peip&roum.  In  the  note  which  accompanied  the 
next  book,  he  asks  Eudemus  to  read  it  and  communicate  it  to 
others  who  can  understand  it,  and  in  particular  to  Philonides 
a  certain  geometrician  whom  the  author  had  met  at  Ephesus. 

The  first  four  books  deal  with  the  elements  of  the  subject, 
and  of  these  the  first  three  are  founded  on  Euclid*8  previous 
work  (which  was  itself  based  on  the  earlier  treatises  by 
Menaechmus  and  Aristaeus).  Heracleides  asserts  that  much 
of  the  matter  in  these  books  was  stolen  from  an  unpublished 
work  of  Archimedes,  but  a  critical  examination  by  Heiberg 
haa  shewn  that  this  is  improbable. 

ApoHonius  begins  by  defining  a  cone  on  a  circular  base. 
He  then  investigates  the  diflerent  plane  sections  of  it»  and 
shews  thai  they  are  divisible  into  three  kinds  of  curves  which 
he  oaUs  ellipses,  parabolas,  and  hyperboks.     He  proves,  the 


propositioii  that^  if  ^  il'  be  the  vertices  of  a  conic  and  if  /^  be 
any  point  on  it  and  PM  the  perpendicular  drawn  from  P  on 
iJ\  then  (la  Iha  mmI  notation)  the  mtio  MF^ :  AM .  MA'  is 


82  THE  rmST  ALKXAlimUAlf  SCHOOL. 

eontUnt  in  an  ellip8e  or  hyiierbola,  and  IIm  r»lio  MT^ :  AM 
in  ennsUnt  in  a  parabola.  These  are  the  eharaeterietie 
properties  on  which  almoet  all  the  rest  of  the  work  is  hased. 
He  next  shews  that,  if  il  be  the  vertex,  /  the  latns  reetom, 
and  if  AM  and  MF  be  the  abscissa  and  ordinate  of  any 
pmnt  on  a  conic  (see  figure  on  last  page),  then  J/P*  is  less 
than,  equal  to^  or  greater  than  I .  AM  according  as  the  conic 
is  an  ellipse,  parabola,  or  h}'perbola ;  hence  the  names  which 
he  gave  to  the  curves  and  by  which  they  are  still  known. 

He  had  no  idea  of  the  directrix,  and  was  not  aware  that 
the  parabola  had  a  focus,  but,  with  the  exception  of  the  propo- 
sitions which  involve  these,  his  first  three  books  contain  most 
of  the  propositions  which  are  found  in  modem  lext-booksb 
In  the  fourth  book  he  develops  the  theory  of  lines  cut 
harmonically,  and  treats  of  the  points  of  intersection  of  systems 
of  conies.  In  the  fifth  book  he  commences  with  the  theory  of 
maxima  and  minima ;  applies  it  to  find  the  centre  of  curva- 
ture at  any  point  of  a  conic,  and  the  evolute  of  the  curve ; 
aud  discusses  the   number  of   normals  which  can  be  drawn  ( 

from  a  puiat  to  a  conic.  In  the  nixth  book  he  treats  of 
similar  conies.  T^h  seventh  and  eighth  books  were  given  up 
to  a  discussion  of  conjugate  diaineten,  the  latter  of  these  was 
eonjecturally  restored  by  El  Halley  in  his  edition  of  1710. 

Tlie  verbose  and  tedious  explanations  make  the  book  re- 
pulsi%'e  to  most  modem  readers ;  but  the  logical  arrangement 
and  reasoning  are  unexceptionable,  and  it  has  been  not  unfitly 
descrilied  as  the  crown  of  Greek  geometry.  It  is  the  work  on 
which  the  reputation  of  Apollonius  rests,  and  it  earned  for  him 
the  name  of  "  the  great  geometrician." 

Besides  this  immense*  treatise  he  wrote  numerous  shorter 
works ;  of  course  the  liooks  were  written  in  Greek,  but  they 
ore  usually  referred  to  by  their  Latin  titles :  those  about  which 
we  now  know  anything  are  enumerated  below.  He  was 
the  author  of  a  work  on  the  iiroblem  '*  given  two  co-planar 
straight  lines  An  and  Bh^  drawn  through  fixed  points  A  and  B\ 
to  draw  a  line  0\%h  from  a  givcm  point  0  outside  them  cutting  - 


APOLLONIUK.  K3 

them  in  a  and  6,  so  that  .Iff  shall  be  to  M  in  a  given  ratio  " : 
he  reduced  the  question  to  seventy-seven  separate  ca^es  and 
gave  an  appropriate  solution,  with  the  aid  of  oonics,  for  each 
case;  this  was  published  bj  E.  Halley  (translated  from  an 
-  Arabic  copy)  in  1706.  He  also  wrote  a  treatise  At  Sections. 
SpaUi  (restored  by  E.  Halley  in  1700)  on  the  same  problem 
under  the  condition  that  the  rectangle  Aa.  Bb  was  given. 
He  wrote  another  entitled  De.  Sectinnf!  Delerminatn  (restored 
by  R.  Simson  in  1749),  dealing  with  problems  such  as  to 
6nd  a  point  P  in  m  ^ven  straight  line  AB  9o  that  PA* 
shall  be  to  PB  in  a  given  ratio.  He  wrote  another  De 
Tartionihus  (restored  by  Vieta  in  1600)  on  the  construction 
of  a  circle  which  shall  touch  three  given  circles.  Another 
work  was  his  De  IndinationibuM  (restored  by  M.  Ghetaldi 
in  1607)  on  the  problem  to  draw  a  line  so  that  the  in- 
tercept between  two  given  lines,  or  the  circumferences  of 
two  given  circles,  shall  be  of  a  given  length.  He  was  also 
.  the  author  of  a  treatise  in  three  books  on  plane  loci,  Dt  Loeis 

I  PlaniM  (restored  by  Fermat  in   1637,  and  by  R.  Simson  in 

1746X  And  of  another  on  the  regular  9oHd$,  And  lastly  he 
wrote  a  treatise  on  uncfoMmstf  ineommewturabfen,  being  a  com- 
mentaiy  on  the  tenth  book  of  Euclid.  It  is  believed  that  in 
one  or  more  of  the  lost  books  he  used  the  method  of  conical 
projections. 

Besides  these  geometrical  works  he  wrote  on  the  methodi  of 
ariikmeticai  calcnlatioti.  All  that  we  know  of  this  is  derived 
from  some  remarks  of  Pappus.  Friedlein  thinks  that  it  was 
merely  a  sort  of  ready-reckoner.  It  seems,  however,  more 
probable  that  Apollonius  here  suggested  a  system  of  numera- 
tion similar  to  that  proposed  by  Archimedes,  but  proceeding 
by  tetrads  instead  of  octads,  and  described  a  notation  for  it. 
f  It  will  be  noticed  that  our  modem  notation  goes  by  hexads, 

a  million  » 10",  a  billion  ^  10»,  a  trillion  »  10"  ke.  It  is  not 
impossible  that  Apollonius  also  pointed  out  that  a  decimal 
system  of  notation,  involving  only  nine  symbolsi  wonld 
fMilitale  rnnerical  multiplioatioiii. 


/ 

I 


H 


84 


TBI  FIRST  ALBXAHDRUN  8CHOQU 


ApoUooiiit  WM  interested  in  MtrcNMNny,  nnd  wrote  n  book 
on  the  siaiiona  and  rt^rtrntiowi  uftkt  planttB  of  which  Ptoleniy 
made  aome  ate  in  writing  the  Almagul.  He  alao  wrote  n 
treatise  on  the  ose  and  theory  of  the  serow  in  statics. 

This  is  a  long  list,  hat  I  should  suppose  that  most  of  these 
works  were  shoK  tracts  on  q>ecial  points. 

like  so  many  of  his  predecessors  he  too  gave  a  oonstraotion 
for  6tiding  two  mean  proportionals  between  two  given  lines,  and 
thereby  duplicating  the  cube.   It  was  as  follows.    Let  OA  and 


OB  be  the  given  lines.  Construct  a  rectangle  OADB^  of  which 
they  are  adjacent  sides.  Bisect  ABiaC.  Then*  if  with  C  as 
centre  we  can  describe  a  circle  cutting  OA  prodoced  in  a  and 
cutting  OB  produced  in  6,  so  that  aDb  shall  be  a  straight  line^ 
the  problem  is  effected.     For  it  in  easily  shewn  that 

Oa.Aa-k-CA^^Cay 

Similarly  Ob.  Bb^CB^^CbK 

Hence  On ,  Aa  =i  Ob ,  Bb. 

That  is,  OaiOb^Bb:  Aa. 

Buty  by  similar  triangles, 

BDiBb^OaiOb'^AaiAD. 

Therefore  OA  iBb^Bb:  Aa^Aa  :  OB^ 

that  is,  Bb  and  Oa  are  the  two  mean  proportionals  between 
OA  and  OB,    It  is  impossible  to  construct  the 


V 


\ 


APOLLOXII'S.      RRATOSTHENES. 


85 


I 


f 


centre  is  C  by  Euclidean  grometry,  hat  Apollonius  gave  a 
mechanical  way  <if  clejicrihini;  it.  This  construction  is  quoted 
hy  several  Arabic  writers. 

In  one  of  the  most  brilliant  |>asHages  of  his  Ai^r^n  hiittO' 
riqn^  Chasles  remarks  that,  while  Archimefles  and  Apollonius 
were  the  most  able  geometricians  of  the  old  world,  their 
works  are  distinguished  by  a  ctrntrast  which  runs  through 
the  whole  subsequent  history  of  geometry.  Archimedes,  in 
attacking  the  problem  of  the  quadrature  of  curvilinear  areas, 
established  the  principles  of  the  geometry  which  rests  on 
measurements;  this  naturally  gave  rise  to  the  inBnitesimal 
calculus,  and  in  fact  the  method  of  exhaustions  as  used  by 
Archimedes  does  not  dilTer  in  principle  from  the  method  of 
limits  as  used  by  Nem  ton.  Apollonius,  on  the  other  hand,  in 
investigating  the  properties  of  ctmic  sections  by  means  of 
transversals  involving  the  ratio  of  rectilineal  distances  and  of 
perspective,  laid  the  foundaticms  of  the  geometry  of  form  and 
position. 

Eratosthenes*.  Among  the  cimtemporaries  of  Archi- 
medes and  Afmllonius  I  may  mention  KrnioHh^n^s.  liom  at 
Cyrene  in  275  b.c.,  he  M-as  educated  at  Alexandria — perliaps 
at  the  same  time  as  Arcliimefles  of  whom  he  was  a  personal 
friend — and  Athens,  and  m'as  at  an  early  age  entrusted  with 
the  care  of  the  university  library  at  Alexandria,  a  post  which 
proliably  he  occupied  till  his  death.  He  was  the  Admirable 
Crichton  of  his  age,  and  distinguishetl  for  his  athletic,  literary, 
and  8cienti6c  attainments :  he  was  also  something  of  a  poet. 
He  lost  his  sight  by  ophtlialmia,  then  ay  now  a  curse  of  the 
valley  of  the  Nile,  and,  refusing  to  live  when  he  was  no  longer 
able  to  read,  he  committed  suicide  in  194  b.c. 

In  science  he  was  chiefly  interested  in  astronomy  and 
geodesy,  and  he  constructed  various  astronomical  instminents 
which  were  used  for  some  centuries  at  the  university.     He 

*  The  works  of  Eratosthenes  esist  only  in  frsgrn^sts.    A  eoiloetion 
of  thns  was  published  hf  O.  Bcmhardy  at  Berlin   in    18S9.: 

pmtoA  hj  B.  Hiller,  Lsipris,  1871 


86  TBK  riBST  ALBXAHmtlAir  8CB0aL. 

■oggettod  the  calendar  (now  known  m  JnlianX  in  whiflh  ovety 

fourth   year   oontains    366   daye;  and  he  determined  the 

obliquity  of  the  ecliptic  as  2S*  61'  W.     He  meaenred  the 

length  of  a  degree  on  the  earth's  surlece^  making  it  to  be 

about  79  mileii,  which  ie  too  long>  by  nearly  10  mileii  and 

thence  calculated    the    circumference  of   the    earth    to   be 

252,000  stadia,  which,  if  we  take  the  Olympic  stadium  ol 

202}^  ymxls,  is  equivalent  to  saying  that  the  radius  is  about  \ 

4,600  miles.     The  principle  used  in  the  determination  is 

correct 

Of  Eratosthenes's  work  in  mathematics  we  have  two  eirtant 
illustrations :  one  in  a  description  of  an  instrument  to  dupli- 
cate a  cube,  and  the  othcir  in  a  rule  he  gave  for  constructing 
a  table  of  prime  numbers.  The  former  is  given  in  many 
books.  The  latter,  called  the  *'  sieve  of  ESratosthenes,"  was  as 
follows:  write  down  all  the  numbers  from  1  upwards;  then 
every  second  number  from  2  is  a  multiple  of  2  and  may  be 
cancelled ;  every  third  number  fn>m  3  is  a  multiple  of  S  and 
may  lie  cancelled ;  every  fifth  number  from  5  is  a  multiple  of  5 
and  may  lie  cancelled ;  and  so  on.  It  has  been  estimated 
that  it  would  involve  working  for  about  300  hours  to  thus 
find  the  primes  in  the  numbers  from  1  to  1,000,000.  The 
labour  of  determining  whether  any  particular  numWr  is  a 
prime  may  be  however  much  shortened  by  observing  that  if  a 
number  can  be  expressed  as  the  product  of  two  factors  one 
must  be  less  and  the  other  greater  than  the  square  root  of  the 
numlier,  unless  the  number  is  the  square  of  a  prime  in  which 
case  the  two  factors  are  equal.  Hence  every  composite  number 
must  be  divisible  by  a  prime  which  is  not  greater  than  its 
square  root. 


The  secoful  century  before  CkruA, 

The  third  century  before  Christ,  which  opens  with  the 
career  of  Euclid  and  closes  with  the  death  of  Apdlonius,  is  the 
most  brilliant  era  in  the  history  of  Greek  mathematics.     But 


♦ 


HrPSICI.ES.      NICOMEDES.  87 

the  great  mathematicianH  <tf  tlint  century  were  geufnelricians, 
and  under  titeir  influence  attention  was  directed  almoat  solely 
to  that  branch  of  niathemnticH.  With  tlie  metboda  they  oaed, 
and  In  trhirh  their  Nucceiwm  were  by  tradition  confined,  it 
wu  hardly  poKsible  to  make  any  further  great  advancv:  to 
fill  Dp  a  few  dctailH  in  a  wurk  that  wraa  completed  in  its 
essential  partti  wafi  all  that  rouhl  be  eflected.  It  waa  not  till 
after  the  lapttn  of  nearly  I  bOO  yeant  that  t)ie  ((enills  of  Descartea 
opened  the  way  to  any  further  progrewt  in  geometry,  and  I 
therefore  pans  over  ttie  numerous  writers  who  followed  Apollo- 
nins  with  hut  slight  mention.  Indeed  it  may  be  said  raaghly 
that  during  the  next  tliuui^nd  yearn  Fappuii  wna  the  sole 
geometriciait  of  great  Hhilily ;  and  during  this  long  period 
almost  the  only  other  pure  matlietnatiL'iaiis  of  exceptional 
geniun  were  Hipparchux  and  I'tolemy  who  laid  tlie  founda- 
tions of   trigonometry,   and    Diopfiantnn   who   laid   tboM  tit 

Early  in  the  necond  centuiy,  circ  180  B.C,  we  find  the 
names  of  three  mathematitians — I{y|)sicles,  Nicomedee,  and 
Dioclea— who  in  their  own  day  were  famous. 

Hypsidea.  The  firat  of  the-se  was  llgptidrt  who  added  a 
fourteenth  book  to  Euclid's  El'm'nln  in  which  the  regular 
solids  were  discusced.  In  anotlier  small  work,  entitled  RUiHgr, 
Hypsiclet  developed  the  theory  tt  arithmetical  progressions 
which  had  lieen  ao  strangely  neglected  by  the  earlier  mathe- 
maticians, a:id  here  for  the  tint  time  in  Orvek  mathematica 
we  find  a  right  angle  divided  in  the  Babylonian  manner  into 
ninety  degrees  ;  posnibly  Eratosthenes  may  haii'e  previously 
estinwted  angles  by  the  number  of  degrees  they  contain,  bat 
this  is  only  a  matter  of  conjecture. 

NlotmiadM.  The  second  was  Nicomedea  who  Invented  the 
carve  known  as  the  condtoid  or  the  shell-shaped  cnrrcL  If 
from  a  fixed  point  S  a  line  be  drkwn  ratting  a  given  fixed 
straight  line  in  Q  and  if  T  be  Uken  on  SQ  so  that  the  length 
QP  is  eonstaot  (say  d),  then  the  locns  a<  />  la  the  ooncboid. 
Its  eqjWttioD  may  bo  p«t  ifl  the  form  rmmwBe$*i.    ItiseMy 


88  THV  riBST  ALKXANDaiAir  8CH00L. 

with  iUaid  to  triaeot*  given  angle  or  todaplioatone«he;Mid 
thia  no  doubt  was  the  caute  of  itn  invention. 

Dipolea.  The  thiid  of  these  mathematicians  was  DMm 
the  inventor  of  the  curve  known  as  the  cUmid  or  the  ivj* 
shaped  curve  which,  like  the  conchoid,  was  used  to  give  a 
solution  of  the  duplication  problem.  He  defined  it  thus :  let 
.  AOA'  and  BOB'  be  two  fixed  diameters  of  a  circle  at  right  angles 
to  one  another.     Draw  two  chords  QQ^  and  RR'  parallel  to  I 

BOB*  and  equidistant  from  it.  Then  the  locus  of  the  inter* 
section  oi  AR  and  QQ'  will  be  the  ciuoid.  Its  equation  can  be 
expressed  in  the  form  y'(2a  —  ;r)  =  a>.  Diocles  also  solved  (bj 
the  aid  of  conic  sectiomi)  a  problem  which  had  been  proposed 
by  Archimedes,  namely,  to  draw  a  plane  which  will  divide  a 
sphere  into  two  parts  whose  volumes  shall  bear  to  one  another  V 

a  given  ratio. 

Forseua.  Zenodoroa.  About  a  quarter  uf  a  century 
later,  say  about  150  &c.,  Ferteut  investigated  the  various 
plane  sections  uf  the  anchor-ring,  and  Ze:HHioru9  wrote  a 
treatise  on  isoperimetrical  figures.  Part  of  the  latter  work 
has  been  preserved  ;  one  proposition  which  will  serve  to  shew 
the  nature  of  the  problems  discussed  is  that  ^  of  segments  of  I 

circles,  having  et|ual  arcs,  the  semicircle  is  the  greatest."  , 

Towards  the  clone  of  this  century  we  find  two  matheom- 
ticians  who,  by  turning  their  attention  to  new  subjects,  gave 
a  fresh  stimulus  to  the  study  of  mathematics.  These  were 
Hipparchus  and  Hero.  ^ 

HipparchUB*.  iiippardius  was  the  most  eminent  of 
Greek  astronomers — his  chief  predecessors  being  Eudoxus, 
Aristarchus,  Archimedes,  and  Eratosthenes.  Hipparchus  is 
said  to  have  been  born  about  160  B.a  at  Nicaea  in  Bithynia; 


*  See  C.  MsnitiuB,  Ilipparehi  im  Araii  rt  Kudaii  phaeHomema  Cosi- 
M^nrurii,  Leipzig,  1894,  and  J.  B.  J.  Delsmbre,  HUtoirr  de  tattrowomU 
aneitune,  FsHa,  1817,  vol.  i,  pp.  106—189.  8.  P.  Tannery  in  his  Re- 
ekerehes  sur  VhUioire  de  VastroHomie  aMciemue,  Paris,  1893,  argues  that 
the  work  of  Hipparchas  has  been  overrated,  but  I  have  adopted  the 
view  of  the  majority  of  writers  on  the  aubject. 


\ 


HtPPARCHUR  W) 

it  is  prolMlilo  that  he  Hpcnt  Mtme  yearn  nt  Alex*iKlri&,  bnt 
Giutlly  lie  took  op  liis  hIxkIc  nt  RI)<mIi»>  wlicr«  be  in«d«  tnont  <rf 
hiB  observations.  DeUmlira  lias  olitninrd  iin  ingenioot  con* 
finnation  of  the  tnulitton  wliich  a.sscrt«>(I  th«t  Hippftrchna 
lived  in  th«  secun<t  ceiiturr  Ix-foreClirist.  Hipparchm  in  nns 
place  sayM  that  the  lonptude  of  n  certain  sti»r  i|  Cuiis  olnenefl 
by  him  was  exactly  00%  and  it  nlionld  be  noted  tl»t  he  wiw 
an  extremply  careful  otjscrvirr.  Xow  in  1 750  it  «■»«  1 16'  i'  10  , 
and,  na  the  first  point  of  Aries  rpf,Tp<les  at  the  nit»  of  .10-2"  « 
year,  the  DbE<Tvatiiin  was  made  about  120  B.C. 

Encept  for  a  short  ctimmfntary  on  a  poem  of  Aratan 
dealing  with  astronomy  all  his  works  are  lost,  but  Ptolemy's 
great  treatise,  the  Almmj'tl,  deHcrib«-<l  below,  was  founded  on 
the  ol»ervations  and  writings  of  HippnrcliUH,  and  from  the 
not«s  there  given  we  infer  that  the  chief  discoverie*  of 
Hipparchus  were  an  follows.  He  determined  the  dnration  ot 
the  year  to  within  mx  minutesof  its  true  value.  Headculatcd 
the  inclination  of  the  ecliptic  and  equator  as  33*  51';  it  was 
actually  at  that  time  'IV  46'.  He  estimated  the  Minoal  pr&- 
cessioii  of  the  e<|UinoxPH  as  .i9" ;  it  is  50*2".  He  stated  the 
lunar  parallax  as  57',  which  in  nearly  correct.  He  worked 
out  the  eccentricity  of  the  solar  orhit  na  1 12\  ;  it  is  veiy 
apprmimately  1/30.  He  determined  the  perigee  and  mean 
motion  of  the  sun  and  of  the  moon,  and  he  calculated  the 
extent  of  the  shifting  of  the  plane  of  the  moon's  motion. 
Finally  be  obtained  the  synodic  perioiis  ot  the  6Te  planets 
then  known,  t  leave  the  details  of  his  obeerrations  and 
calcnlstions  to  writers  who  deid  specially  with  astronomy  snch 
•a  Dekrobra;  bvt  it  may  be  fairly  said  that  this  worii  plocfd 
the  svl^eet  for  tbe  6rst  time  on  a  scientific  basis. 

To  oMuiint  (or  the  Innor  motion  Hipparchus  supposed  tbe 
moon  to  move  with  anifonn  velocity  in  •  circle^  tbe  eMtb 
oocnpying  a  position  near  (bnt  not  at)  the  centre  o(  this  nrele. 
Tbii  IS  eqiivolent  to  saying  that  tbe  orbit  is  on  VfiejtAn  of  the 
Ant  otder.  The  kmgitade  of  tbe  nwoB  obtained  on  this 
bypotbewi  is  eorrcct  to  tbe  first  onler  of  aaaJl  qnantttiea  tar  » 


90 


TUV  riBHT  ALKXAHDaUII  8CBO0L. 


few  ravotutioiis.    To  make  it  oorrect  for  anjr  kngili  ol  timm 


moved  lorweid 
eveetioD.  He 
maimer.  Thie 
be  deCemiiied 

Ur  enabled  hioi 


HipjMunebat  farther  supposed  that  the  apse 

about  y  a  month,  thus  giving  a  correction 

explained  the  motion  of  the  sun  in  a  similai 

theory  accounted  for  all  the  tacts  which  oou 

with  the  instruments  then  in  use,  and  in  parti 

to  calculate  the  details  of  eclipses  with  considerable  accuracy. 

He  commenced  a  series  of  planetary  observations  to  enable 
his  successors  to  frame  a  theory  to  account  forj  their  motions ; 
and  with  great  perspicacity  he  predicted  thit  to  do  this  it 
would  be  necessary  to  introduce  epicycles  of  a  higher  order, 
that  is,  to  introduce  three  or  more  circles  the  centre  of  each 
successive  one  moving  unifuruily  on  the  circuiuference  of  the 
preceding  one.  | 

He  also  formed  a  list  of  1080  of  the  fixed  stars.  It  is  said 
that  the  sudden  appearance  in  the  heavens  [  of  a  new  and 
brilliant  star  called  Iuh  attention  to  the  need  of  such  a 
catalogue;  and  the  appearance  of  such  a  star  during  his 
lifetime  is  confirmed  by  ChiueHC  records. 

No  further  advance  in  the  theory  of  astronomy  was  made 
until  the  time  of  Copernicus,  though  the  principles  laid  down 
by  Hipparchus  were  extended  and  worked  out  in  detail  by 
Ptolemy. 

Investigations  such  as  these  naturally  led  to  irigotuh 
meiry^  and  Hipparchus  must  be  credited  with  the  invention 
of  tliat  subject.  It  is  known  that  in  plane  trigonometry  he 
constructed  a  table  of  chords  of  arcs,  which  is  practically  the 
same  sm  one  of  natural  sines ;  and  that  in  spherical  trigonometry 
he  had  some  method  of  solving  triangles :  but  his  works  are 
lo8t,  and  we  can  give  no  details,  it  is  believed  however  that 
the  elegant  theorem,  printed  as  Euc.  vi,  D,  and  generally 
known  as  Ptolemy's  Theorem,  is  due  to  Hipparchus  and  was 
copied  from  him  by  Ptolemy.  It  contains  implicitly  the 
addition  formula;  for  sin  {A  *  B)  and  cos  (A  *  B);  and  Camot 
shewed  how  the  whole  of  elementary  plane  trigonometry  could 
be  deduced  from  it. 


i 


< 

i 


HERa  91 

I  ought  &Im>  U>  add  that  Hipparchiu  waa  tlM  Gnt  to  in- 
dicate the  position  of  a  place  on  the  earth  by  nMann  at  its 
latitude  and  longitude. 

Hero*.  Tlie  second  of  thpse  nintlipiiinticiann  was  I/ero  </ 
AUxandria  who  pUcml  engineering  and  land-mirveying  cm  » 
icieiitific  basia.  He  wan  a  pnpil  of  Cteiiilian  who  iav«nt«>l 
Wi'eral  ingenious  mncliinen  and  in  nlludcd  to  as  if  he  were  • 
mathematiciitn  of  n<>t«.  It  ix  not  lilcely  that  Hero  flourished 
before  120  rg,  but  the  prwise  period  i>t  which  he  lived  is 
uncertain. 

In  pure  niathenuitics  Hero's  principal  and  most  character- 
istic work  conxlHtK  of  (i)  winio  otcnii'ntaiy  geometrj,  with 
applications  to  the  dt^Urminntion  of  the  areas  of  fields  of  given 
shapes;  (ii)  propositions  on  finding  (he  volumes  of  certain 
solids,  with  applications  to  theAlren,  lislhs,  hanquet'halls, 
and  BO  on ;  (iit)  a  rule  to  find  the  height  of  an  inaccessible 
object ;  aod  (iv)  tables  of  weights  and  measures.  He 
invented  a  solution  of  the  duplicittiiin  problem  which  is 
practically  the  same  as  that  which  Apollonius  bad  already 
discovered.  8onie  coin  men  tntont  Diinlc  that  he  knew  how  to 
solve  a  iiuodratic  equation  even  when  tlie  coeflicients  were 
not  numerical ;  but  this  is  dnuhtful.  He  proved  the  fortnnin 
that  the  arex  of  a  triangle  is  equal  to  [« (■  -  a)  («  -&)(>-  c)j*, 
where  »  is  the  semiperimeter,  and  a,  b,  r,  the  lengths  of  the 
sides,  and  gave  as  an  illustration  a  triangle  whose  aides  wen 
in  the  ratio  l.t :  U  :  15.  He  was  evidently  acquaitated  with  the 
trigonometry  of  Hipparchus.  and  the  values  of  cot  Sv/xaraoom- 
pntcd  for  various  values  of  n,  but  he  nowhere  quotes  ft  fonnoU 
*  Bm  IttrtktKhn  tHT  la  rir  tl  Irt  OHrragrt  rf'if/nm  JTAUraitdrit  by 
T.  H.  UsrtiD  in  tol.  tr  ot  Jlfmoirfi  pHinafi...i  r«eaii»tit£iiiMrTipUom, 
Paris,  1854;  ■«•  also  Canlor,  elispa.  iTtit,  iii.  On  Um  work  mititM 
Dtfinitiota  which  is  sltribDtfd  to  Hero,  iwe  H.  P.  Tannnj,  efasps.  m^ 
in,  and  an  article  tt;  O.  Friedlein  io  BoDcompsgoi'a  Bulletimm  di  UUla- 
fr^Ha,  Usjch,  IBTl,  vol  it,  pp.  93—136.  Rdiliom  of  tbi  sttant  wstks 
of  Ban  wm  pnUiabed  bj  W.  Scbmidt,  LeiptJn,  1H»,  an4  bj  P.  HallMh, 
BoUd.  laM.  As  Englinh  traDilslina  at  lbs  IlwrtitmfiMA  was  p 
t?  B.  Woodsmft  sad  J.  O.  Oraenwood,  Lmidoii,  lUl. 


02 


THK  nBST  ALSXAITDBUN  SCHOOL. 


or  expriMMly  ium  the  value  4if  the  sine ;  it  ia  probable  thai 
the  later  Qreek«  .be  regarded  trigonometry  an  lunniiif  aa 
introduction  to,  and  being  an  integral  fMurt  oi^  aatrooomy. 

The  following  ia  the  uumner  in  which  he  solved*  the  problem 
to  find  tlie  area  of  a  triangle  ABC  the  length  of  whoee  lidee 
are  a,  b,  e.  Let  «  he  the  iieuiiperimeter  of  the  triangle.  Let 
the  inscribed  circle  touch  the  sides  in  D^  JP,  /*«  and  let  O  be 


\ 


I 

I 


I 


itocentre.  On  iM7 produced  take  H  aoihmtCff  ^  JF,thet^tat% 
BU=9.  Draw  OK  at  right  angles  to  Oi!r,  and  CK  at  right 
angles  to  BC\  let  them  meet  in  K.  The  area  ABC  at  ^  is  equal 
to  the  sum  of  the  areas  OBC^  OCA^OAB^^r-^lhr-^ffirmgr, 
that  is,  is  equal  to  BH  .  OD,     He  then  shews  tlmt  the  angle 

*  la  hit  Diopin,  Holtwh,  part  vuu  pp.  8S&— SS7.  It  ahoald  be 
stated  that  toina  critics  think  that  this  is  an  iatsfpoiatioa,  and  is  not 
doe  to  Hera. 


I 


HERO.  9:} 

Oil/^  =  angle  VBK\  henoe  the  triangles  OAF  and  CBK  are 


/.  BCiCK^AFiOF^CHiOD, 
:.  EC  :  CH^CK  lOD^CL.m 
/.  BiliVH^CDiLD. 

:.  BU* :  CU  .  BU^CD  .  BD  :  LD  .  BD^CD  .  BD  :  OL^. 
Henoe 

In  applied  maihematicR  Hero  diacusscxl  the  centre  of 
gravity,  the  five  nimple  machineis  and  the  prohlein  of  moving 
a  given  weight  with  a  given  power;  and  in  one  place  he 
suggested  a  way  in  which  the  power  of  a  catapult  could  be 
tripled.  He  also  wrote  on  the  theory  of  hydraulic  nmchinen. 
%  He  described  a  theodolite  and  cyclometer,  and  pointed  out 
various  problems  in  surveying  for  which  they  would  be  usefuL 
But  the  most  interesting  of  his  smaller  works  are  his  IIvcv- 
/lorura  and  AvrofutTo,  containing  descriptions  of  about  100 
small  machines  and  mechanical  toys,  many  of  which  are  very 
ingenious.  In  the  former  there  is  an  account  of  a  small 
stationary  steam-engine  which  is  of  the  form  now  known 
as  Avery *s  patent :  it  was  in  common  use  in  Scotland  at  the 
beginning  of  this  century,  but  is  not  so  economical  as  the  form 
introduced  by  Watt.  There  is  also  an  account  of  a  double 
forcing  pump  to  be  used  as  a  fire-engine.  It  is  probable  that 
in  the  hands  of  Hero  these  instruments  never  got  beyond 
models.  It  is  only  recently  that  general  attention  has  been 
directed  to  his  disco\'eries,  though  Arago  had  alluded  to  them 
in  his  Hofje  on  Wattw 

All  this  is  very  difierent  from  the  classical  geometry  and 
arithmetic  of  Euclid,  or  the  mechanics  of  Archimedes.  Hero 
did  nothing  to  extend  a  knowledge  of  abstract  mathematiGS ; 
he  learnt  all  that  the  text-books  of  the  day  could  teach  him, 
but  he  was  interested  in  science  only  on  aoooont  of  its  prao- 
tioal  applioalions,  and  so  long  as  his  resolts  w^re  true  he 
eared  nothing  lor  the  kigieal  aoearacj  of  the  prbeeei  bgr  winch 


94  THV  riBST  ALBXAVDBUH  flCHOOL. 


he  MrriTad  at  tham.  Urns  in  findiiig  the  araa  d  a  trleBi^ 
he  took  the  aquaie  root  of  the  prodoet  ol  four  Uiiei.  The 
ckMical  Greek  geometricuuie  permitted  the  nee  ol  the  oqaare 
and  the  cube  of  a  line  beeanae  these  coold  be  repreiented 
geometrically,  but  a  6gure  of  four  dimensiona  is  inconceivable^ 
and  certainly  they  would  have  rejected  a  proof  which  involved 
such  a  conception. 

It  ia  questionable  if  Hero  or  his  contemporaries  were  aware 
of  the  existence  of  the  Rhind  papyrus,  but  it  would  seem  that 
treatises  founded  on  it  and  of  a  similar  character  were  then 
current  in  Egypt,  and  a  text-book  of  this  kind — though 
most  likely  some  eight  centuries  or  so  later  in  date — has  been 
discovered  and  reproduced^.  Doubtless  it  was  from  some  such 
sources  that  Hero  drew  his  inspiration.  Two  or  three  reasons 
have  led  moilem  commentators  to  think  that  Hero^  who  was 
bom  in  Alexandria,  was  a  native  Egyptian.  If  this  be  so^  it 
affords  an  interesting  illustration  of  the  permanence  of  racial 
characteristics  and  traditions.  Hero  spoke  and  wrote  Greek, 
and  it  is  believed  that  he  was  brought  up  under  Greek 
influence;  yet  the  rules  he  gives,  his  methods  of  proof^  the 
figures  he  draws,  the  questions  he  attacks,  and  even  the 
phrases  of  which  he  makes  use,  recall  the  earlier  work  of 
Ahmes. 


Thif  first  century  before  Christ. 

The  successors  of  Hipparchus  and  Hero  did  not  avail 
themselves  of  the  opportunity  thus  opened  of  investigating 
new  subjects,  but  fell  back  on  the  well-worn  subject  ol 
geometry.  Amongst  the  more  eminent  of  these  later  geo- 
metricians wera  Tlieodosius  and  Dionysodorus,  both  of  whom 
flourished  about  50  ac. 

TheodoaiUB.     Theodosius  was  the  author  of  a  complete 

•  The  Akhiuim  papjrruii  by  J.  Baillst  lu  the  M^moint  de  U  mmim 
mrekiQlogiimg/nni^iu  au  Cairt^  vol.  ix,  pp.  1— ee*  Pariih  lesm. 


r 


\ 


CLOSE  or  HIE  riRsT  alexanukian  dchool.       95 

trcfttiee  on  the  geometry  of  the  upbere,  snd  Ot  two  worin  on 
utronomf  ". 

Dionysodonu.  liiottijuidori't  is  known  to  us  only  by 
hia  aolnlkm  of  tl)e  problem  to  divide  k  heminphere  by  »  pUiw 
pftrallel  to  its  tiase  into  two  pnrLs  whoiie  votumes  stwll  be  in 
a  given  ratio.  Like  the  wilutinn  by  Diocles  of  the  simiUr 
problem  for  &  sphere  above  allutletl  In,  it  wu  ABnted  by  tho 
aid  of  conic  nectionst.  Pliny  sayit  that  Dionysodortis  deter- 
mined the  len^h  of  the  nuliuM  of  the  («Hh  appraxitnatety 
•s  43,OOn  siMrlifL,  which,  if  we  tnke  the  Olympic  KtMiimn  a€ 
202}  yards,  is  a  little  lesn  than  5000  milea ;  we  do  not  know 
how  it  WM  obtained.  This  may  lie  compaml  with  the  nmilt 
giTen  by  ErAto«thene«  and  mentioned  nlK>ye. 

End  of  the  Firgt  J/ftiwnrfn«H  School. 

Tim  atl ministration  of  EirtTit  wmk  definitely  nnderteken 
by  Rome  in  30  ac.  The  cloHtng  yenrx  of  the  dynasty  of  the 
Ptolemies  and  the  earlier  yearn  of  the  lioman  ocenpation  of 
the  country  were  marked  by  much  disorder,  drit  and  politicaL 
Hie  studies  of  the  aniversity  were  Batnnally  intermpted,  aiid 
it  is  customary  to  take  this  time  as  th«  cloee  of  the  flnt 
Alexandrian  school. 

■  Tbs  work  on  the  wphtn  «u  nliled  If;  I.  Bamw,  CanMdlO.  ISTi, 
ud  br  E.  NiiM,  Bcrtin,  IHSr  Th«  works  an  wtrawny  vm  pabtWwl 
by  Duypodlu*  In  MT9. 

t  It  li  TtpndoMd  In  B.  Satcr'i  Onthlthlt  irr  m 
ielH^Un,  wteoBi  •dltlcra,  Zarich,  18T1.  |i.  101. 


96 


CHAPTER  V. 

THE  8BOOND  ALEXANDBUIT  HCHOOL^. 

30  aa-641  A.a 

I  cx>]fOi.UDBii  the  Unt  chapter  by  stating  that  the  ftni 
■chool  of  Alexandria  may  be  said  to  have  oome  to  an  end  aft 
about  the  name  time  aa  the  country  lout  its  nominal  inde- 
pendence. But|  although  the  schools  at  Alexandria  snllered 
from  the  disturbances  which  affected  the  whole  Roman  world 
in  the  transition,  in  fact  if  not  in  name,  from  a  republio  to 
an  empire,  there  was  no  break  of  contiuuity ;  the  teaching  in 
the  university  was  never  abandoned;  and  as  soon  as  order 
was  again  established  students  began  once  more  to  flock  to 
Alexandria.  This  time  of  confusion  was  however  contempo- 
raneous with  a  change  in  the  prevalent  views  of  philosophy 
which  thenceforward  were  mostly  neo-platonic  or  neo-pytha- 
goreaii,  and  it  therefore  fitly  marks  the  commencement  of  a 
new  period.  These  mystical  opinions  reacted  on  the  mathe- 
matical  Hchool,  and  this  may  partially  account  for  the  paucity 
of  guud  work. 

Though  Greek  influence  was  still  predominant  and  the 
Greek  language  always  used,  Alexandria  now  became  the  in* 
tellectual  centre  for  most  of  the  Mediterranean  nations  which 
were  subject  to  Rome.      It  should  be  added  however  that 

*  For  sathoritiei,  see  footnote  sbo?e  on  p.  5S.  All  «Utet  given 
hereafter  are  to  be  taken  ae  aaao  domini,  anlew  the  eontrary  is  ezpicMly 
stated. 


I 


HERENUS.      MENELAUK.      NICOMACHUS.  97 

the  direct  connection  witlt  it  i>f  ninny  of  the  mathenwticimiu 
nf  thin  time  is  M  leiwt  (lintlitful,  hut  their  knowledge  wiia 
nltiniat^tjr  oblainnd  from  the  Alexandrian  t«achera,  and  they 
are  tntunlly  drncrilird  ax  of  the  necond  Alextindrian  BchooL 
Such  malhemnlics  an  wen-  taught  at  Konie  were  derived  from 
Greek  Mjurcea,  and  *n  may  thereforr  ronvenlentlj  cownder 
their  extent  in  connection  with  thin  chapter.        • 

The  first  century  after  <.'hria. 

There  is  no  doubt  that  throughout  the  tirat  centaiy  after 
Christ  ([potoetry  continued  to  lie  that  suhjeti  in  science  to 
which  inoHt  attention  wan  devoted.  But  by  thia  time  it  was 
evidc-nt  that  the  geometry  of  Archimedm  and  Apolloiiina  was 
not  capable  of  much  furt)ier  extension  -  and  such  geometrical  * 
treatises  as  were  producpd  consisted  mostly  of  commeittaries 
on  the  writings  of  the  great  mathematicians  of  a  preceding 
age.  In  this  century  the  only  original  wm-ks  of  any  ability 
o<  which  we  know  anj-thing  were  two  by  Berenas  and  one  by 
Menelaus. 

Bereniu.  Menelaiu.  Those  by  i^erfNu*^  J  nf  un,  eirc 
70,  are  on  the  planf  frtiont  of  the  eone  nod  ei/lituirr*,  in  the 
conrao  of  which  he  Ibjh  down  the  fundamental  proposition  of 
timnm'ersals.  That  by  Men'iatu  of  Alrrandria,  circ  9)*,  is 
on  tphtrifal  Irii/onomelry,  investigated  in  the  Enelidean 
method  t.  The  fundamental  theorem  on  whidi  the  subject 
is  based  in  the  relation  between  the  six  segments  of  the  siden 
of  a  spherical  triangle,  formed  by  the  arc  of  a  great  ciide 
which  cats  them  [book  in,  prop.  I].  MeneUna  ftlao  wrote  on 
Um  calculation  of  cbords,  that  its  on  plane  trigonometiy ;  this 
islost 

Nlconuuihiu.  Towards  the  cIom  of  this  oentory,  cira 
100,  IfieonKMehus,  a  Jew,  who  was  bom  at  Ocfssk  in  80 

■  Tb«w  Un  bmi  Hitod  bj  J.  L.  Hnberg.  Ls^sIr.  IMS)  sad  hf 
1.  BaUer,  OiTotd,  1710. 

t  This  WIS  trao>lal«dbrE.HsU«r,Oilbri,  ITM. 


98       THS  SBOOND  ALIXAVDRIAN  BCHOOL 

and  died  eiro.  110,  pablinlied  an  AriikmMie\  whkk  (or  ratfiMr 
the  LaUii  tfmmUtioa  of  it)  remiuiied  for  a  thouiend  jeeri  m 
standArd  Mithority  oa  the  sabjeot.  Oeometrioal  demooatntaoni 
are  here  abandoued«  and  the  woric  it  a  mere  ehunifioatioa 
of  the  results  then  known,  with  namerieal  illnatratioiis :  the 
endenoe  for  the  truth  of  the  propositions  enunciated,  for  I 
cannot  caH  theio  proofs,  being  in  general  an  induction  from 
numerical  instances.  The  object  of  the  book  is  the  study  of 
the  properties  of  numbers,  and  particularijr  of  their  ratios. 
Nicomachus  commences  with  the  usual  distinctions  between 
even,  odd,  prime,  and  perfect^  numbers;  he  next  discusses 
fractions  in  a  somewhat  clumsy  manner;  lie  then  turns  to 
polygonal  and  to  solid  numbers;  and  finally  treats  of  ratio^  § 

proportion,  and  the  progressiona  Arithmetic  of  this  kind  is 
usually  termed  Boethian,  and  the  work  of  Boethius  on  it  was 
a  recognized  text-book  in  the  middle  ages. 


The  second  century  after  Christ 

Theon.  Another  text-book  on  arithmetic  on  much  the 
same  lines  as  that  of  Nicomachus  was  produced  by  Tkeom  of 
Smyrna^  circ  130.  It  formed  the  first  book  of  his  woricf  on 
mathematics,  written  with  the  view  of  facilitating  the  study 
of  Plato's  writings. 

ThjfmAridas.  Another  mathematician  of  about  the  same 
date  was  Thymaridcu^  who  is  worthy  of  notice  from  the  feet 
that  he  is  the  earliest  known  writer  who  explicitly  enunciates 
an  algebraical  theorem.  He  states  that,  if  the  sum  of  any 
number  of  quantities  be  given,  and  also  the  sum  of  every  pair 
which  contains  one  of  them,  then  this  quantity  is  equal  to 
one  (h  -  2)th  part  of  the  ditTerence  between  the  sum  of  these 
pairs  and  the  first  given  sum.     Thus,  if 

X|-t-A:,+  ...  +  x,  =  5, 

*  The  work  ha*  been  edited  bjr  R.  Hoche,  Leipxig,  ia66i 
t  The  Orot-k  teit  of  thcxe  |>sris  which  sre  now  sxiaat.  with  a  Fkeooh 
trsnsUtion,  wss  i«iued  bj  J.  DupuU,  Psrii^  1893. 


i 

i 


i 


I 


PTOLEHr. 


99 


1 


f 
) 


and  if        ^i  -•-  'i  -  «tf     ^  -*-  ^:i  -  'st  ..-t  And  x,  •«-  x.  =  «., 

then  -^1  =  (*t  +  ^4  +  •  ••  +  *■  —  *^)i{^  —  2). 

He  does  tiot  Meem  t«>  have  umvI  a  symbol  to  denote  the  unknown 
quantity,  bat  he  always  represents  it  by  the  same  word,  which 
is  an  approximation  to  syniboliKm. 

Ptolemy*.  About  the  same  time  as  these  writers  Piofmny 
^  sUejrttndria^  who  died  in  168,  produced  his  great  work  on 
astronomy,  which  will  preserve  his  name  as  long  as  the  history 
of  science  endures.  This  treatise  is  usually  known  as  the 
Afmagest :  the  name  is  deri\'ed  from  the  Arabic  title  of  mid' 
9chittii^  which  is  said  to  be  a  corruption  of  fg€yur'nf  [/lo^/xartinf] 
inWa^cs.  The  work  is  founded  on  the  writings  of  Hipparchus, 
and,  though  it  did  not  sensibly  ailvance  the  theory  of  the 
subject,  it  presents  the  views  of  the  older  writer  with  a  com- 
pleteness and  elegance  which  will  always  make  it  a  standard 
treatise.  We  gather  from  it  that  Ptolemy  made  observations 
at  Alexandria  from  the  years  125  to  150;  he  however  was 
but  an  indiflerent  practical  astninomer,  and  the  oliservations 
of  Hipparchus  are  generally  more  accurate  than  those  of  his 
expounder. 

The  work  is  dividnl  into  thirteen  Uioks.  In  the  first  book 
Ptolemy  discusses  various  preliminary  matters;  treats  of  trigo- 
nometry, plane  and  spherical;  gives  a  table  of  chords,  that  is, 
of  natural  sines  (which  is  sulmtaiitially  correct  and  is  probably 
taken  from  the  lost  work  of  Hipparchus);  and  explains  the 
obliquity  of  the  ecliptic ;  in  this  book  he  uses  degrees,  minutes, 
and  seconds  as  measiires  of  angles.  The  second  book  is  de%'oted 
chiefly  to  phenomena  depending  on  the  spherical  form  of  the 

*  See  the  article  Ptoiematm  Ctamditu  by  A»  De  MoriTUi  in  Smith's 
Dktiomarji  of  Greek  ami  Roman  Dioffrttphtf,  London,  1849 ;  S.  P.  Tanneiy, 


i 


Roekenkea  tmr  Vhhioire  ie  Fottronomie  ondemite^  Paris*  189S;  and 
J.  Bw  J.  Delsmhre,  HMoirt  de  rottromnmie  amcienme,  Paris,  1817,  vol.  n. 
An  editfon  of  sU  the  works  of  Ptolemy  which  are  now  extant  was 
yabUdied  si  BAle  in  IMl.  The  Almttfeti  with  varioos  Siiaor  works 
was  edited  hj  M.  Hshaa,  IS  irols,  Paris,  181S-S8^  hot  a  new  editloa 
is  now  in  eovse  of  ptcpsialkNi  kj  J.  L.  Heiterg,  Leipsif,  pert  i^  18W. 

1— \ 


100  THE  SaOOND  AUULAKDRUH  SCHOOL. 


- 


* 


I 


line^  oTy  more  pt 

was  probably  -^ 

jPtolemy  added  ^  \ 


earth:  ha  remarka  that  ^  espUnationa  weold  ba  nach 

•impUfiad  if  tha  aarth  ware  tappoead  to  rotate  on  ita  ajua  onoa 

a  day,  bat  pointa  o^t  that  this  hypothesis  is  inoonsistept  with 

known  facta.    In  tlie  third  book  he  expUins  th^  motion  of  tha 

san  round  tha  earth  by  means  of  axoentrics  and  epicycles :  and 

in  the  fourth  and  fifth  books  he  treata  the  motion  of  the  moon 

in  a  similar  way.    The  sixth  book  is  devoted  to  the  theory  of 

eclipses;  and  in  it  he  gives  3*  8'  30",  that  is  3jYv>  ^  the 

approximate  value  of  w^  which  is  equivalent  to  wking  it  equal 

to  3-1416.     The  seventh  and  eighth  books  oont^  a  catalogue  4 

of  1028  fixed  stars  determined  by  indicating  ihotie,  three  or  * 

more,  that  appear  to  be  in  the  same  straight 

correctly,  lie  on  the  same  great  circle  (this 

copied  from  Hipparchus) :  and  in  another  work 

a  list  of  annual  sidereail  |ihenomena.     The  rekuaining  books 

are  given  up  to  the  theory  of  the  planets. 

This  work  is  a  splendid  testimony  to  thej  ability  of  ita 
author.  It  became  at  once  the  standard  autl|ority  on  astro- 
nomy, and  remained  so  till  Copernicus  .and  ^epler  shewed 
that  the  sun  and  not  the  earth  must  be  regarded  as  the  centre 
of  the  solar  system.  ' 

The  idea  of  excentrics  and  epicycles  on  whicli  the  theories 
of  Hipparchus  and  Ptolemy  are  basr;d  has  been  often  ridiculed 
in  modern  times.  No  doubt  at  a  later  time,  when  more  accu- 
rate  observations  had  been  made,  the  necessity  bf  introducing 
epicycle  on  epicycle  in  order  to  bring  the  theory  into  accord- 
ance witii  tlie  fiicts  made  it  very  complicated,  ^ut  De  Morgan 
has  acutely  observed  that  in  so  far  as  the  ancient  astronomers 
supposed  that  it  was  necessary  to  resolve  every  celestial  motion 
into  a  series  of  uniform  circular  motions  they  erred  greatly,  1 

but  tliat,  if  the  hypothesis  be  regarded  as  a  convenient  way  < 

of  expressing  known  facts,  it  is  not  only  legitimate  but 
convenient.  The  theory  suffices  to  describe  either  tne  angular 
motion  of  the  heavenly  bodies  or  their  change  in  distance^- 
and  as  the  ancient  astronomers  were  concerned  only  with  tha 
former  question  it  met  their  needs :  in  ^ct  it  was  as  good  a  ' 


i 

1 


PTOLEMY.  101 

theory  an  with  their  inMrunients  Knd  knowled^  it  wm  ponible 
to  frume,  (ind  crtrrciponds  to  (he  puprewioii  of  ft  givpn  functiini 
u  A  num  nf  Kinm  nr  coaineii,  ■  method  which  n  of  fm|iKnt  aw 
in  modem  ftnalysiit. 

In  spit*  of  tho  trouble  taken  by  Delwnbre  it  is  aIokm^ 
imponnible  tn  separate  tlie  rcHultx  due  to  Hipparchiu  from 
tliose  due  to  Ptolemy.  But  r>elBnihrc  Mid  De  Slorgsn  agree, 
in  thinking  that  the  ohsenntinnn  quoted,  the  fundamental 
ideas,  and  the  explanation  of  the  apparent  nolar  motion  ara 
due  to  Hipparvhus;  while  nil  the  detailed  expUnatimis  and 
calculations  of  the  lunar  and  planetary  motions  are  due  to 
Ptolemy. 

The  Almayerl  shews  that  Ptolemy  was  a  gemnetrician  of 
the  6rst  rank,  though  it  is  with  the  application  of  geometry 
to  astronomy  that  he  is  chiefly  concerned.  He  was  also 
the  author  of  numerous  other  treatises.  Amongst  these  is  one 
on  ptire  gr.iimrtry  in  which  he  proposed  to  cancel  the  twelfth 
axiom  nf  Euclid  on  pnrallol  lines  and  to  prove  it  in  the  follow- 
ing manner.  Let  the  straight  line  EFOU  meet  the  two 
straight  lines  All  and  CD  so  as  to  make  the  sum  of  the 
angles  BFG  and  FGD  equal  to  two  right  angle*.  It  is  required 
to  prove  that  AB  and  CD  are  parallel.  If  poesible  let  them 
not  be  parallel,  then  they  will  mwt  wheB  prodnoed  saj  »t  M 


"<: 


(or  Sy  Bat  the  angle  J  M?  ia  the  supplement  of  BFG,  a 
is  theraiore  equal  to  FOD :  similarly  the  aa^  FOC  is  eqi 
to  the  an^  BFO.    Henoe  the  onm  of  the  naglea  AFQ  a 


102      THK  SKOOVD  ALEXANDRIAN  8CHOOU 

FOC  tt  equal  to  two  right  angleag  And  the  Iium  BA  aad  DC 
will  thereforo  if  produced  meet  *t  N  (or  M)*  Bot  two  fttimighi 
lines  cannot  encloee  a  sfiace,  therefore  AB  and  CD  cannot 
meet  when  produced,  that  is,  they  are  parallel  OonverMlj, 
if  JJ9  and  C/>  be  parallel,  then  AF  and  CO  are  not  lees 
parallel  than  FB  and  GD\  and  therefore  whatever  be  the 
ftuiu  of  the  angles  A  FO  and  FGC  such  aluo  must  be  the  sum 
of  the  angles  FGD  and  BFO,  But  tlie  suoi  of  the  four  angles 
is  equal  to  four  right  angles,  and  therefore  the  sum  of  the 
angles  BFO  and  FGD  must  be  e<|ual  to  two  right  angles. 

Ptolemy  wrote  another  work  to  shew  that  there  could  not 
be  more  than  three  dimensions  in  space:  he  also  discussed 
orthographie  and  aterewjrapliie  projeciiotut  with  special  refer- 
ence to  the  construction  of  sun-dials.  He  wrote  on  geography, 
and  stated  that  the  length  of  one  degree  of  latitude  is  500 
stadia.  A  book  on  optics  and  another  on  itonnd  are  sometimes 
attributed  to  him,  but  their  authenticity  is  doubtfuL 


The  thii-d  century  after  Christ. 

Pappus.  Ptolemy  had  shewn  not  only  that  geometry 
could  be  applied  to  astronomy,  but  had  indicated  how  new 
methods  of  analysis  like  trigonometry  might  be  thence  de- 
\  doped.  He  found  however  no  successors  to  take  up  the 
work  he  had  commenced  so  brilliantly,  and  we  must  look 
forward  1*^0  years  before  we  find  another  geometrician  of  any 
eminence.  That  geometrician  was  Pappus  who  lived  and 
taught  at  Alexandria  about  the  end  of  the  third  century.  We 
know  that  he  had  numerous  pupils,  and  it  is  proliable  that  he 
temporarily  revived  an  interest  in  the  study  of  geometry. 

Pappus  wrote  several  books,  but  the  only  one  which  has 
come  down  to  us  is  lib  S^wayMyiy*,  a  collection  of  mathe- 
matical papers  arranged  in  eight  books  of  which  the  first  and 
part   of    the   second    have   been   lost.      This  collection 

*  It  liM  beeu  publiHhed  by  F.  Hultuch,  Berlin,  1H76-H. 


r 

i 
1 


PAPPUS.  103 

intends)  to  bp  n  Hynopxin  of  (ircek  mnthnnatic*  logctber 
with  commenta  nnil  Arldilionnl  propoHJtiont  hf  the  editor.  A 
careful  compnrison  of  vorioas  rxtnnt  wnrkn  with  the  account 
given  of  thpm  in  this  liook  !<hi>wn  thxt  it  is  tniiitworthj,  *nd 
we  rely  largely  on  it  for  our  knowledge  of  other  workii  now 
IcMt.  It  is  not  amnged  chronologiciilly,  but  all  the  treatise* 
on  the  Mmp  iiuhj««t  ant  grnupetl  together,  and  it  is  most 
likely  that  it  gives  rou;*hly  the  order  in  which  the  classical 
authors  were  read  at  Alexanilrin.  Probably  the  first  book, 
which  is  now  li»t,  wiu  on  arithmetic.  The  next  four  books 
deal  with  gwimetry  exclunive  of  conic  seclions :  the  sixth  with 
astronomy  including,  M  HubxitHury  Hubjpcts,  optics  and  trigo- 
nometry :  the  aeventli  with  analyniH,  cnnic^  and  porisms :  and 
the  eighth  with  mechanics. 

The  last  two  Ijooks  contnin  a  good  deal  of  original  work 
by  Pappus;  at  the  snnie  time  it  should  l>e  remarked  that  in 
two  or  three  cases  be  ha«  l)een  deteote<l  in  appropriating 
prooh  from  enrlier  authors,  and  it  is  possible  he  may  have 
done  this  in  other  canes. 

Subject  to  this  suspicion  we  may  say  that  Pappus's  best 
work  is  in  geometry.  He  discovered  the  directrii  in  the  conic 
sections,  hut  he  investigated  only  n  few  isolated  properties : 
.  the  earliest  comprdiennive  account  wax  given  by  Newton  and 
Boticovich.  As  an  illustration  of  his  power  t  may  mention 
that  he  solved  [bouk  vii.  prop.  107]  the  prohlem  to  inscribe  in 
a  given  circle  a  triangle  whose  sides  produced  shall  pass 
through  three  collinear  points.  This  ijuettion  was  in  the 
eighteenth  century  generalised  by  Cramer  bj  snppcaing  the 
three  gireo  ])oinls  to  be  anywhere ;  and  wu  considered  » 
difficult  problem.  It  wan  sent  in  1742  as  a  diallenge  to 
Castilkm,  and  in  1776  he  published  a  solntion.  I^gnrnga, 
Euler,  Uiolier,  Fuss,  and  Iiexell  also  gnxe  solntitms  in  1780. 
A  lew  yoM«  Uter  the  pruUem  was  set  to  a  Noapolitaa  lad 
OltaiMM),  who  WM  only  16  bat  who  had  shewn  marked  mathe- 
■atioal  Kbili^,  aad  h»  cxteiKlad  it  to  the  can  of  a  polygoii  at 
n  widm  wbidi  paaa  thimi^  n  fivMi  painta^aodgaTaaMlutNa 


104  THB  81CX>ND  ALBXAllIttUV  8CHOOU 

both  liniple  and  d^gant.  Pooedei  eziended  it  lo  oonioi  €l 
any  qpeciet  and  tabject  to  other  rettrictiona. 

In  meehanict  Pappus  shewed  thai  the  oentie  el  mass  el  a 
triangoUu'  himina  is  the  same  as  that  of  an  inscribed  triangular 
btmina  whose  vertices  divide  each  of  the  sides  of  the  original 
triangle  in  the  same  ratia  He  also  cUscovered  the  two 
theorems  on  the  surface  and  volume  of  a  solid  of  revolution 
which  are  still  quoted  in  text-books  under  his  name:  these  C* 

are  that  the  volume  generated  by  the  revolution  of  a  onrve 
about  an  axis  is  equal  to  the  product  of  the  area  of  the  cmrve 
and  the  length  o(  the  path  described  by  its  centre  of  mass; 
and  the  surface  is  equal  to  the  pruduct  of  the  perimeter  of  the 
cur\'e  and  the  length  of  the  path  described  by  its  centre  of 
mass. 

The  problems  above  mentioned  are  but  samples  of  many 
brilliant  but  isolated  theorems  which  were  enunciated  by 
Pappus.  His  work  as  a  whole  and  his  comments  shew  that  he 
was  a  geometrician  of  power ;  but  it  was  his  misfortune  to 
live  at  a  time  when  but  little  interest  was  taken  in  geometry, 
and  when  the  Hubject,  as  then  treated,  had  been  practically 
exlmusted. 

PoKsibly  a  small  tract^  on  multiplication  and  division  of 
sexagcsinuil  fractions,  which  would  seem  to  have  been  written 
about  this  time,  is  due  to  Pappus. 

The  JourUi  century  after  Ckrisi. 

Throughout  the  second  and  third  centuries,  that  is,  from 
the  time  of  Niconuushus,  interest  in  geometry  had  steadily 
decreased,  and  more  and  more  attention  had  been  paid  to  the 
theory  of  nunilierR,  though  the  results  were  m  no  way  com- 
mensurate with  the  time  devoted  to  the  subject.  It  will 
be  remembered  that  Euclid  used  lines  as  symbols  for  any 
magnitudes,  and  investigated  a  number  of  theorems  about  | 

*  It  wai  edited  by  C.  Henrjr,  Hall«,  187U,  sad  in  valuable  as  au  j 

illustration  of  practical  Greek  arithmetie. 


ARITHMETIC  AND  ALGEBRA. 


10 


numbers  in  *  Hirictlj  scientific  nmnnrr,  hut  he  confined  him 
self  to  cases  where  a  geonietricnl  n^proHf^ntation  was  possihh 
There  are  indications  in  the  m-orks  of  Archimedes  that  he  m-a 
prepared  co  carry  the  sahject  much  further:  hi*  introduces 
niimbf*ri  into  his  geometrical  diHcusHions  and  ilivided  lines  h; 
lines,  but  he  was  fully  occupicfl  by  <»ther  rrsenrcheK  and  hat 
no  time  to  devote  to  arithmetic.  Hero  aliaiid«Yned  the  gee 
metrical  representaticm  of  numliers  but  he,  Nicomachus,  am 
other  later  writers  on  nrithmrtic  did  not  succeevl  in  creatinj 
any  other  symUilism  for 'nun\)»ers  in  g«*noral,  and  thus  miioi 
they  enunciated  a  theorem  they  m*en*  content  to  verify  it  b; 
a  large  numlier  of  numerical  examples.  Tliey  doubtless  kne^ 
how  to  solve  a  quadratic  e(|uation  m-itli  numerical  coefiicientK— 
for,  as  pointed  out  almve,  geometrical  solutions  of  the  equa 
tions  n^  -  5n;  +  c  =  0  ami  nj^  +  ftr  -  r  =  0  are  given  in  Euc.  vi 
28  and  29 — ^but  probably  this  reprenented  their  highi*Ht  attain 
inent. 

It  would  seem  then  that,  in  Kpite  of  the  time  given  to  thei 
study,  arithmetic  and  algebra  had  not  made  any  Kensibl 
advance  since  the  time  of  Archimedes.  The  problems  of  thi 
kind  which  excited  most  interest  in  the  thinl  century  may  b 
illustrated  from  a  cfillectiim  of  (|uestionM,  printed  in  th( 
Palatine  Antliology,  which  mas  made  by  Metrodoras  at  th 
beginning  of  the  next  century,  almut  310.  Some  of  them  an 
doe  to  the  editor,  but  sume  an*  of  an  anterior  date,  and  the^ 
fairly  illustrate  the  way  in  which  arithmetic  was  leading  u| 
to  algebraical  methods.  The  following  are  t3'pica1  examples 
**  Four  pipes  discharge  into  a  cistern  :  one  fills  it  in  cme  day 
anoUier  in  two  days ;  the  third  in  three  days ;  the  fourth  ii 
foar  days:  if  all  run  together  how  noon  will  they  fill  th( 
eSstemt"  *'Deinochares  has  lived  a  fourth  of  his  life  as  i 
boj ;  •  fiftli  M  a  youth ;  a  third  as  a  man ;  and  has  spen 

in  his  dotage:   how  old  is  het"    "Make  i 

^  TpM,  copper,  tin,  and  iron  weighing  60  roinae :  goh 

n  ba  iwo-Uiirds  of  it ;  gold  and  tin  three-fourthi 

IrOB  tlnm4ifU»  oT  it :  find  the  weights  o 


106 


THB  8B0OND  ALXXAWDIUV  SOUOOL. 


the  fold,  copper,  tin,  and  iron  which  are  raqoired.'^  The 
iMt  is  a  namerieal  illoBtration  of  Thymaridas's  theorani  <|BOtcd 
above. 

The  Oerman  commentators  iiave  pointed  out  that  though 
these  problems  lead  to  simple  eqaatious,  thej  can  be  solved  bf 
geometrical  metliods,  the  unknown  quantity  being  reiVresented 
bj  a  line.  Dean  Peacock  also  remarked  that  thej  can  be 
solved  by  the  method  UKcd  in  similar  cases  by  the  Arabians 
and  many  mediaeval  writers.  This  method,  usually  known  as 
the  rt$le  of  faUtt  OMSumption^  consists  in  assuming  any  number 
for  the  unknowQ  quantity,  and,  if  on  trial  the  given  conditions 
be  not  satisfied,  shearing  the  number  by  a  simple  proporticm  as 
ill  i-ule  of  three.  For  example,  in  the  second  problem,  suppose 
we  assume  that  the  age  of  Democlmres  is  40,  tlien,  by  the 
given  conditions,  he  would  have  spent  8§  (and  not  13)  years 
in  his  dotage,  and  therefore  we  have  the  ratio  of  8 J  to  13 
equal  to  the  ratio  of  40  to  his  actual  age,  hence  his  actual  age 
is  00. 

But  the  nioeit  recent  writers  on  the  subject  think  that  the 
problems  were  solved  by  rhetorical  algebra^  that  is,  by  a  process 
of  algebraical  reasoning  expressed  in  wonls  and  without  the 
use  of  any  symbols.  This,  according  to  Nesselniaun,  is  the 
first  stage  in  the  development  of  algebra,  aud  we  find  it  used 
both  by  Ahnies  and  by  the  earliest  Arabian,  Persian,  and 
Italian  algebraists:  examples  of  its  use  in  the  solution  of  a 
geometrical  problem  and  in  the  rule  for  the  solution  of  a 
quadratic  cH|uatii»ii  are  given  later^.  On  this  view  then  a 
rhetorical  algebra  liad  been  gradually  evohed  by  the  Greeks, 
or  was  then  in  proci*&H  of  evolution.  Its  development  was 
however  very  imperfect.  Ilankel,  who  is  no  unfriendly  critic, 
says  that  the  results  attaiiie<l  as  the  net  outcome  of  the  a'ork 
of  six  centuri<*H  on  the  theory  of  numbers  are,  whether  a'e 
look  at  the  form  or  the  sulMtance,  unimportant  or  even  childish, 
and  are  not  in  any  way  the  commencement  of  a  science. 


•4 

I 


Strtf  below,  pp.  iOl^  916. 


DIOPHANTUS.  107 

In  the  niicist  of  thin  flecajing  interest  in  geometry  and 
theM  feel»1e  Attempts  at  algebraic  aritlinietic,  a  iiingle  alge- 
braist of  marked  originality  suddenly  appeared  wlio  created 
what  was  practically  a  new  science.  This  was  Diophantus 
who  introduced  a  system  of  abbre%*iations  for  those  operations 
and  quantities  which  constantly  recur,  though  in  using  them 
he  observed  nil  the  rules  of  grammatical  syntax.  The  result- 
ing science  is  called  by  Nesselmann  njfncnimi^  afg^ra :  it  is  a 
sort  of  shorthand.  Bniadly  speaking,  it  may  lie  said  that 
European  algebra  did  not  ad%'ance  lieyond  this  stage  until  the 
close  of  the  sixteenth  century. 

^lodern  algebra  has  pmgresscfl  one  stage  further  and  is 
entirely  ttjpnhofir  ;  that  is,  it  has  a  language  of  its  own  and  a 
system  of  notation  which  has  no  fib%'ious  connection  with  the 
things  n^presentcnl,  while  the  operations  are  |M*rformed  accord- 
ing to  rertnin  rul(*s  mhich  are  distinct  from  the  laws  of  gram- 
matical construction. 

DiophantUB^.  All  that  we  know  of  Dioithanhis  is  that 
he  lived  at  Alexandria,  and  that  nuist  likely  be  was  not  a 
Greek.  Even  the  date  of  his  career  is  uncertain,  but  probably 
he  flourished  in  the  early  Imlf  of  the  fourth  century,  that 
is,  shortly  after  the  death  of  Pappus.  He  was  84  m*hen  he 
died. 

In  the  above  sketch  of  the  lines  on  which  algebra  has 
developed  I  credited  Diophantus  m-ith  the  invention  of  synco- 
pated algebra.  Tliis  is  a  point  on  which  opinions  differ,  and 
some  writers  believe  that  he  only  systematized  the  knowledge 
which  was  familiar  to  his  contemporaries.  In  support  of  this 
latter  opinion  it  may  be  statefl  that  Cantor  thinks  tliat  there 
are  traces  of  the  use  of  algebraic  symlmlism  in  Pappus,  and 
Friedlein  mentions  a  Greek  papyrus  in  which  the  signs  /  and  O 
are  niipd  for  addition  and  sul»traction  respectively ;  bat  no 
other  direct  evidence  for  the  non-originality  of  Diopluuitns  has 


*  A  critical  cditioD  of  the  eollccted  works  of  fHaphantiM  was  editfd 
kj  8.  P.  Tannefy,  t  vols,  Loptif,  I89S  i  stealso  Dhpkmmiea  a/  Ahmmirim 
ky  T.  L.  Healh,  Oamkridffs.  18R5. 


106  TUB  HBOOND  ALBXAKDRUH  BOUOOU 

been  produoed,  and  no  aneieni  *athor  gives  any  ■uwlioa  lo 
this  opinion. 

Diophantus  wrote  a  dioii  eiaay  on  polygonal  nwnben ;  a 
treatUe  on  algebra  which  has  conie  down  to  on  in  a  mutilated 
condition ;  and  a  work  on  poritms  which  it  loet. 

The  PottfgokuU  Xumhen  contains  ten  propoeitionii  and 
was  probably  his  earliest  work.  In  this  he  abandons  the 
empirical  method  of  Nicomachns,  and  reverts  to  the  old  and 
classical  system  by  which  numbers  are  represented  by  linesi  a 
construction  is  (if  necessary)  made,  and  a  strictly  deductive 
pruof  follows:  it  may  lie  noticed  that  in  it  he  quotes  pro» 
positions,  such  sh  Euc.  ii,  d»  and  ii,  8,  as  referring  to  numbers 
and  not  to  magnitudeti. 

liis  chief  work  is  his  ArithMetie,  This  is  really  a  treatise 
on  algebra ;  algebraic  syinboU  are  used,  and  the  problems  are 
treated  analytically.  Diophantus  tacitly  assumes,  as  is  done  in 
nearly  all  modern  algebra,  that  the  steps  are  reversible.  He 
applies  this  algebra  to  find  ttolutioiis  (though  frequently  only 
particular  ones)  of  tteveral  problems  involving  numbers.  I 
propose  to  consider  succes-sively  the  notation,  the  methods  of 
analysis  employed,  and  the  subjet^t-matter  of  this  work. 

First,  as  to  the  notation.  Diophantus  always  employed  a 
symbol  to  represent  the  unknown  quantity  in  his  equationsg 
but  OS  he  had  only  one  symbol  he  could  not  use  more  than 
one  unknown  at  a  time^.  The  unknown  quantity  is  called 
o  opc^fuk,   and   is   represented   by  r*  or  r*'.     It  is  usually 

printed  as  «.  In  the  plural  it  is  denoted  by  cv  or  «s*^.  This 
symbol  may  be  a  corruption  of  a^,  or  perhaps  it  may  stand  for 
the  word  o-wpoc  a  heapt,  or  possibly  it  may  be  the  final  sigma 
of  this  word.  'Ilie  square  of  the  unknown  is  called  fivro^s, 
and  denoted  by  £*•:  the  cube  Kv^oi,  and  denoted  by  k*  ;  and 
so  on  up  to  the  sixth  power. 

The  coefficients  of  the  unknown  quantity  and  its  powers  are 

*  See,  however,  below,  pp.  111-112,  eiample  (iii),  for  sn   imtsnes 
wliere  there  are  two  uuknown  qa«ntities  in  his  problem, 
t  Hee  above,  p.  4». 


niOPHANTUS.  .  109 

iramberBi  and  *  numerical  coefficient  U  written  imniediatelyafter 

the  qaantitj  it  inulti|ilie8 :  thus  %d -•  Xj  and  f^ca»ssca=:llj& 
An  absolute  tenn  is  regarded  an  a  certain  number  of  units  or 

fifirAt  wbich  are  representefl  by  /a* :  thus  m*5  =  1*  m**^  =\l' 

There  is  no  sign  for  addition  lieyond  mere  juxtaposition. 
Subtraction  is  represented  by  ^  and  this  Hymliol  affects  all  the 
sjmbols  that  follow  it.     Equality  is  represented  l>y  t.     Thus 

rspresents  (>"  +  8x)  -  (rir'  -».  1 )    x. 

Diophantns  also  introduced  a  somewhat  similar  notation 
lor  fractions  involving  the  unknown  quantity,  but  into  the 
details  of  this  I  need  not  here  enter. 

It  will  be  noticed  that  all  these  syniliols  are  mere  abbre- 
viations for  words,  and  Diophantus  reasons  out  his  proofs, 
writing  these  abbreviations  in  the  middle  of  his  text.  In 
most  manuscripts  there  is  a  marginal  summar}*  in  which  the 
symbols  alone  are  used  and  which  is  roally  symbolic  algebra ; 
but  probably  this  is  the  addition  of  some  scribe  of  later  times. 

Tiiis  introduction  of  a  contraction  or  a  symliol  instead  of 
a  word  to  represent  an  unknown  <|uantity  marks  a  gm&ter 
advance  than  anyone  not  acquaint4Hl  with  the  subject  would 
imagine^  and  those  who  have  never  had  the  aid  of  some  such 
abbrsTiated  symbolism  find  it  almost  impossible  to  understand 
complicated  algebraical  prooesHcs.  It  is  likely  enough  that  it 
night  have  been  introduced  earlier,  but  for  the  unlucky  s3'stem 
of  numeration  ado|>ted  by  the  Greeks  by  which  they  used  all  the 
letCen  of  the  alphabet  to  denote  particular  numbers  and  thus 
■Mde  it  impossible  to  employ  them  to  represent  any  number. 

Nextg  aa  to  the  knowledge  of  algebraic  methods  shewn  in 
tlia'book.  Diophantns  ooromencca  with  some  definitions  which 
Ml  dpIaBataon  of  his  notation,  and  in  giving  the 

'*^  lar  wdrnm  Iw  states  that  a  subtraction  multiplied  bj 

list  aa  additioo ;  fay  this  he  means  that  the 

d  .^d im  ike  expansion  of  (a*6)  (e-d)  is 

•  nda  1m  always  takes  care  thai  the 


110 


TUB  SECOND  ALBXANOBUV  SCHOOU 


numbeni  a^h^e^d  are  bo  ehoaen  that  a  u  givatar  than  4  and 
r  it  greater  than  i/. 

The  whole  of  the  work  iteell^  or  at  leaat  ae  inaoh  ae  is  now 
extanti  it  devoted  to  aolviog  problems  which  lead  to  equa- 
tioQa.  It  eontaiiiH  rules  for  solving  a  simple  equation  of  the 
first  degree  and  a  binomial  quadratic.  The  rule  for  solving 
any  quadratic  equation  is  proliably  in  one  of  tlie  lost  books» 
but  where  the  equation  is  of  the  form  oj^  •§•  &r  -f  c  -=  0  lie 
seems  to  have  uiultiplie«l  by  a  and  then  **  completed  tlie 
square''  in  much  the  Maine  way  as  is  now  done:  when  the 
ruots  are  negative  ur  irnitioiuil  tlie  equation  is  rejected  as 
*'  impossible,"  and  even  when  both  routs  are  positive  he  never 
gii'es  more  than  one,  always  taking  tlie  |ioiutive  value  of  the 
square  root  Diopliantus  solves  one  cubic  equation,  uamely, 
a?*  -».  X  =  4x*  +  4  [book  VI,  prob.  19]. 

The  greater  part  of  the  work  is  however  given  up  to  in- 
determinate equatioiiM  lieta-eeii  two  or  three  variables.  When 
the  equation  is  lietwtHfii  two  variables,  then,  if  it  be  of  the 
first  degree,  he  assuuies  a  suitable  value  for  one  variable  and 
solves  the  equation  for  the  other.  Most  of  his  equations  are 
of  the  form  y*  -  Ax*  -¥  Hx  -^  i\  Whenever  A  or  C  is  equal  to 
zero,  he  is  able  to  solve  the  equation  completely.  When  this  is 
not  the  case,  then,  if  J  -  a',  he  assumes  y^  ax  -^  %h\  it  C  ~  t^^ 
he  sssuines  y  =  tnr'i-c;  and  lastly,  if  the  ecmation  can  be  put 
in  the  form  ^  -  {ax  ^  by  +  c\  he  assumes  |y  -  mx :  where  in 
each  cane  tn  lias  some  particular  numerical  value  suitable  to 
the  problem  under  consideration.  A  few  particular  equations 
of  a  higher  order  occur,  but  in  these  he  generally  alters  the 
pit>blem  so  as  to  enable  him  to  reduce  the  equation  to  one  of 
the  aliove  forms.  I 

The  simultaneous  indeterminate  equations  invohing  three 
variables,  or  '* double  equations"  as  he  call^  them,  wliich  he 
considers  are  of  tlie  forms  y*  ==  Aa^  -^  Bx  •¥  C  and  ^^oj^-k-  bx+c. 
If  A  and  a  both  vanish,  he  solves  the  equations  in  one  of  two 
ways.  It  will  lie  enough  to  give  one  of  his  inethuds  which  it 
as  follows :  he  subtracts  and  thus  gets  an  equation  of  the  form 


i 

I* 

I. 


,f 


1 
i 

i. 

i 
.» 

i 


DIOPUANTUS.  Ill 

y  -  *■  =  nu-  +  It  ;  hence,  if  y  *  ;  =  A,  tlipn  y  t  i  =  ("w  +  ii)/A ; 
Kn<l  solvinf;  he  findn  y  nnd  i.  His  tn-nlment  o(  "(loubI« 
equations  "  of  n  liighc-r  order  IimIch  geni*mlity  »nd  dependB  on 
thp  particular  nilinericnl  coiidilionB  of  the  prohlem. 

Limtly,  OS  to  the  loattcr  i>f  Ihc  hook.  The  proH«inB  be 
kttackn  and  tlif  nnalyKia  he  uses  are  no  vuioua  thfit  they 
cannot  ba?  tieocrilted  concJHely  and  I  have  iherefoTR  aelected  flTa 
tj-picnt  )iroblenis  to  illu^tmt«  his  melhiNlH.  What  «emu  to 
strike  hit  critics  niiMt  iit  the  in;:;ennity  with  which  be  wlecta 
OS  hU  unknown  some  quantity  which  lenda  lo  equatkms  anch 
oa  he  can  m>Ivc,  nnil  the  nrlilicea  by  which  he  finda  nnimricol 
•olutions  of  his  equations. 

I  nelect  the  folhiwing  as  charact«riittic  exomplen. 

(i)  Fimljonr  tiHnt^ni,  the  mim  of  evry  arrangemrni  (Arve 
at  A  time  br\i<tj  ji'orw  /  any,  22,  24,  27,  mvi  20  [book  I,  prob.  17J. 

Let  z  be  the  aam  of  all  four  n  urn  Iters ;  henca  the  nam- 
het»  we  *  -  22,  .T  -  24,  »T  -  27,  and  X  -  20. 

.-.  «  =  (;r-22)+(x-24)  +  (x^27)t(«-20). 

.-.   a;-- 31. 

.-    the  nuniben  are  9,  7,   4,  and  11. 

(ii)  Diride  n  nnmhrr,  tiifh  at  13  ichirh  U  As  van  aj  ttto 
wqtutn*  4  and  9,  (>i('>  lico  alhrr  nquara  [book  II,  prob.  10]. 

He  nys  that  since  the  given  squarcn  are  2*  and  3*  he  will 
take  {i  4-  2)*  and  {mx  ~  3)'  us  the  required  aqtUHWi,  and  will 


.-.   (x  +  2)'+(2x-31»=13. 

.-.  x-e/5. 

.".   the  required  aquarex  arc  324/2.')  and  1/25. 

(iii)  Find  ttco  tquare*  mi-A  iia(  thf.  mm  ^  Out  fndmet 
mnd  either  in  n  tqunre  [book   ir,  prob.  29]. 

Let  X*  and  y*  be  the  number*.  Then  x*y  +  ^aiid  a^+a^ 
are  aqaarea.  The  first  will  be  a  aqnare  if  x"  +  I  bp  a  Bqaan^ 
whiob  be  aamncs  nuj  be  taken  eqtMl  to  fft—tf,  baow 


112 


THK  ttlfiOOMD  ALUUMDRIAN  8CBOOU 


X  -  3/4.  He  lias  oow  to  make  9  (^  -i-  1)/I6  a*  square,  lo  do 
this  he  assttuies  tliat  9/  ••-  9  ^  (3^  -  4)',  henoe.y  =  7/34.  There- 
fore the  sqiianM  required  are  9/16  and  49/576. 

It  will  be  rtioollected  that  Diopliautus  had  only  one  symbol 
for  an  unknown  quantity ;  and  in  this  example  he  begins  by 
calling  the  unknowns  ^  and  1,  but  an  soon  as  he  has  found  m 
he  tlien  replaces  the  1  by  the  symbol  for  the  unknown  quan- 
tity,  and  finds  it  in  its  turn. 

(iv)  To  find  a  [rri/ioiia/]  viyki-aHyled  triangle  stccA  lAol  lAe 
Une  biteciimj  au  aeittr  amjle  U  ntiiutuU  [book  vi,  prob.  18]. 

His  solution  is  as  follows.  Let  ABC  be  the  triangle  of 
which  C  Lb  the  right-angle.     Let  the  bisector  AD^^x^  and 


B  O  C 

let  />C  -  3x,  hence  AC  =  Ax.  Next  let  iM^  be  a  multiple  of  3, 
say  3,  .'.  BD^^-^f  hence  Ji^  =  4-4x  (by  Euc.  vi,  3). 
Hence  (4- 4-c)«  =  3«^(4x)«  (Euc  i,  47),  .'.  a:  =7/32.  Multi- 
plying  by  32  we  get  for  the  sides  of  the  triangle  28,  96,  and 
\00.;  and  for  the  bisector  35. 

• 

(v)  A  tttan  buys  x  meaturtB  of  wine^  tome  ai  8  draekmkoe 
a  tiietuure^  the  resi  ai  5.  I/e  jtaye/or  ihtm  a  equate  number  of 
drachuuie,  euck  ihat^  if  AC  be  added  to  tV,  the  reeidiiny  number 
ie  a^.  Find  the  number  he  bought  at  eaeh  price  [book  v, 
prob.  33]. 

The  price  paid  was  2* -60,  hence  Sx>3^-(iO  and 
5x<2''60.  From  this  it  follows  that  x  must  be  greater 
than  11  and  less  than  12. 

Again  x*  -  60  is  to  be  a  square ;  suppose  it  is  equal  to 
(x  -  my  tlien  x  ~  (m*  -i-  60)/2m,  we  have  therefore   ' 


f 


\ 


i 


DIUPUANTUH. 


n<      ,1      <12; 


.-.      19<m<21. 
Diophantns  therefore  asxiimm  that  m  is  eqwU  to  20,  which 
gire>  him  x  =  II};  nnd  mnke^  the  total  cost,  ik^f-SO,  eqwl 
to  73  j  drachmae. 

He  hu  next  to  divide  this  comI  into  two  pMis  which  >haU 
give  the  cost  of  the  K  drachiiuie  menxurps  uid  the  S  d 
ineAflures  nvpectively.     Let  these  parts  Ik-  y  And  ■. 
■   Th™  ^  +  i<721-t>=J. 

_«  K  JS9 


Therefore  r  =     ,  ^  -,  »rui  y  = 


Therefore  (lie  numlver  of  5  drachmae  mmitum  wan  79/1 S,  Uid 
of  8  dmchune  measures  wn*  Sil/ia. 

From  the  enuiiciatiurt  of  tliis  prohlem  it  wonhl  amn 
that  the  wine  was  of  a  poor  ijunlity,  and  Tannery  has 
inReniously  suggi?st<'d  thiit  tite  pricoH  nictitioned  for  nich  a 
wine  are  higher  than  wpc?  usual  until  after  the  end  nf  the 
second  century.  He  therefore  rvjectn  tlie  view  which  was 
formerly  held  that  Diophantus  lii'<-d  in  that  century,  bat  he 
does  not  neeni  to  be  aware  that  IV  Mor^n  had  previously 
ibewn  that  this  opinion  vnn  untenn1)le.  Turnery  inclines 
to  think  that  Diophantus  lived  half  a  centncy  earlier  than 
I  have  supponed. 

I  mentioned  that  Diophantus  wrote  a  third  work  entitled 
Pon$m».  The  book  is  lost,  bat  we  have  the  enunciations  of 
•ome  at  the  propositions  and  though  we  cannot  tell  whether 
tbey  were  rigoranily  |>roved  by  Dit^hantus  they  confirm  onr 
opinion  of  his  ability  and  sagacity.  It  has  been  snggeated 
that  ■ome  of  the  thememi  which  be  assumes  in  his  aritbmetie 
were  proved  in  the  porisms.  Among  the  more  striking  of 
these  resalts  an  the  statements  that  the  differetm  at  two 
eabea  can  bo  always  ezpreNsd  as  the  aun  of  two  wbes ;  tint 
M»  wnmber  «(  the  form  4m  -  I  out  be  ezpmeod  as  the  ■■■ 


114  TUB  8ECOMD  ALBXANUBUM  liCliOQL. 

! 
I 

of  two  aquarBii;  and  that  no  nmnber  of  the  turm  8«^  1  (or 
powiUy  24m  -^  7)  can  be  expressed  as  the  sum  of  three  s^naree: 
to  these  we  may  perhaps  add  the  proposition  that  any  number 
can  be  expressed  as  a  square  or  as  the  sum  of  two  or  three  or 
four  squarea. 

The  writings  of  Diopliautus  exercised  no  perceptible  influ- 
ence on  Greek  mathematics ;  but  his  Artthmeiie,  when  trans- 
lated into  Arabic  in  the  tenth  century,  influenced  the  Arabian  > 
school,  and  so  indirectly  affected  the  progress  of  European  i 
mathematics.  An  imperfect  copy  of  the  original  work  was  j 
discovered  in  1462;  it  was  translated  into  Latin  and  pub-  • 
liiihed  by  Xylander  in  1575  ;  the  translation  excited  general 
interest,  but  by  that  time  the  European  algebraists  had,  on  the 
whole,  advanced  beyond  the  point  at  which  Diophantus  had 
left  off, 

lamblichUB.  /amUichug^  circ.  350,  to  whom  we  owe  a 
valuable  work  on  the  Pytliagurean  discoveries  and  doctrines, 
seems  also  to  have  studied  the  properties  of  numbers.  He 
enunciated  the  theorem  tliat  if  a  number  a'hich  is  equal  to 
the  sum  of  three  integers  of  the  form  3/i,  3m-  1,  3n— 2  be 
taken,  and  if  the  separate  digits  of  this  number  be  added,  and 
if  the  separate  digits  of  the  rt*Hult  be  again  added,  and  so  on, 
then  the  final  result  will  be  C :  for  inst^uice,  the  sum  of  54, 
53,  and  52  is  159,  the  sum  of  the  sepaiate  digits  of  159  is  15, 
the  sum  of  the  separate  digitn  of  1 5  is  6.  To  any  one  confined 
to  the  usual  Ureek  numerical  notation  this  must  have  been  a 
difficult  result  to  prove :  ponsihly  it  was  arrived  at  empirically, 
but  Dr  Gow  thinks  that  it  tends  to  confirm  the  suspicion  tliat 
the  Greeks  possessed  a  symliolism  resembling  the  Arabic 
humeral  notation. 

The  names  of  two  comineutatorH  will  piuctically  conclude 
the  long  roll  of  Alexandrian  mathematicians. 

Theon«  The 'first  of  these  is  Theon  of  Alexandria  who 
flourished  about  370.  He.  was  not  a  mathematician  of 
special  note,  but  we  are  indebted  to  him  for  an  edition  of 
Euclid's  EkmenU  and  a  commentary  on  the  Almagui\  the 


HTPATIA.     THE  ATHEXIAN-  SCH<K>K 


115 


btter*  gir«s  m  gKkt  dekl  of  miwellmneoiis  infomwtion  •boat 
Uw  nanMricsl  methodi  naed  bj  the  Greeks. 

BypftUft.  Tbn  oth«r  wm  lly/nlia  the  dknghter  <A  Theon. 
Sbe  wu  more  dintingutnhed  than  her  tather,  uid  waa  the  Uwl 
Alezkiidrimn  niBtheiDiitician  of  any  general  repulatinn :  she 
vnfte  a  eommentarj'  on  the  Conie*  of  Apolloniiu  and  pussiUjr 
■oim  other  work*,  but  none  of  her  writingH  are  now  estMiL 
EUw  wu  mnrtlered  Kt  the  ioatigBtion  of  the  ChriatiKns  in  415. 
Hw  hte  of  Hypfttia  mnj  aene  to  remind  lu  tlwt  th« 
EMtem  Christiana,  aa  soon  or  they  became  the  dominant  party 
in  the  stAt^  shewed  thcmnelves  bitterly  linatile  to  all  forms  of 
IcMntng  That  very  singleness  of  pnrpoHe  which  had  at  iimt 
ao  nMlerially  aided  their  progreati  developed  into  a  one- 
aidednesa  which  refnsed  tu  ce«  any  good  outside  their  own 
body ;  those  who  did  not  actively  nwist  thpm  were  peraeciited, 
and  tin  manner  In  which  they  carried  on  their  war  against 
tbe<dd  schooht  of  learning  is  pictured  in  the  pages  of  Kingsley'a 
.  noveL  The  final  establishment  of  Christianity  in  the  Eaat 
marks  the  end  of  the  tireek  tcirntific  schools,  though  nominally 
they  continned  to  exist  for  two  hundred  years  more. 


The  Athenian  School  (in  the  fijth  century)^. 

The  hostility  of  the  Eastern  church  to  Ureek  science  ts  far- 
ther illuatratcd  by  the  tall  of  the  later  Athenian  achooL  This 
school  occupien  but  a  smnll  space  in  uur  history.  Ever  since 
Plato's  time  a  certain  number  of  profettional  mathematicians 
hnd  lived  at  Athens ;  and  about  the  year  420  this  school  again 
■ei]nired  eonsider»ble  nputation,  largely  in  consequence  of  the 
■nasraos  atadenta  who  after  the  mnrder  of  Hypatia  migrated 
tkgn  bom  Alexandra.  )U  moat  oelebrated  members  wets 
FhNla^  Da^Msn^  and  Entochu. 

i  with  ttmuBUla  b7  U.  Btlaa  and  psbKsbad  at 


116  THB  8W0HD  AUCXAMDBUH  SCHOOL. 

ProoliUk  FroeiuB  was  bom  at  OontUmtiiiqple  in  Fehraaiy 
412  and  died  at  Athens  on  April  17,  485.  He  wrote  a  eoni- 
mentary  on  Euclid's  EUwmenis^  of  which  that  part^  which  deab 
with  the  first  book  is  extant  and  contains  a  gnsat  deal  of  vain* 
able  information  on  the  history  of  Greek  mathematics :  he  is 
verbose  and  dull,  but  luckily  he  has  preserved  for  us  quotations 
from  other  and  better  authorities.  Proclus  was  succeeded  as 
head  of  the  school  by  Marinoa,  and  Marinus  by  latdoma. 

DamaaciiiB.  Eatocius.  Two  pupils  of  Isidorus,  who  in 
their  turn  subsequently  lectured  at  Athens,  may  be  mentioned 
in  passing.  Damaseiwt  of  Damascus,  circ.  490,  added  to  Euclid's 
EUmenU  a  fifteenth  book  on  the  iniicription  of  one  regular 
solid  in  another.  EutociuM,  circ.  510,  wrote  commentaries  on 
the  first  four  books  of  the  Couiet  of  Apollonius  and  on 
various  work»  of  Archimedes ;  he  also  published  some  examples 
of  practical  Greek  arithmetic.  His  works  have  never  been 
edited  though  they  would  seem  to  de8er\*e  it 

This  later  Athenian  school  was  carried  on  under  great 
difficulties  owing  to  the  opposition  of  the  Christians.  Proclus, 
for  example,  was  repeatedly  threatened  with  death  because  he 
was  "a  philosopher."  His  remark  "after  all  my  body  does 
not  matter,  it  is  the  Hpirit  that  1  shall  take  Mrith  me  when 
I  die,"  which  he  made  to  some  students  who  had  offered  to 
defend  him,  has  been  often  quoted.  The  Christians,  after 
several  ineffectual  attempts,  at  last  got  a  decree  from  Justinian 
in  529  that  **  heathen  learning"  should  no  longer  be  studied  at 
Athens.  That  date  therefore  marks  the  end  of  the  Athenian 
school. 

The  church  at  Alexandria  was  less  influential,  and  the 
city  was  more  remote  from  the  centre  of  civil  power.  The 
schools  there  were  thus  suffered  to  continue,  though  their 
existence  was  of  a  precarious  cliaracter.  Under  these  con- 
ditions mathematics  continued  to  be  read  in  Egypt  for  another 
hundred  years  but  all  interest  in  the  study  had  gone. 

*  It  hsi  been  edited  by  G.  Friedlein,  Leipzig,  1673. 


I 


ROMAN   MATItEMATICfl. 


Roman  Mathematict*. 


\  <ia);ht  not  to  c<inc)uile  thin  piirt  of  the  hJHtory  without 
nnj  mention  of  Itomnn  iiintfiPmnticN,  for  it  wm  thrcmgh  Rome 
that  matliPRinticfi  fint  paMMtl  into  the  currieulnni  nf  mmlipviil 
Europe,  and  in  Home  mII  modom  hi%tiiry  hiM  it«  orifpn.  Therr 
ia  howAvpr  TiM^  little  t>i  wtj*  on  tlie  Kuliject.  Tlie  chief  study  of 
the  place  was  in  fiw:t  thp  nrt  of  ^vemiiient,  whether  by  Ikw, 
by  persunsion,  or  hy  th<wp  ninterinl  means  on  which  nil  govern- 
ment ultimately  rests.  Tliprv  werr  no  donlit  profesw)r«  who 
could  t«ach  the  resultH  of  (Jreek  science  hut  there  w««  no 
demand  for  n  slIiikiI  <if  mat liemn tint.  Italians  who  wished  to 
learn  more  than  the  elpments  of  the  science  went  to  Alex- 
andria or  t«  plnccH  which  drew  their  inspiration  from  Alex- 

The  Ruhjpct  as  tniiRht  in  the  mathematical  schools  at  Rome 
seems  to  have  l)een  cmitinrd  in  aritlinietic  to  the  nrt  of  calcula- 
tion (no  doubt  hy  the  aid  of  the  alMi-UK)  and  perhapa  some  of 
the  easier  parts  of  the  work  of  Nicomochua,  and  in  jreometry 
to  a  few  practical  ruleii ;  though  Home  of  the  arts  fouttded  on  a 
hnowletl^  of  niatheniatim  (especially  that  of  survejing)  wer* 
carried  to  a  high  pitch  of  excellence.  It  M'ould  se*m  aim  that 
special  attention  was  paid  to  the  rppn-sentntion  of  nomhers  hy 
si^s.  The  manner  nf  indicalinj;  nniiitiers  up  to  ten  hy  the 
ase  of  fincen*  mu»t  have  iK-en  in  practice  frmn  quite  earij 
times,  but  ahout  the  first  century  it  had  bera  developed  by 
th«  Romans  into  a  finger-oymliolism  l>y  which  nnmbem  up  to 
10,000  or  perhaps  more  could  he  represented  :  this  would  went 
to  have  1)een  taught  in  the  Roman  schools.  It  ia  described  hj 
Bede  and  therefore  would  seem  to  have  been  known  sa  far 
went  as  Britain;  Jerome  alio  alludes  Xn  it;  tta  ow  haa  itill 
■urrived  in  the  Persian  bazaara. 

I  am  nota^aaint«d  with  any  I^tio  woric  oo  the  princtplea 
of  DMdwaiea,  b«t  titan  wen  numenMU  books  on  tin  pimctical 

■  Tbt  Mhjwl  is  dlMUMd  br  Caator,  abaps.  bxv,  sm,  aad  imi; 
■lB»l7  Bwhri,  pi^  flM-«M. 


118 


TBI  810OND  AUXANDaiAV  SCHOOL. 


side  of  ihetttbject  which  dealt  olabcmlelj  with  Aidiitestend 
and  engineering  problems.  We  nuiy  jndge  what  thejr  were  like 
bf  the  MatAemaiiei  Veiereg^  which  ie  a  colleetion  of  varioiM 
abort  treatiiieii  on  catapults^  engines  of  war,  Ac:  and  bj  the 
fCffOTo^  written  bj  Sextus  Julius  Africanus  about  the  end  of 
the  second  century,  part  of  which  is  included  'n  the  J/aliU- 
maiiei  Veieres^  which  contains,  amongst  other  things,  rules  for 
finding  the  breadth  of  a  river  when  the  opposite  bank  is  occu- 
pied bj  an  eneiuj,  how  to  signal  with  a  semaphore^  kc 

In  the  sixth  century  Boethius  published  a  gecunetry  con- 
taining a  few  propositioua  froui  Euclid  and  an  arithmetio 
founded  on  that  of  Nicomachua;  and  about  the  same  time 
Cassiodorus  discussed  the  foundation  of  a  liberal  education 
which,  after  the  preliminary  trivium  of  grammar,  logic,  and 
rhetoric,  meant  the  quadrivium  of  arithmetic,  geometry,  music, 
and  astronomy.  These  works  were  written  at  Rome  in  the 
closing  years  of  the  Athenian  and  Alexandrian  schools  and 
I  therefore  mention  them  here,  but  as  their  only  value  lies  in 
the  fact  that  they  became  recognized  text-books  in  medieval 
education  I  postpone  their  consideration  to  chapter  viii. 

Theoretical  mathematics  was  in  fact  an  exotic  study  at 
Rome ;  not  only  was  the  genius  of  the  people  essentially  prac- 
tical, but,  alike  during  the  building  of  their  empire,  wliile  it 
lasted,  and  under  the  Goths,  all  the  conditions  were  unfavour- 
able to  abstract  science. 

On  the  other  hand,  Alexandria  was  exceptionally  well 
placed  to  be  a  centre  of  science.  From  the  foundation  of  the 
city  to  it4i  capture  by  the  Mohammedsns  it  was  disturbed 
neither  by  foreign  nor  by  civil  war,  save  only  for  a  few  years 
when  the  rule  of  the  Ptolemies  gave  way  to  that  of  Rome :  it 
was  wealthy,  and  its  rulers  took  a  pride  in  endowing  the 
university :  and  lastly,  just  as  in  commerce  it  became  the 
meeting-place  of  the  east  and  the  west,  so  it  had  the  good 
fortune  to  be  the  dwelling-pUice  alike  of  Greeks  and  of  various 
Semitic  people;  the  one  race  shewed  a  peculiar  aptitude  for 
geometry,  the  other  for  sciences  which  rest  on  measurement. 


fi 


I  • 


> 


I  , 


VJjmE  OF  THE  SECOND   ALCXAXDRIAN  ftCHOOL.      119 

Heiv  too,  howt;vtT,  an  timr  wpnt  nn  t)ip  conditionH  f^rmdnallj 
liecBinp  tnoriR  unfAVouTnbk,  llir  eiidleiw  dJMUMiona  by  the 
OhriKtinnn  nn  tlionli>;;irAl  ilnfnnn"  n'n)  Die  incmning  inaecaritf 
of  tlip  rnipiiT  t^ndinfi  to  divert  nifn'n  thou|cht«  into  other 
chftniiflii. 

EikI  u/  Ihe  •Sfcwif  Ale.ran<lrifiii  School. 

Thr  jim-nrioaH  exiKU'ticr  nml  uiifniitful  hntoiy  of  the  Lut 
two  crnturic^  of  the  wcoiid  Alrxniidrian  iSchfnil  need  no  record. 
In  fi.lt!  M(ihaminn)  di^l,  uml  within  Ifii  jrciim  hiM  sncceMoni 
hud  huIhIu»I  Syria,  PnleNtine,  McHopiitnniiiV  Penia,  iind  ^gypi- 
The  iirecifip  dair  nn  which  Alfxamlrin  fell  iit  doulrtful  hut 
the  mnst  n-linhle  Anth  Itistoriniix  xivn  Itvc  10,  Ctl  -a  date 
which  nl  nny  mte  in  correct  within  Hjihtt^n  mnnthft. 

With  the  full  of  Alrxnndrin  th'>  InnR  hiNlory  of  Urerk 
mBihpniaticH  cnme  to  it  ciHiduMon.  It  iteemn  prolnMe  (hnt  the 
greater  part  of  the  fanioue  uriiverxity  hbmry  and  niuiwuni  had 
been  df^t[iiye<I  hy  the  CliriNtinns  n  hundred  fir  two  handred 
yarn  pn-vinuKly,  and  what  reninined  wan  nnvalued  and  neg- 
lected. Some  two  or  three  yearn  after  the  first  capture  of 
Alexandria  n  iwrioutt  revolt  occurred  ii^  E^'pt,  which  waa 
ultimately  put  down  with  great  severity.  I  ^ee  nn  t«Mon  to 
douht  the  truth  of  the  account  that  after  the  capture  of  the 
city  the  Mnhamnmlanii  destroyed  nuch  university  buildings  and 
collections  bk  were  still  left  it  i^  wiid  that,  when  the  Arab 
commander  ordered  the  library  to  lie  burnt,  the  Qreetti  made 
such  energetic  protests  that  he  conM-ntpd  to  refer  the  matter  to 
the  caliph  Omar.  The  caliph  relumed  thn  answer,  "As  to  the 
books  you  have  mentioned,  if  they  contain  what  la  agreeaUe 
with  the  hook  of  God.  the  liook  of  God  is  sufficient  without 
them ;  and,  if  they  contain  what  is  contrarr  to  the  book  of  Ood, 
there  is  do  need  for  them  ;  an  gire  orders  for  their  deBtructioo." 
The  McooBt  goea  on  to  sajr  tlwl  they  vera  burnt  in  the  pabUe 
faaUM  off  the  city,  ftod  th»t  it  look  six  oiaiithB  to  eonwwi 


120 


CHAPTER   VI. 

THB  BTZAMTIMB  SCBOOk 

641-1458. 


/ 


1 

1 


It  will  be  oonvenient  to  oonsider  tlie  Pjiintintr  Sehool  in 
ocmnection  with  the  historj  of  Greek  matheomtaoM.  After  the 
capture  of  Alexandria  by  the  Mohammedaiu  the  majoritj  of 
the  pliilosopheni  who  previously  had   been  teaching  theie^  ^ 

migrated  to  Constantinople  which  then  herame  the  centre  of  *i 

Greek  learning  in  the  Elant  and  remained  ao  for  800  yeara  \ 

But  though  tlie  history  of  the  Byzantine  school  stretchee  over 
80  many  yenrs- — a  period  about  as   long  as   that  from   the  4 

Norman  Conquest  to  the  present  day  —it  is  utterly  barren  of  ? 

any  scientific  interest ;  and  its  chief  merit  is  that  it  preserved 
for  us  tlie  works  of  the  ditferent  Greek  schools.  The  revelation 
of  these  works  to  the  West  in  the  fifteenth  century  was  one 
of  the  most  important  sources  of  the  stream  of  modem  European 
thought,  and  the  history  of  the  Byzantine  school  may  be 
summed  up  by  saying  that  it  played  the  part  of  a  conduit-pipe 
in  conveying  to  us  the  r(*sults  ot  an  earlier  and  brighter  age.  II 

The  time  was  one  of  constant  war,  and  men's  minds  during 
the  short  intervals  of  peace  were  mainly  occupied  with  theo- 
lo^rical  subtleties  and  pedantic  scholarship.  I  should  not  have 
mentioned  any  of  the  following  writers  had  they  lived  in  the 
Alexandrian  period,  but  in  default  of  any  others  they  may  be 


i 


TH^   nVZANTISE  SCHOOL,  121 

noticed  a*  illuHtratitig  tlie  chnmcter  of  the  nchool.  I  ought 
rIso  perhaps  to  call  tlie  Attention  of  tlie  n-uler  explicitly  to 
the  tmct  thftt  1  am  here  departing  froiii  cltronological  ordei, 
and  that  the  matheinaticiann  mentioned  in  this  chapter  vera 
contempomries  of  tlinsp  cli!icus.w(l  in  the  chapters  derotvd  to 
the  innthemattcfl  of  the  mi-lille  n^eH,  The  Bjnntiiie  «chool 
WBB  so  iHotalecl  that  I  de<>m  this  the  be«t  arrangement  of  thtt 

Hero.  One  of  the  earliest  members  of  the  B^-nuitliH) 
Khool  was  flrra  nf  Ci>n''nHtinnjJr^  circ.  900,  sometimes  called 
the  jrounger  to  distinguish  him  from  Ifero  of  Aleiandria. 
Hero  would  seem  t*i  have  wiitten  on  ge>M)p*y  and  mechanics 
as  applied  to  engines  of  war. 

During  the  (vnth  century  two  emperora,  Leo  VI,  and 
Constantine  VII.,  shewed  con!<ideraltle  intt-rmt  in  antrofwiny 
and  mathematics,  but  the  slimulu*  thun  given  to  the  attidj 
of  these  subjects  wan  only  temporary. 

IteUas.  In  the  eleventh  century  Mifhnrl  PmilH$,  born 
in  1020,  wrote  a  pamphlet*  on  the  tiuadrivium:  it  ia  now  in 
the  National  Lilirnn'  at  Paris. 

In  the  fourternlh  century  we  find  the  namea  of  three 
monks  wlio  paid  attention  to  mathemiitics : 

Plnnndes.  Barlaam.  Ar^yruB.  The  first  irf  tlie  three 
vu  Maximu*  Pliiiiwltiif.  He  wn>i4>  a  c^mimentjuy  oa  the 
first  two  ttooks  of  the  Ariihm'tic  nf  Diopliantna;  k  work  on 
Hindoo  arithmetic  in  which  he  intriMluced  the  use  of  tba 
Arabic  numerals  into  the  Ea-vtem  empire;  Mtd  another  on 
proportions  which  is  now  in  the  National  library  at  Pwia. 
The  next  was  a  Calnhrian  hionk  named  AirttMiN,  who  '«• 
horn  in  1290  and  died  in  tS-IK.  He  wai  the  author  of  s 
work,   /.ogulie,  on   the  Oreek    methods  of  caleolation  iron 

n  wrote  ■  ComprHMmm 
17. 

t  His  aritluaatkal  MMinenta?  was  publiabsd  hf  Inlander.  BUe. 
U7Si  Us  week  «a  Hiiidoa  afilkMUs,  sdilad  Igr  C.  J.  Owfeardl,  «M 


122 


TUB  BYZANTINE  8CB0OL. 


whieb.  we  derive  a  good  deal  of  informaticQ  ms  to  the  way 
in  which  the  Greeks  practicallj  treated  fraciione^.  Barlaam 
leenui  tci  have  been  a  man  of  great  iutelligenoe.  '  He  was  sent 
an  an  aniLasiiador  to  the  pope  at  Avigncjn,  and  acquitted 
hinmelf  creditably  of  a  dilficult  miiuiion ;  while  there  he  taught 
Greek  to  Petrarch.  He  wa8  famous  at  Constantinople  for 
the  ridicule  he  threw  on  the  preprntterous  pretensions  of  the 
monks  at  Mount  Athos  who  taught  that  I  those  who  joined 
them  could,  by  standing  naked  resting  their  beards  on  their 
breasts  and  steadily  reganling  their  stomachs,  see  a  mystic 
light  which  was  the  essence  of  C»od.  Barl$am  advised  them 
to  substitute  the  light  of  reason  for  that  of  their  stomschs — a 
piece  of  advice  which  nearly  cost  him  his  life.  The  last  of 
these  monks  was  Jmae  Aryyrttt^  who  died  in  1372.  He  wrote 
three  astronomical  tracts,  the  manuscripts  of  which  are  in  the 
libraries  at  the  Vatican,  Leyden,  and  Vieniui :  one  on  geodesy, 
the  manuscript  of  which  is  at  the  Escurial :  one  on  geometry, 
the  manuscript  of  which  is  in  the  National  Library  at  Paris: 
one  on  the  arithmetic  of  Nicouiachus,  the  manuscript  of  which 
is  in  the  National  Library  at  Paris :  and  one  on  trigonometry, 
the  manuscript  of  which  is  in  the  Bodleian  at  Oxford. 

Rhabdas.  In  the  fourteenth  or  perhaps  the  fifteenth 
century  SichtJa*  KhalMias  of  Sinjfma  wrote  two  papers  t  on 
arithmetic  wliich  an*  now  in  the  National  Library  at  Pari^ 
He  gave  nil  account  of  the  linger-symbolism  {  which  the 
Uoniaus  had  introduced  into  the  East  and  was  then  current 
there. 

Pachymeres.  Early  in  the  fifteenth  century  Paekymerts 
wrote  tracts  on  arithmetic,  geometry,  and  four  mechanical 
machines. 

MOBChopulUB.  A  few  years  later  iCinuianuei  Mo^hupuluM^ 
who  died  in  Italy  circ.  1460,  wrote  a  treatise  on  magic  squares. 


*  Barlssm'i    lA>gUlie,    edited   by    Dsiypodiua,    was   pabliabml    at 
HtrsMMburK,  157:1;  saotber  edition  urss  isAued  at  Fsri«  in  ICiiO. 
t  Tbey  have  been  edited  by  H.  P.  Tsnueiy,  Psri^  188G. 
X  Hve  above,  p.  117. 


i 


v. 


'I 


MAQIC  SQUARES. 


123 


A  ntagie  mpuirt'  consiflH  of  «  numlNT  of  iDtq^rn  kiTMIgcd  in 
the  fonn  of  ■  *quArn  so  thai  the  flum  of  l\w^  nambera  in  evnry 
row,  in  every  column,  anil  in  cacIi  <lia};i>Ri>l  in  the  SMiie.  It 
the  integers  be  the  conxpcative  nuni)>i>rfl  front  1  to  n',  tb« 
■qakre  is  Mid  to  Ix;  of  the  nih  onlcr,  niul  in  this  cmae  the 
■am  nf  the  numbers  in  nny  ro«,  cohiniii,  or  diiigonMl  !■  equfti 
to  }n(H*4  I).  Tliun  the  lirst  16  iriteftpre,  iirrKnged  in  eitlier 
of  the  fonna  tpsen  beluw,  form  a  iiinpc  MjUitre  of  tbo  fonrth 


11       «       7        R  I    I    S    {  IK      • 


order,  the  sum  of  the  niimliers  in  every  row,  every  column, 
And  each  dini^nal  beinj;  -14. 

In  the  niyntical  phiio5<i]ihy  then  current  certflin  met»- 
phj-itical  ideas  were  often  AK.'uM'inteil  with  pnrticulsr  numberv, 
and  thus  it  wns  natural  (hat  Kuoh  arran^ments  of  nnmber« 
should  attmot  nttention  and  be  ileemc<l  to  poMiem  magickt 
propcrtinL  Tlie  theory  of  the  fomiatinn  uf  magic  Hquami  in 
elegant  nnd  sevrrnt  djstinfpiiiihed  mathrmMicians  Imve  written 
on  it,  but,  though  inten>nl)ng,  I  neeil  luurlljr  say  it  is  not 
useful :  it  in  largely  due  to  I>e  la  Hire  wlio  gave  rule*  tat  thp 
construction  of  a  magic  s(|unre  of  any  order  higher  than 
the  second.  Moschopulvs  iteems  t<»  have  been  the  esriteat 
European  writer  who  attempted  to  deal  with  tbo  mathe- 
matical theory,  but  his  rules  apply  only  to  odd  sqnarva. 
The  astrologers  of  the  fifteenth  and  sixteenth  cmtDriea  wen 
much  impressed  by  such  amnjrements.  In  porticular  the 
famous  Cornelius    Agrippn  (Ut<G— li>.'t5}  constructed   magic 

*  On  the  formation  and  hiibirj  of  nscie  niDsm,  ms  mj  ilntlmtmtieal 
Bttrtatlmu  and  PmblrmA,  Ixindon,  third  cditioD,  IMS,  (Mpt  T.  On 
lbs  work  ol  yowshopaim,  tm  chap,  it  of  8.  OBaltwr'a  OnrUrkH  At 

mucha/l«a.  Leipsin.  187A. 


It4  THS  BTBANTINB  SCHOOL.  I 

■qiiATM  of  the  orden  9»  4,  5,  6,  7,  8,  9  w|iioh  were 
oUied  reepeotively  witli  the  aeveu  Mtmlogiod  ^pUaeto"; 
nauiely,  Batuni,  Jupiter,  Man,  tlie  Hun,  Venua,  Menmry, 
Mid  the  Mocm.  He  Uught  that  a  iiqiiare  jof  one  oell,  in 
which  unity  wan  iii«ert4xl,  n»pn*ieuted  tlie  unity  and  eternity 
of  Gud;  while  th(^  fact  that  a  iquare  of  the  eeoond  order 
could  not  be  convtructed  illuHtrated  the  imperfection  of  the 
four  elements,  air,  earth,  firr,  and  water;  add  later  writem 
added  that  it  wan  aymli«»lic  of  original  tiin.  A  magic  square 
engraved  on  a  silver  plate  was  often  proscribed  as  a  charm 
against  the  plague,  and  one  (namely,  tliat  in  tlie  tirst  dia- 
gram on  the  liiHt  page)  is  drawn  in  the  picture  of  melancholy 
painted  aliout  the  yvar  1 AOO  by  Allirecht  Diirer.  Such  clianns 
are  still  worn  in  tlie  KuMt. 

CoiiNtMntinoplt*  was  captured  by  the  Turks  in  1453,  and 
the  last  Miniiblttnce  tif  a  Gn*ek  school  of  mathematics  tlien 
disappeared.  NumeniUH  Greeks  took  refuge  in  Italy.  In  the 
West  the  nieiiiory  of  Greek  Si*ieiice  liail  vanished  and  even  the 
names  of  all  but  a  few  Ciret'k  writers  were  unknown ;  thus 
the  IsMiks  bn»uglit  by  these  n*fugees  came  as  a  revelation  to 
Kuro|>e,  and,  as  we  sliull  Hee  later,  gave  an  immense  stunulus 
to  the  study  of  science. 


s 


CHAPTER   Vn. 
HTSTKMH   OF    NITMERATION    AND    PKIUITIVK    AKITHHETIC*. 

I  HATH  in  mnnj  plnres  Mlluilt^t  to  the  Greek  method  uf 
RXprrsiiing  numl>pni  in  writing;,  niid  I  hnve  thought  it  best  to 
dpfcr  In  thi*  rhrtpt<-r  thi>  wholn  of  whnt  I  wKnted  to  wty  on  the 
vKriouB  nj-st/'mfi  of  nQmrricnl  nutAiion  which  were  diapUced 
by  Die  wyntpin  intniduct^l  l>y  the  Antlm. 

Fint,  M  to  nynilwIiKm  nnd  1iin|;uii;n>'  The  pUn  of  indi-' 
catinj;  numlifri  hy  ihcdipUof  one  or  tioth  hnndiiiNMimtund 
th»t  wn  find  it  in  univpnuil  ute  nnionj;  Mtrlj  races,  And  the 
memben  of  M  tribes  now  extant  are  able  to  indicate  hy  signs 
nnmbera  At  Iput  an  htjih  on  ten  :  it  in  stated  that  in  scnne 
I  an  gn  ages  the  names  for  the  firet  ten  numben  are  derived  (mm  _ 
the  lingent  used  to  denote  them.  For  Inrger  numbera  w«  soon 
however  reach  a  limit  heyond  which  primiti«v  man  in  ao^ile 
to  count,  ithile  an  far  bh  language  goen  it  ia  well  known  that 
many  tribea  have  no  word  for  any  nom)ier  higher  than  ten, 
and  nome  have  no  word  for  any  number  lioyond  four,  all  higher 

*  The  nihjecl  of  tbiii  chaphT  it  ditcuivtd  hj  Cantor  mkI  b;  Haakcl. 
Sm  iIm  the  Philarnplig  of  Arilhmtlic  Iqr  John  LoUe,  weosd  edhiDa, 
E<Iinbur^b,  isao.  Be-iin  ihew  aathoritiei  the  artiete  on  AritkmtVt 
by  fitorgt  PoKoek  in  ih«  F-tgrlt^tiia  iltinfeOUim,  fart  Stittten, 
London,  IBIS;  E.  B.  Tylnr'a  Primitit*  Cutlart,  LoMloa,  int;  L» 
tifikn   tiamtrmax   H   ftiTilkmeti^tu   ctut    lit   pnplM  4t  raaHffBfH...b7 

~.  H.  Martin,  Bom«.   ISU :   ud  Die  ZnAtMJekm...^  O.  IWailrii. 


126  HTSRMH  or  NUMUUTIOM. 

nnmben  being  expraned  by  the  words  pientj  or  hoop:  bi 
connectkNi  with  this  it  is  worth  ranuurking  that  (as  stated 
above)  the  Egyptians  used  the  symbol  lor  the  woitl  hei^i  to 
denote  an  unknown  quantity  in  algebra. 

The  number  five  is  generally  represented  by  the  open  hand, 
and  it  is  said  that  in  almost  all  languages  the  words  Ave  and 
hand  are  derived  from  the  same  root  It  is  possible  that  in 
early  times  men  did  not  readily  count  beyond  t.y^  and  things 
if  more  numerous  were  counted  by  multiples  of  it.  Thus  the 
Roman  symbol  X  for  ten  probably  represents  two  "  Vs, 
placed  apex  to  apex  and  tieems  to  |ioint  to  a  time  when  things 
were  counted  by  fives*.  In  connection  with  this  it  is  worth 
noticing  that  both  in  Java  and  also  among  the  Axtecs  a  week 
consisted  of  five  days. 

The  members  of  nearly  all  races  of  which  we  have  now 
any  knowledge  seem  however  to  have  used  the  digits  of  both 
hands  to  represent  numbers.  They  could  thus  count  up  to  and 
including  ten,  and  therefore  were  led  to  take  ten  as  their  radix 
of  notation.  In  the  English  language  for  example  all  the 
words  for  nuuibers  higlier  than  ten  are  expressed  on  the 
decinml  system  :  those  for  1 1  and  1 2,  which  at  first  sight 
seem  to  be  exceptions,  being  derived  from  Anglo-Saxon  words 
for  one  and  ten  and  two  and  ten  respectively. 

Some  tribes  seem  to  have  gone  further  and  by  making  use 
of  their  toes  were  accustomed  to  count  by  multiples  of  twenty. 
The  AzU^  for  example,  are  said  to  have  done  so.  It  may  be 
noticed  that  we  still  count  some  things  (for  instance,  sheep)  by 
scores,  the  word  score  signifying  a  notch  or  scratch  made  on 
the  completion  of  the  twenty  ;  while  the  French  also  talk  of 
quatre-vingt,  as  though  at  one  time  they  counted  things  by 
multiples  of  twenty.  I  am  not,  however,  sure  whether  the 
latter  argument  is  worth  anything,  for  I  have  an  impression 
that  I  have  seen  the  word  odante  in  old  French  books ;  and 

*  ftee  slio  ths  Odyaetf,  iv,  418—415,  in  which  appttrentlj  isfeisnes  is 
usds  to  a  lumilsr  ciutoiu. 


I 


SYSTEMS  OF  NUMERATION.  127 


I 


there  is  no  question^  that  neptnnte  and  nonnnie  were  at  one 
time  common  words  for  seventy  and  ninety,  and  indeed  thej 
are  still  retained  in  some  dialects. 

The  only  trilies  of  whom  I  have  n*ad  who  did  not  count  in 
terms  either  of  five  or  of  some  multiple  of  five  are  the  Bolans 
of  West  Africa  who  are  Maid  to  have  counted  by  multiples  of 
seven,  and  the  Maories  who  an*  naid  to  have  counted  by 
multiples  of  eleven. 

Up  to  ten  it  is  com|ianitively  easy  to  count,  but  primitive 
people  find  great  difliculty  in  counting,  higher  numbers; 
apparently  at  first  thin  difliculty  was  only  overcome  by  the 
method  (still  in  use  in  South  Africa)  of  getting  two  men,  one 
to  count  the  units  up  to  ten  on  his  fingers,  and  the  other  to 
count  the  number  of  groups  of  ten  so  fonned.  To  us  it  is 
obvious  that  it  is  equally  effectual  to  make- a  mark  of  some 
kind  cm  the  completion  of  each  group  of  ten,  but  it  is  alleged 
tfi«t  thfi  mpmliem  of  nmny  tribes  never  luocreded  in  oounting 
numbers  higher  than  ten  unless  by  the  aid  of  two  men. 

Most  races  who  shewed  any  aptitude  for  civilization  pro- 
ceeded further  and  invented  fi  way  of  representing  numliers  by 
means  of  pebbles  or  counters  arranged  in  sets  of  ten  ;  and  this 
in  its  turn  developed  into  the  abacus  or  swan-pan.  This  in- 
strument was  in  use  among  nations  so  widely  se|iarated  as  tlie 
Etruscans,  Greeks,  Egyptians,  Hindoos,  Chinese,  and  Mexi- 
cans; and  was,  it  is  believed,  invented  independently  at 
several  different  centres.  It  is  still  in  common  use  in  Rusbia, 
China,  and  Japan. 

In  its  simplest  form  (see  figure  i,  on  the  next  page)  the 
abacus  consists  of  a  wooden  board  with  a  number  of  grooves 
cut  in  it,  or  of  a  table  covered  with  sand  in  which  grooves 
are  made  with  the  fingers.  '  To  represent  a  number,  as  many 
oounters  or  pebbles  are  put  on  the .  first  groove  as  there  are 
unita,  as  many  on  the  second  as  there  are  tens,  and  so  on. 
When  by  its  aid  a  number  of  objects  are  eonnted,  lor  each 


r 

1 
I 


*  8tp,  for  ssamplSt  ▼•  M*  ^  KesiptMi*s  Pfmtiifm„Jk  e^/Hr^  Anlwp^ 

lUi. 


IStl  HVHTBIU  or  MUHUUTION. 


iiUitili 


THE  ARACUiS. 


12d 


object  a  pebble  is  pot  on  the  firRt  groove;  and,  as  noon  as 
there  are  ten  pebb1ei«  there,  tliey  are  taken  oflT  and  one  pebble 
put  on  the  second  groove ;  and  so  on.  ft  was  sometimes,  as 
in  the  Aztec  ^fii/mji,  made  with  a  nondipr  of  parallel  wires 
or  strings  stuck  in  a  pi«H^  of  wof n1  on  which  beads  could  lie 
threaded ;  and  in  that  fonn  is  called  a  swan-pan.  In  the 
namber  represented  in  each  of  the  iiistmmonts  drawn  on  the 
opposite  t*age  there  are  seven  tliouwinds,  three  hundreds,  no 
tens,  and  five  units,  that  i}«,  the  numlirr  is  7305.  Some  races , 
counted  from  left  to  right,  others  fnmi  right  to  left»  Imt  this 
is  a  mere  matter  of  convention. 

The  Roman  alMci  seem  to  have  been  rather  more  ela- 
borate. 'They  contaiined  two  marginal  grooves  or  wires, 
one  with  four  beads  to  facilitate  the  addition  of  fractions 
whose  denominators  were  four,  and  one  with'  twelve  beads 
for  fractions  whose  denominatom  were  twelve :  but  other- 
wise they  do  not  differ  in  principle  from  those  described 
above.  They  were  commonly  made  to  represent  numliers 
up  to  100,000,(KK).  The  Gn^ek  abaci  were  similar  to  the 
Roman  ones.  The  Greeks  and  Romans  used  their  abaci 
as  boards  on  which  they  played  a  game  something  like  Imck- 
gammon. 

In  the  Russian  Uchotii  (figure  ii)  the  instrument  is  impro%*ed 
by  having  the  wires  set  in  a  rectangular  frame,  and  ten  (of  nine) 
beads  are  permanently  threaded  on  each  of  the  wires,  the  wirra 
being  considerably  longer  than  is  necessary  to  hold  them.  If 
the  frame  be  held  horizontal,  and  all  the  beads  be  tom*ards  one 
side,  say  the  lower  side  of  the  frame,  it  is  possible  to  represent 
any  number  by  pushing  towards  the  other  or  u|>per  side  as 
many  beads  on  the  first  wire  as  there  are  units  in  the  number, 
as  many  beads  on  the  second  wire  as  there  are  tens  in  the 
numbefy  and  so  on.  Calculations  can  be  made  somewhat  more 
rapidly  if  the  fire  beads  on  each  wire  next  to  the  upper  side 
be  cokrared  differently  to  those  next  to  the  lower  side^  and  they 
can  be  still  farther  facilitated  if  the  firrti  second,  ...,  ninth 

itcn  in  each  cdnmn  be  respectively  marked  with  qraibob 


130  SYSTEMS  OF  MUMUUTIOH. 

lor  the  numben  1,  2, ...»  9.  Qerbert*  is  laid  to  lisv«  iiitio- 
duoed  the  use  of  tuch  marks,  caUed  apioeii  towmrdi  tho  dote 
of  the  tenth  century. 

Figure  iii  represent*  the  form  of  swan-pan  in  common  nee 
in  China  and  Japan.  There  tlie  development  is  carried  one 
step  further,  and  five  beads  on  each  wire  are  repbused  bj  a 
single  bead  of  a  diiferent  form  or  on  a  different  division,  bat 
apices  are  nut  used.  I  am  told  that  an  expert  Japanese  can 
by  the  aid  of  a  swan-pan  add  numbers  as  rapidly  as  they  can 
be  read  out  to  him.  It  will  be  noticed  tliat  the  instrument 
represeiited  in  figure  iii  on  page  128  is  made  so  that  two 
numbers  can  be  expressed  at  the  same  time  on  it. 

The  use  of  the  abacus  in  addition  and  subtraction  is 
evident.  It  can  be  used  also  in  multiplication  and  division ; 
rules  for  these  processes,  illustrated  by  examples,  are  given 
in  various  old  works  on  arithmetic  f. 

The  abacus  ubviouttly  presents  a  concrete  way  of  representing 
a  number  in  the  decimal  nystem  of  notation,  that  is,  by  means 
of  the  local  value  of  the  digits.  Unfortunately  the  method  of 
writing  numbers  developed  on  differeut  lines,  and  it  was  not 
until  about  the  thirteenth  century  of  our  era  when  a  symbol 
zero  used  in  conjunction  with  nine  other  symliols  was  intro- 
duced that  a  corresponding  notation  in  writing  was  adopted  in 
Europe. 

Next,  as  to  tlie  means  of  representing  numbers  in  writing. 
In  general  we  may  say  that  in  the  earliest  times  a  number 
was  (if  represented  by  a  sign  and  not  a  word)  indicated  by  the 
requisite  number  of  strokes.  Thus  in  an  inscription  from 
Tralles  in  Caria  of  the  date  398  ac.  the  phrase  seventh  year  is 
represented  by  crcoc  |  |  |  1  1  |  |  •  These  strokes  may  have  been 
mere  marks;  or  perhaps  tliey  originally  represented  fingerSi 
since  in  the  Egyptian  hieroglyphics  the  symbols  for  the 
numbers  1,  2,  3,  are  one,  two,  and  three  fingers  respectively, 

•  Hee  below,  p.  144. 

t  For  •imni|»lti  in  R.  Beoord'*  Orommde  of  Artei,  edition  of  1610, 
LonduD,  pp.  :r2»->S63. 


THE  REPRESENTATION  OP  NUMBERS.  131 

• 
thoagh  in  the  later  hieratic  writing  thcfie  sjiiiboh  had  become 

reduced  to  straight  lines.     Additional  symbols  for  10  and  100 

were  soon  introduced  :  and  the  oldest  extant  Egjptiaii  and 

Phoenician  writings  repeat  the  symbol  for  unity  as  many  times 

(up  to  9)  as  was  necessary,  and  then  repeat  the  symbol  for  ten 

as  many  times  (up  to  9)  as  was  nece9Rary,  and  so  on.     No 

specimens  tif  Greek  numeration  of  a  similar   kind   are  in 

existence,  but  there  is  every  reason  to  believe  the  testimony  of 

lamblichus  who  asserts  that  this  was  the  method  by  which 

the  Greeks  first  expressed  numbers  in  writing. 

This  way  of  representing  numbers  remained  in  current  use 
throughout  Roman  history ;  and  for  greater  l>revity  they  or 
the  Etruscans  added  separate  signs  for  5,  50,  kc.  The  Roman 
symbols  are  generally  merely  the  initial  letters  of  the  names 
of  the  numbers ;  thus  c  stood  for  centum  or  100,  si  for  mille 
or  1000.  The  symbol  v  for  5  seems  to  have  originally  repre- 
sented an  open  palm  with  the  thumb  extended.  The  symbols 
L  for  50  and  D  for  500  are  said  to  represent  the  upper 
halves  of  the  symbols  used  in  early  times  for  c  and  Sf.  The 
subtmctive  formn  like  iv  for  liii  are  prolmbly  of  a  later  origin. 

Similarly  in  Attica  five  was  denoted  by  11  the  first  letter 
of  vcWc,  or  sometimes  by  F;  ten  by  A  the  initial  letter  of 
Umi ;  a  hundred  by  H  for  Uarow ;  a  thousand  by  X  for  x^X«oc ; 
while  50  was  represented  by  a  A  written  inside  a  II ;  and  so 
on.  Tliese  Attic  symbols  continued  to  be  used  for  inscriptions 
and  formal  documents  until  a  late  date. 

This,  if  a  clumsy,  is  a  perfectly  intelligible  system ;  but 
the  Greeks  at  some  time  in  the  third  century  before  Christ 
abandoned  it  for  one  which  offers  no  special  advantages  in 
denoting  a  given  number,  while  it  makes  all  the  ofierations  of 
arithmetic  exceedingly  difficult.  In  this,  which  is  known 
from  the  place  where  it  was  introduced  as  the  Alexandrian 
system,  the  numbers  from  1  to  9  are  represented  by  the  first 
nine  ietlen  of  the  alphabH;  the  teHs  from  10  to  90  bj  the 
UBX%  nine  letters ;  and  the  handreds  from  100  to  900  bj  the 
MKk  bIm  lellfliiL    To  do  this  the  OtnAm  wuiled  87  letter^ 


132  ttTttTEMS  OF  NUMIKATION. 

and  as  their  mlpliabet  oantained  onlj  34,  they  re-inierted  two 
letters  (the  dignmuia  aad  koppa)  which  had  flnnerlj  been  la 
it  but  had  becouie  obHolete,  aiid  introduced  at  tfie  end  another 
Hymbc4  taken  frum  the  Phoenician  alpliabeL,  l*hiu  the  ten 
li*ttt*r8  a  to  4  Htoud  respectively  for  the  nuiuben  from  1  to  10  ; 
the  next  eight  letters  for  tlie  uiultiples  of  10  from  20  to  90 ; 
and  the  last  nine  letters  fur  100,  200,  &c.  up  to  900.  Inter- 
mediate nuuiljers  like  11  were  n^preseuted  as  tlie  sum  of  10 
and  1,  that  Ih,  by  the  syiubol  la'.  This  aflbrded  a  notation 
for  all  numbers  up  to  999 ;  and  by  a  system;  of  suffixes  and 
indices  it  was  extended  so  as  to  represent  |  numbers  up  to 
100,0(K),000. 

Tliere  is  no  doubt  that  at  first  the  resultji  were  olitained 
by  the  use  of  the  abacus  or  some  similar  mechanical  method 
and  tliat  the  signs  were  only  employed  to  rejcord  the  result ; 
the  idea  of  operating  with  the  symbols  themselves  in  order  to 
obtain  the  results  is  of  a  later  growth,  and  is  one  with  which 
the  Greeks  never  becaiue  familiar.  The  noniprogressive  cha- 
racter of  Greek  arithmetic  may  be  partly  due  to  their  unlucky 
adoption  of  the  Alexandrian  system  which  eaused  them  for 
most  practical  purposes  to  rely  on  the  abacus,  and  to  supple- 
ment it  by  a  taible  of  multiplications  a'liic|i  was  learnt  by 
heart.  The  results  of  the  multiplication  or  di%jisioii  of  numbers 
other  than  those  in  the  multiplication  table  might  have  lieen 
obtained  by  the  use  of  the  abacus,  but  in  fact  they  were 
generally  gi»t  by  repeated  additions  and  subtractions.  Thus, 
as  late  lis  944,  a  certain  mathematician  wlio  in  tlie  course 
of  his  work  wants  to  multiply  400  by  5  tinds  the  result  by 
ad<lition.  The  same  writer,  when  he. wants  to  divide  6152  by 
15,  tries  all  the  multiples  of  15  until  he  g^ts  to  GOOO,  tliis 
gives  him  400  and  a  remaiiuler  152  ;  he  then  begins  again 
with  all  the  multiples  of  15  until  he  gets  to  150,  and  this 
gives  him  10  and  a  remaintler  2.  Hence  the  answer  Ui  410 
with  a  remainder  2. 

A  few  muthematicians  however  such  a#  Hero  of  Alex- 
andria, Theon,  and  Eutocius  multiplied  and  divided  in  what 


SYSTEMS  OP  NtJMERATION. 


133 


in  essentially  the  same  way  as  we  da 
13  they  proceeded  as  follows. 

ly  X  ci|  =  (c  -»•  y)  (c  -»•  17)  13x18 

=  4(c  +  iy)  +  y(n.||) 
=  p  +  W-  +  X  +  icS 


Thtts  to  multiply  18  hy 

(10  +  3)(10  +  8) 
10(10^8)4^3(10-1^8) 
100  +  80  +  30  +  24 
234. 


I  sQRpect  that  the  lant  step,  in  which  they  had  to  add  four 
nnniljers  together,  wan  obtained  by  the  aid  of  the  almcus. 

Them  however  were  men  of  exc(*ptional  genius,  and  we 
must  recollect  that  for  all  onlinary  purposfMi  the  art  of  calcu- 
lation was  perff>rme«l  only  by  the  use  of  the  abacus  and  the 
multiplication  table,  while  the  term  arithmetic  was  confined 
to  the  theories  of  ratio,  pniportion,  and  of  numbers. 

All  the  systems  here  descriljed  were  more  or  less  clumsy, 
and  they  have  lieen  displaced  among  civilized  races  by  the 
Arabic  system  in  which  there  are  ten  digits  or  symbols, 
namely,  nine  for  the  tirst  nine  numbers  and  another  for  zero. 
In  this  system  an  integral  number  is  denoted  by  a  succession 
of  digits,  each  digit  representing  the  product  of  that  digit 
and  a  power  of  ten,  and  the  number  being  equal  to  the  sum  of 
these  products.  Thus,  by  means  of  the.  local  value  attached 
to  nine  symbols  and  a  symbol  for  nro,  any  number  in  the 
decinuil  scale  of  notation  can  be  expressed.  The  history  of  the 
development  of  the  science  of  arithmetic  with  this  noCatioii 
will  be  considered  below  in  chapter  xi. 


SECOND  PERIOD. 

iCUibtmatits  ■>[  ittt  iflOililt  i^gts  anh  Htnaissanir. 

TMi*  period  biffin*  nbout  ihe  gixlk  mtlnrif,  an-/  may  be  laid 
to  end  tDilK  the  invrnfioH  nf  aitali/tirnl  •/eiimelry  tind  of  lit! 
iiijinilenmal  cnletihi*.  The  chnmrlrriflic  feature  of  tAia  jmriod 
W  tAe  ereatinn  or  drv^/ojiment  nf  nunhrn  arithmetic,  a/t/eOra, 
and  trigonometry. 

In  this  period,  I  connider  firet,  in  cliaj>ter  viit,  the  riae  tjf 
leMniag  in  Western  Eurupe,  and  the  nutthcmatica  irf  the 
middle  sges.  Next,  in  chapter  IX,  I  dixcuu  the  n«tnre  and 
history  of  Hindoo  ftiid  Araliinii  niatbenintics,  and  in  chapter  i 
their  introduction  into  Eurojie.  Then,  in  chapter  zi,  I  trace 
the  subeequent  prognL-»  of  arithmetic  W  the  year  1637. 
Next,  in  chapter  xii,  1  trent  of  tliu  genera)  histoiy  of  mathe- 
malica  daring  the  renoiHiinncc,  from  the  invention  of  printing 
to  Uie  beginning  of  the  seventi^'nth  century,  aay,  frixn  1450 
to  1637;  this  contains  an  account  of  the  comnwiiceinent  td  the 
modem  tRAtment  of  arithmetic,  algebra,  knd  trigonotDetcy. 
Listly,  in  chapter  xiii,  1  consider  the  rerival  of  interest  in 
mechanica,  experimental  methods,  and  pure  geometry  which 
marks  the  last  few  years  of  this  period,  and  aema  «*  a  eon- 
necting  link  between  the  mathematics  of  the  r 
the  Btatbemetics  of  modem  tioie& 


CHAPTER    Vlir. 

THE   HISE  (IF  LEAnNINr;    IN    WF.sTERy   EUROPE*. 
ciuc.  COO-1200. 

KAucntion  in  the  nl^h,  seventh,  nnd  etffhth  eentnriea. 

Tim  first  few  ccnturii's  of  this  nocom)  periid  of  our  hist<iry 
■re  aingiilnrly  Imrrrn  nf  iiiti^rest ;  nnii  iiMlt^  it  irould  1>e 
■tran^  if  we  found  !>cioncc  or  mntlipnintics  ntadiMl  hy  Uiim 
whn  livnl  in  «  condition  of  porpi'tuni  ww.  Droadlj  npoitking 
wo  niay  iMy  thnt  frnm  the  Rixtli  to  the  eighth  centurin  the 
only  places  of  ntudy  in  western  Europe  were  the  Benedictine 
Dionoateries.  We  may  find  there  sotne  nliglit  itttempts  at  ft 
Ktudy  of  literature:  but  the  scivnco  U!<un1ty  tAUght  wm  con- 
fined to  the  vBv  of  the  nlnrux,  ihn  method  of  keeping  Accounts, 
and  A  knowle<lge  of  the  rule  hy  which  the  date  of  Easter  could 
be  det^-rmined.  Nor  was  thin  unreasonable,  for  the  monk  hAd 
renounced  the  world,  and  tliero  was  no  reason  wlij  he  should 
leam  more  science  than  was  rci|uired  for  the  services  of  tbe 
Church  and  his  monasterj.  The  traditions  of  Greek  and  Alex- 
andrian learning  gradually  died  away.  Possibly  in  Rome  and 
a  few  {aTonred  places  copies  of  the  works  of  tbe  great  Orcek 

*  Tbs  nslhMiBtka  of  this  pctiod  faaa  bwa  ihrawtj  ^  Csator; 
tf  B.  Olnthsr,  OwAlefcU  if  wmlktwmlUektn  t/strrrictef  ija  rfw^ 
•dWa  JTHMtsfur, Bcribi,  IWT;  sb4  far  H.  WilMiboru,  KmalMtm  4ir 


138     THB  EI8B  OF  LBARNINO  IN  WnTBSIT  BU10P& 

nmtlieiiiatieiaiM  ware  obUinablo^  though  with  dilHciiltj,  bat 
there  were  no  students,  the  books  were  unvalued,  And  in  tine 
became  very  scarce. 

Three  authors  of  the  sixth  oentuiy — Boethius,  Osssiodoni% 
and  Isidonis — may  be  named  whose  writings  serve  as  a  eoa« 
necting  link  between  the  mathematics  of  classical  and  of 
medieval  times.  As  their  works  remained  standard  text- 
books for  some  six  or  seven  centuries  it  is  necessary  to 
mention  them,  but  it  should  be  understood  that  this  is  the 
only  reason  for  doing  so ;  they  shew  no  special  mathematical 
ability.  It  will  be  noticed  that  tliese  authors  were  contem- 
poraries of  the  later  Athenian  and  Alexandrian  schools. 

Boethina.  Atuciu9  Matdius  Severinus  Boelkitu^  or  as 
the  name  is  sometimes  written  Hoeiiut,  bom  at  Rome  about 
475  and  died  in  52(i,  belonged  to  a  family  which  for  the 
two  preceding  centuries  hod  been  esteemed  one  of  the  most 
illuMtrious  in  Home.  It  was  formerly  believnd  that  he  was 
educated  at  Athens:  this  is  souiewliat  doubtful,  but  at  any  rate 
lie  was  exceptionally  well  read  in  Greek  literature  and  science. 

lioethiuH  would  seem  to  have  wished  to  devote  his  life  to 
literary  pumuits;  but  recognizing  '*tliat  the  world  would  be 
happy  only  when  kings  became  philosophers  or  philosophers 
kings,"  he  yielded  to  the  pressure  put  on  him  and  took  an 
active  share  in  politics.  He  was  celebrated  for  his  extensive 
charities,  and,  what  in  those  days  was  very  rare,  the  care  that 
he  took  to  sc*e  that  the  recipients  were  worthy  of  them.  He 
was  elected  consul  at  an  unusually  early  age,  and  took  ad- 
vantage of  his  position  to  rvform  the  coinage  and  to  introduce 
the  pulilic  use  of  sun-dials,  water-clocks,  dec.  He  reached 
the  height  of  his  prosperity  in  522  when  his  two  sons  were 
inaugurated  as  consuls.  His  integrity  and  attempts  to  protect 
the  provincials  from  the  plunder  of  the  public  officials  brought 
on  him  the  liatred  of  the  Court.  He  was  sentenced  to  death 
while  absent  from  Rome,  seized  at  Ticinum,  and  in  the  bap- 
tistery of  the  church  there  tortured  by  drawing  a  cord  round 
his  head  till  the  eyes  were  forced  out  of  the  sockets,  and 


BOETHIUS.      CAS8IODOB08.  139 

finallj  beaten  to  death  with  cluln  on  Oct.  25,  526.  thich  kt 
iMUt  is  the  Account  Hint  lias  come  down  to  us.  At  a  later 
time  his  miMribi  were  recognized,  nnd  tonibn  and  itatnes  erected 
in  hia  honour  by  the  Btnte. 

BoclhiuR  wAH  the  tait  ttunian  of  note  who  atndied  the 
hui)fung«  and  lil^rature  of  Grrecc,  mid  liin  works  aflbrded  to 
medieval  Europe  some  gUii)|iHO  r>f  tiie  intellectual  llfo  of  the 
old  world.  MiH  importance  in  the  history  of  literature  i»  thus 
very  great,  hut  it  nri«^  merely  from  the  accident  of  the  time 
at  which  he  liced.  After  tlie  intrmluction  of  AriHtotle'ii  works 
in  the  thirteenth  century  liiw  fame  (lii-d  uway,  and  be  ban  now 
sunk  into  an  nlwourity  which  tit  an  pvnt  as  waa  nnce  his 
repntntton.  He  in  best  known  bj'  bis  CoHtolalio,  which  was 
tninMated  by  Alfred  the  (trmt  into  ArifflivSnxon.  for  our 
purpofw  it  IH  suHieicnt  to  n'lte  llmt  the  teaching  of  earl/ 
medieval  mathemnticii  was  mainly  fuundtwl  on  his  geomctiy 
and  arithmetic*. 

His  Gitrnflri/  cnXKnn^!*  of  the  enunciations  (only)  of  the  first 
book  of  Euclid,  ami  of  a  few  M'Ici-tcil  pr<>j>oKitions  in  the  third 
and  fourth  bnokx,  but  with  numerous  practical  applications  to 
finding  area.<i,  Ac  He  adds  an  appendix  with  pniofs  of  the 
first  three  propnaitiuns  to  shew  that  the  enunciations  may  bn 
relied  on.  His  Arithiu'tie  is  foundetl  on  tliat  of  Nicomachna. 
A  text-book  on  music  by  him  was  in  use  at  OxfonI  within 
the  present  century. 

Caasiodonu.  A  few  yean  later  another  Roman,  ifngnut 
AnreHtu  CfMwdomt,  who  was  tmm  about  490  and  died  in 
666,  published  two  works,  D'  Innlilntione  Dieinamut  Litle- 
ramm  and  De  Arlibn*  ne  Dittipfinit,  in  which  not  only  the 
preliminary  triviuni  of  grammar,  logic,  and  riKtaric  were  dis- 
eusspfl,  but  also  the  matbeniatical  (jiiadririum  of  arithmetic, 
geometry,  moaie,  and  aatnmoniy.  These  were  oonsidered 
standard  works  daring  the  micUle  >ges:  the  fenser  wm 
printed '»l  Tenin  in  1739. 


140     TBI  BI8I  or  LKARNIKO  IV  WOTBEN  KOBOHL 

laidomi.  Imdonu,  buhop  of  Seville^  born  in  670  Md 
died  IB  636,  wm  the  author  of  mi  encjelo|iaedie  work  ia  M 
Tolumes  called  Orujiniu^  of  which  the  third  voliuno  it  givm 
up  to  the  quadrivium.     It  was  published  at  Lsipang  ia  1833w 

The  Cathedral  atid  Conventual  Schools^. 

When,  in  tlie  latter  half  of  the  eighth  century,  Charles  the 
Great  liad  established  his  empire,  he  determined  to  promote 
leariiin;]^  so  far  as  he  was  able.  He  began  by  commanding 
that  scliuuls  should  be  opened  in  connection  with  eveiy 
cathedral  and  niuniiMtery  in  hin  kingdom ;  an  order  which  was 
approved  and  materially  assisted  by  the  popes.  It  it  in- 
teresting to  us  to  know  that  this  was  done  at  the  instance 
and  undi*r  the  direction  of  two  Englishmen,  Alcuin  and 
Cleiiient,  who  liad  attached  themselves  to  his  court. 

Alcuin t*  Of  theiie  the  more  prominent  was  Aiettin  who 
was  born  in  Yorkshire  in  735  and  died  at  Tours  in  804.  He 
was  educated  at  York  under  archbishop  Egbert  his  **  beloved 
roaster"  whom  he  succeeded  as  director  of  the  school  there. 
Subsei|uently  he  became  abljot  of  Canterbury,  and  was  sent  to 
Rome  by  OlTa  to  procure  the  pallium  for  archbishop  Eanbald. 
On  his  journey  back  he  met  Charles  at  Parma ;  the  emperor 
took  a  great  liking  to  liim,  and  finally  induced  him  to  take  up 
his  residence  at  the  imperial  court,  and  there  touch  rhetoric, 
logic,  nmtliematics,  and  divinity.  Alcuin  remained  for  many 
years  one  of  the  most  intimate  and  influential  friends  of 
Cliarles  and  was  constantly  employed  as  a  confidential  ambas- 
siulor:  as  such  he  spent  the  years  791  and  792  in  England, 
and  while  there  reorganized  the  studies  at  his  old  school  at 
York.     In  801  he  begged  permission  to  retire  from  the  court 

*  See  Tk€  Schools  of  CfiarUs  the  Great  and  the  Restoratiom  of  Edu- 
cation in  the  Siuth  Century  by  J.  B.  MuUinger,  Loadoa,  1877. 

t  See  the  life  of  Alcuiu  bj  F.  Loreatz,  Hslle,  1829,  transUted  bj 
J.  M.  Slee,  LoudoQ,  1837 ;  Alcuin  nml  tein  Jahrhundert  hj  K.  Weniar, 
Psderbom,  ltf7C;  and  Cantor,  vol  i,  pp.  713—721. 


ALCUIN.  1*1 

M  ns  to  be  able  to  sjiend  the  Inst  yvnn  of  hin  life  in  qaiet : 
with  ilifficultj-  he  obtxinM  Icnvp,  hikI  wrnt  to  the  •hbey  of 
St  Mftrtiii  nt  Tnuni,  of  which  he  IimI  l>n-n  nmilc  h<wl  in  796. 
He  eHtnt>liBti(Hl  &  schmil  in  ronncction  with  the  aIiIifj  which 
Iiecame  very  c«|phrul<^,  aikI  he  reiiminetl  mihI  tftnght  thera 
till  his  fh'Ath  on  Mny  19,  804. 

MoHl  of  thu  rxinnt  writin;!H  of  Ali-uiri  (lent  willi  llteotogy 
or  history,  Imt  Ony  incliidf  a  ralh-ctioii  of  Rritltineticnl  pn>- 
ponition.t  Kuitolile  for  th»  infllruction  of  the  young.  The 
majority  of  the  prupoHitionH  are  easy  pnilrlcnis,  either  determi- 
nate or  iiideteniiinntc,  anil  arv,  I  jin'sutiic,  founded  on  worka 
with  which  he  had  liecome  ncc|uninte<l  wlien  at  Rome.  The 
foliowinj;  is  one  of  the  most  dilticutt.  and  will  give  an  idea  of 
the  character  of  the  work.  If  one  hundre<l  huxhels  of  corn  be 
distributed  among  onu  hundred  people  in  such  a  manner  tint 
each  nian  receiveii  tiiree  huxhels,  each  wonmn  two,  and  each 
ehild  half  a  buHJiel :  how  many  men,  women,  and  children 
were  there  I  The  general  solution  is  {'JO  -  3n)  meu,  Rji  women, 
and  (80  -•  2n)  chihlren,  where  »  may  hnvo  any  of  the  vnlneti 
1,  2,  3,  4,  •!,  C.  Alcuin  only  statt'H  the  ralution  tor  which 
M  =  3  ;  that  in,  he  gives  as  the  antwer  1 1  men,  1 5  women,  and 
74  children. 

This  collection  however  was  the  work  of  a  man  of  excep- 
tional girniun,  and  prolialily  we  nlmll  be  correct  in  laying  tliat 
matheniatics,  if  taught  at  all  in  a  hcIkniI,  wan  generally  con- 
fined to  the  geometry  of  lloetliios  the  uko  of  the  abncun  and 
m u I ti plication  table,  and  piHNibly  the  aritlimetic  of  I3oetliiu8 ; 
while  ext;ept  in  one  of  the^e  scIiooIh  or  iu  a  Benedictine  cloister 
it  was  lianlly  possible  to  get  either  instruction  or  oppiirtunities 
for  atudy.  It  was  of  course  natural  tliat  tlic  works  utted  shoald 
come  from  Roman  sources,  fi>r  Dritain  and  all  the  countries 
iochided  in  the  empire  of  Charles  had  at  one  time  formed  pwrt 
of  the  western  half  of  the  Roman  empire,  and  their  inhabitanta 
continued  for  a  long  time  to  regard  Rome  m  the  centre  of 
dvilintioR,  while  the  higlier  clergy  kept  op  » tolermbly  n 
iaterooane  with  RonMb 


142     THB  B18B  OF  LKABMUfO  IN  WflSTBBV  EOWOn^ 


After  the  death  of  Quurlee  Diany  of  his  aehoob  eoniiieJ 
themselves  to  teaching  Latin,  niusie^  and  theology,  soom 
knowledge  of  which  was  essential  to  the  worldly  suooeM  ol 
the  higlier  clergy.  Hardly  any  science  or  matheniatica  was 
taught,  but  the  continued  exiiiteuce  of  the  schools  ga\-e  an 
opportunity  to  any  teacher  wliofie  learning  or  seal  exceeded 
the  narrow  limits  fixed  by  tradition ;  and  though  there  were 
but  few  who  availed  tlieniiielves  of  the  op|iortunity,  yet  the 
number  of  those  desiring  iuNtruction  was  so  large  that  it 
would  seem  as  if  any  one  who  could  teach  was  sure  to  attract 
a  considemble  audience. 

A  few  scIiooIh,  where  the  teachers  were  of  repute,  be- 
came large  and  acquired  a  certain  degree  of  permanence^  but 
even  in  tliem  the  teaching  was  still  usually  confined  to  the 
trivium  and  quadrivium.  The  former  comprised  the  three  arts 
of  grammar,  logic,  and  rhetoric,  but  practically  meant  the  art 
of  reading  and  writing  Latin;  nominally  the  latter  included 
arithmetic  and  geometry  with  their  applications,  esfiecially  to 
muKic  and  astronomy,  but  in  fact  it  rarely  meant  more  than 
arithmetic  sutlicient  to  enable  one  to  keep  accounts,  music  for 
the  church  services,  geometry  for  the  purpose  of  land-surveying, 
and  astronomy  sutlicient  to  enable  one  to  calculate  the  feasts 
and  fasts  uf  the  church.  The  seven  liljeral  arts  are  enumerated 
in  the  line,  /.liiytici,  tropitt,  ratio;  tiuf/i^rus,  Ioi<m«,  miyuiuM^ 
asira.  Any  student  who  got  beyond  the  trivium  was  looked 
on  as  a  man  of  great  erudition,  Qui  iria,  qui  ^jiUm^  qui  iotum 
Mciltite  vovitf  as  a  verse  of  the  eleventh  century  runs.  The 
special  questions  which  then  and  long  afterju-ards  attracted 
the  best  thinkers  were  logic  and  certain  portions  of  transcen- 
dental theology  and  philosophy. 

We  may  sum  the  matter  up  by  saying  tjliat  during  the 
ninth  and  tenth  centuries  the  mathematics  taught  was  still 
usually  confined  to  that  comprised  in  the!  two  works  of 
Boethius  together  with  the  practical  use  of  the  abacus  and  the 
multiplication  table,  though  during  the  latter  part  of  the  time 
a  wider  range  of  reading  was  undoubtedly  accessible. 


GERBRRT.  143 

Oarbert*.  In  the  t^nth  century  a  man  appoaird  who 
would  in  anj  age  have  been  remarkable  and  who  gave  a  great 
ttimnlus  to  learning.  This  was  Gerh^ri^  an  Aquilanian  by 
birth,  who  died  in  1003  at  about  the  age  of  fifty.  Uis  abilities 
attracted  attention  to  him  even  when  a  lioy,  and  procured  his 
mnoTal  from  the  abliey  school  at  Aurillac  to  the  Spanish 
march  where  he  received  a  good  eflueation.  He  was  in  Rome 
in  97 19  where  his  proficiency  in  music  and  astronomy  excited 
considerable  interest:  but  his  interests  wem  not  confined  to 
these  subjects,  and  he  had  alrearly  mastered  all  the  branches  of 
the  trivium  and  quadrivium,  as  then  taught,  except  logic ;  and 
to  learn  this  he  moved  to  Rlieims  which  archbishop  Adalbero 
had  made  the  most  famous  school  in  Europe.  Here  he  was  at 
onoe  innted  to  teach,  and  so  great  was  his  fame  that  to  him 
Hugh  Capet  entrusted  the  education  of  his  son  Robert  who 
was  afterwards  king  of  France. 

Gerbert  was  especially  famous  for  his  construction  of  abaci 
and  of  terrestrial  and  celestial  glolies ;  he  was  accustomed  to 
use  the  latter  to  illustrate  his  lectures.  The»e  glol)es  excited 
great  admiration  ;  and  he  utilized  this  by  offering  to  exchange 
them  for  copies  of  classical  Latin  works,  which  seem  already 
to  have  become  very  scarce ;  the  lietter  to  effect  this  he  ap- 
pointed agents  in  the  chief  towns  of  Europe.  To  his  efforts  it 
is  believed  we  owe  the  preservatiim  of  several  Latin  works, 
but  he  rejected  the  Christian  fathers  and  Greek  authors  from 
his  library.  In  982  he  received  the  abliey  of  Bobbio,  and 
the  rest  of  his  life  was  taken  up  with  political  intrigues ;  he 
became  archbishop  of  Kheims  in  991,  and  of  Ravenna  in 
998 ;  in  999  he  was  elected  pope,  when  he  took  the  title  of 
Sylvester  II.;  as  head  of  the  Church,  he  at  once  commenced 
an  appeal  to  Christendom  to  arm  and  defend  the  Holy  Land, 
thus  forestalling  Peter  the  Hermit  by  a  century,  but  he  died 

*  Weiflstnboro,  ia  the  work  alresdy  mentioned,  treats  Gabeft  Tcry 
fal|y ;  see  also  Lm  vie  ei  Ub  oenvre$  de  Orrbert^  hj  A.  (Illerip,  Clermont, 
1087 1  Oerberi  voa  ilanl/ae,  by  K.  Werner,  second  eJition,  Vienna,  1881 ; 
iMd  ambtni...Oferm  SMlArsMlica,  edited  by  M.  Bobnov,  BsrUa,  1889. 


144     TBI  RISC  or  LKARNIVQ  UT  WggOMf  BURIAL 

on  Hay  13,  1003,  before  he  had  time  to  eUl^NnAte  hb  pluML 
His  library  i«i  I  believe,  preserved  in  the  Vatican. 

So  remarkable  a  personality  left  a  deep  impress  on  his 
generation,  and  all  sorts  of  fables  soon  b^gan  to  collect  aronnd 
his  memory.  It  seems  certain  that  he  made  a  clock  whidi 
was  long  preserved  at  Magdeburg,  and  an  organ  worked  by 
steam  which  was  still  at  Rheiius  two  centuries  after  his  death. 
All  this  only  tendecl  to  confirm  the  suspicions  of  his  contem- 
poraries that  he  lisul  sold  hiiiis4*lf  to  the  devil ;  and  the  details 
of  his  interviews  with  tliat  gentleman,  the  powera  he  purchased, 
and  his  ellbrt  to  &icape  from  his  bai^in  when  he  was  dying; 
may  be  resul  in  the  pages  of  William  of  Malmesbury,  Orderie 
Vitalis,  and  Platina.  To  these  anecdotes  tlie  first  named 
writer  adds  the  story  of  the  stiitue  inscrilied  with  tlie  words 
**  strike  here,"  which  liaving  uuiused  our  ancestors  in  the  Gesta 
Bomaaarum  has  been  recently  told  again  in  the  EartUy 
Purtjulue, 

Extensive  though  his  in6uence  was,  it  must  not  be  supposed 
that  Gerbert's  wiitings  shew  any  great  originality.  His  mathe- 
matical works  comprise  a  treatisie  on  the  tite  of  the  oLaetts^  one  | 
on  arithmetic  entitled  De  XuMtrorum  Dii^Uione,  and  one  on  j 
yetimrtry.  An  improvement  in  tlie  abacus,  attributed  by  some 
writers  to  BoethiuM  but  which  is  more  probably  due  to  Gerbert, 
is  the  introduction  in  every  column  of  beads  marked  by  different 
clianicters,  oiUed  ci/>iVetf,  for  each  of  the  numbers  from  I  to  9, 
instead  of  nine  exactly  similar  cuunters  or  beads.  These  apices 
were  probiibly  of  Indian  or  Arabic  origin,  and  lead  to  a  repre- 
sentation of  numbers  essentially  the  same  as  tlie  Gobar 
numerals  reproduced  below^,  there  was  however  no  symbol 
for  zero;  the  step  fixim  this  concrete  system  of  denoting 
numljers  by  a  decimal  system  on  an  abacus  to  the  system  of 
denoting  them  by  similar  symbols  in  a'riting  seems  to  us  to  ^ 
be  a  small  one,  but  it  would  appear  that  Gerbert  did  not  make 
it.  His  work  on  geometry  is  of  unetjual  ability  ;  it  includes  a 
few  applications  to  land-surveying  and  the  determination  of 

•  See  below,  p.  lUl. 


TBI  BCTABUSUMENT  OP  UNIVERSITIES.  145 

the  heights  of  inaccessible  oljects,  hut  iiiach  of  it  stfems  to  be 
copied  from  some  pythagorpAn  text-Uiok.  In  the  coarse  of  it 
he  however  solves  one  problem  which  was  of  remarkable 
difficnltj  for  that  time.  The  question  is  to  find  the  sides  of  a 
right-angled  triangle  wlinne  hypothenuse  and  area  are  given. 
He  sajTs,  in  effect,  that  if  these  Utter  lie  denoted  respectively 
by  e  and  A*,  then  the  lengths  of  tlie  two  sides  will  be 

I  {Jt^WW  +  yc«  -  4A«}  and  }  \J<^  +  4 V  -  Ji^  ^^W\. 

Bemelinus.  One  of  Gerljert's  pupils  /^fntWtnif^  published 
a  work  on  the  abacus^  which  is,  there  is  very  little  doulyt,  a 
reproduction  of  the  teaching  of  lierlicrt.  It  is  valuable  as 
indicating  that  the  Arabic  system  of  writing  numbers  was 
still  unknown  in  Europe. 

The  Early  Medieval  Universitiesf. 

At  the  end  of  the  eleventh  century  or  the  beginning  of  the 
twelfth  a  revival  of  learning  took  place  at  several  .of  these 
cathedral  or  monastic  schools  ;  and  in  some  cases,  at  the  same 
time,  teachers  who  were  not  memliere  of  the  school  settled  in 
its  vicinity  and,  with  the  sanction  of  the  authorities,  gave 
lectures  which  were  in  fact  always  on  theology,  logic,  or  civil 
law.  As  the  students  at  these  cc*ntres  grew  in  numbers,  it 
became  possible  and  desirable  to  act  to^^ther  whenever  any 
interest  common  to  all  was  concerned.  The  aksociation  thus 
formed  was  a  sort  of  guild  or  trades  union,  or  in  the  langunge 
of  the  time  a  unirenitoi  maguiramm  el  fchofariitm.  This  was 
the  first  stage  in  the  development  of  the  earliest  medieval 
universities.  In  some  cases,  as  at  Paris,  the  governing  body 
of  the  university  was  fonned  by  the  teachers  alone,  in  others, 
at  at  Bologna^  by  both  teachers  and  students ;  but  in  all  cases 

*  It  is  feprinted  in  011erui*s  sdilaon  of  Oerlwti*s  works,  pp.  Sll—SSSw 
t  8<»i  the  UuifpeniUeB  of  Emropt  in  the  Middle  A^n  bj  H.  BuhAOl, 

Oifnd,  18M;   Die  UmivenMtem  dn  JiitieMUn  Ma  140a  kj  P.  H. 

DsHlfl^  ins  I  and  voL  I  of  Ibt  Onivenitf  ef  Cmmkrid§$  kj  7.  a 

Maltt^sr,  Osabridlfe,  l«7t. 

B.  10 


140     THK  BUS  OP  LBARNIHQ  IV  WtgrOM  SUIOnL 

• 

preoiae  rates  for  the  ooodvet  of  boaiiieiit  and  the  ragvlatfos  of 
the  internal  eoonomy  of  the  guild  were  formnlaled  at  an  earlj 
stage  in  its  history.  The  nmnieipalities  and  nttmerooa  aooio- 
ties  which  existed  in  Italy  sapplied  plenty  of  modeb  for  the 
construction  of  such  rulesi  but  it  is  possible  that  some  of  the 
regulations  were  derived  from  those  in  force  in  the  Moham- 
luethiu  Nchouls  at  Cordova. 

We  are,  aluiost  inevitably^  unable  to  fix  the  exact  date  of 
the  couimeucement  of  these  voluntary  associations,  but  they 
existed  at  Paris,  Bologna,  Salerno,  Oxford,  and  Cambridge 
before  the  end  of  the  twelfth  century:  these  may  be  eon- 
sidensd  the  earliest  uuiventitieH  in  Kurupe.  The  instraction 
given  at  Salerno  and  Bologna  was  mainly  technical — at 
Salerno  in  medicine,  and  at  ikilogna  in  law — and  their  claim 
to  recogiiiticin  as  univfrMitieM,  as  long  as  tliey  were  merely 
technical  scIukiIs,  haH  lieen  disputc^d  ;  the  title  of  university 
wus  generally  accntlited  to  any  tencliiug  body  at  soon  at 
it  was  recH^gnized  as  a  studiufn  generale. 

Although  the  organization  of  tliene  early  universities  was 
independent  of  the  neighliouring  church  and  monastic  schools 
they  seem  in  general  to  have  lieen,  at  any  rate  originally, 
associated  with  such  kcIiooIs,  and  perhaps  indebted  to  them 
for  the  use  of  rooms,  «l'c.  The  universities  or  guilds  (self- 
governing  and  formed  by  teachers  and  students),  and  the 
adjacent  schools  (under  the  direct  control  of  church  or  monastic 
autboritieH)  continuini  to  exi&t  Hide  by  side,  but  in  course 
of  time  the  latter  diminished  in  importance,  and  often  ended 
by  iKHMiiiiiiig  Hubject  to  the  rule  of  the  university  authorities. 
Nearly  all  the  medieval  universities  grew  up  under  the  pro- 
tection of  a  bishop  (or  abbot),  and  were  in  some  matters 
subject  to  his  authority  or  to  that  of  his  chancellor,  from  the 
latter  of  whom  the  head  of  the  university  subsequently  took 
his  title.  The  universities  however  were  not  ecclesiastical 
orgamizations,  and,  though  the  bulk  of  their  members  were 
ordained,  their  direct  connection  with  the  Church  arose  chiefly 
from  the  fact  that  clerks  were  then  the  only  class  of  the 


SABLT  EUROPEAN  UNIVERSITIES.  147 

oomnmiiity  who  were  left  free  by  the  state  to  punae  in* 
lellectual  studies. 

A  univernias  mngigtrcmm  ei  ttcho^arium^  if  successful  in 
attracting  students  and  aoqiiiring  permanency,  always  sought 
Sfiecial  legal  privileges,  such  as  the  right  to  fix  the  price  of 
provisions  and  the  power  to  try  legal  actions  in  which  its 
members  were  concerned.  These  privileges  geiierall)  led  to 
a  recognition  of  its  power  to  grant  degrees  which  conferred 
a  right  of  teaching  anywhere  within  the  kingdom.  The 
university  was  frequently  incorporated  at  or  almut  the  same 
time.  Paris  received  its  charter  in  1200,  and  proliabU 
was  the  earliest  university  in  Europe  thus  oflicinlly  recognized. 
Legal  privileges  were  conferred  on  Oxfonl  in  1214,  and  on 
Cambridge  in  1231 :  the  development  of  Oxford  and  Cam- 
bridge followed  closely  the  preceflent  of  Paris  on  which  their 
organization  was  modelled.  In  the  course  of  the  thirteenth 
century  universities  were  founded  at  (among  other  places) 
Naples,  Orleans,  Padua,  and  Prague ;  and  in  tlie  course  of  the 
fourteenth  century  at  Pavia,  and  Vienna. 

The  most  famous  medieval  universities  aspired  to  a  still 
wider  recognition,  and  the  final  step  in  their  evolution  was  an 
acknowledgement  by  the  pope  or  emperor  of  their  degrees  as  a 
title  to  teach  throughout  Cliristendora— such  universities  were 
doeely  related  one  with  the  other.  Paris  was  thus  recognized 
in  1283,  Oxford  in  1296,  and  Cambridge  in  1318. 

The  standard  of  education  in  mathematics  has  lieen  largely 
6xed  by  the  universities,  and  most  of  the  mathematicians  of 
■absequent  times  have  been  cl<isely  connected  with  one  or 
more  of  them ;  and  therefore  I  may  be  pardoned  for  adding 
a  few  words  on  the  general  course  of  studies*  in  a  university 
in  medieval  timet. 

