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MATHEMATICS 


:AMBRIDG 


Toronto   Simbcrsttj)  JUbrary. 


PRESENTED     BY 


The   University  of  Cambridge 

through  the   Committee  formed  in 
the  Old  Country 

to  aid  in  replacing  the  loss  caused  by  the  Disastrous   Fire 
of  February  the  14th,  18<J(>. 


A    HISTORY 


OF    THE    STUDY    OF 


MATHEMATICS    AT    CAMBRIDGE. 


Hon&Dtt:   C.  J.  CLAY  AND   SONS, 

CAMBRIDGE  UNIVERSITY  PRESS  WAREHOUSE, 

AVE  MARIA  LANE. 


DEIGHTON,  BELL,  AND  CO. 
ILeipjig:   F.  A.  BROCKHAUS. 


A    HISTOEY 


OF   THE   STUDY   OF 


MATHEMATICS  AT  CAMBRIDGE 


BY 


BALL, 


FELLOW   AND   LECTUKEB   OF   TRINITY   COLLEGE,    CAMBRIDGE 
AUTHOR   OF    A   HISTORY    OF   MATHEMATICS. 


(IDamfcrfoge : 

AT     THE     UNIVERSITY     PRESS. 

1889 

[All  Rights  reserved.] 


Qft 

7 


PRINTED   BY   C.    J.    CLAY,    M.A.   AND    SONS, 
AT    THE    UNIVERSITY   PRESS. 


PREFACE. 

THE  following  pages  contain  an  account  of  the 
development  of  the  study  of  mathematics  in  the 
university  of  Cambridge,  and  the  means  by  which 
proficiency  in  that  study  was  at  various  times  tested. 
The  general  arrangement  is  as  follows. 

The  first  seven  chapters  are  devoted  to  an  enume- 
ration of  the  more  eminent  Cambridge  mathematicians, 
arranged  chronologically.  I  have  in  general  contented 
myself  with  mentioning  the  subject-matter  of  their  more 
important  works,  and  indicating  the  methods  of  exposition 
which  they  adopted,  but  I  have  not  attempted  to  give 
a  detailed  analysis %f  their  writings.  These  chapters 
necessarily  partake  somewhat  of  the  nature  of  an  index. 
A  few  remarks  on  the  general  characteristics  of  each 
period  are  given  in  the  introductory  paragraphs  of  the 
chapter  devoted  to  it;  and  possibly  for  many  readers 
this  will  supply  all  the  information  that  is  wanted. 

The  following  chapters  deal  with  the  manner  in  which 
at  different  times  mathematics  was  taught,  and  the 
means  by  which  proficiency  in  the  study  was  tested. 
The  table  of  contents  will  shew  how  they  are  arranged. 
Some  knowledge  of  the  constitution,  organization,  and 


VI  PREFACE. 

general  history  of  the  university  is,  in  my  opinion,  essen- 
tial to  any  who  would  understand  the  way  in  which 
mathematics  was  introduced  into  the  university  curri- 
culum, and  its  relation  to  other  departments  of  study. 
I  have  therefore  added  in  chapter  XI.  (as  a  sort  of 
appendix)  a  very  brief  sketch  of  the  general  history  of 
the  university  for  any  of  my  readers  who  may  not  be 
acquainted  with  the  larger  works  which  deal  with  that 
subject.  I  hope  that  the  addition  of  that  chapter  and  of 
the  similar  chapter  dealing  with  the  organization  of 
studies  in  the  mediaeval  university  will  sufficiently  justify 
me  in  the  use  in  the  earlier  chapters  of  various  technical 
words,  such  as  regents,  caput,  tripos,  prevaricator,  &c. 

I  have  tried  to  give  references  in  the  footnotes  to  the 
authorities  on  which  I  have  mainly  relied.  In  the  few 
cases  where  no  reference  is  inserted,  I  have  had  to 
compile  my  account  from  various  sources.  Of  the  nu- 
merous dictionaries  of  biography  which  I  have  consulted 
the  only  ones  which  have  proved  of  much  use  are  the 
Biographica  Britannica,  six  volumes,  London,  1747 — 66 
(second  edition,  enlarged,  letters  A  to  Fas  only,  five 
volumes,  1778 — 93);  the  Penny  Cyclopaedia,  twenty-seven 
volumes,  London,  1833 — 43;  J.  C.  PoggendorfPs  Biogra- 
phisch-Literarisches  Handworterbuch  zur  Geschichte  der 
exacten  Wissenschaften,  two  volumes,  Leipzig,  1863;  and 
the  new  Dictionary  of  national  biography,  which  at  pre- 
sent only  contains  references  to  those  whose  nariies  com- 
mence with  one  of  the  early  letters  of  the  alphabet. 
To  these  four  works  I  have  been  constantly  indebted : 
I  have  found  them  almost  always  reliable,  and  very  useful, 


PREFACE.  Vll 

not  only  where  no  other  accounts  were  available,  but  also 
for  the  verification  of  such  biographical  notes  as  I  had 
given,  and  often  for  the  addition  of  other  details  to  them. 
No  one  who  has  not  been  engaged  in  such  a  work  can 
imagine  how  difficult  it  is  to  settle  many  a  small  point,  or 
how  persistently  mistakes  if  once  printed  are  reproduced 
in  every  subsequent  account.  In  spite  of  the  care  I  have 
taken  I  have  no  doubt  that  there  are  some  omissions  and 
errors  in  the  following  pages  ;  and  I  shall  thankfully 
accept  notices  of  additions  or  corrections  which  may  occur 
to  any  of  my  readers. 

W.  W.  ROUSE  BALL. 

TRINITY  COLLEGE,  CAMBRIDGE. 
May,  1889. 


ADDITIONAL  ERRATA. 

Page  7,  line  28.    For  seventeenth  read  sixteenth 
„        footnote,  line  1.     Dele   by  John  Norfolk 

line  2.     Dele   in  1445  and  reissued 


TABLE  OF  CONTENTS. 

Chapter  I.    Mediaeval  mathematics. 

PAGE 

The  curriculum  in  arts  of  a  mediaeval  university 2 

The  extent  of  mathematics  read  during  the  twelfth  century.     .         .  2 

The  extent  of  mathematics  read  during  the  thirteenth  century.         .  3 

The  introduction  of  Arab  science  into  Europe.            .         .  4 

The  extent  of  mathematics  read  during  the  fourteenth  century.       .  6 

Cambridge  mathematicians  of  the  fifteenth  century.         ...  9 

Cambridge  mathematicians  of  the  sixteenth  century.        ...  10 

Cuthbert  Tonstall,  1474—1559 10 

Chapter  II.    The  mathematics  of  the  renaissance. 

The  renaissance  in  mathematics 12 

The  study  of  mathematics  under  the  Edwardian  statutes  of  1549.    .  13 

The  study  of  mathematics  under  the  Elizabethan  statutes  of  1570.  13 

Eobert  Eecorde,  1510—1558 15 

The  Grounde  of  artes,  (on  arithmetic)  published  in  1540.    .  15 

The  Whetstone  of  witte,  (011  algebra)  published  in  1556.      .  17 

His  astronomy  and  other  works 18 

John  Dee,  1527—1608 19 

Thomas  Digges,  1546—1595 21 

The  earliest  English  spherical  trigonometry.       ...  21 

Thomas  Blundeville,  died  in  1595 21 

The  earliest  English  plane  trigonometry  (1594).          .         .  22 

William  Buckley,  died  in  1569 22 

Sir  Henry  Billingsley,  died  in  1606 22 

The  first  English  translation  of  Euclid  (1570).            .         .  22 

Thomas  Hill.     Thomas  Bedwell.     Thomas  Hood.  23 


X  TABLE   OF   CONTENTS. 

PAGE 

Richard  Harvey.     John  Harvey.     Simon  Forman.            ...  24 

Edward  Wright,  died  in  1616 25 

The  earliest  treatment  of  navigation  as  a  science.  .  .  26 

Henry  Briggs,  1556—1630 27 

His  tables  of  logarithms 28 

Introduction  of  the  decimal  notation 28 

His  election  to  the  Savilian  chair  of  geometry  at  Oxford.  30 

William  Oughtred,  1574—1660 30 

The  Clavis,  and  his  other  works 30 


Chapter  III.    The  commencement  of  modern  mathematics. 

Characteristics  of  modern  mathematics. 33 

Change  in  the  character  of  the  scholastic  exercises.          ...  35 

Jeremiah  Horrox,  1619—1641 35 

Catalogue  of  his  library 36 

Seth  Ward,  1617—1689 36 

Samuel  Foster.    Lawrence  Eooke 38 

Nicholas  Culpepper.     Gilbert  Clerke 39 

John  Pell,  1610—1685 40 

John  Wallis,  1616—1703 41 

His  account  of  the  study  of  mathematics  at  Cambridge,  1636.  41 

The  Arithmetica  infinitorum 42 

His  Conic  sections,  Algebra,  and  minor  works.    ...  44 

Isaac  Barrow,  1630—1677 46 

His  account  of  the  study  of  mathematics  at  Cambridge,  1654.  46 

Election  to  the  Lucasian  chair  (founded  in  1662).       .         .  47 

His  Lectiones  opticae  et  geometricae 47 

Arthur  Dacres.     Andrew  Tooke.     Sir  Samuel  Morland.  49 


Chapter  IV.    The  life  and  works  of  Newton. 

Newton's  education  at  school  and  college 52 

Discovery  in  1665  of  fluxions  and  the  theory  of  gravitation.     .         .  52 

Investigations  on  expansion  in  series,  algebra,  and  optics,  1668 — 70.  53 

His  optical  discoveries  and  lectures,  1669 — 72.          ....  53 

His  theory  of  physical  optics,  1675. 54 

The  letter  to  Leibnitz  on  expansion  in  series,  1676.  ...  56 

The  Universal  arithmetic ;  the  substance  of  his  lectures  for  1676 — 84.  58 

New  results  in  the  theory  of  equations 58 


TABLE   OF   CONTENTS.  XI 

PAGE 

The  theory  of  gravitation,  1684.     The  De  motu 59 

The  Principia  published  in  1687 60 

Subject-matter  of  the  first  book 60 

Subject-matter  of  the  second  book 61 

Subject-matter  of  the  third  book 61 

His  election  to  parliament,  1689 62 

The  letters  to  Wallis  on  the  method  of  fluxions,  1692.      ...  62 

His  illness  in  1692—94 62 

His  table  of  corrections  for  refraction,  1694 63 

His  appointment  at  the  Mint,  1695,  and  removal  to  London.  .         .  63 

His  Optics  published  in  1704. 63 

The  appendix  on  cubic  curves 64 

The  appendix  on  the  quadrature  of  curves,  fluxions,  &c.    .  65 

The  publication  of  his  Universal  arithmetic,  and  other  works.          .  66 

His  death,  1727 67 

His  appearance  and  character. 67 

The  explanation  of  his  adoption  of  geometrical  methods  of  proof.  69 

His  theory  of  fluxions 70 

The  controversy  with  Leibnitz. 72 


Chapter  V.    The  rise  of  the  Newtonian  school. 

The  rise  of  the  Newtonian  school 74 

Richard  Laughton,  died  in  1726 75 

Samuel  Clarke,  1675—1729. 76 

John  Craig,  died  in  1731 77 

John  Flamsteed,  1646—1719 78 

Richard  Bentley,  1662—1742 80 

Introduction  of  examination  by  written  papers.          .         .  81 

William  Whiston,  1667—1752 83 

Nicholas  Saunderson,  1682—1739 86 

Thomas  Byrdall.     James  Jurin 87 

The  Newtonian  school  dominant  in  Oxford  and  London.          .        .  87 

Brook  Taylor,  1685—1731 88 

Roger  Cotes,  1682—1716 

His  election  to  the  Plumian  chair  (founded  in  1704). 

The  second  edition  of  the  Principia.  ....  89 

The  Harmonia  mensurarum  and  Opera  miscellanea.     .         .  90 

Foundation  of  the  Sadlerian  lectureships 

Robert  Smith,  1689—1768 91 

List  of  text-books  in  common  use  about  the  year  1730.     ...  92 


Xll  TABLE   OF   CONTENTS. 

PAGE 

The  course  of  reading  recommended  by  Waterland  in  1706.     .         .       94 
The  course  of  reading  recommended  by  Green  in  1707.  .         .       95 


Chapter  VI.    The  later  Newtonian  school. 

Characteristic  features  of  the  later  Newtonian  school.      ...  97 

Its  isolation 98 

Its  use  of  fluxions  and  geometry 98 

The  Lucasian  professors. 

John  Colson,  1680—1760 100 

Edward  Waring,  1736—1798 101 

Isaac  Milner,  1751—1820 102 

The  Plumian  professors. 

Anthony  Shepherd,  1722—1795 103 

Samuel  Vince,  1754—1821 103 

Syllabus  of  his  lectures 104 

The  Lowndean  professors.   (Foundation  of  Lowndean  chair  in  1749.) 

Eoger  Long,  1680—1770 105 

John  Smith.    William  Lax 105 

The  lectures  of  F.  J.  H.  Wollaston  and  W.  Farish 106 

Other  mathematicians  of  this  time. 

John  Kowning,  Francis  Wollaston.  George  Atwood.  .  .  .  107 
Francis  Maseres.  Nevil  Maskelyne.  .  .  .  .  ;  .108 

Bewick  Bridge.    William  Frend.     John  Brinckley.           .         .         .  109 

Daniel  Cresswell.     Mies  Bland.    James  Wood 110 

List  of  text-books  in  common  use  about  the  year  1800.  .  .  .  Ill 
Sir  Frederick  Pollock  on  the  course  of  study  in  1806.  .  .  .111 
Experimental  physicists  of  this  time. 

Henry  Cavendish,  1731—1810 114 

Thomas  Young,  1773—1829 115 

William  Hyde  WoUaston,  1776—1828 116 

Chapter  VII.    The  analytical  school. 

Robert  Woodhouse,  1773—1827 118 

Character  and  influence  of  his  works 119 

The  Analytical  Society  :  its  objects 120 

Translation  of  Lacroix's  Differential  calculus.  .  .  .  120 

Introduction  of  analysis  into  the  senate-house  examination  in  1817.  120 

Eapid  success  of  the  analytical  movement 123 

George  Peacock,  1791—1858.  .  124 


TABLE   OF   CONTENTS.  Xlil 

PAGE 

Charles  Babbage,  1792 — 1871. 125 

Sir  John  Herschel,  1792—1871 126 

William  Whewell,  1794—1866 127 

Foundation  of  the  Cambridge  Philosophical  Society.         .         .         .  128 

Text-books  illustrative  of  analytical  methods.           ....  128 

on  analytical  geometry 129 

on  the  calculus 130 

on  mechanics. 130 

on  optics 131 

List  of  professors  belonging  to  the  analytical  school.         .         .         .  132 

Note  on  Augustus  De  Morgan. 132 

Note  on  George  Green 134 

Note  on  James  Clerk  Maxwell. 135 


Chapter  VIII.    The  organization  and  subjects  of  education. 

Subject-matter  of  the  chapter. 138 

The  mediaeval  system  of  education. 

Education  at  a  hostel  in  the  thirteenth  and  fourteenth  centuries.     .  140 

Students  in  grammar 140 

Students  in  arts 142 

Systems  of  lectures.             143 

The  exercises  of  a  sophister  and  questionist.       .         .         .  145 

The  ceremony  of  inception  to  the  title  of  bachelor.     .         .  146 

The  determinations  in  quadragesima.          ....  147 

The  exercises  of  a  bachelor. 148 

The  ceremony  of  creation  of  a  master 149 

The  doctorate 151 

Philosophy  the  dominant  study:  evil  effects  of  this.         .         .         .  152 
The  period  of  transition,  1535 — 1570. 

The  Edwardian  statutes  of  1549 153 

Establishment  of  professorships 154 

The  colleges  opened  to  pensioners 154 

Eapid  development  of  the  college  system. 155 

The  system  of  education  under  the  Elizabethan  statutes. 

The  Elizabethan  statutes  of  1570 155 

Statutable  course  for  the  degree  of  B. A 156 

Statutable  course  for  the  degree  of  M.A 157 

The  professorial  system  of  instruction 158 

Its  failure  to  meet  requirements  of  majority  of  students.    .  158 

Education  of  undergraduates  abandoned  by  university  to  colleges.   .  158 
College  system  of  education  in  the  sixteenth  century.       .        .        .158 


xiv  TABLE   OF   CONTENTS. 

PAGE 

College  system  of  education  at  beginning  of  eighteenth  century.       .  159 

College  tutorial  system 160 

Private  tutors  or  coaches 160 

System  originated  in  the  eighteenth  century.     .         .         .161 

Practice  of  employing  private  tutors  became  general.          .  162 

Chapter  IX.    The  exercises  in  the  schools. 

Subject-matter  of  acts  under  the  Elizabethan  statutes.     .         .         .  164 

General  account  of  the  procedure 165 

Details  of  the  procedure  in  the  eighteenth  century.  .         .         .166 

Arrangement  of  candidates  in  order  of  merit.            ....  170 

The  honorary  optime  degrees 170 

The  moderators's  book  for  1778 171 

Verbatim  account  of  a  disputation  in  the  sophs's  schools  in  1784.    .  174 

Description  of  acts  kept  in  1790  (Gooch's  account).          .         .         .  179 

List  of  subjects  discussed  from  1772  to  1792.             ....  180 

Value  of  the  system.    Eemarks  of  Whewell  and  De  Morgan.    .  -      .  181 

The  pretence  exercises  in  the  sophs' s  schools.     Huddling.        .         .  184 

The  ceremony  of  entering  the  questions  was  merely  formal.     .         .  184 

The  quadragesimal  exercises  were  huddled 184 

The  exercises  for  the  higher  degrees  were  huddled.            .         .         .  184 


Chapter  X.    The  mathematical  tripos. 

The  origin  of  the  tripos,  circ.  1725. 187 

The  character  of  the  examination  from  1750  to  1763.        .         .         .  189 

The  character  of  the  examination  from  1763  to  1779.       .         .         .  190 

The  disputations  merely  used  as  a  preliminary  to  the  tripos.  190 

The  examination  oral. 190 

Description  of  the  examination  in  1772  (Jebb's  account).          .         .  191 

Changes  introduced  in  1779 193 

Two  of  the  problem  papers  set  in  1785  and  1786 195 

Description  of  the  examination  in  1790  (Gooch's  account).       .         .  196 

Institution  of  a  standard  required  from  all  candidates,  1799.  .  198 

Description  of  the  examination  in  1802.  198 

The  problem  papers  set  in  1802 200 

Changes  introduced  in  1800,  1808,  1818. 209 

Changes  introduced  in  1827 211 

Changes  introduced  in  1833 213 

Changes  introduced  in  1838.  ,  213 


TABLE  OF  CONTENTS.  XV 

PAGE 

Changes  introduced  in  1848 214 

Constitution  of  a  Board  of  mathematical  studies.        .         .  215 

Object  of  the  regulations  in  force  from  1839  to  1873.        .         .         .  216 

Origin  of  the  term  tripos. ,  217 


Chapter  XI.    Outlines  of  the  history  of  the  university. 

The  history  is  divisible  into  three  periods.       .....  221 

The  mediaeval  university. 

Typical  development  of  a  university  of  twelfth  or  thirteenth  century.  221 

The  establishment  of  a  universitas  scholarium  at  Cambridge.  .         .  222 

Privileges  conferred  by  the  state  and  the  pope 224 

Similar  facts  about  Paris  and  Oxford 225 

Constitution  of  university  in  thirteenth  and  fourteenth  centuries.  .  226 

The  degree  was  a  license  to  teach.      .         .         .         .         .  226 

The  regent  and  non-regent  houses 227 

The  officers  of  the  university.     ......  227 

Erection  of  the  schools  and  other  university  buildings.    .        .         .  229 

Provision  for  board  and  lodging  of  students 230 

A  scholar  not  recognized  unless  he  had  a  tutor.          .         .  230 

The  hostels 230 

The  colleges 231 

Establishment  of  numerous  monasteries  at  Cambridge.    .         .         .  231 

Chronic  disputes  between  the  university  and  monasteries.  232 

Development  of  municipal  life  and  authority 233 

The  number  of  students. 233 

The  social  position  of  the  students.  ......  234 

Life  in  a  hostel. 235 

Life  in  a  college. 236 

The  amusements  of  the  students.  237 

Strength  of  local  ties  and  prejudices 238 

The  dress  of  the  students  was  secular 239 

Inventory  of  Metcalfe's  goods.    .         .         .         ...         .  239 

The  academical  costume. 240 

Poverty  of  the  mediaeval  university  and  colleges 241 

Steady  development  and  progress  of  Cambridge -Jll 

The  university  from  1525  to  1858. 

The  renaissance  in  England.  242 

In  literature  began  at  Oxford 242 

In  science  and  divinity  began  (probably)  at  Cambridge.     .  242 

The  Oxford  movement  destroyed  by  the  philosophers  there.  242 

History  of  the  renaissance  after  1500  centres  at  Cambridge.  242 


xvi  TABLE   OF   CONTENTS. 

PAGE 

Influence  of  Fisher  and  Erasmus 242 

Migration  of  Oxonians  to  Cambridge.        .         .         .         .         .         .  243 

The  reformation  was  wholly  the  work  of  Cambridge  divines.   .         .  243 

The  royal  injunctions  of  1535. 244 

Endowment  of  professorships. 245 

Eapid  growth  of  the  colleges.  245 

The  Edwardian  statutes  of  1549 245 

The  Elizabethan  statutes  of  1570 245 

Subjection  of  the  university  to  the  crown.          .         .         .  245 

The  university  organized  on  an  ecclesiastical  basis    .         .  247 

Provisions  for  ensuring  general  education.          .         .         .  247 

Eecognition  of  importance  of  making  colleges  efficient       .  247 

The  number  of  students .         .         .  249 

The  social  life  and  amusements  of  the  undergraduates     .         .         .  250 
Prevalent  theological  views  at  Cambridge,  1600—1858.     .         .         .252 

Prevalent  political  views  at  Cambridge,  1600—1858.         .         .         .  252 
Prevalent  subjects  of  study  at  Cambridge,  1600—1858.     .         .         .253 

INDEX  255 


ERRATA. 

Page  14,  line  3.     After  under  insert  the. 
Page  34,  line  8.     For  powers  read  power. 
Page  38,  lines  3  and  5.    For  Bulialdus  read  Bullialdus. 
Page  91,  line  12.     For  seventeenth  read  eighteenth. 
Page  92,  line  4  from  end,  and  page  95,  line  5  from  end.    For  Lahire 
read  La  Hire. 

Page  115,  line  12.    For  His  read  Cavendish's. 

Page  183,  line  20.     For  T.  Bowstead  read  Joseph  Bowstead. 


CHAPTER  I. 
MEDIAEVAL    MATHEMATICS.1 

THE  subject  of  this  chapter  is  a  sketch  of  the  nature  and 
extent  of  the  mathematics  read  at  Cambridge  in  the  middle 
ages.  The  external  conditions  under  which  it  was  carried  on 
are  briefly  described  in  the  first  section  of  chapter  vm.  It  is 
only  after  considerable  hesitation  that  I  have  not  incorporated 
that  section  in  this  chapter ;  but  I  have  so  far  isolated  it  as  to 
render  it  possible,  for  any  who  may  be  ignorant  of  the  system 
of  education  in  a  mediaeval  university,  to  read  it  if  they  feel 
so  inclined,  before  commencing  the  history  of  the  development 
of  mathematics  at  Cambridge. 

The  period  with  which  I  am  concerned  in  this  chapter 
begins  towards  the  end  of  the  twelfth  century,  and  ends  with 
the  year  1535.  For  the  history  during  most  of  this  time  of 
mathematics  at  Cambridge  we  are  obliged  to  rely  largely  on 
inferences  from  the  condition  of  other  universities.  I  shall 
therefore  discuss  it  very  briefly  referring  the  reader  to  the 
works  mentioned  below1  for  further  details. 

1  Besides  the  authorities  alluded  to  in  the  various  foot-notes  I  am 
indebted  for  some  of  the  materials  for  this  chapter  to  Die  Mathematik 
auf  den  Universitdten  des  Mittelalters  by  H.  Suter,  Zurich,  1887 ;  Die 
Geschichte  des  mathematischen  Unterrichtes  im  deutschen  Mittelalter  bis 
1525,  by  M.  S.  Giinther,  Berlin,  1887;  and  Beitrdge  zur  Kenntniss  der 
Mathematik  des  Mittelalters,  by  H.  Weissenborn,  Berlin,  1888. 

B.  1 


2  MEDIEVAL   MATHEMATICS. 

Throughout  the  greater  part  of  this  period  a  student  usually 
proceeded  in  the  faculty  of  arts ;  and  in  that  faculty  he 
spent  the  first  four  years  on  the  study  of  the  subjects  of 
the  trivium,  and  the  next  three  years  on  those  of  the  quad- 
rivium.  The  trivium  comprised  Latin  grammar,  logic,  and 
rhetoric ;  and  I  have  described  in  chapter  viu.  both  how  they 
were  taught  and  the  manner  in  which  proficiency  in  them 
was  tested.  It  must  be  remembered  that  students  while 
studying  the  trivium  were  treated  exactly  like  school-boys, 
and  regarded  in  the  same  light  as  are  the  boys  of  a  leading 
public  school  at  the  present  time.  The  title  of  bachelor  was 
given  at  the  end  of  this  course.  A  bachelor  occupied  a 
position  analogous  to  that  of  an  undergraduate  now-a-days. 
He  was  required  to  spend  three  years  in  the  study  of  the 
quadrivium,  the  subjects  of  which  were  mathematics  and 
science.  These  were  divided  in  the  Pythagorean  manner  into 
numbers  absolute  or  arithmetic,  numbers  applied  or  music, 
magnitudes  at  rest  or  geometry,  and  magnitudes  in  motion 
or  astronomy.  There  was  however  no  test  that  a  student 
knew  anything  of  the  four  subjects  last  named  other  than  his 
declaration  to  that  effect,  and  in  practice  most  bachelors  left 
them  unread.  The  degree  of  master  was  given  at  the  end  of 
this  course. 

The  quadrivium  during  the  twelfth  and  the  first  half  of 
the  thirteenth  century,  if  studied  at  all,  probably  meant  about 
as  much  science  as  was  to  be  found  in  the  pages  of  Boethius, 
Cassiodorus,  and  Isidorus.  The  extent  of  this  is  briefly 
described  in  the  following  paragraphs. 

The  term  arithmetic  was  used  in  the  Greek  sense,  and 
meant  the  study  of  the  properties  of  numbers  ;  and  particularly 
of  ratio,  proportion,  fractions,  and  polygonal  numbers.  It  did 
not  include  the  art  of  practical  calculation,  which  was  generally 
performed  on  an  abacus ;  and  though  symbols  were  employed 
to  express  the  result  of  any  numerical  computation  they  were 
not  used  in  determining  it. 

The   geometry   was   studied   in   the  text-books   either   of 


THE   THIRTEENTH   CENTURY.  3 

Boethius  or  of  Gerbert1.  The  former  work,  which  was  the  one 
more  commonly  used,  contains  the  enunciations  of  the  first 
book  of  Euclid,  and  of  a  few  selected  propositions  from  the 
third  and  fourth  books.  To  shew  that  these  are  reliable, 
demonstrations  of  the  first  three  propositions  of  the  first  book 
are  given  in  an  appendix.  Some  practical  applications  to  the 
determination  of  areas  were  usually  added  in  the  form  of 
notes.  Even  this  was  too  advanced  for  most  students.  Thus 
Roger  Bacon,  writing  towards  the  close  of  the  thirteenth 
century,  says  that  at  Oxford,  there  were  few,  if  any,  residents 
who  had  read  more  than  the  definitions  and  the  enunciations 
of  the  first  five  propositions  of  the  first  book.  In  the  pages  of 
Cassiodorus  and  Isidorus  a  slight  sketch  of  geography  is 
included  in  geometry. 

The  two  remaining  subjects  of  the  quadrivium  were  music 
and  astronomy.  The  study  of  the  former  had  reference  to  the 
services  of  the  Church,  and  included  some  instruction  in  metre. 
The  latter  was  founded  on  Ptolemy's  work,  and  special  atten- 
tion was  supposed  to  be  paid  to  the  rules  for  finding  the 
moveable  festivals  of  the  Church;  but  it  is  probable  that  in 
practice  it  generally  meant  the  art  of  astrology. 

By  the  middle  of  the  thirteenth  century  anyone  who  was 
really  interested  in  mathematics  had  a  vastly  wider  range  of 
reading  open  to  him,  but  whether  students  at  the  English 
universities  availed  themselves  of  it  is  doubtful. 

The  mathematical  science  of  modern  Europe  dates  from  the 
thirteenth  century,  and  received  its  first  stimulus2  from  the 
Moorish  schools  in  Spain  and  Africa,  where  the  Arab  works 
on  arithmetic  and  algebra,  and  the  Arab  translations  of 
Euclid,  Archimedes,  Apollonius,  and  Ptolemy  were  not  un- 
common. It  will  be  convenient  to  give  here  an  outline  of 

1  Prof.  Weissenborn  thinks  that  neither  of  these  books  was  written 
by  its  reputed  author,  and  assigns  them  both  to  the  eleventh  century. 
This  view  is  not  however  generally  adopted. 

2  For  further  details  of  the  introduction  of  Arab  science  into  Europe, 
see  chapter  x.  of  my  History  of  mathematics,  London,  1888. 

1—2 


4  MEDIAEVAL   MATHEMATICS. 

the  introduction  of  the  Arab  geometry  and  arithmetic  into- 
Europe. 

First,  for  the  geometry.  As  early  as  1120  an  English  monk, 
named  Adelhard  (of  Bath),  had  obtained  a  copy  of  a  Moorish 
edition  of  the  Elements  of  Euclid  ;  and  another  specimen  was 
secured  by  Gerard  of  Cremona  in  1186.  The  first  of  these  was 
translated  by  Adelhard,  and  a  copy  of  this  fell  into  the  hands 
of  Giovanni  Campano  or  Campanus,  who  in  1260  reproduced  it 
as  his  own.  The  first  printed  edition  was  taken  from  it  and 
was  issued  by  Ratdolt  at  Venice  in  1482  :  of  course  it  is  in 
Latin.  This  pirated  translation  was  the  only  one  generally 
known  until  in  1533  the  original  Greek  text  was  recovered1, 
Campanus  also  issued  a  work  founded  on  Ptolemy's  astronomy 
and  entitled  the  Theory  of  the  planets. 

The  earliest  explanation  of  the  Arabic  system  of  arithmetic 
and  algebra,  which  had  any  wide  circulation  in  Europe,  wa& 
that  contained  in  the  Liber  abbaci  issued  in  1202  by  Leonardo 
of  Pisa.  In  this  work  Leonardo  explained  the  Arabic  system 
of  numeration  by  means  of  nine  digits  and  a  zero ;  proved  some 
elementary  algebraical  formulae  by  geometry,  as  in  the  second 
book  of  Euclid ;  and  solved  a  few  algebraical  equations.  The 
reasoning  was  expressed  at  full  length  in  words  and  without 
the  use  of  any  symbols.  This  was  followed  in  1220  by  a  work 
in  which  he  shewed  how  algebraical  formulae  could  be  applied 
to  practical  geometrical  problems,  such  as  the  determination  of 
the  area  of  a  triangle  in  terms  of  the  lengths  of  the  sides. 

Some  ten  or  twelve  years  later,  circ.  1230,  the  emperor 
Frederick 'II.  engaged  a  staff  of  Jews  to  translate  into  Latin  all 
the  Arab  works  on  science  which  were  obtainable ;  and  manu- 
script transcripts  of  these  were  widely  distributed.  Most  of 
the  mediaeval  editions  of  the  writings  of  Ptolemy,  Archimedes, 
and  Apollonius  were  derived  from  these  copies. 

One  branch  of  this  science  of  the  Moors  was  almost  at  once 
adopted  throughout  Europe.  This  was  their  arithmetic,  which 

1  See  p.  23,  hereafter ;  and  also  the  article  Eucleides,  by  A.  De  Morgan, 
in  Smith's  Dictionary  of  Greek  and  Roman  biography,  London,  1849. 


JOHN  DE  HOLYWOOD.   ROGER  BACON.        5 

was  commonly  known  as  algorism,  or  the  art  of  Alkarismi,  to  dis- 
tinguish it  from  the  arithmetic  founded  on  the  work  of  Boethius. 
From  the  middle  of  the  thirteenth  century  this  was  used  in 
nearly  all  mathematical  tables,  whether  astronomical,  astrological, 
or  otherwise.  It  was  generally  employed  for  trade  purposes  by 
the  Italian  merchants  at  or  about  the  same  time,  and  from  them 
spread  through  the  rest  of  Europe.  It  would  however  seem 
that  this  rapid  adoption  of  the  Arabic  numerals  and  arith- 
metic was  at  least  as  largely  due  to  the  calculators  of  calendars 
as  to  merchants  and  men  of  science.  Perhaps  the  oriental 
origin  of  the  symbols  gave  them  an  attractive  flavour  of  magic, 
but  there  seem  to  have  been  very  few  almanacks  after  the  year 
1300  in  which  an  explanation  of  the  system  was  not  included. 

The  earliest  lectures  on  the  subjects  of  algebra  and  algorism 
at  any  university,  of  which  I  can  find  mention,  are  some  given 
by  Holywood,  who  is  perhaps  better  known  by  the  latinized 
name  of  Sacrobosco.  John  de  Holywood  was  born  in  Yorkshire 
and  educated  at  Oxford,  but  after  taking  his  master's  degree 
he  moved  to  Paris  and  taught  there  till  his  death  in  1244  or 
1246.  His  work  on  arithmetic1  was  for  many  years  a  standard 
authority.  He  further  wrote  a  treatise  on  the  sphere,  which 
was  made  public  in  1256  :  this  had  a  wide  circulation,  and 
indicates  how  rapidly  a  knowledge  of  mathematics  was  spread- 
ing. Besides  these,  two  pamphlets  by  him,  entitled  respectively 
De  compute  ecclesiastico  and  De  astrolabio,  are  still  extant.  • 

Towards  the  end  of  the  thirteenth  century  a  strong  effort 
was  made  to  introduce  this  science,  as  studied  in  Italy,  into 
the  curriculum  of  the  English  universities.  This  was  due  to 
Roger  Bacon2.  Bacon,  who  was  educated  at  Oxford  and  Paris 

1  This  was  printed  at  Paris  in  1496  under  the  title  De  algorithmo; 
and  has  been  reissued  in  Halliwell's  Eara  matJiematica,  London,  second 
edition,  1841.     See  also  pp.   13 — 15   of  Arithmetical  books,  by  A.  De 
Morgan,  London,  1847. 