The  students  entered  when  quite  young,  sometimes  not 
being  more  than  eleven  oc  twelve  years  old  when  first  coming 

*  For  faller  details  m  Io  their  orKsoitatkm  of  ttodicH,  their  systsfli 
of  niitnietHM,  sad  their  eoottitutioD,  see  mj  HiaUtr^  tf  the  SCw^r  of 
MmiknmHfi  mi  Cnmkri4§u  Csd^bridis,  18(191 

10— S 


148     TBI  RISl  or  LBARKIKO  DT  WKIIBH  MXmom 


into  residence.  It  is  mialeeding  to  describe  them  m  «nder> 
gmduateis  lor  tlieir  age,  their  studies,  the  discipline  to  whieh 
thej  were  sabjected,  and  their  positicm  in  the  nniveruty  shew 
that  they  should  be  regarded  as  schoolboys.  The  first  foar 
yeani  of  their  residence  were  supposed  to  be  spent  in  the 
study  of  the  triviuni,  that  is,  Latin  granunar,  logic,  and 
rhetoric  lu  quite  Hirly  times,  a  considerable  numlier  of  the 
students  did  not  pnigrcss  beyond  the  study  of  Latin  grammar 
—  they  formed  an  inferior  fuculty  and  were  eligible  only  for 
the  degree  of  master  of  grammar  or  master  of  rhetoric — bnt 
the  more  advanced  students  (and  in  later  times  all  students) 
spent  these  years  in  the  study  t>f  the  trivium. 

The  title  of  bachelor  of  arts  was  conferred  at  the  end  of 
this  course,  and  signified  that  the  student  was  no  longer  a 
scliuolboy  and  therefore  in  pupilage.  The  average  age  of  a 
coumiencing  bachelor  may  be  tsken  as  having  been  about 
seventeen  or  eighteen.  Thus  at  Cambridge  in  the  presenta- 
tion for  a  degree  the  technical  term  still  used  for  an  under- 
graduate is  jureiiiif^  while  that  for  a  bachelor  is  vir,  A 
bachelor  could  not  take  pupils,  could  teach  only  under  special 
restrictions,  and  proluAbly  occupied  a  position  closely  analo- 
gous to  that  of  an  undergraduate  now-a-days.  8onie  few 
bachelors  pn>cee<led  to  tht)  study  of  civil  or  canon  law,  but  it 
WAS  AMHumctl  in  theory  that  they  next  studied  the  quadrivium, 
tho  course  for  which  took  three  years,  and  which  included 
about  as  much  science  as  was  to  be  found  in  the  pages  of 
Boethius  and  Isidonis. 

The  de;(ree  of  master  of  arts  was  given  at  the  end  of  this 
course.  In  the  twelfth  and  thirteenth  centuries  it  was  merely 
a  license  to  teach  :  no  one  sought  it  who  did  not  intend  to  use 
it  for  that  purpose  and  to  reside  in  the  university,  and  only 
those  who  had  a  natural  sptitude  for  such  work  were  likely  to 
enter  a  profession  so  ill-paid  as  that  of  a  teacher.  The  degree 
was  obtainable  by  any  student  who  had  gone  through  the 
recognized  course  of  study,  and  shewn  that  he  was  of  good 
moral  character.     Outsiders  were  also  admitted,  but  not  aa  a 


OOUfiSS  AT  A  MEDIEVAL  UNIVERSITY.  149 

matter  of  oonne.  I  may  here  add  that  towards  the  end  of 
the  foarteenth  century  students  began  to  find  that  a  de^^rce 
had  a  pecuniary  value,  and  most  universities  sulisequently 
conferred  it  only  on  condition  that  the  new  luaster  should 
reside  and  teach  for  at  least  a  year.  Somewhat  later  the 
univemities  took  a  further  step  and  liegnn  to  refuse  degrees  to 
thoee  who  were  not  intellectually  qualified.  This  power  was 
assumed  on  the  preceflent  of  a  case  which  arose  in  Paris  in 
1426,  when  the  university  doclin<H]  to  confer  a  degree  on  a 
student — a  Slavonian,  one  Paul  Nicholas — who  had  pcrfitrmcd 
the  necessary  exercises  in  a  very  indifferent  manner :  he  to(»k 
legal  proceedings  to  compel  the  university  to  grant  the  degree, 
b«t  their  right  to  withhold  it  was  established.  Nicholas 
accordingly  has  the  distinction  of  being  tlie  first  student  ever 
"plucked." 

Although  science  and  mathematics  were  recognized  as.  the 
standard  suljects  of  study  for  a  bachelor,  it  is  proliable  that, 
until  the  renaissance,  the  majority  of  tlie  students  devoted  most 
of  their  time  to  logic,  philosophy,  and  theology.  The  subtleties 
of  scholastic  philosophy  were,  dreary  and  barren,  but  it  is 
only  just  to  nay  that  they  provided  a  severe  intellectual 
training. 

We  have  now  arrived  at  a  time  when  the  results  of  Arab 
and  Greeks  science  became  known  in  Europe.  The  history  of 
Greek  mathematics  has  been  already  discussed ;  I  must  now 
temporarily  leave  the  subject  of  medieval  mathematics,  and 
trace  the  development  of  the  Arabian  schools  to  the  same  date; 
and  1  must  then  explain  how  the  schoolmen  becai^e  acquainted 
with  the  Arab  and  Greek  text-books,  and  how  their  introdoo- 
tioo  affected  the  progrMs  of  European  mathematics. 


150 


CHAPTER    IX. 

TBI  MATHEMATICS  OF  TUB  ARABB^ 

Tbb  ■tory  of  Arabian  inaihenuitict  it  known  to  na  in  Ito 
general  oatlineni  but  we  are  at  yet  unable  to  apeak  with  ear- 
taintj  on  many  of  iUi  details.  It  is  however  quite  clear  that 
while  part  of  the  early  knowledge  of  the  Arabs  waa  derived 
from  Qreek  soiiroea,  part  was  obtained  from  Hindoo  works; 
and  that  it  was  on  thane  foundations  that  Arab  science  waa 
built  I  will  begin  by  considering  in  turn  the  extent  of 
mathematical  knowledge  derived  from  these  souroea. 

Extent  of  niatheuiaticM  obtained  from  Greek  Mourcee. 

According  to  tlieir  traditions,  in  themselvea  very  probable^ 
the  scientitic  knowledge  of  the  Arabs  was  at  first  derived  from 

*  TIic  subject  la  diMUrf^ed  at  Icnigth  hy  Cantor,  eUapn.  xxxn — xsxv ; 
bj  Hankc*!,  pp.  17*i— 'ill3 ;  and  bj  A.  vod  Kilmer  iu  Kmllmr$e*ekickU  dt§ 
OritmUs  UHter  dtu  Ckati/em,  Vienna,  1877.  See  aUo  J/meriaaix  peer 
trrvir  a  tkUUnre  comparie  ties  acienct*  matkimatiqiU4  chei  let  Ones  H 
'  l€$  Oriemtaux,  by  L.  A.  ScHiillot,  Paris,  1815-9:  and  the  foUoviaf 
articles  bj  Fr.  W'oepcke,  Snr  Viutroductiom  dt  Varitkmtftique  Imdieume  en 
Occident,  Home,  185{| ;  Smr  rkUtoire  des  sciemces  matkimaliqueB  ekeM  U§ 
OriVNliiMX,  Paris,  1800;  and  ^i moire  aur  lu  pro^ifatiom  des  ek^frm 
•/Jidieiu,  Paris,  1863. 


f 


wtmmmmmmmmmmmmemi'''9^^m^mrHmm 


THE  MATnEMATlL-S  OF  THE  ARABS.  151 


ths  Omk  doctom  who  nttcndrd  tlio  cnliphs  itt  Ritfidiul.  It  ia 
■aid  that  when  (he  AraltJan  ^inqui'mrB  Kfttled  in  townn  thrj 
became  aubject  to  diseases  which  liaH  bi-en  unknnirn  to  tliem 
in  their  life  in  th«  desert.  T1ii>  sluilv  of  medicine  wwi  then 
oonfined  mainly  to  flrepkn  and  JewR,  nnd  many  <rf  these, 
encoaraged  by  the  cnliphn,  settlnl  at  Itii^^ad,  Elamaiiciia,  and 
other  cities;  their  knowlnlgc  of  nil  hmncheii  nf  lefiming  wm 
^xtenMve  and  accumt^  than  thnt  of  the  Amim,  and 
the  teaching  of  the  young,  nn  bn^  oftpn  happened  in  similar 
cases,  fell  into  their  hnndH.  The  intimluction  of  Eumpean 
science  wiia  rendered  the  more  easy  m  rariooa  small  Greek 
•choola  rxinted  in  the  countries  subject  to  the  Aralis:  ther« 
had  for  many  years  been  one  at  Edessa  among  the  Nestorian 
Christians,  and  there  were  otliers  at  Antioch,  Emesa,  and 
aven  at  Damascus,  which  hod  preservnl  the  traditiooa  and 
aonie  of  the  results  of  Greek   learning. 

The  Arabs  soon  remarked  that  the  (ireeka  rvnted  their 
medical  science  on  the  wurks  uf  Hippocrates,  Aristotle,  and 
Gnlen ;  and  these  liooks  were  Iran.iliite^l  into  Arahic  by  order 
of  the  caliph  Haruun  AI  Rnschid  almut  the  year  800.  The 
translation  excited  hu  mui-h  interest  that  his  surcesaor  Al 
Mainun  (813—833)  sent  a  commission  to  Constantinople  to 
obtain  mpica  of  ua  many  scientific  works  as  was  possiMe,  white 
an  emlwssy  for  a  similar  purpose  was  aW  sent  to  India.  At 
the  same  time  a  large  stafT  of  Syrian  clerks  waa  engii];ed,  whose 
duty  it  was  to  translate  the  works  so  ubtnined  intp  A  tabic  and 
Syriac  To  disarm  fanaticisiti  these  clerks  were  ftt  Gnt  termed 
the  caliph's  doctors,  but  in  )*51  they  were  formed  into  V  college^ 
and  their  meet  celcbratnl  member  Hnnein  ilm  Isliak  was 
made  its  6rat  president  liy  the  caliph  MuUwakkit  (847— t>fil>. 
Bonein  and  his  sim  Ishak  ibn  Honein  revised  the  transla- 
tion* before  they  were  finally  issued.  Neither  of  them  knew 
macb  mathematics,  and  several  blunders  were  made  in  IIm 
works  issued  oo  that  subject,  but  another  member  of  the 
oollege,  Tkbit  ibn  Korra,  eliortly  puhtisbed  (reah  < 
whkli  tbenafter  baeuM  the  ataadanl  testa. 


152  THX  MATHElf  ATIOB  OP  TUS  ABABSL 

In  this  way  before  ike  end  of  the  ninth  oentoiy  the  Anhe 
obtained  translationt  of  the  worka  of  Soclid,  Aidumedee, 
Apollonina,  Ptolenij,  and  others;  and  in  some  eaaee  theee 
editions  are  the  only  copies  of  the  books  now  estanL  It  is 
carious  as  indicating  how  completely  Diophantos  had  dropped 
out  of  notice  that  as  far  aa  we  know  the  Arebs  got  no  mann* 
script  of  Ilia  great  work  till  150  years  later,  by  which  timo 
they  were  already  acquainted  with  tlie  idea  of  algebraic  nota-  fl 

tion  and  processes. 

ExtetU  of  inailieinaiics  obtained  from  Hindoo  soiifcet. 

The  Arabs  had  considerable  coiumeroe  with  India,  and  a 
knowledge  of  one  or  both  of  the  two  great  original  Hindoo 
works  on  algebra  had  been  thus  obtained  in  the  caliphate  of 
Al  Mansur  (754 — 775),  though  it  was  not  untU  fifty  or  sixty 
years  later  that  they  attracted  much  attention.  The  algebra 
and  aritliiuetic  of  the  Arabs  were  largely  founded  on  these 
treatises,  and  I  therefore  devote  this  section  to  the  comadera* 
tion  of  Hindoo  mathematics. 

The  Hindoos,  like  the  Chinese,  have  pretended  that  they 
are  the  must  ancient  people  on  the  face  of  the  earth,  and 
that  to  them  all  sciences  owe  their  creation.  But  it  would 
appear  from  all  recent  investigations  that  these  pretensions 
have  no  foundation ;  and  in  fact  no  science  or  useful  art 
(except  a  rather  fantastic  architecture  and  sculpture)  can  be 
traced  back  to  the  inhabitants  of  the  Indian  peninsula  prior 
to  the  Aryan  iuvauion.  Thin  invasion  seems  to  have  taken 
place  at  some  time  in  the  latter  half  of  the  fifth  century  or 
in  the  sixth  century,  when  a  tribe  of  the  Aryans  entered 
India  by  the  north-west  frontier  and  establislied  themselves  as  J 

rulers  over  a  lai^  part  of  the  country.  Their  descendantSy 
wherever  they  have  kept  their  blood  pure,  may  still  be  recog- 
nized by  their  superiority  over  the  races  they  originally  con- 
quered ;  but  as  is  the  case  with  the  modem  Europeans  they 
found  the  climate  trying,  and  gradually  degenerated.     For 


^' 


^ 


ABTA-BHATA.  153 

the  first  two  or  thrra  oentaries  thej  however  retained  their 
intellectual  vigour,  and  produced  one  or  two  writers  of  great 
abilUy. 

V^Arya-Bhata.  The  first  of  these  is  Arya-Bhntn^  who  was 
bom  at  Patna  in  the  year  476.  He  is  freqaently  quoted  hj 
Brahmagupta,  and  in  the  opinion  of  many  commentators  he 
created  algebraic  analysis  though  it  has  been  suggested  that 
he  may  have  seen  Diophantus's  AriihmHic,  The  chief  work  of 
Arya-Bhata  with  which  we  are  acf|UAinted  is  his  Arynhhathitfa 
which  consists  of  mnemonic  verses  rnilMxlying  the  enunciations 
of  various  rules  and  propoRitions.  There  are  no  proofs,  and 
the  language  is  so  obscure  and  concise  that  it  long  rlefi<Hl  all 
efforts  to  translate  it*. 

The  book  is  divided  into  four  fiarts:  of  these  three  are 
devoted  to  astronomy  and  the  elenivnts  of  spherical  trigono- 
metry ;  the  remaining  part  contains  the  enunciations  of  thirty- 
three  rules  in  arithmetic,  algel>ra,  and  plane  trigimometry.  It 
18  probable  that  Arya-Bhata,  like  Brahmagupta  and  Bhaskara 
who  are  mentioned  next,  regarded  himself  as  an  astronomer, 
and  studied  mathematics  only  so  far  as  it  was  useful  to  him  in 
his  astronomy. 

In  algebra  Arya-Bhata  gives  the  sum  of  the  firsts  second, 
•  and  third  powers  of  the  first  n  natural  numbers ;  the  general 
solution  of  a  quadratic  equation ;  and  the  solution  in  integers 
of  certain  indeterminate  equations  of  the  first  degree.     His 
I  solutions  of  numerical  equations  have  been  supposed  to  imply 

that  be  was  acquainted  with  the  decimal  system  of  numera- 
tion. 

In  trigonometry  he  gives  a  table  of  natural  sines  of  the 
angles  in  the  first  quadrant,  proceeding  by  multiples  of  3}\ 

*  A  Banskril  test  of  the  Ar^nhhatki^^  edited  by  U.  Km,  wss 
fuMiihed  St  Lejrden  lo  1R74  %  there  is  sleo  so  article  on  it  bj  the  ssme 
•diier  in  the  /esmef  tf  thi  AHatie  Soeiet^^  Loodoa,  186S,  voL  zz, 
pp.  171—387 :  a  Frendi  trsnalation  by  U  Rodel  of  tbsl  pert  vhfeb  desis 
wHh  algebra  and  trieonometfy  ie  given  in  the  JmumttI  if ilel/faf ,  1879, 
Pteis.  esriw  7»  vuL  no,  M^  tS^-^St. 


154  THK  MATHKMATIGB  OT  THE  ARAML 

defining  m  umt  as  the  temichord  of  doable  the  Mwe.  Aiwiinf 
that  for  the  angle  Sf*  the  aine  is  equal  to  the  cjirciilar  meaireie^ 
he  takes  for  its  valne  225,  t.«.  the  nnmber  of  minutes  in  the 
angle.  He  then  enunciates  a  rule  whieh  is  nearly  nnintelKgible 
but  pnibably  is  the  equivalent  of  the  statement 

sin(n-fl)a-siniui:=sinfia-sin(n-  l)a-jsiniiaeoseee, 

where  a  stands  for  3}*;  and  working  with  this  ionnula  he 
constructs  a  table  of  sinesi,  and  finally  finds  th^  value  of  sin  90* 
to  be  3438.  This  result  is  correct  if  we  take  31416  as  the. 
value  of  Vy  and  it  is  interesting  to  note  that  this  is  the  number 
which  in  another  place  he  gives  for  v.  The  oorrect  trigono> 
uietiical  formula  is 

sin(n  +  l)a-sinfia=sinfia  — siu(n-l)a  — 4sinfiasin*|«. 

Aiya-Bhata  therefore  took  4  sin'  |a  as  equal  to  cosec  a,  £.«.  he 
supposed  that  2  sin  a  »  1  ••-  sin  2a :  using  the  approximate 
values  of  sin  a  and  sin  2a  given  in  his  table,  this  reduces  to 
2  (225)  =  1-1-  449,  and  hence  to  that  degree  of  approximaUon 
his  formula  is  correct.     A   considerable  proportion  of  the 

f  metrical  propositions  which  he  gives  is  wrong. 
Brahxnagupta.  The  next  Hindoo  writer  of  note  is 
Brahntayupla^  who  is  said  to  have  been  bom  in  598  and 
pruliably  was  alive  about  G6(K  He  wrote  a  work  in  verse 
entitled  Brahuu&Sphuta'Siiidlutnia^  that  is,  the  Siddhania  or 
system  of  Brahma  in  astronomy.  In  this,  two  chapters  are 
devoted  to  arithmetic,  algebra,  and  geometry*. 

The  arithmetic  is  entirely  rhetorical.  Most  of  the  problems 
are  worked  out  by  the  rule  of  three,  and  a  large  proportion  of 
them  are  on  the  subject  of  interest. 

In  his  algebra,  which  is  also  rhetorical,  he  works  out  the 
fundamental  propositions  couiiected  with  an  arithmetical  pro- 
gression, and  solves  a  quadratic  equation  (but  gives  only  the 
positive  value  to  the  radical).     As  an  illustration  of  the  pn>> 

*  TlieM  two  chsptera  (cbs|M.  xn  and  svui)  wsis  trsnilstsd  bj  H.  T* 
Colebrooke,  and  paUisbe<l  st  London  in  1817. 


BRAHHAGUPTA.  155 

blemii  given  I  may  ()iiut«  the  follnwing,  which  wbb  reprodneed 
in  slightly  different  farmn  by  varinus  Rubteqnent  wrCtara,  but 
I  replace  the  numben  by  lettenu  "Two  Apes  lired  kt  the 
top  of  K  clifT  of  height  A.  whose  bane  wan  diHtBnt  mA  from  • 
neighbouring  vilUge.  One  descended  the  cliff  «nd  widked  to 
the  village,  the  other  flew  np  a  height  x  and  then  flew  in  a 
straight  line  to  the  villnge.  The  dishinoe  travenied  by  each 
was  the  same,  Pintl  x."  nrnhmagujrta  ffave  the  correct 
answer,  namely  x  —  n(/i/(m  +  2).  In  th«  question  as  enan- 
eiatct]  originally  k=  I  DO,    m  =  2. 

Brahmaguptn  finilH  solutions  in  inte!;era  of  Bereral  in- 
determinate equntionB  of  the  first  degree,  using  the  same 
method  as  that  now  practised.  He  states  one  indeterminate 
equation  of  the  second  degrtf,  namely,  hx*  +  1  =  y*,  and  give* 
as  its  solution  x  ==  *i(/(('  -  n)  and  y  =  (C  +  ")/((*  -  »)-  To  obbun 
this  general  form  he  proved  that,  if  one  Notation  cither  of  that 
or  of  certain  nllied  equations  could  lie  gnessed,  the  general 
solution  could  be  writl^n  dovn  ;  but  he  did  not  exphun  how 
one  solution  could  be  obtained.  Curiously  enongh  this  eiina- 
tion  was  sent  hy  Fermnt  a.1  a  challenge  to  Wallis  and  Loid 
Bronncker  in  the  seventeenth  century,  and  the  latter  fonnd 
the  same  solations  on  Brahniacfupta  had  previously  done. 
Brahmagupta  also  stnted  that  the  equation  y*=N;^—  1  conld 
not  bo  satistied  l>y  integral  values  of  jc  and  y  nnless  h  ounld  be 
eipmsed  as  the  sum  of  the  squares  of  two  integers.  It  ia 
perhaps  worth  noticing  that  the  early  algebraists,  whether 
Oreeks,  Hindoos,  Aralis,  or  Itnlinns,  drew  no  distinction 
between  the  prublems  which  lpd  to  determinate  and  those 
which  led  to  indetermitiate  equations.  It  was  only  after  the 
introduction  of  syncopated  algebra  that  attempts  were  made 
to  give  general  solutions  of  erjuations,  and  the  difficelty  of 
giving  snch  solutions  of  indeterminate  eqnatiuna  other  than 
thoM  of  tlie  first  degree  has  led  to  their  praetical  exclmioii 
from  elementary  algebra. 

In  geometry  Brahmagupta  proved  the  pjthigaraan  propel ty 
ol  a  n^fr«BgM  trian^  (Edc.  1, 47).   He  gaw  nji wmow  he 


166  THK  MATHJCMATIGB  OF  THE  ABABa. 

•  * 

the  ATM  of  a  triangle  and  of  a  quadrilateral  inaeribablo  in  a 
circle  in  terms  of  their  udes ;  and  shewed  that  the  area  of  a 
circle  was  equal  to  that  of  a  rectangle  whoee  sides  were  the 
radius  and  sewiperinieter.  He  was  less  successful  in  his 
attempt  to  rectify  a  circle,  and  his  result  i4  equivalent  to 
taking  VT6  for  the  value  of  v.  He  also  determined  the  snrw 
face  and  volume  of  a  pyramid  and  cone ;  problems  over  which 
Arya-Bliata  had  blundered  badly.  The  next  part  of  his 
geometry  is  almost  unintelligible,  but  it  seems  to  be  an  at» 
tempt  to  tiud  expreHsions  for  several  magnitudes  connected 
with  a  quadrilateral  inscribed  in  a  circle  in  terms  of  its  sides : 
much  of  this  is  wrung. 

It  must  not  be  supposed  that  in  the  original  work  all  the 
propoHitions  which  deal  with  any  one  subject  are  collected 
together,  and  it  is  only  for  convenience  tliat  I  have  tried  to 
arrange  theui  in  that  way.  It  is  impossible  to  say  whether 
the  whole  of  Brahmagupta's  results  given  above  are  originaL 
He  knew  of  Aryai-Bhata's  work,  for  he  reproduces  the  table 
of  sines  there  given ;  it  is  likely  also  that  some  progress  in 
mathematics  had  been  made  by  Arya-Bhata's  immediate  suc- 
cessors, and  that  Brahmagupta  was  acquainted  with  their 
works ;  but  then)  seems  no  reason  to  doubt  that  the  bulk  of 
Biahuuigupta's  algebra  and  arithmetic  is  original,  althou^^ 
perhaps  influenced  by  Diophoiitus's  writings:  the  origin  of 
the  geometry  is  more  doubtful,  proljably  some  of  it  is  derived 

rni  Hero's  works. 
Bhaekara.  To  make  this  account  of  Hindoo  mathematics 
complete,  I  may  depart  from  the  chronological  arrangement 
and  siiy  that  the  only  remaining  Indian  mathematician  of 
exceptional  eminence  of  whose  works  we  know  anything  was 
Bluufkarti  who  was  bom  in  1114.  He  is  said  to  have  been 
the  lineal  successor  of  Brahmagupta  as  head  of  an  astro> 
nomical  observatory  at  Ujein.  He  wrote  an  astronomy  of 
which  four  chapters  have  been  translated.  Of  these  one 
termed  Liiavati  is  on  arithmetic ;  a  second  termed  Hija  Gauita 
is  on  algebra ;  the  third  and  fourth  are  on  astronomy  and  the 


i  . 


BHASKARA.  157 

sphere*;  aoine  of  the  other  chaptora  also  in  voire  mathe- 
matics. This  m'ork  was  I  believe  known  to  the  Arabs  almost 
as  soon  as  it  was  written  and  influenced  their  soltsequent 
writings,  though  they  fniled  to  utilize  or  extend  most  of  the 
discoveries  contained  in  it  The  results  thus  became  in« 
directly  known  in  the  West  Ix^fore  the  end  of  the  twelfth 
century,  but  the  text  itself  was  not  intruduceU  into  Europe 
till  within  recent  times. 

The  treatise  is  in  verse,  but  there  are  explanatory  notes 
in  prose.  It  is  not  clear  whether  it  is  original  or  whether  it 
is  merely  an  exposition  of  the  results  then  known  in  India ; 
but  in  any  case  it  is  nnist  probable  that  Bhaskara  was  ac- 
quainted with  the  Arab  works  which  had  been  writti*n  in  the 
tenth  and  eleventh  centuries,  and  with  the  results  of  Greek 
mathematics  as  transmitted  through  Arabian  sources.  The 
algebra  is  syncopated  and  almost  symbolic,  which  marks  a 
great  advance  over  that  of  Brahmagupta  and  of  the  Arab& 
The  geometry  is  also  superior  to  that  of  Brahmagupta,  but 
apparently  this  is  due  to  the  knowledge  of  various  Greek  works 
obtained  through  the  Arabs. 

Tlie  first  Ixiok  or  Lttnvati  commences  with  a  salutation 
to  the  god  of  visdom.  The  general  arrangement  of  the  work 
may  be  gathered  from  the  following  table  of  contents.  Systems 
of  weights  and  measures.  Next  decimal  numeration,  briefly 
described.  Then  the  eight  operations  of  arithmetic,  namely, 
addition,  subtraction,  multiplication,  division,  square,  cube, 
square-root,  and  cube-root  Reduction  of  fractions  to  a  common 
denominator,  fractions  of  fractions,  mixed  numbers,  and  the 
eight  rules  applied  to  fractions.  The  *' rules  of  cipher,"  namely, 
a^O»a,  0*^:10,  v^«0,  a-hO^oo.  The  solution  of  soma 
simple  equations  which  are  treated  as  questions  of  arithmeUo. 
The  rule  of  fiUse  assumption.     Simultaneous  equations  of  the 

*  8m  ths  srticls  Vifa  GmmiU  in  llie  PcNHy  Cjfelapmedim^  Loiidom 
ISa ;  and  lh«  trandstiont  of  the  Ulmraii  and  the  ll(jM  Qmniim  isMcd 
hj  H.  T.  Oolebiooke,  LoiKion,  1817.  The  ehapCm  on  asltcooiay  aai 
Us  aphste  wmm  sditsd  bjL.  Wilkinson  Galeatta,  184S. 


I 


158  THK  MATflUATIOB  09  THE  ABAB8. 

fini  diigTM  with  applieationii  Solution  of  a  few  qvadiatfa 
oquatioiiiL  Role  of  three  and  eompoliiid  rale  of  threes  with 
Tarioua  cases.  Interest,  disoonnt,  and  partnefshipw  Tiine  of 
filling  a  cistern  by  several  fountains.  Barter.  Arithnetioal 
progressions,  and  sums  of  squariv  and  cubes.  Geometrical  pro- 
gressiona  Problems  on  triangles  and  quadrilaterals.  Approxi- 
mate value  of  V.  Some  trigonometrical  formulae.  Contents 
of  solids.  Indeterminate  equations  of  the  first  degree.  Ijutly 
the  book  ends  with  a  few  questions  on  combinations. 

This  is  the  earliest  known  work  which  contains  a  syste- 
matic exposition  of  the  decimal  system  of  numeration.  It  is 
possible  that  Arya-Bhata  was  acquMinted  with  it^  and  it  is 
most  likely  that  Bralimogupta  was  so,  but  in  Bhaskara's  arith- 
metic we  meet  with  the  Arabic  or  Indian  numerals  and  a  sign 
for  zero  as  part  of  a  well-recugnized  notation.  It  is  impossible 
at  pre84*nt  to  detinitely  trace  these  numerals  further  beck  than 
the  eighth  century,  but  there  is  no  reason  to  doubt  the  assertion 
tliat  they  were  in  use  at  the  beginning  of  the  seventh  century. 
Their  origin  is  a  difficult  and  disputed  question.  I  mention 
below*  the  view  which  on  the  whole  seems  most  probable  and 
perha|Mi  is  now  generally  accepted,  and  I  reproduce  there 
some  of  the  forms  used  in  early  times. 

To_8um  the  matter  up  briefly  it  may  be  said  thai  the 
LUartiii  gives  the  rules  now  current  for  addition,  subtraction, 
multiplication,  and  divitiion,  as  well  as  the  more  common  pro> 
cesses  in  arithmetic ;  while  the  greater  part  of  the  work  is 
taken  up  with  the  discussion  of  the  rule  of  three,  which  is 
divided  into  direct  and  invente,  simple  and  compound,  and 
is  used  to  solve  numerous  questions  chiefly  on  interest  and 
exchange — the  numerical  questions  being  expressed  in  the 
decimal  system  of  notation  with  which  we  are  familiar. 

Bhaskara  was  celebrated  as  an  astrologer  no  less  than  as  a 
mathematician.  He  learnt  by  this  art  that  the  event  of  his 
daughter  Lilavati  marrying  would  be  fatal  to  himself.  He 
therefore  declined  to  allow  her  to  leave  his .  presence,  but  by 

*  See  below,  fi.  190. 


S 


nHASKARA.  159 

Tftj  of  coDwUtion  he  not  only  called  tbe  firat  book  of  hu 
work  bj  her  name,  but  prnpouniird  many  of  hi*  problem*  in 
the  form  of  questionii  iul(]rf?(sed  to  her.  For  example,  "  Lnrely 
and  dear  Lilavatj,  whnse  eyea  are  like  a  fawn'a,  tell  me  what 
•re  the  numbers  resulting  from  IS'i  multiplied  by  12.  If  tbon 
be  skilled  in  multiplication,  whether  by  whole  or  1^  pnrta, 
whether  by  diviRion  or  by  wpnration  of  digita,  tell  roe,  auRpi- 
oiou*  dumtiel,  what  jn  the  quotient  of  the  product  when  divided 
by  the  same  multiplier," 

I  may  add  here  that  the  problems  in  the  Indian  works  pve 
a  great  deal  of  intrrestinj;  informatinn  aliout  the  eocinl  and 
economic  eomlition  of  the  country  in  which  they  were  written. 
Thus  Bhnskara  discuKHen  aume  cgneHtiuns  on  the  price  of  alavea, 
and  incidentally  n-mnrkii  that  a  female  slara  waa  generally 
■opposed  to  l>e  most  valuable  when  16  years  oM,  and  anlxe- 
qaently  to  decrease  in  value  in  inviinwi  proportion  to  the  age ; 
for  instance,  if  when  IG  years  old  she  were  worth  33  niakka^ 
her  value  when  20  would  be  rcpreeentMl  by  (16  m  32)-;- SO 
nishkas.  It  would  a|>pear  that,  as  a  rough  average,  a  female 
■lave  of  16  was  worth  about  tt  oxen  which  had  worked  for 
two  yean.  The  interest  charged  for  money  in  India  varied 
from  3J  to  5  per  cent,  per  month.  Amongst  other  data  thai 
given  will  Iw  found  the  prices  of  provision*  and  labour. 

The  chapter  termed  Hijn  Ganita  cnniDicneea  with  a  Bent«nce 
ao  ingeniously  fmmc*!  that  it  can  In  read  m  the  ennnciatinn 
of  a  religious,  or  a  {ihiiiFSDphicnl,  or  a  mathematical  truth, 
fihaskara  after  alluding  to  his  Lihimti  or  arithmetic  state*  that 
he  intends  in  this  book  to  proceed  to  the  general  operations  of 
analysis.  The  idea  of  the  notation  is  as  follows.  Abbrevia- 
tions and  initials  are  used  for  symbols ;  subtraction  is  indicatMl 
by  a  dot  placed  above  the  coefficient  of  the  quantity  to  be 
Bubtracted  ;  addition  by  jnxtapO!«ition  merely;  but  no  symbula 
■re  used  for  multiplicatinn.  equality,  or  inequality,  tbeas  being 
written  at  length.  A  product  is  denoted  by  tbe  firat  aylUUe 
o(  the  wotd  anbjoined  to  the  hcton,  between  whioli  •  dot  ia 
wtiinea  placed.     In  a  quotient  or  fFaotioa  Um  divisor  is 


160  THK  If  ATHBIUTICB  or  THB  ARABflL 

written  under  the  dividend  witiibQi  a  line  ol  lepnmtion.  The 
two  udm  of  an  eqimtion  are  written  one  under  the  .other, 
eonfueion  being  prevented  bj  the  recital  in  worde  of  all  the 
stope  which  accompanj  the  ofieration.  Various  qrmbols  lor 
the  unknown  quantity  are  used,  but  most  of  them  are  the 
initials  of  names  of  colours,  and  tlie  word  colour  is  often  used 
as  synonymous  with  unknown  quantity;  its  Sanscrit  equivalent 
also  signifies  a  letter,  and  letters  are  sometimes  used  either 
from  the  slpkaliet  or  from  tlie  initial  syllables  of  subjeets  of 
the  problem.  In  one  or  two  cases  symbols  are  used  lor  the 
given  as  well  as  for  tlie  unknown  quantities.  The  initials  of 
the  words  squiire  and  solid  denote  the  second  and  third  powers, 
and  the  initial  syllable  of  square  root  marks  a  surd.  Poly- 
nomials are  arranged  in  powers,  the  absolute  quantity  being 
always  placed  last  and  distinguished  by  an  initial  syllable  de- 
noting kmiwn  quantity.  Most  of  the  equations  have  numerical 
coerticieiits,  and  the  oueflicient  is  always  written  after  the  un- 
known quantity.  Positive  or  negzitive  terms  are  indiscrinii- 
nately  allowed  to  come  first ;  and  every  power  is  repeated  on 
both  sides  of  an  equation,  with  a  sero  for  the  coefficient  when 
the  term  is  aljsent.  After  explaining  his  notation,  Bhaskara 
goes  on  to  give  the  rules  for  addition,  subtraction,  multiplica- 
tion, division,  squaring,  and  extracting  the  square  root  of  alge- 
braical expressions ;  he  then  gives  th^  rules  of  ciplier  as  in  the 
LHavHUi;  solv«*s  a  few  e<|uations;  and  lastly  concludes  with 
some  operations  on  surds.  Many  of  the  problems  are  given  in 
a  poetical  setting  with  allusions  to  fair  damsels  and  gallant 
warriors. 

Fragments  of  other  chapters,  involving  algebra,  trigono- 
metry, and  geometrical  applications,  have  been  translated  by 
Colebrooke.  Amongst  the  trigonometrical  formuhe  is  one 
which  is  equivalent  to  the  equation  c/(sintf)  soos^c/tf. 

1  have  departed  from  the  chronological  order  in  treating 
here  of  Bhaskani,  but  I  thought  it  better  to  mention  him  at 
tlie  same  time  as  I  was  discussing  his  compatriots.  It  must 
be  remembered  however  tliat  he  fiourished  subsequently  to  all 


\ 


[ 


THE  HATIIEMATIca  OF  THB  ABABa.  161 

the  Arab  mnthemnticinna  consJHpred  in  thn  next  section. 
Tlie    works   with    which    the   Amlm    firxt    lieeame    ftCr)u«int«d 

a  thofle  of  Ary&-Bhnbt  %nd  ilmh)nn;;upt*,  and  pprlMpa  of 
their  sDccesHon  SHdhnra  unA  Piu]iiinnal<h» ;  it  i*  doalitful  if 
they  ever  initrin  much  a^n  of  the  ^tv:it  trentiw)  of  Bhniikiira. 

It  ia  )>m)vihle  thnt  the  iittention  of  th^  Antlw  wm  eftl)«d 
to  the  workn  of  (he  tint  two  uf  thrKc  wntj-ni  Iry  the  het  tliitt 
the  Arnlm  mloplcrf  thp  Indinn  nysti-ni  i)(  nritlinictic.  ani)  wi-re 
thuR  \n\  btliMik  nt  th<' niiiDiniiAtii^nl  li'xt-lNmkHftf  the  llindnnn, 
Tiie  AmiM  hnd  kiwnys  hnil  vonHiilmihhi  ciiinmeret*  with  ItidtK, 
and  with  the  eHtaliliiihnipnt  of  their  empire  the  AnMHint  of  trMl« 
nitturally  increMncd ;  nt  that  time,  ftliout  tlie  year  700,  they 
found  the  Hindoo  nierclmnbi  liej^innin;;  tn  nun  the  i<}'Ktem  of 
nameration  with  which  we  are  fnniiliitr  nnd  Hd'ipteil  it  at  once. 
.  This  immediate  acceptance  of  it  wnn  mnde  the  eanier  m»  they 
had  DO  works  of  science  or  literature  in  wliic-h  attother  Byntcni 
was  used,  and  it  is  doubtfal  whether  they  then  powcaaed  any 
but  the  most  primilivB  system  of  notation  for  expressing  num- 
ber*. The  Aralm  (like  the  Hindrxn)  seem  ^m>  to  have  niade 
little  or  no  utte  of  the  alncu^  nnd  thervfiirc  munt  have  6rand 
Greek  and  Koman  metliiwls  of  cj»U'uhlion  e«treinely  lalMNioim. 
The  earliest  definite  dato  SHsif[n«d  for  the  lue  in  Arabia  nt 
the  decimnl  aystem  of  numeration  in  77.1.  In  that  year  eome 
IfttKan  a.-!lHiT» in liciir tallies  were  brought  to  Bagdad,  and  it  is 
almost  certain  that  in  these  Indian  numerals  (including  a  lero) 
were  employed. 

The  devel'tpmeJit  of  mnlhemntiai  i'»  Arahia\ 
In  the  preceding  sectionn  of  thitt  chapter  I  hare  imliested 
the  two  sources  from  which  the  AralM  derived  tlieit  knowledge 
of  piathematioB,  and  have  sketched  out  niughly  tbn  amount  ot 
knowledge  obtained  frvm  each.  We  may  sum  the  matter  np 
by  nying  that  before  the  end  of  the  eighth  ceiitriiy  the  Arabs 
■  A  wofit  by  B.  Bsldi  on  the  litra  ot  wTersI  el  ths  Arab  MSlha- 
■atidsBS  was  printed  in  Booaompagai's  BulUHnm  4i  Wt>«p«||U,  18?!; 

a.  II 


162       THK  MATHKMATICS  OF  THE  ARAHa 

wera  in  poMf— ion  of  a  good  nmnoriGAl  noUtioa  Mid  of 
Brmhiiuigupta*8  work  on  arithmeiie  and  Algebra ;  while  beCura 
the  end  of  the  uinth  centuij  thf^y  were  apqnainled  with  the 
masterpieces  of  Greek  mathematica  in  geometry,  merhanics^ 
and  astronomy.  I  have  now  to  explain  what  ose  they  made 
of  these  materials. 

AlknriamL  The  first  and  in  some  respects  the  most  iUna- 
trious  uf  the  Arabiau  mathematicians  was  MukntMHed  Urn 
Jiusa  Abu  DJr/ar  Ai-KkwdrizmL  Tliere  is  no  common  agree- 
ment as  to  whidi  of  these  names  is  tlie  one  by  which  he  is  to 
be  known:  the  lant  of  theui  refers  to  the  place  where  he  was 
bom,  or  in  connection  with  wliich  he  was  best  known,  and  I 
am  told  that  it  is  the  one  by  which  he  would  have  been 
usually  known  among  his  contemporaries.  I  shall  therefore 
refer  to  him  by  that  name ;  and  shall  also  generally  adopt  the 
corresponding  titles  to  designate  the  other  Arabian  mathema- 
ticians. Until  recently  this  was  almost  always  written  in  the 
corrupt  form  Alkarimui^  and,  though  this  way  of  spelling  it  is 
incorrect,  it  has  been  sanctioned  by  so  many  writers  that  I 
shall  make  use  of  it 

We  know  nothing  of  Alkarismi's  life  except  that  he  was  a 
native  of  KhoraMsan  and  librarian  of  the  caliph  Al  Mamun ; 
and  that  lie  accoin|»unied  a  niiHsion  to  Afglianistan,  and  possibly 
came  back  tliruugh  India.  On  his  return,  about  830,  he  wrote 
an  algebra*  which  is  founded  on  that  of  Brahmagupta,  but  in 
which  some  of  the  |»nx>£s  rest  on  the  Greek  method  of  repre- 
senting numbers  by  lines.  He  also  wrote  a  treatise  on  arith- 
metic:  an  anonymous  tract  termed  Algoriimi  t)e  Xuttiero 
Intloruui^  which  is  in  the  university  library  at  Cambridge,  is 
believed  to  be  a  Latin  translation  of  this  treatise  t.  fiesides 
these  two  works  he  compiled  some  astronomical  tables,  with 
explanatory  remarks ;  these  included  results  taken  from  both 
Ptolemy  and  Bralimogupta. 

*  II  was  publifthed  by  F.  lUMen,  vith  sn  EngUah  tnuulatiou,  Loudoa. 
1831. 

t  It  was  publiithed  bjr  B.  Boacompaspi,  liooie,  1857. 


ALKARIRMI.  16A 

The  ftl^lm  of  Alkftrixmi  hnldn  »  moot  importent  place  in 
thn  history  of  mnthpmnticB,  fur  we  mny  «iy  thi»t  the  MDhee- 
qaent  Amiiian  nnd  the  enrly  rrifdievnl  workH  on  ttlgebrs  were 
foQnded  nn  it,  and  nl«>  tlint  tliroiigb  it  the  Arabic  or  IndiMt 
ij'steni  of  decimnl  nurniTAtirin  wn«  intmduced  into  the  Wot. 
The  work  in  termed  Al-grhr  tc'  I  mtitahafn:  al-yir,  from 
which  the  won)  nlgehm  in  drrivpti,  mny  be  transhited  hf 
Ih'  yrnlonilinn  And  rcfrm  tit  the  fact  that  any  the  iame  magni- 
tude may  l)p  ad<I<xl  to  or  Kulitrai-tt^l  from  both  aiden  of  an 
erjDAtion  ;  nt  muknbnln  mrnnn  tlie  pmcesn  of  HimpliKcatinn 
and  ia  grntrnlly  umhI  in  connection  with  the  comlnnation  of 
like  tcmin  into  a  nin|;Ie  term.  Tlie  unkmiwn  quantity  is 
termed  either  "  the  thing  "  or  "  thp  mot "  (that  iit,  of  a  plant) 
and  from  the  latter  phmse  our  nse  of  the  word  root  as  applied 
to  the  solution  of  an  equation  ia  derired.  The  aqnare  of  the 
unknown  is  called  "the  power."  All  the  known  quantitiea 
are  numbers. 

The  work  is  divided  into  five  parts.  In  the  fimt  Alkariuni 
give^  without  any  proofs,  rules  for  the  nnlatinn  of  qnndratic 
equations,  divided  into  nix  classes  of  the  forma  lu' =  &E,  ni^  —  e, 
bz-  e,  a^  *  hx  =  e,  tuf  +  c^hr,  and  •w'  =  t>x*t,  where  a,  fr,  e 
are  positive  numberK,  and  in  all  the  applications  <i  =  I.  He 
eoDsidpni  only  real  and  pmitive  mots,  but  he  recogniaea  th« 
existence  of  two  root«,  which  tm  far  ns  we  know  waa  never 
-done  by  the  Greeks.  It  is  Bumewhat  curious  tlmt  when  both 
roots  are  positive  he  generally  lakes  only  that  root  which 
is  derived  from  the  negative  value  of  the  radical. 

He  next  give*  geometrical  proofs  of  these  rules  in  a 
manner  analogous  to  that  of  Buclid  ii,  4.  For  example^  to 
mdrv  the  equation  «*  +  I  Ox  =  39,  or  any  equation  of  the  form 
)f  +  px  =  q,  he  gives  two  methods  of  which  'one  is  aa  Eollowa. 
Let  AB  represent  the  value  of  x,  and  construct  on  it  the 
square  ABCt)  (see  figure  on  next  page).  Produce  OA  to  // 
vtA  DCUt  F  »a  %\M.  All  =  Cy=U  (or  \p)\  and  eomplete  the 
figare  >•  drawn  below.  Then  the  areas  AC,  BB,  and  BF 
represent  the  magnitadea  «*,  ix,  and  6x.  Tfaoe  the  leftJiaad 
11— S 


164 


TBI  MATHBMATIGB  OT  THE  4RABB. 


aide  of  the  equatton  is  reprwmnted  by  the  mini  ol  the  m«m  AC^ 
y/i9,  and  if/;  that  it,  by  the  gnomon  ifC<7.|  To  both  aides  of 
the  equation  add  the  aquaro  KO^  the  area  of  which  ia  ^5  (or 
l/i^  and  we  shall  get  a  new  square  whusp»  area  is  fay  hypo> 
thesis  equal  to  39  +  25,  that  is»  to  64  (or  i -«*  |  |i^  and  whose 

H  A 0 


¥ 


side  theroforo  U  8.  The  side  of  this  square  DH  whieh  is 
equal  to  8  will  exceed  Ali  which  is  equal  to  5  by  the  valne  of 
the  unknown  required,  which  therefore  is  3.1 

In  the  third  part  of  the  book  Alkariimi  eonsidera  the 
product  of  (x^a)  and  (x  ^  h).  In  the  foukh  part  he  states 
the  rules  for  addition  and  subtraction  of  IsxpresBions  whieh 
invoh'e  the  unknown,  its  square,  or  its  squara  root;  gives  mlea 
for  the  calculation  cif  square  routs ;  and  concludes  with  the 

theorems  that  ajb »  J^  and  Ja  Jb  =  J^  In  the  Afth 
and  Isiit  part  he  given  suiue  prubleuis,  such,  !for  example^  as  to 
find  two  numbers  wliuse  sum  is  10  and  the  dilTerenoe  of  whose 
squares  is  40. 

In  all  tlieiie  early  works  there  is  no  clear  distinction  between 
arithmetic  and  algebra,  and  we  tind  the  account  and  explana- 
tion of  arithmetical  processes  mixed  up  with  algelHraand  treated 
as  part  of  it.  It  was  from  this  book  then  that  the  Italians 
first  obtained  not  only  the  ideas  of  algebra  but  also  of  an  arith- 
metic founded  on  the  decimal  system.     This  arithmetic 


TABIT  IBN   KORRA.      ALKATAMI.  165 

long  known  as  aJfgoritm^  or  the  art  of  AlkariMmi,  which  served 
to  distingaish  it  from  the  arithmetic  of  Boethius ;  this  name 
remained  in  use  till  the  eighteenth  century. 

Tabit  ibn  Korra.  The  work  commenced  by  Alkarisnii 
was  carried  on  by  Tahit  ihn  Korra^  bom  at  Harran  in  836  and 
died  in  901,  who  was  one  of  the  most  brilliant  and  accom- 
plished scholars  produced  by  the  Arabs.  As  I  have  already 
stated*  he  issued  translations  of  the  chief  works  of  Euclid, 
Apollonias,  Archiinedes,~af'id  Ptolemy.  He  also  wrote  several 
original  works,  all  of  which  are  lost  with  the  exception  of  a 
fragment  on  algi^bra,  consisting  of  one  chapter  on  cubic  equa- 
tions, which  are  solved  by  the  aid  of  geometry  in  somewhat 
the  same  way  as  that  given  later  *. 

Algel>ra  continued  to  de%'elope  very  rapidly,  but  it  re- 
mained entirely  rhetoricaL  The  problems  with  which  the 
Aralw  were  concerned  were  either  the  wilution  of  equations, 
problems  leading  to  equations,  or  properties  of  numliers.  Tlie 
two  most  prominent  algebraists  of  a  later  date  were  Alkayami 
and  Alkarki,  both  of  whom  flourished  at  the  beginning 
of  the  eleventh  century. 

Alkayami.  The  firnt  of  these,  Omnr  Alknyami^  is  notice- 
able for  his  geometrical  treatment  of  cubic  equations  by  which 
be  obtained  a  root  as  tho  abscissa  of  a  point  <»f  intersection 
of  a  conic  and  a  circlet.  The  equations  he  considers  are  of 
the  following  foniis,  where  a  and  e  stand  for  positive  integers, 
(i)  a^  -f  d^x  -  6V.  whose  root  he  sa^-s  is  the  aliscissa  of  a  point 
of  intersection  of  a^  =  6y  and  y*  =  «  (c  -  x) ;  (ii)  a^  -»•  oac"  -  c*, 
whose  root  he  says  is  the  abscissa  of  a  point  of  intersection 
of  «y=fc"  and  y*  -  r  (x  +  #i)  ;  (iii)  x^  *  ojb^ -»•  ^*«  -  **«,  whose 
root  he  says  is  the  abscissa  of  a  point  of  intersection  of 
^= (x  A  a)  (e  -«)  and  x(6  ^  y)  =  6e.  He  givtt  one  biquadratic, 
namely,  (100  -  x*)  (10  -  x)'  s  8100,  the  root  of  which  is  deter- 
mined by  the  point  of  intersection  of  (10-x)ys1Hf  and 
c*  -f  y*  a  100.    It  is  sometimes  said  that  he  stated  that  it  was 

*  8m  btlow«  p.  SSI. 

t  His  tmlin  cm  alftbffa  WM  paUislied  by  Fr.  WospslM.  FlMisp  IMl. 


166  THK  mathhutigb  or  the  araml 

impossible  to  aolye  the  ^luitioii  a^ -»•  y*  «  j^  in  pontiv^  iaWfH% 
or  in  oilier  words  that  the  sooi  of  two  enbes  enn  never  be 
neube;  though  whether  he  gave  nn  eccumte  prool^  or  whether, 
ae  is  more  likely,  the  proposition  (if  enanciated  at  all)  was  the 
result  of  a  wide  induction,  it  is  now  impossible  to  say;  hot 
the  fact  that  such  a  theorem  is  attributed  to  him  will  serve  to 
illustrate  the  extraordinary  prugress  the  Arabs  had  made  in 
algebra. 

AlkarkL  The  other  mathematician  of  this  time  (ctre. 
1000)  wImhu  I  mentioned  was  Aikarki*.  He  gave  expressions 
for  the  sums  of  the  first,  second,  aiid  tliini  powers  of  tlie  first 
n  natural  numbers ;  solved  various  equations,  including  some 
of  the  forms  au^  *  ix^  *  c  s  Q ;  and  discussed  surds,  shewing^ 

for  example,  that  \^8  -i-  v08  »  ^50. 

Even  where  the  methods  of  Arab  algebra  are  quite  general 
the  applicatiooH  are  confined  in  all  cases  to  numerical  problems, 
and  the  algebra  is  so  arithmetical  that  it  is  difficult  to  treat  the 
ubjects  apart.  From  their  books  on  arithmetic  and  from  the 
observations  scattered  through  various  works  on  algebra  we 
may  say  that  the  mctduds  used  by  the  Arabs  for  the  four 
fundamental  processes. were  analogous  to,  though  more  cum- 
brous tlian,  those  now  in  use ;  but  the  probleuis  to  which  the 
subject  was  applied  were  similar  to  those  given  in  modem 
books,  and  were  solved  by  similar  methods,  such  as  rule  ol 
three,  ^c.  Some  mimir  improvements  in  notation  were  intro> 
duoed,  such,  for  instance,  as  the  introduction  of  a  line  to 
separate  the  numc^rator  from  the  denominator  of  a  fraction ; 
and  hence  a  line  between  two  symUils  came  to  be  used  as  a 
'  symlM>i  of  diviHiont.  Alhosseiu  (980-1037)  used  a  rule  for 
testing  the  correctness  of  the  results  of  addition  and  multi- 
plication  by  '*  casting  out  the  niues.*'  Various  forms  of  this 
rule  have  been'given,'~T>ut'  they  all  depend  on  the  proposition 
that,  if  each  number  in  the  question  be  replaced  by  tlie  re- 

*  UU  MlKebrs  wsi  publibh«d  ky  Fr.  Woepcke,  1S53,  aud  bin  arithmstis 
was  trmnitlsted  into  German  by  Ad.  Uocbheim,  UsUe,  1878. 
t  See  below,  p.  347. 


THB  MATHEMATICR  OV  THB  ARABS. 


167 


mainder  when  it  is  divided  by  9,  lind  if  these  remainders  be 
added  or  maltiplied  as  directed  in  the  question,  then  this 
result  when  di\'ided  bj  9  will  leave  the  same  remainder  as  the 
answer  whose  correctness  it  is  desired  to  test  when  divided 
faj  9 :  if  these  remainders  difTer,  there  is  an  error.  The 
■election  of  9  as  a  divisor  was  due  to  the  fact  that  the  remain- 
der when  a  number  is  diviiled  by  9  can  be  obtained  by  adding 
the  digits  of  the  number  and  dividing  the  sum  by  9. 

I  am  not  concerned  with  the  Arabian  views  of  astronomy  or 
the  value  of  their  observations,  but  I  may  remark  in  passing 
that  the  Arabs  accepted  the  theory  as  laid  down  by  Hippar- 
ehus  and  Ptolemy,  and  did  not  materially  alter  or  advance  it. 
I  may  however  add  that  Al  Mumun  caused  tlie  length  of  a 
degree  of  latitude  to  lie  measured,  and  he,  as  well  as  the  two 
mathematicians  to  be  next  named,  determined  the  obliquity  of 
the  ecliptic 

AlbtttegnL  Albn^aai.  Like  the  Greeks,  the  Arabs 
rarely,  if  ever,  employed  trigonometry  except  in  connection 
with  astronomy:  but  they  used  the  trigonomotrical  ratios 
which  are  now  currenr,  and  worked  out  the  plane  trigonometry 
of  a  single  angle.  They  were  also  acquainted  with  the 
elements  of  spherical  trigonometry.  Al^tai^ni^  bom  at 
Batan  in  Mesopotamia  in  877  and  died  at  Dagrlad  in  929, 
was  among  the  earliest  of  the  many  distinguished  Arabian 
astronomers.  He  wrote  the  iScUtwe  of  the  i^tam^^  which  is 
worthy  of  note  from  its  containing  the  discovery  of  the 
motion  of  the  sun's  api>gee.  In  this  work  angles  are  de- 
termined liy  *'  the  seniTchonl  of  twice  the  angle,"  that  is,  by 
the  sine  of  the  angle  (taking  the  radius  vector  as  unity).  It 
la  doubtful  whether  he  wiss  acrquainted  with  the  previous 
introduction  of  sines  by  Arya-Bhata  and  Brahmagupta; 
Giipiiarchut  and  Ptolemy,  it  will  be  remembered,  had  used 
the  chord.  AlbatQgni  was  also  acquainted  with  the  funda- 
mental formula  in  spherical  trigonometry  giving  the  side  of 
a  triangle  in  terms  of  the  other  sides  and  the  angle  induded 
*  It  was  sililsd  h$  Hsgkisiontsnns,  Moiwikeig,  1M7* 


168  THE  MATHSXATIGB  OF  TUK  ARAHa 

by  Umul  BborUy  after  the  death  of  Albatcgni,  AUmigmmt 
who  is  alHO  known  as  Abmi-Wt^a^  bom  in  940  and  di^  in 
99^  introdooed  certain  trigonometrical  function^  and  eon- 
structed  tables  of  tangents  and  cotangents.  He  was  celebrated 
not  oidy  as  an  astronomer — being  the  discoverer  of  the  moon's 
variation — but  as  one  of  the  most  distinguished  geometricians 
of  his  time. 

Alhasen.  Abdnd-gehL  The  Arabs  were  at  first  content 
to  take  tlie  works  of  Euclid  and  Apolkmius  for  their  text- 
books  in  geometry  without  attempting  to  conunont  on  them, 
but  Alhazen^  bom  at  BosKora  in  987  and  died  at  Cairo  in 
1038,  issued. in  1036  a  collection*  of  problems  something  like 
tlie  DiiUi  uf  Kucliil.  l^*Mides  couimentaries  on  the  definitions 
of  Euclid  and  on  the  Ahwujest  Alliazen  alno  wrote  a  work  on 
optics  t,  which  includes  the  earliest  scientific  account  of  atmo- 
spheric refniction.  It  aliio  contains  souie  ingenious  geometry, 
amongst  other  things,  a  geometricjd  solution  of  the  problem  to 
find  at  what  point  of  a  concave  mirror  a  ray  from  a  given 
point  must  be  incident  so  as  to  lie  reflected  to  another  given 
point  Another  geometrician  of  a  slightly  later  date  was 
Abd-algrhl  (circ.  1100)  who  wrote  on  conic  sections,  and  was 
also  the  author  of  throe  small  geometrical  tracts. 

It  was  shortly  after  the  last  of  the  mathematicisns  mentioned 
above  tliat  liliaskara,  the  third  great  Hindoo  mathematician, 
flourished :  there  is  every  reason  to  believe  that  he  was  familiar 
with  the  works  of  the  Arab  school  as  described  above,  and  also 
that  his  writings  were  at  once  known  in  Arabia. 

The  Arab  scIuniIs  ctmtinued  to  flourish  until  the  flfteenth 
century.  But  they  produced  no  other  mathematician  of  any 
except itinal  genius,  nor  was  there  any  great  advance  on  the 
methods  indicaUnl  above,  and  it  is  unnecessary  for  me  to  crowd 
my  pages  with  the  names  of  a  number  of  writers  who  did  not 
materially  aflect  the  progress  of  the  science  in  Europe. 

From  this  rapid  sketch  it  will  be  seen  that  the  work  of  the 

•  It  WM  trsnHlstea  lijr  L.  A.  8<kliUot,  and  pobliabwl  st  Paris  hi  188t. 
t  It  wsa  publubed  st  aolt  in  1572. 


THE  MATHEMATICS  OF  THE  ARABS. 


169 


Arabs  (inclading  therein  writers  who  wrote  in  Arabia  and 
lived  ander  Eastern  Mohammedan  rule)  in  arithmetic,  algebra, 
and  trigonometry  was  of  a  high  order  of  excellence.  Thej 
appreciated  geometry  and  the  applications  of  geometry  to 
astronomy,  bat  they  did  not  extend  the  lioands  of  the  science. 
It  may  be  also  added  that  they  made  no  special  progress. in 
statics,  or  optics,  or  hydrostatics;  though  there  is  abnndant 
evidence  that  they  had  a  thorough  knowledge  of  practical 
hydraulics. 

The  general  impression  left  is  that  the  Aralis  were  quick 
to  appreciate  tlie  work  of  others — notably  of  the  Greek  mastere 
and  of  the  Hindoo  mathematicians — Init,  like  the  ancient 
Chinese  and  Egyptians,  they  were  unable  to  systematically 
develope  a  subject  to  any  considerable  extent  Their  schools 
may  be  taken  to  have  lasted  in  all  for  about  650  years,  and  if 
the  work  produead  be  compared  with  that  of  Orec^  or  modem 
European  wrile^  it  is^  as  a  whoK  leoond-rale  both  in  <|uantity 
and  quality. 


S«p.^- 


A 


.  f 


%..•  • 


C'f^^^ 


►y. 


170 


CHAPTKR  X 

TBI  INTttOOUCriOH  OK  AHABIAV  WOEKII  IMTO  KUBOFB. 

CIBC   1160-1450 

III  the  last  diApter  but  one  I  diaconed  the  developaant  of 
Boropean  mathematics  to  a  date  which  oorrMponda  itNigUy 
with  the  end  of  the  **darfc  ages";  and  in  the  bat  chapter 
I  traced  the  history  of  the  mathematics  of  the  Hindoos  and 
Arabs  to  tlie  name  date.  The  mathematics  of  the  two  or 
three  centuries  that  follow  and  are  treated  in  this  chapter  are 
characterised  by  tlie  introduction  of  the  Arabian  mathematical 
text-books  and  of  Greek  books  derived  from  Arabian  souroesi 
and  tlie  assimilation  of  the  new  ideas  thus  pi-esented. 

It  was  however  from  Spain,  and  nut  frum  Arabia,  that 
Arabian  niatheiuatics  tintt  came  into  western  Europe.  The 
Muurs  had  established  their  rule  in  Spain  in  747,  and  by  the 
tenth  or  eleventh  century  hud  attained  a  high  degree  of  civili- 
sation. Though  their  political  relations 'with  the  caliphs  at 
Bagdiid  were  liomewhat  unfriendly,  they  gave  a  ready  welcome 
to  the  works  of  the  great  Arabian  mathematicians.  In  this 
way  the  Arab  translations  of  the  writings  of  Euclid,  Archi- 
medes, Apollonius,  Ptolemy,  and  perhaps  of  other  Greek 
authoi-s,  together  with  the  works  of  the  Arabian  algebraists, 
wera  read  and  commented  on  at  the  three  great  Moorish 
schools  of  Graimda,  Cordova,  and  Seville.  It  seems  probable 
that  these  works  indicate  the  full  extent  of  Moorish  learning, 


THE  ELEVENTH   AND  TWELFTH  CEMTUBIES.         171 

bat,  u  all  knowledge  wu  jenlounly  gunnled  from  ChriBtiMis, 
it  ia  impossilile  to  RprAk  with  oMtAinly  either  on  this  point  or 
on  tl)At  of  the  time  when  the  Arab  bouks  were  fint  introdaced 
into  B)Niin. 

The  elflVentli  century.  The  pnrliest  Mooriah  writer  at 
tlintioction  of  whum  I  fiiiil  mention  is  Oeber  Ibn  Aphla, 
who  wiu  1x>rn  at  Seville  and  i\ieil  tuwnnlH  tlw  lat(«r  part  of 
the  eleventh  century  at  Cordova.  Hn  wrot«  on  Mtrooomj 
and  trigonninetrj-,  iinil  was  ac<|unmted  with  the  theorem  tliat 
tho  nines  of  the  anglen  of  a  Hphericnl  triangle  are  proportional 
to  the  sines  of  the  opposite. bides*. 

Arzachel|.  Atinthcr  Arab  of  nlmut  the  same  dat«  waa 
Anrtehd,  who  was  living  at  Tuledo  in  lOtiO.  He  angf^esled 
that  the  planets  moved  in  elli[RtcH,  but  hi«  contemjiormnes  with 
Micntifio  iMtolerHn(.-e  declined  to  argue  aliont  a  statement 
which  wa«  oontrar^  to  Ploleniy's  conclusion  in  the  Atttwyt. 

Tbt  twaldb  ctntory-    Uunng  the  wurw)  of  Uw  (weKMi 

Mnlury  M|>)m  vt  Ui»  IhniIir  uml  in  Nptin  wtr*  obtaiiwd  In 
«Mt«m  UhriBt«nduiii.    The  flrnt  nu-p  lo«ardi  pmonrini  ■ 

knowledge  of  Arab  and  Moorish  Mience  waa  taken  bjr  ui 
English  monk,  Adelliard  of  Bath^,  who,  under  the  disgniae 
of  a  Mohamnit'dnn  student,  attended  some  lectnres  at  Coidova 
about  112U  and  obtained  a  copy  of  Kuclld's  fikmeiU*.  Thia 
copy,  translated  into  LAtio,  wu  the  foundation  ot  all  the  edi- 
tions known  in  Europe  till  ld3.'l,  when  the  Grack  text  was 
recovered.  How  rapicUy  a  knowledge  of  the  work  apread  we 
may  judge  when  we  recollect  thut  liefore  the  end  of  the  thir- 
teenth eentarj  Koger  Bacon  was  familiar  with  it,  while  before 
the  close  of  tbe  foarteentli  century  the  tirat'Gve  booka  formed 

■  ilia  irorka  *crc  Iranitslfd  into  Latin  hj  0«f«ni  aad  pubUibid  at 
Naremberii  >>■  1^^- 

t  ti««  a  memoii  bj  U.  HtcioKibiwiihr  id  Bmnampeyira  BMttimm 
di  BitUogntfl*,  KMT,  voL  ii. 

t  On  111*  inflonioa  o[  Adelhard  and  Ben  Ecia,  at*  tha  JUmmdImmgin 
imr  OatkithU  rfcr  Mtktmtlik  ia  Um  ZiitMtht^  J*r  Jf«t4twrt>,  voL 


172   iMTBODUonoii  or  ababiaw  womu  mo  mjiopb. 


part  of  tint  regiiUur  oarricolim  at  tonie^  if  not  aU*  mxd} 
'the  enuucUtioiui  of  Euclid  iiaeni  to  have  beeo  knowa  beiova 
Addhard*!!  time,  and  puMiUy  at  enrly  at  the  jear  1000^ 
tliough  copies  were  rara  Adelhaid  also  procured  a  mana* 
script  of  or  commentary  on  AJkarismi's  work,  which  ha  liko> 
wise  translated  into  L^tin.  He  also  issued  a  text- book  on  the 
use  of  the  abacus. 

Bon  Ezra*.  During  the  same  century  other  translatioot 
of  the  Arab  text-liooks  or  commentaries  on  them  were 
obtained.  Amongst  those  who  were  most  influential  in  intru* 
ducing  Moorish  learning  into  Europe  I  may  mention  Abrakam 
Ben  Ezra.  Ben  Ezra  was  bom  at  Toledo  in  1097,  and  died 
at  Itome  in  1167.  He  was  one  of  the  OMJst  distinguished 
Jewish  rabbis  who  had  settled  in  Spain,  where  it  must  be 
nx-ulk-cted  that  they  were  tolerated  and  e%'en  protected  by 
the  MoorM  on  account  of  their  medical  skill.  Besides  soma 
astronomical  tables  and  au  astrology,  Ben  Esra  wrote  an 
arithmetic  t;  in  this  he  explains  the  Arab  system  of  numera- 
tion with  nine  symbols  and  a  zero,  gives  the  fundamental 
proceMses  of  arithmetic,  and  explains  the  rule  of  three. 

Gerard  I .  Another  European  who  was  induced  by  the 
reputation  of  the  Arab  schools  to  go  to  Toledo  was  Gerard^ 
who  was  born  at  Cremona  in  1114  and  died  in  1187.  He 
translat«fd  the  Arab  edition  of  the  AlmagtMi^  the  works  of 
Alhazen,  and  the  works  of  Alfarabius  whose  name  is  other- 
wise unknown  to  us :  it  is  believed  that  the  Arabic  numerals 
were  used  in  this  translation,  made  in  1136,  of  Ptolemy's 
work.  (Jerard  aliio  wrote  a  short  treatise  on  algorism  which 
exists  in  munuRcript  in  the  Ikxileian  Library  at  OxfonL  He 
was  act|uainted  with  one  of  the  Arab  editions  of  Euclid's 
Eleinenitf  which  he  translated  into  Latin. 

*  8m  footnote  t  on  P*  171. 

t  Au  siislysit  of  it  was  published  by  O.  Terqnsm  in  LiooviUs's 
Jourmal  for  1841. 

I  See  Bonoompagiii's  DtUm  viim  e  delU  open  di  Oktrmrdo  Ct 
Boiue,  lb51. 


LEONARDa  173 

John  Hispalensis.  Amnnj;  the  contemporsrieii  of  G^nrd 
wu  John  //iipattn»iii  of  Seville,  ori;n>in"7  ■  nbH  bat  oon- 
Terted  to  Christianity  And  liaptizrd  ondcr  the  iMune  giiren 
ftbove.  Hp  mnde  tranalAtionii  of  HeremI  Anh  and  lloorish 
woriia,  And  also  wrote  an  algorism  which  cnDt«ina  the  mrliert 
exunplen  of  the  extmclton  tit  tlte  Hqunre  roots  of  nnmben  by 
the  sid  of  the  decimni  nutntion. 

The  thirteenth  century.  Durinf;  the  thirteenth  centnrj 
there  wm  a  revival  <if  IpAmin;;  tbroujjIiDut  Europe,  but  the 
new  learning  whs  I  lielievc  confined  to  «  ver/  limited  clam. 
The  early  yenni  of  this  century  are  mcmoisble  for  the  de- 
velopment of  several  aniverBitien,  and  for  the  appearance  of 
tbrm  remarkable  mathemBticiani — Lronardoof  Pisa,  Jordonni, 
and  Roger  Bacon  the  FranciRcan  monk  of  Oxford. 

LeonBirdo*.  Lfonanio  Fihonneei  (i.e  filios  Bonaccii)  gene- 
rally known  a.1  Ltimnrdo  nf  Piaa,  wax  l>am  at  Piu  in  1179. 
His  father  Bonacci  was  a  merchant,  and  wu  Mrat  by  his 
fellow- towns  men  to  control  the  cuittom'hoiMe  at  Bngia  in 
Barbary ;  there  I>>onHnlo  was  eilucalnl.  and  be  thas  became 
acquainted  witli  the  Aratiic  system  of  naowratton  as  aim 
with  Alkarismi't  work  on  sl;;p)im  which  was  descrilied  in 
the  lant  chapter.  It  would  seem  thnt  Leonardo  was  entnuited 
with  some  duties  in  connection  with  the  cnstom'hoase,  which 
nqnired  him  to  travel.  He  retumetl  to  Italy  fthont  1200, 
and  in  1202  published  a  work  calletl  Atij^rti  H  afmnrhabala 
(the  title  being  taken  from  Alkarismi's  work)  but  generwlly 
known  as  the  tiW  Al<aei.  He  there  explains  the  Arabic 
system  of  numeration,  and  remarks  on  its  great  ad\-Knlagea 
over  the  Roman  system.  He  then  gives  an  accoanl  ot  algebra, 
and  points  out  the  convenience  of  using  geometiy  to  get  rigid 
denionatrationa   of   algebraical    formulae.     He  ahews  how  to 

*  8n  UiB  Lfbrn  umd  Stkri/ini  /.AnurrfM  ifa  PIm  hf  3.  OisnBg, 
Diibeb),  1886:  and  Csnior,  cfaapa  lu.  xui;  m«  also  two  arlielw  %j 
St.  Wnpeka  in  Un  AUi  dilC  Arademia  ponlifieim  <K  aaM-i  Umxl  for 
IMI.  TOl.  uv.  pp.  S4)— 148.  KntI  ol  )>a>iardo**  writinfia  wm«  aUlal 
tmi  pnblubad  li;  B.  BonoomiiagBi  bvtvMB  lb«  yt^n  IIH  aad  IML 


174  mmoDUcnoii  or  aeabun  wobks  nrro  xubor. 


flolve  tiniple  eqiiatioii%  aolvn  m  faw  quadratic  cgqation^  and 
states  some  metliods  for  the  solation  of  indeteffminate  eqii»> 
tioos;  these  rales  are  illustrated  hy  prohlems  oa  namheriL 
All  the  algebra  is  riietorical,  and  in  one  ease  letters  are 
employed  as  algebraical  symbols.  This  work  had  a  wide  eiro4- 
lation,  and  for  at  least  two  centuries  remained  a  standard 
authority  fruiu  which  numerous  writers  drew  their  inspiration. 

The  Liber  Abttei  is  especially  interesting  in  tlie  history  of 
arithmetic  since  practically  it  introduced  the  use  of  the  Arable 
numerals  into  diristian  Europa  The  language  of  Leonardo 
implies  that  they  were  previously  unknown  to  his  countrymen ; 
he  says  that  having  hud  to  spend  some  years  in  liarbary  he 
there  learnt  tlie  Arabic  system  which  he  found  much  mora 
convenient  than  that  used  in  Europe ;  he  therefore  published 
it  '*in  order  that  the  Latin*  race  might  no  longer  be  deficient 
in  that  knowledge."  Now  Leonardo  had  read  very  widely, 
and  liad  travelled  in  Greece,  Sicily,  and  Italy;  and  there  is 
therefore  every  presumption  that  the  system  wss  not  then  com- 
monly employed  in  Europe. 

Tliougli  Leonardo  introduced  tlie  use  of  Arabic  numerals 
into  coinim*n;iar~aflraini,  it  is  pruliahle  tliat  a  knowledge  of 
them  as  current  in  the  East  was  previously  not  uncommon 
among  travellem  and  merclianUi,  for  the  intercourse  between 
ChriMtiauM  and  Molianimedans.  was  vutticiently  close  for  each 
ti>  leum  something  of  the  language  anil  common  practices  of 
the  other.  We  can  aIho  hanlly  suppose  tliat  the  Italian 
merchants  were  ignorant  of  the  method  of  keeping  accounts 
used  by  scime  of  their  best  customers;  and  w^  must  recollect 
too  that  there  were  numerous  Christians  who  had  escaped 
or  been  ransomed  after  serving  the  Mohammedans  as  slavea^ 
It  was  however  Leonardo  who  brought  the  Arabic  system  into 
general  use,  and  by  the  middle  of  the  thirtelenth  century  a 

*  Dean  Pescock  nays  that  the  earlient  known  sppliiaiiibu  of  tbs  w«iid 
Italians  to  dencrilie  tli«  inliabitanlii  of  liaXy  occurs  about  ths  miildls  oi 
Ihs  thirtettntk  century:  by  the  end  of  that  century  it  wae  in  common 


LEONARDO.  175 

large  proportioh  of  the  Italian  merchants  employed  it  hj  the 
•ide  of  the  old  system. 

The  majority  of  mathematicians  must  have  already  known 
of  the  sjTstem  from  the  works  of  lien  Ezra,  (yerard,  and  John 
Hispalensis.  Bnt  shortly  after  the  appearance  of  lieonanlo's 
book  Alphonso  of  Castile  (in  1252)  published  some  astronomical 
tables,  founded  on  oliservations  made  in  Arabia,  which  were 
computed  by  Arabs  and  which,  it  is  generally  lielieved,  were 
expressed  in  Aralnc  notation.  Alphonso's  tables  had  a  wide 
circulation  among  men  of  science^  and  proliably  were  largely 
iDstrumental  in  bringing  these  numerals  into  universal  use 
among  mathematicians.  By  the  end  of  the  thirteenth  century 
it  was  generally  assumed  that  all  scientific  men  would  be 
acquainted  with  the  system :  thus  Roger  fiacon'  writing  in 
that  century  recommends  the  algorism  (that  is,  the  arithmetic 
loonded  on  the  Arab  notation)  as  a  necessary  study  for  theo- 
logians who  ought  he  says  '*  to  abound  in  the  power  of  num- 
bering."* We  may  then  consider  that  by  the  year  1300,  or  at 
the  latest  1350,  these  numerals  were  familiar  both  to  mathe- 
maticians and  to  Italian  merchants. 

So  great  was  Leonardo's  reputaticm  that  the  emperor 
Frederick  II.  stopped  at  Pisa  in  1 225  in  order  to  hold  a  sort 
of  mathematical  tournament  to  test  ]>onardo*s  skill  of  which 
he  had  heard  such  marvellous  accounts.  The  competitors  were 
informed  beforehand  of  the  questions  to  be  asked,  some  or 
all  of  which  were  composed  by  John  of  Palermo  who  was  one 
of  Frederick's  suite.  This  is  the  first  time  that  we  meet 
with  an  instance  of  tliose  challenges  to  solve  particular  pro- 
blems which  were  so  common  in  the  sixteenth  and  seventeenth 
centuries,  llie  first  question  propounded  was  to  find  a  number 
of  which  the  square^  when  either  increased  or  decreased  bj 
5,  would  remain  a  square  Leonardo  gave  an  answer,  which 
is  correct^  namely  41/12.  The  next  question  was  to  find  by 
the  methods  nsed  in  the  tenth  book  of  Euclid  a  line  whose 
length  m  shoold  satisfy  the  equation  n^  -•-  2je*  4-  lObe  «  30. 
LeooMdo  shewed  by  geometry  thai  the  problem  was  iapoa. 


176   nrrBooucnoN  or  arabiam  wobmb  uno  xubope. 

■iblfl^  but  he  gare  an  approxunate  valiia  of  the  root  ol  this 
eqoatioii,  namely,  l-22'7"  42^33*^4*40^  which  ia  oqoal  to 
1-3688081075....  and  ii  eomct  to  nine  plaoea  of  deeiiaali* 
Another  qiieBtion  was  as  follows.  Three  men  A,  M^  C,  poeiBii 
a  sum  of  money  v,  their  sharee  being  in  the  ratio  3:2:1.  A 
takes  away  «,  keeps  lialf  of  it,  and  deposits  the  remainder  with 
D ;  B  takes  away  y,  keeps  two-thirds  of  it,  and  depoeita  the 
remainder  with  D ;  C  takes  away  all  that  is  left,  namely  s^ 
keeps  five-sixths  of  it,  and  deposits  the  remainder  with  />. 
This  deposit  with  D  is  found  to  belong  to  J,  B^  and  C  in 
equal  proportiona  Find  m,  «;  y,  and  z.  Leonardo  shewed 
that  the  problem  was  indeterminate  and  gave  as  one  solution 
«  -  47,  X  =  33,  y  =  13,  t  =  1.  The  other  competitors  failed  to 
solve  any  of  these  questions. 

The  chief  work  of  Leonardo  is  the  Liber  Abaci  alluded  to 
above.     Tliis  work  contains  a  proof  of  tlie  well-known  result 

(a*  +  6«){c*  +  d*)  =  {ac > bdy  ^  (be ^ ady  ^ {ad  +  bef -^ (bd ^ aef. 

He  also  wrote  a  geometry,  termed  Praeiiea  Geomeiriae^  which 
was  issued  in  1220.  This  is  a  good  compilation  and  soma 
trigonometry  is  introduced ;  among  other  propositions  and 
examples  he  finds  the  area  of  a  triangle  in  terms  of  its  sides. 
Subsequently  he  published  a  Liber  Quadraioram  dealing  with 
problems  simiUr  to  the  first  of  the  questions  propounded  at 
the  tournament  t.  He  also  issued  a  tract  dealing  with  deter- 
minate algebraical  problems :  these  are  all  solved  by  the  rule 
of  false  assumption  in  the  manner  explained  above. 

Frederick  II.  The  emperor  Frederick  //.  wlio  was  bom 
in  1194,  succeeded  to  the  throne  in  1210,  and  died  in  1250, 
was  not  only  interested  i»  science,  but  did  as  much  as  any 
other  single  man  of  the  thirteenth  century  to  diiiseminate  a 
knowledge  of  the  works  of  the  Arab  mathematicians  in  weatem 


*  See  Fr.  Woepcke  in  LiouTiUe'e  Journal  for  1854,  p.  401. 
t  Fr.  Wuepcke  in  LiooTille**  Journal  fur  1855,  p.  54,  has  gifsn  aa 
ansljiis  of  LeonsnloV  method  of  treating  problems  on  square  nombsts. 


FREDERICK  II.  JORDANUH.  177 

Eunipe.  The  uni^'ersity  of  Naplen  remains  as  a  rnonvnient 
of  his  maniflcence.  I  have  already  montioned  that  the 
preiienoe  of  the  Jews  had  been  tolerated  in  Spain  on  aocoant 
of  their  medical  skill  and  scientific  knowledge,  and  as  a 
matter  of  fact  the  titles  of  physician  and  algebraist*  were 
lor  a  long  time  nearly  synonymous ;  thus  the  Jewish  physicians 
were  admirably  fitted  both  to  get  copies  of  th«)  Arab  works  and 
to  translate  them.  Frederick  1 1.  made  use  of  this  fact  to  engage 
a  staff  of  learned  Jews  to  translate  the  Arab  works  which  he 
obtained,  though  there  is  no -doubt  that  he  gave  his  patronage 
to  them  the  more  readily  because  it  was  singularly  offensive  to 
the  pope  with  whom  he  was  then  engaged  in  a  quarrel.  At 
any  rate  by  the  end  of  the  thirteenth  century  copies  of  the 
works  of  Euclid,  Archimedes,  Apollonius,  Ptolemy,  and  of 
several  Arab  authors  were  obtainable  from  this  source,  and  by 
the  end  of  the  next  century  were  not  uncommon.  From  this 
time  then  we  may  say  that  the  development  of  science  in 
Europe  was  independent  of  the  aid  of  the  Arabian  schools. 

Jordanust*  Among  Leonardo's  contemporaries  was  a 
German  mathematician,  whose  works  were  until  the  last  few 
years  almost  unknown.  This  was  Jortinnfu  NenwrariuM^ 
sometimes  called  Jordanug  th  Sturmia  or  Teutanicut,  Of  the 
details  of  his  life  we  know  but  little,  save  that  he  was  elected 
'general  of  the  Dominican  order  in  1222.  The  works  enume- 
rated in  the  footnote  X  hereto  are  attributed  to  him,  and  if  we 

*  For  insUnoe  the  reader  may  reoolled  that  in  Dom  Qmireie  (part  n, 
eh.  15),  wheo  SamtoD  Carsnoo  is  thrown  by  the  kniizht  from  his  horps 
and  has  his  ribs  broken,  an  ml§€kriMin  Is  sQinmoned  to  bind  op  his 


t  See  Cantor,  ehaps.  sun,  xuv,  where  references  to  the  authorities 
on  JordaniM  are  eoUeeted. 

X  Prof.  Cartse,  who  has  made  a  epeeial  etody  of  the  subjcet,  coo- 
sidors  that  the  followiog  works  am  doe  to  Jordanos.  Qttmeiria  wti 
4€  TfimupMa,  pobliiiheil  hj  IL  Cartse  io  1897  in  tol.  vi  of  the  miUii- 
osfea  det  Ccpemieua'Vtreim  to  Them ;  De  UeffeHmetri^i  Atitlmeiiem 
llwwwtlniN,  poblished  by  Faber  BUpolensis  at  Fsris  ia  1496,  seeood 
1614;  AI§mUkmm$  ihtmmirmiw$t  poblisbed  hf  J.  Bshtair  al 

IS 


178     INTRODUCTION  OF  AHABtAN   WORKS  INTO   EUltOPK. 

AMHitoe  (iukt  tb«M  vorki  h«ve  not  been  added  to  or  impiovMl 
I9  Hibwqaenl  tttaoMan,  w«  most  artaem  bin  ooa  of  Um  nort 
amineafc  nuutheuMticwna  ot  the  middle  egei. 

Hit  knowledge  of  geometij  ie  illuitimled  bf  bia  />■  7V»m»- 
gtUit  Mid  D*  ItopKrimttrU.  The  tiKMt  important  ot  tbeoe  b 
tbe  A*  Triaagttlit  which  ie  divided  ioto  fbar  booka.  I^  fint 
book,  beeide*  »  few  definitioai,  ooDteine  tbirteen  propoeitkMw 
t»i  triangliH  which  Rto  bated  on  Eoclid'e  UltmetU*.  Tim 
eecund  book  oonteiiut  nineteen  propoeitiona,  mainly  on  the 
ntioe  of  straight  lines  and  the  oomparuoa  of  the  ftreaa  ot 
trianglea ;  for  example^  one  probleiu  i«  to  find  a  point  inddo 
a  triangle  ao  that  the  lines  joining  it  to  tlie  angular  point* 
may  divide  the  triangle  into  three  equal  parts,  Tbe  thinl 
book  contains  twelve  propositions,  uiainly  ouncerning  araa 
and  chords  of  inrcles.  Tbe  fourth  book  contains  twenty 
eight  pn^tuxitions,  partly  ou  regular  polygons  and  partly  on 
m  niisL-elbneous  i|U««tiaiis  audi  an  tlie  duplication  and  trisection 

problems. 

The  Alfforithmtu  VttHotutrattu  contains  practical  nilea  for 
the  four  fundamental  procetiaes,  and  Arabic  numerals  ara 
generally  (but  not  always)  used.  .  It  Ja  divided  into  tea  booki 
dealing  witli  properties  of  uumticra,  primes,  perfiict  numben, 
polygonal  uumbera,  iic,  ntiuH,  powers,  and  tlie  pNgreeaion^ 
it  would  aeem  frutu  it  that  Jorlanua  knew  the  general  exprea- 
siun  fur  the  square  of  any  algebraic  multinoiuial. 

The  Ik  A'ltmerU  IkuU  cotuiista  of  four  books  containing 
Bolutiuiis  of  one  hundred  and  fifteen  pnibleiiia.  8ume  of  theao 
Ivad  to  simple  or  quadratic  equations  involving  mora  than  one 
unknown  quantity.  He  slwws  a  knowledge  of  proporUon; 
but  many  ui  the  demonstrations  of  his  general  propoaitions  ara 
only  numerical  illustratiooa  of  tbenL 

Nurambcrg  Id  IS31;  Dt  Sumtrit  Dutit,  puUiabed  bj  P.  TmdHcIb  In 
IHTU  sad  cdiMd  in  lutfl  with  oomuieDU  bj  U.  Cnrtis  in  toL  txm 
at  \ha  Uriuihri/l/iir  Ui-lktmaUk  umd  Pkfiik;  U*  yemdmtiu,  fMOibti 
hj  P.  Apian  at  Kuieubarg  lu  lua,  uJ  r^iuiuid  at  Vaoioa  in  UW; 
aed  lasUjr  i«o  or  tbr**  Iruta  on  IHtdEmaia  aslraneaty. 


JORDAKU8.  179 

In  aeTerml  of  the  propomiions  of  the  A1gnr%thmn$  and  De 
Jfumerit  Datui  letters  are  employed  to  denote  both  known 
and  unknown  qaantities,  and  thej  are  ancd  in  the  demonstra- 
tions of  the  rules  of  arithmetic  as  well  as  of  algebra.  As  an 
example  of  this  I  quote  the  following  proposition*  the  object 
of  which  is  to  determine  two  quantities  whose  sum  and 
product  are  known. 

Ikilo  mmwtero  per  dmo  4iui»o  if,  qttod  ex  dmctu  umims  ik  mUfrum  prv- 
dueitmr^  datum /merit,  et  mtrmmque  eorum  Hnlnm  etae  neceue  eet. 

Hit  nomenis  datnt  ahe  diuinuB  in  ah  ei  r,  slqae  ex  ah  in  e  fist  d  datns, 
itemqne  es  abc  in  te  fist  e,  ^amstar  itsque  qosdraplam  d,  qai  fit/,  qao 
dempto  de  «  remanent  f^  et  ip«e  erit  qoadnitani  differentise  ah  sd  e. 
Extrsbstor  ergo  rsdis  ex  g,  et  nit  k,  critqae  k  differeutis  ah  sd  e,  enniqns 
sie  h  dstom,  erit  et  e  et  «i6  dstnm. 

Hnine  operstio  fseile  oonstsbit  hoc  modo.  Verbi  gratis  sit  x  dioiras 
ia  naneros  dao«,  stqne  ex  daota  anios  eomm  in  sliam  fist  xxi ;  enios 
qosdmplonif  et  iptnm  evt  Lxxxtin,  tollstur  de  qasdrsto  x,  hoe  est  c,  et 
remsaent  xti,  enios  rsdix  extrshstnr,  qnse  erit  qustnor,  et  ipee  est 
differentis.  Ipos  toUstnr  de  x  et  reliqanm,  quod  est  n,  dimidietnr^ 
sritqoe  medietss  in,  et  ipM  est  minor  portio  et  msior  Tn. 

It  will  be  noticed  that  Jdrdanus,  like  Diophantus  and  the 
Hindoos,  denotes  addition  by  juxtaposition.  Expressed  in 
modem  notation  his  argument  is  as  follows.  Let  the  numbers 
bo  a-f6  (which  I  will  denote  hy  y)  and  e.  Then  y-f  e  is 
given;  h^nce  (y •«•<;)*  is  known;  denote  it  hj  e,.  Again  ye  is 
gtren;  denote  it  by  il;  hence  4yr,  which  is  equal  to  4^,  is 
known ;  denote  it  bj  /  Then  (y  -  c)'  is  equal  to  e  —/^  which 
is  known ;  denote  it  by  g.  Therefore  y  -  c  =  ^fg,  which  is 
known;  denote  it  by  A.  Hence  y-^e  and  y-«  are  known, 
and  therefore  y  and  e  can  be  at  once  found.  It  is  curious 
that  he  should  have  taken  a  sum,  like  a  -i-  6  for  one  of  his 
unknowns.  In  his  numerical  illustration  he  takes  the  sum  to 
be  10  and  the  product  3L 

Save  for  one  instande  in  Leonardo's  writingSi  the  abovB 
works  are  the  earliest  instances  known  in  European  matha- 
matioi  of  syncopated  algebra  in  wbieh  letters  are  need  for 

•  Pk«m  the  Dirl^McritArti^  book  t,pffopilL 

U-4 


180    IllTBOOUCTIOH  or  A&ABIAH  WOBK8  mtO  KUBOPB, 


algehnucal  ajmbolt  It  ii  prohable  thai  the  AlgorMmm§  was 
not  generally  known  until  it  was  printed  in  ISSI^  and  il 
is  dottbtlul  how  far  the  works  of  Jordanos  exercised  anj 
oonsideraUe  influence  on  the  deTelopment  of  algebra.  In 
fact  it  constantly  happens  in  the  history  of  matlienuitica  that 
improvements  in  notation  or  method  are  made  king  before 
they  are  generally  adopted  or  their  advantages  realised.  Thus 
the  same  thing  may  be  discovered  over  and  over  again,  and  it 
is  not  until  the  general  standard  of  knowledge  requires  soma 
such  improvement,  or  it  is  enforced  by  some  one  whose  seal  or 
attainments  compel  attention,  that  it  is  adopted  and  becomea 
part  of  the  science.  Jordanus  in  using  letters  or  symbols 
to  reprenent  any  quantities  which  occur  in  analysis  was  far 
in  advance  of  his  conteiuporaries.  A  similar  notation  was 
tentatively  introduced  by  other  and  later  mathematicians,  but 
it  was  nut  until  it  had  been  thus  independently  disco%*ered 
several  times  that  it  came  into  general  use. 

It  is  not  necessary  to  describe  in  detail  the  mechaniosi 
optics,  or  astronomy  of  Jordanus.  The  treatment  of  mechauica 
throughout  the  middle  ages  was  generally  unintelligent 

No  mathematicians  of  the  same  ability  as  L^eonardo  and 
Jordanus  appear  in  the  history  of  the  subject  for  over  two 
hundred  years.  Their  individual  achievements  must  not  be 
taken  to  imply  the  standard  of  knowledge  then  current,  but 
their  works  were  accessible  to  students  in  the  following  two 
centuries  though  there  were  not  many  who  seem  to  have 
derived  much  benefit  tlien*from  or  who  attempted  to  extend 
the  bounds  of  arithmetic  and  algebra  as  there  expounded. 

During  the  thirteenth  century  the  most  famous  centres  of 
learning  in  western  Europe  were  Paris  and  Oxford,  and  I 
must  now  refer  to  the  more  eminent  members  of  those  schools. 

Holywood*.  I  will  begin  by  mentioning  Jo^tn  de  Uciy^ 
wood,  whose  name  is  often  written  in  the  latinized  form  of 
Sacrobotco,  Uolywood  was  bom  in  Yorkshire  and  educated 
at  Oxford,  but  after  taking  his  master's  degree   he  moved 

*  Bee  Cantor,  ebap.  xlv. 


HOLTWOOD.      ROGER  BACON. 


181 


to  Paris  and  Uught  then  till  his  death  in  1244  or  1246. 
His  lectares  on  nlgorism  and  algebra  are  the  earliest  of 
whidi  I  can  6nd  mention.  His  work  on  arithmetic  was  for 
many  years  a  standard  aathority ;  it  contains  rules,  but  no 
proofs;  it  was  printed  at  Paris  in  1496.  He  also  wrote  a 
treatise  on  the  sphere  which  was  made  public  in  1 256 :  this 
had  a  wide  and  long-continued  circulation,  and  indicates  how 
rapidly  a  knowledge  of  mathematics  was  spreading.  Besides 
these,  two  pamphlets  by  him  entitled  respectively  ih  Comp^Uo 
BeefennMlieo  and  D^  A  utrofabio  are  still  extant. 

Roger  Bacon*.  Another  contemporar}'  of  lieonardo  and 
Jordanus  was  Roger  Bacon,  who  for  physical  science  did  work 
somewhat  analogous  to  what  they  did  for  arithmetic  and 
algebra.  Roger  Bacon  was  bom  near  Ilchester  in  1214  and 
died  at  Oxford  on  June  11, 1294.  He  was  the  son  of  royalists, 
most  of  whose  property  had  been  confiscated  at  the  end  of  the 
civil  wars:  at  an  early  age  he  was  entered  as  a  student  at 
Oxford,  and  is  said  to  have  taken  orders  in  1233.  In  1234 
he  removed  to  Paris,  then  the  intellectual  capital  of  western 
Europe,  where  he  lived  for  some  years  devoting  himself  espe- 
cially to  languages  and  physics ;  and  there  he  spent  on  books 
and  experiments  all  thai  remained  of  his  family  property  and 
his  savtngR.  He  returned  to  Oxford  soon  after  1240,  and 
there  for  the  following  ten  or  twelve  years  he  laboured 
incessantly,  being  chiefly  occupied  in  teaching  science.  His 
lecture  room  was  crowded  but  everything  that  he  earned  was 
spent  in  buying  manuscripts  and  instruments.  He  tells  us 
that  altogether  at  Paris  and  Oxford  he  spent  over  £2000 
in  this  way — a  sum  which  represents  at  least  £20,000  now-a- 
days. 

Bacon  strove  hard  to  replace  logio  in  the  university  curri- 

*  See  Rogn^  Btieom^  m  rie,  Mt  mirragta..  by  B.  Charlet,  Paris,  1861 ; 
sad  ibs  meOBoir  hj  J.  8.  Brewsr,  prefixed  lo  lbs  Operm  tmtiliim.  Rolls 
SsriaiK  London,  ia09:  asonMwIialdepraeiatofyeritieiMBofUisforBNri 
IhsM  werks  is  givsn  la  H^ger  Bttem^  das  Mmo$rmpkie  kj  L.  Bshatidti 
AiV^Mf ,  1871. 


182  mTBODUcrnoii  or  ibabuv  wosn  mo  Muman. 

culom  by  oiatlieipatHml  and  lingniitio  rtudiei^  \m%  the  intuwwM 
of  the  Age  were  too  strong  lor  bim.  Hii  glowing  enkgj  tm 
**  divine  mntbeoiatica  "  which  aboald  form  the  ionndntaon  d  n 
liberal  educmtion  and  which  '*  alone  oan  porge  the  iutelleei 
and  fit  the  student  for  the  acquirement  of  all  knowledge  "  fell 
on  deaf  ears.  We  can  judge  how  small  was  the  amount  of 
geometry  which  was  implied  in  the  quadrivium  when  he  tells  ua 
that  in  geometry  few  students  at  Oxford  read  beyond  Euc  i,  5 ; 
tliough  we  might  perhaps  have  inferred  as  much  from  the 
character  of  the  work  of  Boethius. 

At  last  worn  out,  neglected,  and  ruined.  Bacon  was  per- 
suaded by  his  friend  Qrosseteste,  the  great  bishop  of  Lincoln, 
to  renounce  the  world  and  take  tlie  Francnscan  vows.  The 
society  to  which  he  now  found  himself  confined  was  singularly 
uncongenial  to  him,  and  he  beguiled  the  time  by  writing  on 
scientific  questions  and  perhaps  lecturing  The  superior  of  the 
order  heard  of  this,  and  in  1257  forbade  him  to  lecture  or 
publish  imythiiig  under  penalty  of  the  most  severe  punish- 
ments,  and  at  the  same  time  directed  him  to  take  up  his 
residence  at  Paris,  where  he  could  be  more  closely  watched. 

Clement  1 V.,  when  in  England,  had  heard  of  Bacon's  abili- 
ties, and  in  126G  when  he  became  pope  he  invited  Bacon  to 
write.  The  Franciscan  onler  reluctantly  permitted  him  to  do 
so,  but  they  refused  htm  any  assistance.  With  difficulty 
Bacon  obtained  sufficient  money  to  get  paper  and  the  loan  of 
books,  and  in  the  short  space  of  fifteen  months  he  produced  in 
I2G7  his  Oput  majus  with  two  supplements  which  summarized 
what  was  then  known  in  physical  science,  and  laid  down  the 
principles  on  which  it,  as  well  as  philosophy  and  literature^ 
sliould  be  studied.  He  stated  as  the  fundamental  principle 
that  the  study  of  natural  science  must  rest  solely  on  experi- 
ment ;  and  in  the  fourth  part  he  explained  in  detail  how 
astronomy  and  physical  sciences  rest  ultimately  on  mathe- 
matics, and  progress  only  when  their  fundamental  principles 
are  expressed  in  a  mathematical  form.  Mathematics,  he  saysi 
should  be  regarded  as  the  alphabet  of  all  philosophy. 


ROGER  BACON. 


183 


Tlie  remilU  that  he  AiriTed  at  in  thin  and  his  other  works 
are  nearly  in  accordance  with  modem  ideas,  but  were  too  far 
in  advance  of  that  age  to  lie  capable  of  appreciation  or  perhaps 
eren  of  comprehension,  and  it  m-as  left  for  later  generations  to 
redisooTer  his  works,  and  give  him  that  credit  which  he  never 
experienced  in  his  lifetime.  In  astronomy  he  laid  down  the 
principles  for  a  reform  of  the  calendar,  explained  the  pheno- 
mena of  shooting  stars,  and  stated  that  the  Ptolemaic  system 
was  unscientific  in  so  far  as  it  rented  on  the  assumption  that 
circular  motion  was  the  natural  motion  of  a  planet,  while  the 
complexity  of  the  explanations  required  made  it  improlmble 
that  the  theory  was  true.  In  optics  he  enunciated  the  laws  of 
reflexion  and  in  a  general  way  of  refraction  of  light,  and 
need  them  to  give  a  rough  explanation  of  the  rainbow  and  of 
magnifjring  glasses.  Most  of  his  experiments  in  chemistry 
were  directed  to  the  transmutation  of  metals  and  led  to  no 
result  He  gave  the  coipposition  of  gunpovrder,  but  there  is 
no  doubt  that  it  was  not  his  own  invention,  though  it  is 
the  earliest  European  mention  of  it.  On  the  other  hand  some 
of  his  results  in  these  subjects  appear  to  be  guesses  which 
are  more  or  less  ingenious,  while  eertain  statements  he  makes 
are  certainlv  erroneoua 

In  the  years  immediately  following  the  publication  of  his 
Opnt  mt^us  he  wrote  numerous  works  which  developed  in  detail 
the  principles  there  laid  down.  Most  of  these  have  now  been 
published  but  I  do  not  know  of  the  existence  of  any  complete 
edition.   They  deal  only  with  applied  mathematics  and  physics. 

Clement  took  no  notice  of  the  great  work  for  which  he  had 
asked,  except  to  obtain  leave  for  Bacon  to  return  to  England. 
On  the  death  of  Clement,  the  general  of  the  Franciscan  order 
was  elected  pope  and  took  the  title  of  Nicholas  lY.  Bacon's 
investigations  had  never  been  approved  of  by  his  superiors, 
and  he  was  now  ordered  to  return  to  Paris  where  we  are  told 
he  was  immediately  accused  of  magic :  he  was  eofidemned  in 
1280  to  imprisonment  for  lile^  and  was  released  only  abovi  a 
year  bsfcra  his  deallL 


184  iHTBOoucnoN  or  arabiaw  wommb  una  simonL 

CtainpMiMii*  Thft  only  otbor  mithcwnnHtriin  cf  M*^  €0a» 
tarj  wbom  I  need  mention  it  Giovanni  Cmmpama^  or  in  (ho 
Utiniied  fomi  Campanu$.  m  canon  of  Pmiiil  A  cofj  of 
Adelhutl's  tnuuUtion  of  Euclid's  KUmmU  fell  into  Um 
hands  of  Oanipanus,  who  issued  it  as  his  own*;  he  added 
a  commentary  thereon  in  whidi  he  discussed  the  praperties 
of  a  regular  reentrant  iMsntagon.  He  alKS  hesides  some 
minor  worksi  wrote  tlie  Tktory  of  ike  FlumeU^  which  was  a 
free  translation  of  the  AlmagetL  / 

The  fourteenth  century.  The  history  of  the  fourteenth 
century,  like  that  of  the  one  preceding  it,  is  mostly  concerned 
witli  the  introduction  and  assimilation  of  Arabian  mathe- 
matical text-books  and  of  Greek  books  derived  from  Arabian 
sources. 

Bradwardinef.  A  mathematician  of  this  time^  who  was 
perhaps  sufficiently  influential  to  justify  a  mention  here,  is 
Thanuu  Bfatiwurdifte^  archbishop  of  Canterbury.  Bradwardina 
was  born  at  Chichester  about  1290.  He  was  educaUsd  at 
Merton  College,  Oxford,  and  subsequently  lectured  in  that 
university.  He  was  the  first  European  to  introduce  the 
cotangent  into  trigonometry.  From  1335  to  the  time  of  his 
death  he  was  chiefly  occupied  with  the  politics  of  the  church 
and  state:  he  took  a  prominent  part  in  the  invasion  of 
France,  the  capture  of  Calais,  and  the  victory  of  Cressy.  He 
died  at  Lambeth  in  1349.  His  mathematical  works,  which 
were  probably  written  when  he  was  at  Oxford, .  are  the 
TraciatuM  de  FroportionibuM^  printed  at  Paris  in  1495;  the 
Aritkmeiiea  Speeulaiiva^  printed  at  Paris  in  1502;  the 
G'eotnelria  Speculatica^  printed  at  Paris  in  1511 ;  and  the  Dt 
Qutulratura  Circuii,  printed  at  Paris  in  1495.  They  probably 
give  a  fair  idea  of  the  nature  of  the  mathematics  then  read  at 
an  English  university. 

*  It  was  priuted  by  Batdolt  at  Venico  in  148S.    On  this  work  tea 
J.  L.  Udbeig  in  tlie  ZriUekrift  fUr  Hatkgmaiik,  vol.  ^uv,  ISMI. 
t  8m  Cantor,  voL  u,  p.  102  «l  m^ . 


THE  FOURTEENTH  CKNTURT.  '185 

Oremms**  Nickofaa  OrttmuM  was  another  writer  of  the 
foarteenth  oentary  who  is  said  in  roost  histories  of  mathe- 
matios  to  have  influenced  the  development  of  the  subject. 
He  was  bom  at  Caen  in  1323,  became  the  confidential  adviser 
of  Charles  V.  by  whom  he  was  made  tutor  to  Charles  YI., 
and  subsequently  was  appointed  bishop  of  Lisieux,  at  which 
city  he  died  on  July  11,  1382.  He  wrote  the  AlgorUmus 
PfstporiioMum  in  which  the  idea  of  fractional  indices  is  intro- 
duced, and  in  the  eyes  of  his  contemporaries  was  prominent  as 
a  mathematician  not  less  than  as  an  economist  and  theologian;  • 
bat  I  do  not  propose  to  discuss  his  writings.  The  treatise 
OD  which  his  reputation  chiefly  rests  deals  with  questions  of 
coinage  and  commercial  exchange:  from  the  mathematical 
point  of  view  it  is  noticeable  for  the  use  of  vulgar  fractions 
and  the  introduction  of  symbols  for  them. 

By  tlie  middle  of  this  century  Euclidean  geometry  (as 
azpoanded  by  Campanus)  and  algorism  were  fairly  familiar 
to  all  professed  mathematiciaus,  and  the  Ptolemaic  astronomy 
was  also  generally  known.  About  this  time  the  almanacks 
began  to  acM  to  the  explanation  of  the  Arabic  symbols  the 
nilet  of  addition^  subtraction,  multiplication,  and  division, 
'^de  algurisma"  The  more  important  calendars  and  other 
treatises  also  inserted  a  statement  of  the  rule^  of  proportion, 
illustrated  by  various  practical  questions. 

In  the  latter  half  of  this  century  there  was  a  general 
revolt  of  the  universities  against  the  intellectual  tyranny  of 
the  schoolmen.  This  was  largely  due  to  Petrarch,  who  to 
bis  own  generation  was  celebrated  as  a  humanist  rather 
than  as  a  poet^  and  who  exerted  all  his  power  to  destroy 
scholasticism,  and  encourage  schfilarship.  The  result  of  these 
influences  on  the  study  of  matliematics  may  be  seen  in  the 
ehangea  then  introduced  in  the  study  of  the  quadriviom. 
The  stimulns  came  from  the  nniversity  of  Fkris,  where  a 
■latnte  to  that  efleot  was  passed  in  1366,  and  a  year  or 

*  Sss  IHf  wmihnmtimkgm  SckHfttn  4n  NhMe  Onm§  kf  U.  0art8S» 
1090. 


186  imooucnoN  or  ababiav  womu  mo  iubopk. 


two  later  simikr  ragttktioiM  weie   mmdm 


at   Oxtoid  and 


HMntioned.    We 
of  the  lamn  of 


GaiolNridge ;  aofortonalely  no  text-books  are 

ean  howeTor  form  a  reaaonable  estimate 

mathematical  resding  required,  by  looking'  at  the  statates 

of  the  nniversities  of  Prague,  of  Vienna,  iand  of  Leipug. 

By  the  sUtntes  of  Pragae,  dnted  1384,  <kndidates  for  the 
bachelor's  degree  were  required  to  have  ^read  Holywood's 
treatiae  on  the  spliere,  and  candidates  for  the  master's  degree 
to  be  acquainted  with  the  first  six  books  of  Euclid,  optics, 
•  hydrostatics,  the  theory  of  the  lever,  and  astronomy.  Lectures 
were  actually  delivered  on  arithmetic,  the  art  of  reckoning  with 
tlie  fingerSj  and  the  algorism  of  integers ;  on  almanacks,  which 
probably  meant  elementary  astrology;  and  .on  the  Almageti^ 
that  is,  on  Ptolemaic  astronomy.  There  is  however  some 
reason  for  thinking  that  mathematics  received  far  more  atten- 
tion here  than  was  then  usual  at  other  universities. 

At  Vienna,  in  1389,  a  candidate  for  a  master's  degree  was 
required  to  have  read  five  books  of  Euclid,  common  perspeo> 
tive,  proportional  parts,  the  measurement  of  superficies,  and 
the  Theory  of  the  Planeis,  The  book  lost  named  is  the  treatise 
by  Campanus  which  was  founded  on  that  by  Ptolemy.  This 
was  a  fairly  respectable  mathematical  standard,  but  I  would 
remind  the  reader  that  there  was  no  such  thing  as  '* plucking" 
ill  a  medieval  university.  The  student  had  to  keep  an  act  or 
give  a  lecture  on  certain  subjects,  but  whether  he  did  it  well 
or  badly  he  got  his  degree,  and  it  is  probable  that  it  was  only 
the  few  students  whose  interests  were  matheiuatical  who  really 
mastered  the  subjects  mentioned  above. 

The  fifteenUi  century.  A  few  facts  gleaned  from  the 
history  of  the  fifteenth  century  tend  to  shew  that  the  regula- 
tions about  the  study  of  the  quadrivium  were  not  seriously 
enforced.  The  lecture  lists  for  the  years  1437  and  1438  of 
the  university  of  Leipzig  (founded  in  1409,  the  statutes  of  which 
are  almost  identical  with  those  of  Prague  as  quoted  above)  are 
extant,  and  shew  that  the  only  lectures  given  there  on  mathe- 
matics in  those  years  were  confined  to  astrology.    The  records 


THE  FIFTEENTH  CENTURY.  187 


of  Bologna,  F!Mlaa»  and  Pisa  seem  to  Imply  that  there  also 
aatrologj  was  thn  only  scientific  subject  taught  in  the  fifteenth 
centaiy,  and  even  as  late  as  1598  the  prufe^sor  of  mathematics 
at  Pisa  was  reqatrpd  to  lecture  on  the  Qtirtdripartiium,  an 
astrological  work  purporting  (prohnbljr  falsely)  to  hare  been 
written  by  Ptolemy.  The  only  mathematical  subjects  men- 
iioAed  in  the  registers  of  the  university  of  Oxford  as  read 
there  between  the  years  1449  and  1463  were  Ptolemy's 
astronomy,  or  some  commentary  on  it»  and  the  first  two 
books  of  Euclid.  Whether  most  students  got  as  far  as  this 
is  doubtful  It  would  seem,  from  an  edition  of  Kuc1id*s 
EhmeniM  published  at  Paris  in  1536,  that  after  1452  candi- 
dates for  the  master's  degree  at  that  university  had  to  tako 
an  oath  that  they  had  attended  lectures  on  the  first  six  books 
of  that  work. 

Beldomandi.  The  only  writer  of  this  time  that  I  need 
mention  here  is  Frodoeimo  BeldomnnHt  of  Padua,  bom  ab(»ut 
1380,  who  wrote  an  algoristic  arithmetic,  published  in  1410, 
which  oon tains  the  summation  of  a  geometrical  series;  and 
some  geometrical  works*. 

By  the  middle  of  the  fifteenth  century  printing  had  been 
introduced,  and  the  facilities  it  gave  for  disseminating  know- 
ledge were  so  great  as  to  revolutionize  the  progress  of  science. 
We  have  now  arrived  at  a  time  when  the  results  of  Arab  and 
Greek  science  were  known  in  Kurope ;  and  this  perhaps  then 
is  as  good  a  date  as  can  be  fixed  for  the  close  of  this  period 
and  the  oommencement  of  that  of  the  renaissance.  The  mathe- 
matical  history  of  the  renaissance  begins  with  the  career  of 
Regiomontanus ;  but  before  proceeding  with  the  general 
history  it  will  be  oonvenient  to  collect  togetlier  tlie  chief 
iseta  connected  with  the  development  of  arithmetic  during 
the  middle  ages  and  the  renaissance.  To  this  the  next  chapter 
isdevotedt 

•  Fer  Authsr  dHsils  see  Boaaompacni's  MOHim  4i  Mfiayni^, 


188 


CHAPTER  XL 

TUfi  DKVKLOPMEKT  OP  ARITUMCnC^ 

cittc.  1300-1637. 


Wb  have  aeen  in  the  last  chapter  thai  bj  the  end  of  the 
thirteenth  oentary  the  Arabio  arithmetio  had  been  fairly  iotio- 
duoed  into  £uro|ie  and  was  practiced  bj  the  side  of  the  older 
arithmetio  which  was  founded  on  the  work  of  Boethius.  It  will 
be  convenient  to  depart  from  the  chronological  arrsngement 
and  briefly  to  sum  up  the  subsequent  history  uf  arithmetic^  but 
I  hope,  by  references  in  the  next  chapter  to  the  inventions  and 
iiupruvements  in  arithmetic  here  described,  that  I  shall  be  able 
to  keep  the  order  of  events  and  discoveries  clear. 

The  older  arithmetic  cuiisistea  of  two  parts :  practical  arith- 
metic or  the  art  of  calculation  which  was  taught  by  means  of 
the  abacus  and  pussibly  the  multiplication  table,  and  theoretical 
arithmetic  by  which  was  meant  the  ratios  and  properties  of 
numbers  taught  according  to  Boethius — a  knowledge  of  the 
latter  being  contiued  to  professed  mathematicians.  The  theo- 
retical part  of  this  system  continued  to  be  taught  till  the 
middle  of  the  fifteenth  century,  and  the  practical  part  of  it 

*  hae  the  article  uu  Arithiuetic  bj  G.  Pemcock  io  the  Knc^clopaedia 
}l€iropoiiiaiM,  vol.  I,  LunJoii.  1815;  Ariihmeiicul  itook*  by  A.  Dn 
Murgsu,  Lotulon,  1h47  ;  suJ  sn  article  by  P.  Treutleiu  of  Karlnmhe, 
io  tb«)  •up|ileiu«ot  (|>p.  1 — lUO)  uf  tbe  AbkaHdiuMyen  sur  Uetckiekte  4ef 
Malkematik^  lb77. 


I 


THE  DEVELOrMENT  OF  ARITHMCTrC.     1300-1( 

wkB  ufied  by  the  smnller  tm<)psn)(-n  in  EngUnd*,  ( 
and  Fntncp  till  Ihn  lieginnin»  <rf  the  neventeMith  eent 

Thft  new  AraUiui  arithmptic  wnn  c-tlled  algnritm  < 
nt  Alkfiristni  to  distinguish  it  from  the  nM  or  Bwthi 
mntic:  The  t^^xt-buoks  on  Algorism  coinmencetl  wjtli  tli 
■jRtpm  of  nutAtinn,  And  begun  by  giving  rule«  for  ndtlil 
tndion,  multi|ilicntion,  and  division;  the  principle*  o 
tion  were  then  applied  U>  various  pmctiol  proltlemft 
books  UHUnlly  concluded  with  geneml  ruin)  for  miui 
eomninn  prolilemn  of  commerce.  Algorinm  wna  in  tm 
cnntile  Arithmetic  tliougli  at  first  it  nlno  included  hII 
then  knoim  ah  algclim. 

ThuR  Algebra  hiw  itx  origin  in  arithmetic ;  And 
people  the  term  nuir'itntl  nrilhmftir,  by  which  it  w 
timea  deaignnttxl,  conveyi  a  more  Accnnite  impremii 
object*  and  ntethotU  than  the  more  elaljorate  deKn: 
modem  mathematicianB — certAinly  betb^r  ihAn  the  i 
of  Sir  William  Haniilton  as  the  scienc*  of  pure 
thAt  of  De  Alorfjan  «a  the  calculus  of  BUCCeHSion.  i 
logically  there  is  a  marked  <linlinction  lietween  ai 
And  algebrm,  for  the  former  ia  the  theory  of  discrete  m 
while  the  Utter  is  that  of  continuous  magnitude 
acienlific  distinction  such  an  this  is  of  quite  recent  or 
tlie  idea  of  continuity  wa<t  not  intrudnced  into  mat 
before  the  time  of  Kepler. 

Of  coarse  the  fundamentAl  nlcs  of  this  Algori 
not  At  drst  strictly  provi>d — that  is  the  work  of  i 
thought— but  until  the  middle  of  the  xeventeenth 
there  WM  aome  discussion  of  the  principles  involve 


*  See,  for  iniUnoa.  Chaucer,  TV  itilltr'i  Tab,  t,  >»— SI 
•pnrc.  Tht  Winlrr'i  Tuff.  An  iv.  Sc.  3;  Olhrtta,  Ad  i,  Be.  1. 
mlBdeDtlj  fsniilUr  wllli  psrlj  Prrnch  or  Oernmn  literatan 
wbtUwr  Ibcj  onotala  atij  ivfcivnee>  to  the  an  of  the  abaem. 
•Iwl  the  Eteb«|n*r  (li<riMim  of  the  Hi^h  Coort  of  JuUoe  dnive 
fttMB  Ih*  table  bctoTs  wliicb  tba  judin*  and  olfions  of  Un  eowt 
■at:  this  vaa  coxrad  willi  bUck  cloth  diviJcd  ialo  sqaHW  « 
ty  while  Him*.  Md  mn^amOj  wm  m«4  h  •■  ebeMH.. 


190     THE  DBVCLOPMKVT  OP  ARITHllRia     1800-1687, 

• 

then  very  few  ariUmetieiMM  have  attenpled  to  J wtify  or 
prove  the  proceaaee  oaed,  or  to  do  mora  than  ennneiale  rolee 
and  illttiitrate  their  luie  bj  mamerical  examplte. 

I  have  alluded  frequently  to  the  Arabic  syetem  of  nnoMri- 
cal  notation.  I  may  therefore  conveniently  begin  by  a  few 
noteii  on  the  history  of  the  iiynibolii  now  currents 

Their  origin  is  obscure  and  has  been  much  disputed^.  On 
the  whole  it  seems  probable  tliat  the  symbols  for  the  nnmben 
4,  5,  6,  7,  and  9  (and  pimsibly  8  too)  are  derived  from  the 
initial  letters  of  the  corresponding  words  in  the  Indo-Baetrian 
alpliabet  in  use  in  the  north  of  India  perhaps  150  yearn  before 
Christ ;  tlmt  the  symbols  fur  the  numbers  2  and  3  are  derived 
rcspectiviily  from  two  and  three  parallel  penstrukee  written 
cursively ;  and  stiutkirly  tliat  the  symbol  for  the  number  1 
represents  a  single  penstroke.  Numerals  of  this  type  were  in 
use  in  India  before  the  end  of  the  second  century  of  our  era. 
Tlie  origin  of  the  symbol  for  lero  is  unknown ;  it  is  not 
impossible  tliat  it  was  originally  a  dot  inserted  to  indicate  a 
blank  space,  or  it  may  represent  a  closed  hand,  but  these  are 
mere  conjectures ;  there  is  reason  to  believe  that  it  was  in- 
troduced in'  India  towards  the  cl<i6e  of  the  fifth  century  of 
our  era,  but  the  earliest  writing  now  extant  in  which  it  occurs 
is  assigned  to  the  eighth  century. 

The  numerals  UHcd  in  India  in  the  eighth  century  and  for 
a  long  time  afterwards  are  termed  Devanagari  numeralsi  and 
their  forms  are  shewn  in  the  first  line  of  the  table  given  on 
the  next  page.  These  forms  were  slightly  modified  by  the 
eiistern  Arabs,  and  the  resulting  symbols  were  again  slightly 
modified  by  the  western  Arabs  or  Moors.  It  is  perhaps  pro- 
bable that  at  first  the  Spanish  Arabs  discarded  the  use  of  the 
symbol  for  zero  and  only  re-inserted  it  when  they  found  how 
inconvenient  the  omission  proved.  The  symbols  ultimately 
adopted  by  the  Arabs  are  termed  dobar  numerals,  and  an  idea 

*  S«e  A.  P.  Pihsn.  Si^Mea  de  NSM/rafioM,  Paris,  1860;  Fr.  Woqxks, 
Mm  pntptitjtUioH  des  ekiffre*  IndUus,  Paris,  1863;  A.  C.  BuracU,  South 
Jmdiam  Palatograpkff^  ilanfalore,  1874;  Is.  Tsjlor,  The  Jlpkabei^  Loadan, 
1883;  and  Cantor. 


BISTORT  OF  THE  ARABIC  STNBOU,  101 

of  the  fonna  mtMt  commonly  usrd  may  Ik;  gnlherad  from  ttioM 
printed  in  tho  second  line  of  the  talite  gii«n  below.  From 
Spain  or  Barbnry  the  Qotnr  nuTnenilM  p.vtwd  tntQ  weatem 
Europe.  The  further  erolntion  of  the  foruii  of  the  nyinliols 
to  those  with  which  we  nrv  fnmilinr  ia  indinted  below  hy 
f«c!>iinilc8*  of  the  auniemla  uhhI  nt  different  timea.  All  the 
«etn  of  nnmerals  here  reprcwnted  nre  written  from  left  to 
right  and  in  the  onlcr  I,  3,  3,  4,  5,  6,  7,  8,  9,  10. 


l.\>,'iiA,^.7.<.r.l'' 


li.j.fi..t,,(r.\.t.p.i» 

From  Ibt  Mimnr  of  ike\ 

H-orU.  printel  bj  Cmi-[      \    Z  )    4-    U    A    f*i,8   QAO 
ton  in  HBO.  I  ■    »i      .  > 

Frntn    ■   Scotch    cmlendarl 

for     1*83.    prol^lj     ol\        i.  2,  5,  9^,^i<5.  A,  8»  ^»  lO 

French  orieiii.  ) 

Kmm   16(H)  onward*  thp  ftynilxiln  fmploTed  un  pneUnllj  the 
BMoe  «s  thou  now  in  nsef. 

The  further  evolution  in  the  East  of  the  OoW  nnmerBls 
proceeded  almost  independently  of  European  ioflDence.  There 
■re  minute  differences  in  the  forms  naed  hj  Tsrioos  writers 

*  Tb«  Bnt.  •Mond,  uhI  lonrth  fumplM  ue  tmkta  ftoM  Is.  Trior's 
Alphmtti,  London.  1881.  *ol.  ii,  p.  268;  Um  olhMS  an  taken  Ikes 
Ladii'a  Fhihtoplit  ofAriHimflir.  pp.  lit,  lid. 

t  Sm  tor  etuii|d«  TuB>Ull'a  Dr  ArU  SiiffiHa»H,  Len4aii.  Ifitt) 
or  Booont's  Onmrndt  rf  ArUi,  LoadoD,  IMO,  and  inWM*M  tf  WUU, 
LoodoD,  IHT. 


198     THE  DBVKLOPMENT  0¥  AKITUMime.     1900-1637. 


and  io  aoiiia  eaaei  altemaiivtt  formti  wiUioai  liowever  •nleri^g 
into  tbeiie  deUiLi  we  may  nj  thai  tbm  nniiiMmb  they  oommoiily 

eiuployed  finally  took  the  form  ihewn  above,  bat  the  symbol 
tliere  given  for  4  in  at  the  prenent  time  generally  written 
cursively. 

Leaving  now  the  history  of  the  symbok  I  proceed  to 
discuss  their  introduction  into  general  use  and  the  develop- 
ment of  alguristic  arithmetic  I  have  already  explained  how 
men  of  science,  and  particularly  astronomersi  had  beoome 
acquainted  with  the  Arabic  system  by  the  middle  of  the 
thirteeuUi  century.  The  trade  of  Europe  during  the  thirteenth 
and  fourteenth  centuries  was  mostly  in  Italian  hands,  and  the 
obvious  advantages  of  the  algoristic  system  led  to  its  general 
adoption  in  Italy  for  mercantile  purposes.  This  change  was 
not  etfected,  however,  without  considerable  opposition  :  thuS| 
an  edict  was  issued  at  Florence  in  1 2U9  forbidding  bankers  to 
use  Arabic  numerals,  and  in  1348  the  authorities  of  the  uni* 
versity  of  Paidua  directed  that  a  list  should  be  kept  of  books  for 
sale  with  the  prices  marked  **uou  per  cifras  sed  per  literas 
claras." 

The  rapid  spread  of  the  use  of  Arabic  numerals  and  arith- 
metic tliruugli  the  n«t  of  Europe  seeius  to  liave  been  quite  as 
largtfly  due  to  the  makers  of  almanacks  and  calendars  as  to 
merchants  and  men  of  science.  These  calendars  had  a  wide 
circulation  in  medieval  times.  They  were  of  two  distinct  types. 
Some  of  them  were  composed  with  special  reference  to  eccle- 
siastical purposes,  and  contained  the  dates  of  the  different 
festivals  and  fasts  of  the  church  for  a  period  of  some  seven  or 
eight  years  in  advance,  as  well  as  notes  on  church  rituaL 
Nearly  every  monastery  and  church  of  any  pretensions 
possessed  one  of  these,  and  several  specimens  are  still  ex- 
tant. ThoHe  of  tlie  second  type  were  written  specially  for 
tbe  use  of  astrologers  and  physician^  and  some  of  them 


194     THE  OBVELaPMINT  OF  ABITHMRia     1900-1687. 

one  ledger,  and  oq  tbm  delitor  side  in  anoihar ;  ihni^  if  dotfi 
be  aold  to  A,  A*%  aecooni  it  debited  with  Uie  lirioe^  and  Uie 
stock  book,  eontaimng  the  tranaactioni  in  oloth,  it  eiedited 
with  the  amount  lokL  It  was  they  too  who  arranged  the 
problems  to  which  arithmetic  could  be  applied  in  different 
classes,  such  as  rule  of  three,  interest,  profit  and  loss,  kc 
They  also  reduced  the  fundamental  operations  of  arithmetic 
*'to  seven,  in  reverence"  says  Fkcioli  *'of  the  seven  gifts  of 
the  Holy  Spirit :  namely,  numeration,  addition,  subtraction, 
multiplication,  diviiiion,  roiidng  to  powers,  and  extraction  of 
roots. "  Bnihmagupta  had  euumerattxl  twenty  processes 
besides  eight  suUiidiary  ernes,  and  had  stated  that  ^a 
distinct  and  several  knowledge  of  these "  was  **  essential  to 
all  who  wished  to  be  calculators";  and,  whatever  may  be 
thought  of  Pacioli's  reason  for  the  alteration,  the  consequent 
simplification  of  the  elementary  processes  was  satisfactory. 
It  may  be  added  that  arithmetical  schools  were  founded  in 
various  parts  of  Germany,  enpecially  in  and  after  the  four- 
teenth century,  and  did  much  towards  familiarizing  traders 
in  northern  and  western  Europe  with  commercial  algoristio 
arithmetic. 

The  operations  of  algoristic  arithmetic  were  at  first  very 
cumbersome.  The  chief  improvements  subiiequently  intro- 
duced into  tlie  early  Italian  algorism  were  (i)  the  simplifica- 
tion of  the  four  fuudameutal  processes :  (ii)  the  introduction 
of  signs  for  addition,  subtraction,  equality,  and  (though  not 
so  important)  for  multiplication  and  division :  (iii)  the 
invention  of  logarithms  :  and  (iv)  the  use  of  decimals.  I  will 
consider  these  in  succession. 

(i)  In  addition  and  subtraction  the  Arabs  usually  worked 
from  left  to  right.  The  modern  plan  of  working  from  right 
to  left  is  said  to  have  been  introduced  by  an  Englishman 
named  Garth,  of  whose  life  I  can  find  no  account.  The  old 
plan  continued  in  partial  use  till  about  1600 ;  even  now  it 
would  be  more  convenient  in  approximations  where  it  is 
necessary  to  keep  only  a  certain  number  of  places  of  decimala 


MULTIPLICATION.  1 95 

The  Indians  and  Arabs  had  seyeral  systems  of  mnltipli- 
eation.  These  were  all  somewhat  laborious,  and  were  made 
tbe  more  so  as  mnltipHcation  tables,  if  not  anknown,  were 
at  any  rate  used  but  rarely.  The  operation  was  regarded 
as  one  of  considerable  difficulty,  and  the  test  of  the  accuracy 
of  the  result  by  "casting  out  the  nines "  was  invented  by  the 
Arabs  as  a  check  on  the  correctness  of  the  work.  Various 
other  systems  of  multiplication  were  sulwe<|uently  emplojred 
in  Italy,  of  which  several  examples  are  given  by  Pacioli 
and  Tartaglia;  and  the  use  of  the  multiplication  table — at 
least  as  fisr  as  5  x  5 — became  common.  From  this  limited 
table  the  resulting  product  of  the  multiplication  of  all 
numbers  up  to  10  x  10  can  be  deduced  by  what  was  termed 
the  regula  ignavi.  This  is  a  statement  of  the  identity 
(5  -I-  a)  (5  -f  6)  =:  (5  -  a)  (5  -  6)  -f  10  (a  -f-  6).  The  rule  was  usually 
enunciated  in  the  following  form.  Let  the  number  five  be 
represented  by  the  open  hand  ;  the  number  six  by  the  hand 
with  one  finger  closed  ;  the  numlier  seven  by  the  hand  with  two 
fingers  closed  ;  the  number  eight  by  the  hand  with  three  fingers 
closed ;  and  the  number  nine  by  the  hand  with  four  fingers 
closed.  To  multiply  one  number  by  another  let  the  multiplier 
be  represented  by  one  hand,  and  the  number  multiplied  by  the 
other,  according  to  the  above  convention.  Then  the  required 
answer  is  the  product  of  the  number  of  fingers  (counting  the 
thumb  as  a  finger)  open  in  the  one  hand  by  the  number  of 
fingers  open  in  the  other  together' with  ten  times  the  total 
number  of  fingers  closed.  The  system  of  multiplication  now 
in  use  seems  to  have  been  first  introduced  at  Florence. 

The  difiiculty  which  all  but  professed  mathematicians  ex- 
perienced in  the  multiplication  of  large  numbers  led  to  the 
invention  of  several  mechanical  ways  of  eflbcting  the  process. 
Of  these  the  most  celebrated  is  that  of  Nlipier's  rods  invented 
in  1617.  In  principle  it  is  the  same  as  a  method  which  had 
been  long  in  use  both  in  India  and  Persia,  and  which  has 
been  described  in  the  diaries  of  several  travellers  and  notably 
ia  the  TrmfAqfSir  John  CUrdin  tit  Pmim,  London,  16M. 


196    THB  DBVBU>pH>NT  OT  ABiraHma    1S0&-1637. 

To  an  thtt  WKthoi  m  tuusbor  at  raeUngoli  r  aUpi  at  howi, 
wood,  metkl,  or  cardboMd  mm  prapkrad,  vil  oMih  of  them 
divided  bjr  erou  Uum  iaiu  dum  little  aqiures;  •  lUp  bning 
genMmlly  »boat  three  incbea  long  Mid  m  thinl  at  «a  incfa 
acroM,  In  the  t<9  «)!»»  oite  of  the  digiti  ia  engnved. 
Mid  the  reeotu  of  multiplying  it  hy  2,  3,  4,1  5,  6,  7,  8,  Mtd 
9  M«  reapectively  entered  in  the  eight  lower;  acjtMrieo :  when 
tbo  rMult  ii  e  nnmlier  of  twu  digits,  the  teii^ligit  is  written 
ftbove  mkI  to  the  left  of  the  unit-digit  Mid  sepunited  frooi  ii 
bj  *  diHgunaJ  line.  The  8li]M  are  uaoftlljf  Hrniii;;^]  in  k  box. 
Figure  i  below  r^HVHeota  nine  such  elipa  •ido  Ly  aide  :  Sgure  ii 


1      a    ^450     TSOO 

■  al  4  _»,ja]o^\]o  ^  _o 

^  bI^  i8  _p  ^4  'al^aj^  _o 
6^'e'o|'5^^s]'o'6  o 
a^al^i^ol^a'-'e^     o 

oV»7T^acl''6''4'>3'a»Tn 


TTT 


FiipiTe  iii. 


Figoie  L  Fittuia  ii. 

■hews  the  seventh  slip^  which  ia  supposed  to  be  taken  out 
of  the  box  and  put  by  ilselL  Suppuw  we  lifuh  to  multipljr 
2985  by  317.  The  process  «•  elfiMiled  by  the  itae  of  these  slips 
ia  as  follows.  Tlie  slips  headed  S,  9,  6,  and  ft  are  token  out 
uf  the  liox  and  put  side  by  side  as  shewn  inifigure  iii  above^ 
The  result  of  uiultiplyiog  2985  by  7  nwy  be  written  tlina 
2965  i 

7  i 

35  I 

56 


MULTIPLICATION.  197 

Now  if  the  reader  will  look  at  the  iteventh  line  in  figure  iii, 
he  will  see  that  the  upper  and  lower  rows  of  figurca  are  renpec- 
tivelj  1653  and  4365 ;  moreover  these  are  arranged  by  the 
diagonals  so  that  roughly  the  4  m  under  the  6,  the  3  under 
the  5y  and  the  6  under  the  3  ;  thus 

16     5     3 
4     3    6     5. 

The  addition  of  these  two  numbers  gives  the  rec|uirefl  result. 
Henoe  the  result  of  multiplying  by  7,  1,  and  3  can  be 
soccessiTely  detennimxl  in  this  way,  and  the  required  answer 
(namely  the  product  of  2985  and  317)  is  then  obtainKi  by 
addition. 

The  whole  process  was  written  as  follows. 

2985 

20895  /  7 
2985  /  1 
8955   '3 

946245 

The  modification  introduced  by  Xapier  in  his  Rahdologin^ 
published  in  1617,  consisted  merely  in  replacing  each  slip  by  a 
prism  with  square  ends,  which  he  calle<l  "  a  rod,"  each  lateral 
face  being  divided  and  marked  in  the  same  way  as  one  of  the 
slips  above  descrilwd.  These  nnln  nnt  only  (HTonomixed  space, 
but  were  easier  to  handle,  and  were  arranged  in  such  a  way  as 
to  facilitate  the  operations  n*quirrd. 

If  multiplicaticn  was  considered  difficult,  division  was  at 
first  r^arded  as  a  feat  which  could  lie  performed  only  liy 
skilled  mathematicians.  The  method  commonly  emplojred  by 
the  Arabs  and  Persians  for  the  division  of  one  number  by 
another  will  be  sufficiently  illustrated  by  a  concrete  instance. 
Suppose  we  require  to  divide  17978  by  472.  A  sheet  of 
paper  is  divided  into  as  many  vertical  columns  as  there 
are  figures  in  the  number  to  be  divided.    The  number  to 


/ 


198  rax  VKWBjoeMxm  or  AuraMma    1300-1687. 

be  diTidfd  w  written  U  Ite  top  and  tU  diTiaor  at  ths bottoai 
th*  flnt  digit  at  Mcb  niunber  baing  pUeed  st  the  kfthand 
■Ida  of  the  p^wr.    Then,  btking  the  Mt^wad  oolnmn,  i  will 


1797a  17978  17973 

j^i j_^ 

S078                  5978 
2__i_ ^J 

3878  3878 
6 _^^^=L  — 

3  a     1      B  3     8     I    T 

J_i^ 

6     t     8 

5     6 

5      8 

ii 

4     7      2 

4  7     2                          4     7     2 
4     7   _a_ 2*^A —  —  2 

O  0      3  0     3      8 


Fifnra  it 


Figura  iii. 


go  into  1  no  times,  b«ic«  the  first  figura  in  the  dividend  is  0, 
which  u  written  under  the  last  figure  at  the  divtaur.  Thia  it 
(«pnMeQt«d  in  figura  i.  Next  (aee  figuro  ii)  re-writ«  the  472 
imnied lately  above  ita  former  puuition  but  shifted  una  plac«  to 
tha  right,  iund  cancel  the  old  ligurea.  Then  4  will  go  into  17 
tour  times  ;  but,  as  on  trial  it  is  found  that  4  is  loo  big  fur  the 
first  digit  of  tlie  dividend,  3  is  selected  ;  3  is  therefore  written 
below  the  last  digit  of  the  divisor  and  next  to  the  digit  of  the 
dividend  last  found.  The  provcss  of  multiplying  the  divisor 
bjr  3  and  Hubtrsctiug  from  the  number  to  be  divided  ia 
indicated  in  figure  ii,  and  shews  that  the  remainder  u  3818. 
A  similar  process  is  then  repeated,  that  is,  47:1  is  divided  into 


DIVISION.  199 

3818,  Ehflwing  th»t  the  quotient  is  38  and  the  remundor 
42.  Thin  ia  repmcnted  in  figttre  iii,  whicli  abewi  the  wbd« 
operation. 

The  method  dencribed  above  nerer  t&atiA  mach  faivmir 
in  Italy.  The  present  Byst^m  was  In  ase  there  as  early  as  tho 
banning  of  the  fourteenth  century,  but  the  metbod  generally 
raiployed  was  that  known  an  the  galley  or  tcralA  lystein. 
The  following  eiample  from  TarUglia,  in  which  it  u  nqoirad: 

49 
0590 
1  3  3  n  (15 

844 


to  divide  1330  by  84,  will  iPrve  to  illustrats  thh  nwtbod :  the 
arithmetic  given  by  Tartaxltn  is  Hliewn  nliove,  where  numbere 
in  thin  type  are  supposed  to  be  scratched  otit  in  the  eonrse  ot 
the  work. 

Tlie  prooevs  is  as  follows.  First  write  the  84  bmeath  tits 
1330,  as  indicated  below,  tlien  84  will  go  into  133  eoe^  heiMa 
the  first  figure  in  the  quotient  is  1.  Now  1  hSkS,  vhieh 
snbtmcted  from  13  leavea  ft.  Writ«  this  ftbove  tbe  13,  and 
cuncel  the  13  and  the  8,  and  we  have  as  tbe  malt  o(  tbe 
fint  step 

5 
1330(1 

84 

Next,  1  X  4  <=  4,  which  subtracted  from  53  leavet  49.  ItueK 
Uie  49,  and  c*acel  tbe  53  and  the  4,  and  w«  ban  m  tbe  rnoA 


59 
1330(1 


which  afaewe  •  cwoainder  490. 


SOO    THB  DBVUAPMBMT  or  AKmilCTia    1800-16S7. 


in  the  quctUent 
bav* 


tg«n 


4 

59 
1SS0(15 

844 

8 

Then  5  x  8  =  40.  and  this  subtmcted  from  49  Imrm  9.  Inaeri 
the  9,  and  caiioel  the  49  and  the  8,  and  we  have  the  foUowing 
result 

49 

5  9 

1  3  3  0  (  IS 

844 

8 

Next  5  X  4  »  30»  and  this  subtracted  from  90  leaves  70.  In- 
sert the  70,  and  cancel  the  90  and  the  4,  and  the  final  resnlti 
shewing  a  remainder  70,  is 

7 

49 
590 

I  3  3  0  (  15 
84  4 

8 


The  three  extra  seros  inserted  in  Tartsglia's  work  are  un- 
necessary, but  they  do  not  affect  the  resulti  as  it  is  evident 
that  a  figure  in  the  dividend  may  be  shifted  one  or  more 
places  up  in  the  name  vertical  column  if  it  be  convenient 
to  do  so 

The  medieval  writers  were  acquainted  with  the  method 
now  in  use,  but  considered  the  scratch  method  mora  simple. 
In  some  cases  the  latter  is  very  clumsy  as  may  be  illustrated 


THK  DEVELOPMENT  OP   ARITMMEnc.      1300-16S7.     !Q1 

ij  the  following  ezKmple  inken  from  PndoU.     Tba  olgeei  is 
to  divide  23400  Uj  100.     The  rrsult  in  obtuned  thu 


04  0 
03  4  00 
3  3  4  0  0  (  234 
10000 

1  00 
■      1 

The  galley  method  wiu  niml  in  Indin,  and  the  ftaliuts 
may  have  derived  il  thence.  In  Italy  it  becttme  obanleta 
Romewhefe  nboat  1600;  but  it  continued  in  ptutiid  km  for 
«t  least  another  century  in  other  countries.  I  ahoald  add  that 
Napier's  rodn-  can  be,  and  sometimes  were,  aied  to  obtain  tho 
result  of  dividing  one  number  by  another 

(ii)  The  signa  -•■  and  -  to  indicate  addition  and  sabtrao- 
tion*  occur  in  Widmnn'R  arithmetic  published  in  I4S!>,  bat 
were  first  bmught  into  general  notice,  at  any  rate  as  symbols 
of  operation,  liy  Stifel  in  I5S4.  I  believe  I  am  ctnrect  in 
saying  that  Vieta  in  1591  was  the  first  well-known  writer 
who  used  these  signs  consistently  throughout  his  work,  and 
that  it  was  nr>t  until  the  beginning  of  the  seventeenth  century 
that  they  became  recogniEiil  and  well-known  syHbols.  The 
sign  ~  to  denote  e(]ualityt  woe  introduced  by  Reconl  in 
1557. 

(iii)  The  invention  of  logarithms^,  withont  which  many 
of  the  numerical  calculations  which  have  eonstaotly  to  be 
made  wotdd  be  practically  impossible,  was  doe  to  Napier  of 
Uerchistoun.  The  first  public  announcement  of  the  discovety 
was  made  in  his  Mirifid  LoyariAmorHm  Cnnonia  Dtter^io, 
pablished  in   1614,  and  of  which  an  English  t 

■  See  Mow,  pp.  S19,  SIS,  S30,  S31. 
t  B«a  below,  p.  aao. 

X  8n  tb«  attid*  on  Lo^itritJIiii  Id  Ihi 
aintii  edition;  mta  sImi  below,  pp.  S4»— SU. 


tot     THE  DBVBLOPMSNT  Or  ABITHllKna     lSOO-1637. 


iasufld  in  the  following  year ;  bnt  ho  had  privntdy  ooomiani- 
cated  a  saimnary  ol  hit  malU  to  Tycho  Braho  as  aaiiy  aa 
1594.  In  this  work  Napier  explains  the  natore  of  logarithnia 
by  a  comparifKm  between  corresponding  terms  of  an  arith- 
metical and  geometrical  progression.  He  illustrates  their  uss^ 
and  gives  tables  of  the  logarithms  of  the  sines  and  tangents 
of  all  anglt^  in  the  first  quadranti  for  differences  of  every 
minute,  calculated  to  seven  places  of  decimals.  His  definition 
of  the  logarithm  of  a  quantity  ft  was  what  we  should  now 
express  by  lO'lug^  (107m)*  This  work  is  the  more  interesting 
to  us  as  it  is  the  first  valuable  contribution  to  the  progress 
of  mathematics  which  was  uiade  by  any  British  writer.  The 
method  by  which  the  logarithms  were  ciUculated  was  explained 
in  the  dmsirueito^  a  posthumous  work  issued  in  1619 :  it 
seems  to  have  been  very  laborious  and  depended  either  on 
direct  involution  and  evolution  or  on  the  formation  of  geo- 
metrical means.  The  method  by  finding  the  approximate 
value  of  a  oonvergrnt  series  was  introduced  by  Newton, 
Cotes,  and  Euler.  Napier  had  determined  to  change  the 
base  to  one  which  was  a  power  of  10,  but  died  before  he 
could  effect  it. 

The  rapid  recognition  throughout  Europe  of  the  advantages 
of  using  logarithms  in  practical  calculations  was  mainly  due  to 
Briggs,  who  was  one  of  the  earliest  to  recognize  the  value  of 
Napier's  invention.  Briggs  at  once  realized  that  the  base  to 
which  Napiers  higarithms  were  calculated  was  very  incon- 
venient; he  accordingly  visited  Napier  in  16 16,  and  urged 
the  change  to  a  decimal  base,  which  was  recognized  by  Napier 
as  an  improvement.  On  his  return  Briggs  immediately  set 
to  work  to  calculate  tables  to  a  decimal  base,  and  in  1617  he 
brought  out  a  table  of  logarithms  of  the  numbers  from  1  to 
1000  calculated  to  fourteen  places  of  decimals. 

It  would  seem  that  J.  Biirgi^  independently  of  Napier, 
had  constructed  before  1611  a  table  of  antiloganthms  of  a 
series  of  natural  numbers:  this  was  published  in  1620. 
In  the  same  year  a  table  of  the  logarithms,  to  seven  places  of  ' 


THE  DEVELOPMENT  OF  ARITHMETIC.      1300-1637.      203 

decim«l8,  of  ttie  sinn  >nd  tADj^nU  of  nnglcs  in  the  first 
quadrant  wnn  hraaclit  out  by  Edmund  Canter,  one  of  the 
tin-sham  Ircturen.  Four  yean  later  the  Intter  mathenwticiMi 
introdncrd  a  ninipie  form  of  Elkle-rulp,  or  an  he  called  it  a 
"line  of  numbera,"  which  provided  a  iDoclianical  method  for 
finding  the  product  nf  two  numbers.  In  the  year  last  men- 
tioned (1634)  Bri^H  published  tnbleH  of  the  logarithms  of 
additional  numbers  nnd  of  various  trigonontetrical  functions. 
Hia  logarithms  of  the  nntural  nunibera  are  equal  to  those  to 
the  Invm  10  when  mulliplicd  Uy  10",  and  of  the  sines  nf  angles 
to  Ihom  to  the  liosu  HI  ilirn  multipliH  hy  10".  Tl»  calcu- 
lation of  the  logarithm!)  of  70,000  numlwra  which  bad  lieen 
omitted  by  Brigqs  from  his  tables  of  1624  was  performed  by  . 
Adrian'Vlacq  and  published  in  1(>2S:  with  this  addition  the 
table  gave  ihc  logarithmn  of  all  numltcra  from  I  to  101,000. 

The  Arithm'iien  /^tgarilhmicn  of  Briggs  and  Vlacq  are 
snb«tantially  the  same  ns  the  existing  tables:  part*  have 
at  difTerent  times  be<'n  reralcalnted,  but  no  taMea  of  an  equal 
tsnge  and  fulness  entirely  founded  on  fresh  computations  have 
been  published  since.  These  tables  were  supplemented  by 
Briggs's  TrigonotHrlrifa  Jirilnnniea,  which  contains  tables  not 
only  of  the  logarithm!)  of  the  trigonometrical  fanctiona,  but 
also  of  their  natural  values  :  it  wn!<  published  posthumously  in 
1653.  A  table  of  loRarithms  to  the  luise  e  of  the  numhna  ' 
from  1  to  KkiO  and  of  the  sines,  tangcnta,  and  secants  of 
angles  in  the  first  quadrant  wan  published  liy  John  Speidell  at 
London  as  early  as  1619,  but  of  course  these  wen  not  so 
useful  in  practical  calculations  as  those  to  the  base  10.  By 
1630  tables  of  logarithms  were  in  general  u9& 

(iv)  The  introduction  of  the  decimal  notation  for  (nctions 
is  also  (in  my  opinion)  due  to  Briggs.  Stevinus  had  in  1965 
used  a  somewhat  similar  notation,  for  he  wrote  a  number 
such  as  25-379  either  in  the  form  25,  3'  7"  0*",  or  in  tho  form 
25®307®9®i  Napier  in  1617  in  his  e^y  on  rods  had 
adopted  the  former  notation  ;  and  RndolfT  had  uwd  a  eome 
what  ainular  notation.    Biiip  also  employed  dwiaMl  bmitiaaa. 


S04     TAB  DBVBU>PMBNT  OP  AERHMBna     1SOO-16S7. 

writing  141'4  m     q  .     But  the  Above-mentioned  writers  bed 

employed  the  notation  onl j  «t  a  oonciae  way  of  stating  resttlti^ 
and  made  no  use  of  it  as  an  operative  form.  Tlie  same  nota- 
tion oocars  however  in  the  tables  published  by  Briggs  in  1617, 
and  would  seem  to  have  been  adopted  by  him  in  all  his  works; 
and.  though  it  is  difficult  to  speak  with  absolute  certainty.  1 
have  myself  but  little  doubt  that  he  there  employed  the 
symliol  as  an  operative  form.  In  Napier's  posthumous  Com- 
giruetio  pul>lislied  in  1G19  it  is  defined  and  used  systematically 
as  un  ofnTative  fonn,  and  as  this  work  was  written  after 
eonsultntion  with  HriftgK,  about  1GIJV~C,  and  probably  was 
revised  by  the  latter  lii*fore  it  was  isMued«  I  think  it  confirms 
the  view  that  the  invention  is  duo  to  Briggs  and  was  com- 
municated by  him  to  Napier.  At  any  rate  it  was  not  em- 
ployed as  an  operative  form  by  Nspier  in  1617,  and,  if  Napier 
were  then  acf|uainted  with  it,  it  must  lie  supposed  that  he 
regarded  its  use  as  unMuitable  in  ordinary  arithmetic.  Before 
the  sixteenth  century  fractions  were  commonly  written  in  the 
sexagesinial  notation*. 

In  Napier's  work  of  1619  the  pciiiit  is  written  in  the  form 
now  adopted,  but  Briggs  underlined  the  decimal  figures,  and 
would  have  printed  a  number  such  as  25*379  in  the  form 
25379.     Sulisequent  writers  added  another  line  and  would 

have  written  it  as  25|379 ;  nor  a-as  it  till  the  beginning  of  the 

eighteenth  century  tliat  the  current  notation  was  generally 
employinl,  and  even  yet  the  notation  varies  slightly  in 
dillerent  countries.  A  knowledge  of  the  decimal  notation 
becaiiie  geueiul  among  practical  men  with  the  introduction  of 
the  French  decimal  staudanJs. 

*  For  examples,  see  above,  pp.  100,  104, 176. 


CHAPTEK  XII. 

TUB  HATIIEMATK-H  OF  THE    HEN  A IH8ANCK*. 
ciKc.  1450-1637. 

Thr  liut  chAptpr  is  ft  (IiKTi-wiion  from  the  chronohipaU 
ftirftngetnent  to  which,  ns  fitr  an  p(is.Hi)i)r.  I  hnre  thmqghont 
adhered,  hut  I  trunt  by  rpfcr^nces  in  this  clinpter  to  keep  the 
order  of  erpntn  nnil  di!<o<iverieH  clmr.  I  n-tum  now  to  ttw 
general  hitit<iry  nf  ninthfmfttioi  in  wf^tern  Kiirope.  Mothe- 
m&ticiann  hnd  bftrcly  aMiniihitnl  the  knnwlrflge  olitftined  frnm 
the  Amtn,  includin;;  their  tninHlnlion!i  nf  Rreek  •TTitcm,  when 
the  rrfugppH  whti  ejumpnl  fmm  Ci)nHtnntinnple  after  the  fnti  of 
the  enitem  em|iirp  hrwusht  the  tiri^Einid  *nrkn  Mid  the  tnidt- 
tions  of  (!r»>ek  scieniw  into  Italy.  Tliui  by  Ihft  middle  of  the 
fifteenth  rentury  the  chief  rpnuIlN  of  Gr(«k  and  Aralrian 
matheiiiaticH  were  a«ce.<«.«il>le  to  Eumponn  Htndenta. 

The  invention  nf  printing  nlxiut  Ihnt  time  renderrd  the 
din-semi nation  of  dis<^vrries  coniparitively  eiwy.  It  is  almnitt 
a  truism  to  remark  that  until  printing  wan  introduced  a  writer 
appralnl  to  a  vpry  limitetl  class  of  rrnilern,  liot  we  are  perhapa 
apt  to  forget  that  when  a  me<licval  writfr  "pabtinhed"  a  wo?k 
the  results  were  known  to  only  a  frw  of  his  contemponiriM. 
Tbia  had  not  been  the  oue  in  claiwical  timei,  tor  then  and 

*  When  DO  other  nlncnees  arn  ginn,  ice  |«rta  sti,  iltt,  Str,  mat 
Iba  cati;  cfaaptm  ot  put  rr  ot  Canlor's  ror(<«iiiif«a;  <m  Um  Haliaa 
mathmiatieiani  of  tbla  period  nee  bIwi  Ouil.  Libri,  miMr*  4m  tHncn 
mmtklmttitmn  n  /lalir,  *  toU.  Parte,  18BI1-1U1. 


too  THE  lUTHSMATICB  OP  THI  BKNiMHBAIIffB. 

uutil  the  fourth  oentary  of  our  era  Alezandria  wm  tha  noo^ 
niied  oentre  for  the  reoeptioD  and  diuemhuiiioii  of  now  vorin 
and  diflooveriet.  In  medieval  Europe  on  the  other  hand  theia 
was  no  eommon  centre  through  which  men  of  icienoe  eonld 
communicate  with  one  another,  and  to  this  cause  the  dow 
and  fitful  development  of  medieval  mathematict  may  be  partly 
ascribed. 

The  intitxluction  of  printing  marks  the  beginning  of  the 
modem  world  in  science  as  in  politics ;  for  it  was  contempo- 
raneous with  the  assimilation  by  the  indigenous  European' 
school  (which  was  bom  from  scholasticism*  and  whose  hbtoiy 
was  traced  in  chapter  viil)  of  the  results  of  the  Indian  and 
Arabian  schools  (whose  history  and  influence  were  traced  in 
chapters  ix  and  x)  and  of  the  Greek  schools  (whose  history 
was  traced  in  chapters  ii  to  v). 

The  last  two  centuries  of  tliis  period  of  our  history,  which 
may  be  described  as  the  renaissance,  were  distinguished  by 
great  mental  activity  in  all  branches  of  learning.  The  creation 
of  a  fresh  group  of  universities  (including  those  in  Scotland) 
of  a  somewhat  less  complex  type  tlian  the  medieval  universities 
above  described  testify  to  the  general  dcsir^  for  knowledge. 
The  discovery  of  America  in  1492  and  the  discussions  that 
preceded  the  Reformation  flooded  Europe  with  new  ideas 
which,  by  the  invention  of  printing,  were  widely  disseminated  ; 
but  the  advance  in  mathematics  was  at  least  as  well  marked 
as  tliat  in  literature  and  that  in  politics.  \ 

During  the  first  parC  of  this  time  the  attention  of  mathe- 
maticians was  to  a  large  extent  concentrated  on  syncopated 
algebra  and  trigonometry :  the  treatment  of  these  subjects 
is  discussed  in  the  first  section  of  this  chapter,'  but  the  relative 
importance  of  the  mathematicians  of  this  po-iod  is  not  very 
easy  to  determine.  The  middle  years  of  the  ilenaissance  were 
distinguished  by  the  development  of  symbolic  algebra :  this  it 
treated  in  the  second  section  of  this  chaptei^.  The  close  of 
the  sixteenth  century  saw  the  creation  of  the  science  of  dyna- 
mics :  this  fonns  the  subject  of  the  first  section  of  chapter  JUL 


BEOIOHONTAMUa  Wl 

Aboot  the  tinnie  time  anil  in  the  early  yfun  of  the  aeventeenth 
centnrf  conaidetmble  ntt^ntion  wan  paid  to  pure  gemnetry : 
this  forms  the  subject  of  the  Bccond  nection  of  clwpt«r  Xlll. 

The  developtnent  of  ti/ncojxited  algebra  and  trigonometry. 

Rflglomontantis*.  Aimnigst  the  mknjr  distinguished 
writers  of  this  time  Johann  N'-jiomrmfaHiM  wm  the  earliest 
and  one  of  the  most  able.  He  wivi  l>om  at  KiiniKsberg  on 
June  6,  U3G,  and  died  nt  Uouio  on  July  6,  I47fi.  His  real 
name  won  Johnjin''Ji  ifulUr,  but,  following  the  cnstom  of  that 
time,  he  issued  \i\%  publications  under  a  Latin  pseudonym 
which  in  his  cose  was  taken  from  Ins  birthplocfl.  To  his 
friends,  bis  neighltoure,  ond  his  trodenpeople  he  may  hare 
been  Johannes  Miiller,  but  the  literary  and  scientific  world 
knew  hira  as  Kegioniontanus,  junt  as  they  knew  Zepeinik  ■• 
Copernicus,  and  Schwarzerd  as  Melanchtbon.  It  seeras  as 
pedantic  as  it  is  confuting  to  refer  to  an  author  bj  his  actual 
name  when  he  is  universally  recognited  onder  Miother :  I 
shall  therefore  in  all  cases  an  far  as  powHible  us«  that  title 
only,  whether  latinized  or  not,  by  which  a  writer  is  genemlly 

IlegiomontAnun  studied  mathematics  at  the  nniverstty  of 
Vienna,  then  one  of  the  chief  centres  uf  mathematical  stndien 
to  Europe,  under  Turbach  who  was  prufessor  there^  His 
first  work,  done  in  conjunction  with  Purlmch,  consisted  of  an 
analysis  of  the  AlmaqrM.  In  this  the  trigonometrical  fonctions 
tine  and  emine  were  used  and  a  table  of  natural  sinea  was 
introduced.  Furboch  died  before  the  book  was  finished;  it 
was  finally  published  at  Venice,  but  not  till  1496.     As  sooo  m 

■  Hii  Uf«  «■■  «TiltcD  b;  P.  OkMcndi,  The  Hsinw,  SMond  sditioa, 
IS59.  His  Icltern,  which  afford  maeh  Tslokl-le  intonoatiaa  «■  lbs 
ustbamatim  ot  fala  tim*.  were  eollrcted  and  vclited  hjC.Q.  Ton  Hnr, 
NarembarR,  ITM.  An  acooDDl  of  hia  work*  will  h«  foand  te  lUflmmm 
brm.  Ha  t*i"<»"  farViu/fr  irt  Carmitiu.  bj  A.  tlagln,  OnalM, 
U74 1  BM  olw  Conlor,  elwp.  ta. 


208  THK  MATUKMATIGB  OP  THI  EDfAOHAMOB. 

this  was  oomplalad  RflgiomooUniu  wrote  a  work  on  Mtrology; 
which  contains  some  Mtnmomical  teUes  and  a  Uble  of  aaUural 
tangente :  this  was  published  in  1490. 

Leaving  Vienna  in  1462,  Regiomontanns  travelled  for 
some  time  in  Italy  and  Germany ;  and  at-last  in  1471  settled 
for  a  few  years  at  Nuremberg  where  he  established  an  obser* 
vatory.  opened  a  printing-press,  and  probably  lectured.  Three 
tracts  on  astronomy  by  him  were  written  here.  A  mechanical 
eagle,  which  Happed  its  wings  and  saluted  the  Emperor 
Maximilian  I.  on  his  entry  into  the  city,  bears  witness  to 
his  mechanical  ingenuity  and  was  reckoned  among  the  marvels 
of  the  age.  Thence  Regiomontanus  moved  to  Rome  on  an 
invitation  from  Sixtus  IV.  who  wished  him  to  reform  the 
calendar.  He  was  assassinated,  shortly  after  hb  arrival,  at 
the  age  of  40. 

Regiomontanus  was  among  the  first  to  take  advantage  of  | 

the  recovery  of  the  original  texts  of  the  Qreek  matliematical 
works  in  order  to  make  himself  acquainted  with  the  methods 
of  reasoning  and  results  there  used;  the  earliest  notice  in 
modem  Europe  of  the  algebra  of  DiophantUM  is  a  remark  of 
his  that  he  hod  seen  a  copy  of  it  at  the  Vatican.  He  was 
also  well  read  in  the  works  of  the  Arab  mathematicians. 

Tlie  fruit  of  this  study  was  shewn  in  his  De  Triauguii$ 
written  in  1464.  This  is  the  earliest  modem  systeihatic 
exposition  of  trigonometry,  plane  and  spherical,  though  the 
only  trigonometrical  functions  introduced  are  those  of  the  sine 
aud  cosiue.  It  is  divided  into  live  books.  The  first  four  are 
given  up  to  plane  trigonometry,  and  in  particular  to  determin- 
ii  g  triangles  from  three  given  conditions.  The  fifth  book  is 
devoted  to  spherical  trigonometry.  The  work  was  printed  at 
Nuremberg  in  1533,  nearly  a  century  after  the  death  of 
Regiomontanus. 

As  an  example  of  the  mathematics  of  this  time  I  quote  one 
of  his  propositions  at  length.  It  is  required  to  determine  a 
triangle  when  the  difference  of  two  sides,  the  perpendicular  on 
the  base,  and  the  difference  between  the  segments  into  which 


REGIOMONTANU& 


209 


the  bsfie  ia  thus  divided  are  given  [book  ii,  prop.  23].     The 
following  is  the  solution  given  bj  Regiomontanns. 

• 

8il  Islis  IriAiigaliM  ABO,  cujtm  dtio  laters  AB  ti  AO  differenlts 
habesnl  noU  HO,  dnetaqiie  perpendieoUri  AD  daoram  csAimni  BD  el 
DO,  dillemilia  tit  EO  :'hae  dnme  differentifte  mnt  dmtae,  et  ipsa  peipeii- 
dieahtfit  AD  data.  Pieo  qood  omnia  Uterm  triaogoli  noCa  eonelodentar. 
For  aiiem  lai  el  eentat  hoc  pn>blem«  al>iolTemus.  I>eior  ergo  diffcreDtia 
latcffVB  al  8,  diffcieotia  CMamii  12,  ti  perpendicnlarii  10.  Pooo  pro 
Wsl  aaaai  lam,  el  pro  ai^grvicato  lat«niia  4  ret,  use  proportio  bssis  sd 


eofifferiem  latermn  eflt  at  IIG  td  XrE,  wilicet  nnias  ad  4.  Erit  ergo  BD 
\  rei  minas  6,  sed  AB  erit  3  nn  dcroptiH  }.  Duco  A  Bin  ae,  produeantur 
4  eeoMna  et  2|  demptU  0  rebuR.  Item  BD  in  sc  facit  |  ecn^aii  et  36 
miniM  6  vehaa:  haio  addo  qoadratnm  dc  10  qui  cat  100.  Collifnintur  | 
ern«a«  et  ISO  minnii  6  n*bna  arquales  videlicet  4  cenftibuH  et  2|  dcmptia 
6  rebaa.  RcKtaarando  itaqne  dcfectafl  ct  aoferendo  ntmbiqar  aeqnalia, 
qnemadmoilani  are  ipm  praecipit,  liabcmna  eenauii  aliquot  aeqnalea 
nnroero^  iinde  eognitio  rei  patebit,  et  indc  tria  latera  triangnli  more  sao 
inootoieet. 

To  explain  the  language  of  the  proof  I  should  add  that 
Regiomontanus  always  calls  the  unknown  quantity  re$^  and 
its  square  censfts  or  zenmu  \  but  though  he  uses  these  technical 
terms  he  writes  the  words  in  full.  He  commences  by  saying 
that  he  will  solve  the  problem  by  means  of  a  quadratic  equa- 
tion (per  artem  rei  et  census) ;  and  that  he  will  suppose  the 
diffeienoe  of  the  sides  of  the  triangle  to  be  3,  the  differsnoe 
of  the  segments  of  the  base  to  be  12,  and  the  altitude  of  the 
triaBgle  to  be  10.    He  then  takes  for  his  vnkiMywii  quantitj 

&  14 


210 


THI  MATHIlUTiOS  OP  THS  BKNillWAWOlL 


(oiuun  ran  or  x)  the  bwie  of  the  triaii|^  and  theraftm  tha 
•mn  of  the  udet  will  be  iae.  Tliereforo  BD  will  be  eqoal  to 
|ap-6  (I  rei  miniifG),  and  J  J?  will  be  equal  to  8«-t  (>  vm 
dempiis  I) ;  hence  J  IP  (iii?  in  ae)  will  be  4aE^  4- S|  -  te  (4  oeiiMis 
et  2|  demptis 6  rebus), and ^I)" will  be }«'••- 56 -  6«.  ToBJ^ 
he  adds  AL^  (quadratum  de  10)  which  is  100,  and  slates  that 
the  sum  of  the  two  is  equal  to  AB*.  This  he  says  will  give 
the  value  of  x*  (census),  whence  a  knowledge  of  x  (cognitio  rei) 
can  be  obtained,  and  the  triangle  determined. 

To  express  this  in  the  language  of  modern  algebra  we  have 

AG^-'AB'^DG^^DB'. 


Therefore 
Hence 


but  by  the  given  numerical  conditions 

il(?-ilir  =  3  =  l(/)^-/)^), 
.-.    AG^AB=-i(DG^DB)^ix. 
il2r=2x-|,  and  BD^\x^^. 
(2«-})«=(|x^6)U100. 

From  which  x  can  be  found,  and  all  the  demenU  of  the 
triangle  determined. 

It  is  worth  noticing  that  Regiomontanus  merely  aimed  at 
giving  a  general  method,  and  the  numbers  are  not  chosen  with 
any  special  reference  to  the  particular  problem.  Thus  in  his 
diagram  he  does  not  attempt  to  make  GE  anything  like  four 
times  as  long  fui  Cr//,  and,  since  x  is  ultimately  found  to  be 

equal  to  ^^21,  the  point  D  really  falls  outside  the  base.  Tha 
order  of  the  letters  ABG^  used  to  denote  the  triangle^  is  o( 
course  derived  from  the  Greek  alphabet. 

Some  of  the  solutions  which  he  gives  are  unnecessarily 
complicated,  but  it  must  be  remembered  that  algebra  and 
trigonometry  were  still  only  in  the  rhetorical  stage  of  develop- 
ment,  and  when  every  step  of  the  argument  is  expressed  in 
words  at  full  length  it  is  by  no  means  easy  to  realise  all  that 
is  contained  in  a  formula. 


REGIOMONTANU&     PURBACH.     CUSA. 


211 


It  will  be  observed  from  the  above  example  that  Regiomon- 
tanus  did  not  hesitate  to  apply  algel>ra  to  the  solution  of  geo- 
metrical problems.  Another  illustration  of  this  is  to  be  found 
in  his  discussion  of  a  question  which  appears  in  Brahmagupta's 
Siddhanta.  The  problem  was  to  construct  a  quadrilateral, 
having  its  sides  of  given  lengths,  which  should  be  inscribable 
in  a  circle.  The  solution^  given  by  Regiomontanus  was 
effected  by  means  of  algebra  and  trigonometry.  ' 

The  AfgoriihmfiM  D^monMraUts  of  Jordanus,  described 
above,  which  was  first  printed  in  1534,  was  until  recently  uni- 
versally attributed  to  Regiomontanus.  This  work,  which  is 
ooncemed  with  algebra  and  arithmetic,  was  known  to  Hegio- 
montanus  and  it  is  possible  that  the  text  which  has  come  down 
to  us  contains  additional  matter  contributed  by  him. 

Regiomontanus  was  one  of  the  most  prominent  mathema- 
ticians of  his  generation  and  I  have  dealt  with  his  works  in 
some  detail  as  typical  of  the  most  advanced  mathematics  of 
the  time.  Of  his  contemporaries  I  shall  do  little  more  than 
mention  the  names  of  a  few  of  those  who  are  best  known; 
none  were  quite  of  the  first  rank  and  I  should  sacrifice  the 
proportion  of  the  parts  of  the  subject  were  I  to  devote  much 
space  to  them. 

Pnrbaoht-  I  may  begin  by  mentioning  George  Purback, 
first  the  tutor  and  then  the  friend  of  Regiomontanus,  bom 
near  linx  on  May  30,  1423,  and  died  at  Vienna  on  April  ^ 
1461,  who  wrote  a  work  on  planetary  motions  which  was 
published  in  1460;  an  arithmetic,  published  in  1511 ;  a  table 
of  eclipses,  published  in  1514;  and  a  taUe  of  natural  sine^ 
published  in  1541. 

Onsal*  Next  I  may  ^nention  Nieolae  von  Cuea^  who  was 
bom  in  1401  and  died  in  1464.    Although  the  mm  of  a  poor 

*  II  was  pablinhsd  by  C.  O.  nm  Marr  at  Nuremberg  in  1786. 

t  His  life  was  written  by  P.  Gaeeendi^  The  Hague,  eeoona  sditioB, 
1€M. 

t  His  life  was  wfiltcB  by  F.  A.  Sefaaipff,  Tftbiiifen,  1871;  ao4  Us 
•oOssM  works.  sdiM  Ij  H.  Brtri*  WM  pubUaheA  al  Bile  in  IMIu 

14— S 


212         TBI  lUTflSMATICB  OP  THE  MtWAIMAIICf. 

fiahermfta  and  witlioat  infliienoe^  ho  roia  r»pMlly  in  the  okmht 

and  in  spite  of  being  **m  rBfonner  before  the  veComiotMMi  * 

become  m  cardinoL     His  motbemoticol  writings  dool  with  the  •  I 

reform  of  the  calendar  and  the  quadrature  of  the  eirolo:  in 

the  latter  problem  his  constroetion  is  equivalent  to  taking 

I  {J'S  -¥  Jii)  as  the  value  of  w.     He  argued  in  favour  of  the 

diurnal  rotation  of  the  earth. 

Ohnquet.  I  maj  also  here  notice  a  treatise  on  arith- 
metic, known  as  Le  7'ripariy\  by  Nicolas  Ckuquei^  a  bachelor 
of  medicine  in  the  university  of  PariS|  which  was  written  in 
1484.  This  work  indicates  that  tlie  extent  of  mathematics 
then  taugbt  was  somewhat  greater  than  was  generally  believed 
a  few  years  ago.  It  ooiitaius  the  earliest  known  use  of  the 
radical  sign  with  indices  to  mark  the  rout  taken,  3  for  a 
square-ruot,  3  for  a  cube-ruot,  and  so  on ;  and  also  a  definite 
statement  of  the  rule  of  signs.  The  words  plus  and  minus  are 
denoted  by  the  contractions  p,  iu.     The  work  is  in  French. 

Introduction t  of  signs -r  and-.  In  England  and 
Germany  algorists  were  less  fettered  by  precedent  and  tradi- 
tion than  in  Italy,  and  introduced  some  improvements  in 
notation  which  were  hardly  likely  to  occur  to  an  Italian.  Of 
these  the  most  prominent  were  the  introduction,  if  not  the 
invention,  of  the  current  symbols  for  addition,  subtraction, 
and  equality. 

The  earliest  instances  of  the  regular  use  of  the  signs  -i- 
and  -  of  a'hich  we  liave  any  knowledge  occur  in  the  fifteenth 
century.  Johanna's  Wiiiman  of  Eger,  bom  about  1460,  matricu- 
lated at  Leipzig  in  1480,  and  probably  by  profession  a  physi- 
cian, wrote  a  Mtrcantiie  Arithmetic,  published  at  Leipzig  in 
1489  (and  modelled  on  a  work  by  Wagner  printed  some  six 

*  See  an  article  by  A.  Marre  in  BonoompsKni't  BuiUtimo  di  ^{^•' 
gr«{fia  for  18S0,  vol.  \iu,  pp.  5.>5— 659. 

t  8ee  articles  by  P.  TreuUein  (Die  dtuUeke  Cou)  in  the  Abkamdlmmfem 
tur  Getekickte  der  Maihematik  for  1879 ;  by  Pe  Morgan  in  the  Cambridge 
Pkilimupkieal  TraHtactious,  1871,  voL  ii,  pp.  203—912;  and  by  Bon- 
oompagni  in  the  BuiUtimo  di  bibliogrt^/ia  for  1876,  voL  n,  pp.  186—210. 


iirntoDUCTioN  or  siqnh  for  plus  AMD  mNCa    813 

or  seven  yt-an  earlier):  in  thin  \took  tbcM  tigna  *ra  nwd 
merely  wi  niftrks  Biftnifying  exceiM  or  deficiency ;  the  corre- 
■ponding  nse  of  tli«  word  nurpluN  or  oveipliw*  «»  onco 
comtnon  Mid  ■<!  still  ivtninM]  in  coninienn. 

It  is  notiM^lile  tlint  tlie  oigns  ^ni-mlly  occur  only  in 
]>ractical  mercantile  ijueNtionii :  hcnre  it  ImW  lieen  cot^ectured 
that  they  were  originnlly  warehtmse  nrnrkn.  S"me  kinds  ol 
goodn  were  sold  in  s  sort  of  wuoilen  chrxt  cnlled  a  lai/^,  which 
wlien  full  woa  np|>.-trently  expected  to  weigh  rouglily  either 
three  or  four  cutii^.m  ;  if  one  of  these  cnaen  were  a  little 
lighter,  sny  5  llm..  tlinn  fuur  centners  Widnian  descrilicB  it  ■■ 
weighing  4e  -■llln.:  if  it  were  5  lln.  henvier  than  the  nomml 
weight  it  id  dencriliett  na  weighing  Ak — |_5tha.  Thcsymbda 
ftre  used  an  if  they  would  lie  fiiunlinr  to  his  reiulen;  ftnd 
there  are  mme  slight  re<vu>ns  for  thinking  that  thene  marici 
were  chnlked  on  the  che^t.s  ns  they  came  into  the  wamhooses. 
We  infer  that  the  more  usunl  case  was  for  a  chest  to  weigh 
a  little  less  tlmn  its  n-put«d  weight,  and,  as  the  sign  —  placed 
between  two  numl)erH  whs  a  conimctn  Rymbol  to  signify  aoine 
connection  l>etit'een  them,  thnt  seetiis  to  have  been  taken  aa 
the  standard  ca,ie,  while  the  vertical  l>*r  was  originally  a 
small  mark  supenulded  on  the  sign  -  tw  distinguish  the  two 
symltols.  It  will  Im  oliser^ecl  that  the  i-ertical  line  in  th« 
symhol  for  excess  |irint*<<l  aliove  is  somewhat  shorter  than 
the  horizonUtl  line.  This  is  also  the  v^ruw  with  Stifel  and  nnMt 
of  the  early  wrileni  who  used  the  syi»i)ol:  some  presaea  con- 
tinued to  print  it  in  thi.t,  its  rorliest  ft>rm,  till  the  end  o(  ths 
seventeenth  century.  Xylander  on  the  other  band  in  1575 
has  the  verticnt  hnr  much  longer  than  tb«  hofuontal  lin^ 
and  the  symbol  is  something  like  -f . 

Another  conjecture  is  that  the  synilml  for  j>lu4  is  derived 
fram  the  Latin  abbreviation  <f-  for  'I ;  while  that  for  Mtntu  is 
obtained  from  the  bar  which  is  oft«n  used  in  ancient  mana- 
•eripta  to  indicate  an  omiaslon  or  which  b  written  over  th» 

•  BMrM(aLedt.SZT,MMn,BBdIlbMalLt.MM4L 


tl4  Tqi  If ATHBHATICa  OF  THB  RBMAiatAirOI. 


eoQtraeted  form  of  m  woid  to  signify  that  certain  leltaw  have 
been  left  out.  Thb  riew  hat  been  often  Mipported  on  a  ptioci 
groonda,  bat  it  haa  recently  found  powerful  adTocalet  in 
Profetaori  Zangmeiiiter  and  Le  Pkige  who  also  coniider  thai 
the  introduction  of  these  aymbob  for  plus  and  minus  may  be 
referred  to  the  fourteenth  century. 

These  explanations  of  the  origin  of  our  symbob  for  piu9 
and  fAtfius  are  the  most  plausible  that  have  been  yet  advanced, 
but  the  quetttion  is  difficult  and  cannot  be  said  to  be  solved. 
Another  suggested  derivation  is  that  -i-  is  a  contraction  of  V 
the  initial  letter  in  Old  Oemian  of  plus,  while  —  is  the  limiting 
form  of  m  (for  niinus)  when  written  rapidly.  De  Morgan* 
proposed  yet  another  derivation:  the  Hindoos  sometimes 
used  a  dot  to  indicate  subtraction,  and  this  dot  might  he 
thought  have  been  elongated  into  a  bar,  and  thus  give  the 
sign  for  minus;  wliile  the  origin  of  the  sign  for  pius  was 
derived  from  it  by  a  superadded  bar  as  explained  above :  but 
I  take  it  that  at  a  later  time  he  abandoned  this  theory  for 
what  has  been  called  the  wareliouse  explanation. 

I  should  perhaps  here  add  that  till  tlie  close  of  the  six- 
teenth century  the  sign  -f  connecting  two  quantities  like  a  and 
b  was  also  used  in  the  iieuse  thiit  if  a  were  taken  as  the  answer 
to  some  queMtion  one  of  the  given  conditions  would  be  too  little 
by  b.  This  wua  a  relation  which  constantly  occurred  in  solu* 
tions  of  «|Uf»8tionH  by  the  rule  of  false  sHsumption. 

Lastly  I  would  repeat  again  that  these  signs  in  Widman  are 
only  alil>n*viationM  and  not  MynibolM  of  operation ;  he  attached 
little  i>r  no  importance  to  them,  and  no  doubt  would  have 
been  amazed  if  he  hod  bet-n  told  that  their  introduction  was 
preparing  the  way  for  a  revolution  of  the  processes  used  in 
algebra. 

The  AlyorithmuM  of  Jordanus  was  not  publiahed  till  1534  ; 

Widman's  work  was  hardly  known  outside  (Germany ;  and  it 

is  to  Pacioli  that  we  owe  the  introduction  into  general  use 

of  syncopated  algebra;  that  is,  the  use  of  abbre\iations  for 

*  8m  hU  Ariikmetieai  Hooks,  London,  1S47,  p.  19. 


PACiou.  215 

certain  of  the  more  common  algebraical  quantities  and  opera- 
tions, Irat  where  in  using  them  the  mles  of  syntax  are  ob- 


Paeioli*.  LueoM  Padotl,  sometimes  known  as  Luca9  di 
BurgOf  and  sometimes,  but  more  rarelj,  as  Lfteat  Paciolus^  was 
bom  at  Bnrgo  in  Tuscan j  about  the  middle  of  the  fifteenthv 
centary.  We  know  little  of  his  life  except  that  he  was  a 
Franciscan  friar ;  that  he  lectured  on  mathematics  at  Rome^ 
Pisa,  Venice,  and  Milan ;  and  that  at  the  last  named  city  he 
was  the  first  occupant  of  a  chair  of  mathematics  founded  by 
Sfona :  he  died  at  Florence  about  the  year  1510. 

His  chief  work  was  printed  at  Venice  in  1494  and  is 
termed  SumnM  de  ariihmetiea^  geametria^  proporzioni  e  pro- 
porzianalUa.  It  consists  of  two  parts,  the  first  dealing  with 
arithmetic  and  algebra,  the  second  with  geometry.  This  was 
the  eariiest  printed  book  on  arithmetic  and  algebra.  It  is 
mainly  based  on  the  writings  of  Leonardo  of  Pisa,  and  its 
importance  in  the  history  of  mathematics  is  largely  due  to  its 
wide  circulation. 

In  the  arithmetic  Pacioli  gives  rules  for  the  four  simple 
processes,  and  a  method  for  extracting  square  roots.  He  deals 
pretty  fully  with  all  questions  connected  with  mercantile 
arithmetic,  in  which  he  works  out  numerous  examples,  and  in 
particular  discusses  at  great  length  bills  of  exchange  and  the 
theory  of  book-keeping  by  double  entry.  This  part  was  the 
first  systematic  exposition  of  algoristic  arithmetic  and  has  been 
already  alluded  to  in  chapter  xi.  It  and  the  similar  work  by 
Tartaglia  are  the  two  standard  authorities  on  the  subject. 
Most  of  the  problems  are  solved  by  the  method  of  false  assump- 
tion, but  there  are  several  numerical  mistakes. 

The  following  example  will  servo  as  an  illustration  of  the 
kind  of  arithmetical  problems  discussed. 

IlNQr for  1440 dooaU al  Yenioe 8400 sugar  kiaves,  whose  aMt  vei^t 
is  7100  HIV  I  IpigrasafeslotlMsgvDttpcreeBli;  to  the  waighsis  sad 

•  8m  H.  SteigBiallsr  la  Um  ZtiUekHfi  flr  MtkemaHk^  1080,  «qL 
;  also  Lttri,  vol.  m,  ppu  lli— 14S;  sad  Osator,  sba^  Lvn. 


216  THB  MATHEMATICS  OF  TBI  REVAISSAlfOI. 

canvas,  and  in  feet  to  Ibe  ordinary  paekera  in  Ibe  whole,  i  daoala;  lor 
Um  tax  or  octroi  dutj  on  the  firat  amount,  1  doeat  per  eent ;  aflerwaida 
for  dn^  and  tax  at  the  offiee  of  exports,  8  dneate  per  eent. ;  for  vritinf 
direetiona  on  the  hoxet  and  booking  th^  paieage,  1  doeal ;  ibr  the  baik 
to  Rimini,  19  dueate ;  in  compiimouta  to  the  captains  and  in  drink  lor 
the  crews  of  armed  barks  on  several  ooeasions,  1  dncats ;  in  expenses  fSor 
provisions  for  myself  and  servant  for  one  month,  6  dueats ;  for  expensea 
for  several  short  joum«!}'s  over  land  here  and  there,  for  barbers,  for 
washing  of  linen,  and  of  boots  for  myself  and  servant,  1  dneat ;  upon  my 
arrival  at  llimini  I  pay  to  the  captain  of  the  port  for  porf  duee  in  the 
money  of  that  city,  9  lira ;  for  porters,  disembarkation  on  land,  and 
carriage  to  the  magazine,  6  lire  ;  as  a  tax  upon  entrance,  4  soldi  a  load 
which  are  in  number  32  (sach  beiiiK  the  custom) ;  for  a  booth  at  the  fair, 
4  soldi  per  load ;  I  further  find  that  the  measures  used  at  the  fair  are 
different  to  those  used  st  Venice,  snU  that  140  lire  of  weight  are  there 
equivaleut  to  lUO  at  Venice,  and  that  4  lire  of  tlieir  silver  coinage  are 
equal  to  a  ducat  of  gold.  I  ask  theiefure,  at  how  mnch  I  must  sell  a 
hundred  lire  lUmini  in  order  that  I  may  gain  10  per  cent,  upon  my 
whole  adventure,  and  what  is  the  sum  which  I  must  receive  in  Venetian 
money  ? 

In  the  algebra  he  discusses  in  some  detail  simple  and 
quadratic  equations,  and  prohlems  on  numbers  which  lead  to 
such  equations.  He  mentions  the  Arabic  classification  of  cubic 
equations,  but  adds  timt  their  solution  appears  to  be  as  im- 
possible as  tlie  quadrature  of  the  circle.  The  following  ia  the 
rule  he  gives*  for  solving  a  quadratic  equation  of  the  form 
a:*  ••-  X  =  a :  it  is  rhetorical  and  not  syncopated,  and  will  serve 
to  illustrate  the  inconvenience  of  tliat  method. 

"Si  res  et  census  numero  coaequantur,  a  rebus 
diniidio  sumpto  censum  producers  debes, 
addereque  numero,  cujus  a  radice  totiens 
tolle  semis  rerum,  census  latusque  redibit" 

He  confines  his  attention  to  the  positive  roots  of  equationa. 

Though  much  of  the  matter  described  above  is  token  from 
Leonardo's  Liter  AUicif  yet  the  notation  in  which  it  is  expressed 
is  superior  to  thait  of  Leonarda     Pacioli   followa   Leonardo 

•  Edition  of  1494,  p.  145.        | 


PAaoLi.  217 

nnd  the  Arnba  in  calling  the  unknown  quftstitj  the  Aim/,  in 
Ibitian  rona^hcnco  algolini  wns  simtetimM  Vnowil  mi  Um 
coaxic  art— or  in  Lntin  t'k,  nn<)  rminctinM^  denotes  it  by  <»  or 
R  or  /{j.  He  cntlit  the  square  of  it  ivimim  or  i/mnu  ftiid  Kme- 
timea  denotes  it  hj  ee  or  Z ;  similurly  the  cnbe  of  it,  or  eubn, 
in  noinelinies  represented  hy  rn  or  (.' ;  the  fourth  power,  or 
cKnm  di  txiuo,  is  written  either  nt  length  or  m  ee  (/i  en  or  as 
ee  re.  It  may  be  notiecd  thnt  alt  hJK  c<)uKtio»ii  are  uummoal 
■o  tliat  he  did  not  riw  to  the  coneeptiDn  of  representing  known 
quantities  liy  lett^rt)  an  Jonlanus  had  done  nnd  m  is  tlie  GHse 
in  modem  slgel>ni :  liut  Liiiri  given  two  iiiHtMices  in  which  in 
a  proportion  be  rvpreseiitJ*  a  number  by  a  letter.  He  indicstes 
addition  nnd  efjuality  by  the  initial  letters  of  the  words  jiltu  . 
and  an/niilii,  but  he  generally  evades  the  introduction  of  « 
■ym)K>l  for  mimu  by  writing  hh  (iunntitie4  on  that  aide  of  the 
equation  which  makes  them  positive,  though  in  a  few  places 
he  denotes  it  by  iTi  for  miniu  or  by  ttf.  for  lUmpttu.  This  is  a 
commcncctnent  of  syncopntc<I  algebra. 

There  is  nothing  Etnking  in  the  results  he  anivcs  at  in 
the  second  or  geometrical  pnrt  of  the  work  ;  nor  in  two  other 
tracts  on  geometry  which  he  wrote  nnd  which  were  printed 
•I  Venice  in  IMtt  nnd  1.709.  It  may  be  noticed  however 
that  like  Itt^nmonlnnus  he  applied  algnltrn  to  aid  him  in 
investigating  the  geontetricat  pnipertifs  of  figarcn. 

The  following  prublem  will  illustrate  the  kind  of  geome- 
trical questions  he  ntt.icknt.  The  radius  of  the  inscribed  circle 
of  a  triangle  is  4  inches,  and  the  Hegments  into  which  one  tide 
is  divided  by  the  point  of  contact  arc  6  indies  and  8  inches 
respectively.  Determine  the  other  nicies.  To  eolve  this  U  u 
sufficient  to  remark  that  r«^  A  >^*(«-n)(«-i)(«-e)  which 
givea  4«  -  JiT(§ -  li)  7 cVw,  hence  #  =  21;  therefore  tho 
required  sides  are  31-6  and  21  -  8,  that  ia,  IS  and  IS.  But 
Pacioli  Riaken  no  use  of  titeno  formulae  (with  whidi  he  woa 
acquainted)  hut  gives  an  elalmrate  geometricaj  constntetioB 
and  then  uses  algebra  to  lind  the  lengths  of  Torioaa  segmaita 
(tf  the  lines  be  wants.    The  woric  ia  too  loi^  for  Ma  to 


218  THB  MATHOIATIGB  6^  THB  ■UfAlfWAllCB. 

VBprodiioe  bare^  bat  the  following  moalymM  iff  it  wiD  afbid 
•offident  matoriab  for  its  reprodaetion.  ijet  ABC  be  tbo 
triani^  Df  E^  F  the  points  of  oonUet  of  the  sides,  end  O 
the  centre  of  the  given  circle.  Let  H  be  tbcl  point  of  inter- 
section of  OJSr  end  DF^  and  K  that  of  OC  abd  DE.  Let  L 
and  if  be  the  feet  of  the  perpendiculars  drawn  from  E  and 
F  on  BC.  Draw  JP/»  parallel  to  iiJSr  and  cutting  BC  in  P. 
Then  Pacioli  determines  in  succession  the  magnitudes  of  the 
following  lines :  (i)  OB,  (ii)  OC,  (iil)  FD,  (iv)  FU,  (t)  ifZl, 
(vi)  iTAT.  He  then  forms  a  quadratic  equation  from  the 
solution  of  which  he  obtains  the  values  of  MB  and  MD. 
Similariy  he  finds  the  values  of  LC  and  LD.  He  now  finds 
in  succession  the  values  of  EL,  FAf,  EP  and  LP ;  and  then 
by  similar  triangles  obtains  the  value  €i  AB  which  is  13. 
This  proof  was,  even  sixty  years  later,  quoted  by  Cardan  as 
"incomparably  simple  and  excellent,  and  the  very  crown  of 
mathematics."  I  cite  it  as  an  illustration  of  the  involved  and 
inelegant  methods  then  current.  The  problems  enunciated  are 
very  similar  to  those  in  the  De  Triatigulu  uf  Regiomontanus. 

Leonardo  da  Vinoi.  The  fame  of  Leonardo  da  Vinei  as 
an  artist  has  overshadowed  his  claim  to  considemti^  as  a 
mathematician,  but  he  may  be  said  to  have  prepared  the  way 
for  a  more  accurate  conception  of  mechanics  and  phyiiics,  while 
his  reputation  and  influeuce  drew  some  attention  to  the  sub- 
ject; he  was  an  intimate  friend  of  PaciolL  Leonardo  was 
the  illegitimate  son  of  a  lawyer  of  Vinci  in  Tuscany,  was  born 
in  1452,  and  died  in  Prance  in  1519  while  on  a  visit  to 
Francis  I.  Several  manuscripts  by  him  were  seized  by  the 
French  revolutionary  armies  at  the  end  of  the  last  century, 
and  Venturi,  at  the  request  of  the  Institute,  reported  on  those 
concerned  with  physical  or  mathematical  subjects^. 

Leaving  out  of  account  Leonardo's  numerous  and  important 
artistic  works,  his  mathemsticol  writings  are  concerned  chiefly 

*  Ettai  §ur  let  ouvraget  phiftieo-wtatkimatiquet  de  Lionardde  Vimei^  hj 
J.-B.  Vsniori,  PsrtB,  17U7.  See  also  the  memoir  bj  Pr.  Woepeke,  Rome. 
1S56. 


TMK   MATnitMATlat  OF  THE  REVARRANCB.  210 

with  mw:hiin>c»,  hydrnulic*.  nnil  itpticn — lii«  concluMoim  being 
Diually  bns^  on  pxprrimciil.i.  Ilin  trentment  of  h^dnnlica 
■nd  optics  invnlven  l>ut  little  niKtIifnmticfl.  The  mechanics 
contain  ntimeroun  anil  Rcrinuti  f  rmn  ;  the  be*t  portions  ara 
thone  denling  wilh  tlic  rquiljlinuin  of  n  Icvf-r  under  *nj  fornix 
the  laws  of  friction,  the  stnliility  of  n  body  M  Affected  by  the 
poaition  of  ita  centrp  of  gmvity,  the  ntmiprth  of  tiFMna,  Mid 
the  orbit  of  a  pnrticlr  ondpr  n  cenlml  fnrcn ;  he  nlm  trmteil  % 
few  easy  problemH  by  virtual  nMinipnU,  A  knowledge  of  the 
triangle  of  forces  in  ncciutionnlly  attribute]  to  him,  but  it  in 
prubablf  that  hiii  views  on  the  subject  were  nomewhat  indefi- 
nite. Itroiulty  Rpeaking  we  may  tay  that  his  mathemattcal 
work  is  unfinished,  and  conHiHtft  largely  of  suggestions  which 
h«  did  not  Hincuiw  in  <lelsil  and  could  not  (or  at  any  mt« 
did  not)  verify. 

Diirer.  Albreehl  VHrfr*  WM  another  artiit  of  the  same 
time  who  wan  also  known  aa  a  mntbematician.  He  was  bom 
at  Hun-mberg  on  May  21.  1471,  and  die<l  there  on  April  6, 
1528.  His  chief  mathematical  work  wa!«  iMoed  in  1525  and 
contains  a  discussion  of  perHpective,  ftome  geometry,  and  eer^ 
tain  f(raphica1  solution!!:  Ijttin  translations  of  it  were  iasned 
in  1532,  1555,  and  1605. 

CopemiOOB.  An  account  of  Xirnln'in  Cop^miemi,  bom  at 
Thorn  on  Feb.  19,  1473  and  died  at  Frauenberg  on  May  7, 
1543,  and  his  conjectura  thiit  the  earth  and  planeta  all  re- 
volved ronnd  the  sun  belong  to  aHtrvnnniy  rather  than  to 
mathematiot.  I  may  however  a<l<l  that  Copemicns  wrata  on  . 
trigonometry,  his  results  being  publisher!  as  a  taxt-book  at 
Wittenberg  in  1542;  it  is  clear  though  it  contains  nothing 
new.  It  is  evident  from  this  and  his  nntrotKMaj  that  be 
was  well  read  in  the  literature  of  matlinnatics,  and  waa 
himself  a  mathematician  of  considerable  power.  I  describe 
his  statement  as  to  the  motion  of  the  earth  as  a  conjectoro 
because  he  advocated  it  only  on  the  gronad  that  it  gave  a 

*  Sm  Darer  ab  Ualhf»mtiker  bjt  H.  BtaigmaUM, 


220  Tm  MATHIlUTlCa  or  TBI  BlWAIWiAWCl, 


simple  azpUuiAiioa  of  natiifml  phwMimiina.  { Galileo  in  1M2 
wM  the  fint  to  trj  to  supply  anything  like  a  proof  el  this 
hypothesis.  l 

By  the  beginning  of  the  sixteenth  eentnry  the  printing- 
press  h^pok  to  be  active  and  many  of  the  worin  of  the  earlier 
mathematicians  became  now  for  the  first  time  accessible  to  all 
students.  This  stimulated  iuquiiy,  and  before  the  middle  of 
the  century  numerous  works  were  issued  which,  though  they 
did  not  include  any  great  diacaveries,  introduced  a  variety 
of  small  improvements  all  tending  to  make  algebra  more 
analytical 

Record.  The  sign  now  used  to  denote  equality  was  in- 
truduced  by  Roberi  Jieeard^.  Record  was  bom  at  Tenby  in 
Pembrokeshire  about  1510  and  died  at  London  in  1558.  He 
entered  at  Oxford,  and  obtained  a  fellowsliip  at  All  Souls 
College  in  1531 ;  thence  lie  migrated  to  Cambridge,  where  he 
took  a  degree  in  medicine  in  1545.  He  then  returned  to 
Oxford  and  lectured  there,  but  finally  settled  in  London  and 
became  physician  to  Eklward  VI.  and  to  Mary.  His  prosperity 
must  have  been  short-lived,  for  at  the  time  of  his  death  he 
was  con6ned  in  the  King's  Bench  prison  for  debt. 

In  1540  he  published  an  arithmetic,  term<fd  the  Grounde  of 
ArieSf  in  which  he  employed  the  signs  -i-  to  indicate  excess 
and  —  to  indicate  deficiency ;  '*  -i*  whyche  lietokeneth  too 
mudie,  as  this  line  —  plaine  without  a  crosse  line,  betokeneth 
too  little."  In  this  liook  the  equality  of  two  ratios  is  indi- 
cated by  two  equal  and  parallel  lines  whose  opposite  ends 
are  joined  diagonally,  ex,  yr,  by  Z .  A  few  years  later,  in 
1557,  he  wrote  an  algebra  under  the  title  of  the  WkeUione 
of  IViiie,  This  is  interesting  as  it  contains  the  earliest  intro- 
duction of  the  sign  -  for  equality,  and  he  says  he  selected 
that  particular  symbol  because  than  two  parallel  straight 
lines  "noe  2  thynges  can  be  nioare  equalle."  M.  Charles 
Henry  has  however  pointed  out  tluit  this  sign  is  a  recognised 

*  See  pp.  15—19  of  mj  Ui$torp  of  the  Stud^  qf  UatkematicM  mi 
Cambridge^  Cambriclge,  I88tl. 


RUDOLFF.     RIESE.     STHFEU  221 

abhreviiitHm  for  th«  won]  ^H  in  medieval  nrnnuAcripU;  and 
this  would  seem  to  indicate  a  more  pmhahle  origin.  In  this 
work  Record  ahewed  how  the  nqoam  root  of  an  algebraical 
expression  could  be  extracted. 

He  also  wrote  an  astronomy.  These  works  give  a  clear 
▼lew  of  the  knowledge  of  the  time. 

RndoUll  Riese.  Almut  the  same  time  in  Germany, 
Rudolff  and  Riese  took  up  the  subjects  of  algebra  ami 
arithmetic.  Their  investigations  form  the  basis  of  StifeFs  well 
known  work.  ChriMoff  RnMff*  published  his  algebra  in 
1525;  it  is  entitled  />t«  Comr,  and  is  founded  on  the  writings 
ol  Faeioli  and  perhaps  of  Jonlanus.  RudolflT  introduced  the 
sign  of  ^J  for  the  square  root,  the  symliol  being  a  comiption  of 
the  initial  letter  of  the  word  radir^  similarly  ^J^JJ  denoted 
the  cube  root^  and  J^  the  fonrth  root.  Adam  Ri^$e\  was  bom 
near  Bamberg,  Bavaria,  in  1489  of  humble  parentage,  and  after 
working  for  some  years  as  a  miner  set  up  a  nchool ;  he  died 
at  Annaberg  on  March  30,.  1559.  He  wrote  a  treatise  on 
practical  geometry,  hut  his  most  important  book  was  his  well 
known  arithmetic  (which  may  be  dcscrilied  as  algebraical) 
issued  in  1536  and  founded  on  Pacioli*s  work.  Riese  used  the 
qnnhols  -i*  and  ~ . 

Siifelt.  The  methods  used  by  RudolflT  and  Riese  and 
their  results  were  brought  into  general  notice  through  Stifel's 
work  which  had  a  wide  circulation.  Mirkarl  Sii/d^  mmietimes 
known  by  the  Latin  name  of  ik{ffelhi9^  was  liom  at  Emlingen 
in  I486  and  died  at  Jena  on  April  19,  1567.  He  was 
originally  an  Augustine  monk,  but  he  accepted  the  doctrines 
of  Luther  of  whom  he  was  a  personal  friend.  He  tells  us 
in  his  algebra  that  his  conversion  was  finally  determined  by 
noticing  that  the  pc^  Leo  X.  was  the  beast  mentioned  in  the 

*  Sas  E.  Wmppler,  GewekiekU  der  demUchen  Algehm  im  sv  JmMmn* 
ilfrlr,  Zwickau,  1887. 

t  Bas  two  works  by  B.  BsrlH,  Veher  Adam  iti>«f,  Aonsbflrg,  18H; 
sad  DU  Com  ron  Adam  Riem,  Anoaberg,  1880. 

X  Ths  aalhoriliss  en  Blifd  an  givn  by  Csolor,  ehsf.  um. 


228  THB  MATHSMATIGB  OF  TBI  RKVAUaiirGB. 


Revdation.  To  thew  thb  it  was  only  neoeHaiy  to  add  up  tho 
numbers  represented  by  the  letters  in  Leo  deoinins  (the  m  bed 
to  be  rejected  since  it  clearly  stood  for  mf$Ur%um)  and  the 
result  amounts  to  exactly  ten  less  than  666»  thus  distinetly 
implying  tlmt  it  was  Leo  the  tenth.  Luther  aocepted  his 
conversion,  but  frankly  told  him  he  had  better  dear  his  mind 
of  any  nonsense  about  the  number  of  the  beast. 

Unluckily  for  himfielf  Htifel  did  not  act  on  this  adrioe. 
Believing  that  lie  hnd  discovered  tlie  true  way  of  interpreting 
tiie  biblical  prophecies,  he  announced  that  the  world  would 
oonie  to  an  end  on  Oct.  3rd,  1533.  The  peasants  of  Holsdorf, 
of  which  place  he  was  pastor,  aware  of  his  scientific  reputation 
accepted  his  assurance  on  this  point.  Home  gave  themselves 
up  to  religious'  exercises,  others  wasted  their  goods  in  dissipa- 
tion, but  all  abandoned  their  work.  When  the  day  foretold 
had  passed,  many  of  the  peasants  found  themselves  ruined: 
furious  at  having  been  deceived,  they  seized  the  unfortunate 
prophet,  and  he  was  lucky  in  finding  a  refuge  in  the  prison  at 
Wittenberg,  from  which  he  was  after  some  time  released  by 
the  penional  intercession  of  Luther. 

Htifel  wrote  a  small  treatise  on  algebra,  but  his  chief 
mathematical  work  is  his  Arithfneiiea  hiii^gra  published  at 
Nureuiburg  in  1544,  with  a  preface  by  Melanchthon. 

The  first  two  books  of  the  ArUhmeliea  ItUeyra  deal  with 
surds  and  incomuiensurables,  and  are  Euclidean  in  form.  The 
third  book  is  on  algebra,  and  is  noticeable  for  having  calli'd 
general  attention  to  the  German  practice  of  using  the  signa 
-I-  and  ~  to  denote  addition  and  subtraction.  There  are  faint 
traces  of  these  signs  being  occasionally  employed  by  Stit'el 
as  symbols  of  operation  and  not  only  as  abbreviations ;  thia 
application  of  them  was  apparently  new.  He  not  only- 
employed  the  usual  abbreviations  for  the  Italian  words  which 
represent  the  unknown  quantity  and  its  powers,  but  in  at. 
least  one  case  when  there  were  several  unknown  quantities, 
he  represented  them  res|iectively  by  the  letters  A^  B^  C,  kc. ;. 
thus  re-introducing  the  general  algebraic  notation  whiiUiL  hsdL 


STIFEU      TARTAOLIA.  223 

hlten  into  disuse  since  the  time  o(  JordAfliiB.  It  ued  lo  be. 
■aid  that  Stifel  wm  the  real  inventor  of  logmritbniB,  bnt  it  is 
now  certain  thnt  this  opinion  was  due  to  »  tni>i4>prehMiaioa  of 
■  ptissage  in  which  he  compares  geometric^  Atid  »rithmetiGal 
progicsiiionB.  Stifel  is  said  to  have  indicated  »  fonnols  tor 
writing  down  the  coefficients  of  the  variova  terms  in  tba 
expansion  of  (1  -tx)'  if  thooe  ia  the  expuuKHi  ol  (1  +a:)*~* 
were  icnown. 

In  1553  Stifel  brought  out  an  edition  of  RndoKTsifM  Com 
in  which  he  intnxluced  an  improvemeot  in  the  ftlgebraie 
notAtion  then  current.  The  nymbola  at  that  tioM  onlinarilf 
used  for  the  unknown  quantitjr  and  its  powen  wera  lettera 
which  atood  for  abbreviations  of  the  words.  Among  tboas 
frequentlj  adopted  were  R  or  J^  for  mdix  or  n»  («),  £  or 
C  for  tentiu  or  c«nnw  (r?),  C  or  A'  for  e»btu  {af),  ka.  Thus 
^  *bx  —  i  would  have  been  written 

I  ^  p.  5  A  R).  4  ; 
where  p  ntAods  for  plus  and  m  for  minus.     Other  letton  and 
sjrmbols   were  also  employed:    thus  Xylander  (1575}  wonkt 
have  denoted  the  above  exprCKsion  hj 

a  notation  similar  to  this  was  sometimes  used  bj  Vieta  and 
even  hj  Perrost.  The  advance  made  hy  8tiCd  was  that  ho 
introduced  the  symbols  \A,  \AA,  \AAA,  for  the  nnknown 
qaantity,  its  square,  and  its  cube,  which  shewed  at  a  glanee 
the  relation  between  thent. 

Tartaglia.  Sieeoto  Foitfana,  generally  known  aa  XieM— 
Tarlaglia,  that  is,  Nicholas  the  stammerer,  wm  born  at 
Brescia  in  1500  and  died  at  Venice  on  December  14,  1B57. 
After  the  capture  of  the  town  by  the  French  in  \S\i  nwat 
of  the  inhabitaots  took  refuge  in  the  cathednl,  and  wen 
there  massacred  by  the  sotdiers.  His  fathtt-,  who  was  » 
poatal  messenger  at  Brescia,  was  amongst  the  killed.  The 
boy  himself  had  his  skall  split  throuj^  in  Uitm  pbwe^ 
vkile  his  jaws  and  his  palate  were  ovt  ap«i|  ht  *■«  kft 


224  THB  MATHKMATICa  Or  TUB  RBMAiatarOB. 

for  dead,  but  his  moUier  got  {nto  the  oatbedrml*  and  flndiag 
hioi  still  alivtt  nuuiaged  to  curry  him  ofll  Deprived  of  ell 
reaooroee  she  recollected  that  dogs  when  woonded  elwejs 
licked  the  injured  pUce^  end  to  that  remedy  he  attribated 
his  nltimate  recovery,  but  the  injury  to  his  palate  produced 
an  impediment  in  his  speedi  from  which  he  received  his 
nickname.  His  mother  managed  to  get  sufficient  money  to 
pay  for  his  attendance  at  echoed  for  fifteen  days,  and  he 
took  advantage  of  it  to  steal  a  copy-book  from  which  he 
subsequently  taught  himself  how  to  read  and  write;  bat 
so  poor  were  they  tliat  he  tells  us  he  could  not  aflTord  to 
buy  paper,  and  was  obliged  to  mip^ke  use  of  the  tombstones 
as  slates  on  which  to  work  his  exercises^ 

He  commenced  his  public  life  by  lecturing  at  Veronal 
but  he  waa  appointed  at  some  time  before  1535  to  a  chair 
of  nmtlieniatics  at  Venice  where  lie  was  living  when  he 
became  famous  through  hisi  acceptance  of  a  challenge  from 
a  certain  Antonio  del  Fiori  (or  Florido),  Fieri  had  learnt 
from  his  iiiaster,  one  Scipionti  Ferreo  (who  died  at  BoUigna 
in  1526),  an  empirical  Molution  of  a  cubic  equation  of  the 
form  jr*  +  9^ » r.  This  s«>lution  was  previously  unknown  in 
Europe,  and  it  in  prolmble  that  Ferreo  had  found  the  result 
in  an  Arab  work.  Tartaglia,  in  answer  to  a  nv^uest  from 
Colla  in  1530,  stJited  that  he  could  efiect  the  wilution  of  a 
numerical  etiuatiun  of  the  form  x^  -f  yxi^  "=  r.  Fieri  believing 
that  Tartaglia  was  an  impostor  challenged  him  to  a  contest. 
Accoixliiig  to  this  cliullenge  each  of  them  was  to  depoiiit  a 
certain  stake  with  a  notary,  and  whoever  could  solve  the 
most  problems  out  of  a  collection  of  thirty  pnipounded  by 
the  otluT  was  to  gut  the  stakes,  thirty  days  lieing  allowed  for 
the  solution  of  the  questions  proposed.  Tartaglia  was  aware 
that  his  lulversary  wiis  ac(|uaiiited  with  the  solution  of  a  cubic 
e<|uation  of  some  iNirticular  form,  and  suspecting  tliat  the 
questions  proposed  to  him  would  all  depend  on  the  solution  of 
such  cubic  e<iuations  set  himself  the  problem  to  find  a  general 
solution,  and  certainly  discovered  how  to  obtain  a  solution  of 


I 


TARTAOLtA.  2i5 

some  if  not  a\\  culiic  n)uittii>ii8.  Hii  aolation  is  believed  to 
hnve  depended  on  a  gFonirtricnl  ciiTiHtniction*,  but  led  to  the 
formuU  which  is  often,  but  unjustly,  <I(iicril>ed  u  C»rd«n'(i. 

When  the  cunU^t  took  place  all  the  i|ue«tiona  pmp>jwd 
to  Tarbv^lia  weiv,  u  he  had  nu^tpected,  reducible  to  the  nlaticm 
at  a  cubic  eq^untion,  and  he  nucceeileil  n-ithin  two  houn  in 
bringing  them  to  particular  ciutnt  at  the  w|U«tion  ^ -nix^r,  of 
which  he  knew  tlie  solution.  His  opponent  failed  to  mIva 
any  of  the  problems  prdpotted  to  him,  which  as  a  matter  of 
fact  were  all  reducible  to  numerical  e<iuations  of  the  form 
x'*-pi'  =  r.  Titrtnglin  was  therefore  the  conqueror  j  he  sub- 
eequently  compose*!  Rome  vltbpm  conimemorBtive  of  his  tHctorj. 

The  chief  work*,  of  Tnrtaglia  >i^  <u  follows.  (i)  His 
jVom  neifHza,  published  in  1537  ;  in  this  lie  invenlifiated  tb« 
fall  of  bodies  under  gntvity  ;  and  he  determined  the  rwige  of 
a  projectile,  stating  that  it  was  a  maximum  wImii  the  angle  M 
projection  was  4.5',  but  thin  seenin  to  have  been  a  Inckj 
guess,  (ii)  An  arithmetic,  publi^the^l  in  two  pnrts  in  ir)56. 
(iii)  A  treatise  on  numU-rs,  published  in  fonr  pnrts  in  1560, 
and  sometimes  treHted  as  a  continuatitm  of  the  arithmetic: 
in  thia  he  shewed  how  the  ooetlicients  of  x  in  tlie  expansion  of 
(1  4-z)'  could  be  cntcul&ted,  by  the  une  of  an  arithmetical 
triangle  t.  from  those  in  the  expansion  of  (1  +x)""'  for  the 
cues  when  n  is  equal  to  3,  3,  -I,  n,  or  G.  It  is  pomible  Uiat 
he  also  wrote  a  treatise  on  nlgebra  and  the  scdution  of  culiio 
equations,  but  if  so  no  copy  is  now  extant.  The  other  works 
were  collected  into  a  single  edition  and  republished  at  Venice 
in  1C06. 

The  trmtine  on  arithmetic  and  numlKra  in  ime  of  tha  chief 
authorities  for  our  knowledge  of  the  early  ItaliMi  algorani. 
It  is  verboae,  but  gives  a  clear  account  of  the  diflerent  eritb- 
ntetical  methods  then  in  use,  and  has  numerons  historicel  notes 
which.  OB  far  M  we  can  judge,  are  reliable,  and  are  nltimaleljr 
the  aotboritiea  for  many  of  the  statements  in  the  last  chapter. 
It  eontains  an  immense  number  of  qnestioas  on  vwvrj  kind 
•  Baabetow,  p.131.  f  See  below,  VpL  IM— 4. 

1.  U 


226  TUB  MATUOIATIGB  OF  THB  RBNAUHUICGB. 

of  problem  which  would  be  likely  to  oooor  in  mewiantile 
arithinetio,  and  tliere  are  ■e\'eiml  aUmmpU  to  firmnie  algebimical 
foniialae  aoitable  for  parUcular  iMoblema. 

Theae  problems  give  incidentally  n  good  deal  of  infonnation 
as  to  the  ordinary  life  and  commercial  costoms  of  the  time. 
Thus  we  find  tliat  tlie  intermt  deinatuled  on  first  class  security 
in  Venice  ranf{ed  from  5  to  12  per  cent  a  year;  while  the 
interest  on  coniuiercial  transactions  ranged  from  20  per  cent, 
a  year  upwanls.  Tartaglia  illustrates  the  evil  effects  of  the 
law  forbidding  usury  by  the  aiiaiuier  in  wliicli  it  was  evaded 
in  farming.  Fanners  wiio  were  in  debt  were  forced  by  tlieir 
creditoni  to  sell  all  their  crops  iniiiiediately  after  the  harvest ; 
tlie  market  being  thus  glutted,  the  price  obtained  was  very 
low,  and  the  money-lenders  purchased  the  com  in  open  market 
at  an  extremely  cheap  rate.  The  farmers  then  had  to  liorrow 
their  siHxl-corii  on  condition  that  they  replaced  on  equal 
quantity,  or  paid  the  then  price  of  it,  in  the  month  of  May, 
wlion  c«»ni  was  diNirent.  Again,  Tartnglio,  who  had  been  asked 
by  the  iiingiHtniteH  at  Verona  to  frame  for  them  a  sliding  scale 
by  which  the  price  of  bread  would  be  fixed  by  that  of  com, 
enters  into  a  ciiHCUssion  on  the  principles  which  it  was  then 
supposed  should  regulate  it.  In  another  place  he  gives  the 
rules  at  that  time  current  for  preparing  medicines. 

Pocioli  had  given  in  his  arithmetic  some  problems  of  on 
amuMing  cliarocter,  and  Tartaglia  imitated  him  by  inserting  m 
large  collection  of  mathematical  puzzles.  He  half  apologizes 
for  introducing  them  by  Haying  that  it  was  not  uncommon  at 
desMert  to  propose  arithiiieticul  questions  to  the  oouipany  by 
way  of  anmsement,  and  he  therefore  oddn  some  suitable 
problems.  He  gives  several  questions  on  how  to  guess  a 
number  thought  of  by  one  of  the  company,  or  the  relationships 
caused  by  the  marriage  of  relatives,  or  difliculties  arising  from 
inconsistent  bequests.  Other  puzzles  are  sunilar  to  the  follow- 
ing. "Three  beautiful  ladies  have  for  husbands  three  men, 
who  are  young,  handsome,  and  gallsnt,  but  also  jealous.  The 
party  are  travelling,  and  find  on  the  bank  of  a  river,  over 


TARTAOLIA.      CARDAN,  2Z7 

which  they  hitve  lo  pAna,  n  HniAtI  Ixmt  which  gmi  hold  no  mora 
thnn  two  pemonx.  How  cnn  thej'  pass,  it  being  ai^med  that, 
in  order  to  Avoid  ncnndnl,  ni>  wuinan  xhAll  be  left  in  th« 
nncietf  of  a  man  unlcsfl  her  huslmnd  in  pr^«entl''  "A  nhip, 
carrying  ah  luuscngera  fift^vn  TurkH  nnd  fifteen  ChritttiuM, 
encountora  n  Htorm  ;  nnd  the  pilot  drrliirc.'i  that  in  order  to 
Mve  the  ship  and  crew  one-hnlf  of  the  pitMengera  mant  be 
thrown  into  the  hca.  To  choose  the  victiiiM,  th«  pnaacngwa 
are  pinced  in  a  circle,  nnd  it  is  ngrent  thnt  e«'ery  ninth  nwn 
shall  )»  cn.<t  overlKHird,  nvkoiiin;;  from  a  liertain  point.  Id 
what  manner  must  they  he  armn}^,  ho  that  the  lot  nwj  fall 
exclusively  Uj-on  the  Turksl"  "Tlirpe  men  robbed  a  gentle- 
man of  a  vane  containing  21  ouncen  of  hnlsnni.  Whilitt  mnning 
away  they  met  in  a  wood  with  a  glns.VBeller  of  whom  in  a 
great  hurry  they  purvhasnl  three  vessels.  On  reaching  a  [riace 
of  safety  they  wish  to  divide  the  ln>oly,  hut  they  6nd  that 
their  vessels  contain  5,  II,  and  l-t  ounceH  respectively.  How 
can  they  divide  tlie  linlHnm  into  ei|ual  portionst" 

Thene  prohlenis— wmc  of  which  iirc  of  oriental  origin — 
form  the  basis  of  the  collections  of  ninthmiMtical  recreationi 
by  Biichet  de  ^li'/iriac,  OEanaiii,  and  Montncia*. 

Cardant.  The  life  of  Tartaglia  wns  emlntterod  by  a 
quarrel  with  hin  contenipiirary  Cardan  who  published  Tartaglia'a 

•  Bolulions  of  these  and  other  niniiUr  problpmi  tre  given  In  my 
Malhemnlftat  Rrrrrntimu  and  Prahlfnu,  chspn.  I,  n.  On  Bachet,  tee 
below,  p.  315.  Jitrqiut  Otanam,  bora  St  BoDiii:nrai  iu  ISM  and  died  in 
1717,  left  numeniaiwoiki  of  which  one,  worth  mentkniinit  here,  iahiaiW- 
rri'iiliaiMii'iffci'iu(i;iifir(phjr>i7i(ri.3ToIiunm,  Purl ii,  1806.  Jtanfltlrnme 
Uimlutla,  born  at  Ljons  in  172S  and  died  in  Palis  in  1799,  edited  and 
miwd  Otanam'i  mathematical  recreationi.  Rii  history  of  allempla  to 
■qnan  the  circle.  1734,  and  hiitorf  oF  mathrmatica  to  Iha  and  of  the 
KTcntecDlfa  eeplnij  io  3  votamex,  1766,  are  inlenatinf  and  vahabls 

t  Then  ■■  an  admirable  araoani  of  his  life  in  tb*  Kemprltt  tiefrmfkU 
ffn/raU,  by  V.  aardoa.  Cardan  left  an  BQtobioftnpliy  of  wliiek  an 
analjrria  by  B.  Motley  was  pobliihcd  in  two  rolDinei  in  London  la  UM. 
All  Cardaa'a  ptinlad  works  were  eoUected  by  8ponia,aad  faUiihsd  la 
W  votaM^  Iijou,  1669;  Um  worim  on  arilhaxlia  and  niuw^n  «m 


228 


THE  MATHEMATICS  OP  THE  EBMAIBBAilCB. 


of  Milan,  and  was 


•olatioa  of  a  enbie  oqoatioo,  whidi  he  had  obtained  nndar 
a  pledge  of  eeciecj.  Oiroiwmo  Carda/fn  ijraa  bom  ai  Favia 
on  Sept.  24.  1501.  and  died  at  Rome  on  Scipt  21.  1578.  Hia 
career  is  an  account  of  the  most  extraordimirj  and  inconsistent 
acts.  A  gambler,  if  not  a  murderer,  be  wes  also  an  ardent 
student  of  science^  solving  problems  whicn  had  long  baffled 
all  investigatioii ;  at  one  time  of  his  life  ne  was  devoted  to 
intrigues  which  were  a  scandal  even  in  the 'sixteenth  century, 
at  another  he  did  nothing  but  rave  on  astrology,  and  yet  at 
another  he  declared  that  philosophy  was  the  only  subject 
worthy  of  man's  attention.  His  was  thd  genius  that  was 
closely  allied  to  madnesR. 

He  was  tlie  illegitimate  son  of  a  lawyer 
educated  at  the  universities  of  Fsvia  and  Psdua.  After 
taking  Im  degree  he  coDimenced  life  as  a  doctor,  and  practised 
hiM  profession  at  Sacoo  and  Milan  from  1524  to  1550;  it  was 
during  this  period  that  he  studied  mathematics  and  published 
his  chief  works.  After  spending  a  year  or  so  in  France. 
Scotland,  and  England,  he  returned  to  Milan  as  professor  of 
science,  and  sliortly  afterwards  was  elected  to  a  chair  at  Pavia. 
Here  he  divided  his  time  between  debauchery,  astrology,  and 
nieclianics.  His  two  sons  were  as  wicked  and  passionate  as 
himself:  the  elder  was  in  1560  executed  for  poisoning  his 
wife,  and  about  the  same  time  Cardan  in  |i  fit  of  rage  cut  off 
the  ears  of  the  younger  who  had  committed  some  offence; 
for  this  scandalous  outrage  he  suffered  no  punislimeut  as 
the  pupe  Gregory  XIII.  granted  him  protection.  In  1562 
Cardan  moved  to  Bologna,  but  the  scandals  connected  with 
his  naihe  were  so  great  that  the  university  took  steps  to 
prevent  his  lecturing,  and  only  gave  way  under  pressure  from 
Home.  In  1570  he  was  imprisoned  for  heresy  on  account  of 
his  liaviiig  publLnhed  the  horoscope  of  Christ,  and  when 
released  he  found  himself  so  generally  detested  that  he  deter* 
mined  to  resign  his  chair.     At  any  rate  he  left  Bologna  in 

oonlaioed  in  the  fourth  voIqsm.  It  is  ssid  that  there  are  in  the  Vstieaa 
seversl  msouecript  note-books  of  his  which  have  not  been  jet  edited. 


f 

I 


CARnAN.  2S9 

1571,  «nd  shortly  »ft*rwnrds  imncd  Ut  Komv.  OnUn  wm 
tlie  inoHt  diHtiiiguishcd  astrologer  of  his  time,  and  when  bo 
settled  Kt  Rume  ho  received  a  penguin  in  order  to  secure  his 
services  as  astrologpr  to  the  papnl  court.  This  proved  fatal  to 
him  fur,  having  forctuUI  that  lie  Hliould  die  on  a  particular 
day,  he  fett  ubiigrd  to  commit  suicide  in'order  to  keep  up  hia 
repulAtion — so  at  leant  the  story  runs. 

The  chief  niatliptnntical  work  of  Cnrdan  w  the  Ar$  Magna 
pu))Iish<>d  at  Nurcmlierg  in  15(5.  Cnrdnn  wsamnch  int«re8l«d 
in  the  cont«i4t  lietween  Tartaglia  and  Fiori,  and  as  he  had 
already  liegan  writing  this  book  he  nnkcd  Tartaglia  to  com- 
municate his  niethod  of  solving  a  culne  isqoation.  Tartaglia 
refused,  whereupon  Cardan  ahnwd  him  in  the  most  violent 
terms,  but  shortly  afterwards  wrote  saying  that  a  certain 
Italian  nobleman  hod  heard  of  Tartaglia's  fAnw  and  was  most 
anxioQs  to  meet  him,  and  )?e;;f,'e<)  him  to  <!Oine  to  Milan  at 
once.  TartAglta  came,  and  tlioiigh  he  found  no  nolileman 
awaiting  him  at  tiio  end  of  bin  journey,  lie  yielded  to  Canlan's 
impirtunity  and  gave  him  the  rule  he  wanted,  Cardan  on  lits 
aide  taking  a  solemn  oath  that  )ie  would  never  reveal  it^  and 
would  not  even  commit  it  to  writing  in  such  a  way  tlint  after 
his  death  any  one  could  underxtand  iL  Tlie  rule  is  given  in 
some  duggcrel  verses  which  am  still  extant.  Cardan  asHcrta 
that  he  was  given  merely  the  mtull,  and  that  ho  obtained  tho 
proof  himst-lf,  Init  this  i«  doubtful.  He  sreins  to  have  at  WKO 
taught  the  metlirwl,  and  one  of  his  pupilx  Ferrari  reduced  the 
e()uation  of  the  fourth  degree  to  a  uubiu  and  so  solved  it. 

When  the  An  Ua.jna  wbm  published  in  1549  the  Weaeh  U 
faith  was  made  manifest.  Tartaglin  not  nnnatnr»llj  waa  vmy 
angry,  and  after  an  acrintonious  controvcnty  he  aent  a  diallenge 
to  Okidan  tu  take  |Mirt  in  a  mathematical  dneL  The  pre- 
liminaries were  settled,  and  the  place  of  meeting  waa  to  be  a 
cerUin  church  in  Milan,  hut  when  the  day  arrived  GanlMi 
failed  to  appear,  and  sent  Ferrari  in  his  stMHL  Both  aides 
claimed  the  victory,  though  I  gather  that  Tartaglia  WM'the 
nrara  ancoevfal;  at  any  nie  hia  opponents  bnric*  i^  the 


230  THS  MATUKMATiai  OP  THS  ElliAIHK4WCR> 

meetings  and  he  deemed  hinaelf  lortmuite  in  eeeaping  with  hie 
life.  Not  only  did  Ckrdan  racoeed  in  hie  freod,  bat  modem 
writeni  have  often  attribated  the  adlatlon  to  him,  ao  that 
TartagUa  has  not  even  that  posthiimoaa  repatation  whidi  at 
least  is  Us  doe. 

The  Ar$  Maywk  is  a  great  advance  on  any  algebra  pi^ 
vioimly  published.  Hitherto  algebraists  had  confined  their 
attention  to  those  roots  of  equations  which  were  positive. 
Cardan  disciuuied  negative  and  even  imaginary  rootS|  and 
proved  tliat  the  latter  would  always  occur  in  pairs,  though  he 
declined  to  commit  himself  to  any  explanation  as  to  the 
meaning  of  these  ^  sophistic "  quantities  which  he  said  were 
ingenious  though  uselessL  Most  of  his  analysts  of  cubic  equa- 
tions seems  to  have  been  original ;  he  shewed  that  if  the  three 
routs  were  real,  Tartaglia's  solution  gave  them  in  a  form 
which  involved  imaginary  quantities.  Except  for  the  somewhat 
similar  researches  of  Bouiljelli^  a  few  years  later,  the  theory 
of  imaginary  quantities  received  little  further  attention  from 
nuithematiciaiis  until  Euler  took  up  the  matter  after  the  lapse 
of  nearly  two  ceiituries.  Clauss  first  put  the  subject  on  a 
systematic  and  scientific  basis,  intrmlucod  the  notation  of 
complex  variables,  and  used  the  symbol  t,  which  had  been 
introduced  by  Euler,  to  denote  the  square  root  of  (—  1) :  the 
modem  theory  is  chiefly  Isised  on  his  researches. 

Cardan  establihlied  the  relations  connecting  the  roots  with 
the  coefiicients  of  an  equation.  He  was  also  aware  of  the 
principle  tliat  underlies  Descartes's  '*  rule  of'  signs,"  but  as  he 
followed  the  custom,  then  general,  of  writing  his  equations  as 
the  equality  of  two  expressions  in  each  of  which  all  the  terms 
were  positive  he  was  unable  to  express  the  rule  concisely.  He 
gave  a  method  of  approximating  to  the  root  of  a  numerical 
equation,  founded  on  the  fact  that,  if  a  function  have  opposite 
signs  when  two  numbers  are  substituted  in  it,  the  equation 
obtained  by  equating  the  function  to  zero  will  have  a  root 
between  these  two  numbers. 

*  Bee  bduw,  p.  234. 


CARDAN.      KEIIRAIlt.  '  231 

Cftrdnn'R  nnlution  of  a  quMlratic  equntion  in  geometrical 
and  lubeUntmlly  the  mmr^  nx  that  given  h;  Alkkrimni  Hn 
•olution  of  It  cubic  equation  i»  aIko  ^^motriciil,  and  nuj  be 
i]lufitra(«d  bj  tlir.  following  cane  wliich  he  ^ven  in  chi>[>t«r  xl 
To  solve  the  equation  r*  +  Gx  =  20  (or  any  equation  of  the  fonn 
^  +  qx~  r),  take  two  cubes  noch  that  the  rectangle  under  thdr 
respective  edf^  ih  2  (nr  J?)  and  the  difference  of  tlieir 
Totumen  in  20  (or  r).  Then  x  wit]  Ik-  <>quAl  to  Uie  diflerence 
between  the  edges  of  tlii;  cubes.  To  verify  thin  he  (irrt  given  • 
geometrical  lemma  to  shew  that,  if  from  a  line  AC  a  portion 
CB  bo  cut  off;  then  the  cube  on  jf  5  will  be  lew  than  tlie 
difference  between  the  cubes  on  AC  and  BC  by  three  timea 
the  right  parallclopiped  whow  edges  are  rOTpectirelj  equal  to 
AC,  BU,  and  ^B— this  statement  is  equi^ident  to  the  alge- 
braical identity  (n-6)'  =  n'-6'- 3n6  {n  -  4)— and  the  tad 
that  X  satisfies  the  equation  is  then  obvioiu.  To  obtain  the 
lengths  of  the  e<Iges  of  the  two  cubes  he  haa  only  to  aolve 
a  quadratic  equation  for  which  the  geometrical  aolntion  pre- 
rionsly  given  sutfict^. 

like  all  previouN  mnthematicians  he  gives  neparnte  proofi 
of  his  rule  for  tlie  different  forms  <if  equations  which  can  fall 
under  it  Thus  he  proven  the  rule  inciependently  for  equa- 
tions of  the  form  :^  +  px  =  i,  x'-p^  +  q,  x'*pz  +  q  =  0,  aod 
z*4f  =  p\  It  will  be  noticed  that  with  geometrical  proofa 
this  wax  the  natural  course,  but  it  does  not  appear  that  be  was 
aware  that  the  resulting  formulae  were  genend.  The  eqaations 
he  considen  are  numerical. 

Shortly  after  Canlan  came  a  nundwr  of  mathematiciana 
who  did  good  work  in  fleveloping  the  Rubject,  bnt  who  are 
hardly  of  sufficient  importance  to  require  detailed  mention 
here.  Of  thme  the  most  cclebrat«tl  are  perhapa  Ferrari  and 
Rheticna 

Farrmri.  Andm-iro  Fermro  usually  known  m  Femwi, 
whose  naine  I  have  alreaily  mentioned  in  oonnectkm  with  the 
solution  of  a  biquadratic  equation,  was  bora  at  Bologna  on 
Feb.  2,  1933,  and  died  on  Oct.  5,  1565.     Hia  paranU  wei« 


232  TUB  M ATI1K1CATIG8  OP  TUB  RKVAUBAllCBi 


poor  Mid  he  wm  takon  into  CSutian**  •onrko  m  «a  oriMid  boy, 
but  was  allowed  to  attend  his  master^s  leotvres^  and  snbse- 
quentljr  became  his  must  oelebraled  pupil.  He  is  described  as 
^'a  neat  nM^  litfJe  fellow,  with  a  bland  voics^  a  eheerfol  iace, 
and  an  agreeable  short  none,  fund  of  pleasure^  of  great  natural 
liuwers  '^  bat  **  with  the  temper  of  a  fiend.**  His  manners  and 
nuDierouM  acoumpliiiliments  procured  him  a  place  in  th«  service 
of  the  cardinal  Ferrando  Qonzagai  where  he  managed  to  make 
a  fortune.  His  diwiipations  told  on  Ids  health,  and  he  retired 
in  1565  to  Uologiia  where  he  began  to  lecture  on  mathematics. 
He  was  puiHuned  the  Mime  year  either  by  his  sister,  who  seems 
to  liave  been  the  only  person  for  wlioui  lie  had  any  alTection, 
or  by  her  paramour. 

Such  wurk  as  Ferrari  produced  is  iucurpurated  in  Gkrdan's 
An  Jfagna  or  Bumbelli's  AUfAra^  but  nothing  can  be  defi- 
nitely asHignod  to  him  except  the  solution  of  a  biquadratic 
equation.  Colla  had  pn>poHed  the  sulutiou  uf  the  equation 
X*  -f  Gj.^  •«-  36  ~-  GOx  as  a  challenge  to  matheuiaticiaus :  this  par- 
ticular equatiun  had  probably  been  fuund  in  some  Arabic 
work.  Nuthing  is  knuwn  alx>ut  the  history  of  this  problem 
except  that  Ferrari  succeeded  where  Tartaglia  and  Cardan 
had  failed. 

RheticUB.  (Jeory  Joachim  Bheiieuit^  bom  at  Feldkirch  on 
Feb.  15,  1514  and  died  at  Kaschau  on  Dec.  4,  1576,  was 
prufessor  at  Wittenlierg,  and  subsequently  studied  under 
Copernicus  whose  wurks  were  produced  under  the  direction  of 
Rheticus.  Rheticus  constructed  various  trigunometrical  tables 
some  uf  which  were  published  by  his  pupil  Otho  in  1596. 
These  were  sulwequently  completed  and  extended  by  Vieta 
and  PitiHcus,  and  are  the  bonis  uf  those  still  in  use.  Rheticus 
alsu  fuund  the  values  of  sin  2$  and  sin  30  in  terms  of  sin  0 
and  cos  0,  and  was  aware  that  trigunometrical  ratios  might  be 
defined  by  means  uf  the  ratios  uf  the  sides  of  a  right-angled 
triangle  without  intit>ducing  a  circle. 

I  add  heit)  the  mimes  uf  sunie  other  celebrated  mathema- 
ticians of  alN>ut  the  same  time,  thougli  their  wurks  are  now 


TIIK  HATIIEMATICH  nr  TIIK  KEKAISHAHCE.         233 

of  littln  value  to  uny  snve  antiquariktiH.  FmiolKIU 
Manrolyoas,  Iwm  nt  Messinn  of  (;rp*-k  pkrcnU  in  149-1  and 
Aifti  in  1575,  tninitlaterl  nunirrouH  lyttin  and  Oraok  matho- 
nialicnl  wnrkit,  and  cJincu.vieit  the  mnim  regarded  as  iiectiann  oC 
a  cone  :  hin  workn  were  {lubliHlird  al  Veniiv  in  1575.  J6U1 
Boirel.  born  in  1493  and  died  at  UrcnuMo  in  1572,  wrote  an 
algplim,  founded  on  that  of  Stifel ;  and  ft  history  of  the 
quadrature  of  the  circle:  hin  workn  werr  puMisbed  at  Lyons 
in  \Mt9.  WUhelm  Xylnnder,  linm  nt  AnKxbarg  on  Dpc  26, 
in.^3  and  died  im  Feb.  10,  1.^76  Ht  lleidelliei^g,  wltm  iiincn 
lASK  ho  hiid  lieen  profcwwtr,  1irouj;hl  out  an  edition  of  tha 
wnrkn  of  PkcIIun  in  ITiSG  ;  an  edition  of  Euclid's  El-nfnt*  in 
1562;  an  edition  of  the  Arilfim-He  of  I>iophantna  in  1575; 
and  some  minor  worki  which  were  eiillncted  and  puMixhcd  in 
1577.  Federigo  Commiuidino,  Imm  at  Urbino  in  1509 
and  died  there  on  Sept.  3,  1575,  pnlilishrd  a  trannlatinn  of  thn 
wnrkti  of  Archimcilc*  in  ISflS  ;  nelectinii!i  from  ApoDonitiH,  and 
PappUN  in  I5ri6  ;  an  edition  of  Kudid'n  EUmenl»  in  1573;  and 
nelectionn  from  AriHtarchuN,  Ptolemy,  Hero,  and  nippnn  in 
1^74:  all  lieinp  accnmpanied  )>jr  commentarien.  jMSt^tlM 
Pelfltier,  bom  at  1e  Mann  nn  July  I't,  1517  and  died  at  Parix 
in  July  1582,  wrote  text^liookB  on  algebrk  and  geometry : 
modt  of  the  renultn  of  Ktifel  and  Carrlan  are  included  in  the 
former.  Adrian  Bomanns,  bom  at  IxMvain  on  8epL  29, 
1561  and  died  on  May  4.  1625,  prnfesnor  of  mathematica  and 
medicine  at  the  university  of  Jjouvain,  wna  the  first  to  provo 
the  usual  formula  for  sin  {A  +  R\  And  lastly,  Bartholonwas 
Pitisoiu,  l>om  oi>  Aug.  34,  1561  and  died  at  Heidelberg, 
where  he  waa  profewmr  of  matheniatic^  on  Jnly  3,  1613, 
published  his  rrigimontrtty  in  1599;  thin  eont«ins  the  expres- 
sions for  nn  {A±B)  and  cne  (jl  +  ^  in  terms  of  the  trigowK 
metrical  ratioe  of  A  and  B. 

About  this  time  also  severml  text-books  won  produced 
which  if  they  did  not  extend  the  boundAiiM  o(  Uie  Bahjeok 
systematiaed  it.  In  particular  I  may  mentkM  Unm  hj  B— i 
■Dd  BombellL 


234  TUB  MATUKMATICS  OF  TUK  RKNA18SAIICB. 

Barnw^.  F^li&r  Rawuis  wm  bom  ai  Calh  in  PioMdy  fai 
1515.  and  wan  killed  at  Paris  at  the  nuMncre  of  8t  Barthi^ 
lomew  on  Aug.  24.  1572.  He  was  educated  at  the  nniTenily 
of  Paria.  and  on  taking  hiii  degree  be  astoniahed  and  channed 
the  anivenitjT  with  the  brilliant  declamation  he  ddiverad  on 
the  theeia  that  everything  AriatoUe  had  taaglit  waa  falaa  He 
lectured — for  it  will  be  remembered  that  in  early  daya  there 
were  no  profeaiiorB — first  at  le  ^lana.  and  afterwarda  at  Pkria ; 
at  the  latter  he  founded  the  fint  chair  of  mathematics. 
Besidea  aome  worka  ou  philosophy  he  wrote  treatiaea  on 
arithmetic,  algebra,  geometry  (founded  on  Euclid),  astronomy 
(founded  on  the  works  of  Copernicus),  and  physics  which  were 
long  regarded  ou  the  continent  as  the  standard  text-books  on 
these  subjects.  They  are  collected  in  an  edition  of  his  worka 
published  at  fi&le  in  1539. 

'  Bombelli.  Closely  following  the  publication  of  Cardan's 
grciit  work,  JCa/aeiio  JlottUttlli  publiMlied  in  1572  an  algebra 
which  is  a  systematic  exposition  of  the  knowledge  then  current 
ou  the  subject.  lu  the  preface  he  traces  the  history  of  the 
subject,  and  alludes  to  Diophantus  who,  in  spite  of  the  notice 
of  llegiomontanus,  was  still  unknown  in  Kurope.  He  discusses 
radicals,  real  and  imaginary.  He  also  treats  the  theory  of 
equiitious,  and  shews  that  in  the  irreducible  case  of  a  cubic 
equation  the  ruots  are  all  real ;  and  he  remarks  that  the 
problem  to  trisect  a  given  angle  is  the  same  as  that  of  the 
solution  of  a  cubic  equation.  Finally  he  |^ve  a  large  collection 
of  problems. 

Bombelli's  work  is  noticeable  for  his  use  of  symbols  which 
indicate  an  approach  to  index  noUition.  Following  in  the 
steps  of  Stifel,  he  introduced  a  symbol  vi;  for  the  unknown 
quantity,  ^  for  its  square,  vi/  for  its  cube,  and  ho  on,  and 
thei*efore  wrote  a^ -^  bx-A  as 

\  \tj  p.  b  \lj  UL  i, 

*  HtM  the  uionugrmphs  by  Ch.  Wsddiugton,  Parts,  1855;  and  by 
C.  Desmsze,  Paris,  1864. 


THK   nEVEUtPMENT  OP  At.UEBRA.  235 

Ulevinui  in  IMG  employed  ©i  ®i  ©.■■■'•>  •  simil'M'  *»/» 
•nd  Biiggestc^,  tlintigh  he  did  not  une,  a  corresponding  noUtion 
for  fractional    in^icefu     He  would    have   written   the   ahowv 


j  l®  +  5©-4G> 

But  whcth^  the  nynibols  went  more  or  less  con«'onient  they 
were  still  only  ahl>reviiitioBH  for  wotdH,  and  were  ulgect  to 
«ll  the  nilcH  cf  Nyntnx.  They  merely  nfTorded  k  aort  of  short- 
hand liy  which  the  vAtioun  stepe  and  n.-gulm  could  bo  gxprewwd 
oonciBcIy.  Tlic  next  ndvance  was  the  ci'cation  of  aymboUo 
algebra,  and  the  chief  credit  of  that  ia  due  to  Vieta. 


ITit  dgvetopmenl  of  lynibolic  atgebra. 

Wc  have  now  rmched  a  puint  lieyund  which  any  con- 
ndcrable  development  of  nlyebra,  no  long  as  it  wan  strictly 
syncopated,  could  Imnlly  pTX>ceed.  It  is  ertdent  that  Stifcl 
and  Bi>mbeili  and  other  writent  of  the  sixteenth  century  had 
introduced  or  we™  on  the  point  of  introducing  some  rf  the 
idcBs  of  Hyniliolic  algebra.  But  ro  fnr  wt  the  credit  of  in- 
venting Bymlxtlic  nigebra  can  be  put  down  to  any  one  man 
wfl  may  perlmps  ansign  it  to  Vieta,  whild  wq  may  (ay  that 
Harriot  and  DeHcuries  did  more  than  any  other  writers  to 
bring  it  into  general  use.  It  moflt  be  remembered  however 
that  it  took  time  liefore  all  thene  innovations  became  genendly 
known,  and  they  were  not  familiar  to  mathematidans  until  the 
lapse  of  some  years  aft«r  they  had  been  published. 

Vieta*.  FrancUeiu  TiX-t  (/Voafou  fUU)  was  born  ia 
1540  at  Fontcnay  near  la  Koehelle  and  died  in  Paris  in  1603. 
He  was  btooght  np  as  a  lawyer  and  practised  for  aome  tiOH 


236  TUK  MATUKMATICB  OW  THK  EKMAUUVOT 

ai  Iha  FarisiMi  bar;  be  iban  beeame  a  nenbar  of  Iba  pio- 
vindal  parliament  in  Brittanj ;  and  finally  in  1680  tbnwgh 
the  influence  of  the  duke  De  Rohan  be  was  made  nuurter  of 
rw|ae^  an  office  attached  to  the  parliament  at  Buria;  the 
nsMt  of  his  life  was  spent  in  the  public  service.  He  was  a 
firm  believer  in  the  right  divine  of  kingSi  and  probably  a 
sealous  catholic  After  1580  be  gave  up  most  of  bis  leisure 
to  mathematicHi  though  his  great  work  In  ArUm  Anatyiieam 
Iwagoye  in  which  he  explained  how  algebra  could  be  implied 
to  the  solution  of  geometrical  problems  was  not  published  till 
1591. 

His  mathematical  reputation  was  already  considerable^ 
when  one  day  the  ambassador  from  the  Low  Countries  re- 
marked to  Henry  IV.  that  France  did  not  possess  any 
geometricians  capable  of  solving  a  problem  which  had  been 
propounded  in  1593  by  his  countryman  Adrian  Romanus  to 
all  the  mathematicians  of  the  world  and  which  required  the 
solution  of  an  equation  of  the  45th  degree.  The  king  tliere- 
U|iou  suuuiioncd  Victa,  and  informed  him  of  the  challenge. 
Vieta  saw  that  the  equation  was  satisfied  liy  the  chord  of  a 
circle  (of  unit  radius)  which  subtends  an  angle  2r/45  at  the 
centre,  and  in  a  few  minutes  he  gave  back  to  the  king  two 
solutions  of  the  problem  written  in  pencil.  In  explanation  of 
this  feat  I  should  odd  that  Vieta  hud  previously  discovered 
how  to  form  the  equation  connecting  sin  u$  with  sin  $  and 
cos^.  Vieta  in  his  turn  asked  Komanus  to  give  a  geometrical 
construction  to  descrilje  a  circle  which  should  touch  three 
given  circles.  This  was  the  problem  which  Apollonius  had 
treated  in  his  De  Tactionibtis,  a  lost  book  which  Vieta  at 
a  later  time  conjecturally  restored.  Hoiuanus  solved  the 
problem  by  the  use  of  conic  sections,  but  failed  to  do  it  by 
Euclidean  geometi'}*.  Vieta  gave  a  Euclidean  solution  which 
so  impressed  llomauus  that  he  travelled  to  Fontenay,  where 
the  French  court  was  then  settled,  to  make  Vieta's  acquaint- 
ance— an  acquaintanceship  which  rapidly  ripened  into  warm 
friendsliip. 


YIETA.  237 

Henry  wnn  much  utmck  witK  the  nhilJtj  ibewn  bj  Viet* 
in  this  ntAttcr.  Tlie  Spaniiinls  hod  at  thftt  time  •  cipher 
oonlAJning  nearly  600  chitmcU-ri)  which  was  periodical! j 
changed,  and  which  they  brlie^rd  it  wna  imponilile  to  de- 
cifdier.  A  deHpnlch  having  Ixwn  int«rcepted,  the  king  gave 
it  to  VietA,  and  anked  him  to  try  to  read  it  and  find  the  key 
(o  the  Hynteni.  VieU  sDCceeded,  and  for  two  yean  the  French 
used  it,  greatly  to  thoir  profit,  in  the  war  which  waa  then 
raging.  So  convinced  wa*  Philip  II.  that  the  cipher  eouM 
not  be  dincover^  thnt  when  he  found  hu  plans  known  he 
complained  to  the  jiope  that  the  French  were  nnng  ■orce;y 
againnt  hini,  "contrary  to  the  practice  of  the  Christian 
faith." 

Viebi  wrot«  numerous  workn  on  algebra  and  geometry. 
The  mofit  important  are  the  In  Artftn  Anedjflicam  tingoge, 
Toara,  1591  ;  the  Siijiplrmenttim  flrinnrlrMUi  and  a  collection 
of  geometrical  prolilenis,  Tount,  IS93  ;  and  the  De  A'Hmeront 
Pofulntiim  Ji'iuJiiliott*,  Paris,  1600 :  all  of  these  were  printed 
for  private  circulation  only,  Init  they  werB  collected  by  F.  van 
Schuoten  and  pu)>lished  in  one  volume  at  Leyden  in  1646. 
Vieta  also  wrote  the  De  ^Equalionnm  Rteognitiaite  et  £mem- 
datiotu  which  was  pnlilished  after  his  death  in  1615  by 
Alexander  Anderson. 

The  /n  Arl'in  is  the  earliest  work  on  nymlmlic  algebra.  It 
also  introduced  the  use  of  letters  for  Imth  known  and  anknown 
(ponitivc)  (guantitien,  a  notation  for  the  powers  of  quantities 
and  enforced  the  advantage  of  working  with  bomogcneons 
equationa.  To  this  an  appendix  called  LogiHien  Speeiota  was 
added  on  addition  and  multiplication  of  algelmuca]  qnantities, 
and  on  the  powers  of  a  binomial  up  to  the  sixth.  Vieta 
implies  that  he  knew  how  to  form  the  coefficienta  of  theae  six 
eipansions  liy  means  of  the  arithmetical  triangle  as  TartagUa 
had  previovsly  done,  but  Pascal  gave  tbo  general  role  tor 
fbrming  it  for  any  order,  and  Stifel  had  already  indiealed  tbe 
method  in  the  expansion  of  (1  -*-x)*  if  thaw  in  the  expaoaiaa 
at  {I  *m)*-^  were  kaowB;  Newton  was  tin  Int  to  ^  th* 


288  THE  lUTHDIATICB  OP  THE  MDrAUBAVCC 

genenl  ezpramion  lor  the  coeffieiani  of  «f  in  the  mrpMwinn  cf 
(1  -f  «)*.  Another  appendix  known  as  ZeUiiea  on  the  eolation 
of  equations  was  subsequently  added  to  the  Jn  ArUtm. 

The  In  Artem  is  memorable  lor  two  impruveroents  in 
algebraic  notation  which  were  introduced  here^  though  it  is 
probable  tliat  Vieta  took  the  idea  of  both  from  other  authors. 

One  of  these  improvements  was  that  he  denoted  the  known 
quantities  bjr  the  consonants  B,  C,  />,  Ac.  and  the  unknown 
quantities  bj  the  vowels  A^  E^  /,  ire.  Thus  in  anj  problem 
he  was  able  to  use  a  number  of  unknown  quantities :  in  this 
particular  point  he  seems  to  have  been  forestalled  by  Jordanus 
and  by  Stifel.  Tlie  present  custom  of  using  ^he  letters  at  the 
beginning  of  the  alphabet  a,  6,  c,  iSrc.  to  represent  known 
quantities  snd  those  towards  the  end,  ^  yi  ^  Ac.  to  represent 
the  unknown  quantities  was  introduced  by  DjeHCsrtes  in  1637. 

The  other  improvement  was  this.  Till  this  time  it  had 
been  the  cuKtom  to  introduce  new  syniliolH  to  represent  the 
square,  cube,  Ac.  of  quantities  which  hod  already  occurred  in 
the  equations ;  thus,  if  R  or  N  stood  for  x^  Z  or  C  or  Q  stood 
for  Q^^  and  C  or  K  for  x*,  Ac.  So  long  as  this  ^as  the  case  the 
chief  tui  vantage  of  algebra  was  that  it  afforded  a  concise  state- 
ment of  i-esultM  every  statement  of  which  was  reasoned  out^ 
But  when  Vieta  used  A  to  denote  the  unknowli  quantity  x^  he 
sometimes  employed  A  quatlrahu^  A  euinut^ ...  ito  represent  a^, 
2*,  ...,  which  at  once  shewed  the  connection  between  the 
different  powers;  and  later  the  successive  powers  of  A  were 
commonly  denoted  by  the  abbreviations  Aq^  Ac^  Aqq^  Ac.  Thus 
Vieta  would  have  written  the  equation 

ZBA* -- DA  ^  A*  =  Z, 

wi  B  3  in  A  qtuni,  -  D  pfatto  in  A-k-A  eubo  ae^uatur  E  tolido. 
It  will  lie  oliserved  that  the  dimensions  of  the  (Constants  (B^  D^ 
and  E)  are  chosen  so  as  to  make  the  equation  homogeneous : 
this  is  characteristic  of  all  his  work.  It  will  be  also  noticed 
that  he  does  not  use  a  sign  for  equality :  and  iq  fact  the  parti- 
cular sign  =  which  we  use  to  denote  equality  wSs  employed  by 


\ 


him  U)  repn-fcnt  "  tho  ilifTrrpnce  Iietwefti."  Vieta's  notKtion  is 
not  ao  convetiiont  &s  that  jirrviouHly  dmxI  hf  Btitd,  Bmnbelli 
and  St^viiiun,  but  it  was  more  gcneraily  iidopted ;  occmioimI 
inRtnncea  (if  an  nppronch  la  index  notntion,  anch  M  A\  Are 
aoid  to  occur  in  Vieta's  workn. 

Thene  two  stppn  were  nlmost  eMsential  to  aiij  farther  pro- 
grvf»  in  nlgolim.  In  Ixith  of  tlipm  Victn  lind  Itecn  f6mit«11ed, 
but  it  wnn  bin  goml  luck  in  enipbasinnf;  their  importance  to 
be  the  mennH  of  ninking  tliem  generally  known  »t  ft  timewlien 
opinion  wns  ripe  for  nuch  an  ndt'nnce. 

The  De  .f^qualtniiiim  H'mgnit'ume  «(  Enmutalion^  »  montlj 
on  the  theury  of  equntioni.  Vieln  here  shewed  that  the  fimt 
nieniljcr  of  nn  nlgebmicAl  e<)UAtinn  ^{r)-  0  could  bareratved 
into  linenr  fnctors,  nnd  explnined  how  tlte  coeffieientit  of  x  . 
could  be  exprexRpd  as  functionx  of  thp  root*.  He  aim 
indicnlrd  how  from  a  given  equation  another  conid  be 
obtniiicd  whone  roots  were  ei{ual  to  those  of  the  original 
increase*]  by  a  given  quantity  or  multiplied  by  a  given 
quantity :  nnd  he  uted  this  method  to  gpt  rid  ot  the  eoefficient 
of  z  in  a  quadratic  equation  and  of  the  coefficient  ot  ;^  in  a 
cubic  equation,  and  wan  tbun  enabled  to  give  the  general 
algebraic  solution  of  both. 

His  solution  of  a  cubic  equation  is  nn  follows,  first  reduce 
the  equation  to  tbe  form  3?  +  Sn'x  =  20'.  Next  lot «  s=  «»/y  -  |f, 
and  we  get  y*  -v-  26^  =  a'  which  Ih  a  (juadratic  in  gp*.  Hence  y 
can  Iw  found,  and  therefore  x  can  be  determined. 

His  solution  of  a  biquadratic  is  nimilar  to  that  known  as 
Ferrwi's,  and  ettsentiaily  os  follows.  He  finttgot  rid  of  the 
term  involving  x*,  thus  reducing  the  equation  to  tbe  fcnn 
x'  *  aV  +  fr*x  -  c*.  He  then  took  tbe  t«rnui  Innrfving  ^  and  ■ 
to  the  right-hand  side  of  the  equation  and  added  i^  +  {y*  to 
each  Btdc^  so  that  the  equation  became 

He  then  chose  y  so  that  the  right-hand  side  of  4Ue  eqialHy  ia 
a  perfect  sqakie.    Sabatilsttng  this  nUye  «(  y,  be  WM  aUt 


S40  TBI  lUTHBMATICS  OP  THE  RSHAIflSAVCI. 

to  Uka  tlie  aquara  root  of  holh  udm^  and  thiit  otUia  ivo 
qoadratie  equattons  lor  a^  ^"^  ^  whidi  caa  be  aolvod. 

The  De  NumeroM  Poieitatum  Bemduium§  dealt  with  wn»a* 
rical  eqoationB.  In  this  a  method  for  approzimatiiig  to  the 
values  of  positive  roots  is  given,  but  it  is  prolix  and  of  little 
nae,  though  the  principle  (which  is  similar  to  that  of  Newton's 
rule)  is  correct.  Negative  roots  are  uniformlj  rejected.  This 
work  is  hardlj  worthy  of  Vieta's  reputation. 

Vieta's  trigonometrical  researches  are  included  in  various 
tracts  which  are  collected  in  van  Schooten's  edition.  Besides 
some  trigonometrical  tables  he  gave  the  general  expression  for 
the  sine  (or  chord)  of  an  angle  in  terms  of  the  sine  and  cosine 
of  its  subiuultiples :  Delambre  considers  this  as  the  completion 
of  the  Arab  system  of  trigonometry.  We  may  take  it  then 
tliat  from  this  time  the  results  of  elementary  trigonometry 
were  familiar  to  mathematicians.  Vieta  also  elaborated  the 
theory  of  right-angled  spherical  triangles. 

Among  Vieta's  miscellaneous  tracts  will  be  found  a  proof 
that  each  of  the  famous  geometrical  problems  of  the  triaection 
of  an  angle  and  the  duplicaticm  of  the  cube  depends  on  the 
solution  of  a  cubic  equation.  There  are  also  some  papers 
connected  with  an  angry  controversy  with  Claviusi  in  1594, 
on  the  subject  of  the  reformed  calendar,  in  which  Vieta  was 
not  well  advised. 

Vieta's  works  on  geometry  are  good  but  they  contain 
nothing  which  requires  mention  here.  He  applied  algebra 
and  trigonometry  to  help  him  in  investigating  the  properties 
of  figures.  He  also,  as  I  have  already  said,  laid  great  stress 
on  the  desirability  of  always  working  with  homogeneous 
equations,  so  tluit  if  a  square  or  a  cube  were  given  it  should 
be  denoted  by  expressions  like  a'  or  6'  and  not  by  terms  like 
iM  or  n  which  do  not  indicate  the  dimensions  of  the  quantities 
they  repreHcnt  He  had  a  lively  dispute  with  Scaliger  on  the 
latter  publishing  a  solution  of  the  quadrature  of  the  circle^ 
and  Vieta  succeeded  in  shewing  the  mistake  into  which  his 
rival  liad  fallen.     He  gave  a  solution  of  his  own  which  as  far 


VIETA.      aiRARO.  241 

MB  it  guc*  IH  corrvct,  nrid  HUt«d  tlmt  tliu  nrc»  of  A  squkra  in  to 
that  of  the  circunisc rilling  circle  m 

^!  -  J(i  *  Jl)  «  -Jli  *  J(i  *  A))  ■■•«<•■»/■ :  1- 
This  is  one  of  the  r.-irlicBt,  nttt^nipts  to  (iml  tlic  v^ao  of  a-  hj 
moans  of  nn  infinibr  Rfrios.  Hr  wa-i  wrll  Aci^uaintMl  with  the 
extant  writings  of  the  (Jrwk  p^mctricinnjt,  and  introduced  tho 
CQriouB  custom,  which  during  the  scvnnt«'cnth  and  eighteenth 
centuries  liccame  fnshiunaliie,  of  mb^ring  loat  classical  workB. 
He  himiwlf  produced  n  conjectural  re?<tomtion  of  the  D€ 
Taetionifmt  of  Apollonius. 

Glrord.  Vieta'ii  rcNulti  in  trigonometry  and  the  theory 
of  e<|uationfi  »crc  extended  liy  Aflxrt  Girtird,  a  Dutch  mathe- 
matician, who  was  bom  in  Lorraine  in  1592  and  died  on 
Dec  9,  1632. 

In  tfi'26  Ciranl  puMixIml  at  the  Hague  a  short  trratiao  on 
trigonometry,  to  which  were  appended  talilcn  of  tho  values  of 
the  trigonometrical  functiona.  Tliis  work  contains  the  earliest 
hm  of  the  aMircvintions  *tn,  (ntt,  fc  for  sine,  tangent,  and 
secant  The  nupplcmental  trinngW  in  upherical  trigonometry 
ant  also  diiKunted ;  their  properties  necm  to  have  been  dis- 
covered by  (iirnrd  and  Snell  at  al)Out  the  Mnip  time,  Ointn) 
aim  gave  the  expression  for  the  nri'ti  <if  a  spherical  triangle 
in  terms  of  the  spherical  excewt — this  wm  discovered  inde- 
pendently by  Cavatieri.  In  1(>37  Gimrd  Inmigfat  out  an 
edition  of  Maroloia's  Gcometr}'  with  conHiderahle  addi^ona 

Gintrd'H  algebraical  inveHtigations  arc  contained  in  his  Inveti- 
tion  nmiirlk  en  VnUjibre  publinhnl  at  AniHteidaRi  in  1629 :  this 
contains  the  earliest  use  of  brackets ;  a  geometrical  interpiv- 
tation  of  the  nf^tive  sign  ;  the  statement  that  the  nnmber  of 
roots  of  an  algebraical  ec|nation  is  e<iasl  to  its  degree;  the 
distinct  ncognition  of  iniaginai^  roots ;  and  probaUj  implies 
also  a  knowledge  that  the  firat  niemlicr  of  so  algebraical  eqo*- 
tion  ^(z)  -  0  could  bo  resolved  into  linear  factors.  OirH'd's 
ioTesUgations  were  unknown  to  ni«it  of  his  eoatsraporsrie^ 
and  eTercised  no  appreciable  inSnenoe  ob  thi  dTslopMSirt 
k 

\* 


i4S  TUB  HlTHEUATICH  OF  TBK   BEMAI8&AHCE. 

Th»  invention  uf  logHritlinui  by  Nkpier  of  UrrchisbMin  in 
1614,  Mid  tbeir  introduction  into  Engluid  by  Briggi  and 
othen,  b«ve  been  ^Irewly  mentioned  in  chapter  xi.  A  few 
wonit  on  tboe  umhematicinns  Duy  be  here  added. 

Napisr*.  JoAn  A'a/iUr  was  born  at  Mercliistoun  in  1550 
•nd  died  on  April  4,  1617.  He  spent  uuet  of  his  time  on  tbe 
family  entate  newr  EdinbuqjU,  and  touk  an  iwtive  part  in  the 
political  and  religiuuH  cunt  rove  nies  of  the  day  ;  the  businees 
of  his  life  waa  to  bhew  that  the  pope  wan  anti-Christ,  but  hia 
favourite  amusement  wan  the  study  of  matlieiuatics  and  Hcienock 

As  soon  as  the  use  of  enpunenta  became  couiuioo  in  algebra 
the  introduction  at  lugarithniM  would  nnturally  follow,  but 
Napier  reasoned  out  tlie  result  without  the  um  uf  any  symbolic 
notation  to  assist  )iiiii,  and  the  invention  of  lugarithnii  was  the 
result  of  Die  etTurt-s  of  nuny  years  with  a  view  to  abbreviate 
tlw  processes  of  nmlliplii^tiun  and  division.  It  ia  likely  that 
Kapler's  atleution  may  have  been  partly  direcUxl  to  the 
desirability  of  fauilitatin;;  computations  by  the  stupendous 
arillmictical  effotts  uf  Mime  of  hia  cuntempomriea,  who  seem 
to  have  taken  a  keen  pleoBuru  in  surpassing  one  anothtr  in 
the  ext4;nt  to  which  they  curried  multiplications  and  diviaionx. 
The  trigonometricul  tablt-M  by  Itheticuu,  which  were  published 
in  1596  and  1613,  were  calculated  in  a  moat  laburious  way; 
Vieta  biiiMicIf  delighted  in  arithmetical  calculations  which 
must  have  taken  days  of  liuni  work  and  of  which  the  results 
ofl4-n  served  no  uwfut  purpuHO :  L.  van  tieulen  (1539-1610) 
practicallj'  devoted  hit  life  to  finding  a  numerical  approximation 
lo  the  value  of  ■-,  liuully  in  1610  obtaining  it  correct  to 
35  places  uf  deciniatH:  while,  to  cit«;  one  mure  instance, 
P.  A.  Catildi  (154(^-1626),  whu  is  chieHy  knuwn  for  his 
invention  in  1CI3  of  the  form  of  continued  fnictiomi,  must 
liave  spent  yean  in  nuuierical  calculatiuns. 

•  Sm  the  J/rMuin  u/  N.,pitr  by  Uuk  Napier,  Edioburgh,  1834.  in 
edition  at  all  hia  worki  vsa  Uaued  at  Edinburub  ia  IKtS.  A  biblioKrapbj 
of  hli  wntiuiia  is  appeoded  to  a  trsDilatioa  o(  the  Conilrwlio  b;  W.  B. 
llacdonald,  Rilitil>u>Kli,  IRMU. 


NAPIER.      BRions.  243 

In  rcpinl  la  NnpirrH  othrr  work  I  nwy  iiptm  mcntiiin 
thnt  in  liiii  R'lM'J'xjin,  publislioti  in  1GIT,  he  intnidvcnl  an 
impnivw!  fonn  of  mA  by  tlie  use  of  which  the  pnidact  of  two 
numlxTH  can  )ic  found  in  m  mcchnnicnl  wAjr,  or  the  qnotient 
of  iHiB  numlier  by  nnotlicr.  He  also  inrrnled  two  other  roH« 
cnlled  "virjfulw"  liy  which  MjUBrp  onil  cnhc  rnotfl  mn  lio 
cxtmctml.  I  nhoulil  add  thnt  in  Bphcrical  trigonometry  he 
diHcovcred  cprtnin  forinulnb  known  s.s  Napier's  Minlagien,  nnd 
enuncinted  tlic  "  rule  of  circular  pirtn "  for  the  aolatioii  of 
right-Anglcd  Hpherical   triangles. 

Briggs.  Tlie  name  of  Bri^gii  in  inRcpnnibly  Kmociiited 
with  the  liintorj-  of  lugnrilhnis.  //<rnry  I>ri^»*  wm  bom  nesr 
llnlifax  in  l-nGI  :  he  was  educated  at  8t  John's  College, 
Cand)ridgc,  took  h\n  dcgrre  in  I5HI,  nnd  o)>twned  ■  fcllownhip 
in  1.588:  ho  was  circtcvl  to  the  OrcKimm  pmfcmonhip  of 
geometry  in  159C,  and  in  1GI9  or  11)20  hocftine  Saviliim 
profranor  nt  Oxford,  n  chnir  which  he  held  nntti  his  death 
on  Jim.  20,  1631.  It  may  ins  intrrratinK  to  add  thni  the 
chair  (if  g<>om<-try  f<>un<lc4  by  Sir  ThoninH  Urathum  was  tlie 
farlieHt  profcs-soreliip  of  ninthematicH  eHtaMixhed  in  Great 
ItriUiin.  Some  twenty  yenni  earlier  Sir  Henry  Savile  had 
given  at  Uxfurd  open  tertumf  on  Greik  geometiy  and  geo- 
mctricinnN,  and  in  Hi  ID  he  endowed  the  chairs  ol  geometry 
and  astntuomy  in  thnt  university  which  are  atill  aanociated 
with  hin  nanto.  Both  in  London  and  al  Oxford  Briggn  was 
the  first  occupant  of  the  chair  of  geometry.  He  began  hin 
lectures  at  0;ifi>r(l  with  tlie  ninth  pnipiHitiim  of  the  fimt  iiook 
of  Euclid  :  thnt  being  Uie  farthest  point  to  which  Uavilo  had 
lieen  able  to  carry  his  audiences.  At  Cambridge  the  Lacastaa 
chair  was  establinhcd  in  1663,  the  earliest  occnpuits  bejng 
Barrow  and  Newtoa 

The   almost   immediate   adaption   throagbiMt   Evnpe   of 

logarithms  for  astronomical  and  otiier  calculations  waa  watnly 

the   work   of    Briggs,   who   undertook   the   tedious   work,  of 

*  8m  pp.  tl—tO  of  mj  lIMarj  af  Ikt  St»if  »/  MsMmsUc*  al  (taf 

»rWf#,  Csmbridf^  IBSB. 

Ift— \ 


844  THE  MATHKHATICH  Of  TUB  UHAiaaAKCR. 

aalcuUtinf  utd  prepnriug  Ublea  of  loffiuithau.  Atnoagal 
otben  be  ounviim'd  Kuplur  uf  the  wlvMilage*  of  Napier'* 
dioouvMy,  SDii  tbe  Hpnaul  of  the  uae  of  lotpu-ilhuu  van 
raudered  man  niiid  by  the  Kui  uid  repuUtion  uf  Kepler 
»bo  by  his  ULIm  uf  16:25  and  1629  biuuglit  tbom  into  vugua 
iu  Uenuany,  while  Cavaliuri  in  1624  and  likliuuiicl  Wiugale 
in  1626  did  a  uutilar  uvrvice  for  Italiuii  and  Fn-uob  niatlie- 
luaticiana  roapectivdy.  Brigga  aiHU  was  iiistruuieiital  iu 
briiiipng  intu  ooiiuuou  usu  tbu  uiotbud  uf  luiij;  diviaiuu  now 
generally  ewployal. 

Harriot.  Tkiiiiuu  Ilarriol,  wfau  waa  burn  at  Onfunl  id 
I&60,  aiul  diixl  iu  Lomluu  un  July  3,  1621,  did  a  grt»t  d«al 
tu  extend  and  codify  iho  tlieury  uf  etjuutions.  Tlie  early  part 
uf  bis  life  waa  apiriit  id  Aiuurica  with  8ir  WiUUir  llalfigb : 
while  tbere  lia  nuulc  ibo  finl  survey  of  Virginia  and  North 
Caroliiui,  tlie  niupu  of  tliew  being  subttequrntly  prusi-utMl  tu 
QuoL-n  Elixttbetb.  On  liiit  n^'tum  tu  England  he  itottlcd  iu 
Luiuloii,iLncI  gave  up  niuul  uf  bin  time  to  iuutb(.-maticul  ntudits. 

The  majority  uf  tlie  propuiitiuns  I  bnv«  aauigned  to  Vieta 
are  to  be  fuund  in  Hnrriot'H  writings,  but  it  is  uncertain 
whether  they  were  discuvi^red  by  hiin  indcpundeiitly  uf  Vieta 
or  not.  In  any  case  it  is  prubublu  that  Viutii  bad  not  fully 
reoliied  all  that  won  cuntaincd  in  the  propusitionn  be  liad 
eniincintuL  Some  uf  tbu  con!MH|uencea  of  tliesr,  with  ext«n- 
HiunH  and  a  syitteuiuliu  i-xpo»itioii  uf  tbti  theory  of  equatioiui, 
were  given  by  Harriot  iu  biii  Arlit  Aaalifliaie  I'nixui,  which 
WHM  tirHt  print«l  in  I6J1.  The  Projcit  ia  more  analytical 
than  any  algebra  t'uit  pn-ceded  it,  and  marlcs  an  advance  buth 
in  BymholiHrn  and  notatioii,  but  Ufgalive  lUid  imaginary  ruota 
are  rejeclML  It  wah  widely  rrud,  and  proved  one  of  the  mutit 
powerful  iiuttrunii.'uia  iu  bringing  analytical  methud.H  into 
'  general  uae.  JIarriut  wua  the  first  to  ime  the  aigUH  >  and  < 
to  represent  gn«ter  llinn  and  le^t  than.  When  lie  denoted 
the  unknuwa  quantity  by  ii  he  repreM-nted  a'  by  aa,  a'  by  aaa, 
and  KU  on.  Thia  iHadi&Llnct  i  ui  prove  men  t  on  Vieta's  notation. 
The  tuuue  iiymbulidm  was  ubed  by  Wallis  as  Ute  as  1685,  but 


HARRIOT.      nuaHTREO.      ,  S45 

concanvntly  with  tlie  miMlirn  index  notation  which  wan 
introrfucpd  by  DpscarU's.  I  npwl  not  ntlode  to  ths  other 
inTmligntions  nf  Hnrriot,  lu  thejr  nre  comfMntiTelj  of  mwll 
importnncfi ;  »xtract8  from  wtme  of  tliotn  w«ra  pnhlinhed  hy 
S.  P.  nignu.!   in   1833. 

Ooghtred.  Anmnie  thosf  who  contriliuted  to  the  general 
lulogition  in  Rtiglnnd  of  thpw  vHrioux  improvenirnU  Mid  ad- 
tliti»iiii  to  nl^ri^tn  and  Algebra  wm  Wiffinm  Ougkhfd*,  who 
wa.<i  Ixim  At  Euin  od  .Mnrch  n,  IR'H,  itnd  died  at  hia  vicr-rags 
of  Albur}'  in  Suirry  on  June  HO,  1660  r  it  ia  nometinm  nud 
thnt  the  cnuw  of  his  drnth  wna  the  excitement  and  delight  which  , 
he  expcrienctni  "At  henring  the  lloUBe  of  ComiiMMM  [or  Con- 
vention] lind  voti-d  the  King's  return  "  ;  n  recent  critic  iwldB 
that  it  nhould  Ijc  remembered  "  by  way  of  exrane  tliat  he 
[Oughtird]  wan  tlieii  eiglity-ntx  ycAra  old,'  bat  perhaps  the 
Rtory  ia  aufficimtlj-  diwredit^^  by  the  date  of  hia  death. 
Oughtre<I  wnx  educated  At  Eton  and  King's  CSoll^e,  Cam- 
bri<ij,'e,  of  tho  Initer  of  which  colleges  he  w»  a  fellow  and  for 
Nome  timn  matheniAtical  lecturer. 

Hi«  Clneiii  .}fathfmntim'  publi-^hed  in  1631  in  a  good  aja- 
temntic  t^-xt-lKMik  on  nriUimetic,  and  it  cimtoinit  practically  all 
Ihnt  wiui  then  known  on  the  Hubject.  In  thia  work  bo  intro- 
ductal  the  Hyiiibnl  x  for  multiplication.  Me  alao  intralnced  the 
aynibol  :  :  in  proportion :  previously  t^i  bin  time  a  pmportim 
■inch  on  a  :b  -  e:d  wan  UHunlly  writtrn  ti»  n-b—e  —  d;  he 
df^noled  it  liy  n  .b  i:e .  d.  WnDJH  nays  that  Bome  foond 
fault  with  the  book  on  account  of  the  atjle,  but  that  they 
only  di.iplayecl  their  own  incompetence,  f or  Oaghtred'a  "wonh 
be  alwayn  full  but  not  ntlundant."  Pell  makea  a  aonMwhat 
dmilar  remark. 

Oughtretl  aim  wrote  A  trcatbipon  trigonomettjpahUalMdin' 
1657,  in  which  ahtncviaiionn  for  Jtimr,  emnii*.  Ack  were  employed. 
Thin  waa  really  an  important  advance,  hut  the  worin  of  Oiiwd 


■  See  pp.  3D— 31  of  vaj  Hilort  of  IV  Rtittf  tf  J 
CamhrUfe,  Cunbridee,  IBH9.    A  eomptcts  edition  of  OattbuTt  Wdrka 
WM  pabtidted  at  Oilord  in  1G77.  ' 


S46  TBI  lUTHElf ATICB  OP  THE  RDfAUBAVCI. 

and  Ovghtredy  in  whidi  thej  were  naed,  were  n^gieeted  end 
■oon  fofgolteiiy  end  it  wen  not  until  Enter  reintrodnoed  eiwi- 
trnctioiu  for  the  trigonometricnl  functions  that  thej  were 
generally  adopted. 

We  niay  say  roughly  that  henceforth  elementary  arith* 
ttieticy  algebra^  and  trigonometry  were  treated  in  a  manner 
which  is  not  Hulwtantially  different  from  that  now  in  uee ;  and 
tliat  the  Hulmequent  improveiuents  introduced  were  additiomi  to 
the  BubjecU  an  then  knoa-n,  and  not  a  re-arrangement  of  them 
on  new  foundations. 


The  otigin  of  the  tiiotv  ootnnion  egmboU  in  algAra. 

It  may  be  convenient  if  I  collect  he:«  in  one  place  the 
iicattered  remarks  I  have  made  on  the  introduction  of  the 
various  symbols  for  the  more  common  operations  in  algebra*. 

The  later  Greeks,  the  Hindoos,  and  Jordanus  indicated 
addition  by  mere  juxtaposition.  It  will  be  observed  that 
this  is  still  the  custom  in  arithmetic,  where,  for  in- 
stance, 2}  stands  for  2  •»•  |.  Tlie  Italian  algebraists,  when 
they  gave  up  expressing  every  operation  in  words  at  full 
length  and  introduced  syncopated  algebra,  usually  denoted 
jJuM  by  its  initial  letter  P  or  p,  m  line  being  sometimes  drawn 
through   the   letter  to  shew  that  it  was  a  contraction  or  a  , 

symbol  of  operation  and  not  a  quantity.     The  practice,  how-  ^ ' 

ever,  was  not  uniform ;  Pacioli,  for  example,  sometimes  denoted 
plus  by  f*,  and  sometimes  by  e,  and  Tartaglia  commonly 
denoted  it  by  ^  The  German  and  Englisli  algebraists  on 
the  other  luuid  introduced  the  sign  +  almost  as  soon  as  they 
used  algorism,  but  they  spoke  of  it  as  irty/iuii*  additamm  and 
employed  it  only  to  indicate  excess,  they  also  used  it  with 

*  Sm  also  two  artielet  bj  C.  Heniy  in  the  Joiie  and  Julj  namben  of 
the  Revue  Arek^otogique,  1879,  voL  xxxvii,  pp.  324 — SSS,  vol.  xxxvui, 
pp.  1—10. 


•• 


AI/IEBRAIC  SYMBOIA.  247 

ft  npeciat  meaning  in  solutions  )>y  ihe  method  of  lKl«e  anump- 
tion.  Widmiin  uh<h1  it  tm  »n  ablirevintion  for  excns  in  1489 : 
by  1630  it  wnn  part  of  the  recognized  notAtion  of  algebra,  and 
wu  aItc  n»ed  im  n  nymliul  of  operation. 

Siiblraefinn  van  in{tic»t<^  liy  Diophantns  by  mi  inverted 
»nd  truncated  ^.  Tiie  Hindoos  denoted  it  by  a  dot.  The 
Italian  algebmiiits  when  they  introduced  ttyncopated  algebr* 
generally  denoted  miniiir  by  J/  or  m,  a  line  being  MMnetimeii 
drawn  through  the  letter;  but  the  pmctice  wm  not  uniform 
— Pacioli,  for  example,  denoting  it  nometimea  by  Si,  and 
■ometiraea  by  il«  for  drmptnn.  The  Gemian  and  English 
algebmiKtit  introduced  the  present  nj-nilio)  which  they  dewri bed 
a*  fiyHum  unhlrartoriim.  It  in  most  likely  that  the  vertical 
bar  in  the  nymbol  for  plu<i  wa.i  superimposed  on  the  (lymbol 
for  minuH  to  distinguish  the  two.  It  may  be  noticed  that 
Pacioli  and  Tartaglia  found  the  sign  -  already  used  to  " 
denote  a  diviHion,  a  ratio,  or  a  proportion  indifferentty.  The 
present  Hifrn  for  minun  wa.i  in  general  uite  liy  about  the  year 
1630,  and  was  then  employed  an  a  Hymliol  of  operation. 

Vieta.  Rchooten,  and  uthem  among  their  contemporaries 
employed  the  sign  =  written  between  two  qnantitin  to  denote 
the  diRerenco  between  them;  t\mna=b  means  with  them 
what  we  denote  by  a  -  6.  On  the  other  hand  Barrow  wrote  — : 
for  the  Rame  purpose.  I  am  not  aware  when  or  by  whom 
the  cun^nt  syndiol  ->-  was  first  used  with  this  signification. 

Oughtred  in  1631  uned  the  sign  x  to  indicate  mMftiptim- 
lion;  Harriot  in  1631  denoted  the  operation  by  a  dot; 
Descart^n  in  1637  indicated  it  by  juxtApoiition.  I  am  not 
aware  of  any  8ym))ols  for  it  which  were  in  pnTions  om. 
Leibniti  in  1686  employed  the  sign  —  to  denote  nraltiplica- 
tion. 

DivitioH  wan  ordinarily  denoted  by  the  Arab  way  of 
writing  the  qnantitien  in  the  form  of  a  fimction  by  roeana  at 
a  line  drawn  between  them  in  any  of  the  fcrma  u—t,  mfh,  at 

r.    Onghtred  in  1631  employed  a  dot  to  denote  ntfaerdiviaioa 


S48  THB  MATUBMATIGB  OP  THK  BBHAlBaAirOI. 

or  *  ratio.    LBibniti  in  1686  ewployad  the  lign  s^  to  denolo 


I  do  not  know  when  the  colon  (or  STmbol:)  wae  int 
introduced  to  denote  a  ratio^  hot  it  occun  in  a  work  hj 
Clairaut  pabliahed  in  1760.  I  believe  that  the  cnrrent  eymb^  * 
for  diviiiion  •?-  is  only  a  combination  of  the  —  and  the  tjnnbol  : 
for  a  ratio^  it  waa  uiied  by  Johann  Heinrich  Rahn  at  Znrieh 
in  1659,  and  by  John  Pell  in  London  in  1668.  The  Hymbol  ~ 
was  ttHed  by  Barrow  and  other  writers  of  bin  time  to  indicate 
continued  proportion. 

Tlie  current  symbol  for  ^qtiaiiiy  was  introduced  by 
Record  in  1557 ;  Xylander  in  1575  denoted  it  by  two 
parallel  vertical  lincH ;  but  in  general  till  the  year  1600  the 
word  was  written  at  length ;  and  from  then  until  the  time  of 
Newton,  say  alwut  1680,  it  wsh  more  frequently  represented 
by  Gc  or  by  X  than  by  any  other  symbol.  Elither  of  these 
latter  signs  was  u-sed  as  a  contraction  for  the  first  two  letters 
of  the  word  atfqualu. 

The  symbol  :  :  to  dencite  proportion,  or  the  equality  of 
two  ratios,  was  introduced  by  Oughtred  in  1631,  and  was 
brought  into  common  use  by  Wallis  in  1686.  There  is  no 
object  in  having  a  symbol  to  indicate  the  equality  of  two 
ratios  which  is  diflfereut  from  that  used  to  indicate  the 
equality  of  other  things,  and  it  is  better  to  replace  it  by  the 
sign  =. 

The  sign  >  for  ii  greater  than  and  the  sign  <  for  it  iesg  ^ 

iKan  were  introduced  by  Harriot  in  1G31,  but  Oughtred 
simultaneously  invented  the  symbols  ID  and  — 3  for  the 
same  purpose ;  and  these  latter  were  frequently  used  till  the 
beginning  of  the  eighteenth  century,  ex,  yr.  by  Barrow. 

The  synibuls  =¥  for  t#  tiol  equal  io^  ^  u  not  greater  tkan^ 
and  '^  for  is  tiot  Ifiu  than  ara  of  recent  introduction,  and 
I  believe  are  ran*ly  used  outside  Great  Britain. 

The  vinculum  was  intruduced  by  Vieia  in  1591 ;.  and 
Imickets  were  first  used  by  Girard  in  1G29. 

Tlie  symUil  J  to  dt*m»t4f  th»  R4|uare  root  was  introducMl  by 


AI/IEIIRAIC   HTMRnm.  249 

Rniltilfrin  1 526;  n  nimilnr  notAtion  hntl  Iteen  lutfd  hf  Bhukus 
and  b}'  Chuqut-t. 

The  <liir?reRt  nipthods  of  rrprcrtenting  the  power  to  wliich 
•  mngnitudc  vm*  misefl  huve  Ijwn  nIreMljr  briefly  Alludfd  to. 
'nic  mrlieNt  known  ntt«mpt  to  fmnii>  n  HjriHbnIic  notation  wim 
made  by  Bombelli  in  1-173  when  hn  rppre»»ent«l  the  anknnwn 
qnanlity  by  ^^  it^  tM|Dar«  by  \ij,  lU  cnlie  hj  \i},  tec  In 
1586  Stet-inuH  use*!  0,  0.  •  ic.  in  a  nimikr  w»y;  and 
mggeat«d,  thoii;;h  )ie  did  not  aw>,  n  cnrrcHponding  notation 
(or  fmctianal  indicm.  In  1591  Vietn  improved  on  thin  by 
denoting  the  difTcrent  powers  of  A  by  A,  A  fittrf.,  A  ewA,,  ic, 
M>  that  he  cniild  inilicnle  tlio  powpm  nf  different  magnitndm ; 
Hnrriot  in  1G31  further  impmved  on  Vietn's  notation  by 
writinj;  nn  for  a',  aan  for  n',  &c,  and  tliiit  remained  in  afe  (or 
fifty  yenn  concurrently  with  the  index  notation.  Tn  1634 
P.  Herigonus,  in  his  Ciirtiis  mnl/urmnliau  pnbliiihed  in  fire 
volumes  nt  Paris  in  1634-1637,  wrote  n,  a3,  oS, ...  (or  a,  <^, 
a'.... 

The  idea  nf  nsing  expnnenlH  to  mnrk  the  power  to  which 
a  quantity  wag  raised  wns  due  to  Descartes,  and  wan  intra' 
duced  by  him  in   10.17;   Imt  he   uk«I  only  poaitii-e  integral 

indicen  n',  n',  n*, AVnIliii  in  ICiQ  explained  the  meaning 

of  negative  and  fractionnl  indices  in  expremions  nnch  an  n'', 
n\  ice;  the  latter  conception  having  lieen  foreshadowed  1^ 
Ocf>Nmu<  and  peHiapx  by  Mtevinuii.  Finally  the  idea  ot  an 
index  unrcstriclei)  in  magnitude,  nuch  an  tlie  n  in  the 
exprcsaion  a',  is,  I  believe,  doe  to  Newton,  and  wh  intra- 
duc*<d  by  him  in  connection  with  the  hinomtal  theorem  in 
the  letten  for  LeibnilE  written  in   1678. 

The  nynibol  oo  for  infinity  wan  finit  employed  liy  Wallin  in 
16.55  in  hin  Arilhatetien  Infintlorum ;  hat  doen  not  occur 
again  until  1713  when  it  in  used  ia  James  Benwaili'a  An 
Citnjitelittwii.  Thin  nign  won  nnmcttmea  emplqjed  by  the 
Romonn  to  denote  the  nnmlirr  1000,  and  it  haa  hem  eonjee- 
tared  that  thin  kd  to  ita  iM'ing  applin)  to  represent  say  ¥erj 
large  nnmlHT. 


250 


THK  MATHBMATIGB  OP  THB  BSNAI8BAN0B. 


Thera  are  bot  few  tpecUl  aymbolji  in  tri(BOiioiiieUy,  I  mmj 
however  add  here  the  following  note  which  fontaini  all  thai  I 
have  been  aUe  to  learn  on  the  inbject  The  earrent  aexa- 
gesimal  diviHion  of  angleii  ia  derived  from  |  the  Babjrioniaaa 
through  the  Greeks.  The  Babylonian  unit  angle  was  the 
angle  of  an  equilateral  triangle ;  following  their  osnal  praetioe 
this  was  divided  into  sixty  eqnal  parts  or  ^egreesy  a  degree 
was  subdivided  into  sixty  equal  parts  or  minutes,  and  so  on : 
it  is  said  that  60  was  assumed  as  the  base  of  the  system  in 
order  tliat  the  number  of  degrees  corresponding  to  the  circum- 
ference of  a  circle  should  lie  the  same  as  the  number  of  days 
in  a  year  which  it  is  alleged  was  taken  (at  any  rate  in  pracUoe) 
to  be  360. 

Tlie  word  irtii«  was  used  by  Uegioniontanus  and  was 
derived  from  the  Arabs:  the  terms  gecaui  and  tangeni  were 
introduced  by  Thomas  Finck  (bom  in  Denmark  in  1561  and 
dit*d  in  1646)  in  his  Geonketriae  Rotundi^  Hale,  1583:  the 
wonl  citnecani  was  (I  believe)  first  used  by  Rheticus  in  his 
OjiUH  PtUatiiiiuii^  1596:  the  terms  cmine  and  cttitiuyttU  were 
first  employed  by  £.  Gunter  in  his  Canon  Trianguiorut^ 
London,  1620.  'llie  abbreviations  jriii,  laM,  tee  were  used  in 
1626  by  Girard,  and  those  of  co*  and  eoi  by  Oughtred  in 
1657 ;  but  these  contractions  did  not  come  into  general  use 
till  Killer  iie-introduced  them  in  1748.  The  idea  of  trigono- 
metrical JnticiioiM  originated  with  John  Bernoulli,  and  this 
view  of  the  subject  was  elalx>rated  in  1748  by  Euler  in  his 
Inirvtlttciio  in  Autilgttin  htfinitonim. 


CHAPTER  XIII. 

THE  CLOSE  OF  THE   RENAISSANCB*. 
CIRC.  1586-1637. 

Thk  closing  ypnn'  of  tlic  n-nniftsanw  were  imtrkt^  by  a 
rcvivnl  of  inUrc^t  in  ncnrly  nil  lirancliRs  of  nmthenifttieM  ftnd 
Ncicnce.  An  for  an  pure  nintlicnrntic!)  in  concenml  ve  luii'e 
ftlmuly  Norn  thnt  during  the  IakI  Imlf  of  the  iiixteenth  emtniy 
their  hud  t>wn  a  jtrent  wlvance  in  nlgel>rft,  theory  of  eqm- 
tionn,  and  trijronoinetry ;  nn<I  we  Htinll  iihortlj  aee  (in  the 
Kecond  nection  of  thiH  chapter)  that  in  th«  ettrly  part  ot  the 
HevcntMinth  centnry  Komc  new  jirocess^'s  in  geometij  were 
ini-entM.  If  howpvcr  we  turn  U>  nppliiil  nMthemftticn  it  ij> 
impossiblo  not  to  be  Htrack  by  tlir  fnct  that  eren  an  late 
a.1  the  middle  or  end  of  the  nixtprnth  cnntnij  no  marked 
progreu  in  the  theory  hod  lieen  made  tram  tha  time  o( 
Arehiiaeden.  Staticn  (of  nolidii)  and  hydroxtatici  remained  in 
much  the  Blate  in  which  he  had  left  them,  while  dynamics  Aa 
a  ncrence  did  not  exist.  It  wna  Sterinan  who  gaTo  the  6nt 
impalae  to  the  renewed!  rtndy  of  ntnlicit,  and  Galileo  who  laid 
the  foundation  of  dynamic*;  and  to  their  works  the  firat 
•action  of  thia  chapter  is  devoted. 

■  8m  loobMle  to  ohaptar  in. 


252  THS  CWm  OP  THK  BBNAUaiHOI. 


The  devetopmeni  of  me^nia  and  experimmUat  m§ikod9. 


Stdviniis*.  Simon  Sievinua  was  born  at  BnigM  in  154^ 
and  ditfd  at  ibe  Hague  earlj  in  the  aeventeenth  oenioiy.  We 
know  very  little  of  his  life  aa\*e  that  he  was  originally  a 
merchant's  clerk  at  Antwerp^  and  at  a  later  period  of  his  Ufe 
wa8  the  friend  of  Prince  Maurice  of  Orange  by  whom  ho  was 
made  qiiarter-inaster-general  of  the  Dutch  army. 

To  his  contemporaries  he  was  best  known  for  bis  works  on 
fortifications  and  military  engineering,  and  the  principles  he 
laid  down  are  said  to  be  in  accordance  with  those  which  are 
now  usually  accepted.  To  the  general  populace  be  was  also 
well  known  on  account  of  bin  invention  of  a  carriage  which 
waM  propelled  by  sails ;  this  ran  on  the  sea-shore,  carried 
twenty-eight  people,  and  easily  outntripped  horses  galloping 
by  the  side:  his  model  of  it  was  destroyed  in  1802  by  the 
French  when  they  invaded  Holland.  It  was  chiefly  owing 
to  the  influence  of  Stevinus  that  the  Dutch  and  French  began 
a  proper  system  of  book-keeping  in  the  national  accounts. 

I  have  already  alluded  to  the  introduction  in  his  Arltk- 
iiWic,  published  in  1585,  of  ex|K>nents  to  mark  the  power  to 
which  quantities  were  raised  :  for  instance  he  wrote  Sj:'-5x-i-1 
as  3 0  -  5 0+  1  ©.  HiH  notation  for  decimal  fractions  was 
of  a  similar  character.  He  further  suggested  the  use  of 
fractional  (but  not  negative)  exponents.  In  the  same  book  he 
like  wise  sugj^ted  a  decimal  system  of  weights  sml  measures. 

He  alno  published  a  geometry  which  is  ingenious  though  it 
dues  not  contain  nuiny  results  which  were  not  previously 
known  :  in  it  some  theorems  on  perspective  are  enunciated. 

*  An  SDslysii  of  his  works  is  given  in  the  iiUtoire  drt  teiettet§ 
maihemtttiqut^t  et  pfty$ique§  chet  le*  litlget  by  L.  A.  J.  Qaetelet,  Brussels,  | 

IHGC,  pp.  144 — ICS;  see  also  Noiice  hittoriqne  •ut  la  vie  ei  les  omrmgeg 
de  SteriHus  by  J.  V.  Gothsls,  Brussels,  1641 ;  end  Let  iraraux  de  Sierimmt 
by  M.  Sieicben,  Brussels,  184C.  The  works  of  Stevinos  were  eoUecled 
by  Suell,  traiiMUt<^  into  Latin,  soil  poblished  at  Leydeu  in  lOOm  under 
tlie  title  Hifpamnemnia  Malkeauitiea. 


STEVINUS,  253 

It  is  hvwcvcr  on  hin  Stnlirt  nnJ  U^lmMitlin  pnlili«lir<l  (in 
FIciimbIi)  nt  lipyHrn  in  I.VG  tlinl  IiJn  fiiniv  will  rmt  In  tlii* 
work  he  vnuncintoi  tlie  triniigle  of  forat— ft  thGoroin  wliidi 
8oma  think  wim  fimt  pn)poamlp<i  by  Tjcuiwrdo  d*  Vinci. 
Stflvinas  rcganla  tliis  oh  the  fumUmi'iitnl  propmittun  of 
the  HuUjcct;  prpviuun  U>  the  pulilicntjon  uf  hiii  work  tlie 
science  of  sintics  hiul  rpNt«(l  on  the  tlicury  ot  the  1«>'er,  hut 
sulMc*|uetitty  it  becAnic  uhu»1  to  cniiinience  by  proving  the 
potwiliility  of  wprrstetitinj;  fore™  liy  Htmiglit  linnt,  itnd  bq 
of  redacih-^  many  tlieorvms  to  giMnietriad  propositionH, 
and  in  particular  tu  uhtnining  in  tlint  way  a  pnicrf  of  tite 
pnralU-logmm  (which  is  equivalent  to  tlie  trinnglo)  of  forcCH. 
tjteviiiuB  iH  not  clfiar  in  liis  arrangement  of  the  various 
pmpoHitioiiB  or  in  tlirir  logical  sequence,  and  the  new  trint- 
ment  of  the  subject  wax  not  definitely  ntablisltcd  lifforc 
tlic  Appenntnce  in  IGt^T  of  Varignon's  work  on  niFclianic& 
Bt«viiiuM  hIho  found  the  force  which  muHt )«  exort«il  nlung  tlie 
line  of  greatest  Hlnpc  to  support  a  given  weight  on  nn  inclinetl 
pisiifl— a  problem  the  iiolution  of  which  hud  licen  ktng  in 
dispute  He  further  dJKtinguishea  Ijetwcen  staUc  and  nn- 
BtAble  (>qutlitirinm.  In  hydraxtatiuii  ho  discniHea  the  )|iioition 
of  the  pressure  which  a  fluid  can  exercise,  «nd  explains  the 
M>«nlied  hydrostatic  paradox. 

His  niethoal  *  of  Knding  the  renolviTd  part  of  *  ^l^ca  in  « 
given  direction,  wt  i1)ustratc<l  by  the  case  ol  a  weight  renting 
on  an  inclined  plane,  is  a  good  Hpecinien  of  his  work  kikI  ia 
worth  quoting. 

Ue  takes  a  wedge  AllC  whpKo  Ihwc  AB  is  horiamital  [and' 
w!iu«e  nidcs  ItA,  tlC  are  in  the  ratio  2  to  I^  A  thre«d 
connecting  a  nnmbor  of  Hniall  pi]ua)  eijuidistant  weiglits  idplacn) 
over  the  wedge  as  indicated  in  the  figure  overpnge  (which  1 
repradace  from  hia  demonstration)  m  tlutt  the  namber  tt 
these  weights  on  BA  U  U>  the  number  on  BC  in  the  bmim 
proportion  m  A<1  is  to  JtC  [this  is  always  possibla  it  th» 
dimensions  of  the  wedge  be  property  cboMn,  and  be  plaws 
■  UgfomitnmU  JtMthtmatiM,  nLn,4t  SIMtm,  piepi  If. 


254 


THE  CLOfIB  OP  THE  RENAI88AMCB. 


four  weigfaU  roftting  on  BA  and  two  on  BC\  Wo  noy 
replace  tiieiie  weights  by  a  heavy  nnilDrm  chain  TSLYT 
withoai  allering  his  aigument.      He  says,  in  elEKti  thai 


O  T 


experienoe  shews  ihut  such  a  chain  will  remain  al  rest:  if 
not,  we  could  obtain  perpetnal  motion.  Thus  the  effect  in 
the  direction  BA  of  the  weiglit  of  the  part  TS  of  the  chain 
must  balance  the  eflect  in  the  direction  BC  of  the  weight 
of  the  part  TV  of  the  chain.  Of  c<Mirm*  BC  may  be  vertical, 
and  if  so  the  above  statement  is  equivalent  to  saying  that  the 
effect  in  the  direction  BA  of  the  weight  of  the  chain  on  it  is 
diminished  in  the  proportion  of  BV  to  7/^,  in  other  words  if  a 
weight  W  reslM  on  an  inclined  plane  of  inclination  a  the 
component  of  IK  down  the  line  of  gruatcst  slope  is  IK  sin  a. 

Stevinus  was  somewliat  dogmatic  in  his  statemeutSi  and 
allowed  no  one  to  ditler  from  his  conclusiouM,  "and  those,** 
says  he,  in  one  place?,  "  who  cannot  see  this,  may  the  Author 
of  nature  liave  pity  upon  their  unfortunate  eyes,  fur  tlie  fault 
is  not  in  the  thing,  but  in  the  sight  which  we  are  unable  to 
give  them." 

Galileo*.      Just  as   tlie   modem   treatment    of    statics 

*  See  the  biogrsphj  of  OalUeo,  bj  T.  H.  Martin,  Psris,  18Ca  Theis 
U  also  a  life  bj  Sir  Dand  Brewster,  London,  1841 ;  and  a  long  notioe  bj 
Libri  in  the  fourth  volume  of  his  iiUloire  de$  iciemce$  wiaikfmaiiqutM  em 
Italic.    An  edition  of  Galileo's  works  was  issued  in  16  Yolumes  by 


GALILEn.  255 

originnteii  willi  Ktevinun,  w  tlip  foundnliiHi  <if  Ibo  acicnon  of 
dynnmiiat  in  ilut-  to  Gnlilfvx  Onlilnt  f/iilUei  wm  liom  nt  Pi» 
on  Feh.  lit,  1564,  nnd  died  ocrnr  Florence  on  Jan.  8,  1642: 
His  fatlier,  m  piwir  (iMcendant  of  nn  old  and  nol>1n  Florantina 
house,  WAR  himwlf  m  fair  miitlieniiit[c.'ian  nnd  *  good  mnaiciMi. 
.  Gftlileo  wu  4^lucnt«d  nt  the  moiiivit«ry  of  VallombrosM  when 
hia  liternry  Aliility  nnd  mecluinic»l  in);^-iittity  Mttracled  con- 
■tdentblo  nttention.  Ilr  won  pprsuiwlcd  to  ticconw  a  novitinto 
of  the  order  in  1580,  Imt  Ium  fntlicr,  wlio  inteiMled  htm  to  bo 
k  doctor,  nt  oncn  reiiiovpil  Iiim,  rihI  srnt  him  in  IS8I  to  the 
nnivenity  of  Pim  to  atudy  medirinir.  It  wbh  there  thtit  ho 
noticnl  that  tlie  f;rrAt  >ironm  lump,  which  still  linngs  from  tlw 
roof  of  the  cathndral,  performed  its  oHcillntiona  in  e<]n«l  times, 
and  indcpL-ndcntly  of  whether  the  oscillntions  were  IaT;ge  or 
■mall — A  fact  whicli  ho  venficd  Ity  counting  hiit  palte.  He 
had  been  liithcrto  kept  in  ignomnec  of  matliematicn,  but  ons 
day,  by  i^hnncc  hmring  a  lecture  on  geometry,  he  wan  no 
faacinated  by  the  science  that  tliencefurward  In  devoted  all 
hia  leiHure  to  its  Htudy,  nnd  finally  l»e  f;ot  leave  to  diHconltnuo 
bin  medical  xtudieH.  He  left  tlin  nniveniity  in  158$,  and 
almoat  immeilintely  cummenccd  Iiih  urj^iiial  researches. 

He  published  in  IUSI  nn  account  ot  the  hydrwtatic 
balMice,  and  in  I  b66  an  cscny  on  the  ccntn-  of  gravity  in  BolidH. 
The  fame  of  tlicxo  works  secured  for  him  the  appuinUuent 
to  tiie  ntathematicnl  chair  at  Pisa — the  Hlipend,  as  was  then 
the  ease  with  most  profesnonihipa,  being  vwy  suiall.  Daring 
the  next  thre«  years  he  cnrricl  un,  from  the  loaning  tower,  that 
series  of  experiments  on  falling  Ixxtien  which  estaMiahed  the 
fint  principles  of  dynamics.  Unfortunately  the  manner  in 
which  he  pruRiulgated  his  discoveries  aiid  the  ridicule  be 
threw  on  tboed  who  ojipeeed  him  gave  nob  annatuiml  ofience^ 
•od  in  1591  he  was  obliged  to  resign  bis  paailioo. 

B,  AlUri,  tloroice,  1S43-1»>6.  A  good  1DS117  of  Ui  hllcrs  oa  raiiooi 
malheniBlinl  nbjnU  ban  been  lince  duoorerad,  aad  a  bsw  a«d  eoM> 
plrte  cdilioa  is  now  Mof  braed  bj  Uw  llaliaa  OowwMft,  lknM% 


t56  TUB  OLOUE  OP  TUI   ItEHAIWUHCB. 

At  thu  time  he  lauiu*  tw  hava  bven  luuoh  tuunpend  \iy 
want  uf  itMMMjr.  Influenoe  wh  however  exerted  oo  bin  behaif 
with  the  TeaeliMi  aeiuite,  «uil  ho  wm  ftppuinted  proEewur  *t 
Padii»,.«  chair  which  he  held  fur  eightoen  yeara  <IS92-I610>. 
Hii  leataree  tUera  aeeiii  to  have  been  chiedy  oo  niechauica  and 
hydraatMtics,  and  tlie  Hubataiice  of  them  in  contained  in  hi« 
tnatiM  Ml  nwehaiiicti  wliiuh  wiw  puliIUIted  in  1612.  In  tlieae 
lecture*  he  ivpeatcd  his  Pisiin  experiiiiL-ntii,  and  demonatnitod 
that  falling  budies  did  nut  (lui  wan  then  belieted)  dtvoeitd  with 
velocititx  pruportionul  ainongst  other  tliiiigii  tu  their  weights. 
He  further  shewed  tliat,  if  it  were  assunicd  tliat  they  descended 
with  a  lUiifunuly  acceleruted  iiiotiuii,  it  wan  puNtiblti  to  deduce 
the  retAtionx  connecting  velocity,  space,  luid  time  which  did 
actually  exist.  At  a  bt^-r  dut«,  liy  ubHeri-ing  the  tiuiea  of 
descent  of  budien  tiliiliiig  down  inclined  planes,  he  ahowed  tliut 
this  hyputhosis  was  true,  lie  itlso  pruved  thut  the  putb  uf  a 
pnijectitu  was  a  paraliuln,  and  in  iluing  su  implicitly  used  thu 
principles  hiid  down  in  the  tiret  two  liiws  uf  luotiaii  us 
enunciiit«d  by  Newt«u.  He  gave  an  uccumtii  delinitiun  uf 
uiumentuui  which  some  writers  luve  tliuugbt  may  be  taken  lo 
imply  a  recognition  uf  the  truth  uf  the  third  law  of  motioa 
The  laws  of  motion  ure  however  nuwhcre  enuriciut«d  in  a 
preciae  and  detinite  form,  and  (iulileu  must  be  regarded  rsther 
iM  preparing  the  way  fur  Newtou  tliim  aa  being  himself  tbo 
creutor  uf  tlie  science  uf  dynuniica. 

In  stutien  he  laid  down  the  principle  tliat  in  mivcliines  what 
WHS  gained  in  power  wiu  lost  in  upcetl,  and  in  the  luuue  ratio. 
In  the  statica  of  solids  he  found  the  furce  which  cau  support 
A  given  weight  on  an  inclined  pliuie  ;  in  hydroHtatics  he  pro- 
IKwndeil  the  luore  eli-meiitury  theurvniH  on  pnwjure  and  vu 
ttuating  bodifH;  while  among  hydruttaticul  instruments  bo 
used,  and  pnrha|M  inventt-d,  the  thonuumeter,  Lliuugh  in  m 
•oniewluit  imperfect  furm. 

It  is  however  as  an  uatruuomor  lliat  must  people  regard 
tiali]eu,«nd  though  strictly  H[>uaking  his  astivnomical  researches 
lie  outside  tlie  subJKt  matter  uf  this  book  it  may  be  iutcnest* 


GALILEO.  237 

ing  to  give  the  lending  fncU.  It  wits  in  the  Hpring  nf  I6<)9 
that  Galileo  hcnrd  tliat  a  tulic  coiitAiiting  IrciRoa  had  heea  nuMle 
by  nn  upticinn,  H.  LippcrMheim  or  l.ippcrehey,  of  Middkburg, 
which  servrd  to  nmgniry  objects  Reen  throflgh  iL  Thin  g»Te 
him  the  clue,  and  he  constructed  a  Hescopc  of  that  kind  which 
Htill  brarn  hin  name,  and  of  which  an  onlinnty  npcnt-gliwa  u  ah 
oxmnpli-.  Within  a  few  months  ho  hnd  produced  inntramenta 
which  were  capnble  of  magnifying  thirty-two  dtameten,  and 
within  a  ycnr  he  hnd  mnde  and  pnhlishrd  obaen'stioni  on  tho 
mlar Epotx,  th<r  lunar  mountains,  JupiUr'n  Entellitea,  the  phase* 
of  Venus,  and  Saturn's  ring.  The  discoveiy  of  the  raicroscnpa 
followed  nntarally  from  that  of  tlie  tclfwopa  Honour*  and 
emoluments  were  nhowered  on  him,  and  he  «'■■  enabled  in  1610 
to  give  up  his  professorship  and  retire  to  Florence.  In  1611  he 
paid  a  temporary  viHit  to  Home,  and  <<xhihit«d  in  the  ganlena 
of  the  Vatican  tho  new  worlds  revealed  by  the  telcacope. 

It  would  tteem  that  Galileo  had  alwn/s  believed  in  the 
Oopcrnican  f>yatem,  but  wan  afraid  of  prnmutgating  it  on 
account  of  the  ridicule  it  excited.  The  existence  of  Jupiter's 
Mtcllites  BPcmed  however  to  make  it»  truth  almost  certain,  and 
he  now  lioldly  preached  it.  The  orlhodox  party  resented  his 
action,  and  on  KeK  24,  1616,  (hp  lii<|iii>^iticni  declared  titat  to 
Bupprate  the  sun  the  centre  of  the  solar  system  was  absurd, 
heretical,  and  contrary  to  Holy  Scripture.  Theedictot  March 
5,  1616,  which  carried  this  into  cflect  haH  never  been  repealed 
though  it  hint  been  long  tdtcitly  ignored.  It  is  well  known 
that  toward!^  the  middle  of  the  Miventoenth  ceotatj  the 
Jesuits  evaded  it  by  tnuting  the  theory  as  an  bypotbens  from 
which,  though  fal«c,  certain  n^utts  would  follow. 

In  January  1632  Galileo  published  his  dialogncs  on  the 
system  of  the  world,  in  which  in  clear  and  forcible  langnage 
he  expounded  the  Copenican  theory.  In  theses  iqipMently 
through  jmlousy  of  Kepler's  fame,  he  does  not  so  nodi  as 
mention  Kepler's  laws  (the  first  two  of  which  had  been  pnb- 
liafaed  tn  1609,  and  the  third  in  1619);  he  rqflcia  Kepler's 
hypotbeaiB  that  the  tides  m«  cMised  by  the  attractioa  ol  tba 


258 


THB  CL08B  Or  THE  RKNAI88AV0I. 


moooy  and  irieii  to  ezpUun  th«ir  exitienoe  (which  hm  alWyt 
is  «  oonfirmaiioii  of  the  Copernican  hypothesis)  hj  the  ttate- 
ment  that  di£bieiit  parte  of  the  earth  rotate  with  differani 
▼dodtiee.  He  was  more  saocesafiil  in  shewing  that  mechaniosi 
prindples  would  aoooont  i^k  the  fact  that  a  stone  thrown, 
straight  up  would  fall  again  to  the  place  from  which  it  was 
thrown — a  fnct  which  previously  had  been  one  of  the  diief 
difficulties  in  the  way  of  any  theoiy  which  supposed  the  earth 
to  be  in  motion. 

The  publication  of  this  book  was  approved  by  the  papal 
censor,  but  suKstantially  was  contrary  to  the  edict  of  1616. 
Galileo  was  summoned  to  Rome,  forced  to  recant,  do  penance, 
and  was  released  only  on  promise  of  obedience.  The  docu- 
ments recently  printed  shew  that  he  was  threatened  with  the 
torture,  but  that  there  was  no  intention  of  carrying  the  threat 
into  effect. 

When  released  he  again  took  up  his  work  on  mechanics, 
and  by  1636  had  finished  a  book  which  was  published  under 
the  title  DUconti  intonioadue  nuove  scienze  at  Leyden  in  1638. 
In  1637  lie  last  his  sight,  but  with  the  aid  of  pupils  he  con- 
tinued his  experiments  on  mechanics  and  hydrostatics,  and  in 
paKicular  on  the  posKibility  of  using  a  pendulum  to  regulate 
a  cluck,  and  on  the  theory  of  impact. 

An  anecdote  of  this  time  has  been  preserved,  which  may 
or  may  not  be  true,  but  is  sufficiently  interesting  to  bear 
repetition.  According  to  one  version  of  tlie  story,  Galileo 
was  one  day  interviewed  by  some  members  of  a  Florentine 
guild  who  wanted  their  pumps  altered  so  as  to  raise  water  to 
a  hei;;ht  which  wius  greater  than  thirty  feet ;  and  thereupon  he 
remarked  that  it  might  be  desirable  to  first  find  out  why  tlie 
water  ruse  at  all.  A  bystander  intervened  and  said  there  was 
no  difiiculty  about  that  because  nature  abhorred  a  vacuum. 
Yes,  said  Galileo,  but  apparently  it  is  only  a  vacuum  which  is 
le.Hs  than  thirty  feet.  His  favourite  pupil  Torricelli  was 
present,  and  thus  had  his  attention  directed  to  the  question 
which  he  subsequently  elucidated. 


OALILEa      BACON.  S59 

Okltleo's  work  rnny  I  think  be  btirly  Humnimt  np  bj  mying 
thnt  hiti  researches  on  nicchnnicn  are  deserving  of  high  pniiae^ 
Mid  thnt  they  ore  iiiemorable  for  clearly  cnDnciKting  tho  ^l 
thnt  Bcienco  must  lie  founded  on  lawn  obtnincd  bj  experiment; 
his  Ntronumicnl  ultservations  and  his  dnlnctionn  tlicrcfrotn 
were  alra  exccllont,  aim]  were  exjMtundi-d  inth  n  liU^nry  skill 
which  IcnvuH  nothing  to  be  desired,  hut  though  he  prodnccd 
■onie  of  the  evidence  which  placed  the  Copcmican  theorj  on  ii 
Mtisfnctory  Utm  he  <lid  not  himself  make  nnj  specul  advsac« 
in  the  theoty  of  nstninoiiiy. 

Franolt  Baoon*.  Tlio  nccemity  of  tn  experimeabtl 
foandntton  for  wienco  wm  aIbo  ndvocated  with  oonttdefsUo 
effect  by  Gnliloo's  contempomry  Fmnei»  Bacon  (LonI  VeniUm), 
who  wur  born  at  I»mlon  on  Jan.  33,  1561,  »nd  died  on 
April  9,  I62C.  Ho  was  rducntf>d  at  Trinity  Coltege,  Cam- 
bridge. His  career  in  pulitics  and  at  the  bftr  culminated  in 
hin  liccoroing  lord  chancellor  with  the  title  of  Lord  Vernlam : 
the  stary  of  his  auhseqnpnt  degradation  for  accepting  bribes  is 
well  known. 

His  chief  work  is  the  A'unini  Onja^wm.  pnltlished  in  1620, 
in  which  he  layi  down  the  principles  which  should  guide  tbom 
who  are  making  enperinientc  on  which  they  pn^xMe  to  found 
a  theory  of  any  branch  of  physics  or  applied  mathematica.  Ho 
gave  rules  by  which  the  results  of  intluction  could  be  tested, 
hnaty  generalization!)  avoided,  and  expenmenta  uned  to  check 
one  another.  The  influence  of  thin  treatise  in  the  eight<H^nth 
century  wa*  grvat,  but  it  is  pmbable  thnt  during  the  preceding 
century  it  was  little  reatl,  and  the  remnrk  repeated  by  xev-eikl 
French  writers  that  Bacon  and  Descartes  an  the  craaton  o( 
modem  philosophy  tests  on  a  misapprehension  <tf  Bacon's 
influence  on  his  contempoiaries :  any  detailed  account  of  this 
book  belongs  howeTer  to  the  history  of  scienti&o  ideas  ratlwr 
than  to  that  of  mathynatics. 

*  See  his  lifo  bjr  J.  Spcddisg.  London,  1813-H.  Ito  tot  sdilin  oT 
his  works  U  Lhsl  bj  EUK  Bpadding,  sod  Usath  ul  I  fvfaMM,  Urnkm, 
MODad  sdition,  1670. 


260 


THK  CLOU  or  THM  BSVAIIfUMOE. 


Befoie  leaving  the  mljed  of  mpfikd  ^■fhemiHci  I  may 
add  a  few  woide  on  the  writings  of  Onldinni^  Wrfghti  nod 
SnelL  I 

Ooldlnilt.  J/abakkuk  OuUinuit  borai  nt  8t  Onll  on  Jnno 
12, 1577,  and  died  at  Orits  on  Nov.  3^  1643^  was  of  Jewiih 
dcHount  but  wan  bruught  up  an  a  proteKtanl:  he  wan  converted 
to  llouian  catholiciHiu  and  became  a  Jesuii  when  he  took  the 
chrUtiau  name  of  Paul,  and  it  was  to  him  that  the  Jesuit 
colleges  at  Rome  and  Qriitz  owed  their  n^atlieinsticai  reputa- 
tion. Hie  two  theorems  known  by  the  i|aiue  of  Pappus  (to 
which  I  have  alluded  above)  were  publwhed  by  Quidinusin  the 
fourth  book  of  his  De  Ceniro  GrainUUu,  Vienna,  1635-1642. 
Not  only  were  the  rules  in  question  taken  without  acknow- 
lodgment  ftxim  Pappus,  but  (according  to  Sloutucla)  the  proof 
of  them  given  by  Guldinus  was  faulty,  though  he  was  success- 
ful  in  applying  them  to  the  determination  of  tlio  volumes  and 
surfaces  of  certain  solids.  The  theorems  were  however  pre- 
viously unknown,  and  their  enunciation  excited  considerable 
interest.  { 

Wri^t*.  I  may  here  also  refer  to  EduMrd  Wriyhi^  who 
is  worthy  of  mention  for  liaving  put  the  art  of  navigation 
on  a  scientific  basis.  Wright  was  bom  in  Norfolk  about  1560, 
and  died  in  1615.  He  was  educated  at  Caius  College,  Cam- 
briilge,  of  which  society  he  was  subsequently  a  fellow.  He 
seems  to  have  been  a  good  sailor,  and  ho  ^lad  a  special  talent 
for  the  construction  of  instruments.  About  1600  he  was 
elected  lecturer  on  mathematics  by  the  East  India  Company ; 
he  then  settled  in  liundon,  and  shortly  afterwards  was  ap- 
|iointed  mathematical  tutor  to  Henry,  Prince  of  Wales,  the  son 
of  James  I.  His  mechanical  ability  may  bo  illustrated  by  an 
orrery  of  his  coiuilrucuon  by  which  it  was  possible  to  predict 
eclipses :  it  was  shewn  in  tho  Tower  as  a  cuiiosity  as  late  as 
1675. 

In  the  maps  in  use  before  the  time  of  Gerard  Mercator  a 

•  See  pp.  25—97  of  mj  Huior^  ^  ike  Study  of  Mtaihemaiie$  «l 
Cambridge,  Csmbridge,  1889. 


PJ 

o 


WRionr.    nNELU  t$t 

degree,  whether  tif  latitude  or  htngitude,  find  been  repreaentfd 
in  mU  cate*  hy  the  name  length,  nnil  the  courm  to  bo  panued 
by  A  VF!isrl  wnH  mnrked  on  the  ninp  hy  a  Htraigttt  line  jmning 
the  ports  of  nirival  and  drpiirlum.  Mercnior  hwd  men  that 
this  led  to  conHidemlilc  emira,  nnd  had  rpiilixed  that  to  make 
tliis  method  of  tracing  the  course  nf  n  xhtp  at  all  Bocnrat«  the 
itpacp  afi^igned  an  the  map  to  a  degree  of  tatttude  ought 
gradually  to  increase  as  the  latitude  increased.  Using  thin 
principle,  he  had  empiricnily  constructed  some  chart^  which 
were  pnblinhed  attoat  ISOO  or  IJiTO,  Wright  wet  hinwelf  the 
problem  to  determine  tlin  theory  on  which  sncb  maps  should 
be  dmwn,  and  succeeded  in  (liKcorrring  the  law  nf  the  scale  of 
thci  map^  though  hin  rule  is  strictly  correct  for  small  am  only. 
The  result  was  published  in  the  second  edition  of  Blnndeville'« 

In  1399  Wright  publi^ihed  his  Cerinin  Emn  in  Ifariga- 
turn  Deltr.tfd  nnd  Correrfeil,  in  which  he  explained  thf  tbeoij 
and  inRerted  n  table  of  meridionn)  parls.  TheraasoniDg  shews 
cnnaiderablo  g(^>nietrical  power.  In  the  course  of  the  work 
he  giveA  the  declinations  of  thirtytwo  star*,  explains  the 
phenomena  of  the  dip,  parallax,  anil  refrnction,  and  adds  a 
(Able  of  magnetic  declinations ;  he  nisumt^  the  earth  to  lie 
stalinnnry.  In  the  following  ymr  he  pulilifihpti  Mime  mapn 
constructed  on  Ins  principle.  In  these  the  northemmoat  point 
of  Au<itmlia  ia  shewn  :  the  latitude  of  I^nilon  is  taken  to  be 
51'  32'. 

SnelL  A  cont^mporaiy  of  Ouildinus  and  Wright  waa 
WilUbrod  Snetl,  whone  name  is  still  well  known  through  bis 
discovery  in  1619  of  the  Inw  of ,  n-fmction  in  optics.  Snell 
was  bom  at  Leydrn  in  1581,  occupied  a  cbnir  of  mathematics 
at  the  univcrmty  thet*,  and  died  them  on  Oct  50,  1626.  He 
was  one  of  those  infont  prodigies  who  oocanionally  ajipear,  and 
at  the  age  of  twelve  he  is  said  to  have  lieen  aeqwunted  with 
the  stAndard  mathematical  works,  I  will  bera  only  add  that 
in  geodeay  he  laid  down  the  principle  for  iluterubung  the 
length  of  the  arc  of  a  meridian  from  the  mMrarMnent  nt  any 


THE  CLOfiB  OP  THK  BtVAIBBiirCI. 


base  line^  and  in  spharical  trigonometrj  he  JSaoomni  tfM 
pn^wrties  of  the  poUr  or  svpplementel  triangla 


Retrival  of  iutereii  in  pure  geomdry. 

The  close  of  the  iitxte.*nth  oentary  was  marked  not  obIj  hj 
the  attempt  to  found  a  theory  of  dyiiamics  Uaaed  oo  kwa 
derived  from  experimeuti  but  aliio  by  a  rehired  interest  ia 
geometry.     TIiih  wan  Uirgely  due  to  the  influence  of  Kepler. 

Kepler^.  Jakann  Kepier^  one  of  the  founders  of  modem 
astronomy,  was  bom  of  humble  parents 'near  Stuttgart  oo 
Dec.  27,  1571,  and  died  at  Ratislxui  on  Nov.  15,  1630.  He 
was  educatMl  under  MsHtlin  at  Tflbingen;  in  1593  he  was 
appointed  professor  at  Qrats,  where  he  made  the  acquaintmnce 
of  a  wealthy  widow  whom  he  married,  but  found  too  late  that 
he  had  purchaHed  his  freedom  from  pecuniary  troubles  at  the 
expense  of  domestic  happiness.  In  1599  he  accepted  an  ap- 
pointment as  assistant  to  Tycho  Brahe,  and  in  1601  suc- 
ceeded hiM  master  as  astronomer  to  the  emperor  Rudolph  11. 
But  his  career  was  dogged  by  bad  luck ;  first  his  stipend  was 
not  paid  ;  next  his  wife  went  mad  and  then  died,  and  a  second 
marriage  in  1611  did  not  pro%'e  fortunate,  although  this  time 
he  hail  taken  the  precaution  to  make  a  preliminary  selection 
of  eleven  girls  whose  merits  and  demerits  he  carefully 
analysed  in  a  paper  which  is  still  extant;  while  to  complete 
his  discomfort  he  was  expelled  from  his  chair,  and  narrowly 
escaipi^  condemnation  for  heterodoxy.  During  this  time  he 
dcpeud<Kl  f«>r  his  income  on  telling  fortunes  and  casting 
honiscopes,  for,  as  he  says,  "  nature  which  has  conferred  upon 
every  aninud  the  means  of  existence  has  designed  astrology  as 

*  See  JokuHH  Krppltr^s  Lehtn  uttd  U'irkem^  by  J.  L.  E.  von  Brsit«eh  veii, 
Htattgart,  IHSI ;  and  R.  Wolfs  Gesekickte  lUr  Astromtmie,  lluuicb.  lM71. 
A  complete  edition  of  Kepler's  vorki  was  pablithed  hj  C.  Friaeh  at 
Frankfort  in  8  volomet  lS5tt-7l;  and  an  aoaljais  of  ilia  matbematical 
part  of  his  ehief  work,  the  liarmoHice  if aifufi,  is  gi? en  bj  Clisales  in  his 
Afer^  khtaHqme.    Bm  also  Cantor. 


KEPLER.  263 

an  adjunct  and  ally  to  aRtronom}*."  He  seems  however  to 
have  had  no  scrapie  in  charging  heavily  for  his. services,  and 
to  the  surprise  of  his  contemporaries  was  found  at  his  death  to 
have  a  oonsiderahle  hoard  of  money.  He  died  while  on  a 
journey  to  try  and  recover  for  the  benefit  of  his  children  some 
of  the  arrears  of  his  stipend. 

In  de5cribing  Galileo's  work  I  alluded  briefly  to  the  three 
laws  in  astronomy  that  Kepler  had  di.scovered,  and  in  connec- 
tion with  which  his  name  will  lie  always  a!»4iciate<l ;  and  I 
have  already  mentioned  the  prominent  part  he  Uiok  in  bring- 
ing logarithms  into  general  use  on  the  continent  Tliese  are 
familiar  facts,  but  it  is  not  known  ho  generally  that  Kepler  was 
also  a  geometrician  and  algebraist  of  considerable  power ;  and 
that  he,  Desargues,  and  perhaps  Galileo  may  be  considered  as 
forming  a  connecting  link  lietween  the  mathematicians  of  the 
renaissance  and  those  of  modem  times. 

Kepler's  work  in  geometry  consists  rather  in  certain 
general  principles  enunciated,  and  illustrated  by  a  few  cases, 
than  in  any  sjrstematic  exposition  of  the  subject.  In  a  sliort 
chapter  on  conies  inserted  in  his  Pamftftomena^  published  in 
1604,  he  lays  down  what  has  lieen  called  the  principle  of 
continuity;  and  gives  as  an  example  the  statement  that  a 
paraliola  is  at  once  the  limiting  case  of  an  ellipse  and  of  a 
hyperbola;  he  illustrates  the  same  doctrine  by  reference  to 
the  foci  of  conies  (the  wonlybcfi^  was  introduced  by  him);  and 
he  also  explains  that  parallel  lines  should  be  regarded  as  meet- 
ing at  infinity.  He  introduced  the  use  of  the  eccentric  angle 
in  discussing  properties  of  the  ellipse. 

In  his  SUtr^ottifiriti^  which  was  published  in  1615,  he  deter- 
mines the  volumes  of  certain  vessels  and  the  areas  of  certain 
surfaces,  by  means  of  infinitesimals  instead  of  by  the  long  and 
tedious  method  of  exhaustions.  These  investigations  as  well 
as  those  of  1604  arose  from  a  dispute  with  a  wine  merchant  as 
to  the  pn^r  way  of  gauging  the  contents  of  a  cask.  This 
use  of  infinitesimals  was  objected  to  by  Guldinus  and  other 
wnten  as  inaccurate^  but  though  the  methoda  of  Kqder  are 


264 


THB  CI/)6X  OP  THE  miWAIMAIIOK> 


not  aliogeilier  tret  from  objeeiion  he  waa  aubsUntuJlj  eorraeli 
and  by  applying  the  kw  of  contanuity  to  inflniteafmali  he 
prepared  the  way  for  Oai^ieri's  method  of  indivisihleii  and 
the  infinitesimal  calcalus  of  Newton  and  Leibniti. 

Kepler^s  work  on  astronomy  lies  outside  the  scope  of  this 
book.  I  will  mention  only  tliat  it  was  founded  on  the  ob- 
servations of  l^cho  Brabe*  whose  assistant  he  was.  His  three 
laws  of  planetary  motion  were  the  result  of  many  and  laborious 
efforts  to  reduce  the  phenomena  of  the  Molar  system  to  certain 
simple  rules.  The  first  two  were  published  in  1609.  and  stated 
that  the  planets  describe  ellipses  round  the  sun,  the  sun 
being  in  a  focus;  and  tliat  the  line  joining  the  sun  to  any 
planet  sweeps  over  equal  areas  in  equal  times.  The  tliird  was 
published  in  1619,  aud  stated  that  the  squares  of  the  periodic 
times  of  the  planets  are  proportional  to  the  cubes  of  the  major 
axes  of  their  orbits.  The  laws  were  deduced  from  observa- 
tions on  tlie  motions  of  Mam  and  the  earth,  and  were  extended 
by  analogy  to  the  other  planets.  I  ought  to  add  that  he 
attempted  to  explain  why  these  motions  took  place  by  a  hypo- 
thesis which  is  not  very  diiferent  from  Descartes's  theory  of 
vortices  Kepler  also  devoted  considerable  time  to  the  eluci- 
dation of  the.  theories  of  vision  and  refraction  in  optics. 

While  the  conceptions  of  the  geometry  of  the  Greeks  were 
being  extended  by  Keplel-,  a  Frenchman,  whose  works  until 
recently  were  almost  unknown,  was  inventing  a  new  method 
of  investigating  the  subject — a  methiid  which  is  now  known 
as  projective  geometry.  This  was  the  discovery  of  Desargues 
whom  r  put  (with  some  hesitation)  at  the  close  of  this  period, 
and  not  among  the  mathematicians  of  modem  times. 

Deaargueat*  Geranl  Destirtjues^  born  at  Lyons  in  1593, 
and  ilivd  in  1 GG2,  was  by  profession  an  engineer  and  arelutect, 
but  he  gave  some  courses  of  gratuitous  lectures  in  Paris  from 

*  For  an  aocouut  of  Tyclio  Brabe,  bom  at  Knndatmp  in  154C  and 
ak*a  at  l*rague  in  ICOI,  aem  hi«  life  by  J.  L.  E.  Dreyer,  RilinburKh,  lH90. 

t  Sctf  OrHvrrt  tie  VfMri/uf*  by  M.  Poudra,  ^  vols.,  Paris,  1864 ;  and 
a  note  in  the  BiUioikeca  Matkemalica^  1885,  p.  HO. 


I62G  to  nlnnt  1630  which  mnile  it  givjit  imprvwuon  upon  hu 
conU>inp>mricx.  Itoth  DcHcnrtrNt  nnil  Pnscn)  hud  n  high  opininn 
of  his  work  nnd  Ahililic^nnfl  both  iniwie  connidenible  ow  of 
the  iheoreniH  hp  hiwl  pnuncidted. 

In  1636  Dfriarguen  issued  a  work  on  penpective ;  botmnct 
of  his  r(->icnrchcn  wrtre  cmlindied  in  his  nnwllan  pmieft  on 
conict,  publishf^l  in  )C-19,  k  copy  of.  which  wm  diacovered 
hj  ChiuJcK  in  1815.  I  Ukn  the  following  sumiiMry  of  it  (mm 
C.  Taylor's  work  on  conies  Dcwrjiups  commencM  with  m 
ntftt^^inent  of  the  doctrint>  of  continuity  an  laid  down  1^ 
Knpler :  thus  the  points  nt  thp  oppoxjltt  ends  of  «  Btraight 
line  ate  regnrded  am  coinciilpnt,  imrallel  lines  are  treat«l  m 
meeting  at  n  point  at  inlii)ity,  and  p.-inill<-l  planes  on  a  line  at 
infinity,  while  a  straight  line  may  In-  considered  an  ■  circle 
whone  centre  is  at  inlinity.  Tlie  theory  of  inrolation  of  idx 
poiota,  with  its  special  coses,  in  laid  down,  and  the  projective 
property  of  pencils  in  involution  is  established.  The  theory  of 
polar  lines  is  expnundnl,  and  its  annlngue  in  Rpnce  suggest^, 
A  tangent  is  defined  as  the  limiting  cn.<u>  of  a  secant,  and  an 
asympt4)lo  as  a  tnngent  at  infinity^  Desnrgnes  slirwn  that  thn 
lines  which  Join  four  points  in  a  plane  dct«miine  throe  paira 
of  tines  in  involution  on  any  imnsvennl,  nod  from  any  c»nic 
through  the  four  points  another  pair  of  lines  can  be  olitaincd 
which  are  in  involution  with  any  two  of  the  fomiftr.  Ho 
proves  that  the  poinfn  of  intersi-clinn  of  the  diagonals  and 
the  two  pairs  of  opposite  siilm  of  any  (]undrilAter^  inscribed 
in  a  conic  are  a  conjugate  triad  with  resprct  to  the  conic,  and 
when  one  of  the  three  points  is  nt  ioGnity  its  polar  is  m 
diameter  ;  hut  he  fsils  to  explain  tlie  cnse  in  which  the  qnnd- 
rilateral  is  a  parallelogram,  although  he  had  formed  the  con- 
ception of  a  straight  line  which  was  wholly  at  inlinity.  Thn 
book  therefiire  may  be  fairly  said  to  contain  the  fanthunental 
theorems  on  involution,  homology,  poles  and  polan,  and  per- 
Elective. 

The  influence  exerted  by  the  lecturra  of  Da— tgnea  on 
DeaoartcM,  Pancnl,  Mid  the  French  geometrkiMM  ct  ttte  Ml«U- 


taa 


THB  CLOBX  or  THB  BKWAiaBAVaL 


teenth  oeniuiy  wm  ooluiidarable ;  bai  the  suljaet  of  pwjaotif 
geometry  aoon  fell  into  oblivion,  ehiefly  beoanae  the  nnnlytienl 
geometry  of  Deacartet  wm  ao  much  more  powerful  as  a  method 
of  proof  or  diiioo%'ery. 

The  researches  of  Kepler  and  Desargnes  will  serve  to 
remiiul  us  that  as  the  geometry  of  the  Greeks  was  not  capable 
of  much  further  extension,  mathematicians  were  now  beginning 
to  M*ek  for  new  methods  of  investigation,  and  were  extending 
tlie  conceptions  of  geometry.  The  invention  of  analytical 
geometry  and  of  the  infiniti^siinal  calculus  temporarily  diverted 
attention  from  pure  ge<mietry,  but  at  the  beginning  of  the 
present  century  there  was  a  revival  of  interest  in  it,  and  since 
then  it  has  been  a  favourite  subject  of  study  with  many 
matlieinaticianH. 


Mathematical  knowledge  at  the  close  of  the  reHaissanee. 

Tlius  by  the  beginning  of  the  seventeenth  century  we  may 
say  that  the  fundamental  principles  of  arithmetic,  algebra, 
theory  of  ei|uationis  aiid  trigonometry  had  lieen  laid  down,  and 
the  outlines  of  the  subjects  as  we  know  tliem  had  been  traced. 
It  muHt  be  however  remembered  that  there  were  no  good 
elementary  text-books  on  these  subjects ;  and  a  knowledge  of 
them  was  therefore  confined  to  those  who  could  extract  it  from 
the  ponderous  treatises  in  which  it  lay  buried.  Though  much  of 
the  modern  algebraical  and  trigonometrical  notation  had  been 
intror]uce<l,  it  was  not  familiar  to  mathematicians,  nor  was  it 
even  univenuilly  accepted  ;  and  it  was  not  until  the  end  of  the 
seventeentli  century  that  the  language  of  these  subjects  was 
definitely  fixed.  Considering  the  absence  of  good  text-books  I 
am  inclined  rather  to  admire  the  rapidity  with  which  it  came 
into  universal  use,  tlian  to  cavil  at  the  hesitation  to  trust  to  it 
alone  which  many  writers  shewed. 

If  we  turn  to  applied  mathematics  we  find  on  the  other 
hand  tluit  the  Kcii'nce  of  statics  had  made  but  little  advance  in 


THE  CL08B  OF  THE  RENAISSANCE. 


267 


the  eighteen  centuries  that  had  elapsed  since  the  time  of 
Archimedeis  while  the  foundations  of  dynamics  were  laid  by 
Galileo  only  at  the  ckwe  of  the  sixteenth  century.  In  fact^  as 
we  shall  see  later,  it  was  not  until  the  time  of  Newton  that  the 
science  of  mechanics  was  placed  on  a  satisfactory  liasis.  The 
fundamental  conceptions  of  mechanics  are  difficult,  but  the 
ignorance  of  the  principles  of  the  subject  shewn  by  the  mathe- 
maticians of  this  time  is  greater  than  would  have  been  antici- 
paied  from  their  knowledge  of  pure  mathematics. 

With  this  exception  we  may  say  that  the  principles  of 
analytical  geometry  and  of  the  infinitesimal  calculus  were 
needed  before  there  was  likely  to  be  much  further  progress. 
The  former  waa  employed  by  Descartes  in  1637,  the  latter  was 
biTented  by  Newton  some  thirty  or  forty  years  later:  and 
their  introduction  may  be  taken  as  marking  the  commence- 
■wni  of  the  period  of  modem  mathematica. 


THIRD  FSRIOD. 
jnolttni   ilAatt)rm.-iiiti. 

ne  hiiditrg  of  modtm  miiUirmiilies  Inyiiu  reilk  tke  inrrmlioH 
of  nrwi/yini/  •jeimtelry  nru/  ihn  injiuitrgimal  eafnUtu.  The 
malhrHiatint  u  /nr  more  eomjilex  ihaii  Hint  produced  i'm  «M«r 
of  the  prtotiling  perimls :  but,  Jurinif  the  reettUiteAttk  ami  evjk- 
Imulh  txnturiai,  it  may  ht  gen'rally  drterihed  lU  thnrtKleriztid 
Ay  the  tknlo/im'-nt  of  attnli/i-U,  at%H  ilt  aiijiHentvm  lo  the  ;•&•- 
a  t(f  nntnre. 


I  continue  tlic  chnmnlogicnl  nrmnf^nicnt  of  Iho  Hnlijf^ct. 
'  ChMptcr  XV  tx>ntAin3  the  bistmy  of  the  forty  yvmn  fnHn  1635 
to  1675,  iind  Hn  account  of  tlic  mathematical  duicor(>ri<?fl  of 
Dpscartcts  CavAlicri,  PancaI,  WkIIis,  FrroMt,  and  Hnygens. 
ChapUr  XVI  in  given  up  to  a  di.scawtion  of  Nowton'a  rcnmrchcs. 
Chapter  xvii  containH  an  accwint  of  the  works  of  Leibnits  and 
hit  followen  during  the  fintt  half  of  the  eighteenth  centarj 
(including  D'Alemliert),  and  of  the  cuntemponuy  Englifib 
Hchool  to  the  drath  of  Mac-laurin.  Tlin  mtricH  of  Enler,  Ia> 
grange,  lAplacc,  and  their  contemporariea  brm  tbo  salgeet- 
matter  of  chapter  xvili. 

Lastly  in  chapter  xix  I  havci  addrd  Rome  notca  on  a  fcw  of 
tbo  mathnnaticiana  of  recent  tiracN ;  Imt  I  exdnde  all  detailed 
raference  to  living  writ«rt,  and  partly  bccaaM  of  ikin,  partly 
for  other  reaaons  there  given,  the  account  of  coatemponiy 
mathemalica  doea  Dot  profeaa  to  cover  the  ralgoct. 


CHAPTER    XIV. 

THE   ms-mRV   OF   MODERN    UATHEHATICa 

Tub  division  bctwwn  this  period  and  that  treated  in  the 
ImI  iiix  chapten  i^  hy  no  mtuiii!!  m  well  rittOncd  m  ttmt  which' 
DcpantUM  the  history  of  Grei'k  mfttlioniatics  from  tlie  imtlie- 
matics  of  the  middle  ages.  The  mcthoiU  of  AiMljviM  used  in 
the  iievent^enth  century  and  the  kind  of  prohlema  attacked 
changed  but  gnulunlly ;  and  the  mathematicians  at  the  begin- 
ning of  thin  [Miriod  were  in  immediate  relations  with  thoN  at 
the  end  uf  that  lant  con.ttderpd.  For  tlii^  rataon  khiki  writcni 
lui\'c  divided  the  hiHtory  of  ruathemnlics  Into  two  part*  only, 
treating  the  twhoolmcn  an  the  linen!  HUcceHSon  of  the  Greek 
■nathcmaticianN,  and  dating  the  creation  of  modem  mathe- 
matioi  from  the  introductiun  of  the  Arab  test-book*  into 
Enrope.  The  division  I  have  given  in  I  think  more  con- 
venient, fur  the  introduction  of  analytical  geumetiy  and  of 
the  infinitesimal  calculus  rfvotu  lionized  the  development 
of  the  subject,  and  therefore  it  nccnis  preferaUe  to  take 
their  invention  as  marking  the  ooinmenccment  of  modem 
nkathematicH. 

The  time  that  has  elapxod  ninco  thcHC  methods  were  in- 
vented has  been  a  period  of  incesaant  int«llectna]  actintj  in 
all  departmenU  of  knowledgv,  and  the  prognM  made  in  niaUie- 
■natica  has  liecn  imroenna  Tlie  greatly  mitended  range  of 
knofwMgo,  tha  mMi  of  uateiiak  to  be  aiailiiiwl.  Um  wimmm 


272 


THE  HinOBY  OP  MODBBN  MATHXIUTIOHL 


of  penqpeoUv«^  and  even  the  ecboes  of  old  oontitiveniee 
bine  to  incroMe  tlie  diffieulties  of  an  aatbor.  Am  however  the 
leading  facte  are  generally  known,  and  the  works  pnbliihed 
during  this  time  are  accessiUe  to  any  stadent,  I  misy  deal 
more  concisely  with  the  lives  and  writings  of  modem  matho- 
uiaticians  than  with  those  of  their  predeceaaorsy  and  confine 
myself  mora  strictly  than  before  to  those  wlio  have  materially 
affected  the  progress  of  the  subject. 

To  give  a  sense  of  unity  to  a  histoiy  of  mathematics  it  is 
uecesaary  to  treat  it  chronologically,  but  it  is  possible  to  do 
this  in  two  ways.  We  may  discuss  separately  the  develop- 
ment of  different  branches  of  mathematics  during  a  certain 
period  (not  too  long),  and  deal  with  the  works  of  each  mathe- 
matician under  such  heads  as  they  may  fall  Or  we  may 
describe  in  sucoe^on  the  lives  and  writings  of  the  mathema- 
ticians of  a  certain  period,  and  deal  with  the  development  of 
difiereut  subjects  under  the  heads  of  those  who  studied  them. 
Personally,  I  prefer  the  latter  course;  and  not  the  least 
advantage  of  this,  from  my  point  of  view,  is  that  it  adds  a 
human  interest  to  tlie  narrative.  No  doubt  as  the  subject 
becomes  more  complex  this  course  becomes  more  dillicult,  and 
it  may  be  tliat  when  the  history  of  mathematics  in  the  nine- 
teenth century  is  written  it  will  bo  ueccssary  to  deal  separately 
with  the  separate  branches  of  the  subject,  but,  as  far  as  I  can, 
I  continue  to  present  the  history  biugraphically. 

lloughly  s|ieaking  we  may  say  that  five  distinct  stages  in 
the  histury  of  modern  mathematics  can  be  discerned. 

First  of  all,  there  is  the  invention  of  analytical  geometry  by 
Descartes  in  1G37 ;  and  almost  at  the  same  time  the  intro- 
duction of  the  method  of  indivisibles,  by  the  use  of  which 
areas,  volumes,  and  the  positions  of  centre's  of  mass  can  be 
determined  by  summation  in  a  manner  analogous  to  that 
effected  now-a-days  by  the  aid  of  the  integral  calculus.  The 
method  of  indivisibles  was  soon  superseded  by  the  integral 
calculus.  Anal}lical  geometry  however  maintains  its  position 
as  part  of  the  necessary  training  of  every  mathematician,  and 


THE   HISTORY  OF  MODERN   MATHEMATICS.  273 

for  all  purposes  of  rexenrch,  is  incomparably  mora  potent  than 
the  f^eometry  of  tlip  nncientn.  The  Inlter  is  Htill  no  donbt 
an  admimble  intellectuftl  tmining.  And  it  freqaently  aflbrdu 
an  cipgant  (IflmonHtmtinn  of  wimo  pi-upoKition  the  truth  of 
which  in  alretwly  knnwn,  but  it  r»H|uirwt  a  special  promlun 
for  every  pnrticulnr  problem  nttackfil.  TIip  former  on  the 
othor  hand  lAyii  down  a  few  Hiiiiple  rulex  \tf  which  any 
property  can   be  at  once  proved  <ir  disproved. 