2  See  Roger  Bacon,  sa  vie,  ses  ouvrages...  by  E.  Charles,  Paris,  1861; 
and  Roger  Bacon,  eine  Monographic,  by  Schneider,  Augsburg,  1873.     The 
first  of  these  is  very  eulogistic,  the  latter  somewhat  severely  critical.     An 


6  MEDIEVAL   MATHEMATICS. 

and  taught  at  both  universities,  declared  that  divine  mathe- 
matics was  not  only  the  alphabet  of  all  philosophy,  but  should 
form  the  foundation  of  all  liberal  education,  since  it  alone 
could  fit  the  student  for  the  acquirement  of  other  knowledge, 
and  enable  him  to  detect  the  false  from  the  true.  He  urged 
that  it  should  be  followed  by  linguistic  or  scientific  studies. 
These  seem  also  to  have  been  the  views  of  Grosseteste,  the 
statesmanlike  bishop  of  Lincoln.  But  the  power  of  the  school- 
men in  the  universities  was  too  strong  to  allow  of  such  a 
change,  and  not  only  did  they  prevent  any  alteration  of 
the  curriculum  but  even  the  works  of  Bacon  on  physical 
science  (which  might  have  been  included  in  the  quadrivium) 
were  condemned  as  tending  to  lead  men's  thoughts  away  from 
the  problems  of  philosophy.  It  is  clear,  however,  that  hence- 
forth a  student,  who  was  desirous  of  reading  beyond  the 
narrow  limits  of  the  schools,  had  it  in  his  power  to  do  so  :  and 
if  I  say  nothing  more  about  the  science  of  this  time  it  is 
because  I  think  it  probable  that  no  such  students  were  to  be 
found  in  Cambridge. 

The  only  notable  English  mathematician  in  the  early  half 
of  the  fourteenth  century  of  whom  I  find  any  mention  is 
Tlwmas  Bradwardine l,  archbishop  of  Canterbury.  Bradwardine 
was  born  at  Chichester  about  1290.  He  was  educated  at 
Merton  College,  Oxford,  and  subsequently  lectured  in  that 
university.  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  (i)  the  Tractatus  de  proportionibus, 
printed  at  Paris  in  1495;  (ii)  iheArithmeticaspeculativa,  printed 

account  of  his  life  by  J.  S.  Brewer  is  prefixed  to  the  Bolls  Series  edition 
of  the  Opera  inedita,  London,  1859. 

1  See  vol.  iv.  of  the  Lives  of  the  Archbishops  of  Canterbury,  by  W.  F. 
Hook,  London,  1860—68 ;  see  also  pp.  480,  487,  521—24  of  the  Aperg u 
historique  sur... geometric  by  M.  Chasles  (first  edition). 


THE   FOURTEENTH   CENTURY.  7 

at  Paris  in  1502  ;  (iii)  the  Geometria  speculative!,,  printed  at  Paris 
in  1511 ;  and  (iv)  the  De  quadratures  circuli,  printed  at  Paris 
in  1495.  They  probably  give  a  fair  idea  of  the  nature  of  the 
mathematics  then  read  at  an  English  university. 

By  the  middle  of  this  century  Euclidean  geometry  (as 
expounded  by  Campanus)  and  algorism  were  fairly  familiar  to 
all  professed  mathematicians,  and  the  Ptolemaic  astronomy  was 
also  generally  known.  About  this  time  the  almanacks  began 
to  add  to  the  explanation  of  the  Arabic  symbols  the  rules  of 
addition,  subtraction,  multiplication,  and  division,  "  de  al- 
gorismo."  The  more  important  calendars  and  other  treatises 
also  inserted  a  statement  of  the  rules  of  proportion,  illustrated 
by  various  practical  questions ;  such  books  usually  concluded 
with  algebraic  formulae  (expressed  in  words)  for  most  of  the 
common  problems  of  commerce.  Of  course  the  fundamental 
rules  of  this  algorism  were  not  strictly  proved — that  is  the 
work  of  advanced  thought — but  it  is  important  to  note  that 
there  was  some  discussion  of  the  principles  involved. 

I  should  add  that  next  to  the  Italians  the  English  took  the 
most  prominent  part  in  the  early  development  and  improve- 
ment of  algorism1,  a  fact  which  the  backward  condition  of  the 
country  makes  rather  surprising.  Most  merchants  continued 
however  to  keep  their  accounts  in  Roman  numerals  till  about 
1550,  and  monasteries  and  colleges  till  about  1650  :  though  in 
both  cases  it  is  probable  that  the  processes  of  arithmetic  were 
performed  in  the  algoristic  manner.  No  instance  in  a  parish 
register  of  a  date  or  number  being  written  in  Arabic  numerals 
is  known  to  exist  before  the  seventeenth  century.  /  4»  -V *• 

In  the  latter  half  of  the  fourteenth  century  an  attempt 
was  made  to  include  in  the  quadrivium  these  new  works  on 
the  elements  of  mathematics.  The  stimulus  came  from  Paris, 
where  a  statute  to  that  effect  was  passed  in  1366,  and  a  year 
or  two  later  similar  regulations  were  made  at  Oxford  and  Cam- 

1  An  English  treatise  by  John  Norfolk,  written  about  1340,  otill 
extant.  It  was  printed  in- 11 16  atrd  rciooucd  by  Halliwell  in  his  Ear  a 
mathematica,  London,  second  edition,  1841. 


8  MEDIAEVAL  MATHEMATICS. 

bridge ;  unfortunately  no  text-books 1  are  mentioned.  We  can 
however  form  a  reasonable  estimate  of  the  range  of  mathe- 
matical reading  required,  by  looking  at  the  statutes  of  the 
universities  of  Prague  founded  in  1350,  of  Vienna  founded 
in  1364,  and  of  Leipzig  founded  in  13892. 

By  the  statutes  of  Prague3,  dated  1384,  candidates  for  the 
bachelor's  degree  were  required  to  have  read  Holywood's 
treatise  on  the  sphere,  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 
the  fingers,  and  the  algorism  of  integers ;  on  almanacks,  which 
probably  meant  elementary  astrology;  and  on  the  Almagest, 
that  is  on  Ptolemaic  astronomy.  There  is  however  some  reason 
for  thinking  that  mathematics  received  there  far  more  attention 
than  was  then  usual  at  other  universities. 

At  Vienna  in  1389  the  candidate  for  a  master's  degree  was 
required4  to  have  read  five  books  of  Euclid,  common  perspec- 
tive, proportional  parts,  the  measurement  of  superficies,  and 
the  Theory  of  the  planets.  The  book  last  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" 
in  a  mediaeval  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  mathematical  who  really 
mastered  the  subjects  mentioned  above. 

1  See  p.  81  of  De  V organisation  de  Venseignement...au  moyen  age  by 
C.  Thurot,  Paris,  1850. 

2  The  following  account  is  taken  from  Die  Geschichte  der  Mathematik, 
by  H.  Hankel,  Leipzig,  1874. 

3  See  vol.  i.  pp.  49,  56, 77,  83, 92,  108, 126,  of  the  Historia  universitatis 
Pragensis,  Prag,  1830. 

4  See  vol.  i.  p.  237  of  the  Statuta  universitatis  Wiennensis  by  V.  Kollar, 
Vienna,  1839:  quoted  in  vol.  i.  pp.  283  and  351  of  the  University  of 
Cambridge,  by  J.  Bass  Mullinger,  Cambridge,  1873. 


THE   FIFTEENTH   CENTURY.  9 

At  any  rate  no  test  of  proficiency  was  imposed ;  and  a  few 
facts  gleaned  from  the  history  of  the  next  century  tend  to 
shew  that  the  regulations  about  the  study  of  the  quadrivium 
were  not  seriously  enforced.  The  lecture  lists  for  the  years 
1437  and  1438  of  the  university  of  Leipzig  (the  statutes  of 
whieh  are  almost  identical  with  those  of  Prague  as  quoted 
above)  are  extant,  and  shew  that  the  only  lectures  given  there 
on  mathematics  in  those  years  were  confined  to  astrology.  The 
records1  of  Bologna,  Padua,  and  Pisa  seem  to  imply  that  there 
also  astrology  was  the  only  scientific  subject  taught  in  the 
fifteenth  century,  and  even  as  late  as  1598  the  professor  of 
mathematics  at  Pisa  was  required  to  lecture  011  the  Quadri- 
partitum,  a  spurious  astrological  work  attributed  to  Ptolemy. 
According  to  the  registers2  of  the  university  of  Oxford  the 
only  mathematical  subjects  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  Euclid  published  at  Paris  in  1536,  that  after  1452  candidates 
for  the  master's  degree  at  that  university  had  to  take  an  oath 
that  they  had  attended  lectures  on  the  first  six  books  of  Euclid. 

The  only  Cambridge  mathematicians  of  the  fifteenth  century 
of  whom  I  can  find  any  mention  were  Holbroke,  Marshall,  and 
Hodgkins.  No  details  of  their  lives  and  works  are  known. 
John  Holbroke,  master  of  Peterhouse  and  chancellor  of  the 
university  for  the  years  1428  and  1429,  who  died  in  1437,  is 
reputed  to  have  been  a  distinguished  astronomer  and  astrologer. 
Roger  Marshall,  who  was  a  fellow  of  Pembroke,  taught  mathe- 
matics and  medicine ;  he  subsequently  moved  to  London  and 
became  physician  to  Edward  IV.  John  Hodgkins,  a  fellow  of 
King's,  who  died  in  1485  is  mentioned  as  a  celebrated  mathe- 
matician. 

1  See  pp.   15,  20  of  Die,  Geschichte  der  mathematischen  Facultdt  in 
Bologna  by  S.  Gherardi,  edited  by  M.  Kurtze,  Berlin,  1871. 

2  Quoted  in  the  Life   of  bishop  Smyth  (the  founder  of  Brazenose 
College)  by  Ralph  Churton,  Oxford,  1800. 


10  MEDIEVAL   MATHEMATICS. 

At  the  beginning  of  the  sixteenth  century  the  names  of 
Master,  Paynell,  and  Tonstall  occur.  Of  these  Richard  Master, 
a  fellow  of  King's,  is  said  to  have  been  famous  for  his  know- 
ledge of  natural  philosophy.  He  entered  at  King's  in  1502, 
and  was  proctor  in  1511.  He  took  up  the  cause  of  the  holy 
maid  of  Kent  and  was  executed  for  treason  in  April,  1534. 
Nicholas  Paynell,  a  fellow  of  Pembroke  Hall,  graduated  B.  A. 
in  1515.  In  1530  he  was  appointed  mathematical  lecturer  to- 
the  university.  The  date  of  his  death  is  unknown. 

Cuthbert  Tonstall1  was  born  at  Hackforth,  Yorkshire,  in 
1474  and  died  in  1559.  He  had  entered  at  Balliol  College, 
Oxford,  but  finding  the  philosophers  dominant  in  the  university 
(see  p.  243),  he  migrated  to  King's  Hall,  Cambridge.  We  must 
not  attach  too  much  importance  to  this  step  for  such  migrations 
were  then  very  common,  and  his  action  only  meant  that  he 
could  continue  his  studies  better  at  Cambridge  than  at  Oxford. 
He  subsequently  went  to  Padua,  where  he  studied  the  writings 
of  Regiomontanus  and  Pacioli.  His  arithmetic  termed  De  arte 
supputandi  was  published  in  1522  as  a  "farewell  to  the  sciences " 
on  his  appointment  to  the  bishopric  of  London.  A  presenta- 
tion copy  on  vellum  with  the  author's  autograph  is  in  the 
university  library  at  Cambridge.  The  work  is  described  by 
De  Morgan  in  his  Arithmetical  Books  as  one  of  the  best 
which  has  been  written  both  in  matter,  style,  and  for  the  his- 
torical knowledge  displayed.  Few  critics  would  agree  with  this 
estimate,  but  it  was  undoubtedly  the  best  arithmetic  then  issued, 
and  forms  a  not  unworthy  conclusion  to  the  .mediaeval  history 
of  Cambridge.  It  is  particularly  valuable  as  containing  illus- 
trations of  the  mediaeval  processes  of  computation.  Several 
extracts  from  it  are  given  in  the  Philosophy  of  arithmetic  by 
J.  Leslie,  second  edition,  Edinburgh,  1820. 

That  brings  me  to  the  close  of  the  middle  ages,  and  the 
above  account — meagre  though  it  is — contains  all  that  I  have 


1  See  vol.  i.  p.  198  of  the  Athenae  Cantabrigienses  by  C.  H.  and  T. 
Cooper,  Cambridge,  1858 — 61. 


TONSTALL.  11 

been  able  to  learn  about  the  extent  of  mathematics  then  taught 
at  an  English  university.    About  Cambridge  in  particular  I  can 
give  no  details.     The  fact  however  that  Tonstall  and  Recorde, 
the  only  two  English  mathematicians  of  any  note  of  the  first 
half  of  the  sixteenth  century,  both  migrated  from  Oxford  toj 
Cambridge  in  order  to  study  science  makes  it  probable  that  it] 
was  becoming  an  important  school  of  mathematics. 


CHAPTER   II. 

THE    MATHEMATICS    OF   THE   RENAISSANCE. 
CIRC.  1535—1630. 

THE  close  of  the  mediaeval  period  is  contemporaneous  with 
the  beginning  of  the  modern  world.  The  reformation  and  the 
revival  of  the  study  of  literature  flooded  Europe  with  new 
ideas,  and  to  these  causes  we  must  in  mathematics  add  the 
fact  that  the  crowds  of  Greek  refugees  who  escaped  to  Italy 
after  the  fall  of  Constantinople  brought  with  them  the  original 
works  and  the  traditions  of  Greek  science.  At  the  same  time 
the  invention  of  printing  (in  the  fifteenth  century)  gave 
facilities  for  disseminating  knowledge  which  made  these  causes 
incomparably  more  potent  than  they  would  have  been  a  few 
centuries  earlier. 

It  was  some  years  before  the  English  universities  felt  the 
full  force  of  the  new  movement,  but  in  1535  the  reign  of  the 
schoolmen  at  Cambridge  was  brought  to  an  abrupt  end  by 
"the  royal  injunctions"  of  that  year  (see  p.  244).  Those 
injunctions  were  followed  by  the  suppression  of  the  monas- 
teries and  the  schools  thereto  attached,  and  thus  the  whole 
system  of  mediaeval  education  was  destroyed.  Then  ensued  a 
time  of  great  confusion.  The  number  of  students  fell,  so  that 
the  entries  for  the  decade  ending  1547  are  probably  the  lowest 
in  the  whole  seven  centuries  of  the  history  of  the  university. 

The  writings  of  Tonstall  and  Recorde,  and  the  fact  that 
most  of  the  English  mathematicians  of  the  time  came  from 
Cambridge  seem  to  shew  that  mathematics  was  then  regularly 
taught,  and  of  course  according  to  the  statutes  it  still  con- 


THE   MATHEMATICS   OF   THE   RENAISSANCE.  13 

stituted  the  course  for  the  M.A.  degree.  But  it  is  also  clear 
that  it  was  only  beginning  to  grow  into  an  important  study, 
and  was  not  usually  read  except  by  bachelors,  and  probably 
by  only  a  few  of  them.  The  chief  English  mathematician 
of  this  time  was  Recorde  whose  works  are  described  im- 
mediately hereafter;  but  John  Dee,  Thomas  Digges,  Thomas 
Blundeville,  and  William  Buckley  were  not  undistinguished. 

The  period  of  confusion  in  the  studies  of  the  university 
caused  by  the  break-up  of  the  mediaeval  system  of  education 
was  brought  to  an  end  by  the  Edwardian  statutes  of  1549  (see 
p.  153).  These  statutes  represented  the  views  of  a  large  number 
of  residents,  and  it  is  noticeable  that  they  enjoined  the  study  of 
mathematics  as  the  foundation  of  a  liberal  education.  Certain 
text-books  were  recommended,  and  we  thus  learn  that  arith- 
metic was  usually  taught  from  Tonstall  and  Cardan,  geometry 
from  Euclid,  and  astronomy  from  Ptolemy.  Cosmography  was 
still  included  in  the  quadrivium,  and  the  works  of  Mela, 
Strabo,  and  Pliny  are  referred  to  as  authorities  on  it. 

The  Edwardian  code  was  only  in  force  for  about  twenty 
years.  Fresh  statutes  were  given  by  Elizabeth  in  1570,  and 
except  for  a  few  minor  alterations  these  remained  in  force  till 
1858.  The  commissioners  who  framed  them  excluded  mathe- 
matics from  the  course  for  undergraduates — apparently  because 
they  thought  that  its  study  appertained  to  practical  life,  and 
had  its  place  in  a  course  of  technical  education  rather  than  in 
the  curriculum  of  a  university.  These  opinions  were  generally 
held  at  that  time1  and  it  will  be  found  that  most  of  the 
English  books  on  the  subject  issued  for  the  following  sixty  or 
seventy  years — the  period  comprised  in  this  chapter — were 
chiefly  devoted  to  practical  applications,  such  as  surveying, 
navigation,  and  astrology.  Accordingly  we  find  that  for  the 
next  half  century  mathematics  was  more  studied  in  London 
than  at  the  universities,  and  it  was  not  until  it  became  a 

]  See  for  example  vol.  i.  pp.  382 — 91  of  the  Orationes  of  Melanchthon, 
and  the  autobiography  of  Lord  Herbert  of  Cherbury  (born  in  1581  and 
died  in  1648)  which  was  published  in  London  in  1792. 


14  THE   MATHEMATICS   OF   THE   RENAISSANCE. 

science  (under  the  influence  of  Wallis,  Barrow,  and  Newton) 
that  much  attention  was  paid  to  it  at  Cambridge.  ^ 

It  must  however  be  remembered  that  though  under^liza- 
bethan  statutes  mathematics  was  practically  relegated  to  a 
secondary  position  in  the  university  curriculum,  yet  it  re- 
mained the  statutable  subject  to  be  read  for  the  M.A.  degree. 
That  was  in  accordance  with  the  views  propounded  by  Ramus1 
who  considered  that  a  liberal  education  should  comprise  the 
exoteric  subjects  of  grammar,  rhetoric,  and  dialectics ;  and  the 
esoteric  subjects  of  mathematics,  physics,  and  metaphysics  for 
the  more  advanced  students.  The  exercises  for  the  degree  of 
master  were  however  constantly  neglected,  and  after  1608 
when  residence  was  declared  to  be  unnecessary  (see  p.  157)  they 
were  reduced  to  a  mere  form. 