In  the  tnviid  place,  we  have  the  invention,  snow  thirty 
yenni  later,  of  the  Auxional  or  difTfrentinl  cnlcnluK,  Wherever 
a  quantity  changes  nccontinf;  to  some  continuous  law  {and 
mwt  things  in  nature  ilo  so  change)  the  differential  calculus 
enables  un  to  inennure  its  ral«  of  increiL-<e  or  decrease;  and, 
from  its  rate  of  increase  or  decrease,  the  integral  calculus 
enables  us  to  tind  tlm  urisiiint  quantity.  Formerly  eveiy 
separate  function  of  t  such  as  {\  *  xY,  log  (It  a:),  sina^ 
tan~'  r,  &c,  could  lie  exjiniideil  in  ascending  powers  of  z  only 
by  means  of  such  special  pnxvdure  ai  was  suitable  for  that 
particular  problem  :  but,  by  the  aid  of  the  calcalnn,  the  expan- 
sion of  any  function  of  r  in  ancendini;  power*  of  '  is  in  geneiKl 
reducible  to  one  rulo  which  covent  all  cane*  alike.  So  a^in 
the  theory  of  maxima  and  minima,  the  determination  of  the 
lengths  of  cuneH,  and  the  arnax  enclosed  by  them,  the  debr- 
minntion  of  surfacen,  of  volumes,  and  of  centree  of  mam,  and 
many  other  problems  are  each  n^Iucible  to  a  sinf^  rule.  T\m 
theories  of  difTerential  equations,  of  the  calculus  of  variatiofui, 
of  finite  diflierencea^  Ac,  are  the  developments  of  the  ideaa  Of 
the  calculus. 

These  two  subjects — analytical  gpometry  and  the  calculus — 
became  the  chief  instruments  of  further  prograM  ia  mathe- 
matics. In  both  of  them  a  sort  of  machine  was  coostracted : 
to  solve  a  problem,  it  was  only  neccwwry  to  put  in  the  par- 
ticular function  dealt  with,  or  the  equation  of  the  partienlar 
curve  or  surface  considered,  and  on  performing  certain  rimple 
t^Mrations  tlie  rwult  came  oaL  Tbe  ralidity  of  the  procea 
waa  prand  otiee  for  all,  and  it  was  no  loOj 


18 


274 


THK  HUrrOBY  OP  MODBBN  MATHIIUTICaL 


invent  lome  ipecial  method  lor  everj  Mpumte  f onetaooi  ewrvt^ 
or  Burface. 

In  the  ikird  pUioe,  Huygent,  following  Qalileo^  laid  the 
foundation  of  a  aatialadtory  treatment  of  dynamioi^  and 
Newton  reduced  it  to  an  exact  science.  The  latter  mathe- 
matician proceeded  to  applj  the  new  analytical  methods  not 
onl}'  to  numerous  problems  in  the  mechanics. of  solids  and 
fluids  on  tlie  earth  but  to  the  solar  system:  the  whole  of 
mechanics. terrestrial  and  celestial  was  thus  brought  within 
the  domain  of  mathematics.  There  is  no  doubt  that  Newton 
UHed  the  calculus  to  obtain  many  of  his  results,  but  he  seems 
to  have  thought  that,  if  his  demonstrations  were  established 
by  the  aid  of  a  new  science  which  was  at  that  time  generally 
unknown,  his  critics  (who  would  not  understand  the  fluxional 
calculus)  would  fail  to  realize  the  truth  and  importance  of 
his  discoveries.  He  therefore  determined  to  give  geometrical 
pruofH  of  all  luH  resultn.  He  accordingly  cast  the  Fritaeifna 
into  a  gtHuiietrical  form,  and  thus  presented  it  to  the  world  in 
a  languaige  which  all  men  could  then  understand.  The  theory 
of  mechanics  was  extended  and  was  systematized  into  its 
modern  form  by  Lagrange  and  Laplace  towards  the  end  of  the 
eighteenth  century. 

In  the  fourth  place,  we  may  say  that  during  this  period 
the  chief  branches  of  physics  have  been  brought  within  the 
scope  of  mathematics.  This  extension  of  the  douuun  of  mathe- 
matics was  commenced  by  Huygens  and  Newton  when  they 
propounded  their  theories  of  light;  but  it  was  not  until  tlie 
bc*gi lining  of  this  century  that  sutiiciently  accurate  oliservations 
were  made  in  most  physical  subjects  to  enable  niatheinatical 
reasoning  to  be  applied  to  them. 

Numerous  and  far-reaching  conclusions  have  been  obtained 
in  physics  by  the  application  of  mathematics  to  the  results  of 
olMcrvations  and  experiments,  but  we  now  want  some  more 
simple  liy|iothe»e8  from  which  we  can  dtnluce  those  laws  which 
at  present  fonn  our  8tarting-|)oint.  If,  to  take  one  example, 
we  could  say  in  what  electricity  consisted,  we  might  get  aome 


THB   HI8T0RT  OF  MODERN   MATHEMATICS.  J75 

rimple  laws  or  hypotheses  from  wliicli  liy  the  kid  of  ni&the- 
mnties  aII  the  o1xier\'pil  phfiioiiirna  coalH  be  deducMl,  in  the 
mnie  way  nn  Nrwtoii  dnlucfd  nil  Die  rpNults  of  phynicnl  Mtro- 
Roniy  froTD  llio  law  of  gmviinlion.  All  tinea  of  mearch  neeni 
mmrpovpr  to  imlicftlc  tliat  thpre  M  i»n  intintato  connection  l»- 
twp«-n  t)ifl  difft^rent  bmnchefi  tif  phyNicn,  c.,'/.  Itetwfcn  light,  hntt, 
eliuticity,  rlectricity,  and  magnetifiin.  T)i0  ultimate  expUn»' 
tion  of  thiB  and  of  the  iMdingfact^in  physics  aeemtt  to  demand 
aotudyof  moleculnr physics;  »  knowlwlgeof  nioleoiUirplijrBiai 
in  its  turn  wrems  tit  miuirp  some  thi^ry  nn  to  the  constitation 
of  matter ;  it  woul'l  further  appear  that  the  key  to  the  consti- 
tution of  matter  is  to  lie  fouiul  in  cliemintry  or  chemicml 
ptiysics.  Ko  the  matter  Etnnds  at  present;  the  connection 
between  the  d  itTcrenl  )imnchi>s  of  phy^iics,  and  the  fantUmental 
laws  of  those  branches  (if  Ihnre  be  any  simple  ones),  are  riddles 
which  arc  yet  Dnmlvnl.  This  hislnry  does  not  pretmid  to 
treat  of  problrms  which  are  now  the  subject  of  inrestigation : 
the  fact  mfsa  that  mathemnticfti  physics  is  mainly  the  creMtion 
of  the  nineteenth  century  wouM  exclmie  all  detailed  discuMion 
of  the  nubjrct. 

Fifthly,  this  period  han  seen  an  immense  extension  of  pure 
mathematics.  Much  of  this  ix  the  croatiim  of  oompaiBtively 
recent  times,  and  I  regani  the  ilelnils  of  it  m  outside  the  limil* 
of  this  book  thou];h  in  chapter  xix  I  have  allowed  myself  to 
mention  some  of  the  snbjectA  discns.vd.  The  ntosl  striking 
features  of  this  extension  an*  the  dcvelopmenta  of  higher 
geometry,  of  higher  arithmetic  or  the  theory  of  nnmbeni, 
of  higher  alf^hra  (including  the  theory  of  forms),  and  of 
the  theory  of  equations,  also  the  discussiwi  of  fancticms  *4 
doable  and  multiple  periodicity,  and  notably  the  crastioa  of 
a  theory  of  f-anctions. 

This  hanty  summary  will  indicate  the  suhfeots  treated  and 
the  limiutionn  I  have  imposed  on  myselt  The  hlstofj  of  the 
origin  and  growth  of  analysis  and  its  opidicatioD  to  Uw 
material  nniverse  eoinea  within  my  parrieir.  The  sxtensioat 
in  tits  latter  half  of  th«  nineteenth  oentny  Vt  p«n  — th» 
IS— a 


276 


THE  HI810BY  OF  MODBRW  MATHIMATIOHL 


matia  and  ol  Uie  applicatioii  of  maihumafioi  to  phyiimi 
pniUeuii  open  m  new  period  which  lies  beyond  the  Umiti  of 
thii  book ;  and  I  allude  to  thene  irabjeoU  onlj  ao  iar  as  thsf 
may  indicate  the  directions  in  which  the  fnture  history  of 
mathematics  appears  to  be  developing. 


. 


CHAPTER    XV. 


HIRTORT  or  MATIIKMATIfS  FROM  DESCARTtt  TO  HUYOEKS*. 
CIBC.  1635-1675. 


I  rKoposx  in  this  clinptrr  tii  conKitlor  tlm  hwtory  tif  matbe- 
tnatiCK  during:  Die  f'irtjr  ypaiK  in  the  inidrllo  of  the  m\-enteenth 
century.  1  regnnl  Dpsi^irtoH,  Cnv«)irri,  Pnscal,  Wnllts,  Fenniit, 
«nfl  Huygens  aa  tho  lni<ting  nintlieninticiitnf  <rf  thin  time. 
I  hIiaII  trvnt  tlictn  in  thnt  onlpr,  and  I  kIhH  conclude  with 
*  lirief  IJRt  of  the  mom  cniini*nt  rtminining  mMthenwliciAiu 
of  the  wine  date. 

I  hnre  nirrndy  xtAted  t)mt  t)>e  nintheHinticiAna  of  thia 
period — and  the  rrninrk  applies  more  pitrtirulArly  to  Descnrtea, 
FbkaI,  nnd  Fcnnat— were  largely  inRucnred  hy  the  teaching 
of  Kepler  nnd  DeEargueN,  and  I  would  repent  again  tiuA  I 
regnni  tliese  latter  and  <!alilcct  an  forming  ft  connecting  link 
I>elween  the  writers  of  the  renaiwAnnco  ftnd  thoM  of  modem 
limes,  t  nhould  nlm  add  that  the  mnthcniatidang  considend 
in  tliiH  chapter  were  content  porn  ries,  and,  although  I  hat-e  tried 
to  place  them  roaglily  in  nuch  an  order  that  Uieir  chief  works 
ahall  conra  in  a  chmnolt^cal  arrangement,  it  ia  emential  to 
remember  that  they  were  in  relation  one  with  the  other,  and 
in  general  were  aciinainted  with  one  anutber'a  wercbea  aa 
aoon  a«  them  were  poliliahed. 

*  Brr  Cantor,  part  it,  to),  n.  pp.  S99-*H :  elbar  MlhntttiM  Ibr 
the  BallieniaUdaM  of  Ihii  period  ara  wraUmMil  to  Ike  faolMlaai 


278     MATIUUIATIOI  FBOM  DUOAIITU  TO  UUTQKIiaL 


t . 


Daioartei^.  Subject  to  tlie  above  ramarks  we  uaj 
■ider  Descartett  as  Uie  fint  of  the  aiodem  eehool  of  matbe- 
matics.  Bene  DueairUM  iraa  bom  near  Toars  on  March  51, 
1596,  aiid  died  at  Stockhobn  on  Feb.  11, 1650 :  thus  he  was  a 
ountouipurary  of  Galileo  and  Desargues.  Hit  lather,  who  as 
the  name  implies  waH  uf  a  good  family,  wan  aocustomed  to 
speml  lialf  the  year  |kt  Rennes  when  the  local  pariiament, 
in  which  he  held  a  commiiision  as  councillor,  was  in  sewsion, 
and  the  rest  of  the  time  on  his  family  estate  of  Lee  Cariee 
at  La  Haye.  Rend,  the  second  of  a  family  of  two  sons  and  one 
daughter,  was  sent  at  the  age  of  eight  years  to  tlie  Jesuit 
School  at  La  Fleche,  and  of  Uie  admirable  discipline  and 
education  there  given  he  speaks  must  highly.  On  account  of 
his  delicate  health  he  was  permitted  to  lie  in  bed  till  late  in 
the  mornings ;  this  was  a  custom  which  he  always  followed,  and 
when  he  visited  Pascal  in  1G47  he  told  him  that  the  only  way 
to  do  good  work  in  mHthemaitics  and  to  preserve  his  health  was 
never  to  allow  anyone  to  make  him  get  up  in  the  morning 
before  lie  felt  inclined  to  do  so :  an  opinion  which  I  chronicle 
for  the  beiietit  of  any  schoolboy  into  whose  liamis  this  work 
may  fall 

On  leaving  school  in  1612  Descartes  went  to  Paris  to  be 
introduced  to  the  world  of  fanhion.  Here  through  the  medium 
of  the  Jesuits  he  made  the  acquaintance  of  Mydorge  and 
renewed  his  schoolboy  friendsliip  with  Mersenne,  and  together 
with  them  he  devoted  the  two  years  of  1015  and  1616  to  tlie 
study  of  niatheuuitics.  At  that  time  a  man  of  position 
usually  entered  either  the  army  or  the  church;  Descartes 
choso  the  former  profession,  and  in  1617  joined  the  army  of 


*  Bve  La  lU  iU  Uttcarttt  by  A.  Bsilkl,  2  vols.,  iVu-itf,  1691,  which 
i«  Kummiu-ized  in  vul.  i  of  K.  Fijicher*ii  iSetchiehte  der  ueuem  PhilotophU^ 
Muuich,  1878.  A  tolerably  complete  sccouDt  of  Descartetf't  matbematicsl 
sod  pbyiiical  iuventigstiotui  in  ^ivcD  in  Eracb  and  Gruber'b  KncyeloiMidie^ 
Tbe  luoiftt  oouifilete  edition  of  hi«  works  is  tliut  bj  Victor  Cousin  In 
U  vuU.,  Paris,  lH24-ti.  Home  minor  papers  ^ubseqaeatlj  diseovcisd 
were  printed  by  F.  de  Careal,  Pariii,  185U. 


Prinro  MaariL-e  of  Omngo  then  At  Urcftn.  Walking  DinHigh 
tho  HtrecU  thero  ho  saw  a  plarard  in  Dutc)i  which  excited  liia 
cnrioBity,  find  Mtj>pping  the  first  pn-sser  asked  him  to  tmiulnlo 
it  into  cither  French  ur  LAtin.  Tlio  Etrnnger,  wliu  hApp^ned 
to  be  Ifumc  Boccknian,  the  head  of  tlie  Dutch  Collrga  at  Dart, 
offered  to  do  no  if  Descftrtcs  would  annwer  it:  the  plMOtrd 
being  in  fact  a  challrii^  to  all  the  wurhl  to  mire  «  certain 
geometrical  pn>blrm.  Dcwnrt^^n  worked  it  out  within  ■  few 
houns  and  a  warm  friendship  between  him  »iid  Beeckmiin  wm 
the  tvault.  This  unexpected  teitt  of  hJH  mAthem«tieal  «ttun> 
mentH  mndc  the  unconj^ninl  life  nf  the  annj  tliataHtcfHl  to 
him,  but  under  fnmily  inlluenra  and  trndition  lie  remained  m 
■oldier,  and  wiui  per!iuade«l  at  the  comnicnctinient  of  the  thirtj 
jrean'  war  to  volunlt-er  under  Count  de  llncquoy  in  the  army,- 
of  Bavaria.  He  continued  all  thix  time  to  occupy  hii  leisnra 
with  nialheiuatical  studies,  and  waa  accustomed  to  dato  the 
limt  ideas  of  his  new  philosophy  and  of  his  analytical  geouiotry 
from  three  dreams  which  he  experienced  on  the  night  of  Nov. 
10,  1619,  at  Xeubcrg  when  campaigning  on  Uic  Danube  He 
regarrled  this  n»  the  critical  day  of  hin  life,  and  one  which 
dctormined  his  whole  future. 

Ho  resigned  his  commission  in  the  npring  of  1621,  and 
spent  the  next  five  years  in  travel,  during  moat  ot  which  time 
he  continued  to  study  pure  mntheniatic!!.  In  1626  we  find 
him  settled  at  FarLi  "a  little  well-built  fii^re,  modestly  clad 
in  green  taffety,  and  only  wearing  swonl  and  feather  in  token  of 
his  quality  as  a  gentleman."  l>uring  the  tint  two  years  there 
he  interested  himself  in  general  society  and  spent  hU  leisnte  in 
thccvnstmctinnof  <^tical  inNlnimcnla;  bat  these  parraita  were 
merely  the  relBxations  of  one  who  failed  to  find  in  philoaophy 
that  theory  of  the  aniverae  which  he  waa  connneed  finally 
awaited  him. 

In  1628  Cardinal  dc  Beniltc,  the  founder  of  the  Oratoriana, 
met  DeiicaHea,  and  waa  no  much  impnwsrd  l^  hi>  enaveraa 
tion  that  he  urged  on  him  the  duty  of  devoting  hia  life  to 
the  •saminatifMi  ot  Iratb.    DoMvtes  agreed,  ud  Um  bettor 


280     MATHEMATICS  PtKOM  DBSOARTn  TO  HUTOBNB. 

to  Meuro  hinuelf  finon  intermpiion  movwl  lo  Holkuid,  Aim 
at  the  height  of  its  power.  There  for  twenty  yean  he  lived, 
giving  up  all  hui  time  lo  philosophy  aiul  matheineticai  Scieaee^ 
he  sayiy  may  be  compared  to  a  tree,  metaphysics  is  the  root| 
physics  is  the  trunk,  and  the  three  chief  branches  are  me- 
chanicsy  medicine,  and  morals,  these  forming  the  three  apf^ica- 
tions  of  our  knowledge,  namely,  to  the  external  world,  to  the 
huiuan  body,  and  to  the  conduct  of  life. 

He  spent  the  first  four  years,  16:S9  lo  1633,  of  his  sUy  in 
Holland  in  writing  Le  Monde  which  embodies  an  attempt  to 
give  a  physical  theory  of  tlie  univeme ;  Init  finding  that  its 
publication  was  likely  to  bring  on  him  the  Itostility  of  the 
church,  and  having  no  desire  to  pose  as  a  maKyr,  he  abandoned 
it:  the  incomplete  manuscript  was  published  in  1664.  He 
then  devoted  himself  lo  composing  a  treatise  on  universal 
science ;  this  was  published  at  Leyden  in  1637  under  the  title 
Diucours  de  la  uieihode  pour  bien  coaduire  sn  raUon  et  chereker 
ia  verite  dans  !e$  sciences,  and  was  accompanied  with  three 
appendices  (which  posMibly  were  not  issued  till  1638)  entitled 
La  Dutptrlque,  Les  Jlettores^  and  La  O'eouUtrie :  it  is  from  tlie 
last  of  these  that  the  invention  of  anslytical  geometry  dates. 
In  1641  he  published  a  work  called  Mediiattones  in  which 
he  explained  st  some  length  his  views  of  philosophy  as 
sketched  out  in  the  Dlscours.  In  1644  he  issued  the  Fritu:ijna 
PhUosoi»kiae^  the  greater  part  of .  which  was  devoted  to 
physical  science,  especially  the  laws  of  motion  and  tlie  theory 
of  vortices.  In  1647  he  received  a  pension  from  the  French 
court  in  honour  of  his  discoveries.  He  wpfnt  to  Sweden  on 
the  invitation  of  the  Queen  in  1649,  and  died  a  few  mouths 
hiX^T  of  iniiammation  of  the  lungs.  [ 

In  appearance,  Descartes  was  a  small  man  with  large  head, 
projecting  brow,  prominent  nose,  and  black  liair  coming  down 
to  his  eyebrows.  His  voice  was  feeble.  In  disposition  he  was 
cold  and  selfiMh.  Considering  the  range  of  his  studies  he  was 
by  no  means  widely  read,  and  he  despised  both  learning  and 
art  unless  something  tangible  could  be  extracted  ther^rom. 


DESCA  RTKS  2S1 

lie  npvpr  mnrriml  anil  left  no  descendntilM,  though  he  hvl  one 
illegitimate  Hao^hter  who  tli<^  young. 

An  to  l)iii  philonnphicAl  thpnrics,  it  will  be  nnffidcnt  to  my 
thnt  he  (liscUKwd  the  sniiie  pmhlemH  which  have  been  <leliat«d 
for  the  last  two  (hoDRnnd  yearii,  ant]  jiroltiibly  will  lie  dehntcd 
with  e(]nitl  xf»\  two  thouwind  years  iienee.  It  h  httrdly  nece»- 
mrjr  to  say  thnt  the  problems  themwlvra  arc  of  importance 
nnti  interest,  but  from  the  nature  of  tlie  cAM  no  Nolntion  ever 
offered  ia  capable  either  of  rigid  pniof  or  (A  Hinpniof;  all 
that  car)  be  effected  in  to  make  one  explanation  more  prob- 
able than  another,  and  whenever  >  philr>s<>[Jier  like  DeMartea 
believe*  that  he  biM  at  last  finally  nttletl  »  qDesttun  il  has 
been  pomiblc  fur  hin  HueccKson  to  point  out  the  fallacy  in 
bin  nwniniptions.  I  have  rejid  miinewhoro  that  phikwt^iy  ban 
alwayn  been  chiefly  engaged  with  the  in<«r-reIations  tA  (iwl, 
Nature,  and  ^lan.  The  earlieiit  philosophers  wer«  Gireka 
who  occupied  themsetven  mainly  with  th«  rdatSona  between 
Gnd  and  Natun-,  and  dealt  with  Man  separately.  The 
Christian  Church  was  so  alfsorlieil  in  the  relation  of  (!«d  to 
Man  an  to  entirely  neglect  Nature.  F^rwlly  medeni  philo- 
sophers concern  themiielves  chiefly  with  the  rehitiona  between 
Man  and  Nntuiv.  AVbetber  this  ix  a  correct  histAficai 
generaliutinn  of  the  viewn  which  have  been  suocenrively 
prrtvalent  I  do  not  care  to  diseu«8  here,  hat  the  atatoment  aa 
to  the  ncopo  nf  modem  philmophy  marks  the  limitationa  of 
Desoirtea'i  writiiign. 

Descart«ffi's  chief  contributionH  to  mathenwtica  were  hia 
analytical  geometry  and  hia  theory  of  rorticee,  and  it  is  on 
hia  researches  in  connection  with  the  former  of  tbeee  snbjedM 
that  hia  rvpuUition  rests. 

Analytical  geometry  doea  not  conHist  merely  (aa  is  aome- 
timen  loosely  said)  in  the  application  of  algebra  to  geometry : 
that  had  been  done  by  Archimedes  and  many  otbera,  and  had 
beconie  the  asoal  method  of  procedure  in  the  vorka  ol  tlw 
mathematiciana  of  the  sixteenth  century.  Tlie  grast  adranea 
Mde  liy  Uewwici  w«a  that  be  saw  that  a  point  in  «  plana 


282     MATUCMATlCtl  PBOM  DBUCAKTSU  TO  HUYOBMa 


*  ccmkl  be  oompletdy  deiermined  if  ita  di«Uitce%  wmj  m  mmI  f^ 
from  two  fixed  lines  drawn  at  right  angles  \a  the  plane  wera 
given,  with  the  convention  familiar  to  us  af  to  the  interpre- 
tation of  positive  and  negative  values;  and  that  though  an 
equation  /(as,  y)  =  0  was  indeterminate  and  could  be  satisfied 
by  an  infinite  number  of  values  of  x  and  y,  yet  these  values 
of  X  and  y  determined  the  co-ordinates  of  a  number  of  points 
which  form  a  curve,  of  which  the  equation y(ar,  y)=  0  expresses 
some  geometrical  property,  tliat  is,  a  property  true  of  the 
curve  at  every  point  on  it.  Descartes  asserted  tliat  a  point  in 
sfiace  could  be  simihurly  determined  by  three  coordinatoii  but 
he  c^ntiued  his  attention  to  plane  curves. 

It  was  at  ouce  seen  that  in  order  to  investigate  the 
properties  of  a  curve  it  was  sufficient  to  select,  as  a  definition, 
any  characteristic  geometrical  property,  and  to  express  it  by 
means  of  an  equation  between  the  (current)  coordinates 
of  any  point  on  the  curve,  tliat  is,  to  translate  the  definition 
into  the  language  of  analytical  geometry.  The  equation  so 
obtained  contaiiiiH  implicitly  every  property  of  the  curve,  and 
liny  particubtr  property  can  be  deduced  from  it  by  ordinary 
algebra  without  troubling  about  the  geometry  of  the  figure. 
This  may  have  been  dimly  recognized  or  foreshadowed  by 
earlier  writers,  but  Descartes  went  further  and  pointed  out 
the  very  iiupurtaint  facts  that  two  or  more  curves  can  be 
referred  to  one  and  the  same  system  of  coordinates,  and  that 
the  points  in  which  two  curves  intersect  can  be  determined  by 
finding  the  roots  common  to  their  two  equations.  I  need  not 
go  further  into  detaiLi,  for  nearly  everyone  to  whom  the  above 
is  intelligible  will  liave  read  analytical  geometry,  and  is  able 
to  appreciate  the  value  of  iU  invention. 

iVscartes's  GconUtr'ni  is  divided  into  three  books:  the 
first  two  of  these  treat  of  analytical  geometry,  and  the  third 
includes  an  analysis  of .  the  algebra   then   current      It   is  1 

somewliat  difficult  to  follow  the  reasoning,  but  the  obscurity 
was  intentional.  "  Je  n'ai  rien  omis,"  Hays  he,  **qu*4  dessein... 
j'avois  pr€vu   que  certtines  gens  qui  se  vantent  de  s^voir 


DESCARTES.  283 

tout  n'numicnt  ^rm  manqnt^  d«  dire  qai;  jo  n'«voiii  rien  ^fvit 
qu'ils  n'eiisM;nt  s^-u  auparovnnl,  si  jc  mo  fnsio  renHa  Man 
intelligible  pour  cux." 

Tli«  firxl  lionk  commence  with  nn  explnnntian  of  the 
principles  of  nnalyticnl  <;ivmctry,  niiil  conUinH  «  Hiwiiwion 
of  n.  certAin  prublcm  wliidi  IiwI  been  pn)puiindwl  hy  Pnpptia  in 
the  HovenUi  liook  of  hin  Swayoiyij  and  of  which  wnw  pnrUcaUr 
cwtcii  hw)  Ijopn  onnsidcrcd  bjr  Euclid  niid  ApoltonitiB.  Tho 
gnnf'ral  tlicorein  had  bnfHi?<l  pinvi'ma  ^T^Mnotriciitivv  mhI  it 
WK  in  the  Attempt  to  nolve  it  tliat  L>c»ciirtM  wm  led  to  tho 
iiiventicm  of  nnnlytical  geometry.  TIte  full  onunclntion  ut 
tliD  prublem  i«  mtlicr  involved,  but  the  hmmI  importMit 
ante  in  to  find  the  Incut  of  it  point  Huch  Ihttl  tho  prwluct 
of  the  perpend iculnrH  un  in  given  atmight  lines  ahsll  be  in  » 
conHtnnt  ntio  to  the  product  of  thn  pcrpendicuhn  on  m 
other  given  straight  lines.  The  ancientM  hud  solved  this 
geomctricAlly  for  the  caw  m-l,  n  :  ],  and  the  case  md, 
n  -  2.  Pappus  hiul  further  stab^  that,  if  m  =  n  =  2,  the  loctti 
was  s  cunic,  liut  he  gavo  no  proof ;  DtucmUm  nlm  failed  to 
prove  this  liy  pun;  geuinctry,  but  he  Hhcwod  that  tho  atm 
was  repreocntod  by  an  equation  of  tho  second  dttgrce,  thst  lit, 
was  a  conic ;  subsequrntly  Newton  gave  an  elegani  solntkin  of 
the  problem  by  pure  geometry. 

In  the  BiMiond  liook  Uescnrtj^it  divides  curves  into  two 
claHsea ;  namely,  geotiietricnl  and  niechnaiotl  curves.  He 
defines  gconietrieal  curvcH  an  tlxno  wlijch  C«n  bo  generated 
by  tho  inb;n«;ction  of  two  lines  ouch  HKiiing  parallel  to  one 
coordinate  axis  with  "comniunHnrable"  velocitiea;  by  which 
term  he  nienna  that  tly/tlr  is  an  algel>raiail  function,  w^  fur 
example,  is  the  case  in  the  ellipse  and  the  eiaaoid.  He  gmIIs 
m  cnrre  mechanical  when  the  ratio  of  the  veloeitiea  of  tlieee 
lines  is  "incomnienenrahle" ;  by  which  term  he  ineuis  that 
djtldx  is  a  transcendental  function,  as,  for  ezMnple,  is  the  c«ae 
in  the  cycloid  and  the  qusidratrix.  Descartes  oooliBed  his 
dtscoRsioa  to  algebraical  curves,  and  did  not  trmt  of  ttia 
theory  of   mechanical   corves.      The  clswiUcstJoil  into  nlgO' 


SM     MATUItHATICH  FHUH   DKtiCAIITEll  TO   HUTOBNtl. 

brvicsl    and    truucendental    curvea    now    luiul 
Newton*. 

DeMutes  •!«>  p&id  particular  attention  to  tbe  tbeoc;  *4 
tim  tuigonU  to  curvvs— aa  purluipa  tuiglit  be  iaferrad  frou 
bU  ayataui  Ot  cUuilicutiuii  Juat  alludtvl  to.  The  tlien  currant 
deflnition  of  a  tangent  at  a  puiiit  wan  a  Htraigbt  line  through 
tbe  point  aucb  that  betwtuii  it  anil  the  curve  no  other  Htmight 
line  could  be  drawn,  tluit  it,  tite  alraiglit  line  uf  cluaeat  cnn- 
Uct.  DoKvtat  proposed  to  sulntituto  for  tbia  a  Ktatenienl 
eqnivaleut  to  thji  ubsertiou  tliat  tbti  tangent  is  tbe  limiting 
puiitioii  uf  tlw  Bucaiit ;  >Vniiiil,  and  at  u  lutor  (Into  .Maclaurin 
Mid  Ijigrange,  adu|)U.-d  tliLn  dftinition.  Bar mw,  followed  by 
Newton  and  Leibnitz,  L-unaiderud  a  curve  aa  tbe  limit  uf  an 
iliKcribed  polj^n  when  tbe  Kidea  1n.-umii(:  iiideUuitoly  amall, 
and  Htatod  tliat  a  utile  of  tlie  piilj'gun  wlieu  prtNlu(.-ed  became 
ill  tlw  limit  a  tangent  tu  the  curve.  Kobervul  uu  tlie  other 
liand  defined  a  taii^'iit  at  a  point  an  the  directiuu  of  luution 
at  that  inntAnt  of  a  puint  wliioli  was  di-aoribiug  the  curv& 
'lliu  niiultii  are  the  luunt'  whichever  dulinitiun  is  HeleutMl,  but 
tlie  cuiitrovemy  a»  to  whit:b  detinitiun  waa  the  correct  uue  wan 
none  the  lesB  lively.  DcHcnrt^.i  illu.ttnit*.-d  his  tlivory  by  giving 
tlie  general  rule  for  drawing  tangents  and  nunual.s  to  a  ruutetto. 

Tlie  luctlioal  uu-d  by  Ut^si-artcs  to  liiid  the  tangent  or 
iiomial  at  any  puitit  of  u  given  cur\e  wiih  Bub^tiuitially  aa 
fuUowa.  He  tlet^rttiiiied  the  centre  and  rodiuu  uf  a  circle 
wbicb  aliuuld  cut  tiie  curve  in  twu  cuniHVUtive  poiuta  tber«. 
Tbe  tangent  to  Uie  circle  at  that  puint  will  be  the  retiuirud 
tangent  to  the  curve.  In  iiiudeni  tcxl-buuks  it  is  usual  to 
expnwa  tlie  condition  tlial  two  uf  tlie  puinlii  in  which  a 
atraigbt  line  (aucb  ita  g~  nwe  +  c)  cuta  tbe  curve  aluill  cuincide 
with  tlie  given  puint :  tliin  enablea  um  tu  determine  >ii  and  c, 
and  tbua  the  equation  uf  tbe  taiigiMit  there  in  deteniiined. 
Deacartea  however  did  nut  venture  to  du  this,  but  aelecting  a 
circle  as  tbe  simplest  curve  and  un«  to  wliicb  he  knew  how  to 
diaw  a  tangent,  be  au  fixed  hia  circle  na  to  make  it  touch  tbe 
•  tim)  beluK,  p.  3S0. 


DESCARTES.  285 

given  cnrre  lit  the  point  in  quivtian  And  thus  radnced  the 
pTvliIeni  to  lirowiiij;  a  tnngriit  to  a  circle.      I  ilioiild  note  in    - 
paming  that  he  only  applied  thin  methii'l  to  curves  which  are 
Bjrmnietricfil  about  an  axtn,  and  he  took  the  centre  of  the  circle 
on  the  oxia. 

The  oljscure  ntjie  delibcratoly  adopted  bj  DencMtes  di- 
mini.'iheil  the  circulation  and  iniTnedinU>  apjireciation  of  these 
books ;  I)ut  a  LAtin  traniilAtion  of  them,  with  exptanatorj 
noUw,  was  prepared  by  F.  de  Itcaune,  and  an  edition  of  thi^ 
with  a  cummentarf  by  F.  van  Schooten,  issued  in  I6A9,  waa 
widely  read. 

The  third  book  of  tli<-  Oromilrir  contninn  an  analjMB  of 
the  alfreitra  then  current,  and  it  lion  aOected  the  language 
of  the  fiubji^t  bj  tixin;:  the  caitoin  of  employing  the  letters  at 
the  beginning  of  the  niphnliet  to  deni>t«  known  qaantitien,  and 
those  at  the  end  of  the  alphaliet  to  denote  unknown  quan- 
tities*. Dewarten  further  introdncwt  the  Rfitein  of  indicM 
now  in  use ;  very  likely  it  wnn  original  on  hia  p«rt,  but 
I  would  here  remind  the  nmder  that  the  aaggestion  had 
been  made  by  previooB  writer*,  though  it  had  not  been 
generally  adopted.  It  w  <ioabtful  whether  or  not  Descartea 
recognifwd  timt  his  lettem  might  represent  ftny  quantities, 
positive  or  negntive,  and  that  it  was  suRicient  to  prove  a  ptD- 
position  for  one  general  owe.  Ho  was  the  enrliest  writ«'  to 
realize  the  advnntAge  to  be  obtained  by  taking  all  the  trnns 
of  an  equation  to  one  side  of  it,  though  Ulifel  and  Harriot 
had  sometimes  employed  that  form  by  choice.  He  realised  the 
meaning  of  negative  quantities  and  used  them  freely.  In 
this  book  he  made  use  of  the  rule  for  finding  a  limit  to  the 
number  of  positive  and  of  n^ative  roots  of  an  algebraical 
equation,  which  is  still  known  by  his  name ;  and  introduced 
the  method  of  indoterminate  coeffinentn  for  the  solution  of 
equations.     He  believed  that  he  bad  given  a  method  bywhi^ 


S86     MATHnUTIGB  PBOM  DBaCAKTU  TO  BUTQimL 


algebraical  aquaiioiis  ol  any  order  oould  be  aolved,  b«i  In  lUa 
be  was  miHtakeu,  He  inade  use  ol  tbe  nietbod  of  indetonni- 
nate  ooeffieientii. 

Of  tbe  two  other  appeodioea  to  the  Dueoun  ooe  waa 
devoted  to  opiia.  The  chief  interest  of  this  oousiata  ia  the 
statement  given  of  the  law  of  refraction.  This  appears  to  have 
been  taken  from  Snell's  work,  though  unfortunately  it  is 
enunciated  in  a  way  which  might  lead  a  reader  to  suppose 
that  it  is  due  to  the  researches  of  Descartes.  Descartes 
would  seem  to  have  repeated  Snell's  experiments  when  in 
Paris  ia  1626  or  1627,  and  it  is  possible  that  be  subsequently 
forgot  how  much  lie  owed  to  the  earlier  investigations  of 
Hnell.  A  large  part  of  the  optics  is  devoit*d  to  determining 
tlie  bettt  sliape  for  the  lenties  of  a  telescope,  but  the  mechanical 
ditticulties  in  grinding  a  surface  of  glass  to  a  required  form 
are  ho  great  as  to  render  these  investigations  of  little  practical 
use.  Deflcart4:^  seems  to  have  lioen  doubtful  whether  to 
regard  the  rays  of  light  as  proceeding  from  the  eye  and  so  to 
speak  touching  tlie  object,  as  the  U  reeks  liod  done,  or  as 
proceeding  from  the  object,  and  so  atlecting  the  eye;  but, 
since  he  considered  the  velocity  of  light  to  be  infinite,  he  did 
not  deem  the  point  particularly  important. 

The  other  apfiendix,  on  tneieoTB^  contains  an  explanation 
of  nuniennis  atmospheric  phenomena,  inclucling  the  rainbow ; 
the  explanation  of  the  latter  is  necessarily  incomplete,  since 
DeHcartes  was  unacquainted  with  the  fact  that  the  refractive 
index  of  a  subntance  is  different  for  lights  of  different  colours. 

Descartes'tt  physical  theory  of  the  universe,  embodying  most 
of  the  results  contained  In  his  earlier  and  unpublished  Zs 
MotuUf  is  given  in  his  Prineipia^  1644,  and  rests  on  a  meta- 
physiad  bonis.  He  counnenees  with  a  discussion  on  motion ; 
and  then  lays  down  ten  laws  of  nature,  of  which  the  first  two 
are  almost  identical  with  the  first  two  laws  of  motion  as 
given  by  Newton ;  the  remaining  eight  laws  are  inaccurate.  He 
next  proceeds  to  discuss  the  nature  of  matter  which  he  regards 
as  unifonn  in  kind  though  there  are  three  forms  of  it.     He 


DESCARtES.      CAVAUERl.  287 

I  thnt  the  mnttor  of  tho  unircrw  mnnt  he  in  motion, 
and  that  the  motion  muNt  rcHult  in  a  numtx^  of  roriieen.  H« 
ntntes  that  the  aun  is  the  centre  of  nn  iinmenK  whirlpool  of 
thia  mntter,  in  which  the  plnnetB  Kont  and  are  iiwept  roand 
\ikrt  Rtrnws  in  «  whirlpool  of  wnt«r.  Ench  phnet  is  snppowd 
to  lie  the  centre  of  it  nrcondrtiy  whirlpool  i>y  which  itii  nntvliitnt 
»re  cnrried  :  thrf«  secondary  whirlpools  tuv  Mpponed  to  pro- 
duce vftriationa  of  dennity  in  the  currounding  mntiam  which 
constttnt«  the  primary  whirlpool,  and  so  rniiM  the  planeta  to 
move  in  ellipRf!!<  and  not  in  circles.  All  tlipee  amumptions  are 
arbitral^,  and.  unsiipporteil  hy  any  invpstigatmn.  It  i*  not 
difficult  to  pn>vi>  that  on  hi^  hypnthi>!>i4  tlte  Min  wovM  Im  in 
the  centre  uf  tliene  ellipM-s  ami  not  at  a  foriia  {aa  Kppler  had 
shewn  van  the  ca.«e),  anc)  that  the  weight  of  a  hodj  at  every 
placn  on  the  surface  of  the  earth  e;(cept  the  equator  wnaM  act 
in  a  direction  wliich  was  not  vertical ;  but  it  will  be  mfficinnt 
hero  to  my  that  Newton  in  the  srcomi  hook  of  hia  Prineipia, 
1687,  considemi  the  theory  in  detail,  anil  shewed  that  ita  con- 
nequencen  am  not  only  inconsistent  with  each  of  Kepler's  lawi 
and  with  the  fundamental  lawn  of  meclianica,  Irat  are  aim 
at  variance  with  the  lawn  of  nature  assumed  by  Dcacartes. 
Rtill,  in  spite  of  ita  cnuleness  and  ita  inherent  defecta,  the 
theory  of  vortices  marks  a  fresh  em  in  astronomy,  for  it  waa 
an  attempt  to  pxplain  the  phenomena  of  the  whole  nnirersA  hy 
the  same  mechanical  lawn  which  cxjieriment  shews  to  be  tra«« 
on  the  earth. 

CttTalieii*.  Almost  contempomnpoasly  with  the  publica- 
tion in  I6n7  of  De»caTt«8'R  getmtetry,  the  principles  of  the 
integral  calculus,  so  far  as  they  are  concerned  with  nimmation, 
were  being  worked  out  in  Italy.  Thin  waa  eflected  bj  what 
was  called  the  principle  of  indivisibles,  and  waa  the  invention 
of  Oavalieii.     It  waa  applieal  by  him  and  hit  eonlempofariea  ^ 

■  Cavaliari'i  lir«  has  bcvn  irriltcn  h;  T.  Frisi,  Uilan,  1T7B;  by  F.    ' 
Pttiui,  HOaa,  IMS ;  b;  Gibrio  l^iola,  Milan,  1M4 1  sad  bj  A.  Favwa, 
Boto«aa,  IBHB.     Aa  analirfi  of  bis  «c«ks  i»  itlwa  in  Haifa's  //MafN 
dn  IMmm,  Pvfai,  18W-8,  vol.  IT,  Vf-  09-90. 


S88     MATHKMATIGB  FROM  DBCARTB  TO  HUTGimL 


to  nmneroiis  proUaoM  oonneoted  with  the  qnadratttfeof 
and  mirlaoeai  tlie  daterminatloii  of  voluinei^  aod  tlie  pim^iiHt 
at  oentrea  of  uiam.  It  lerved  the  nme  porpoie  m  the  todiom 
method  of  exhAOBtions  used  hy  the  Oreeka;  in  prineiple  the 
mathodji  are  the  same,  bat  the  notation  of  indivisibles  is  OMive 
concise  and  convenient  It  was,  in  its  torn,  superwded  at  the 
beginning  of  the  eighteenth  oentujry  bj  the  integral  calcnlos. 

Boitavetitura  Caralieri  was  bom  at  Milan  in  1598|  and  died 
at  Bologna  on  Nov.  27, 1647.  He  became  a  Jesuit  at  an  early 
age;  on  the  reoommendation  of  the  Order  he  was  in  1629  made 
profesHor  of  mathematics  at  Bologna;  and  he  continued  to 
occupy  the  chair  there  until  his  death.  I  have  already 
mentioned  Cavalieri's  name  in  connection  with  the  introduc- 
tion of  the  use  of  logarithms  into  Italy,  and  have  alluded  to 
hiH  diHCOvery  of  the  expresiuon  for  the  area  of  a  spherical 
triangle  in  terms  of  the  spherical  excess.  He  was  one  of  the 
most  influential  mathematicians  of  his  time,  but  hb  subsequent 
reputation  rests  mainly  on  his  invention  of  the  principle  of 
indivinibles. 

The  principle  of  indivisibles  had  been  used  by  Kepler  in 
1604  and  1615  in  a  somewhat  crude  form.  It  was  first  stated 
by  Cavalieri  in  1629,  but  he  did  not  publish  his  results  till 
1635.  In  his  early  enunciation  of  the  principle  in  1635 
Cavalieri  asserted  tliat  a  line  was  made  up  of  an  infinite 
number  of  points  (each  without  magnitude),  a  surface  of  an 
infinite  number  of  lines  (each  without  breadth),  and  a  volume 
of  an  infinite  number  of  surfaces  (each  without  thickness).  To 
meet  the  objections  of  Guldinus  and  others,  the  statement 
was  recast,  and  in  its  final  form  as  used  by  the  mathematicians 
of  the  seventeenth  century  it  was  published  in  Cavalieri's 
EjoereUaiioi^i  Geameirieae  in  1647 ;  the  third  exercise  ia 
devoted  to  a  defence  of  the  theory.  This  book  contains  the 
earliest  demonstration  of  the  properties  of  Pappus*.  Cava- 
lieri's works  on  indivisibles  were  reissued  with  his  later 
corrections  in  1653. 

*  See  abofs,  pp.  104, 260. 


CAVALIER!. 


289 


The  method  of  indivisibles  rests,  iD  effect,  on  the  ussanip- 
tion  that  any  magnitude  may  be  divided  into  an  in6nite 
number  of  small  quantities  which  can  lie  made  to  liear  any 
required  ratios  («»x.  gr,  etjualitv)  one  to  the  other.  The 
analysis  giren  by  Cavalieri  is  harrily  worth  quoting  except 
as  being  one  of  the  first  steps  taken  towanis  the  formation  of 
an  infinitesimal  calculus.  One  example  will  suffice.  Huppose 
it  be  required  to  find  the  area  of  a  right-angled  triangle.  Let 
the  base  be  made  up  of  or  contain  n  points  (or  indivisibles), 
and  similarly  let  the  other  side  contain  na  points,  then  the 
ordinates  at  the  successive  points  of  the  base  will  contain 
Oy  2a, ... ,  fia  points.  Therefore  the  number  of  points  in  the 
area  is  a  -i-  2a  -i- ...  -i-  nn ;  the  sum  of  which  is  \n*a  f  Jua.  Since  n 
is  very  large,  we  may  neglect  Jiia  as  inconsiderable  compaml 
with  JaVi,  and  the  area  is  }  (na)  n,  that  is,  \  x  altitude  x  lisse. 
There  is  no  difliculty  in  criticizing  such  a  proof,  Imt,  although 
the  form  in  which  it  is  presented  is  indefensible,  the  substance 
ol  it  is  eorrect 

It  would  be  misleading  to  give  the  above  as  the  only 
■pedmen  of  the  method  of  indivisibles,  and  I  tlierefore  quota 
another  example,  taken  from  a  later  writer,  which  will  fiurly 


illustrate  the  use  of  the  method  when  modified  and  corrected 
by  the  method  of  limita.  Let  it  be  required  to  find  the  area 
outside  a  parabola  APC  and  bounded  by  the  curve,  the 
tangent  at  il^  and  any  diameter  DC.    Complete  the  panJMo- 


290     MATHKHATIC8   rRQH    DESCARTfS  TO   BITTORMS. 

gram  ABCD.  Divide  AD  into  n  ojtul  p«rta,  let  JJ/  f^mtMii 
r  of  tbeiu,  Mtd  )e>t  MN  betlie(r+  l)lh  put.  Draw  Sffamd 
JfQ  pMullel  to  AB,  wkil  dnw  I'li  panUlel  to  Alt.  Then, 
when  M  bncomea  JaJetiaiUiIy  Ur|{e,  tlie  cun-LlineRr  are*  APCD 
will  In  tlw  limit  uf  the  huiu  of  kll  parallulc^rama  lika  Ptf. 
Now 

Knm.FN.m.Tt^BD  =  21P.ilff:DC.AD. 
But  I9  tbo  propertim  of  the  parabola 

MP:DC=AU':M>'=i':u\ 
mkI  HX.AD'^X.n. 

Henoe  MP.iIJf:DC.AD^r':t^. 

Tlicrefbra  area  P.V-.nrtn.  BD^r':  «■ 

Tlierefon^  ullim&tely, 

»rMil/>C/>  :aiva/IZ>=  ]■-»  3'  + ... +(n- !)■:«■ 
=  i«{«-l)(2«-l):«- 
wliiL-h,  in  the  limit,  =1:3. 

It  in  periiapa  wurth  nittii.-iii|{  that  Cav»iieri  mm!  hit  mio- 
ceaaora  always  uiul  tlio  luethiHl  to  find  the  ratio*  of  two  areu, 
volutnes,  or  magiiitudt-s  of  the  HAuie  kind  and  dimenHiona,  that 
is,  they  never  thuui;ht  uf  an  ari'a  as  cMutainiug  ho  many  units 
.  of  area.  Tlie  i<tca  of  coiiipaving  a  ningiiitude  with  a  unit  of  the 
naitie  kinil  ceeuia  to  have  liet^r)  due  to  Wallia. 

It  in  evident  that  in  its  direct  forui  the  method  ia  a|K 
plicaUe  to  oidy  a  few  curves.  Cavalieri  proved  that,  if  m  La 
a  pOHitive  integer,  then  the  liii>it,  whi-n  n  w  infinite,  of 
(I-  +  2-+. ..+«")/»""  is  1/(PH+1J,  which  hi  equivalent  to 
KayiuK  tliat  he  fuund  the  inti'grul  to  i  of  ar"  tntia  j;  =  0  to 
x=  1  ;  lie  alito  (li&rU''si-d  the  quiulniture  of  the  hyperlnla. 

Puottl'.     Aniiitij;  the  contemporaries  of  DcscurteH  none 

■  See  P-xtal  hj  i.  Berlr&Dil,  Purii,  IH9I ;  auil  PoKal  uin  Ltlat 
uad  ttint  KSmp/e.  Iv  J.  O.  Drcj.lurff,  l^eipiif;,  IBTO.  l-UKsl'i  life, 
wriltcn  by  hit  uitar  Hme  IViii-r,  «>■  editrd  bf  A.  1>.  FaugSre,  Parw, 
IHIC,  and  baa  lurmed  ibu  baaia  fur  oeveraJ  work*.  Au  editiuu  ol  bU 
wriliDKa  waa  puUialuJ  id  i  TOliuuca  at   Ihe  Ilaipie  ia   1779,  aaooiul 


PASCAL.  201 

displajed  greater  nnturnl  gcniun  than  Pnxcal,  but  bu  mathe- 
Rialtcal  Fvputntion  wwin  mure  on  what  he  might  hare  done 
than  on  what  he  actually  rffpclwl,  iw  during  a  cnnoHlprahte 
part  of  hix  lifn  he  ilrrnirtl  it  his  duty  In  ilevot«  hia  whoki 
time  tn  n^lig-iouii  fXcrciMPM. 

ItlauK  I'lumil  wan  l>om  nt  Clcnmmt  on  Jan«  19,  1G23,  and 
dinl  at  Paris  on  Aug.  19,  lfiG2.  Mix  fntlipr,  a  Inoil  jndgp  at 
Clrrmont  nn<l  Iiimiwlf  of  sonip  scinntiHc  niputation,  niovrd  to 
Paris  in  16:11,  partly  to  prospcntn  lii.i  nun  Bcientific  ntudipK, 
partly  to  carry  nn  the  nlurDtion  of  IiIh  only  son  who  had 
already  di-xplaynl  exceptinnnl  nliility.  Fnxcal  was  kept  at 
home  in  onI<*r  to  enxure  his  not  heing  overworked,  and  with 
the  name  otijoct  it  wn-i  directed  tlint  hin  education  should  Iw 
at  first  confined  to  the  study  of  Inngnagm  and  should  not 
include  any  math  em  alien.  This  nntumlly  excited  tho  boy'a 
curiosity,  and  one  day,  l)eing  then  twelve  years  old,  he  ankcd  in 
what  geometry  conni.stod.  His  tutor  replied  that  it  wan  the 
science  of  con  struct  in  j;  exact  ligur«n  and  of  ilrt^rmining  the 
proportions  lictween  their  ilifTerent  jwvrts.  Fo.'wnl,  itimnlated 
no  doubt  by  tho  injunction  against  rending  it,  ga\-e  up  hix 
play-time  to  this  new  atudy,  and  in  a  few  weeks  lind  disoorerrd 
for  himself  many  propi-rties  of  Kgures,  and  in  particular  the 
proposition  tluit  the  sum  of  the  nngteii  of  ft  triangle  in  equal 
to  two  right  angles.  I  have  reml  somewhere,  bat  I  cannot 
lay  my  hand  on  the  authority,  that  his  proof  merely  oon> 
sisted  in  turning  the  angnlar  points  of  a  triangular  piece  of 
paper  over  no  as  to  meet  in  the  centra  of  the  inacribed 
circle ;  a  similar  demonstration  can  be  got  by  turning  the 
angular  pointa  over  so  as  to  meet  at  the  foot  ot  the  pei^ 
pendicular  diswn  from  the  biggest  angle  to  the  apparit« 
aide.  His  father  struck  by  this  dispti'y  of  ftbilitj  g»ve  him  « . 
copy  of  Eoclid'fl  SlenvtnU,  a  book  which  hiKal  read  with 
andity  and  aoon  mastered. 


292     MATHnUTiCB  nU>M  DB0ARR8  T( 


BUTOmi 


At  the  age  of  fbuiteea  ha  waa  admit  md  to  the  waeklj 
meetingi  of  Roberval,  Meraemie^  Mjdorge^  and  othar  VnmA 
geometriciani ;  from  which,  ultimately,  the  Fkendi  Academy 
aprung.  At  aixteen  FkMcal  wrote  an  eaHaylon  conic  aectiona; 
mid  in  1641,  at  the  age  of  eighteen,  he  constructed  the  llrrt 
arithmetical  machine,  an  instrument  which  eight  years  later 
he  further  improved.  His  correspondence  frith  Fermat  about 
this  time  shews  tliat  he  was  then  turning  his  attention  to 
analytical  geometry  and  physics.  He  repeated  Torricelli's 
experiments,  by  which  the  presMure  of  the  atmosphere  could 
be  estimated  as  a  weight,  and  he  confirmed  his  theory  of  the 
cauHC  of  liarom^trical  variations  by  obtaining  at  the  same 
instant  readings  at  diffeltet.  altitudes  on  the  hill  of  Puy-de- 
I>6me. 

In  1650,  when  in  the  midst  of  these  researches,  Pascal 
suddenly  aliandoned  his  favourite  pursuits'  to  study  religion, 
or  aM  he  siiyM  in  his  Petuteit  **to  contemplate  the  greatness 
and  the  misery  of  man  " ;  and  about  tlie  same  time  he  per- 
suadiHl  the  younger  of  his  two  sisters  to  enter  the  Port  Royal 
society. 

In  1 653  he  had  to  administer  his  father's  estate.  He  now 
took  up  his  old  life  again,  and  made  several  experimenta  on  the 
pressure  exerted  by  gases  and  liquids :  it  was  also  about  this 
period  that  he  invented  the  arithmetical  triangle,  and  together 
with  Femiat  created  the  calculus  of  probabilities.  He  waa 
meditating  marriage  when  an  accident  again  turned  the 
current  of  his  thoughts  to  a  religious  life.  He  was  driving  a 
four-in-hand  on  Nov.  23,  1651,  when  the  horses  ran  away ;  the 
two  leaders  dashed  over  the  |iarapet  of  the  bridge  at  Neuilly, 
and  Pascal  was  saved  only  by  the  traces  breaking.  Always 
somewhat  of  a  mystic,  he  considered  this  a  special  sunmions  to 
aliondon  the  world.  He  wrote  an  account  of  the  accident  on 
a  small  piece  of  parchment,  which  fbr  the  rest  of  his  life  he 
wore  next  to  his  heart  to  perpetually  remind  him  of  his 
covenant;  and  shortly  moved  to  Port  Royal  where  he  con- 
tinued  to  live  until   his  death   in    1663.     Constitutionally 


PAHCAL.  293 

delicMte,  he  htid  injarvd  Iiih  hmlth  by  his  inccsnnt  rttidy; 
from  the  Age  uf  Hcvt-nteon  or  cighhK-n  hn  snflpred  from 
inaomnm  and  acDto  dyHprpHia,  ami  at  tlie  time  of  his  Heath 
was  phyiiically  worn  oat. 

His  fnniuua  I'rveiiieinl  Lrlfrn  dirvctrd  ngsinst  tlte  JenniU, 
uid  his  Penult,  wrrc  written  towardH  tlin  close  of  hiit  life^  and 
nn:  the  first  example  nf  that  fiiii^liM  fumi  which  is  character- 
ifltic  of  tlie  Ix^t  French  lilernturc.  Tlie  only  iiMtlmmttical 
work  that  he  pruduced  nft^r  retiriiis  to  Port  Royal  was  the 
GWMyon  the  cycluid  in  1058.  He  wiw  Hufferiiig  fnHniTccp- 
li^fwnesH  And  Umth-ache  when  tlie  idea  occurred  to  hitn,  and  to 
his  surprixc  his  twth  imiiiediatcly  ceased  to  acbc  Itcgarding 
this  as  a  divine  intimation  to  proceed  with  the  problem,  he 
worked  incesNantly  for  eight  days  at  it,  and  completed  a 
tolerably  full  account  of  the  geometry  of  tlte  cycloid. 

I  now  proceed  to  consider  his  mathematical  woifca  in 
rather  greater  dclAil. 

His  early  es-sny  on  l\\K  i/fnm':lry  of  conia,  written  in  1639 
but  not  publi-slicd  till  1779,  KeeniH  to  have  Ijccn  founded  on 
llie  teaching;  of  Dc-snrgucs.  Two  of  the  reenlts  arc  important 
as  well  a"  interesting.  Tlie  first  of  tliexe  ii  the  theorem  known 
now  na  "  Pascal's  theorem,"  namely,  that  if  a  hexagon  be 
inscribed  in  a  conic,  the  points  of  intersection  of  the  oppodta 
■ides  will  lie  in  a  straight  line.  The  second,  whidi  is  rmlly 
due  to  £)esargueH,  in  that  if  a  qusflrilaterat  be  inacribed  in 
a  cnnic,  and  a  straight  line  be  drawn  cutting  the  ndea 
taken  in  order  in  the  points  A,  B,  C,  and  />,  and  the  conic  in 
PmadQ,  then 

PA .  PC :  PB.  PI)  =  QA.QC:  QB.  QD. 

Pascal  intrcslunwi  his  ariUinitlienl  triatigU  in  1653,  bat  no 
account  of  his  method  was  printed  till  1665.  Tba  triangle  ia 
oonstructed  as  in  the  figure  on  the  next  page,  each  borisontal  line 
being  formed  from  the  one  above  it  by  making  ererj  nombw  , 
in  it  equal  to  tlie  sum  of  thorn  alxn-e  and  U>  the  Mt  of  it  in 
the  row  immediately  above  it ;  ex.  ^.  the  barth  ii«mfaar  in 


204     MATUKMATICH  PKOM  DBSCABTBI  TO  BUTQimL 

the  fourth  line^  namely,  9(^  is  equal  to  l-f3-f6-flO.  The 
Qumbeni  in  each  line  are,  what  are  now  called,  fymrmU 
numbem.  Thoie  in  the  firei  line  are  called  nuubeni  of  the 
first  order;  those  in  the  aeoond  liiie^  natural  numbeni  or 
numbem  of  the  Meoond  order ;  thoiie  in  the  Uiird  line  unmben 
of  the  third  order,  and  no  on.  It  in  eaidly  shewn  that  the 
iMth  number  in  the  nth  row  is  (ta-i-  m-  2)!/(i»-  l)!(n  — 1)1 


1 

1 

1 

y 

•  •• 

a 

3 

y 

•  •• 

s 

y 

/lO 

IS 

••• 

4y 

4, 

20 

as 

1 
1... 

/i 

15 

as 

70 

1 
i  — 

/. 

• 

• 

• 

• 

RuKsal's  aritliiuetical  triangle  (to  any  required  order)  is 
got  by  drawing  a  diagonal  downwards  from  right  to  left  as 
in  the  figure.  The  numbers  in  any  diagonal  give  the  ooefll* 
cients  of  the  expansion  of  a  binomial :  for  example^  the  figures 
in  the  fifth  diagonal,  namely,  1,  4,  G,  4,  1,  are  the  coefficients 
in  the  expansion  (a  ••-  h)K  Pascal  used  the  triangle  partly  for 
this  purpose,  and  partly  to  find  the  numbers  of  combinations 
of  M  things  taken  ti  at  a  time,  which  he  stated  (correctly) 
to  be  (yi  -h  I) (h  -h  2)  (u  -h  3)  ...  m/(fi»  -  n)! 

Perhaps  as  a  mathematician  Pafical  is  best  known  in 
connection  with  his  correspondence  with  Format  in  1654  in 
which  he  laid  clown  the  principles  of  .the  ihttory  of  prolxtbUitieM, 
This  correspondence  arose  from  a  pixiblem  proposed  by  a 
gamester,  the  Chevalier  de  Mere,  to  Pascal  who  communicated 
it  to  Fermat.  Tlie  problem  was  this.  Two  players  of  equal 
skill  want  to  h^ve  the  table  liefore  finishing  their  game.  Their 
scores  and  the  number  of  points  which  constitute  the  game 


PASCAU  105 

bcinft  gitrnt  it  in  draircd  to  find  in  wlmt  proportion  thej 
Hhoald  divide  the  Htnkps.  Piuicnl  and  FertlMl  tigravd  <itl  Uw 
Wiawcr,  but  gave  different  proofs.  The  following  is  »  tnuml*- 
tion  of  Poscnl's  sulutioii.     That  of  Fcrniat  ia  gi^-en  \Mw. 

The  folloviiig  is  mj  mrthod  Tor  ilctermining  lfa«  (live  of  each  rbfer, 
«hm.  for  eimmple,  two  playcn  jilty  a  (-amc  of  ihtea  poiata  and  each 
plajiT  han  Kinkrd  33  pinlolc*. 

tiap[i0TC  that  the  first  plajpr  liaa  gninni  tvo  pointa,  and  (he  Mcnnd 
player  one  point;  they  hnvn  now  to  pla;  (or  a  point  on  thia conditioa, 
UiBt,  if  the  first  plajcr  cun.  be  take*  all  Ihv  money  which  ii  at  itake, 
naniel;,  (it  pinlolcs ;  vhik',  if  tbc  ^rrond  player  cnin,  each  plmyer  haa  two 
poinlB.  Ml  thai  Ihcy  are  on  term'  of  ei|ualily.  an^.  if  th<7  Icare  oft  pla7- 
inR.  eacb  oocht  to  take  H'J  piitolc*.  Thn<i,  if  lht>  flnt  plajer  ttain,  thai 
64  pintolci  brtoDR  to  him.  anJ.  if  be  Idh.  then  32  piilolei  belong  to  him. 
If  therefore  the  pinyen  do  not  urinh  to  pis;  thin  same,  bnt  to  aeparate 
without  plajine  it,  the  &n>t  plaj-ct  would  t»j  Id  the  aeonid  "  I  am  ccrtam 
of  33  piitole*  even  if  1  toK  Ihin  game,  and  an  for  tbc  othn  33  tMitoka 
perhaps  I  ahalt  bate  Ihcm  and  perbap*  yon  will  bave  them ;  the  ehaneea 
are  eijoal.  Let  tii  then  ditida  Iheaa  32  pistoles  equallr,  and  give  me  al*o 
the  33  pistoles  of  which  I  am  certain."  Thus  the  first  player  will  hara 
4h  pistoles  aud  the  aecoad  IC  plxtolcs. 

Kelt,  anppoao  that  tbc  first  player  has  twined  two  poinli  and  Iha 
■eoond  player  none,  and  that  they  are  about  to  play  for  a  point;  Ifca 
condition  then  is  that,  if  the  fint  player  gain  Ihii  p^nt.  be  acentca  tha 
Kame  and  takes  the  64  pistoles,  and,  it  the  second  player  gain  this  point, 
then  the  players  will  be  In  the  lithatioa  already  examined,  in  which  tha 
first  player  in  entitled  to  44  pistoles  and  the  second  to  16  pistole*, 
ThuH,  if  Uwy  do  not  wish  to  play,  the  flr^t  player  wanM  aaj  t«  the  aeoond 
"  If  I  gain  the  point  I  gain  61  pistoles;  if  I  losa  it,  I  an  colilled  to 
4H  pistoles.  Qi>c  mo  then  tin  4"  pistoles  of  wliioh  I  am  eartain,  tad 
divide  the  other  lA  rqually,  since  our  chances  of  gaining  tha  point  an 
eiiQal."  Thus  the  first  player  will  ha>e  66  pistoles  and  Ifaa  aeooad  pUjW 
8  pistoles. 

Finally,  sopposo  that  the  first  ptayrr  has  gained  ona  point  and  tba 
aecond  player  none.  II  tbey  proered  In  play  for  a  point,  the  condition  la 
thai,  if  Ibe  first  player  gain  it,  the  players  will  he  la  Ibe  tltiMtioa  Aral 
•uunined,  in  whieh  Ibe  Hist  player  is  entitled  tn  SS  pisMaa ;  If  tha  Bnt 
player  lost  Dm  point,  each  player  lias  then  a  point,  aadaacb  la  aBlIlM 
to  39  pistolea.  Thns.  it  they  do  not  wish  to  play,  tha  Bnt  plqnr  wmU 
aajto  lb«  amad  "Uin  me  the  B3  pirtola  of  whieh  I  am  eaiUin  sad 
A*Ma  Ibe  remaEnder  Pl  tha  M  piatoln  eqaally,  thai  le,  JMIm  U  wlilw 


296     MATBEMATlCa  PROM  DUCAHTKH  TO  HUYOKini 
aqoAlJ^.'*    Tbm  lilt  flirt  piiqnr  wUI  bafllit  wmh  of  H  —a  I»  pirtolM, 

Paical  proceeds  next  to  ooniiider  the  biiiuIm'  ptphleni  when 
the  game  U  wud  by  whoever  first  obtains  m-¥u  pointsi  and 
one  pUyer  has  m  while  the  other  has  m  points.  The  answer 
is  obtained  by  using  the  arithmetical  triangle.  The  general 
solution  (in  which  the  skill  of  the  players  is  unequal)  is  given 
in  many  mudem  text-books  on  algebra  and  agrees  with 
PaMcal's  result,  tluMigh  of  course  the  notation  of  the  latter 
is  dilTereiit  and  less  convenient. 

Pascal  made  a  most  illegitimate  use  of  tlie  new  theory  in 
the  seventh  cliapter  of  his  Peuseat.  In  eflfect^  lie  puts  his 
argument  that,  as  the  value  of  eternal  liappiness  must  be 
infinite,  then,  even  if  the  probability  of  a  religious  life 
ensuring  eternal  happiness  be  very  small,  still  the  expectation 
(which  is  measunxl  by  the  pmduct  of  the  two)  must  be  of 
sufficient  magnitude  to  make  it  worth  while  to  be  religious. 
Tlie  argument,  if  worth  anything,  would  apply  equally  to 
any  religion  which  proiiiiHed  eternal  liappintwH  to  those  who 
accepted  its  doctrines.  If  any  conclusion  may  be  drawn  from 
the  statement  it  is  the  undc&irability  of  applying  mathematics 
to  questions  of  morality  of  which  Home  of  the  data  are 
necessarily  outside  the  range  of  an  exact  science.  It  is  only 
fair  to  add  tliat  no  one  hod  more  contempt  than  Pascal  for 
those  who  clianged  their  opinions  according  to  the  prospect  of 
material  benefit,  and  this  isolaUHl  passage  is  at  variance  with 
the  spirit  of  his  writings. 

The  last  mathematical  work  of  Pascal  was  tliat  on  the 
eydoid  in  1658.  The  cycloid  is  the  cun-e  traced  out  by  a 
point  on  the  circumference  of  a  circular  hoop  which  rolls  along 
a  straight  line.  OaliKH>,  in  iri30,  luul  Uten  the  first  to  call 
attention  to  this  curve,  the  shape  of  which  is  particularly 
graceful,  and  had  suggested  that  the  arches  of  bridges 
should  be  built  in  this  form*.     Four  years  later,  in  1634, 

*  The  bridge,  by  Eiwez,  soroM  the  Cam  in  the  grounds  of  Trinity 
College,  Cambridge,  has  cycioidsl  arches. 


PASCAL.     WALLIS. 


297 


Roberval  foand  the  area  of  the  cycloid ;  DencarieK  Unnighi 
Utile  of  this  Holutioii  and  defied  him  to  find  its  tangents,  the 
same  challenge  being  alno*  nent  to  Fermat  who  at  pnco  solved 
the  problem.  Several  questions  connected  with  the  curve,  and 
with  the  sarfficn  and  volume  generated  by  its  revolution  about 
its  axis,  base,  or  the  tangent  at  its  vertex  were  then  proposed 
by  various  mathematicians.  These  and  some  analogous  ques- 
tions, as  well  as  the  positions  of  the  centres  of  the  mass  of  the 
solids  formed,  were  solved  by  Pascal  in  1658,  and  the  results 
were  issued  as  a  challenge  to  the  world.  Wallis  suoooeded  in 
solving  all  the  questions  except  those  connected  a'ith  the 
centre  of  mass.  Pascal's  own  solutions  wcn^  eflfectcd  by  the 
method  of  indivisibles,  and  are  similar  to  those  which  a 
modern  mathematician  would  gi^'e  by  the  aid  of  the  integral 
caleulua  He  obtained  by  summation  what  are  equivalent  to 
the  integrals 

/sin  ^d^    /sin'  ^<^     /^  >tin  ^d^ 

one  limit  being  either  0  or  ^w.  He  also  investigated  the 
geometry  of  the  Archimedean  spiral.  These  researchcfi, 
according  to  D'Alemliert,  form  a  connecting  link  between 
the  geometry  of  Archimedes  and  the  infinitesimal  calculus 
of  Newton. 

Wallis^  John  Wal!i$  was  bom  at  Ashford  on  Nov.  22, 
1616,  and  died  at  Oxford  on  Oct.  28,  1703.  He  was  educated 
at  Felstead  school,  and  one  day  in  his  holidays,  when  fifteen 
years  old,  he  happened  to  see  a  book  of  arithmetic  in  the 
hands  of  his  brother ;  struck  with  curiosity  at  the  odd  signs 
and  symbols  in  it  he  borrowed  the  book,  and  in  a  fortnight^ 
with  his  brother's  help,  had  mastered  the  subject.  As  it  was 
intended  that  he  should  be  a  doctor,  he  was  sent  to  Emmanuel 
College^  Cambridge^  while  there  he  kept  an  "act**  on  the 
doctrine  of  the  circulation  of  the  blood — this  is  said  to  have 

*  Set  B^  Hi$torf  of  th€  .Vhvfy  o/  If Af  ArsMl/et  «t  Cmmhrt4fie^  pp.  41— 
I6w    An  editkn  of  WsDis^s  naUieoislical  works  was  paUitlml  in  Ihrss 
at  Oxfofd,  I698.W. 


SOB     IUTUKMATIC8  nOM  DUOAITIGB  TO  BUTOimk 

been  the  fini  oeoeaioii  in  Evrope  on  whieh  thk  theoiy  wan 
publicly  niAinUined  in  a  du^Uiion.  Hie  interaeU  however 
oentred  on  mathemalict. 

He  WM  elected  to  a  fellowship  at  Qneenef  College^  Ckni* 
bridge,  and  iiobseqiientlj  took  ordeni  bat  on  the  whole 
adhered  to  the  Puritan  party  to  whom  he  rendered  great 
amiiMlanoe  in  deciphering  the  royalist  despatches.  He  however 
joined  the  moderate  Presbyterians  in  signing  the  remonstrance 
against  the  execution  of  Charles  I.,  by  which  he  incurred  the 
lasting  hostility  of  the  Independents.  In  spite  of  their  oppo- 
sition»  he  was  appointed  in  1649  to  the  Savilian  chair  of 
geometry  at  Oxford,  where  he  lived  until  his  death  on  Oct.  28^ 
1703.  Besides  liis  mathematical  works  he  wrote  on  theology, 
logic,  and  philoM4>hy ;  and  was  the  first  to  devise  a  system  for 
teaching  deaf-mutes.  I  confine  myself  to  a  few  notes  on  his 
more  important  mathematical  writings.  They  are  notable 
partly  for  the  introduction  of  the  use  of  infinite  series  as  an 
ordinary  part  uf  analysis,  and  partly  for  tlie  fact  tliat  they 
revealed  and  explained  to  all  students  the  principles  of  the  new 
met  hods  of  analysis  introduced  by  his  cuntem|iorarie8  and 
immediate  predecessors. 

In  1G55  Wallis  published  a  treatise  on  eouie  weciUmt  in 
which  they  were  define<l  analytically.  I  liave  already  men* 
tioued  tliat  the  Gtotartrie  of  Descartes  is  both  ditficult  and 
obscure,  and  to  many  of  his  contemporaries,  to  whom  the 
method  was  new,  it  must  have  been  incomprehensible.  This 
work  did  siimething  to  make  the  method  intelligible  to 
all  mathematicians:  it  is  the  earliest  book  in  which  these 
curves  are  considered  and  defined  as  cur\'es  of  the  second 
degree. 

The  most  important  of  Wallis's  works  was  his  Ariihmeiiea 
Infinltortini^  which  was  published  in  165(i.  In  this  treatise 
the  methods  of  analysis  of  l>escartes  and  Cavalieri  were 
syntematized  and  greatly  extended,  but  their  logical  exposition 
is  open  to  criticism.  It  at  once  became  the  standard  book 
on  the  subject,  and  is  constantly  referred  to  by  subsequent 


WALLia  299 

writcrH.  It  iH  prcWrd  l>y  n  nhort  tmct  on  conic  Huctjnru. 
lie  coniDici)c<-a  1>y  proving  the-  Uw  of  iiiiliccst;  iibpwB  that 
a^,  X-',  «-•  ...  n'[>n-si'ut  I,  I/j-,  1/j^  ... ;  Hint  *1  rcptmniU  the 
w|unro  rout  of  x,  tltnt  i^  n-prmciits  the  culw  root  of  2*,  mmI 
generally  tlint  z~'  reprcitpnU  tlio  n'cipruvnl  of  x",  umI  tliMt 
«'■'  rrprvscnts  tW  ijth  nutt  of  *'". 

Lmving  tlio  nunirrnuH  a1ge)>nvicn1  npplicAtionii  uf  thia  dis- 
covery he  ni-xt  proceiils  to  find,  liy  tlie  nti-tliud  of  indit-iHiltln^ 
the  iircA  encluar<l  bct«'ecn  the  curve  y  ^  z",  the  nxiH  of  x,  am] 
any  ordinnU}  x  —  ti ;  nn<l  lie  provcH  tlint  the  ratio  of  this  arm 
to  that  of  the  pamllelogmm  on  the  nnmc  b*Me  and  of  tlie 
same  altitude  is  e<|UHl  to  the  ratio  1  :ni  +  1.  He  apparently 
assumed  that  the  same  reitult  would  Im  tntn  also  for  the 
cuno  y-(ur",  where  a  i«  any  constant,  and  m  any  nnmlier 
penitivc  or  negative;  liut  he  only  diHcutisei  the  caac  of  the 
parabola  tn  which  m=2,  and  that  of  the  hyperbol*  in  which 
*n  ^  -  1  :  in  the  tatter  case  his  interprelAtion  of  the  result  ia 
incorrect.  He  then  hIibwh  tlint  Rimilnr  reau1t«  might  tie 
written  down  for  any  curve  of  the  form  y  -  2»m"  ;  and  hcnco 
that,  if  the  ordinate  y  of  a  cunn  can  l>e  expanded  in  powern 
of  the  uleciivvt  ^  ita  qaadraturo  can  lie  determined:  thus  ho 
says  that,  if  the  equation  of  a  curve  were  y  =  a^  +  *'+a?+  .,,, 
its  area  would  lie  r*  Jj:"*  Jx'+  ....  He  then  appliea  this 
to  tl»  qnadntturo  of  the  cunes  ^-{x-i^,  jr  =  («-«^', 
y=(«-ar^',  ^^{x-r'y,  Ac.  taken  between  tlie  limita  x=0  and 
z  -  1 ;  and  nhewn  that  the  arean  nte  rcs|iectively  1,  {,  Jg,  f\f, 
ic.  Ho  next  considers  curves  of  the  form  y  =  *"*  mh!  estab- 
lishra  tho  theorem  that  the  area  bounded  by  the  onm,  the 
axis  of  z,  and  the  ordinate  x=  1,  is  to  the  areA  of  the  lectMlgle 
on  the  same  bmc  and  of  the  name  altitude  asasim-l- 1.     This 

is  equivalent  to  finding  the  value  of   i   x''''dx.    He  illastfrntes 

this  by  the  parabola  in  which  m  =  2.  He  Htaten,  but  doe*  not 
proTt^  the  cofimponding  result  for  a  cnrre  of    the  fbnn 

WalUs  shewed  woaidemble  inffnaity  in  ndndng  tlw 


SOO     MATUdUTICH  nOM  MBSCABTKH  TO  HUTQIIUL 


equAiioiM  of  eonrw  to  the  fonnt  given  ahove^  biil|  as  lie  wan 
onecqiuunted  with  the  btnomial  theorem,  he  eoald  noi  eftet 
the  qoadnttttre  of  the  circle,  whose  equation  is  ysi(«  — a^, 
•inoe  he  was  unable  to  expand  this  in  powers  of  as.  He  Isid 
down  however  the  principle  of  interpolation.  ThuS|  as  the 
ordinate  of  the  circle  ^  =  (x  —  a^  is  the  geonietrical  mean 
between  the  ordinates  of  the  curves  ^=(x— o^*  and  ^»{x^af)\ 
it  luight  be  suppused  (hat,  as  an  approximation,  the  area  of 


the  semi-circle 


j\x^x^)dx. 


which  is  !«*,  might  be  taken  aa 


the  geometrical  mean  between  the  values  of 

j  {x-t^fdx    and     i  (x^a^'dx^ 

tliat  is,  1  and  }  ;  this  is  equivalent  to  taking  4^1  or  3*26  ... 
as  the  value  of  w.  fiut,  Wallis  argued,  we  have  in  fact  a 
series  1,  \,  ^^,  ^}^, ...,  and  therefore  the  term  interpolated 
between  1  and  I  ought  to  be  so  chosen  as  to  obey  the  law 
of  this  series.  This,  by  an  elaljorate  method,  which  I  need 
not  descrilje  in  detiul,  leads  to  a  value  for  the  interpolated 
term  which  is  equivalent  to  talking 


=:  2 


2.2.4.4.6.6.8.8... 
1.3. 3. 6. Sir. 7. 9...' 


Tlie  subsequent  mathematicians  of  the  seventeenth  century 
constantly  used  interpolation  to  obtain  results  which  wo 
sliould  attempt  to  olitain  by  direct  aual^'sis. 

In  this  work  also  the  formation  and  properties  of  con- 
tuiued  fractions  are  discussed,  the  subject  having  been 
brought  into  prominence  by  Brouncker's  use  of  these  frao- 
tions. 

A  few  years  later,  in  1659,  Wallis  published  a  tract  con* 
taiuing  the  solution  of  the  probleuis  on  the  cycloid  which  liad 
been  proposed  by  Pascal.  In  this  he  incidentally  explained 
how  the  principles  laid  down  in  his  Ariiktuttica  IfifiHitorum 
could  be  used  for  the  recti licatiou  of  algebraic  curves;  and 


WALUR.  301 

gkve  a  eolntion  of  the  problem  to  rectiff  th*t  semi'Cnhiciil 
[Mralnin  x'=ai^,  which  hud  lieen  (tincuvered  in  1657  by  his 
pupil  Willinm  Nei).  This  wnx  the  first  OM  in  «bich  Uw 
ien^h  of  A  curved  line  wivi  delcmiined  hy  rtMtbpmaticii,  and 
rince  all  attempts  to  rectify  the  ellipw  nnd  hjperbnln  htiH 
Iieen  (newHHsrilj)  ineffectunl,  it  hud  liet-n  previooily  supposed 
that  no  carves  oould  be  rvctified,  wi  indeed  DencArtes  had 
definit«>ly  luwertcd  to  lie  the  cohc.  The  cycloid  wm  the  seoond 
curve  rectified  ;  thi^i  was  done  by  Wren  in  1658. 

Elarly  in  1 658  a  Himilnr  iliscorery,  independent  of  thnt  of 
Neil,  WAS  made  by  van  Ileunict*,  and  thiH  wm  publishH  I^ 
van  Schootrn  in  hU  edition  of  Descnrtps's  Gfomrtria  in  1659. 
■  Van  Heumct'B  method  is  ns  fullows.  He  nippoMn  the  curve 
to  be  referred  to  recUngulnr  n%r» ;  if  thin  be  so,  and  if  {x,  y) 
be  the  oMirtlinaten  of  any  point  nn  it,  nnd  »  the  length  of  the 
nnm)al,  and  if  another  [loint  whime  c<M>rdinateH.Ara  {x,  ^)  be 
taken  such  that  i; :  A  -  n  :  y,  where  A  in  a  constant ;  then,  if  d» 
be  the  element  of  the  length  of  the  rei|uire(]  curve,  we  have  by 
similar  trian;;lpii  tin  :ilx-  «  :  y.  Therefore  kh  =  t/dx.  Hence^ 
if  the  area  of  the  locus  of  the  point  {r,  ^)  can  be  found,  the 
fir«t  cune  can  be  rectifies!.  In  this  way  van  Hearagt  eflected 
the  rectification  of  the  carve  y'^ox*;  but  added  that  the 
rectification  of  the  parabola  y'^ru  is  impossible  since  tfe 
reqnimi  the  quadrature  of  the  hyperholn.  Tlie  milntimin  given 
by  Neil  and  Wallin  are  oomewhat  niniilnr  to  that  given  t^  van 
Heumct,  though  no  general  rule  in  enuncintcd,  and  the  analysis 
in  clumsy.  A  third  method  was  nugg<Mted  liy  Fermat  in  16G0, 
but  it  is  inel^Ksnt  and  laborious. 

The  theory  of  the  collision  of  bodim  was  propounded  by 
the  Royal  Society  in  16C6  for  the  connideration  of  mathe- 
maticiana  Wallis,  Wren,  and  Haygenn  sent  oonvci  and 
similar  solutions,  all  depernling  on  what  b  now  called  the 
cooaerration  ftf  momentum ;  but,  while  Wren  and  Hnygens 
confined  their  theory  to  perfectly  elastic  bodies  Wallis  aaa> 

*  On  *■»  HMiaH,  tm  the  SlUiofftrM  JVaOtBalfca,  UB7,  loL  t. 


802     MATBCMATIGB  FROM  0BWAETB  TO  HUTOmL 

■iderad  aho  imperfectly  eUstie  bodies.  Thie  win  lolloived  in 
1669  by  a  work  on  iiUtioi  (centres  of  grmTityXand  in  1670  by 
one  on  dynainics :  tbeie  provide  a  convenient  eynoptit  of  vbat 
was  then  known  on  the  subject. 

In  1685  Wallis  published  an  Al^drm,  preceded  by  a 
historical  account  of  the  development  of  I  the  subjecti  whieh 
contains  a  great  fleal  of  valuable  information.  The  second 
edition,  issued  in  1693  and  forming  the  second  volume  of  his 
Opera^  was  coiuiiderably  enlarged.  This  algebra  is*noteworthy 
as  containing  the  first  systematic  use  of  formulae.  A  given 
magnitude  is  here  represented  by  the  nunierical  ratio  which 
it  bears  to  the  unit  of  the  same  kind  of  magnitude:  thus, 
when  Wallis  wants  to  compare  two  lengths  he  regards  each  as 
containing  so  many  units  of  length.  This  perliaps  will  be 
mode  clearer  if  I  say  that  the  relation  between  the  space 
descrilied  in  any  time  by  a  |»article  moving  with  a  uniform 
velocity  would  lie  denoted  by  Wallis  by  the  formula  s  » W, 
where  $  is  the  nuiulier  representing  the  ratio  of  the  space 
deHcriUnl  to  the  unit  of  length  ;  while  previous  writers  would 
liave  denoted  the  same  relation  by  stating  what  is  equivalent* 
to  the  prupoiiitiuu  «| : «,  =  v,  f , :  v,<^  It  is  ciiriouK  to  note  that 
Wallis  n'jecUHl  S8  absurd  the  now  usual  )dea  of  a  negative 
number  oh  bi*iug  less  tlum  nothing,  but  accepted  the  view  that 
it  is  tMimetliing  greater  than  infinity.     The  latter  opinion  may 


lie  tenable  and  not  inconsistent  with  tlie 
hardly  a  more  Miniple  one. 

Format  t.    While  Descartes  wss  laying 


former,  but  it  is 


the  foundations  of 


i 


analytical   geometry,   the    same   subject  Was  occupying  the 


*  See  ex,  gr.  Newton's  Prineipia,  bk.  i.  Met.  i,  lemma  10  or  11. 

t  The  bent  edition  of  Fennat'ii  worlu  is  that  in|  4  volmnea,  edited  by 
8.  P.  Tannery  sud  C.  Henry,  and  published  by  the  French  goveroment; 
vol.  1.  ItfUl;  vol.  11,  1S94;  voL  ui,  ISiMi;  and  vol.  iv,  1901.  Of  earlier 
editionH,  I  may  mention  one  of  his  papers  and  corfespondenee,  printed 
by  his  nephew  at  Tonlonse  in  two  volumes,  1670  and  1679:  of  whieh 
a  summary,  with  notes,  was  published  by  E.  Brsatinne  si  Toulooae  hi 
1S53,  and  a  reprint  was  issued  at  Berlin  in  1861.     . 


\ 


FGRMAT.  803 

Att«ntion  of  nnnther  nnd  not  Irsa  distin;rituh«d  FrenchniMi. 
Tliis  was  Feriiiat.  Pi'nx  Je  Fermnl,  who  wan  bom  near 
MontAulnn  ia  1601,  nnd  dinl  nt  CoKtrt-H  on  Jan.  12,  I6GS, 
was  the  Bon  at  a  lent  her  merchant ;  he  was  edncato)  at  home; 
in  1631  he  olitnined  the  post  nf  ix>uncjlh>r  for  the  local  parlia- 
ment at  TonlonHe,  and  he  discharged  the  duties  of  the  office 
with  Gcnipulnus  accumcy  and  fidelttf.  Tliere,  devoting  mon>> 
of  hia  leiiure  to  mAtliemnttcs,  he  spent  the  remainder  of  hia 
life — a  life  which,  hut  for  a  nomewhnt  Acrimtmioaa  diKpat« 
with  Dencart«H  on  the  validity  of  certain  analyaia  naed  hj  the 
latter,  was  unruffled  hy  any  event  wJiich  call»  for  special  notice. 
Tlie  digpute  wan  chiefly  due  to  the  olwctirity  of  Dencartm, 
but  the  tact  nnd  courtesy  i>f  Fennat  l>roUf{ht  it  to  a  friendly 
conclusion.  Fermat  wiv>  a  pioal  wholnr  and  amniied  himneK 
hy  conjecturnlly  restoring  the  work  of  Apolloniui  on  plane 
loci. 

Except  a  few  isolnted  pn|>erR,  Femmt  pahlished  nothing  in 
hiR  lifetime,  nnd  ^ve  no  Bj-stematic  exposition  of  hiH  methMli. 
Some  of  the  mont  Htriking  of  hiH  rcMultfi  were  found  after  hw 
death  on  loune  nhects  of  paper  or  written  in  tha  margins  of 
works  which  he  hail  read  and  annotated,  and  are  anaccom* 
panied  liy  any  proul  It  is  thus  HOmcwhnt  difficult  to  eatimatfl 
the  datrn  and  originality  of  jiiit  work.  He  was  conatitntion- 
ally  modest  nnd  retiring,  and  dnen  nut  Reeui  to  have  intended 
hi»  papera  tu  Ite  pulilinhe^l.  It  is  prulnble  that  be  revised 
his  notea  as  occasion  re<|uire«l,  and  that  his  published 
works  represent  the  final  fonn  of  his  researches  and  there- 
fore cannot  lie  dated  much  earlier  than  1660.  I  shall 
consider  septtrately  (i)  his  investigations  in  the  tbeocr  ol 
numbera ;  (ii)  his  use  in  geometry  of  anal/siB  and  of  infini- 
tesimals; and  {iii)  his  method  of  treating  qoestimn  of  piob*- 
hility. 

(i)  Thi  tkeoty  of  numhT*  appears  to  have  been  the 
favourite  study  of  Fennat  He  prepared  an  edition  of  Dio- 
phantus,  and  the  noten  arul  comments  UMraaa  ccntain 
nomerfus  thnorans  of   connderalile  elegawMi     Host  of  tha 


aOi     1UTHBIUTIC8  FBOM  OBIOARm  TO  HUTOUa 


proob  of  Fennat  are  kMt»  and  it  is  poaiibb  that  toiBe  of 
were  not  rigoroiui — an  imluctioii  by  analogj  and  tlie  iataition 
of  genius  sufficing  to  lead  him  to  correct  resalia.  The  ioUov* 
ing  ezaiiiples  will  illustrate  these  investigations. 

(a)  If  /»  be  a  prime  and  •  be  prime  to p,  then  •'^'-1  is 
divisible  by  j^  that  is,  a^*-  \  sO  (mod.  p).  A  proof  of 
tliifl,  first  given  by  Euler,  is  well  known.  ▲  more  general 
theorem  is  that  a<M«l-.l  =0  (mod.  m)»  where  •  is  prime  to 
ft  and  ^  (ft)  is  the  number  of  integers  less  than  n  and  prime 
to  it 

(b)  A  prime  (greater  than  3)  can  be  expressed  as  the 
diflferenoe  of  two  square  integers  in  one  and  only  one  way. 
Feniiat's  proof  is  as  follows.  Let  ti  be  the  prime^  and  suppose 
it  equal  to  a^-y**  that  is,  to  (x-i-y)  (^-y).  Now,  by  hypo- 
tlieKi-H,  the  only  integral  fact4>r8  off  m  are  u  and  unity,  hence 
x-hy-M.and  ;r>ys  1.  Solving  these  equations  we  get 
X -  |(h -»-  1)  and  y  =  |(h -  1). 

(r)  He  gave  a  proof  off  the  statement  made  by  I>i<^»hantus 
that  tlie  MUiH  off  tlie  Hquares  off  two  integers  cannot  be  off  the 
ffonii  4n  -  I  ;  and  he  atldtd  a  corullary  which  I  take  to  mean 
that  it  is  impassible  that  the  product  off  a  square  and  a  prime 
of  the  ffumi  4t«  -  1  [even  iff  multiplied  by  a  number  prime  to 
the  latter],  can  bo  either  a  square  or  the  sum  off  two  squares. 
¥W  example,  44  is  a  multiple  off  11  (which  is  of  the  form 
4  X  3  -  1)  by  4,  hence  it  cannot  lie  expressed  as  the  sum  of  two 
squamt.  He  also  stated  tliat  a  number  of  the  form  o'-»-6\ 
wliere  a  is  prime  to  6,  cannot  be  divided  by  a  prime  of  the 
form  4m  ~  1. 

(f/)  E\'ery  prime  off  the  ffbrm  4m  •»- 1  is  expressible,  and 
that  in  one  way  only,  as  the  sum  off  two  squares.  This  problem 
wsH  first  sol\*cd  by  Kuler  who  sliewed  tluit  a  number  of  the 
form  2"  (4m  ^\)  can  be  always  expressed  as  the  sum  of  two 


«4 


(e)     Iff  a,  &,  e  be  integers,  sudi  that  a^^6*»c*. 

cannot  be  a  square.     lAgrange  gave  a  solution  of  thia. 

{/)    The  determination  of  a  number  x  snch  that 


PERM  AT.  303 

mity  be  m  cjunro,  when)  »  is  n  given  integor  which  ia  not  tt 
aquitre.     Lngrnngc  gave  n  solution  i)t  thin. 

iff)  Tliere  in  on\j  ono  integral  Holution  ot  the  equation 
x*  +  2  =  ^ ;  Nnd  there  nrc  un]y  two  itttpgntl  KJationK  of  Uw 
etjiMtion  ar*  +  4  -  jf*.  The  re))uiml  mlutioriN  *re  (i\-idetit1j  lor 
the  (irat  cr|nation  x  -  A,  and  for  the  xecond  rqiution  x  =  2  and 
X  -  1 1.  ThiH  question  wax  tssueil  as  a  challisngc  to  the  &igliah 
loatheninticianR  Wallis  and  Digby. 

<A)  No  inf«gral  valup*  of  «,  y,  =  can  be  found  to  Mtisfy 
the  equation  x"  *  y"  = ;',  if  n  be  an  integer  greater  than  2. 
This  proposition*  has  nctjuircil  pxtrannhnary  cclebritj  from 
the  fact  that  no  general  demonRtration  of  it  haa  been  given, 
but  there  is  no  reason  tji  doubt  that  it  is  tme. 

Probably  Fermat  diitcovered  itx  truth  ftnit  tor  the  can 
n  =  3,  and  then  for  the  caxe  n  -  4.  Hih  proof  far  the  former 
of  thef«  ouieN  is  Inst,  but  that  for  the  latter  n  extant,  and  a 
eimilar  proof  for  the  cane  of  n  =  3  wan  j^veA  1^  Enter.  Tbene 
proofs  <]epend  apm  xhewing  that,  if  three  !ntq(rKl  ^-alooi  of 
X,  y,  >  cnn  be  found  which  Kntisfy  the  equation,  then  it  wilt  he 
pumilile  to  find  tliree  other  and  smaller  inters  which  aim 
satinfy  it:  in  this  way  finally  we  shew  Ihnt  (he  pquatinn  mnife 
be  satisfied  by  thrre  vatuca  wliicti  obviouxly  tlo  not  sntinfy  it 
Thus  nn  integral  Kotution  is  possible.  It  would  aeem  that  this 
method  w  inapplicable  to  any  cases  except  thoaeof  n^Sand 
i*=4. 

Fermat'n  dincoiTry  of  the  general  theorem  was  made  later. 
A  proof  can  be  given  on  the  asBumption  that  a  nuinber  can- be 
t«>iolved  into  prime  (complex)  factors  in  one  and  only  one 
way.  The  aaaumption  baa  lieen  made  by  nome  writ«n,  but  it 
ia  not  universally  true.  It  ia  poasible  "  ~ 
iODM)  erroneoas  nupponition,  )nit  it  is 
complexes,  and,  ud  the  whole,  it  aeems  moat  lilcely  that  ha 
diMOTered  a  rigorous  demonstration. 

*  On  Ibis  enriou  prapoiilion,  Me 
PnMtmt,  tUi4  ediUon,  pp-  M~48. 


306     MATaJKHATICtf  FROM  OttlCASTfii  TO  UUTattf& 


Ia  18S3  Legendre  obUined  a prool  Cor  Uie  oam  cf  nat; 
in  1832  Lejeune  Dirichlet  gare  one  for  » » 14,  uid  ia  1840 
Luu^  and  Lebesgne  gave  proob  for  ti  =  7.  Tbe  proportion 
appeani  to  be  tme  univemally,  and  in  1849  Konuner,  hj 
nieana  of  ideal  primes,  proved  ii  to  be  ao  for  all  niunben 
except  thoee  (if  any)  which  aatisfy  three  eonditiona.  It  is  not 
certain  whether  any  number  can  be  found  to  satisfy  these 
cuuditious,  but  there  is  no  number  less  than  100  which  does 
sa  The  proof  is  complicated  and  difficult,  and  there  can  be 
no  doubt  is  based  on  considerations  unknown  to  Fenuat.  I 
may  add  that,  to  prove  the  truth  of  the  proposition  when  m  is 
greater  than  4  obviously  it  is  sufficient  to  confine  ourselves  to  i 

cases  a'here  m  is  a  prime,  and  the  first  step  in  Kummer's  j 

demonstration  is  to  shew  that  one  of  the  numbers  as^  yi  s 
must  be  divisible  by  n. 

The  following  extracts,  from  a  letter  now  in  the  ani- 
versity  library  at  Leyden,  will  give  an  idea  of  Format's 
methods ;  the  letter  i^  undated,  but  it  would  appear  that,  at 
the  time  Fenuat  wrote  it,  he  had  proved  the  proposition  (A) 
above  only  for  the  case  when  n  s  3. 

Je  ne  mVn  nervis  su  eommencemetii  qoe  pour  demooirer  let  pnipo- 
•itioim  negatives,  comiue  par  exemple,  qu'il  n*y  s  socQ  nomlMne  aiolndrs 
de  Touit^  qu*un  multiple  de  3  qui  aoit  compoe^  d'un  quarr6  et  du  triple 
d'un  autre  quarr^.  Qu'U  n'y  a  sucun  trisngle  reetsngle  de  uombiw  dont 
Tsire  soit  un  numbre  quarr^*.  Ls  prcuve  ■•  fkit  par  4ra>MYV  ^  «^ 
ddi/Mirov  en  oette  maniere.  8'U  y  auoit  aucun  triangle  rectangle  en 
nombres  entien,  qui  eubt  Hon  aire  eiigale  k  un  quarr^,  il  jr  auroit  na 
autre  triangle  moindre  que  celny  la  qui  auroit  la  meame  propriety.  8'il 
y  en  auoit  un  second  moindre  que  le  premier  qui  eust  la  meeme  pro- 
priety il  J  en  auroit  par  un  pareil  raisonnement  nn  troieieme  moindre 
que  ce  Boouud  qui  auroit  la  meame  propriety  et  en&n  un  qnatrieme,  un 
ciuquieme  ete.  a  rinlini  en  descendant  Or  eat  il  qn'eatant  donn^  un 
nombre  il  n*j  on  a  point  inlinia  en  descendant  moindres  qoe  oelnj  la, 
j'entena  parler  touajoura  dea  nombrea  entiera.     D'ou  on  oonelud  qu*U  eat  Ij 

done  impoasible  qu*il  j  ait  aucun  triangle  rectangle  dont  I'aire  aoit  j 

quamS.     Vide  foliu  poet  aeiiuena. . . . 

Je  Au  longtempa  aana  pounoir  appUquer  ma  methode  anz  qoeations 
aflirmatiues,  parce  que  le  tour  et  le  biata  pour  j  venir  eat  beauoonp  plot 


FERMAT. 


307 


maUiac  que  oclny  doni  je  me  sere  nai  ncj^atives.  IV  sorte  que  Ion  qa*il 
me  folai  demonstter  qae  toat  nombie  premier  qui  fmrfNume  de  raiiil6  no 
maltiplc  de  4,  est  compcw6  de  deaz  qoarrei  je  me  trvaoay  en  belle  peine. 
Mmis  en  fin  one  meditation  diYenes  foit  reiter^  me  donn*  leu  lamicm 
qui  me  manquoicnt.  Et  Igs  qncstions  affirmatiufs  passerent  par  ma 
mcthodc  k  Tajde  dc  quelqnes  noaae anx  principcn  qo*il  j  fallniit  joindre 
par  necesiiit^.  Ce  prof^reii  de  mon  raisonnemcnt  en  cen  qnestionn  affir- 
matiYcs  cstoit  tel.  Hi  nn  iiombre  premier  ptin  k  diiicrction  qni  rarpanse 
de  Tonit^  on  mnltiple  dc  4  n*eiit  point  composd  de  denz  quarrcx  it  j  aura 
on  nombre  premier  dc  mcsme  nature  moindre  que  le  douiic ;  et  eniiuite 
nn  troiBieme  encore  moindiv,  ctc.<  en  dciicendant  a  Tinfini  Ju^quet  a  ee 
q«ie  uouR  arriviex  an  nombre  5,  qui  est  le  moindre  de  tous  oenz  de  oette 
nature,  lc«|uel  il  8*en  suiTroit  n'estre  pas  compost  de  deux  quarrcz,  ee 
qn*il  est  pourtont  d'ou  on  doit  infcrer  par  la  deduetion  k  Timpossible  que 
tons  oonz  de  cctte  nature  sont  par  consequent  eomposei  de  3  quanes. 
11  y  a  infinies  questions  dc  oette  espece. 

Mais  il  7  en  a  quelqucs  autrcs  qui  demandent  de  nouYeauz  principcs 
pour  7  appliqucr  la  dcsccnte,  et  la  recherche  en  est  qnelques  fois  si  mal 
ais^,  qu*on  n*7  pent  Ycnir  qu*auee  une  peine  extreme.  Telle  e^t  la  ques- 
tion sniuante  que  Bachct  sur  Diophante  aToQe  n'avotr  jamais  pen  demon- 
strer,  sur  le  snject  de  laquvlle  M'.  Descartes  fait  dans  une  de  see  lettret 
la  mesme  declaration,  jus«|ues  la  qu*il  confesse  qu'il  la  jnge  si  diflieile, 
qn*il  ne  toit  point  dc  Yoye  p^r  la  resoudre.  Tout  nombre  est  quarr^, 
on  compose  de  deux,  de  trois,  on  de  quatre  quarrcx.  Je  ra7  enfin  ranges 
sous  ma  methode  ct  je  demonstre  que  si  nn  nombre  donn^  tt*estoit  point 
de  oette  nature  il  y  en  auroit  un  moindre  qui  ne  le  seroit  pas  non  plus, 
puis  un  troisieme  moindre  que  le  second  de.  k  Tinfini,  d*on  Ton  infere 
que  tons  Ics  nombrcs  sont  de  cette  nature.... 

J*ay  ensuite  eonsiderc  cerUines  questions  qni  bien  que  negatiTcs  ne 
restent  pas  de  rcceuoir  trcs-grande  difBcultr  la  methode  pour  y  pratiquer 
la  desoente  estant  tout  a  fait  diucrse  des  prcoedentes  oomme  il  sera  ais£ 
d*esprouuer.  Telles  sont  les  sninantes.  II  n'y  a  aucun  cube  diuisiblc 
en  deux  cnbes.  II  n*y  a  qu'un  seul  qnarre  en  entiers  qui  angmentr  du 
binaire  fawse  on  eube,  ledit  quarrC*  est  S5.  II  n'y  a  que  denx  quarrex  en 
entiers  letquds  angmentcs  de  4  fjMsent  cube,  lesdite  qnarrestool  4 
et  121.... 

Apr68  auoir  ooom  tontee  ees  questions  la  pinpari  de  diuerses  (tic) 
aalare  et  de  dilferente  lav<m  do  demonstrer,  j*ay  pass^  a  llnuentioo 
dea  regies  genendes  poor  resoudre  les  equations  simples  el  doubles  de 
Diophante.  On  propose  par  exemple  3  qaaiT.-i>79o7  etgaox  a  on  qoarr6 
(hoe  crt  ifar+7967  «  qnadr.)  J*ay  one  regie  generale  pour  resoudre 
eelle  aqoalkm  m  elle  crt  posalhia,  oq  deeoovfir  too  impossibilil^.  £1  , 
aiorf  an  tow  lee  eni  el  m  loot  aombtn  IobI  dee  qnnig  qw  dee  oailte. 
Ob  fnfom  oeMt  efMlta  doobia  Sr-i-S  tl  ftv'ff  loggni  iha<awBi  ^^^^ 


308     M ATUEMATIUI  FBOM  DMUCmMH  TO  BUTOIlia 


qotni.  Backel  m  gioriia  an  am  aammmttAtm  mu  DiophASlt  A*mmk 
ttouH  «M  itgU  Ml  deojL  MS  partkolkn.  Jo  U  doaat  fmank  m  loate 
Mrto  de  cm.    Bt  dsteniiiiM  |Mur  ngU  ai  tlU  Ml  fOMibIt  ini  boa..., 

Voila  Mminairaneni  lo  oonto  do  mm  locihoffohM  iw  lo  o^|ool  im 
nombiM.  Jo  no  Taj  oocrit  quo  poiM  qno  j'o|i|itclioiido  quo  lo  loWr 
d'ootondiool  do  mottoo  oa  long  UhiIm  om  domonitrotiooo  ol  om  Mt  JiodM 
no  luauqiiora.  En  tout  mo  oetio  ipdiMtion  Mrairo  aia  ovauoato  poor 
irouTer  d'eox  mMiuM  oe  quo  jo  ii*Mteiis  point,  prindpalomont  ai  M'.  do 
Carcani  ct  Fr6niele  lour  font  part  do  qoelquM  demonstrationa  |iar  la 
deacento  que  jo  leur  aj  cnuoyoM  our  lo  oojoot  do  quolqaM  propooatkNia 
neitatiiiea.  El  pent  ettre  la  pwtcritA  mo  aMura  gr^  do  luy  avoir  fail 
oonnoiatni  quo  kn  auci«siM  n*ont  pas  tout  aMU,  at  wtto  relation  ponrra 
paiuier  dans  I'Mprit  do  ooui  qui  vUinUrout  apfM  nioj  poor  traditio  lani- 
psdis  ad  fiUos,  coiume  parle  lo  grand  Chancelier  d'Angletorre,  auinani  la 
aentiment  et  la  deuino  duqud  j'a^joustono^,  mnlti  perlranaibnni  ol  augo- 
mtur  scientia. 


(ii)  I  next  proceed  to  mention  Feniiat'a  use  in  geometry 
of  atialt/ils  atul  of  \i\fiu\itMiuujil*,  It  would  aeeni  froui  his 
correHpoudeiice  tlmt  he  had  tliought  out  the  priaciplea  of 
analytical  geometry  for  himself  before  reading  Descartea'a 
GeottUtrie,  and  had  realized  that  from  the  equation  (or  as  he  ( 

calls  it,  the  "  specific  property  ")  of  a  curve  all  its  properties  ^ 

could    be    deduced,     liis  extant    papers  on  geometry   deal  i 

however  mainly  with  the  application  of  infinitesimals  to  the 
determination  of  the  tangents  to  curves,  to  the  quadrature  of 
curves,  and  to  questions  of  maxima  and  minima;  probably 
these  papers  are  a  revision  of  his  original  manuscripts  (which 
he  destroyed)  and  were  writt4*n  about  1663,  but  there  is  no 
doubt  that  he  was  in  possession  of  the  general  idea  of  his 
method  for  finding  maxima  and  minima  as  early  as  1628  or 
1629. 

He  obtained  the  subtangent  to  the  ellipse,  cycloid,  cissoid, 
conchoid,  and  quadratrix  by  makuig  the  ordinates  of  the 
curve  and  a  straight  line  the  same  for  two  points  whose 
abscissae  were  x  and  x-e;  but  there  is  nothing  to  indicate  i 

that  he  was  aware  that  the  process  was  general,  and,  though 
in  the  course  of  his  work  he  used  the  principle,  it  is  probaUe 


\ 


FERMAT. 


309 


thAt  he  never  separated  it»  so  to  speak,  from  the  symbols  of 
the  particular  problem  he  was  considering.  The  first  definite 
statement  of  the  method  was  doe  to  Barrow*  and  was  pub- 
lished in  16G9. 

Fermat  also  olitained  the  areas  of  paralmlas  and  hyper- 
Iwlas  of  any  order,  and  determined  the  centres  of  mass  of  a  few 
simple  laminae  and  of  a  paraboloid  of  revolution.  As  an 
example  of  his  method  of  solving  thene  questio.is  I  will  quote 
his  soluti«>n  of  the  pniblem  to  find  the  area  between  the 
parabola  y's^,  the  axis  of  x,  and  the  line  x^a.  He  sayn 
that,  if  the  several  ordinates  at  the  points  for  which  x  is 
equal  to''#i,  a(1-«),  a(1-r)', ...  be  drawn,  then  the  area 
will  be  split  into  a  number  of  little  rectangles  whose  areas  are 
respectively 

Tlie  sum  of  these  is  /i*  <•*  if/{l  -  (I  -  <?)'} ;  and  by  a  sulwidiary 
proposition  (for  lie  was  not  acquainted  with  the  binomial 
theorem)  he  finds  the  limit  «»f  this,  when  e  vanishes,  to  lie 

iP  ^'*  *^^  theorems  last  menticmed  were  published  only 
after  his  death  ;  and  pnibably  they  were  not  written  till  after 
he  had  rrad  the  works  of  Cavalieri  and  Wallis. 

Kepler  had  remarked  that  tlie  values  of  a  function  imme- 
diately adjacent  t4i  and  «m  either  side  of  a  maximum  (or 
minimum)  value  must  be  equal.  Fennat  appliefl  this  principle 
to  a  few  examples.  Thus,  to  find  the  maximum  xwXvlq  of 
X  (a  -  «),  his  method  is  essentially  equivalent  to  taking  a  con- 
secutive value  of  x^  namely  x— «  where  e  is  very  small,  and 
putting  x(o*«)«>(x  — «)((•- X  4- «).  Simplifying,  and  ulti- 
mately  putting  esO,  we  get  a;=  \a.  This  value  of  m  makes 
the  given  expressioti  a  maximum. 

(iii)  Fermat  must  share  with  Fiscal  the  honour  of  having 
founded  ike  cAeory  of  probabiiiiieM.  I  have  already  mentioned 
the  problem  proposed  to  Fkecal^  and  which  he  oommvnicated 


^% 


810     XATHKMATIC8  raOM  DBOABTBI  TO  HUTOIini 


to  Feniuit^  and  have  (here  given  PMcel'e  aolotioii.  FenmCli 
aolatian  depends  on  the  theory  of  comhiimtiotMi  and  will  be 
sufficiently  illustrated  by  the  following  example  the  snbstanee 
of  wliicli  is  taken  from  a  letter  dated  Aug.  24,  1654,  whidi 
oocum  in  the  oorrespondenoe  with  PkncaL  Fermat  discnssw 
the  cane  of  two  phkyem,  A  and  B^  where  A  wants  two  points  to 
win  and  B  three  pointH.  Then  the  game  will  be  oertainly 
decided  in  tlie  oourHe  of  four  trials.  Take  the  letters  a  and  A 
and  write  down  all  the  combiiwtions  thai  can  he  formed  of 
tour  lettem.  These  combinations  are  16  in  number,  namely, 
ofUMf  aaabf  aaba^  aabb;  nbtia^  abab^  fMnt^  abbb;  iooo,  bamb^ 
babOf  babb ;  bbeui^  bbab^  bbba^  bbbb.  Now  every  combination  in 
which  a  occurs  twice  or  oftener  represents  a  case  favourable  to 
A,  and  every  combination  in  which  b  occurs  three  times  or 
oft4'ner  repreitentA  a  case  favourable  to  ^.  Thus,  on  counting 
tlieiii,  it  will  lie  found  that  there  are  1 1  cascH  favourable  to  il, 
and  5  castas  favourable  U)  7/ ;  and,  since  these  cases  are  all 
equally  likely,  J's  chance  of  winning  the  game  is  to  ffn  chance 
as  1 1  is  to  5. 

Tlie  only  other  pru!>lein  on  this  subject  which,  as  far  as 
1  know,  attnictiHl  the  attention  of  Feniiat  was  also  proposed  to 
him  by  Pascal  and  was  as  ft>llows.  A  person  undertakes  to 
throw  a  six  with  a  die  in  eight  throws ;  supposing  him  to  have 
uiiule  thriH)  throws  without  success,  what  porticm  of  the  stake 
slioukl  lie  be  allowefl  to  take  on  condition  of  giving  up  his 
fourth  throw  t  Feniiat*s  reasoning  is  as  folio wa  Tlie  chance 
of  success  is  },  so  that  lie  should  lie  allowed  to  take  \  of  the 
stake  cm  condition  of  giving  up  his  throw.  But^  if  we  wish  to 
estimate  the  value  of  the  fouKh  throw  before  any  throw  is 
made,  then  the  lirst  throw  is  woKh  I  i*  the  stake;  the  second 
is  worth  J  of  what  remains,  that  is,  ^g  of  the  stake ;  the  third 
throw  Is  worth  J-  of  what  now  reiiiains,  tliat  is,  ^Yr  ^'  ^^® 
stake  ;  the  fourth  throw  is  worth  I  of  what  now  remains,  thai 
is  Y'?9g  of  the  stake. 

Fermat  does  not  seem  to  have  carried  the  matter  much 
further,  but  his  correspondence  with  Pascal  shows  that  his 


FERMAT.      HUTOEXS. 


811 


▼iewB  on  ilie  fundamental  principles  of  the  subject  were  iic- 
curate :  those  of  Pascal  were  not  altogether  correct. 

Fennat's  reputation  is  qnite  unique  in  the  histoiy  of 
science.  The  problems  on  numbers  which  he  had  proposed 
long  defied  all  efforts  to  solve  them,  and  many  of  them  yielded 
only  to  the  skill  of  Euler.  One  still  remains  unsolved.  This 
extraordinary  achievement  has  ovemhadowed  his  other  work, 
but  in  fact  it  is  all  of  the  highest  order  of  excellence,  and  we 
can  only  regret  that  he  thought  fit  to  write  so  littla 

Hnygena^.  Chrufttan  Iluyg^nt  was  bom  at  the  Hague 
on  April  14, 1629,  and  died  in  the  same  town  on  June  8, 1695. 
He  generally  wrote  his  name  as  Hugens,  but  I  follow  the  usual 
custom  in  spelling  it  as  above:  it  is  also  sometimes  written 
as  Huyghens.     His  life  was  uneventful 

In  1651  he  published  an  essay  in  which  he  shewed  the 
Mlacy  in  a  system  of  quadratures  proposed  by  Gr^goire  de 
Saint-Vincent  who  was  well  versed  in  the  geometry  of  the 
Greeks  but  had  not  grasped  the  essential  points  in  the  more 
modem  methods.  Tliis  essay  was  followed  by  tracts  on  the 
quadrature  of  the  conies  and  the  approximate  rectification 
of  the  circle. 

In  1654  his  attention  was  directed  to  the  improvement  of 
the  telescope.  In  conjunction  with  his  brother  he  devised 
a  new  and  lietter  way  of  grinding  and  polishing  lensea 
As  a  result  of  Uiese  improvements  he  was  able  during  the 
following  two  years,  1655  and  1656,  to  resolve  numerous 
astronomical  questions  \  as  for  example  the  nature  of  Satum*s 
appendage.  His  astronomical  oli8er\'ations  required  some 
exact  means  of  measuring  time,  and  he  was  thus  led  in  1656 
to  invent    the  pendulum  clock,   as  described  in   his  tract 


*  A  new  sdltioB  ol  all  Hi^TRens^s  works  and  eoiwpoodencs  is 
Mug  bmcd  si  Ihs  Hsgos,  188a,  Ae.    An  6sril«r  cdilioii  of  Ms  works 
fNiUliilMii  in  six  vohnaca,  four  al  Lsydsn  in  17S4  snd  two  at 
in  1798  (a  lifi  by  s*Ofav«sads  is  praiidi  lo  Hm  im 
I  fcis  sBisiiHis  eotif  spendwas  wss  pukliriMi  ai  ^ki^ 


18». 


sit     lUTHUUTlCS   FBOX   DiSlCABTES  TO   HOTaCNS. 

Harolai/imtit,  16SR.  The  Uwe-pieoM  pravigoal;  in  bm  hwl 
beat  bftbuHM-oloclca. 

In  the  yeu  16S7  Huygena  wrote  »  amall  work  on  tbe 
calculus  of  profaabilitie*  founded  on  the  oorreepondenoe  of 
Phc«1  Bad  Pemut.  He  spent  &  couple  of  yean  in  England 
about  thia  timeu  Hia  reputntion  waa  now  ao  great  that 
in  1665  l4Niii  XIV.  offerMi  him  a  peniiion  if  he  would 
lire  in  Ktfia,  which  accordingly  then  became  his  place  of 
reaidenoe. 

In  1CC8  he  sent  to  the  Koyal  Society  of  Tendon,  in 
answer  to  a  problem  tliey  Imd  proposed,  a  nietnoir  in  which 
(•imultaneoualy  with  W&llia  and  Wren)  lie  proved  by  ex- 
periment tliat  the  momentum  in  a  certain  diroction  before 
the  colliiuon  of  two  Wliea  is  etjual  to  the  momentum  in 
that  direction  after  the  coUixioo.  This  wan  one  of  the  points 
in  niechanicH  on  wliiL-h  Dt>scart4«  hud  be<-n  mintaken. 

The  raoHt  iiiipurtAut  of  Huygciis'it  work  was  hin  Ilurolo- 
g!um  OfcHlalurium  published  at  Paris  in  1673.  Tlie  first 
chapter  in  devot«<I  to  pifndulum  clucks.  The  neojnd  chapter 
contains  a  complete  account  of  the  dtMceiit  of  heavy  liudini 
under  their  own  weights  in  a  vacuum,  either  vertically  down 
or  on  smooth  curvts.  Ainun^pit  otlier  prupuHitions  he  shews 
that  the  cycloid  ih  tautochronom.  In  the  third  cliapler  ha 
defines  evolutes  and  involutes,  proves  some  of  their  mure 
elementary  propertit.-H,  and  illustrutes  hiit  methods  by  finding 
tlie  evolutes  of  the  cycloid  and  the  [larabola.  These  are  the 
earliest  instances  in  which  the  envelupe  of  a  muving  Une  was 
(leterndncd.  In  the  fourth  chapter  he  sohes  the  problem  of 
the  compound  pendulum,  und  xhews  tlwt  the  centres  of  oscil- 
lation and  sus|Mtnnioii  nre  iiiterc)iang*'uble.  In  the  fifth  and 
lost  chapter  hedisi^UMics  again  the  theory  of  clocks,  points  out 
that  if  tlie  liob  of  tlie  pendulum  weru  mode  by  means  of 
cycloidal  checks  to  osciilute  in  a  cycloid  the  oscillations  would 
be  isochronous  ;  aiid  tiiiinheH  by  Hhewing  that  the  oentrifugiil 
force  on  a  body  which  moves  round  a  circle  of  nulius  r  with 
a  uniform  velocity  tt  varies  dinvtiy  as  v*  and  inversely  as  r. 


HUTGENS. 


813 


This  work  contains  the  first  attempt  to  apply  dynamics  to 
bodies  of  finite  sise  and  not  merely  to  particles. 

In  1675  Huygens  pro|iofied  to  regulate  the  motion  of 
watches  by  the  ufie  of  the  balance  spring,  in  the  theory  of 
which  he  had  been  perhaps  anticipated  in  a  somewhat  am- 
bigaouB  and  incomplete  statement  made  by  Hooke  in  1658. 
Watches  or  portable  clicks  had  lieen  invented  early  in  the 
sixteenth  century  and  by  the  end  of  that  century  were  not 
rery  uncommon,  but  they  were  clumsy  and  unreliable,  being 
driven  by  a  main  spring  and  regulated  by  a  conical  pulley  and 
Tei^  escapement ;  moreover  until  1687  they  had  only  one 
hand.  The  first  watch  whoHC  motion  was  regulated  by  a 
balance  spring  was  made  at  Paris  under  Huygens's  directions, 
and  presented  by  him  to  Louis  XIV. 

The  increasing  intolerance  of  the  Catholics  led  to  his 
return  to  Holland  in  1681,  and  after  the  revocation  of  the 
edict  of  Nantes  he  refused  to  hold  any  further  communica- 
tion with  France.  He  now  devoted  himself  to  the  con- 
struction of  lenses  of  enormous  focal  length :  of  these  Uiree 
of  focal  lengths  123  ft,  180  ft,  and  210  ft  were  sulisequently 
given  by  hiiii  to  the  Royal  Society  of  London  in  whose 
possession  they  still  remain.  It  was  about  this  time  that  ho 
discovered  the  achromatic  eye-piece  (for  a  telescope)  which  is 
known  by  his  name.  In  1689  he  came  from  Holland  to 
England  in  order  to  make  the  acquaintance  of  Newton  whono 
Prificipin  had  been  published  in  1687.  Huygens  fully  recog- 
nised the  intellectual  merits  of  the  work,  but  seems  to  have 
deemed  any  theory  incomplete  which  did  not  explain  gravita- 
tion by  mechanical  causes. 

On  his  return  in  1690  Huygens  published  his  treatise  on 
tight  in  which  the  undulatory  theory  was  expounded  and  ex- 
plained. Most  of  this  had  been  written  as  early  as  1678. 
The  general  idea  of  the  theory  had  been  suggested  by  Robert 
Hooke  in  1664,  but  he  had  not  investigated  its  consequences 
in  any  detail.  This  paMication  falls  outside  the  years  con- 
nderad  in  this  chapter,  but  here  it  may  be  briefly  said  tJhat^ 


814     lUTHBMATICS  WBOU  DBOABTBI  T^  HUfOEMa 


aooording  to  the  wave  or  unduklory  Iheiiffy,  wpmom  k  iDed 
with  AD  eztremeljr  ram  etlier«  and  li^t  it  eanaed  bj  *  MfiM 
of  wavM  or  vibratioiii  in  ihiietber  whiohj  are  lei  in  BKitioii 
by  the  pulsatiooa  of  the  laminons  body.  Vtm  this  hypothem 
Huygena  deduced  the  laws  of  reflexion  and  refractions  es- 
pUined  -  the  |)lienoQiena  of  doable  refraction,  and  gave  a 
ooiiAtniction  for  the  extraordinary  ray  in  biaxal  eryvtahi; 
while  he  found  by  experiment  the  chief  phenomena  of 
polariiation. 

'Ilie  immense  reputation  and  unrivalled  powers  of  Newton 
led  to  diabelief  in  a  theory  which  he  rejected,  and  to  the 
general  adoption  of  Newton'ii  emission  theory ;  but  it  should 
be  noted  that  Huygens's  explanation  of  some  phenomenal  such 
as  the  colours  of  thin  plates,  was  inconsistent  with  the  results 
of  experimenta,  nor  was  it  until  Young  and  Wollaston  at  the 
beginning  of  this  century  revived  the  undulatory  theory  and 
modilied  some  of  its  details  and  Fresnel  elaborated  their  views 
that  its  acceptance  could  be  justified. 

Besiden  tliesie  works  Huygens  took  |iart  in  most  of  the 
controversies  and  challenges  which  then  played  so  laige  a  part 
in  the  niatheuiatical  world,  and  wrote  several  minor  tracts. 
In  one  of  these  he  investigated  the  form  and  properties  of  the 
catenary.  In  another  he  stated  in  general  terms  the  rule  for 
finding  maxima  aud  minima  of  which  Femiat  had  made  use^ 
antl  shewed  tliat  the  suhtangent  of  an  algebraical  curve 
/(jT,  y)  =  0  was  equal  to  js^/y^»  where y^  is  the  derived  function 
of  /(x,  y)  regarded  jis  a  function  of  y.  In  some  posthumous 
works,  issued  at  I^eytlen  in  1703,  he  further  shewed  how  from 
the  focal  lengths  of  the  component  lenses  the  magnifying 
power  of  a  telescope  could  be  determined ;  and  explained  some 
of  the  phenomena  connected  with  halos  and  parhelia. 

I  should  add  that  almost  all  his  demonstrations,  like  those 
of  Newton,  are  rigidly  geometrical,  and  he  would  seem  to  have 
made  no  use  of  the  differential  or  iluxional  calculus,  though  he 
admitted  the  validity  of  the  methods  used  therein.  Thus,  even 
when  first  written,  his  works  were  expressed  in  an  archaic 


BACHET.      UGRSENKE.  31fi 

Ungnoge,  nnd  pc^li(v[n  received  Ipu  ntt^ntion  than  thnr 
inlrinsic  tuorita  desrrred. 

I  )mvc  now  traced  the  development  «[  miithenmtiai  for  • 
pcricMl  which  we  niny  tAke  mughlj'  an  dntiiig  from  1635  to 
1G75  under  the  influence  of  DescArt«>s,  Cavalieri,  I^scftl, 
Wttllifl,  Fermnt,  Miid  Ilujrgens.  The  Hfo  of  Newton  partly 
overlaps  this  prriod:  big  wi,rl(«  ami  inflncnc«  are  considered 
in  the  next  chnpt«r. 

I  niny  dinmiHs  the  remaining  mathemoticlana  of  thii 
time*  with  compAntively  nlight  noticn.  The  mont  eminent 
of  them  are  Jiark*!,  Harrow,  Hmnnch-r,  Collins,  fh  la  t/irt, 
dn  Monhire,  Frhiicle,  Jnmn*  Or^gortf,  IIhoIm,  Uwtde,  Xtchaltui 
Mercator,  3ftrnm>ir,  Ptil,  Raiirrval,  Ro^mrr,  Jtnlh,  Sainl- 
Vineenl,  Sluir,  Torrir'lli,  Tur/iimfutnium,  mn  SehooUn,  rirnin*, 
and  IFrwii.  In  the  following  noten  I  have  nrmnf^  the  above- 
mentioned  mathemnticianx  so  that  an  far  an  poMiiUe  their 
chief  contributinns  nhnll  come  in  chronological  order. 

Bachet.  CUtud*  (iMftard  Bnchrt  rir  Me^rine  wm  born  at 
Bonrg  in  1581,  and  died  in  1638.  He  wrtite  the  PnbUmn* 
ptaimnlii,  ni  which  the  6nit  edition  wan  iMoed  in  1613,  a 
■ocon^  and  enlargrd  edition  wan  l)rou;;ht  oet  in  1624;  thin 
contain*  an  interesting  collection  of  arithmetical  trick*  and 
questionn,  many  of  which  are  quoted  in  my  ifnllt^nuUienl 
Rfrrt-ntionn  and  Pmhl'mM,  He  alao  wrote  Ltit  iUmKnU 
ttrithmrtitpifi,  which  exisU  in  manuwript ;  and  a  trannlation 
of  the  AriOintrtie  of  DiophantuR.  Bnchet  wan  the  eariieat 
writer  who  dixcuKned  the  aolution  of  indeterminate  eqaationa 
by  mrAnx  of  continued  fnwtionn- 

Henenne.  Marin  MrmnxnT,  bom  in  1588  and  died  at 
Pnrifl  in  1648,  waa  a  Franciiwan  friar,  who  made  it  hia 
businens  to  be  acquainted  and  correspond  with  the  French 
mathematicians  of  that  date  and  many  of  their  foreign  ooo- 
teiDpotariea.     In  1634  he  pnbliahod  a  trannlatioD  of  Galilao'a 

■  Nota  on  •evenl  of  tbMc  mnlhanalidui  will  ta  hmi  la  a 
Holtao'a   JTalkaMriMt    Dirtitrntn^    aW  Traeti.   S  Til— iia,  UwJhifc, 

uis-ins. 


310     MATHK1UTIC8  mOlf  DBOAETIB  TO  HUTOnm 


meohanict;  in  1644  ha  iaraed  hit  Cogiiaim  PhfaJm-MmiU' 
moliooy  bj  which  he  ib  best  known,  onnUining  nn  neoonni 
of  aome  experiments  in  phydoe;  ha  also  wrote  n  efnoiMb 
of  mathemnticsy  whidi  was  printed  in  1664. 

Tlie  preface  to  the  CoyiUUa  contalni  a  Btateaient  (poidbly 
dae  to  Feruiat)  that,  in  order  that  2^—1  may  be  prime^  the 
only  values  of  /i^  not  greater  than  257,  which  are  possible  are 
1,  2,  3,  5,  7,  13,  17,  19,  31,  67,  127,  and  257 :  the  nomber  67 
is  probably  a  misprint  for  61,  With  this  correction  the 
statement  appears  to  be  true,  and  it  has  been  verified  for 
&U  except  nineteen  values  of  p:  namely,  71,  101,  103,  107, 
109,  137,  139,  149,  157,  163,  167,  173,  181,  193,  199,  227, 
229,  241,  and  257.  Of  these  values,  Mersenne  asserted  tliat 
p  =  257  makes  2'  —  1  a  prime,  and  that  the  other  values  make 
2'  —  1  a  composite  number.  The  verifications  for  the  cases 
when  p^^lt  89,  127  apparently  rest  on  long  numerical 
calculstiouM  made  by  single  computators  and  not  published ; 
until  those  demonstrations  have  been  confirmed  we  may  say 
that  twenty-two  cases  still  await  verification  or  require 
further  investigation.  The  factors  of  2''—  1  when  /i  =  G7  and 
/>=:89  are  not  known,  the  calculations  meraly  showing  that 
the  resulting  numbers  could  not  be  prima  It  is  most  likely 
that  these  reHults  are  particular  coses  of  some  general  theorem 
on  the  subject  which  remains  to  be  discovered. 

The  tlieiiry  of  perfect  numbers  depends  directly  on  that  of 
Mersienne's  numbers.  It  is  probable  that  all  perfect  numbers 
are  included  in  the  formula  2^~*(2^~1),  where  2''-l  is  a 
prime.  Euclid  proved  tliat  any '  number  of  thui  form  is 
perfect;  Euler  Mliewed  tliat  the  formula  includes  all  even 
perfect  numbers ;  su  J  there  is  reason  to  believe — though  a 
rigid  demonstration  is  wanting — that  an  odd  number  cannot 
be  perfect  If  we  assume  that  the  last  of  these  statements  is 
true,  then  every  perfect  number  is  of  the  above  form.  Thus, 
if  p=2,  3,  5,  7,  13,  17,  19,  31,  Gl,  then,  by  Mersenne's  rule, 
the  corresponding  values  of  2^*  —  1  are  prime ;  they  are  3,  7,  31, 
127,8191,131071,524287,2147  483647,2.^5843009213693951; 


ROBERVAL.      VAK  XCHtKITEN.      NAIMT-VIKCENT.      317 

Knd  the  corresponding  perfect  numbenmro  6,  28,  406,8128, 
33550336,8589869056, 13743860132t<,  23058*5008139932128, 
«nd  2658455991569831741651092615953843176. 

Roberval*.  Gill't  Perum'ur  (lU)  RiArrrnl,  born  Kt  Roboi^ 
vaI  in  1603  mid  died  >t  Parin  in  1675,  dewribed  himnclf  from 
Uio  place  of  hin  birt)>  ns  dc  Holtcrval,  a  scignorial  title  to 
which  he  had  no  right.  He  dincnssc*!  the  natnra  id  thfl 
tangcnU  to  cartTs,  solved  some  of  the  <asier  qDMtiowi 
conuoct«d  with  the  cycloid,  gpnerali/^l  Archimcdea'a  theo- 
rems on  the  fijiiml,  wrote  oil  mn'ltanics,  and  on  the  method 
of  inttivinihleB  which  hv  rendiTr<l  nnirc  prrciae  and  logical. 
He  was  a  professor  in  the  university  q{  Pnria,  and  in  corre- 
spondence with  nearly  all  the  leatling  ntathematicians  of  his 

Van  Sohooten.  Fmn»  mn  Sr^>ten,  to  whom  we  owe  an 
edition  of  Vieta's  works,  siiccecdrt]  his  father  (who  had  tattght 
niatliematics  to  Huygrn!!,  Hudde,  and  iSlui'^)  as  |irutc)wor  at 
Leyden  in  1646  :  he  brought  out  in  1659  a  Latin  translation 
of  Descartra'a  Gittmelriit ;  and  in  1657  a  mllcctioii  of  mathe- 
matical exercises  in  which  ho  recommendrd  tlie  use  of  co- 
ordinates in  space  of  three  d i mention g :  he  (lied  in  1661. 

Saint-Tincentf .  Griijinf.  Jt  Saint-  Ki>i'via,'a  Jesuit, Ixini 
at  Bruges  in  1584  and  died  at  Ghent  in  I66T|  discovered  the 
expansion  of  log  (1  *-x)  in  ascending  powen  r4  x.  AlthoDgh 
a  circlc-squarer  he  l*  worthy  of  mention  for  the  nDmeroas 
iheorcins  of  int^re^t  which  he  discovered  in  his  search  after 
the  impossible,  and  Mnntucla  ingenioosly  remarks  that  "no 
one  ever  Bi{Uar«l  the  circle  with  so  much  ability  or  (except  for 
his  principal  object)  with  so  much  nnccesB."  He  wrote  two 
books  on  the  subject,  one  published  in  1647  and  the  other  in 
1666,  which  cover  some  two  or  throe  thoDsand  cknelj  printed 
pages;    the   fallacy  in   the   quadrature  was  pointed   out   by 

*  A«omp(e(aeditiDao(  his  works  was  IndoiM  in  tka  oU  IffaNn* 
of  tha  koAaaj  of  Bcinwe*  pnbliiliBd  ia  1II9S. 

t  B««L.A.J.4«s4eM'aa>M*^rf<*KtaMM(*«taA4fM,BmMls, 


318     lUTHBIUTIGB  FROlf  DBM^AKTBtt  TO  HUYOIMi 

Hnygent.  In  the  fernier  work  he  aied  indiTinblei ;  aa  6Mliar 
work  entitled  nsonmata  Maikemaiien  paUiJied  in  1694 
oontaini  a  clear  aooount  of  the  method  of  eihanitionii,  whieh 
is  affiled  to  several  quadiatorei^  notaUy  that  of  the  hyper- 
bola. 

Tbrricelli*.  Evangditia  Tvrnadli^  bom  at  Faensa  on 
Oct  15,  1608  and  died  at  Florence  in  1647,  wrote  on  the 
quadrature  of  the  cycloid  and  couics;  the  theory  of  the 
barometer;  the  value  of  gravity  found  by  observing  the 
motion  of  two  weights  connected  by  a  string  passing  over  a 
fixed  pulley;  the  theoiy  of  projectiles;  and  the  motion  of 
fluids. 

Hudde.  Johann  Uudde^  burgomaster  of  Amsterdam, 
was  bom  there  in  1633  and  died  iu  the  same  town  in  1704. 

He  wrote  two  tracts  in  1G59:  one  was  on  the  reduction  of 

equations  which  have  equal  roots ;  in  the  other  he  stated  what 

is  equivalent  to  the  proposition  that,  ii/(x^  y)»0  be  the  alge- 

braical  equation  of  a  curve,  then  the  subtaugent  is  "y  ^/^» 

but  being  ignorant  of  the  notation  of  the  calculus  his  enuncia- 
tion is  involved. 

Fr6niclet.  Bermml  FrcnicU  de  Besttf^  bom  in  Fkris  ciro. 
1605  and  died  in  1670,  wrote  numerous  |Mipers  on  combina- 
tions and  on  the  tlieory  of  nuniljcrs,  also  on  magic  squares. 
It  may  be  interesting  to  vAd  that  he  challenged  Huygens  to 
solve  the  following  system  of  equations  in  integers,  a:'-f  j^^s", 
a*  ..  H^  -»- 1\  X -  y  =^u -  V :  a  solution  was  given  by  M.  P6pin 
in  1880. 

De  Laloubdre.  AtUaine  de  Laloubere,  a  Jesuit,  bom  in 
Languedoc  in  1600  and  died  at  Toulouse  in  1664,  is  chiefly 
celebrated  for  an  incorrect  solution  of  PascaTs  problems  on 
the  cycloid,  which  he  gave  in  1660,  but  he  has  a  better  claim 

*  His  mathenisticsl  writing*  were  published  at  Florenee  in  1644, 
under  the  liUe  Open  OeoHUirica, 

t  Fr^nide*s  nuAoeUsneous  works,  edited  by  De  la  Hire,  wen  pob- 
ia  the  Mmoires  de  fAcadtmit,  vol  v.  1091. 


MERCATOR.      BARROW. 


319 


to  distinction  in  having  been  the  first  mathematician  to  stndj 
the  properties  of  the  helix. 

N.  Mereator.  Xieholas  Mercaior  (sometimes  known  as 
Kanffmanv^  was  bom  in  Holstein  about  1620,  bat  remded 
most  of  his  life  in  England:  he  went  to  France  in  1683, 
where  he  designed  and  constructed  the  fountains  at  Versailles, 
but  the  payment  agreed  on  was  r^sed  unless  he  would  turn 
Gktholic:  he  died  of  vexation  and  poverty  in  Fsris  in  1687. 
He  wrote  a  treatise  on  lo^rithms  entitled  LogarUhnuhUd^nioa 
published  in  1668,  and  discovered  the  series 

log  (I  +«)  =  «- J«*+  Ja^- JaJ*+...; 

be  proved  this  by  writing  the  equation  of  a  hyperbola  in  the 
form 

y  =  = —  =  l-«-»-a:*-a^+ .... 
^     1  +x 

to  which  Wallis's  method  of  quadrature  could  be  applied. 
The  same  series  had  been  independently  discovered  by  Saint- 
Vincent. 

Barrow^.  I§aac  Barrmo  was  bom  in  London  in  1630, 
and  died  at  Cambridge  in  1677.  He  went  to  school  first  at 
Charterhouse  (where  he  was  so  troublesome  that  his  father  was 
heard  to  pray  that  if  it  pleased  God  to  take  any  of  his  children 
he  could  best  spare  Isaac),  and  subsequently  to  Felstead.  He 
completed  his  education  at  Trinity  College,  Cambridge ;  after 
taking  his  degree  in  1648,  he  was  elected  to  a  fellowship  in 
1649,  he  then  resided  for  a  few  years  in  college,  but  in  1655 
be  was  driven  out  by  the  persecution  of  the  Independents. 
He  spent  the  next  four  years  in  the  East  of  Europe,  and  after 
many  adventures  returned  to  England  in  1659.  He  was 
ordained  the  next  year,  and  appointed  to  the  professorship 
of  Greek  at  Cambrklge.  In  1662,  he  was  made  professor  ci 
geometry  at  Oresham  College,  and  in  1663^  was  selected  as  the 
fiwt  occupier  ol  the  Lqeasian  chair  at  Oambcidge,   Heresigned 

•  His  msthfwstfwil  votks,  sdiisi  lij  W.  WIwwbII,  ms  isHNi  al 
CMWdgiiaMQl 


S20     MATUKMATial  rBOM  DttCASIW  TO  HVTailia 


the  hilar  to  hk  popO  Newton  in  1669  wImmw  enperior  nhOiliei 
lie  reoogniied  and  frankly  acknowledged,  fbr  the  wieinrter 
of  his  life  he  devoted  himself  to  the  stndy  oC  divinity.  He 
WAS  appointed  master  of  Trinity  College  in  1679,  and  held  the 
post  until  his  death*     • 

He  is  described  as  'Mow  in  statute^  lean,  and  of  a  pale 
eumplexion,''  slovenly  in  his  dress,  and  an  inveterate  smoker. 
He  was  noted  for  his  strength  and  courage^  and  once  when 
travelling  in  the  East  he  saved  the  ship  by  his  own  prowees 
f  mm  capture  by  pirates.  A  ready  and  caustic  wit  made  him 
a  favourite  of  Charles  II.,  and  induced  the  courtiers  to  respect 
even  if  they  did  not  appreciate  him.  He  wrote  with  a  sus- 
tained and  Homewliat  stately  eloquence,  and  with  his  blamelesB 
life  and  scrupulous  conscientiousness  was  an  impressive  per- 
sonage of  the  time. 

His  earliest  work  was  a  complete  edition  of  the  Elemettit 
of  Euclid  which  he  iMSUcd  in  1G55,  lie  published  an  English 
translation  in  1G60,  and  in  1G57  an  edition  of  the  Data.  His 
lectures,  delivered  in  1664,  1655,  and  1666,  were  published  in 
1 683  under  the  title  L^ctionet  MtUKeuMtlea^ :  these  are  mostly 
on  the  metaphysical  basis  for  mathematical  trutlis.  His 
lectures  for  1667  were  published  in  the  same  year,  and  suggest 
the  analysis  by  which  Archimedes  was  led  to  his  chief  results. 
In  1669  he  issued  his  Lectionet  Opiicaa  €i  Geauuiricae;  it  is 
said  in  the  preface  tliat  Newton  revised  and  correoted  these 
lectures  adding  matter  of  his  own,  but  it  seems  probable  from 
Newton's  remarks  in  the  ttuxional  controversy  that  the 
additions  were  contined  to  the  parts  which  dealt  with  optics : 
this,  which  is  his  most  important  work  in  mathematics,  was 
republished  with  a  few  minor  alterations  in  1674.  In  1675 
he  published  an  edition  with  numerous  comments  on  the  firet 
four  iKJoks  of  the  Conies  of  Apollonius,  and  of  the  extant 
works  of  Archimedes  and  Theodosius. 

In  the  optical  lectures  many  problems  connected  with  the 
reflexion  and  refraction  of  light  are  treated  with  ingenuity. 
The  geometrical  focus  of  a  point  seen  by  reflexion  or  refraction 


ItARROW. 


321 


is  defined ;  and  it  is  explained  that  the  image  of  an  object  is  the 
locos  of  the  geometrical  foci  of  every  point  on  it  Barrow  also 
worked  oat  a  few  of  the  easier  properties  of  thin  lenses; 
and  consideFablj  simplified  the  Cartesian  expUnation  of  the 
rainbow. 

The  geometrical  lectnres  contain  some  new  ways  of  deter- 
miniiig  the  areas  and  tangents  of  o^irves.  The  most  celebrated 
of  these  is  the  method  given  for  the  determination  of  tangents 
to  carves^  and  this  is  sufficiently  important  to  require  a  detailed 
notice  becaose  it  illustrates  the  way  in  which  Barrow,  Hadde^ 
and  Sluse  were  working  on  the  lines  suggested  by  Fermai 
towards  the  methods  of  the  differential  calculus.  Format  had 
observed  that  the  tangent  at  a  point  Pon  a  curve  was  determined 
if  one  other  point  besides  P  on  it  were  known ;  hence,  if  the 
length  oC  the  subtangent  i^r  could  be  found  (thus  determining 


the  point  1%  then  the  line  TP  would  be  the  required  tangent 
Now  Barrow  remarked  that  if  the  abscissa  and  ordinate  at  a 
point  Q  adjacent  to  /*  were  drawn,  he  got  a  small  triangle  PQR 
(which  he  called  the  differential  trianglci  because  its  sides  PS 
and  PQ  were  the  difierences  oC  the  abscissas  and  ordinates  oC 
P  and  9)b  io  thai 

TJfiMP^QRiXP. 

To  find  QR  i  RP  he  supposed  that  x^  y  were  the  eoordinates  of 
P^  and  m^%  p^m  those  oC  Q  (Barrow  aetuaUy  used  p  hxr  m 
and  M  for  y  but  I  alter  theee  to  agree  with  thd  modem 

a.  '  >\ 


dtt     MATHBIf  ATICS  r|U>ll  DUCAms  TO  HUTOmL 

pimctioe).' '  SubsliUting  thm  eoonUnatfls  of  Q  in  the  «|wilkai 
of  the  curve,  and  n^leeting  the  aquares  and  higher  powen  of 
€  and  a  as  oompared  with  their  Ant  powem,  he  obtained  •  : «. 
The  ratio  a/«  wan  sabaeqaently  (in  acooidanoe  with'  a  eogges* 
Uon  made  bj  Hloie)  termed  the  anguhw*  ooeflicient  oC  the 
tangent  at  Uie  point. 

Barrow  applied  thin  method  to  the  corvee  (i)  a^  ip^-^jTi^^'Vl 
(ii)  Q^^^  =  f^\  (iii)  3^'¥^  =  rxjff  called  la  jfo/and^;  (iv) 
y  »  (r  -  x)  tan  ^'2r^  the  quadrairix ;  and  (v)  y  =  r  tan  var/3n 
It  will  be  HUliicient  here  if  I  take  as  an  illustration  the  simpler 
case  uf  the  parabola  y'^s/Nc  Using  the  notation  given 
above,  we  have  for  the  point  /*,  f^^yx;  and  for  the  point 
P,  (y  -  a)*  =  |i  (x  -  e).  Subtracting  we  get  2ay  —  a*s  jie.  But^ 
if  a  be  an  infinitesimal  quantity,  a'  must  be  infinitely  smaller 
.  and  therefore  may  be  neglected  when  compared  with  the 
quantities  2iiy  and  pe.  Hence  Say  ~/>f,  tliat  ii,  e  :  a  =  2y  :  ^. 
Therefore  TM :  y  ^  «  :  a  =  2y  :  ;>.  Hence  TJf  =  2y*/|>  =^  2x. 
Tliis  is  exactly  the  procedure  of  the  differential  calculus, 
except  that  we  tliere  have  a  rule  by  which  we  can  get  the  ratio 
a/e  or  dy/dw  directly  without  the  labour  of  going  through 
a  calculation  similar  to  the  above  for  every  separate  case. 

Brouncker.  IF»//iaiii,  VUeouiU  Brow^ker^  one  of  the 
founders  of  the  Royal  Society  of  London,  bom  about  1620, 
and  died  on  April  5,  1684,  was  among  tlie  most  brilliant 
mathematicians  of  this  time,  and  was  in  intimate  relations 
with  Wallis,  Fermnt,  and  other  leading  mathematicians.  I 
mentioned  above  his  curious  reproduction  of  Brahmagupta's 
solution  of  a  certain  indeterminate  equation.  Brouncker 
proved  that  the  area  enclosed  between  the  equilateral 
hyperbola  xy  =  1,  the  axis  of  x,  and  the  ordinates  x=l  and 
X  =  2,  is  equal  either  to 

1  1  1  ,,111 

1.2*3.4*6.6'"      '^'^^-S^i-i*- 

He  also  worked  out  other  similar  expressions  for  different 
Mrema  bounded  by  the  hyperbola  and  straight  lines.     He  wrote 


1 


\ 


BBOUNCKER.  JAMES  GREGORT. 


323 


on  the  rectification  of  tlie  parabola  and  of  the  cycloid*.  It  is 
noticeable  that  he  aiied  infinite  aeries  to  expreiis  qoantitiea 
whose  values  he  could  not  otherwise  determine.  In  answer  to 
a  request  of  Wallis  to  attempt  the  quadrature  of  the  circle  he 
shewed  that  the  ratio  of  the  area  of  a  circle  to  the  area  of  the 
circumscrilied  square,  that  is,  the  ratio  v  :  4  is  equal  to  the 
ratio 

I      r     3*     6*     7* 


1  -1-2  4-  2  4-2  -1^  2  4.../ 


1. 


Continued  fractions!  had  been  introduced  by  Gataldi  in  his 
treatise  on  finding  the  sc|uare  ruotA  of  numliers,  publiKhed  at 
Bologna  in  1613,  but  Brouncker  seems  tf>  have  been  tlie 
earliest  writer  to  investigate  their  properties. 

James  Gregory.  Janvui  Grt^ory^  bom  at  Drumoak  near 
Alierdcen  in  1638,  and  die<l  at  Edinburgh  in  Octolier,  1675,  was 
successively  professor  at  St  Andrews  and  Edinlnirgh.  In  1660 
he  published  his  Optica  Pramoia^  in  which  the  reflecting 
telescope  known  by  his  name  is  described.  In  1667  he  issued 
his  Vera  Ciradi  et  Ilypfrbolati  QundrtUnra^  in  which  he  shewed 
how  the  areas  of  the  circle  and  hyperliola  could  be  obtained  in 
the  form  of  infinite  convergent  series,  and  here  (I  believe  for 
the  first  time)  we  find  a  distinction  drawn  between  convergent 
and  divergent  series.  This  work  contains  a  remarkable  geo- 
metrical pniposition  to  the  eficct  that  the  ratio  of  the  area  of 
any  arbitrary  sector  of  a  circle  to  that  of  the  inscribed  or 
circumscribed  regular  polygons  is  not  expressible  by  a  finite 
number  of  algebraical  terms.  Hence  he  inferred  that  the 
quadrature  of  a  circle  was  impossible :  this  was  accepted  by 
Montucla,  but  it  is  not  conclusive,  for  it  is  conceivable  that 
some  particular  sector  might  be  squared,  and  this  particular 
sector  might  be  the  whole  circle.     This  book  contains  also  the 

*  Ob  thess  iBvesiisftiioiiii,  sss  his  pspert  in  the  PhfkmpMemi  Trmm* 
meU^m,  London.  166S,  1672, 167S,  and  1S7S. 

t  Ob  lbs  hitloiy  oT  ooBliaaed  frsotions  sss  papefs  kj  8.  Ofinther  and 
A.  FSfaio  is  Booeomfspd*s  BmUeUm  Wllf^rn^s,  Boom,  lS7i«  ^«l. 
va,  ppw  SIS,  4SI,  SSS. 


i 


384     lUTUIMATlOB  raOM  DBBCABtB  TO|  HUTOim. 


Mriiosi  enaneMtioa  oC  the  expAukMia  in  aariM  of  do  a^  coi  j% 
■an**  m^  and  oai*>  a  It  was  reprinted  jn  1668  with  mi 
appendix,  Oeamuirum  Pam^  in  whidi  Oregofy  explained  kew 
the  vvdumea  of  aolids  of  revolntion  ooold  bei  determined.  In 
1671,  or  petiiapa  earlier,  lie  eatablialied  the  tjieorani  thai 

^»tan^-itan'^-^|tan^«^..., 

I 

the  roiult  being  true  only  if  ^  lie  between  —  \w  and  \w.  Thia 
is  the  tbeoreoi  on  which  many  of  the  aubaeqaent  caiculataona 
of  approxiniatioiia  to  the  numeral  value  of  v  have  been 
baaed. 

Wren.  Sir  CkruiapKtr  Wrtn  was  bom  at  Knoyle,  WQt- 
shire,  on  Oct  20,  1632,  and  died  in  London  on  Feb.  25,  172^ 
Wren's  reputation  as  a  mathematician  has  been  ovenhadowed 
by  his  fame  aa  an  architect^  but  he  waa  ^vilian  profeaaor 
of  aatninomy  at  Oxford  from  1661  to  1673,  and  for  some 
time  president  of  the  Royal  Society.  Together  with  Wallis 
and  Huygens  he  investigated  the  laws  of  collision  of  bodies ; 
he  also  discovered  the  two  systems  of  generating  lines  on  the 
hyperboloid  of  one  sheet,  though  it  is  probable  that  he 
confined  his  attention  to  a  hyperboloid  of  revolution*. 
Besides  these  he  wrote  papers  on  the  resistance  of  fluids,  and 
the  motion  of  the  pendulum.  He  was  a  friend  of  Newton 
and  (like  Huygens,  llooke,  H  alley,  aud  others)  had  made 
attempts  to  shew  that  the  force  under  which  the  planets  move 
varies  inversely  as  the  square  of  the  distance  from  the  sun. 

Wallis,  Brouiicker,  Wren,  and  Boyle  (the  last-named  being 
a  chemist  and  physicist  rather  than  a  mathematician)  were  the 
leading  philosophers  who  founded  the  Royal  Society  of  London. 
The  society  arose  from  the  self-styled  "  indivisible  college  "  in 
London  in  1C45 ;  most  of  its  members  moved  to  Oxford  during 
the  civil  war,  where  Hooke,  who  was  then  an  assistant  in 
Boyle's  laboratory,  joined  in  their  meetings ;  the  society  waa 
formally  constituted  in  London  in  1C60;  and  waa  incorporated 
on  July  15,  1662.     The  Accademia  dei  lancei  was  founded 

*  Bee  tbi:  PhUmopkkal  TrojiMiclicwf,  London,  1669. 


I 


HOOKE.      COLLIWS.      FELL  826 

tn  1603,  the  French  AcAdeni^  in  IG6$,  ud  the  Berlin 
AcuHeiny  in   1700. 

Rookfl.  /tolH-rt  Hook':,  hom  «t  Freshwater  on  Julj  18, 
1C35,  nnd  ilici  in  London  on  March  3,  1703,  was  educated  at 
Westminster,  nnd  Christ  Church,  Oxford,  and  in  1G65  became 
proffsmr  of  geometry  at  Gresham  College,  a  poat  which  he 
occupind  till  hin  death.  He  in  still  known  by  the  law  which 
hf>  diiicovered  tlint  the  ten-iion  exerted  hj  a  atretdwd  string  ia 
(within  certain  limitn)  proportional  to  the  extenMon,  or,  in 
other  wordR,  that  the  stn-ss  ia  pn>portional  to  .the  Htrain. 
He  invpnt«l  and  discusHcd  the  conic-.d  pendnlnm,  and  was  the 
tint  to  Htate  explicitly  that  the  motions  of  the  hearenly  bodies 
were  merely  dynamical  prohlem*.  He  wofi  aa  jealona  as  he  waa 
Tuin  and  irriuble,  and  accused  botli  Newton  and  Hnygena  t^ 
unfairly  appropriating  hin  resulbi.  Like  Hnjgens,  Wren,  and 
Halley  he  made  etTortn  to  find  the  law  of  force  nnder  whid) 
the  planets  more  about  the  sun,  and  he  believed  tite  law  to  be 
that  of  the  inverse  s>inare  of  the  distance.  He,  tike  Hnygena, 
discovered  that  the  small  oscillationx  of  a  cmled  spiral  spring 
were  practically  isoclirvnous,  and  was  thus  led  to  recommend 
(ponsihly  in  165)*)  the  use  of  the  Imlnnce-spfing  in  watchca;  he 
hod  a  watch  of  thin  kind  miule  in  London  In  1675,  it  was 
finished  junt  three  months  Inter  than  a  nimilar  one  made  in 
Fkris  nnder  the  directions  of  Huygens. 

Collins.  Jo/iH  Col/in*,  liom  near  Oxford  on  March  5, 
1635,  and  died  in  London  on  Nov.  10,  1683,  was  a  man  tii 
great  natural  ability  but  of  slight  education.  Beiugderoted  to 
mathematics  he  sprnt  hin  spare  time  in  corienpoudenoe  witb 
the  leading  mathematicians  of  the  time  for  whom  be  waa 
always  ready  to  do  anything  in  hin  power,  and  he  has  beea 
described— not  inaptly— as  the  English  Hersenne.  To  hin 
we  are  indebted  for  much  information  on  the  details  «l  the 
dincoreriei  of  the  period*. 

Pell.  Another  mathematician  who  devoted  a  enuodenUe 
■  Bw  Um  CwTfTJaw  BpiitpSemm.  and  S.  P.  B%mrn  OmrmpmdiHn 
^  SeknUJk  Mm  if  th*  Stttumuh  Cemlmrjf,  OifaiA,  IHL 


7 


MATHEIUTIOB  PROll  DBCAVnS  TO  HUTIim. 

part  of  his  itineto  making  known  the  diaeovoiiM  of  odMn^nnd 
to  eorreqiondenoe  with  leading  mathemnticinne  was  Mkm  PidL 
Pell  was  born  in  Sunez  on  Blarch  1/ 1610,  and  died  in  Tjondon 
on  Dec.  10,  IC85.  He  was  educated  at  THnity  Opilege, 
Cambridge ;  he  oocapied  in  mooeHsion  the  mathematical  chain 
at  Amsterdam  and  Breds ;  he  then  entered  the  English  diplo- 
matic aenrioe;  but  finally  settled  in  1661  in  London  where 
he  spent  the  last  twenty  years  of  his  life.  His  chief  works 
were  an  edition,  with  considerable  new  matter,  of  the  Alyebrm 
by  Branker  and  Rhonius,  London,  1668;  and  a  table  of  square 
numbers,  Loudon,  1672. 

Blase.  Xene  /Vtiitfow  Waliker  de  Sfute  (Sitmtu)^  canon  of 
Li^ge,  bom  on  July  7,  1622,  and  died  on  Blarch  19,  1685, 
found  for  the  subtangent  of  a  curve  y  (ob^  y)  »  0  an  expression 

which  is  equivalent  to  -  y  ;4^  /  /  ;  he  wrote  numerous  tracts*, 

and  in  particular  diflcussed  at  aome  length  spirals  and  points  oC 
inflexion. 

ViviaaL     Vitieenzo  Viviani^  a  pupil  of  (lalileo  and  Tor- 
ricelli,  1x>m  at  Florence  on  April  5,  1622,  and  died  there  on 
Sept  22,  1703,  brought  out  in  1659  a  restoration  of  the  lost 
book  of  Apollonius  on  conic  sections ;  and  in  1701  a  restora- 
r  tion  of  the  work  of  Aristaeus.     He  explained  in  1677  how  an 
i  anglo  could  be  trisected  by  the  aid  of  the  equilateral  hyperbola 
(or  the  conchoid.     In  1692  he  propuRed  the  problem  to  con- 
struct  four   windows   in  a  hemispherical  vault  so   that  the 
rpniAiiider  of  the  surface  can  be  accurately  determined  :  a 
celebrated  problem  of  which  analytical  solutions  were  given 
by  WalliH,  Leibnitz,  David  Gregory,  and  James  BemoullL 

TsohimhaUBeiL  Khreti/ried  Walther  van  Tm:himhauten 
was  born  at  Kinlingswalde  on  April  10,  1631,  and  died  at 
Dremlen  on  Oct.  1 1,  1708.  In  1682  he  worked  out  the  theoiy 
of  caustics  by  reflexion,  or,  as  they  were  usually  called,  cata- 
caustics,  and  shewed  that  they  were  rectiflable.     This  was  the 

*  Some  of  his  pspeni  were   pobliiihed  by  Le  Psige  in  vol.  svn  of 
BoaeompMgni'a  IfulUtimo  di  billioyraJUi^  B«fia«  \%^* 


\ 


ISCHtRNHADSEK.      DE   LA    HIRE.      ROEUElL     ROLLB.     S27 

aecond  case  in  which  the  envelope  of  n  moring  line  waa 
determined.  He  conHtnicted  burning  miiron  of  great  power. 
The  trftniifornuition  by  which  he  rpinuvnl  reitun  iDtOTnedwte 
temia  from  a  givpn  algebraical  ecjuation  if  well  known :  it 
WU  published  in  the  Aetii  Emdifomm  for  ll?83. 

Se  la  Hire.  n.7yi/«  IM  la  //.>  (or  lAtAirr),  bom  in  Pkria 
on  llareh  IS,  1640,  and  di«l  tbcre  im  April  31,  1719.  wrote 
on  graphicnl  inpthodii,  1C73  ;  on  the  conic  Mction*,  IGS5  ;  a 
treatise  on  epicycloida,  1694;  one  on  ronlettfa,  1703;  nnd 
lastly  another  on  concboi  Jn,  1 708.  His  work*  on  conic  acctKHW 
Mid  epicycloids  were  founded  on  the  teaching  of  Dtnargnen, 
whose  favourite  pupil  he  van.  He  al<M  traniUted  the  naayof 
Monchoputun  on  magic  nquarvs,  and  collected  many  of  the 
theorems  un  them  which  were  previously  known:  this  was 
pablinhed  iu   1705. 

Roemer.  Otof  Rormtr,  bom  at  Aurhnin  on  SepL  25, 
1644,  and  died  at  Cc^nhi^n  on  Sept.  19,  1710,  waa  the  first 
to  Bieanura  the  velocity  of  light :  this  wm  done  in  1670  by 
meann  of  the  eclipaea  of  Jnptter'i  satellites.  He  broagfat  the 
trannit  and  ninral  circle  into  common  un^  the  aUaamnth 
having  Inen  previously  generally  employed,  and  it  waN  on  his 
recommendation  that  aittronomical  obnervationa  of  stars  were 
mbnequently  made  in  general  on  the  meridian.  He  waa  also 
the  firnt  to  intmdnce  micrometera  and  rending  microacopea  into 
an  obaervatocj.  He  alao  deduced  from  the  propertiea  tA 
epicycloids  the  form  of  the  teeth  in  toothed-wheeh  heat  fitted 
to  aecure  a  nniform  motion. 

Rolla.  ^nfM  RolU.  1)om  at  Aml>ert  on  April  31,  IG-ll, 
and  died  in  Parin  on  Nov.  8,  1719,  wrote  an  algelira  in  1689 
which  contain!  the  theorem  on  the  poaitinn  of  the  iDote  of  an 
c<)aation  which  ia  known  by  bin  name.  He  paUiahed  in  1696 
atre»ttMon  theaulatinnii(Jr*liMtiDni,  whether  detrnninato  or 
indetenninate,  and  he  pndnced  aevenU  other  ninor  worln. 
He  taught  that  the  differential  caknlna,  which,  as  we  ahall  see 
later,  had  bnn  introdoced  towards  the  doae  of  the  seventeenth 
fleBtw7,  waa  nothing  bat  a  collection  of  ingunlona  fallarira 


828 


CHAPTER  XVL 

THX  UrS  AND  WORKS  OP  MlinOII^. 


Tbm  iniitliPfTintii*inTiff  oontiderad  in  tho  ImI  fhapjiHr 
menoed  the  creation  of  tlioae  proeeiea  which  dinlingQiah 
modem  matheniatice.  The  extraordinary  ahilitiea  of  Newton 
enaUed  him  within  a  few  years  to  perfect  the  more  elementary 
of  those  processes,  and  to  distinctly  advance  every  hranch  of 
mathematical  science  then  studied,  as  well  as  to  create  some 
new  subjects.     Newton  was  the  contemporary  and  friend  of  ! 

Wadlii,  Huygens,  and  others  of  those  mentioned  in  the  last 
cliapter,  but,  though  most  of  his  mathematical  work  was  done 
between  the  years  16G5  and  1G86,  the  bulk  of  it  was  not  li 

printed — at  any  rate  in  bouk-forni — till  some  years  later.  i 

1  propoNe  to  discuss  the  works  of  Newton  somewhat  more  j 

fully  than  those  of  other  mathematicians,  partly  because  of 
the  intrinsic  importiince  of  his  discoveries,  and  partly  because 
this  l)ook  is  nuiinly  intended  for  Englisli  readers  and  the 
development  of  matliematicM  in  Great  Britain  was  for  a 
century  entirely  in  the  hands  uf  the  Newtonian  schooL 

*  Newton  *ii  hfe  and  works  are  diioaiuuMl  in  Thi  Memoin  of  Seteiom^  by 
D.  Urewstor,  2  voluuies,  Edinburgh,  §econd  edition.  18C0.  An  edition  of 
most  of  Newton'n  works  was  publislied  by  8.  Horsley  in  5  Yolumet, 
London,  1771M(5;  and  a  bibliograpliy  of  them  was  issued  by  G.  J. 
Gray,  Cambridge,  ItisH;  sec  also  tlic  eataloKue  of  the  Portsmouth 
Collection  of  Newton*s  papers,  CaiubridKS,  1888.  My  Kua^  am  tAtf 
grnetUt  coulrMt*,  and  hitlury  uf  Sftrttm't  Priueipm^  London,  1893,  aMJ 
be  Mlto  consulted. 


I 


\ 


THE  LIFE  AND  WORKS  OF   NEWTON.  329 

Itane  Nrtrttin  was  liom  in  IJiicoln.ihire  tinr  Gnntham  on 
Dec.  25,  l<it2,  and  ctini  At  Kemington,  rjondon,on  M»rch  30, 
1727.  He  WM  fducAted  at  Trinity  Collp;^  Cttmbridgp,  Mid 
livpd  tliere  from  IfiGl  till  IG96  during  which  time  he  produced 
thn  balk  of  his  work  in  niathematics ;  in  1696  he  wiu  ap- 
point^ to  n  vnluftlde  Govemnient  office,  and  moved  to  London 
where  he  resided  till  hin  death. 

Hin  fnther,  who  h«d  died  shortly  before  Xewton  w»  born, 
was  A  jeoman  fanner,  and  it  wan  inlendnl  tltftt  Newton  ehovM 
carr7  on  the  pnteninl  farm.  He  was  lu-nt  to  Kchool  at  Qnuitham, 
where  hin  learning  and  nKwhanical  proficiency  excited  Aonie 
attention  ;  and  w*  one  instance  of  hi"  ingenuity  T  may  mention 
that  he  constructe<i  a  clock  worked  by  water  which  kept  veiy 
fair  time.  In  IG56  he  returned  home  to  learn  tlie  buHuiem  of 
a  farmer  under  the  Ruirfance  of  an  old  family  aen-ant.  Newton 
however  spent  moHt  of  hiR  time  aolving  prublema,  making 
experinienta,  or  deviting  mechanical  modelti ;  his  mother 
noticing  thia  twntilily  resolved  to  find  aome  moR  congenial 
occDpation  for  him,  and  his  uncle,  hiiving  been  himaelf 
educated  at  Trinity  Colle^,  CamlnHdge,  recommended  that 
he  should  Im  nent  there. 

In  1661  Newton  accordingly  entered  u  a  rabumr  at 
Trinity  College,  where  for  the  firxt  time  lie  fbnnd  himaelf 
among  Burronndinga  which  were  likely  to  develnpe  hii  powen^ 
He  aeema  however  to  have  had  but  little  interect  for  general 
aociety  or  fur  any  purttuita  Hare  ttcience  and  mathematioi,  and 
he  complained  to  his  frienda  that  he  found  tiie  other  nnder- 
gradnates  disorderly.  Luckily  he  kept  a  diary,  and  we  can 
thui  form  a  fair  iilcn  of  (he  counie  of  education  of  the  moat 
advanced  ttudenta  at  an  P^ngliah  nniversity  at  that  tiiiM  He 
had  not  read  any  mathematics  liefore  coming  into  residence, 
but  waa  acquainted  with  SanderNon'a  Iakjk,  which  was  then 
freijnently  read  aa  preliminary  to  mathemotica.  At  the  b»i 
ginning  of  his  firat  October  tenu  he  happened  to  atroll  down 
to  Stoarbrtdge  Fair,  and  there  picked  up  a  book  on  aatrokigj, 
but  oonld  not  undendnnd  it  on  acoonnt  of  tlM  poMnhc^  m& 


880  THI  UPI  AND  WOBU  OP  VBW10V. 

jkrigoiioiiMtrjr.  He  ih&ntdn  bought  a  Bndidt  and 
prised  to  And  how  obviom  the  proporitiope  ■eemeJ,  He  thire> 
upon  reed  Oagfatred's  Clava  end  Deioertee^e  OSomHrU^  the 
latter  of  which  he  managed  to  master  hj  himself^  thoQ|^  with 
some  difBcalty.  The  interest  he  felt  in  the  sabject  led  him  to 
take  up  mathematics  rather  than  chemistij  as  a  serious  study, 
HiM  suhnequent  mathematical  reading  as  an  undergraduato 
was  founded  on  Kepler*s  OpiicB^  the  works  of  Vietai  van 
Scliooten's  MiseeHanieg^  Deacartes's  GScmHrUf  and  Wallis*s 
Arithmeiiea  Infinitofum :  he  also  attended  Barrow's  lectures. 
At  a  later  time  on  reading  Euclid  more  carefully  he  formed  a 
high  opinion  of  it  as  an  instrument  of  education,  and  he  used 
to  express  his  regret  that  he  had  not  applied  himself  to 
geometry  before  proceeding  to  algebraic  analysis. 

There  is  a  manuscript  of  his,  dated  May  28,  1665,  written 
in  the  same  year  S3  that  in  which  he  took  his  B.A.  degree^ 
which  is  the  earliest  documentary  proof  of  his  invention  of 
fluxions.  It  was  about  the  same  time  that  he  discovered  the 
binomial  theorem^. 

On  account  of  the  plague  the  college  was  sent  down  during 
partM  of  the  years  1665  and  1666,  and  for  several  months  at 
this  time  Newton  lived  at  home.  This  period  was  crowded 
with  brilliant  discoveries.  He  thought  out  the  fundamental 
principles  of  his  theory  of  gravitation,  namely,  that  every 
particle  uf  matter  attracts  every  other  particle,  and  he  sus-  ] 

pected  that   the  attraction  varied   as  the  product  of  their  j 

iiia&ses  and  inversely  as  the  square  of  the  distance  between 
them.     He  also  worked  out  the  fluxional  calculus  tolerably  |1 

completely:  thus  in  a  manuscript  dated  Nov.  13,  1665,  he 
used  fluxitHis  to  find  the  tangent  and  tlie  radius  of  curvature 
at  any  point  on  a  curve,  and  in  October,  1666,  he  applied 
them  to  several  problems  in  the  theory  of  equations.  Newton 
communicated  these  results  to  his  friends  and  pupils  from  and 
after  1669,  but  they  were  not  published  in  print  till  many 
years  lateh     It  was  also  while  staying  at  home  at  tliis  time 


I 


i 


\ 


Newton's  views  on  OBAvrrr,  166(1. 


331 


thiit  be  devised  some  imtrompntfi  for  grinding  Imwes  to  [Htr- 
ticnl&r  fomiN  other  than  nphericnl,  and  perhn]>«  he  decoinpoMd 
•oUr  light  into  difTerrnt  colaun. 