I  think  it  will  be  found  that  in  spite  of  this  official  dis- 
couragement the  majority  of  the  English  mathematicians  of  the 
early  half  of  the  seventeenth  century  were  educated  at  Cam- 
bridge, even  though  they  subsequently  published  their  works 
and  taught  elsewhere. 

Among  the  more  eminent  Cambridge  mathematicians  of  the 

1  See  p.  346  of  Ramus;  sa  vie,  ses  ecrits,  et  ses  opinions  by  Ch. 
Waddington,  Paris,  1855.  Another  sketch  of  his  opinions  is  given  in 
a  monograph  of  which  he  is  the  subject  by  C.  Desmaze,  Paris,  1864. 
Peter  Ramus  was  born  at  Cuth  in  Picardy  in  1515,  and  was  killed  at  Paris 
at  the  massacre  of  St  Bartholomew  on  Aug.  24,  1572.  He  was  educated 
at  the  university  of  Paris,  and  on  taking  his  degree  he  astonished  and 
charmed  the  university  with  the  brilliant  declamation  he  delivered  on  the 
thesis  that  everything  Aristotle  had  taught  was  false.  He  lectured  first  at 
le  Mans,  and  afterwards  at  Paris ;  at  the  latter  he  founded  the  first  chair 
of  mathematics.  Besides  some  works  on  philosophy  he  wrote  treatises 
on  arithmetic,  algebra,  geometry  (founded  on  Euclid),  astronomy  (found- 
ed on  the  works  of  Copernicus),  and  physics  which  were  long  regarded 
on  the  continent  as  the  standard  text-books  on  these  subjects.  They  are 
collected  in  an  edition  of  his  works  published  at  Bale  in  1569. 
Cambridge  became  the  chief  centre  for  the  Bamistic  doctrines,  and  was 
apparently  frequented  by  foreign  students  who  desired  to  learn  his  logic 
and  system  of  philosophy :  see  vol.  n.  pp.  411 — 12  of  the  University  of 
Cambridge,  by  J.  Bass  Mullinger,  Cambridge,  1884. 


RECORDE.  15 

latter  half  of  the  sixteenth  century  I  should  include  Sir  Henry 
Billingsley,  Thomas  Hill,  Thomas  Bedwell,  Thomas  Hood, 
Richard  Harvey,  John  Harvey,  and  Simon  Forman.  These 
were  only  second-rate  mathematicians.  They  were  followed  by 
Edward  Wright,  Henry  Briggs,  and  William  Oughtred,  all  of 
whom  were  mathematicians  of  mark:  most  of  the  works  of  the 
three  last  named  were  published  in  the  seventeenth  century. 


After  this  brief  outline  of  my  arrangement  of  the  chapter  I 
return  to  the  Cambridge  mathematicians  of  the  first  half  of  the 
sixteenth  century. 

The  earliest  of  these — if  we  except  Tonstall — and  the  first 
English  writer  on  pure  mathematics  of  any  eminence  was 
Recorde.  Robert  Eecorde1  was  born  at  Tenby  about  1510. 
He  was  educated  at  Oxford,  and  in  1531  obtained  a  fellowship 
at  All  Souls'  College ;  but  like  Tonstall  he  found  that  there  was 
then  no  room  at  that  university  for  those  who  wished  to  study 
science  beyond  the  traditional  and  narrow  limits  of  the  quadri- 
vium.  He  accordingly  migrated  to  Cambridge,  where  he  read 
mathematics  and  medicine.  He  then  returned  to  Oxford,  but 
his  reception  was  so  unsatisfactory  that  he  moved  to  London, 
where  he  became  physician  to  Edward  VI.  and  afterwards  to 
Queen  Mary.  His  prosperity  however  must  have  been  short- 
lived, for  at  the  time  of  his  death  in  1558  he  was  confined  in 
the  King's  Bench  prison  for  debt. 

His  earliest  work  was  an  arithmetic  published  in  1540 
under  the  title  the  Grounde  of  artes.  This  is  the  earliest 
English  scientific  work  of  any  value.  It  is  also  the  first 
English  book  which  contains  the  current  symbols  for  addition, 

1  See  the  Athenae  Cantabrigienses  by  C.  H.  and  T.  Cooper,  two  vols. 
Cambridge,  1858  and  1863.  To  save  repetition  I  may  say  here,  once 
for  all,  that  the  accounts  of  the  lives  and  writings  of  such  of  the  mathe- 
maticians as  are  mentioned  in  the  earlier  part  of  this  chapter  and  who 
died  before  1609  are  founded  on  the  biographies  contained  in  the  Athenae 
Cantabrigienses. 


16  THE   MATHEMATICS   OF  THE    RENAISSANCE. 

subtraction,  and  equality.  There  are  faint  traces  of  his  having 
used  the  two  former  as  symbols  of  operation  and  not  as  mere 
abbreviations.  The  sign  =  for  equality  was  his  invention. 
He  says  he  selected  that  particular  symbol  because  than  two 
parallel  straight  lines  no  two  things  can  be  more  equal,  but 
M.  Charles  Henry  has  pointed  out  in  the  Revue  archeologique 
for  1879  that  it  is  a  not  uncommon  abbreviation  for  the  word 
est  in  medieval  manuscripts,  and  this  would  seem  to  point  to  a 
more  probable  origin.  Be  this  as  it  may,  the  work  is  the  best 
treatise  on  arithmetic  produced  in  that  century. 

Most  of  the  problems  in  arithmetic  are  solved  by  the  rule 
of  false  assumption.  This  consists  in  assuming  any  number 
for  the  unknown  quantity,  and  if  on  trial  it  does  not  satisfy 
the  given  conditions,  correcting  it  by  simple  proportion  as  in 
rule  of  three.  It  is  only  applicable  to  a  very  limited  class  of 
problems.  As  an  illustration  of  its  use  I  may  take  the  follow- 
ing question.  A  man  lived  a  fourth  of  his  life  as  a  boy;  a  fifth 
as  a  youth;  a  third  as  a  man;  and  spent  thirteen  years  in  his 
dotage :  how  old  was  he1?  Suppose  we  assume  his  age  to  have 
been  40.  Then,  by  the  given  conditions,  he  would  have  spent 
8§  (and  not  13)  years  in  his  dotage,  and  therefore 
8f  :  13  =  40  :  his  actual  age, 

hence  his  actual  age  was  60.  Recorde  adds  that  he  preferred 
to  solve  problems  by  this  method  since  when  a  difficult  question 
was  proposed  he  could  obtain  the  true  result  by  taking  the 
chance  answers  of  "such  children  or  idiots  as  happened  to  be  in 
the  place." 

Like  all  his  works  the  Grounde  of  artes  is  written  in  the 
form  of  a  dialogue  between  master  and  scholar.  As  an  illus- 
tration of  the  style  I  quote  from  it  the  introductory  conversa- 
tion on  the  advantages  of  the  power  of  counting  "  the  only 
thing  that  separateth  man  from  beasts." 

Master.  If  Number  were  so  vile  a  thing  as  you  did  esteem  it,  then 
need  it  not  to  be  used  so  much  in  mens  communication.  Exclude 
Number  and  answer  me  to  this  question.  How  many  years  old  are 
you? 


RECORDE.  17 

Scholar.     Mum. 

Master.  How  many  days  in  a  week?  How  many  weeks  in  a  year? 
What  lands  hath  your  father?  How  many  men  doth  he  keep?  How 
long  is  it  sythe  you  came  from  him  to  me  ? 

Scholar.     Mum. 

Master.  So  that  if  Number  want,  you  answer  all  by  Mummes. 
How  many  miles  to  London?... Why,  thus  you  may  see,  what  rule 
Number  beareth  and  that  if  Number  be  lacking,  it  maketh  men  dumb, 
so  that  to  most  questions,  they  must  answer  Mum. 

Recorde  also  published  in  1556  an  algebra  called  tlie  Whet- 
stone ofwitte.  The  title,  as  is  well  known,  is  a  play  on  the  old 
name  of  algebra  as  the  cossic  art:  the  terra  being  derived  from 
cosa,  a  thing,  which  was  used  to  denote  the  unknown  quantity 
in  an  equation.  Hence  the  title  cos  ingenii,  the  whetstone  of 
wit.  The  algebra  is  syncopated,  that  is,  it  is  written  at  length 
according  to  the  usual  rules  of  grammar,  but  symbols  or  con- 
tractions are  used  for  the  quantities  and  operations  which  occur 
most  frequently.  In  this  work  Recorde  shewed  how  the  square 
root  of  an  algebraical  expression  could  be  extracted — a  rule 
which  was  here  published  for  the  first  time. 

Both  these  treatises  were  frequently  republished  and  had  a 
wide  circulation.  The  latter  in  particular  was  well  known,  as 
is  shewn  by  the  allusion  to  it  (as  being  studied  by  the  usurer) 
in  Sir  Walter  Scott's  Fortunes  of  Nigel.  To  the  belated 
traveller  who  wanted  some  literature  wherewith  to  pass  the 
time,  the  maid,  says  he,  "returned  for  answer  that  she  knew  of 
no  other  books  in  the  house  than  her  young  mistress's  bible, 
which  the  owner  would  not  lend ;  and  her  master's  Whetstone, 
of  Witte  by  Robert  Recorde."  So  too  William  Cuningham1 
in  his  Cosmographicall  glasse,  published  in  1559,  alludes  to 

1  William  Cuningham  (sometimes  written  Keningham)  was  born  in 
1531  and  entered  at  Corpus  College,  Cambridge,  in  1548.  The  Cosmo- 
graphicall glasse,  is  the  earliest  English  treatise  on  cosmography. 
Cuningham  also  published  some  almanacks,  but  his  works  have  no 
intrinsic  value  in  the  history  of  the  mathematics'.  He  practised  as  a 
physician  in  London,  under  the  license  conferred  by  his  Cambridge 
degree. 

B.  2 


18  THE   MATHEMATICS   OF   THE   RENAISSANCE. 

Recorde's  writings  as  standard  authorities  in  arithmetic  and 
algebra  :  in  geometry  he  quotes  Orontius  and  Euclid. 

Besides  the  two  books  just  mentioned  Recorde  wrote  the 
following  works  on  mathematical  subjects.  The  Pathway  to 
knowledge,  published  in  1551,  on  geometry  and  astronomy;  the 
Principles  of  geometry  also  written  in  1551;  three  works  issued 
in  1556  on  astronomy  and  astrology,  respectively  entitled  the 
Castle,  Gate,  and  Treasure  of  knowledge ;  and  lastly  a  treatise 
on  the  sphere,  and  another  on  mensuration,  both  of  which  are 
undated.  He  also  translated  Euclid's  Elements,  but  I  do  not 
think  that  this  was  published. 

In  his  astronomy  Recorde  adopts  the  Copernican  hypothesis. 
Thus  in  one  of  his  dialogues  he  induces  his  scholar  to  assert 
that  the  "earth  standeth  in  the  middle  of  the  world."  He 
then  goes  on 

blaster.  How  be  it,  Copernicus  a  man  of  great  learning,  of  much 
experience,  and  of  wonderful  diligence  in  observation,  hath  renewed 
the  opinion  of  Aristarchus  of  Samos,  and  affirmeth  that  the  earth  not 
only  moveth  circularly  about  his  own  centre,  but  also  may  be,  yea  and 
is,  continually  out  of  the  precise  centre  38  hundred  thousand  miles :  but 
because  the  understanding  of  that  controversy  dependeth  of  profounder 
knowledge  than  in  this  introduction  may  be  uttered  conveniently,  I  will 
let  it  pass  till  some  other  time. 

Scholar.  Nay  sir  in  good  faith,  I  desire  not  to  hear  such  vain  phan- 
tasies, so  far  against  common  reason,  and  repugnant  to  the  consent 
of  all  the  learned  multitude  of  writers,  and  therefore  let  it  pass  for 
ever,  and  a  day  longer. 

Master.  You  are  too  young  to  be  a  good  judge  in  so  great  a  matter : 
it  passeth  far  your  learning,  and  theirs  also  that  are  much  better  learned 
than  you,  to  improve  his  supposition  by  good  arguments,  and  therefore 
you  were  best  to  condemn  nothing  that  you  do  not  well  understand: 
but  another  time,  as  I  said,  I  will  so  declare  his  supposition,  that  you 
shall  not  only  wonder  to  hear  it,  but  also  peradventure  be  as  earnest 
then  to  credit  it,  as  you  are  now  to  condemn  it. 

This  advocacy  of  the  Copernican  theory  is  the  more  remark- 
able as  that  hypothesis  was  only  published  in  1543,  and  was 
merely  propounded  as  offering  a  simple  explanation  of  the  phe- 
nomena observable :  Galileo  was  the  first  writer  who  attempted 


DEE.  19 

to  give  a  proof  of  it.  It  is  stated  that  Recorde  was  the  earliest 
Englishman  who  accepted  that  theory. 

Recorde's  works  give  a  clear  view  of  the  knowledge  of  the 
time  and  he  was  certainly  the  most  eminent  English  mathe- 
matician of  that  age,  but  T  do  not  think  he  can  be  credited  with 
any  -original  work  except  the  rule  for  extracting  the  square 
root  of  an  algebraical  expression. 

Another  mathematician  only  slightly  junior  to  Recorde  was 
Dee,  who  fills  no  small  place  in  the  scientific  and  literary  records 
of  his  time,  and  whose  natural  ability  was  of  the  highest  order. 
John  Dee1  was  born  on  July  13,  1527,  and  died  in  December 
1608.  He  entered  at  St  John's  College2  in  1542,  proceeded 
B.A.  in  1545,  and  was  elected  to  a  fellowship  in  the  following 
year.  On  the  foundation  of  Trinity  College  in  1546,  Dee  was 
nominated  one  of  the  original  fellows,  and  was  made  assistant 
lecturer  in  Greek — a  post  which  however  he  only  held  for  a 
year  and  a  half.  During  this  time  he  was  studying  mathematics, 
and  on  going  down  in  1548  he  gave  his  astronomical  instru- 
ments to  Trinity. 

He  then  went  on  the  continent.  In  1549  he  was  teaching 
arithmetic  and  astronomy  at  Louvain,  and  in  1550  he  was 
lecturing  at  Paris  in  JEnglish  on  Euclidean  geometry.  These 
lectures  are  said  to  have  been  the  first  gratuitous  ones  ever 
given  in  a  European  university  (see  p.  143).  "My  auditory  in 
Rheims  College"  says  he  "was  so  great,  and  the  most  part  elder 
than  myself,  that  the  mathematical  schools  could  not  hold  them; 
for  many  were  fain  without  the  schools  at  the  windows,  to  be 
auditors  and  spectators,  as  they  best  could  help  themselves 
thereto.  I  did  also  dictate  upon  every  proposition  besides  the 

1  There  are  numerous  biographies  of  Dee,  which  should  be  read  in 
connection   with   his   diaries.     Perhaps  one  of  the  best  is   in   Thomas 
Smith's  Vitae...illustrium  virorum.    A  bibliography  of  his  works  (seventy- 
nine  in  number)  and  an  account  of  his  life  are  given  in  vol.  n.  pp.  505-9 
of  the  Athenae  Cantabrigienses. 

2  Here,  and  hereafter  when  I  mention  a  college,  the  reference  is  to  the 
college  of  that  name  at  Cambridge,  unless  some  other  university  or  place 
is  expressly  mentioned. 

2—2 


20  THE   MATHEMATICS.  OF   THE   RENAISSANCE. 

first  exposition.  And  by  the  first  four  principal  definitions 
representing  to  their  eyes  (which  by  imagination  only  are 
exactly  to  be  conceived)  a  greater  wonder  arose  among  the 
beholders,  than  of  my  Aristophanes  Scarabseus  mounting  up  to 
the  top  of  Trinity  hall  in  Cambridge."  The  last  allusion  is  to 
a  stage  trick  which  he  had  designed  for  the  performance  of  a 
Greek  comedy  in  the  dining-hall  at  Trinity  and  which,  unluckily 
for  him,  gave  him  the  reputation  of  a  sorcerer  among  those  who 
could  not  see  how  it  was  effected. 

In  1554  some  public-spirited  Oxonians,  who  regretted  the 
manner  in  which  scientific  studies  were  there  treated,  offered 
him  a  stipend  to  lecture  on  mathematics  at  Oxford,  but  he 
declined  the  invitation.  A  year  or  so  later  we  find  him 
petitioning  queen  Mary  to  form  a  royal  library  by  collecting 
all  the  dispersed  libraries  of  the  various  monasteries,  and  it  i» 
most  unfortunate  that  his  proposal  was  rejected. 

On  the  accession  of  Elizabeth  he  was  taken  into  the  royal 
service,  and  subsequently  most  of  his  time  was  occupied  with 
alchemy  and  astrology.  It  is  now  generally  admitted  that  in 
his  experiments  and  alleged  interviews  with  spirits  he  was  the 
dupe  of  others  and  not  himself  a  cheat.  His  chief  work  on 
astronomy  was  his  report  to  the  Government  made  in  1585 
advocating  the  reform  of  the  Julian  calendar  :  like  Recorde  he 
adopted  the  Copernican  hypothesis.  The  Government  accepted 
his  proposal  but  owing  to  the  strenuous  opposition  of  the 
bishops  it  had  to  be  abandoned,  and  was  not  actually  carried 
into  effect  till  nearly  two  hundred  years  later. 

During  the  last  part  of  his  life  Dee  was  constantly  in 
want,  and  his  reputation  as  a  sorcerer  caused  all  men  to  shun 
him.  The  story  of  his  intercourse  with  angels  and  experi- 
ments on  the  transmutation  of  metals  are  very  amusing,  but 
are  too  lengthy  for  me  to  cite  here.  His  magic  crystal  and 
cakes  are  now  in  the  British  Museum. 

He  is  described  as  tall,  slender,  and  handsome,  with  a  clear 
and  fair  complexion.  In  his  old  age  he  let  his  beard,  which 
was  then  quite  white,  grow  to  an  unusual  length,  and  never 


DIGGES.      BLUNDEVILLE.  21 

appeared  abroad  except  "in  a  long  gown  with  hanging 
sleeves."  An  engraving  of  a  portrait  of  him  executed  in  his 
lifetime  and  now  in  my  possession  fully  bears  out  this  de- 
scription. No  doubt  these  peculiarities  of  dress  added  to  his 
evil  reputation  as  a  dealer  in  evil  spirits,  but  throughout  his 
life  .he  seems  to  have  been  constantly  duped  by  others  more 
skilful  and  less  scrupulous  than  himself. 

Among  the  pupils  of  Dee  was  Thomas  Digges,  who  entered 
at  Queens'  College  in  1546  and  proceeded  B.A.  in  1551. 
Digges  edited  and  added  to  the  writings  of  his  father  Leonard 
Digges,  but  how  much  is  due  to  each  it  is  now  impossible  to 
say  with  certainty,  though  it  is  probable  that  the  greater  part 
is  due  to  the  son.  His  works  in  24  volumes  are  mostly  on  the 
application  of  arithmetic  and  geometry  to  mensuration  and  the 
arts  of  fortification  and  gunnery.  They  are  chiefly  remarkable 
as  being  the  earliest  English  books  in  which  spherical  trigo- 
nometry is  used1.  In  one  of  them  known  as  Pantometria  and 
issued  in  1571  the  theodolite  is  described:  this  is  the  earliest 
known  description  of  the  instrument2.  The  derivation  is  from 
an  Arabic  word  alhidada  which  was  corrupted  into  atJielida 
and  thence  into  theodelite.  Digges  was  muster-master  of  the 
English  army,  and  so  engrossed  with  political  and  military 
matters  as  to  leave  but  little  time  for  original  work;  but 
Tycho  Brahe3  and  other  competent  observers  deemed  him  to  be 
one  of  the  greatest  geniuses  of  that  time.  He  died  in  1595. 

Thomas  Blundeville  was  resident  at  Cambridge  about  the 
same  time  as  Dee  and  Digges — possibly  he  was  a  non-collegiate 
student,  and  if  so  must  have  been  one  of  the  last  of  them.  In 
1589  he  wrote  a  work  on  the  use  of  maps  and  of  Ptolemy's 
tables.  In  1594  he  published  his  Exercises  in  six  parts, 
containing  a  brief  account  of  arithmetic,  cosmography,  the  use 
of  the  globes,  a  universal  map,  the  astrolabe,  and  navigation. 

1  See  p.  40  of  the  Companion  to  the  Almanack  for  1837. 

2  See  p.  24  of  Arithmetical  books  by  A.  De  Morgan,  London,  1847. 

3  See  pp.  6,  33  of  Letters  on  scientific  subjects  edited  by  Halliwell, 
London,  1841. 


22  THE   MATHEMATICS   OF  THE   RENAISSANCE. 

The  arithmetic  is  taken  from  Recorde,  but  to  it  are  added 
trigonometrical  tables  (copied  from  Clavius)  of  the  natural 
sines,  tangents,  and  secants  of  all  angles  in  the  first  quadrant; 
the  difference  between  consecutive  angles  being  one  minute. 
These  are  worked  out  to  seven  places  of  decimals.  This  is  the 
earliest1  English  work  in  which  plane  trigonometry  is  intro- 
duced. 

Another  famous  teacher  of  the  same  period  was  William 
Buckley.  Buckley  was  born  at  Lichfield,  and  educated  at 
Eton,  whence  he  went  to  King's  in  1537,  and  proceeded  B.A. 
in  1542.  He  subsequently  attended  the  court  of  Edward  VI., 
but  his  reputation  as  a  successful  lecturer  was  so  considerable 
that  about  1550  he  was  asked  to  return  to  King's  to  teach 
arithmetic  and  geometry.  He  has  left  some  mnemonic  rules  on 
arithmetic  which  are  reprinted  in  the  second  edition  of  Leslie's 
Philosophy  of  arithmetic,  Edinburgh,  1820.  Buckley  died  in 
1569. 

Another  well  known  Cambridge  mathematician  of  this 
time  was  Sir  Henry  Billingsley,  who  obtained  a  scholarship  at 
St  John's  College  in  1551.  He  is  said  on  somewhat  question- 
able authority  to  have  migrated  from  Oxford,  and  to  have 
learnt  his  mathematics  from  an  old  Augustinian  friar  named 
Whytehead,  who  continued  to  live  in  the  university  after  the 
suppression  of  the  house  of  his  order.  The  latter  is  described 
as  fat,  dirty  and  uncouth,  but  seems  to  have  been  one  of  the 
best  mathematical  tutors  of  his  time.  Billingsley  settled  in 
London  and  ultimately  became  lord  mayor ;  but  he  continued 
his  interest  in  mathematics  and  was  also  a  member  of  the 
Society  of  Antiquaries.  He  died  in  1606. 

Billingsley's  claim  to  distinction  is  the  fact  that  he 
published  in  1570  the  first  English  translation  of  Euclid.  In 
preparing  this  he  had  the  assistance  both  of  Whytehead  and  of 
John  Dee.  In  spite  of  their  somewhat  qualified  disclaimers, 
it  was  formerly  supposed  that  the  credit  of  it  was  due  to  them 

1  See  p.  42  of  Arithmetical  books  by  A.  De  Morgan,  London,  1847. 


BILLINGSLEY.      HILL.      BEDWELL.      HOOD.  23 

rather  than  to  him,  especially  as  Whytehead,  who  had  fallen 
into  want,  seems  at  the  time  when  it  was  published  to  have 
been  living  in  Billingsley's  house.  The  copy  of  the  Greek 
text  of  Theon's  Euclid  used  by  Billingsley  has  however  been 
recently  discovered,  and  is  now  in  Princetown  College, 
America 1 ;  and  it  would  appear  from  this  that  the  credit  of 
the  work  is  wholly  due  to  Billingsley  himself.  The  marginal 
notes  are  all  in  his  writing,  and  contain  comments  on  the 
edition  of  Adelhard  and  Campanus  from  the  Arabic  (see  p.  4), 
and  conjectural  emendations  which  shew  that  his  classical 
scholarship  was  of  a  high  order. 

Other  contemporary  mathematical  writers  are  Hill,  Bedwell, 
Hood,  the  two  Harvey  s,  and  For  man.  They  are  not  of 
sufficient  importance  to  require  more  than  a  word  or  two  in 
passing. 

Thomas  Hill,  who  took  his  B.A.  degree  from  Christ's 
College  in  1553,  wrote  a  work  on  Ptolemaic  astronomy  termed 
the  Schoole  of  skil :  it  was  published  posthumously  in  1599. 

Thomas  Bedwell  entered  at  Trinity  in  1562,  was  elected  a 
scholar  in  the  same  year,  proceeded  B.A.  in  1567,  and  in  1569 
was  admitted  to  a  fellowship.  His  works  deal  chiefly  with  the 
applications  of  mathematics  to  civil  and  military  engineering, 
and  enjoyed  a  high  and  deserved  reputation  for  practical  good 
sense.  The  New  River  company  was  due  to  his  suggestion. 
He  died  in  1595. 

Thomas  Hood,  who  entered  at  Trinity  in  1573,  proceeded 
B.A.  in  1578,  and  was  subsequently  elected  to  a  fellowship,  was 
another  noted  mathematician  of  this  time.  In  1590  he  issued  a 
translation  of  Ramus's  geometry,  and  in  1596  a  translation  of 
Urstitius's  arithmetic.  He  also  wrote  on  the  use  of  the  globes 

1  See  a  note  by  G.  B.  Halsted  in  vol.  n.  of  the  American  journal  of 
mathematics,  1878.  The  Greek  text  had  been  brought  into  Italy  by 
refugees  from  Constantinople,  and  was  first  published  in  the  form  of  a 
Latin  translation  by  Zamberti  at  Venice  in  1505 :  the  original  text 
(Theon's  edition)  was  edited  by  Grynasus  and  published  by  Hervagius  at 
Bale  in  1535. 


24  THE   MATHEMATICS   OF   THE   RENAISSANCE. 

(1590  and  1592),  and  the  principles  of  surveying  (1598).  In 
1582  a  mathematical  lectureship  was  founded  in  London — 
probably  by  a  certain  Thomas  Smith  of  Gracechurch  Street — 
and  Hood  was  appointed  lecturer.  His  books  are  probably  tran- 
scripts of  these  lectures  :  the  latter  were  given  in  the  Staples 
chapel,  and  subsequently  at  Smith's  house.  Hood  seems  to  have 
also  practised  as  a  physician  under  a  license  from  Cambridge 
dated  1585. 

Richard  Harvey,  a  brother  of  the  famous  Gabriel  Harvey, 
was  a  native  of  Saffron  Walden.  He  entered  at  Pembroke 
in  1575,  proceeded  B.A.  in  1578,  and  subsequently  was  elected 
to  a  fellowship.  He  was  a  noted  astrologer,  and  threw  the 
whole  kingdom  into  a  fever  of  anxiety  by  predicting  the  terrible 
events  that  would  follow  from  the  conjunction  of  Saturn  and 
Jupiter,  which  it  was  known  would  occur  011  April  28,  1583. 
Of  course  nothing  peculiar  followed  from  the  conjunction  ;  but 
Harvey's  reputation  as  a  prophet  was  destroyed,  and  he  was 
held  up  to  ridicule  in  the  tripos  verses  of  that  or  the  following 
year  and  hissed  in  the  streets  of  the  university.  Thomas  Nash 
(a  somewhat  prejudiced  witness  be  it  noted)  in  his  Pierce  penni- 
lesse,  published  in  London  in  1592  says,  "Would  you  in  likely 
reason  guess  it  were  possible  for  any  shame-swoln  toad  to  have 
the  spet-proof  face  to  outlive  this  disgrace?"  Harvey  took  a 
living,  and  his  later  writings  are  on  theology.  He  died  in 
1599. 

John  Harvey,  a  brother  of  the  Richard  Harvey  mentioned 
above,  was  also  born  at  Saffron  Walden :  he  entered  at  Queens' 
in  1578  and  took  his  B.A.  in  1580.  He  practised  medicine 
and  wrote  on  astrology  and  astronomy — the  three  subjects 
being  then  closely  related.  He  died  at  Lynn  in  1592. 

Simon  Forman1,  of  Jesus  College,  born  in  1552,  was  another 
mathematician  of  this  time,  who  like  those  just  mentioned 
combined  the  study  of  astronomy,  astrology,  and  medicine  with 
considerable  success ;  though  he  is  described,  apparently  with 

1  An  account  of  Forman's  life  is  given  in  the  Life  of  William  Lilly, 
written  by  himself,  London,  1715. 


WRIGHT.  25 

good  reason,  as  being  as  great  a  knave  as  ever  existed.  His 
license  to  practise  medicine  was  granted  by  the  university,  and 
is  dated  1604.  He  was  a  skilful  observer  and  good  mathema- 
tician, but  I  do  not  think  he  has  left  any  writings.  He  died 
suddenly  when  rowing  across  the  Thames  on  Sept.  12,  1611. 

With  the  exception  of  Recorde,  Dee,  and  Digges,  all  the 
above  were  but  second-rate  mathematicians ;  but  such  as  they 
were  (and  they  are  nearly  all  the  English  mathematicians  of 
that  time  of  whom  I  know  anything)  it  is  noticeable  that  with- 
out a  single  exception  they  were  educated  at  Cambridge.  The 
prominence  given  to  astronomy,  astrology,  and  surveying  is 
worthy  of  remark. 


I  come  next  to  a  group  of  mathematicians  of  considerably 
greater  power,  to  whom  we  are  indebted  for  important  contri- 
butions to  the  progress  of  the  science. 

The  first  of  these  was  Edward  Wright  \  whose  services  to 
the  theory  of  navigation  can  hardly  be  overrated.  Wright  was 
born  in  Norfolk,  took  his  B.A.  from  Caius  in  1581,  and  was 
subsequently  elected  to  a  fellowship.  He  seems  to  have  had  a 
special  talent  for  the  construction  of  instruments;  and  to 
instruct  himself  in  practical  navigation  and  see  what  improve- 
ments in  nautical  instruments  were  possible,  he  went  on  a 
voyage  in  1589 — special  leave  of  absence  from  college  being 
granted  him  for  the  purpose. 

In  the  maps  in  use  before  the  time  of  Gerard  Mercator  a 
degree  whether  of  latitude  or  longitude  had  been  represented 
in  all  cases  by  the  same  length,  and  the  course  to  be  pursued 
by  a  vessel  was  marked  on  the  map  by  a  straight  line  joining 
the  ports  of  arrival  and  departure.  Mercator  had  seen  that 
this  led  to  considerable  errors,  and  had  realized  that  to  make 
this  method  of  tracing  the  course  of  a  ship  at  all  accurate  the 

1  See  an  article  in  the  Penny  Cyclopaedia,  London,  1833 — 43 ;  and  a 
short  note  included  in  the  article  on  Navigation  in  the  ninth  edition  of 
the  Encyclopaedia  Britannica. 


26  THE   MATHEMATICS   OF  THE   RENAISSANCE. 

space  assigned  on  the  map  to  a  degree  of  latitude  ought 
gradually  to  increase  as  the  latitude  increased.  Using  this 
principle,  he  had  empirically  constructed  some  charts,  which 
were  published  about  1560  or  1570.  Wright  set  himself  the 
problem  to  determine  the  theory  on  which  such  maps  should 
be  drawn,  and  succeeded  in  discovering  the  law  of  the  scale  of 
the  maps,  though  his  rule  is  strictly  correct  for  small  arcs  only. 
The  result  was  published  by  his  permission  in  the  second  edition 
of  Blundeville's  Exercises.  His  reputation  was  so  considerable 
that  four  years  after  his  return  he  was  ordered  by  queen 
Elizabeth  to  attend  the  Earl  of  Cumberland  on  a  maritime  ex- 
pedition as  scientific  adviser. 

In  1599  Wright  published  a  work  entitled  Certain  errors 
in  navigation  detected  and  corrected,  in  which  he  very  fully 
explains  the  theory  and  inserts  a  table  of  meridional  parts. 
Solar  and  other  observations  requisite  for  navigation  are  also 
treated  at  considerable  length.  The  theoretical  parts  are  cor- 
rect, and  the  reasoning  shews  considerable  geometrical  power. 
In  the  course  of  the  work  he  gives  the  declinations  of  thirty- 
two  stars,  explains  the  phenomena  of  the  dip,  parallax,  and 
refraction,  and  adds  a  table  of  magnetic  declinations,  but  he 
assumes  the  earth  to  be  stationary.  This  book  went  through 
three  editions.  In  the  same  year  he  issued  a  work  called  The 
liav en-finding  art.  I  have  never  seen  a  copy  of  it  and  I  do  not 
know  how  the  subject  is  treated.  In  the  following  year  he 
published  some  maps  constructed  on  his  principle.  In  these 
the  northernmost  point  of  Australia  is  shewn :  the  latitude  of 
London  is  taken  to  be  51°  32'. 

About  this  time  Wright  was  elected  lecturer  on  mathe- 
matics by  the  East  India  Company  at  a  stipend  of  .£50  a  year. 
He  now  settled  in  London,  and  shortly  afterwards  was  ap- 
pointed mathematical  tutor  to  prince  Henry  of  Wales,  the  son 
of  James  I.  He  here  gave  another  proof  of  his  mechanical 
ability  by  constructing  a  sphere  which  enabled  the  spectator  to 
forecast  the  motions  of  the  solar  system  with  such  accuracy 
that  it  was  possible  to  predict  the  eclipses  for  over  seventeen 


BRIGGS.  27 

thousand  years  in  advance :  it  was  shewn  in  the  Tower  as  a 
curiosity  as  late  as  1675.  Wright  also  seems  to  have  joined 
Bedwell  in  urging  that  the  construction  of  the  New  River  to 
supply  London  with  drinking  water  was  both  feasible  and 
desirable. 

As  soon  as  Napier's  invention  of  logarithms  was  announced 
in  1614,  Wright  saw  its  value  for  all  practical  problems  in 
navigation  and  astronomy.  He  at  once  set  himself  to  prepare 
an  English  translation.  He  sent  this  in  1615  to  Napier,  who 
approved  of  it  and  returned  it,  but  Wright  died  in  the  same 
year,  before  it  was  printed:  it  was  issued  in  1616. 

Whatever  might  have  been  Wright's  part  in  bringing 
logarithms  into  general  use  it  was  actually  to  Briggs,  the 
second  of  the  mathematicians  above  alluded  to,  that  the  rapid 
adoption  of  Napier's  great  discovery  was  mainly  due. 

Henry  Briggs1  was  born  near  Halifax  in  1556.  He  was 
educated  at  St  John's  College,  took  his  B.A.  degree  in  1581,  and 
was  elected  to  a  fellowship  in  1588.  He  continued  to  reside  at 
Cambridge,  and  in  1592  he  was  appointed  examiner  and 
lecturer  in  mathematics  at  St  John's. 

In  1596  the  college  which  Sir  Thomas  Gresham2  had 
directed  to  be  built  was  opened.  Gresham,  who  was  born  in 
1513  and  died  in  1579,  had  been  educated  at  Goiiville  Hall, 
and  had  apparently  made  some  kind  of  promise  to  build  the 
college  at  Cambridge  to  encourage  research,  so  that  his  final 
determination  to  locate  it  in  London  was  received  with  great 
disappointment  in  the  university.  The  college  was  endowed 
for  the  study  of  the  seven  liberal  sciences ;  namely,  divinity, 
astronomy,  geometry,  music,  law,  physic,  and  rhetoric. 