LftBving  out  detailn  and  inking  roand  munbera  only,  bh 
reuaning  kt  this  time  on  tho  thpory  of  gnntation  iemm 
to  have  been  as  follows.  He  RUHprctnl  that  tbe  force  irhich  - 
retained  the  moon  in  itn  orbit  aliout  thr  earth  wa*  tbe  muiw  aa 
terraBtrial  grarity,  and  to  verify  thin  hypotheaia  be  proeeeded 
thns.  He  knew  that,  if  a  utone  were  allowed  to  fall  near  tb* 
■orfaoa  of  the  e*rth.  tbe  attnwtion  of  tbe  earth  (that  is,  tha 
woight  of  the  Btooe)  caiued  it  to  move  thnmgh  16  feet  in 


one  necond.  The  moon'ii  orbit  relative  to  the  fl«rth  in  nearly  a 
circle ;  and  aa  a  roDf;h  approximation,  taking  it  to  be  ao,  he 
kn«w  the  dintanoe  of  the  moon,  and  tlicrrfore  the  length  of  it« 
path  ;  be  also  knew  the  time  the  moon  took  ttt  go  once  round 
it,  namely,  a  month.  Hence  he  could  enstly  find  ita  Telocity  at 
any  pnint  toch  as  M.  He  could  thpivfore  doA  tbe  dintancn 
MT  through  which  it  wnnld  move  in  tbe  next  necond  if  it 
were  not  palled  hy  the  earth'*  attraction.  At  the  end  of  that 
second  it  was  however  at  M',  and  tlicrefore  the  earth  mnat 
have  pulled  it  through  tbe  distance  TM'  in  one  second 
(amnming  the  direction  of  the  earth's  pull  (O  he  eonstant). 
Now  he  and  several  (Aysicintn  of  the  time  had  eaqjoetand 
boin  Kepler's  third  Uw  that  tbe  attraction  %4  tbe  aHth  m  • 


382  THB  UrS  AMD  WOSKB  OW  NIWTM. 


body  woald  be  fomid  to  deereoie  m  ibo  bodj 
tuiiier  away  from  the  oartb  in  a  proportioii  inTonofy  as  ibo 
•qiiaro  oC  the  diaUnee  from  tbe  oentie  of  tbe  earth* ;  if  tbk 
were  tbe  actual  law  aud  if  gravity  were  the  eole  loroe  wbaeh 
retained  the  moon  in  ita  orbits  then  TM'  ehoold  be  to  16  Caet 
in  a  proportion  which  was  invendy  as  tbe  aqnare  of  tbe 
distance  of  Uie  moon  from  the  centre'  of  the  earth  to  tbe 
•quara  of  the  radios  of  the  earth.  In  1679,  when  he  repeated 
the  investigation,  TJi'  was  foand  to  have  the  valne  which  was 
required  by  the  hypothesiii  aud  the  verification  was  complete; 
but  in  16G6  his  estimate  of  the  distance  of  the  moon  was 
inaccurate,  and  when  he  made  the  calcuUUon  he  found  that 
TJ/'  was  about  one-eighth  less  than  it  ought  to  have  been  on 
his  hypothesis. 

This  discrepsncy  does  not  seem  to  have  shaken  his  faith  in 
the  belief  that  gravity  extended  as  far  as  the  moon  and 
varied  invensely  as  the  square  of  the  distance;  but^  from 
Whinton's  notes  of  a  conversation  with  Newton,  it  would 
seem  that  Newton  inferred  that  some  other  force — probably 
Descartes's  vortices — acted  on  the  moon  as  well  as  gravity. 
This  statement  is  confirmed  by  Pemberton's  account  of  tlie 
investigation.  It  seems  moreover  that  Newton  already  be- 
lievetl  finiily  in  the  principle  of  universal  gravitation,  that  i^ 
that  every  particle  of  matter  attracts  every  other  particle^  and 
suMpected  that  the  attraction  varied  as  the  product  of  their 
masses  and  inversely  as  the  square  of  the  distance  between 
them :  but  it  is  certain  that  he  did  not  then  know  what  tlie 
attraction  of  a  spherical  mass  on  any  external  point  would  . 

be,  and   did  not  think  it  likely  that   a  particle  would   he  I 

attracted  by  the  earth  as  if  the  latter  were  concentrated  into 
a  single  particle  at  its  centre. 

On  his  return  to  Cambridge  in  1G67  Newton  was  elected 
to  a  fellowship  at  his  college,  and  permanently  took  up  his 
residence  there.  In  the  early  part  of  1669,  or  perhaps  in 
1668,  he  revised  Barrow's  lectures  for  him.     The  end  of  the 

*  Aa  Mrgameni  leading  to  this  resoU  is  giYen  bek>w  on  p.  Ml. 


\ 


THE   Urit   AKD  W0KK8  OF   NEKTON.  333 

tonrt«enth  iMtare  in  known  to  hnve  lircn  written  liy  Newton, 
bat  how  mDch  of  the  rent  is  due  to  hU  nuggeKtiuns  cniinot  now 
be  determined.  As  noon  m  this  wa.i  fininliol  ha  wna  miikal  by 
Bmtuw  and  Collins  tu  edit  Knd  add  nottv  tii  ■  tronnlAtion  of 
Kinckhnpien's  Afgrhra ;  he  consented  to  do  this,  but  on 
condition  thnt  his  iinnie  !t)iould  not  nppenr  in  the  nintbT.  In 
1670  he  bIw  Itegnn  a  syateinntic  exposition  of  hif  nnalyniH  )tj 
infinite  serins  '•'•c  object  of  which  wiia  to  exprew  the  nnliimto 
of  K  cane  in  nn  intinilo  nlf^cbraiciU  xerics  every  lenn  of  which 
conid  be  int^-gmteii  by  Wallis'i  rule,  hin  nwult*  on  thin  Hubjoct 
htul  been  conimunicKtrd  to  Bnirow,  GiliinM,  and  utben  in 
1605.  Tliii  WM  never  finiJihcd :  tiie  fnigment  whk  publtHlird 
in  1711,  hut  the  xubHtAnce  of  it  hnd  hivn  priiil^^l  rut  iin 
appendix  to  the  Optie*  in  1704.  These  works  were  only  the 
fruit  of  Newton's  leisure ;  most  of  Iuh  time  during  these  two 
yearn  being  given  op  to  optical  rescArches. 

In  October,  1669,  Barrow  resigned  the  Lucnsian  chair  in 
favour  of  Newton.  During  his  tenure  of  the  pmffvisnrHhip, 
it  waa  Newton's  practice  to  lecture  publicly  once  a  week,  for 
from  half-an-hour  to  an  hoor  at  a  time,  in  one  tcnn  of  each 
year,  probably  dictating  his  lectures  nn  rapidly  as  they  could 
be  taken  down ;  and  in  the  week  following  the  lecture  to 
devote  four  hours  to  appointments  which  he  gave  to  stndcnta 
who  wished  to  conie  to  hid  rooms  to  discuss  the  results  of  the 
previous  lecture.  He  never  repeat«l  a  courHO,  which  uHUnlly 
ounsiBte<l  of  nine  or  t^-n  lectures,  and  geiiemlly  the  lecturen  of 
one  coun«  began  from  the  point  at  which  the  preceding courao 
had  ended.  The  manuscripts  of  his  lectures  for  seventeen  out 
of  the  first  eighteen  years  of  his  tenure  are  entnnt. 

Wlien  first  appointed  Newton  chose  optics  for  the  subject 
of  bia  lectures  and  reneardies,  and  before  the  end  of  16G9  he 
had  worked  out  the  detAils  of  his  discovery  <if  the  decom- 
poaitioB  of  a  imy  of  white  light  into  rays  of  different  colonre 
«  of  a  pram.  The  complete  explanation  of  the  theory 
r  followed  fivm  this  diaooveij.  TbeM  discoveries 
I  U«  Mbjeokmatter  of  the  lecMiM  which  be  delivend 


8S4  .     TUB  UR  AND  WOlUCtt  OT  mBWTOM. 

M  TjumMiin  iMulenor  in  the  ymn  1M9,  1670,  ami  1671»  Hm 
chief  new  rernilU  wera  embodiwi  in  a  paper  eoniniwuealed 
to  the  Bojal  Society  in  Febmarja  1672,  and  enheeqaently 
paUiahed  in  the  PkUo§oykieal  TruBmuiioHt,  The  mannicripi 
of  his  original  lecturee  waa  printed  in  1729  under  the  title 
LediuneM  Opiieae,  This  woric  is  divided  into  two  book%  the 
first  of  which  contains  four  sections  and  the  second  fivei 
Tlie  first  section  of  tlie  first  book  deals  with  the  decomposition 
of  solar  light  by  a  prism  in  conseciuence  of  the  unequal  re- 
fraugibility  of  the  rays  that  compose  it^  and  a  description 
of  his  experiments  is  added.  The  Hocoiid|  section  contains  an 
account  of  the  metlKMl  which  Newtun  in\'euted  for  tlie  deter* 
mining  tliu  cueificieiits  of  refraction  of  diflerunt  bodies.  This 
is  done  by  making  a  ray  pans  through  a  prism  of  the  material 
so  Uiat  the  deviation  is  a  minimum ;  and  he  proves  that^  if  tlie 
angle  of  tho  prisiii  be  %  and  the  deviation  of  the  ray  bo  ^  the 
ntfnictive index  will  bo  sin  |(t -i- £) cosec  Jt.  The  third  section 
is  t»n  refractions  at  plane  surfaces ;  he  here  sliews  tliat  if  a  ray 
paiis  through  a  prism  with  minimum  deviation,  the  angle  of 
inciileiice  is  equal  to  the  angle  of  emergence — most  of  this 
section  is  devoted  to  geometrical  solutions  of  different  problems. 
The  fourth  Hection  contains  a  discussion  of  refractions  at  curved 
surfaces.  Tlie  second  book  treats  of  his  theory  of  colours  and 
of  the  rainbow.  | 

By  a  curious  chapter  of  accidents  Newton  failed  to  correct 
the  chromatic  aberration  of  two  colours  by  means  of  a  couple 
of  prisms.  He  therefore  abandoned  the  hope  of  making  a 
refracting  telescope  which  should  bo  achr>niatic,  and  instead 
designed  a  reflecting  telescope,  probably  on  the  model  of  a 
small  one  which  he  had  made  in  1GG8.  The  form  he  used  is 
that  still  known  by  his  name;  the  idea  of  it  was  naturally 
suggested  by  Gregory^s  telescope.  In  1672  he  invented  a 
reflecting  microscope,  and  soine  years  later  he  invented  the 
sextant  which  was  re-discovered  by  J.  Hadley  in  1731. 

His  professorial  lectures  from  1673  to  1683  were  on 
algebra  and  the  theory  of  equations,  and  are  described  below ; 

\ 


\ 


NEWTON'S  VIEWS  ON  OPTICS,   1609-1675.  335 


but  much  of  hin  time  daring  Uicho  years  war  occapicd  with 
other  investigations,  and  I  may  remark  that  througliout  his 
life  Newton  must  have  devoted  at  least  as  much  attention  to 
chemistry  and  theology  as  to  mathematics,  though  his  conclu- 
sions are  not  of  sufficient  interest  to  require  mention  here. 
His  theory  of  colours  and  his  deductions  from  his  optical 
experiments  were  attacked  with  amsidoralilc  vehemence  by 
Pardies  in  France,  Linus  and .  Lucas  at  Liege,  Huokc  in 
England,  and  Huygens  in  Paris;  hut  his  opponents  were 
finally  refuted.  The  correspondence  which  this  entaih^  on 
Newton  occupied  nearly  all  his  leisure  in  the^ears  1672  to 
1675,  and  proved  extremely  diHtanteful  to  him.  Writing  on 
Dec.  9,  1675,  he  says,  "I  was  so  persecuted  with  discussions 
arising  out  of  my  theory  of  liglit,  that  1  blamed  my  own 
imprudence  for  parting  with  so  sulistantial  a  blessing  as  my 
quiet  to  run  after  a  shadow."  Again  on  Nov.  18,  1676,  he 
observes,  '*!  see  I  have  made  myself  a  slave  to  philosophy; 
but,  if  I  get  rid  of  Mr  Linus's  Imsiness,  I  will  resolutely  bid 
adieu  to  it  eternally,  excepting  what  I  do  for  my  private  satis- 
faction, or  leave  to  come  out  after  me ;  for  I  see  a  man  must 
either  resolve  to  put  out  nothing  new,  or  to  become  a  slave  to 
defend  it."  The  unreasonable  dislike  to  have  his  conclusions 
doubted  or  to  be  involved  in  any  correspondence  about  them 
was  a  prominent  thut  in  Newton's  character. 

Newton  was  deeply  interested  in  the  question  as  to  how 
the  effects  of  light  were,  rpally  produced,  and  by  the  end  of 
1675  he  had  worked  out  the  corpuscular  or  emission  theory 
— a  theoiy  to  which  he  was  fierhaps  led  by  his  researches  on 
the  problem  of  attraction.  Only  three  ways  have  been  sug- 
gested in  which  light  can  be  produced  mechanically.  Either 
the  eye  may  be  supposed  to  send  out  something  which,  so  to 
speak,  feels  the  object  (as  the  Greeks  believed) ;  or  the  object 
peroeired  may  send  out  something  which  hits  or  affects  the 
eye  (aa  assumed  in  the  emisnon  theoiy);  or  there  may  be 
■ome  medium  between  the  ey«  And  the  dgect,  and  the 
may  oanse  ■ome  change  in  the  form  or  oooditiKMaL  ^ 
iDlorveniiig  medium  and  th«a  aSeei  ^bia  v|^  \9«k  ^<i^s»  ^^^ 


886  THB  uwm  AMi>  woRKa  or  Hiwroir, 

HujgBDft  miiipoiied  in  tlie  wave  or  yndwlatoty  tiMOiyX  I^ 
will  be  eoongh  here  lo  my  thai  on  either  of  the  two  hitler 
theories  all  the  ob^-ioiiii  phenomena  of  geomelrioal  option  eneh 
aa  reflexioo,  relrection«  dre.,  can  be  acooonted  far.  Within 
the  preient  oeutury  crucial  experiments  have  been  devised 
which  give  different  results  according  as  one  or  the  other 
theory  is  adopted ;  ail  these  experiments  agree  with  the  re- 
sults of  the  uiidulatory  theory  and  differ  from  the  results  of 
tlie  Newtonian  theoiy :  the  latter  is  therefore  untenable. 
Until  however  the  theory  of  interference,  suggested  by 
Young,  was  *  worked  out  by  Fr&inel,  the  hypothesii  of 
Huygens  failed  to  account  for  all  the  facts,  and  even  now 
the  properties  which,  under  it,  have  to  be  attributed  to  the 
intervening  medium  or  ether  involve  difficulties  of  which  we 
still  seek  a  solution.  Hence  the  problem  as  to  how  the  effects 
of  light  are  really  produced  cannot  be  said  to  be  finally 
solved.  It  should  be  noted  that  Newton  nowhere  expresses 
an  opinion  that  the  corpuscular  theory  is  true,  but  always 
treats  it  as  an  hypothesis  from  which,  if  true,  certain  results 
would  follow.  It  would  moreover  seem  that  he  believed  the 
wave  theory  to  be  iotrinsically  more  probable,  and  it  was  only 
the  difficulty  of  explaining  diffraction  on  that  theory  that  led 
him  to  reject  it :  this  difficulty  was  removed  by  FresneL 

Newton's  corpuscular  theory  was  expounded  in  memoirs 
couimuuicated  to  the  lloyal  Society  io  December,  1G75,  which 
are  suUitaiitially  reproduced  in  lus  Opiic*^  published  in  1704. 
In  the  latter  work  he  dealt  in  detail  with  his  theory  of  fits  of 
easy  reflexion  and  transmisnion,  and  the  colours  of  thin  plates, 
to  which  he  added  an  explanation  of  the  colours  of  thick 
plates  [lik.  11,  part  4]  and  oljscrvatious  on  the  inflexion  of 
light  [bk.  111]. 

Two  letters  written  by  Newton  in  the  year  1G76  are 
sufficiently  interesting  to  justify  an  allusion  to  them.  Leibniti^ 
who  liad  been  in  London  in  1673,  had  communicated  some 
results  to  the  Royal  Society  which  he  had  supposed  to  bo  new, 
but  which  it  was  pointed  out  to  him  had  been  previously  proved 
by  JUoutoii,     This  led  to  a  correspondence  with  Oldenburg^ 


\ 


NEWTON  ON  EXPRESSIONS  IN  SBRIEK,  1676.       337 

the  aecreiary  of  the  Society.  In  1674  Leibniti  wrote  mjing 
that  he  ponesaed  **  general  analytical  methods  depending  on 
infinite  series."  Oldenburg  in  reply  told  him  that  Newton 
and  Gregory  had  niied  such  series  in  their  work.  In  answer 
to  a  reifuest  for  information  Newton  wrote  on  June  13,  1676, 
giving  a  brief  account  of  his  method,  Imt  adding  the  expan- 
sions of  a  binomial  (that  is,  the  binomial  theorem)  and  of 
nn^'a;;  from  the  latter  of  which  he  deduced  that  of  sinae^ 
this  seems  to  be  the  earliest  known  instance  of  a  reversion  of 
series.  He  also  inserted  an  expression  for  the  rectification  of 
an  elliptic  arc  in  an  infinite  series. 

Leibnits  wrote  on  Aug.  27  anking  for  fuller  details ;  and 
Neirton  in  a  long  but  interesting  reply,  dated  Oct.  24,  1676, 
and  sent  through  OldenlHirg,  gives  an  account  of  the  way  in 
which  he  had  lieen  led  to  some  of  his  results. 

In  this  letter,  Newton  begins  by  saying  that  altogetlier  he 
had  used  three  methods  for  expansion  in  series.  His  first  was 
arrived  at  from  the  study  of  the  method  of  interpolation  by 
which  Wallis  had  found  expressions  for  the  area  of  a  circle 
and  a  hyperbola.   Thus,  by  considering  the  series  of  expressions 

(1  -a^*,  (1  -OE*)!,  (1  -x')^  ...,  he  deduced  by  interpolations  the 
law  which  connects  the  successive  coefficients  in  the  expansions 

of  (1  -a*)*,  (1  -as*)',  ... ;  and  then  by  analogy  obtained  the 
expression  for  the  general  term  in  the  expansion  of  a  binomial, 
tliat  is,  the  binomial  theorem.  He  says  tliat  he  proceeded  to  test 

this  by  forming  the  square  of  the  expansion  of  (1  ^sfy  which 
reduced  to  1  -  a^ ;  and  he  proceeded  in  a  similar  way  with 
other  ezpansionsL     He  next  tested  the  theorem  in  the  case 

of  (1-a^'  by  extracting  the  square  root  of  l-a^9  muire 
ariihmeiieo.  He  also  used  the  series  to  determine  the  areas  of 
the  drde  and  the  hyperbola  in  infinite  series,  and  found  that 
the  resolta  were  the  same  as  those  he  iiad  arrived  at  by  other 


Having  estabUshed  this  rstolti  he  then  diaoarded   the 
■Mlhod  of  interpolalton  in  aeries,  and  enpkgred  his  fainoimal 


838 


THE  UR  AND  WORKS  OW  mCWTOV. 


theorem  lo  eifMVM  (when  pOMible)  the  onUnate  of  m  owe  fa 
an  infinite  teriee  in  aaoending  power*  of  the  ■herfwi,  niid  Umm 
by  Walliii'e  method  he  obtained  expreeiioni  in  an  ialaite 
■eries  for  the  areas  and  arcii  of  oorves  in  the  manner  deeeribed 
in  the  appendix  to  hia  Opiic$  and  his  Ih  Amafyd  pmr 
E^uatioHeM  Xumero  Ttrmimorum  iMfiniiag*.  He  atatee  that 
he  had  employed  this  second  method  before  the  pbgoe  in 
1665-66,  and  goes  on  to  say  Uiat  he  was  then  obliged  to 
leave  Cambridge,  and  subseqaently  (presumably  on  his  retom 
to  Cambridge)  he  ceased  to  pursue  these  ideas  as  he  found 
tliat  Nicholas  Mercator  had  employed  some  of  them  in  his 
Loyarithmo-^tckniea^  published  in  1668 ;  and  he  supposed  that 
the  remainder  had  been  or  would  be  found  out  before  he 
hiuiHclf  was  likely  to  publish  lus  discoTories. 

Newton  next  explains  tliat  he  had  also  a  third  metliod,  of 
which  (he  Hays)  he  had  aliout  1669  sent  an  account  to  Barrow 
and  Collins,  illuHt rated  by  applications  to  areas,  rectification, 
cultature,  «l'c.  This  was  the  method  of  fluxions;  but  Newton 
gives  no  deKi^riptioii  of  it  hon%  though  he  addn  some  illustrations 
of  iu  UH\  The  first  illuKtratioii  is  on  Uie  quadrature  of  the 
curve  represented  by  the  equation 

y  =  tuT  (6  4-  ca:")', 

which  he  says  can  l)e  efl*ected  as  a  sum  of  (m-k-  \)ln  terms  if 
(lit  +  1  )/ii  be  a  poAitive  integer,  and  which  he  thinks  cannot 
otherv/iRo  be  effected  except  by  an  infinite  series  f.  He  also 
gives  a  list  of  other  forms,  which  are  immediately  integraUe, 
of  which  the  chief  are 

TL       -*.•    TJa 5i»   a--> (a  ♦&«•  +  «*)*». 

a  +  6x*  +  cx*"     a +  64*  + ex**'  ^  ' 

«-"-» (a  +  &r-)^l  (c  +  €iLi:-)->,   a*-"""  (a  +  &je»)  (<?  +  d!a*)H; 

where  m  is  a  positive  integer  and  n  is  any  number  whatever. 

*  See  KnOow,  pp.  353.  S5S. 

t  Thin  ii  not  po^  the  integration  is  possible  if  p+(Bi  +  l)/s  bs  an 


1 


OORRESPONDENCB  WITH  LEIBNITZ,  1676-1677.       339 

Lttfitly  he  points  out  that  the  area  of  any  carve  can  be  eauilj 
determined  approximately  by  the  metliod  of  interpolation 
deiicribed  below  in  discuming  hifi  Xfethodti*  Differ^niiaiU, 

At  the  end  of  his  letter  Newton  alludes  to  the  solution  of 
the  "inverse  problem  of  tangents,"  a  subject  on  which  LeilmiU 
had  asked  for  information.  He  gives  formulae  for  reversing 
any  series,  but  says  that  besides  these  formulae  he  has  two 
methods  for  solving  such  questions  which  for  the  present  he 
wilt  not  descrilie  except  by  an  anagram  m-hich  being  read  is 
as  follows,  **Una  methodus  consistit  in  extractione  fluentis 
quantitatis  ex  aequatione  simul  involvente  fluxionem  ejus: 
altera  tantum  in  assumptione  seriei  pro  quantitate  qualibet 
incognita  ex  qua  caetera  commode  derivari  possunt^  et  in 
oollatione  terroinorum  homologorum  aequationis  resnltantia, 
ad  eruendos  terminos  assuniptae  seriei." 

He  implies  in  this  letter  that  he  is  worried  by  the 
questions  he  is  asked  and  the  controversies  mised  about 
every  new  matter  which  he  produces,  which  shew  his  rashnest 
in  publishing  **quod  umbram  captando  eatenus  perdideram 
quietem  meam,  rem  prorsus  substantialem." 

Leibnits,  in  his  answer,  dated  June  21,  1677,  explains  his 
method  of  drawing  tangents  to  curves,  which  he  says  proceeds 
**  not  by  fluxions  of  lines  but  by  the  differences  of  numbers  ** ; 
and  he  introduces  his  notation  of  dx  and  dy  for  the  infini- 
tesimal differences  betweeh  the  coordinates  of  two  consecutive 
points  on  a  curve.  He  also  gives  a  solution  of  the  problem  to 
find  a  curve  whose  subtangent  is  constant,  which  shews  that 
ha  could  integrate. 

In  1679  Hook^  at  the  request  of  the  Royal  Society,  wrote 
to  Newtoo  expressing  a  hope  that  he  would  make  further 
communications  to  the  Society  and  informing  him  of  various 
fmdB  then  recently  discovered.  Newton  replied  saying  that 
he  had  abandoned  the  study  of  philosophy,  but  he  added  that 
the  earth's  diurnal  motioa  might  be  proted  by  the  experiment 
of  obwifing  the  deviation  from  the  perpeiMUealar  of  a  tioiie 
dropped  from  m  heif^i  to  the  gnwDd— aa  experiment  whiek 


I 


.840  THE  UR  AMD  WOEK8  OW  KIWTOV. 


WM  MifanqoeoUj  niade  by  tlie  Sodety  ami  ■ucoeeded.  Hoolw 
in  kia  letter  mentioiied  Pfcanl'i  geodetical  reaearalMs;  in 
these  Picard  uaed  a  value  of  the  radioB  of  the  earth  whieh  k 
■ubstantially  correct  This  led  Newton  to  repeat,  with  Picanfi 
daU|  hia  calculations  of  16fi6  on  the  lunar  orbit^  and  he  thoa 
verified  his  supposition  that  gravity  extended  as  far  as  the 
nxx>n  and  varied  inversely  as  the  square  of  the  diitancei  He 
then  proceeded  to  consider  the  general  theory  of  motion  under  ] 

a  centripetal  force,  and  demonstrated  (i)  the  equable  descrip- 
tion of  areas,  (ii)  that  if  an  ellipse  were  described  about  a 
focus  under  a  centripetal  force  the  law  was  that  of  the  inverse 
square  of  the  distance,  (iii)  and  conversely,  that  the  orbit  of 
a  particle  projected  under  the  influence  of  such  a  force  was  a 
conic  (or,  it  may  be,  he  thought  only  an  ellipse).  Obeying 
his  rule  to  publiMh  nothing  which  could  land  him  in  a  scien- 
titic  coiitruvefMy  these  reHults  were  locked  up  in  his  note- 
books, and  it  was  only  a  Kpecitic  question  addressed  to  him  { 
f^ve  years  later  tliat  led  to  their  publication. 

The  UnivertuU  ArUhnmtie^  which  is  on  algebra,  tlieory  of 
equations,  and  miscellaneous  problems,  contains  the  substance 
of  Newton's  lectures  during  the  years  1G73  to  1683.  His 
manuscript  of  it  is  still  extant;  Whiston*  extracted  a  some- 
what reluctant  permission  from  Newton  to  print  it,  and  it  was 
published  in  1 707.  Amongst  several  new  theorems  on  various 
points  in  algebra  and  the  theory  of  equations  Newton  here 
enunciates  the  following  iinportjint  results.  He  explains  that 
the  equation  whose  roots  are  the  solution  of  a  given  problem 
will  liave  an  many  roots  as  there  are  different  possible  cases ; 
and  he  considers  how  it  happens  that  the  equation  to  which 

•  William  WkittoH,  born  in  Leioestershire  on  Dec.  9,  1067,  edacatod 
At  Clare  CoUetse,  Csiu  bridge,  of  which  society  he  was  s  fellow,  and  died 
in  London  on  Aug.  22,  1752,  wrole  Heveral  works  on  astronomj.  He 
acted  as  Newton'n  deputy  iu  the  Lucasian  chair  from  1699,  and  in  170S 
tuooeeded  him  as  profesiior,  bat  he  was  expelled  in  1711,  mainly  for 
theological  reasons.  He  was  sooceeded  by  Nioholas  Saanderson,  the 
bUnd  mathematician,  who  was  bom  iu  Yorkshire  in  1682  and  died  al 
CiiriMi'a  College,  Cambridge,  on  ^v^iil  Vl«  Vl^'i. 


NEWT0N*8  LECTURES  ON   ALGEBRA,  1G73-1683.     Ml 


a  problem  lead^  may  contnin  roots  which  do  not  satiiify  the 
original  question.  He  extends  Descartes's  mle  of  signs  to 
give  limits  to  the  number  of  imaginary  roots.  He  uses  the 
principle  of  continuity  to  explain  how  two  real  and  unequal 
roots  may  become  imaginary  in  passing  through  equality, 
and  illustrates  this  by  geometrical  considerations;  thence 
he  shews  that  imaginary  roots  must  occur  in  pairs.  Newton 
also  here  gives  rules  to  find  a  superior  limit  to  the  positive 
roots  of  a  numerical  equation,  and  to  determine  the  approxi- 
mate values  of  the  numerical  root&  He  further  enunciates 
the  theorem  known  by  his  name  for  finding  the  sum  of  the  nth 
powers  of  the  roots  of  an  equation,  and  laid  the  foundation  of 
the  theory  of  symmetrical  functions  of  the  roots  of  an  equation. 
The  n^ost  interesting  theorem  contained  in  the  work  is 
his  attempt  to  find  a  rule  (analogous  to  that  of  Descartes  for 
real  roots)  by  which  the  number  of  imaginary  roots  of  an 
equation  can  be  determined.  He  knew  that  the  result  which 
he  olitaincd  was  not  universally  true,  but  he  gave  no  proof  and 
did  not  explain  what  were  the  exceptions  to  Uie  rule.  His 
theorem  is  as  follows.  Suppose  the  equation  to  be  of  the  nth 
degree  arranged  in  descending  powers  of  x  (the  coeflk-ient  of 
^  being  positive),  and  suppose  the  n  -i- 1  fractions 


1. 


n     2     n  -  1  3 
n-.\  V   n-22'**' 


2 


•  -'in-r 


1 


to  bo  formed  and  written  below  the  corresponding  terms  of 
the  equation,  then,  if  tlie  square  of  any  tenn  when  multiplied 
by  the  corresponding  fraction  bo  greater  than  the  product  of 
the  terms  on  each  side  of  it,  put  a  plus  sign  above  it :  other- 
wise put  a  minus  sign  above  it,. and  put  a  plus  sign  above 
the  first  and  last  terma  Now  consider  any  two  consecutive 
terms  in  the  original  equation,  and  the  two  symbols  written 
above  them.  Then  we  may  have  any  one  of  the  four  toUowin|r 
:  (a)  the  terms  of  the  same  rign  and  the  synbob  of  the 
sign ;  (/i)  the  terms  of  the  same  sign  and  the  symbob 
of  opposiie  signs;  (y)  the  terms  of  opposite  signs  and  Uub 


i 


342  THB  LlfK  AND  WORKH  OT  lllWtOV. 

•yiubob  of  the  mna  sign;  (t)  the  tenui  of  opporite 
and  the  sjinbola  of  oppoHile  tigiiM.  Then  ii  htm  been  ehewn 
that  the  number  of  negative  ruotii  will  not  exceed  the  nnaher 
of  ceMci  (e)y  and  the  number  of  poMitive  rootii  will  not  exceed 
the  number  of  canes  (y) ;  and  therefore  the  number  of  imagi* 
nary  roots  is  not  less  than  the  number  of  cases  (fi)  and  (t). 
In  other  words  the  number  of  changes  of  signs  in  the  row  of 
symbols  written  above  the  equation  is  an  inferior  limit  to  the 
number  of  imaginary  ruots.  Newton  however  asserted  that 
''you  may  almost  know  how  many  roots  are  impossible**  by 
counting  the  clianges  of  sign  in  the  series  of  symbols  formed 
Hs  above.  That  is  to  say  he  thought  that  in  general  the 
actual  number  of  poHitive,  negative  and  imaginary  roots  could 
bo  got  by  the  rule  and  not  merely  superior  or  inferior  limits 
to  these  numbers.  But  though  he  knew  that  the  rule  was 
not  universal  he  could  not  find  (or  at  any  rate  did  not  state) 
wluit  were  the  exceptions  to  it:  this  theorem  was  subse- 
quently discuHiiod  by  Campbell,  Maclaurin,  Euler,  and  other 
writers;  at  last  in  18G5  Sylvester  succeeded  in  proving  the 
general  result*. 

In  August,  1684,  Halley  came  to  Cambridge  in  order  to 
consult  Newtou  about  the  law  of  gravitation.  Hooke,  Huygens, 
llalley,  and  Wren  had  all  conjectured  that  the  force  of  the 
attraction  of  the  sun  or  earth  on  an  external  particle  varied 
inversely  as  the  S4|uare  of  the  distance.    These  writers  seem  to  i 

have  independently  shewn  that,  if  Kepler's  conclusions  were 
rigorously  true,  as  to  which  they  were  not  quite  certain,  the 
law  of  attraction  must  be  that  of  the  inverse  square.  Prob- 
ably their  argument  wais  as  follows.  If  r  be  the  velocity  of  a 
planet,  r  the  radius  of  its  orbit  taken  as  a  circle,  and  T  its 
periodic  time,  v -^  "Iwr'iT,  But,  if  y  be  the  acceleration  to  Uie 
centre  of  the  circle,  we  have  f  =  r'/r.  Therefore,  substituting 
the  above  value  of  v, /- 4s^r/7''.  Now,  by  Kepler's  third 
law,  T*  varies  as  r^ ;  hence  f  varies  inversely  as  r*.     They 


*  See  the  Proceedingt  of  the  Loudom  Muihtmatical  Socuig^  1865^ 
fol.  I,  no.  2. 


NEWTthN'8   ItK   M'lTr,    I08+.  3W 

could  not,  however,  dnluoo  from  tho  Inw  Hie  orbita  of  the 
plftncts.  Hitllcy  explained  that  their  intiiitigMtiara  wen 
■topped  by  tlicir  innhility  to  solve  thin  proltWi,  and  Mk«(l 
Newton  if  ho  conld  find  nut  whnt  the  orbit  of  a  planet  would 
be  if  the  law  of  Atti-nction  wore  that  of  the  inverse  ■quara. 
Newton  inimpdiivtflly  replied  that  it  was  an  ellipae,  and  - 
proniised  to  send  or  write  out  afresh  the  demonstration  of 
it  which  he  hod  found  in  lGi3.  This  wns  Kiit  in  No^-enibcr, 
1684. 

Instigated  l>y  Hnlley,  Newt<»n  now  returned  to  the  problem 
of  gnvitation  ;  and  licfure  the  nutunu)  of  1GS4,  he  had  worked 
out  the  Bubxlance  of  pnipositiona  I — 19,  21,  30,  32 — 39  in  the 
first  book  of  the  /'riudjiM.  TlieMC,  together  with  notm  on  the 
laws  of  motion  and  various  leiiim»s,  were  reiwl  for  his  lectures 
in  the  MichiwItnaH  Trrm,  1684. 

In  Novemlier  Hi»!ley  received  Newton's  promised  oomran- 
nication,  which  probably  contistcd  uf  the  Hubstance  of  pro- 
piMitionn  1,  11,  and  either  pmponition  17  or  the  first  corollary 
of  proposition  I.t ;  thereupon  Hnllcy  again  went  to CambridgD 
wherv  he  Haw  "  a  curiou<i  treatise,  D'  Moln,  drawn  up  since 
August."  Alost  likely  this  contAined  Newton's  manuscript 
notes  of  the  lectures  alxne  nlludnl  to:  tlie:tc  notes  are  now 
in  the  university  library  and  are  headed  '*IM  Jtolt  Cor- 
jxiruiH."  Halley  begge<l  that  the  results  mi;;ht  be  published, 
and  finally  secured  a  pmmise  that  they  should  be  sent  to  the 
Koyal  Society :  they  were  accordingly  coinmunicnted  to  the 
Society  not  Uter  than  February,  1685,  in  ttie  paper  De  J/ofw, 
which  contains  the  snbstAncc  of  the  following  pntpewtions  in 
the  I'rineipia,  buuk  I,  props.  1,  4,  G,  7,  10,  II,  10,  17,  33; 
book  II,  props.  2,  3,  4. 

It  seoms  also  to  have  been  due  to  the  influence  and  tact  of 
Ualley  at  this  visit  in  November,  IG84,  that  Newton  undertook 
to  attack  the  whole  problem  of  gravitation,  and  practically 
pledged  himmilf  to  publisli  his  resalta :  thew  are  contained  in 
tlw  Principia.  As  yet  Newton  had  not  determined  the  attno* 
tion  of  a  si^ierical  body  on  an  external  pointt  nor  had  he 


344  TUI  Un  AMD  WOEKS  Or  VBWTOII. 

gJcalaled  the  deUik  of  iho  planeUiy  motioiM  •vm  if  the 
members  of  the  aolar  ajstem  ooiakl  be  niganied  ae  poiati.  The 
fint  problem  was  solved  in  1685»  probably  either  in  Jaaaaiy 
or  February.  ''No  •ooner,'*  to  quote  from  I>r  Qlsishfr^ 
address  on  the  bioentenary  of  the  publication  of  the  Frime^rim^ 
"  had  Newton  proved  this  superb  theorem — and  we  know  from 
his  own  words  that  he  had  no  expectation  of  so  beautiful  a 
result  till  it  emerged  from  his  mathematical  investigatiooi— 
than  all  the  mechanism  of  the  universe  at  once  lay  spread  before 
him.  When  he  discovorvd  the  theorems  that  form  the  first 
three  sections  of  book  i,  when  he  ga%'e  them  in  hLi  leciurss  of 
1681,  he  was  unaware  that  the  sun  and  earth  exerted  their 

• 

attractions  as  if  they  were  but  points.  How  different  must 
these  propositions  have  iieemed  to  Newton's  eyes  when  he 
realized  that  these  results,  which  he  liad  believed  to  be  only 
approximately  true  when  applied  to  the  solar  system,  were 
really  exact !  Hitherto  they  had  been  true  only  in  so  far  as  he 
could  regard  the  sun  as  a  point  compared  to  the  distance  of 
the  planets,  or  the  eurth  as  a  point  compared  to  the  distance 
of  the  moon — a  distance  amounting  to  only  about  sixty  times 
the  eaith*8  radius — but  now  they  were  mathematically  true,  ex-  j 

cepting  only  for  the  slight  deviation  from  a  perfectly  spherical 
form  of  the  sun,  earth  and  planets.  We  can  imagine  the  effect 
of  this  sudden  transition  from  approximation  to  exactitude  in 
stimulating  Newton's  mind  to  still  greater  efforts.  It  was  now 
in  his  power  to  apply  mathematical  analysis  with  absolute 
precision  to  the  actual  problirms  of  astronomy." 

Of  the  three  f undameutal  principles  applied  in  the  Pfineipia 
we  may  say  that  the  idea  tliat  every  particle  attracts  every 
other  particle  in  the  uuivene  was  foniied  at  least  as  early  as 
IGGG  ;  the  law  of  e<|uable  description  of  ar^as,  its  consequences, 
and  the  fact  tliat  if  the  law  of  attraction  were  that  of  the 
inverse  square  the  orbit  of  a  particle  about  a  centre  of  force 
would  be  a  conic  wene  proved  in  1679 ;  and  lastly  the  dhioovery 
tliat  a  sphere,  wIioho  density  at  any  point  depends  only  on  the 
distance  from  tlie  centre,  attracts  an  external  point  as  if  the 


\ 


\ 


Newton's  riuxciriA,  1685-1687.  345 

whnt«  masM  wer«  collrcletl  nt  it8  centre  wM  iiumIs  in  lfi8S, 
It  WH  tb»  last  discovery  that  rnnhlnl  him  to  «pply  the  finl 
two  princi[iles  to  thi'  phenomena  nf  hodii^  of  finite  rise. 

The  draft  of  the  fir^t  liuok  of  the  /'ritifi/Hi  wm  finiiibed 
before  the  summer  of  168.'!,  Iiut  the  correctioiiB  and  MMitions 
took  some  time,  and  tlie  Ixxik  won  not  pn-smted  to  the  KojJ 
Society  until  April  28,  1686.  Tliin  iNKik  is  given  vp  to  the 
considaroktitHi  of  the  motion  of  pnrticle.t  or  hotlien  in  free  iiiace 
either  in  known  orbita,  or  nnder  the  action  of  known  force*, 
or  under  their  mutual  altnwtion ;  nii<i  in  particular  ho  in- 
dicntea  how  the  eflectx  iif  disturbing  forvcK  may  Iw  calculated. 
In  it  hImo  Newton  grnernliziii  the  law  »f  attraction  into  a 
Btatemenl  that  every  particle  of  matlrr  in  the  nnivorse  att«^icta 
everj-  other  particle  with  a  force  which  variw  directly  ait  the 
product  of  their  niawea  and  invcnvly  hh  the  xqnare  of  the 
distntice  between  them ;  and  he  thence  dedace*  tlie  law  of 
attraction  for  npherical  Nhelln  of  con^tnnt  denaity.  Hm  book 
in  prefaced  l»y  an  introduction  on  tltc  science  of  dynamjca, 
which  detinen  the  liniitd  of  matiieinatical  in^Totigatioii.  Uia 
object,  he  aaya,  in  to  apply  mathemntini  to  the  pbenomeiih 
of  nature ;  among  thene  phenuniEna  motion  ia  on«  of  Uw 
moRl  important;  now  motion  w  the  effect  of  ioro^  and, 
though  he  does  not  know  what  in  the  nature  or  origin  of 
force,  still  many  of  its  effects  can  lie  measured ;  and  it  is 
these  that  form  the  subject-matter  of  the  work. 

The  second  book  of  the  Principin  was  coin[deted  by  the 
Mummer  of  1686.  TItia  book  trratt  of  motion  in  a  resisting 
medium,  and  of  hydrtwtatiot  and  liydrodynamics,  with  special 
applications  to  waves,  tides,  and  aconstii.-H.  He  conclDdca  it 
by  showing  that  the  Cartesian  theory  of  Tortiom  wu  in* 
GonaiHtent  both  with  the  known  facta  Mid  with  the  Usrs  al 


I 


The  next  nine  or  ben  months  were  devoted  to  the  third 
bo<A.  Probably  for  this  he  had  originally  no  Materiab  ready. 
Ue  eonmenoea  by  discussing  when  and  how  far  It  ia  JMti- 
fiaUe  to  oonatnicl  hypotbeMS  or  UMoriaa  In  eMMaib  >*st 


346  TUB  Un  AMD  WOKKH  Off  NKWIOlf. 

known  phenomenA.  He  prooeeds  to  apply  Uio 
obtained  in  the  tint  book  to  tbe  chief  pUenooiena  of  the 
■oUr  syiteni,  and  to  determine  the  roiiniw  and  diitancei  of 
tbe  pUneta  and  (whenever  sufficient  data  existed)  of  their 
satelliteik  In  particular  the  motion  of  the  ukmmi,  the  Yarions 
inequalities  therein,  and  the  theory  of  the  tides  are  worked 
out  in  detail.  He  also  investigates  the  theory  of  oonietSi 
shews  that  they  belong  to  the  soUr  system,  explains  how 
from  three  observations  the  orbit  can  be  determined,  and 
illustrates  his  results  by  considering  certain  special  cometic 
The  third  book  as  we  Imve  it  is  but  little  more  than  a  sketch 
of  what  Newton  had  tinally  proposed  to  hiuiself  to  accomplish; 
his  original  scheme  is  among  the  *' Portsiuouth  papers,"  and 
his  notes  sliew  that  he  continued  to  work  at  it  for  ifome  yearn 
after  the  publication  of  the  first  edition  of  the  PrineijAa :  the 
most  interesting  of  his  memoranda  are  those  in  wliich  by 
nie;ins  of  Huxious  he  has  curried  his  results  beyond  the  point 
at  which  he  was  able  to\ranslate  them  into  geometry^. 

The  denionst  rut  ions  throughout  the  work  are  geometrical, 
but  Ut  nuidi'rn  of  ordiiuiry  ability  are  rendered  unnecessarily 
ditlicult  by  the  ul>>ence  of  illustrutions  and  explanations,  and 
by  the  fact  tliut  no  clm^  is  given  to  the  method  by  which 
Newton  urrivi'd  ut  his  results.  The  reuson  why  it  was  pre- 
sentetl  in  u  geometrical  form  uppeurs  to  have  tjeen  tliat  the 
intinitesimul  calculus  was  then  unknoa-n,  and,  had  Newton 
used  it  to  tlenionstrute  results  which  were  in  themselves 
opposed  to  the  prevalent  philosophy  (»f  the  time,  the  contro- 
versy us  to  the  truth  of  his  results  would  liuve  been  bumpered 
by  a  dispute  concerning  the  vuliility  of  the  methods  used 
ill  proving  them,  lie  therefore  cast  the  whole  reasoning 
into  u  geometricul  sliupe  which,  if  soniewhut  longer,  can  at 
any  rate  Xtc  nuMle  inti'lligilile  to  ull  nuithemutical  students. 
80  closely  did  he  folltiw  the  lines  of  Grt*ek  geometry  tliat  he 
constantly   us4h1  graph icul   methods,  and   represented   forces, 

*  For  s  fuller  acooant  of  the  friueipia  tee  my  Ktsajf  on  ike  9eH€tU^ 
eoHttHts,  aud  hutorff  0/  Setrtou^i  Principia,  Loudoo,  1S93. 


\ 


nkwton's  i-rixviI'Ia.  347 

velocitieit,  snci  other  niagnitudt^  in  tlic  Enclidean  wajt  liy 
■trvglit  lines  {kjc,  ijr.  InKik  i,  lemma  10),  nnd  not  bj  n  oerbiin 
number  of  untU.  llic  \aXivT  and  in<Nleni  )»ieth«l  linil  be«n 
introduced  by  Wsllin,  And  miiHt  have  been  familiar  to 
Newton.  The  elTect  «f  liis  confininj;  Itiix^elf  rigimiaslj  to 
cliuu<i<»l  geumetry  is  that  the  I'riucijiin  i*  written  iu  »  - 
lAiigunge  which  in  aivhnic,  even  if  not  unfAniilittr. 

The  ndoptiim  of  grunietriual  methiidH  in  the  i'ritiripia  tov 
[lurpnfws  nf  (lemonntratiun  dooK  nut  indiott^  *  preforpiKO  on 
Ncwtoii'H  p«rt  for  geometry  over  annlyHiN  iwf  »n  iniitniment  * 
uf  reHcftrch,  fur  it  is  known  nuw  tlmt  Newtuii  axed  tite  HaxiunMl 
cnlcutuB  in  the  fint  in>tUincc  in  finding  Nome  uf  tlie  tlioun-nus 
specialty  those  tiiwanlit  the  end  of  InniIi  i  and  in  Imok  li ; 
Mid  in  fact  one  of  the  moNt  iroporbint  uscn  nf  tlut  cnknlas  in 
stnt«<1  in  lionk  II,  lemma  2.  Itut  it  id  unly  junt  to  rara»rk 
that,  at  the  time  uf  it«  publication  and  for  ncftrly  «  century 
aflcm-HrdM,lheiliffen.'ntial  and  Huxionnl  cnkul as  went  mtt  fully 
dcvelnpeil  and  «lid  not  jkhscks  llio  itnnie  8U]icriority  over  the 
method  ho  adoptttd  which  they  do  nuw ;  and  it  in  »  inatt^n-  for 
antonishment  that  when  Ncwtuii  did  enigdoy  tlw  calculUH  ho 
wu  able  to  uw  it  to  w  good  an  effect.  The  ability  nhewn  in 
the  tnuiHlation  in  a  few  nraiilliH  of  thoimn^  ho  numenniii  and 
of  Ml  great  complexity  into  the  Innguiigi-  of  the  getniiptiy  of 
Archimedes  and  Apnllonius  in  I  luppuHe  unparalletod  in  tlie 
history  of  mathematics. 

The  priiitin;|r  of  the  work  wan  nlow  and  it  waa  not  finally 
paliliBhe<i  till  tite  sumiiieruf  IGSt.  The  whole  cunt  waa  borne 
by  Halley  who  also  corrected  the  pr^titU  and  even  put  hin  own 
reacarches  on  nna  iiile  Ui  press  the  printing  forward.  Tiro 
concisenesK,  ab«enc«;  of  illustralioms  and  Kynlhetical  character 
of  the  book  restricted  the  nuniliers  of  llHxe  who  were  able  to 
appreciate  its  vniuo  ;  and,  though  nearly  all  competent  critics 
admitted  tho  validity  uf  tlie  omclusionn,  mme  little  time 
elapsed  before  it  aBected  the  current  beliefs  at  educated  men. 
I  shoald  be  inclined  U>  uy  (but  on  this  point  opiniom  diSer 
widely)  that  within  ten  yeara  ol  its  publicataon  it  wh  gens- 


848  .  TUK  UR  AMD  WORKH  or  V 


KW10V. 


vmlly  aooeptad  in  Britain  •■  giving  a  correoi  aeoonni  of  tlie 
lawn  of  tho  univenie ;  it  wiis  dmilarly  aoooploil  within  abont 
twenty  yean  on  tlie  oontinenti  except  in  Fianoo  wboiv  the 
Carteeiau  hypoihesis  held  ite  ground  until  Voltaire  in  1738 
took  up  the  advocacy  of  the  Newtoniau  theory. 

Tlie  man'uBcript  of  the  PrtMcipia  waa  finithed  hy  1688. 
Newton  de%-otMl  the  remainder  of  that  year  to  hiii  paper  on 
phyKical  opticH,  the  greater  part  of  which  is  given  up  to  the 
aubject  of  diffraction. 

Ill  1087  Jaiiiea  II.  liaviug  tried  to  force  the  univeraity  to 
admit  aa  a  nuister  of  arta  a  Iloinan  Catholic  prieat  who  refuaed 
to  take  tlie  outha  of  aupremacy  aud  allegiance,  Newton  took 
a  prominent  part  iu  reaiatiiig  the  illegal  interference  of  the 
king,  and  waa  one  of  the  deputation  aeut  to  London  to  protect 
the  righta  of  the  univeraity.  Tlie  active  part  taken  by 
Newton  in  thia  utfair  l«d  to  hia  being  in  1689  elected  member 
for  the  uiiiverhity.  Thia  parliament  only  laated  thirteen 
moiitliM,  and  on  ita  diaaoluiion  he  gave  up  hia  aeat.  He 
waa  8uljH04|uuntly  returned  iu  1701,  but  he  never  took  any 
prominent  part  in  politica. 

Oil  111!!  coming  back  to  Cambridge  in  1690  he  resumed  his 
mathematical  atudiea  and  com^Kpoiideiice,  but  probably  did 
not  lecture.  The  two  lettora  to  Wallia,  in  which  he  explained 
hia  method  of  fluxioiia  and  tluenta,  were  written  in  1692  and 
published  in  1693.  Toa'ards  the  close  of  1692  and  throughout 
the  two  following  yeara  Newton  had  a  long  illneaa,  aufiering 
from  inaomnia  and  general  iiervoua  irritability.  Perhapa  he 
never  quite  regained  hia  elasticity  of  mind,  and,  though  after 
hia  recovery  he  aliewed  the  same  power  in  solving  any  queation 
propounded  to  him,  he  ceaaed  thenceforward  to  do  original 
work  on  hia  own  initiative,  and  it  waa  somewhat  difficult  to 
atir  him  to  activity  in  new  subjecta. 

In  1694  Newton  began  to  collect  data  connected  with  the 
irregularitiea  of  the  moon'a  motion  with  the  view  of  reviaing 
the  part  of  the  Priticipia  which  dealt  with  tliat  subject.  To 
render    the    obaervationa    more    accurate    he    forwarded    to 


\ 


J 


THR   LIFE  AND  WORKS  OP  NEWTOH.  349 

FUinsteed  *  a  tntile  of  conrctionn  for  rrf  mction  which  he  had 
{ireviuiiMly  nuule.  This  vns  not  pulilinhnd  till  1731,  when 
Halley  cnnimnnicaU^I  it  to  the  Itoynl  SociH>7.  The  original 
cnlcuUlionN  of  Npwt4>n  nnil  the  paprni  mnneeted  with  it  am 
in  the  Portsmouth  collt-ction,  and  nhi^w  that  Newton  ohtAined 
it  liy  (inilinf;  thi-  pnth  of  a  ray,  hy  mennn  of  quadratonw,  in  a 
mnnnrr  p<)uivalrnt  to  the  iu>luti'in  of  a  diffrrenlial  nquntinn. 
An  mi  illuMlrntiiin  uf  Newttm'R  i^niiiK  I  may  mrntinn  that 
even  aa  late  aa  IT-'U  Ruler  fniloi)  U  miIvc  the  name  problem. 
In  1TK3  lAplnre  f^ikve  a  rule  for  conittructing  such  a  table, 
and  hi)>  reHulli  nK^ee  itulntAntinlly  with  those  of  Newton. 

I  do  not  luiiposo  that  Newt^m  would  in  any  raae  have 
produced  much  more  original  work  after  his  illnena;  Imt  hi* 
appuintnient  in  IGUG  oh  wnrden,  anil  hh  promotion  in  IG99 
to  the  mnntrmhip  of  the  Mint  at  a  Hnlnry  of  £1500  a  year, 
brought  bin  KJentific  inveHtigalinn'i  to  nn  end,  though  it  wai 
only  after  this  tliat  many  of  lii«  previous  investigation*  were 
published  in  the  form  of  iMiokii.  In  1696  ho  moved  to  London, 
in  1701  he  re?<igrie<l  the  Lucaatnn  chair,  and  in  1703  he  wm 
elected  president  of  the  Itoyal  Society. 

In  ITOJ  Newton  published  hia  Oj>lie»  which  containa  thv 
renulta  of  the  papers  already  mentioned.  To  the  Ant  edition 
of  this  book  were  appended  two  minor  works  which  have  no 
Hpccial  connection  with  opticn ;  one  lieing  on  cnliie  curveii,  the 
otlter  on  the  quadrature  of  cur\es  and  on  flnxioM.  Both  of 
them  were  olit  manuncript^  with  which  his  friends  and  pnpils 
were  familiar,  but  they  were  here  published  urbt  tt  orH  for 
the  first  time. 

■  Jolim  HamitrrA,  bam  kt  Detbj  in  1646  and  died  at  OieniwUi  ia 
171S,  <raa  one  of  iIm  mort  diiiingaiabed  aitroDomen  ol  Ibis  scr,  and 
tbc  flnl  aitronomoT-rojal.  He!>iiilr>  maeh  Taluable  work  ia  astnmomr 
Im  inTeul«d  tltc  ojuleni  (pabliaha)  in  IfMIl)  of  drawing  nu^  hj  ym- 
jeeting  the  narface  of  the  apliere  on  an  enTclopiog  enoe,  whidi  can  Ibsn 
b*  anwiappcd.  HU  lUe  b;  B.  F.  Bailj  was  pubUabad  la  IionlaB  In 
16U,  bal  Tariooa  italeiDMiU  in  il  sbonld  ba  read  wU  bf  tide  wilk 
tlKM  in  Btewatcr'a  life  ol  Nawlon.  FlaauUad  ass  taewaifad  aa  aa- 
treiMOV-nTBl  b^  Kdmaad  Ualiay  (no  briow,  pp.  NO— 1). 


S50  THE  UPS  AMD  WORKS  OP  MSWTON. 

The  fint  of  theae  mppendioet  it  entitlad  £nmmmrmiio 
rum  Teriii  Oniiuut^;  the  object  iieenui  to  be  to  flloetiAte  the 
ime  of  analytical  geometry^  amd  m  the  applioatioo  to  eoaies 
wan  well  known  NewUm  velected  the  theur)*  of  cubioa 

He  begins  with  Home  general  theorooia,  and  daariSee 
carves  according  ah  their  equations  are  algebraical  or  traaa- 
oendental:  the  former  being  cat  by  a  straight  line  in  a 
nuuilor  uf  pointH  (rml  or  iiiiagiuary)  eqaal  to  tlie  degree  of 
tlie  curve,  the  latter  lieing  cut  by  a  straight  line  in  an  infinite 
nuuilier  of  pointn.  Newton  then  sliews  that  uuuiy  of  the 
uiOHt  important  pruperties  of  conies  have  their  analogues  in 
the  theory  of  cubicn,  and  he  discusHes  the  theory  of  asymptotes 
aiMl  curvilinear  diameters. 

After  these  general  theorems  he  commences  his  detailed 
examiimtion  of  cuhicH  by  (Hiinting  uut  thiit  a  cubic  must  have 
at  \viini  one  tva]  point  ut  infinity.  If  the  asymptote  or  tangent 
at  this  point  U*  at  a  finite  distiinoe,  it  may  lie  taken  for  the 
axi.s  of  y.  This  asymptote  will  cut  the  curve  in  three  points 
alto;(t'thiT,  t>f  which  at  U'ltst  tw<»  are  at  infinity.  If  the  third 
point  U*  ut  a  finite  diMUuice,  then  (by  one  of  his  general  tlaHirems 
on  asyiiipttitrs)  the  e4|uation  can  be  written  in  the  form 

where  the  axi*8  of  x  and  y  are  the  asymptoteM  of  tlie  hyperbola 
which  in  the  locus  of  the  mid<lle  points  of  all  chords  drawn 
panillel  Ui  the  axis  of  y\  while,  if  the  thini  point  in  which 
this  asymptote  cuts  the  curve  lie  also  at  infinity,  tlie  equation 
can  be  written  in  the  form 

xy  -  €u^  +  h^  -I-  ca;  -f  </. 

Next  he  takes  the  case  where  the  tangent  at  the  real  point 
at  infinity  is  not  at  a  finite  distance.  A  line  parallel  to  the 
direction  in  which  the  curve  goes  to  infinity  may  be  taken 
as  the  axis  of  y.  Any  such  line  will  cut  the  curve  in  three 
points  altogether,  of  which  one  is  by  hypothesis  at  infinity,  and 

*  Go  tbu  work  snd  its  bibliogrsphj,  see  my  memoir  in  Ibe  2VttHM«tMNMi 
o/ihf  LomloH  Malhemalieal  Society,  l^l,  \ot  xxn,  p|i.  104—143. 


\ 


MBWTON  OK  CUBIC  CURVEa  S51 

one  in  neoeBsarilj  at  a  finite  diHUnce.  He  then  nhewii  that,  if 
the  remaining  point  in  which  thin  line  cntR  the  carve  he  at  a 
finite  dintance,  the  equation  can  be  written  in  the  form^ 

y*  =  aj^  -k- bji^  ■¥  ex -^  d ; 

whihs^  if  it  be  at  an  infinite  distance,  the  eciuation  can  ho 
written  in  the  fomi 

y  =  rtje*  +  hat? -i- ex -i- fL 

Any  cubic  w  therefore  reducible  to  one  of  four  charac- 
teriHtic  forniA.  Each  of  th(*se  fonns  w  thf*n  diwuHnrd  in  detail, 
and  the  pONsibility  of  the  existence  of  double  points,  inolated 
ovalis  ke.  in  worked  out.  Tlie  final  result  is  that  in  all  there 
are  seventy-eight  possible  fornis  which  a  cubic  may  take.  Of 
these  Newton  enumerated  only  seventy-two;  four  of  the  re- 
mainder were  mentioneil  by  Stirling  in  1717,  one  by  Nicole 
in  1731,  and  one  by  Nicholas  Bernoulli  alw>ut  the  same  time. 

In  the  course  of  the  work  Newton  states  the  remarkalile 
theorem  that^  just  as  the  shallow  of  a  circle  (cast  by  a  luminous 
point  on  a  plane)  gives  rise  to  all  the  conies,  so  the  sharlows  of 
the  curves  represented  by  the  equation  y*  =  rw^  +  ftn;*  4-  ex  +  <f 
give  rise  to  all  the  cubics.  This  remained  an  unsolved  puzzle 
until  1731,  when  Nicole  and  Clairaut  gave  demonstrations  of 
it:  a  better  proof  is  that  given  by  Murdoch  in  1740,  which 
depends  on  the  claHsification  of  these  curves  into  five  species 
according  as  to  whether  their  points  of  intersection  with  the 
axis  of  X  are  real  and  unequal,  real  and  two  of  them  (k|aal 
(two  cases),  real  and  all  equal,  or  two  imaginary  and  one  real. 

In  this  tract  Newton  also  discusses  douUe  points  in  the 
plane  and  at  infinity,  the  description  of  carves  satisfying 
given  conditions,  and  the  graphical  soluUon  of  problems  by 
the  use  of  carves. 

The  second  i^ipendix  to  the  OpiicM  is  entitled  De  Qmndr^ 
turn  Cnrramm,  Most  of  it  had  been  oommanicated  to 
Barrow  Ih  1668  or  1669»  and  probably  was  familiar  to 
Newton's  papils  and  friends  from  that  tisM  oiiwardsL    U 


/ 


862  THB  UVE  AMD  WORKB  Of  MKWTOII. 

TImi  bulk  of  the  Unit  |Murt  b  a  lUlamaiit  of  VmwUmfk 
lutttlHid  of  effecting  the  qiuulmtiire  and  rectification  of  onrvoe 
by  means  of  infinite  aeries :  it  i«  noticeable  ae  containing  the 
earliest  use  in  print  of  literal  indices,  and  also  the  first 
printed  statement  of  the  binomial  theorem,  bnt  these  novelties 
are  introduced  only  incidentally.  Hie  main  object  is  to  give 
rules  for  de%*elupiug  a  function  of  x  in  a  aeries  in  ascending 
powers  of  ar,  so  as  to  enable  mathematicians  to  effect  the 
quadrature  of  any  curve  in  which  the  ordinate  y  can  be  ex- 
presHed  as  an  explicit  algebraical  function  of  the  slisciwse  x, 
Wallis  had  shewn  how  this  quadrature  could  be  found  when  y 
was  given  as  a  num  of  u  number  of  multiples  of  powers  of  ae, 
and  Newton's  rules  of  expaimioii  here  estaUislied  tendered 
putMible  the  siuiilar  quadrature  of  any  curve  whose  ordinate 
can  bu  expreMMHl  as  the  huui  of  an  infinite  nunilier  of  such 
teniiH.     In  Uiim  way  he  eU'ectM  the  quadniture  ot'  the  curves 

y-,-:v  *=(-•*«•)».  y=(-»^A  »=(iz£y. 

but  naturally  the  renults  are  expressed  as  infinite  series.  He 
then  proceeds  to  curves  whose  ordinate  is  given  as  an  implicit 
function  of  the  alMcissa ;  and  he  gives  a  method  by  which  jf 
can  lie  expri*8sed  as  an  infinite  series  in  ascending  powers  of  oe; 
but  the  application  of  the  rule  to  any  curve  demands  in  general 
such  coiiiplicaU^d  uuuierical  calculations  as  to  render  it  of  little 
value.  He  concludes  this  part  by  shewing  that  the  rectification 
of  a  curve  can  be  etifected  in  a  somewhat  similar  way.  His 
process  is  equivalent  to  finding  the  integral  with  regard  to  x 
of  (1  -t-y*)^  in  the  fonn  of  an  infinite  series.  I  should  add 
that  Newton  indicates  the  importance  of  determining  whether 
tlie  series  are  convergent — an  observation  far  in  advance  of 
his  time— but  he  knew  of  no  general  test  for  the  purpose; 
and  in  fact  it  was  not  until  Qauss  and  Cauchy  took  up  the 
question  that  the  necessity  of  such  limitations  was  commonly 
recognized. 

The  part  of  the  appendix  which  1  have  just  described  is 


\ 


Newton's  methud  or  KLUXioMit.  853 

proctioilljr  tlic  finiita  nn  Ncwlon'x  iniiiiiiscrijit  De  AimJi/**  pir 
Eipintionrt  Xiimtro  Ttrmintmun  fiifinilnn,  which  wan  snbHO- 
qaently  |>rinU!d  in  1711.  It  U  tuiiii  that  this  wait  origtnallj 
intended  to  form  An  appendix  to  Kinckhuyspn'a  Alg^rtt, 
which,  iM  I  hnve  nlre.-uly  luid,  he  «t  one  time  int«ndixl  to 
edit.  The  nulntnnce  of  it  wnn  coinniunicAtoal  to  Dihtow,  Mid 
hy  hilt)  U>  Collinx,  in  Irttera  of  July  31  nnil  Aog.  13,  1G69; 
niitl  a  aummarf  of  part  of  it  was  included  in  the  letter  t)t 
Oct.  21,   1676,  annt  to  Leibnitz. 

It  should  be  ruiul  in  connection  with  Newton'a  Methoditt 
Differential  if,  hIho  published  in  1711.  ScMin  additioniJ 
theorema  are  there  given,  nnd  he  discuKHW  his .  metliod  of 
intcrpoUtion,  wliich  had  been  briefly  described  in  tite  letter 
of  Oct  24,  1 676.  The  principle  ih  this.  If  y«^(x)  be  k 
function  of  x  and  if  when  x  in  .luearivsiveljr  pnt  equal  to 
n,,  a,, ...,  the  values  of  y  1»  known  and  )>e  A,,  Ag,...,  then  A 
pArsliola  wIiiMw  equation  ih  y  - p *  qx  +  tx'  *■ ...  can  be  drawn 
through  the  points  (n,,  A,),  (t,.  &,),...,  and  the  ordinate  of  this 
parabola  niny  be  takfin  as  an  approxiinntion  to  the  ordinate  of 
the  curve.  ITic  flegtre  of  the  parabola  will  of  conne  be  one 
less  than  the  namber  of  given  points.  Newton  point*  oat 
that  in  this  way  the  areas  of  any  cnrvco  can  be  tqipnHUDuUdy 
de  terra  ined. 

The  Kccond  part  of  thin  appendix  to  the  Ojitiei  conlaina  » 
description  of  Newton's  method  of  fluxiunH.  This  is  lie«t  oon- 
flidcred  in  connection  with  Newton's  iiianuwript  on  the  same 
subject  which  waH  publi-<hcd  by  John  Colnon  in  1736,  and  of 
which  it  IM  a  summary. 

The  fluxional  cnlculuH  is  one  form  of  the  infinitesimal 
calculus  exprensod  in  a  certain  notntion.jufitsM  the  differential 
calcnluM  is  another  aspect  of  the  same  calculus  exprssnod  in  a 
different  notation.  Newton  aMumed  th.tt  all  geotnetriea]  mag- 
nitudes might  be  conceived  aN  generated  by  continuoos  motion; 
thos  a  line  may  bn  considered  an  generated  by  the  motioa  of  » 
point,  a  inrface  by  that  of  a  line^  a  solid  by  that  of  a  sarfiwe, 
a  plane  angle  by  the  rotation  of  a  line,  and  ao  on.    The 


354  THE  UPC  AMD  WOEU  OP  IIKWTON. 

qoaatity  Urns  generated  wm  de6iied  bjr  him  ae  the  §mmA  er 
flowing  quantity.  The  velodtjr  of  the  moving  magnitude  wne 
defined  as  the  flnxion  of  the  fluent.  This  aeeme  to  be  the 
earliest  definite  reoognition  of  the  idea  of  a  continaonsfuetion, 
though  it  liad  been  foreshadowed  in  some  of  Napiei^e  papenL 

Newton's  treatment  of  the  subject  is  as  follows.  Theve 
are  two  kinds  of  problems.  The  object  of  the  first  is  to 
find  the  fluxion  of  a  given  quantitji  or  more  geneimlljr  ''the 
relation  of  the  fluents  being  given,  to  find  the  relation  vt  their 
fluxions.**  TIais  is  equivident  to  differentiation.  The  object 
of  the  second  or  invene  method  of  fluxions  is  from  the  fluxion 
or  some  relations  involving  it  to  determine  the  fluent^  or  more 
generally  *'an  equation  being  proposed  exhibiting  the  relation 
of  the  fluxioiu  of  quantities,  to  find  the  relations  of  those  quan- 
tities, or  fluents,  to  one  another*.*'  This  is  equivalent  either  to 
integration  which  Newton  termed  the  method  of  quadrature, 
or  to  the  solution  of  a  differential  equation  which  was  called 
by  Newton  the  inverse  method  t»f  tangents.  The  methods 
for  solving  these  problems  are  discussed  at  considerable  length. 

Newton  then  went  on  to  apply  these  results  to  questions 
connected  with  the  maxima  snd  minima  of  quantities,  the 
method  of  drawing  tangents  to  curves,  and  the  curvature  of 
cur\'eH  (namely,  the  determination  of  the  centre  of  curvature, 
the  radiuH  of  curvature,  and  the  rate  at  which  the  radius  of 
curvature  increases).  He  next  considered  the  quadrature  of 
curves,  and  the  rectification  of  cur\'est.  In  finding  the  maxi- 
mum and  niinimum  of  functions  of  one  variable  we  regard 
the  change  of  Kign  of  the  difference  between  two  consecutive 
values  of  the  function  as  the  true  criterion :  but  his  argument 
i»  that  when  a  quantity  increasing  has  attained  its  maximum 
it  can  liave  no  further  increment,  or  when  decreasing  it  has 
attaine<l  its  miniiuuui  it  can  have  no  further  decrement ;  con- 
sequently the  fluxion  must  be  equal  to  nothing. 

*  Colson's  edition  of  Newton's  msnoHcript,  pp.  xxi,  xxii; 
t  Ibid,  pp.  xxil,  xxiii. 


\ 


Newton's  method  of  fluxions.  355 

It  has  lioen  reiiiarkod  that  ncitlier  Ncwtuii  nor  Leilinits 
produced  a  calculufi,  that  is  a  clasMificd  collection  of  rales ;  atid 
that  the  problems  they  diftcussed  were  treatcf!  from  first  prin- 
ciples. That  no  doubt  iH  the  usual  sequence  in  the  history  of 
such  diHCOvericfs  though  the  fact  is  frequently  forgotten  by 
subsequent  writers.  In  this  case  I  think  the  statement,  so 
far  as  Newton's  treatment  of  the  cliflTerentinl  or  fluxi^nal 
part  of  the  calculus  is  omcerned,  is  incorrect,  as  the  forcgiing 
account  sufficiently  shews. 

If  a  flowing  quantity  or  fluent  were  represented  by  jc, 
Newton  denoted  its  fluxion  by  i^  the  fluxion  of  x  or  second 
fluxion  of  X  by  X,  and  so  on.  Hiniilarly  the  fluent  of  x  was  de- 
noted liy  U|»  or  sometimes  by  x  or  [^].     The  infinitely  small 

part  by  which  a  fluent  such  as  x  increased  in  a  small  interval  of 
Ume  measured  by  o  was  called  the  moment  of  the  fluent ;  and 
its  value  was  shewn*  to  be  ^.  Newton  adds  the  important 
remark  that  thus  we  may  in  any  problem  neglect  the  terms 
multiplied  by  the  second  and  higlier  powers  of  o,  and  we  can 
always  find  an  e<|uation  between  the  coordinates  x;  y  of  a 
point  on  a  curve  and  their  fluxions  x,  ^,  It  is  an  application  of 
this  principle  which  constitutes  one  of  the  chief  valuen  of  the 
calculus;  for  if  we  desire  to  find  the  effect  produced  by 
several  causes  on  a  system,  then,  if  we  can  find  the  effect  pro- 
duced by  each  cause  when  acting  alone  in  a  very  small  time, 
the  total  eflfect  produced  in  that  time  will  be  equal  to  the  sum 
of  the  separata  eflfects.  I  should  here  note  the  fact  that  Vince 
and  other  English  writers  in  the  eighteenth  century  used  i  to 
.  denote  the  increment  of  x  and  not  the  vekicity  with  wliich  it 
increased ;  that  is,  i  in  their  writings  stands  for  what  Newton 
would  have  expressed  by  dbo  and  what  Leibniti  would  have 
written  as  c£cL 

I  need  not  discuss  in  detail  the  manner  in  which  Newton 
treated  the  problems  above  mentioned.  I  will  only  add  that^ 
in  qiite  ef  Uie  form  of  his  definition,  the  introduetion  into 


S^  THE  UVE  AMD  WOBK8  OT 

;  gUMueUy  of  the  idea  of  tinia  wm  evaded  by  sappoii^g  ilyifc 
Home  qoaatii/  (*r.  jfr.  the  absdaia  of  a  point  on  a  curve) 
iuf ;reaied  equaU/ ;  and  the  required  molta  then  depend  en 
Uk)  rate  at  which  otiier  quantities  (e^  ^.  the  ordinate  or 
racliuH  of  curvature)  iucrease  relativd/  to  the  one  so  chuiien^. 
The  fluent  no  chmieu  ia  wliat  we  now  call  the  independent 
variable ;  its  fluxion  wan  termed  the  '*  principal  fluxiou " ; 
and  of  ooune,  if  it  were  denoted  by  x^  then  st  was  constanti 
and  oonaequeutly  x  ^  0. 

There  in  no  question  that  Newton  Ubea  the  method  of 
fluxions  in  16GG,  and  it  is  practically  certain  that  accounts  of 
it  were  comuiuiiicatcd  in  manuscript  to  friends  and  pupils 
from  and  after  1G69.  The  manuscript,  froiii  which  most  of 
the  above  summary  has  been  taken,  is  believed  to  have  been 
written  between  1G71  and  1677,  and  to  liave  been<in  circula- 
tion at  Ciiiiibridge  from  that  time  onwards,  though  of  course 
it  i8  passible  that  parts  were  rewritten.  It  was  unfortunate 
that  it  wixa  not  published  at  once.  Htrangers  at  a  distance 
naturally  judged  of  the  method  by  the  letter  to  Wallis  in 
1G92,  or  by  the  TracUUn4  lU  Quadritiura  CurvaruM^  and  were 
not  aware  that  it  liad  been  so  completely  de%'e1oped  at  an 
earlier  date.  This  was  the  cause  of  numerous  misunder- 
standings. 

At  the  Slime  time  it  nmst  be  added  that  all  mathematical 
analysis  was  leading  up  to  the  ideas  and  luethods  of  the  infi- 
nitesimal calculus.  Foreshadowings  of  the  principles  and 
even  of  the  language  of  tluit  calculus  can  be  found  in  the 
writings  of  Napier,  Kepler,  Cavalieri,  Pascal,  Femiat, 
Wallis,  and  Barrow.  It  was  Newton's  good  luck  to  come 
at  a  time  when  everything  was  ripe  for  the  discovery,  and 
his  ability  enabled  liim  to  construct  almost  at  once  a  com- 
plete calculus. 

The  notation  of  the  fluxional  calculus  is  for  most  purposes 
less  convenient  than  that  of  the  ditierential  calculus.  The 
latter  was  invented  by  Leibnitz  probably  in  1675,  certainly  by 
*  Colson*M  edilion  of  Newton's  msouneript,  p.  20. 


\ 


/. 


THE  LIFE  AND  WORKS  OP  KBWTOK.  S57 

1677,  luid  was  inil>lished  in  1684,  some  nine  years  before  the 
earl ient  |Hrinted  acooant  of  NewtonV  method  of.  fluxions.  But  / 
'the  question  whether  the  general  idea  of  the  calculus  expressed  / 
in  that  notation  was  obtained  by  Leibnitz  from  Newton  or 
whether  it  was  invented  independently  gave  rise  to  a  long  and 
bitter  controversy.  The  leading  facts  are  pven  in  the  next 
cha|>ter.  The  question  is  one  of  considerable  difficalty,  bat  I 
will  here  only  say  that  from  what  I  have  read  of  the  voluminoas 
literature  on  the  question,  I  think  on  the  whole  it  points  to 
the  fact  that  Leibnitz  obtained  the  idea  of  the  differential 
calculus  from  a  manuscript  of  Newton's  which  he  saw  in 
1675  or  perhaps  1676.  I  lielieve  however  that  tlie  prevalent- 
opinion  is  that  the  inventions  were  independent. 

The  remaining;  events  of  Newton's  life  require  little  or  nO 
comment.  In  1 705  he  was  knighted.  From  tliis  time  onwan  n 
he  devoted  much  of  his  leisure  to  tlieology,  and  wrote  at  gre^t 
length  on  prophecies  and  pmlictions,  subjects  which  had  alwi^ys 
been  of  interest  to  him.  His  UnirtrmU  AnthmetU  was  pub- 
lished by  Whiston  in  1 707,  and  his  A  nnlytU  by  Infinite  SetHen 
in  1711 ;  but  Newton  had  nothing  to  do  with  the  preparaMon 
of  either  of  these  for  the  press.  His  evidence  before  the  House 
of  Commons  in  1714  on  the  determination  of  longitude  at  sea 
marks  an  important  epoch  in  the  history  of  nangation. 

The  dispute  with  licibnitz  as  to  whether  he  had  derived 
the  ideas  of  the  differential  calculus  from  Newton  or  invented 
it  independently  originated  aliuut  1 708,  and  occupied  much  of 
Newton's  time,  espccisUy  lietween  tlie  jrears  1709  and  1716. 

In  1709  Newton  was  persuaded  to  allow  Cotes  to  prepare 
the  loiig-talked-of  second  edition  of  the  Prinrifnn:  it  was 
issued  in  March  1713.  A  third  edition  was  |niblished  in  1726 
ondm*  the  direction  of  Henry  Pemberton.  In  1725  Newton V 
health  began  to  faU.  He  died  on  March  20,  1727»  and  eight 
days  later  was  buried  with  state  in  Westminster  Abbeyi 

Hit  chief  works,  taking  them  in  their  order  of  poblkaation, 
aro  the  i'^mei/ria,  published  in  1687;  the  OpCsca  <^^kw«9«»)^< 
dien  on  ciiKs  tmnm^  the  iftMNhmHire  tmd  T«d^i^«s^«^  4 


/ 


\ 
\ 


SS8  THE  UFI  AVD  WORU  OT  MIWTOM. 

hjf  ikt  um  of  infiniis  mtmi^  and  the  m§Aod  ^  JUmUmti^ 
pobliiilud  in  1704;  the  UHivmat  Ariikmeiie.  pnbliriied  in 
1707;  the  ifna/yMf  per  Seriet^  Flweumn^  kc^  and  the 
MtthoAu  Differ€niiali9,  poUinhed  in  1711;  the  Ueiimm 
Optieae^  publiiibed  in  1729 ;  the  Method  of  Fluauohs^  ke.  (that 
in,  NewUm^$  Manuetripi  oh  Jimxi^iu)^  translated  bj  J.  CSobun 
and  published  in  1730 ;  and  the  GeotMetria  Anal^Uea^  printed 
in  1779  in  the  first  volume  of  Horsley's  edition  of  Newton's 
works. 

Ill  appearance  Newton  was  short,  and  towards  the  dose  of 
his  life  rather  stout,  but  well  set,  with  a  square  lower  jaw, 
urown  eyes,  a  broad  forehead,  and  rather  sliarp  featnret. 
Him  liair  turned  grey  before  he  was  tliirty,  and  remained  thick 
and  white  as  silver  till  his  death. 

Am  to  hiM  manners,  lie  drensed  slovenly,  was  rather  languid, 
a  ad  was  often  so  absorlied  in  his  own  thoughts  as  to  be 
a. #y tiling  but  a  lively  companion.  Many  anecdotes  of  his 
ex..reiue  absence  of  mind  when  engaged  in  any  investigation 
lia%*e  lieeii  prPM'rved.  Tlius  once  when  riding  home  from 
Griiitliam  lie  dismounted  to  lc»ul  his  horse  up  a  steep  hill, 
wlitiu  lie  turned  at  the  top  to  remount  he  found  that  he  had 
the  bridle  in  his  hand,  while  his  horse  had  slipped  it  and  gone 
away.  Again  on  the  few  occasions  when  he  sacrifioed  his  time 
to  iMitertJiiii  liirf  frieiuls,  if  he  left  them  to  get  more  wine  or  for 
any  rfimilar  reason,  he  would  as  often  as  not  be  found  after  the 
lapse  t»f  some  time  working  out  a  problem,  oblivious  alike  of 
his  expoctiiiit  guests  and  of  his  errand.  He  took  no  exercise, 
indulged  in  no  amusements,  and  worked  incessantly,  often 
spending  eighteen  or  .  'neteen  hours  out  of  the  twenty-four  in 
writing. 

In  character  he  was  religious  and  conscientious,  with  an 
exceptionally  high  standard  of  morality,  having,  as  Bishop 
liuriiet  said,  "  the  whitest  soul "  he  ever  knew.  Newton  was 
always  perfectly  straightforward  and  honest,  but  in  lus  con- 
troversies with  Lc*ibnitz,  Hooke,and  others,  though  scrupulously 
JuBt,  he  woH  not  generous ;  and  \t  ^Qu\d  «mui  that  he  frequently 


THE  LIFE  AND  WORKS  OF  NEWTON.  359 

took  oflenoe  at  a  chance  expression  when  none  was  intended. 
He  modestly  attributed  his  discoveries  largely  to  the  admirable 
work  done  by  hb  predecessors ;  and  once  explained  that,  if  he 
had  seen  farther  than  other  men,  it  was  only  because  he  had 
stood  on  the  shoulders  of  giantic  He  summed  up  his  own 
estimate  of  his  work  in  the  sentence,  "  I  do  not  know  what  I 
may  appear  to  the  world ;  but  to  myself  I  seem  to  have  lieen 
only  like  a  boy,  playing  on  the  sea-shore,  and  diverting  myself, 
in  now  and  then  finding  a  smoother  pebble,  or  a  prettier  shell 
than  ordinary,  whilst  the  great  ocean  of  truth  lay  all  undis- 
covered before  me."  He  was  morbidly  sensitive  to  being  in- 
volved in  any  discussions.  I  believe  that,  with  the  exception  of 
his  papers  on  optics,  every  one  of  his  works  was  published  only 
under  pressure  from  his  friends  and  against  his  own  wislies. 
There  are  several  instances  of  his  communicating  papers  and 
results  on  condition  that  his  name  should  not  be  published : 
thus  when  in  16GU  he  luid,  at  Collins  s  request,  solved  some 
problems  on  harmonic  series  and  on  annuities  which  had 
previously  liaflled  investigation,  he  only  gave  permission  that 
his  results  should  l)e  published  "so  it  lie,"  as  he  says,  ** without 
my  name  to  it :  for  I  see  not  what  there  is  desirable  in  public 
esteem,  were  I  able  to  acquire  and  maintain  it:  it  would 
perhaps  increase  my  acquaintance,  the  thing  which  I  chiefly 
study  to  decline." 

In  mathematical  power  he  has  never  been  surpassed :  of 
this  his  extant  works  are  the  only  proper  test.  Perhaps  the 
most  wonderful  single  illustration  of  his  powers  was  the  com- 
position in  seven  months  of  the  first  book  of  the  Principia, 

As  specific  illustrations  of  his  ability  I  may  mention  his 
solutiona  of  the  problem  of  Pappus,  of  John  Bemoulli't 
challenge,  and  of  the  question  of  orthogonal  trajectories.  The 
problem  of  Pappus,  here  alluded  to,  is  to  find  the  locus  of  a 
point  such  that  the  rectangle  under  its  distances  from  two 
given  straight  lines  shall  be  in  a  given  ratio  to  the  rectangle 
nnder  ifti  diatanoeB  froni  two  other  given  stmiiiSB^  Vr«»^ 
Many  gMNneirieiani  fran  the  tine  ot  K.^tdG^Mocen^n^Na^w^ 


S60  THC  UrE  AND  WORKS  OF  NIWTOir.  I 

find  a  geometrical  wdutioii  and  had  failed,  Imt  wImI  had  i 

proved  iniuiperable  to  hiii  fMredeoeMon  aeens  to  hate  pva>  j 

Mated  little  diflkul^  to  Newton  who  gave  an  elegant  deiBon- 
Btration  that  the  loeus  was  a  conie.  Qeometvj,  laid  JMgrukgp 
when  recommending  the  iitudy  of  analytii  to  hit  popibi  it  a 
strong  bow,  but  it  h  one  which  only  a  Newton  can  follj 
utilise.  As  another  eiuuiiple  I  may  mention  that  in  1696 
John  Bemonlli  cliallenged  mathematicians  (i)  to  determine  t 

the  brachiBtochroiie,  and  (ii)  to  find  a  cur\'e  such  that  if 
any  line  drawn  from  a  fixed  point  O  cut  it  in  P  and  Q  then 
OI^-t-OQ*  would  be  constant.  Leibnits  solved  the  fimt  of 
these  questions  after  an  interval  of  rather  more  than  six 
months,  and  then  suggested  they  should  be  sent  as  a 
challenge  to  Newton  and  others.  Newton  received  the 
problems  on  Jan.  29,  1G97,  and  tlie  next  day  gave  the 
complete  solutions  of  both,  at  the  same  time  generalising 
the  second  question.  An  almost  exactly  similar  case  occurred 
in  17 IG  when  Nea*ton  was  asked  to  find  the  orthogonal 
trajectory  of  a  family  of  curves.  In  Hve  hours  Newton 
solved  the  problem  in  the  form  in  which  it  was  propounded 
to  liim  and  laid  clown  the  principles  for  finding  tnijectori(*s. 

It  is  almost  impossible  to  describe  tlio  effect  of  Newton's 
writings  without  Is'ing  Huspected  of  exaggeration.  But,  if 
the  Htate  of  mathematical  knowledge  in  1GG9  or  at  the  death 
of  PitHcal  or  Fennat  lie  compared  with  what  was  known 
in  1G87,  it  will  be  seen  how  immense  was  the  advance.  In 
fact  we  may  nay  tlmt  it  took  mathematicians  half  a  century  or 
more  liefore  they  were  able  to  aH.similate  the  work  which 
Newton  had  produced  in  those  twenty  years. 

In  pure  geometry,  Newton  did  not  establish  any  new 
methodii,  but  no  modem  writer  has  shewn  the  same  power 
in  using  those  of  classical  geometry.  In  algebra  and  the 
theory  of  et|uatioiis,  lie  introduced  the  system  of  literal 
indices,  estiiblished  the  binomial  theorem,  and  created  no  in- 
considerable piirt  of  the  tli(*ory  of  equations :  one  rule  which 
hff  eaunviuUHl  in  this  subject  t«uia\ut^i\  \.\VV  a  few  years  ago 


ftUHMARY  OF  XEWTON'S  INVEKTinATIONfl.  361 

an  Qiiflolwd   riddle  which    had  overtaxed   the   mources  of 
Y  Hueceeding  matheniaiiciaris.     In  analytical  g«H>meiry,  he  intro- 

duced the  modem  clasHification  of  curven  into  alj[»eliraical  and 
tranflcendental ;  and  eHtahlished  many  of  the  fundamental 
properties  of  aayniptotefl,  multiple  pointn,  and  isolated  loops, 
illustrated  hy  a  discussion  of  cuhic  curves.  The  fluxional  or 
infinit€9«imal  calculus  was  invented  by  Newton  in  or  before 

g    ^ .   the  year   1666,  and  circulaterl   in   manuscript   amongst  his 

friends  in  and  after  the  year  1669,  though  no  account  of  the 
metliod  was  printed  till  1693.  The  fact  that  the  results  are 
now-a^lajTs  expressed  in  a  different  notation  has  led  to  Newton's 
investigations  on  this  subject  lieing  somewhat  overlooked. 

Newton  further  was  the  first  to  place  dynamics  on  a 
satisfactory  liasis,  and  from  dynamics  he  deduced  the  theory  of 
statics:  this  was  in  the  introduction  to  the  Prineipia  pulv 
lishod  in  1687.  The  theory  of  attractions,  tlie  application  of 
the  principles  of  mechanics  to  the  solar  system,  the  creation  of 
physical  astronomy,  and  the  establishment  of  the  law  of 
universal  gravitation  are  wholly  due  to  him  and  were  first 
published  in  the  same  work.  The  particular  questions  con- 
nected with  the  motion  of  tlie  earth  and  moon  were  worked 
out  as  fully  as  was  then  possible.  Tlie  theory  of  hydro- 
dynamics was  created  in  the  second  Ixiok  of  the  Priticijpia^ 
and  he  ailded  considerably  to  the  theory  of  hydmstatics  which 
may  be  said  to  have  l)een  first  discussed  in  modem  times  by 
Pascal.  The  theory  of  the  propagation  of  waves,  and  in  par- 
ticular the  application  to  determine  the  velocity  of  sound,  is  due 
to  Newton  and  was  published  in  1687.  In  geometrical  optics, 
he  explained  amongst  other  things  the  decomposition  of  liglit 
and  the  tlioory  of  the  rainljow;  he  invented  the  reflecting 
telesoc^  known  by  his  name,  and  the  sextant.  In  physical 
opticii  he  suggested  and  elaborated  the  emission  theoiy  of 
light 

The  abowliiii  does  not  exhaust  tkesuhjects  be  invesUs^id^itfk.. 
hit  it  will  serve  to  illustrate  Iww  inM\L«\  ^aaVSokK^fl^'owBffxt 
the  hminrj  ol  nmUiiiiiialica.    On  \m  'imfiMiil^  w^ 


/ 


862  TBI  UtI  AND  WOBKS  OF  XEWTOK. 

efliKta,  it  wilt  be  eiKtu^  U>  ijuote  the  mowrkii  of  two  or  three 
of  thoee  wbu  ware  iiulMM|UftDtly  concerned  with  the  eubject- 
mfttter  of  the  iVutef^iVi.  LagrAuge  described  the  /Viunpta  h 
the  gntXe»t  pruductiun  of  the  humui  mind,  uid  uid  ha  felt 
clued  *t  Mich  ui  illiiKtr&tion  uf  what  uian'i  intellect  might  be 
'  cepaUe.  Id  deecribiiig  the  effect  uf  hia  own  writings  ukd 
tlione  of  I^pUce  it  van  a  fuvuurite  renuirk  of  liis  timt  Newton 
wus  nut  onljr  tlie  gn-utest  genius  that  luut  ever  existed  bat  ha 
wiia  also  tlie  must  fortuiiutt-,  fur  iis  tlwre  is  but  uae  univen*:,  it 
okit  Lu|^n  but  to  one  iiiim  in  the  world's  luitory  to  be  the 
iiilerpivter  of  its  Ihwh.  Lnpluce,  who  is  in  general  very  sparing 
of  \m  pnuw,  inukph  uf  Newton  the  one  extvption,  and  the 
words  in  which  he  eimmerut*^  the  caustfH  which  "  will  alwayn 
aMHurn  to  the  i'riiicipiii  a  preeniincnce  uliuve  all  the  other  pro- 
fluctitins  of  bmiiBii  ^(duus  "  luve  Iiit-n  uft4'n  tjuoted.  Not  less 
rMiuirkuljlt'  is  tlie  humngi-  rendered  by  Ouusn :  fur  other 
gnitt  luathenmticiiiiiH  ur  philuHnpheni,  he  useil  the  epithets 
nuigiiUK,  ur  cluruH,  ur  cluri!>.siiima  ;  for  Newtun  nlune  he  kept 
till-   pivlix    BUUinms.  ,    Kiiudly    Uiut,    wliu    liud   iiuuir  n  npecial 

Htwiy  itf    Newtuii's    wurkH,  huuih  up  his  reituirks  l>y  saying, 
"vtimnit'  ip^iliiL'trf  et  euiniiiu  experiiileiilutt-ur  Newton  est  HAni    - 
('gilt ;  piir  \n  ri'uiiiini  dw  civ  deux  gi<iireM  de  g^iiies  k  Innr  [Jus 
liuut  degn'-,  it  est  sans  oxeniple." 


\ 


CHAPTER  XVII. 

LEIBNITZ   ANn  THE   MATIIEMATrclANH  OP  THE  FIRHT 
HALF  OF  THE   EKIHTEKSTH  CENTUBT*. 

I  HAVK  briefly  tmcnl  in  the  Ia.tt  ehapUr  the  luturp  nnd 
rxtcnt  of  NfiUin'n  con  triUut  ions  to  wipnee.  Mmlmi  Minlyrin 
in  howpver  <l«"riviH!  dirpctlj-  from  tiip  worku  of  licibiiitx  mid 
tlie  elfter  Bpmoullis ;  anri  it  i*  immaterial  to  tn  whrtWr  the 
fnndAmpntAl  ideivn  of  it  wore  ohtninnl  \iy  tlipm  from  Npwtoit, 
or  discuvcrpd  intlpppnHpntly.  TIh'  En^linh  mathrnuiticianH  of 
Ihp  yeara  connidprpt)  in  thin  rhnptiT  continued  to  uw  thn 
Iiuif^itS^  AX''  notntinn  ot  Newton:  they  nrn  than  MMnewhnt 
diKtinct  (mm  their  ctinlinpntnl  i:^>nten)poruiFii,  mm!  I  hnre 
thrrrfom  jtroujied  tlipiu  tii^'tlter  iti  ii  HPction  by  themnelveti. 

Iieibnitt  and  the  BernonUit. 

Ziaibnitst-  (lollfrird  Withrtm  f^ilyniti  (nr  Z»i6mz)  wu 
hom  At  1ieipr.ig  on  June  :>I  (O.  8.),  1646,  nnd  died  nt  HtuioTer 
on  Not.  14,  lil6.  His  father  dird  Wforr  he  wm  aix,  ftnd  U)« 
tenchinK  "t  the  school  to  which  he  wiw  then  went  wm  ineffi- 
cient, Imt  his  indastry  tnumpheil  over  nil  difficnltim  ;  by  tlw 
time  lie  wiu  twelve  he  hud  tau){ht  himself  In  reiwl  lAtin  ranily, 
mmI  hftd   began  Greek;   tuid   liefore  he  wiu   twenty  be   h«d 

*  Hee  Caobn-,  toI.  ni ;  other  kDtl>oritieii  tot  Uw  MllwmtUeifcai  et 
the  period  urt  m«nlinfied  in  the  tontnole*. 

t  Be«  the  lite  ol  Leibniu  b;  O.  E.  Uahimaer,  S  niliiMai  and  • 
•Mnt,    Bmlaa,  1943  and    1846.    Leibniti 
beett    oollcded    ud  adibd  by  0.  J.   Oerhnnll  ia  7  i 
aad  BaOa,  IB4»-fiS. 


S64  LiiBNrnL 

nuuitered  the  ordinAiy  text-booka  on  umUhf&auttie^  pliik^ 
■opiijr,  theology,  and  Uw.  Kefuaed  the  degree  of  doelor  of 
kwa  at  Leipiig  by  tboiie  who  were  Jealona  of  his  joath  and 
leamiiig,  he  moved  to  Nareinberi^.  An  etaajr  whieh  he  thero 
wrote  on  the  atudy  of  Uw  waa  dedicated  to  the  Elector  of 
MainZy  and  led  to  hia  appointment  by  the  elector  on  a  oomniia- 
aion  for  the  reviaion  of  aouie  atatutea,  from  which  he  waa 
aubnequently  promoted  to  the  diplomatic  aervic&  In  the 
latter  cupncity  he  supported  (unsuooesiifully)  the  daima  of  the 
Geniiaii  candidate  for  the  crown  of  Poland.  The  violent 
seizure  of  variouH  amall  placea  in  Alaace  in  1760  excited 
univeraol  alarm  in  Germany  aa  to  the  deaigna  of  Louia  XIY ; 
and  Leibnitz  drew  up  a  scheme  by  which  it  waa  propoaed  to 
oflfer  CSermaa  co-operation,  if  France  liked  to  take  Egyp«  and 
use  the  possession  of  tliat  country  as  a  basis  for  attack  against 
Holland  in  Asia,  provided  France  would  agree  to  leave  Oer- 
uiany  unciisturbecl.  This  beairs  a  curious  resemblance  to  the 
Miniilar  plan  by  which  Napoleon  I  propos<H]  to  attack  England. 
In  IG72  Lt'ibnitz  went  to  Paris  on  the  invitation  of  the 
French  govenunent  to  explain  the  deUiils  of  the  scheme,  but 
nothing  came  of  it. 

At  Paris  he  met  Huygens  who  was  then  residing  there, 
and  their  conversation  led  lieibnitz  to  study  geometry,  which 
he  descrilied  as  opening  a  new  world  to  him ;  Uiough  as  a 
matt4*r  of  fact  he  had  previously  written  some  tracts  on 
various  minor  points  in  mathematics,  the  moat  important 
being  a  paper  on  combinations  written  in  l(iG8,  and  a  descrip- 
tion of  a  new  calculating  machine.  In  January,  IG73,  he  waa 
sent  on  a  political  mission  to  London,  wliere  he  stopped  some 
montliH  and  nuule  the  acquaintance  of  Oldenburg,  OollinSi  and 
others :  it  was  at  this  time  that  he  communicated  tlie  memoir 
to  the  Koyal  Society  in  which  he  was  found  to  luive  lieen 
fonnitalled  by  Mouton. 

In  IG73  tlie  elector  of  Mainz  died,  and  in  the  following  year 

lieibnit^  entered  the  service  of  the  Brunswick  family ;  in  1676 

he  again  visited  London,  and  iUeu  u\ov«d  \a^  ll«a\over^  where, 


\ 


LE1BN1T3S.  365 

till  his  death,  he  occupied  the  well-paid  pjHt  of  lihrarian  in 
the  ducal  library.  His  pen  was  thenceforth  employed  in  all  the 
political  iDattem  which  affected  the  Hanoverian  family,  and  his 
services  were  recognized  by  honours  and  distinctions  of  ^*arious 
kinds :  his  memoranda  on  the  various  political,  iiistorical,  and 
tlieological  questions  which  concerned  the  dynasty  during  the 
forty  years  from  1673  to  1713  form  a  valuable  contributicm  to 
the  history  of  that  time. 

Leibnitz's  appointment  in  the  Hanoverian  scr^'ice  gave 
him  more  time  for  his  favourite  pursuits.  He  used  to 
assert  that  as  the  first-fruit  of  his  incrpaMxl  leisure  he 
invented  the  differential  and  integral  calculus  in  1674,  fiut 
the  earliest  traces  of  the  use  of  it  in  his  extant  note-books 
do  not  occur  till  1675,  and  it  was  not  till  1677  that  we 
find  it  developed  into  a  c«msistent  system :  it  was  not  pul>- 
lished  till  1684.  Nearly  all  his  mathematical  papers  were 
produced  within  the  ten  years  fnnn  1682  t4i  1692,  and  m<Mt  of 
them  in  a  journal,  called  the  Acta  Entffifoniwi^  which  he  and 
Otto  Mencke  had  founded  in  1682,  and  which  had  a  wide 
circulation  on  the  continent. 

Leibnitz  occupies  at  least  as  large  a  place  in  the  history  of 
philosophy  as  he  does  in  the  history  of  mathematicM.  Mcmt  of 
his  philosophical  writings  were  composed  in  the  last  twenty  or 
twenty-five  years  of  his  life  ;  and  the  point  as  to  whether  his 
views  were  original  or  whether  they  were  appropriated  from 
Spinoza,  whom  he  vinited  in  1676,  is  still  in  question  among 
phikMophers,  though  the  evidence  seems  to  point  to  the  origin- 
ality of  Leibnitz.  As  to  Leilmitz's  system  of  philosophy  it 
will  be  enough  to  say  that  he  regarded  the  ultimate  elements 
of  the  universe  as  individual  percipient  beings  whom  he  called 
monads.  According  to  him  the  monads  are  centres  of  force, 
and  substance  is  force,  while  qmce,  matter,  aiid  motion  are 
nerdy  phenomenal:  finally  the  existence  of  God  is  inferred 
from  the  existing  harmony  among  the  monads.  His  services 
to  litoimtnre  were  almost  at  considerable  as  those  to  philosophy ; 
in  pwtienhr  I  may  sini^ls  <mi  Us  averthhiw  of  tbe  ilisa 


366  LOiiNrnL 

prevalent  belief  ihai  Hebrew  wm  tbe  priiueval  hngnefe  el 
the  hunum  nioei 

In  1700  tbe  Academy  of  Berlin  waa  created  on  bk  adviec^ 
and  he  drew  np  tbe  firMt  body  of  statates  lor  it  On  tbe 
aoceasion  in  1714  of  bis  mastery  George  I,  to  tbe  tbrone  of 
England,  Leibniti  waa  thrown  aside  asj  a  nseleM  tod; 
he  was  forbidden  to  come  to  England;  iumI  tbe  last  two 
years  of  his  life  were  spent  in  neglect  and  disbonoor. 
He  died  at  Hanover  in  1716.  He  was  overfond  of  money 
and  pehioiial  distinctions;  was  uiiscnipolpiiSi  as  might  be 
expected  of  a  professional  diplomatist  of  tl^t  time ;  but  poa- 
scKsed  singularly  attractive  mamiern,  and  sill  who  once  came 
under  the  charm  of  his  personal  presence  remained  sincerely 
attached  to  him.  His  mathcMiiatica)  reputation  was  largely 
augmented  by  the  eminent  pusitiou  tliat  he  occupied  in 
diplomacy,  philosophy,  and  literature;  and  the  power  thence 
derived  was  considerably  increaMcd  by  his  influence  in  tbe 
management  of  the  Ada  J^rwliiorutn.         | 

The  last  years  of  his  life — from  1709  to  1716— were  em- 
hit  t«Ted  by  the  long  controversy  with  John  Keill,  Newton, 
and  others  as  to  whether  he  liad  discoverra  the  differential 
calculus  independently  of  Newton's  previous  investigations  or 
whether  he  had  derived  the  fundamental  idea  from  Newton 
and  merely  invented  another  notation  for  it..  The  controversy* 
occupies  a  place  in  the  scientific  history  of  i  the  early  years  of 

*  The  cane  iu  fkvoar  of  the  iudtipeudent  iuvWntion  bj  Leibnits  is 
Htaied  in  (ierhardt'ii  LeihuizrHt  wMthematitcke  Ktkh/leM ;  and  in  the 
third  volume  of  M.  Cantor's  iie$ehichu  drr  Muikeutatik.  The  argnuents 
on  tlie  other  vide  are  piven  in  U.  Slouian*e  IstibnitzeM  Amtprttck  nuf 
die  KrjiuiiMHif  tUr  DijferetuitilreckHUMg,  LeipziKi,'  1857,  of  which  an 
Engliidi  traniflation,  with  additions  by  Dr  Sloni^n,  waa  published  at 
Cambridge  in  1800.  A  summary  of  the  evidence  wiU  be  found  in  O.  A. 
Gibtiou's  memoir,  Proceediugt  of  the  KdiHbunjk  Matkematieal  Socieig, 
vol.  XIV,  1896,  pp.  148 — 174.  The  history  of  the  invention  of  the  eal- 
cuius  is  given  in  an  article  on  it  in  the  ninth  edition  of  the  Emqf- 
eiopaedia  BritaHHtea,  and  in  P.  Mansion's  Es^uitM  de  Fkistoire  dm 
calcul  imfiniiethukU  Osnd,  1881. 


\ 


LEiBNny^  367 

the  eighteenth  century  quite  dispniportionate  to  its  true 
importance,  but  it  so  materially  aflectcd  the  hintory  of  mathe- 
matics  in  western  Europe,  that  I  feel  obliged  to  give  the 
leading  facts,  though  I  am  reluctant  to  take  up  so  much  space 
with  questions  of  a  personal  character. 

The  ideas  of  the  infinitesimal  calculus  can  lie  exprcfised 
either  in  the  notation  of  fluxions  or  in  that  of  difl<*rrntialK. 
Tlie  former  was  used  by  Newton  in  I  GOG,  and  communicated 
in  manuscript  to  his  friends  and  pupils  from  IGG9  fmwanls, 
but  no  distinct  account  of  it  was  printe<l  till  1 G93.  The  earliest 
use  of  the  latter  in  the  note-books  of  Leibnitz  may  be  probably 
referred  to  1G75,  it  was  employe«l  in  the  Iett4*r  sent  to  Newton 
in  1677,  Aiid  an  account  of  it  was  printed  in  the  memoir  of 
1684  described  below.  There  is  no  question  that  the  diffe- 
rential notation  is  due  to  Leibnitz,  and  the  sole  question  is  as 
to  whether  the  general  idea  of  the  calculus  was  taken  from 
Newton  or  discovered  independently. 

The  case  in  favour  of  the  independent  invention  by 
Leibnitz  rests  on  the  ground  that  he  pulilished  a  description 
of  his  method  some  years  before  Newton  printed  anything  on 
fluxions,  that  lie  always  alluded  to  the  discovery  as  being  his 
own  invention,  and  that  for  some  years  this  statement  was 
unchallenged ;  while  of  course  there  must  be  a  strong  pre- 
sumption that  he  acted  in  good  faith.  To  rebut  tliis  case  it  is 
necessary  to  shew  (i)  that  he  saw  nome  of  Newton's  papers  on 
the  sulgect  in  or  before  1G75  or  at  least  1677,  and  (ii)  that  he 
thence  derived  the  fundamental  ideas  of  the  calculus.  The 
fact  that  his  claim  was  unchallenged  for  some  years  is,  in 
the  particular  circumstances  of  the  case,  immaterial 

That  Leibnitz  saw  some  of  Newton's  manuscripts  was 
always  intrinsically  probable;  but  when,  in  1849,  C.  J. 
Gerhardt*  examined  Leibnitz's  papers  he  found  among  them 
A  manuscript  copy,  the  existence  of  which  had  been  previously 
WMOspeetedv  in  Leibnitx's  handwriting  of  extracts  from 
Newton's  Ih  Analfri  per  BfmUumm  Jfumero  Terminorwm 
*  CiwlwiiHi  ItfftalrfWi  wmikemmiiteke  Mkrfflca^'^^v^*^* 


/ 


368  UUAKITZ. 

l^fiMtM  (whieh  was  printMl  in  ilie  At  Q^mAmhurm 
in  1704)|  U^getlier  with  noien  on  tbeir  expnanion  in  ilis 
ciifleronlial  notation.  The  qoention  of  the  date  ni  whioh 
thow  extnustc  were  made  in  therefore  all  important.  It  ia 
known  tliat  a  copy  of  Newton's  nianniicript  had  been  tent 
to  Ttfcliirnhattiien  in  May,  1675,  and  an  in  that  year  he  and 
Leibniti  were  engaged  together  on  a  piece  of  work,  it  is  not 
iiupoHsible  that  these  extracts  were  made  then*.  It  is  also 
puMiiilile  that  they  may  have  been  made  in  1676,  for  Leibnits 
disciuwed  the  question  of  analysis  by  infinite  series  with 
Collins  and  Oldenburg  in  that  yesr,  and  it  is  ^  ^rwri  pioljable 
that  they  would  have  then  shewn  him  the  manosoript  of 
Newton  on  that  subject,  a  copy  of  which  was  possessed  by  one 
or  both  of  them.  On  tlie  other  hand  it  may  be  supposed  that 
Leibnitz  iuade  the  extracts  from  the  printed  copy  in  or  after 
1704.  Leibnitz  shoKly  before  his  death  admitted  in  a  letter 
to  Conti  tliat  in  167G  Collins  had  shewn  him  sume  Newtonian 
papers,  but  implied  that  they  were  of  little  or  no  value — pre- 
sumably he  referred  to  Newton's  letters  of  June  13  and 
Oct.  24,  1C76,  and  to  the  letter  of  Dec.  10,  1672  on  the 
method  of  tangents,  extracts  from  which  accompanied  t  the 
letter  of  June  13 — but  it  iM  curious  tlmt,  on  the  receipt  of 
thc«e  letters,  Leibnitz  should  liave  made  no  further  inquiries, 
unless  he  was  already  aware  from  other  sources  of  the  method 
followed  by  Newton. 

Whether  Leibnitz  made  no  uho  of  the  manuscVipt  from 
which  he  had  copied  extracts,  or  whether  he  had  previously 
invented  the  calculus  are  questions  on  which  at  this  distance 
of  time  no  direct  evidence  is  available.  It  is  however  worth 
noting  that  the  uupublLshed  Portsmouth  Papers  sliew  tlmt, 
when,  in  1711,  Newton  went  carefully  into  the  whole  dispute, 
he  picked  out  this  maimscript  as  the  one  wliicli  had  probably 
somehow  fallen  into  the  hands  of  Leibnitz  {.     At  that  time 

*  Blunum,  English  trsniiUiion,  p.  34. 

t  Oerhanll,  vol.  i,  p.  91. 

^  Catalogue  of  ForUmouth  Payen^  pp.  zvi,  z?n,  7,  S. 


LEIBNITZ.  369 

there  was  no  direct  evidence  that  Leibniti  had  seen  thia 
mnuttficripi  before  it  was  printed  in  1704,  aiid  accordingly 
Newton's  conjecture  was  not  published ;  but  Gerhardt's  dis- 
covery of  the  copy  niade  by  Leibnitz  tends  to  confirm  the 
accuracy  of  Newton's  judgment  in  the  matter.  It  is  said  by 
some  that  to  a  man  of  Leibnitz's  ability  the  manuscript^ 
especially  if  supplemented  by  the  letter  of  Dec.  10,  1672, 
would  supply  sufficient  hints  to  give  him  a  clue  to  the  methods 
of  the  calculus,  though  as  the  (luxional  notation  is  not  em- 
ployed in  it  anyone  who  used  it  would  have  to  invent  a 
notation ;  but  this  is  denied  by  others. 

There  wns  at  first  no  reaHoii  to  suHiiect  the  ^nnI  faith  of 
Leibnitz;  and  it  was  not  until  the  appearance  in  1704  of  an 
anonymous  review  of  Newton's  tract  on  quadrature,  in  which 
it  was  implied  that  Newton  had  borrowed  the  idea  of  the 
fluxioiial  calculus  from  I^eibnitz,  that  any  responsi1>le  mathe- 
matician^ questioned  the  statement  that  fieibnitz  had  invented 
the  calculus  independently  of  Newton.  It  is  universally 
admitted  that  there  was  no  justification  or  authority  for  tlie 
statements  made  in  this  review,  which  was  rightly  attributed 
to  Leibnitz.  But  the  subsequent  discussion  led  to  a  critical 
examination  of  tlie  whole  question,  ami  doubt  was  exprpssed 
as  to  whether  Leilniitz  had  not  derived  the  fundamental  idea 
from  Newton.  Tlie  case  against  lieilmitz  as  it  appeared  to 
Newton's  friemis  was  summed  up  iii  the  Commereium  EpiMo- 
/icMiif  issued  in  1712,  ami  detailed  references  are  given  for  all 
the  facts  mentioned. 

No  such  summary  (with  facts,  dates,  and  references)  of 
the  case  for  Leibnitz  was  issued  by  his  friends;  but  John 
Bernoulli  attempted  to  indirectly  weaken  the  evidence  lij 
Attacking  the  personal  character  of  Newton:  this  was  in  a 
letter  diOed  June  7,  1713.  The  charges  were  false,  and, 
when  pressed  for  an  explanation  of  them,  Bemonlli  uott 
■olemnly  denied  having  written  the  letter.     In  aooepUng  the 

•  la  16W  DaJUisg  haa  sssussa  LsikuU  of  pisgkrism  froBS  Itewlen, 
Wl  Diriilkr  was  Bol  a  fsfson  of  aiMb  infortsnes. 


/ 


S70  LBIBNITX. 

denial  Newton  added  in  a  private  letter  to  Um  tlie  ioikmi^g 
remarks  which  are  interesting  as  giving  Newton's  aooooni  el 
why  he  was  at  last  induced  to  take  anj  part  in  the  eon- 
trovemy.  **  I  have  never/*  said  he^  *Vgrasped  at  fiune  amoQg 
foreign  nationii,  but  I  am  very  desirous  to  preserve  my  dia- 
racter  for  honesty,  which  the  author  of  that  epistle^  as  if  hy 
tlie  authority  of  a  great  judge,  liad  endeavoured  to  wrest  from 
me.  Now  Uiat  I  am  old,  I  have  little  pleasure  in  mathematical 
studies,  and  I  have  never  tried  to  propagate  my  opinions  over 
the  world,  but  have  rather  taken  care  not  to  involve  myadf 
in  disputes  on  account  of  them." 

Iieibnitz*s  defence  or  explanation  of  his  silence  is  given  in 
the  following  letter,  dated  April  9,  1716,  from  him  to  CSontL 
*'  Pour  repoudre  de  point  en  point  k  Touvrage  public  centre 
raoi,  il  falloit  un  autre  ouvrage  aussi  grand  pour  le  moins  que 
eelui-la  :  il  falloit  entrer  dans  un  grand  detail  de  quantity  de 
minuties  posneeM  il  y  a  trento  k  quarante  ans,  ckmt  je  ne  me 
mmvenois  ^(*r« :  il  me  falloit  cheix*lier  mes  vieilles  lettres, 
doiit  plusieum  ,He  Hont  perdui^s,  outre  que  le  plus  souvent  je 
ii'ai  point  garde  les  minutes  des  niienneH :  et  les  autres  sont 
cnHevelies  dans  un  grand  tan  de  papiem,  que  je  ne  ponvois 
debrouiller  qu'avec  du  tenipn  et  de  la  patience ;  mais  je  n*en 
aviiiH  guere  le  loisir,  ^tant  charge  pr6ientement  d'occupations 
d'une  toute  autre  nature." 

The  death  of  Leibnitz  in  171G  only  put  a  temporary  stop 
to  the  controversy  which  was  bitterly  debated  for  many  years 
later.  The  question  is  one  of  ditliculty ;  the  evidence  is 
conHicting  and  circumstantial ;  and  every  one  must  judge  for 
hiniH(*lf  which  opinion  seems  most  reasonable.  Essentially  it 
is  a  case  of  Jjeibiiitz's  word  against  a  number  of  suspicious 
details  pointing  against  him.  His  unacknowledged  possession 
of  a  copy  of  part  of  one  of  Newton's  manuscripts  may  be 
explicable;  but  the  fact  that  on  more  than  one  occasion  he 
deliberately  alteretl  or  added  to  important  documents  {er,  gr, 
the  letter  of  June  7,  1713,  in  the  CharUi  Voiafig^  and  that  of 
April  8,  171G,  in  the  AeUi  Eniditur%im\  liefore  publishing 


\ 


LEIBNITZ. 


371 


them.  Mid,  what  is  worse,  that  a  material  date  in  one  of  his 
manuscripts  hasi^ii  falsifted*  (1675  being  altered  to  1673), 
makes  his  own  testimony  on  the  subject  of  little  value.  In 
spite  of  this,  I  believe  the  majority  of  niorlem  writers  would 
accept  the  view  iliat  proliably  licibnitz's  invention  of  the 
calculus  was  independent  of  that  (»f  Newton,  but  on  the 
whole,  I  think  it  likely  that  Leibnitz  renil  parts  or  the  whole 
of  Newton's  manuscript  De  Anafyin  before  1677 ;  how  much 
he  was  assisted  by  it  it  is  more  difficult  to  say.  It  must  be 
recollected  that  what  he  is  alle^^ed  to  have  received  was  rather 
a  number  of  suggestions  than  an  account  of  the  calculus ; 
and  it  is  possible  that  as  he  did  not  publish  his  results  of 
1677  until  1684,  and  that  as  the  notation  and  subsequent 
development  of  it  were  all  of  his  own  invention,  he  may 
have  been  led,  thirty  years  later,  to  minimize  any  assistance 
which  he  had  obtained  originally,  and  finally  to  consider  that 
it  was  immateriaL 

If  we  must  confine  ourselves  to  one  system  of  notation 
then  there  can  lie  no  doubt  that  that  which  was  invented  by 
Leibnitz  is  better  fitted  for  most  of  the  purposes  to  which  the 
infinitesimal  calculus  is  applied  than  that  of  fluxions,  and 
for  some  (such  as  the  calculus  of  variations)  it  is  indeed 
almost  essential.  It  should  lie  rememliered  however  tliat  at 
the  beginning  of  the  eighteenth  century  the  methods  of  the 
infinitesimal  calculus  had  not  Iteen  S3'stematized,  and  either 
notation  was  equally  good.  The  development  of  that  calculus 
was  the  main  work  of  the  mathematicians  of  the  first  half  of' 
the  eighteenth  century.  The  differential  form  was  adopted  by 
continental  mathematicians.  The  application  of  It  by  Enler, 
Lagrange,  and  Laplace  to  the  principles  of  mechanics  Ui4 
down  in  the  /Viiurtpia  was  (he  great  achievement  of  the  hist 
half  of  that  century,  and  finally  demonstrated  the  superioritj; 
of  tbe  diflbrcntial  to  tbe  fluxionAl  calcolua.  The  tnuislatioo  ojf 
tbe  Primeipia  into  the  langnage  of  modem  analysis  and  tbe 

•  OMtar,  iHw  adfoeaiss  LsihnHa's  slaias,  IhiidDi  UmA  «ka 
Mil  be  Isksn  to  be  LAiriU'^  MTt;  w 


872  Lnmnn. 

ilUiiig  in  ol  tbe  deUak  of  the  Newtoniui  tiieoiy  fay  Ilia  aid  of 
that  analysis  wara  afleoted  by  Titplaoa 

The  contioveny  with  Leibnits  was  regarded  in  Rngland  aa 
an  attempt  by  foreignen  to  defmod  Newton  of  Ilia  eiedit  of 
his  invention,  and  the  qoesUon  was  complicated  on  both  sides 
by  national  jealousies.  It  was  therefore  natniml  thoagh  it  was 
unfortunate  that  in  England  the  geometrical  and  flnzional 
methodfl  as  used  by  Newton  were  alone  studied  and  employed. 
For  more  tlian  a  century  the  English  school  was  thus  out 
of  touch  with  continental  mathematicians.  The  conse- 
quence was  that,  in  spite  of  the  brilliant  band  of  scholars 
formed  by  Newton,  the  improvements  in  the  methods  of 
analyHiii  gradually  etfected  on  the  continent  were  almoat 
unknown  in  Britain.  It  was  not  until  1820  that  the  value 
of  analytical  methods  was  fully  recognised  in  England,  and 
tlijit  Kewton'H  countrymen  again  took  any  large  share  in  the  I 

development  of  luatheniatics. 

Iit!aving  now  this  long  controversy  I  come  to  the  dis- 
cuMiiion  of  the  mathematical  papeni  produced  by  Leibnits, 
all  the  more  importunt  of  which  were  published  in  the  Acta 
EruiiUiMTuitL  They  are  mainly  concerned  with  applications 
of  the  iiitiiiitesimal  calculus  and  with  various  questions  on 
mechanics. 

The  only  papers  of  tirst-rate  importance  which  he  produced 
are  those  on  the  ditFereatial  calculus.  The  earliest  of  theae 
waH  one  publiHlied  in  the  Acta  Erudiiorum  for  October,  1684, 
in  which  lie  enunciated  a  general  metliod  for  tioding  mi^if^mii 
and  iiiiiiima,  and  for  drawing  tangents  to  curves.  One 
invenie  problem,  namely,  to  find  the  curve  whotie  subtangent 
w  constant,  wiis  also  discUBsed.  The  notation  is  the  same 
as  that  with  which  we  are  familiar,  and  the  differential 
coetticients  of  sc^  and  of  products  and  quotients  are  determined. 
In  1G8G  he  wrote  a  paper  on  the  principles  of  the  new 
calculus.  In  both  of  these  papers  the  principle  of  continuity 
is  explicitly  asHumed,  while  his  treatment  of  the  subject  is 
baaed  on  the  use  of  infinitesimals  and  not  on  that  of  the 


LEIBNITZ. 


373 


limiting  vtUue  of  ratios.  In  aniiwcr  to  some  oljectiona  which 
were  raised  in  1694  by  Bernard  Nieuweniyt  who  asserted  that 
dy/cir  Mftood  for  an  unmeaning  quantity  like  0/0»  Leibnits 
explained,  in  the  same  way  as  Barrow  had  previously  done, 
that  the  value  of  dyjdx  in  gef>metry  could  be  expressed  as 
the  ratio  of  two  finite  quantities.  I  think  that  Leibnitz's 
statement  of  the  olijects  and  methods  of  the  infinitenimal 
calculus  as  contained  in  these  papers,  which  are  the  throe 
most  important  memoirs  on  it  that  he  produced,  is  somewhat 
obscure,  and  his  attempt  to  place  the  subject  on  a  meta- 
physical basis  did  not  tend  to  clearness;  but  the  fact  that 
all  the  results  of  modem  mathematics  are  exprcsned  in  the 
language  invented  by  Leibnits  has  proved  the  best  monument 
of  his  work. 

In  16^6  and  1692  he  wrote  papers  on  osculating  curves 
These  however  contain  some  bad  blunders;  as,  for  example, 
the  assertion  that  an  osculating  cirole  will  necessarily  cut 
a  curve  in  four  consecutive  points:  this  error  was  pointed 
out  by  John  Bernoulli,  liut  in  his  article  of  1692  Leibnits 
defended  his  original  asserti<m,  and  insisted  that  a  cirele 
oould  never  cross  a  curve  where  it  touched  it. 

In  1692  Leibnits  wrote  a  memoir  in  which  he  laid  the 
foundation  of  the  theory  of  envelopes.  Tliis  was  further 
developed  in  another  paper  in  1694,  in  which  he  introduced 
for  the  first  time  tlie  terms  *' coordinates  **  and  '^axes  of 
coordinates." 

Leibnits  also  published  a  good  many  papers  on  mechanical 
subjects;  Imt  some  of  them  contain  mistakes  which  shew 
that  he  did  not  understand  the  principles  of  the  subject. 
ThuS|  in  1685,  he  wrote  a  memoir  to  find  the  pressure  exerted 
by  a  sphere  of  weight  W  placed  between  two  inclined  planes 
of  complementary  inclinations,  placed  so  that  the  lines  of 
greatest  slope  are  perpendicular  to  the  line  of  the  intersection 
of  the  planesi  He  asserted  that  the  pressure  on  each  plane 
must  ooosifll  of  two  eomponentsi  ''unum  quo  dediviter 
descendere  lendit,  mltemm  qui*  planum  dedive  premit."  He  • 
fuftber  miA  fhaX  isr  meiaphysiQal  fWMUs^abia  woa.  dl  ^dub  V«^ 


874 

prenttiw  nvftt  be  equal  to  W.  Heoee^  if  M  end  JP  be  Ike 
required  praeMiren^  and  e  end  (v-e  the  ifidineftiniM  of  llie 
pUneii,  he  finds  thai 

/f=iir(l-ttna^coea)  aiid  iT^) ir(l -coee-frine). 

The  tnie  valueii  are  if^lfcosa  and  Jt^Wuam,  Never- 
theless soine  of  his  papers  on  mechanics  are  valuable.  Of 
these  the  most  important  were  two^  in  1689  and  1694,  in 
which  he  solved  the  problem  of  finding  an  isoohronops  curve ; 
one,  in  1697,  on  the  curve  of  quidcest  descent  (this  was  the 
problem  sent  as  a  challenge  to  Newton);  and  two,  in  1691 
and  1692,  in  which  he  stated  the  intrinsic  equation  of  the 
cur\'e  assumed  b}*  a  flexible  rope  suspended  from  two  pointsi 
tliat  is,  the  catenary,  but  gave  no  proof.  This  last  problem 
had  been  originally  proposed  by  Galilea 

In  1689,  tliat  Im,  two  years  after  the  Prifteipia  had  been 
publishod,  he  wrote  on  the  movements  uf  the  planets  which 
liu  Htatird  were  produceil  by  a  motion  uf  the  ether.  Not  only 
were  the  equatioiiM  of  motion  which  he  obtained  wrong,  but  1^ 

bib  deductions  from  tlieni  were  not  even  in  ac€»rdance  witli  j^ 

his  own  axioms.  In  another  memoir  in  1706,  that  is,  nearly 
twenty  years  after  the  Principui  had  Ijeen  written,  he  admitted 
tliat  he  had  made  some  mistakes  in  his  former  paper  but 
adhered  to  his  previous  conclusions,  and  summed  the  nmtter 
up  by  saying  **it  is  certain  that  gravitation  generates  a 
new  force  at  each  instant  to  the  centre,  but  the  centrifugal 
force  also  generates  another  awsy  from  the  centre.... The 
centrifugal  force  may  l>e  considered  in  two  aspects  according 
as  the  movement  is  treated  as  along  the  tangent  to  the  curve 
or  as  along  the  arc  of  the  circle  itself."  It  seems  clear  from 
this  paper  that  he  did  not  really  understand  the  principles  uf 
dynamics,  and  it  is  lianlly  necessary  to  consider  his  work  on 
the  bubject  in  further  detail.  Much  of  it  is  vitiated  by  a  con- 
stant confusion  lietween  momentum  and  kinetic  energy :  when 
the  force  is  *'  piisMive  '*  he  uses  the  first,  which  he  calls  the  vU 
muriua,  as  tlie  measure  uf  a  force ;  when  the  force  is  '*  active" 
he  uses  the  Jatter,  the  double  of  which  he  calls  the  vis  viva. 


ji 


\ 


\ 


LEIBNITZ. 


375 


The  series  quoted  f>y  lipilHiiiz  conipriNe  those  for  «*, 
log(l-i-x),  sinjc,  versa*,  and  tan^'x;  all  of  thetie  had  Ijeen 
previously  published,  and  he  rarely,  if  ever,  added  any 
demonstrations.  Leilmitz  (like  Newton)  recognized  the  im- 
portance of  James  Grpgory's  remarks  on  the  necessity  of 
examining  whether  infinite  series  are  convergent  or  divergent^ 
and  proposed  a  test  to  distinguish  series  whose  terms  are 
alternately  positive  and  negative.  In  1693  he  explained  the 
method  of  expansion  by  indeterminate  coefficients,  though  his 
applications  were  not  free  from  error. 

To  sum  the  matter  up  briefly,  it  seems  to  me  that  lieibnitz's 
work  exhibits  great  skill  in  analysis,  but  much  of  it  is  un- 
finished, and  when  he  leaves  his  symbols  and  attempts  to 
interpret  his  results  he  frequently  commits  blunders.  No 
doubt  the  demands  of  politics,  philo80|)hy,  and  literature  on 
his  time  may  have  prevented  him  from  elaborating  any 
problem  completely  or  writing  a  systematic  exposition  of 
his  views,  though  they  are  no  excuse  for  the  mistakes  of 
principle  which  occur  in  his  papers.  Some  of  his  memoirs 
contain  suggestions  of  methods  which  have  now  become  valu- 
able means  of  analysis,  such  as  the  use  of  determinants  and  of 
indeterminate  coefficients:  but  when  a  writer  of  manifold 
interests  like  Leibnitz  throws  out  innumerable  suggestions, 
some  of  them  are  likely  to  turn  out  valuable ;  and  to  enume- 
rate these  (which  he  did  not  work  out)  without  reckoning  the 
others  (which  are  wrong)  gives  a  false  impression  of  the  value 
of  hia  work.  But  in  spite  of  this,  his  title  to  fame  rests  on  a 
sure  basis,  for  by  his  advocacy  of  the  diflerential  calculus  his 
name  is  inseparably  connected  with  one  of  the  chief  instru- 
ments of  analysis,  just  as  that  of  Descartes — another  philoso- 
pher— is  with  analytical  geometry. 

Leibnitz  was  but  one  amongst  several  continental  writers 
whose  papers  in  the  Ada  Kmdiiantm  familiarized  mathe- 
niaticians  with  the  use  of  the  differential  calculus.  The  most 
important  of  tlwse  were  James  and  John  Bemovlli,  both  of 
wlioai  wen  warm  friends  and  admirers  of  Leibnita^  anfi  \i^ 


S76  JAMU  BKBNOUUX 

their  devoted  advooAcy  kis  repoUlion  is  largely  doei  No4 
only  did  they  take  a  prominent  paK  in  nearly  eveiy  niath^ 
matical  quention  then  diiicasiied,  bat  nearly  all  the  leadii^ 
mathematicians  on  the  continent  during  the  first  half  of  the 
eighteenth  century  came  directly  or  indirectly  under  the 
influence  of  one  or  both  of  them. 

The  BemouUiii*  (or  as  they  are  sometimes,  and  perhaps 
more  correctly,  called  the  Bemouillia)  were  a  family  of  Dutch 
origin,  who  were  driven  from  Holland  by  the  Spanish  peraecu- 
tion^s  and  finally  nettled  at  Bsle  in  Switierland.  The  first 
member  of  the  family  who  attained  distinction  in  mathematics 
was  James. 

James  Bemoollit-  Jacvh  or  Jame^  Beriwulii  was  bom 
at  Bale  on  Dec.  27,  1654 ;  in  1687  he  was  appointed  to  a 
cliair  of  mathematics  in  the  university  there ;  and  occupied  it 
until  his  death  on  Aug.  16,  1705. 

He  was  one  of  the  earliest  to  realize  how  powerful  as  an 
instrument  of  analynis  was  the  infinitesimal  calculus,  and  he 
applies]  it  to  several  probleniH,  but  he  did  not  himself  invent  f  i 

any  new  pruccHM^.  His  grtMit  influence  was  uniforuily  and 
succesKfully  exerted  in  favour  of  the  use  of  the  differential  cal- 
culus, and  his  lessons  on  it,  which  were  written  in  the  form  of 
two  essays  in  1G91  and  are  published  in  the  second  volume  of 
his  works,  shew  how  completely  he  had  even  then  gmsped  the 
principles  of  the  new  analysis.  Tliese  lectures,  which  contain 
the  earliest  use  of  the  tenii  integral,  were  the  first  publislied 
attempt  to  conKtruct  an  integral  calculus;,  for  Leibnitz  had 
tn^jitcd  each  problem  by  itself,  and  had  liot  laid  down  any 
general  rules  on  the  subject. 

The  must  important  discoveries  of  James  Bernoulli  were 

I 

i 

*  See  Uie  account  in  the  Allgemeitu  Dtut$ch€\  Dioffrapkitf  vol.  n, 
Leip'iig,  1S76,  pp.  470—483.  I 

t  See  the  ^ioge  by  B.  de  Fontenelle,  Paris,  1766;  also  Ifontoda'S 
HUtoire,  vol.  u.  A  cuUected  edition  of  the  vorka  of  Jamea  BerooolU  was 
publiahed  iu  two  folumea  at  Geneva  in  1744,  and  an  aoooant  of  his  Ufa 
ii  pri'lized  to  the  fint  volume. 


v 


JAMRS  AND  JOHN   BERNOULU. 


S77 


his  solution  of  the  prohlem  to  find  an  isochronous  carve ;  his 
proof  that  the  construction  for  the  catenary  which  harl  hoen 
given  hy  Leibnitx  was  correct,  and  his  extenMion  of  this,  to 
strings  of  variable  density  and  under  a  central  force  ;  his  de- 
termination of  the  form  taken  hy  an  elastic  rod  fixed  at  one 
end  and  acted  on  by  a  given  force  at  the  other,  the  fihntiea ; 
also  of  a  flexible  rectangular  sheet  with  two  sidon  fixed  hori* 
xontally  and  filled  with  a  heavy  liquid,  the*  fintmria;  and 
lastly  of  a  sail  filled  with  wind,  the  r^farin.  In  169G  he  offered 
a  reward  for  the  general  solution  of  isoperimetrical  figures, 
that  is,  of  figures  of  a  given  Hp(*cie»  and  given  perimeter 
which  shall  include  a  maximum  area:  his  own  wilution, 
publiHhed  in  1701,  is  corn*ct  ns  far  an  it  goes.  In  IfiOS 
he  published  an  essay  on  the  diflerential  calculus  and  its 
applications  to  geometry.  He  here  invcHtigated  tlte  chief 
pn>pertie8  of  the  equiangular  spiral,  and  especially  noticed  the 
manner  in  m-hich  various  ciirv(>s  deduced  from  it  repnirluced 
the  originsl  curve:  struck  by  this  fact  he  l^egged  that,  in 
imitation  of  Archimedes,  an  e<iuiangular  spiral  should  Is) 
engraved  on  his  tonilistone  with  the  inscription  efuiem  numero 
mniata  reswryo.  He  also  brought  out  in  1695  an  edition  of 
Descartes's  Gemnefru!.  In  his  Am  Conj^tandi^  published  in 
1713,  he  entablished  the  fundamental  principles  of  the  calculus 
of  probabilities;  in  the  counse  of  the  work  he  defined  the 
numbers  known  by  his  name*  and  explained  their  use,  he  also 
gave  some  theorems  on  finite  differences.  His  higher  lectures 
were  mostly  on  the  theory  of  series ;  these  were  published  by 
Nicholas  Bernoulli  in  1713. 

John  Bemonllit.    John  BemtmtU^  the  brother  of  James 

*  A  biUiogrsphy  of  Bemoolli's  Numbcrt  has  been  inTen  bj  G.  S.  Ely, 
in  the  American  Jwrnnt  of  Jlalkematie*,  IBM),  toI.  v,  pp.  238— £15. 

t  D'Alsmbert  wrote  a  eukvifttie  ^lofte  oo  the  work  sod  infloenee  of 
John  Bemoalli,  bai  he  cxplieitlj  lefniied  to  deal  with  bin  private  life  or 
qosirels;  see  also  MootiMla*s  iiUtmrt^  vol.  n.  A  eoUeeted  edition  of  the 
works  of  Mm  BcmoidU  was  publislMd  at  Ooncva  in  four  volomeo  in  174S, 

his  cofwspondMMW  with  LeUioits  was  poUkhed  fai  two  volumss  al 
fai  ITtf. 


878  JOHN  BBEMOULU. 


BernouUi,  wm  bora  at  BAle  on  Aug.  7»  1M7»  and  dfad 

there  on  Jul  !»  1748.     He  oocnpied  (hb  ehnir  of  nuillin- 

niniios  ni  Qroningen  from  1695  to  1705 ;  and  at  Bile^  when 

he  auooeeded  his  brother,  from  1705  to  174&    To  all  who 

did  not  acknowledge  his  merits  in  a  manner  commensnrate 

with  his  own  view  of  them  he  behaved  most  uiyostly :  as  an  l\ 

illustration  of  his  chanicter  it  may  be  mentioned  that  he  ' 

attempted  to  substitute  for  an  incorrect  solution  of  his  own 

on  the  problem  of  isoperimetrical  cur\'es  another  stolen,  from 

his  brother  James,  while  he  expelled  his  son  Daniel  from  his 

houHO  for  obtaining  a  prize  from  the  French  Academy  which 

he  luul  expected  to  receive  liinuielf.    He  was  however  the  most 

succesiiful  teacher  of  his  age,  and  had  the  faculty  of  inspiring 

bin  pupilH  with  almoMt  as  pasMioiiate  a  seal  for  mathematics  as 

he  felt  hiiiiHclf.     Tlie  general  adoption  on  the  continent  of  the 

diflerential  rather  tlian  the  fluxioual  notation  was  largely  due 

tu  hiK  influence. 

Leaving  out  of  account  bin  innumerable  controversies,  the 
chief  discoveries  of  John  Bernoulli  were  the  exponential  cal- 
culus, the  treatment  of  trigonometry  as  a  branch  of  analysis, 
the  coiulitioiiM  for  a  geodesic,  the  detennination  of  orthogimal 
trajectories,  the  solution  of  the  bracliist4Mrhrone,  the  statement 
tlmt  a  my  of  Hglit  traversed  such  a  path  that  l^ftds  was  a 
uiiiiiuiuiu,  and  the  enunciation  of  the  principle  of  virtual  work. 
I  lielieve  that  he  was  the  first  to  denote  the  accelerating  effect 
of  gravity  by  an  algebraical  sign  y,  and  he  thus  arrived  at  the 
formula  v^  =  2ffh :  the  same  result  would  have  been  previously 
expressed  by  the  proportion  V|' :  r,*  =  A, :  A,.  Tlie  notation  ^ 
to  indicate  a  function^  of  x  wais  introduced  by  him  in  1718, 
and  displaced  the  notation  X  or  (  pn»pased  by  him  in  1698 : 
but  the  general  adoption  of  symbols  like  j]  F^  ^  ^,...  to 
repri'sent  functions,  seems  to  be  mainly  due  to  Euler  and 
Lagrange. 

*  On  the  tueaning  atwigned  at  firnt  to  the  void  /tutttiom  see  s 
sots  bj  M.  Cautor,  VluUrmtdiaire  de$  matktmaticietu^  January  1896, 
vol.  III.  pp.  23—23. 


\ 

\ 


JOHN   BERNOULLI.  379 

Several  memhcrn  of  the  Mune  family,  hot  of  a  younger 
generation,  enriched  mathematics  by  their  teaching  and 
writings.  The  nnost  important  of  these  were  the  tliree  sons  of 
John ;  namely.  Nicholas,  Daniel,  and  John  the  yoanger ;  and 
the  two  sons  of  John  the  yoanger,  who  bore  the  names  of 
John  and  James.  To  make  the  account  complete  I  add  here 
their  respective  dates.  Xichilaf  Bemmdti^  the  eldest  of  the 
three  sons  of  John,  was  bom  on  Jan.  27,  1695,  and  was 
drowned  at  St  Petersburg  where  he  was  professor  cm  July  26, 
1726.  Daniel  Bemoufli^  the  second  son  of  John,  was  bom  on 
Feb.  9,  1700,  and  died  on  March  17,  1782;  he  was  professor 
first  at  St  Petersburg  and  afterwards  at  Bale,  and  shares  with 
Euler  the  unique  distinction  of  having  gained  the  prise  proposed 
annually  by  the  French  Academy  no  less  than  ten  times:  I  refer 
to  him  again  a  few  pages  later.  John  Bemouili^  the  youngiT, 
a  brother  of  Nicholas  and  Daniel,  was  bom  on  May  18,  1710, 
and  died  in  1790;  he  also  was  a  professor  at  Bale.  He  left 
two  sons,  John  and  James :  of  these^  the  formeri  who  was  liom 
on  Dec  4,  1744,  and  died  on  July  10,  1807,  was  .astronomer- 
royal  and  director  of  mathematical  studies  at  Berlin ;  while 
the  latter,  who  was  bom  on  Oct.  17,  1759,  and  died  in 
July  1789,  was  successively  professor  at  Bale,  Verona,  and 
St  Petenliurg. 

Hie  development  of  analysis  on  the  continent. 

Leaving  for  a  moment  the  English  mathematicians  of  tlie 
first  half  of  the  eighteenth  century  we  come  next  to  a  numlier 
of  continental  writers  who  Imrely  escape  mediocrity,  and  to 
whom  it  will  be  necessary  to  devote  but  few  words.  Their 
writings  mark  the  steps  by  which  analytical  geometry  and  the 
difierential  and  integral  calculus  were  perfected  and  made 
familiar  to  mathematicians.  Nearly  all  of  them  were  pupils 
of  one  or  other  of  the  two  elder  Bemoullisi  and  they  were  so 
nearly  eontempormrica  thai  it  is  diflkult  to  arrange  them 
dumiol0gically>    The  inosi  enioent  of  theni  are  Crmm9\  dk 


880  L*liQimTAL.     VABIQIIOM. 

Oua^    de    Ifomiamri^   Fagmmo^    fUc^^iUfi^    Jficoh^    An^wl, 
Bieeaii^  Saurin^  mad   Varigtum,  ' 

VBoniUL  GuUiaume  t^nm^oU  AtUmm  tUctpikd.  Mmr- 
quis  de  Si-Ifetme^  born  At  PteU  in  1661,  and  died  IImto  on 
Fell.  2y  1704,  w«s  among  the  eaiiient  popilii  of  John  Bomonlli, 
who,  in  1691,  spent  aome  months  at  rH6spital*s  hoose  in 
Paris  for  the  purpose  of  teaching  him  the  new  caknlua.  It 
seems  strange  but  it  is  substantially  true  that  a  knowledge  of 
the  infinitesimal  calculus  and  the  power  of  using  it  was  then 
oonlined  to  Newton,  Leibiiits,  and  the  two  elder  fiemoullis — 
and  it  will  be  noticed  tlmt  they  were  the  only  mathematicians 
wlio  soh'ed  the  more  ditKcult  problems  then  proposed  as  chal- 
lenges. There  was  at  that  time  no  text-book  on  the  subject| 
and  tlie  credit  of  putting  together  the  tirst  treatise  whidi 
explained  the  principles  and  use  of  the  method  is  due  to 
THospital :  it  was  published  in  1696  under  the  title  Amdymde^ 
injiiiitikenl  pf.tiu.  This  contains  a  partial  iu%'estigation  of 
the  limiting  value  of  the  ratio  of  functions  which  for  a  certain 
value  of  the  variable  Uike  the  indeterminate  form  0  :  0,  a 
problem  holved  by  John  Bernoulli  in  1704.  This  work  luui 
a  wide  circulation,  it  bnmglit  the  differential  notation  into 
general  use  in  France,  ai.d  helped  to  make  it  known  in 
Eun>|)e.  A  supplement,  containing  a  similar  treatment  of 
the  integral  calculus,  t<igetlier  with  additions  to  the  differential 
calculus  which  had  bt;en  made  in  the  following  lialf  century, 
was  published  at  Pkris,  1754-6,  by  L.  A.  de  Bougainville. 

L'Hospital  took  part  in  most  of  the  challenges  issued 
by  Leibnitz,  the  Bernoullis,  and  other  continental  mathe- 
maticians of  the  time ;  in  particular  he  gave  a  solution  of 
the  brachistochrone,  and  investigated  the  form  of  the  solid 
of  least  resistance  of  which  Newton  in  the  Priftcijna  liad 
stated  the  result  He  also  wrote  a  treatise  on  analytical 
conies,  which  was  publislied  in  17U7,  and  for  nearly  a  century 
was  deemed  a  standard  work  on  the  subject. 

Varignon*.    Pierre    Variynint^  born  at  Caen  in  1654,  and 
*  See  the  Huge  by  B.  de  Foutcnelle,  Paris,  176G. 


DE  HONTHORT.    NICOLE.     PARG14T.    SAURIN.    DE  QUA.      381 

died  in  PaiHs  on  Dec.  22,  1722,  war  nn  intimfit«  friend  ot 
Newton,  LeilinitK,  miiI  the  Benioullis,  And,  after  rUospttal,  wm 
theeArlitwt  and  niont  powtrful  advocate  in  Fntnceof  thenae  of  ' 
thfl  dilTerentisI  cnlculoB.  He  roaliEed  t)ie  necnmtj  of  obtaining 
»  test  for  fxniiTJniug  tlie  convcrgpncy  of  aeriea,  bnt  the 
annlyticnl  difficulticH  were  beyond  his  powcni.  He  aimpllfied 
the  proofs  of  man;  of  tlie  leading  pnipoHittona  in  mechanin, 
and  in  1687  f^ast  tlie  tiratinent  of  the  nuhjeet,  baring  it  on 
the  composition  of  forces.  Hia  works  were  pabliahed  at  niri* 
in  1725. 

Da  Montmort.  Hloolo.  Pirrrf  Raijmond'h  ifoiUmort, 
bom  at  Pariit  on  Oct.  27,  IG7R,  and  died  there  on  Oct^  7, 
1719,  wait  interested  in  the  subject  of  finite  dtKrencea.  fl« 
determined  in  1713  the  nnm  of  u  temiB  uf  a  6nil«  aerien  ol 
the  form 

a  theorem  whicli  Heeins  to  have  lieen  independently  n- 
dincovcred  by  Chr.  tioldbnch  in  1718.  Frattpttt  Xieoln,  who 
was  bom  at  Paris  on  Dec.  23,  1683,  and  died  there  on 
Jan.  18,  1758,  pnblinhed  bin  Tmife  dn  ciJciU  eki  dijfftnmen 
Jinira  in  1717;  it  contAinn  rules  lioth  for  forming  diflervncea 
and  for  efTi-cting  the  suiiiination  of  K'^en  Hpririi.  Deaidra  thin, 
in  170fi,  he  wrote  a  work  on  nmlel(4-n,  especintlj  spherical 
epicycloids:  and  in  172!)  ami  17^1  he  puhliHhod  menioini  on 
Newton's  es-iay  on  curves  of  the  thinl  ilegreo. 

Pftrent.  Baoiin.  De  Ona.  Antoiwi  PttrttU,  bom  at 
Paris  on  SepL  Ifi,  IGCG,  and  diet]  there  on  Sept  26,  171G, 
wrot«  in  1700  on  analytical  geometry  of  three  dimensiana.  . 
Hifl  works  were  cotlectevl  and  published  in  three  Tolnmea  at 
Paris  in  1713.  Jom/A  Sanria,  bom  at  Gourtatson  in  1659, 
and  died  at  Paris  uo  Dec  29,  17.17,  was  the  &nt  to  shew  how 
tbe  tangent*  at  the  multiple  points  of  cnrre*  eovld  be  deter- 
mined 1^  analym  Jmn  Paul  r/<i  Gua  df  Jfi/tM,  was  born  at 
Oarcuaonne  in  1713,  and  died  at  IWis  on  June  1^  IT^.    %» 


88S  CRAlin.      RIOCATI.      rAOVAVa 

pobliBbed  ia  1740  a  work  on  matiytiofi  gaonwlijr  la  wUflb  ht 
applied  it|  without  the  aid  of  the  differantial  eAlottlii%  lo  lad 
the  tangentM,  allymptolei^  and  various  lingular  points  oC  an 
algebraical  cur%'e ;  and  he  further  shewed  how  singuhur  points 
and  isolated  loops  were  affected  hj  conical  projectioiL  He 
gave  the  proof  of  Descartesfs  rule  of  signs  which  is  to  be 
found  in  most  modem  works :  it  is  not  clear  whether  Descartes 
ever  proved  it  strictly,  and  Newton  seems  to  have  regarded  it 
ms  obvious. 

Cramer.  Gabriel  Crauier^  bom  at  Geneva  in  1704,  and 
died  at  Bagnols  in  1752,  was  professor  at  Geneva.  The  work 
by  which  he  is  best  known  is  his  treatise  on  algebraic 
curves*  publislied  in  1750,  which,  as  tar  as  it  goes,  is  fairiy 
complete ;  it  contains  the  earliest  demonstration  that  a  curve 
of  the  nth  degree  iH  in  general  determined  if  |  m  (m  -i-  3)  points 
on  it  be  given  :  this  work  in  still  sometiiiien  read.  Besides 
thiH,  he  tHJitcHl  the  works  of  the  two  elder  BemouUis;  and 
wnit4i  on  the  physicul  caiuse  of  the  spheroidal  sliape  of  the 
planets  and  the  nuition  of  tlieir  apneH,  1 730,  and  on  Newton's 
treatment  of  cubic  curves,  174G. 

Riccati.  Jaeopo  FraiiceMO^  Count  Riecati^  bora  at  Venice 
on  May  28,  1C7G,  and  died  at  Treves  on  April  15,  1754,  did 
a  great  de4il  to  disseminate  a  knowledge  of  the  Newtonian 
pliiloMophy  in  Italy.  Besides  the  equation  known  by  his 
muiic,  certain  cases  of  which  he  succeeded  in  integrating,  he 
discussed  the  question  of  the  possibility  of  lowering  the  order 
of  a  given  iliiferential  equation.  His  works  were  published  at 
Treves  in  four  volumes  in  1758.  He  Imd  two  sons  who  wrote 
on  several  minor  points  connected  with  the  integral  calculus 
and  differential  equations,  and  applied  the  calculus  to  several 
mechanical  questions :  these  were  Viticenzo^  who  was  bom  in 
1707  and  died  in  1775,  and  GiardnfUf^  who  was  bom  in  1709 
and  died  in  1790. 

Fagnano.  Giulio  Carlo,  Count  Fagnauo^  and  MarquU  de 
7'o0r/it,  lM>m  at  Hinigaglia  on  Dec.  6,  1682,  and  died  on  Sept  26, 

*  8ee  Cantor,  chapter  cxn. 


PAGNAXO.     CLAIRAUT.  S83 

I766y  may  be  mid  to  have  been  the  first  writer  who  directed 
attention  to  the  theory  of  elliptic  functions.  Failing  to  rectify 
the  ellipee  or  hyperbola,  Fagnano  attempted  to  determine  arcs 
whose  difference  should  be  rectifiabla  He  also  pointed  oat 
the  remarkable  analogy  existing  between  the  integrals  which 
represent  the  arc  of  a  circle  and  the  arc  of  a  leroniscate. 
Finally  he  proved  the  formula 

»=2tlog{(I-.')/(l+t)} 

where  i  stands  for  %/—  1.  His  works  were  collected  and 
published  in  two  volumes  at  Posaro  in  1750. 

It  was  inevitable  that  Home  matheniaticians  should  object 
to  methods  of  analyMis  founclfnl  on  the  intiniteKimal  calculus. 
Tlie  most  prominent  of  these  were  Virtani^  D^  fa  I/ir^^  and 
JiofU^  whose  names  were  mentioned  at  the  close  of  chapter  XT.' 

80  far  no  one  of  the  school  of  Leibnitz  and  the  two  elder 
Bemoullis  had  shewn  any  exceptional  ability,  but  liy  the  action 
of  a  numlier  of  second-rate  writers  the  methods  and  language 
of  analytical  geometry  and  the  differential  calculus  had  become 
well  known  by  about  1740.  The  clone  of  this  school  is 
marked  by  the  appearance  of  Ctatraut,  D'Alembert^  and  Daniel 
jBernauUi,  Their  lives  overlap  the  period  considered  in  the 
next  chapter,  but,  though  it  is  difficult  to  draw  a  sharp  dividing 
line  which  shall  separate  by  a  definite  date  tlie  mathematicians 
there  considered  from  those  whose  writings  are  discussed  in 
this  chapter,  I  think  that  on  the  whole  the  works  of  tliese 
three  writers  are  liest  treated  here. 

Olairant.  AtexU  Claunh  Clairaui  was  bom  at  Paris  on 
May  13,  1713,  and  died  there  on  May  17,  1765.  He  belongs 
to  the  small  group  of  children  who,  though  of  exceptional 
precocity,  survive  and  maintain  tlieir  powers  whrai  grown  up. 
As  eariy  as  the  age  of  twelve  he  wrote  .a  memoir  on  four 
geomeirioa^  carves^  but  his  first  important  work  was  a 
treatisu  on  iortaoas  curves  puUisbed  when  he  wsa  t&|SB^la«L 
— a  work  whidi  proenred  for  Um  admiiMMa  \a  ^^m^ 


384 


0L4I] 


Aoidfliiij.  In  1781  he  gave 
noted  by  Newton  that  all  coi 
projeotione  of  one  of  five  pni 
In  1741  CUiimttt  went  on  n 
the  length  of  a  meridinn 
on  hU  retarn  in  1743  he  pol 
de  la  lerre.  This  is  founded  onl 
it  had  lieen  shewn  tliat  a  niassl 
rotation  about  a  line  through  i( 
tlie  mutual  attraction  of  its 
spheroid.  This  work  of  Claii 
spheroids  and  contains  the  proof] 
rating  effect  of  gravity  in  a  pi 


ol  ilM  iMi 
of  the  thiid  oider 


itifie  expedition  to 

{on  the  earth's  surlMM^  and 

his  TUorU  de  Imfyum 

paper  by  Maclaurin,  where 

homogeneous  fluid  set  in 

itre  of  mass  would^  under 

iclei^  take  the  form  of  a 

treated  of  heterc^geneous 

»f  his  formula  for  the  aocde- 

if  Utitude  /y  namely, 


y»(7{l4^($i»i|-€)sin«/}, 


gravity,  m  the  ratio  of  the 
equator,  and  c  the  ellipticity 
In  1849  Stokes*  shewed 
whatever  was  the  interior 
Lh  provided  the  surface  was 
ellipticity. 


where  O  is  the  value  of  equatori^ 
centrifugal  force  to  gravity  at  tl 
of  a  meridian  aection  of  the  (sai 
that  the  same  result  was  tru^ 
constitution  or  density  of  the  in 
a  spheroid  of  equilibrium  of  smi 

Impressed  by  the  power  oflgeometry  as  shewn  in  the 
writings  of  Newtou  and  Mshaurin,  Clairaut  abandoned 
analysis,  and  his  next  work,  the  t'A^orie  de  la  lune^  published 
in  1752,  is  strictly  Newtonian  inlcharacter.  This  contains  the 
explanation  of  the  motion  of  tlA  apse  which  had  previously 
puzzled  astronomers,  and  which  piairaut  had  at  first  deemed 
so  inexplicable  that  he  was  on  tl  le  point  of  publishing  a  new 
hypothesis  lui  to  the  law  of  at  raction  when  it  occurred  to 
him  to  caiTy  the  approximation  to  the  third  order,  and  he 
thereupon  found  that  the  result  was  in  accordance  with  the 
oljservations.  This  was  follow<  J  in  1754  by  some  lunar 
tabli*s ;  Clairaut  subsequently  >  rote  various  papers  on  the 
orbit  of  the  moon,  and  on  the  m(  tion  of  comets  as  affected  by 

*  Heo  Cambridge  Pkiloiophicui  Tru  Uiaethma,  ncl,  nu,  fip.  Vtt-^^M, 


/ 


d'alembert.  d85 

the  pertarlmtioD  of  the  planrls,  [mrticul»rly  on  the  path  of 
U*l  ley's  comet 

Hin  growing  popularity  in  wocicty  himlereit  bin  neientiBe 
work:  "engag^"  Hnyn  Boenat,  "i  rim  loupen,  k  det  Wll«*^ 
entralne  par  un  gout  >-if  pour  ten  frnimeH,  vonlMnt  nllier  le 
pUiHir  a  MM  trnvnux  ordinnireis  Jl  [icrtlit  le  repoAi  l>  iMnU^ 
enfin  la  Tie  a  l'»ge  dc  cinqunnte-deux  anx." 

D'AIembert*.  JmnJr.-ltanH  D' Alntil>'rl,  wm  bnm  tX 
Prtria  on  Nov.  16,  1717,  nnd  diwi  there  on  <X-t.  29,  17C3.  Ha 
una  the  jllegi(iinnt«  child  of  the  chevalier  [)pHt<iucbes.  Being 
Bbftndonpd  by  Hih  mother  on  the  Ht«pH  of  the  little  church  t^ 
St  Jenn-le-Rond  which  then  nentled  under  the  grcftt  porch  vi 
Niitre  Dame,  he  wiut  tAken  Ut  the  parinh  ccrnimmwry,  who^ 
following  the  uHunI  pmctice  in  Huch  cnsen,  gtvfl  him  the 
christian  name  of  Jran-le-Rond  :  I  do  not  know  hy  wbnt  titlo 
be  Huheeqnently  HsaQined  the  right  to  prefix  <h  to  fau  lume. 
He  wsH  bonnlcd  out  l>y  the  parinh  with  the  wife  of  •  gluier 
in  a  briaII  way  of  buxinew  who  lived  near  the  cathedral, 
and  here  he  necmn  to  have  found  n  real  home  tbongfa  a 
hnmbie  one.  Hin  father  appears  to  hnve  looked  after  him, 
and  paid  for  hia  going  to  a  srhool  witere  he  obtained  a  fair  . 
mathematical  education. 

An  eiway  written  liy  him  in  173«  on  the  integral 
calculus,  and  another  in  1740  on  "ducki  and  dmkee"  or 
ricochelJt  attract«<l  nome  attention,  and  in  the  nme  year 
he  wan  elected  a  meniher  of  the  French  Academy ;  this  was 
probably  due  to  the  influence  of  hin  father.  It  in  to  bis 
credit  that  he  absolutely  refoned  to  leave  hin  adopted  mother 
with  whom  be  continued  to  live  until  her  death  in  1757.  It 
cannot  ho  said  that  she  sympathized  with  his  soccess  for,  at 
the  height  of  his  fame,  she  renionstnited  with  him  tor  wasting 
his   talents  on   such    work :    "  vous   no   seres   jamais  qa'nil 

*  Condorcel  sod  J.  Bsatien  hsT*  left  iikclchM  of 
his  liteTW7  >o^  hare  been  publiitwd,  bol  there  ii  n< 
e(  fail  sticetUia  writiiitpt.    Boow  papcn  and  Mien,  diseuvMWl 
tivdy  rsMBtly,  w««  iMbUibsd  hr  &  Hcsc)  «k  V«Aa  ^™n. 


886  D^ALSiiBcrr. 

philoMophe,"  mud  ahe^  ^et  qu'esi-oe  qa*vii  pkiloaophet  c^tt*  mu 
tutt  qui  satoormente  pemUni  im  vm^  poar  qa'oa  pMrIa  de  lai 
loraqatl  n'j  ierm  pliuk* 

Stmrly  all  hiii  mathenuitiaU  works  ware  prodooed  dnril^( 
the  yearn  1743  to  1754.  The  lirrt  of  tlieae  was  hk  TVwttf  0U 
djfHamiqne^  pabliiilied  in  1743,  id  which  he  eniincialea  the 
principle  known  by  his  name,  namely,  thai  the  **  internal 
forces  of  inertia  "  (that  is,  tlie  forces  which  resist  acceleration) 
must  be  equal  and  opposite  to  the  forces  which  produce  the 
acceleration.  Tliis  niay  be  inferred  from  Newton's  second 
reading  of  his  third  law  of  motion,  but  the  full  conseqnences 
luuk  not  been  realixed  previously.  The  application  of  this 
principle  enables  us  to  ol>tain  the  ditTerential  equations  of 
motion  of  any  rigid  system. 

In  1744  D'AleinU^rt  publislicd  his  Traiie  de  tequilibrt 
ei  du  motivrmeni  de»  JtuitltH^  in  which  he  applies  his  principle 
to  fluids:  this  led  to  partial  differential  equations  which  he 
wttM  then  unable  to  solve.  In  1745  he  developed  that  part 
of  the  subject  which  dealt  with  the  motion  of  air  in  his 
Theorie  ijeitertUfi  iUs  tvii/«,  and  this  again  led  him  to  partial 
differential  e«|uations:  a  necoiid  edition  of  this  in  1746  was 
dedicati^d  to  Frederick  the  Great  of  Prussia,  and  procured  an 
invitation  to  Berlin  and  the  offer  of  a  pension;  he  declined 
the  fonaer,  but  subsequently,  after  some  pressing*  pocketed 
his  pride  and  the  latter.  In  1747  he  applied  the  differential 
calculus  to  the  prublein  uf  a  Wbrating  string,  and  again 
arrived  at  a  partial  differential  equation. 

Hiri  analyHis  had  three  times  brought  him  to  an  equation 
of  the  form 

and  he  now  succeeded  in  shewing  that  it  was  satisfied  by 

where  ^  and  ^  are  arbitrary  functions.    It  may  be  interesting 
io  give  hiH  solution  which  was  published  in  the  transactions 


\ 


d'alehrert.  387 

of  the  Berlin  Academy  for  1747.    Ho  Iirpios  hy  "wying  tlw*,  if 
ba  denoted  by  p  »nd  —  hy  7,  then 

du  —  pdx  +  71  ft. 
Bat,  liy  tlip  givi-ii  p<|UAti(m,  ^  =  7*i  »"•'  therefore prft  +  7rf«  iii  . 
nlsn  Hn  (txAct  diflcfviitiitl :  di^ioU.-  it  liy  itr. 
Thi'trfoTO  ttr  ^  jffl  *  q-lj-, 

Henoe     i/it +  (/c^  (/xlc^ft/r) +  </Kft4^  T'/z)  .  (;>+f)(f6!  + tft), 
una         ./m  -  rfe  =  (/Htr  +  7'/()  -  (/-ft  +  qdr)  ^  (;>  -7)  (</«-<l(). 
ThuK  M  +  B  mDKt  br  n  function  of  r  +  t,  «nd  ¥  -  r  mnrt  be  » 
(unction  of  «  -  /,     Wc  may  thrreforo  [lul 

Mid  H-v-2ili{x-l). 

Ilenop  u=^(x  +  0  +  f  (i    0- 

D'Alenihitrt  added  that  tlie  cuiidilioiii  til  the  phjM'^ 
proliI«m  uf  A  vilirating  Miring  di-nianil  thiit,'when  x  —  0,  « 
Mbould  vanish  for  <^l  vaIueh  of  (.     Hence  identtadlj 

Axsaming  th*t  luitli  functionn  con  Im*  expuided  in  intcgrml 
powpnt  of  I,  thiH  rdinifTH  thnt  thny  Hhould  oontain  tmly  odd 
poven.     Hence 

f(-0--*{0  =  *(    ')■ 

Theirfore  h  ^  ^  (x  4^  ()  +  ^  (.c  -  (). 

Enter  now  took  the  mutter  up  and  thawed  that  Iha 
equation  of  tlie  fomi  of  the  string  was  „  ^"^  iji  *'"'  ^^^ 
the  gener&l  integral  wm  u  ^  ^  («  -  r/)  -i-  ^  (x  4-  at),  where  ^  and 
tfr  are  arbitrary  functions. 

The  chief  remaining  contribnlionn  of  D'Alembeti  tA 
mathematica  were  on  phjMcal  «rtn)iMnn)  \  «qm8n&l  <>^  "^^ 

«.— 1.. 


888 


DANIEL  BERNOUIXI. 


proooisioo  of  tlie  equinoxei^  and  on  varUtiiiiii  ia  the  obliqvitj 
of  the  ecHpiio.  Theiie  were  eolleoied  in  hii  Sftiimm  du  tmtmdt 
pulilidied  in  three  volumee  in  1764. 

During  tlie  Utter  part  of  hia  life  he  me  mnialj  ooonpied 
witli  the  gnmt  French  encjclopeedia.  For  this  he  wrote  the 
intruductioiiy  and  uianieruos  pliiloHophical  |  end  nmtJiemeticel 
articleH :  tlie  lieMt  are  thuHO  on  geometry  aiid  on  probabilitiee. 
IliM  Mtyle  iM  brilliant,  but  not  poliMhed,  and  faithfully  reflecte 
hiH  diameter,  which  was  bold,  honest,  and  frank.  He  defended 
a  Hcvere  criticinni  which  h«  liad  offered  on  Home  mediocre  work 
by  tlie  remark,  '* j'aiiiie  inieux  etre  inci vil  i|tt'eiinuy£ " ;  and 
with  his  dislike  of  sycophants  and  bores  it  is  not  surprising 
tliat  during  his  life  he  had  more  enemies  tlian  friemis. 

Daniel  Bemonlli*.  Daniel  HeniouiH^  whose  name  I 
mciitiuiied  alxivc,  and  who  was  by  far  jtlie  ablest  of  the 
younger  lk*riiouUis,  was  a  contemporary  and  iiitiiiiato  friend 
of  Euler,  whose  W4irks  are  mentioned  in  the  next  chapter. 
Daniel  Bernoulli  was  born  on  Feb.  9,  1700,  and  died  at 
Bale,  when;  he  wss  professor  of  natural  philosophy,  on  March 
17,  1782.  He  went  to  St  Petersburg  in  1724  as  professor 
of  iiiiitlieiuatics,  but  the  roughness  of  the  social  life  was 
distasteful  to  him,  and  he  was  not  sorry  when  a  temporary 
illness  in  1733  allowed  him  to  plead  his  health  as  an  excuse 
for  leaving.  He  then  retunuHl  to  Bale,  and  held  successively 
chairs  of  medicine,  metaphysics,  and  natural  philosophy  there. 


*  The  only  sccouut  of  Daniel  BeraouUi*i  life  with  which  I  am 
acquainted  is  the  rioffr  by  his  friend  Condoroet.  Marie  Jean  JtUoine 
Nicotat  Curiiat,  Marquis  de  CuHdtnxeU  was  born  in  I'icardy  on  Kept.  17, 
1743,  and  fell  a  victim  to  tlie  republicau  terrorists  on  March  2tf,  17M. 
Ue  was  secretary  to  the  Academy  and  is  the  author  of  uumeroos  eloges. 
He  is  perha|)s  more  celebrated  for  hia  studies  in  philosophy,  literature, 
and  politics  than  in  mathematics,  but  his  mathematical  treatment  oC 
probabilities,  and  his  discussion  of  diffeivntial  equations  and  finite  dif- 
ferences, shew  an  ability  which  might  have  put  him  in  the  tirst  rank  had 
he  concentrated  his  attention  on  uuithematics.  He  sacrificed  himself 
in  a  vain  cflort  to  guide  the  revolutionary  torrent  into  a  eonstitntiooal 
cbannel. 


\ 


J 


MATHEUATICIANS  OF  THE   ENQLIHH  SCHOOL.        389 

His  earliest  mntheiiiAticnl  work  whk  the  Exereilaliont* 
publislipd  in  1721,  whirh  contAinH  n  luilution  of  the  diflbrptitinl 
f><|uiiti(tn  |int]MK>fHl  \iy  lUrc-Ati.  Two  ycnr*  Ut«r  lie  ptrintni 
nut  for  th(!  first  time  tlip  frc<)ui>nt  dciiralrility  ut  rcmhing  « 
conipDund  inntion  into  miitiuiis  of  tmnHintion  and  ntotions  of 
roUtion.  His  chief  work  ib  hiH  //^/rorfymmt^"*  pnlilinhwl 
in  1 7M :  it  rpM>ni1>lm  lAgrnngp's  .VMtaiqn*  annlt/tujnt  in 
bcinj;  Kmingpd  eo  thnt  oil  thi-  rMultit  nra  conw><iiipnces  nf 
n  Hin;;h-  priiicijilc,  nniiiely.  in  thin  cn.v<>,  the  conRerv«tion  (4 
cnrrjiy.  TIiiM  w»w  fnllnwpd  liy  n  nirnuiir  on  the  theory  of 
the  tides  to  which,  conjuinllj-  with  nicmoirx  Iqr  Ealer  »nd 
Mncinurin,  n  prize  wns  nwAnIe<1  Uy  the  French  AcMdeinjr: 
thrw  thire  nienuiire  contain  all  tlint  wivt  done  on  this  ttobject 
brtwoen  tlie  puhlicntion  of  Newton's  Priuei/iin  Mid  the 
inTentigiitions  of  fjUplncc.  ib-moulli  nim  wrote  n  large 
numlier  of  pnpors  an  vitrioun  nipchnniotl  (|iMHtionti,  esprciidly 
on  proMrnin  connected  with  vilirntin;;  Rtrinffn,  and  the  nuln- 
tions  given  liy  Tnytor  Anil  liy  D'AIemlierL  He  n  the  e*rlte«rt 
writer  who  Attpmpl«l  to  fonnulnto  a  kini'ttc  theory  of  gww*, 
And  Iw  applied  thn  idp»  Ui  mplnin  tlip  inw  usnciKtH  with  the 
nAiiipn  of  lloylp  and  MnHottr. 

The  Eniflinh  mutfieniiiticiaiin  of  the  eiykteentk  eenturg. 

I  have  rFspncd  a  nntii^  of  the  Rnf:linh  mathenMticiAnii 
who  niccenlod  Newton  in  onirr  that  thR  memhrrx  nf  tho 
EnKlinh  school  ntny  lie  all  treated  together-  It  w»«  almoNt  « 
matter  of  course  that  the  Bn;;)iHh  Rhonld  at  linit  have  Adnpted 
the  notation  of  Newton  in  the  infinit^ttinial  calculiia  in  pre- 
fen>nce  U>  that  of  I^ilinitz,  and  ciniieijiiently  the  En){liKh 
Rchool  would  in  Any  rase  have  developed  un  noniewhat  difletent 
line*  to  that  on  the  continent  whr-re  a  knowledge  uf  the  in* 
finitesinial  calcnlan  was  derived  nolely  from  Leibnitz  and  Uie 
Bemonllia.  But  thin  neparatiun  into  two  distinct  Khooh 
brcMne  rery  marked  owing  to  the  action  of  Leibniti  and 
John  Bemonlli,  which  wan  natamlty  nwntnd  Itj  ^v^VmIx 


390  lUVIO  QUBOOBT.     HAWR. 

friemb :  and  to  for  forty  or  fifty  y«uni|  to  tho  aivtttal  MnA* 
VAUtage  of  boUi  iiidei|  the  quarrel  niged.  The  kediBg  aieoifaert 
of  the  Eiigliiih  Mchool  were  Coiu^  Ihmuivrt^  DiiUm^  Dmmd 
Oreyofy^  litdUjf^  Jiaelanriu^  Simprnm^  and  Tm^flor.  I  may 
however  again  remind  my  readers  that  as  we  approach  modem 
tini€*ii  the  iiamber  of  capable  matliematicians  in  Britain, 
Krauce,  Qenuany,  and  Italy  becomes  very  considerable^  but 
that  in  a  popular  sketch  like  this  book  it  is  only  the  leading 
men  whom  I  pro|HiBe  to  mention. 

To  David  Uregory,  Halley,  and,  Ditton  I  need  devote  but 
few  words. 

David  Oregory.  David  Orftforjf^  tl^e  nephew  of  the 
James  Gregory  mentioned  above^  bom  at  Aberdeen  cm 
June  24,  IGGl,  and  died  at  Maidenliead  on  Oct.  10,  1708, 
was  appointed  profesaur  at  Edinburgh  in  1684,  and  in  1C91 
was  on  Newtcin's  recoiiiuieiidation  elected  Savilian  professor 
at  Oxfunl.  His  chief  works  are  one  on  geometry,  issued  in 
1C84 ;  une  ou  optics,  publislied  in  1G95,  which  contains 
[p.  98]  the  earliest  suggention  of  the  possibility  of  making  an 
achromatic  cunibinatiun  of  lenses ;  and  one  on  the  Newtonian 
geometry,  physics,  and  astronomy,  issued  in  1702. 