Briggs  was  appointed  to  the  chair  of  geometry.  He  seems 
at  first  to  have  occupied  his  leisure  in  London  by  researches  on 

1  See  the  Lives  of  the  professors  of  Gresham   College  by  J.  Ward, 
London,  1740.     A  full  list  of  Briggs's  works  is  given  in  the  Dictionary  of 
national  biography. 

2  See  the  Life  and  times  of  Sir  Thomas  Gresham,  published  anony- 
mously but  I  believe  written  by  J.  W.  Burgon,  London,  1845. 


28  THE   MATHEMATICS   OF   THE   RENAISSANCE. 

magnetism  and  eclipses.  Almost  alone  among  his  contempo- 
raries he  declared  that  astrology  was  at  best  a  delusion  even  if 
it  were  not,  as  was  too  frequently  the  case,  a  mere  cloak  for 
knavery.  In  1610  he  published  Tables  for  the  improvement  of 
navigation,  and  in  1616  a  Description  of  a  table  to  find  the  part 
proportional  devised  by  Edw.  Wright. 

In  1614  Briggs  received  a  copy  of  Napier's  work  on 
logarithms,  which  was  published  in  that  year.  He  at  once 
realized  the  value  of  the  discovery  for  facilitating  all  practical 
computations,  and  the  rapidity  with  which  logarithms  came 
into  general  use  was  largely  due  to  his  advocacy.  The  base 
to  which  the  logarithms  were  at  first  calculated  was  very 
inconvenient,  and  Briggs  accordingly  visited  Napier  in  1616, 
and  urged  the  change  to  a  decimal  base,  which  was  recog- 
nized by  Napier  as  an  improvement.  Briggs  at  once  set  to 
work  to  carry  this  suggestion  into  effect,  and  in  1617  brought 
out  a  table  of  logarithms  of  the  numbers  from  1  to  1000  calcu- 
lated to  fourteen  places  of  decimals.  He  subsequently  (in  1624) 
published  tables  of  the  logarithms  of  additional  numbers  and  of 
various  trigonometrical  functions.  The  calculation  of  some 
20,000  logarithms  which  had  been  left  out  by  Briggs  in  his 
tables  of  1624  was  performed  by  Vlacq  and  published  in  1628. 
The  Arithmetica  logarithmica  of  Briggs  and  Vlacq  are  sub- 
stantially the  same  as  the  existing  tables:  parts  have  been 
recalculated,  but  no  tables  of  an  equal  range  and  fulness  entirely 
founded  on  fresh  computations  have  since  been  published. 
These  tables  were  supplemented  by  Briggs's  Trigonometrica 
Eritannica  which  was  published  posthumously  in  1633. 

The  introduction  of  the  decimal  notation  was  also  (in  my 
opinion)  due  to  Briggs.  Stevinus  in  1585,  and  Napier  in  his 
essay  on  rods  in  1617,  had  previously  used  a  somewhat  similar 
notation,  but  they  only  employed  it  as  a  concise  way  of  stating 
results,  and  made  no  use  of  it  as  an  operative  form.  The  nota- 
tion occurs  however  in  the  tables  published  by  Briggs  in  1617, 
and  was  adopted  by  him  in  all  his  works,  and  though  it  is 
difficult  to  speak  with  absolute  certainty  I  have  myself  but 


BRIGGS.  29 

little  doubt  that  lie  there  employed  the  symbol  as  an  operative 
form.  In  Napier's  posthumous  Construct™  published  in  1619 
it  is  defined  and  used  systematically  as  an  operative  form,  and 
as  this  work  was  written  after  consultation  with  Briggs,  and 
was  probably  revised  by  him  before  it  was  issued,  I  think  it 
confirms  the  view  that  the  invention  was  due  to  Briggs  and 
was  communicated  by  him  to  Napier.  At  any  rate  its  use  as 
an  operative  form  was  not  known  to  Napier  in  1617.  Napier 
wrote  the  point  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.  Later  writers  added  another  line 
and  wrote  it  25 1379  ;  nor  was  it  till  the  beginning  of  the  eight- 
eenth century  that  the  notation  now  current  was  generally 
employed. 

Shortly  after  bringing  out  the  first  of  his  logarithmic  tables, 
Briggs  moved  to  Oxford.  For  more  than  two  centuries — 
possibly  from  the  time  of  Bradwardine — Merton  had  been  the 
one  college  in  that  university  where  instruction  in  mathematics 
had  been  systematically  given.  When  Sir  Henry  Savile  (born 
in  1549  and  died  in  1622)  became  warden  of  Merton  he  seems 
to  have  felt  that  the  practical  abandonment  of  science  to  Cam- 
bridge was  a  reproach  on  the  ancient  and  immensely  more 
wealthy  university  of  Oxford.  Accordingly  about  1570  he 
began  to  give  lectures  on  Greek  geometry,  which,  contrary 
to  the  almost  universal  practice  of  that  age,  he  opened  free 
to  all  members  of  the  university.  These  lectures  were  pub- 
lished at  Oxford  in  1621.  He  never  however  succeeded  in 
taking  his  class  beyond  the  eighth  proposition  of  the  first  book 
of  Euclid.  "Exolvi,"  says  he,  "per  Dei  gratiam,  domini  audi- 
tores ;  promissum ;  liberavi  fidem  meam ;  explicavi  pro  men 
modulo,  definitiones,  petitiones,  communes  sententias,  et  octo 
priores  propositiones  Elementorum  Euclidis.  Hie,  annis  fessus, 
cycles  artemque  repono." 

In  spite  of  this  discouraging  result  Savile  hoped  to  make 
the  study  a  permanent  one,  and  in  1619  he  founded  two  chairs, 
one  of  geometry  and  one  of  astronomy.  The  former  he  offered 


30  THE   MATHEMATICS   OF   THE   RENAISSANCE. 

to  Briggs,  who  thus  has  the  singular  distinction  of  holding  in 
succession  the  two  earliest  chairs  of  mathematics  that  were 
founded  in  England.  Briggs  continued  to  hold  this  post  until 
his  death  on  Jan.  26,  1630. 

Among  Briggs's  contemporaries  at  Cambridge  was  Oughtred, 
who  systematized  elementary  arithmetic,  algebra,  and  trigono- 
metry. William  Oughtred1  was  born  at  Eton  011  March  5, 
1574.  He  was  educated  at  Eton,  and  thence  in  1592  went  to 
King's  College.  While  an  undergraduate  he  wrote  an  essay  on 
geometrical  dialling.  He  took  his  B.A.  degree  in  1596,  was 
admitted  to  a  fellowship  in  the  ordinary  course,  and  lectured 
for  a  few  years;  but  on  taking  orders  in  1603  he  felt  it  his 
duty  to  devote  his  time  wholly  to  parochial  work. 

Although  living  in  a  country  vicarage  he  kept  up  his 
interest  in  mathematics.  Equally  with  Briggs  he  received  one 
of  the  earliest  copies  of  Napier's  Canon  on  logarithms,  and  was 
at  once  impressed  with  the  great  value  of  the  discovery. 
Somewhat  later  in  life  he  wrote  two  or  three  works.  He 
always  gave  gratuitous  instruction  to  any  who  came  to  him, 
provided  they  would  learn  to  "write  a  decent  hand."  He 
complained  bitterly  of  the  penury  of  his  wife  who  always 
took  away  his  candle  after  supper  "whereby  many  a  good 
motion  was  lost  and  many  a  problem  unsolved " ;  and  one  of 
his  pupils  who  secretly  gave  him  a  box  of  candles  earned  his 
warmest  esteem.  He  is  described  as  a  little  man,  with  black 
hair,  black  eyes,  and  a  great  deal  of  spirit.  Like  nearly  all  the 
mathematicians  of  the  time  he  was  somewhat  of  an  astrologer 
and  alchemist.  He  died  at  his  vicarage  of  Albury  in  Surrey 
on  June  30,  1660. 

His  Clavis  mathematica  published  in  1631  is  a  good  syste- 
matic text-book  on  algebra  and  arithmetic,  and  it  contains 
practically  all  that  was  then  known  on  the  subject.  In  this 
work  he  introduced  the  symbol  x  for  multiplication,  and  the 

1  See  Letters... and  lives  of  eminent  men  by  J.  Aubrey,  2  vols.,  London, 
1813.  A  complete  edition  of  Oughtred's  works  was  published  at  Oxford 
in  1677. 


OUGHTRED.  31 

symbol  ::  in  proportion.  Previously  to  his  time  a  proportion 
such  ac  a  :  b  =  c  :  d  was  written  as  a  —  b-c-d,  but  he  denoted 
it  by  a  .  b  ::  c  .  d.  Wallis  says  that  some  found  fault  with  the 
book  on  account  of  the  style,  but  that  they  only  displayed  their 
own  incompetence,  for  Oughtred's  "words  be  always  full  but 
not  redundant."  Pell  makes  a  somewhat  similar  remark. 

A  work  on  sun  and  other  dials  published  in  1636  shews 
considerable  geometrical  power,  and  explains  how  various  astro- 
nomical problems  can  be  resolved  by  the  use  of  dials.  He  also 
wrote  a  treatise  on  trigonometry  published  in  1657  which  is 
one  of  the  earliest  works  containing  abbreviations  for  sine, 
cosine,  <kc.  This  was  really  an  important  advance,  but  the 
book  was  neglected  and  soon  forgotten,  and  it  was  not  until 
Euler  reintroduced  contractions  for  the  trigonometrical  func- 
tions that  they  were  generally  adopted. 

The  following  list  comprises  all  his  works  with  which  I  am 
acquainted.  The  Clavis,  first  edition  1631;  second  edition 
with  an  appendix  on  numerical  equations  1648;  third  edition 
greatly  enlarged,  1652.  The  circle  of  proportion,  1632;  second 
edition  1660.  The  double  horizontal  dial,  1636  ;  second  edition 

1652.  Sun-dials   by   geometry,   1647.     The   horological  ring, 

1653.  Solution  of  all  spherical  triangles,  1657 '.    Trigonometry, 
1657.     Canones  sinuum  etc.,  1657.     And  lastly  a  posthumous 
work  entitled  Opuscula  mathematica  hactenus  inedita,  issued  in 
1677. 

Just  as  Briggs  was  the  most  famous  English  geometrician 
of  that  time,  so  to  his  contemporaries  Oughtred  was  probably 
the  most  celebrated  exponent  of  algorism.  Thus  in  some 
doggrel  verses  in  the  Lux  mercatoria  by  Noah  Bridges,  London, 
1661,  we  read  that  a  merchant 

"may  fetch  home  the  Indies,  and  not  know 
what  Napier  could  or  what  Oughtred  can  do." 

Another  mathematician  of  this  time,  who  was  almost  as 
well  known  as  Briggs  and  Oughtred,  was  Thomas  Harriot  who 
was  born  in  1560,  and  died  on  July  2,  1621.  He  was  not 


32  THE   MATHEMATICS   OF   THE   KEXAISSANCE. 

educated  at  either  university,  and  his  chief  work  the  Artis 
<ui(il  i/ticae  praxis  was  not  printed  till  1631.  It  is  incom- 
parably the  best  work  on  algebra  and  the  theory  of  equations 
which  had  then  been  published.  I  mention  it  here  since  it 
became  a  recognized  text-book  on  the  subject,  and  for  at  least 
a  century  the  more  advanced  Cambridge  undergraduates, 
including  Newton,  Whiston,  Cotes,  Smith,  and  others,  learnt 
most  of  their  algebra  thereout.  We  may  say  roughly  that 
henceforth  elementary  arithmetic,  algebra,  and  trigonometry 
were  treated  iu  a  manner  which  is  not  substantially  different 
from  that  now  in  use ;  and  that  the  subsequent  improvements 
introduced  are  additions  to  the  subjects  as  then  known,  and 
not  a  re-arrangement  of  them  on  new  foundations. 

The  work  of  most  of  those  considered  in  this  chapter — 
which  we  may  take  as  comprised  between  the  years  1535  and 
1630 — is  manifestly  characterized  by  the  feeling  that  mathe- 
matics should  be  studied  for  the  sake  of  its  practical  applications 
to  astronomy  (including  astrology  therein),  navigation,  mensura- 
tion, and  surveying;  but  it  was  tacitly  assumed  that  even  in 
these  subjects  its  uses  were  limited,  and  that  a  knowledge  of  it 
was  in  no  way  necessary  to  those  who  applied  the  rules  deduced 
therefrom,  while  it  was  generally  held  that  its  study  did  not 
form  any  part  of  a  liberal  education. 


CHAPTER  III. 
THE   COMMENCEMENT   OF  MODERN  MATHEMATICS. 

IN  the  last  chapter  I  was  able  to  trace  a  continuous 
succession  of  mathematicians  resident  at  Cambridge  to  the  end 
of  the  sixteenth  century.  The  period  of  the  next  thirty  years 
is  almost  a  blank  in  the  history  of  science  at  the  university, 
but  its  close  is  marked  by  the  publication  of  some  of  the  more 
important  works  of  Briggs,  Oughtred,  and  Harriot.  We  come 
then  to  the  names  of  Horrox  and  Seth  Ward,  both  of  whom 
were  well-known  astronomers;  to  Pell,  who  was  later  in 
intimate  relations  with  Newton;  and  lastly  to  Wallis  and  to 
Barrow,  who  were  the  first  Englishmen  to  treat  mathematics 
as  a  science  rather  than  as  an  art,  and  who  may  be  said  to  have 
introduced  the  methods  of  modern  mathematics  into  Britain. 
It  curiously  happened  that  in  the  absence  of  any  endowments 
for  mathematics  at  Cambridge  both  Ward  and  Wallis  were 
elected  to  professorships  at  Oxford,  and  by  their  energy  and 
tact  created  the  Oxford  mathematical  school  of  the  latter  half 
of  the  seventeenth  century. 

The  middle  of  the  seventeenth  century  marks  the  beginning 
of  a  new  era  in  mathematics.  The  invention  of  analytical 
geometry  and  the  calculus  completely  revolutionized  the  de- 
velopment of  the  subject,  and  have  proved  the  most  powerful 
instruments  of  modern  progress.  Descartes's  geometry  was 
published  in  1637  and  Cavalieri's  method  of  indivisibles,  which 
is  equivalent  to  integration  regarded  as  a  means  of  summing 
series,  was  introduced  a  year  or  so  later.  The  works  of  both 
B.  3 


34   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

these  writers  were  very  obscure,  but  they  had  a  wide  circula- 
tion, and  we  may  say  that  by  about  1660  the  methods  used  by 
them  were  known  to  the  leading  mathematicians  of  Europe. 
This  was  largely  due  to  the  writings  of  Wallis.  Barrow 
occupies  a  position  midway  between  the  old  and  the  new 
schools.  He  was  acquainted  with  the  elements  of  the  new 
methods,  but  either  by  choice  or  through  inability  to  recognize 
their  power$  he  generally  adhered  to  the  classical  methods.  It 
was  to  him  that  Newton  was  indebted  for  most  of  his  instruc- 
tion in  mathematics;  he  certainly  impressed  his  contemporaries 
as  a  man  of  great  genius,  and  he  came  very  near  to  the 
invention  of  the  differential  calculus. 

The  infinitesimal  calculus  was  invented  by  Newton  in 
1666,  and  was  among  the  earliest  of  those  discoveries  and 
investigations  which  have  raised  him  to  the  unique  position 
which  he  occupies  in  the  history  of  mathematics.  The  calculus 
was  not  however  brought  into  general  use  till  the  beginning 
of  the  eighteenth  century.  The  discoveries  of  Newton  mate- 
rially affected  the  whole  subsequent  history  of  mathematics, 
and  at  Cambridge  they  led  to  a  complete  rearrangement  of  the 
system  of  education.  It  will  therefore  be  convenient  to  defer 
the  consideration  of  his  life  and  works  to  the  next  chapter. 

The  chief  distinction  between  the  classical  geometry  and 
the  method  of  exhaustions  on  the  one  hand,  and  the  new 
methods  introduced  by  Descartes,  Cavalieri,  and  Newton  on 
the  other  is  that  the  former  required  a  special  procedure  for 
every  particular  problem  attacked,  while  in  the  latter  a  general 
rule  is  applicable  to  all  problems  of  the  same  kind.  The 
validity  of  this  process  is  proved  once  for  all,  and  it  is  no 
longer  requisite  to  invent  some  special  process  for  every  sepa- 
rate function,  curve,  or  surface. 

Another  cause  which  makes  it  desirable  to  take  this  time 
as  the  commencement  of  a  new  chapter  is  the  change  in  the 
character  of  the  scholastic  exercises  in  the  university  which 
then  first  began  to  be  noticeable.  The  disturbances  produced 
by  the  civil  wars  in  the  middle  of  the  seventeenth  century 


HOKROX.  35 

did  not  affect  Cambridge  so  severely  as  Oxford,  but  still 
they  produced  considerable  disorder,  and  thenceforward  the 
regulations  of  the  statutes  about  exercises  in  the  schools 
seem  to  have  been  frequently  disregarded.  The  Elizabethan 
statutes  had  directed  that  logic  should  form  the  basis  of  a 
university  education,  and  that  it  should  be  followed  by  a 
study  of  Aristotelian  philosophy.  The  logic  that  was  read  at 
Cambridge  was  that  of  Ramus.  This  was  purely  negative 
and  destructive,  and  formed  an  admirable  preparation  for  the 
Baconian  and  Cartesian  systems  of  philosophy.  The  latter 
were  about  this  time  adopted  in  lieu  of  a  study  of  Aristotle, 
and  they  provided  the  usual  subject  for  discussions  in  the 
schools  for  the  remainder  of  the  seventeenth  century,  until  in 
their  turn  they  were  displaced  by  the  philosophy  of  Newton 
and  of  Locke1. 

I  shall  commence  by  a  very  brief  summary  of  the  views  of 
Horrox  and  Seth  Ward,  and  shall  then  enumerate  some  other 
contemporary  astronomers  of  less  eminence.  I  shall  next 
describe  the  writings  of  Pell,  Wallis,  and  Barrow ;  and  it 
will  be  convenient  to  add  references  to  a  few  other  mathemati- 
cians the  general  character  of  whose  works  is  pre-newtonian. 


Jeremiah  Horrox2 — sometimes  written  Horrocks — was  born 
near  Liverpool  in  1619;  he  entered  at  Emmanuel  College  in 
1633,  but  probably  went  down  without  taking  a  degree  in 
1635  or  1636;  he  died  in  1641.  From  boyhood  he  had 
resolved  to  make  himself  an  astronomer.  No  astronomy  seems 
then  to  have  been  taught  at  Cambridge,  and  Horrox  says  that 
he  had  chiefly  to  rely  on  reading  books  by  himself.  He  had 
but  small  means;  and  desiring  that  his  library  should  contain 
only  the  best  works  on  the  subject  he  took  a  great  deal  of 

1  See  p.  69  of  On  the  Statutes  by  G.  Peacock,  London,  1841. 

2  See  his  life  by  A.  B.  Whatton,  second  edition,  London,  1875.     The 
works  of  Horrox  were  collected  by  Wallis  and  published  at  London  in 
1672. 

3—2 


36   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

trouble  in  selecting  them.  The  list  he  drew  up,  written  at 
the  end  of  his  copy  of  Lansberg's  tables,  is  now  in  the  library 
of  Trinity  and  sufficiently  instructive  to  deserve  quotation. 

Albategnius.  J-  Kepleri  Tabulae  Eudolphinae. 

Alfraganus.  Lansbergii  Progymn.  de  Motu  Solis. 

J.  Capitolinus.  Longomontani  Astron.  Danica. 

Clavii  Apolog.  Cal.  Rom.  Magini  Secunda  Mobilia. 

Clavii  Comm.  in  Sacroboscum.  Mercatoris  Chronologia. 

Copernici  Revolutiones.  Plinii  Hist.  Naturalis. 

Cleomedes.  Ptolemaei  Magnum  Opus. 

Julius  Firmicus.  Regiomontani  Epitome. 

Gassendi    Exerc.    Epist.    in    Phil.  Torquetum. 

Fluddanam.  Observata. 

Gemmae  Frisii  Radius  Astronomicus.  Rheinoldi  Tab.  Prutenicse. 

Cornelii  Gemmae  Cosmocritice.  Comm.  in  Theor.  Purbachiu 

Herodoti  Historia.  Theonis  Comm.  in  Ptolom. 

J.  Kepleri  Astron.  Optica.  Tyc.  Brahagi  Progymnasmata. 

Epit.  Astron.  Copern. Epist.  Astron. 

Comm.  de  Motu  Martis.  Waltheri  Observata. 

This  list  probably  represents  the  most  advanced  astronomical 
reading  of  the  Cambridge  of  that  time. 

In  spite  of  his  early  death  Horrox  did  more  to  improve 
the  lunar  theory  than  any  Englishman  before  Newton  :  and  in 
particular  he  was  the  first  to  shew  that  the  lunar  orbit  might 
be  exactly  represented  by  an  ellipse,  provided  an  oscillatory 
motion  were  given  to  the  apse  line  and  the  eccentricity  made 
to  vary.  This  result  was  deduced  from  the  law  of  gravitation 
by  Newton  in  the  thirty-fifth  proposition  of  the  third  book  of 
the  Principia.  Horrox  was  also  the  first  observer  who  noted 
that  Venus  could  be  seen  on  the  face  of  the  sun  :  the  obser- 
vation was  made  on  Nov.  24,  1639,  and  an  account  of  it  was 
printed  by  Hevelius  at  Danzig  in  1662. 

Seth  Ward1  was  born  in  Hertfordshire  in  1617,  took  his 
B.A.  from  Sidney  Sussex  College  in  1637  at  the  same  time 
as  Wallis,  and  was  subsequently  elected  a  fellow.  In  his 

1  See  his  life  by  Walter  Pope,  London,  1697;   and  Letters and 

lives  of  eminent  men  by  J.  Aubrey,  2  vols.,  London,  1813. 


WARD.  37 

dispute  with  the  prevaricator  in  1640,  he  was  publicly  re- 
buked for  the  freedom  of  his  language  and  his  supplicat  for 
the  M.A.  degree  rejected,  but  the  censure  seems  to  have  been 
undeserved  and  was  withdrawn.  He  was  celebrated  for  his 
knowledge  of  mathematics  and  especially  of  astronomy;  and 
he  was  also  a  man  of  considerable  readiness  and  presence. 
While  residing  at  Cambridge  he  taught,  and  one  of  his  pupils 
says  that  he  "brought  mathematical  learning  into  vogue  in  the 
university... where  he  lectured  his  pupils  in  Master  Oughtred's 
Clavis." 

He  was  expelled  from  his  fellowship  by  the  parliamentary 
party  for  refusing  to  subscribe  the  league  and  covenant.  On 
this  Oughtred  invited  him  to  his  vicarage,  where  he  could 
pursue  his  mathematical  studies  without  interruption.  His 
companion  on  this  visit  was  a  certain  Charles  Scarborough,  a 
fellow  of  Caius  and  described  as  a  teacher  of  the  mathematics 
at  Cambridge,  of  whom  I  know  nothing  more. 

In  1649  Ward  was  appointed  to  the  Savilian  chair  of 
astronomy  at  Oxford  and,  like  Wallis  who  was  appointed  at 
the  same  time,  consented,  with  some  hesitation,  to  take  the 
oath  of  allegiance  to  the  commonwealth.  The  two  mathe- 
maticians who  had  been  together  at  Cambridge  exerted  them- 
selves with  considerable  success  to  revive  the  study  of 
mathematics  at  Oxford ;  and  they  both  took  a  leading  part  in 
the  meetings  of  the  philosophers,  from  which  the  Royal 
Society  ultimately  developed.  Ward  proceeded  to  a  divinity 
degree  in  1654,  and  subsequently  held  various  ecclesiastical 
offices,  including  the  bishoprics  of  Exeter  and  Salisbury.  He 
died  in  January,  1689. 

Aubrey  describes  him  as  singularly  handsome,  though 
perhaps  somewhat  too  fond  of  athletics,  at  which  he  was  very 
proficient.  Courteous,  rich,  generous,  with  great  natural 
abilities,  and  wonderful  tact,  he  managed  to  make  all  men 
trust  his  honour  and  desire  his  friendship — a  somewhat  as- 
tonishing feat  in  those  troubled  times. 

He  wrote  a  text-book  on  trigonometry  published  at  Oxford 


38   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

in  1654,  but  he  is  best  known  for  his  works  on  astronomy. 
These  are  two  in  number,  namely,  one  on  comets  and  the 
hypothesis  of  Bulialdus  published  at  Oxford  in  1653  ;  and  the 
other  on  the  planetary  orbits  published  in  London  in  1656. 
The  hypothesis  of  Bulialdus,  which  Ward  substantially  adopted, 
is  that  for  every  planetary  orbit  there  is  a  point  (called  the 
upper  focus)  on  the  axis  of  the  right  cone  of  which  the  orbit  is 
a  section  such  that  the  radii  vectores  thence  drawn  to  the 
planet  move  with  a  uniform  motion  :  the  idea  being  the  same 
as  that  held  by  the  Greeks,  namely,  that  the  motion  of  a 
celestial  body  must  be  perfect  and  therefore  must  be  uniform. 

Other  astronomers  of  the  same  time  were  Samuel  Foster, 
Laurence  Rooke,  Nicholas  Culpepper,  and  Gilbert  Clerke.  I 
add  a  few  notes  on  them. 

Samuel  Foster1,  of  Emmanuel  College,  who  was  born  in 
Northamptonshire,  took  his  B.A.  in  1619,  and  in  1636  was 
appointed  Gresham  professor  of  astronomy,  but  was  shortly  ex- 
pelled for  refusing  to  kneel  when  at  the  communion  table :  he  was 
however  reappointed  in  1641,  and  held  the  chair  till  his  death, 
which  took  place  in  1652.  He  wrote  several  works,  of  which 
a  list  is  given  on  pp.  86-87  of  Ward's  Lives :  most  of  them  are 
on  astronomical  instruments,  but  one  volume  contains  some 
interesting  essays  on  various  problems  in  Greek  geometry. 
Foster  took  a  prominent  part  in  the  meetings  of  the  so-called 
"indivisible  college"  during  the  year  1645,  from  which  the 
Royal  Society  ultimately  sprang. 

Foster  was  succeeded  in  his  chair  at  Gresham  College  by 
Rooke.  Laurence  Rooke1,  who  was  born  in  Kent  in  1623,  took 
his  B.A.  in  1643  from  King's  College.  He  lectured  at  Cam- 
bridge on  Oughtred's  Clavis  for  some  time  after  his  degree.  Like 
Foster  he  took  a  leading  part  in  the  meetings  of  the  indivisible 
college :  being  a  man  of  considerable  property  he  assisted  the 
society  in  several  ways,  and  in  1650  he  moved  to  Oxford  with 


1  See  the  Lives  of  the  professors   of  Gresham  College  by  J.  "Ward, 
London,  1740. 


ROOKE.      CULPEPPER.      CLERKE.  39 

most  of  the  other  members.  In  1652  he  was  appointed  pro- 
fessor of  astronomy  at  Gresham  College,  and  in  1657  he  ex- 
changed it  for  the  chair  of  geometry,  which  he  held  till  his 
death  in  1662.  His  lectures  were  given  on  the  sixth  chapter 
of  ,Oughtred's  Clavis,  which  enables  us  to  form  an  idea  of  the 
extent  of  mathematics  then  usually  known.  A  list  of  his 
writings  is  given  in  Ward :  most  of  them  are  concerned  with 
various  practical  questions  in  astronomy. 

Nicholas  Culpepper,  of  Queens',  who  was  born  in  London 
on  Oct.  18,  1616,  entered  at  Cambridge  in  1634  and  died  on 
Jan.  10,  1653,  was  a  noted  astrologer  of  the  time.  He  used 
his  knowledge  of  astronomy  to  justify  various  medical  remedies 
employed  by  him,  which  though  they  savoured  of  heresy  to  the 
orthodox  practitioner  of  that  day,  seem  to  have  been  fairly 
successful.  It  is  doubtful  whether  he  was  a  quack  or  an 
unpopular  astronomer.  I  suspect  he  has  a  better  claim  to  the 
former  title  than  the  latter  one,  but  I  give  him  the  benefit  of 
the  doubt.  His  works,  edited  by  G.  A.  Gordon,  were  published 
in  four  volumes  in  London  in  1802. 

Gilbert  Clerke,  a  fellow  of  Sidney  College,  was  born  at 
Uppingham  in  1626,  and  graduated  B.A.  in  1645.  He  lectured 
for  a  few  years  at  Cambridge,  but  in  1655  was  forced  to  quit 
the  university  by  the  Cromwellian  party.  He  had  a  small  pro- 
perty in  Norfolk  and  lived  there  till  his  death.  His  chief 
mathematical  works  were  theDeplenitudine  mundi,  published  in 
1660,  in  which  he  defended  Descartes  from  the  criticisms  of 
Bacon  and  Seth  Ward ;  an  account  of  some  experiments 
analogous  to  those  of  Torricelli,  published  in  1662;  a  com- 
mentary on  Oughtred's  Clavis,  published  in  1682;  and  a 
description  of  the  "spot-dial,"  published  in  1687.  He  was 
a  friend  of  Cumberland  and  of  Whiston.  He  died  towards  the 
end  of  the  seventeenth  century. 


The   three  mathematicians  to  be   next   mentioned — Pel], 
Wallis,  and  Barrow — were  men  of  much  greater  mark,  and 


40   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

in  their  writings  we  begin  to  find  mathematics  treated  as  a 
science. 

John  Pell1  was  born  in  Sussex  on  March  1,  1610:  he 
entered  at  Trinity  at  the  unusually  early  age  of  thirteen, 
and  proceeded  to  his  degrees  in  regular  course,  commencing 
M.A.  in  1630.  After  taking  his  degree  he  continued  the 
study  of  mathematics,  and  his  reputation  was  so  consider- 
able that  in  1639  he  was  asked  to  stand  for  the  mathe- 
matical chair  then  vacant  at  the  university  of  Amsterdam; 
but  he  does  not  seem  to  have  gone  there  till  1643.  In  1646 
he  moved,  at  the  request  of  the  prince  of  Orange,  to  the 
college  which  the  latter  had  just  founded  at  Breda.  In 
1654  he  entered  the  English  diplomatic  service,  and  in  1661 
took  orders  and  became  private  chaplain  to  the  archbishop  of 
Canterbury.  He  still  however  continued  the  study  of  philo- 
sophy and  mathematics  to  the  no  small  detriment  of  his  private 
affairs.  It  was  to  him  that  Newton  about  this  time  explained 
his  invention  of  fluxions.  He  died  in  straitened  circumstances 
in  London  on  Dec.  10,  1685. 

He  was  especially  celebrated  among  his  contemporaries  for 
his  lectures  on  the  algebra  of  Diophantus  and  the  geometry  of 
Apollonius,  of  which  authors  he  had  made  a  special  study.  He 
had  prepared  these  lectures  for  the  press,  but  their  publication 
was  abandoned  at  the  request  of  one  of  his  Dutch  colleagues. 
In  1668  he  issued  in  London  a  new  edition  of  Branker's  trans- 
lation from  the  Dutch  of  Khonius's  algebra,  with  the  addition 
of  considerable  new  matter:  in  this  work  the  symbol  -^  for 
division  was  first  employed.  In  1672  he  published  at  London 
a  table  of  all  square  numbers  less  than  108.  These  were 
his  chief  works,  but  he  also  wrote  an  immense  number  of 

1  See  the  Penny  Cyclopaedia,  London,  1833 — 43.  The  custom  which 
prevailed  amongst  the  more  wealthy  classes  of  obtaining  as  soon  as 
possible  the  horoscope  of  a  child  enables  us  to  fix  the  date  of  birth  with 
far  greater  accuracy  than  might  have  been  expected  by  those  unacquainted 
with  the  habits  of  the  time.  Pell  for  example  was  born  at  1.21  p.m.  on 
the  day  above  mentioned. 


PELL.      WALLIS.  41 

pamphlets  and  letters  on  various  scientific  questions  then  de- 
bated: those  now  extant  fill  nearly  fifty  folio  volumes,  and  a 
competent  review  of  them  would  probably  throw  considerable 
light  on  the  scientific  history  of  the  seventeenth  century,  and 
possibly  on  the  state  of  university  education  in  the  first  half  of 
that  century. 

The  following  are  the  titles  and  dates  of  his  published 
writings.  On  the  quadrant,  2  vols.,  1630.  Modus  supputandi 
ephemerides,  1630.  On  logarithms,  1 631.  Astronomical  history, 
1633.  Foreknower  of  eclipses,  1633.  Deduction  of  astronomical 
tables  from  Lansberg's  tables,  1634.  On  the  magnetic  needle, 
1635.  On  Easter,  1644.  An  idea  of  mathematics,  1650. 
Br anker's  translation  of  Rhonius's  algebra,  1668.  A  table  of 
square  numbers,  1672. 

The  next  and  by  far  the  most  distinguished  of  the  mathe- 
maticians of  this  time  is  Wallis.  John  Wallis1  was  born  at 
Ashford  on  Nov.  22,  1616.  When  fifteen  years  old  he  hap- 
pened 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  had  mastered  the 
subject.  It  was  intended  that  he  should  be  a  doctor,  and  he 
was  sent  to  Emmanuel  College,  the  chief  centre  of  the  academical 
puritans.  He  took  his  B.A.  in  1637;  and  for  that  kept  one 
of  his  acts,  on  the  doctrine  of  the  circulation  of  the  blood — 
this  was  the  first  occasion  on  which  this  theory  was  publicly 
maintained  in  a  disputation. 

His  interests  however  centred  on  mathematics.  Writing 
in  1635  he  gives  an  account  of  his  undergraduate  training. 
He  says  that  he  had  first  to  learn  logic,  then  ethics,  physics, 
and  metaphysics,  and  lastly  (what  was  worse)  had  to  consult 
the  schoolmen  on  these  subjects.  Mathematics,  he  goes  on, 
were  "scarce  looked  upon  as  Academical  studies,  but  rather 

1  See  the  Biographia  Britannica,  first  edition,  London,  1747 — 66,  and 
the  Histoire  des  sciences  mathematiques  by  M.  Marie,  Paris,  1833—88. 
Wallis's  mathematical  works  were  published  in  three  volumes  at  Oxford, 
1693-99. 


42   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

Mechanical... And  among  more  than  two  hundred  students  (at 
that  time)  in  our  college,  I  do  not  know  of  any  two  (perhaps 
not  any)  who  had  more  of  Mathematicks  than  I,  (if  so  much) 
which  was  then  but  little  ;  and  but  very  few,  in  that  whole 
university.  For  the  study  of  Mathematicks  was  at  that  time 
more  cultivated  in  London  than  in  the  universities."  This  pas- 
sage has  been  quoted  as  shewing  that  no  attention  was  paid  to 
mathematics  at  that  time.  I  do  not  think  that  the  facts  justify 
such  a  conclusion;  at  any  rate  Wallis,  whether  by  his  own 
efforts  or  not,  acquired  sufficient  mathematics  at  Cambridge  to 
be  ranked  as  the  equal  of  mathematicians  such  as  Descartes, 
Pascal,  and  Fermat. 