Halley.  Edmnitd  /Ai//^y,  liom  in  London  in  1G56,  and 
die<l  at  (Ireenwich  in  1742,  was  educated  at  St  Paul's  School, 
I^ndon,  and  Queen's  College,  Oxford,  in  1703  succeeded 
Wullis  as  Savilian  prufetisur,  and  subsequently  in  1720  was 
.app«>iiited  iistronoiner  royal  in  succeHsion  to  Flamsteed  whose 
HUtonii  Citt'lfstiti  Rritannica  he  edited  in  1712  (first  and 
imptirfect  edition).  Halley 's  name  will  be  recollected  for 
the  generous  manner  in  which  he  secured  the  inmiediate 
pulilicaition  of  Newton's  Princtfiia  in  1G87.  Most  of  his 
original  work  wiis  on  astronomy  and  allied  subjects,  and  lies 
outside  the  limits  of  this  book ;  it  may  be  however  said  tliat 
the  work  is  of  excellent  4|Uiility,  and  both  Lalande  and  Afairan 
speak  of  it  in  the  highest  terms.  Halley  conjecturally  restored 
the  eighth  and  lost  liook  of  the  conies  of  Apollonius,  and  in 
1710  brought  out  a  magnificent  mlition  of  the  whole  work: 


\ 


DtTTOK.      TAYLOR.  391 

he  Klao  edib>d  the  works  oE  Serpnun,  thoi#  ot  MeneUni,  and 
■ome  of  the  minor  works  of  Apolloniun.  He  wm  in  his  tnm 
micceedpd  »t  Greenwich  as  «Rtrononier  royal  by  Bmdlejf*. 

Ditton.  ffiimiAiy  Dillon  wiw  Imnt  at  SalinlHiry  on 
May  39,  167r>,  nnd  died  in  I/)ndon  in  1715  at  Christ'ii 
Honpitid  where  he  was  matheiimticnl  master.  He  doe*  not 
neeni  U>  have  paid  mnch  attention  to  nintheiuatica  until  he 
Clime  to  London  alMut  1705,  and  hin  enrly  death  was  a  diatinct 
low  to  Eiighsl)  Kcience.  He  pulitishe<]  in  I'OC  a  textJtook  on 
flunioiis;  this  nnd  another  Eimilar  work  by  William  Jmiea 
which  wax  issue«I  in  1711  uccupini  in  Knelitnd  much  the  aame 
place  that  rHcispiUl's  trentisedi<l  in  France;  in  1709  DitUm 
i)Hued  nn  algflmi ;  and  in  1712  a  trentisc  on  perspective.  He 
also  wrote  numerous  pn|)erH  in  the  I'hilnntpkieal  TmtttaHioHt; 
he  wan  the  earliest  writer  tii  attempt  to  explain  the  phe- 
nomenon of  capillarity  on  riiatheronticnl  principles ;  and  he 
invented  a  method  for  finding  the  longitude  which  h«  been 
since  utied  on   various  occasionii. 

Taylort.  /Intik  Taytor,  bum  at  Rclmontofi  on  Aug.  18, 
lCe».y  and  dioi  in  London  un  Dec.  2'J,  1731,  wait  educated  at 
8t  John's  (Villege,  Caiubridge,  and  was  among  tiie  most  en- 
thusiaxtic  of  Newton's  admirers.  From  the  year  1712  ouwaida 
he  wrote  numerous  paper*  in  the  yAi'/oMi^tcn/  TrauMKlUm* 
in  which,  among  other  thin^,  he  discusMd  the  motion  of 
projectiles,  the  centre  uf  oscillation,  and  the  forms  taken  fay 
liquids  when  miord  by  capillarity.  In  1719  be  resigned  the 
■ecrrtaryship  of  the  Royal  Society  and  abandoned  the  study 

•  Jnmn  llrndlrg.  bom  in  r.lou«e!ilrn>l)iT«  in  1699,  and  dimi  in  1761, 
wan  the  niiwt  ilUtincni^hni  ■■trannnxr  ot  the  fin>t  half  of  tht  ei)|ht«Dlk 
cmtnrr.  Amonn  hi*  man  impnrt«tit  diieoTerir*  were  the  csplaaatiaa 
of  a<lnmt>mkal  aberration  |17M|,  the  ctinw  of  nnlalioD  (ITM),  and  Us 
nnpiricsi  rormola  for  oiimctiuu*  lor  rcrraction.  Il  is  pmfaaps  doI  too 
naeh  to  *aj  Ihst  ha  van  tht  Dnt  anlronomer  who  miula  the  art  of  oWn- 
ing  part  of  a  melliodieal  (ciencp. 

t  Ad  a(«aiinl  of  bin  lih  b;  Hir  William  Voong  la  prcflied  te  llw 
CvmlrmfliilHiPkilmofhiea:  Ihi*  ma  priotH  at  Lonisa  bi  IfM  for  private 
•iKalatiDa  and  ia  now  pilrrnwlj  rare^ 


S9S  TATU>R. 

€t  maiheniAlioH.  His  Murliest  work,  and  Uiat  hj  whieh  ht  k 
geneimlljr  luiowii,  b  hii  Mukodut  ImcrwmttUarmm  /KnMto  «l 
Invena  pobUiiliad  in  London,  in  1715.  This  onntoim  {jf'^  f] 
m  imiuf  of  the  well-known  theorem 

by  which  a  funetion  of  a  tingle  variable  ean  be  expanded 
in  powers  of  it.  He  docH  not  oomiider  tlie  eonvergenqr 
of  the  iieriea,  and  the  proof  which  involves  nameroos  asramp- 
tiona  is  not  worth  rppmducing.  The  work  alno  inclodee 
several  theorenm  on  interpoUtion.  Taylor  was  the  earliest 
writer  to  c1t*al  with  theorems  on  the  eliange  of  the  inde- 
pendent varialile;  he  was  perhaps  the  first  to  realise  the 
poHHibility  of  a  calculus  of  operation,  and  just  as  he  denotes 
the  tith  diflerential  ooetlicient  of  y  by  y.,  ho  he  uses  y.|  to 
n*prp8eiit  the  integral  of  y;  lastly  he  iM  UMually  recognized 
as  the  creator  of  the  theory  of  finite  differences. 

Tlie  applicationH  of  the  calculuH  to  variouH  questions  given 
in  the  JfethtKitut  have  hunlly  received  that  attention  they 
doKerve.  The  most  important  of  them  iH  tlie  theory  of  the 
trotiHrerKe  %'ibrations  of  strings,  a  problem  which  had  liaffled 
previtms  invent igatom.  In  this  investigatitm  Taylor  shews 
that  the  number  (if  half -vibrations  executed  in  a  second  is 

wyl(DPILN), 

where  L  is  the  length  of  the  Htring,  ^V  its  weight,  P  the  weight 
which  Ktrctches  it,  and  D  the  length  of  a  seconds  pendulum. 
Tliis  is  cornH;t,  but  in  arriving  at  it  he  asHumes  tlmt  every 
point  of  the  string  will  pass  through  its  position  of  equili- 
brium  at  the  same  instant,  a  restriction  which  D'Alembert 
subsequently  shewed  to  be  unnecessary.  Taylor  also  foand 
the  form  which  the  string  assumes  at  any  instant. 

The  JffihiMitts  also  contains  the  earliest  determination  of 
the  (litTerentiail  equation  of  the  path  of  a  ray  of  light  when 
traversing  a  heterogeneous  medium ;  and,  assuming  that  the 
deunity  of   the   air  depends  only  on  its  distance  from   the 


\ 


TAYLOR.      COTES.  393 


mrth'n  Hnrfncp,  Taylor  nhlnint^  hy  mennH 
approximnU^  fomi  of  tlip  purve.  Tin*  form  of  the  cntaimrj  Mid 
the  detprmi nation  of  the  centren  of  oRcillntion  mkI  pemUMion 
Kra  sImi  discumted. 

A  tivAtise  on  pen|>ectire,  by  Taylor,  pnUished  in  1719, 
contniriH  (ho  enrlinit  f^ni>ral  enancintion  of  tha  principkt  of 
vanJBhirif!  poititH  ;  though  the  idpn  of  Tnniiiliing  ptMntu  for  hori- 
«>nt«]  ftnil  pnnitlel  liiipn  in  a  picture  huni;  in  ft  verttcfti  pUn« 
bud  bwn  enuncint<<d  byUujdo  Ulnlrli  in  his  Pmptvtivat  f.ibri, 
Pim,  IGOO,  And  by  Stevinnn  in  hii  Scinffrfphui,  Leydsn,  1G08.  . 

Cotes.  Unt/'r  Ciil'ji  was  bom  nmr  Leicester  «mi  Jnly  10, 
1682,  and  died'  at  Cnmbrid^  on  June  5,  1716.  He  wm 
fdncated  at  Trinity  Oollffn',  Cninbridgp.  of  which  Mcirty  he 
wu  «  fellow,  nnd  in  1 706  wns  electa*)  to  the  newly<re»l«d 
Plumian  chair  of  utronomy  in  the  nnivpnity  of  CMnbridge. 
From  i70n  U>  171-1  his  time  wns  mainly  occa pied  tn  editing 
the  necond  edition  of  the  Priii^i/nn.  The  rniwrk  at  Newton 
that  if  only  Cotes  had  lived  "we  mieht  have  known  tiome- 
thinf;'  indicat«>s  the  opinion  of  his  abilitin  held  byntostof 
his  content  (lorarien. 

Cntes'n  writinfpi  were  collected  nnd  pablishrd  In  172S 
andpr  th«  titles  Ilarmonin  .Vrnnimnnn  Hid  Oper»  Jfite^- 
latint.  His  lectures  nn  hydrostfttio  were  pnblishpd  in  1738. 
A  Inr^  part  of  the  llnntmnin  Mrnmirnmni  is  given  np 
to  the  decom position  nnd  integration  of  rfttionil  al^bnicnl 
exprfWHionfl :  that  part  which  <ln«ls  with  the  theory  of  partial 
fractions  was  left  unfinished,  but  wns  completml  by  Dentoivre. 
Cotnt's  tbrorpni  in  trigonometry,  which  depends  on  forming  the 
quadratic  factors  of  r*  -  1,  is  well  known.  The  propmition  that 
"  if  from  a  fixed  point  0  a  line  Iw  drawn  cutting  a  curve  in 
9).  9t>  --<  Q>t  ""d  '^  point  P  lie  taken  on  the  line  no  that  the 
reciprocal  of  OP  in  the  arithmetic  mean  i>f  the  reciprocals  ol 
OQi.OQt,  ■■■,0Q„  then  the  locus  of  /'will  be  a  strmight  tine  " 
in  also  dufi  to  Cotes.  The  title  of  tlie  book  was  derived  fmm 
the  latter  theorem.  The  Optm  Mue*lhw^  eontaiiH  »  paper 
on  the  method  for  determining  the  mnnt  prohtfale  rcMilt  Ina 


S94  OOTEB.     DIMCMTBI. 

A  ttuiuber  of  ohiervAtioiis :  Ihis  wm  the  eeriieii  atlwnpl  to 
fnuue  A  theory  of  erroni.  It  alio  containai  esBejs  OA  NewUmVi 
JietAodw$  I^ifertidialiSf  ou  the  conaitructioii  of  ifMm  hj  the 
lueUiod  of  diflerencea,  od  the  descent  of  a  body  imder  gravity, 
«Mi  the  cyck>idal  penduluui,  and  on  projectiles. 

Demoivre.  Abraham  J^emaipn  (moce  conecUy  written 
AH  tie  Molvrt)  wiiM  bom  at  Vitry  on  May  26,  16C7»  and  died 
in  London  on  Nov.  27,  1754.  Hin  parents  came  to  Kngland 
when  lie  was  a  boy,  and  his  education  and  friends  were  alike 
Engliiili.  His  interest  in  tlie  higher  mathematics  is  said  to 
liave  originated  in  his  coming  by  cliunce  across  a  copy  of 
Newton's  Pruticipia.  From  tlie  tioye  on  him  delivered  in  1754 
before  the  French  Academy  it  would  seem  that  his  work 
as  a  teaclier  of  mathematics  had  led  him  to  the  house  of 
the  Karl  of  Devonshire  at  the  instant  when  Newton,  who 
had  asked  peniiission  to  present  a  copy  of  his  work  to  the 
eiirl,  watt  coming  out.  Taking  up  the  book,  and  charmed 
by  the  far  reaching  conclusions  and  the  apparent  simplicity 
of  the  reusiiuiii;;,  Dt*iiioivre  thought  nothing  would  be  easier 
than  to  iiiaiMti*r  the  subject,  but  to  his  surprise  found  that 
to  follow  the  argument  overtaxed  Ids  powers.  He  however 
liought  a  copy,  and  as  he  had  but  little  leisure  he  tore 
out  the  pages  in  order  to  carry  one  or  two  of  them  loose  in 
his  pocket  so  that  he  could  study  them  in  the  intervals  of  his 
work  HM  a  teacher.  8uUsei|uently  he  joined  the  Royal  Society, 
and  liecaiiie  intimately  connected  w^th  Newton,  Halley,  and 
other  mathematicians  of  the  Englisli  school.  The  manner 
of  Ills  deiith  lias  a  certain  interest  for  psychologists.  Shortly 
Ix^fore  it,  he  declared  that  it  was  necessary  for  him  to  sleep 
some  ten  minutes  or  a  quarter  of  an  hour  longer  each  day 
than  the  preceding  one :  the  day  after  he  had  thus  reached  a 
total  of  something  over  twenty-three  hours  he  slept  up  to  the 
limit  of  twenty-four  hours,  and  then  died  in  his  sleep. 

He  is  best  known  for  having,  together  with    Lambert, 

created  tliat  part  of  trigonometry  which  deals  with  imaginary 

t/uuiitiiien.     Two  theorems  on  this  \iart  of  the  subject  are  still 


\ 


DEMOIVRE.      HACLAURIN.  39S 

connected  with  hia  iMDie,  iiKnielf,  that  whicli  MHerto  that 
Mtmtx  +  iconnx  in  one  of  tlie  values  of  {HJnx-i- ioosx)",  and 
that  which  givps  the  variouH  i)ua(lnitii!  fiicton  of  i"  -  3/»c"  +  I. 
His  chief  wi>rk!4,  uther  than  numerous  papers  in  the  Pliiltt- 
topiiral  Traiiffirlioii*,  were  Thf  Dortriar  nf  Chaneta  publuhfd 
in  IT18,  ami  the  J/u'-'V/.iii'n  J..>./yfi'cn  published  in  1730.  In 
the  fonner  the  theory  of  recurrini;  Neries  wan  firHt  K>veii,  and 
the  theory  uf  partial  fractiouH  whii'h  Cutes'n  prpm»ture  death 
had  Irft  untinixhecl  wa.<<  iiimplelea),  while  the  rule  for  finding 
the  probability  uf  a  coiiipuuiid  event  wax  enunciated.  Tlw 
Utter  book,  l>esidi-s  the  tri^fotionietricnl  propottitions  mentioned 
above,  containM  Rome  theoremii  in  nstninomy  but  lliey  an 
treated  aa  probleiiis  in  nnalyHix. 

Maclaoiin*.  C''>/in  .Ifnclnnrin,  who  wim  bom  at  KtlnuMlan 
in  Argj'llshin-  in  February  IfiOS,  anil  ilie<l  at  Yiirk  on  Jane  14, 
I74C,  wan  educat«il  at  the  university  uf  Claagow  ;  in  1717, 
he  wan  elected,  at  the  early  age  of  ninetMin,  profeiwor  at 
mathenwticR  at  Aberdeen  ;  and  in  I7^ri,  he  waw  appointed  thtt 
deputy  of  the  malhenintieal  profeflwir  at  l^inbni^h,  and  nlti-' 
nintely  f>ucc<«(lett  hint :  there  wan  sciTne'dlffimlty  in  aecurinK  a 
Htipend  Cor  a  deputy,  and  Xrwton  privately  wrote  offering  (o 
bear  the  cost  so  aji  to  enable  Hie  univenuty  to  tiecure  tlie 
serricen  of  MacUurin.  Maclaurin  took  an  active  p«rt  in 
oppoxing  the  advance  uf  the  Young  Pretender  in  1745;  on 
the  appnnrh  of  the  HighlanderH  he  fled  to  York,  bat  the 
expoNure  in  the  trencheo  at  Rdinbursh  and  the  privations  h« 
endured  in  hin  escape  proved  fatal  to  hioL 

Hia  chief  works  are  his  (leantrtrui  Orgttnien,  Lnndoo, 
1720  ;  his  Dr.  fAnfnrum  Gtom^lrifttrtim  Pm/iriftittititta, 
Ixmdon,  1720;  his  Trrf>ti«!  on  flnximut,  h>linbnnth,  1743; 
hin  Algrlm,  lyindon,  174H,  and  his  AeenHitt  <^  XmHon't 
Oineorrrirj',   London,   1748. 

The  first  aeclion  of  the  (imt  part  of  the  O^imtftriti  Organint 
in  on  onnim ;  the  second  on  nralal  citbien ;  the  third  on  other 


306  MACUiUEiir. 


and  on  quAitioi;  and  the  ioarth  ■eetioo  fa  oa  famnil 
propertiai  of  carvea.  Newton  had  shewn  that^  if  two  ai^gka 
hounded  hy  iitraight  lines  torn  roond  their  reiqpeetive  nuniaita 
MO  that  the  point  of  intenection  of  two  of  these  lines  moves 
along  a  stnught  line,  the  other  point  of  intersection  will 
deHcribe  a  conic ;  and,  if  the  firut  point  move  along  a  oonic^  the 
Meoond  will  describe  a  quartic  Maclaurin  gave  an  analytical 
diMCUMaion  of  the  general  theorem,  and  shewed  how  by  thu 
method  various  curves  could  be  practically  traced.  Thfa  work 
contaioH  an  elaborate  discttiMion  on  curves  and  their  pedalsi 
a  branch  of  gi*ometry  which  he  had  created  in  two  papers 
publittlied  in  tlie  PhUoiopkieal  Trafuadions  for  1718  and 
1719. 

The  tiecoiul  part  of  the  work  is  divided  into  three  sections 
and  all  appendix.  Tlie  first  nection  contains  a  proof  of  Golesfs 
theureui  above  alluded  to;  and  also  the  analogous  theorem 
(duicovered  by  hiuiself)  that,  if  a  straight  line  OPii\...  drawn 
through  a  fixed  point  O  cut  a  curve  of  tlie  nth  degree  in  u 
points  y,,  I\y,..<t  <^d  if  the  tangents  at  1\^  Pt*"-  <^ut  a  fixed 
line  Ox  in  points  ^,,  ^ «,...,  then  the  sum  of  the  reciprocals 
of  the  diMtancen  O^i,  0^,,...  is  constant  for  all  positions  of 
the  line  OJ\J\.,.,  These  two  theorems  are  generalizations  of 
thijtie  given  by  Newton  on  diameters  and  asymptoteH.  Either 
is  deducible  from  the  other.  In  the  scHX>nd  and  third  sections 
tlicKe  theorems  are  applied  to  conies  and  cubics ;  most  of  the 
harmonic  properties  connected  with  a  qiiodrilateml  inscribed 
in  .a  conic  are  deteimined ;  and  in  particular  the  theorem  on 
an  inscrilied  hexagon  which  is  known  by  the  name  of  Pascal 
is  deduced.  Pascal's  essay  was  not  publislied  till  1779,  and 
the  earliest  printed  enunciation  of  his  tlieorem  was  tlu&t  given 
by  Maclauriii.  Amongst  other  propositions  he  shews  tlu&t, 
if  a  quaidrilateral  be  inscribed  in  a  cubic,  and  if  the  points 
of  intersection  of  the  opposite  sides  also  lie  on  the  curve,  then 
the  tangentji  to  the  cubic  at  any  two  opposite  angles  of  the 
qu!ulrihit4*nil  will  nu*et  on  the  curva  In  the  fouKh  section 
he  considers  some  theorems  on  central  force.   Tlie  fifth  section 


\ 


MACLAURIN.  307 

contains  itnme  thporcniH  on  thp  HpRcription  o(  cnrvn  thmagh 
^ren  points.  Onr  of  thr-ne  (which  inrluHes  PwgmI's  mi  a  pmr- 
ticolnr  cme)  in  thnt  if  n  polygon  lie  (iefomied  tm  that  while 
efkch  of  its  sitlps  p.-uwn  througli  a  fixrd  pMiit,  itii  Miglcfl  (mve 
onff)  dpMribfi  respwlivHy  cnrvcs  of  tlic  Nith,  nUi,  jAh,... 
degrera,  then  shftll  tlif  rrmaining  nn^'le  dpscriho  a  corve  of  tlie 
dpgrpo  Sntri/i-...;  hut,  it  the  given  pointn  be  oollineiu',  the 
resulting  cone  will  Iw  only  of  the  dcgn*  mitfi....  Thi«  ewHjr 
WM  rpprintcd  with  tulditinns  in  thn  PhiftartiJiieal  Tmtuaetimu 
for  1735. 

The  Trrniifi-  of  Fliuiom  publiKlipd  in  1742  wm  the  lirai 
logical  *nd  xyRl^-nintic  expusitiim  of  tlip  method  of  8wioM. 
The  cause  of  itn  publicntinn  wiw  an  attack  hj  Berke1«]r  OQ  the 
principtrs  of  the  infinitt^imal  cnlcolnn.  In  it  [art.  761,  p.  610] 
Aliiclaurin  gave  a  proof  of  thr  theorem  that 

/(r)=/(0).^-(0)  +  ^/"(O)t.... 

This  WAR  obtAined  in  tho  manner  given  in  nrnny  modcni  text- 
books by  MNsaming  that  /{x)  can  )k  pipandod  in  a  (omt 
like 

thcti  nn  dilTcreiitiating  iind   putting  j:  =  0  in  the  Buccesnive 

iTtiulta,   the   valucH  of    A.,  A am  olitaincfl :   but  he  did 

not  invcHtigat^r  thr  coiivrrgency  of  the  Hcrien.  The  rcault  liad 
been  previouxly  given  in  17'30  by  Jamca  Htiriing  In  his 
Melhottitg  Diffurettlia/u  [p.  I02J,  ami  of  oosnie'is  at  oace 
dodncible  from  Taylor's  thoon-m.  Maclaurin  also  here  enun- 
ciated [art^  .150,  p.  3«!>]  tlie  imporlAnt  thoortm  tliat,  if  ^{x)  be 
positive  and  dociraBo  as  r  incrcasen  from  x=n  loxciao,  then 
theaerin 

*(<.)t*(«+l)  +  *(«  +  2)+... 

is  oDnvergent  or  divergent  aa  |     ^  (2)  die  i«  finite  or  infinita 

He  abo  gave  the  correct  theorf  of  maxima  and  i 
ralea  for  finding  and  diacriminating  mnltiple  f 


898  MACLAUElir. 

This  troaUae  b  however  eapecUIly  valuable  lor  the  nlv- 
tiou  it  contaiiis  of  numeroiu  problenia  in  geomeifj,  ilftftie^ 
the  theory  ol  attractioiiai  and  Mtronomy.  To  nlve  iheia 
Maclaurin  reverted  to  cUasical  methods,  and  so  powerfyl  did 
these  processes  seem,  when  used  by  him,  t^iat  Clairant  after 
reading  tlie  work  alsuidoned  analysiis  and  attpicked  the  problem 
of  the  figure  of  the  earth  again  by  pure  geoipetry.  At  a  later 
time  tliis  part  of  the  book  was  described  by  Lagrange  as  the 
'^  chef-d'a'uvre  de  gtoni^trie  qu'on  peut  coinparer  k  tout  ce 
qu'Arcliiuittie  nous  a  lainsfS  de  plus  beau  et  de  plus  ing^nieux."* 
Maclaurin  also  determined  the  attraction  ttf  a  homogeneous 
ellipsoid  at  an  internal  point,  and  gave  soiAe  theorems  on  its 
attraction  at  an  external  point ;  in  effecting  this  he  introduced 
the  conception  of  level  surfaces,  that  is,  surfaces  at  every 
point  of  which  the  resultant  attraction  is  |ierpeiidicular  to  the 
surface.  No  further  advance  in  the  theory  pf  attractions  was 
mode  until  Lagrange  in  1773  introduced  |  the  idea  of  the 
potential  ^[aclaurin  also  shewed  that  a;  spheroid  was  a 
possible  form  of  equilibrium  of  a  mass  of  homogeneous  liquid 
rot^itiiig  about  an  axis  pulsing  through  itsi  centre  of  mass. 
Filially  lie  diticussed  the  tides :  this  part  hoq  been  previously 
publiKhed  (in  1740)  and  liad  received  a  prizej  from  tlie  French 
Academy. 

Among  Maclauriii's  iiiiiiDr  works  is  liis  4^yebra^  publinhed 
in  1748,  and  founded  on  Newton's  Universal  Arit^tmetic.  It 
contains  the  results  of  some  early  papers  of  Maclaurin; 
notably  of  two,  written  in  1726  and  1729,  An  the  number  of 
imaginary  roots  of  an  etjuation,  suggested  by  Newton's 
theorem;  and  of  one,  written  in  1729,  containing  the  well- 
known  rule  for  finding  eijual  roots  by  mean^  of  the  derived 
equation :  in  this  book  negative  quantitiesl  are  treated  as 
being  not  less  real  than  pasitive  quantities  To  this  work  a 
treatise,  entitled  De  Linearum  G'eomriricdru^n  ProprletaiibuM 
OeMralibus,  was  added  as  an  appendix ;  besides  the  paper  of 
1720  above  alluded  to,  it  contains  ismie  additional  and  elegant 
theorems.     Maclaurin  also  produced  in  1728  on  exposition  of 


1 


HACI.AI'RtN.      STEWART.      SlUPSON.  399 

thfl  NrwUiiiiaii  pliiliHopliy,  which  is  iiiwirf«nr»t«l  in  Um*  pnrt- 
huninuH  work  priiitr^t  in  \7il*.  Ahmnt  the  liwt  pnpnr  ha 
wm(«  «wt  nno  printpH  in  the  I'hUntinjiliiml  TraHmetimv  for 
1743  in  which  he  iU^u-us.'hhI  fnun  it  ninth i-nnttiail  point  of  view 
the  fonii  of  a  bcp's  cell. 

.^lltcll«urin  WAX  anr  of  tho  moit  ahk  matheniKticitinN  <if  tho 
nphtpf-nth  c^rntur^,  hut  hiH  influrnce  nrt  the  pni;;reM  of  llritiNh 
innthcnmlic*  wan  on  the  whnte  unfortunalo. .  By  liiniwlf 
iiluind<>ninc  thf  ukc  liofh  t>f  NnntyitiH  ami  nf  thn  inKnilrHiinat 
cnlntlun  hr  inducr<l  Newtmi's  murilrynn-n  to  confine  Ihon- 
etIvpk  til  Nf^wton'R  nicthodN,  nnd  it  wm  not  nntil  aliunt  I8'J0, 
when  the  dilTerpntinl  calculus  was  intniHdccd  into  Ihn  Cam- 
brifl;,'^  curriculum,  that  Kn^Iii«h  niAthcrnaticiaiiH  marie  an/ 
general  use  of  the  more  powerful  methods  of  modern  analyitia. 

Stewart.  Maclaurin  wnn  Kuccee^hil  in  bit  chair  at 
Edin)>ur|;h  by  his  pupil  A/niflm'e  S'ru-nrl,  ))om  at  Kothcsay  in 
1717  and  di<Hl  at  Edinburgh  on  Jim.  'A  I78>%  a  nwtbe- 
matictati  of  considerable  power,  to  wlxiin  1  allude  in  pKKning 
for  hill  theorems  on  the  pmlilem  of  three  bodieH  and  for 
his  diKCussinn,  treated  by  transverwii!!  and  involution,  of  the 
properties  of  the  circle  and  Htmight  line. 

Simpson*.  The  Inst  meinlier  of  the  Kngliitli  nchool  wlnmi 
I  need  mention  here  in  Thomw  Simjuton,  who  wait  ham  in 
Leicexterahirv  on  Aug.  20,  1710,  and  died  on  May  14,  1761. 
His  father  was  a  weaver  and  he  owed  his  e<luration  to  kin 
own  effort*.  His  mathematical  interestH  were  first  aroused  by 
the  solar  eclipse  which  took  plocn  in  17'J4,  and  with  tlte  aid 
of  a  fiirtune- telling  pedler  he  nia-stereil  Cocker'it  Arithm/ttif  mA 
the  elements  of  algebra.  He  then  fpiTe  up  his  weaving,  and 
became  an  unher  at  a  school,  and  by  constant  and  laborious 
effortn  improved  his  mathematical  education  so  that  by  1735 
he  was  able  to  solve  several  question*  which  had  been 
feoently  propoaed  and  which  involved   the  infinilimmal  cal- 

■  A  tkcleh  ot  SiopsoD'*  U[^  with  a  Ubliognfliy  af  Ua  writia^  hj 
3.  B«n«  and  C.  Hnttoa  wan  pnUishvd  in  Ijtmdea  in  17H:  a  short 
MSBioif  It  also  pteOiHI  lo  the  later  ediliona  of  bii  wort  oa  flasioKa. 


400  BiimoN. 

calttiL  He  nest  moved  to  London,  and  in  174S  was  iqipointed 
proloMor  of  niathematioi  at  Wodwidit  a  post  wUeh  he  eo»- 
ttnued  to  occupj  till  his  death. 

The  works  published  bj  Simpson  prove  him  to  have  besn 
a  man  of  extraordinary  natunU  genius  and  extreme  indastty. 
The  most  important  of  them  are  his  Fluximu^  1737  and  1750, 
with  numerous  applications  to  physics  and  astronomj;  his 
LtiwM  o/Chatyce  and  his  EtmitfB^  1740 ;  his  theoiy  of  AnnuHim 
and  JievertioHs  (a  branch  of  niatliematics  that  is  due  to  James 
Dodfion,  died  in  1757,  who  was  a  master  at  Christ's  Hoiqpital, 
London),  with  tables  of  tlie  value  of  lives,  1742;  his  Du- 
BertiUimu^  1743,  in  which  the  figure  of  the  earth,  the  force 
of  attraction  at  the  surface  of  a  nearly  spherical  body,  the 
theory  of  tlie  tidcK,  and  the  law  of  astronomical  refraction 
are  discuiwed;  his  Algebra^  1745;  his  Geometrjf^  1747;  his 
Trigofwmeiry^  1748,  in  which  he  introduced  the  current  ab- 
breviations for  the  trigonometrical  functions ;  his  Select  Exur- 
einfir,  1752,  containing  the  solutions  of  numerous  problems  and 
a  theory  of  guimery ;  and  lastly,  hiM  MUceliatieouM  TracU,  1 754. 

The  work  last  mentioned  consists  of  eight  memoirs  and 
these  contain  his  best  known  investigations.  The  first  three 
papers  are  on  various  problems  in  astronomy ;  the  fourth  is  on 
the  theory  of  mean  oljHervations ;  the  fifth  and  sixth  on  problems 
in  fluxions  and  algebra ;  the  seventh  contains  a  general  solution 
of  the  isoperimetrical  problem ;  the  eighth  contains  a  duicussion 
of  the  third  and  ninth  sections  of  the  Priiteifna^  snd  their 
applicatitiii  to  the  lunar  orbit.  In  this  laitt  memoir  Simpson 
obtained  a  differentijil  e«iuation  for  the  motion  of  the  a]iHe  of 
the  lunar  orbit  similar  to  tliat  arrived  at  by  Clairaut,  but 
instcjul  of  Hohing  it  by  succi*SHive  ap|»roxinuitions  he  deduced 
a  general  solution  by  indeteniiinate  coeflicieuts.  The  result 
agrees  with  that  given  by  Clairaut.  Simpson  first  solved 
this  pn>blem  in  1747,  two  years  later  tlian  the  publication  of 
Clairaut's  memoir,  but  the  solution  was  discovered  inde- 
pendently of  Ch&iraut's  researches  of  which  Simpson  first  heard 
in  1748. 


CHAPTER  XVIII. 

LAGRANQK,    LAPLACE,  AND  THEIR  CONTEXPORARIER. 
CIBC.  1740-1830. 

The  Innt  chnpttr  ointAtnii  the  hixtory  of  tvo  aepante 
■choolD — the  cftntinpntftl  and  thf  RritJHh.  In  the  ewlj  jmn 
(rf  the  eighteenth  ct- ntur>-  thp  En;;liHh  mc-IhuI  Appntred  vigonMH 
And  fruitful,  )itrt  cWndence  mpidly  set  in,  nnd  nfter  the  drathi 
of  Maclaurin  nnd  Simpson  no  British  ninlheniiiticisn  appenml 
who  is  at  all  compirahle  In  the  continental  mathematicians  of 
the  latter  half  of  the  ei};ltteentli  centurr.  Thia  fact  is  partly 
explicable  bj  the  isolation  of  the  Hchool,  [nrtlj  Itjr  jtn  tendency 
to  rely  too  excluaivelj*  on  geoniflrical  and  Hnxional  meihodN. 
Some  attention  was  however  pien  to  practical  leience,  hnt, 
except  for  a  few  reinarkn  at  the  end  <rf  thin  chapter,  I  do  not 
think  it  nrcesxarj'  to  dincuvA  EngllNli  matliematici  in  detail, 
until  about  It^'JO  when  analyticnl  inethiMlii  ajrain  canw  into 
vogue, 

Un  the  continent  under  the  influeiire  nf  Jolm  Bemonlli 
the  calcului  had  become  an  initmment  of  great  analytical 
power  ex pTVMMl  in  an  arlmirabln  notation— rtmI  for  practical 
applicationH  it  Si  impoflsihle  to  over-en liniat«  the  value  of  a 
good  notation.  The  nubject  of  mechanics  remained  however 
in  much  the  condition  in  which  Newt^m  had  left  It*  nntil 
D'Alemhertt  by  making  une  of  the  differential  calctilna,  did 
■otnething  to  extend  iL  Universal  gravitation  aa  enunciated 
iu  the  Priitcipia  was  accepted  as  an  eslahlisbed  fact,  but  the 
geotnetrkal  methods  adopted  in  proving  it  werv  difficnii  to 
follow  or  to  BM  in  analogoas  problems ;  Midaarint  Bhasiyia^ 


402     LAQBANQl,  LAPLACI;  AVD  THSIR  OOmTPIKWIAlfM. 

and  Claimat  maj  be  regaided  m  Umi  InH  nuUliMuilidMM  «l 
distinction  who  employed  tliem.  htuMy  the  NewtooiMi  theory 
of  light  wen  generally  received  as  correct 

The  leading  mathematicians  of  the  era  on  which  we  are 
now  entering  are  Euler,  Lagrange,  f^place,  and  L^gendre. 
Briefly  we  may  say  that  Euler  extended,  summed  up^  and 
completed  tlie  work  of  his  predecessors ;  while  Lagrange  with 
almost  unrivalled  skill  developed  the  infinitesimal  calculus 
and  theoretical  mechanics,  and  presented  them  in  forms 
similar  to  those  in  which  we  now  know  them.  At  the  same 
time  Laplace  made  some  additions  to  the  infinitesimal  cal- 
culus, and  applied  that  calculus  to  tlie  theory  of  universal' 
gravitation ;  he  also  createtl  a  calculus  of  probabilities. 
lii*geudn*  invented  spherical  liarmotiic  analysis  and  elliptic 
inU*gnils,  and  added  to  the  theory  of  numbeni.  The  works 
of  tliene  writers  are  Htill  standard  authorities.  I  shall  con- 
tent inyH4*lf  with  a  mere  sketch  of  the  chief  discoveries 
eniliucliecl  iii  tlieui,  referring  anyone  who  wishes  to  know 
ni€»re  to  the  works  tlieninelves.  Lagrange,  lAplace,  and 
fiegendre  creuU'd  a  French  school  of  mathematics  of  which 
tlie  younger  members  are  divided  into  two  groups;  one 
(including  Poinson  and  Fourier)  began  to  apply  mathematical 
analysis  to  physics,  and  the  other  (including  Monge^  Camot, 
and  Poncelet)  created  modern  geometry.  Strictly  speaking 
some  of  the  great  mathematicians  of  recent  times,  such  as 
iJaUHs  and  Abel,  were  contemporaries  of  the  mathematicians 
hist  named ;  but,  except  for  this  remark,  I  think  it  con- 
venient to  defer  any  consideration  of  them  to  the  next 
chapter. 

The  dfVflopiMnt  of  imalynM  and  viechmiics. 

Euler*.     Ltanihani  Kuhr  was  born  at  lisle  on  April  lA, 
1707,  and  died  at  St  Petersburg  on  Sept  7,  1783.     He  was 

*  The  chief  focU  in  Eul«r*t  life  sra  giveu  by  N.  Fuiu,  and  s  IimI  of 

Bul«r*«  writings  is  |>nitiMd  to  his  Vorrt$foiuUii€et  S  vols,  Bt  Pstersboig; 

JifS:f,    Ealer*d  ciarlior  works  am  cUmoussmI  by  Csntor,  climptors  csi,  ciiii. 


EUtEB.  403 

the  M>n  of  n  Luthnmn  minister  wlm  liad  nettled  «t  B>le^  nnd 
was  piIucaU^  in  liin  nntive  tuwii  uriilcr  the  dirpction  q(  Juhn 
Il<<ni<>iilli,  with  whose  xonn  Unniel  niifl  Nicholu  he  formed  » 
lifelong  frienilihip.  When,  in  17'2'',  tlie  yoanger  Bemonllis 
went  to  ICuRsin,  on  the  iiivitittion  of  tlie  <^mpmii,  they  pro- 
cun>d  X  plftce  tliere  f>ir  Roler,  which  in  173^  he  exvhnnjcetl  f<ir 
the  chnir  of  nmrheninticn  then  Micntnl  liy  Daniel  Itennwlli. 
TIte  Hfyerity  nf  the  dininto  Airi«t/Hl  hin  ejMif^t,  aikI  in  I73S 
he  loHt  th<>  iiM>  <if  line  eye  coiu|]|eleIy.  Iti  1741  lie  moved  to 
Rcrlin  At  till!  rtvgiirHt,  nr  mther  nmiiimiiil,  of  Fmlerick  the 
Cirpwl ;  liere  he  ntnved  till  I7CG,  when  hn  returned  to  Ramia, 
nnd  w«B  sucm'e<leil  nt  Borlin  Uy  LHKrniige.  Within  two  or 
thrm  yearn  of  Iiih  Koing  Imck  to  Ht  Petenburg  he  hemmo 
Mind :  liut  in  i>[>ite  nf  thin,  nnd  atthnugh  Hin  houwe  tojiPther 
with  mnny  of  hw  pnperH  were  Imnil  in  1771,  he  rncant  Mid 
impruved  ni<Ht  of  liii>  enrlier  wnrkn.  lie  died  of  apoplexy  in 
1783.     He  wnn  marrird  twice. 

I  think  we  nwy  lum  op  Enler'n  work  by  myinK  th*t  he 
cr*nt«l  «  jfoo"!  deAl  of  nnnlyxtii,  nnd  rpi'iHed  klmoat  nil  the 
limnchm  iif  pure  matheinatica  which  were  then  known,  filling 
up  the  delAilR,  adding  proof's  imil  nrmnFring  the  whole  in  n 
conRintent  form.  Such  work  is  very  imporlAnt,  nnd  it  in 
fortunate  for  Hcience  when  it  falls  into  linndu  m  c 
thotc  of  Evler. 

Euler  wrote  an  immenxe  number  of  memotra  on  all 
kindN  of  mnthemnticnl  nubjecU.  His  chief  workn,  in  which 
many  of  the  re«alu  of  earlier  memoira  are  embodied,  are  an 
followfi. 

In  the  Arst  pliMe,  he  wrot^  in  I74R  his  liitndmelia  in 
Anatt/nn  tufijiiliirum,  which  wan  intenditd  to  lerve  aa  an 
introduction  to  pnre  analytical  mathematica.  Thin  in  divided 
into  two  parta. 

Hie  fint  part  of  the  Anali/n*  /nfinilantUt  contains  Um 
bulk  of  the  matter  which  ia  to  be  found  in  modem  text-liotte 
et*,  and  cxra.     Mo  onnpleU  edillaa  of  Enln'*  aimim  hM  bM*  pA- 


//I 


404     LAOIUKQl,  LAFLAC^  AMP  THTOI  OQimniPOEAmm, 

on  algebi«i  theory  of  egoationi,  and  trigonomeirj.  In  Iho 
algebni  he  paid  particnhur  attention  to  the  ezpanrion  oC  vaiione 
fiinctiont  in  aeriee»  and  to  the  minunation  of  given  aeriee ;  and 
pointed  oat  explicitly  that  an  infinite  ■eriea  cannot  be  eaiely 
employed  onleM  it  is  convergent.  In  the  trigonometry,  mmh 
of  which  ia  founded  on  P.  C.  lia^'er's  Arithmeiie  t^f  S'nm 
which  had  been  published  in  1737,  Eoler  developed  the  idea 
of  John  Bernoulli  that  the  iiubject  was  a  branch  of  analyea 
and  not  a  mere  appendage  of  astronomy  or  geometry :  he  also 
introduced  (contemporaneously  with  Simpson)  the  current 
abbreviations  ibr  the  trigonometrical  functions,  and  shewed 
tliat  the  trigonometrical  and  rxponential  functions  were^con- 
nected  by  the  relation  cos  B-k-ininO-e**, 

Here  too  [pp.  85,  90,  93]  we  meet  the  symbol  «  used  to 
denote  the  Ijane  of  the  Napierian  logarithms,  namely,  the 
incuiumenKurable  number  2*71828...,  and  the  symbol  w  used 
to  denote  the  incumnienHurable  number  3*1 4159....  The  use 
of  a  Hingle  symbol  to  denote  the  number  3*71828...  seems  to 
lie  due  to  Coten,  who  denoted  it  by  M;  Euler,  in  1731,  denoted 
it  by  e.  To  the  best  of  my  knowledge,  Newton  had  been 
the  first  to  employ  the  literal  exponential  notation,  and  Euler, 
using  the  form  a',  had  taken  a  as  the  base  of  any  system  of 
logarithms :  it  is  pmbable  that  the  choice  of  «  for  a  particular 
base  was  detennined  by  its  being  the  vowel  consecutive  to  & 
The  use  of  a  single  symbol  to  denote  the  number  3*14159... 
appears  to  have  been  introduced  about  thc^  beginning  of  the 
eighteenth  century.  W.  Jones  in  170G  represented  it  by  v,  a 
syuiljol  which  had  been  used  by  Oughtred  in  1647  and  by 
Harrow  a  few  years  later  to  denote  the  periphery  of  a  circle. 
John  Ik^rnoulli  represented  the  number  hye;  Euler  in  1734 
denoted  it  by  />,  and  in  a  letter  of  1736  (in  which  he  enunciated 
the  theorem  that  the  sum  of  the  squares  of  the  reciprocals  of 
the  natural  numbers  is  ^t*)  he  used  the  letter  c;  Chr. 
Goldbach  in  1742  used  w ;  and  after  the  publication  of  Euler^a 
AiuiiydU  the  symbol  w  was  generally  employed. 

The  numbers  e  and  w  would  enter  into  mathenmtical 


EULER.  405 

AnKtyHiH  frnm  whxlever  n'uto  thp  solypct  wan  Mpprtmched.  The 
latter  rrprpaentfl  among  otiier  things  tlic  ratio  of  the  circnni- 
fercnce  of  a  circle  to  itn  diamehr,  liul  it  is  a  mere  accident 
that  that  is  taken  for  its  drfinitinn.  Do  Morgan  in  the  BtutgH 
of  Pitratlor^s  tells  an  anecdote  which  illuHlimtes  how  little  the 
usual  definition  sujjgpKU  its  rpnl  oriirin.  He  was  explaining 
to  an  actuary  what  was  the  chance  ttiat  at  the  end  nt  a  given 
time  a  certAin  pn>piiKion  of  sumo  group  of  pcNtple  would  ho 
alive ;  and  quot4>d  the  actuarial  formula  involving  w,  which,  in 
answer  to  a  rinestion,  he  cxplainnl  sUioil  for  the  ratio  of  the 
circumference  of  a  circle  to  iU  diameter.  Hi»  ac<)aaintanca 
who  had  so  far  lixtx-nei)  to  the  explanation  with  interest  inter- 
rupted him  and  explained,  "  My  dear  friend,  that  mint  be  » 
delusion ;  what  can  a  circle  have  tn  do  irith  the  number  of 
people  alive  at  the  end  of  a  given  timet" 

The  second  part  of  the  Annfi/iit  InJiniUtnm  is  on  aiM- 
Ijtical  geometry.  Euler  mmmencerl  tlii*  p«rt  hj  dividing 
carves  into  algebraical  and  tranHrendental,  and  established  a 
variety  of  pnipositionn  which  are  true  (or  all  algebraical 
carves.  He  then  applinl  these  to  the  general  equation  of  the 
second  degree  in  two  dimensions,  iihewe<l  that  it  represents 
the  various  conic  nectjons,  and  deduced  moet  of  their  proper- 
ties from  the  general  equation.  He  also  ronstflered  the  classi- 
fication of  cubic,  qnartic,  and  other  algebraical  curves.  He 
next  discussed  the  r|ue!>tion  as  U>  what  surfaces  are  represented 
by  the  general  equation  of  the  second  degree  in  three  dirnen- 
sions,  and  how  they  may  he  discriminated  one  from  the  other: 
some  of  these  surfaces  had  not  been  previously  in\-estigated. 
In  the  ooume  of  this  analysis  he  laid  down  the  mlea  for  the 
transformation  of  enordinatee  in  space.  Here  also  wa  find  the 
earliest  attempt  to  bring  the  curvature  of  snrbces  within  the 
domain  of  mathematics,  and  the.  first  complete  dlsemsioii  of 
tortuous  curves. 

The  Awlywi  Infinitomm  was  followed  in  1755  by  the 
IntlitiUwnm  Catfuli  D^firvtUtnliM  to  which  it  was  intended  aa 
■n  introduction.  This  is  the  Gnt  text-boolt  mi  tin  diftrentid 


406     LAURANQi;  LAFUkOB,  AMD  TMICIK  aniTKimNUBm. 

caIcuIiis  which  hM  aaj  ckim  to  be  regaided  m  eottpbtfl^  and 
it  may  be  wid  that  maoy  modem  iroatinee  on  the  salijeei  are 
baied  cm  it ;  at  the  lame  time  it  ahould  be  added  that  the 
expoeitiou  of  the  principles  of  the  ambjeot  in  often  prolix  and 
ohicure^  and  aonietimes  not  altogether  accurate. 

Thin  Bcriee  of  works  wm  completed  by  the  publication  in 
three  volumes  in  1768  to  1770  of  the  /nslftlMlioNds  Caladi 
IniegraiU  L*i  which  the  results  of  sevenU  of  EuUnt's  eariier 
memoirs  on  tlie  «iime  subject  and  on  differential  equations  are 
included.  This,  like  the  similar  treatise  on  the  differential 
calculus,  summed  up  what  was  then  known  on  the  subject, 
but  many  of  the  theorems  were  recast  and  the  pruofo  improved. 
The  Ueta  and  Gamma^  functions  were  invented  by  Euler  and 
are  discussed  here,  but  only  as  illustrations  of  methods  of 
reduction  and  integration.  His  treatment  of  elliptic  integrals 
is  superficial ;  it  was  suggested  by  a  theorem  given  by  John 
Laiidrn  in  tlit*  PhiioMo^ical  Traiymdiinu  for  1775  connecting 
the  arcs  of  a  liyiwrbola  and  an  ellipse.  Euler's  works  that 
form  this  triliigy  have  gone  through  numerous  subsequent 
editions. 

The  classic  prublenis  on  isoperimetricsl  curves,  the  brachis- 
toclimue  in  a  resisting  medium,  and  the  theory  of  geodesies 
(all  of  which  had  lieen  suggested  by  his  master  Jolin  Ber- 
noulli) had  engage<l  Kuler's  attention  at  an  eariy  date ;  and 
in  solving  them  he  was  led  to  the  calculus  of  variations.  The 
general  idea  of  this  was  laid  down  in  his  CurvaruM  J/iriximt 
Jfinimive  Propi-ieiaie  Oautlenlium  invrntio  A'onri  ac  FacUis 
published  in  1744,  but  the  complete  develtipment  of  the  new 
calculus  was  first  effected  by  l^igrange  in  1759.  The  method 
used  by  f^grange  is  described  in  Kuler's  integral  calculus,  and 
is  the  same  as  that  given  in  most  modem  text-books  on  the 
subject. 

In  1770  Euler  published  his  Aideituny  zur  Algebra  in  two 
volumes.     A  French  translation,  with  numerous  and  valuable 

*  The  hitftorjr  of  th«  GiAmms  (anetioo  is  given  in  a  monograph  bj 
.fiirmel  la  the  Miemoirts  de  la  »ueUU  tUs  «cieNcr«,  Boiilsaox,  1886. 


^/ 


\ 


KULKK.  407 

•dilitioiM  hjr  I^fintn;^,  ww  timuf^hl  oat  in  1791 ;  «nd  » 
trmtise  cm  Aritliiiietic  lij-  Eulir  was  npprndt'il  to  it  The  firil 
voIdtdp  trf«tJi  of  deUrminat*  alsrbrn.  Tht«  ODnUina  one  dl 
the  Fftrlicwt  Attempts  to  place  tho  futidamriitiJ  procewfw  on  a 
Bcientitic  hmis:  the  samp  suhjecl  had  attrncted  D'Alemhert'B 
attention.  Thi<t  work  nlso  includes  the  pniof  at  the  binamiMl 
theorem  for  an  unrestricted  index  which  in  »till  Itnnwn  liy 
Euler'n  name;  the  proof  is  founded  on  the  principle  ol  ttie 
pennaneiict^  of  equivalent  forms,  but  Euler  made  no  »ttem|lt 
to  investigate  the  convcrgency  of  the  xeries :  that  he  should 
have  omitted  this  essential  »tep  is  the  moiv  cnrioUK  as  he 
had  himself  rec(^LEe<l  the  necessity  of  conndering  the  con- 
Tcrgcncy  of  inlinite  series:  Vandennonde^  proof  given  ia 
1764  suSeni  from  the  name  defect. 

The  necond  volome  of  thn  algebra  treata  of  indetenntDate 
or  Diophantine  algebra.  ThiH  contaimt  the  mlntiama  of  eonw 
of  the  proltlemn  proposed  by  Fcrmnt,  and  which  had  hitherto 
remained  unsolved. 

As  illuHtrnting  the  Himplicity  and  directnens  of  Ealer"* 
methods  I  give  the  Rulnttance  of  his  rlemonKtration*,  allnded 
to  above,  that  all  even  perfect  nnmber>  arc  included  in  Euclid's 
formula,  2'"'/'.  wher«  p  stands  for  2"  —  1  and  in  a  primof. 
Let  jV  be  an  even  perfect  numlier.  iV  ix  ei-en,  hence  it  can  be 
written  in  the  form  2"'a,  where  a  is  not  divisible  by  2.  IT 
is  perfect,  that  is,  is  e<|nal  U>  the  sum  of  ail  its  integral 
BulidiviKora ;  therefore  (if  the  number  ibielf  he  reckoned  aa 
one  of  ita  dirisora)  it  in  e^ual  to  half  the  mm  of  all  iti  in* 
legTwl  divisors,  which  we  may  denote  by  2'V.  Hincn  iff^SIf, 
we  have  2  *  2'-'a  =  S2«-'«=  S2«-' x  Sa. 

.-.  2"o  =  (l+2  +  ...+2'-')S«-(2--l)»i, 

tberefom  >  :  S«  =  2'- I  :  2'=f>  ip-t- 1.     Henoe  ■  =  An  and 

*  Cammmlatirmrf  AriUmuiieat  Cvllfrtae,  Ht  Pi>Mi>un,  Hit,  voL  n, 
p.  All,  art.  107:  Sjlmln  pablishtd  ao  ualjns  of  Ihs  ■ 
Nmtwre,  Dm.  in.  1B87,  vol  iiim.  p.  163. 

t  Boa  tt,  98  j  we  above,  p.  t\t. 


408     LAQRANOB,  LAPUIOB.  AUD  TUBIK  OONTSMrOBABmL 

lm^k(p-^  I);  and.  sanoe  Um  imlio  j»  :  j» -f  1  in  ui  ito  lowtil 
teniMy  k  miuit  be  a  positive  integer.  Now,  mnleM  Xsl,  we 
have  l^K  fit  and  Xp  at  imcton  ot  Xp ;  nufeover^  if  |»  be  noi 
prime,  tkere  will  be  other  faeton  alaa  Henee,  anle«  X»  1 
and  j»  be  a  prime,  we  have 

Ikpm  l-k-k-k-p-k-Xp-k  ...^{k-t-  l)(/i-ft>  I) •»..... 

Bat  thin  in  inoonsisteut  with  the  reralt  SAp  =  St  =  X  (ji  -i- 1). 
Hence  X  muHt  be  equal  ti>  1  and  p  must  be  a  prime.  There- 
fore a  =  |i,  theiYfore  A' -  2*->  a  -^  2*->  (2*-  I).  I  may  add  the 
oorolhury  that  Mince  |/  is  a  prime,  it  follows  that  m  is  a  prime; 
and  the  determination  of  what  values  of  m  (lf>«s  than  257) 
make  p  prime  falls  under  Mer§enne's  rule. 

The  four  works  mentioned  above  comprise  most  of  what 
Euler  produced  in  pure  mathematics.  He  also  wrote  nameroos 
memciirs  on  nearly  all  the  subjects  of  applied  mathenuitics  and 
mathematical  physics  then  studied :  tlie  chief  novelties  in 
them  are  as  follows. 

In  the  mechanics  of  a  rigid  system  he  determined  the 
general  equations  of  motion  of  a  body  about  a  fixed  point, 
which  are  ordinarily  written  in  tlie  form 

and  he  gave  the  general  equations  of  motion  of  a  free  body, 
which  are  usually  presented  in  the  form 

J-  (mu)  -  nu6,  +  fiiir^,  =  JT,  and    -^  - k^'f^  4-  A,'*j  =  ^ 

He  also  defended  and  elaborated  the  theory  of  **  least  action  " 
which  had  been  propounded  by  Maupertuis  in  1751  in  his 
EstMi  de  ctMrnioioyie  [p.  70]. 

In  hydrodynamics  Euler  established  the  general  equations 
of  motion,  which  are  conmionly  expressed  in  the  form 

^  dp  ^  y     du        du       du        du 
pdx^  *       di        ilx       dy         ^  * 

At  the  time  of  his  death  he  was  engaged  in  writing  a  treatise 


\ 


EULEH.  40A 

on  hydroniTChKnim  ill  which  thntiTntiti<>nt(if  theinihject  woaM  . 
have  hwn  enmplptcljr  rpcwtl. 

Hin  most  impnrtAnt  works  on  iiNtnmiimr  «re  h»  Thmrin 
A/ol'iata  Plnnrbirum  ft  Cnrrufitruin,  puMislipd  in  1744;  hw 
TM*oria  Mottui  f.unarii,  puhltKlml  in  It^'i'l  :  nnd  his  Tkforia 
Atotiiitm  Lnnru".  pulitinliFd  in  1772.  Tn  thevp  he  KtUckfd  the 
proh1pmorthn?e1io()i(-R:  hnnupposed  thr)NxIycnn)ii<lciTH,(''2.(rr. 
the  moon),  to  carry  three  rectanjutliir  ftKPs  with  it  In  it«  motion, 
the  ftxe*  tno»ing  pnmllel  to  theniM-lvpH,  nnd  to  them  nxe*  nil 
tlie  niotioTia  were  referred.  TJtiii  method  is  not  convenient,  but 
it  wii.1  from  Euter'd  renulte  that  Mnyer*  coriNlrticted  the  luniir 
lAblefi  for  which  hin  widow  in  1770  recrived  £.'iOno  frnm  the 
English  pnrlianient,  nnd  in  recof^ition  of  Euler's 
■nni  of  £.100  was  nlno  voted  mt  an  honomrium  to  him, 

Eulerwumnch  int«re»t«d  in  optioi.  In  1746  he  d 
the  reUtive  merit*  of  the  emiKHion  And  nndulfttory  theorjea  of 
light;  he  on  the  whole  preferred  the  lAtt«r.  In  1770-71 
he  pubtinhed  hin  optioil  renenrcheii  in  three  volnme)!  nnder 
the  title   Diitjttrim. 

He  ftlHO  wrote  on  clemcntAry  work  on  phjxics  imd  thv 
fundnmentAt  principles  of  niHthetniitical  philosophy.  This 
originAted  from  An  invitAtion  ho  receivi-d  when  ho  Bmt  went 
tn  Berlin  to  give  Inmons  on  phynicH  to  the  prtncefw  of  Anhalt' 
DcMAii.  These  lectum  were  pu)ilif<hi>d  in  1768-1773  in 
three  volumes  onder  the  title  Lftlrr».,.iniT  ^nrfqnen  aujaU 
d«  pkifwiqti*....  And  for  hnlf  a  cento rj  remMned  a  stand«rd 
trefttifte  on  the  subject. 

■  Of  conrae  EoleKn  mngiiificent  workn  were  not  the  only 
text-honks  contAining  originnl  mntter  pmduoed  At  this  time: 
Amongst  numerous  writers  I  would  npecinlly  single  out  Danvl 
Bemotilli,  fiiinpmm,  Lamtiert,  Bhnul,  Tr*tab/eg,  nnd,  Arbogiul 
■I  hkving  inSuenced  the  development  of  nuthrawtioi.    To 

*  Jokama  TatiAt  Uw/rr,  bom  in  Wortembeii  in  ITU  sad  dM  In 
I76S.  WM  dinetor  ol  the  Bnitluh  obMrrstor?  si  U«(ti^M.  IfoM  at  Ui 
BHUotn,  other  thsa  hli  Idbm  tsUoi,  wgra  poUiiilKd  fa  17TX  Mdv  Dh 
liUeOpn  Imiila. 


I 

I 

410    LAQSANQK.  LAPLAOK^  AHD  THBIB  OCUfTBMrOSABIBi 

the  two  fint^nentioned  I  have  alwmiy  ftlllubd  ia  thm  lagt 
chmpier. 

Lambert^.  Jokann  ffeinriek  iMmheri  wan  born  at  Mdl- 
haoaen  on  Aog.  38, 1728,  and  died  at  Berlin  on  Sept  2S,  1777. 
He  wan  the  aon  of  a  unall  tailor,  and  had  to  rely  on  hia  own 
efforts  for  his  education ;  from  a  clerk  in  eouie  iron-woilu,  he 
got  a  place  in  a  newspaper  olBoe,  and  tubaequently  on  the 
recommendation  of  the  editor  he  was  appointed  tutor  in  a 
private  family  which  secured  him  the  use  of  a  good  library  and 
sufficient  leisure  to  use  it.  In  1759  he  settled  at  Augsburg, 
and  in  1763  removed  to  Berlin  where  he  was  given  a  small 
pension  and  (inally  made  editor  of  the  Prussian  astronomical 
almanack. 

Lambert's  most  important  works  were  one  on  optics,  issued 
in  1759,  which  suggested  to  Arago  the  lines  of  investigation  he 
sub6e(|uent1y  pursued  ;  a  treatise  on  perspective,  published  in 
17r>9  (to  which  in  1768  an  appendix  giving  practical  appli- 
cations was  added);  and  a  treatise  on  comets,  printed  in  1761, 
containing  the  well-known  expression  for  the  area  of  a  focal 
sector  of  a  conic  in  tenus  of  the  chord  and  the  bounding 
radii.  Bedsides  these  lie  communicated  numerous  fiapers  to 
the  Berlin  Academy.  Of  these  the  most  important  are  his 
memoir  in  17<)8  on  transcendental  magnitudes,  in  wliich  he 
provinl  that  w  is  inconunensurable  (the  proof  is  given  in  Le- 
gend re's  (/tomeirif^  and  is  there  extended  to  v^) :  his  paper  on 
trigonometry,  reoil  in  1 768,  in  which  he  developed  Denioivre's 
theon^us  on  the  trigonometry  of  complex  variables,  and  intro- 
duced the  hyperljolic  sine  and  cosine  t  denoted  by  the  symbols 
sinh  X,  cosli  x:  his  essay  entitled  analytical  obsen'ations,  pub- 

*  See  Lamhtrt  uaeh  teinem  LebtH  und  Wirken  by  D.  Uuber,  fisksa 
lK2il.  Most  of  Laiubert*M  meiuoin  are  collected  in  his  Deiinlge  zum 
Gthraucke  tier  Math^uMtikt  published  in  four  volumes,  Berlin,  176*S- 
177«. 

t  These  functions  are  said  to  have  been  previously  suggested  bj 
F.  C.  Mayer,  see  I  fie  Lehre  von  den  UyperbtlfunktioHen  by  H.  Giintber, 
Halle,  1081,  and  Utitriige  zur  OetckU-hU  der  ueueren  Matkematik^  Ans- 
Imeb,  1H8L 


LAMIIEKT.    BtZOVT.  TKEHHLEV.   ARBOOAHT.  LAUKAWIK    411 

lUhed  in  1771,  wKidi  in  the  earliest  Httempl  to  fonn  functiotuU 
cqnatinris  by  pxprpssitig  the-  given  prnprrtieH  in  tlie  Innguitgn 
of  the  differentinl  calculus  "id  thru  int^-gnting :  lutly  hiit 
piiper  un  via  vivn,  puMinhecl  in  l7H.t,  in  which  for  the  fintt 
time  lid  expiTA.srd  NewUm's  srconc)  Inw  of  motion  in  tlw  no- 
tntion  of  the  ditt'erenlinl  ciilculuR. 

BfixoQt.  Trembley.  Arbogast.  Of  the  vther  niAtbe- 
ni«tici*ns  Almre  mentioned  I  here  ndil  n  few  wonts,  filif.nnt 
Bfitnil,  liorn  nt  Nemoure  on  Mnrch  31,  1730,  nnd  died  on 
Sept.  *27,  1783,  besides  numerous  minor  works,  wroto  » 
ThroTiK  gettimle  rfw  iqiuifinnn  ali/fhri^w*,  puMinhed  nt  Patis 
in  1779,  which  in  pnrticukr  cimtAitied  much  new  and 
vkluiil)ie  mntter  on  the  theory  of  elimination  And  syni- 
metrical  functions  of  the  rnntn  of  on  e(]UKtion :  he  naed 
determinAiitR  in  ft  paper  in  the  tlUlmrr,  th  Caat'iimir.  myth, 
1764,  but  did  not  trent  of  the  gcneml  theiiry.  ./mn  Trrmbtty, 
bom  Mt  Geneva  in  1740,  jtnd  died  on  Kept.  18,  1811,  con- 
tributed Ut  the  development  of  diffofntial  cquHtionN,  finite 
diflerenceN,  And  the  calculuK  of  prolMbilitiea.  Limiii  FmH'^in 
Antoiu*  Arbnffiuit,  Imrn  ill  Alunco  on  Oct.  4,  MM,  aihI  <lie<l 
At  Htnunburg,  where  he  wns  pnifessnr,  im  April  8, 18fl3,  wrote 
on  neriea  And  thn  d^^Hvatireti  known  by  hin  niune:  be  wwt  the 
firat  writer  tn  Repnrate  the  nymlii>l.<<  of  npi>mtian  from  th(«e 
of  quantity. 

I  do  not  wisli  to  crowd  my  pn;^  with  An  nocount  of  thoTC 
who  have  not  diMtinctly  niivAuccd  the  subject,  but  1  hare 
mentions]  the  aIjovc  wnten>  liecnuNc  their  nnntea  Are  still  well 
kooii-n.  ■  We  m»y  hownver  wiy  that  the  discoreriee  of  Euler 
and  LAgrangc  in  the  Buhjectn  which  Ihey  treAtod  were  m>  com- 
plete And  fnr-Tvsching  that  what  their  lew  gifl«d  coiitempo- 
nnm  added  is  not  of  nuffictent  importAnce  to  require  tnentMn 
in  a  book  of  this  nature. 

lAfruigfl*.     Jt»fph  Louit  L«ifritni/«,  the  greatest  mathe- 

*  SumnviM  ot  the  Hfe  sod  work*  of  LanraniTe  are  ghw  hi  Um 
Mmgtith  Cfdcf<M*i»  ani)  th«  A.M-yirlA^nlfa  Brilanmit^  ^dUtkyUtei^ 
tt  vhicA  1  han  ntad*  emndwabh  om  ;    '     ~  '  ~"~ 


412     LAQBlMOg.  LAPLAOB.  AHO  THBIB  QOllTBMfOBUUBi 

matacian  of  the  eighteentli  eentorji  was  bom  at  Tiria  ob 
Jan.  25,  1736.  and  died  at  Auris  on  April  10,  181S.  Hk 
father,  who  had  charge  of  the  Sarflinian  military  cheiti  was 
of  good  social  position  and  wealthy,  but  before  hie  eon  grew  vp 
he  had  lout  most  of  his  property  in  specolationa^  and  young 
Lagrange  had  to  rely  for  his  position  on  his  own  abUitiea.  He 
was  educated  at  the  college  of  Turin,  but  it  was  not  ontil  he 
was  seventeen  that  he  shewed  any  taste  for  mathematics :  his 
interest  in  the  subject  being  first  excited  by  a  memoir  by 
Halley^,  across  which  he  came  by  accident.  Alone  and 
unaided  he  threw  himHelf  into  mathematical  studies,  at  the 
end  of  a  year's  incessant  toil  he  was  already  an  accomplishMl 
mathematician,  and  was  made  a  lecturer  in  tlie  artillery 
school. 

,  The  first  fruit  of  Lagrange's  labours  here  was  his  letter, 
written  when  he  was  still  only  nineteen,  to  Euler  in  which 
he  solved  the  iHoperimetrical  problem  which  for  mure  than 
lialf  a  century  luul  been  a  subject  of  discussion.  To  effect 
the  solution  (in  which  he  sought  to  determine  the  form 
of  a  function  so  tlmt  a  formula  in  which  it  entered  should 
satisfy  a  certain  condition)  he  enunciated  the  principles  of  the 
calculus  of  varijitions.  Euler  recognized  the  generality  of  the 
method  adopted,  and  its  superiority  to  that  used  by  himself ; 
and  with  rare  courU^sy  he  withheld  a  paper  he  had  previously 
written,  which  covered  some  of  the  same  ground,  in  order  that 
the  young  Italian  might  have  time  to  complete  his  work,  and 
claim  the  undisputed  invention  of  the  new  calculus.  Tlie 
name  of  this  branch  of  analysis  was  suggested  by  Euler. 
This  memoir  at  once  placed  Lagrange  in  the  front  rank  of 
mathematicians  then  living. 

In  1758  Lagrange  (nitablished  with  the  aid  of  his  pupils 
a  society,  which  was  subseciuently  incorporated  as  the  Turin 

grmphjr  of  bin  writingii.  Lagrange's  work*,  tsdiied  by  MM.  J.  A.  Berrei 
and  U.  Darboux,  were  publiHbed  in  14  volum««,  l>arit,  1867-ieOi. 
Delambre*!  account  of  bin  life  in  printed  in  tbe  flril  volume. 

*  On  lh«  f  icellenct)  uf  ihn  modarn  algvbra  in  certain  uptaaol  prcbtemst 
J^Miffvjfhical  iytfUMicfiuNf,  1(»U3,  vul.  svui,  p.  WiO. 


\ 


41S 

Academy,  und  in  th«  five  volamm  of  iU  tmnMutioai,  nmMlly 
known  M  tlif  AfiMr/lniifft  Taurinmiiin,  laimt  ot  his  f«rijr 
irritini^B  nrp  U>  be  found.  Many  (if  theae  Kre  vlabonite 
■nemoin.  The  fimt  volume  coiitAin^  n  mpmoir  on  the  theory 
of  the  propasAtioii  of  Hound ;  in  ihia  he  indicates  m  mtatake 
mftde  by  Newton,  obtains  the  general  ilifTeivntial  eqntition  for 
the  motion,  antl  inte-;rates  it  for  motiiin  in  a  straight  line. 
This  volume  also  containn  the  complete  Hnlutjon  of  the  problem 
of  a  Ntrin^  vibratint;  tninftveriely  ;  in  thin  paper  he  points  ont 
a  lack  of  generality  in  the  Hotationn  previoaaly  given  by 
Taylor,  D'AJpniliert,  and  Kaler,  and  arrives  at  the  coucliuion 
that  the  form  of  thn  curve  at  any  time  (  in  given  \yf  the 
equation  y  ^onin  ntzsin  iit  The  article  conclndea  with  a 
Biaatvrly  discusiion  of  echoes,  ImaLs,  and  componnd  aoandx. 
Other  articlen  in  this  volume  nrv  on  recarring  Kriea,  proba- 
bilities, and   fh*"  calculus  of  variations. 

The  necond  volume  contains  a  long  paper  emhodjing  the 
resnlta  of  several  memoira  in  the  lirst  volume  on  the  theoiy 
and  notation  of  the  calculuH  of  vnriationft ;  and  he  illontntm 
its  uw  by  cleilucinE  tlie  principle  of  leant  action,  and  bj 
nolutions  of  various  problems  in  dynnmics. 

The  thirrl  volume  includes  the  solution  of  leTeml  dynamical 
problems  by  means  of  the  calculus  of  variations  ;  some  papen 
on  the  inte^ml  calculus;  a  solution  of  Fermat's  proMem 
mentioned  above,  to  dm)  a  numlter  x  which  will  make 
(]^n+  1)  a  square  where  j;  is  a  given  integer  which  is  not  a 
eqnare ;  and  the  general  differential  e<)DntinaB  of  motion  for 
three  bodies  moving  under  their  mutual  attractions 

In  1761  Lagrange  stood  without  a  rival  as  the  foremoat 
mathematician  living;  but  the  unceasing  Inbosr  ot  the  pre- 
ceding nine  years  hod  seriously  affected  hut  health,  and  the 
docton  refused  to  be  responsible  for  liis  reaaon  or  life  tmlem 
he  would  take  rest  and  exercine.  Although  his  health  wm 
temporarily  rentored  his  nervous  system  never  qdite  reoonerad 
ita  tone,  and  henceforth  he  oonstanlly  sufferad  fron  ■ttaefct  vt 
proConnd  melancholy. 


414     LAORANOB.  LAFLACB,  AND  THBIB  OONTHOOBARUBl 

Tlie  next  wurk  be  pmduoed  was  in  1764  on  the  libimtfaMi 
of  the  UMMMi,  luicl  lui  expUujitiim  lui  to  why  the  aiMiie  iaoe  was 
alwayn  turned  to  the  earth,  a  prubleni  which  he  treated  fay  the 
aid  of  virtual  work.  Uin  mlution  in  enpedally  intetfinting  m 
containing  the  germ  of  the  idea  of  geueraliaed  equations  of 
motion,  equatioiw  which  lie  fimt  foniudly  proved  in  1790i 

He  now  started  to  go  on  a  visit  to  London,  hut  on  the 
way  fall  ill  at  Paris.  Tliere  he  was  received  with  marked 
honour,  and  it  was  witli  regret  he  left  the  brilliant  society 
of  that  city  to  return  Ut  his  provincial  life  at  Turin.  His 
further  stay  in  Piedmont  was  liowever  sliort.  In  1766 
Kuler  left  llcrliii,  and  Frederick  the  (Jreat  immediately  wn»te 
expressing  the  wish  of  'Hhe  greatest  king  in  Europe*'  to 
liave  ''the  greatest  matlienuittcian  in  Kun»|ie'*  nwident  at 
his  court.  Lagrange  acirept4*tl  the  iitTrr  and  K|M»nt  tlie  next 
tw«*nty  ymrs  in  PruMsiJi,  wlieiv  \w  prislucetl  not  oidy  Um^ 
long  s«*rii*H  of  1111*111011*8  puliH-slied  in  tlit*  l)<*rliu  and  Turin  traiiM- 
lU'tiouM  but  liiH  nioiiuiiienUil  work,  the  .l/miiii^M^  mmlt^tiqu^, 
HiH  ivNidiMici^  ut  Itin'lin  i^miiii«*iictHl  with  an  unfortunate  mis- 
Uike.  Finding  ni«»st  of  liis  col  leagues  married,  and  assured  by 
tlirir  wives  tliait  it  was  the  only  way  Ui  lie  liappv,  he  married ; 
liin  wife  stjun  died,  but  the  union  wau  not  a  happy  one. 

I  ^grunge  was  a*  favourite  of  the  king,  who  used  frequently 
to  dise«iurse  Ui  liiiii  on  the  advantages  of  perfect  regularity  of 
lif&  The  lc\sson  went  home,  and  thenceforth  Lagrange  studied 
his  mind  and  Isidy  as  though  they  were  machines,  and  fouml 
l»y  experiment  the  exact  amount  of  work  which  he  was  able 
to  do  without  breaking  down.  Evei-y  night  he  set  himself  a 
definite  task  fcir  the  next  day,  and  on  completing  any  branch 
of  a  subject  he  wrote  a  short  analysis  to  see  what  points  in  the 
demonstnitions  or  in  the  subject-matter  were  capable  of  im- 
provement. He  always  thought  out  the  subject  of  his  papers 
before  he  Wgan  to  compose  them,  and  usually  wrote  them 
straight  oH'  without  a  single  erasure  or  correction. 

His  mental  activity  during  these  twenty  years  was  amaxing. 
Not  only  did  lii^  produce  his  splendid  MeeuHique  unnlt^iiquf^ 


\ 


LAOIUNOE.  415 

but  b«  cnntribut^I  bptwcen  (ine  and  two  Imndml  pnpen  to 
the  Aciuleiiiies  of  IterlJii,  Turin,  iitid  Fans.  Sviue  (^  time 
nre  really  trentispn,  nnil  nil  wilhuul  exception  an  at  m 
hijHi  order  of  excdlpnvc.  Kxcept  for  h  shiMt  time  when  he 
won  ill  he  prwlucvil  i>ii  an  nvrnige  nUiut  our-  mnmoir  li  nmnth. 
Of  tlie»<e  I  nol«  tlie  fnlloning  as  nniiiriK  tlw  mont  important. 

Pintt,  Ins  contributions  t^i  tin-  fnurtli  and  fiftli  vnlumcM, 
176fi-ir7.-(,  of  the  Mi^rriranen  Tanriiieimii  of  which  tlie 
must  im|MirlAnt  wan  the  one  in  1771  in  wliich  be  diiKUsmd 
how  nunieniUH  aNtmiiutiiicnl  <i)iw-rvntions  xhouhl  be  cumhiiie<l 
M>  ax  tu  give  Ibe  iiwnI  j>ni)>able  result.  And  later,  hia  con- 
tributions Ui  tlio  linit  twii  vnluniea  17^4-17}<'i  of  the  tnns- 
octiiina  of  the  Turin  Aciwleniy;  b>.tfie  first  of  which  he 
contributed  n  [uijht  im  tlie  prpHNure  exert<>d  liy  fluicla  in 
ntotion,  and  to  the  iiei.-und  nn  nrtide  on  intftfnttiun  by  infinite 
8i'ri<-K,  ami  the  kind  of  pnibleniH  fur  wliich  it  ia  nuitable. 

McNit  of  the  nieiiioira  sent  to  Pari''  were  on  astrmUHnical 
questions,  and  among  these  T  ought  particulnriy  to  mention 
hJH  memoir  on  the  Jovian  syHti^ni  in  I7G6,  Iiin  ewiay  on  the 
problodi  of  three  Imlim  in  1772,  liis  work  on  the  Mcnlar 
equation  iif  the  niuon  in  177^,  and  his  trentise  nn  cnmetary 
perturlMtiimH  in  1778.  Tliet^  were  all  tvritten  nn  sahjecta 
pro[K«rd  hy  the  French  Acatlemy,  and  in  each  caae  the  priae 
watt  awardeil  to  hitn. 

Tlie  greater  numlier  <if  his  pa)>erB  during  this  time  were 
however  contributed  to  the  Berlin  Acnilemy.  Several  of 
them  tieni  with  qvestions  on  nlifhi-a.  In  particalar  I  may 
mention  the  fiilloiring.  (i)  llix  discUBsion  at  the  Krintion 
in  integers  of  indeterminate  quadratics,  17(i9,  and  genendlj 
of  indeterminate  equations,  1770.  (ii)  Hia  trad  on  the 
theory  of  elimination,  1770.  (iii)  His  memcnra  on  a  general 
pTDoess  for  Holving  an  algebraical  equation  of  any  degrei^  1770 
and  1771 ;  this  method  fails  hr  equations  of  an  order  «bora 
the  fourth,  because  it  then  involves  the  sotuttoo  of  Ml  eqtM- 
tjon  of  higher  dimensions  than  the  one  proponed,  bat  it  gins 
all  the  mlotioiM  of  his   pntdecesnoni  as  modiBcatJumi  of  m 


416    LAOSANQB.  LAPLACE,  AHD  THUB  OOimifraBABl 

tingle  priiiciiile.  (iv)  The  oomplele  euliatioii  dl  m  hiiUMiim 
equation  id  Miy  degree^  ihie  ie  nonfained  ia  the  ■Mmoifs  hit 
mentioned,  (v)  lAstly,  in  1773.  hie  treatment  di  delenain- 
ante  of  the  second  and  third  order,  a:id  ol  invariantai 

Several  ol  his  early  papers  a|i|o  deal  with  qneetions  eon* 
nected  with  the  neglected  but  idngularly  feednating  enljeet 
of  the  theory  of  numben.  Among  these  are  the  following, 
(i)  Hiii  proof  of  the  theorem  that  every  integer  which  ie  not 
a  tiquare  can  be  expressed  as  tlie  sum  of  two^  three^  or  foor 
iuUfgral  squares,  1770.  (ii)  His  proof  of  Wilson's  theorem  that 
if  n  be  a  prime,  tlieu  |n  - 1  + 1  is  always  a  multiple  of  n, 
1771.  (ill)  His  meiiioir^"^  1773,  1775,  and  1777,  which 
give  the  deiiioiistnitious  of  several  results  enunciated  fay 
Fenuat,  and  not  previously  proved,  (iv)  And  lastly  his 
method  for  determining  the  factors  of  nuuibers  of  the  form 
«*  +  iiy*. 

Tliere  are  sImo  numerous  articles  on  various  points  of 
aiMlytical  ytotntiry.  In  two  of  them,  written  rather  later,  in 
1792  iind  1793,  he  reduced  the  equations  of  the  quadrics  (or 
conicoMlH)  to  tlieir  cuiioiiical  forms. 

During  the  years  from  1772  to  1785  he  contributed  a  long 
Meriini  of  iiieiiioirM  which  created  the  science  of  diffrreuiiai 
tquiUioita^  lit  uny  ratt^  u.h  fur  as  piirtial  differential  equations 
an*  coiic(*riuHl.  I  do  nc»t  think  that  any  previous  writer  had 
tkfiit*  any  thing  lieyoiid  cuiisidering  equations  of  soiiie  particular 
form.  A  large  part  of  these  results  were  collected  in  the 
Nccoud  edition  of  Euler's  integral  calculus  which  was  pub- 
liKhtHl  hi  1794. 

Lugrunge'K  papers  on  u^chtinicn  require  no  separate  men-  I 

tion   here   as   the   results  arrived   at  are  embodied   in   the 
Mreanique  analyllque  which  is  described  below. 

Lastly  there  are  numerous  memoirs  on  problems  in 
oMtrttwHHy,  Of  these  the  most  important  are  the  following, 
(i)  On  the  attraction  of  ellipsoids,  1773:  this  is  founded  on 
Maclaurin's  work,  (ii)  On  the  secuUr  equation  of  the  moon, 
1773;   also   noticeable   for  the  earliest  introduction   of   the 


uaiusaG.  417 

Mm  (it  the  potcntinl  Thn  pntentiKl  (if  k  body  tA  any  point 
is  the  Bum  of  the  miiwi  nf  pvpry  elentent  of  the  body  when 
divided  l>y  its  distance  front  thfi  point.  Lagntnge  shewed 
that  if  the  potential  of  a  body  at  an  eitermal  point  were 
known,  the  attraction  in  any  direction  cogid  I*  at  once  found. 
The  theory  of  the  potential  was  elnliorated  in  m  paper  aent 
to  Berlin  in  1777.  (iii)  On  the  ntotion  of  the  nndei  of  a 
planet's  orbit,  1774.  (iv)  On  the  HUhility  of  the  planetaiy 
orbits,  1770.  (v)  Two  memoirs  in  which  the  method  ot 
determining  the  orbit  of  a  comet  fnmi  three  ulMen'atiuna  n 
completely  worliM)  out,  177f*  and  1783:  this  has  not  indeed 
proved  pmclicidly  available,  but  his  spteiti  of  calculating  tha 
perturliationN  by  nieann  of  mechniiicnl  (|Uiwlrature!t  has  formed 
thelioMitKif  moHt  sulne<)Uent  re-searches  on  the  subject,  (vi)  His 
determination  nf  the  siecular  and  periodic  variations  nf  the 
elements  of  the  planein,  17)^1-1784  :  the  upper  limits  amigned 
for  these  agree  clowty  with  those  obtainetl  later  by  Lererrier, 
am)  l^f^rtniijp  proceeded  as  far  as  the  kmiwledge  then  pua-' 
nessed  of  the  maKses  of  the  planets  pemn(t<^l.  (vii)  Thre« 
niemmrn  on  the  method  of  interpolntion,  1793,  1792,  and 
1 793 :  the  part  of  finite  diflVrences  dealing  thnvwith  i«  now 
in   the  same  staf^  aa  that  in   which    Ijngran<:;e  left  it. 

Over  anil  alwive  these  variou*  pn|irn,  he  eiimpn»e«l  hin  Rreat 
trealise,  the  MrraiiiqMf  niiafytiqnf.  In  this  he  lays  dnwn  the 
lawuf  virtual  work,  and  from  that  om>  fundamental  priitciple, 
1>y  the  aid  of  the  calculuN  of  varialinns,  di>dnceH  the  whole 
of  mechanics,  both  of  solids  and  fluids.  Tlw  object  of  tha 
book  is  to  tthew  that  the  subject  is  implicitly  included  in  a 
single  principle,  and  to  give  general  formulae  from  which  any 
particular  remit  can  lie  obtained.  The  method  of  generalind 
coordinates  by  wbich  he  obtained  thin  iriiult  is  pertiapa  tha 
RKMt  brilliant  result  of  bis  onalyxis.  Instead  of  following  tlw 
motion  of  each  individual  part  of  a  matt-rial  nyslMii,  aa 
lyAlembert  and  Enler  had  done,  be  shewed  that,  if  we  drtar- 
nine  ita  oonfignration  by  a  sufficient  number  of  variaMea 
whoM  number  is  the  same  as  that  of  the  dq;ran  ol  (iwJum 


418     LAQRAMOB.  LAPLACK.  AND  THBIB  OOMTIIIFORARinL 


poMesied  by  thesyalemytlieiitliekiiieUoaiid  pnttintiri 
of  tlie  tytlem  can  be  exprened  in  terniA  ol  thm^  vmrinhleii  and 
tbe  differential  eqoaiionii  of  motion  tbenoe  deduced  fay  linilile 
differentiation.  For  example,  in  dynamics  of  a  rigid  ayiiteni 
he  replaoen  tlie  consideration  of  the  particolar  problem  hj 
the  general  equation  which  is  now  usually  written  in  the  form 

ddT    BT    dV    ^ 
*--T  +  —  =  0. 

Amongst  other  minor  theorems  here  given  I  may  mention  the 
proposition  tliat  tlie  kiuetic  energy  imparted  by  given  impulses 
to  a  material  system  under  given  constraints  is  a  maximum, 
and  the  principle  of  least  action.  All  the  analysis  is  so 
elegant  tliat  Sir  William  Rowan  Hamilton  said  the  work 
could  lie  only  described  as  a  scientific  poem.  It  may  be 
interesting  tu  note  that  Lagrange  remarked  that  mechanics 
was  really  a  branch  of  pure  uiatliematics  analogous  to  a 
geometry  of  four  diiiienHions,  namely,  the  time  and  the  three 
coonlinaites  of  the  point  in  space ;  and  it  is  said  tliat  he  prided 
himself  that  from  the  beginning  to  the  end  of  tlie  work  there 
was  not  a  single  diagnini.  At  first  no  printer  could  be  found 
who  would  publish  the  book ;  but  Legeiidre  at  last  persuaded 
a  Paris  Hriu  to  undertake  it,  and  it  was  issued  under  his 
supervision  in  1788. 

In  1787  Frederick  died,  and  Lagrange,  who  had  found 
the  climate  of  Berlin  trying,  gladly  accepted  the  offer  of 
Louis  XVI.  tu  migrate  tt>  Paris.  He  received  similar  inviti^ 
tioiis  from  Spain  and  Naples.  In  France  he  was  received  with 
every  mark  of  distinction,  and  special  apartments  in  the  Li>uvre 
were  preparcnl  for  his  reception.  At  the  lieginning  of  his 
residence  hen*  he  was  seized  with  an  attack  of  iiielancholy, 
and  even  the  priiiteil  copy  of  his  JJicaHiqw,  on  which  he  had 
worke<l  fur  a  quarter  of  a  century  lay  for  more  than  two  years 
unopened  on  his  desk.  Curiosity  as  to  the  results  of  the 
French  revolution  first  stirred  him  out  of  his  lethai^%  a 
eurhtsUy   which    soon    tumetl    to    alarm    as  the   revolution 


LAORAKnE.  419 

develnpetl.  It  was  about  tlip  nntuft  time,  1793,  that  the  nn- 
KCCOunlAble  snHness  of  his  Hfe  and  hJH  timidity  nio\-ed  the 
ntmpassion  of  a  youn;;  girl  wlio  innintcil  iin  imurying  him,  and 
proved  n  devoted  wife  to  whom  he  liecanie  waniily  attached. 
AlthdUf-h  tlie  dpcroe  of  Oct4)l»rr,  U9\  which  ordered  all 
fiin-ignpnt  to  Irave  Frnnce,  Gprcinlly  rxeiniited  him  by  tmmi^ 
hi>  wn.<i  prFpnrin<;  to  c-scnpe  vheii  he  w,-ui  oflered  thfi  precJdeiK^ 
of  the  ci>nin)iHtion  for  the  reform  of  wei^fhta  anil  meMnrei^ 
The  clioice  of  the  unitH  finally  srleet^^  was  targnly  dae  tu  biui, 
and  it  wan  mninly  owing  to  bin  inDuence  that  the  deciniMl 
nil hili  virion  was  necepted  by  the  commission  of  1799. 

Though  f.A;;mnge  hmt  iteterminnl  to  escape  from  Prance 
while  there  was  yet  time,  he  was  never  in  any  danger;  and 
the  diSerent  revolutionary  govemmenti  (and  at  a  later  time 
Napoleon)  loiuled  fiim  with  honours  and  difitlnctionA.  A 
striking  testimony  to  the  n^pecl  in  which  he  wax  held  waa 
nhewn  in  1796  when  the  Freneli  commii^uy  in  Italy  waa 
onlerrd  to  attend  in  fall  state  on  Lagrange's  father,  and 
tender  the  con^^tulnlions  of  the  ivpublic  on  the  achierementa 
of  his  son,  who  "hnd  done  hnnour  Ut  all  mankind  hy  hi* 
genius,  and  whcni  it  wa.o  the  special  gloty  of  Piedmont  to 
have  prodocerf."  It  may  l>e  iwlifed  that  Napoleon,  when  he 
attained  power,  wannly  cncouniged  Rcientific  atudies  in  France 
and  was  a  liberal  benefactor  of  tbem. 

In  Wn-")  Lngrnngn  wa.1  appoint*^  to  a  mathematicKl  chair  . 
at  the  newly-cstnblished  Ecole  normnle,  which  enjoyed  only  a 
brief  existence  of  four  months.  His  lectures  here  were  quite 
elementary  and  contain  nothing  of  any  special  importance,  bat 
they  were  published  liecause  tlie  profewora  had  to  "pledge 
themaelves  to  the  n-prenentatiTn  of  (he  people  and  to  eikdi 
other  neither  to  rem]  nor  to  repeat  frt>m  memory,"  and  the 
diwwiinirs  were  ordered  to  be  taken  down  in  ahorthand  in 
order  to  enable  the  deputiea  to  Me  bow  the  piirfwoffe  so- 
quitted  themselree. 

On  the  entabliafament  ot  the  Ecole  polyteduik|ae  in  179T 
I«gnnge  wwa  nwde  ■  ptofawwt  -,  inA  Va»  Vata^wt  *■**  «» 


420     UOKANai,  UiPLACK.  AND  TBEIB  COl 

dMcribed  hf  nwthemutici&nit  who  iuul  the  good  fortune  to  he 
kbie  to  Attend  tbeiii,  an  altuant  perfect  both  in  fonm  ukI  lufttter. 
B^nniag  witli  ttie  uiert»t  eleiueiitii  he  led  hU  he«rera  on 
until,  Mlmust  unknuwu  tu  thcinselveii,  theyj  wera  tbeuuielvea 
cjctending  the  bounds  of  tlie  subject :  abuva  kU  he  ■nij>reued 
on  bia  puiuU  the  uIvAnb4;c  uf  always  aidng  general  nietbodi 
expnsHt^  in  a  Hymiuetriciil  niiUition. 

His  lecture*)  on  the  difletvritini  cuIcuIub  form  the  ba.iu  of 
hin  Thiorit  dn /attctiont  anali/liqHra  wbiub  was  publuhed  in 
1797.  Thia  work  in  the  exteusion  of  an  idea  contained  in  a 
paper  be  had  wnito  tlie  Derlin  Meiuoinin  l773,andit»oltj««t 
ii  lu  Hulmtitutn  for  Iho  ihfferentiul  cuIcuIuh  a  group  of  theoreiua 
InmhI  on  the  develupment  uf  at^liniic  functiuna  in  nerim.  A 
Miniewluit  similar  niethud  luul  lieen  prvviouHly  used  by  John 
Landen  in  hin  KivUimt  Auul^tin,  publislbed  in  lAindon  iu  17&8. 
Ijigntn;^  bi'lieved  ilmt  ho  i-ouM  tliuH  get  rid  of  thiMe  diHi- 
L-ultiiit,  coiinec-t««l  with  the  um*  »f  iiiliuitely  liirgv  or  iiitiiiit4^ly 
Biuall  i|uantities,  wlikli  philuwphent  prufesned  tiinef  in  the  uhuuI 
treatuient  of  tlie  dilTert'iitiul  cuk-ulus,  Tlie  liuok  is  divided 
into  tliree  parts ;  of  these  the  lirHt  treats  of  the  gt^-nend  theory 
uf  functii<nH,  and  gives  an  atgcbniic  proof  uf  Taylor'*  theorem, 
the  validity  of  which  is,  however,  open  to  (|ut«lion  ;  the  uocond 
deaU  with  applications  to  geometry;  and  the  third  with 
applications  to  liw.-chaiiicH.  Aliotlier  treatise  on  the  xauie 
linen  wan  liia  Le^otu  mir  te  calcut  iltn  fuiicluiiu,  issued  in  IS04. 
These  works  uuiy  be  considered  as  the  slarting-point  for  the 
researches  of  Caucby  and  Jacohi. 

At  a  later  pcrio<l  Lnf^an^'e  reverted  to  the  use  of  iuKni- 
tAsimali  in  pn4ereiice  to  founding  the  diflerenlial  calculus  on 
n  atudy  of  algi'ltraic  forms :  and  in  the  preface  to  the  second 
edition  of  the  J/rratiiVyw,  which  wa.i  issued  in  1811,  he 
justities  tlic  entpliiyu)ciit  of  infiniteHiuiuls,  and  concludes 
by  saying  that  **  wheii  we  hare  grasped  the  apirit  uf  the 
inKnitesiuml  melhijil,  and  have  verified  the  exactness  of  its 
reaults  either  by  tlie  geometrical  method  of  prime  and  ultininte 
ratiiis  or  by  the  analytical  nictbud  of  di'rivml  functionx,  we 


\ 


LAOKANUE.  4S1 

in*y  Mnpli>7  infinitely  «iiiftll  quKntitieH  m  a  sure  »nd  valoablB 
menriH  of  BhorU-niiig  nnd  Himptifjring  nitr  pnraft.'' 

Um  RimlutioH  d<!i  iqnnlwnt  numeri'jw*,  poblmbnl  ill 
1798,  wdfl  iilHit  tlio  fniit  nf  hii  Icclurra  nl  tlin  Polytrclinic 
In  IhJH  he  givus  tlie  inrtlinl  of  spproxinutting  to  thn  rrni  niota 
of  An  equation  by  nipnnii  of  cuntinui-<l  f ravtionH,  mhI  muiicifttcH 
wsvcral  otlicr  tlii'oreniB.  In  a  note  nt  tho  end  lie  >hew«  how 
Femint's  tlw^>mn  tlintn'"'-!  ==  0<nnxl/j),  wtwre/' w*  primo 
and  a  is  prime  to  /i,  niny  Im^  nppliitl  to  give  the  complete 
algeltmicnl  wilutioii  <if  nny  )iinoMii;iI  eijuntion.  Ife  mIho  here 
explninti  liow  tilt-  rquntton  wlioso  riiots  are  tlin  wiuMtw  <il  the 
diflercntrH  of  the  rootft  of  the  ori^nnl  tHguntion  mny  lie  used 
tio  AS  to  give  coiiHidemlilo  infunnNtion  bh  ta  the  pcHition  ukI 
nnture  of  tlioxc  rootx. 

The  theory  of  thr  planctnry  niMionM  hnd  formod  the  auhjcd 
of  aoRic  of  tha  moHt  rpninrkKlilc  of  Ijngrnngc's  Berlin  pnpeni. 
In  tfOC  the  Huhjwt  wiu  ruijirnrd  l>y  I'liiHson  who  in  m  pitper 
rend  heforo  the  Fn-nch  Acwlrmy  Hhewwi  that  Lngninge'a 
forniulno  led  to  ccrtjiin  limitM  for  the  staltility  of  the  oHrilM. 
IjigrpMigi',  who  wiis  present,  now  diwUHted  tlie  wlule  mibjcct 
afreidi,  nnd  in  n  memoir  communicttetl  t<>  llie  Aewleniy  in 
HfOif  fxphiinnl  how  by  tlie  varialiuriof  krhitrary  coiiMtnntBtlio 
periodical  and  Hccular  incijualitieH  of  any  system  ot  mutiMlly 
interacting  budics  cuutd  be  detemnned. 

In  IHIO  LagrHnge  cumnienced  a  tliorougli  revisioo  of  the 
ifecaniqiit  aiut/f/fiqtt^,  but  he  wna  able  tn  complete  only  aboul 
twu-thirds  uf  it  before  his  death. 

In  appeamnce  he  wan  of  mi-dium  height,  and  alightty 
formed,  with  pale  blue  eyes,  and  a  colourless  cimiplcxiun.  In 
character  he  was  nervotu  and  tintid,  he  detentrd  oontroveny, 
and  to  avoid  it  willingly  allowed  olltcni  to  take  the  credit  for 
what  he  had  himwlf  done. 

Ijtgrange'a  inleresta  were  esttentially  thmie  ol  a  rtudent  at 
pure  mathematics:  he  sought  and  ohtnincrl  far-reaching abatract 
resnita,  and  waa  content  to  lean  the  applioationa  to  otbon. 
Indeed  no  intxmsiderable  part  «f  the  diseovniaa  ol  U&^m^ 


422     LAQRAMOB,  LAPLACB,  AHD  TBBIB  OOimifPORAmUBl 

contemporaiy,  lApbci^  coniiliitii  of  the  appUoilioa  of  llio 
LBgrangum  fomiuke  to  the  facte  of  nataro;  lor  exAOiple^ 
lAphM»'8  ooncliuikMui  on  the  velocitj  of  aoaiid  and  the  eeciiler 
acoeleration  of  the  moon  ere  implicitly  involved  in  Ij^prmngeTe 
rfMalia.  The  only  difficolty  in  nnderatanding  Legrmnge  Is 
that  of  the  aubject-matter  and  the  extreme  generality  of  hia 
proeeiHieH;  but  hui  analysis  in  ''as  lucid  and  luminoua  aa  it 
is  symmetrical  and  ingenious.- 

A  recent  writer  speaking  of  Lagrange  says  truly  that 
he  took  a  prominent  part  in  the  advancement  of  almost 
every  branch  of  pure  mathematics.  Like  Diophantus  and 
Fenuat  he  pusnessed'a  special  genius  for  the  theory  of  numliersi 
and  in  this  subject  he  gave  solutions  of  many  of  the  problems 
which  Irod  been  pro|iosed  by  Fenuat,  and  added  some  theorems 
of  his  own.  He  created  the  calculus  of  variations  To  him 
too  the  theory  of  differential  equations  is  indebted  for  its 
positicin  OS  a  science  rather  than  a  collection  of  ingenious 
artifices  for  the  solution  of  particular  problems.  To  the 
calculus  of  finite  differences  he  contributed  the  formula  of 
interpolation  which  bears  his  nam&  But  above  all  he  im- 
presHed  on  mechanics  (which  it  will  be  remembered  he  con- 
sidered a  bnmch  of  pure  mathematics)  that  generality  and 
completeness  towards  which  his  labours  invariably  tended. 

Laplace^.  Pit-rrt  Simon  Lujilact  was  bom  at  Beaumont- 
en- Auge  in  Normandy  on  March  23,  1749,  and  died  at  Paris 
on  March  5,  1827.  He  was  the  son  of  a  small  cottager  or 
perhaps  a  farm-labourer,  and  owed  his  education  to  the  interest 
excited  in  some  wealthy  neighbours  by  his  abilities  and 
eng2iging  presence.  Very  little  is  known  of  his  early  yearS| 
for  when  he  became  distinguisheil  he  had  the  pettiness  to  hold 
himself  aloof  both  from  his   relatives   and  from  those  who 

*  The  foUowing  account  of  Laplace's  life  and  writinga  is  mainly 

Coundixl  on  tlie  articlcM  in  the  KHtjIuh  Cyclopaedia  and  the  KHejfclopaedia 

Britanhica,    LapUee*«  works  wvie  pubhahed  in  Heven  voluiuea  by  the 

>*jnem'h  goveniinent  in  1813-7;  and  a  new  edition  with 

mddiUouMl  matter  was  Issued  ai  Vai'is  vu  lix  xolunies,  187a-ei. 


\ 


LAPLACE.  423 

lind  nKsixUvl  liiiii.  It  would  Bwm  Ihnt  fmni  a,  papil  lie  tM-oune 
kn  iinlicr  in  llin  Bcliiml  nt  DpAnniniit ;  )>ut,  li&^'in);  proctiird  ■ 
IctU-r  i>f  introduction  to  D'Alcnilicrt,  lie  wrnt  tn  Pnrin  to 
faah  liii  fortune.  A  pnprr  on  ttir  principle  of  iiirchnniot 
excitofl  D'AIonilirrt'it  intrrrat,  niid  on  liit  rccommcncliitioii 
ft  plncc  in  tlio  militnry  iichool  wivh  uirpivcl  to  lAplacc. 

Secure  of  A  compcti-ney,  lyiplacc  now  tlircw  hiniticif  into 
nrifpnnt  rrHcnrcli,  nnil  in  tin"  nrxt  wvcnt<^n  yennt,  1771— ITt'T, 
he  produced  much  of  lii*  original  work  in  AHtronomj.  Thii 
csmnionced  with  a  memoir,  reiul  Mdtv  the  French  AcMlemf 
in  1773,  in  which  he  shewnl  thnt  thr  ptftnclarj  motions  wont 
«tablo,  and  carried  the  pro')f  us  far  as  the  cube*  of  the  eccen- 
tncities  ami  inctinntionH.  This  was  followoi)  ity  several  pspen 
on  points  in  th(^  integral  calculun,  finit<.-  diHcrences,  diffrranti«l 
oquittions,  and  nstranoiny. 

During  the  yennt  1784'1T87  he  produced  Rome  niemoira 
of  (exceptional  power.  Prominent  among  tlienc  is  one  read 
in  1764,  and  reprintnl  in  the  thini  voIun)e  of  the  Mmtntqnr. 
rrlerif,  in  which  he  completely  determined  the  attraction  of  « 
spheroid  on  a  particle  outJiido  it.  This  it  itiPmorHble  Eor  tlie 
intniduction  into  analyitis  of  spherical  harmonica  or  LaplMo'i 
coefficients,  and  also  for  the  development  of  the  nso  of  tho 
potential ;  a  nftme  first  given  by  Green  in  1838. 

If  the  coordiiiat««  of  two  points  Im'  (r,  fi,  «)  ami  (r,  f!,  «'), 
and  if  .r'^r,  then  the  reciprocal  of  the  dintniKfi  Iwtween  them 
can  lie  expanded  in  powers  of  rfr,  and  the  riMpcctire  copflicieiits 
are  Laplace's  coefficients.  Their  utility  ansen  from  the  fact  that 
every  function  of  the  coordinates  of  a  p>int  on  a  aphera  can  be 
espamled  in  a  series  of  them.  It  should  be  stated  that  the 
similar  coefiicients  for  space  of  two  dimenxion^  together  with 
■ome  of  their  properties,  hod  been  previously  giren  by 
L^endre  in  n  pAper  mnt  to  the  French  Academy  in  1783. 
Legendre  had  goiNl  reason  to  cnnipliun  of  the  way  in  which  he 
wan  treated  in  thin  matt«r. 

Thia  paper  is  also  remarkable  for  the  development  a( 
the  ide*    irf    the    potential,    which   waa  »ygwiySaiw&  ^xa«. 


/ 


424     LAORANQK,  LAPLAC%  AND  TUKIR  OONTBIIFOEAEUBIL 

I^gFMige*  wlio  liad  oied  it  in  Ua.meiiioin  of  1779^  17779 
and  1780.  Laplace  iihDwed  that  the  potential  always  Mitirfei 
the  dilTerential  equation 

and  fMi  tliiH  rpHult  hia  Mulmeciuent  work  on  attractiona  wm 
iMMed.  Tlie  quantity  ^Y  han  been  termed  the  ounoentratioa 
of  K,  and  it«  value  at  any  point  indicatoH  tlie  exoew  of  thb 
value  of  V  tliere  over  itn  uiean  value  in  the  neighbourhood  of 
the  point.  Laplace's  ei|uation»  or  the  more  general  fona 
TT-  —  4vyi,  appeam  in  all  branclics  of  uiatlieiiuitical  physica 
According  to  8ome  writera  tliiH  follows  at  oiioe  from  the  fact 
that  V  is  a  scalar  operator ;  or  the  equation  may  represent 
analytically  some  general  law  of  nature  which  has  not  been  yet 
reductKl  to  words;  or  possibly  it  might  be  regarded  by  a 
Kantian  as  the  outward  sign  of  one  of  the  necessary  forms 
through  which  all  phenomena  are  perceived. 

This  memoir  was  followed  by  another  on  pliuietary  in- 
equalities, which  wiiH  presented  in  three  sections  in  1784,  1785, 
and  17H6.  This  deals  mainly  with  the  explanation  of  tlie 
''gr(*at  ine4|uality"  of  Jupiter  and  Saturn.  Laplace  shewed 
by  general  considerations  that  the  mutuid  action  of  two 
planets  could  never  largely  atTect  the  eccentricities  and  in- 
clinations of  their  orbits ;  and  that  the  peculiarities  of  the 
Jovian  system  were  due  to  the  near  approach  to  commen- 
surability  of  the  mean  motions  of  Jupiter  and  Saturn  :  furtlier 
developments  of  these  theorems  on  planetar}'  motion  were 
given  in  his  two  memoirs  of  1788  and  17H9.  It  was  on  these 
data  that  Delambro  computed  his  astronomical  tables. 

The  year  1787  was  rendered  memorable  by  Laplace's  expla- 
nation and  analysis  of  the  relation  betwi^en  the  lunar  accelera- 
tion  and  the  S4*cular  changes  in  the  eccentricity  of  the  earth's 
orbit :  this  investigation  completed  the  proof  of  the  stability 

*  See  the  UuHttiH  of  the   New  York  Msthenuiliesl  Soeiety,  18M, 
rol.  I,  pp.  C6 — 71. 


\ 


LAPLACE.  425 

of  the  wIioIp  Nnlar  Hjitt^m  on  tlie  Awitimptinn  thAt  it  oottMAa  ol 
«  collection  of  rigid  Imdics  iimving  in  h  vacnnm.  All  the 
menioint  nNtve  «IIu<!cH  to  were  prwo-ntcH  lo  the  Frandi 
Acrtdemy,  Aiid  thpy  are  printotl  in  the  MtMoira  primula 
pnr  flirvrra  wttvifM. 

Ijiplnce  now  set  liimNcIf  the  tvtk  lo  wiita  «  work  which 
should  ■' offer  A  complete  Holution  of  tho  grCKt  mechMiicnl 
prohlem  prcKonted  by  the  fwiUr  sj-stem,  and  bring  th«or]r  to 
coincide  xo  closely  with  observiitlon  thnt  einpiricMl  eqnntions 
shiiold  no  longer  fintl  a  pliico  in  antrononiical  tMUrfl."  Tlie 
nvult  is  emliodiod  in  the  Erponilion  tin  Kt/^ihtte  Hn  monde  nnd 
thn  MrcitHiifH'!  rrlaitK. 

The  former  wuh  puhliiihrd  in  ITS').  And  gires  »  gcncnil 
explanation  of  the  phcnomenn,  liul  oinibt  all  details.  It  con- 
tAins  a  Hiinimary  of  the  hiatory  of  antronontjr :  ihi"  snmmiuj 
procumi  fitr  ita  author  the  honour  of  ndmimion  to  tho  forty 
of  the  French  Acndemy ;  it  in  commonly  cHteenied  one  of  the 
master-piece*  of  French  literature,  though  it  is  not  altogether 
relialilc  for  thu-  later  periods  of  which  it  treata. 

T)ic  nebular  h}'pntheais  wan  kpreenunciAt4Hl*.  According  to 
this  hypothtsig  the  »)liir  Hyiitem  has  btvn  evoU-e<l  from  a  globular 
mass  of  incamlcHccnt  gas  rotating  round  an  axia  thmugh  ita 
centre  of  maivi.  An  it  cooled,  thin  niawi  contracted  and  suceemiTe 
rings  broke  off*  from  its  outer  ealge.  Thnte  lings  in  their  tsm 
cooled,  and  finally  oondcmied  into  the  plancta,  while  the  san 
rvprcswntH  Uie  central  core  which  to  Htill  left.  Certiun  eorrec- 
tionii  re<ioired  liy  modem  science  were  added  hy  Vi.  Roche,  and 
recently  the  theory  has  liccn  distniRHcd  critically  by  R.  Wolf. 
Some  of  the  arguments  against  the  hypothcMs  are  gii-en  in 
Faye's  Orvjinr  rfii  moiide,  Parin,  IK84,  wliere  an  ingeniool 
ihodiliGation  of  the  liypotheHis  is  ptopof«d,  by  which  the  author 
attempts  to  explain  the  peculiar) tie*  of  the  kxial  rotation 
of  Xeptune  and  Uranus,  and  the  retn^rade  motioa  ot  the 
Mlellites  of  the  latter  planet.  The  subject  is  one  id  great 
■  Ob  Ibe  UMor7  ot  Um  MbalM  hnwtbaus  m  Tfa  rkOU  Umbrtrm^ 


426     LAQRANQK,  LAPLACE,  AMD  TUBIR  OONTEHraEABlBl 


diffieuliyi  but  prohaUy  modenn  opinioii  is  indinad  to  aooepi 
the  nebular  hjrpotheiUB  m  a  vtru  mMJOi  though  reeiyiiiing 
that  otlier  eauaet  also  (and  notably  meteoric  aggrefatiooa 
and  tidal  friction)  have  oontribnted  to  the  development  of 
the  planetary  synteui.  The  idea  of  the  nebular  hypothene 
had  been  outlined  by  Kant*  in  1755,  and  he  had  aliio  nig- 
gested  meteoric  aggregations  and  tidal  friction  as  causes 
affecting  the  formation  of  tlie  soUr  system:  it  is  probable 
that  Laplace  was  not  aware  of  this. 

According  to  the  rule  published  by  Titius  of  Wittemberg 
in  1766 — but  generally  known  as  Bode's  law,  from  the  fact 
tliat  attention  wan  called  to  it  by  Johanii  Elert  Bode  in 
1778 — the  dintances  of  the  planets  from  the  sun  are  nearly  in 
the  ratio  of  the  numbers  0  +  4,  3^-4,  6+4,  12  -i-  4,  drc.,  the 
(m  +  2)th  term  being  (2*  x  3)  •••  4.  It  would  be  an  interesting 
fact  if  this  could  lie  deduced  from  the  nebular  or  any  other 
liy|)otheseH,  but  so  far  as  I  aiu  aware  only  one  writer  has 
made  any  serious  attempt  to  do  so,  and  his  conclusion  seems 
Ut  Im)  that  the  law  is  not  sufKciently  exact  to  be  more  than 
a  cuiivenieiit  means  of  remembering  the  general  result. 

I^place's  analytical  discussion  of  the  solar  system  is  given 
in  his  Mtcaniqtte  eelesU  published  in  tive  volumes.  An  aiialysui 
of  the  contents  is  given  in  the  Enyluh  C*fch>itaeilitJL  The  fimt 
two  voluiiieii,  |»ublislied  in  1799,  contain  methods  for  calcu- 
lating the  motions  of  the  planets,  determining  their  figures, 
and  resolving  tidal  problems,  llie  third  and  fourth  volumes, 
publisluHi  in  1802  and  1805,  contain  the  application  of  these 
methods,  and  several  astronomical  tables.  'Hie  fifth  volume, 
published  in  1825,  is  mainly  historical,  but  it  gives  as 
appt*n<Hce8  the  results  of  F^place's  latest  researches.  Laplace's 
own  investigations  emlKxlied  in  it  are  so  numerous  ami 
valu.ible  that  it  is  rt*grettalile  to  have  to  add  that  many  results 
are  appropriat4w|  frtini  writers  with  scanty  or  no  acknowledg- 
ment, and  the  c*«mclusioiis— which  have  Is'en  di^icribed  as  the 

*  8««  AtfNf*«  Caamoffouy,  edited  by  W.  Usttie,  OUsguw,  1900. 


I.APLACK.  427 

orgMnued  rexnlt  of  »  century  of  pntirnt  Uiil — «re  frecinnntlj 
mentionnl  an  if  they  wi-re  iluc  l<>  LApl.vx*. 

The  matter  of  itto  Mfraniqi"-  cflfft'  in  rxcellrnt,  liat  it  U 
liy  nil  iDPiinH  wwy  readinp.  lliot,  »hi>  nwistcd  LnpliKV  in 
rcviHing  it  for  tlio  prcFw.  snyH  tliat  I^nplacr  liimwlf  wm  fre- 
quently unnltip  to  recovor  the  (It-tnils  in  tlie  clMin  of  rpMioning, 
Am),  if  sntisGed  tlint  the  coiicIuMonit  were  ctirrect,  he  wwt 
content  to  inxert  the  constantly  recurring  fnrmniA  "  II  est  mw 
k  »i»ir."  The  itfritnifu'  efl'tirr  in  not  only  the  tUMimlation  ol 
tlio  Pritifi/iin  into  the  l»iigiiiige  of  tlie  difTerentinl  cnlculan, 
but  it  completes  pnrtn  of  which  Newton  hifl  been  anahlo  to 
till  in  the  detrtils.  K.  F.  Tiwwmndu  rwerit  work  may  lie 
taken  ax  the  modem  prencntntiun  of  dynamical  aatrononiy  mi 
classical  lintst,  but  Lnplace'ii  treatise  will  always  renkain  a 
standard  authority. 

lAplace  went  in  state  to  Iwg  Napoleon  to  accept  a  copy  of 
hiH  work,  and  the  followiTi;;  act^unt  of  the  interview  is  well 
authenticated,  and  bo  chnrncteri.Htic  of  all  the  parties  concvmed 
that  I  quote  it  in  full.  Someone  had  told  Napoleon  tliat  the 
book  contained  no  mention  of  the  name  uf  (iod ;  Napoleon, 
who  was  fond  of  putting  cmbarnvwing  igueMttons,  received  it 
with  the  remark,  "  M.  Laplace,  they  tell  me  you  have  written 
this  large  book  on  the  Hyntem  of  the  univen<e,  and  havo  never 
e»cn  mentioned  iU  Creat^ir,"  Iviptitce,  who,  though  the  rnont 
supple  of  jioliticianM,  was  as  stiff  ^h  a  martyr  on  every  point  of 
his  philosophy,  drew  himself  up  and  answen-d  blontly,  "  Jo 
n'avais  pas  liexuin  de  cette  hypothexe-l^"  Napoleon,  greatly 
amused,  told  this  reply  to  Lagrange,  who  excbunwd,  "  Ah ! 
c'est  line  belle  bypotht«e  ;  i,-a  expliquc  Ijeauconp  de  choMa." 

In  1812  laplace  issued  hiti  Thinrie  aiialj/tiqiM  dea  probo' 
6tKfe**.  Hie  theory  is  sUted  to  In-  only  commoo  amse  ex- 
pressed in  mathematical  language.  The  method  of  estimating 
Um  ratio  of  the  nomber  of  favourable  cases  to  the  whole 
number  of  ponsihle  canes  harl  been  indicsM  bjr  La|dao« 
■  A  sainniMj  of  Laplus*!  rrMniUnt  b  gifrt  In  As  Mtida  m 
FivlwbUilj  in  th«  BHtfclnfnUi*  Utmpatilaiim. 


4S8     LAURANQB,  LAPLACB»  AMD  TUKIR  OONTEHraRAEinL 

in  A  IM4per  writleo  in  1779.  It  ooositta  in  tronling  IIm 
Mioctswiive  valuoi  of  any  fanciion  m  the  ooeflfeiQnAs  in  IIm 
expttUBum  of  another  function  with  relerenoe  to  n  diflbrent 
varwiile.  The  hitler  in  therefore  cmllod  the  geoeniting  ftmetion 
of  the  former.  Ijiplaee  then  ahewii  how,  by  means  of  interpohi* 
tion,  thette  cuelficientt  may  be  determuied  from  the  generating 
functioiL  Next  lie  attackg  the  convene  pniblem,  and  from  the 
coetiicienU  he  finds  the  generating  function;  this  b  eflfecCed  by 
the  solution  of  an  equation  in  finite  ditfereuces.  The  nietliod 
is  cumbi*niouie,  and  in  consequence  of  the  increased  power  of 
aiiJilysis  is  now  rarely  used. 

This  treatise  includes  an  expusitioii  of  the  nietlmd  of  hsist 
squanw,  which  is  a  remarkable  testimony  to  Laplace's  com- 
mand over  the  processes  of  analysis.  The  method  oi  least 
8(|UJires  for  the  combination  of  numerous  observations  had 
becMi  given  empirically  by  UausH  and  LegiMidn*,  bi.t  the  fouKh 
chapter  of  this  work  contains  a  formal  proof  of  It,  on  which 
the  whole  of  the  tlifory  of  ern>rs  luis  been  since  U%sed.  This 
was  eH*ecti*d  only  by  a  most  intricate  analysis  specisUy  invented 
for  the  purp«»s(%  but  the  fonn  in  which  it  is  pi^esented  is  so 
miiigrc  and  unsatisfactory  that  in  Hpite  of  the  unifonn  accuracy 
of  the  results  it  was  at  one  time  questioned  whether  Laplace 
had  actually  gone  through  the  difiicult  work  he  so  briefly  and 
often  incorrectly  indicates. 

In  1819  Laplace  published  a  pt>pular  account  of  his  work 
on  probability.  This  book  liears  the  Maine  relation  to  the 
T/teorie  tlfH  probabiliift  that  the  S»fiUti»he  du  uwnde  does  to 
the  Meraniqiie  ciUste. 

Amongst  the  minor  disco veri(*s  of  Laplace  in  pure  nmtlie- 
matic8  I  may  mention  liin  discussion  (simultaneously  with  Van- 
deniioiide)  of  the  general  theory  of  determinanU  in  1772  ;  his 
proof  that  every  e4|uation  of  an  even  degree  nmst  have  at  least 
one  real  f|Uii(lratic  factor ;  his  reduction  of  the  solution  of  linear 
difierential  ecjuatioiis  to  definite  integrals ;  and  hLs  solution  of 
the  linear  pai*tial  difi'ereiitial  equation  of  the  second  order.  He 
was  als«i  the  tirst  to  consider  the  difficult  problems  uivolved  in 


• 


LAPLACE.  4S9 

eqaKtioiiN  of  mixntdilTrrpncm,  am)  to  pr<iv«thKt  the  Mention  o( 
RR  e(|uatiun  in  finite  ditTun'nccH  of  the  lint  degree  and  the 
second  order  miifht  \m  idwnjs  <il)tninei]  in  the  fiinn  vt  r 
continan]  fmctimi,  ItesidpH  these  origiiml  riiMDverieN  be 
determined,  in  hit  tlieory  uf  |tr«bnl>itities,  thp  TAlnra  of  ft 
nDinl>pr  ot  t)ie  more  L-omnioii  definite  inte;;mli :  Ami  in  the 
Bvne  btiok  ^ftve  the  genemi  proof  of  the  thnHwiii  Pniittci«ted  ' 
hjr  hagmngf.  for  the  devehipinent  of  nny  implicit  (unction  in 
n  xerieM  hjr  mennH  nf  diHerentinl  ciieflicienta. 

In  theorrticitl  phyRica  the  theory  of  cAplllarj  attrwction 
in  doe  to  Ijiptnce  who  ncceptnl  tlin  idea  pmponmled  by 
H&Dksliee,  in  lite  /'A i7(ur>/>/i irn/  Tmiuiirliinu  for  1709,  thftt 
the  phenornennn  was  duo  to  it  force  of  nttmctiini  which  wm 
insennilile  nt  HcnNJlile  dJatnuceH.  The  part  which  (kftln  with 
the  action  of  a  solid  on  n  litgaid  mid  the  mutual  action  ii(  two 
tiqui<lit  won  not  worked  out  thoniugldy,  hut  ultimately  wm 
completed  by  Uausn :  Neumann  Inter  filled  in  a  tew  detailit. 
In  I8fi2  Lfinl  Kelvin  (Sir  William  Thomson)  shewed  that,  if 
we  assume  the  niolecular  constitution  of  matter,  tlie  lawi  of 
capillary  attractiun  can  he  deduced  fniin  the  Newtonian  law  ot 
gmvitation. 

I^ploce  in  1816  wm  the  first  to  point  out  explicitly 
why  Newton 'x  theory  of  vibratory  motion  gave  on  inoomet 
value  for  the  velocity  of  «iund.  The  actual  velocity  la 
greater  than  tliat  calculated  liy  Newton  in  conaec|nencc  o(  the 
heat  developed  by  the  Rudden  compression  nf  the  wr  which 
increnaes  the  einnticity  and  therefore  the  velocity  of  the  aonnd 
transmitt^.  Laploce'ii  invent  i  gut  ions  in  jimcticol  phyaiea  wer« 
confined  to  thoM  cnrrie<l  on  by  him  jointly  with  LavMiuer  in 
the  yoara  li''2  to  1784  on  the  specific  heat  of  voriooa  bodioi. 

replace  •eemit  to  have  regnnled  aruilysin  merely  on  «  mouu 
of  attacking  phynical  proldenui,  though  (he  ability  with  whicli 
he  invented  the  necnwry  analynis  in  aIniOHt  pbenonienal.  Aa 
long  aa  his  results  were  true  he  took  but  little  trooUe  to  ex< 
plain  the  at«pa  by  which  he  arrived  at  them  ;  he  nerer  atodM 
d«f[anc«  or  symmetiy  in  hin  procemeo,  and  it  waa  awttefart 


430     LAQRAKOK.  LAPLACB,  AND  THKIB  OOmWMrORAMVBL 

for  him  if  be  ooold  by  any  meuis  aolve  the  putienbr  ^aattfon 
he  waa  diHCiutiiiig. 

It  would  h*ve  been  well  for  I^plaoe**  r^tatioii  if  he  h«l 
been  oonteai  with  his  icieiitifio  work,  but  above  all  things  he 
coveted  social  faiue.  The  skill  and  rapidity  with  which  he 
managed  to  change  his  politics  as  occasion  required  would  be 
amusing  had  they  not  been  so  servila  As!  Napoleon's  power 
increased  Laplace  aliaudoued  his  republican  principles  (which, 
since  they  luid  faithfully  reflected  the  opinions  of  the  party  in 
power,  had  themselves  gone  through  numerous  changes)  and 
begged  the  flnit  consul  to  give  him  the  pont  of  minister  of  the 
interior.  Napuleoii,  who  desired  the  {lupport  of  men  of  scienci^ 
agreed  to  the  propotuU ;  but  a  little  less  than  six  weeks  saw 
the  close  of  Laplace's  political  career.  Napoleon's  memo- 
niudiuii  on  his  dismissal  is  as  follows.  '*  G^metre  de  premier 
rang,  I^place  ue  tarda  piis  k  se  niontrer  adiiiiuiNtnitear  plus 
que  iiic<1ioci*e ;  des  son  premier  travail  nous  reconnumes  que 
nous  nims  etioiis  troiiipe.  Lupluce  ne  Hiiistssait  aucune  ques- 
tion suuH  son  veritable  pt»int  de  vue :  il  clierchait  des  sub- 
tilites  piirtout,  n'avait  (|ue  des  idees  probl^iiiatiques,  et  portait 
eiitin  Tesprit  des  '  infiniuient  petits'  j usque  dams  Tadminis- 
tnitiou."  I 

Althougli  Laplace  was  removed  from  office  it  was  desirable 
to  retaiiii  his  allegiance.  He  was  accordingly  raised  to  the 
senate,  and  to  the  third  volume  of  the  J/rcanique  ctleMte  he 
prefixed  a  note  that  of  all  the  truths  therein  contained  the  most 
precious  to  the  author  was  the  declaration  he  thus  made  of  his 
devotion  towards  the  peace-maker  of  Europe.  In  copies  sold 
after  tlie  restoration  this  was  struck  out^  In  1814  it  was 
evident  that  the  empire  was  falling ;  Laplace  hastened  to 
t4*nder  his  services  to  the  Bourbons,  and  on  the  restoration 
w:is  rewarded  with  the  title  of  marquis :  tli(^  contempt  tliat  his 
mure  honest  colleagues  felt  for  his  conduct  in  the  matter  may 
be  read  in  the  pages  of  Paul  Louis  Courier.  His  knowledge 
was  useful  on  the  immerous  scientific  commissions  on  which 
/i#f  8er%'i%l,  and  probably  accounts  for  the  manner  in  which  his 


\ 


LAPLACE.      LEOEN'DRE.  431 

politicnl  itwinccrily  wjw  overlfmkpd ;  l>ut  tlie  petliiMws  of  hi> 
chnractor  mu^t  nol  mnkc  an  fi>rg«t  Kow  grrat  were  his  services 
to  science. 

Tliat  Ijnpliwo  WON  vhId  nnd  fwlHsli  in  not  denied  hy  hw 
wnmiest  lulniirem  ;  hin  ciiniluct  Ui  thr  )M>ni-fiict'ira  of  hin  youth 
And  hiH  imliticnl  friends  wns  an<;ral<-ful  nnd  cantemptibh ; 
while  \t\*  B|)prnprialiiin  of  thi-  reMuIbi  iif  tlioM  who  were  cont- 
pimtively  unknown  wems  to  Ik-  well  estahlislied  And  is 
nbnoluUiIy  indefensible— of  thos*  wlioni  he  thus  tremted  three 
Eubsequently  rose  to  diiitinction  (l^genilre  and  Foiiri<^r  in 
France  nnd  Young  in  En;;lnnd)  and  never  forgot  the  injaatic« 
of  which  they  hail  lieen  the  victims.  On  the  other  Hide  it  may 
lie  naif)  that  O'n  xnine  questions  he  shewed  indepentlence  of 
character,  and  he  never  concealed  his  views  on  religion, 
philosophy,  or  science,  however  distasteful  they  mi^t  he  to 
the  nuthorities  in  p>wer ;  it  should  be  also  atMcd  tliat  twrardi 
the  closfl  of  hifi  life  and  espn-ially  to  the  work  of  hin  pnpih 
Ijiplocc  was  Inth  genemui  and  appreciative,  kihI  in  one  ctue 
Muppreened  n  paper  iif  hin  own  in  nnter  that  a  pupil  might  linvo 
the  Bole  credit  of  the  invcstif^tion. 

Legendre.  Aftrinn  Mnrir  f^gtwlrr  wan  l«m  at  ToulouM 
on  Sept.  18,  17.t3,  and  dieil  at  Paris  on  Jan.  10,  1833.  Tlie 
hauling  events  nf  hin  life  arc  very  Hiniple  and  may  lie  sonimml 
op  briefly.  Iln  wan  edacatcd  at  tlie  Mazarin  Gollege  in  IVrvi, 
appointed  professor  at  the  military  Hchnol  in  Pftris  in  1777, 
was  a  member  of  the  Anglo-French  cummission  of  1787  to 
connect  Greenwich  and  Paris  geodetically ;  ier»«d  on  several 
of  the  public  commissions  from  1792  to  lt<IO;  was  made  a 
prafewuir  at  the  Normal  school  in  1795;  and  snbaequently 
held  a  few  minor  gni-pmment  app>intment.^  17»e  inflnence 
of  lAplacn  was  st<>Adily  e!(ert«>d  ngainx*.  hi*  obtaining  ofEoe 
or  pnldic  recognition,  ahrl  Ix'gendre,  who  was  a  Umid  student, 
accepted  the  obscurity  to  which  the  hostility  of  his  colleagiie 
condemned  him. 

L^ndr«'s  analysis  in  of  a  high  onler  of  cxcellenee  and  is 
secnml  only  to  that  produced  hy  f  Agmnge  and  I<iplufl%  Umm^ 


r 


432     MQRANOK.  LAPLACE,  AHO  THBR  OOlimfHMtAmm. 


it  ui  not  so  originaL  HU  ehief  works  mn  his  Oimmfii'i$t  Vk 
7%€orU  dot  nombrtm^  his  Cideui  tniqfralf  mad  his  ftwdiwM 
ellipiiques.  These  mdude  the  resalU  of  his  Tarious  papen  on 
tiieiie  MnbjectA.  Betddes  these  he  wrote  a  treatise  which  gave 
tlie  mle  for  the  method  of  least  squares,  and  two  groups  of 
niemoinii  one  on  the  theory  of  attraction^  and  the  other 
on  geodetical  operations. 

The  niemoini  on  attractions  are  anidysed  and  discussed  in 
Todhunter's  Uittory  of  the  Tkeone$  of  A  ttraetioH.  The  earliest 
of  Uiese  nieiuoin,  presented  in  1783,  was  on  the  attraction 
of  spheroidn.  Tliis  coiitaiuH  the  introduction  of  L^ndreVi 
cuefficientH,  which  are  siHuetiiues  called  circular  (or  aonal) 
hannonics,  and  which  are  particular  cases  of  Laplace's  oo- 
eflicieiits ;  it  ultio  includes  the  Molution  of  a  problem  in  which 
the  potential  im  used.  The  second  memoir  was  conmiunicated 
ill  17H4,  and  is  on  tliu  form  of  equilibrium  of  a  masn  of 
roUititig  liquid  which  in  approximately  Hpherical.  Tlie  third, 
writteu  in  178G,  ih  on  the  attraction  of  oonfocal  ellipsoids. 
Till)  fourth  iM  on  the  fif^re  which  a  fluid  planet  would  assume, 
and  its  law  of  density. 

Hm  papers  on  geodesy  are  three  in  number  and  were 
presented  to  the  Academy  in  1787  and  1788.  The  most  im- 
portant result  is  that  by  wliicli  a  spherica)  triangle  nuiy  be 
tre2it<H]  OS  plane,  providcMl  certain  corrections  ara  applied  to  the 
angles.  In  connection  with  this  subject  he  paid  considerable 
attention  to  geodesies. 

The  method  of  least  squares  was  enunciated  in  his  Nouvdleg 
uitthuife*  published  in  180G,  to  which  supplements  were  added 
in  IS  10  and  1820.  Gauss  independently  had  arrived  at  the 
same  result,  had  used  it  in  1795,  and  published  it  and  the 
law  of  facility  in  1809.  Laplace  was  the  earliest  writer  to 
give  a  proof  of  it:  this  was  in  1812. 

Of  the  other  books  produced  by  Legendre,  the  one  roost 
widely  known  is  his  tUtinenUde  ytometrie  which  was  published 
in  1794,  and  was,  at  one  time,  widely  adopted  on  the  con- 
tinent as  a  substitute  for  Euclid.     The  later  editions  contain 


I-ETJENnRE.  433 

tho  elcnic-nU  if  Irigimonirtry,  nnil  pniofs  iif  tlio  irnCionAlitj 
uf  9-  KnH  x'.  An  npifndix  nn  tlic  ilifiicult  <]aeiition  of  Uw 
thp<>i7  of  pnntllpl  lin«>H  wns  insard  in  Il:i03,  Mid  in  lioond  up 
witli  moHt  uf  the  HuliH^ucnl  editions. 

HiH  Tlirorif  '/'«  nomhrrt  wjw  j)ubliHhi.-<l  in  1798,  And  np- 
peiwliccfl  were  Added  in  It^lR  nnd  lti2''>:  (fm  thml  edition, 
iMucd  in  two  rolumcH  in  lt^30,  includrs  llie  reNulto  of  his 
vnrioUN  InU^r  papers,  And  nlill  mnniiis  n  Rtnnilitnl  work  un  the 
sultJpcL  It  nmy  bo  NaicI  that  he  heiv  cnrriiil  the  nalgcct  mi 
fnr  u  wM  poHsiblo  by  the  npplicatioti  of  ordinnry  nigctm;  but 
ho  did  not  renliic  tlint  it  tnif;ht  1>e  rrgnnled  nii  «  higher 
iun(hmi-tic,  And  ho  form  n  distinct  nnbjrct  in  nMlhcmatioi. 

nw  Uw  of  ()uwln>tic  reciprocity,  wliich  connccto  nny  two 
Olid  primes  wna  tint  proved  in  this  hook,  but  t)ie  result  hitd  been 
enunciated  in  m  memoir  of  1785.  Gauiw  ci)lrd  the  propoMtion 
"  the  gcin  uf  arithmetic,"  and  no  \pst  tlinii  nix  Hepnrate  prooli 
are  to  tie  found  in  hin  workn.  Tlio  theorem  in  as  follows.  If 
p  be  a  prime  nnd  n  )>e  prime  to  />,  then  wa  know  tliat  the 
remaindiT  when  n"''""  i>i  divided  by  p  in  citlter-t-  1  or—  1. 
L^endre  denoted  thin  remainder  by  (njp).  When  the  re- 
mainder in  +  I  it  in  pnuiblo  to  find  a  Fiquara  number  whidi 
when  dirideii  by  p  It-aves  a  rcmuinder  »,'  that  is,  ft  ia  ft 
qvadnttic  residue  of  p ;  when  the  nnnninder  in  —  1  tliere  exists 
no  each  square  number,  and  n  is  a  non-rexidoe  of  p.  "Ow 
law  of  qnadnttic  reciprocity  is  eicpreswd  by  the  theorem  that* 
if  A  and  6  be  any  o<ld  primes,  then 

(«/*){*/'■)-{- 1)'"-"'*-"; 

thus,  if  &  be  a  retidne  of  rt,  then  n  is  alxu  a  retddue  of  6,  nnlea 
both  of  the  primes  a  and  6  are  of  the  form  4m  +  3.  In  other 
words,  if  ft  and  b  be  odd  primes,  we  know  that 

«K*-il  =  *l  (■wid6),an<lWC-'>  =  *l  (moda); 

and,  by  L^endre's  law,  tlie  two  amltif^ilJeM  will  be  either 
both  positive  or  both  negative,  unless  a  and  A  ftre  both  o(  Uw 
form  4m  +  3.     Thus,  if  one  odd  primu  be  a  wn^nAMb  dL 


i34     LAtiRANQK,  LAPLACK.  AND  TUKIR  OOHRMrORASmL 

uioilier»  tlien  the  latter  will  be  a  iUNiiHreaid«e  ol  IIm  fomar. 
OauMi  and  Kunimer  have  BulNequently  pvoved  aimilar  laws  of 
cubic  and  Itiqiiadimtie  reciprocity;  and  an  important  himndi  of 
tlie  theory  of  numbers  has  been  based  on  these  researches. 

Tfiis  work  also  contains  the  useful  theorem  by  which, 
when  it  is  possible,  an  indeterminate  equation  of  the  second 
degree  can  lie  reduced  tu  the  form  aa^  •¥  bff  h-  e:;*  -  0.  Lpgendre 
here  diHcuitaed  tlie  forms  of  numbem  which  can  be  expressed 
as  the  sum  of  three  squares;  and  he  proved  [art.  404] 
that  the  number  of  primes  less  than  n  is  approximately 
ti/(log.n  -1083G6). 

The  Ejixrcicf4  de  eaicul  tHiiyral  was  published  in  three 
volunieti,  1811,  1817»  1826.  Of  these  the  third  and  most  of 
the  first  are  devoted  to  elliptic  functions;  the  bulk  of  this 
being  ultimately  included  in  the  Fonriions  eUipiiquet.  The 
contents  of  the  remainder  of  the  treatise  are  of  a  miscel- 
LuieriUH  cimnwtor ;  they  include  integration  by  series,  definite 
integrals,  and  in  particular  an  elaborate  discussion  of  the 
Beta  and  the  Gaiimia  functions. 

Tlie  Traiie  iUm  foncliom  tliipitgu^M  was  issued  in  two 
volumes  in  1825  and  1826,  and  is  the  most  important  of 
Legend  re's  works.  A  third  volume  was  added  a  few  weeks 
before  his  d(*ath,  and  contains  three  nH^moirs  on  the  researehes 
of  Aliel  uiid  Jacobi.  Legendre's  investigations  had  commenced 
with  a  paper  written  in  1 786  on  elliptic  arcs,  but  here  and  in 
his  other  papers  he  tn*ated  the  subject  merely  as  a  problem  in 
the  iiitegnil  calculus,  and  did  not  see  that  it  might  lie  con- 
sidered OS  a  higher  trigonometry,  and  so  constitute  a  distinct 
branch  of  analysis.  Tables  of  the  elliptic  integrals  were 
constructed  by  hiiii.  The  modern  treatment  of  the  subject  is 
foundtnl  on  that  of  Abel  and  Jacobi.  The  superiority  of 
their  iiietluids  was  at  once  recognized  by  Legendro,  and  almost 
the  last  iwi  of  his  life  was  to  recommend  those  discoveries 
which  he  knew  would  consign  his  own  labours  to  comparative 
oblivion. 

Thb  may  serve  to  remind  us  of  a  fact  which  f  wudi  to 


PFAFF.      MOXOE.  435 

■pecinlly  niiphnsizc,  lintiii'ljr,  that  Ghusk,  Abel,  Jnoobi,  itnd  cunra 
olhcn  of  the  mntlirmntk-ian^  alluilirii  Ui  in  the  next  chftpicr 
wcro  cont4-mpon«rieM  nf  the  mcml>iTtt  of  the  French  nclHxtl. 

Pfaff.  I  niny  here  mention  another  writer  wIk»  iJho  made 
A  Hjiecinl  Mtailj-  of  ihe  integral  cnlculuK.  Tills  wm  JrAitiM 
Fri'il'-rirk  I'/nff,  Uini  nt  Ktuttgurt  i>n  Dec  22,  1765,  aiid 
ilinl  at  Hallc  <m  April  i\,  lf3-'i,  who  ww  drscrihed  by 
lApl.-ice  an  the  mmt  rmim'nt  niatliematiciMi  in  Gemwny  nt 
the  Iwginning  iif  this  century,  a  doncription  which,  had  it  imt 
I>een  for  (imihs'ii  exiMtence,  would  have  Ix'cn  tnw  enoDgh- 

Pfair  wiM  the  pn-curtKjr  of  the  German  school,  which 
under  Gauss  and  hix  fulloweni  largely  determined  the  lines 
on  which  innthcmaticH  developed  during  the  nineteenth 
century.  Uc  wok  an  intinint«  friend  of  Oauw,  and  in  fact 
Ihe  two  matheinatieiann  liveil  t^ipelher  at  Hclrmtadt  dnring 
the  year  1798,  after  Uanioi  had  lini&hed  his  anirerrity  coarae. 
PfaCTs  chief  wurk  wan  his  (iinliniNhed}  OiiqHigitionea  Anaty- 
ticne  on  the  integral  ralculun,  publishei)  in  1797;  and  his 
mont  important  nirnioirK  were  either  on  the  calculns  or  on 
flifferential  e<|aationn:  on  the  Utter  subject  hin  paper  rowl 
before  the  Berlin  Academy  in  I8I4  is  noljceable. 

The  creation  of  vtodern  geoinetrtf. 
^Miilo  Euler,  LAgmnge,  Laplace,  and  Lcgcndre  were  per- 
fecting analynis,  tlic  Ricmhen  of  Another  group  of  Firnch 
niathematicinnn  were  eitending  the  mngn  of  geometry  by 
methotH  nimilar  to  tlione  pre^-iounly  i)S4>d  by  DeKargues  and 
PaKcal.  The  rtivival  of  the  study  of  syntbetic  geometry  is 
largely  duo  to  Poneelet,  but  the  subject  is  alxo  amociated  with 
the  naiuca  of  Mongc  and  L.  Camot ;  itx  great  derekqiment  in 
more  recent  timcn  in  mainly  duo  to  Steiner,  von  StAudt,  mmI 
Cremona. 

Monga*.     (fiuyxin/  ilanya-wwt  bom  at  BeMune  on  Hay  10, 

■  Sec  Enai  Uttoriqat  nr  la  tnirnu  ..Jr  MoKgt,  bf  P.  P.  C  Dapia, 

Puia,  1819 :  alKi  Ow  Nctict  ftiXon'fw  nr  Uenft  ky  B.  BikM^  Aafa, 


436  TUS  CBKATION  Or  MOOKEN  QBOIUTKY. 

1746,  and  diixi  »t  Fkriii  oo  July  S8,  1818.  He  was  the  aon 
of  a  small  pedlar,  and  waa  educated  in  the  vdiools  of  the 
Oratorianiii  in  one  of  which  he  iiulMequently  became  an  vaher. 
A  plan  of  Beaune  wkidi  he  liad  made  fell  into  the  hands 
of  an  officer  who  recommended  the  military  authorities  to 
admit  him  to  their  traiiiing-ifchool  at  Mesidres.  His  birth 
however  precluded  his  receiving  a  commisiiion  in  the  army, 
but  his  attendance  at  au  annexe  of  the  school  where  surveying 
and  drawing  were  tiiught  was  tolerated,  thougli  he  was  told 
tliat  he  was  not  sulKoieiitly  well  born  to  be  allowed  to  attempt 
problems  which  i*e4uired  calculation.  At  lust  his  importunity 
caiue.  A  pkui  of  u  fortress  having  to  be  drawn  from  the 
data  supplied  by  certain  obisier\'atioiis,  he  did  it  by  a  geo- 
metrical construction.  At  first  the  officer  in  chai*ge  refused 
to  receive  it,  Ijecuuse  etiquette  required  tliat  not  less  tluui  a 
certain  tiuie  hIiou1<1  lie  Ui»ed  in  making  such  dm  wings,  but  tlie 
superiority  of  the  nietluid  over  tliat  then  Utught  was  so 
obvious  that  it  was  accepted;  anil  in  1708  Monge  was  made 
piiifossor,  on  the  undenitaiiding  that  the  results  of  his  descrip- 
tive geometry  wei-e  to  be  a  military  secret  confined  to  officers 
alx>ve  a  i*ert4iin  rank. 

In  1780  he  was  appointed  to  a  chair  of  mathoniatics  in 
Paris,  and  this  with  some  provincial  appointments  which  lie 
held  gave  him  a  comfortable  income.  Tlie  earliest  paper  of 
any  special  importance  which  he  connnunicated  to  the  French 
Acmlrmy  was  one  in  1781  in  which  he  discussed  the  lines  of 
curvature  drawn  on  a  surface.  These  had  been  first  considered 
by  £u1er  in  1760,  and  defined  as  those  normal  sections  whose 
curvature  was  a  maximum  or  a  minimum.  Monge  treated 
them  as  the  locus  of  those  points  on  the  surface  at  which  suc- 
cessive normals  intersect,  and  thus  obtained  the  general  differ- 
ential e<[uation.  lie  applied  his  results  to  the  central  quadrics 
in  1795.   In  17dG  he  published  his  well-known  work  on  statics. 

Monge  eagerly  embnicecl  the  doctrines  of  the  revolutiim. 
In  1792  he  liecame  minister  of  the  marine,  and  assisted  the 
couiimitcii  of  public  safety  in  utilizing  science  for  the  defence 


\ 


HONfiE,  437 

of  thp  rrpuhlic  When  tin-  Tcrrori»ls  ol>tain«I  power  he  wiw 
daiiounm],  nnd  (hiIj'  ptcAprd  the  pailliitiiir  l>f  ft  hastjr  flight. 
On  hh  n'tum  in  1794  lie  wai"  nimle  n  pnifpfwor  nt  the  tihort- 
livrd  Nnniinl  school  where  lie  fpvve  lecture*  on  de«criptive 
deometry  ;  tlie  notes  of  these  wen*  puhli»Jir<l  antler  the  regula- 
tion ftlxive  nlluded  to.  In  I  TOG  lie  weni  to  ItAljr  nn  the 
roving  coiiimiuion  which  wixk  «'nt  wilti  orilerK  to  compel  the 
vrtrioHH  Italinn  towns  to  offer  pictun's,  iionlptare,  or  other 
wnrkn  of  art  that  they  might  pitssi^s  a-i  a  present  or  in  liea 
of  crHitrilmtiiinn  to  the  Freneh  republic  fur  removal  to  Pariii. 
In  1798  iie  nccppted  a  niittsinn  to  Itoiiie,  nn<l  after  execating 
it  j()inr<l  Napoleon  in  Kgypt  Thence  nfter  the  na\-al  and 
military  victories  of  England  he  ewnped  to  France. 

>lon^  then  Bettlr<l  clown  nt  Pnrii,  and  urniinwde  protestor 
at  the  Polj-teclinic  whool,  where  he  gave  Icctore*  cm  de- 
Bcriptive  geinnetry;  these  were  palilislied  in  1800  in  the  form 
nf  a  text-book  entitled  GfomAri'-  iV-trriptirt.  This  work  con- 
tAtnx  propoditions  on  the  form  anil  relntive  position  of  f^imetri- 
cal  figures,  deduced  I>y  the  use  of  tmnsversalit.  The  theory  of 
perspective  ia  coniidered;  this  includes  the  art  of  reprenentinf; 
in  twit  dimenNions  Rwimetrienl  ohjeets  which  are  of  three  , 
dinienmon!!,  a  prolilem  which  Mon^  uMially  mlved  Ity  the  aid 
of  two  diaj;rams,  one  IjeinR  the  plan  and  the  other  the  eleva- 
tion. Mimge  also  discussed  the  qunRlion  ns  to  whether,  if  in 
Molvinf;  a  problem  certain  subsidiary  quniiiitieo  intriMlnced  t« 
facilitaU^  the  solution  liecomo  imni^nnry,  the  validity  of  the 
Molnlion  is  therehy  impnired,  and  he  shewed  tliat  the  rmnit 
wonid  not  Ik-  aHectw).  On  the  re^itomlion  Ike  was  d^irived 
of  his  "ffice«  and  hoTioiTrs,  a  degmdntioii  which  preyed  on  hii 
mind  and  which  he  <tid  not  Inn);  survive. 

Mmt  of  his  miscellnneiiuH  juipers  are  embodied  in  bin 
worlts  Ap/ilimtioH  th  ral-jfhr*  A  In  gimnHri'  published  in  1805, 
and  Appltentitta  da  Faiuifi/m!  it  In  gf&melrir,  the  fmirth  edition 
of  which,  pablished  in  1819,  was  revise*)  by  him  jort  before 
hia  death.  It  cnntainit  among  other  results  hi*  M»l«th>o  of  k 
pMiisI  dificrentinl  eqnatinn  of  the  wcmnA  enAffv. 


488  CABNOT.     POHCBLCT. 

Camoi^  Laxan  NiekoUu  MargueriU  Cmrm^  bom  at 
Nolay  ou  Hay  13, 1753^  and  died  at  Magdebnig  oo  A«g.  SS; 
1823,  was  educated  at  Bui*gundy,  and  obtained  a  coimiwinn 
ia  the  engineercoqpe  of  Cond^  Although  in  the  amy,  he 
continued  his  mathematical  studies  in  which  he  islt  great 
int^nrnt  His  first  work,  published  in  1784,  was  on  machines: 
it  contains  a  statement  which  forpshadows  the  principle  u( 
euer^  as  applied  to  a  falling  weight,  and  the  earliest  proof  of 
the  fact  tliat  kinetic  energy  is  lost  in  the  collision  of  im- 
perfectly elastic  Ijodies.  On  the  outbreak  of  the  revolution 
in  17H9  he  threw  himnelf  into  politics.  In  1793  he  was 
elected  on  the  committee  of  pulilic  safety,  and  the  victories 
of  the  French  army  were  largely  due  to  his  powers  of 
organization  and  enforcing  dincipline.  He  continued  to 
occupy  a  prominent  place  in  every  successive  form  of  govern- 
ment till  179G  when,  having  oppOHed  Napoleon's  eoap  tteUU^ 
he  had  to  Hy  from  France.  He  took  refuge  in  Geneva,  and 
there  in  1797  issued  his  La  tnHaphyniqnt  du  ealcul  infinitrttimal. 
In  1802  he  a-ssisted  Napoleon,  Imt  his  sincere  republican 
convictions  were  inconsistent  with  the  retention  of  ollice.  In 
1 H03  he  pruduced  his  GtouuirU  de  ^Hmtioti.  This  work  deals 
with  prujtH^tive  rather  than  descriptive  geometry,  it  also  con- 
tains an  elaborate  discussion  of  the  geometrical  meaning  of 
negative  rutits  of  an  algebraical  equation.  In  1814  he  offered 
his  services  ti*  tight  for  France,  though  not  for  the  empire; 
and  on  the  restoration  he  was  exilcnl. 

Poncelett.  J^nn  Victur  PonceJ^i^  born  at  Metz  on  July  1, 
1788,  and  died  at  Paris  on  Dec.  22,  1867,  held  a  commission 
in  the  French  engineers.  Having  been  made  a  prisoner  in  the 
Fn*ncli  n^trtMit  fi-uni  Moscow  in  1812  he  occupied  his  enforced 
leisure  by  writing  the  Traitt  de^  jiropriHeM  projeciives  det 
JiyureMf  published  in  1822,  which  was  long  one  of  the  best 

*  See  the  ilwje  by  Arsgo,  which,  like  most  obilusiy  noiiees,  is  a 
psmio^ric  rather  than  an  impsrtisl  biography. 

t  See  Im  vit  et  Ui  ouvntye*  de  Pouetlet  by  I.  Dadion  and  C. 
P^B,  1809. 


\ 


THE   DK\'ELOPllt\T  OF    PHTRlCfL  430 

known  text-buolcH  on  iiioJern  geuinetry.  iiy  nieMiH  of  pro- 
jectiun,  recipirxntinn,  and  honiolu;;<>us  li;-unit  he  eKtnblisheil 
all  the  chief  pnipcrtit^  of  conicn  nru)  ijuiHlric!*.  He  alxo  timl^ 
the  theory  of  puly^DnH.  Hix  tmLtixe  on  pmctical  mechjinica  in 
1826,  hin  memoir  on  wnt«r-milla  in  I82G,  ntiH  his  report  on 
the  English  machinery  snd  tools  exhilitteil  at  the  Intcmattonsl 
Exhilntion  held  in  I^nclon  in  11^51  deser\«  mention.  He 
contributed  numerous  articles  to  Crelle'*  journal ;  the  ntoot 
valuable  uf  Utcs*^  deal  with  tlin  explanation,  by  tlic  aid  ot  the 
doctrine  of  continuity,  uf  imaginary  solution!  in  geometrical  - 

TJie  tterelopnunt  of  nuithenmtieitl  jihyncK. 

It  will  Im!  noticed  that  I^grange,  I^place,  and  [.egendre 
muNtly  occupied  theniwlves  with  analysis,  geometry,  aitd  astro- 
nomy. 1  am  inclinMl  H>  regard  Cnuchy  and  the  Krench  mathe- 
mnticinns  of  the  present  day  as  belonging  in  «  different  Hcbanl 
of  thought  to  thiLkeiinsidered  in  this  chapter  and  I  place  them 
aniuiigHt  ntodeni  mathematicians,  Imt  I  tliink  tliat  Fourier, 
PoiwMn,  and  tlie  majority  of  their  contnmpomriesare  the  line»l 
BucceHsorH  of  I^tgraiige  and  I^place.  .If  this  liew  be  cnrrect,  it 
would  teem  that  the  hiter  memlwrt  of  the  French  school 
devoted  tliemnelvM  mainly.  ti>  the  npplicittiun  of  matlieaiatical 
analysis  to  physics.  Uefurc  cimsidrring  thew  niathematicians 
I  may  mention  the  distinguiKhcd  English  exfieriniental  physic- 
ists who  werctlieir  contempuraHe>i,  and  whose  nteritH  ha\-eonly 
recently  received  an  Mleijuate  recognitiim.  Chief  among  these 
are  Cavendish  an<l  Young. 

CftTendiflh*.  The  Honourable  llrari/  CartndiA  was  bom 
at  Nice  on  Oct.  10.  1731,  and  died  in  fj^ndon  on  Feb.  4,  1810. 
Hia  tastes  for  scientific  research  and  mathematics  were  formed 

■  An  aceooDt  of  bis  life  hj  0.  WilKon  will  be  bond  fai  (be  Snt 
ntDM«  of  Ibe  pafalicationa  of  the  CsTendUh  SociplT.  London,  1>V1.    ~ 
Khetrieal  Rnmirhrt  were  edited  b;  J.  C.  Kaivell,  sad  r  '"  ' 
•  in  187». 


410  0AVSNDI8H.     RUMFORa     TOUVa 

at  Cambridge,  wliere  he  resided  fatMu  1749  to  17ftS.  Heemled 
experiinental  electricity,  And  wm  one  of  the  earliest  writers  to 
treat  chemistry  as  ao  exact  science.  I  mention  him  hei^  on 
account  of  his  experiment  in  1798  to  determine  the  density  of 
the  earth,  by  estimating  its  attraction  as  compared  with  that 
of  two  given  lead  balls :  tlie  rrault  is  that  tlie  mean  density  of 
the  earth  is  about  fi\'e  and  a  half  times  that  of  water.  This 
experim(*ut  waM  carried  out  in  accordance  with  a  suggestion 
which  IumI  bei»n  Unit  made  by  John  Micliell  (1724-1793),  a 
fellow  of  QuecHH*  College,  Cambridge,  who  had  died  before  he 
was  able  to  carry  it  into  effect. 

Romford*.  iS'tV  Benjamin  T7unn$on^  CoiimI  Run^ord^  bom 
at  Concord  on  March  26,  1753,  and  died  at  Auteuil  on  Aug. 
21,  1815,  was  of  English  descent,  and  fought  on  the  side  of  the 
loyalists  in  the  American  War  of  secession :  on  the  conclusion 
of  peace  lie  settled  in  England,  but  subse<|uently  entered  the 
service  of  liavaria  where  liis  |iowers  of  organization  proved 
of  great  value  in  civil  as  well  as  military  affairs.  At  a  later 
periu'l  lie  again  resided  in  England,  and  when  there  founded 
the  Hoyal  Institution.  Tlie  majority .  of  his  papers  were 
coiniiiunicaUHl  to  the  Uoyal  Society  of  I^ondon  ;  of  these  the 
most  important  is  his  memoir  in  which  he  shewed  that  heat 
and   work  are  mutually  convertible.  > 

Yountf  t-  Among  the  must  eminent  p^iysicists  of  his  time 
was  Ttiunuut  Yottny,  who  was  born  at  Milyerton  on  June  13, 
177.'),  and  diinl  in  London  on  May  10,  1829.  He  seems  as  a 
boy  to  have  been  soiiiewliat  of  a  prodigy,  being  well  read  in 
modern  languages  and  literature  as  well  as  in  science ;  he 
always  k(*pt  up  his  literary  tastes  and  it  wiis  he  who  in  1819 
first  suggested  the  key  to  decipher  the  Egyptian  hiero- 
glyphics, which  J.  F.  Champollion  used  so  successfully.    Young 


*  An  edition  of  Rnmford'M  work*,  edited  by  Oeorge  Ellis, 
panied  by  a  Lioffraphy  waa  published  by  the  Xmericsii  Aeademy  of 
Hcivncrm  at  Uoi»toii  in  1872.  I 

f  Hi*  ciillt«t4*d  wotkt  aud  a  vueuxoVc  fm  V\>a  i^C«  were  published  by 
O.  i'tmsaek,  -I  %*ulumes,  Londoa,  \Hlk^.  \ 


\ 


\ 


TOi:!Ja.      DALTON.  *♦! 

WM  deslinni  to  )n>  n  ilixtor,  nml  nftor  nttendine  Irvturm  ftt 
EilinburRli  iincl  li.ittiii^n  c-n(rrcrl  nt  PjiiiimnuH  Cutlpj^, 
Caniliriil<::r,  fn>ni  which  ho  tunic  Iiih  ^c«nt'  in  17!)9;  And  U>  ■ 
his  stajr  at  the  niiirertity  lie  nttriliutnl  inoch  lif  htn  future 
dJHtinctinn.  His  mmlicnl  carper  »ns  not  pnrticiiUrly  «ic- 
crasful,  niict  his  fnvDurit<^  nrnxini  that  n  iiiPHicMl  (liM^mMia  is 
onlj  A  ImUnce  of  pnilmhilitit^  wns  not  nppmHntecl  hy  hin 
pittients  who  hmknl  fur  certjvinty  in  retnni  for  their  tec. 
Fnrtonatfly  hiH  private  tiicniiK  wpre  :iiiiple.  Several  pnpc™ 
contrihutnl  to  various  )r>Amed  Bocietira  ttom  1798  onwanlo 
prove  him  to  liave  lieen  »  iiiatheinnticinn  uf  considerable 
power;  hut  the  reseArchen  witich  have  iinniortAlucetl  hin  nAme 
AW  thow  by  which  he  Inid  down  t}ie  laws  of  interference  of 
WAVPH  and  of  liRht.  and  was  thuM  aWf  to  tHUggent  the  meAna  Iqr 
which  the  chief  ditficidties  then  felt  in  the  way  of  the  Accepts 
Ance  of  the  undulalory  theory  of  ligiit  could  lie  o\-ercome, 

Dalton*.  Anothpr  dtstinKuinhH  writer  of  the  iwtiiD 
period  was  Jnhn  DitJlnn,  who  wat  Uim  in  CamlieriAnd  on 
SopL  S,  176C,  and  tlit-d  nt  MaiieheHtor  on  JiiTy  27,  1(*44. 
DAlton  invnti^ated  the  (otiHinn  of  vapours,  And  the  law  of 
thp  expansion  nf  a  gns  under  chan'.'es  of  tempemtare.  Hn 
aim  fonndrd  thft  atomic  theorj-  in  clieniiNtry. 

It  will  he  gathcre<l  from  (liene  iioten  tliat  Uie  English 
school  of  physicists  at  the  Ix-^^niiiiiK  of  thii*  century  wero 
mostly  concerned  with  the  experimental  Hide  of  the  suhject. 
But  in  fact  n<i  sntisfactiiry  theory  could  be  formed  witliont 
some  similar  careful  determination  of  tlie  fiwts.  Hie  moat 
eminent  French  physicists  of  the  sanie  time  were  Fourier, 
PoisEon,  Ampere,  ami  Fresnel.  Their  method  of  trefttinj; 
the  subject  is  more  mathenintical  than  that  of  tlteir  BngliMh 
Gont^Tnponriea,  and  the  two  firnt  namni  wrre  distinjtaishMl 
for  general  msthentAtical  ability. 

■  8n  the  Urmoir  «/  IMtiim  Iir  R.  A.   Hmith,  hmAan,   IBM;  taA 
W.  C.  Hmi7'i  MCDMit  in  lbs  C'swWM  Soeittf  T 
IBM.  • 


448  FOUMER. 

Fsmlar*.  The  firat  of  thne  Fnnch  [^raieuta  vm  Jemi 
BaptiaU  Jottfik  FimriKr,  wlw  wan  born  at  Auserra  on  March 
31,  17CA,  Mtd  died  »t  Farix  on  May  16,  1830.  He  tu  Uw 
■un  of  A  tailor,  and  was  educatctl  by  the  BenedicUnea.  The 
ouuiiniiwions  in  thn  xcientific  ooqM  of  the  army  were,  aa  is 
still  tho  cane  in  RuHnia,  n-Her^ed  for  thfute  of  good  birth,  and 
IwinK  thuM  iiieligibltt  lie  nccfptml  a  military  lecturealiip  un 
matJitMoiitic-H.  He  took  a  prominent  part  in  hiH  own  district 
in  prtiniotiuK  the  revniitliuii,  and  was  rewordMl  by  on  ap 
pointnient  in  1795  in  thtr  Noruinl  hcIiouI,  and  iiubsM|uently  by 
a  chair  at  the  Pulyti>c'lini('  mIiuijI, 

Fourier  wi>nt  with  Nupuleun  on  hia  eaHt«m  expedition  in 
I70K,  and  was  nwih'  governor  uf  I^wer  EgypL  Cut  off  from 
Fruiicf-  hy  the  English  H<-i-t,  ho  urgnniu-d  the  workiihupH  on 
which  t)t»  Fr«ncb  army  hiul  to  rely  for  their  nmnitioiui  of 
war.  He  alw  curitributj^l  Heceral  mathematical  pnpers  tu  the 
E^ypliiui  iHHtituti-  which  Nupoh-on  founded  at  Cuiru  with  a 
rifw  of  wfukeiiing  Knj^liHh  intlui-ncc  in  the  EoHt.  AfUr  tlie 
Ilritixh  victimcN  anil  the  (iipitulutiun  iif  tlii^  Ft-pncb  under 
(ifiu-ral  Menou  in  1S01,  Fnurii-r  rrturniHl  to  France  and  waa 
niiwl«  pn-f<'<'t  of  Crciiolile.  and  it  was  while  tht-re  tliat  lie 
Miiule  his  i>x|icriiiu'nt.s  un  the  propitiation  of  heal.  He  moved 
to  I'nriH  in  }H\C,.  In  ]S3:2  hi-  puhli^hnl  hia  rkiittv  amtlffti^ut 
ilr  III  vhidrur,  in  whiL-h  be  Xmivn  hiH  n'liHiiiking  un  Nt-wton'a 
l»w  of  iiHilinf;,  miiiit'ly,  that  the  lluw  of  heat  between  two 
oiljaiH-nt  moln-uh-s  is  |iro{)i>rtiiinal  to  (lie  infinitely  Hmull 
dill'en'ncf  of  their  li'iuiHTuturett.  Tn  thia  work  he  sliewH  tliat 
any  fuiie^nn  of  a  vuriidile,  whether  continuous  iir  diacon- 
tinuourt,  eitn  1m>  expundcd  in  a  aeries  of  nines  of  multiples  of 
the  variable ;  a  result  which  ih  cunstantty  useil  in  modern 
nnalysia.  fjigrange  bad  fjiven  particular  cases  of  tlio  thiiiivui 
and  had  implietl  tbut  th<-  method  was  general,  but  he  liad  nut 
puntui-<l  the  Kuhject.  Diriehlet  was  the  first  to  give  a  satis- 
factory dt-nioiiKtration  uf  it. 

'  An  aiiiliiHi  of  hia  works,  editfd  h]  U.  TMAonv.  «•»  vnhliihal  iu 
S  vuluuMM,  J'aiia.  ItttM.  In'Jli. 


FOURIER.      SADI  CARNOT.      pftlSRON.  443 

Ponrier  left  nn  uiiRniNliecI  wnrk  on  dcUTtnirmt«  fqaaliotis 
which  wjw  ttlitnl  \>y  Nnvipr,  nnd  pul>liHbe<I  in  1851 ;  thin 
cnnlniiiH  iHucli  orifpnnl  nmtt^^r,  in  jmrticuliir  there  in  it  <lr>ninn- 
stmtiun  of  Fonricr'n  tliouniii  on  tlie  pmitiitn  of  the  nntH  of 
An  nlgi^braicAl  cquntion.  Iwigrnnfre  hiwl  Hhewn  how  the  rootn 
of  nn  iilgrbrnicnl  njunlion  might  Im'  Efpnrntotl  by  mrana  of 
Anothrr  rtjuntion  wh(Hi>  rooU  wcrn  Ihc  squAfM  of  thr  ilifTpr- 
encvs  of  the  niot.s  of  (hn  originnl  cH|uiition.  BwUn,  in  If07 
nnd  181 1,  had  i-nuncinti-d  tiw  l)im)n>iii  gpnentllf  known  by 
the  nnme  of  Fourier,  but  th«  (lemanKtrntion  wm  not  idtof^her 
HntiKfitctory.  Fourier's  pniof  i«  the  Bnnie  an  th«t  untiAlly  gi^'en 
in  tcxt-bnoks  on  tlie  theory  of  rfjuntions.  Hie  finni  anlotion 
of  the  prultleni  wan  given  in  1 829  by  Jacques  Ch«rlM  Francois 
Stunn  (I80.V185?i), 

Sadi  Cnmot*.  Among  Fourier'n  cont^mponirip*  who 
were  inlcreHliil  in  Uh*  theory  of  bent  the  mcwt  <-mim>nt  wiw 
•S'fb/i  Caruot,  n  Hon  of  tlir  eminent  geonietrician  mentioiind 
nlM>ve.  8wll  Carnot  wn.i  imrn  nt  Vans  in  17%,  nnd  died 
there  of  diolem  in  Aiiguit.  1833;  he  wiwi  »n  officer  in  tb« 
Preiicli  am)y.  In  I8J-I  hf  iKHunl  n  short  work  entitled 
Jtr/U-rwHH  mir  la  fniintniicf  tnolrif  <hi  /»»  in  which  ho 
nttempted  to  (lel«rniinr  in  wimt  wny  he*i  produced  itit 
mccluiniml  rllect.  He  mndc  Ihe  mistAke  of  mMumiiig  tbnt 
hmt  WAS  innteriAl,  but  bin  emtny  mny  Iw  tflkcn  nit  initinting 
the  modem  theory  of  thermfNlynniiiics. 

Poiasont.  Simfnn  D'.Hi»  /'oi^oit,  1<om  at  Pilhi%-iera  on 
June  21,  1781,  And  dinl  nt  Pnriit  on  April  '2^,  1840,  in  nlmont 
eqtinlly  di)itin;^iiihed  for  Un  ApplicntionH  of  innthemntics  to 
mechnnicM  nnil  to  pliyMict.  His  fnlher  hnd  been  n  prJvAtA 
mktier,  nnd  on  bin  retirement  wai  given  tumn  nmnll  adminin- 

*  A  Kketch  of  hii  life  and  an  Enuliih  transUtion  of  his  K/JUrfmm  wn* 
pab)iii>icd  bj  H.  H.  Thnnlon.  London  and  New  York,  IWNX 

t  Mrnioin  nf  Poiiwon  will  im  Tunnd  in  lh«  KnrgeUfMtdim  Dn'laanJM, 
the  Tmnitarliniu  of  th,  Rognt  Aitroimmieal  Sorirt^,  ToL  V,  aad  Ango'a 
ilofrt,  vd.  n :  Dm  latter  eontaino  a  htUioRraphj  nf  Pnlaaoa**  paptn  aaal 


i44  poisaoii. 

traUve  pcMt  in  his  native  vWmgp :  whan  the  nvolvlfaMi  hnkm 
out  he  appeMw  to  have  MMimed  the  goveminent  of  the  pbee, 
and,  being  left  undisturbed,  beenme  a  penon  of  ■one  loeal 
iiuportanoe.  The  boy  was  put  out  to  nune^  and  he  need  to 
tell  how  one  day  hit  father,  coming  to  eee  him,  found  that  the 
nurne  hod  gone  out,  on  pleasure  bent,  haling  left  him  sus- 
pended by  a  Hiiiall  ourd  attached  to  a  nail  fixed  in  the  walL 
Thin,  slie  explained,  was  a  neceiisary  precaution  to  prevent 
him  from  perishing  under  the  teeth  of  the  various  animals 
and  aniumlculae  that  roomed  on  tlie  floor.  Poimon  used  to 
add  that  hin  gymnastic  effortM  carried  him  incessantly  from 
one  side  to  the  other,  and  it  was  thus  in  his  tenderest  infancy 
tluit  he  commenced  those  studieH  on  tlie  pendulum  that  were 
to  occupy  so  large  a  part  of  his  mature  age. 

He  was  cnlucatiHl  by  Ium  father,  and  destined  much  against 
his  will  U»  lie  a  doct4»r.  His  uncle  tiffered  to  teach  him  the  art; 
and  liegau  by  making  liim  prick  the  veins  of  cabliage-leaves 
with  a  laiict*t.  When  pt^rfect  in  tliiM,  he  was  allowed  to 
put  f»n  blisters;  but  in  aliiio^tt  the  first  case  he  did  this  by 
hims4*lf,  the  {Mitieut  dietl  in  a  few  hours,  and  though  all  the 
iiHHlicnl  pnictitioners  of  the  place  ii.Hsun'd  him  that  *'the  event 
was  a  very  common  one"  he  vowed  he  would  lm%'e  nothing 
nion*  to  do  with  the  profi^Hsion. 

Poisson,  on  his  return  home  after  this  adventure,  dis- 
covered amongst  the  otticial  papers  sent  to  his  father  a  copy 
of  the  questions  s4*t  at  the  Polytt*clinic  scluiol,  and  at  once 
fount!  his  canvr.  At  the  age  of  seventeen  he  enten-d  tlie 
Polytechnic,  and  his  abilities  excited  the  interest  of  Lagrange 
and  Liipli'ice  whose  friendship  he  retaine<l  to  the  end  of  their 
lives.  A  memoir  on  finite  difierences  which  he  wrote  when 
only  eighteen  was  reporttnl  on  so  favourably  by  Legendre  that 
it  was  onlered  Ut  lie  published  in  the  Reeueil  lieg  aavatUs 
/ttaHyettt,  As  soon  as  he  liad  finished  his  course  he  was  made 
a  lecturer  at  the  school,  mid  ho  continued  through  his  life  to 
hold  various  government  scientiHc  |iost.s  and  profftHsorKliips. 
lie   wwi   sumewliat   of    a   mjcVaXv^V;  \x\vA    ti^uxaiued    a   rigM 


PoissoN.  445 

repuhlicAii  till  IS15  when,  with  n  view  to  nmbiiig  nnother 
pmpirv  imjiiw«il)lt-,  Iw  joinol  tlip  IcfptinHsl.s.  Ilo  Uwilt  Imiw- 
ever  no  Mctivp  pnrt  in  politicn,  an<1  miulc  the  Ktucly  of 
nintlinnnticH  hiJi  nnrnKCiiiriit  n»  wril  iu<  hiN  huBtlirox. 

Hin  worknnnil  nirtiHiin  bi-r  lietwn-n  tlinvaml  tivar  hundrT«l 
ia  nuiiihpr.  Tlit>  chief  trratiiicii  whicli  he  wrote  were  hit  Tmili 
dr  iitrrimiqw' fjiu\>\i*\m\  m  two  volunieN,  1811  itiid  It^.tS,  which 
WAN  long  n  KUtndanl  work  ;  Itin  Throrii'  nanvrll'.  df.  Fartiim 
eaiiittair',  183]  :  h\%  Tkiorif  mnthemntiqii*  dt  In  rhnlmr, 
1835,  t€)  which  n  Hupjih'ment  wiut  niltlt-tl  in  18.17;  mhI  hid 
Pffh^nlir-s  mir  tn  jir'Jinhilili  */»■»  jwjfm'-nU,  1837.  He  IimI 
intcmlnl,  if  he  hntt  lived,  to  write  n  work  which  Khoald  onver 
nil  nintheninticnl  phyKiCH  nnd  in  which  Ihf  n>><nltc  of  the  throe 
bookii  IakI  nntiied  would  have  lN<en  inctiqiornted. 

Of  hii  mcmnin  on  t)ie  Huliject  of  |iure  nMlheiimtica  the 
tiKKt  iniportAnt  nrc  thoso  on  definite  integwK  ami  Foarier'H 
wirirtt,  their  npplicntion  tn  phj-sicul  pnililemK  iiHiHlitating  one 
of  his  chief  clnimR  to  diHtinctiim ;  hii  amny  on  tl»e  caIcuIuk 
of  varialinnii;  and  hin  pH[KrH  on  the  prolnliiiity  uf  the  nicnn 
results  of  olKer^'nti'innt. 

PerhapH  the  moxt  reinnrknhln  of  liiH  mi'nioira  in  npplied 
ntnUiemnl irs  are  thivv  <m  the  Ihexry  nf  iilectnMtntioi  tuid 
mngiietixni,  which  oHgiiintnl  n  nnw  liranch  of  nrnthenwticitl 
phj-sics:  he  snpp<«nl  that  the  rosults  were  due  tn  the 
attractioiiH  nn<l  repulsimw  of  iniptndt-ndilc  pnrticle*.  The 
inoHt  important  of  thime  on  physical  astroiiiimy  are  the  two 
read  in  1806  ([)rint«d  in  1809)  un  the  secular  inoqualities  of 

*  AmoDR  raiii«i>ii's  conlemporBrii's  who  iludied  niccbuiiea  unl  of 
«fai>M  worki  Iw  nude  one  I  nutj  mention  Limit  roinmil,  who  ww  lioni 
In  PiuiB  oa  Jan.  3,  ITTT,  uhI  died  then  on  Dec  5.  lHa>.  In  hii  Klnlffvc 
twbliBbnl  in  1M)3  he  trcatr*!  thn  iiubjrcl  withont  au;  axpljoit  wfcreaca 
^o  dTnamic :  the  tlitoiy  of  couples  ia  largely  due  to  him  (1M6),  aa  also 
Uia  notion  ot  a  bodj  in  npnce  noder  Iha  aclion  of  no  fonaa. 

t  Sn\htJ<ninialpalfifrhnique  from  1813  to  IWM.  and  Om  llfmajm 
4t  CafiuUmit  lor  IfSS;  Iha  iltmoin.  4t  Vicaitmit,  18SI|  aai  Um 
Comai-ititei  if  Uwtpi,  IB3T  ao^  followios  foan.  Moat  o(  hk  wdBJii 
~        n  Um  three  penodicali  ben  mentiDnaL 


446  P01880K.     AMPlaiK.     PEBNIL.     BIOT. 

the  meui  motiopii  of  the  pUnetii  and  on  the  varintion  of 
arUtnuy  oonsianU  introduoed  into  the  aoiatioiDii  of  qneetione 
on  mechanics ;  in  theiie  Pbiiaon  diaouaeeB  the  qoesUon  of  the 
utability  of  the  planetary  orbitt  (which  Lagfmnge  had  alraady 
proved  to  the  tint  degree  uf  approximation  for  the  diatorhing 
liMTceii),  and  ahews  that  the  reault  can  be  extended  to  the  third 
order  of  amall  quantitlea :  theae  were  the  memoini  which  led 
to  Lugmiigo'a  laiiioua  memoir  of  1808.  Pbiaaon  alao  publiahed 
a  paper  in  1821  on  the  libration  of  tlie  moon ;  and  another  in 
1827  on  the  motion  of  the  earth  about  its  centre  of  gravity. 
Ilia  moat  inipurtunt  niemoira  on  the  tlieory  of  attraction  are 
one  in  1829  on  the  attraction  of  aplieroida,  and  anotlier  in 
1835  on  the  attniction  of  a  huiii<igent*ou8  ellipsoid:  the 
aubatitutiun  of  tlie  correct  equation  involving  the  potential, 
namely,  V^K^-4a'p,  for  Laplacii'a  form  of  it,  T^K^O,  waa 
tii-st  publiahed^  in  1813.  Lastly  I  may  mention  hia  memoir 
in  1825  on  the  theory  of  wavea. 

Ampere t*  Andre  Marie  Ampere  waa  bom  at  Lyona  on 
January  22,  1775,  and  died  at  Maraeillea  on  June  10,  1836. 
He  was  widely  read  in  all  brancliea  of  lemming,  and  lectured 
and  wrote  on  many  of  theui,  but  after  the  year  1809,  when  he 
waa  made  professor  of  analysis  at  tlie  Polytechnic  school  in 
Paris,  he  contincnl  himself  almost  entirely  to  mathematics  and 
acieiice.  Ilia  pJiiM^rs  on  the  connection  bi*tween  electricity 
and  magnetism  were  written  in  1820.  According  to  hia 
theory,  propimncliKi  in  1820,  a  molecule  of  umtter  which  cun 
be  magnet izchI  ia  traversed  by  a  clost^d  electric  current,  and 
magnetization  is  produced  by  any  cause  which  makes  the 
direction  of  theses  curri*nts  in  the  different  molecules  of  the 
iKxIy  appruiicb  parallelism. 

Fresnel.  Blot.  A  uyufUin  Jean  Frtgnei^  born  at  Broglie 
on  ^lay  10,  1788,  and  died  at  VilleHl'Avruy  on  July  14, 
1827,  was  a  civil  engineer  by  profession,  but  he  devoted  hia 
leisure  to  the  study  of  physical  optics.    The  undulatory  theory 

*  In  the  Bulletin  det  scifuces  of  the  SocUti  philomatiqne. 

t  See  C.  A.  Valaon's  Elude  tur  la  vie  et  lee cmvragte  d*Jmpiret  Ljona,  1885. 


V 


FRESXEL.      UiriT.     ARAGO.  U7 

(if  li{;1it,  whicti  Hookr,  Huygcns  nnri  Enirr  hiul  BupportcH  (Hi 
li  priori  grounds,  hnd  Uvii  IhvuI  oh  rxperiiwcnt  liy  the 
rexenrches  nt  Young.  Frosnpl  dniucoil  tlic  niathrin«tic«I 
conspqupiici^  of  these  exporiiiii-ntM,  iind  explained  the  phnio- 
mena  of  interfirrnre  Ixilh  of  onliiiniy  nnil  polMrianl  light 
FrcKtiplH  fricnil  and  coliti'iiiporary,  J'lm  Htplitle  Itial,  who 
was  Ifom  nt  Pnri.s  on  April  21,  ITU,  and  dird  tlwre  in 
U*62,  rrquirt^  «  woitl  or  two  in  passing.  Mi«t  iA  his 
ntftlhemntical  work  wna  in  cnnnr-clioii  will)  the  Huliji-ct  of 
optics  And  Mpcciftlly  tho  polnrizHtion  of  light.  Hiii  syHtenMtic 
works  worp  pmduced  within  thr  yennt  lfi05  iind  1817;  » 
■election  of  IiIb  tnorv  vMlualile  memoirs  wm  pnblhthed  in  PtuiB 
in  IB.^8. 

AragO*.  fntnqoig  Jran  Douihiiqrre  Af^'jo  wsh  bom  at 
KKtagel  in  thn  Fyrrn«w  on  Fih.  L'fl,  \'t>^^,  imt\  dird  in  Fkrin 
on  Oct.  2,  1853.  He  wa'4  olucatnl  nt  the  pDlyl«chnic  sohool, 
Purii,  nnd  we  giilhcr  inim  Va*  nutoliiography  tlint  Imwevor 
distinguished  were  the  pnifcssnre  nf  thitt  inKtitutJon  they  were 
rem&rknbly  jncnpnlile  of  inipftrting  their  knowledge  or  nuun- 
tninlng  <li<tcipline. 

In  I^^OI  Ar&gn  wu  tnnde  secrvtary  to  the  oliserviiloty  nt 
Paris,  nnd  from  18()6  to  1^^09  he  was  engagi-<l  in  meaKuring  » 
meridinn  Arc  in  onler  to  detifniiiiic  the  cxnet  length  nf  » 
metre,  lie  wan  then  apjiointi^l  to  a  leading  pmt  in  the 
uhnervatory,  given  n  res)deiie«-  tlierv,  and  niftdc  a  prufeNHor  At 
the  Polyterhnic  Rchool,  where  he  enjoyed  A  inArknl  kucgcss 
as  a  lecturer.  Ho  HulKtefjuenlly  gave  )io|iaUr  lectnm  on 
ANtronomy  %hict»  were  l>oth  lucid  and  accurate,  a  coinlitMlion 
of  qualities  which  was  mrer  then  than  now.  He  reoi^nited 
the  national  olKcr^atory,  the  nianHgenient  of  which  lisd  long 
been  inefficient,  but  in  doing  thin  Ium  want  of  tMct  And  caBrtoty 
raiacd    many   annecewiary   difficulties.     He  ntnainGd   to  the 

*  Ango't  wOTk%  vhich  ineladc  (logn  «n  man;  of  Um  iMdioit  —tht 
malioBni  of  the  laat  fin  or  ail  tentane*.  b»te  bMn  ctlllid  faf  IL  J,  A. 
BwTnl  and  puUlshcd  in  foartflen  toIdidm,  Pari*,  18H-T.  !■  mMo- 
bia||Tapii7  is  pceflied  t«  th*  6nt  toIiudo. 


MS  ARAOa 

end  m  oonsiiiieiit  itsfiablicaii,  and  tAUst  the  em^  ttHtU  of  18S1^ 
though  half  bUnd  and  dying,  he  raugned  his  punt  as  aatranoner 
rather  than  take  the  oath  uf  allegiance.  It  is  to  the  eiedit  of 
Napoleon  IIL  that  he  gave  directiona  that  the  oM  man  ■bonld 
be  in  no  way  disturbed,  and  slmuld  be  left  free  to  lay  and  do 
what  he  liked. 

Arago'H  earlicHt  ph}'8ical  researches  were  on  the  pressure 
of  steaui  at  different  teiuperatureM,  and  the  velocity  of  soand, 
1818  to  1822.  His  magnetic  observatioiui  mostly  took  place 
from  1823  to  1826.  He  discovered  what  has  been  called 
rotatory  maguetiHm,  and  the  fact  that  must  bodies  could  he 
magnetized :  thene  dii)co\'erit's  were  completed  and  explained 
by  Faraday.  He  warmly  MUpportetl  Fresiiel's  optical  theories^ 
and  the  two  pliilotiuphers  conducted  tugetlier  those  experi- 
ments on  the  iMilurization  of  light  which  led  to  the  inference 
that  the  viliratioiiH  of  the  luminiferoun  ether  were  traiutvene 
til  the  diiYctioii  of  motion,  and  that  polarization  comusted  iu 
a  resolution  of  n*ctilint*ar  motion  into  components  at  right 
angles  to  ciich  other.  The  siibiietiuent  invention  of  the 
polariscope  and  discovery  of  rotatory  polarization  are  due  ta 
Arago.  The  general  idea  of  the  experimenUU  determination 
of  the  velocity  of  light  in  the  manner  8ub8(H|uently  effected 
by  Fizeau  and  Fouc*iiult  was  HUggesU^d  by  him  in  1838,  but 
his  failing  eyt^sight  prevented  his  arranging  the  details  or 
making  the  experiments. 

It  will  lie  noticinl  tliat  some  of  the  hist  members  of  the 
French  school  were  alive  at  a  comparatively  recent  date,  but 
nearly  all  their  niathematicid  work  was  done  liefore  the  year 
1830.  They  are  the  direct  succ-essors  of  the  French  writers 
who  llourishe<l  at  the  commencement  of  the  nineteenth  cen- 
tury, and  seem  to  have  Imhsu  out  of  touch  with  the  great 
German  mathematicians  of  the  early  part  of  it  on  whose 
researches  nmch  of  the  liest  work  of  the  century  is  based ; 
they  arc  thus  phiced  here  though  their  writings  are  in  some 
canes  of  a  later  date  than  those  of  (jauss,  Abel,  Jacobi,  and 
other  ijjatliematicians  of  reccut  times. 


\ 


\ 


IVORT.      THK  CAMBItlDOE   ANALYTICAL  HCHOOL.      449 

The  iiiti-odiiclion  of  ttnul'jxU  info  Kiiglamf.    ■ 

Tlir  coiiipMe  iiuiUtinn  of  llic  Kii(;lis|i  Mlinail  ftnd  ilii 
Hcrntiim  tn  ;;mii)(-triml  iiicI)iix!n  an-  l\w  mint  limrlcnl  fratarat 
in  iU  hwtiiry  tiurin);  ttip  lattvr  hiilf  of  tito  oiglitwntli  oiitury  ; 
unil  the  uliwnce  «f  nny  ci>n*i(lpnil>Ic  coiitriliution  li>  U»« 
advnnn'mciit  <if  inntlicnintiml  wiener  vns  a  nntnrnl  cnn- 
wtjupiiw.  Oni!  fVHult  <>f  tliiH  win  tliftt  tin?  etirrjgr  of  Gnglifh 
mfii  of  wioncc  wnn  Inrgety  iIgvoU->I  (o  pnu-tirAl  iJiyaicfi  and 
pmclicat  RKtronumy,  which  wrre  in  ciiiisrtiiKiicc  ntadird  in 
AritAin  pnrhn]Hi  more  thnii  plHewhrir. 

Ivory.  AlmoHt  the  only  Knj;li!>li  ninthrmMicifui  itt  tlic 
Irgiiiniii;!;  of  tliiH  century  wlio  usnl  nnnlyticnl  ntrthodit  nnd 
whose  work  requirrw  mention  here  is  Ivory,  lo  whom  tho  crle- 
Imted  Ihetirrm  in  nttractinnn  in  due.  S!r  Jnm^»  /mry  wnn 
liom  in  Dundee  in  1765,  nnd  dimi  on  Kfpt  21,  1K43.  Aft«r 
gnuluAtiitg  Kt  St  Andrews  }ic  hccnmr  the  iimniiging  pKrtnrr 
in  A  flax-Kpinning  cumpnny  in  K»rfnrshin>,  liut  continued  to 
drvote  RiORt  of  his  leisure  to  mnthenmlieK.  In  1804  hn  was 
ninde  profi-swr  at  the  Royal  Mililnry  Cdlege  at  ?Iariow, 
which  was  sultnequently  moved  lo  Snnclliiirst ;  lie  wan  knighted 
in  1831.  He  contrilnttefi  nunieruui  impcnt  to  the  PhUo- 
fOfJiiml  TraHMfi-ti"!".  the  rnnst  rcmnrknble  lietng  tlnwe  on 
■ttractioiii^  In  one  of  lliew,  in  ]t*0%  he  "hewed  how  the 
attraction  of  a  homogeneoUH  ellipsoirl  im  nil  external  point  is 
a  multiple  of  thnt  of  nnother  elligKoid  on  an  intemnl  piint : 
tire  latter  can  )>e  easily  olitainrd.  He  criticised  Laplace's 
solution  of  tli«  method  of  leant  w|uai-en  with  annec«xM»7 
hitt<;mcHs,  and  in  tennN  which  shewed  that  he  had  failed  to 
nndcT^tand  iL 

The  Cambridge  AnalTUool  School.  Towarda  the  clow 
of  t)»e  laiit  century  the  nrore  thoughtful  mcmlien  iif  tlie 
Cambridge  school  of  mntltemntics  began  to  reongniise  that  their 
violation  from  their  conlinentnl  omtemporaries  was  a  aerioas 
evil.  The  earlint  attempt  in  England  to  expUin  the  notaUon 
and  methods  of  the  calculus  as  used  on  the  contitimit  waa  Aa^ 

.  ■» 


450    KlfiB  OP  TUB  CAMBKIDUB  ANALTTICU.  BCIIOOIi^ 


t4>  Woodhumiey  who  slandM  out  as  the  apostle  of  the  new 
nient.  It  in  ckmbtful  if  he  could  have  hnnight  the  analjftaeel 
luetbocU  into  v^gue  by  hiuuielf ;  but  hiai  views  weie  entluwi- 
Mftically  adopted  by  tliree  stodentfli  Peeoock,  Rebbege^  and 
Heracliely  who  succeeded  in  ceriyiug  out  the  reforms  he  had 
suggested.  In  a  Imuk  which  will  fall  into  the  hands  of  few 
but  EiigliKli  mulers  I  nuiy  be  pardoned  fur  making  qiaoe 
for  a  few  remarks  on  these  four  mathematicians*.  The 
original  stuiiulus  came  from  French  sources  and  I  therefore 
place  tliese  reiiuirkh  at  tlie  cIohc  of  my  account  of  the  French 
school,  but  I  should  odd  that  the  Englisli  mathenuiticians  of 
this  century  at  once  struck  out  a  line  independent  of  their 
French  contemporaries. 

Woodhooae.  Boberi  WixMouse  was  bom  at  Norwich  on 
April  28,  1773  ;  wiim  educated  ut  Caius  College,  Camliridge,  of 
which  society  he  wus  suljM.H|Ucntly  a  fellow  ;  was  Flumian  pro- 
fessor in  the  university ;  and  continued  to  live  ut  Cambridge 
till  hiH  death  «»n  December  23,   1827. 

Woodliou.s«*H  earliest  work,  entitled  the  Principles  q/* 
Analfftictit  CatctUuiiun,  was  published  at  Cambridge  in  1803. 
In  this  he  explaincHl  the  differential  notation  luid  strongly 
pressed  the  employ  men  t  of  it,  but  he  severely  criticized  tlie 
nietli«Mls  ustnl  by  c*ontinentjd  writers,  and  their  constant 
asKuniptioii  of  nonevideut  principles.  This  was  followed  in 
1809  by  a  trigonometry  (plane  auul  spherical),  and  in  1810 
by  a  liistoricad  tivatis^^  4in  the  calculus  of  variations  and 
is4i{ieriii)etrical  pixiblems.  He  next  producetl  lui  astronomy ;  of 
which  the  tirst  liouk  (usually  liuund  in  two  volumes)  on  practical 
and  descriptive  astrououiy  was  i^suetl  in  1812,  and  the  second 
liouk,  containing  an  aicamnt  4»f  the  treatment  of  physical 
iLstrunoniy  by  Laplace  and  other  c*ontinental  writers,  was 
iiisued  in  1818.  All  these  works  deal  criticadly  with  the 
scientiHc  foundation  of  the  subjects  oinsidered — a  point  which 
is  m»t  unfre4|uently  neglected  in  modem  text-book& 

*  The  following  account  is  oondeniad  from  my  History  of  ike  Stmdg 
qf'J/a/Atmatics  ut  Cambridge^  CamYktV^A^^* 


\ 


RISE  OP  THK  CAMRRIIXiE   ANALTTICAL  SCHOOL.     451 

A  nwfi  like  WfrndhnoNC,  of  xcrupulims  Ixmnur,  aniTermll; 
rcspectml,  a  tmined  logician,  nn<l  with  n  caustic  wit,  wu  wpll 
fittc*!  U>  inlrodaco  a  new  ij-stem ;  mh)  the  fact  thnl  when  Ik 
firMt  callcil  attention  lii  thn  continc-nlnl  niinlj*MH,  he  oxpoxrd 
the  unsoundness  of  snnic  of  the  usual  mcthfxla  of  c«tnl>liHhin); 
it,  more  likn  An  i)p[Mincnl  thnn  a  pnrtiain,  wiw  ax  pilitie  m  it 
wait  lionest.  Woodhousc)  did  not  Fxrrcisn  much  influence  on 
Iho  majority  of  his  contflniporari™,  and  th"  movement  might 
havo  diij  away  for  the  time  licing,  if  it  had  not  been  for  tlic 
advocacy  "f  Pr-acocic,  ilnWmfp-,  and  Herschel,  who  formed  an 
Analytical  Society,  with  the  object  of  advocating  the  gencml 
UGo  in  the  university  of  annlj-ticnl  methodH  and  of  tite  diflc- 
rcntinl  notiilion. 

Peacock.  G'/ynje  I'mrittk,  who  wac  the  most  infltienlial  of 
the  early  niendK-rn  of  the  new  school,  was  Ikmti  at  I)enlan  on 
April  9,  1791.  He  was  nlucated  at  Trinity  College,  Caiu- 
bridge,  of  which  society  he  was  NulHequcntly  a  fellow  and 
tutor.  The  eHtablishment  of  tlic  nniversity  obaenittory  was 
mainly  due  to  hi.n  efforLi,  and  in  1836  lie  wn«  appmnted  to  the 
Lowndcnn  profcMisoi^hip  of  astronomy  and  gomietry.  In  1839 
he  wai  made  ilean  of  Ely,  and  rcHidetl  there  till  his  death  on 
lfo\.  »,  I8.j8.  Althougli  Pencnck'ii  inttnenoe  on  Engliwh 
niatltcmaticians  wiu  considendilc  he  lin-t  left  but  few  me- 
moriuln  of  his  work ;  but  I  may  note  tliitt  Iiih  report  on  recent 
progress  in  analysis,  1633,  commenced  thosi-valDaldcHummaries 
of  Bcietilific  progress  which  enrich  many  of  the  annual  volume* 
of  the  Tran»nclion»  of  the  llritish  A.-»ii>«^'i;ttiun. 

Babbage.  Another  impurtnnt  mcmlierof  tlx!  AiMlytiad 
8ociety  was  Cfiarlt*  /i'thbuyr,  who  was  Imm  al  Tutnea  on 
Dec.  26,  1792;  Ite  «ntereil  at  Trinity  College,  Cambridge,  in 
1810;  BubNe<inently  liecame  Lucasinn  prufetMtr  in  the  nniver- 
mty;  and  died  in  London  on  Oct.  ]><,  ]>>1\.  It  wan  be  who 
gave  the  name  to  the  An>l)-ticsl  Society,  wliick  be  stated  was 
formed  to  advoeale  "  tlie  principtex  of  pure  t^iun  «8  oppoMd 
to  the  dol-nge  of  tbe  nniveraitj."  In  1820  the  Astronomiod 
Society  waa  fonnded  mainly  through  hia  efiurt^  and  at  a  latar 


452    KIKB  OP  TUB  CAMBRIDQB  ANALYTICAL  KHOCM* 


time,  1830  to  1832,  ha  took  m  promiiieiii  put  in  the  ftMUMbtSon 
of  the  British  AjiHOciation.  He  will  be  rimenifaerad  ibr  his 
luatheniatical  memuani  on  the  oalculus  of  fmictioiiiii  and  hki 
inveution  uf  «n  analytical  machine  which  ooakl  not  only 
perfonn  the  ordinary  pnioeiiaes  of  arithnietij:  bat  eoukl  tabu- 
late the  values  of  any  function  and  print  the  resultii. 

HanohaL  The  third  of  tlmeie  who  |  lielped  to  bring 
analytical  luethodt  into  general  use  in  England  was  tlie  son 
uf  Sir  William  Herschel  (1738-1 822),  th^  most  illustrioiis 
astronomer  of  tlie  latter  lialf  of  tlie  lastj  century  and  the 
creator  of  modem  stellar  astronomy.  Sir  John  Frederick 
Willtatn  Ihrtchel  wa»i  boni  on  March  7,  1792,  educated 
at  8t  John's  College,  Cambridge,  and  died  on  May  11, 
1871.  His  earliest  original  work  was  a  paper  on  Cotes's 
theoi*c*iu,  and  it  was  followed  by  others  on  mathematical 
auiilysis,  hut  Ills  clesire  to  complete  his  father's  work  led  ulti- 
maU*ly  to  his  taking  up  nstronumy.  His  papers  on  light  and 
astronomy  contain  a  clear  exposition  of  the  principles  which 
underlie  the  umtlicniatical  treatment  of  those  subjects. 

In  1813  the  Analytical  Society  published  a  volume  of 
meiii4>irH,  of  which  the  preface  and  the  first  fiaper  (on  continued 
products)  are  due  to  Babbage;  and  three  years  later  they 
issutnl  a  translation  of  Lacroix's  Tniite  eUinentaire  c/u  ctUeul 
difftre.utitl  et  tlu  calcul  inftt/rtiL  In  1817,  and  again  in  1819, 
the  diircrential  notation  was  used  in  the  university  examina- 
tions, and  after  1820  its  use  was  well  established.  The 
Analytiuil  Society  followi*d  up  this  rapid  victory  by  the  issue 
in  1820  of  two  volumes  at  examples  illustrative  of  the  new 
method ;  oue  by  Peacock  on  the  diHerential  and  integral 
calculus,  and  llu*  other  by  licrschel  on  the  calculus  of  finite 
ditreiH'iices.  Since  then  English  works  on  the  infinitesimal 
calculus  liave  aluindoned  the  exclusive  use  of  the  fiuxioiud 
notiition.  It  should  be  noticed  in  pjissing  that  Lagrange  and 
Laplace,  like  the  majority  of  other  modem  writers,  employ 
lH>th  the  fiuxioiml  and  the  dificrential  notation  ;  it  was  the 
excluidvo  adoption  of  the  former  IVubA*  nraa  so  ham^ring. 


\ 


RISE  OF  THE  cAMBRtnoE  a>;ai,\tical  STHOOI-   453 

Amongnt  thoH(>  wlio  mnlprinlly  amninU^I  in  extending  the 
use  of  the  new  (itialysis  wcrp  Willinm  WltewrfT  <I794-I8fi0) 
And  Grorcp  BiddHi  Airy  (IKOI  1832),  In-th  Fellows  of  Trinity 
Coltegp,  Ginibridge,  Tlip  former  iraunl  in  1819  »  work  on 
mprhnnic!!,  nnd  the  latter,  who  wns  a.  pupil  of  Peacock,  pnb- 
lislied  in  1826  his  Trartf,  in  which  the  new  method  wm 
Applied  with  great  nuccesK  to  rnriotu  phyniCAl  pmblems.  The 
efibrtN  of  the  society  were  nnpplempntnl  l>y  the  rapid  paUica- 
tion  of  good  (est^hookn  in  which  nnalj-iiii  wm  freely  naed. 
The  employment  of  analytical  methods  xpread  (rom  CMnbridgo 
over  the  rest  of  Britain,  Mid  by  1830  tbesw  n 
into  general  nan  tiiere. 


CHAPTER  XIX.     ! 

MATHSHATICa  OF  THE   NINITEENTB   CBNTUBT. 

.  Thi  nineteeotli  c«iitury  Km  iie«ii  the  creiHtian  of  numerwu 
nftw  deputiuMitii  of  pun-  uwtlieiiiaticN — notably  of  m  thaary 
of  numben,  ur  hiK)""*  oritlnuelic ;  uf  theorie*  of  fonnit  and 
gniupB,  or  A  hig)ifr  al<;i!brii;  uf  tluHiriea  of  functions  of 
multiple  periodicity,  ur  h  higher  trigonometry;  and  of  k 
geuerftl  theory  of  fu  net  limit,  euibracin^;  extviiHive  regiuiu  of 
higher  knalysis.  The  dewlopinentt  of  synthetic  and  ana- 
lytical geometry  liuve  uIho  pnujticiilly  creuted  new  Hubjecta. 
Further  the  appliiution  uf  mathenmtics  to  physical  prubleiitii 
Ikas  revolutionized  the  fouiulutions  and  treutiuctit  uf  tliat 
NubjecC 

Ni'W  developijicntM,  HUi'h  as  these,  may  hi:  taken  as 
opening  »  new  perind  in  thi-  hislury  of  the  Huhjcvt,  and  I 
recognize  that  in  the  futun-  n  writer  who  divider  the  history 
uf  mat  hematics  as  I  have  duiie  wuuld  piuperly  treat  tlie 
mathematics  of  tht-  .seventeciilh  and  eijjhieeiitli  centuries 
0:1  forming  one  pc'riix],  and  would  treat  the  matheniatica  uf 
tite  nineteenth  century  an  cominencing  a  new  period.  Thut 
however  wuuld  imply  a  tolenihly  complete  and  ByKteiuatic 
account  uf  the  <levelopjtteut  of  the  Hul>jui;t  in  tlie  nineti-entli 
century.  But  evidently  it  is  impossible  fur  nie  to  discus.^ 
ttde<|untely  the  mathciuutics  of  u  time  ho  near  to  us,  and  the 
works  of  inatlieuiuticiuns  Eume  uf  whum  are  living  and  Kome 
uf  whom  1  have  met  and  known,  tlence  1  make  nu  attempt 
tu  give  A  cuinplete  ai;couitt  of  U\o  uvutbematics  of  the  nine- 
tteutU  cvntury,   hut  aa  a  »orV  ot  u.vV^''"^^*  ^  *^*  ^vaxSva-t 


MATHEMATICS  OF  THE  NINETEENTH  CENTURY.     45  > 

chaptera  I  mention  the  more  striking  features  in  the  hiNtory 
of  recent  pure  mathenmticM,  in  which  I  include  theoretical 
dynamics  and  astronomy;  I  do  not  pro|io8e  to  discuss  in 
general  the  recent  application  of  mathematic8  to  ph}7iics. 

In  only  a  few  canes  do  I  give  an  account  of  the  life  and 
works  of  the  matlieniaticians  mentioned  ;  hut  I  have  added 
brief  notes  alx>ut  some  of  those  to  whom  the  development 
of  any  branch  of  the  Kubject  is  chiefly  due,  and  an  indi- 
cation of  that  part  of  it  to  which  they  have  directed  most 
attention.  Even  with  these  limitations  it  has  lieen  very 
difficult  to  put  together  a  connectefl  acc«>unt  of  the  mathe- 
matics of  recent  times ;  and  I  wish  to  repeat  explicitly  that 
I  do  not  suggest,  nor  do  I  wish  my  readers  to  supptise,  that 
my  notes  on  a  subject  give  the  names  of  all  the  chief  writers 
who  have  studied  it.  In  fact  the  quantity  of  matter  produced 
has  been  so  enormous  that  no  one  can  expect  to  do  more  than 
make  himself  acquainted  with  the  works  produced  in  some 
special  bmnch  or  branches.  As  an  illustration  of  this  remark 
I  may  add  that  the  committee  appointed  by  the  \\oyn\ 
Society  to  report  on  a  catalogue  of  periodical  literature  esti- 
mated, in  1900,  that  more  than  1,500  menHiirs  on  pure 
mathcnnatics  were  now  issued  annually,  and  more  than 
40,000  a  year  on  scientific  subjects. 

Most  histories  of  mathematics  do  not  treat  of  the  work 
produce<l  during  this  century.  Tlie  chief  exceptions  with 
which  I  am  acquainted  are  a  short  dissertation  by  H.  Hankel, 
entitled  IHe  EnifricMutig  tier  Jfaihfmatik  in  d^n  Utzin^ 
Jnkrkund^tieVf  Tubingen,  1885 ;  the  eleventh  and  twelfth 
volumes  of  Marie's  Ilisioire  de$  utime^n  in  which  are  some 
notes  oh  mathematicians  who  were  bom  in  the  last  century; 
Oerhardt's  Ge$ekiehie  d^r  Mnthematik  in  D^nt^Mnnd^  Munichy 
1877 ;  a  Disconrw  on  the  professors  at  the  Horfoonne  by 
O.  Hermite  in  the  Butleiin  de$  nei^nc^^  mntkemaiiquenj  1890 ; 
F.  C  Klein's  Leciwret  an  Maihemaiiew  (Evanston  Colloquium), 
New  York  and  London,  1894;  and  E.  Ijimpe's  Di«  rnne 
M^Memmiik  in  den  Jakren  1884-1899,  Beriio,  1899. 


/ 


456     lUTHEMATICB  OF  THE  NIinmiNTB  CnTOET.  * 

A  few  hMtoriM  td  tho  developmeiit  o(  fMurtiotthur  ml^Mte 
have  been  written --such  as  thos^  by  Isaso  Todhonter  om 
the  theories  of  attraction  and  on  the  cahmlus  of  proba- 
bilities— while  the  annual  volumes  of  the  British  Amod^tiaa 
contain  a  number  of  reports  on  the  progress  in  several 
different  branches  of  modem  mathematics ;  one  or  two  sanular 
ret>orU  (and  nuUbly  one  in  1857  by  J.  L.  F.  Bertrand  on  the 
develupnient  of  mathematical  analysis)  have  been  presented 
to  the  French  Acsdemy.  The  ninth  edition  of  the  Fne^o- 
paetlla  Britauniea  alHO  contains  some  important  memoin. 
Tlie  KucyklitiMidie  tier  MntkeuuUiMcken  IVisMMckt^Up  which 
Li  now  in  course  of  insue,  aims  at  representing  tlie  present 
state  of  knowledge  in  pure  and  appli(*d  mathematics,  and 
doubtless  it  will  to  a  large  extent  supersede  the  reporte 
hen*  mentioncHl. 

1  have  found  tlieno  authorities  and  tlieHO  reports  useful, 
but  I  have  derived  uiuHt  OMsistance  in  writing  this  chapter 
fi-oiii  the  obituary  notices  in  the  proceedings  of  various  learned 
Societies,  foreign  as  well  as  BritiMli ;  I  am  also  indebted 
to  inforniutiun  kindly  furniHlied  mo  by  various  friends,  and  if 
I  do  not  further  dwell  on  tliiM,  it  is  only  that  I  would  not 
mnnii  to  make  them  renponsible  for  my  errors  and  omissions. 

A  p«iricMl  of  exceptional  inti*llectual  activity  in  any  subject 
in  uMually  followed  by  one  of  coiiipurutive  Mtagnation;  and 
after  the  deatlis  of  Lagrange,  Laplace,  Legendre,  and  PoL*t!«on, 
the  Fn*iicli  school,  which  liiul  occupiinl  so  prominent  a  position 
at  the  U*ginning  of  this  century,  ceased  for  simie  years  to 

.  produce  uiucli  nuw  work.  Ktiine  of  the  mathematicians  whom 
I  inti'ud  to  mention  first,  Ciauss,  Aliel,  and  Jacobi,  were 
contempiiraries  of  the  later  years  of  the  French  matlienuiticians 
just  named,  but  tht*ir  writings  appear  to  me  te  bc*long  te  a 
different  school,  and  thus  art^  properly  placed  at  the  beginning 
of  a  fn*sh  cliapt4*r. 

There  is  no  mathematician  of  this  century  whose  writings 

iiaiv  hiul  a  greater  effect  iXmu  iVvwa  \A  \^>wMa\  wvw  U  it  oa 


\ 


OAUSR.  457 

only  one  branch  nf  the  Hrirncc  thnt  hin  inllapnce  hw  l«ft  » 
permanent  mnrk.  I  rniinnt  lltcrrforp  coiitnienc*  my  Mccoant 
of  the  mAthematics  of  rcopnt  (inic>(  lirttpr  thiM  l>y  detcrjInnK 
very  briefly  his  more  important  rewarcbe". 

Gbubs*.  A'lfV  Fri-^lrifl-  Gniitt  wnn  Ixim  nt  Uninswirk  on 
April  eS,  1777.  find  ilini  nt  (iottinRPn  on  FpK  23,  IMS.  Ifiit 
father  vns  n  hrioklnyer,  nnd  nauR!!  wnn  indebted  for  a  liberal 
e^lacation  (nincli  nf^inxt  tlic  will  of  Ids  jmifntii  who  wiiiheil 
to  profit  by  hiN  wages  an  a  lalniurer)  to  tli<>  notice  which  hii 
talento  procurwl  from  the  rci^nifiK  duke,  fn  1792  br  wa«  Kent 
to  thn  Caroline  Collet*,  nnil  by  179r>  profetmo™  ami  puptia 
idike  admitted  that  he  knpw  all  that  th«  fomwr  conbl  \nKh 
him ;  it  was  while  therp  thnt  hp  inrewtipatrtl  th*  nirtbod  of 
leant  s«|uareti,  and  proved  by  induction  the  law  of  qundnilic 
recipriH'ity.  Thence  hi>  went  to  (lilttiniren,  wher<^  he  Mtudied 
under  Kiuitner :  many  of  his  dixcoveries  in  |Ik>  theory  of  num- 
bers were  made  while  a  student  here.  In  I79H  he  returned 
to  Bmnswicic,  where  he  earned  a  Hoinewlint  prrcariouH  liveli- 
hooi]  by  private  tuition. 

In  1799  Gnuns  publinheii  his  demon itration  that  every 
algebraicnl  equation  has  a  n<ot  of  the  form  n  +&i ;  a  theorem 
(if  which  A]toK''therhegnvethrerdiNtinrt  pnxifi<.  In  1801  thia 
was  followed  by  liin  Di*q'ii"ili'niriiArHhm''lietjf.,  which  is  printed 
ai  the  fimt  volume  of  hin  collected  workn.  Tlic  fprntcr  pnri 
of  it  bad  lieen  s*'nt  to  the  French  Aradcmy  in  the  preceding 
year,  and  h«d  been  rejected  in  n  manner  which,  «\'en  if  tho 
book  bad  been  w  worthless  at  the  referee-*  Iwljeved,  wonbl 
havo  \Kvn  unjustifiable ;  GauHS  wiw  deeply  hart,  and  hia 
r«lactnnce  to  pubtisb  bis  inimtigations  may  be  pnrtly 
ftttrilmtAblfl  to  this  unfnrtnnale  incident. 

The  next  dixcovery  tA  Gauss  was  in  a  totally  different 

■  GaaM'i  mll«et«d  worh*.  eilitcd  t?  E,  J.  SeherinR.  wn«  hmcd  hy  tha 
Ro^l  Soeidj  of  06ltinitea  In  7  volnineii.  lACt-TI.  A  lama  aAoiint  ol 
additional  natter  bw  bren  linM  pnbliibcd,  and  toiyn  laiy  be  expected, 
aee  tol.  *ni  o(  hii  work*.  1900,  BDd  two  noten  bj  K.  C.  Ktdn,  timthtmm' 
Nie*«  Animlem,  1999,  tdI.  li.  pp.  1S8— 133,  and  1900,  tcL  un,  pp^  W-.4B. 


458     MATHEMATICB  OP  THE  MimTJUCMTH  CUTUBT. 

defMuriDient  of  matbetiiatiGiL  The  alnenoe  of  any  pbuiet  bi  tiM 
Hpnce  between  Man  and  Jupiteri  where  BodeVilaw  wonMhave 
led  olmerven  to  expect  one^  had  been  long  remarked,  bnt  It 
was  not  till  1801  that  any  one  of  the  nunerooa  groap  of 
minor  planeti  wliich  occupy  that  space  was  observed.  The 
discovery  was  made  by  O.  Piani  of  Palermo;  and  was  the 
more  intereHting  as  its  announcement  ooenrred  simultaneoosly 
with  a  publication  by  Hegel  in  which  he  severely  eriticiaed 
aMtrunomers  for  not  paying  nuMre  attention  to  philosopliy,  a 
Mcieiicey  said  lie,  which  would  at  once  have  shewn  them  that 
there  could  not  possibly  be  more  than  seven  planets,  and  a 
Htudy  of  wliich  would  therefore  have  prevented  an  absurd 
waste  of  time  in  looking  for  what  in  the  nature  of  things 
could  never  lie  found.  Hie  new  planet  was  named  Ceres,  but 
it  wiiM  seen  under  c^mlitions  which  appeared  to  render  it  im- 
practicable to  forecant  its  orbit.  The  obnervations  were  fortu- 
nately coiiiiiiunicated  to  Gauss;  he  calculated  its  elements, 
and  his  aiialynis  proved  him  to  be  the  first  of  theoretical  astro- 
uoiiiers  no  less  than  the  greatest  of  '^arithmeticiana" 

'Hie  attention  excited  by  these  investigations  procured  for 
him  in  1807  the  uH'er  of  a  chair  at  8t  Petersburg,  which  lie 
d(H;lined.  In  the  same  year  he  was  appointed  director  of  the 
GottiiigiMi  observatory  and  professor  of  astronomy  there. 
These  otlict^s  he  ret«*iined  t4>  his  death  ;  and  after  his  ap- 
pointment he  never  slept  away  from  his  obeiervatory  except 
Oil  one  occasitiii  when  he  attended  a  scientitic  congress  at 
])erliiL  His  lectures  were  singularly  lucid  and  perfect  in 
form,  and  it  is  said  that  he  used  here  to  give  the  analysis  by 
which  he  hail  amved  at  his  various  results,  and  which  is  so 
conspicutmsly  al«s4*nt  from  his  published  demonstrations;  but 
fur  fear  his  auditors  should  lom*  the  thread  of  liLs  discourse,  he 
never  williiij^ly  permitted  them  to  take  notes. 

I  have  already  mentioned  liauss*s  publications  in  1799, 
1601,  aiul  1802.  For  some  years  after  1807  his  time  was 
iimiiily  occupied  b}'  work  connected  with  his  observatory.  In 
180U  he  publishetl  at  Hamburg  his  Throria  Jfotug  Corporum 


I 


\ 


QAUHS.  459 

CoelegUum^ »  troatiMe  which  oontrilnitecl  largely  to  the  improve- 
ment of  practical  aMtrotiomy,  and  introduced  the  principle 
of  curvilinear  triangulation :  and  on  the  Rame  suhjecC,  hat 
connected  with  obM'rvationfi  in  general,  we  have  hifi  memoir 
Tkearia  CombinntionU  Obs^rvationum  ErroribuM  Mimmh 
Obnoxia^  with  a  second  part  and  a  supplement. 

Somewliat  later,  he  took  up  the  HuUjcct  of  f^eodesj,  acting 
from  1821  to  1848  as  scientific  adviser  to  the  Danvih  and 
Hanoverian  governments  for  the  sar\'e3r  then  in  progress: 
his  papers  of  1843  and  186G,  (^^iber  Geyntftanda  iter  hiihem 
Geodfin^^  contain  his  researches  on  the  suhject. 

Gauss's  researches  on  electricity  and  magnetism  date  from 
about  the  year  1830.  His  6nit  paper  on  the  theory  of 
magnetism,'  entitled  Infnmfas  Vis  Jfagn^ltra^.  Terr^$tri$  nd 
Menturam  Abiofuiam  Retocata^  was  published  in  1833.  A  few 
months  afterwards  he,  together  with  W.  E.  Welier,  invented 
the  declination  instrument  and  the  bifilar  magnetometer; 
and  in  the  same  year  they  erected  at  Gottingen  a  magnetic 
olMervatory  free  from  iron  (as  Humlmldt  and  Arago  had 
previously  done  on  a  smaller  scale)  where  they  made  magnetic 
observations,  and  in  particular  shewed  that  it  was  practicable 
io  send  telegraphic  signals.  In  connection  with  this  observa- 
tory Gauss  founded  an  association  with  the  object  of  securing 
continuous  oliservations  at  fixed  times.  The  volumes  of  their 
publications,  RetuUai^.  ana  d^.r  B^obarhiung^n  d^  Magneti- 
wehen  Vertins  for  1838  and  1839,  contain  two  important 
memoirs  by  Gauss,  one  on  the  general  theoiy  of  earth-mag- 
netism, and  the  other  on  the  theory  of  forces  attracting 
aeoording  to  the  inverse  square  of  the  distance. 

Gauss,  like  Poisson,  treated  the  phenomena  in  electro- 
statics as  due  to  attractions  and  repulsions  lietween  imponder- 
able particles.  Loitl  Kelvin  (then  William  Thomson)  in  1846 
shewed  that  the  effects  might  also  be  supposed  analogous 
to  a  flow  of  heat  from  varioos  sources  of  electricity  properly 
distributed. 

In  eiectrodynamica  Gauss  arrived  (in  1835)  at  a  fcmlv 


460     MATHEMATICS  OF  THS  NIKRKIirra  CUTUBT. 

equivalent  to  that  given  faj  W.  K  Weber  of  CKilliageA  ia 
184d,  namely,  that  the  attraetioo  between  two  eleetrited 
particles  «  and  t'  wlioBe  distance  apart  ie  r  depends  on  their 
reUtive  motion  and  position  according  to  the  tonnvhi 

QauHS  however  held  that  no  hypothesis  was  satlilsctory  whidi 
rested  cm  a  formula  and  was  not  a  oonMequence  of  a  physical 
conjecture,  and  as  he  could  not  frame  a  plausible  physical 
conjecture  he  abandoned  the  subject. 

8uch  conjectures  were  proposed  by  Riemann  in  1858^  and 
by  C.  Neumumi,  now  of  I^ipzig,  and  £.  BeUi  (1823-1892)  of 
PiHa  in  1808,  but  Helniholtz  in  1870,  1873,  and  1874  shewed 
that  they  were  untenable  A  simpler  view  which  regards  all 
electric  jind  magnetic  phenomena  as  stresses  and  moticms  of  a 
material  elantic  medium  had  Ijoeu  outlined  by  Michael  Faraday 
(17Ui-18G7),  and  was  elaborated  by  James  Clerk  Maxwell 
(1831-1879)  of  Cambridge  in  1873;  the  Utter,  by  tlie  use  of 
generalized  cuordi nates,  wan  able  to  deduce  the  consequences, 
and  the  agreement  with  experiment  is  clone.  Maxwell  con- 
cludtnl  by  Hliewing  that  if  the  medium  were  the  same  as  tlie 
so-called  luniiniferuuH  ether,  the  velocity  of  light  would  be 
<H|Uul  to  the  ratio  of  the  electromagnetic  and  electrostatic 
units,  and  Hubsequent  ex|M'rimentM  have  tendcnl  to  confinn 
this  conclusion.  The  theories  pi*eviously  current  had  aasunied 
the  existence  of  a  simple  elastic  solid  or  an  action  beween 
matter  and  ether. 

The  above  and  other  electric  theories  were  classiHed  by 
J.  J.  Thomson  of  Cambridge,  in  a  n*port  to  the  British 
Association  in  1885,  into  those  not  founded  on  the  principle 
of  the  conservation  of  energy  (such  as  those  of  Ampere, 
Gnissniann,  Stefan,  and  Korteweg);  those  which  rest  on  as- 
sumptions ccmcerning  the  velocities  and  positions  of  electrified 
particles  (such  as  those  of  Gauss,  W.  E.  Weber,  Kiemann, 
and  R.  J.  E.  Clausius) ;  tliosa  which  require  the  existence  of 
a  kind  of  energy  of  which  we  have  no  other  knowledge  (such 


\ 


fiAl'ss.  Ml 

lu  tlie  tli<^>ry  nf  C.  NruniAon) ;  those  wlik-li  rest  nn  HynMnkftl  - 
consMlrrntionH  liut  in  which  no  ncciunt  is  Ukoii  uf  tin  KCtinn 
of  tho  dielrctric  fnuch  m  the  tht-ory  of  F.  K.  Npunmnn) ;  »nd 
finnlly  those  whicli  m^t  on  dynaniicnl  coHHiilcnitiimH  nnd  in 
which  the-  nctiun  of  Iho  rlii-Ii-ctric  is  ci)rf<iiliTfld  <niicli  A"  JUx- 
well's  th<-ory).  In  tliR  report  thew  theorir*  ure  tleNeribnl, 
criticiznl,  nnd  aHnpnnil  with  the  resuKs  «»f  eitperiinentii, ' 

CinuEM'H  n-Kt-ftrchcw  on  optics,  nnd  p»pcciftlly  vn  i^t«iiiN 
of  Ifiiihs,  were  pul)lishe<i  in  1840  in  liin  Duiiilrijicht  (ThI'T- 
tnehnnijru. 

From  thin  Hkctch  it  will  lie  M*n  thnt  the  Rnmml  wrterrd 
by  Gaqu's  nnrfttflicx  wns  extmiinliTinrily  wide,  Kiid  it  nmy  lin 
Added  Ihnt  in  niAtiy  cn»^  hit*  inx-esligntiunx  MT^dl  t^i  inilialc . 
new  linen  of  work.  He  wah  however  the  IftHt  of  tltc  grmt 
mitthcniitticinnH  whose  intcrcHtN  were  nearly  anivcmnl :  since 
hia  time  tlie  lit4-mlun'  <if  nio^t  lirnni-heH  of  mnthcmaticH  htm 
grown  Ml  fast  that  mathematicians  have  been  farced  to 
npeciftiize  in  aunic  pnrticrular  department  or  depnrtnientit.  I 
will  now  mention  very  briefly  wiine  of  the  moat  important  of 
Ilia  diNcnTerirs  in  pure  math cma lie*. 

Ilix  moKt  crlebrrehfl  work  in  |>urc  mntliemnlicH  in  th« 
Du^nifilioiiri  AriUtntr.licni:  which  hn»  provod  a  MtArtiiift  point 
for  twvcnti  valuable  investigntionx  on  (he  thrary  of  nainlierH. 
This  timtisc  iiihI  r.iegpndra'a  Thenri'  dfn  uomhm  remain 
Btandanl  workn  on  the  theory  of  naml>FrH ;  bat,  juHt  an  in 
hiM  discuHKion  of  elliptic  function."  ijegendr«  failr<I  to  riae 
to  the  conception  of  a  new  Hubject,  nnd  confined  hinuelf  to 
rcganlint;  their  tlicory  aa  a  chapter  in  the  int^ral  caleulua, 
HO  he  treated  the  theory  of  namlH-n  a-n  a  chapter  in  algeliTK. 
Uauw  however  realized  that  tlie  thrurj-  of  diNcrete  magnitodea 
or  higher  arithmetic  was  of  a  different  kind  from  that  of 
Gontinuoaa  magnitodeif  or  algebrs,  and  ho  introdnoed  a  new 
notation  and  new  methods  of  analysis  of  which  anbaeqnent 
writem  hare  generally  availed  themnelvea.  The  theory  of 
nambers  may  be  divided  into  two  main  dinsioiu,  naiikely, 
Uie  tkaory  of  cottgmencea  and  the  theory  ol  fotms.    Both 


462     lUTHBMATIQI  OF  THS  KIimBBIfTH  CEMTUBT. 

I 

divinoos  were  dincoiMed  by  Gmuml  In  fMurtietthur  tiM  JH»- 
quUiiitmes  Ariiktmetieae  introduoed  the  modem  theory  of 
oongruenceif  of  the  lint  end  Heoond  onkrii  end  to  thie  Oenei 
reduced  indeterminate  enalyiiiii.  In  it  4hN>  he  dieooaed  the 
eolation  of  binomial  equations  of  the  form  s^a  1 :  this  in^tilveB 
the  celebrated  theorem  tliat  it  ie  poeaible  to  conetmct  by 
elementary  geometry  regular  polygons  ol  whidi  the  number 
of  sides  is  2**  (2*  ••- 1 ),  where  m  and  m  are  {nt^^ers  and  3*  -i- 1  is 
a  prime;  a  diiicovery  he  liad  made  in  1796.  He  developed 
the  theory  of  ternary  quadratic  forms  involving  two  indeter- 
minates.  He  also  investigated  tlie  theory  lof  determinantSi  and 
It  was  on  Gauss's  mults  tliat  Jaoobi  basJMl  his  researches  on 
that  subject 

The  theory  of  functions  of  double  periodicity  had  its  origin 
ill  the  ditKX>veries  of  Abel  and  Jacobi,  w^ich  I  describe  later. 
Both  the^ie  mathematicians  arrived  at  the  theta  functions, 
which  i*lay  no  large  a  part  in  the  theory  of  tlie  subject.  Gauss 
however  luul  iiiclepeudently,  and  indeed  at  a  far  earlier  date, 
discovered  these  functions  and  some  of  their  properties ;  having 
bet^ii  led  to  them  by  certain  integraln  which  occurred  in  the 
Detenninatio  AUr€iciiouis,  to  evaluate  which  he  invented  the 
transformation  now  associated  with  the  name  of  JacobL 
Though  Gauss  at  a  later  time  commu^iicated  the  fsct  to 
Jacobi,  he  did  not  publish  his  researches;  they  occur  in  a 
series  of  note-books  of  a  date  not  later  than  180t(,  and  are 
included  in  his  collected  works. 

Of  the  remaining  memoirs  in  pure  mathematics  the  most 
remarkable  are  those  on  the  theory  of  biquadratic  residues 
(wherein  the  notion  of  complex  numbers  of  the  form  a  -¥  Id 
was  first  introduced  into  the  theory  of  numljers)  in  which  are 
included  several  tabli»s,  and  notably  one  of  the  number  of 
the  classes  of  binary  quadratic  forms;  that  relating  to  the 
proof  of  the  theorem  that  every  numerical  equation  has  a  real 
or  imagimiry  root;  that  on  the  summation  of  series;  and 
lastly  one  on  interpolation :  his  introduction  of  rigorous  tests 
tor  the  couvergency  of  infinite  series  in  worthy  of  attention. 


\ 


HAUNK.  463 

Specially  noticfKl)!^  also  am  Iiik  iiivrstipitioni  »m  hj-jipr- 
goonietric  sprips—tbew  contAin  a  diNOUMHion  of  thr  (!amma 
fanctinn ;  thin  subjpct  Into  ninn-  lioctnitc  one  of  coiiHideraliln 
impDrtAncr,  and  hait  Iipcn  wrilti-n  on  by  (among  uthem) 
Kuntmcr  and  Kicmann.  Finally  wc  have  tlm  impnriMit 
mcnwir  on  tho  cunfumial  n<prvsrntali<>n  of  oiic  surface  upon 
anotlinr.  in  which  the  renuIlM  given  1>y  Tjagnngc  for  ftarfacvH 
of  rovoluliiin  arc  ^nerallKfl  for  nil  iturfnccn.  It  wtnild  MTni 
alno  that  Gnuicg  liiul  diwnvc-r«il  Minic  iif  the  pniperltos  of 
<]uat4>mi<ins,  though  tliene  invi'stigntionn  were  not  pnliliithcd 
until  a  fow  yean  ago. 

In  thn  theory  of  attractions  we  Imvc  a  paper  on  the 
attraction  of  h<imi>gencnuH  ell)|wni<h ;  the  alreAdy-mentionerf 
memoir  of  I?3!>,  Jlf'jrni'hie  /^/irtntzi-  in  /ttsifkuttg  nu/  die 
im  vrlthrlftt  VfrhnUnitiK  rf"  QntvtmU  H'.r  Bttl/rrnvj,  on  llic 
tlicoT}'  of  forces  attracting  accinxling  to  the  in^-emo  M|nare  of 
the  distance;  and  the  ineinoir,  Oft'rminado  AltrarlimiU,  in 
which  it  Ih  shewn  that  the  Heculiir  Tarintiona,  which  tho  ele- 
n»ent»  of  the  orliit  nf  a  planet  experience  fnHn  the  attraction 
of  another  planet  which  disturlis  it,  are  the  same  a*  if  themau 
of  the  disturbing  planet  were  distnliut«d  »\w  its  orlnt  into 
an  elliptic  ring  in  hqcIi  a  manner  that  eijunl  maMtex  of  the  ring 
would  corrwpond  to  arc»  of  the  orbit  dpscriljed  in  equal  times. 

The  greftt  mnKters  of  modcni  arialyHis  an-  Ijij^range,  lAplace^ 
and  Gaasn,  who  were  eontempornrien.  It  in  int^'resting  to  noto 
the  marked  contrast  in  their  styles.  Ijtgrange  ia  perfect  both 
in  form  and  matter,  be  is  careful  tu  explain  hia  procedure, 
and  thciugb  hin  nrguments  are  general  they  are  eaKf  to  folkiw. 
Laplace  on  tlie  otlier  hand  explains  nothing.  In  indifferent 
to  Htyle,  and,  if  satisticii  that  his  remilU  ar«  correct,  in 
content  to  leave  them  either  with  no  proof  or  with  a  faulty 
on&  Gauss  is  as  exact  and  elegant  as  Lagrange,  but  even 
DioTB  difficult  to  follow  than  Laplace,  for  he  mnovea  every 
tnce  of  the  analysis  by  which  he  reached  hin  reaalU,  and 
stodies  to  give  &  proof  which  while  rigorous  shall  be  as  concise 
■ad  syatbetieal  as  possible^ 


464     MATHBMATIGS  OP  THB  NINBTBUITa  CSKTORT. 

Diriohlet*.  One  of  Gaiiai'tt  pupilii  to  whom  I  nwy  hfttm 
mlliide  is  Lejeuiie  Diricklei,  whoiie  inasleriy  oxpoaitioa  of  the 
discoverieti  of  Jaoobi  (who  wan  hU  fftther-in-law)  and  of  Uaun 
liEtt  uiuiuly  overnhadowed  hui  own  original  invfut.igationi  on 
HUiiilar  HuhjecU.  /V/«fr  O'uUav  Ltjeuue  Dirieklti  was  bom  at 
Diirea  on  FcU.  13,  1805,  and  died  at  Gtittingen  on  May  5» 
1859.  llo  held  micccshively  profeMnoroliijMi  at  Bretlan  and 
IWrlin,  and  on  GauiM'ii  death  in  185i>  wan  appointed  to  anioceed 
hiui  an  prufettiior  of  ilie  higher  niatlieinatics  at  Giittingen. 
He  intended  to  lininh  GauHn^ii  incomplete  works,  for  which  he 
was  mhuirably  fitted,  but  hiit  early  death  prevented  this ;  he 
imiduced  however  bcveiid  uienioirs  wliich  Imve  considerably 
facilitated  the  couipivhension  of  some  of  Gauss's  more  abstruse 
nieth<NlH.  Of  Uirichlet  s  original  rescHirches  the  most  celebrated 
are  that  on  the  establishment  of  Fourier's  Theorem,  and  that 
in  the  theory  of  numbers  on  asymptotic  laws,  (that  is,  laws 
which  Approximate  more  closi*ly  to  accuracy  as  the  numbers 
concerned  become  larger). 

l^te  Theory  of  Xumbers^  or  Hiyher  Ariihtnetic,  The  re- 
searches of  Gauss  on  the  theory  of  numbers  were  continued 
or  supplemented  by  Jacotti^  who  tirst  proved  the  law  of  cubic 
reciprocity ;  discussed  the  theory  of  residues ;  and,  in  his 
Canon  Arithtneticuit^  gave  a  table  of  residues  of  prime  roots. 

Eisensteint.  The  subject  was  next  taken  up  by  Ferdinaudi 
(JtUthold  Eist'Hsft'in,  a  proft^ssor  at  the  university  of  Berlin/ 
who  was  born  at  IkTlin  on  April  IG,  1823,  and  died  there 
on  Oct.  11,  1852.  The  solution  of  the  problem  of  the  re- 
presentation of  numbers  by  binary  quadratic  forms  is  one  of 

*  Hia  workii  were  ihttuvd  in  two  voluDiea,  Berlin,  1889,  1897.  His 
leotureii  oo  tlie  theory  of  uuiubcm  were  edited  by  J.  \V.  R.  Dedekind, 
third  edition,  Brunswick,  1879-Sl :  his  investigations  on  the  theoiy  of 
the  potential  Lave  been  edited  by  F.  Gnibe,  second  edition,  Leipzig, 
1887.  There  is  a  note  on  some  of  his  researches  by  C.  \V.  BorchardI 
in  CuU*\  Journal^  vol.  lvii,  l8uU,  pp.  91— 9*i. 

t  For  a  sketcli  of  his  life  and  researches  aee  Abkaudluu^n  zur 
Oetchichtt  tUr  Malhtmatik,  1895,  p.  14S  et  $eq. 


\ 


THE  TIIEORV  OF   NUMnF.BS.  465 

the  grrnt  achicvpiiK^nts  of  tiauss,  nnri  th(-  ftindAincntal  princi- 
ples upiHi  which  the  trr'Ht  I  Ill-lit  tif  Kuih  quwlions  mt  were 
given  liy  him  in  the  IHa^Hifitimira  Arilhwt'tf^tf.  (!»■»  tlipre 
addml  ttmw  rcswitx  rfUtiti^r  h>  lemary  quiulnitiG  (omiR,  bnt 
the  gmerni  extenxinn  from  (wo  Ut  thrro  indeUrniinAles  was 
the  work  nf  EixenHtriii,  wlm,  in  liis  niemnir  K'ne-  Thtorrme.  d*r 
hij&rren  ArUhnfJik,  defined  the  ordiiinl  nrif)  generic  chftractcrt 
of  trmi(r)-qnadratic  fnmiH  of  nn  vneteti  df term i mint ;  ftnd,  in 
the  oise  of  definite  fnniis,  n<uignril  thr  wrij^hl  of  Anj'  nnler  or 
genns ;  Init  he  Hid  not  consider  forms  of  nn  r^'en  determinant, 
nor  give  any  demonntnitionH  of  hin  work. 

Eisenstein  atno  ctmMderrd  the  theorems  relftting  (<>  the 
ponnibility  of  repn-Ki-nting  n  iiundier  a,t  n  huRI  uf  squami,  and 
■hewed  ihnt  thft  gi>iirml  theorf-in  wha  limited  to  eight  M|unres. 
Tlip  solutionn  in  the  caws  of  two,  four,  and  nix  aqaares  may  he 
obtuined  hy  meanH  nf  elliptic  functions,  hut  the  caneii  in  which 
the  namber  of  KqunreK  in  uneven  involve  special  ptvcennes 
peculiar  to  the  theorj-  of  numlien.  Eifienxt«in  gave  the  mAm- 
lion  in  the  case  of  three  squar*^.  He  also  left  a  rtatnnent 
of  tlie  solution  he  hud  obtained  in  the  ca.<ie  nf  five  nquam* ; 
but  his  mtults  were  pulilinhe^l  without  proo^  and  applj  only 
to  numbers  which  are  not  divisible  by  a  Hquare. 

Henry  Smitht.  One  of  the  most  original  and  powerful 
mathematicians  of  the  school  founded  by  tiauM  was  Ileniy 
Smith.  I/'uryJohn  Slrphm  Smith  was  born  in  l.ondon  on 
Nov.  S,  162G,  and  died  at  Oifon)  on  Feb.  9,  1883.  H«  was 
educated  at  Rugby,  and  at  Bnlliol  College,  Oxford,  of  which 
latter  xociety  he  was  a  fellow;  and  in  I8GI  he  was  elected 
Savilian  pn>fesw>r  ot  geometry  at  OxfnnI,  where  he  resided 
till  his  death. 

*  CrrUr't  JoKnmt,  toL  hit,  IMT.  p.  3K8. 

t  Smith'!  enllecM  msthcmBtical  work*,  ntited  bjr '-  W.  L.  OlaMter, 
sad  pretsord  b;  a  Uognphiesl  a«Uh  and  other  pspas.  wsra  pabHahtd 
in  1  voIddcs,  Oiloid,  18M.  Tb«  foUowins  aoeotnit  b  aM'a«*«J  i^<» 
Um  obitnai?  Dotke  In  Un  Monlblj  aotieM  of  Um  AtCwcwfcal  HcwWr. 
18B4,  vf.  lBft-l*9. 

1.  *» 


4G6       MATHBMATICB  OF  THS  NINRIIinH  OEMTUBT. 


The  subject  in  oonneetioii  with  which  fimitbVi  bmm  b 
»pecUlly  aiisociated  U  the  thecNy  of  nooiben^  and  to  thb 
he  devoted  the  yearn  from  1854  to  1864.  The  ramlte 
of  hia  historical  researches  were  given  in  his  report  published 
in  parts  in  the  TraasaeiiuHM  of  the  British  Association  from 
1859  to  18G5;  this  report  contains  an  account  of  what  had 
been  done  on  the  subject  to  that  time  together  with  some 
additioiuU  matter.  The  chief  outcome  of  his  own  original 
work  on  the  subject  is  included  in  two  memoirs  printed  in  the 
PhUoMopfiical  7*ratuactio9iM  for  1861  and  1867 ;  the  first  being 
on  linear  indeteniiinate  equations  and  congruences,  and  the 
second  on  the  orders  and  genera  of  ternary  quadratic  forms. 
In  the  latti^r  memoir  demonstrations  of  Eisenstein's  results 
and  thoir  (^xtennion  to  ternary  quadratic  foniis  of  an  even 
determinant  were  supplied,  and  a  complete  classification  of 
ternary  quadratic  fonns  was  given. 

Smith,  however,  did  not  coutiiie  himself  to  the  case  of  three 
in<let4*i'iiiiiiute8,  but  succ4M»ded  in  establishing  the  principle's  on 
which  the  <*xten.sion  to  the  general  ca&e  of  m  indetemiinates 
depends,  and  obtained  the  general  formulae;  thus  efiecting 
the  givatest  advance  made  in  the  subject  since  the  publication 
<if  Gauss's  work.  In  the  account  of  his  methods  and  results 
wliirli  ap|M*anH]  in  the  Proceedhuj*  of  the  Royal  Society*, 
Siiiitli  ii*iiiark«Hl  that  tlie  theorems  Mating  to  the  representa- 
tion of  iiuiiiImts  by  four  M|uares  and  other  simple  quadratic 
forms,  art*  de«lucil>le  by  a  uniform  method  from  the  principles 
then)  inclicattMl,  as  also  are  the  theorems  relating  to  tlie  repre- 
siMitutioii  of  imiiiU^rs  by  six  and  eight  squares.  He  then 
prot*tH*«l(Hl  to  Miy  that  as  the  series  of  theorems  relating  to  the 
n*pii*s4*ntatioii  of  nuniliers  by  sums  of  squares  ceases,  for  the 
re}iS4in  assigtuHl  by  KiM^iistein,  when  the  numlier  of  squares 
surpass4*s  eight,  it  was  desirable  to  complete  it.  Tlie  results 
for  even  squart^s  were  known.  The  principal  theorems  re- 
lating to  the  case  of  live  squares  had  been  given  by  Eisenstein, 
but  he   hod   considered  only  those  numbers  which  are  not 

•  St-e  ruJ.  xm.  im>4,  VP-  VM— 'a»,iu^N<A.xH\,V^RAxV^  197— SOB. 


\ 


THE  THEonr  OP  XUMRERfL  467 

divisibU'  Uy  ft  squnrp,  niiil  he  hml  iint  i-ongidem)  tW  ciwe  of 
Hcvpn  8(|UniTH.  Smitli  liprc  r<>iii]<lct<'(I  tlir  (^nncuttion  uf  the 
theorems  for  tlip  castr  of  K\p  xquiii-i's,  niiil  ttdded  the  corrr- 
nponding  llipomiis  for  the  cnsp  of  seven  si|Ui«reN. 

This  paper  was  llie  ixviviiiii  of  a  dr.iiimtic  incitlont  in  tho 
history  of  niathemnticH.  Kourtren  ycnn"  lattr,  in  iRiiorance  of 
Hmith'fi  work,  the  rloiiionntmlioii  bikI  cuniplelion  of  Eicenstf^in'a 
iheorrmH  for  tive  wjaarex  were  set  hy  the  fVcnch  Aaulemj  an 
the  Huhject  of  their  "(Jniml  prix  diii  scipho-*  m«th<'niaUqiiP8." 
Smith  wrote  out  the  (lemniisimiiun  of  his  fiennml  theorems  m 
far  AS  waa  re<]uir«<I  to  prove  the  n^ults  in  the  Hpecta)  cue  of 
five  »)UAreH,  nnil  only  a  mnntli  after  hiH  deatli,  in  March  1883, 
the  prizn  was  awarded  to  liim,  another  prize  lieing  rIho  awarded 
to  II.  Minkowski  of  Elonn.  J<o  episode  eoaM  iHingout  in  « 
nH)rc  striking  li^lit  the  extent  of  SniitliH  rexearchei  tlian  that 
a  i|Uefltinn  of  which  he  had  jpven  the  Holulion  in  1867  an  a 
oonilliirj'  from  jceneral  fnnnulne  whioli  ),'overne(l  llie  whole 
class  of  investigntiiiiis  to  «  hii-h  it  iM-lonf^eil  aliouhl  liave  \wrn 
regnrdnl  hy  the  Frenth  Academy  as  one  whone  solution  was 
of  Kuch  dilhculty  and  importance  n»  to  lie  worthy  of  their 
great  jirize.  It  has  l>een  alxo  a  matter  of  comment  that  Ihey 
should  hnve  known  mt  little  of  ciintem[iorftry  Knglish  and 
German  rrsearvhcK  <in  the  suliject  la  to  lie  unaware  that  the 
result  of  the  prolileiii  they  were  pniposiiig  wan  then  Ijing  In 
tlieir  own  Ulirary. 

Among  iSmith's  other  iiivpKligalionK  1  nuty  specially  men- 
tion his  geometrical  memoir  Siir  qwl'iiim  itrubifrnm  mAJflfa 
el  biqiiiwimtiq'"!',  for  which  in  It^Gt*  lie  was  awarded  the 
.Sleiner  prin-  of  the  Iterlin  Acndrmy.  In  a  paper  which  he 
contrilnited  lo  the  >(((■  of  the  Acawlemiti  dei  Lincei  for  1877 
he  eslnhliahed  >  very  remarkahle  analytical  i«lation  connecting 
the  modular  equation  of  order  nand  the  theory  of  liinal^r  quad- 
ratic fonns  belnnging  to  the  |iositive  clelemiinant  «.  In  thin 
paper  the  modular  curve  is  repmented  analytically  by  a  curve 
in  such  a  manner  as  to  present  an  actual  geometrical  image  ol 
the  complete  syntems  of  the  reduced  qnaalratie  forms  beloiigiBg 


44)8     MATHEMATICS  OP  THK  KINRIINTB  CUTUBT. 

to  tbci  deterniiiuuit^  and  a  geouetrioal  uiieqMVtotioii  k  givi^M  to 
the  idbtM  of  '^claHBy"  ^  iHiuivaltmce,"  aiid  **  reduced  fom.*  H« 
wa«  akto  the  author  of  important  papem  in  which  hewiooeeded 
in  extending  to  complex  4uadratic  foruM  many  of  Gauflt*«  inveeti- 
gationa  reUting  to  real  4uadratic  forms.  He  was  led  bj  his 
reHearclu*s  on  the  theory  of  numlieni  to  the  tlieory  of  elliptae 
functions,  and  tlie  results  he  srrived  at,  especiallj  on  the 
thi*ories  of  the  theta  and  omega  functional  are  of  importance. 

Tlie  theory  of  nuutberg^  as  treated  to^Uy,  may  he  taad  to 
originate  with  Gauss.  I  have  already  mentioned  very  briefly 
the  investigations  of  Jacobi^  Dirichlet^  EUetutUin^  and  ilenry 
Smith.  I  content  myself  with  adding  some  notes  on  the 
suljsequent  development  of  certain  liranelies  of  the  tlieoiy. 

The  distribution  of  primes  lias  been  discussed  in  paKicular 
by  <V.  /'.  A  UiemanH,  J.  J.  Sylvtuter,  and  P.  L.  Tchehychef* 
(1H21-1894)  of  St  Petersburg.  Itii>mann's  short  tract  on 
the  number  of  primes  which  He  between  two  given  numbers 
afloitls  a  striking  instance  of  his  analytical  powers.  Legendre 
had  previously  shewn  that  the  number  of  primes  less  than  n  is 
appruxiiiiately  fi/(log,n  -  1 08366) ;  but  Riemann  went  further, 
and  this  tract  and  a  memoir  by  Tchebycheff  contain  nearly  all 
tliat  lijis  lx*en  done  yet  in  connection  with  a  problem  of  so 
obvious  a  cluinicter  that  it  has  suggested  itself  to  all  w^o 
have  c«>nHidt*riMl  the  theory  of  numbers,  and  yet  which  over- 
taxed the  powers  even  of  Ijugronge  and  Gauss. 

Tlie  partition  of  numbers,  a  problem  Ui  which  Euler  had 
paid  considerable  attention,  liiis  been  treated  by  A,  Cayley^ 
J,  ./.  St/fiyt<(*'i;  and  /*.  A.  MaeMahon, 

The  repnvsentation  of ^lumljers  in  special  foniis,  the  possible 
divisors  of  numbers  of  s|H*ciHed  forms,  and  general  theorems 
concerned  with  the  divisors  of  numbers  liave  been  discussed  by 
J,  LiouvUfe  (l80D-i882),  the  editor  from  1836  to  1874  of  the 
well-known  mathematical  journal,  and  by  •/.  ir.  L,  Glauher 
of  Cambridge. 

*  Tcliebychers  collected  works,  edited  bj  H.  Markoff  and  N.  Sonia, 
Mn  in  coane  of  issue  at  8t  Pstersliurg ;  vol  i  was  publishad  in  1899. 


\ 


MATHEMATICS  OV  THE  NINETEKNTH  CEAVrURY.      469 

The  Gtmcrpitfm  of  klcal  primes  in  duo  iu  £.  A*.  Kumm^r 
(1810>1893)  of  Ik»r1in,  niid  hiR  invrKtigations  wcrr  cfNitinurtl 
hy  t/.  ir.  A*.  D^iekiml^  tho  i*«iitor  of  l)iriclilct*M  workif. 
B,  B,  Knmnirr  hIwi  c*xt4*nfli*fl  (fauKM*ii  tlif^oromii  on  qufidmiic 
rosifiupM  to  rpHiduoH  of  a  higlirr  onler. 

Hk*  Hahject  of  i|uiwlnitic  binomialii  Iiah  lN*on  ntudicd  liy 
A,  Is.  Vw&hy ;  of  temary  aiul  quadratic  fomiH  hj  L.  Krtmeehtr* 
(1823-^1891)  of  llfrlin;  and  of  ti*rnary  fomiM  liy  C.  iiermiie 
(18S2-190I)  of  Paris. 

The  moHt  common  ioxi-UMikii  am,  pi*riia|NS  that  hy  O.  B. 
MathewH,  Cambridge,  1892  ;  that  hy  E.  I^cas,  Paris,  1891  ; 
and  that  by  E.  Calion,  Paris,  1900.  Tuterest  in  problems 
oonnrctod  witli  the*  thctiry  f>f  nunilicrs  scrms  n*rently  to  have 
flagged,  and  possibly  it  may  Ije  fouiHl  liercafter  that  the 
subject  is  approached  lietter  on  other  linni. 


Th^  iheory  of  funeiiont  ofdouhUe  nmi  mfJliiJe  ffenotiieiiff 
is  anotlior  subject  to  which  much  attention  has  been  paifl 
during  this  century.  I  have  already  mentioned  that  as  i^rly 
as  1808  (tauKs  hail  discovered  the  tlieta  functions  and  some 
of  their  properties,  liut  his  investigations  remained  for  many 
years  ccmcealed  in  his  note-books ;  and  it  was  to  the  researches 
made  between  1820  und  1830  by  Aliel  and  Jacolyi  tliat  the 
modem  development  of  the  subject  is  due.  Their  treatment 
of  it  has  completely  superseded  that  used  by  Legendre,  and 
they  are  justly  reckoned  as  the  creators  of  this  liranch  of 
mathematics. 

Abelt.  Aieln  iienriek  AM  was  bom  at  FindoC  in  Norway 
on  Aug.  5,  1802,  and  died  at  Aremlal  on  April  6,  1829,  at 
the  age  of  twenty-six.     His  memoirs  on  elliptic  fonctions, 

•  8«e  the  nmllfiim   of   the   New  York   (Amsrican)    MaUiemstMsl 
Boeicty,  vol.  i,  1991-i,  pp.  ITS— 184. 

t  The  life  of  Abd  1^  C.  A.  Bjerkues  was  published  at  Stockholai  in 
1880.  Two  editions  of  Abel's  works  have  been  pabUshcd,  of  wlneh  lbs 
ImI,  edited  bj  8jkw  and  Lit  and  issued  at  Christlaiua  in  two 
te  1881,  is  tbs  ome  eomplete. 


470       MATUBMATICK  OF  TUB  NIHREBHTU  CBNTUBT. 


paUiaiied  in  CrdU§  Jawmal^  treat  the  ml^JMt  fraai 
the  point  of  view  of  the  theory  of  eqnntions  nnd  nlgebraie 
forma,  a  treatment  to  which  his  researches  naturally  led  him. 

The  important  and*  very  general  resnlt  known  at  Abel'a 
theorem,  which  wan  MabHeqnently  applied  by  Riemann  to  the 
theory  of  tnuiscendenUl  functiona,  waa  aent  to  the  French 
Academy  in  1826,  but  (mainly  tlirough  the  inaction  of  Oauchy) 
wan  not  print«!d  until  1841 :  ita  publication  then  waa  due  to 
eiiquirieH  luude  by  Jacobi,  in  cunaequence  of  a  atatement  on 
the  MubJGct  by  II.  Holmbue  in  hia  edition  of  AbeFa  works 
iaaued  in  1839.  It  in  far  frum  eaay  to  atato  Abel'a  Theorem 
intelligibly  and  yet  ct>iicisely,  but,  broadly  apeaking,  it  may  be 
deKcrilMHl  an  a  theorem  fur  evaluating  the  auni  of  a  number  of 
intogrulK  which  liave  the  suiiue  integrand  but  different  liuiita 
— tlie.se  liinitH  being  the  rootti  of  an  algeljraio  equation.  The 
theorem  gives  the  huiii  of  the  integrals  in  terma  of  the 
coiiKtunts  occurring  in  this  ec|uation  and  in  the  integrand. 
We  may  r(*gard  the  inverse  of  tlie  iiiti'gral  of  this  integrand 
HH  a  nt*w  tnuisceiidental  function,  and  if  so  tlie  theorem 
funiislieri  a  prupi*rty  of  this  function.  For  instance,  if  Abel'a 
tluHireni  Ije  applied  to  the  integrand  (\-j^)~h  it  fumialiea 
the  addition  theorem  for  the  circular  (or  trigononietriciil) 
functions.  The  name  of  Abelian  function  haa  been  given  to 
the  higher  traiLscendents  of  multiple  periodicity  which  were 
first  discussed  by  Alx*l. 

Alx*l  criticized  the  use  of  infinite  series,  and  discovertnl 
the  well-known  tlii*orem  which  furnishes  a  test  for  the  validity 
of  the  n*sult  obtained  by  umltiplying  one  infinite  series  by 
another.  As  illustrating  his  fertility  of  ideas  I  may,  in  passing, 
notice  his  celebratcHl  demonstration  that  it  is  impossible  to 
express  a  roiit  of  the  general  quintic  equation  in  tenus  of 
ita  coefiicients  by  means  of  a  finite  number  of  radicals  and 
rational  functions;  this  theorem  was  the  more  important 
since  it  definitely  liniitecl  a  fie!«i  of  mathenmtica  which  had 
previoualy  attn&cted  imnieix>us  writers^  I  should  add  tliat  thia 
theorem   luul   been   euuuc\&UA  a;!^  eaxV]  li^^  V\^>^  Vi<|  Paolo 


\ 


JAa)Bi. 


471 


Raffini,  an  Italian  physician  practising  at  Modena ;  hut  I 
helieve  that  the  proof  he  gave  wan  doBciont  in  generality. 

JacoU^.  Carl  Gtuffnv  Jarnh  Jacobi^  liom  of  Jewish 
parents  at  Potsdam  on  Dec.  10,  1804,  and  died  at  lierlin  on 
Feb.  18, 1851,  was  educated  at  the  university  of  lierlin,  where 
he  obtained  the  degree  of  doctor  of  phil(i6«>|>liy  in  1825.  In 
1827  he  became  extraordinary  professor  of  mathematics  at 
Konigsberg,  and  in  1829  was  promoted  to  lie  an  ordinary 
professor;  this  chair  he  occupied  till  1842,  when  the  Prussian 
government  gave  him  a  pension,  and  he  moved  to  Berlin 
where  he  continued  to  live  till  his  death  in  1851.  He  was  the 
greatest  mathematical  teacher  of  his  generation,  and  his 
lectures,  though  somewhat  unsystematic  in  arrangement, 
stimulated  and  influenced  the  more  able  of  his  pupils  to  an 
extent  almost  unprecedented  at  the  time. 

Jaoobi's  most  celelnated  investigations  are  those  on  elliptic 
functions,  the  modem  notation  in  which  is  suljstantially  due 
to  him,  and  the  theory  of  which  he  established  simultaneously 
with  Abel  but  independently  of  him.  Jacobi*s  results  are 
gi%'en  in  his  treatise  on  elliptic  functions,  published  in  1829, 
and  in  some  later  papers  in  Crelh^n  Jonrnn!^  they  are  earlier 
than  Wfrierstrass*s  researches  which  are  menticmed  lielow. 
The  correspondence  between  Legendrc  and  Jacobi  on  elliptic 
functions  lias  been  reprinted  in  the  first  volume  of  Jacoln^s 
collected  works«  Jacobi,  like  Aljel,  recognized  tliat  elliptic 
functions  were  not  merely  a  group  of  theorems  on  integraticm, 
lioi  that  they  were  types  of  a  new  kiinl  of  function,  namely, 
one  of  double  periodicity  ;  hence  he  paid  particular  attention 
to  the  theory  of  tlie  theta  function.  The  following  passagof 
in  which  he  explains  this  view  is  sufficiently  interesting  to 
doserre  textual  reproduction : — 

*  See  a  J.  Oerhaidt*!  Oem-kickif  4er  Mmtkfmmtfk  9m  PfmttehUn^^ 
llmifeh,  1877.  Jaeobi*8  collected  works  were  edited  by  DirichK  > 
volomea,  Berlin,  184S-71,  and  aceooipenied  bj  a  biogrmplqr,  18iit ;  a 
new  edition,  under  the  euperviskm  of  C.  W.  Borehardt  and  K.  Weisftlrait, 
leoed  at  B«lia  la  7  volume^  INHI-IWI. 

t  8te  Ids  mOmM  workf,  voL  1, 1881.  p.  87. 


472       MATUEMATIOi  Or  TUB  NlNKTEKlTllI  CBMTUET. 


B  quo,  earn  miivMMUB,  qiM  AogI  pototl. 
Mu4jtieMB  dooet,  fiuurlioMt  •Hiptir—  bob  •Uk  adBBBMnii 
ImiMeiidenlibiit,  qoM  qtubuadAia  gMdeni  ckgaatiit,  iatiaam  plBiihi 
illM  aat  maioriboi,  aad  spedem  qaAndAm  iU  iBcui  pirfbeti  •!  ahiohili. 

• 

Among  Jaoolii'ii  other  'inveiitigBtkiiis  I  nwy  qpeoBlly  iiQgb 
out  kin  paperv  on  det4)nuuiBnU|  which  did  a  great  deal  to  bring 
them  into  general  iMe ;  and  particuUrly  hiM  invention  of  the 
JacoUaii,  tliat  in,  of  liie  fuiictioiial  detenninant  formed  by  the 
H*  fMirtial  diflereutial  coe!ficieni«  of  the  first  order  of  n  given 
f  unctionii  of  m  independent  variaUcii.  I  oaght  alio  to  mention 
hiH  |>a|ieni  on  Abeliau  trauRcendentu ;  his  inveKtigationa  cm  the 
theory  of  numlierM,  to  which  I  liave  already  alluded ;  bin  im- 
portant mcuioira  on  the  tlieory  of  differential  equaticmHi  both 
ordinary  autl  pai-tial ;  his  dei'elopment  of  tlie  calcalus  of 
vAriatioiiH;  and  his  contributions  to  the  problem  of  three 
bodies,  and  oilier  {Mirticular  dyimiiiical  probl<^ms:  most  of 
the  results  of  the  resi^arches  last  named  are  included  in  his 
VorifntuHtjen  idttr  Dynamik, 

Riemann*.  Hvortj  FrieJrU-k  HenJtanl  liieuuutn  was  bom 
at  l)nrs4*lciiz  on  Sept.  17,  182G,  and  died  at  Sefasciitm  July  20, 
1 80G.  lie  studied  at  Gtittingen  under  Gauss,  and  subhet|uently 
at  Ucrlin  under  Jacobi,  Dirichlet,  Steiner,  and  Elsenstein,  all 
of  whom  were  professors  thert^  at  the  same  time.  In  spite 
of  poverty  and  sickness,  lie  struggled  to  pursue  his  researches. 
In  1857  he  was  niiule  pn>fessor  at  Ciiittingenj  general  recog- 
nition of  his  powers  soon  followed,  but  in  18G2  his  health 
began  to  give  way,  and  four  years  later  he  died,  working,  to 
the  end,  cheerfully  and  courageously.  I 

liieiiiann  must  be  (!sti*emed  one  of  the  most  profound  and 
brilliant  niatlieniaticiaiis  of  his  time.  The  an^ount  of  matter 
he  produced  is  small,  but  its  originality  ind  poa^er  are 
manifest — his  investigstions  on  functions  and  on  geometry, 
in  particular,  initiating  developments  of  great  importance. 

*  Uiemanu**  Cdllvcted  works,  edited  by  U.  Weber  aild  prefaced  by  an 
account  of  bit  life  by  DcdekiDd,  were  publUbed  at  Leip2^ig,  aecood  edition, 
189:^,    AuoUwr  short  biography  of  KieiuMUU  has  beenl  written  by  E.  J. 
JUcbersng,  Gutlingeu,  1H4«7.  \ 


J 


MATHEMATK^  OF  THK   \IN'»nTK\TII  CENTURV.      +73 

His  earlipst  paper,  written  in  1850,  war  mi  algebraic 
funi-tionii  of  n  complex  variftMo :  thin  pivR  riMt  to  a  now 
method  of  tmitins  tlir  tlirory  of  functioiiH.  The  fiei-ctupmrnt 
of  thJH  nielhiil  ia  npivinlly  <)uc-  Ui  the  Uuttinpm  Bchwil  with 
which  Iho  namr-B  of  Iticmniin  nnd  Klein  «re  «»  ckwely  rmio- 
ciat«d.  In  ISM  Iti«mann  wroU-  kin  eelr1ir«te<l  memoir  nn 
the  hypotheKCH  on  which  geometry  is  founded  ;  to  thin  HDhject 
1  allude  lielow.  Thin  whh  Huccee<led  liy  rnenKMn  on  elliptic 
functions  And  on  the  diKtrihutian  of  prinm:  tlteiie  iiAva 
been  nlrmdy  iiM-iitiiinif].  fAKlly,  in  muttipio  periialic  fane- 
tionR,  il  in  Imntly  Ido  mucli  to  my  that,  in  hin  memoir  in 
Bnrrhnrtll'»  Jaurnid  fur  tV.'iT,  he  did  fur  the  AlieliAn  fnnctiuna 
whnt  Al^l  had  dune  fur  (he  elltplic  functioMK. 

EUifitir  ami  AhrHaii  FHiuiiutif,  or  lliijhrr  TriitoimmKlrt/*. 
I  have  Already  nlludnl  to  tlw  renearches  of  Iffftntfrr,  Umit*, 
AM,  Jni-obi  and  Ririitniin  on  elliptic  and  Aliclian  (unctionn. 
The  naliject  han  licen  aIko  diHcuKMsl  hy  (Aniitng  otlier  wril<-m) 
J.  a.  Nf^tJiain  (XnXG^Xt^Hl)  of  K<)ni^ii>>c>n:,  who  wrote  (in 
IK44)  on  the  hypcrvlUptic  or  duuhlc  thetA  fundjon  mhI  on 
functionH  of  two  vAriAbleH  with  fnur  periods ;  A.  Gojtti 
(1813-1847)  of  Berlin,  who  dincuiwedt  hyperelliptic  fniic- 
tionii;  L.  Krmi'^ter*  (It<23~ie91)  of  Berlin,  who  wrote  on 
elliptic  Tunctionn;  /,.  Kiinigubfryrr^  uf  Heidellierg,  who  dia- 
GUHsed  the  trwrisfbrniation  of  the  double  tltet*  fnni^ion ; 
/'.  BriomAi  (I824-1S9J)  of  Home,  who  wrote  on  dliptic 
«nd   hyperelliptic   fuRctionii ;    ifenry  Umith  61  Oxfbn),  wbo 

*  8«e  the  iDlrodaclion  lo  y.lliflUrhe  Ftmrii'itim  bj  A.  BniM|wr, 
Mcond  edition  (ed.  bj  F.  lliillcr),  Htllc,  lN90i  and  OtttirUt  itfr 
Thenrit  irt  fliiptiichfH  TrmiKtmleHlfn,  by  I..  Kiinimbemer,  Laipiif, 
1879.  On  the  hinloiy  of  Abrlian  (nuctinnn  tm  the  Tmmmtllom  ^  Ike 
BritUh  Aaecialian,  toL  Lini,  LondoD.  IN9T.  pp.  3411— SMI. 

t  See  CrrlU'i  Jimnutt.  voL  xi(T.  IS47.  pp.  377— SIS  1  ui  oUtMiy 
»Mct,  hj  Jaoobi,  i*  pm  on  pp.  313 — 3IT. 

;  nil  ooUrated  worki  in  4  toIbhi**,  edited  by  X.  BsbmI,  an  nvr 
In  courM  of  paUicAtion  at  Lcipiift.  XKUi,  Ac. 

I  8m  Ua  iMtom,  pablidbcd  at  LetpaiB  in  IH7I. 


474     lUTHUUTIOi  OF  THE  NINETEENTH  OINTVBT. 

dJKiWMtl  thtt  trsnofomimtion  theory,  the  tbeU  and  omega 
fnpctkuu,  and  Mrtain  fuuctiuns  of  the  modulus;  A.  Ca^Uif 
of  Ounbridg«,  whu  was  the  finit  to  work  out  (in  1845)  the 
theory  of  doubly  infinite  products  and  detenuine  their  perio- 
dicity, and  who  hu^  written  at  Ifiigth  on  the  connection 
between  tlie  meArcliPs  of  I..egcndre  und  Jocobi ;  and  C.  Urr- 
mite  (1822-1901)  of  fiiris,  whuHc  reitearclies  are  toiHtty 
ooncemed  with  the  tranHforuiatioii  tlieury,  and  the  higher 
develupinent  of  the  thetA  functions. 

Weientntn*.  The  Hubject  of  higher  trigtinometry  was 
put  on  »  tomewliut  dilFercut  footing  by  the  n-searchat  uf 
WeierHtnuts.  A'nrl  Writmlnuii,  horn  in  Westphalia  on 
OctubiT  31,  1815,  and  died  at  Berlin  on  Fubruarj-  19,  1897, 
wiui  one  of  tlie  grcHitciil  tiiiithcuuiticiuns  of  the  nineteenth 
century.  He  took  no  purt  in  public  uflkirs ;  his  life  was 
uneventful ;  aiul  he  Hpeiit  tlw  IhkI  forty  yeunt  uf  it  at  Berlin, 
where  be  was  prufc»ur. 

With  two  hnuicliL's  uf  pure  niHtheiiiutics — i>lliptic  and 
Abelian  functions  und  the  theory  of  functiuos— bis  name  is 
iitsejMrulily  Limiiti-U')!.  His  citrlier  researcheu  on  elliptic 
fuiictiuiiti  n-lutMl  lu  the  thctn  functions,  which  he  treated 
under  u  niudifitil  fiinn  in  which  they  are  expres.-'ilile  iu 
powfrw  uf  the  uiudulu.-^.  At  a  liittT  period  he  developed  a 
rtM-thod  for  treating  all  elliptic  functions  in  a  sj-muieti^cal 
luiinner.  lie  wili  naluniliy  led  to  this  nietliod  by  his  re- 
seiirelii-s  on  the  ^p'lieral  theory  of  functions,  which  cuonlinated 
und  euinpriMil  various  lines  of  invest  i<;ii  I  ion  previously  treated 
indeiieiideiitly.  In  piirlieuhir  he  constructed  n  th<.Mry  of 
unifunu  aiudytie  func-tiuns.  The  repreucnlalion  uf  functions 
by  inhnito  pruduel.t  and  M-ries  also  claiuie«)  his  especial 
attention.  Besides  functions  he  also  wrote  or  lectured  on 
tlio  nature   td   the   uaauiuptiona   niiule   in   analj^is ;  un    llie 

•  W.'icr«tn>»i'<i  colktlrd  wuik>  un  uuw  in  ouunc  ul  i»iic.  Berlin. 
lVJt,*tc.:Mketcbe4t.fUiii Garter  by  O.  UiUmtXeUeraiKl  bjH.  foiacarA 
•ra  tiid-n  iu  .lrl>i  Mutk<malie.i,  lttU7,  tul.  ill,  pp.  7U— M,  aud  ItW, 
rol.  tUI,  pp.  1— IB. 


MATHEMATICS  OF  THK  NINETEENTH  CENTUBT.     475 

micnias  of  vnrintiiinH ;  nml  on  tlip  lh«Niry  of  minima  narfiiw*. 
His  mclhorls  bit  noticpnldo  for  thfir  wSde-rmwhinj;  awl 
gonoral  cJinmcUr.  RcwTit  invr«ti(;iiliiinM  on  plliptic  futiHionn 
have  bc«n  largely  tiasprf  on  Wpiith trass's  iiictliodn. 

AniDng  otlicr  prominriit  nintlipmntipinnn  wlio  Iihtc  recrotly 
writton  on  thin  nnhject,  1  mny  mciitioii  tli^  imn>m  of  G.  If. 
IM/Aen*  (1844-1899),  nn  officer  in  tlic  French  army,  whow* 
invG9<tigation>i  irprr  Ini^-ly  foundni  on  Weientnuis'it  work ; 
F.  C.  XletH  of  Giitlin^n.  who  has  written  on  Ahelian  func- 
tions, elliptic  modular  functions,  and  hyfMTplliplic  fnnclions; 
//.  A.  .Sefiimrz  nt  Berlin;  //.  Il't^r  of  Stmsxlmrg;  J/.  A'Sth^r 
of  EHangen;  W.  SMht  i,{  Aix-la-Oiapelle;  /'.  G.  A*rv>A«itiM«  of 
Rerlin  ;  nnil  ./.  W.  /,.  Uhixlwrr  nf  Cambriilge,  who  has  in 
pnrticular  developed  the  theory  of  the  wtji  fanction. 

The  moHt  nsuiil  text-lKMikx  <if  lo-<lay  oi>  Mliptk  functions 
are,  perhngn,  thftse  liy  J.  Tannery  and  J.  Mitlk,  4  volnmes, 
Paris,  18'J3-]90I;  liy  1'.  K.  Ap)M<ll  am)  K.  I^tcour,  Vmrw, 
1896;  l>y  H.  \Vel>er,  Brunxwick,  IttUl  ;  and  liyG.  H.  Halphen, 
3  TDlames,  PhHn,  lt«K6-lt(9I. 

Thf,  Tki^ry  of  F.iiiri!.,„f.  I  have  alrrady  mentioned 
that  the  modem  theory  of  functionH  is  lan^fly  due  to  Wder- 
RtrnH.-i.  It  is  a  singularly  attractive  Huhject,  and  pnimiww  to 
prove  an  important  and  fnr-Tvachin|;  lirnnch  of  niathemnticH. 
Historically  it  may  lie  naid  to  have  Iieen  initiated  .1>y  A. 
Cnitrhy,  who  laid  the  foundation)!  i>f  the  tlmiry  of  synectio 
functioHK  of  a  complex  variahle.  Work  on  thme  lines  was 
continued  l>y  J.  fAouvi/le,  who  wnit«  cliietly  on  doubly  periodic 
fnnction.t.  Tliese  invwtigations  were  extended  and  connected 
in  the  work  by  A.  Itriot  and  J.  C.  Jimtqiitl,  and  sulMcqncntly 
were  further  developed  \>j  C.  Ilmnife  {18'-"2-l90l). 

Next  f  may  refer  to  the  researches  on  the  tlieory  of 
algebmic  functions  which  have  their  origin  in  G.  F.  B. 
Aiemann'a  paper  of  IS.^O;  in  continuation  of  wiiich  //.  A. 

'  A  (keteli  of  H^alphen'a  life  and  wotka  U  %nta  in  IJmnilU't  Jomrmml 
for  1S89,   pp.   U5— SS9,  and  In  the  CompUi  RtmAn*,   IStO,  raL  n. 


476    MATUisiiATiai  or  tuk  ninetkbntu  cbhtuet. 

Sekwan  of  Berlin,  etUbliiihed  aooimlely  certaiii  theoraBi  cf 
which  the  proolii  given  by  Rienuuin  were  open  to  objection. 
8ubiieqaenily  /'.  C  JkUin  of  Gottingen,  oonnecled  Hienuyw's 
theory  of  fiuictionii  with  the  theory  of  groapii  nnd  wrote  on 
autoiuorpliio  and  uiodular  functionn;  //.  Paineare  of  Buii|  nbo 
wrote  on  autuniorphic  funcUous,  and  on  the  general  theory 
with  Bpecial  applications  to  ditferetitial  equations ;  and  qnite 
recently  Cr.  Paiuleci  of  PariH  has  written  on  onifomi  funo- 
tions ;  and  A',  itenwel  of  Berlin  on  algebraic  functions. 

I  'liave  already  said  that  the  work  of  WeiemtraHS  shed  a 
new  light  oil  the  whole  Hubject.  His  theory  of  analytical 
functions  has  been  developed  by  M.  G.  JlUiafj^Lejfier  of 
Stocklioliu,  one  of  the  mout  distinguished  of  living  nuithe- 
niaticiiiiis ;  and  t\  Ut^rmilr^  J\  £.  Ap^trfi^  C.  £,  Pieard^  and 
£,  GourstU^  uH  of  PariN,  have  ulsu  written  on  special  brandies 
of  the  general  theory. 

As  U*xt-boi>kH  I  iiiay  mention  A.  li.  Forsyth's  Theory  of 
Funciiont  of  a  Comjiiex  VaruiUe^  sccoiul  edition,  Cambridge, 
IIKX);  />iV  Funktionntheot'if'  by  J.  Petersen,  Copeiiiiageii,  1898; 
ALft^  Theortm  by  Ji.  F.  JUker,  Cauubridge,  1897;  Des 
Foturtioiis  uhjebrhjHfH  by  P.  E.  Appell  and  £.  Goursat,  Pauris, 
1895  ;  Tlie  Theory  of  Fuiiciions  by  J.  llarkness  and  F.  Morley, 
Loiuloii,  1893;  and  iterhaps  C.  Neuuianii's  Vorletntajen  iiber 
HiemaniCt  Tfieorit  dcr  AbtVuclmn  hUeyrule^  second  edition, 
Leipzig,  1884. 

Higher  Afyehni,  The  theory  of  numbers  may  be  con- 
sidered as  a  higher  arithmetic,  and  the  theory  of  elliptic 
and  Alx;liaii  functions  as  a  higher  trigonometry.  The  theory 
of  higher  algebra  (including  the  theory  of  e<|UatiouM)  lias  also 
attracted  coiiHiderable  attention,  and  was  a  favourite  subject 
of  study  of  the  uiathemuticiaiiH  whom  I  propose  to  mention 
next,  though  the  interests  of  these  writers  were  by  no  means 
limited  to  this  subject. 

Cauchy*.     AuyuMtin   LouU  Cauehtf^  the  leading   repre- 

•  Sec  Im  rit  et  Ut  Iniruiix  dt  CaucK\|\i>|  'U.XiXwsci^^'^^NswftWi^ Paris, 


\ 


CAUCHY.  477 

8entAti%'e  of  the  Freiicli  hcIiooI  of  analysis  in  the  nineteenth 
century,  was  Ijorn  at  Parin  on  Aug.  21,  1789,  and  dierl  at 
Soeanx  on  May  25,  1857.  He  was  educated  at  the  Poly- 
technic school,  the  nursery  of  «>  many  French  mathematicians 
of  that  time,  and  adopted  the  profession  of  a  ci%il  engineer. 
His  earliest  mathematical  paper  was  one  on  polyhedra  in 
181 1.  Legendro  thought  so  highly  of  it  that  he  asked  Cauchy 
to  attempt  the  solution  of  an  analogous  prohlem  which  had 
hafiled  pre%*ious  investigators,  and  his  advice  was  justified  by 
the  success  of  Cauchy  in  1812.  Memoirs  on  analysis  and  the 
theory  of  numbers,  presented  in  1813,  '1814,  and  1815,  shewed 
that  his  ability  m-as  not  confined  to  geometry  alone :  in  one 
of  these  papers  he.  generalized  some  results  which  had 
been  established  by  Gauss  and  Legendro ;  in  another  of 
them  he  gave  a  theorem  on  the  number  of  values  which 
an  algebraical  function  can  assume  when  the  literal  con- 
stants it  contains  are  interchanged.  It  wan  the  latter 
theorem  that  enabled  Aljel  to  shew  that  in  general  an 
algebraic  equation  of  a  degree  higher  than  the  fourth  cannot 
be  solved  by  the  use  of  a  finite  nnmlier  of  purely  algebraical 
expressions. 

To  Cauchy  and  Gauss  we  owe  the  scientific  treatment  of  v 

series  which  have  an  infinite  number  of  tenns,  and  the  former 
established  general  rules  for  investigating  the  oonvergency  and 
divergency  of  such  series.  It  is  only  a  few  works  of  an  earlier 
date  that  contain  any  discussion  as  to  the  limitations  of  the 
series  employed.  It  is  said  that  lAplace,  who  was  present 
when  Cauchy  read  his  first  paper  on  the  subject^  was  so  im- 
pressed by  the  illustrations  of  the  danger  of  employing  such 
aeries  without  a  rigorous  investigation  of  their  oonvergency 
that  he  put  on  one  side  the  work  on  which  he  was  then 
engaged  and  denied  himself  to  all  visitors,  in  order  to  see 
if  any  of  the  demonstration  given  in  the  earlier  volumes  of 
the  Jf^eofitfiM  elbsf^  were  invalid;  and  he  was  fortunale 

186a.  A  eoHvMs  sditiMi  of  hb  works  is  now  bting  issosd  by  lbs  PkcBsb 


478     MATHEMATICS  Of  TUB  NIinETBBllTH  ODITimT. 

snoogh  tofiiidtluitiiomalemlerronluidlieQiiUiiisiBlradaMd. 
Tlie  troaiment  oi  Merifti  Mid  off  iUe  ffaudaoienUl  oooeepUoiMi 
of  the  calcaliiii  in  mofit  off  the  text-buolM  then  cnnent  wne 
based  on  Baler's  works,  and  to  any  one  timined  to  aeenmte 
habita  off  thought  was  not  free  from  objection.  It  is  one  of 
the  chief  merits  off  Cauchy  that  he  phMsed  thoMO  subjects  on  a 
logical  foundation. 

On  tlie  restoration  in  1816  the  French  Academy  waa 
purgtxl,  liiid,  in  Npite  of  the  indigiuition  and  scorn  off  French 
scientific  iMjciety,  Cauchy  HOCt»pted  a  seat  which  was  prucured 
fur  him  by  the  expulHion  of  Monge.  He  was  also  at  tlie  same 
time  iiuide  pnift^sor  at  the  Polytechnic ;  and  his  lectures  there 
on  algebraic  anulysin,  tlie  calculus,  and  tlie  theoiy  of  curves 
were  publiHlunl  as  U^xt-lioukM.  On  tlie  revolution  in  1830  he 
went  into  exile,  uiul  was  first  appoiiitetl  profesHor  at  Turin, 
wlieiii*t)  he  HfMiii  iiuivchI  t4)  Prague  to  undertake  the  education 
t>f  tlu*  C«»iiite  (le  C/liiiiiilionl.  Ho  retunuHl  to  Fnuice  in  1837  ; 
liiul  ill  1848,  and  u^^aiii  in  1851,  by  Hpecial  tli.sp«*iiMatitiii  of  tlie 
eiii|M*n>i*  wiis  ullowetl  to  iKVupy  u  cliiiir  of  iiiatlieiiiatics  witliout 
taking  the  oath  of  allegiance. 

liin  lU'tivity  was  prudigiouH,  ami  from  18341  to  1859  he 
published  in  the  traiisiictioiis  of  the  Academy  or  tlie  Compteg 
Jiftttitm  over  (»00  original  iiiemoim  and  aliout  150  reports. 
They  cover  an  extnu>nliiiarily  wich*  raiigtf  of  Hubjects,  but  are 
of  very  uiie€|ual  merit. 

Aiiuiiig  the  IIIOI11  iiiip«irtant  of  his  ri*m*arclu«  are  the 
diHcuHMioii  of  t4*HtH  for  the  convergt*ncy  of  m*rifM;  the  deter- 
mi'iatioii  of  the  iiuiiiU^r  of  real  aiul  imaginary  niots  of  any 
i..^eliraic  4H|uatioii ;  his  method  of  calculating  theiie  roots 
appitixiiiiately ;  Iuh  theory  «if  the  Hymmetric  functions  of  tlie 
C(M*tticieiitH  of  e4|iiatioiis  of  any  tiegree  ;  his  ik  priori  valuation 
of  a  quantity  less  than  the  leiist  ditt'erence  lietween  the  roots 
of  an  equation;  his  papt^ni  on  detenninants  in  1841  which 
assitkted  in  bringing  tlieiii  into  general  use ;  and  his  investiga- 
tioiiu  on  the  theory  of  numbers.  Cauchy  also  did  sonietliing 
to  rfduce  the  art  of  determining  definite  integrals  to  a  science. 


\ 


CAUCHY.      AHOASn.      HAMILTON.  479 

but  thiH  branch  of  the  integml  cnlculuH  slill  rniMin!!  without 
much  HjKtrm  or  mctho'l :  thp  ruli^  for  fiiidiAg  the  princifMl 
TAlaes  of  inlfgralH  was  enuncintH  by  him.  The  ntlculun  of 
residues  wnw  hix  invention.  His  pmnf  of  Taylor's  theorem 
wrmn  to  hftve  iirip-inntnl  fn)m  n  iHwupwion  of  the  doulile 
periodicity  of  elliptic  funclion.s.  The  mennii  of  nlirwing  * 
connection  liptwe*>n  diirrrent  hmnclies  of  a  subject  by  giving 
imaginnry  vnlui^  to  independent  varinbles  i«  largely  Hoe  to 
him.  He  also  gnve  a  direct  analytical  method  for  determining 
planetary  ine<|Dalitie)i  «>f  long  period.  To  phynicH  he  con- 
tributed memoirs  on  waves  and  on  the  quantity  of  light 
reflected  fnim  the  Hurfneex  of  nielaln,  as  well  al  Other  pApem 

Argand.  T  may  mention  here  the  name  of  Jntn  Snhrrt 
Argau'l  who  was  liom  at  Geneva  on  .luly  '22,  171)8,  and  died 
cire,  IPa.'i.  In  hiM  A'«/ii,  iiLiued  in  IfOB,  he  gave  a  geometrical 
reprenenlntion  of  a  eoniplex  nomlier,  ami  applied  it  U>  shew 
that  every  nlgebrnii-  e>|iintion  hn.^  a  root:  this  was  prior  to 
the  memi<irH  of  (.iauHs  miil  Ciurhy  on  tlie  MHie  subject,  Init 
the  eswi}'  did  not  attract  much  ntli-ntion  when  it  was  firat 
pabljsbed. 

An  earlier  demonstratiim  that  ^/(-  1 )  may  be  inter- 
preted to  indicate  ]ierpeit<lieulnrity  in  iwn-dinH'nHitmal  spAce, 
and  even  the  cxtenMoii  of  the  idr.t  In  ihrMMlimennional 
space  by  a  melh(Nl  fnreshadiiwing  the  usn  of  iiualfmions  had 
been  giien  in  a  memoir  by  C.  Wcssel,  prcspnte<l  to  the  Copen- 
hagen Academy  of  8cience)i  in  March,  1797;  other  memoirs 
on  the  same  subject  hiul  been  publishetl  in  tlie  /'Ai/oso/iAicn/ 
TmtimiftioHii  for  18*16,  and  liy  II.  Kiihn  in  the  TrtinmtetumM 
for  U-W  of  the  St  Petersburg  Academy* 

Bamiltonf.     In  the  opinion  of  w>mc  wril«n,  the  theory 

■  B«  W.  W.  Beman  in  Ihe  Proerrdingi  of  ttit  AmeHrmm  J-neimUtK 
/or  fkr  AdnHtrmfKt  of  .Seicnrv,  tdL  un,  1897. 

t  8ea  Uw  life  «(  HaniiltoD  (with  a  iMiognphy  of  Us  wrili^a)  br 
R.  P.  OraTo,  tbiM  volamra.  l>ublin.  1KH3-H9 :  the  Irwli^  b«U  ars 
ghsn  in  an  artida  in  the  Karlk  Briihh  Brtitw  for  1886. 


o 


480     MATHEMATICS  Or  THB  XntRKnTH  CMTUBI. 


of  qaatemkiiui  will  be  ulttiiiAldy  wtaeincd  one  oC  the  gMtti 
diflooveriett  uf  the  ninetoeiith  oeniary  in  pim  ■■fhwnilica . 
that  difloovery  in  due  to  iSliV  IVHiiam  Xowtm  timmiiiam^  wlw 
was  born  of  Hootch  parentii  in  Dublin  on  Aog.  4,  1M6^  and 
died  there  on  Sept.  2,  18G5.  Hie  educntion,  whieh  was 
carried  on  at  home,  Heenia  to  have  been  singularly  discnniTO : 
under  tlie  influence  of  an  uncle  who  was  a  good  linguist  he 
first  devoted  luniself  to  linguistic  studies;  by  the  time  he 
was  seven  he  could  rend  Jjitiuy  Greek,  French,  and  German 
with  facility;  and  when  thirteen  he  was  able  to  boast  that 
he  was  familiar  with  as  many  languages  as  he  had  lived  yean. 
It  was  about  this  time  that  he  came  across  a  copy  of 
Newton's  UuittriMl  Ar%thnuiie\  this  was  his  introduction 
to  muderu  analysis,  and  he  soon  mastered  the  elements  of 
analytical  geometry  and  the  calculus.  He  next  read  the 
J*rincipM,  and  the  four  vulumes  tlit*n  published  of  Laplace's 
Micaniqne  ctieste.  In  the  latter  he  detected  a  mistake, 
and  his  paper  on  the  subject,  written  in  1823,  attracted 
conMiderable  attention.  In  the  following  year  he  entered 
at  Trinity  College,  Dublin :  his  university  career  is  unique, 
for  the  chair  of  iiHtronoiiiy  liec<iming  vacant  in  1827,  wliile 
he  was  yet  an  undergraduate,  he  was  asked  by  the  electors 
to  stand  for  it,  and  was  elected  unanimously,  it  being  under- 
stood that  he  should  Ije  left  free  to  pursue  his  own  line  of 
study. 

His  earliest  pa|HT  on  optics  was  written  in  1823,  and 
published  in  1828  under  the  title  of  a  Tfteoty  qf  SyMUma  of 
Hay*^  to  which  two  supplements,  written  in  1831  and  1832, 
were  afterwards  oddtnl ;  in  the  latter  of  these  the  phenomenon 
of  conical  n^fruction  is  predictwl.  This  was  followed  by  a 
pa|ier  in  1827  on  the  principle  of  Yaryiny  Jc<tOM,and  in  1834 
and  1835  by  memoirs  on  a  Getteral  Melhodin  Dynamics  i  the 
subject  of  tliiH>ieticHl  dynamics  being  pruperly  treated  as  a 
branch  of  pure  mathematics.  His  lectures  on  QuiUemion* 
were  published  in  1852;  some  of  his  results  on  tliis  subject 
would  seem  to  have   been   previously  discovered  by  Gauss, 


\ 


J 


HAMtLTOX.      nRASRMAXX.      BOOLE.  481 

but  thcMC  were  unknown  nnd  anpubliMhctl  until  lunj;  after 
HaniilUni'it  clentli.  Aii)i>ii(:st  ]■»  otlicr  )>ii|K'ni,  I  nuiy  MpocUlly 
mention  one  i>n  thn  (orTii  or  tlic  wilution  of  tlie  gnneral 
algelimic  equation  of  the  Kftli  (lefjiw,  which  conRnneil  Abel's 
conclusion  that  it  cnnnnt  )>e  ex|in-s.seil  )i^  A  finite  nnniber  of 
purely  nlgehraicnl  exprrssinnH ;  one  on  fluctunting  fanctinns; 
one  on  the  hodi^raph ;  anil  lastly  one  on  the  namerical 
Rolntinn  of  rlitTerenttal  equntions.  His  Kl-m-nU  t<f  Qunttmuttu 
was  issued  in  1866:  of  tliii*  a  competent  authority  Myn  that 
the  Rictliodn  of  nnalyHiH  then;  given  xhcw  as  great  an  advance 
over  thoHe  of  analytical  geometry,  iis  the  latter  shewed  over 
those  of  Euclidean  geometry.  In  inore  recent  timea  the 
Buhjet't  has  l)cfn  further  deieloped  liy  P.  (J,  Tnit  of  Ktlinlwrgh. 

Hamilton  was  painfully  fastidious  on  what  he  pobliiiheil, 
and  he  left  a  Ini^  collection  of  mniiuncrijitH  which  are  now 
in  (he  library  of  Trinity  CoUege,  Dublin,  some  <if  which  it  \» 
to  be  hoped  will  lie  ultimately  printed. 

GrasBmann*.  Tlie  i<lea  of  non^immutative  algebras 
and  of  quatemiotia  seems  to  hare  oecurn'd  In  (irawtmann 
and  Boole  at  about  the  ttnme  time  as  to  Hamilton.  l/enwttiH 
Giinlhrr  Grammann  was  liom  in  Stettin  on  April  15,  1809, 
and  died  there  in  1S77,  He  was  professor  at  the  gymnasiam 
at  Stettin.  }Iis  researches  on  non-com  mutative  algebras  are 
containol  in  his  Auml-hiKmrfJ'/ir',  tint  putilished  in  IH4  and 
enlarged  in  1862.  Tlie  .scientific  trentnient  of  the  fundamental 
principles  of  algebra  initiated  by  Hamilton  and  OrasMiiann, 
was  continunl  by  De  Morgan  and  Boole,  and  nubneqnently  waa 
further  developed  )>y  H.  Hankel  in  his  votk  on  complexei^ 
1667,  and,  on  somewhat  different  lines,  by  0.  Cantor  in  his 
memoirs  on  the  theory  of  irrationals,  1671 ;  the  discusnoa  is 
however  no  technical  that  I  am  unable  to  do  more  than  allude 
to  it.  Graasniann  also  investigated  the  prgpertin  of  bonw- 
loidal  hyper-sp»c& 

Boole.    ff«or?«  /too/0,  bom  at  Lincoln  en  Nov.  3, 1815, 

*  Hh  eollMtod  worlu  in  3  nilinwa,  edited  by  F.  Bncsi.  sn  now  la 
ooHM  of  toM  •!  Ufaig,  18H,  *e. 


482     1IATUEMATIC8  Or  THK  MlNITBIimi  GUITUET. 

mmI  died  At  Cork  on  Deo.  8,  1864,  independently  invented  m 
ByHtooi  off  non-conunutative  algebra.  From  lib  nemoire  on 
linear  tranidomiatuNu  part  of  the  tlieory  oC  invariante  hae 
developed. 

Onloia*.  A  new  development  off  algebra — ^the  theory  oC 
groups  of  Kubetitutionn — was  snggeiited  by  EvmriM€  OalaUf  who 
proiuited  to  be  one  of  tlie  moHt  original  niathematieiant  oC 
tlie  iiineU>entli  century,  bom  at  Fariai  on  Oct  26,  1811,  and 
killtMl  in  a  tluel  on  May  30,  1832,  at  the  early  age  of  20. 

The  uuidt^ni  tlieury  of  groupH  originated  with  tlie  treat 
UM*nt  by  CliiloiM,  Caucliy,  and  J.  A.  tii^rret;  their  work  it 
niaiuly  concerned  with  finite  diiicontinnous  lubstitution 
grou|M.  TliiH  lino  of  inveMtigation  haH  lieen  punuied  by 
C.  Jonlun  of  Pariii  and  fi.  Netto  of  Straiwburg.  Tlie  pro- 
blcMii  of  o|H*ruti(iiiii  with  diHcoiitinuouM  groups,  with  applica- 
tiiiiiM  to  the  tliiHiry  of  fuiictioiiis  lia«  lM*«*ii  furtlu«r  taken  up 
by  (unions  oth«*i*H)  K.  CI.  KniU'iiiuM  of  IkTlin,  F.  C.  Klein 
of  (i«ittiiiK<*ii|  Hiiil  \V.  lluriiMiilo  formerly  of  diiibriilge  and 
now  of  ttriHMiwicii. 

De  Morgan  t*  Atiyusius  De  XforyaH,  lioni  in  Madura 
(Miulnut)  in  June,  180C,  and  died  in  London  on  March  18, 
1H71,  wiis  etlucateil  at  Trinity  College,  CAinbridge,  but  in 
the  then  Mtute  of  the  law  wa«  (an  a  unitarian)  ineligible  to 
a  fellowMliii).  In  1828  he  iMH^iiiie  professor  at  the  newly- 
eslublished  uiiiverHity  of  London,  which  in  the  saiiie  iniiti<- 
tiitioii  tot  that  now  known  as  University  College.  There, 
thmugh  hiH  works  and  pupils,  he  exercised  a  wide  influence 
«iii  Kii^lihli  iimth(*iiiiiticiuns.  The  Ix>iidon  Matheniatioal 
S'STii^ty  was  largely  his  creation,  and  he  took  a  prominent 
|Nirt  ill  the  |ir4K*i*e< lings  of  the  Uoyal  Astronomical  Society, 
lie  was  (le«*|)ly  n*ail  in  the  philosophy  and  history  of  mathe- 
iiiaticH,  hut  th<*  results  are  giv<*n  in  scatten*d  articl(*s;  of 
thes4*    i    hav4*    iiuule    considerable    use    in    this    book.     Hiii 

*  On  hilt  iuvciitiKutioiis,  uco  the  edition  of  bis  works  with  an  intro* 
duotioii  by  K.  Ticurd,  Tsris,  1HSI7. 

t  Hill  Ufi*  WArt  written  by  bis  widow,  U.  E.  i)«  Morgan,  LfNukm,  letM. 


\ 


I 


DE   MOROAS.      CATLEV.  483 

incnioin  on  tlio  fuundntion  of  algebra;  hbi  trratise  on  Um 
diflerpntinl  calculuH  puhlislin]  in  11^12,  n  work  of  ^rmt  Ability 
■ml  notiMwble  for  the  rignniuH  treatiiipnl  of  inHnite  Hiin ; 
and  h'\K  Krticl<^  on  tlic  cnlculuN  of  functionn  And  on  thn  theory 
of  nruhnbilitieH  nn  worlliy  of  Tipectnl  not«.  The  Article  on 
the  cnlculua  of  functions  conttiinti  nn  ini'Mtif(ation  of  the 
principle  of  syinbolic  miKoning,  but  the  np[dicfttinns  dp*! 
with  the  Holution  of  functional  t>(|antionB  rather  th«n  with 
thp  ^neral  thwiry  of  functions. 

Cayley*.  Another  Knglinhman  who  will  he  si  ways 
rsl<%nml  one  of  the  gn-nt  mntliPnintirinDA  of  thin  prolific 
centur)-  war  Arihur  CH;/f'ff.  Cnyley  wiw  Imrn  in  Surivy  on 
Aug.  IR,  1821,  And  nftrr  nlucntinn  at  Trinity  College,  Cmm- 
Inidge,  wBs  CfkUed  to  the  bnr.  But  bin  interesitii  centmd  on 
mnthrmnticR ;  in  180;i  h*-  wnn  eltvlt^l  KitillftHitn  Profwwor  at 
Cnnibridgi*,  «n<l  lip  Bpent  them  the  rest  of  hia  life,  lie  died 
on  Jun.  2C>.  180.1. 

Cnylpy's  irritingn  dial  with  consifiemble  parts  of  mndem 
pure  niAlheinnticM.  I  hare  nlrendy  inontionnl  hia  writingH  on 
the  partition  of  nuinlx'ni  and  on  rlliptic  (nnctiunN  trenKd  from 
■lacobi's  point  of  view:  bin  lnt<-r  writings  on  elliptic  fanc- 
tioun  dealt  mainly  with  the  theory  of  transformation  and 
the  modular  p<{uatiiin.  It  in  however  by  bin  tnventigations 
in  AnAlyticftl  geometry  and  on  higher  algelira  that  he  will  be 
best  remenil>ered. 

In  analytical  groinetry  the  mnception  of  what  is  called 
(perhaptt,  not  very  hnppily)  the  aimftile  is  due  to  Cayley.  As 
stat^l  by  hinixetf,  the  "  theory,  in  ctFect,  is  that  the  metrical 
properties  of  a  figum  ar^  not  the  properties  of  the  figare 
considered  prr  nc.but  its  properties  when  considered  in 
connection  with  another  figure,  namely,  the  conic  termed  the 
absolute";  Itence  metric  properties  can  be  mbjected  to  de> 
Kcrii'tive  treatment  He  contributed  largely  to  the  general 
theory  of  curt-es  and  MUrfnceit;  his  work  ratting  on  the 
*  Bis  coUcded  works    ia   13  Tolomcs  were  immti  si  CsabtUtji. 


484    MATHKiuTics  OF  THK  NiNnnina  cnmmT 


aapamptioD  of  the  neoenarily  close  oonneqtfaMi  bei< 
Imucd  aiid  gBOiiietrical  opermtioiuak 

In  higher  algebm  the  theory  of  uiYaryuiia  k  due  to  Objkj: 
his  ten  clanical  memoim  oii  binary  and  tenmry  Coring  and  hie 
researches  on  matrices  and  nou-conunntativjs  algebras  mark  an 
epoch  in  the  development  of  the  snbject.     | 

Sylvester.  Another  teacher  of  the  same  time  was  Jamm 
JottjJi  Sytcesier^  bom  in  Liindon  on  Sept.  3;  1814,  and  died  on 
March  15,  1897.  He  too  was  educated  it  Osmbridge^  and 
while  there  formed  a  life-long  friendship  with  Gayley.  like 
Cayley  he  was  called  to  the  bari  and  yet  preserved  all  his 
interests  in  mathematics.  He  held  professorships  successively 
at  Woolwich,  Baltimore,  and  Oxford.  He  had  a  strong 
personality  and  was  a  stimulating  teacher,  but  it  is  difficult 
to  descnbe  his  writingH,  for  they  are  numerous,  disconnected, 
and  (liscurslve.  i 

On  the  theory  of  numljers  HylvcHter  wrote  valuable  papers 
oil  the  (listriliution  of  priiiicH  and  on  the  partition  of  numbers. 
On  anjilysis  ho  wrote  on  the  calculus  and  on  ditferential 
equations.  But  perhaps  his  favourite  study  was  higher 
algebra,  and  from  his  numerous  memoirs  on  this  subject  I 
may  in  particular  single  out  those  on  canonical  forms,  on  the 
theory  of  contra  variants,  on  reciprucants  or  differential  in- 
variants, and  on  the  theory  of  equations  (notably  on  Newton's 
rule).  I  may  also  odd  tliat  he  created  the  language  and 
notation  of  considerable  parts  of  those  subjects  on  which  he 
wrote. 

The  writings  of  Cayle}*  and  Sylvester  stand  in  mariEed 
contrast :  Cayley's  are  methodical,  precise,  formal,  and  com- 
plete ;  Sylvester'H  are  impetuous,  unfinished,  but  none  the 
lesK  vigorous  and  stimulating.  Both  mathematicians  found 
the  greatest  attraction  in  higher  algebra,  and  to  both  that 
subject  in  its  modem  form  is  deeply  indebted. 

Lie*.     Anotlier  great  analyst  of  the  nineteenth  century, 

*  Se«  the  obituary  noiioe  by  A.  R.  Poriyth  in  the  Ytw  Book  of  tko 
Xoyal  Sociil^,  London,  1901. 


\ 


I 


485 

to  whom  I  must  Mllurfe  lierc,  in  Marhtu  SnfiAHt  Lv,  hom  on 
Dec  12,  1943,  uml  ilidl  on  Frh.  IH.  IFI99.  Lie  wm  edncntnl 
At  ChrintiMniA,  whence  he  nlitAJned  n  trnvflling  KchiJAnthip, 
mhI  in  the  rouno  of  hin  jimrnryn  itinile  thf  )»c(]iMintAnc«  of 
Klein,  DnrliouK,  Anil  Jortlnn,  to  whiMO  inffuonca  his  siilMe- 
quent  cnrcer  in  Inrf^lj'  <lup. 

In  1K70  h«  diHcovrnvI  the  tntnisfiimiAtion  by  which  k 
sphere  cnn  lie  made  to  ciirrvxpnrKi  (o  m  ftrftight  line,  And,  by 
the  URf  of  which,  tlfruTrniN  on  A^-;:n-gnlni  of  lines  can  In 
trnnHlntol  into  tlteorr-nts  on  nf:j::re^U^  of  wplicn-fl.  Thin  wah 
followed  hy  It  thniH  on  lite  llimry  of  tniigcntiAl  tmimformationa 
for  npncc 

In  1H73  he  lircnnK-  profcwHir  nt  Chrislinnin.  IliH  eAriirat 
rvBn»rch«^  l«"PC  wpiv  on  the  rrlntion*  Iwtweeii  dilTerentiAl 
diUAlionn  »nd  inKnitp«ii.>nl  IrnnHfonnnlinnH.  This  iiatunitly 
ltd  him  to  the  general  theory  of  linitc  continuous  gnnipH  of 
sulistilutions ;  tlw  rpsultH  of  his  invest i^ntiann  on  thin  subject 
Ate  embodied  in  hi"  Tkrnrir  r/rr  Tmii'j'ormalionfgrHppnL, 
IjeipriR.  3  vo^lme^  1  liSS- 1  MM.  Hr  proceeilral  next  to 
conKtder  the  theory  of  infinite  rontinuouH  groups,  nnd  it  in 
expeeted  thnl  bin  eoncIuHionii  on  this  xubjecl  will  be  puhlixbed 
shortly.  Alioat  IK79  IJe  turned  hin  Attention  i"  aliflerrntiiU 
geometry  ;  a  syntemntic  expnilion  of  this  is  in  course  of  issue 
in  bis  fttanurlri'!  rUr  lUriihrntigftraHKjbrmnfioneiu 

Lie  NCeniK  to  \v\w  lieen  dixsppninted  And  soored  hj  the 
■tW'iicr  of  Any  geneml  recognition  of  tbe  vAlue  of  his  reaultt. 
ItepatAtion  rxnie,  but  it  cnnie  slowly.  In  1K86  be  moi'ed  to 
Leipzig,  Ami  in  ll:<9S  Imck  to  CbriKtiAiiin,  where  a  post  hAd 
been  created  for  him.  He  bnNNlcd  tiowerer  over  whAt  he 
deemed  was  the  undue  neglrel  of  the  put,  snd  the  lii^ipineaa 
of  tbe  Inst  decade  of  his  life  was  much  Afleded  by  it> 

Bo  mAny  other  writen  hnve  trettted  the  saliject  <A  Higher 
Algrhm  (including  therein  the  theory  of  fornw  maA  the  theoij 
of  equAtions)  tbKt  it  is  difficult  to  saniinKriHi.  tbsir  concluioM 
or  to  single  oat  individBAls. 


486     MATUfilUTICB  or  THI  MUllTIBim  guituet. 


The  oon  vergeocj  of  leriM  hM  bean  diawwwi  hfJ.i»^ 
(1801-ia59)  of  Zorich,  V.  L.  F.  Arimt^  (1822*1900)  tho 
necreUiy  of  the  French  Aoedemj ;  £.  Ji.  Kumitmr  (1810-1803) 
of  Berlin ;  V.  Dini  of  Pisa ;  A.  Priugtheim  of  Munich*;  mmI 
Sir  (Jwrye  Gabriel  JSiakeei  of  CMubridge^  to  whom  the  well- 
known  theorem  on  tlic  critical  valuee  of  the  sums  of  periodio 
lerioH  Ih  due. 

On  the  tlieory  of  gruupii  of  iiubiititutioiis  I  have  alreadj 
mentioned  the  work,  on  the  one  hand,  of  Galoi«|  Cauchy, 
Serrct,  Jordan,  and  NetUi,  and,  on  the  other  hand,  of  Fro- 
beniuM,  Klein,  and  Burusido  in  connection  with  dinoontinuomi 
groups,  and  that  of  Lie  in  connection  with  continuoua 
groapM. 

I  may  also  mention  the  following  writem.  C*.  ir.  Bar^ 
eharJi*  (1817-1860)  of  Berlin,  who  in  particuhu-  diacnaaed 
generating  functions  in  the  theory  of  equations,  and  arithmetic* 
geometric  means.  C.  ihnniU  (1822-1901)  of  Paris,  who  dis- 
cussed the  theory  of  associated  covariants  in  binary  quantics, 
the  theory  of  ternary  qualities,  and  who  applied  elliptic 
functions  to  tind  a  solution  of  the  quintic  equation  and  of 
Lame's  differential  equation.  Eurico  BtHi  (1823-1892)  of 
Pisa  and  /*.  nrionchi  (1824-1897)  of  Rome,  both  of  whom 
discussed  biliary  quantics.  S»  //.  Arou/wid,  who  developed 
symbolic  methods  in  connection  with  the  invariant  theory  of 
qualities.  J\  J.  Oonian^  of  Erlaugen,  who  lias  written  on  the 
theory  of  equations,  the  theories  of  groups  and  forms,  and 
shewn  tliat  there  are  only  a  tinite  number  of  concomitants  of 

*  Ou  the  retfearchen  of  lUabe,  Bertrand,  Kunimer,  Dini,  and  Prings- 
h«im,  see  the  liulUtin  of  tlie  New  York  (American)  Mathcmaiioil 
Society,  vol.  ii,  18U2-3,  pp.  1—10. 

t  Stokeii*H  coUected  matheiuatical  and  pkjriiical  papers  in  Si  Toliunes, 
were  iiMUed  at  Cambridge,  1880,  18h3. 

X  A  collected  edition  of  hia  worka,  edited  by  O.  Hettner,  waa  iaaued 
at  Berlin  in  1888. 

i  An  edition  of  hia  work  on  invariaiitu  (detenninanta  and  binaiy 
foriutt),  edited  by  G.  Kerachenateiiier  waa  i!<!»ued  at  Leipxig  in  three 
fdoniea,  1883,  1887,  1893. 


\ 


XATHEXATICS  OF  THE   MNETKKNTH  CENTURY.      487 

qiuntics.  R.  F.  A.  Clebick*  (1833-1872)  of  Giitlingen,  wIm 
independently  investignteH  the  theory  of  bintuy  forma  in 
■om«  papers  collecUxl  and  puUishcd  in  lh71  ;  lie  idso  wn>t« 
on  Abolian  functions.  P.  A,  MiK.\fii/iiii,  nn  officer  in  tlw 
British  nnny ,  who  tins  written  on  the  conncvlion  of  sym- 
metric functions,  invnrinnts  and  covnrinntM,  the  concomitanU 
of  binary  for'niK,  nnd  combinatory  nnnlj-His.  /'.  C.  Klein  ot 
Gottingen,  who,  in  addition  to  his  rmrArcliP",  alrcndy  men- 
Uonctl,  on  functionH  nnd  on  finite  tliscontinuuuH  gruiipH,  Ium 
written  on  diilerential  etiuations.  .1.  H.  F'irmj/th  of  Camliridge, 
who  has  (icvelopcil  the  iheury  of  invariants  and  the  genera) 
theorjr  of  differential  e<|uationN,  temnriants,  and  qunlcmnrtants. 
U.  Paiiileri  of  TAris,  who  lins  written  on  the  theory  *4 
diScrcntial  equations.  And  lastly  D.  llUbrrt  of  Gottin(p>n, 
who  bos  tre«tid  the  thciry  of  Imniogcni-uus  forms. 

No  account  nf  contemporary  writings  on  higlter  algebra 
would  Iw  complete  without  a  reference  to  the  admirable  text- 
bookD  pro«)uced  hy  G.  Salmon,  provost  of  Trinity  College, 
Dublin,  in  his  Ili^/k'-.r  Algebra,  and  by  J.  A.  Senct  (lfil9- 
Ig^S),  pivfewior  at  the  Rorlxmne,  Paria,  in  bis  6'oHra 
ifAlyibre  mpirieurr,  in  wiiicli  the  chief  discoveries  o<  their 
respective  authors  are  em)x)die<l.  An  admirable  historical 
summary  of  the  theory  of  the  complex  variable  is  given  in 
the  Vprliwungm  iihtr  dU  eom/At-xen  Znhleii,  Leipzig,  lf67  by 
H.  Hankel  (1839-1873)  of  Tubingen. 

Analylienl  Groin'irg.  It  will  l)e  convenient  next  t«  otll 
attention  to  another  division  of  pure  niathematica — analy- 
tical geometry —which  has  been  greatly  devehqwd  in  recent 
years.  It  has  been  studied  by  a  hont  of  inwiem  writers,  Imt 
I  do  not  propose  to  ttmcrihe  their  investigations,  and  I  shall 
content  myself  by  merely  mentioning  the  nanm  of  the  Mlow- 
ing  mathematicians 

*  In  aeooilDt  of  fan  lite  snd  work*  u  prialrd  in  Um  ilalktmallKht 


488     MATUBIIAT1C8  OP  THI  mNETEENTH  CEMTDET. 


1846)  both  of  Dublin,  were  two  of  the  eariieet  Britiih  wrileis 
in  thu  oeiitury  to  take  up  the  snbject  of  aoAljtieel  geometry, 
bat  they  worked  nminly  on  linett  already  stndied  by  olheim. 
Frevh  develupuienUi  were  iutroduoed  by  Juiiuit  MdektrX 
(1801-18G8)  of  Bonn,  who  devoted  hiniaelf  especially  to  the 
study  of  algebraic  curvee^  of  a  geometry  in  which  the  line 
is  the  element  in  space,  and  Ur  the  theory  of  congruenoes 
and  complexes;  his  equations  connecting  the  singularities 
of  curves  are  well  known :  in  1847  he  exchanged  his  chair 
for  one  of  physicts  and  his  subset|uent  researches  were  on 
spectra  and  magnetiKm. 

The  majority  of  the  memoirs  on  analytical  i^metry  by 
A.  CajfUy  and  by  Henry  Smith  deal  with  the  theory  of  curves 
an€l  surfaces ;  the  most  remarkable*  of  those  of  L,  O.  iletm 
(1811-1874)  of  Munich  are  on  the  plane  geometry  of  curves; 
of  those  of  t/.  6'.  Darbuux  of  Paris  are  on  tlie  geometry  ol 
surfaiccH ;  and  of  tlio-se  of  6*.  //.  liatpktn  of  Paris  are  on  tlie 
singularitieH  of  surfaces  and  on  tortuous  curves.  The  singu- 
laritivH  of  curves  and  Hurfaoes  have  also  been  considered  by 
//.  U.  Zeuthtn  of  Copenhagen,  and  by  //.  C  //.  SchaUri^  of 
Hauibuq^.  The  theory  of  tortuous  curves  has  been  discussed 
by  J/.  Niither  of  Erlangeu ;  and  H.  F.  A.  CleUck  of  Gottingen 
has  applied  AlMfl's  theorem  to  geometry. 

Among  more  recent  text-lxMks  on  analytical  geometry 
are  J.  O.  Darlioux's  llteorie  ytneraU  dtg  Murjwcttt^  and  Zes 
ityuietiUM  ortlu*goiuuijc  tt  its  cuontonnees  ctirviiiyueM;  R.  F.  A. 
Clebsch*8  Vui'Usnnytu  iiber  G'touteirie^  edited  by  F.  Lindenuum; 
and  G.  Salmon's  Conic  ^Sections,  Gtometry  of  Three  UiuietunoHs^ 

*  tioo  luH  Trfaiitr  uh  iomt  uttr  (Jeomeirical  Meikodt,  Iiondon,  1878. 

t  St-e  bin  ouUccUmI  wurks  otliUtl  kjr  JvlUtt  and  Hsughtou,  Dublin, 
1880. 

X  l'liu:ker*H  collectetl  work«  in  two  volutueM,  edited  kj  ▲•  HuhfltnflJMS 
and  F.  Puck«;U,  w«rv  publiidiiMl  at  Leip^g,  1875,  l89d. 

I  HcbuberI'M  lecturer  Iiayo  U,*«*u  pttUulied  by  F.  Lindeuisna,  two 
rolunit^  Leipzig,  187u,  18U1. 


\ 


MATHEMATICS  OF  THE  NIWETEENTH  CEKTURT.      489 

Mid  UighfT  Plane  Cunf» ;  in  which  the  chief  disoorerie*  of 
these  writers  lire  eiiilKulipH. 

Fin»lly  I  mnj  allarle  to  the  extension  of  the  mlyect-nwttw 
of  nnnlf  licnl  peoiiictry  in  the  writinpi  iif  //.  G.  (iroMmitHH  in 
1M4  itml  1X63,  G.  F.  H.  Hirmnan  in  IC-il,  A-  Caykg,  wnl 
othera,  by  the  intruduction  of  the  i(lf»  of  flpacc  oC  h  ilinwii- 
aioiiK. 

Aii'ihjna.  Atiinng  thine  win i  haw  extcwle"!  the  mnge  of 
«n>l}-siM  (incluilirig  the  enlculuH  nntl  iliffhrcnlj»l  eqM«tiun«)  or 
wliom  it  i»  difficult  to  plaro  in  uny  of  the  precedinR  dOepirica 
»ra  tho  following,  whom  I  mention  in  nljikMiictiatl  onier. 
P.  E.  A}i/irlf  of  PiiriH;  J.  L.  F.  firtmufi  i4  P*riB;  O'.  itool« 
of  Cork  ;  A.  L.  Caurhif  of  Paris;  J.  G.  Onrhcux  of  Paria; 
A.  It.  Fiirtt/th  of  Cnmhridgc;  >'.  6'.  Frobe'iiut  of  Berlin; 
lazarw,  Furha  of  Berlin  ;  G.  II.  IlalfA'-H  of  Parw ;  C.  G.  J. 
Jaeobi  of  Berlin ;  C.  JontnH  of  Pnrin ;  L  KuHigtb/rgtr  of 
Ileidelbersi  Sv}Aus  K<tr»hrrli*  (18.(3-1891)  (rf  Btockholm; 
M.  S.  Lit!  of  Leijixig;  //.  Poau^ri  of  Pwis;  G.  F.  B. 
AurHrtidi;  //.  A.  .Sfhtntrz  of  Berlin;  J.  J.  ^y/cvffer;  and 
A'.  H>t«n«fmM  of  Berlin,  who  developed  tha  calcnlas  of 
vkriationH.  * 

Si/iilhflir  Gtmnrtry.  The  writers  I  have  ittentimied  kboTO 
mcMtly  concerned  theniKelvcM  with  xnalyiUH.  I  will  next 
descriljc  some  of  the  mon>  important  works  produced  in  thia 
centDiy  on  synthetic  ge<imelryt. 

Modem  synthetic  geonietry  ntay  be  nnid  to  liave  had  its 
origin  in  the  works  of  Monge  in  lAW,  Cnmot  in  1803,  Mid 
Poncelet  in   182*2,   but   thctie   only   foreshadowed   the   great 

■  See  tba  Bnlltlin  dn  tiemfn  imUhfrnaliqnn,  TOl.  XT,  pp.  911— SSff. 

t  The  Aptrfu  hutarijHe  «ir  rorifinr  tt  U  Mrilnpfttttut  Jn  mithoia 
ra  ^mftrit  bj  M.  Chmnles,  I'srin,  woond  rdiUoo,  IBTS,  aad  JNr 
ifwlhtlivht  Ofomelrie  im  Jllfrikum  «arf  in  rfrr  Kranit  bf  Th.  HcTi^ 
Bttmufciirg,  1886.  aonUin  itHewrting  ■nmniariCToHlW  hhtMy^Cmui— tai% 


400     MATHEMATICS  OF  TUC  NIMRSKNTH  GUITUET. 

extaoaioD  it  was  to  rooeive  in  Germaaj,  of  wUoh  HtninBr  awl 
yon  Staudt  are  perhaps  the  best  known  expooenta. 

Bteiner^.  /ocoft  ^S^euier,  *«  the  greatest  geooielffieian  riaee 
the  time  o£  Apolloniiis,"  was  bom  at  Utsensdorf  on  Mareh  18^ 
1796,  aiul  died  at  Bern  on  April  l»  18G3.  Ub  fiither  was  a 
peasant,  and  tlie  boy  had  no  opportunity  to  learn  reading  and 
writing  till  t)ie  age  of  iourteen.  He  subsequently  went  to 
Heidelberg  aud  thenoe  to  Beriin,  supporting  himself  by  giving 
lessons.  His  SyitenuUueke  EiUwiektluH^en  was  published  in 
1832,  and  at  once  uiade  his  reputation:  it  contains  a  full 
discussion  of  the  principle  of  duality,  aud  of  the  projective 
and  homugraphic  relations  of  rows,  pencils,  J:c.,  based  on 
metrical  properties.  By  the  influence  of  Crelle,  Jaoobi,  and 
the  von  Huniboldts,  who  were  impressed  by  the  power  of  this 
work,  a  chair  of  geometry  was  created  for  Steiner  at  Berlin, 
and  he  continued  to  ciccupy  it  till  his  death.  The  most  im- 
portant of  his  other  reiiearches  are  contained  in  papers  which 
appeared  originally  in  Crelte*t  Jountai,  and  are  embodied  in 
his  Sy  idtetUche  Geonieirie :  these  relate  chiefly  to  properties 
of  algebraic  curves  and  surfaces,  pedals  and  roulettes,  and 
maxima  and  minima;  the  discussion  is  purely  geometricaL 
Steiner's  works  ma}'  be  considered  as  the  classical  authority 
on  reccnit  synthetic  geometry. 

Von  Staudt.  A  system  of  pure  geonietr}',  quite  distinct 
from  that  ex|>oundMl  by  Steiner,  was  proposed  by  Earl 
Geoiy  ChrUtuiH  von  Siamii^  bom  at  Rothenburg  on  Jan.  24, 
1798,  and  died  in  18G7,  who  held  the  chair  of  mathematics  at 
Krlangeii.  In  his  (Jfouulrit  dtr  Layt*,  published  in  1847,  he 
constructed  a  system  of  <;eoiaetry  built  up  without  an}*  refer- 
ence to  numljer  or  magnitude,  but,  in  spite  of  its  abstract 
form,  he  succeeded  by  means  of  it  alone  in  c^stablishing  the 
non-metrical  projcH;tive  properties  of  figures,  discussed  imagin- 
ary points,  lines,  aiul  planes,  and  even  obtained  a  geometrical 

*  Htciuer*!!  collected  workii,  edited  by  WeietntrsaA,  were  iiiraed  in  two 
Yolmues,  lierliu,  1881-8*2.  A  Hkelcli  of  his  life  is  contsiued  in  Ihe  Kraa- 
Nrrmiy  an  Sieintr  by  G.  F.  Geiser,  SdiaffhauaeD,  1874. 


\ 


MATHEMATICS  OF  THE  NINETEENTH  CENTURY.     491 

definition  of  a  number:  these  view's  were  farther  elaborated 
in  his  Beiirdfje  z^tr  Geomefrui  der  Lng^.^  1856-1860.  This 
geometry  is  carious  and  brilliant,  and  has  been  used  by 
Colmann  as  the  basis  of  his  graphical  statics. 

As  usual  text-books  on  83'nthetic  geometry  I  may  mention 
M.  Chasles's  TraiU  de  geometrifi  snjtrrieurf^  1852;  J.  Steiner's 
Variesungen  iifmr  tyniheti8^.he  G^onuiri^f  1867 ;  K  Cremona's 
Elemenit  de  geonteirie  prnffrtir^^  Eii;;lish  translation  by 
C.  Leudesdorf,  Oxfonl,  second  edition,  1893 ;  and  Th.  Heye's 
Geomeirie  der  Lagr^  Hanover,  1866-1868,  English  translation 
by  T.  F.  Holgate,  New  York,  part  1,  1898.  A  good  presen- 
tation of  the  mortem  treatment  of  pure  geometry  is  contained 
in  the  Introduzione  ad  una  florin  geomftrica  deiU  curve  piane^ 
1862,  and  its  continuation  Pretiminari  di  una  teorin  geo- 
wmtriea  delfe  euperfieie  by  Luigi  Cremona,  of  the  Polytechnic 
School  at  Rome. 

The  di  (Terences  in  ideas  and  methods  formerly  obser\*ed  in 
•naljrtic  and  synthetic  geometries  tend  to  disappear  with 
their  further  development. 

Xon^EttrJidfnn  Geom^frg.  Here  T  may  fitly  add  a  few 
worrls  on  recent  investigations  on  the  foundations  of  geometry. 

The  questirm  of  the  truth  of  the  assumptions  usually 
made  in  our  geometry  had  lieen  considered  liy  J.  Haccheri 
as  long  ago  as  1733;  and  in  more  recent  times  had  been 
discussed  by  N.  I.  Loliatschewsky  (1793-1856)  of  Kasan, 
in  1826  and  again  in  1840;  by  Ctauss,  perhaps  as  early  as 
1792,  certainly  in  1831  and  in  1846;  and  l»y  J.  Bolyai 
(1802-1860)  in  1832  in  the  appendix  to  the  first  volume  of 
his  father's  T^ntamen;  but  Riemann's  memoir  of  1854 
attracted  general  attention  to  the  subject  of  non-Euclidean 
geometry,  and  the  theory  has  been  since  extended  and 
siropliAc^  by  various  writers,  notably  by  A.  Cayley  of 
Ounbridge,  E.  Beltrami*  (183^-1900)  of  Pbvia,  byH.UT. 

•  A  list  of  BeUnaTt  wriliBgR  Is  tfnn  fai  the  Ammtt  H  mmiemmikm. 


492    MATuniATics  OP  TBI  MiiimsirrH  cdituet. 

TOQ  Hdmholis  (1831-1894)  of  Berlin,  liy  F.  a  Kim  cf 
GoUing^,  and  by  A.  N.  WhiteiiMd  of  (SMnbrMga  in  hit 
Uuivenud  Algebra,  The  mabjeci  b  «>  technical  Ihat  I  confine 
niyiielf  to  a  hare  iiketch  o£  the  arganient^  finom  which  the 
'idea  in  derived. 

That  a  space  of  two  dimenoionii  iihoald  have  the  geometrical 
prupertieift  with  which  we  are  made  familiar  in  the  studj  of 
elementary  geometry,  it  in  neceiMary  that  it  should  be  possible 
at  any  place  to  construct  a  figure  congruent  to  a  given  figure ; 
and  this  is  so  only  if  the  product  of  the  principal  radii  of 
curvature  at  every  point  of  the  space  or  surface  be  constant. 
There  are  three  species  of  surfaces  which  possess  this  property: 
namely,  (i)  spherical  surfaces,  where  the  product  is  poniti^'e ; 
(ii)  plane  surfaces  (which  lead  to  Euclidean  geometry),  whero 
it  is  zero ;  and  (iii)  what  Beltrami  has  called  pseudo-spherical 
surfaces,  where  it  is  negative.  Moreover,  if  any  surface  be 
bent  without  dilution  or  con  traction,  the  measure  of  curvature 
remains  unaltered.  Thus  these  three  species  of  surfaces  are 
types  of  thriHs  kinds  on  which  congruent  figures  can  be  con- 
structed. For  instance  a  plane  can  be  rolled  into  a  cone,  and 
the  system  of  geometry  on  a  conical  surface  is  similar  to  that 
on  a  plauie. 

These  kinds  of  space  of  two  dimensions  are  distinguished 
one  from  the  other  by  a  simple  test.  Through  a  point  of 
spherical  space  no  giKxletic  line — a  geodetic  line  being  defined 
as  the  shortest  distance  between  two  points — can  be  drawn 
parallel  to  a  given  geodetic  line.  Through  a  point  of  Euclidean 
or  plane  space  one  and  only  one  geodetic  line  (i.e.  a  straight 
line)  can  lie  drawn  parallel  to  a  given  ginidetic  line.  Through 
a  point  of  pseudo-spherical  space  more  than  one  geodetic  line 

*  For  referencen  see  my  Maihtmaiical  RrcreatioMM  and  ProUewu^ 
LoDtlun,  IHlMi,  chup.  z.  A  historical  nummary  of  the  treatment  of  noo- 
Euclidean  Reometry  i«  giwn  in  J.  Frinchaurs  EUmtMt§  tier  oXmAmteu 
Geometries  Leipzig.  187U ;  aud  a  report  by  G.  B.  Ualnted  on  prugreas  in 
the  subject  is  printed  in  Science,  N.  S.  voL  z,  New  York,  Itl99,  pp.  545— 
5J7. 


\ 


NON-EUCLIDEAN  OEOMETRT. 


493 


can  be  drawn  fiarallel  to  a  given  geodetic  line,  and  all  these 
lines  form  a  pencil  whose  vertical  angle  in  constant. 

It  might  be  thought  that  we  have  a  demonstration  that  oar 
space  is  plane,  since  through  a  given  point  we  can  draw  only 
one  straight  line  parallel  to  a  given  straight  line.  This  is  not 
10^  for  it  is  conceivable  that  our  means  of  observation  do  not 
permit  us  to  say  with  absolute  accuracy  whether  two  lines  are 
parallel ;  hence  we  cannot  use  this  as  a  means  to  tell  whether 
our  space  is  plane  or  not.  A  better  test  can  be  deduced  from 
the  proposition  that  in  any  two^limensional  spboe  of  uniform 
curvature  the  sum  of  the  angles  of  a  triangle,  if  it  differ  from 
two  right  angles,  will  tliflfer  by  a  quantity  proportional  to  the 
area  of  the  triangle.  Hence  it  may  happen  possibly  that, 
although  for  triangles  such  as  we  can  measure  the  diflerence 
is  imperceptible,  yet  for  triangles  which  are  millions  of  times 
bigger  there  would  be  a  sensible  diflerence. 

If  space  be  spherical  or  pseudo-spherical,  its  extent  is  finite; 
if  it  be  plane,  its  extent  is  infinite.  In  regard  to  pseudo- 
spherical  space,  I  should  add  that  its  extent  may  lie  in6nite, 
if  it  be  constructed  in  space  of  four  dimensions. 

In  the  preceding  sketch  of  the  foundations  of  non-Euclidean 
geometry  I  have  assumed  tacitly  that  the  measure  of  a  distance 
remains  the  same  everywhere.  Klein  has  shewn  that,  if  this 
lie  not  the  case  and  if  the  law  of  the  measurement  of  distance 
lie  properly,  chosen,  we.  can  obtain  three  systems  of  plane 
geometry  analogous  to  the  three  systems  mentioned  abova 
These  are  called  respectively  elliptic,  parabolic,  and  hyperbolic 
geometries. 

The  above  refers  only  to  hyper-space  of  two  dimensions. 
Naturally  there  arises  the  question  whether  there  are  diflTerent 
kinds  of  hypor-spaoe  of  three  or  more  dimensions.  Riemann 
shewed  thai  there  are  three  kinds  of  hyper-apace  of  three 
dimensions  having  properties  analogous  to  the  three  kinds  of 
hyper-qwce  of  two  dimensions  already  discussed.  These  are 
differantiated  by  lbs  test  whether  at  every  point  no  geodeCioal 
or  gas  gsodctical  SMJMS^  or  a  hsdcnl— of  fpodetfral 


494     MATHKlfATICS  OP  THI  imnBmilTH  CINTDET. 


surfaoet  oui  be  drawn  |MuraUel  to  m  given  tbifaee:  n  gwwtotknl 
surface  being  defined  at  snob  tbat  every  ^eodetie  Una  joining 
two- points  on  it  lies  wboUjr  on  tbe  snrfaoeJ 

Aieehauieg.  I  tball  conclude  tbe  cbapter  with  a  lew  noCea 
— more  or  kflt  diKCumive— on  brancbeii  of  matbematies  of  a 
lent  abstract  character  and  concerned  witli  firoblenis  tbat 
occur  in  nature.  I  commence  by  mentioning  the  sabjcet  of 
meclianicfi.  Tlie  subject  .  may  be  treated  graphicaUy  or 
analytically.  j 

Orapkict,  In  the  science  of  graphics  rules  are  laid  down 
for  Holving  various  pn»bleiiis  b}'  the  aid  of  die  drawing-board : 
the  iikmIph  of  calculation  which  are  pt^niiissible  are  considered 
ill  modem  prujtH;tive  geometry,  and  tlie  subject  is  closely 
connect4Hl  with  tliat  of  modern  geometry.  This  method  of 
attiu-kiii;;  questioiiH  Iiom  licen  hitherto  applied  chiefly  to 
pitibleiiis  in  iiu*chiinic8,  elasticity,  and  electricity ;  it  is 
enptvijilly  useful  in  engineering,  and  iii  that  subject  an 
aveni;^*  <lniu>(litHiimii  ouglit  to  1m)  able  to  obtain  approximate 
solutions  of  iiHMt  of  the  equjitionSy  diflernfjiitial  or  otherwine, 
with  which  he  is  likel}'  to  bt)  concerned,  which  will  not 
involve  errors  greater  than  would  have  to  be  allowed  for  in 
any  case  in  eon.sec|ueiice  of  our  imperfect  knowledge  of  the 
structure  of  the  materials  employed. 

The  thcH>ry  may  be  said  to  have  originated  with  Poncelet's 
work,  hut  I  believe  that  it  is  only  within  the  last  twenty 
years  that  systematic  expositions  of  it  ha^e  been  published. 
Among  the  liest  known  of  ::uch  works  I  may  mention  the 
iiraphUche  Statik^  by  (*.  CulManu,  Zurich,  187ri,  rect*ntly 
f*<lite<l  by  W.  Hitter;  the  Lfzioni  di  $tatica  yntfiat^  by 
A,  Facuro^  Padua,  1H77  (Fn*nch  translation  annotated  by 
P.  Trrrier  in  2  volumes,  1879-85);  the  Valcttlo  yrafico^  by 
L,  Cremuna,  Milan,  1879  (English  translation  by  T.  H.  Beare, 
Oxford,  1889),  which  is  largely  founded  on  Miibius's  work; 
La  ittatique  yrapliiqtie^  by  M.  Xrvy,  Paris,  4  volumes,  1886- 
S8;  imd  La  siatica  yrafica^  by  C.  Sairotti^  Milan,  1888. 


\ 


GRAPHICAL  MECHANICa  49^ 

The  general  character  of  the^ie  Yiooks  will  be  mifHciently 
illustrated  bj  the  following  note  on  the  contentii  of  Calmann'n 
work.  Culmann  commences  with  a  description  of  the  geo- 
metrical representation  of  the  foar  fundamental  processes  of 
addition,  subtraction,  multiplication,  and  dinsion;  and  pro- 
coeds  to  evolution  and  involution,  the  latter  lieing  effected  bj 
the  use  of  equiangular  spiral  He  next  shews  how  the  quan- 
tities considered — such  as  volumes,  moments,  and  moments  of 
inertia — may  be  represented  by  straight  lines ;  thence  deduces 
the  laws  for  combining  forces,  couples,  drc. ;  and  then  explains 
the  construction  and  use  of  the  ellipse  and  ellipsoid  of  inertia, 
the  neutral  axis,  and  the  kern  ;  the  remaining  and  larger  part 
of  the  book  is  devoted  t^i  shewing  how  geometrical  drawings, 
made  on  these  principles,  give  the  solutions  of  many  practical 
problems  connected  with  arches,  bridges,  frameworks,  earth 
pressure  on  walls  and  tunnels,  «i-c. 

Tlie  subject  has  been  treated  during  the  last  twenty  years 
by  numerous  writers  especially  in  Italy  and  liermany,  ami 
applied  to  a  large  numlx*r  of  problems.  But  as  T  stated  at 
the  beginning  of  this  chapter  that  I  should  as  far  as  possible 
avoid  discussion  of  the  works  of  living  authon  I  content 
myself  with  a  bare  mention  of  the  subject. 

Clifford*.  I  may  however  afld  here  a  brief  note  on 
Cliflbrd,  who  was  one  of  the  eariiest  English  matliematicians  of 
the  latter  half  of  this  century  to  advocate  the  use  of  graphical 
and  geometrical  methods  in  preference  to  analysis.  WiUiam 
Kingdan  Clifford^  bom  at  Exeter  on  ]!klay  4,  1845,  ami  died  at 
Madeira  on  March  3,  1879,  was  educated  at  Trinity  College, 
Cambridge,  of  which  society  he  was  a  fellow.  In  1871  he  was 
appointed  professor  of  applied  mathematics  at  University 
College,  London,  a  post  which  he  retained  till  his  death.  His 
remarkable  felicity  of  iilustration  and  power  of  seizing  analo- 
giea  made  him  one  of  tto  mcMt  brilliant  ezpoandeni  oC  mathe- 

*  For  flMhsr  dotsils  of  Olilbra*t  life  sad  work  iw  the  aatfaoriliM 
quoM  \m  tiw  aiMo  on  him  in  the  tHetlcmr^  €f  StMAmmk  flV%f»  »»«%-> 


/ 


496  ANALmCAL  MBCHAMICa 

nmUoal  prineiplflt.  Hb  liealth  iukd  in  1876|  when  tho  writor 
of  tliis  book  andertook  his  work  for  m  few  MoatlM;  C9iSiid 
then  went  to  Algeria  and  returned  at  the  end  of  the  jear,  bat 
only  to  break  down  again  in  187&  His  most  important  works 
are  his  Tkeory  of  BiquatenUatu,  On  ik$  CUutifioiUiam  ff  Zoei 
(unfinished)^  and  Tke  TAeory  qf  Grapks  (unfinished).  His 
CauMiieal  Du§eetiaH  of  a  BUmaHU*§  Sur/ae^  and  the  iSZemsnCf 
of  Dynamic  also  contain  much  interesting  matter. 

Anaiytieal  MeckanicM.  I  next  turn  to  the  question  of 
mechsnics  treated  analytically.  The  knowledge  of  mathematical 
mechanics  iif  auiids  attained  hy  tlie  great  mathematicians  of  the 
U&Hi  century  may  be  sai€l  to  be  summed  up  in  the  admirable 
XifeuniqtuB  antifytique  by  Lagrange  and  TraUi  lU  mieaniqiM 
by  PoiMHOiiy  and  the  application  of  the  results  to  astronomy 
forms  tlie  subject  of  Laplace's  Mtamique  cHeUe,  These  works 
have  been  already  deHcribed.  The  mechanics  of  fluids  is  more 
diflicult  tlian  that  of  nolids  and  the  theory  is  less  advanced. 

Theoretical  Staticty  especially  the  theory  of  iJke  poieniiai 
and  attractions^  has  received  considerable  attention  from  the 
mathematicianH  of  thin  century. 

I  liave  already  uieiitioiied  that  the  introduction  of  the  idea 
of  the  potential  is  due  to  Lagrange,  and  it  occurs  in  a  memoir 
of  a  date  an  early  as  1773.  The  idea  was  at  once  grasped  by 
Laplace  who,  in  his  memoir  of  178-1,  used  it  freely  and  to 
whom  the  credit  of  the  invention  was  formerly,  somewhat 
unjustly,  attributed.  In  the  sauie  memoir  Laplace  also 
extended  the  idea  of  zonal  liarmonio  analysis  which  had 
been  introduced  by  Legendre  in  1783.  Of  Gau$tf$  work  on 
attractions  1  liave  alreiuly  spoken.  The  theory  of  level 
surfuc4*s  and  lines  of  force  is  largely  due  to  CKasirs  who 
also  deU^nniuiHl  the  attraction  of  an  ellipsoid  at  any  external 
point.  I  may  also  hero  mention  the  BaryctntrutckeM  Calcul 
published  in  1826  by  A.  F.  SfUbitis^  (1790-18G8)  who  was 
one  of  the  best  known  of  Gauss's  pupila 

*  Ui«  collected  works  were  pubHth«d   at   Loipsis  in  4  volomss, 
lba5-7. 


ORERN.  497 

Oreen^.  Gtnrtje  Green  was  one  of  the  earliest  writera  of 
tills  century  who  investigated  further  the  properties  of  the 
potential.  Green  was  born  near  Nottingham  in  1793  in  a 
huniYile  condition  in  life,  and  died  at  Cambridge  in  1841. 
Although  self-educated  he  contrived  to  get  access  to  various 
mathematical  books,  and  in  1827  wrote  a  paper  on  the  po- 
tential— in  which  the  tenn  was  first  introduced — proved  its 
chief  pniperties,  and  applied  the  results  to  electricity  and 
magnetism.  This  contains  the  imporUint  theorem  now  known 
by  his  name.  Tliis  remarkable  paper  was  seen  by  some  neigh- 
bours who  were  able  to  appreciate  the  power  shewn  in  it :  it 
was  published  by  suliscription  in  1828,  but  does  not  seem  to 
have  attracted  much  attention  at  first.  Similar  results  were 
independently  establishcxl,  in  1839,  by  Gauss  to  whom  their 
general  dissemination  was  due. 

In  1832  and  1833  Green  presentcMl  papers  to  the  Cambridgo 
Philosophical  Society  on  the  (M|uilibrium  of  fluids  and  on 
attracti<ms  in  space  of  n  dimensions,  and  in  the  latter  year  his 
memoir  on  the  motion  of  a  fluid  agitated  by  the  vibrations  of 
a  solid  ellipsoid  was  read  before  the  Royal  Society  of  Edin- 
burgh. In  1833  he  entered  at  Caius  College,  Cambridge,  and 
was  subsequently  elected  to  a  fellowship.  He  then  threw 
himself  into  original  work,  and  produced  in  1837  his  papers  on 
the  motion  of  waves  in  a  canal,  and  on  the  reflexion  and 
refraction  of  sound  and  light.  In  the  latter  the  geometrical 
laws  of  sound  and  light  are  deduced  by  the  principle  of  energy 
from  the  undulatory  theory,  the  phenomenon  of  total  reflexion 
is  explained  physically,  and  certain  properties  of  the  vibrating 
medium  am  deduced.  He  also  discussed  the  propagation  oC 
light  in  any  crjrstalline  medium. 

Theoretical  Dynnmiat^  which  was  cast  into  its  modem 
form  by  Jacol>i,  has  been  studied  by  most  of  the  writera  aliove 
mentioned.  I  may  also  here  repeat  that  the  principle  oC 
^'Varying  Action"  wm  eiabofmlcd  by  Sir  William  Hamilton 

•  A  eoUeeM  •SA&a  M  Ontm^u  works  wm  poblMMd  al  GtabrMge 
to  1871. 


498     MATUBMATIGS  OF  TBI  NHfRBKHTH  CDITUmT. 


in  18379  and  the  ^  HaniiltoniMi  equatiooft*  wara  gmm  im 
1855;  and  I  may  further  call  attention  to  the  djnaotteal 
invesUgationi  of  J.  £.  £.  Bour  (1832-1866)  and  of  J.  K  F. 
Bertrand  (1822-1890X  both  of  Pkria.  The  um  of  geneffmliied 
cooidinateti,  introduced  by  Lagrange,  has  now  beoonie  the 
customary  means  of  attacking  dynamical  (as  wM  as  OMUiy 
physical)  problems. 

As  usual  text-books  I  uuty  mention  those  on  particle 
and  rigid  dynamics  by  £.  J.  Kouth,  Cambridge;  Legmu  mtr 
rintegraiion  dett  equatiotis  differttUleUet  de  la  mueoMiquiB  by 
P.  Pkinlov^,  Paris,  1895,  and  Vi^kiegration  det  equaiionsde  la 
mtcanique  by  J.  Graindon^  Brussels,  1889.  Allusion  to  the 
treatiMe  on  Natural  PliiloHophy  by  Sir  William  Thomson  (now 
Liird  Kelvin)  and  P.  O.  Tait  iuay  be  also  here  made. 

On  the  mechanics  of  fluids,  liquids,  and  gases,  apart  from 
the  physical  theories  on  which  they  rest,  I  propose  to  say 
nothing,  except  to  refer  to  the  memoirs  of  Green,  Sir  George 
Stokes,  Lord  Kelvin  (lietter  known  as  Sir  William  Thomson), 
and  von  Helniholtz.  The  fascinating  but  diflicult  theory  of 
vortex  rings  is  due  to  the  two  writers  last  menti<m(*d.  One 
prolileui  in  it  liiis  lieen  also  conMidei^  by  J.  J.  Thomson,  but 
it  is  a  subject  which  is  an  yet  ljcyon€l  our  powers  of  analysis. 
Tlie  subject  of  sound  may  lie  treated  in  connection  with 
hycirodynsmics,  but  on  this  I  wouki  refer  the  reader  who 
wishes  for  further  infonnation  to  the  work  publislied  at 
Cambridge  in   1877  by  Loni  Kayleigh. 

T/foreliad  AntroHwny  is  included  in,  or  at  any  rate 
closely  connected  with,  theoretical  <lynamics.  Among  those 
who  in  this  century  have  devoted  themselves  to  the  study 
of  theoretical  astronomy  the  name  of  Gauiu  is  one  of  the  most 
prominent ;  to  his  work  I  have  already  alluded. 

Bessel*.     The  liest  known  of  Gauss's  contemporaries  was 

*  See  pp.  3G— «sa  of  A.  M.  Clerke^ii  UUtory  of  AtiroHomy,  Edinborgh, 
1887.  Besnel's  oollecled  works  and  eomsspondeoee  have  been  sdited  by 
It.  Engelinnnu  and  publinhed  iu  four  voluiues  at  Leipzig,  187»— 82. 


\ 


THEORETICAL  ASTRONOMY.  499 

Frifdrieh  WUhdm  Besttef^  who  was  bom  at  Minden  on 
Jul  J  22,  1784,  and  died  at  Konigslier^  cm  March  17,  1846. 
Besael  commenced  hiH  life  an  a  clerk  on  board  ship,  but  in 
1806  he  became  an  assistant  in  the  otwervatory  at  lilienthal, 
and  wa8  thence  in  1810  promoted  to  be  director  of  the  new 
Pmraiian  observatory  at  Krmigsberg,  where  he  continued  to 
live  during  the  remainder  of  his  life.  Bessel  introduced  into 
pure  matliematics  tliose  functions  which  are  now  callcfl  by 
his  name,  this  was  in  1824  though  their  Uf«e  is  indicated  in  a 
memoir  seven  years  earlier ;  but  his  most  notable  achievements 
were  the  reduction  (given  in  his  Fundamenta  Agfnmamiae^ 
Konigslierg,  1818)  of  the  Greenwich  olwen-ations  by  Brmdiey 
of  3,222  stars,  and  his  determination  of  the  annual  parallax 
of  61  Cygni.  Bradley's  olisen-ations  have  been  recently 
reduced  again  by  A.  Anwers  of  Berlin. 

Leverrier^.  Among  the  astronomical  events  of  this 
century  the  discover}'  of  the  planet  Neptune  by  Leverrier  and 
Adams  is  one  of  the  *"  most  striking.  Urbain  Jean  Jo$fpk 
Leverrier^  the  son  of  a  petty  Government  employ^  in  Nor- 
mandy, was  bom  at  St  L6  on  March  11,  1811,  and  died  at 
Paris  on  Sept.  23,  1877.  He  was  educated  at  the  Polytechnic 
school,  and  in  1837  was  appointed  as  lecturer  on  astronomy 
there.  His  earliest  researches  in  astronomy  were  communi- 
cated to  the  Academy  in  1839 :  in  these  he  calculated,  within 
much  narrower  limits  than  Laplace  had  done,  the  extent 
within  which  the  inclinations  and  eccentricities  of  the  planetary 
orbits  vary.  Tlie  independent  discovery  in  1846  by  Leverrier 
and  Adams  of  the  planet  Neptune  by  means  of  thedisturlianoe 
it  produced  on  the  orbit  of  Uranus  attracted  general  attention 
to  physical  astronomy,  and  strengthened  the  opinion  as  to  the 
universality  of  gravity.  In  1855  Leverrier  succeeded  Arago 
•S  director  of  the  Paris  observatory,  and  reoi^niaed  it  in 

*  For  foiiher  details  of  his  Ufe  tee  B«rtfmad*s  ^loff  io  voL  xu  of  the 
Uiwmifu  it  VmemSimUi  and  for  so  aeoowil  of  his  work  see  AdaaM*s 
iB  voL  sxxn  of  liis  Jfeaf %  ItfeClcM  of  liis  Boys! 


^%-^ 


500     MATHBIUTIGB  OF  THK  tflNCTIEirra  OtMTUBT. 

•ceowUaca  with  the  reqiiireneiiU  of  Biodeni  Mtrononiy. 
Leverrier  now  aet  himself  the  task  of  diwrwiMng  the  theo- 
retical inveitigiAttoiiB  of  the  pUnetary  motions  and  of  reviri^g 
all  tables  which  involved  them.  He  lived  just  long  enough  to 
sign  the  last  proof-sheet  of  this  work. 

Adams*.  The  co-discoverer  of  Neptune  was  John  C^mA 
AdatnSf  who  was  bom  in  Ckimwall  on  June  5,  1819,  educated 
at  8t  John's  College,  Cambridge,  subsequently  i^ipointed 
Lowndean  profesHor  in  the  University,  and  director  of  the 
OlMer\'atory,  and  who  died  at  Caiubridge  on  Jan.  21,  1892. 

There  are  three  important  problems  which  are  specially 
associated  with  the  name  of  Adams.  The  first  of  these  is  his 
discovery  of  the  planet  Neptuno  from  tbe  perturbations  it 
produced  on  the  orbit  of  Uranus :  in  point  of  time  this  was 
slightly  earlier  than  Leverrier's  investigation. 

The  Kecund  is  hip  memoir  of  1855  on  the  secular  aoodera- 
tion  of  the  iuoun*s  mean  motion.  Laplace  had  calcuUted  this 
on  the  hypothesiti  tliat  it  was  caused  by  the  eccentricity  of 
the  earth's  orbit,  and  had  obtained  a  result  which  agreed 
substantially  with  the  value  deduced  from  a  comparison  of  the 
records  of  ancient  and  modem  eclipses.  Adams  shewed  that 
certain  terms  in  an  expression  ha&d  been  neglected,  and  that 
if  they  were  taken  into  account  the  result  was  only  about 
one-half  that  found  by  Laplace.  The  correctness  of  the 
calculations  of  Adams  was  denied  by  Plana,  Pont^couLint,  and 
other  continental  astronomers,  but  Delaunay  in  France  and 
Cay  Icy  in  England  veritied  the  work. 

The  third  investigation  connected  with  the  name  of  Adams, 
is  his  determination  in  1SG7  of  the  orbit  of  the  Leonids  or 
shooting  stars  which  weiv  cs|iecially  conspicuous  in  November, 
18GG,  and  whose  |KTiod  is  about  thirty-three  years.  H.'  A. 
Newton,  of  Yale,  had  sliewn  that  there  were  only  five 
possible    orbits.     Adauns    calculated   the    disturbance   which 

*  Adsms'g  collected  papers,  with  a  biography,  wars  intted  in  2 
volumes,  Csmhridge,  lb96,  1900. 


TnEORmrAL  astronomt.  501 

would  be  prodncrd  by  thn  plnnrts  on  the  motion  of  the  node 
of  the  orbit  of  B  Hwartn  of  inotpors  in  p;»ch  of  there  cue*,  and 
fonnil  that  thiH  iliKtiirlBtiic«  ftgreixl  with  ohwrvation  for  one 
of  the  po<isiblf<  orbits,  but  for  noiip  of  thr  othern.  Hence 
the  orbit  wan  known. 

Othrr  well-known  aNtronomcrH  of  thi"  wntnry  are  G.  A.  A. 
Wrtun  (1781-1861).  wbnsc  work  on  tlip  motion  of  the  moon  wmn 
publi-.h«l  in  1«32;  Connl  P.  C.  D.  Ponlf^nlant  (\~i5-.\87\); 
C.  E.  D'taiiTuiij  (18lfi-lS72).  whose  w  -rk  on  the  luniw  theo^ 
iiiiltcAt«fl  the  lirnt  niptluHl  yft  nuygPstiKi  for  the  analytical 
in>'e.«tignttims  of  thn  wholn  problem,  nnd  whone  (incampleto) 
lunar  tables  are  amonj;  the  astronomical  achievements  of 
thin  century;  P.  A.  /An»rii"  (179i-l«74),  head  of  the 
olnerratory  at  Gotha,  who  compilnl  the  lunar  tables  published 
in  J^ondon  in  18-i7  which  are  ^ttill  uiml  in  the  prepAration  of 
the  Nautical  Almanack,  and  elnborat«(l  the  methods  employed 
for  the  determination  of  lunar  and  planetary  perturbatkms ; 
and  /'.  /*.  riwrrmrf  (184S-1P9C)  of  P«ri^  whoae  itffanique 
citrate  i%  now  the  Htanitanl  authurity  on  dynamical  astronomy. 

Anion):;  living  matliematiciAnN  I  may  mention  the  follow- 
ing writen.  f;.  II'.  //i/f,  until  recently  <m  the  staff  of  tlw 
A  mrriran  EiArmfrin,  who  (in  I  f  S4)  detcrminetl  Ibe  inequalitiei 
of  the  miMin'H  motiun  dui-  to  the  non-Hphericnl  figure  of  the 
earth — an  investigation  which  completed  Delantuty's  lunar 
theiiryt.  Hilt  ban  alno  dealt  with  the  Mcnlar  motion  of  the 
moou'H  perigee  and  the  motion  of  n  planet's  perigee  under 
certain  cimditiona;  and  hivi  written  on  the  analytical  tlieoiy 
of  the  motion  of  JupiU>r  and  Saturn,  with  a  riew  to  the 
prppamtinn  nf  tnbW  of  their  positions  at  any  given  time, 
■fimon  A'ftvroiiib,  superintendent  of  the  American  Epk^-merii, 
who  re-examii^  the  Greenwich  olnervationa  from  the  earlieat 
timrn,  applied  the  renotts  to  the  lunar  theory,  and  revined 

*  For  an  xconnt  of  his  aomenmi  mcnioira  Mt  lh«  Timnmiilem  wf 
Ikr  /TojmI  Socittf  (/  Ijondon  for  I87«-77. 

t  Od  mant  drvdopmenl  nf  Ui«  Innar  ihearj,  wtt  th*  Tfmtmtttam 
tftkt  atia»k  AttatUthm,  nl.  uv,  ImihIdk.  l«M,>.«kK. 


SOS     lUTBUATIOB  OP  THK  HIHCTBIIITH  OIHTimT. 

HanMn't  tabka.  O.  U.  Dairwm  ol  Ounbridge^  wImi 
written  on  the  effect  of  tkiee  on  \iiooiM  spheraidii  tiM  d^ 
velopment  of  planetary  sjatema  by  neana  of  tidal  frietionv  ^^ 
mechanlca  of  meteoric  awamu^  kc^  and  U.  Poimemri  of  Btfi^ 
who  haa  discussed  the  difficult  problem  of  three  bodies^  and 
the  form  assumed  by  a  mass  of  fluid  under  its  own  attraction. 
The  treatise  on  the  lunar  theory  by  E.  IK.  Browu^  Cambridgo^ 
189G.  and  a  report  (printed  in  the  Bepori  ^  ihs  Briiisk 
A*m)ciation,  I»ndon,  1899,  voL  LXU,  pp.  121— -159)  by  M.  T. 
Whittaker  on  researches  connected  with  the  aolution  of  the 
problem  of  three  bodies,  contain  valuable  accounts  of  recent 
progress  in  the  lunar  and  planetary  theoriea 

Within  the  last  half  century  the  results  of  spectrum 
analysis  have  lieen  applied  to  determine  the  constitution  of 
the  heavenly  liodies,  and  their  directions  of  motions  to  and 
from  the  eaKh.  The  early  history  of  spectrum  analysis  will 
be  always  sKsoeiated  with  the  names  of  G.  R,  Kirehkoff 
(1824-1887)  of  Berlin,  of  A.  J.  Aiiyiirum  (1814-1874)  of 
Upsala,  and  of  Sir  (ieorye  G.  Siokes  of  Cambridge,  but  it  per- 
tains to  optics  rather  than  to  astronomy.  How  unexpected 
was  the  application  to  astronomy  is  illustr.ited  by  the  tact 
tliat  A.  Conite  in  1842,  when  discufwing  the  study  of  nature, 
regn^tted  the  waste  of  time  due  to  some  astronomers  paying 
attention  to  the  fixed  Ktars,  since,  he  said,  nothing  could 
poM-sibly  lie  learnt  aljout  them;  and  indeed  a  century  ago  it 
would  liavi*  seemed  incn'dible  that  we  could  investigate  the 
cheiiiicAl  constitution  of  worlds  in  distant  space. 

During  the  last  few  years  the  range  of  astronomy  haa 
been  still  fuKher  extended  by  the  art  of  photography.  To 
what  new  developments  this  may  lead  it  is  as  yet  impossible 
to  say. 

MaiheiUiUical   Phyncs,     An    account   of    the   history   of 

mathematics  and  allied  sciences  in  this  century  would  not  be 

other   tha.i   misleading  if   there   were   no   reference  to  the 

Application  of  mathematics  io  u^iwxtiYouA  v^blems  in  heat. 


MATHEMATICAL  PHTSICa 


503 


elasticitjr,  liglit»  electricity,  and  other  phyHicnl  fm1>ject«.  The 
hifitorj  of  matlieinatical  physicH  is  liowovcr  so  extenHive  that 
I  could  not  pretend  to  do  it  juMtice,  evitn  were  itA  considera- 
tion properly  included  in  a  history  of  mathematics.  At  any 
rate  I  consider  it  outside  the  limits  I  have  laid  down  for 
myself  in  this  chapter.  I  alNindon  its  discussion  with  regret 
liecause  the  Camhridge  school  has  played  a  prominent  part  in 
its  develf^menty  as  witness  (to  mention  only  three  or  four  of 
thoee  concerned)  the  names  of  Hir  George  G.  Stokes  professor 
from  1849,  liord  Kelvin  (Sir  William  Thomson),  J.  Clerk 
Maxwell  (1831-1879)  professor  from  1871  to  1879,  Lord 
Rayleigfa  professor  from  1879  to  1884,  and  J.  J.  Thomson 
lirolesaor  from  1884.  It  is  however  interesting  to  note  that 
the  ad>-anoe  in  oor  knowledge  of  physics  is  largely  due  to 
the  application  to  it  of  mathematics,  and  every  year  it 
becomea  more  difficult  for  an  experimenter  to  make  any  mark 
in  the  tubjeei  nnlen  he  is  alao  a  mathematician. 


-i 


605 


INDEX. 


AbftCQii,  defleription  of,  127-30 

—  ref.  to,  3,  2U,  GO,  117,  132,  137, 
144.  145,  189 

Abdal-gehl,  168 
Abel,  469-71 

—  r«f.  to,  402,  431.  44».  4o6,  402, 
409,  471,  473,  477,  481 

Abers  theorem,  470,  488 
Abclian  fanctioo^  400,  434,  402, 

470,  473,  473-5.  470,  487 
Aberration  (Mtronomicsl),  391 
Aba  Djefar;  tet  Alkarifinii 
Abal-Wafa;  9ft  Alhnxjaiii 
Academj,  Plato*s.  44 

—  the  French,  292,  32.>,  4C7 

—  the  Berlin,  32r.,  366 
Aecademia  dei  Ijincei.  324 
Achilles  and  tnrtoijie,  paradoi,  33 
Action,  leant.  408,  413,  418 

—  Tarring.  497 
Adalbero  of  Kheims,  143 
Adams,  J.  C,  500-1.    ref.  to,  499 
Addition,  procemet  of.  194 

—  ■ymbolt  for,  6,  107,  109,  159, 
179,  201,  212^,  217,  220,  221, 
222,  223.  234,  246 

Adelbaid  of  Bath,  171 

—  ref.  to,  184 
Africanas,  Jalias,  118 
Agrippa,  Comeliot,  ref.  lo,  128 

iroca,  3-8.    ref.  to,  76^  94,  106 
O.  B.,  458 
liin  papjTfDa,  94 
Albatagni,  187 
Albftri  oo  OftUleoi,  26S 
AlbosJMii,  168 
Akofai,  \¥^\ 


Alexander  the  (treat.  49,  53 
Alexandria,  nniversitjr  of,  53,  95,. 

96.  117.  119 
Alexandrian  librarr.  53,  85.  119 

—  Schools,  chapters  it,  ▼ 

—  svnihols  for  namhprs,  131 
Alfarabins.  ref.  to.  172 
Alfred  the  Great,  ref.  to.  139 
Algelira.    Trvatnl  geometricallj  bj 

Euclid  and  his  whool.  60-2. 105. 
DeTelopment  of  rhetorical  and 
syncopated  algebra  in  tlie  foarth 
crntiirj  after  Christ.  10-V13. 
Discuiwcd  rhetorically  bjr  the 
Hindoo  and  Arab  niathemati- 
cianit.  chapter  ix;  bj  the  earlj 
Italian  writers,  cliapter  x;  and 
Tacioli.  216.  Introduction  of 
svncopated  algebra  bj  Khaskara, 
m  KM);  Jordanas,  178-9; 
Ketnomontanns,  209-10;  Record, 
2*20-1;  Sttfel,  222-3;  Cardan, 
2:10-1 ;  llombelli.  231 ;  and  Ste- 
tIiius,  235.  Introduction  of  ajm- 
b(»lic  algebra  bj  Victa,  237-40; 
Girard,  241 ;  and  Harriot,  244. 
DcTcloped  bj  (amongst  others) 
Descartes,  285;  Wallis,  302; 
Newton,  341-2;  and  Eole^ 
406-7.  Recent  extensions  01, 
476-87 
Algebra,  deftnitiona  of,  189 

—  earliest  problema  in,  105 

—  earliest  theorem  in,  98M9 

—  higher,  476-87 

—  hiatorical  defdopmeol,  ldS-7 

—  historiet  of,  62,  302 

—  offifia  of  Im,  188 


/ 


S06 


IHDIZ. 


Mgebm,  qrmboU  in,  916-40 
AlgetnU.  «qiMtioiis;   m«   8im|iU 

•qnalloiM,  Qiuulimtie  «qiuUioiii| 

Ac 
AlgebriiU,  177 
AlgoriMD.  165,  178,  181,  186,  189, 

194.  925 
Alhaien,  168.    lef.  to,  179 
AlhoMeiii.  166 
Alkariani,  169<4 

—  ref.  to,  179,  179,  189,  991 
AlkArki,  166 

Alkajami,  165-6 
Ai-Khwirixmi;  tt  Alk^riaml 
Allinau,  G.  J.,  ref.  to,  19,  14.  19. 

97,  SOl  99,  95,  49 
Almaisest,  the,  99-101 

—  ref.  to,  H4,  89,  114,  159,  169, 
Km,  167.  168,  170,  171,  179, 
177,  183,  184,  186.  187,  907, 
993 

Almanacks,  185,  192-9 

Al  MamoD,  Caliph,   ret   to,  151, 

102 
Al  Mansur,  Caliph,  ref.  to,  159 
AlphuDHO  of  CaHtile,  175 
AlphuDso's  tables,  175 
Al  Raschid,  CaUph,  ref.  to,  151 
Amaaiii  of  £g>'pt,  ref.  to,  16 
America  discovery  of,  206 
Ampere,  446.    ref.  to,  460 
Amyclas  of  Athens,  48 
Analysis.  Cambridge  Hchool,  449-58 

—  higher,  489 

—  in  synthetic  geometry,  45 
Analytical  geometry,  origin  of,  979, 

281-4,  308;  on  development  uf, 

see  cliapters  xv-xix 
Anaxagoras  of  Clazomenae,  36 
Anaximander,  18 
Anchor  ring,  48,  88 
Andermm  on  Vieta,  237 
Angle,  Mexagesinial  division,  4,  250 

—  triricctioii  of  36,  3«J,  88,  240, 
326 

Augstriim,  502 
Angular  coeDieient,  322 
Auliarmonlc  ratios;  te  Geometry 

(Modern  Synthetic) 
Aiitliulogy,  Pahitine,  64.  105 
Antioch,  Grvt^k  School  at,  151 
AatipLo,  37-^ 


in 

ApioM^  180.  144 
Apofiea,  Mw'a.  167 
ApoUonina,  79-OS 

—  lef.  to,  54.  91,  1U»  11*1  119. 
165,  168L  170L  177.  t9ii»  t8ii» 
941,  988,  808.  89QL  991^  Sfiib 
890,  490 

Appall,  P.  B.,  475b  476,  489 
Apee,  BotioD  of  lonar,  904,  400 
Arabio  nnmerala,  191,   188^  158, 

158,    161,   164.   179,   174.   175. 

190-8 

—  origin  of;  WK  191 

Arabs.  Mathematioa  of,  ehaplir  n 

—  iutroduoed  into  Chhui,  9 

—  mtrodaced  into  Eorope,  cluipi  s 
Arago,  447-8 

—  ref.  to,  98,  410^  448,  459,  499 
Aratua,  48,  89 

Arbogast,  411.    ret  to,  409 
Archimedean  minrora,  68 

—  screw,  67 
Archimedes,  66-9 

^  ref.  to,  54,  65, 81, 83,  85, 88, 99, 
103,  116, 159, 165b  170, 177. 988, 
251,  267,  297,  890,  877,  898 

Archippus.  80 

Archytas,  90-9 

—  ret  to,  29,  88,  44,  46 
Area  of  triangle,  91,  99 
Areas,  conservation  of,  968 
Argand,  J.  U.,  479 
Argyrus,  122 
Aiistaeus,  50 

—  ret  to,  49,  59,  79,  81.  896 
Aristarchus,  64-6.    ref.  to, .«.  998 
Aristotle,  50-1 

—  ref.  to,  13,  14,  99,  54,  189. 161« 
231 

Aritttoxenus.  23 

Arithmetic.  PrimitiTO,  chapter  vu. 
Pre-lielleuic,  2-5.  Pythagoraan, 
28-30.  Practical  Greek,  60, 104, 
116,132,133.  Theory.of,  tn^ted 
geometrically  by  moat  of  theGieek 
mathematicians  to  tha  end  of  the 
first  Alexandrian  school,  60;  and 
thencefor^-ard  treated  empiricallj 
(Uoethian  arithmetic)  by  most  of 
the  Greek  and  European  niathe- 
maUeVana  to  the  end  of  the  i6or- 


\ 


INDEX. 


507 


ieenth  ceniary  after  Cbrisi,  98, 
132-3,  188-9.  AlRoriRtio  ftriih- 
metic  invented  bj  the  Hindoos, 
158;  adopted  bjr  the  Armbe,  161, 
164;  and  aned  ninee  the  four- 
teenth centaiy  in  Knrope,  179, 
174,  19(^;  development  of 
European  arithmetic,  180a-]6S7, 
chapter  » 

Arithmetic,  Higher;  ft  Nomben 
Theory  of 

Arithmetical  machine,  S98, 361, 459 

—  problems,  64,  75,  76 

—  progresMonii,  30,  79,  87,  159 

—  triangle,  225,  237,  293-4 
*A^M^r^«i|t  signification  of,  60 
Aronhold,  8.  H.,  486 

Arts,  bachelor  of,  148 

—  master  of,  14^9 
Aiya-Bhatta,  153-4 

—  ref.  to,  156,  158,  161,  167 
Aryan  invasion  of  India,  152 
Arzachel,  171 

Assumption,  role  of  fklse,  106, 157, 

176,  214,  215 
Assurance,  life,  400 
Astrology,  158,  186-7,  262 
Astronomical  Society,  London,  451, 

482 
Astronomy.  Descriptivcastronomy 
outside  range  of  work,  vi.    Esriy 
Greek  theories  of,  17, 18,  24,  36, 
48,  64,   65,  78,    85.    Hcientific 
astronomy  founded  by  Hippar- 
chus,  89-90;  and  developed  by 
Ptolemy  in  the  Almagent,  99-101. 
Studied  by  Hindoos  and  Arab^ 
153,    154,   156,   157,    167,    171. 
Modem   theoiy  of,   created  by 
Copernicus,  219;    Galileo,  256, 
257 ;  and  Kepler,  264.    Physical 
astronomy  created  by  Newton, 
chapter    zvt.       Developed     by 
(amongst  others)  Claiimui,  384; 
Lagrange,  415,  416-7;  LaphMe, 
494-7;  and  in  recent  times  by 
Gauss  %xA  oUiers»  chapter  ztx 
Asymptotes,  theory  of«  850 
Athens,  School  of,  chapter  m 
_  Second  Sdiool  of;  115-6 
AthoB,  M  oni,  199 

thmy  !■  diMialiy,  441 


Atomistic  School,  34 

Attains,  80 

Attic  Mymbols  for  numbers,  131 

Attraction,  theories  of,  330-9,  840, 
343-5,  384.  398.  416.  493,  439, 
446,  419,  456,  463,  496,  497 

Anstralia,  msp  of,  961 

Autolycns,  64 

Anwers,  A.,  499 

Avery's  steam-engine,  98 

Bahbs^,  451-9.    ref.  to,  450. 459 
Babylonians,  mathematics  ol^  5,  6 
Bachelor  of  arts,  degree  of,  148 
Bachet,  315 

-  ref.  to,  227,  307,  308 
Bacon,  Francis,  259.    ref.  to,  806 
Bacon,  Roger,  181-3 

—  ref.  to,  171,  173,  175 
Baillet,  A.i  on  Descartes,  278 
Baillet,  J.,  on  Akhroim  papyrus,  94 
Baily,  11.  F..  on  FUmsteed,  349 
Baker,  H.  F.,  on  Abel's  Thaonm* 

476 
Baldi  on  Arab  mathematiea,  161 
Ball,  W.  W.  R.,  ref.  to,  90,  89. 

123,   147,   920,  943,   945,    960, 

297,  305,  815,  828,  846,    850, 

450,  492 
Barlaam,  121-2 

Barometer,  invention  of,  999,  818 
Barral  on  Arago,  447 
Banow,  319-92 

—  ref.  to,  54,  95,  943,  947,  948, 
284,  309,  330,  332,  838,  888» 
3.11,  353,  356,  373,  404 

Bastien  on  D'Alembert,  886 
Beare,  T.  H.,  on  graphics,  494 
Beaone,  De,  ref.  to,  285 
Bede  on  finger  qrmbolism,  117 
Beeckman,  I.,  ref.  to,  979 
Beldomandi,  187  . 
Beltrami,  E.,  491,  499 
Beman,  W.  W.,  479 
Benedictine  monasteries,  187*  141 
Ben  Ezra,  179.    ref.  to,  175 
Berkeley  on  the  calenlna,  tn 
Berlet  on  Riese,  991 
Berlin  Academy,  866 
Bemelinaa,  145 

Bemhardy  on  Etatdethcnes,  85 
BenMwUi,  Oukl^ 


/ 


508 


imnuL 


BanoolU.  DmM,  raf.  to,  17^  401 
BenouUi,  Jmmm,  sre-T 

—  nt  t<s  S4«.  SM,  87« 
BemouUI,  JaniM  II.,  879 
BtrooulU,  John,  877-11 

—  ifi .  to,  850,  8611,  860,  868.  878. 
875.  879,  880,  889,  401,  408. 
4fM.  406 

Bemoulli,  John  II.,  879 
Bernoulli,  John  III.,  879 
Bernoulli,.  Nichohui,  379 
^  rel.  to,  851,  877.  408 
BeruoulU'a  Numbers,  877 
Bi*rnoullif,  the  /oun^er,  879 
Bertrmnd.  9KI0,  456,  486,  489,  499. 

499 
B«rulle,  Cardinal,  ref.  to,  279 
He^wX,  4H8-9 
BeiiMerii  functiout,  499 
Beta  fuuctiun,  406,  iU 
Betti,  R.,  460,  4h6 
BeviH  and  Hulton  on  Simpson,  899 
B^zout,  411 
BhaMkara,  15<*>-60 

—  ref.  to,  153,  161,  168 
Bija  Ganita,  156.  15U-60 
Binoiiiiul  i-quatiouN,  416,  4*il,  469 
Binomial  theorem,  223,  925,  887. 

352    407 

Biot.  447.     ref.  to,  362,  427 

Biquadratic  equation,  )65,  829, 
232,  239 

Bi<(u;idratic  reciprocity,  434 

Hit|UHdratic  reMitlues,  462 

Bjerkii'eM  on  Abel,  469 

Bobyniu  uu  Ahme«,  3 

Biickh  on  Babylonian  measures,  2 

Bode*s  law,  426,  458 

Boethiau  aritliuietic;  tre  Arith- 
metic 

BovtbiuM,  138  9 

—  ref.  to,  98,  118.  141,  142,  144. 
148,  182,   1H8 

lioetiuri;   gee  BoethiuH 

Bologna,   university  of,  145,  146, 

187 
Bolyai.  J..  491 
Bonibelli,  234-5 

—  tef.  to.  230,  232,  239,  249 
Bonacci;  »e^  I^eonardo  of  Pisa 
BoncompaKui,  ref.  to,  9,  162,  173, 


Book  >8tyiiig.  in.  Hi.  m 

Boolo.  O^Tm-l  8^110^481, 

Booth.  J..  488 

Bonhonit,  488.    lit  l8|  484,  «n 

Boml.  888 

BoMovkh.  108 

BosMit  on  Ckinnt.  886 

BoQgaiBTUlo,  Do.  880 

Bonqnet.  Briot  tad,  478 

Boor,  J.  R«  E..  498 

BoyU.  894,  889 

Bmchistorhrone^    tht^   990^  tT4. 

878,  880,  406 
BrackeU,  intiodiMlioa  ot  841, 948 
Bradley,  891.    itt  to^  499 
Bradaardine,  184 
Brahmagupta,  l»4-8 

—  ret  to,  158,  157.  158.  181. 188. 
167.  194,  211,  892 

Branker,  826 

Bretwhneider,  lef.  to,  18, 85, 48, 88 
BreitHchwert  on  Kepler.  969 
Bi^'aer  on  Koger  Baoou.  181 
Brewster,  D.,  ref.  to,  254,  828,  849 
BrigK*.  ^43-4.    ref.  to,  908,  908. 

2U4 
Brio>cIii,  F.,  473,  486 
Briot  and  Biouquet,  475 
BrisMiu  on  Mouk*,  435 
BritiHh  Association,  451,  456 
Brouucker,  LonI,  322-3 

—  ref.  to,  155,  324 
Brown,  K.  W.,  502 

Brunei  on  (iamma  function,  408 
Bryso,  32,  .H8 
liubnov  on  Gerbert,  148 
liucquoy,  De,  rvf.  to,  279 
Itudan,  443 

ButTon  on  Archimedes,  68 
Bull  problem,  the,  75-6 
BarK'i,  J..  202,  203,  204 
liuniell  on  uumeimls,  190 
Burnet  on  Newton,  358 
Hurnnide,  W.,  482,  486 
Byxantine  School,  chapter  vi 

Calien,  E.,  on  numbers,  489 
Calculatinic  machine,  292.  804.489 
Calculation;  tee  Arithmetie 
Calculus,  IntlniteHinuil,  278.  858-7. 
.S66-73,   376,  379-83^881,  897. 


\ 


index: 


509 


Calcalus  of  Opemiionii,  SM,  411 
^  of  Vftrifttionis  406,  418,  418, 

472   4H!I 
K^alendans  17,  86,  18.^.  198-8,  818 
Cftmliridge,    QDivenitj    of,    186, 

449-63,  508 
CampaniM,  184.    ref.  to,  186 
CMnpbell,  348 
Cantor,  G.,  481 
Cantor,  M.,  ref.  to,  vii,  8,  6,  3,  9, 

13,  14,  19,  89,  80,  35,  40,  58,  64, 

66,  91,  107,  117,  185,  137,  140, 

150,    173,    177,   ItK),  805,   807. 

815,   881,  868,   863,   866,  871, 

878,  888 
CapRt,  Hugh,  ret  to,  143 
OapilUritT,  391,  489,  415 
CareaTi,  308 
Cardan,  887-31. 

—  icf.  to,  68,  818,  885,  888,  888 
Careil  on  Hciieartes,  378 
Gamot,  Laxare,  438 

—  ref.  to,  90,  408,  435,  489 
Gamot,  Sadi,  443 

Cartes,  Dee;  $ee  Descartea 
Cart«8ian  vortices,  887,  338,  345, 

348 
Cassiodoms,  139.    ref.  to,  118 
Catacaastics,  386 
Castillon   on    P)sppas*s   problem, 

108 
Cataldi,  848,  383 
Catenary,  374,  377,  393 
Cathedral  Schools,  the,  140-8 
Cauchj,  476-9 

—  ref.  to,  358,  480,  439,  469,  470, 
475.  479,  488,  486,  489 

Canftties  are  rcctifiable,  886 

Cavalieri,  887-90 

^  fvf.  to,  841,  844,  864,  8n,  896, 

809,  356 
CaTendish,  H.,  439-40 
Cajley.  483-4 
^  ref.    to,    468,   474,    488,   489, 

491,  500 
Censo  dl  cenao,  817 
CensQS,  809,  817,  883,  838 
Centras  of  mass,  76, 104, 860, 888, 

808,  800 
Gentrifngal  force,  818  • 

Cera,  the  planet,  456 


Chaldean  mathematics,  3,  9 
Oiambord,  Comte  de,  ref.  to,  478 
Champollion,  ref.  to,  440 
Cbanccllnr  of  a  aniversitj,  146 
Chsrdin,  8ir  John,  ref.  to,  195 
Chark*H  the  Oirat,  140,  141,  148 
Charles  I.  of  England,  ref.  to,  298 
Charles  II.  of  England,  ref.  to,  320 
Charles  V.  of  France,  ref.  to,  185 
Charles  VI.  of  France,  ref.  to,  185 
Charles,  E.,  on  Roger  Bacon,  181 
Chasles,  M.,  ref.  to,  63,  85,  868, 

26\  48t),  491,  496 
Chaucer,  ref.  to,  189 
Chinese,  enrl.T  mathcmaties,  9-10 
Chios,  Hchool  of,  33 
Christians  (Eastern   Cboreh)  op- 

pospd  to  (treck  science,  115, 116, 

119 
Chnqiiet,  818.    rrf.  to,  849 
Cicrro,  ref.  to,  69 
Ciphers;  tre  Namerals 
Ciphers,  discoveries  of,  837,  89H 
Circle,  qaadratare  of,  (or  sqnaring 

the).  27.  32,  36,  39;  also  9e€  w 
Circular  harmonics,  438 
Cis!K>id.  88 
Clairaut,  383-5 

-  ref.  to,  248.  351,  898,  400,  408 
Clausius,  R.  J.  E.,  460 
Clavia^  840 

Clehwh,  R.  F.  A..  487,  488 

Clement,  ref.  to,  140 

Clement  IV.  of  Rome,  ref.  to,  188, 

183 
Clerk  Maxwell;  tee  Maxwell 
Clerke,  A.  M..  498 
CHfford,  W.  K.,  495-6 
Clocks,  23r»,  312.  313 
Cocker's  arithmetic,  399 
CoefKcient,  angalar,  388 
Colebrooke,  ref.  to,  154,  157,  160 
CoUa,  331,  833 
Collins,  J.,  385 

—  ref.  to,  383,  338,  353,  859,  864, 
368 

CoUisioB  of  bodies,  801,  818,  884 
Colours,  theoi7  <<  381,  888,  884 
Goboa  on  Nawton's  floxioBa,  858, 

854,  855,  856^  858 
Comets,  884 
Cowmodiao,  888.    icC  to.  64 


/ 


SIO 


DfDBX. 


CfftftTiitiminiMw,  XmIU  oBt  SI 
ComoMieiiuii  •piitoUMiii«  869 
CofniiUx  aoniMn,  i80,  469;  479, 

488 
Comptoi  wiftblM,  S30 
Comto,  A.,  raf.  to,  MS 
CkMichoid,  87 

CondoiQel,  888.    rat  to,  IMS 
Cone,  Mdioua  of,  49 

—  uurUce  of,  78,  166 

—  volume  ol^  47,  78,  166 
CkMigrueuoM,  461,  46i,  466»  488 
Couio  Seciiont  (Ueomeiiical).  Dis- 

cuHfletl  bj  most  of  the  Oraek  geo- 
metriciAnii  after  MenaeohmniL 
49 ;  eii|it*oiAlIy  b/  Euclid,  68 ;  ana 
Apolloiiiuii,  80-9;  iuteivet  in, 
revived  by  writingii  of  Kepler, 
268;  and  DejiannieB,  265;  and 
nubitequenUy  by  I'aHcal,  898 ;  and 
Maclaurin,  895,  3U6.  Treatment 
of  by  modern  synthetio  geome- 
try, 435-9,  489-91 

Conies  (Analytical).  Invention  of 
by  Deitcarteii,  281-5,  and  by  For- 
mat, 308;  treated  by  WalliM,  298, 
and  £uler,  405;  recent  exten- 
HiouM  of,  489 

Conicoids,  72,  73,  405,  416 

Couou  uf  Alexandria,  66,  67,  72 

Cooiiervation  of  energy,  389,  418, 
438,  4IH) 

ConHtantiuc  VII.,  the  Emperor, 
121 

Constantinople,  Call  of,  124 

Conti,  368,  370 

Continued  fractions,  242,  328,  421, 
429 

Continuity,  principle  of,  264,  265, 
341,  372,  43U 

CoutravariaiitH,  484 

Conventual  Schooln,  140-5 

Convergenoy,  323,  352,  875,  381, 
397,  404,  462,  477.  478,  486 

CoordinaWs,  281-2,  878 

—  generalized,  414,  417,  498 
Copernicus,  219 

—  ref.  to,  90,  100,  207,  234,  257, 
258 

Cordova,  School  of,  146,  170,  171 
Cornelias  Agrippa,  ref.  to,  128 
Corpuscular  theory  of  h^V%  S3l&-% 


tl7 


CoeiiM,  I«7.  Ml,  iOf  ,  t46^ 
Coijr,  ante  for,  SM 
Oor^jr.  ante  tat.  9H 
Coseio  ait,  il7 
CoUagent,  91.  168,  ia4.  Ml, 
Cotanynte,  table  oi;  168 
CoCaa,  898 

—  rat  to.   Ml,  887.   SSi^  4M, 
459 

Courier  oa  liaplaoa,  480    . 

Cousin  oa  Dascartea,  S78 

Cramer,  O.,  882 

Crelle,  refl  to^  49U 

Cremona,  L.,  486,  491,  494 

Crystallography,  94 

Ctesibua,  91 

Cuba,  217 

Cube,    dnpUcatiiMi   of,   81-1,   M^ 

48-4,  46,  49-50,  84,  86,  88,  91, 

240 

—  origin  of  problem,  48 
Cubio  curves,  Newtoa  oa,  85(^1 
Cubic  equaUous,  78,  110, 165.  »4. 

225,  231.  234,  289 
Cubic  reciprocity.  434.  464 
Culmann  ou  graphics,  491,  494-6 
Curtxe,  M.,  ref.  to,  177.  186 
Curvature,  lines  of.  486 
Curve  of  quickest  descent.  860^  874, 

878,  380,  406 
Curves,  areas  of;  aet  Quadimtara 
Curves,  classification  of,  888,  850^ 

405 
Curves  of  the  third  degree,  850-1 
Curves,  rectification  of,  801-2,  828, 

826.  338,  852,  854 
Curves,  tortuous,  383,  405,  488 
CuHS,  Cardinal  da,  211-2 
Cycloid,  293,  296-7,  300,  801,  812 
Cyzicenus  of  Athens,  48 
Cyzicus,  School  of,  chapter  ui 

D'Alembert,  385-8 

~  ref.  to,  297,  877,  892,  401,  407. 

413,  417 
Dalton,  J.,  441 
Daniascius,  116 

Damascus,  Oraek  Sehool  at,  161 
Darbonz,  412,  442,  485,  488,  489 


\ 


\ 


•  I 


I 

»■ 

i  • 


r  • 
i 


INDEX. 


511 


i 


I^Mypodiat  on  Theodosioji,  M 
T>8  Beaaoe,  ref.  to,  383 
II9  Btnille,  Cardinal,  ref.  lo,  379 
I>8  Bonpunnlle,  880 
?)e  Buc«|oo7,  ref.  to,  379 
l)e  Careil  on  Defleartcl^  378 
ledmal  fractions,  304,.  3S3 
>ecimal  nnmeration,  74>5,  83-4, 

153.  158,  161,  164,  173,  175-6, 

190-3 
Deeimal  point,  2113-4 
Decimal  meaimrea,  301,  353,  419 
De  Condoroet,  388 
Dedekind,  ret  to,  464,  469,  473 
Defective  numben,  80 
De  Fontenello,  ref.  to,  876 
Degree,  length  of,  86,  95, 167,  384, 

447 
Degrees,  angular,  4,  87 
I>e  Gua,  381-3 
De  Kemnten,  ref.  to,  137 
De   la    Hire,    337.    ref.   to,  133, 

818 
De  LalonMre,  818 
Delambre,  88,  89.  99,   101,    340, 

413 
Delannaj,  501.    ref.  to,  500,  501 
De  rHospiUl,  380.    ret  to,  891 
Ddian  problem;  te  Cuba 
De  MalTes,  3R1-3 
De  MM,  ref.  to,  394 
De  Miziriae,  315 

—  ref.  to,  337,  307,  306 
Demoeritus,  34 

Demoivre,  394-5.    ref.  to,  398, 410 
De  Montniort,  381 
Da  Moigan,  A.,  483-3 

—  ref.  to,  54,  58,  63,  99,  100, 101, 
118,  188,  S13,  314,  405,  481 

Da  Morgan,  8.  E.,  483 
Denptos  for  minua^  310,  317 
Denifle,  P.  H.,  ref.  to,  145 
De  Rohan,  ref.  to,  336 
Deaargnes,  364-6 

—  ref.  to,  363,  3n,  378,  398,  837. 
435 

Peacsrles,  378-87 

—  vei:  to,  57,  87,  385,  338,  345, 
S47»  349,  359.  365,  367,  373, 
377*   397,   398.  801,  808,  807, 

'   ,    808,  880.  883,  841,  875,  8n.  883 
/»—  nl8  of  rigM  of,  385,  841,  883 


Descartes,  Tortices   of;   see   Car- 
tesian vortices 
De  Slaze,  336 

—  ref.  to,  317,  331,  333 
Desmaie  on  Ramus,  334 
Destouchcs,  ref.  to,  3H5 
Determinants,  375,  411.  416,  438, 

463,  465,  473,  478,  486 
Dcvanagari  numerals,  190,  191 
Devonshire,  Earl  of,  ref.  to,  d'.M 
Didion  and  Dupin  on  Poooelet.  438 
Difference  between,  sign  for,  339 
Differences,  Finite,  381,  893,  417, 

422,  429 

—  mixed,  439 

Differential  calculus;  see  Calculoa 
Differentia]  equations,  883,  386-7, 

406,   411,   416,   435,   473,    481, 

484,  485,  4H6,  4m7,  489 
Differential  triangle,  the,  831 
Differentials,  .^19,  430 
Diffraction,  314,  836.  441,  447 
Digbj.  SOS 
Dini,  U.,  486 
Dinocrates,  53 
Dioostratus  of  Cjzieus,  49 
Diocles,  88.    ref.  to,  95 
Dionjsius  of  Tarentum,  80 
Dionvsodorus,  95 
Diophantus,  107-14 

—  ref.  to,  39,  73,  87,  131,  153, 
153,  l.'(6,  308,  333,  331,  804, 
307,  315,  433 

Directrix  in  oonica.  83,  103 
Dirichlei,  Lejeune,  464 

—  ref.  to,  306,  443,  468,  .469,  471, 
472 

Distance  of  sun,  65 

DiKturbing  forces,  845,  415,  500, 

501 
Ditton,  H.,  891 
DiviMon,    proccasea   of,    197-301, 

314 

—  symbols  for,  150,  166,  347 
Dodecahedron,  diseoveiy  of,  33 
Dodson  on  life-assurmaoe,  400 
Don  Quixote,  177 
DosithMts,  66,  70,  78,  74 
DooUa   entrr,    book*kaepiag   k9% 

198,  315,  353 
Doable  iheto  ffnwIioM;  aw  KlUt-  '\ 
tie 


/ 


sia 


IIIDKZ. 


Di^ydoffff  oa  I>^ma1«  MO 
Prajw  oo  Tjrelio  Bnbt,  9M 
DnUUcr,  869 
Dapin,  tet:  to,  4S5,  4S8 
Daplmtiou  of  cnbe;  see  Cabo 
Dapnis  on  Thuoo,  1W     - 
D'UrliAn  on  ArkUrahui,  M 
DArer.  919.  •  ref.  to^  124 
l>)'iiaiuie«;  «€tf  Mtschanks 

e,  s^iuIk>I  for  2'71(f98.../40l 
EanlwU,  Arvhbiitbop,  r«r.  lo,  140 
Earth,  deunitj  of,  440 

—  dimensions  of,  M^  US,  ]l»l,  447 
Eccentric  auKle,  203 

EcUpMtt  foivtuid  by  Thales,  17 
Ecliptic,  oblic|iiit/  of,  Htl,  99 
Edeiif^ft,  Greek  Sehool  at,  151 
Edward  VL  of  England,  ref.  to,  220 
Egbert,  Arcbbiidiop,  ref.  to,  140 
Eg>'ptian  uiatheuiatics,  chap,  i,  94 
Eiaeiilohr,  ref.  to,  3,  6,  8 
EiiieONteiu,  4G4-5 

—  ref.  to,  465,  464i,  467,  468,  472 
EUiitic  string,  tension  of,  325 
KhiKtica,  377 

Eleatic  School,  33 
Electricity,  445,  4.V.)-riO 
Eleuientfi  of  Euclid;  trr  Euclid 
Eliniinutiuu,  tlieury  of,  411,  415 
Elizabeth  of  EngUnd,  ref.  to,  244 
Ellipse,  area  of,  72 

—  recti  dcation  of,  3M3 

Elliptic  function!*,  4tl6,  134,  461, 

465,     I6H,    460  71,   473.    473-5, 

476,  479,  4H3,  4H6 
Elliptic  geoiuetr)',  493 
Elliptic  orbits  of  planet^  171,  264, 

310,  Mi 
Ellis,  O.,  on  Kuniford,  440 
Ellis,  K.  li.,  on  Fr.  Bacon,  259 
Ely  on  Bcrnoalli's  numbers,  377 
Emef>s,  Gteek  School  at,  151 
Emission  theory  of  light,  335-6 
Energy,  conservation  of,  3ti9,  418, 

418,  438,  460 
Encstrom,  ref.  to,  285 
Engel,  F.,  on  Grassmaun,  481 
Engelmann  on  Bes^el,  498 
Enneper,  A.,  ref.  to,  473 
Envelopes,  812,  327,  873 
EpiebMrmttM,  30 


EpifOiiM,  94 
EpiiTclM,  99.  991  160 
'iSpieyeloida,  897,  891 
£(|ualiiy.  qrmbuk  lor.  8^  199^  991, 
217.  290,  989,  949 

—  origin  of  ayvM,  999 

s,  mcnningi  of ,  990,  989.  949 
£i|ttationa;  tee  Sinipln  fiHoat, 

Qoadmtie  cgnationi,  de. 
Equations,  diffeimitial,  8k9,  89^T» 

406,  411,  416,  485,  499 

—  indeterminatn.  110,  189.  188b 
827.  415 

—  uumbsr  of  ruols,  487,  477 

—  position  of  roots,  ttO^  8979 
841-2,  382,  421,  448 

—  roots  of,  ittiagtuAij,  980 

—  roots  of,  negative,  280 

—  theory  of,  241,  840-2,  404,  490^ 
476,  484,  485,  496 

Equiangular  spiral,  877,  498 
Eratosthenes,  85-6 

—  ref.  to,  44,  87,  88,  95 
Errtirs,  theory  of,  .H94,  40U,  415^ 

428,  432.  419,  457 
Ersch  and  Gruber  on  Desenrtot. 

278 
Esflei,  ref.  U\  296 
EtiKT,  luminiferous,  314,  886,  460 
Euclid,  54-64 
~  ref.  to,  44,  69,  79,  98,  104, 182, 

165,    168,    170,   177,  288,   820; 

see  also  below 
Euclid's  KlemeHlt,  55-^ 

—  ref.  to,  114,  116.  118,  189.  152, 
165,  168,  170,  171,  172.  175, 
177,  178,  182,  184,  185,  199^ 
187,  233,  234,  291.  820,  830, 
432 

Euc.  ai.  12,  l*tolemy*s  proof  of^ 
101 

ref.  to,  15,  182 
raf.  to,  83 
ref.  to,  25 
ref.  to,  15 
ref.  to,  33 
ref.  to,  15 
ref.  to,  25 
ref.  to,  16,  17,  95, 
991 
ret  to.  27 
ref.  to,  27 


Euc. 

I.  5. 

— 

12. 

— 

13. 

— 

15. 

—■ 

23. 

— . 

26. 

—m. 

29. 



32. 

^^ 

I. 

44. 

— 

1. 

45. 

\ 


1 

•    1 

• 
•        • 

1 

Boe.  t,  47. 

ref.  to,  7,  10,  85^ 

1 

26-7,  29,  41, 155 

\                       ^  1.  48. 

ref.  to,  7,  25,  89 

—  n,  i. 

ref.  to,  27 

—  M.  •• 

ref.  to,  108 

—  n,  6. 

ref.  to,  60 

*  \ 

—  n,  6. 

ref.  to,  60 

—  11,  a 

ref.  to,  108 

y     1 

—  n.  IL 

ref.  to,  47,  60 

r 

—  u.  14. 

ref.  to,  27,  60 

—  m,  18. 

ref.  to,  32 

—  ni,  81. 

ref.  to,  16,  41 

.    j       —  m,  85. 

ref.  to,  32 

—  ▼. 

ref.  to,  46 

■  • 

—  n,  3. 

ref.  to,  15 

• 

—  n,  4. 

ref.  to,  15,  27 

» 

—  yi.  17. 

ref.  to,  27 

' 

—  n,  25. 

ref.  to,  27 

• 

—  Ti,  28. 

ref.  to,  60,  105 

—  ▼!,  29. 

ref.  to,  60,  106 

.      • 

—  n,  D. 

ref.  to,  90 

—  n,  86. 

ref.  to,  407 

•:      !      — X- 

ref.  to,  50,  88 

j      —  It  L 

ref.  to,  47 

1      —  1,  8. 

ref.  to,  50 

1      —  1,  117. 

reC  to,  62 

—  M,  18. 

ref.  to,  82 

—  zn,  8. 

ref.  to,  41,  47,  48 

—  xn,  7. 

ref.  to,  47 

—  XII,  10. 

ref.  to,  47 

—  Ult,  l-l». 

ref.  to,  47.  58 

—  zni,  8-18. 

ref.  to,  58 

:      —  zm,  18-18. 

ref.  to,  59 

—  XIT. 

ref.  to,  87 

—  !▼. 

ref.  to,  116 

Euclidean  spaoe,  492 

Eodemiu,  18,  18.  18,  48,  80,  81 

EodoxoB,  46-8 

—  nf.  to,  88,  44,  88,  81,  8R 

Baler,  408-8 

—  ref.   to,   108,  803,   880,   848, 

850,  804,   805,   811,   848,  849, 

871,  878,   879,   889,   403,  418, 

418,  417,  485,    488,   447,  468, 

478 

BoiTtM  oC  MetBpoolom,  44 

Boloeiiiis  116. 

rat  to,  88^  188 

Efwtioii,80 

Bfolatei^  818 

. 

1       Bientfkt,  88,  100 
.1       Bieewifi  MUDbtrib  80 

*• 

BiJmiwr,  O0«l  of,  108 

□L  518 

Ezhaiutiooa,  metbod  ^  47,-  88, 

288 
Expansion  of  btnomiml,  887,  851^ 

407 

—  of  eo§  (if^IOt  288 

—  of  eo§  X,  824 

—  of  oo8'>  X,  824 

—  of  e*.  875 

—  of/Cjt'i.AK  892    . 

—  of  /  (X).  397 

—  of  log  (1+x),  317.  819,  878 

—  of  Bin  [A  ^  h),  233 

—  of  tin  X,  824,  337,  875 

—  of  gin-'  X,  334,  337 
^  of  Un-*  X,  824,  875 

—  of  vera  X,  375 

Expuneion  in  eerieti,  352,  375,  881, 

392,   397,   404,   462,  470,  477, 

478,  463 
Ezperimentfl,  necesmtj  of,  25,  78, 

182,  259,  441 
Ezponential  culenlnis  878 
Ezponentii,  160, 185,  235, 289, 244, 

249,  252,  285,  299,  852,  404 

Fftber  Stapalenrit  oo  Jordanuii,  177 
FabricioB  on  ArehytM,  80 
Facility,  law  of,  432 
Fagnano,  882-^ 
FalM  amomption,   role  ol,   106, 

157,  176,  214,  215 
Faraday,  ref.  to,  448,  460 
Faugire  on  Paaeal,  290 
Favaro,  A.,  ref.  to,  8,  287,  828,  494 
Faye  on  nebular  hypotbeaia,  425 
Fermat,  302-11 

—  ref.  to,  83,  155,  228,  TH,  284, 
293,  294,  301,  812,  321,  822, 
856,  860.  407,  418,  416,  422 

Ferrari,  231-2.    ref.  to,  229,  889 

Ferreo,  224 

Fibonaoci;  ««e  Leonardo  of  Pte 

Figorato  nambera,  294 

Finck,  250 

Finger  iiymboli«B,  117,  122,  126, 

180 
Finite  diifmDeea,  882,  417,  488, 

429 
FMfi,  824,  825,  229 
Fire  ongiiM  inrentfld  IvHtm,  88 
FiadMr  on  Dateartea,  878 
FHv,  Ihiofi  aooated  bj,  186,  18T 


/ 


I 


614 


IHDBXi 


FiicM,  nt  to^  448 
Flamrtatd,  »49 

—  raf.  to.  890 
Florido.  884.  888^  SS8 
FliMnU,  880.  888,  847,  848.  864-7. 

891.  897 
Flaxioiua  eftlenlu.  878. 858-7. 897 

—  eoBtroYtny.  887.  888-78 
FlauoDm    880,    888»    847.    848^ 

854-7.  891.  897 
Foeni  of  a  eonio.  89^  888 
FontMiA;  U€  TaitaglU 
FonteneUe,  de.  ref.  to,  878 
Force,  componeDt  of .  in  a  givoo 

direetioD,  258-4 
Forces,  {Muralleloi^aiii  U^  51,  868, 

881 

—  triangle  of;  919.  258,  881 
Forms  in  algebra,  485-7 

—  in    theory   of  nnmbert,  481, 
4C5-9 

Forsyth,  A.  U.,  476,  481,  487,  489 
Foncault.  ref.  to.  448 
Fourier,  442-3 

—  ref.  to.  402.  431.  439,  446 
Fourier's  theorem,  442,  4C4 
Frcetionn,    ouuiinuMl,    242,    888, 

421,  429 

—  symbol  for.  150,  1G6,  185,  247 

—  treatment  of,  3.  4,  5,  76,  808, 
204 

Francis  I.  of  France,  ret  to,  218 
Frederick  II.  of  Oermaiiy,  176-7 

—  ref.  to,  175 
Frederick    the  Great  of 

ref.  to.  386,  403.  414,  418 
French  Academy.  292,  467 
Frdnicle,  318.  ref.  to,  308 
Fresnel,  446-7.    ref.  to,  314,  836. 

448 
Friedlein,  O..  ref.  to,  83,  91,  107. 

116,   125.  139 
Friiich  oil  Kepler.  262 
Frischauf   on  absolute  geomeiiy, 

492 
Frisi  on  CaTalieri,  287 
Frobeuiufl,  475,  482,  486,  489 
Fochs.  489 
Functions,  notation  for,  878 

—  theory  of,  473,  474,  475-6,  482. 
483 

Fus^  ref.  to,  103,  402 


flahndi,  ti^  888 
Oalaoa  Anl^rtea.  88 
Oaki^  nt  to,  181 
OalOn^  854-8 

—  wtL  to^  78^  980, 161,  881^  88^ 
977.  878.  988,  888.  874 

Oallaj  ^yttom  of  difiita,  188-881 

Qaloia,  489.  488 

Qamma  fwMtion,  408^  484.  888 

Oarth,  i«f.  to,  184 

Gaiaendi,  i«f.  to.  807.  8U 

Oanaa.  467-88 

—  i«f.  to,  980,  859,  888.  40^  480^ 
429,  482,  438,  435,  448,  481^ 
465,  466,  468,  469,  478,  478. 
477,  479,  480.  491.  497,  498 

Oober  ibn  ApUa,  171 

Oeiaer  on  Steiner,  490 

Oelon  of  Syraeuae,  74 

Oeminas^  lef.  to,  18 

Oeneralised  cooidinatea,  414,  417, 
460,  498 

Generating  lines.  824 

Geodesies,  878.  406,  488 

Geodesy,  261,  459 

Geometrical  progressions^  80.  81, 
71.  75,  158 

Geometry.  Egyptian  Gcomaiiy. 
5-8.  CUssical  Syntbetie  Goo- 
metry,  discussed  or  osad  bj 
nearly  all  the  mathematiciana 
considered  in  the  first  poriod, 
chapters  n-v;  also  by  Newton 
and  his  school,  ohaptera  iti, 
zvn.  Arab  and  Mediefal  Goo- 
metry,  founded  on  Greek  works, 
chapters  Tin,  n,  z.  ^  Goometry 
of  the  renaissance;  eharaeterised 
by  a  free  use  of  algebra  and  trigo- 
nometry, chapters  xn,  xm.  Ana- 
lytical Geometry,  272.  281-8; 
discussed  or  used  by  nearly  all 
the  mathematicians  considered 
in  the  third  period,  chapters 
ziv-ziz.  Modem  Synthetic  Geo- 
metry, originated  with  Desaignes, 
264-5;  continued  by  Pascal. 
293;  Maclaurin,  396;  Monge. 
Camot,  and  Poncelet.  435-9; 
recent  derelopment  of,  489-91 

Geometry,  origin  of,  5-6 

Gaorge  I.  of  England,  ref.  lo,  868 


''■i 


Ocnrd,  ITl.  ref.  to,  171.  I7E 
Gcrbnt  (StItoIct  II.),  H.l-fi 
OcrhiTdt,   nt.  to,   131,  S63,  S6e, 

367,  368.  309,  45.1.  471 
Ont»  Bomannmrn.  H4 
GbtrUldi  on  ApollonmR.  S3 
Gibson  on  orifcin  ol  olenla*.  SGC 
OiMinii  OD  LFoaKidn.  113 
Oirard,  241.    rcf.  to,  'iiS.  248.  ISO 
Olushn-,  344,  4^%  408,  47S 
Oloben.  143 

OnomoD!<  or  odd  namben,  24 

Oohar  nanwralu,  144,  190.  191 

Ooldbftch,  381,  401 

Ooldcn  KTCtion,   th«.  46,  47.  39 

aom.^n.  CnHinal,  ref.  to,  233 

Or.pel.  A..  473 

OnrdHn,  P.  A.,  4Wi 

Gore  on  nebuUr  fajpothwia.  4M 

U<ith«I>  on  Sterinno,  2.13 

GoarMl,  E.,  on  ranciians,  47fi  . 

Gov,  ref.  to.  3.  6,  13.  53.  64,  M, 

114 
QraindoTiw,  J.,  ref.  to,  4nH 
Onnmir.  MDdpntH  in.  148 
OrantuU.  Scliool  of,  170 
Orapbiol  metbodn,  RO.  346.  494-A 
Gnummknn.  481.     ref.  to,  460,  489 
Gravcf  on  Hamiltoo,  4711 
OnTcsandr.  ■',  on  HnTRhco*,  911 
Oraiitj,  ecniTM  ol,  76.  104,  860, 

388,  303.  309 

—  Uw  of,  331,  331-3,  340,  34>-«, 
384 

—  HT-mbol  tor,  378 
On;  OD  Newton**  wr 
Orotcr  than,  snnbol  I 
Greateat  eomnoa  meamro.  61 
Greek  MieDce,  25 

Okcd,  497.     nt.  to,  49S 
Greenwood  on  Hero,  91 
GTcgoi7  XQI.  of  Koma,  328 
Otagofj,  Darid,  390.    ret.  to,  IM 
Onpti7,  Jamet,  333-4 

—  nf.  to,  SSt,  337,  373 
GnahMB,  Wr  Tho*.,  ref.  to,  143 
OraMBtols,  Bbibop,  ret.  to.  im 
Gnmr^  tbrara*  of,  482,  483 
Onbe  on  DirichlBt.  4U 

Go*.  d«,  S8I-8 

I  LribDita.  m» 


inf[»,  338 


Qnldinn%  260.     ret.  lo,  MS.  SM 
OnnnowdCT,  inTHtlion  of,  IBS 
Oimtcr.  F...  303,  250 
OJinlhcr.  S..  12S,  137,  MS,  410 

ItHdlej,  ref.  to,  SU 
HiillfT,  390-1 

-  ref.   to.  79,  88,  97.   834,  UX 
343,  S47.  349,  385,  8M,  411 

Halma,  M..  n(.  to.  99,  llfi 
Halphrn.  U.  H.,  476,  488,  489 
Haloled,  G.  B.,  on  hypngoorngtiy, 

493 
namilton.  Sir  Wm.,  479-81 

-  Tft.  to.  IS9.  418,  481,  497 
Hand  n*fd  todrnote  Dtci,  188,  111 
Hnnkel.   ref.  (o,   13,   IS,  K,  tl, 

106.   117,  lli,   160,   455,   Ml, 

4''7 
Hmn-vn.  fiOl.     nl.  to,  fiOS 
HBrkne>=i>.  J.,  on  tnnetioni,  476 
Hiinnonic  nnaljnis,  423,  4S3.  490 
Harmonic    nlKMi;     itt    OeomiUf 

(Modern  Smthetie) 
Hnrmonic  -nin,  SO,  44S 
llaronn  AI  KaiKliKl,  icf.  to,  ISl 
Harriot.  241-S 

-  r-r.  to,  33S,  147,  148;  IBS 
Ilaslie  <iD  Kant,  420 
Haoghloo  on  UaeCnIla^  488 
IfankxbFf!  on  eapilUri^,  419 
HfHp  tor  unknown  omnbo',  fi.  108 

126 

li«)rT  oC, 
tlentii.  D.  0„  o 
lleatb.  T.  U,  On  Diophantn^  107 
ne'.;el,  ref  to,  456 
HeibcK.  ref.  to,  54,  M.  71;  SI,  97, 

99.  107.  184 
Bclii,  .119 
Hclmbolti,  von,  r£  to,  4«0, 491. 

498 
Henr?  IV.  of  Fnoee,  mT.  to,  SM 
Heni7  of  Walca,  mT.  to,  M» 
Heni7,  C,  nT.  to,  104.  990.  S«C, 

302.  385 
H<^nr7,  W.  C,  «■  IMUm,  441 
HeuKl  K..  47>,  470 
Heraclc>d«a,  8l 
BeriKonM,  119 
Hennite,  rat   to,  <H,  «•.  ««. 

478,  47S,  4M 


/  / 


S16 


ursipaL 


fUnu/tiMMB  of  AIImm^  48 
Ibro  of  Aloundiia,  91-4 

—  nt  u^  100^  ua,  iM^  its 

Bmo  of  CoMUPtiBopK  ^^ 
Herodotus,  nf.  to,  8.  6 
HondMl,  Sir  John,  468 

—  ict  to.  450 
HmeM,  8ir  Wtllum,  458 
Hene.  488 

Hettner  on  Borahardt,  486 

Henradt,  Tan,  801 

Hiero  of  SpaeoM,  67,  77 

Hieroglypbies,  Egyptian,  440 

Hilbert,  D.,  487 

Hill,  O.  W..  501 

HiUer  on  EratoathouM,  85 

Hindoo  mathematicn,  158-61 

Hipparahns,  88-91 

—  lef.  to,  70,  87, 91,  99, 101, 187 
HippasuB,  22,  SO 

Hippiat.  8C-7 

Uippocratea  of  Chios,  89-44 

—  ref.  to,  38,  66 
Uippooratet  of  Cos,  89.  151 
Hire.  De  la.  327.    nf.  to,  12S,  818 
Uiittory,  melLods  of  writUig,  278 
Hoche  on  Nicomachus,  98 
Ilochheiin  on  Alkarki,  166 
Hodograph,  481 

Hoefer,  r«f.  to,  19 
Hoffmann  (on  Euc.  i,  47),  86 
Holgate  on  Beys,  491 
Hulmboe  on  Ab«l,  470 
Ilolywood,  180-1.    ref.  to,  186 
Huuiogeueitj.  VieU  on,  237,  238 
Homology,  2C6 
Honeiu  ibn  Isbak,  151 
Hooke,  325 
-^  mL  to.  313,  335,  339,  842,  858, 

447 
Honley  on  Newton,  328 
HoHpital,  r,  380.    ref.  to,  891 
Huber  on  Lambert,  410 
Hudde,  318.    ref.  to,  317,  321 
Hugens;  ate  Huygens 
Hultsch,  ref.  to,  64,  91,  92,  102 
Homboldt,  469,  490 
Hutton,  ref.  to,  235,  899 
Huygens,  311-6 

--  /e/.  to,  274,  277,  801,  817,  818» 
324,  3'M,  335,  336,  342,  864,  Ul 
Hujgbena;  see  Huygens 


llovtOB,  881 ;  DPAkiihsrt^  888; 

IfadaoriB,   898;    Knlsr.    888; 

and  r^uBlasOi  498 
HjdfostatMS.    Dofslopad  If  Ar» 

ehimsdeStTt;  bj8tofiwu,888: 

hj  Galitoo.  856;  bj  Faaoal,  898; 

by  Mewton,  f61}  and  by  Kaki; 

408  i 

Hypatia,  118.    fat  to,  118 
HypsrboUe  gsonetiy,  498 
HypsrboUe  triflonometiy,  410 
Hyperboloid  of  ono  shesi,  884 
Hypwr-eUiptiof^inetioiis;  sMEUIplk 

functions 
Hyper-gsometHo  series,. 468 
Hyper-geometiy,  491-8 
87 


lambliehos,  114.  ref.  to,  19, 80, 181 
Imaginary  nnn^bera,  230,  ^  478, 

479 
lucommensnrables,  28,  82,  50,  88 
Indeterminate  eoefficienta,  875 
Indeterminate  forms,  880 
Indian  mathematics,  chapter  B 
Indian  numerals,    121,  133,  158, 

158,    161,   164,   172,   174,  178^ 

190-8 
—  origin  of,  190-1 
Indices.  160.  185,  285,  239,  848, 

249,  252,  286,  299.  852,  404 
IndiTUible  College,  824 
Indivisibles,  method  of,  264,  887* 

90.  317 
Inductive   arithmetic,  98,  188-8^ 

188-9 
Inductive  geometry,  7-8,  10 
Infinite  Mries,  difficulties  in  eon* 

necUon  with,  33,  823,  852,  878, 

381,   397.  404,  462,    470,    477, 

483 
Infinite  series,  quadrature  of  eoriM 

in,  299.  823,  837-8,  852-8 
Infinitesimal  calculus;  $ee  Galenlns 
Infinitesimals,  use  of,  268-4,  480 
Infinity,  symbol  for,  849 
Instruments,  mathematical,  81, 87, 

45 
Integral  calculus;  •#«  Calcnina 
\n\arl«x«ntft^  ^soaDcL^  oC^  814, 888, 


\ 


\\ 


I 


1    \ 

\ 

J     < 

m 

1 
1     -  i 

1 

, 

J\ 

') 

If 

•J 


i 

■  • 

! 


\ 


^  X 


\    \ 


IntcqwIntioD.  method  of,  900,  S37, 

8.^3.  809,  417.  m 
IntarUnlK.  i»i,  VM,  480,  487 
lOToluln,  31i 
Imrolulinn :  fK  Orometij  (Mnleni 

HfttlhclK) 


Iibak  ibo  Honpin,  Ifil 
Iiidonig  ot  Atlico*.  116 
IiidnrniorScTilU,  110.    rF. to,  118 
IrachiODODS  curre,  371,  377 
iKiperimclrical  problem,  IM,   S77< 

376,  400,  413 
1*017,  **^ 

Jacobi.  471-3 

-  >«r.  to.  420,  tM,  UK,  IfiS.  ir>i, 
4C8,  4ri9,  170,  473,  473,  474, 
483.  4W,  490,  497 

jKobmni>.  173 

Janm  I.  of  Enpliitid,  ref.  to,  3fiO 

imlaet  II.  of  England,  rcf.  lo^  848 

Jelletl  on  »>cCuUii(;h,  488 

Jerome  on  linger  ajinboliiim,  117 

Jews.  acionoB  of,  8.  173,  177 

John  of  Palenno,  175 

John  Ui»pak:n-i9, 173.    let.  to,  17G 

JoDc*.  Wtn.,  .VJl.  401 

Jordftn,  C.  4X3,  18.1,  ine,  489 

JonUnn'.  177-«l 

—  ret.  10.  173,  311,  314.  317,  SS3, 
"1.  aifi 


KuIncT,  467 

Kant,  reC.  to.  424,  436 

EaoffmanD  lor  Mereatot),  S19. 338 

Keill.  36fi 

KclTin,  Lofd.  439.  45n,  198,  BOS 

Kemptcn,  da,  137 

Kepler,  363-1 

—  ref.  10, 189.  814,  357.  265,  3M, 

177,  387,  S88.    309,   S30^    SSI. 

US,  SM 
Sephr'i  lawi,  1ST,  M4,  387,  Ml, 

S4S 
Setn  00  AfTk-Bhate,  lU 
Kcnclwoateiiicr  on  Oonlaii,  48n 
KmtW,  118 


M7 

Kinckfanyaen,  Rf.  lo,  SSS,  SH 
Kinf:>:le7  00  HrpatU,  lU 
KirclihnlT.  603 

Klein,  F.  C.  455,  4S7,  471.  47fi, 
4Tii.  IMJ,  48.1.  IM,  487,  493, 491 
KniH-lic  on  P^oel^^  115 
K5nii.'i<l>cr|ier,  L.,  478,  409 
Korlcwcd,  460 

KDnHleTfki,  S.,  489 

Krciiirr  am  Arab  aeienM,  IM 

KroiKfker,  L-.  469.  4TS 

Kiihn.  479 

Knmmer,  Xtt.  434,  46S,  469,  488 

Kiiniubcrg  on  Eadona,  46 

Laenur  on  elliptk  tanetiona,  47( 
Iacioii.  4'i3 
LaKiaDKC.  411-39 

—  ref.  «f.  57,  103.  r4.  SM.  SOS, 
sno.  303,  371,  378,  S89,  398,  403, 
inc,  IDT.  437, 43s,  439,  443.  414, 
lin,  4.S3.  456,  4S3.  4C8,  498.  438 

Lahire.  3^.    ret  to,  131,  «M 

Laloubire.  318 

Lambert,  110-1.    nf.  to.  SM 

Lam^.  306,  486  _, 

Lninpo.  ref.  to.  455 

l^nilcn.  406.  430 

Uplnce.  423-31 

—  rrf.  lo.  274,  319.  369,  371,  SR9, 
40':,  123.  431,  433,  43.i,  439,  444, 
41tl,  4 1%  *Mt,  453,  456.  463,  477. 
480,  4'.Ki,  499.  600 

Laplikcc'a  coeffieienta,  433,  433 
Latilado,  introdnctaon  of.  18,  91 
Lavoinicr.  4W 
Law.  tacKltr  of.  148 
Leatt  acliOD.  408,  418,  418 
LcKAi  common  nintti[ria,  61 
Lewt  square*,  418,  431,  449,  45T 
Lebeseoe.  806 
Legrndre,  431-4 

—  lef.  lo,  806.  403,  418.  433,  438, 
431,  i3.%  439.  444,  45ft,  461,  468, 
16!>.  471.  478.474,477.  4M 

Leftendre'a  «o«IDeiMita,  433,  4S3 
Leibniti,  863-76 

—  ref.  to.  347.  364,  384,  816,  838, 
837.  339,  SSS,  8U,  US,  SCr,  IM, 
BfiO.  376.  ITT,  180.  n> 

Leipiia.  nBhmito  tl,  IW 
L»rm»  DMihM;  m  DiikfeM 


i  \ 


\i} 


618 


UTMOL 


I- 


IiHDttiMtttot  48 

818,  891,  884 
Lm  YL  of  Conitantinopto,  181 
Lm  Z.  of  Borne,  Slifal  oa,  981 
Laodamua  of  Athens,  48 
Leon  of  Athene,  48 
Leonardo  de  Vinei,  918-9 

—  ref .  to,  968 
Leonardo  of  Pi«i,  178-8 

—  ref.  to,  69,  918,  916 
Leonide  (ahooting  atari),  800 
Le  Paige,  914.  896 

Lealie  on  arithmotie,  195,  191 
Less  than,  symbol  for,  944,  948 
Letters  in  diagrams,  91,  40 

—  to  indicate  maguitodM,  61, 160, 
179,  929,  938 

Lendppas,  84 

LeudeMlorf  on  Cremona,  491 

Lever,  principle  of,  68 

Leverner,  499-Ml.    ret  to,  417 

L6vy  un  graphics,  494 

Lexvll  on  Pappus's  problem,  108 

L'HuMpital.  3H0.     ref.  to,  891 

Lhulivr,  103 

Librution  of  moon,  414,  446 

Libri,  ref.  to,  905,  915,  917.  954 

Lie.  444-5 

—  ref.  to,  409,  486,  489 
Life  aHsurancc,  400 

Light,  ph^Hical  theories  of,  68, 986, 
813-1,  335-6,  4UU,  441,  446-7, 
4117 

—  velocity  of,  286.  397.  418,  460 
LiUvati.  the.  156-60 
Limiting  values,  3H0 

LimiU.  method  of.  989,  990 
Lindemaun.  39,  488 
Lines  of  curvature,  436 
Lintearia.  377 
Linus  of  Liege.  335 
Liouville,  468,  475 
Lippershey,  2.>7 
Lobatschewsky,  57,  491 
LogarithuiB,  201-3.  223.  219-8,  988 
London  Mathematical  Society,  489 
Longitude.  91.  357.  391 
Lorentz  on  Alcuin.  140 
Loria,  ref.  to,  ziiv 
Lonis  XIV.   of   France,    lef.    to^ 
SrJ,  313,  3Ci 


Loidi  XYL  of  rmmm.  wtL  K 

LncM  di  Baigo;  set  PlwioH 
Loom  of  Li4fa,  888 
Loetaa,  nl.  lo,  99 
Lnnea,  qoadmUare  of*  41-8 
Lather,  rot  to,  891,  898 
I4N^80 


MaoCixllagb,  488 

Maodonald  on  Napte,  849 

MaeUuria,  898-9 

—  ref.  to,  884,  849,  884.888,4ftl, 

416 
UacMahon,  P.  A.,  468,  487 
Magiti  squares,  198-A,  818,  88T 
BXaguettsm,  445-8. 448,469-80^488 
Mauran,  890 
Mnlvea,  de,  881-9 
Mamercns,  18 
Mandryatus,  18 
Manitius  on  Hipparehua,  88 
Mamdon  on  the  ealeoloa,  888 
Maps,  944.  960-1 
Marcellus,  68,  78 
Marie,  ref.  to,  66,  987,  466 
Marinus  of  Athens,  116 
Mariutte,  389 

Msrkotl  on  Tchebyeheff.  468 
Marulois,  241 
Marre  on  Chuqaet,  919 
Martin,  ret  to^  91.  196,  964 
Mary  of  England,  ref.  to,  998 
Madcheroni,  59 
Mass,  oentrea  of.  76, 104.  960^ 

309,  309 
Master,  degree  of,  148 
Mii»tlin,  269 

Mathematici  Veterea,  the,  118 
Mathews,  O.  B..  on  numbers. 
Matter,  constitution  of.  976 
Matthiessen,  59 
Maupertuis,  408 
Maurice  of  Orange,  lef.  to.  968L 

979 
Maurolyeus,  933 
Maxima  and  minima,  dctenuaa- 

tion  of.  309,  314,  854,  879,  897. 

490 
Maximilian  I.  of  Oermanj,  908 
MazweU,  J.  C.  439,  460,  481,  608 
lU^«i,  F.  Q.^  404«  410 


r 


\ 


i» 


INDEX. 


519 


i 


I 


Umytr,  J.  T.,  409 

Mechanics.  .Diiciisfled  Iqr  Arehjr- 
Us,  31;  Aristotle,  51;  Arcm- 
medes,  76;  and  Pappas,  104. 
DeTelopment  of,  hy  Bterinas  and 
Galileo,  253-6;  and  bj  Hnj^ens, 
Sltt--3.  Treated  djnamieallj  by 
Newton,  914  ei  aeq.  Subsequently 
extended  by  (among  others) 
D*Alembert,  Madaarin,  Eolsr, 
Lagrani*e,  Laplace,  and  PoissoB, 
chapters  ZTn,  ZTin.  Recent  woik 
on,  494^ 

Medicine,  Greek  practitioners,  ISl 

Medieral  nniTersities,  145-9 

Melsnchthon,  ref.  to,  907,  999 

Melissos,  83 

Mensechmian  triads,  49 

Menaechmas,  49-50 

—  ref.  to,  88,  55,  79,  81 
Menelaus,  97.    ref.  to,  891 
Menge  on  Euclid,  54 
Menoo,  General,  ref.  to,  449 
Mercantile  arithmetic,  161, 174-^ 

18&-201,  919,  915 
Mercator,  G.,  260-1 
Mercator,  N.,  819.    ref.  to,  888 
Mercator*8  fyrojection^  961 
M«r«,  de,  ref.  to,  994 
Mersenne,  815-6 

—  ref.  to,  978,  292,  408 
Meteoric  aggregations,  496 
Mfton.  86 
Metrodoms,  105 
M4siriac  815 

—  lef.  to,  927.  807,  808 
Michel],  J..  440 
Microscope,  957,  884 
MiU's  Logic  ref.  to.  45 
Milo  of  Tarentom,  90,  99 
Minkowski,  467 

Minos,  King,  ref .  to,  44 
■Minns;  tt  Subtraction 
-»  symbols  for,  5,  107.  109,  159, 

901,  912-4,  917.  920,  921,  922, 

928.  917 

—  origin  of  symbol,  912-4 
Mittag.Leffler,  474,  476 
Mdbins,  496.    ref.  to,  494 
Mohammed,  ref.  to,  119 
Mohammed  ibn  Mnaas  Ht  hSkt^ 


Moivre,  de,  894-5.  ref.  to,  898, 410 
Molk  on  elliptic  functions,  478 
Moments  in  theory  of  fluxions,  855 
Monastic  mathematics,  187-49 
Monge,  435-7 
~  ref.  to,  403,  478»  489 
Montmort,  de.  881 
Montncia,  927 

—  ref.  to,  260,  317,  328,  876,  877 
Moon,  secular  acceleration  ol^  429, 

500 
Moors,  mathematics  of,  170-5 
Morgan,  A.  de;  «^r  De  Morgan 
Morley,  F.,  on  functions,  476 
Morlcy  on  Cardan,  997 
Moscbopulus,  192-4.    ref.  to*  897 
Motion,  laws  of,  956,  986 
Mouton,  336,  364 
Miiller;  *te  Begibmontanns 
MulUnger,  ref.  to,  140.  145 
Multiple  poinU,  351,  381,  889 
Multiplication,    processes    of^    4, 

109,  132,  138,  195-8 

—  symbols  for,  947 
Murdoch,  851 

Murr  on  Begiomontanns,  907.  911 
Music,  in  the  quadririum,  94,  95, 

118,  137-43 
Musical  progression,  80 
MnUwakkil,  Caliph,  lef.  lo,  151 
Mydorge,  ref.  to,  978.  992 

Napier  of  Merehistonn,  242-8 

—  ref.  to,  201,  202,  201,  856 
Napier,  Mark,  ref.  to,  242 
Napier*s  rods,  195-7 

Naples,  uniTenity  of,  147,  177 
Napoleon  I.,  864,  419.  497,  480, 

437,  438,  449 
Napoleon  III.,  448,  478 
Naucrates,  81 
Narier  on  Fourier,  448 
Narigation,  science  of,  261 
Nebular  hypothesis,  42S-« 
NegatiTs  sign,  5, 107, 109, 159,901, 

912-4,  217,  220,  221,  922,  947 

—  geometrical  interpretalioii,  941 
Neil,  801 

Nencleides  of  Athens,  48 
Neptune,  the  pUnd,  499,  500 
Nesselmann,  nt  to,  59,  §2,  106 
Netlo.  E^  489,  486 


I 


590 

lUnwu,  0,  Ul,  MO,  Mi.  m 
MtODtuii,  r.  K.,  Ml 
KawemlL  &.  SOI 
MawtcNi,  B.  A^  gf  Tml^  800 
NawloB,   bMo,  elutptar  xn  (mi 
tabia  of  MDtanto) 

—  »f.  to.  GT,  Tt,  BS.  lOS,  Ma.  tn, 
SW,  3iS.  lU,  149,  StW,  SU, 
M7,   374.  373,  «M,    903,    BU, 

514.  330.  334.  BOS,  aW,  >67. 
SIM,  3UU,  370,  STI,  S79,  374, 
381,   Mi,   Ml,   886,    SM,    SW, 

aui,  3'j4,  aus,  SM,  km,  400, 

401,   403,   4U1,   411.    41B,    487, 
il'J,  413,  4M,  4tM 
Ne»tuii-*  I'liHcipla,  341-8,  S57 

—  nf.  lo,  «5tl,  374,  -.Ml,  ao-j,  ais, 

515,  B7I,  StO,  3H1,  SM,  StM, 
t!l3.  Ml,  400.  4U1,  411,  4i7, 
439,  4W 

Klcbolu  IV.  of  Boom.   mL  Ia, 

183 

KicboMi,  Pm],  raL  to,  149 
Nicholu  BhkbJu  of  Smyruft.  Ill 
Nicole,  381.    n-f.  hi,  3&1 
Nicooucliiu,  97-8 

—  raf.  to,  117,  118,  11^  1S9 
,    Nieomedi!*,  87 

NieutelM  of  Akuuaiw,  Ii6 
Nieuwenlyt,  379 

Ninm,  culing  out  Um,  1G6,  lU 
NiiM,  icr.  to,  tit,  US 
Huiuule  foi  niiii'tjr,  137 
HoD-lfiicliilun  g«(Muatr7,  491-4 
Nuttar,  U.,  473,  488 
M  Bin  ben,  drfecttv*,  3t) 

—  eicemivi',  SO 

—  flduratc,  3114 

—  yalM,  S>J,  81,  816-7 

—  poift-oDHi,  3v,  iva 

Numbsri,  1I1C017  of.  TmttiUMit 
of  hy  Pylliagunii,  38-80;  hj 
Euclid,  01-3;  by  l>iophuitUM, 
113-1;  by  Fetmat,  SU3-8;  bj 
Eukr,  407-H ;  by  Lasraii||c,  41S, 
41V;  by  Liitcnilre,  433-1;  bjr 
Gaunit  Niid  utlivr  lUktbeuuiUoiuu 
«f  nHxaX  tiiiieii,  457,  4«l-3, 
4t>4-U,  470,  477.  47H,  48!l,  484 

NDnwnlii,  •j-iiiUiU  lur,  13-1-33, 
144,    IM,    ICI,   174,    179.   188- 


_  rftSSa,!! 
0S>,  Ml.  Id,  149 

oidudMic  tu,  UT.  m. 

Ollwii  oa  (telwt,  14S,  III 
OlUUno  on  ?uou'i  _  •  ■  - 
"■iph.irf.hs 

OpMitIoB*.nlMilwal3SI,«a.4U 

OppAit,  nt  to,  ■ 

OptiM  toaonwIiiMll.  DiMOMillf 
(unumc  otbcn)  KmoOA,  M ;  ?•*■ 
pu^  103;  AilMMn,  IW;  IbigN 
tteeon.  laS;  BmU,  iOl;  Dcmw- 
Um.  380;  lUnraw.  530;  H«>toB, 
33»-4:  QkOM,  461;  ud  Sir 
WilliM)  HMoUtoo,  480 

—  fpbyMeal),63,a86,SlS-4,MS-«, 
408,  441,  446-7,  497 

Onlcric  Vibilu,  nt.  to,  144 

Orexmiu,  185.    nl.  to,  849 

OriecUlioD  of  Egyiitun  ri  iinilw. 


Orlcani 


ruty  of,  147 


Orrery,  48.  78,  360 
OitcilLitioQ,  Mutw  «(.  aia,  MI 
OKulaling  oircia,  379 
Otfao,  £13 
Oughued,  945-6 

—  Kf.  to,  147,  348,  350,  130.  404 
Oiroid,  nniieimi^  ol,  186.  181 
Ouuutto.  937 

w,  v«lu«  ol,  6,  8,  69.  70.  100,  IH, 
ISti.  \M,  341,  343,  SUO,  ■» 

—  incoDimeiuuTabilily  tt,  lO,  tM, 
410,  433 

—  iultotluvlioD  ol  aymbul,  404-6 
I'ashyawiva,  133 

Pwioli,  315-8 

—  ref.  to,  194,  195,  301,  118,  S91, 
336.  316,  317 

l>Miolua;  tt  PkdoU 
PadDft,  nniv«nity  o^  117,  107.  IM 
l>>iulevi:-,  a.,  470,  487,  498 
I'klittiD*  Autholofcy,  64,  106 
I'RplHia,  1U3-4 

—  Kl.  \A,  U.  W,  «b,  U,  W,  T«^ 


IKDBX. 


5S1 


i 


M,  83,  87,  i07,  SeO^  S8S,  188» 
859 
PtoiboU,  erolnle  of,  818 

—  quadntiire  of,  70-1,  888-90^ 
899,  809 

—  toctifiontion  of,  801 
Parmbolio  geometry,  498 
Parmllel    lines,    101-8,   888»    488, 

492-3 
Parallelognm  of  foieee,  51, 888, 881 
Pardies,  385 
Pftrent,  381 
Pluris,  nnWcnity  of,  145, 148, 147, 

185,  187 
PanDeniflcn,  38 
PMcal,  890-7 

—  lef.  to,  57,  237,  865,  377,  878, 
8011,  310,  311,  315,  356,  860, 
361,  396,  435 

Pftvia,  oniveraity  of,  147 
Pteeock,  451 

—  ref.  to,  125, 174, 188,  440,  450» 
452 

Pedals,  896,  490 

Pdetier,  883 

Pell,  825.    ref.  to,  248 

PembertoD,  lef.  to,  332,  857 

PeDdQloiB,  motion  of^   855,  856, 

311,  812,  825,  444 
Pentagrmm-ttar,  the,  8(^-1 
Pdpin  on  Fr4niele'e  problem,  818 
Perfect  nomben^   80,  61,    816-7, 

407,  408 
Pirier  on  Peseal,  890 
Perseoe,  88 

P^peetiTe,  252,  265,  898 
IVtcr  the  Hermit,  ref.  to,  148 
Petersen  on  fonctions,  476 
Petrareh,  122,  185 
Phalereos,  58 
Petri  on  Cosa,  811 
Pfair,  435 

Pbere^ydes  of  Syroa,  19 
Philip  U.  of  Spain,  ref.  to,  887 
Philippaa  of  Athens,  48 
Pbilolans,  88,  80,  48 
Philonidaa,  81 
PhikpoBns,  48 

PhikMophji  tvsatnent  of^  881 
PboenieMUi  BMllisnatiea,  1-8 


Fhyriea,! 
8;  alio 


«f  mAjmIs 


Piassi  of  Palermo,  458 

Picard,  C.  E.,  476 

Pieard,  K^  482 

Picard,  J.,  340 

IMhan  on  numerals,  190 

Piola  on  CaTslieri,  287 

Pisa,  oniversity  of,  187 

PitiKcns,  233.    ref.  to,  232 

Plana,  501.    ref.  to,  500 

Planetary  motions,  24,  48,  64,  84, 
90,  lUO,  171,  219,  857-8,  264, 
887,  374,  417,  424-6,  458,  468, 
499-502 

—  sUbility,  417,  484,  446 
Planets,  astrological,  124 
Plannden,  121.    ref.  to,  198 
Platina,  ref.  to,  144 
PUto,  44-6 

—  ref .  to,  22,  29,  30,  87,  58,  67 
Pliny,  ref.  to,  95 

Plucker,  488 

Plos;  $et  Addition 

^  symbols  for,  5,  107,  109,  150, 

179,  201,  212-4,  817,  890^  8S1« 

828,  C46 

—  origin  of  symbol  +,  818-4 
natarch,  ref.  to,  16 
POckels  on  Plikcker,  488 
Poincar^,  474,  476,  489,  508 
Poinsot,  445 

Point,  Pythagorean  def.  of^  85 
Poisson,  443-6 

—  ref.  to,  402,  421,  489,  456,  459, 
496 

Polar  triangle,  241,  268 
Polarization  of  light,  314,  447, 448 
Poles  and  polars;   nee  Geonetiy 

(Modem  Synthetic) 
Polygonal  numbers,  29,  108 
Polygons,  regular,  468 
Polyhedrons,  regnlar,  98,  87,  58^ 

87.  116 

—  semi-regolar,  74 
Ponoelet,  43H-9 

~  ret  to,  104,  409,  485,  488,  494 
Pontfeonlant,  501.    ret  lo^  500 
Porisms  of  Euclid,  68 

—  of  Diophantns,  118 
Pori-Boyal,  society  of,  8984 
Potential,  the,  417,   488-4»  481; 

446,  464^  486^  497 


/ 


522 


INDKX. 


¥omm^  origlB  of  kn^  40 
Powm;  «M  EipoiMnU 
FniOM^  onivmfty  of;  UT.  IM 
I'rwkri  OB  Cavftlkri,  M7 
Pvetendor,  the  Yoong.  nl.  to,  MS 
Prime  and  ollim*te  ratios,  430 
Primes,  61,  63,  S16 

—  dUtribtttioB  of,  48S-4»  468,  47t, 
484 

PrioKsheim  on  eonveigeiieT,  486 
l^riuting,  invention  of,  305^  806 
ProbabUitiet,    theory   of,    99fr^ 
809-10.   SIS,   877,   8U8-4,  885, 
400,  411,  418,  415,  487-8,  488, 
41U.  457,  488 
Proclus,  116 

—  r«r.  to,  18,  15,  18,  88,  56 
Prcduct,  symbols  for,  847 
Proinvssious,  ariihmeticsl,  80,  78, 

67,  158 

—  geomctricsl,  80,  61,  71,  75,  158 

—  musical,  80 
Projectiles,  225.  856 
Proportion,  symbols  for,  845,  848 

—  treatmeut  by  Euclid,  60-1 
PbiUuH,  181.     ref.  to,  833 
pM'udohphorical  B|»ace,  488 
PtolcmieM,  dynasiy  of,  53,  85,  118 
PtoWmy,  UU-J08 

—  ref.  to,  70,  84,  87.  88,  90,  158, 
ItS'i,  IG5,  107,  170,  171,  178. 
177,  183,  184,  1H6,  lb7.  807, 
833;   also  §ee  Almagest 

Pulley,  theory  of,  81.  76 
Purbacb,  811.     ref.  to,  807 
Puzzli*«i,  33-4,  61,  226-7,  315 
Pyramid,  surface  of,  73,  156 

—  volume  of,  47,  73,  156 
PytbaKoraH,  19-30 

—  ref.  to,  3,  62 
PytliaKoivaii   ^k*hool,    the,    18-38. 

rvf.  to,  41,  55,  114 


Quadratic  e4|uatioiis,  60,  81,  105, 

no,  154-5,  163-4,  216 
QuadrMtic  reciprocity,  433,  457 
Quadratic  rebiduen,  433,  468 
Qaadratriz,  86,  37,  49 
Quadrature  of  circle;    see  Circle, 

also  tee  w 
—  cone,  73,  156 


808.  818,  987-4.  8it-4 

—  oUipsi^  78 

—  limea,  41-8 

—  parabnlni^  f8-l« 
808 

—  snbere,  88,  78 
Quadriea,  78,  40S.  418 
Qmulrilatenl,  nrra  of ,  Iff 
Qnadrivioai,    84,  118,   181.   U8^ 

140,  148,  148,  185,  188 
Quantios,  486 
Qnmrtie  eqoatioo,  165,  888*  888. 

838 
Qoatemions,  468,  478,  480,  481 
Quetelet,  lef.  to,  858,  817 
Quiutio  equmtion,  470.  477.  481, 

486 
Quipus;  $€€  AbaeiM 
Quotient;  ut  DivisioB 

—  symbols  for,  158,  166,  847-8 

Kaabe  on  conver|{ency,  486 
Rabdoligia,  the,  197,  848 
Badical,   symbob   for,   160,   818^ 

221,  248-8.  289 
llahn,  248 
Rainbow,  explanation  ol^  188,  888, 

321,  333,  334 
lUloigh,  Sir  Walter,  nf .  to,  844 
lUmus,  234 
Uaididall,  ref.  to,  145 
Itatdolt  on  Campanula  184 
Ratio,  symbols  for,  845^  848 
Rifttionaf  numbers,  Euclid  on,  61 
Raylcigb,  Lord,  488,  508 
Recent  mathematics,  chapter  sn 
Reciprocants,  484 
Record,  220-1 

—  ref.  to,  180,  181.  801,  848 
RiTreations.  mathematical,  886-7. 

315 
BeciiUcation  of  corres,  801-8, 888. 

326,  338,  352,  854 
Recurring  series,  895,  418 
Reductio  ad  absurdum,  41 
Ri>duction  in  geometry,  41 
Iteformation,  the,  206 
RefracUon,    183,    861,    886.    814, 

320,   334,    836,    848,    891,  461, 

480,  497 

—  atmosp!iorie,  168 


i 


J 


INDEX. 


528 


^ 


RcgiomonlADat,  207-11 

—  raf.  top  167,  317,  218»  384,  850 
fi«gDla  ignari,  195 
Benaimance.  the  mathemiitMs  of, 

ehaptera  zn,  ziii 
BfM  aiied  for  unknown  qoantity, 

1(»,  309,  317,  329,  338 
Ileflidae^  theory  of,  43S-4«  46^ 

4&I 
Besistanoe,  solid  of  l^ast,  880 
Berersion  of  serieii,  837,  839 
Reye  on  modem  geometty,  489« 

491 
Rhabdas,  183 

Rhcticnt,  833.    ref.  to,  843,  850 
Rhetorical  alRoHra,  106,  109,  154» 

173,  179,  809,  816 
Rhind  papyrus,  the.  8-8 

—  ref.  to.  10,  94,  106 
Rhonias,  886 

RiceatI,  883.    lef.  to,  389 
Riemann,  478-3 

—  lef.  to,  57.  460.  468,  468,  470, 
478,  47o,  476,  489,  491,  498 

Riesc,  881 

Rigaud,  lef.  to,  343,  885 
Ritter  on  Culmann,  494 
Roberral,  817.    ref.  to,  884,  898, 

897 
Roche  on  nebolar  hypothesis,  485 
Rodct,  ref.  to,  8.  158 
Rods,  Napier*s.  l?l5-7,  848 
Roemer,  387 
Rohan,  ref.  to,  836 
RoUe.  837 
Roman  mathematics,  117-8 

—  symbols  for  number*,  181 
Romanus  of  LouTain,  838 

—  ref.  to,  836 

Rome,  mathematics  at,  117-8 
Roots  of  equations,  imaginary,  880, 
478 

—  negatiTe,  880 

—  number  of,  457,  478 

—  origin  of  term,  168 

^  position  of,  885, 887, 841-8, 888, 
481,  448 

—  synunetrieal  fooetioiia  of,  841, 
411,  478 

Boota,  ■qnaie,  eobe^  Ae^  iOO,  818, 
881,  849,  899 

7 


Rowm  on  Alkarismi,  168 
Rosenhain,  J.  O.,  478 
Routh  on  mechanics,  498 
Royal  loBtitution  of  London,  440 
Roval  Society  of  London,  884   . 
Rudolff,  281.    ref.  to,  888 
Rudolph  11.  of  Uermany,  lel.  to, 

Runini.  471 
Rumford,  Count,  440 

Hacchcri,  491 

Saint- Mernne;  $ef  L'Hospital 

Saint- Vincent,  317-8 

—  ref.  to.  311,  319 
Sairotti  on  graphics,  494 
Kalenio,  nnivcnvity  of,  146 
Salmon,  4H7,  488 
Sanderson's  Logic,  889 
Sardou  on  €!ardan,  887 
Saunderson  of  Cambridge,  840 
Saurin,  381 

Sarilc.  Sir  Hen.,  848 
Scali)^T.  340 
Scharpff  on  Cnn,  811 
Schering,  ref.  to,  457.  478 
Schmidt  on  Hero,  91 
Schneider  on  Roger  Bacon,  181 
Schocnflies  on  Plucker,  488 
Schoucr  on  Jonianus,  177 
Schools  of  Char]e^  140-5 
Schooten,  Tan,  317 

—  ref.  to,  237,  840.  885,  880 
Schubert,  H.  C.  H.,  488 
Schwarz,  H.  A.,  475,  476,  489 
ScorcH,  things  counte<l  by,  186 
Scratch  system  of  diTision,  199- 

201 
Screw,  the  Archimedean,  67 
Secant,  168,  341,  850,  400,  404 
Section,  the  golden,  46,  47,  59 
Secular  lunar  acceleration,  600 
Svdillot,  ref.  to,  9,  loO.  168 
Septante  for  scTcnty,  187 
Sercnus,  97.    ref.  to,  891 
Series;  §ee  Eipansion 

—  rerersion  of,  837,  889 
Serret,  413,  488,  486.  487 
SsTille,  School  of,  170 
Sezagesimal  angles,  4,  850 
Sexagesimal  Craetioiis,  100,  ITS 
BeitaDt,  invvotioo  of;  884 


524 


iin>BX. 


■*Or»vMMkU  OD  HiqrgliMMb  811 
BhdMiMMt,  nL  to,  109 
Sigiks,  niU  oi.  lOmiO 
Bimpla  equalioiM,  106^  110 
Bimplieiiu.  raf.  to,  49 
SimpMO,  TImniim,  899-100 

—  riff,  to,  401,  401 

BimaoB,  Hobert,  55.    nf.  to.  88 
Bioe,  90,  97,  99,  1J8-4,  158.  187, 

207.  i41,  215,  250,  400,  401 
Sin  X,  Mries  for,  824,  887,  875 
Bin'i  X,  lerkfli  for,  824,  887 
Bines,  table  of,  70 
SixiiM  IV.  of  Rome,  i«f.  to.  206 
Blee  on  Aleuin,  140 
Bliderole,  203 

Slonum  on  cnlculoa,  866,  868 
Bluftius;  see  Bluze,  de 
Sluze.  de,  82G 

—  ref.  to,  317,  821.  1^ 
Smith,  Henrj,  465-8 

—  ref.  to,  468,  473,  488 
Smith,  11.  A.,  on  Dalton,  441 
Snoll.  261.    rcf.  to,  252,  286 
SocraUm,  ref.  to,  41 

Solid  of  least  rctivtanoe,  880 
SoHd»;  iff  Polyhedrons 
Souin  on  Tchebycheff,  468 

.  SophiritM,  the.  36 
Sound,  Telocity  of,  413,  422,  429 
Spanish  mathematics,  170-5 
SpeUding  on  F'  incis  Bai-on,  859 
Speidell  oi  log:^ithm8,  203 
Sphere.  Murfoce  and  volume  of,  60 
Hphereii,  voluuieti  of,  47 
Spherical  excess,  241 
Spherical  harmonics,  423,  432 

'  Spherical  space,  492-3 
Spherical  trigouoiuetry,  167,  288 
Spheroids,  Archimedes  on,  72,  78 
Spinoza  and  Leibnitz,  365 
Spiral  o(  Archimedes,  72 
Spiral,  the  equiangular,  377,  495 
Spouius  on  Cardan,  227 
Stiuaro  root,  symbols  for,  160, 212, 

221,  219.  2119 
S4|uarcs.  table  of,  2 
Stiuaring  the  circle;  see  Circle 
Stohl,  W.,  475 
Staigmiilkr,  ref.  to,  215,  219 
BupuleuMu  00  Jordanoa,  177 


stMi.  Uiit  oi;  801  liii  ill. 

Btatici,  Mt  lltalMBias 

BUQdl,   vom  480.    Nt  K  tti» 


fltiiatn  aiigiML  Bim^%  88 
Btafaa.  460 

BtMohtB  OQ  BtcfiwM,  988 
SteliMr,  490 

—  i«r.  to,  488,  478.  490.  481 
Btttinachnaidcr  oo  AnuielMl,  171 
BteTinna,  858-4 

—  ref.  to,  78.  80S^  888|  888»  8i0» 
898 

Stewart,  Maltbew,  899 

Btifel,  221-8 

~  ref.  to,  801,  818.  888.  884.  887, 

238,  885 
BtifTeUas;  m»  Btifal 
BtirUng,  851,  897 
Stobaeua,  ref.  to,  65 
Stokes,  G.  G.,  486,  498.  8021  808 
Strabo,  ref.  to,  2,  44 
String,  vibrating,  theory  of,  888-7, 

389.  392,  413 
Studium  generale,  146 
Sturm,  ref.  to,  443 
Style  or  gnomon,  18 
Subtangent,  308,  818,  881.  888 

—  constant,  339.  872 
Subtraction,  prueessea  ol^  194 

—  symbols  fur,  5,  107,  109,  159. 
201,  212-4,  217,  220,  921,  828, 
247 

Suidas,  ret  to,  18 

Sun,  distance  and  radina  of^  88.  86 

Bundiab.  18 

Supplemental  triangle,  841,  268 

Surds,  symbols  for,  160,  218^  221. 

248,  299 
Suter  on  Dionysodonii^  95 
Swan-pan;  see  Abacus 
Sylow  and  Lie  on  Abel,  460 
Sylvester  11.,  143-5 
Sylvester.  484 

—  ref.  to,  342,  407,  468,  489 
Symbohc  algebra,  107 
Symbols,  algebraical,  246-50 

—  trigonometrical.  250 
Symmetrical  functions  of  roota  of 

an  etpiation.  341,  411,  478 
Syncopated  algebra,  107 
Synthetio  geometry;  tee  Geomalij 


"V' 


/ 


1 

f 


INDBJL 


525 


Tbbil  ilm  Korrm,  IftS.    i«f.  to,  161 

l*«u,  481,  496 

Tuii;ent  (geometrical),  S84-5,  817, 

S21-2 
Tangent  (trigoDoiiietrieal),  168,S41, 

»0.  400,  404 
Tan~*jr,  leriee  for,  824,  875 
Tanneiy,  J.,  on  elliptle  fenetkiDt, 

475 
Tannery,  8.  P.,  ref.  to,  19,  28,  86, 

62;  88,  91,  99,  118,  122,  802 
Tartaglia,  22^7 

—  lef.  to,  195,  199,  216, 229,  280, 
232,  237,  240,  247 

Tartalea;  $ee  Tartaglia 
Tantochronooe  enrre,  312 
Taylor  (Brook),  891-8 

—  ref.  to,  889,  413 

Taylor's  tlieoiem«  392,  897,  420, 

479 
Taylor,  C^  on  coniet,  266 
Taylor,  It.,  on  nnmeimU,  190, 191 
TaTlor,  T.,  on  I^jthagoraR,  80 
Tchebyelieff,  468 
Teleecopet,  267,  811,  313, 814, 828, 

834 
Ten  at  radii;  »ee  Decimal 
Tension  of  elastic  string,  826 
Terqoem  on  Ben  Esra,  172 
Terrier  on  graphics,  494 
Tetrad,  Pythsgorcan,  24 
Tbales,  14>7.    ref .  to,  8 
Thaiias  of  Athens,  48 
Theaetctnn,  60.    ref.  to,  48,  66,  69 
Theano,  ref.  to,  20 
Theodoros  of  Cyrene,  82,  88,  44, 

60 
Theodosias,  94.    ref.  to,  820 
Theon  of  Aleiandria,  114-6 

—  ref.  to,  66,  67,  182 
Theon  of  Smyrna,  98 
Tbeophrastas,  ref.  to,  18 
Thermodynamics,  418 
Thermometer,  indention  of,  266 
Theta   ftinctions,   462,  468,   469, 

471.  478-4. 
Theodhis  of  Athens,  48 
Thomsoii,  J.  J.,  460,  498,  608 
Tbonaon,  Sir  Benjamin,  440 
Thoawoo,  Sir  William;  $t9  Kidwim 
Thtm  bodiea,  pmMsfli  oC;409, 416, 

678,602 


Thurston  on  Camot,  448 
Thymaridas,  9^-9.    ref.  to^  106 
Tidal  friction,  426,  602 
Tides,  theory  of,  267-8,  889,  898^ 

426,  602 
Timaens  of  Locri,  82,  44 
Tisserand,  427,  501 
Titios  of  Wittemberg,  426 
Todhnnter,  ref.  to,  432,  466 
Tonstall,  191 
Torricelli,  318 

—  ref.  to,  258,  292,  826 
Tortaons  cnrres,  883,  406,  488 
Toschi,  382-3 
Trajectories,  860,  878 
TransTCfMls,  97 

Trembley,  411 

Trentlein,  ref.  to,  178,  188,  212 

Triangle,  area  of,  91,  92 

—  arithmeUcal,  226,  237,  298^ 
Triangle  of  forces,  219,  253,  881 
Triangnlsr  nnmbers,  29 
Trigonometrical  functions,  90,  97, 

99,  153-4,  156,  167-8,  207,  241, 
245,  250,  378,  400,  404,  470 

Trigonometrical  symbols,  origin  of^ 
2o0,  400,  404 

Trigonometry.  Ideas  of  in  Rhind 
papyrus,  8.  Created  by  HIppar- 
chus,  90;  and  by  Ptolemy,  99. 
Considerrd  s  part  of  astronomy, 
and  treated  as  such  by  the  Greeks 
and  Arabs,  167-8.  Hindoo  works 
on,  153-4,  156.  160.  Treated  by 
most  of  the  mathematicians  of 
the  renaissance,  chaptera  zn,  xnL 
Development  of  by  John  Barw 
noulli,  378;  DemoiTrs,  894-6; 
and  Euler,  404;  and  Lambert, 
410 

Trigonometry,  addition  fonnnlaa, 
90,  233,  470 

Trigonometry,  Hi^ier;  aee  EUiplie 
functions 

Triple  triangle,  the,  20 

Irisection  of  angle,  86, 89, 88, 240, 
Y26 

Tririum,  the,  118,  189,  142,  148 

Tsehimhansco,  826.     ref.  lo^  868 

Tsehotfl;  see  Abaeoa 

^eho  Braha,  202,  282,  864 

T^lor,  K.  a,  ni:^  186 


i 


ftS6 


iMDn. 


Ujeia,  IM 

Undolatofy  Tkaonr  ((Mk^*  '^^-^ 

896,  4(MI.  441.  446-7 
UniveniUM.  »«ikval,  14A-7 

—  eonioilnm  aI,  147-9,  lM-7 
UnivMriitieii  of  muUiMUMt,  906 
Unknown  qoanUlj,  word  or  lymbol 

for,  6.  106,  166,  160,  166,  60!^ 
617,  323.  623,  264,  686,  666 
Urban,  d*.  on  ArUUrohiu,  64 

Yftlaon.  ref.  to.  446.  476 

Van  Ceulen,  242 

Vandermond«,  407.  426 

Van  Uoonuit,  801 

VaniKhiug  points,  898 

Van  SchouUun,  817 

->  ref.  tu,  287.  240,  265,  680 

Variation,  lunar,  166 

Variations,  calculus  of,  406,  412, 

413.  445,  472.  475,  489 
Varignon.  380-1.    ref.  to,  258 
Velaria.  37*^ 

Venturi  on  Letmanlo  da  Vinci,  216 
Vem  X,  series  for,  375 
Verulam.  Lonl.  259.    ref.  to,  806 
Vibrating. htring.  386-7,  389.  892, 

413 
Vieuiui.  university  of,  147,  166 
VieU.  235-41 

—  ref.  to,  83.  201,  223.  232.  235, 
242,  244,  247,  249.  317.  330 

Viga  Oanita,  150-7,  159-00 

Vince.  ref.  to,  355 

Vinci.  Leonardo  da,  218-9 

—  ref.  to.  253 

Vinculum,  intrudurtion  of,  246 
Virtual  work,  389,  413,  417,  436 
Vis  mortua.  374 
Vis  viva.  374 
Vitaliii,  ref.  to,  144 
Vitruviuri,  ref.  to,  77 
Viviaui,  326 
Vlacq,  203 

Voltaire  on  Newton,  848 
Von  Breitachwert  on  Kepler,  262 
Von  Uelmholtz,  400,  492,  498 
Von  Huuiboiat.  459,  490 
Von  Murr,  ref.  to.  207,  211 
Von  SUudt,    490.    ref.    to,    466, 
49U 


646 


WiiMingUi  —  B— MWi  WH 
WagMT,  |6I6 
Wallis,  697-666 

—  ret  to,  64,  166,  644,  t66. 
677,  S9(K  606,  606,  611,  619, 
826,  664,  826.  626,  8«|,  6t6» 
887.  847.  646.  886.  666 

Wapidor  JMi  RndoUr,  661 
Watdias,  iBWitioa  oi;  616,  666 
Watt,  ret  to,  98  ^ 
Wave  Theory  (Optica),  618^  6t6» 

409.  441.  446-7 
Weber,  HL,  472,  476 
Weber,  W.  E.,  459,  460 
Weierstrasa,  474-6 

—  ref.  ui  471.  476.  469.  460 
Weissenb<^  ref.  to,  187,  146 
Werner,  ref.  to.  140.  148 
Wessel,  479 

Weyr,  ref,  to,  8,  6 
Whewell,  W.,  458 
Wliiston,  340.    ref.  to,  882,  657 
Wliitebeiul,  A.  N.,  492 
Whittaker^  E.  T.,  502 
Wiihnan,  212.    r«f.  to,  201,  647 
Wilkinson  on  Bliaskara,  157 
William  of  Malmesboiy,  ref.   to. 

144 
Williamson  on  Endid,  54 
Wilson  on  Cavendish,  439 
Wilson's  Theorem.  416 
Wingate,  B.,  244 
Woei^cke.  ref.  to,  68,  150,  166, 176. 

170.  190.  218 
Wolf,  202,  425 
WoUaston,  ref.  to,  814 
Woodcroft  on  Hero,  91 
Woodliouse,  450-1.    ref.  to,  450 
Work,  virtual,   389,  413,  417,  486 
Wnn.  324 

—  ref.  to.  301,  312,  824,  846 
Wright,  2G0-1 


Xanophanes,  38 

Xylauder,  238 

—  ref.  to,  114,  121,  218,  288. 

Tear,  duration  of,  17,  66^  66 
Young,  Thoft.,  440-1 


IHDBX. 


527 


^  »45. 


K 


i.  SM, 


Tooog;  TImm.*  rat.  lo^  814,  tSC, 

ai,  489,  447 
Toang,  Sir  Wm^  oa  Ti^lor,  891 


ZftBgiiieitlcr,  raf.  lo^  914 


Zcnims,  909,  917,  998,  988 
ZeH>,  qrmbol  for,  19(^1 
ZeU  function,  475 
Zeathen,  59,  79,  80,  488 
Zeaiippot,  66 

ZicRler  on  RcgiomonUnot,  907 
ZoniU  liAnnoniet,  489 


143 


357 


947 

r.  to. 


173, 


1^50 
488 


I 

I 

/ 
/ 


t  r«im»  ar  4.  jm*  c  w.  cut.  «t  nn  viiifmRV 


A  SHORT  ACCOUNT  OF  THE 

HISTOKY  OF  MATHEMATICS 
bt  w.  w.  rouse  ball 

[Third  Edition.    Pp.  ixif  +  327.    Price  10«.  net] 

MACUniLAN  AND  CO.   hn.,  LONDON  AND  HEW  I'OBK. 


iThis  hook  gives  an  ncconnt  of  the  lives  nnd  discoveries  of ' 
those  nwtlieaiatidnns  to  whom  the  development  of  the  sabfecC 
is  mainly  due.  Hie  use  of  techniotlities  hiu  lieen  Avoided 
and  the  work  is  intelligible  to  imy  one  ncqnftintrd  with  the 
elements  of  niallteniBtira. 

The  author  commences  with  an  account  of  the  origin  and 
progresa  of  Ur^ek  mathematics,  fixHn  which  the  Alexandrian, 
the  Indian,  and  the  Arab  schools  mnj  lie  said  to  have  arisen. 
Nest  the  mnthcnmlici  of  medieval  Europe  and  the  rennisaance 
are  'Icscrifaeil.  Tlie  latter  part  of  the  liook  is  devoted  to  tho 
history  of  modem  mnthematics  (beginning  with  the  inventioa 
of  analjtical  s^ometrj  and  the  infinitesimal  calculus),  the 
account  of  which  is  brought  down  to  the  present  time. 


Thi«  eicellrnt  nnmrnarj  of  lh«  hiitorj  of  nuthfituitici  iinp|ilies  a 
want  ohieb  ha^  loni;  been  frit  in  Ihi*  eaantrj.  Tlie  eitremelj-  dirGcDlt 
qneKtiini,  how  far  on'-h  a  work  lOiould  be  tnhnienl,  han  be*n  mjItcJ  with 
gteat  iact....Tbe  work  cunUina  nisn.v  i-alnalile  hinlx,  snil  ■■  thiirouRtily 
nadslile.  Tbe  biuurspliiei,  whicb  iiielade  thoM'  of  nunl  of  the  men  wbo 
pIsTcd  important  pxttn  in  the  deTelnprnrnt  of  ealtara.  sre  fnii  and  KenersI 
CDougti  to  ialereiit  the  otdinsry  reader  a"  well  ai  the  apreialiat.  It*  Tains 
to  the  latter  ii  much  incraued  by  lh«  nnmemuii  refereooea  to  anlhoritJM^ 
a  good  table  of  eonlcnia,  and  a  fall  and  areuiab]  indei. — Tht  Saturiag 

ilr  Bsll'a  book  Khoitld  ueet  with  a  htartj  weleome,  lor  thnoith  «• 

poa»tM  other  hietotica  of  apeeial  branches  of  malhenulic^  this  is  the  finl 

■ertoo*  attempt  that  baa  been  made  in  tbe  EoRluh  langnaRS  to  give  a 

qrslitBatie  aoooont  of  tbe  origin  and  derelopiDent  c(  the  acieote  aa  a 

-  lAola.    It  ia  wiillen  too  in  an  aKraeUve  atj-la.    TeehniwUtiee  ara  not 

''      '  ■•and  th*  work  ia  inlanpanad  with  baomphiesi 

'         at  tba  8«DBnl  iwder.    Thna  tht 

laUha  MVMdftwilll  ihaaaia, 


A  vMlth  of  MlboiiliM,  oAm  Ayr  inm  iiBfitiart  wilk 
A  work  rach  m  this  oilnmolv  forviiabto ;  aai 
■iftwinaHft  hafo  icmob  to  bo  grmtcral  lor  tko  VMI  mmommi  of  Ia* 
lomuUioD  whiAh  Iim  baao  ooodMBoed  iolo  thk  aliofft  mooommL.,jM  a 
•unrey  of  to  wide  extent  it  is  of  oooreo  impoMiblo  to  ^dfo  oaiythiBf  k«l 
m  bore  eketcb  of  the  ¥00000  Uoee  of  veeeorebt  ond  thio  dreooMtoooo 
tends  to  render  o  norrotiTe  lenippy.  It  eoys  mneh  for  Mr  Boll's  descrip* 
tivo  skill  that  bis  bistoiy  reods  mors  like  o  oontinnoos  stoij  tbon  % 
series  of  merely  oonsecotivo  sununories. — The  Aoadtmff, 


We  con  heartily  reoommond  to  our  wisthemsficml  reoders,  and  to 
others  also,  Mr  Ball's  iiUtoiy  of  ^athem^iicM.  Tho  history  of  what 
miKht  be  sappoeed  a  diy  subject  is  tokl  in  the  pleasaotest  and  most  read- 
able style,  and  at  the  name  time  there  is  evidenos  of  tho  most  carefdl 
research. — The  Ohttrrator^, 

All  the  salient  points  of  mathematical  hi«to^  are  gifcn,  and  many 
of  the  results  of  recent  aotiqnarian  nhieareh ;  but  it  must  not  be  imagined 
that  the  book  is  at  all  Uq-.  Od  the  oontraiy  tho  bioipraphical  sketches 
frequently  cootsin  smutting  auecdotes,  and  many  of  the  theorems  men- 
tioned are  xtty  clearly  explained  so  as  to  bring  them  within  the  grasp  of 
thotse  who  are  only  acquainted  with  elementaiy  mathwmstics. — Smture. 

Le  Ktyle  de  M.  Ball  est  clair  st  Elegant,  de  nombrenx  spervos  rendeni 
Cscile  de  suivre  le  fil  de  son  exposition  et  «le  frequentes  citations  permet- 
tent  4  celui  qui  le  devire  d*spprt>foudir  les  recherches  que  Tsuteur  n'a  po 
qu*ettleurer....Cet  ouvrage  pourra  deveuir  trett  utile  comme  manuel  d*his- 
toire  des  math^'Uiatique^  pour  les  rtudiants,  et  il  ne  sera  pas  deplace  dans 
les  biblioth^ques  dee  savants. — Bibtioiheca  Mathrmatica, 

The  author  niodeiitly  describes  his  work  as  a  compilation,  but  it  is 
thoroughly  well  digested,  a  due  proportion  is  observed  between  tho 
Tarious  partH,  aud  when  occasion  demands  he  does  not  hesitate  to  give 
an  indepcndeut  judgmeut  on  a  disputed  point.  His  verdicts  in  such 
instances  appear  to  us  to  be  generally  souud  and  reasonable...  To  many 
readers  who  have  nut  the  courage  or  tlie  opportunity  to  tackle  the 
ponderous  volumes  of  Montuda  or  the  (mostly)  ponderous  treatises  of 
Germsn  writers  on  specisl  periods,  it  may  be  somewhat  of  a  surprise  to 
tind  what  a  wealth  of  human  iuteicht  attaches  to  the  history  of  so  '*dry" 
a  subject  as  mathematics.  We  ars  brought  into  contact  with  many  re- 
markable men,  some  of  whom  liave  played  a  great  part  in  other  fields,  as 
the  names  of  Qerbert,  Wren,  Leibnitz,  Descartex,  Pascal,  D'Alembert, 
Csrnpt,  smong  others  may  testify,  and  with  at  least  one  thorough  black- 
guard (Cardan) ;  and  Mr  Ball's  pages  abound  with  quaint  and  amusing 
touches  churacteristie  of  the  authors  under  consideration,  or  of  ths  times 
in  which  they  lived. — Manchttter  Guardian. 

There  can  be  no  doubt  that  the  author  has  dons  his  work  in  a  very 
excellent  way.... There  is  no  one  interested  in  almost  any  part  of 
mathematical  science  who  will  not  welcome  such  sn  exposition  as  the 
present,  st  once  popularly  written  aud  exact,  embracmg  the  entire 
subject.... Mr  Ball's  work  is  destined  to  become  a  standard  one  on  the 
subject.  — riie  Ula$tfow  Herald. 

A  most  interesting  book,  not  only  for  those  who  are  mathematicians, 
hat  for  the  much  larger  cifc\e  ol  t^osa  whA  cats  to  trace  the  eonrse  of 
geaenl  scientific  progress.    U  ia  ^nUau  Vn  vm^  %  ^wj  ^%v  ^wsa  who 
bmve  unlj  an  eleiuenUry  acquaVnUncft  ^*^>^^*^^^5^2^,'*i'^=*^ 
ererj  jmgo  oomething  of  i^T^orsi  mXfctwX— 1V«  OxJ^tA  ^a^V^^t. 


A  PRIMER  OF  THE 

HISTORY  OF  MATHEMATICS 
Bt  w.  w.  rouse  ball, 

[pp.  iv  +  146.     Price  2».  neL] 
MACMILLAN   ASD  CO.   Lm,   LONDON    AND  KBW  TOBK. 


Tint)  book  contAJna  a  sketch  in  popular  Inngnage  of  tho 
history  of  mnthemntim  ;  it  inctuden  mune  notice  of  th«  live* 
anil  suiTOuiidinfpi  of  those  to  whom  the  ilevelopment  of  the 
nuhject  is  mainly  due  (»  well  ai  of  their  ^' 


This  Primer  in  wrill*D  in  lh«  an-i-MM*  kItIc  with  which  the  ■olhor 
hui  mvle  mil  ae<iUKint«d  in  hii  preiina'  mutii  ;  and  we  are  nun  that  all 
reaipn  tt  it  will  b«  leail;  to  uy  that  Mr  Bull  linn  meceeited  in  the  htm 
b<>  hu  formpd.  that  "  it  nuiT  nnl  be  Dnioh-nntins  "  eftn  to  thow  who 
mn  iinBcqaainted  with  (he  Iniilini;  facta.  II  in  jint  the  book  to  give  an 
inlFlUcrnt  ynnng  Btnilent.  uii)  ohonld  tllitTe  him  on  to  the  prnval  of 
ytr  BsH'a  "  Shnrt  AcCoUDl."  The  prexent  work  ii>  not  a  mere  T^thattjli  at 
tbut,  thnnch  nalarall;  moat  or.vhni  ia  here  civcn  will  be  fonnd  in  eqniia- 
lent  rami  m  the  larner  work.. ..The  choice  of  malarial  appears  to  a*  la  ba 
such  as  shonld  lend  intercxl  to  the  ulndr  of  matliemali«ii  and  incrraiie  it* 
edacalioiial  vatue,  which  ha«  been  the  aathur>  aim.  The  book  (piea  weU 
into  the  pocket,  and  ia  eicellenlly  printed.— 7h'  »ea4tmif, 

XVe  have  here  a  new  instance  nf  Mr  lloate  Dall'*  *ki1l  in  KiriiiR  in  ft 
small  space  an  intcllii.'ible  wcunnt  ot  ■  lame  Hubject  In  137  pases  wa 
have  a  nkelch  oftbe  proBreu  of  raalhrinalica  from  the  earliest  reourda  up 
to  the  iniddW  of  thin  cvDiarj.  anil  jet  it  ii>  inlen-ilini;  to  read  and  bjr  no 
mean*  a  mere  catalogue Tht  Manehfttr  RiifirrfMa. 

It  ia  not  often  Ihut  a  rcTicvernf  mnthemalieal  works  can  eonfeis  that 
be  hna  r«nd  one  «f  Ibem  Ihroiish  from  coTer  tn  rarer  vilhoal  abatement 
of  interest  or  rntiL'ne.  Dot  Ihal  ix  true  nf  Mr  Ilnnse  Ball'i  wonderfttDT 
ealertaininKlillle'-HiKtirTiif  Maihematip<',"ir)iichweheartil;reconimena 
to  even  tho  quite  ra<limviitnT7  mathematician.  The  capable  mathentatical 
ma'ter  will  not  fai!  to  find  a  dnzen  inlereating  facts  Uieiain  to  nrunn  hia 
leaching. — Thr  SalurAaif  ICeritw. 

A  fascinatinx  little  tolarae.  irhich  ahooM  be  in  the  baodi  of  all  wha 
do  not  posaesH  lbs  more  claburale  Ilhtnry  of  MmlMrmatia  by  the  aaoM 
author.— TAf  Jl-iUirmaliral  Gattttr. 

This  excellent  sketch  ahoald  be  in  the  hands  of  emr  (todenl,  whether 
he  is  stndjinK  malhematica  or  no.  In  most  tat^s  there  is  an  anlbrlnnaia 
lack  of  knowledKe  upon  this  anbjeel,  and  we  velcoma  anythinn  that  will 
help  to  aap^tly  the  dvGcienc?.  The  Prinier  ia  written  in  ■  eoneise,  lucid 
and  caaj  manner,  and  inrea  tbe  miler  a  Reoeral  Idea  of  tba  progrw  of 
mathematica  that  ia  both  interesting  and  initrvctira. — Tttt  CamhrUft 

Mr  Ball  haa  not  been  deterred  hj  the  niateiNt  and  necMa  of  hit 
larger  ■' History  of  Mathematica"  from  pghlinbtng  ■  itmh  BompiBdi— 
1b  aboni  a  qnarter  of  the  ■pau...,0l  oonrae,  «bal  bi  MW  alaM  te 
_  . .  ....„.,..    ._.  ..    *-»ai,««^'S-i-.s«-».. 


loMvMti  of  tlw  lOiloiy  or  %  MitMt  kikb  iiiliMl  to  ili  alii^r.  < 
iacwMtt  ito  tdnmlfci— I  lai— .    We  can  jwagiiw  aa  ballir  callMatfa  far 

a  oftiiiAil  pariiMl  of  ~ 


TIm  anthor  baa  4om  food  aarrioa  to  mathamatidana  hj  ^■^I'f  im  | 

work  in  this  special  field.. ..The  Primer  ^ivea,  in  a  lirief  eonmaM,  the  I 

hiatoiy  of  the  advanee  uf  thia  hraneh  of  leienoe  when  under  Oieek  Uifln*  I 

ence,  anring  the  Middle  Ages,  and  at  the  Kenaiiianee,  and  then  Roea  on 
to  deal  with  the  introduction  of  modern  analTtiJ*  and  ita  recent  develop- 
mentii.  It  refere  to  the  life  and  work  of  the  lead«fm  of  mathewatJeal 
thouffht,  adds  a  new  and  enlarged  Talne  to  weU*known  pioblema  hy 
treating  of  their  inception  and  hietorj,  and  lighta  np  with  a  warm  and 
penonal  intere«t  a  bcience  which  eonie  of  ite  detraeton  have  dared  to 
call  doll  and  cold. — Thr  Educatiomtl  Retiew. 

It  is  not  too  much  to  aajr  that  thia  little  work  ahonld  be  in  the  poseca- 
aion  of  every  mathematical  teacher. . .  .Tlie  Primer  gives  in  a  small  compaaa 
the  leading  e%'eDt«  in  the  development  of  mathemaiics. ...  At  the  same  tuna, 
it  is  no  drj  chronicle  of  facts  and  theorems.  The  biographical  skelehea 
of  the  greiit  workers,  if  short,  are  pithy,  and  often  amusing.    Well-known 

Eopoiutious  will  attain  a  new  interest  for  the  papil  as  he  traces  their 
story  lung  before  the  time  of  Euclid.—  The  Jourmai  of  Kdticatitm* 

This  is  a  work  which  all  who  apprehend  the  value  of  **  mathematics  " 
ahoulil  read  and  study...,  and  thoite  who  wish  to  learn  how  to  think  will 
find  advantage  in  reading  it. — The  Knglisk  Mechanic. 

Mr  Bair«  book  should  serve  as  an  admirable  introduction  to  the 
aubjvct.  It  supplies  a  very  obvious  want,  and  should  be  welcome  to  all 
teachers  who  denin:  to  teach  their  subject  in  an  intelligent  manner.— Jibe 
ScoUutan. 

The  subject,  so  fur  as  our  oa*n  lan»;uage  is  concerned,  in  almost  Mr 
Ball's  own,  aiid  those  who  have  no  leisure  to  read  his  former  work  wiU 
find  in  this  Primer  a  luKhly-readuble  and  iuMtruvtive  chapter  in  the  history 
of  education.  The  condensation  has  been  skilfully  done,  the  reader*a 
interest  beiu^  suatained  by  tlie  introductitm  of  a  good  deal  of  far  from 
tedious  detail. —  The  Gltugov  Herald. 

Mr  W.  W.  House  Ball  is  well  known  as  the  author  of  a  veiy  clever 
history  of  mathematics,  besides  useful  works  on  kindred  subjects.  Uia 
latest  production  is  A  Primer  of  the  Ilislury  of  MalhrmaticSf  a  book  of 
one  huiidivd  and  forty  pagt-s,  giving  in  non*tc€hnical  language  a  full« 
concise,  and  readable  narrative  of  the  development  of  the  science  from 
the  days  of  the  Ionian  CI  reeks  until  the  present  time.  Anyone  with  a 
leanin}{  towards  algebraic  or  geometrical  studies  m'ill  be  intensely  inter- 
ested in  this  account  of  pro;{ress  from  primitive  usages,  step  by  step, 
to  our  proent  elaborate  liy^temrt.  The  livex  of  the  men  who  by  their 
research  and  discovery  lielpetl  along  the  good  work  are  described  briedy, 
but  gtuphica]ly....The  Primer  nhould  become  a  standard  text*book. — The 
Literary  World, 

The  book  affords  to  students  in  our  high  schools  and  colleges  a  means 
of  gaining,  with  a  small  eipenditure  of  time,  a  sulUciently  complete 
hist«iry  of  the  mathematical  subjinrts  they  are  studying,  to  give  them 
a  much  greater  appreciation  of  and  interest  for  such  subjects. — Sciemce, 

This  is  a  capiul  little  sketeh  of  a  subject  on  which  Mr  Ball  la  an 
Mckuowlcd^ed  authority,  and  o(  which  too  httle  is  generally  known, 
ilr  ISaH,  moreover,  wriiea  ea»UY  and  w«Vl,  wvA\a»  >\x«  vtv  <^  saying  what 
be  luM  to  «ay  in  tu  intereslinn  aX^W— TKe  ScWm\  Qw«.T4\a». 


\ 


> 


MATHEMATICAL 


RECREATIONS  AND  PROBLEMS 
bt  w.  w.  rouse  ball. 

[Third  Edition,    Pp.  xil  +  276.    Price  7*.  net.] 

r 

HACMILLAN  AND  CO.,  Ltd^  LONDON  AND  NEW  YORK. 


This  work  is  divided  into  two  pnrt^,  the  finit  on  mathe- 
matical  recreations  and  puzzles,  the  second  on  some  problems  of 
historical  interest ;  but  in  both  parts  questions  which  involve 
advmnccd  mathematics  are  excluded. 

The  mathematical  recreations  include  numemus  elementary 
questions  and  paradoxes,  as  well  as  problems  such  as  the  pro- 
position tliat  to  colour  a  map  not  more  than  four  colours  are 
neccMsarr,  the  explanation  of  the  efloct  of  a  cut  on.  a  tennis 
ball,  the  fifteen  puzzle,  the  eight  queens  problem,  the  fifteen 
school-girls,  the  construction  of  mapc  squares,  the  theory  and 
history  of  mazes,  and  the  knight's  path  on  a  chess-board. 

The  second  part  commencen  with  a  sketch  of  the  history 
of  three  classical  problems  in  f^P'^metry  (namely,  the  duplication 
of  the  culje,  the  trisection  of  an  angle,  and  the  quarlrature  of 
the  circle)  and  of  astrology.  The  last  three  chapters  are 
devoted  to  an  acc<»unt  of  the  hypotheses  as  to  the  nature  of 
space  and  mass,  and  the  means  of  measuring  time. 


i  .  V 

M  fffvUflBs  fi 

#  nm%  mad 


Mr  Ball  hsii  slmidT  aitainf«l  a  ponition  in  the  front  rank  of  writers 
on  rahjcctx  connected  with  the  hiiitoiy  of  mathctnaticff,  and  thin  brochore 
will  mdd  another  to  hiii  pucccwie«  in  thin  fieM.    In  it  he  ha^  collected  a 
mam  of  information  hearinir  opon  matter^  of  nore  grncral  interest, 
written  in  a  ntvle  which  in  eminently  rradable,  sod  at  the  vame  time 
exact.     He  has  done  his  work  m*  thoron^hlT  that  he  han  lefl  few  ears  for 
other  gleaners.    The  nature  of  the  work  is  completelr  indicated  to  the 
mathematical  student  by  itii  title.     Does  he  wmnt  to  rerire  his  acqoaint- 
anoe  with  the  Probi^tmeM  riuhniiB  ti  D^lerfnhln  of  Baehetor  the  P/rriO' 
tiom$  Mnthimati^meM  et  Phn^i^tiet  of  OzanaoB?   Lei  him  take  Mr  Ball  for 
his  eonpanion,  and  be  wifl  have  the  cream  of  these  works  pot  before  him 
with  a  wealth  of  tUnstratioa  qnite  ddightfol.    Or,  eomiii^  tA  tPMi^x^wwfc 
times,  he  will  have  fall  and  aocutata  dV«t«M\fm  ^  ^^dM  tSNw»>^^g«=^^_ 
•ChhMss  rings,*  •the  ftftem  B^MMA-vtaAa  v>^M^"^^  ^_^ 
AstfeJent  siMes  k  davQied  l5  MoomAa  «ft  iDM^ 


tlw  ThiM  ClMsiflAl  PioUmm}  lb«ra  U  alto  a  Mil  dbttak  of 


f:# 


mmI  ialemliiig  oqUiiim  of  Um  pwnat  lUto  of  oor  knowMft  of  Imw- 
■paeo  and  of  tho  oooilitttiion  of  matter.  Tliia  oBOBMialioA  taiHy 
indifatfn  Khft  w%^f  ^^nA\mA^  imi  ||  tnffieiaall j  tlatot  whal  tht  nadw 
maj  axpael  to  find.  HoKov^r  for  Um  om  of  toidiri  who  aaaj  vkh 
to  pursue  tlie  Mvoral  heads  fnrther,  Mr  Bali  gives  detsiied  lefsMneeo 
to  the  soorese  from  whenoe  he  has  derived  hU  intormation,  Theeo 
Jimikematical  Bicreatiom  we  can  eoaunend  as  snited  lor  aiathematieiana 
and  eqnallj  for  others  who  wish  to  while  awsj  an  oecssinnil  hoar.— - 
Tht  Aeudemy. 

The  idea  of  writing  vome  sach  aoeonnt  as  that  before  us  most  havo 
been  present  to  Mr  BslPs  mind  when  he  was  collecting  the  material 
which  he  has  so  skilfullj  worked  up  into  his  UUtorjf  of  MatkemmiUt, 
We  think  this  because... manj  bits  of  ore  which  would  not  suit  tho 
earlier  work  find  a  fitting  niche  in  this.  Howaoerer  the  case  majr 
be,  we  arv  sure  that  non-matliematical,  as  well  as  msthematical, 
readers  will  derive  amusement,  and,  we  venture  to  think,  profit  withal, 
from  a  peruHal  of  it.  The  author  has  gone  venr  exhaustively  ov«r 
the  ground,  and  has  left  us  little  opportunitjr  of  adding  to  or  eorrecting 
what  he  has  thun  repruduced  from  bis  note-books.  The  work  before 
us  itf  divided  iuto  two  parts:  mathematical  recreations  and  matho. 
maticsl  problems  and  speculations.  All  these  matters  are  treated  lucidly, 
and  with  tfuilicient  detail  for  the  ordinary  reader,  and  for  othere  there  la 
ample  blorc  of  n;k*iviiceii....Our  snalysiA  sliows  how  great  an  extent  of 
ground  iH  covered,  and  the  account  is  fully  pervaded  bj  the  attractive 
charm  Mr  Ball  knows  so  well  how  to  infuse  iuto  what  many  persons 
would  look  upon  as  a  dry  subject. — Nature. 

A  fit  hequel  to  itn  author**  valuable  and  interesting  works  on  tho 
history  of  mathematics.  There  is  a  fancination  about  this  volume  which 
reituUii  from  a  happy  combination  of  puzzle  aud  paradox.  There  is  both 
milk  for  babes  and  strong  meat  for  grown  meu....A  great  deal  of  the 
information  is  hardly  accensible  in  any  English  books;  and  Mr  Ball 
would  deserve  the  giatitude  of  matliematicians  for  having  merely  col*' 
lected  the  factt*.  But  he  has  presented  them  with  such  lucidity  and 
vivacity  of  style  that  there  is  not  a  dull  page  in  the  book;  and  he  has 
added  minute  and  full  bibliographical  references  which  greatly  enhance 
the  value  of  liis  work. — The  Cambridge  Review. 

I^Iatheinaticiaus  with  a  turn  for  the  paradoxes  and  puzzles  connected 
with  uumbi-r,  H|)aoe,  and  time,  in  which  their  science  abounds,  will 
delight  in  Mathematical  Uecreatiuus  and  Problems  of  Past  and  Pre§eni 
Times.— The  Times. 

Mathematicians  have  their  recreations;  and  Mr  Ball  sets  forth  tho 
humours  of  mathematicii  in  a  book  of  deepest  interest  to  the  clerical 
reader,  and  of  no  little  sttractiveneM  to  the  layman.  The  notes  attest 
an  enormous  amount  of  research. — The  SatioMti  Observer. 

Mr  Ball  hss  produced  a  book  of  extreme  and  all  but  nniqne  inteical 
to  general  readers  who  dabble  in  science  as  well  as  to  prolresed  mathema- 
ticians. — The  Scottish  Leader. 

Mr  BskUt  to  a'bom  we  are  already  Vni\%>AfedlQx  two  excellent  Historiea 
of  AfaiLeniatics.  haa  jusl  produced  %  V»\;«^^ /w^a.  \«k  ^h»|^ 
Wmuaied  by  those  who  enjoy  ^^^^^.  ^^JJl^^ 


h 


/ 


ermaki,  and  voziks— old  and  new ;  and  it  will  be  ttraiiRe  if  eren  the 
iboet  leeroed  do  mil  find  foniething  fresh  in  the  assortment — The 

Mr  Roose  Ball  has  the  tme  gift  of  story-telling,  and  he  writes  so 
lileasantlj  that  though  we  enjor  the  falness  of  his  knowledge  we  are 
tempted  to  forj^et  the  eonoideraHle  amonnt  of  labour  involTed  in  the 
preparation  of  his  book.  He  gives  as  the  history  and  the  mathematics 
of  many  problems... and  where  the  limits  of  his  work  prercnt  him  from 
dealing  fally  with  the  points  raified,  like  a  tme  worker  ne  Rires  as  ample 
references  to  original  merooirfi....Tbo  book  is  warmly  to  be  recommended, 
and  shoald  find  a  place  on  the  shelves  of  erenr  one  interested  in  mathe- 
matics and  on  thoi«e  of  erery  pablio  library. — fhf  Mtinche$ter  Gtiardian, 

A  work  which  will  interest  all  who  delight  in  mathematics  and 
mental  exercises  generally.  The  student  will  often  take  it  np.  as  it 
eontnins  many  problems  which  puzzle  eren  clever  people. — The  Engli$h 
Mecluinie  amd  World  of  Science, 

This  is  a  book  which  the  general  reader  should  find  as  interesting 
as  the  msthcmatician.  At  all  ercnts,  an  intelligi*nt  enjoyine«jt  of  its 
contents  presupposes  no  more  knowledge  of  mathematics  than  is  now^i- 
days  possessed  by  almost  ereryboily. — The  Athtnanm, 

An  exceedingly  interesting  work  which,  while  appealing  more  directly  to 
those  who  are  somewhat  matliemstically  ihcKned.  it  is  at  the  same  tima 
ealcnhtted  to  interest  the  general  reader.... Mr  Ball  writes  in  a  highly 
interesting  manner  on  a  fascinating  subject,  the  result  being  a  work 
which  is  in  erery  respect  excellent. — The  Mechnnicnl  World. 

E  am  lirro  muito  interessante,  consagrado  a  reereios  mathematicos, 
algnns  dos  quaes  sfto  muito  bellos,  e  a  problemas  interessantes  da  mesma 
seiencia,  que  nio  exige  para  ter  lido  grandes  conhecimentos  mathematicos 
e  que  tem  em  grao  elcTado  a  qnalidade  de  instruir,  deleitando  ao  mesmo 
tempo. ^-vToMma/  de  $ciencion  wmthemMicoB,  Coimbra, 

The  work  is  a  very  judicious  and  suggestive  compilation,  not  meant 
mainly  for  mathematicia*is,  yet  made  doubly  valuable  to  them  by  copious 
references.  Tbe  style  in  the  main  is  so  compact  and  clear  that  what  is 
central  in  a  long  argument  or  process  is  admirably  presented  in  a  few 
words.  One  great  merit  of  this,  or  any  other  really  good  book  on  such  a 
subject,  is  its  soggcstiveness ;  and  in  running  through  its  pages,  one  is 
pretty  sure  to  think  of  sdditional  problems  on  the  same  general  lines.— 
Bulletin  of  the  Sete  Yjrk  Jinthemotical  SoeiHy, 

A  book  which  deserves  to  be  widely  known  by  those  who  are  fond  of 
solring  puzxles...and  will  be  found  to  contain  an  admirable  classified 
collection  of  ingenious  questions  capable  of  mathematical  analysis.  As 
the  author  i'^  himself  a  skilful  roathemfttician,  and  is  careful  to  add  an 
analysis  of  Ano>>t  of  the  propositions,  it  may  easily  be  believed  that  there 
ia  food  for  study  as  well  as  amusement  in  his  pages.... Is  in  every  wnj 
wortl^  of  praise.— TAe  School  Guardian, 

Once  more  the  author  of  a  5*orf  Hi$tonf  of  Jtathemuitict  and  a 
Hiitoqf  of  the  Study  of  J^athemotic*  nt  Cambridye  gives  evidence  of  the 
width  of  his  reading  and  of  his  skill  in  compilatioii.  Prom  the  elemen* 
tary  arithmetical  poasiM  which  were  known  in  tho  sixteenth  and  seven* 
tcenth  eentoriea  to  those  modem  ones  the  mathematical  dfsensslMa  e& 
wUck  haa  taied  the  energise  oC  the  ableBlin.Tea{U«iSMK,^n!n^ 
hnen  bean  left  nnffqprasmtod.  Tbi  MWinin  «lllua«seQM^% 
an  MtoM  with  giwi  tnlntai„.,TWa  Vow>l  W  %  m^^Mtm 
" UtanMn.— Tha  OsfsHL  Meittf^M- 


A  HISTORY  OF  THE  STUDY  OF 

MATHEMATICS  AT  CAMBRIDGB 
bt  w.  w.  rousb  ball. 

[Pp.  xvl  +  264.    Price  6m.] 

THE  UNIVERSITT  PRESS.  CAMBBIDOK. 


Thm  work  contains  an  aocount  of  the  devetopment  oC  the 
•tody  of  niathematics  in  the  university  of  Camhridge  from  the 
twelfth  century  to  the  middle  of  the  nineteenth  oentaryi  and  % 
description  of  the  meanH  by  which  proficiency  in  that  study 
was  tested  at  various  times. 

The  first  part  of  the  book  is  devotcjd  to  an  ennmermtioo  of  *  ' 

the  more  eminent  Cambridge  mathematicians,  arranged  chrono- 
k>gica]ly :  tlie  subjcHrt-nrntter  of  their  more  important  works  is 
stated,  and  the  uiethoda  of  exposition  which  they  used  are  in- 
dicated. Any  reader  who  may  wish  to  omit  details  will  find 
a  description  of  the  cliaracteristic  features  of  each  period  in 
the  introductory  paragraplis  of  the  chapter  concerning  it. 

The  Hccond  piirt  of  the  book  treats  of  the  manner  in  which 
matheniutics  won  taught,  and  of  the  exercises  and  examinations 
retiuiriHl  of  Htudt*ntH  in  Huccessive  generations.  A  sketch  is 
given  of  the  origin  and  hihtory  of  the  Mathematical  Tripos; 
thiH  includt*s  the  Kubstamce  of  the  earlier  |N|rts  of  the  author's 
work  on  that  subject,  Cambridge,  1880.  To  exiihun  the 
relation  of  inatheiiiaticH  to  other  departments  of  study  a  brief 
outline  of  the  general  history  of  the  university  and  of  the 
organization  of  e«luc;ition  therein  is  added. 


The  preHeiit  vtiluuie  iit  very  plfSMsnt  readini;,  atid  tliouKh  much  of  it 
neceHKiirily  s)i)>eAl8  only  to  luttthematicianii,  tliere  are  partu — e.tf.  the 
chapters  on  Newton,  ou  tbd  growth  of  the  tripos,  sod  on  the  history  of 
the  aniverrtity — which  a«e  full  of  interest  for  a  general  reader.... The  book 
is  well  written,  the  style  is  crivp  and  clear,  and  there  is  s  bumoroas 
appreciation  uf  some  of  the  curious  old  regulations  which  have  been 
superseded  by  time  and  chsuge  of  cuHtom.  Though  it  seems  light,  it  must 
reprertent  an  ezteuKive  ntudy  and  inveHtigation  on  the  part  of  the  author, 
the  eftsential  results  of  which  are  skilfully  given.  We  can  most  thoroughly 
commend  Mr  BslPs  ?olume  to  all  readers  who  are  interested  in  mathe- 
matics or  in  the  growth  and  the  position  of  the  Cambridge  school  of 
mathematicians. — The  Manchetter  (SuardiaM.  | 

Void  on  lirre  dont  Is  \eclurs  vi\sv^t«  \o>A  d'^^td  le  regret  que  des 
Itnirsax  soalogues  n*aient  pss  itik  UVls  v^tu  \oa\fc^^«sY.«s\j%^aSM^«m^^ 
Mvee  aaUnt  de  soin  el  de  c\sftfe....Tou\*%\«%  v^^^  ^^>wt%ww^^\ 
wivement  int^ress^.— llulltflia  des  scieactt  «a\Wwi«rt^a^t%, 


\ 


/ 


A  book  of  plcAPftiit  and  oneftil  ffcadiiiff  for  both  hiftorUnt  and  msUio- 
BuUidaiM.  Mr  Ball's  prerionii  renearehen  iolo  this  kiml  of  histoiy  ha?* 
already  efitablitb«d  his  repntaiioD,  and  the  book  is  worthy  of  tha  re- 
potation  of  its  aathor.  It  is  mors  than  a  detailed  aoeoant  of  the  rise 
and  pf  ogress  of  mathematics,  for  it  iiiTolTes  a  very  eiaet  history  of  the 
UniYersity  of  CambridRS  from  its  fonodatioii. — The  Edmftimml  Tim€$. 

Mr  Ball  is  far  from  ooDfining  his  narratiTe  to  the  partiealar  setenee 
of  which  he  is  himself  an  acknowledged  master,  and  his  aeeoant  of  the 
study  of  mathematics  becomes  a  series  of  biosraphical  portraits  of 
eminent  professors  and  a  record  not  only  of  the  intelleetnal  life  of  the 
ftiU  hot  of  the  manners,  habits  and  discussions  of  the  great  body  of 
Cambridge  men  from  the  sixteenth  centnry  to  onr  own.... He  has  shown 
how  the  University  has  jn«(lificd  its  liberal  rrpntstion,  and  how  amply 
prepared  it  was  for  tlie  larger  freedom  wliich  it  now  enjoys. — The  thtilff 
liew*. 

Mr  Ball  has  not  only  given  os  a  detailed  sceonnt  of  the  rise  and 
progress  of  the  science  with  which  the  name  of  Cambridge  is  generally 
associated  but  has  alMi  written  a  brief  but  reliable  and  interesting  history 
of  the  nniversity  itself  from  its  foundation  down  to  recent  tiroes.... The 
book  is  pleasant  reading  alike  for  the  mathematician  and  the  student  of 
history.  ^.Sr  Jiime$*a  Gazette, 

A  very  handy  and  ralnable  book  containing,  as  it  does,  a  vast  deal 
of  interesting  information  which  conld  not  without  inconceivable  troaUa 
be  found  elsewhere.... It  is  verr  far  from  forming  merely  a  mathematical 
biographical  dictionary,  the  growth  of  mathematical  science  being  skil- 
fiilly  traced  in  connection  with  the  sncces<ire  names.  TlH>re  are  probably 
very  few  people  who  will  be  able  thoroughly  to  appreciate  the  author^s 
laborious  research«*s  in  all  sorts  of  memoirs  and  transartions  of  learned 
societies  in  order  to  un«*art1i  the  material  which  he  has  mo  agreeably  con- 
densed.... Along  with  this  there  is  much  new  matter  which,  while  of  great 
interest  U»  uiathematicians.  and  more  esp«>ciaI1.T  to  men  brought  up  at 
Cambridge,  will  be  found  to  throw  a  good  deal  of  new  and  important 
light  on  the  history  of  education  in  general. — The  Gln»^w  Herald, 

Exreedinglr  interesting  to  all  who  care  for  mathematics.... After 
giving  an  account  of  the  chief  Cambridge  Mathematicians  and  their 
works  in  chronological  order,  Mr  Rou.<e  Ball  gor9  on  to  deal  with  the 
history  of  tuition  and  examinations  in  the  University. ..and  recounts  the 
steps  by  which  the  word  **  tripos  **  changed  its  meaning  **  from  a  thing 
of  wood  to  a  man,  from  a  man  to  a  speech,  from  a  speech  to  two  sets  of 
verses,  from  vrrses  to  a  sheet  of  coarse  foolscap  paper,  from  a  paper  to  a 
list  of  names,  and  from  A  list  of  names  to  a  system  of  examination." 
Never  did  word  undergo  so  many  alterations. — The  tJtentry  ITorlrf. 

In  giving  an  account  of  the  development  of  the  study  of  mathematics 
in  the  University  of  Cambridge,  and  the  means  by  which  mathematical 
pro6cienry  was  tested  in  successive  generations.  Mr  Ball  has  taken  the 
novel  plan  of  devoting  the  first  half  of  his  book  to... the  mors  eminent 
Cambridge  mathematicians,  and  of  reserving  to  the  second  part  an 
account  of  how  at  various  times  the  subject  was  taught^  and  how 
the  result  of  its  study  was  tested.... Very  intclresting  informattoa  is  given 
about  the  work  of  the  students  during  the  diHsrenl  periods,  witli 
■pecimens  of  ptobleai*papeni  as  llur  baek  aa  1^01.  T^  \s>^l  ^^y 
c^Jeyablo,  and  kHw  a  wp^til  «»i  mniite  9fii«A  ^ 
Mlboritie0whttftMM4^fldmttei«aid^«ll^9M     "' 


AN  ESSAY  ON    I 

THE  OEMK8IS,  C0MTKNT8,  AMD  HI8T0RT  OP 

NEWTON'S  ^^PRINCIPIA'* 
bt  w.  w.  rouse  ball. 


[Pjk  X  -h  175.    Price  6$.  net] 

ICAOIflLLAN  AND  CO.  Ltd..  LONDON  AND  NEW  TOBK.  \l' 


This  work  contaiiui  an  account  of  the  successive  diaooT 
of  Newton  on  gravitation,  the  uiethoda  he  used,  and  the  hislorj 
of  his  researcheH. 

It  oomnienceH  with  a  review  of  the  extant  authoritiea 
dealing  with  the  subject.  In  the  next  two  chapters  the  in- 
vestigations made  in  1 606  and  1679  ai«  discussed,  some  of  the 
documents  dealing  therewith  being  here  printed  for  the  first 
time.  The  fourth  chapter  is  devoted  to  the  investigationa 
made  in  1684  -.  these  are  illustrated  by  Newton*s  professorial 
lectures  (of  which  the  original  manuscript  is  extant)  of  that 
autumn,  and  are  summed  up  in  the  almost  unknown  memoir 
of  February,  1085,  which  is  here  reproduced  from  Newton's 
holograph  cop}*.  In  the  two  following  chapters  the  details  of 
the  pre|»aratiun  from  168.5  to  1687  of  the  Prineipia  are 
di'scribed,  and  an  analyHis  of  the  work  is  given.  The  seventh 
cliapter  ctiuipriHeM  an  account  of  the  researches  of  Newton  on 
gravitation  subsequent  to  the  publication  of  the  first  edition 
of  the  J'rinctpia,  and  a  sketch  of  the  histor}*  of  that  work. 

Ill  the  last  chapter,  the  extant  letters  of  1678-1679  be- 
•twet*ii  Houke  and  Newton,  and  of  those  of  1686-1687  between 
Halley  and  Newton,  are  ivpriiited,  and  there  are  also  notes  on 
the  extant  vorreH|)ondeiice  concerning  the  production  of  the 
sectmd  and  thini  etlitions  of  the  Pnncipia, 

For  the  eMsj  which  we  have  before  ut,  Mr  Ball  should  reoeif«  ths 
tbrnnhg  of^H  tUone  to  whom  the  nam«  ol  ^«>n\o\\  t«eiIU  the  memoiy  of  a 
gremt  man.     The  Pn'scijiiu ,  W«\OiM Wm^%\iAVvBi%m«i^>MiiMSQX  «V^tiW\csn£% 


\ 


» 


> 


life,  ifl  also  to-daj  Um  dassio  of  oor  mathmnatical  wiiiingii,  and  will  b« 
•0  for  fODie  iime  to.  ooiiie....Tlie  Talna  of  the  preacni  work  is  alio  en- 
baoeed  1^  the  fact  that,  besides  containing  a  few  as  yet  wnpablished 
letters,  there  are  collected  in  its  psges  qootations  from  all  docaments, 
thus  forming  a  complete  sonunary  of  cTerytbing  that  is  known  on  the 
subject. ...'rhe  aatfaor  in  so  well-known  a  writer  on  anything  connected 
with  the  history  of  mathematics,  that  we  need  make  no  mention  of  the 
thorooghness  of  the  essaj,  while  it  wonid  be  saperflnoos  f or  ns  to  add 
that  from  beginning  to  end  it  is  pleaMmtlj  written  and  delightful  to  read. 
Those  well  acquainted  with  the  Prineipia  will  find  much  that  will 
interest  them,  while  those  not  ro  fully  enlightened  will  learn  much  by 
reading  through  the  account  of  the  origin  and  history  of  Newton's 
greatest  work. — Sntmre, 

An  Euatf  on  Setrton*$  Prineipia  will  suggest  to  many  something  solely 
mathematical,  and  therefore  wholly  uninteresting.  No  inference  could 
be  more  erroneous.  The  book  certainly  deals  largely  in  scientific  techni- 
eahties  which  will  intenrst  experts  only ;  but  it  also  contains  much 
historical  information  which  might  attract  many  who,  from  laziness  or 
inability,  would  be  Tcry  willing  to  take  all  its  mathematics  for  granted. 
Mr  Ball  carefully  examines  the  evidence  bearing  on  the  development  of 
Newton's  great  discovery,  and  supplies  the  resder  with  abundant  quota- 
tions from  contemporary  authorities.  Not  the  least  interesting  portion 
of  the  book  is  the  appendix,  or  rather  appendices,  containing  copies  of 
the  original  documents  (mostly  letters)  to  which  Mr  Uall  refers  in  his 
historical  criticisms.  Several  of  these  bear  upon  the  irritating  and 
nnfoonded  claims  of  Hooke. — The  Atkenamin, 

• 

La  savante  monographic  de  >I.  Ball  est  rMig^  avee  bcaoooupde  soin, 
el  4  pinsienrs  ^rds  elle  |ieut  p-jtru  de  modele  pour  des  ^rits  de  la  mem« 
nature. — BiMiotheea  Mathemntiea, 

Newton's  ^incipia  has  world-wide  fame  as  a  classic  of  mathematical 
science.  But  those  who  know  thoroughly  the  contents  and  the  history 
of  the  book  are  a  select  company.  It  was  at  one  time  the  purpoi^e  of 
Mr  Ball  to  prepare  a  new  critical  edition  of  the  work,  accompanied  by  a 
prefatory  history  and  notes,  and  by  an  analytical  commentary.  Matue* 
maticians  will  regret  to  hear  that  there  is  no  profipect  in  the  immediate 
futnre  of  seeingthis  important  book  carried  to  completion  by  so  compe- 
tent a  hand.  They  will  at  the  aame  time  welcome  Mr  Ball's  E^mm  on 
ike  Prineipia  for  the  elucidations  which  it  gives  of  the  process  by  which 
Newton's  great  work  originated  and  took  form,  and  also  as  an  earnest  oC 
the  completed  plan. — The  Scot$man, 

In  this  essi^  Mr  Ball  presents  ns  with  an  account  highly  interesting 
to  mathematicians  and  nature!  philosophere  of  the  origin  and  history  of 
that  remarkable  product  of  a  great  geuins  Philoeophiae  SaturatiB  Prin* 
eipia  Mathematiea,  'The  Mathematical  Principles  of  Natural  Philosophy,' 
better  known  .by  tlie  short  term  /Vinci»ia....Mr  Ball's  essi^  is  one  of 
extreme  interest  to  students  of  physicsJ  science,  and  it  is  snre  to  be 
widel(y  read  and  greatliy  appreciated. — Tk€  Gtatgim  Herald. 

To  his  vrell-kDowB  and  scholar^  treatises  on  the  Hieiarw  of  JlsCAe* 
flMNcr  Mr  W.  W.  Boose  Ball  has  added  An  Emmw  on  NewUm^t  PHmeipim. 
Ntwloa'a  Primipim,  as  Mr  Ball  jnsUy  obasrvea,  m  the  dasata  oC  Eas^S&dk. 
■athamatical  writiogs;  and  thia  aMEnaL^VoaBmaiQms^  «BA^\ite«&an^ 


•r  %  mHML  fJillMi  or  M«««Mi'k  mat  vorii^vliiili  Mr  Mllrili 


■•  tifl  hm  mm  tioatwpiatrf.    It  It  maah  to  to  hoped  UmI  1m  will  mot 
01  hfa  inleatioo,  lor  ao  BatfA  imth— ■tfaiin  It  Ukflljr  lo  do  tlM  «o» 
Wttar  or  ia  o  moto  Mvoraat  niril....Ii  It  lonnimmy  A  mr  tkol  Mr  BoO 
boo  o  oooiplHo  knowfodico  of  l^t  Mbjcel.    Ho  writeo  with  oa  oooo  OMi 
Ihot  010  laio.— rAtf  Seoititk  LmMt. 


Lt  volomo  do  M.  Room  Boll  nmfvrmo  loul  oo  ooo  Ton  po«l  dlriwr 
■OToir  Mir  lliUlolio  doc  Primeifet ;  o'imI  d'oUlmin  Virof lo  d*nB  oipril 
oloir,  Joaicitui,  ot  mii\ioditiWt.^ilmUetim  iIm  AKe<rN«M  HofMoMl/fOM. 

Mr  Ball  hoa  put  into  mimII  tpoco  o  vonr  froot  deol  of  lalonoUBg 
-  Bottor.  oud  his  book  ou^ht  to  me«>t  with  m  wide  elreolotiOB  onioog  lovon 
of  Newton  and  the  Prineipia, — Th§  Academff, 

Adnirere  of  Mr  W.  W.  Bonie  Boll**  Short  Attamit  nf  tkt  HiUorf  tf 
IlalhrmaticM  will  be  ^Ud  t<>  r«)Ci*ive  a  detailed  ttndjr  of  the  hietoiy  of  the 
Frimciph  from  tlie  name  hand.  Thin  book,  like  its  ptedeceoior,  givoa  % 
rery  Incid  account  of  its  Mubject.  We  ftnd  in  It  an  aoeoonl  of  Newloo'a 
iBvesliKatiimii  in  bin  earlier  yearn,  which  are  to  lonie  extent  ooUeeted  in 
the  tract  <!«'  Molm  <tlie  germ  of  the  iViarijiJa)  the  text  of  whieh 
Mr  UouMe  llall  k'iveii  u*  in  full.  In  o  later  clia|4er  there  is  a  f^ 
analyMiH  of  the  Principia  ituvlf,  and  after  that  an  aciunnt  of  the  prepara- 
tion of  the  second  aiid  third  editions.  l*robablj  the  part  t4  the  book  whieh 
will  Im*  found  iiioi»t  intcreiititiK  by  the*  Keuvral  reader  is  the  account  of  the 
eorreiipoudence  of  Newton  with  Hooke,  and  with  llalley,  aboot  ibo 
conteutit  or  the  piiMicatiou  of  the  Principia.  This  corrcspondenoe  is 
giTen  in  full,  so  far  an  it  is  recoverable.  Hooke  does  not  appi«r  to 
advantage  in  it.  He  socuHes  Newton  of  stealing  his  ideas.  His  vain  and 
envious  diti]KMition  made  his  own  merits  appear  great  in  his  eyes,  and 
be-dwarfed  the  work  of  others,  ko  that  he  seems  to  have  believed  that 
Newton's  great  performance  was  a  mere  ex|>anding  and  editing  of  the 
ideas  of  Mr  Hooki- — ideas  which  were  meritorious,  but  after  all  mere 
guexses  at  truth.  This,  at  all  eventti,  is  the  most  charitable  riew  we  can 
take  of  his  cituduct.  llalloy,  on  tlie  contrary,  appi'ars  as  a  man  to  whom 
we  ought  to  feel  moHt  grateful.  It  almost  seems  as  though  Newton*s 
pliysicul  inrti^'ht  and  extraordinary  mstheuuitical  |iowers  might  have  been 
largely  waMti*d,  at  was  iVsail'H  ran}  genius,  if  it  had  not  been  for  HaUe^*a 
single>hearted  anJ  self- forgetful  effortA  to  got  from  his  friend's  genma 
all  he  i*ould  for  the  enliKhteunient  of  men.  It  aas  probably  at  his  sug* 
g«*Htion  tlittt  the  writing  of  the  Principia  was  undertaken.  When  tha 
work  wiiH  |in's«'nt<il  to  the  IU>yul  Si>ciety,  they  undert4K>k  its  publication, 
but,  U'ing  without  the  ninvnitary  funds,  the  exiiense  fell  upon  Halley. 
When  Newton,  stung  by  Hooke*s  accusations,  wuhed  to  withdraw  a  part 
of  the  work,  Hulley's  tact  was  ret|uired  to  avert  the  catastrophe.  All  tho 
drudgery,  worr}*,  and  expense  fell  to  his  sliare,  and  was  accepted  with 
the  most  Kenerous  i^ood  nature.  It  will  be  seen  that  both  the  technical 
student  and  the  general  reader  may  find  much  to  interest  him  in 
Mr  Itouse  Hairs  book. — The  Manche$ter  Outtrdian, 

Une  histoire  tr^s  bien  fuite  de  la  gendse  du  livre  immortel  do  Newton. 

...Le  livre  de  M.  Hall   est  une  monographie  pr^oieuse  sur  nn  point 

important  de  Thistoire  des  math^matiqnes.    n  contribnera  4  aocroftre,  it 

o'est  possible,  la  gloire  de  Newton,  en  rfirdlant  4  beanconp  do  leetenn, 

Mvee  quelle  merveilleuie  rapidity  VVUuaXit  tJtsoifiS^Vfa  lAs^^tta  i^  4laH  4  la 

edeaett  ee  JDOoument  immortel.  Vm  Priaeipio.— Ma\h«%V%« 


#  9 


NOTES  ON  THE  HISTORY  OF 

TRINITY  COLLEGE,   CAMBRIDGE 

Bt  w.  w.  rouse  ball. 

[pp.  xir  +  183.     Price  2».  Gd.  net.] 
MACMILIAS   AND  CO.   I.ti...    J^SDON   ASD  KEW  TOBE. 


ml  voliiiiini...tmtiiut 


Thin  nindnit  >tii]  Ooprrti'mliiiR  Utlte  Tnltitni 
morv  for  itn  Kubji-ct  Ihun  nmny  of  tlw  morp  fnri 
of  llie  Hparalc  «.>llrEri>  of  llip  Kosli'li  iiniT>i> 
extrcmclr  rcmdBb'r,  ■nd  Irulv  infonuiog  rlinpliT''  it  iptrt  the  nmdra  K 
Tcrj  riviil  Hccnniit  ■!  nuce  of  llie  orii-iii  and  dL-trlnplDMit  of  llw  Uoinr- 
Filj  of  Cambtidi^F,  of  tlie  ri>e  m,n-\  m»>\n»\  rapreinaf  of  thr  collcftr*,  ot 
Kidr'h  Hall  a<  loiinH«l  )it  EilwarJ  It.  <if  tlif  ■n|>f>re«>iail  of  Ktmt'«  Hall 
hj  Hrni.r  VIII  on  Ih^vmhvr  IT,  I'M.  tltr  fomnlAliuii  of  Trinilj  Collene 
by  tojal  charter  on  I)cccnil«r  19.  and  the  iml>*G<illriit  fiirtitneii  of  the 
prEniier  collvec  of  Camliri>l).'e.  The  Rnhji'ct  iir  in  a  WV"  treated  nndn  th* 
niifcoiiiive  lieadn  of  the  cuIIi-kc.  but  this  a  gaite  auboiilinate  to  Um 
handlinc  and  cliaiarti'Tisaliiin  of  the  iu><jeet  nnrlct  lonr  gnM  period^- 
namclj.  that  diirini:  the  Middle  .\ee\  that  dnrint;  the  Benai»aDoe,  II  it 
nndpr  the  Elizaliellian  KlaliilcK,  bd>1  that  dnring  the  taut  half^^Dtnrj-. 
The  collocFn  art>«  from  the  detrrtninatinn  of  the  CniTerthy  ta  |ire*«iit 
(titilenlH  irho  sere  tjtj  youns  from  H?ekiiie  loilthnE.  whrtber  nnder  llw 
vinaof  one  or  other  nt  the  relJRJDa'  ork'ni — ^aelienin'tanee  which  Rhom 
thin  Univenit7  to  hari'  been  an  eiivntiallT  la/  coqvinition.  Eatlj  in 
the  Kiiteenth  rtnlni^  Die  Call<'i:e  hrul  ahwirbe-l  all  the  meinbeni  of  the 
Univenitj,  and  h''Dcetoilh  tbe  Cniveriilv  *R"  little  more  than  the 
dwiite-uiaatinc  bodi-  In  stnilent*  who  lirrd  and  mured  and  had  tbeir 
educational  bcine  uodcT  the  co1leKe«....Tfae  Uiiiii-r<il7  flnalljr  took  tbe 
form  ol  in  atrgreEate  if  ■eparole  and  independent  coipotmlinnK,  vitb  A 
federal  etmstitntion  annlotioiii  in  ■  roach  aort  of  nj  tu  that  of  Um 
United  Statet  of  America,  and  diderrut  from  aimilw  eotporatiofl*  at 
Faria  b;  the  fart  that  the<e  laltvr  were  alvaji  aobjcct  to  Dnimllr 
■nperviB ion.... There  ia  a  good  aceoanl  of  the  tOort^^of  Koina  an  to 
reassert  the  UniTcnily  at  the  eipenat  of  th«  ColWI.  HD  ^ifSrlM 
btsini  Ur  fiali:a  ^k  wiU  la^t  (town  till  ha  hU  Md%A«»^w(«>^>a«. 
M  mi,—Tht  auiiow  HtT*ld, 


66    10    G     £80       «M. 


H  k  %  rfga  of  Ibt  IJMii^  Mil  •  iiy  itf rfiatwy  — i,  wiw,.^  talMr  I       41 

tMlm  tht  tiooblii  to  mmki  tin  hiitocf  of  hii  ooJlMe  Irnnini  lo  hii  wrafli- 

Oondteiiif  tU  Iftck  of  food  book*  Aboal  tlw  Uaiimitia^  v«  M^^ 
Ifr Ball  that  iMbMbeMi  food  MKwigli  to iMriBlfDrAkfiirdiiltb*  noadl 
iM  modetl^  caUo  his  book  only  •«Notot,'*  jH  Hit  ^BOmmOf  wmMtii.     . 
and  there  It  plenty  of  information,  at  woQ  as  ahnndaats  of  gM  stories. 

Ifr  Ball  has  pot  not  only  the  pupils  for  whooi  he  eoapiled  these 
notes,  hut  the  large  world  of  Trinity  men,  under  a  great  ohugation  by    ^ 
thie  eompendiotts  but  lucid  and  intertsting  hiatorv  of  the  soeisty  to  _ 
whose  service  he  is  devoted.    The  value  of  hie  contribntion  to  our  know-  ' 
led|{e  is  inen*aicd  1^  the  cztrswe  simplicity  with  which  he  tells  his  stoiy, 
and  the  \'m  sUggeiitiTe  details  which,  without  much  comment,  he  has 
selected,  with  admirable  discernment,  out  of  the  wealth  of  materiale  al 
his  disposal.    Hin  initial  account  of  the  development  of  the  University  is 
brief  but  extremely  clear,  pn'ienting  us  with  facta  rather  than  thecries, 
but  Ctotsblitthiuv',  with  much  di«tinctuvss,  the  essential  difference  betm^on 
the  hostels,  out  of  which  the  more  niodem  colleges  gniw,  and  thai 
moosHtic  life  whidi  poorer  students  were  often  tempted  to  join. — Th§ 
GuurdiaH. 

• 

An  interestitog  and  valuable  book... It  is  de«eribed  by  its  author  as 
**  little  more  than  an  orderly  transcript "  of  what,  as  a  Fellow  and. Tutor 
of  the  College,  he  has  been  accustomed  to  tell  his  pu|»ils.  But  while  it 
does  not  pretend  either  to  the  form  or  |o  the  exhauttiveneis  of  a  set 
histoiy,  it  is  seholsrly  enough  to  rank  as  an  authority,  and  far  more 
interesting  au«l  readsble  thsn  most  acaJemio  histories  are.  It  gives  an 
instructive  slurtch  of  the  development  of  the  University  and  of  the 
psrticuhir  history  of  Trinity,  uotiog  its  riie  and  policy  in  the  earlier 
centuries  of  itH  esistence,  until,  under  the  misrule  of  Bentley,  it  came 
into  s  state  of  disorder  which  nearly  reitulted  in  its  diMolution.  The 
subsequent  nse  of  the  College  and  its  position  in  what  Mr  Ball  calls  the 
Victorian  rvnaissanoe,  are  drawn  in  lines  no  le«s  suggestive ;  and  the 
book,  as  a  whole,  cannot  fail  to  be  welcome  to  every  one  who  is  closely 
interested  in  the  progress  of  the  College. — The  Scoismam. 

Mr  Ball  has  succeeded  very  well  in  giving  in  this  little  volume  just 
what  sn  iutelligent  undergraduate  ought  and  probably  often  does  desire  to 
know  about  the  buildings  and  the  history  of  his  College... The  debt  of  the 
** royal  and  religious  foundation**  to  Henry  VIII  is  explained  with 
fulucHH,  and  there  is  much  interesting  matter  as  to  the  manner  of  life 
and  the  expenses  of  students  in  the  sixteenth  century. — The  J/oa- 
Chester  Guardian. 


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