Wallis  was  elected  to  a  fellowship  at  Queens',  commenced 
M.A.  in  1640,  and  subsequently  took  orders,  but  on  the  whole 
adhered  to  the  puritan  party  to  whom  he  rendered  great  assist- 
ance 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 — a  fact  which  when  he 
subsequently  lived  at  Oxford  did  something  to  diminish  his 
unpopularity  as  a  mathematician  and  a  schismatic. 

There  was  then  no  professorship  in  mathematics  and  no 
opening  for  a  mathematician  to  a  career  at  Cambridge  ;  and  so 
Wallis  reluctantly  left  the  university.  In  1649  he  was  ap- 
pointed to  the  Savilian  chair  of  geometry  at  Oxford,  where  he 
lived  until  his  death  on  Oct.  28,  1703.  It  was  there  that  all  his 
mathematical  works  were  published.  Besides  those  he  wrote 
on  theology,  logic,  and  philosophy ;  and  was  the  first  to  devise 
a  system  for  teaching  deaf-mutes.  I  do  not  think  it  necessary 
to  mention  his  smaller  pamphlets,  a  full  list  of  which  would 
occupy  some  four  or  five  pages :  but  I  add  a  few  notes  on  his 
more  important  mathematical  writings. 

The  most  notable  of  these  was  his  Arithmetica  infinitorum, 
which  was  published  in  1656.  It  is  prefaced  by  a  short  tract 
on  conic  sections  which  was  subsequently  expanded  into  a 
separate  treatise.  He  then  established  the  law  of  indices,  and 


WALLIS.  43 

shewed  that  x~n  stood  for  the  reciprocal  of  xn  and  that  xplq 
stood  for  the  qth  root  of  xp.  He  next  proceeded  to  find  by  the 
method  of  indivisibles  the  area  enclosed  between  the  curve 
y  =  xm,  the  axis  of  x,  and  any  ordinate  x  =  h;  and  he  proved 
that  this  was  to  the  parallelogram  on  the  same  base  and  of  the 
same  altitude  in  the  ratio  1  :  ra+  1.  He  apparently  assumed 
that  the  same  result  would  also  be  true  for  the  curve  y  =  axm, 
where  a  is  any  constant.  In  this  result  ra  may  be  any  number 
positive  or  negative,  and  he  considered  in  particular  the  case  of 
the  parabola  in  which  ra  =  2,  and  that  of  the  hyperbola  in  which 
m  =  —  1 :  in  the  latter  case  his  interpretation  of  the  result  is 
incorrect.  He  then  shewed  that  similar  results  might  be 
written  down  for  any  curve  of  the  form  y  =  Haxm-,  so  that  if 
the  ordinate  y  of  a  curve  could  be  expanded  in  powers  of  the 
abscissa  x,  its  quadrature  could  be  determined.  Thus  he  said 
that  if  the  equation  of  a  curve  was  y  =  x°  +  xl  +  x2  +  . . .  its  area 
would  be  x  +  Jo;2  +  ^x3  +  —  He  then  applied  this  to  the  quad- 
rature of  the  curves  y  =  (l-c2}0,  y  =  (1  -  x2)1,  y  =  (l-x2)2, 
y  =  (1  —  cc2)3,  &c.  taken  between  the  limits  x  =  0  and  x  =  1 :  and 
shewed  that  the  areas  are  respectively 

1  2  8  1  6      J^n 

*i       "3'       TT>       ^T>  <BC< 

He  next  considered  curves  of  the  form  y  —  xm  and  established 
the  theorem  that  the  area  bounded  by  the  curve,  the  axis  of  x, 
and  the  ordinate  x  =  1,  is  to  the  area  of  the  rectangle  on  the 
same  base  and  of  the  same  altitude  as  m  :  m  +  1.  This  is  equi- 

fl   - 
valent  to  finding  the  value  of   I  xmdx.     He  illustrated  this  by 

J  o 
the  parabola  in  which  m  =  2.     He  stated  but  did  not  prove  the 

corresponding  result  for  a  curve  of  the  form  y  =  xp^Q. 

Wallis  shewed  great  ingenuity  in  reducing  curves  to  the 
forms  given  above,  but  as  he  was  unacquainted  with  the 
binomial  theorem  he  could  not  effect  the  quadrature  of  the 

circle,  whose  equation  is  y  =  (\-  x2y',  since  he  was  unable  to 
expand  this  in  powers  of  x.  He  laid  down  however  the  prin- 
ciple of  interpolation.  He  argued  that  as  the  ordinate  of  the 


44   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

circle  is  the  geometrical  mean  between  the  ordinates  of  the 
curves  y  =  (1  —  or2)0  and  y  =  (1  —  x2)\  so  as  an  approximation  its 
area  might  be  taken  as  the  geometrical  mean  between  1  and  §. 
This  is  equivalent  to  taking  4^/f  or  3-26...  as  the  value  of  TT. 

But,  he  continued,  we  have  in  fact  a  series  1,  f,  T8y,  |4,  

and  thus  the  term  interpolated  between  1  and  §  ought  to  be 
so  chosen  as  to  obey  the  law  of  this  series.  This  by  an 
elaborate  method  leads  to  a  value  for  the  interpolated  term 
which  is  equivalent  to  making 

2.2.4.4.6.6.8.8.. 


7T 


_ 

=  2 


1.3.3.5.5.7.7.9.. 


The  subsequent  mathematicians  of  the  seventeenth  century 
constantly  used  interpolation  to  obtain  results  which  we  should 
attempt  to  obtain  by  direct  algebraic  analysis. 

The  Arithmetica  infinitorum  was  followed  in  1656  by  a 
tract  on  the  angle  of  contact;  in  1657  by  the  Mathesis  univer- 
salis;  in  1658  by  a  correspondence  with  Fermat;  and  by  a 
long  controversy  with  Hobbes  on  the  quadrature  of  the  circle. 

In  1659  Wallis  published  a  tract  on  cycloids  in  which 
incidentally  he  explained  how  the  principles  laid  down  in  his 
Arithmetica  infinitorum  could  be  applied  to  the  rectification 
of  algebraic  curves :  and  in  the  following  year  one  of  his 
pupils,  by  name  William  Neil,  applied  the  rule  to  rectify  the 
semicubical  parabola  x3  =  ay2.  This  was  the  first  case  in  which 
the  length  of  a  curved  line  was  determined  by  mathematics, 
and  as  all  attempts  to  rectify  the  ellipse  and  hyperbola  had 
(necessarily)  been  ineffectual,  it  had  previously  been  generally 
supposed  that  110  curves  could  be  rectified. 

In  1665  Wallis  published  the  first  systematic  treatise  on 
Analytical  conic  sections.  Analytical  geometry  was  invented 
by  Descartes,  and  the  first  exposition  of  it  was  given  in  1637  : 
that  exposition  was  both  difficult  and  obscure,  and  to  most  of 
his  contemporaries,  to  whom  the  method  was  new,  it  must  have 
been  incomprehensible.  Wallis  made  the  method  intelligible 
to  all  mathematicians.  This  is  the  first  book  in  which  these 


WALLIS.  45 

curves  are  considered  and  defined  as  curves  of  the  second  degree 
and  not  as  sections  of  a  cone. 

In  1668  he  laid  down  the  principles  for  determining  the 
effects  of  the  collision  of  imperfectly  elastic  bodies.  This  was 
followed  in  1669  by  a  work  on  statics  (centres  of  gravity),  and 
in  1670  by  one  on  dynamics :  these  provide  a  convenient 
synopsis  of  what  was  then  known  on  the  subject. 

In  1686  Wallis  published  an  Algebra,  preceded  by  a  his- 
torical account  of  the  development  of  the  subject  which  contains 
a  great  deal  of  valuable  information  and  in  which  he  seems 
to  have  been  scrupulously  fair  in  trying  to  give  the  credit  of 
different  discoveries  to  their  true  originators.  This  algebra  is 
noteworthy  as  containing  the  first  systematic  use  of  formulae. 
A  given  magnitude  is  here  represented  by  the  numerical  ratio 
which  it  bears  to  the  unit  of  the  same  kind  of  magnitude  :  thus 
when  Wallis  wanted  to  compare  two  lengths  he  regarded  each 
as  containing  so  many  units  of  length.  This  will  perhaps  be 
made  clearer  if  I  say  that  the  relation  between  the  space  de- 
scribed in  any  time  by  a  particle  moving  with  a  uniform 
velocity  would  be  denoted  by  Wallis  by  the  formula  s  =  vt, 
where  s  is  the  number  representing  the  ratio  of  the  space  de- 
scribed to  the  unit  of  length;  while  previous  writers  would 
have  denoted  the  same  relation  by  stating  what  is  equivalent  to 
the  proposition  sl  :  s2  =  vltl  :  V2t2:  (see  e.g.  Newton's  Principia, 
bk.  i.  sect,  i.,  lemma  10  or  11).  It  is  curious  to  note  that 
Wallis  rejected  as  absurd  and  inconceivable  the  now  usual  idea 
of  a  negative  number  as  being  less  than  nothing,  but  accepted 
the  view  that  it  is  something  greater  than  infinity.  The  latter 
opinion  may  be  right  and  consistent  with  the  former,  but  it  is 
hardly  a  more  simple  one. 

I  have  already  stated  that  the  writings  of  Wallis  pub- 
lished between  1655  and  1665  revealed  and  explained  to  all 
students  the  principles  of  those  new  methods  which  distinguish 
modern  from  classical  mathematics.  His  reputation  has  been 
somewhat  overshadowed  by  that  of  Newton,  but  his  work  was 
absolutely  first  class  in  quality.  Under  his  influence  a  brilliant 


46   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

mathematical  school  arose  at  Oxford.  Tn  particular  I  may 
mention  Wren,  Hooke,  and  Halley  as  among  the  most  eminent 
of  his  pupils.  But  the  movement  was  shortlived,  and  there 
were  no  successors  of  equal  ability  to  take  up  their  work. 

I  come  next  to  Barrow,  the  earliest  occupant  of  the  Lucasian 
chair  at  Cambridge.  Isaac  Barrow1  was  born  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  entered  at  Trinity  in  1644,  took  his  bachelor's  degree  in 
1648,  and  was  elected  to  a  fellowship  in  1649,  at  the  same  time 
as  his  friend  John  Ray,  the  famous  botanist.  He  then  resided 
for  a  few  years  in  college,  where  he  took  pupils.  It  was  for  two 
of  them  that  he  translated  the  whole  of  Euclid's  Elements  : 
this  remained  a  standard  English  text-book  for  half  a  century 
(see  p.  84).  In  1655  he  was  driven  out  of  the  country  by 
the  persecution  of  the  Independents.  A  few  months  before,  in 
1654,  he  delivered  a  speech  from  which  I  quote  the  following 
passage  as  it  throws  some  light  on  the  study  of  mathematics  at 
Cambridge  at  that  time. 

Nempe  Euclidis,  Archimedis,  Ptolemaei,  Diophanti  horrida  olim 
nomina  jam  multi  e  vobis  non  tremulis  auribus  excipiunt.  Quid  memo- 
rein  jam  vos  didicisse,  arithmeticae  ope,  facili  et  instantanea  opera  vel 
arenarum  enormes  numeros  accurate  computare,  etiarnsi  illas  non  tantum, 
ut  fit,  maris  littoribus  adjacerent,  sed  etiam  ingenti  cumulo  quaquaversus 
ad  primum  mobile  et  extremas  Mundi  oras  pertingerent :  rem  vulgo 
miram  et  arduam  creditu,  at  vobis  effectu  facilem  et  expeditam?  Quid, 
quando  Geometries  subsidio,  non  solum  terrarum  longe  dissitos  tractus, 
sed  et  patentissimas  Cceli  regiones  emetiri  nostis,  interim  ipsi  quietem 
agentes,  nee  loco  omnino  cedentes,  ad  praelongas  regulas  catenasve  im- 
menso  spatio  applicandas  ?  Quid  referam  alios,  sublimibus  alis  ingenii 

1  A  very  appreciative  account  of  the  academical  life  and  surroundings 
of  Barrow  by  W.  Whewell  is  prefixed  to  vol.  ix.  of  A.  Napier's  edition  of 
Barrow's  works,  Cambridge  1859.  Another  account  of  his  life  is  given  in 
the  Lives  of  the  professors  of  Gresham  College  by  J.  Ward,  London,  1740. 
Barrow's  lectures  were  edited  by  W.  Whewell,  Cambridge,  1860. 


BARROW.  47 

supremum  sethera  consceudentes,  astrorum  vestigiis  presse  inhserere, 
paratos  districtirn  dicere,  quam  inagna,  et  quam  alta  sunt ;  quantum  sui 
circuli,  et  quo  tempore  conficiant,  et  qualem  orbitam  describant,  quasi 
non  cum  nobis  in  hisce  terris,  sed  cum  superis  in  palatio  Dei  omnipo- 
tentis  aetatem  transigerent  ?  Sane  de  horribili  monstro,  quod  Algebram 
nuncupant,  domito  et  profligate  multi  e  vobis  fortes  viri  triumpharunt : 
permulti  ausi  sunt  Opticem  directo  obtutu  inspicere;  alii  subtiliorem 
Dioptrices  et  utilissimam  doctrinam  irrefracto  ingenii  radio  penetrare. 
Nee  vobis  hodie  adeo  mirabile  est,  Catoptrices  principia  et  leges  Mecha- 
nicaa  non  ignorantibus,  quo  artificio  magnus  Archimedes  Romanas  naves 
comburere  potuit,  nee  a  tot  seculis  immobilem  Vestam  quomodo  stantem 
terrain  concutere  potuisset. 

Barrow  returned  to  England  in  1659,  and  in  the  following 
year  he  was  ordained  and  appointed  to  the  professorship  of 
Greek  at  Cambridge;  in  1662  he  was  also  made  professor  of 
geometry  at  Gresham  College.  In  the  same  year  a  chair  of 
mathematics  was  founded  at  Cambridge  under  the  will  of 
Henry  Lucas,  of  St  John's  College,  one  of  the  members  of 
parliament  for  the  university,  and  Barrow  was  selected  as  the 
first  occupant1  of  it. 

His  lectures,  delivered  in  1664,  1665,  and  1666,  were  pub- 
lished in  1683  under  the  title  Lectiones  mathematicae :  these  are 
mostly  on  the  metaphysical  basis  for  mathematical  truths.  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  Lectiones  opticae  et  geometricae,  which 
is  his  most  important  work.  In  the  part  on  optics  many 

1  The  successive  professors  were  as  follows.  From  1664  to  1669, 
Isaac  Barrow  of  Trinity;  from  1669  to  1702,  Sir  Isaac  Newton  of 
Trinity  (see  chapter  IV.) ;  from  1702  to  1711,  William  Whiston  of  Clare  (see 
p.  83 ) ;  from  1711  to  1739,  Nicholas  Saunderson  of  Christ's  (see  p.  86) ; 
from  1739  to  1760,  John  Colson  of  Emmanuel  (see  p.  100)  ;  from  1760  to 
1798,  Edward  Waring  of  Magdalene  (see  p.  101);  from  1798  to  1820, 
Isaac  Milner  of  Queens'  (see  p.  102) ;  from  1820  to  1822,  Robert  Woodhouse 
of  Caius  (see  p.  118) ;  from  1822  to  1826,  Thomas  Turton  of  St  Catharine's 
(see  p.  118  n.) ;  from  1826  to  1828,  Sir  George  Biddell  Airy  of  Trinity  (see 
p.  132);  from  1828  to  1839,  Charles  Babbage  of  Trinity  (see  p.  125); 
from  1839  to  1849,  Joshua  King  of  Queens'  (see  p.  132);  who  was 
succeeded  by  the  present  professor,  G.  G.  Stokes  of  Pembroke. 


48      THE   COMMENCEMENT   OF   MODERN   MATHEMATICS. 

problems  connected  with  the  reflexion  and  refraction  of  light 
are  treated  with  great  ingenuity.  The  geometrical  focus  of  a 
point  seen  by  reflexion  or  refraction  is  defined  ;  and  it  is 
explained  that  the  image  of  an  object  is  the  locus  of  the 
geometrical  foci  of  every  point  on  it.  A  few  of  the  easier  pro- 
perties of  thin  lenses  are  also  worked  out,  and  the  Cartesian  ex- 
planation of  the  rainbow  is  simplified.  The  geometrical  lectures 
contain  some  new  ways  of  determining  the  areas  and  tangents 
of  curves.  The  latter  is  solved  by  a  rule  exactly  analogous  to 
the  procedure  of  the  differential  calculus,  except  that  a  separate 
determination  of  what  is  really  a  differential  coefficient  had  to 
be  made  for  every  curve  to  which  it  was  applied.  Thus  he  took 
the  equation  of  the  curve  between  the  coordinates  x  and  y1, 
gave  x  a  very  small  decrement  e  and  found  the  consequent 
decrement  of  y,  which  he  represented  by  a.  The  limit  of  the 
ratio  a/e  when  the  squares  of  a  and  e  were  neglected  was 
defined  as  the  angular  coefficient  of  the  tangent  at  the  point, 
and  completely  determined  the  tangent  there. 

Barrow's  lectures  failed  to  attract  any  considerable  audi- 
ences, and  on  that  account  he  felt  conscientious  scruples  about 
retaining  his  chair.  Accordingly  in  1669  he  resigned  it  to  his 
pupil  Newton,  whose  abilities  he  had  been  one  of  the  earliest 
to  detect  and  encourage.  For  the  remainder  of  his  life  Barrow 
devoted  most  of  his  time  to  the  study  of  divinity.  In  1675  he 
issued  an  edition  in  one  volume  of  the  works  of  Archimedes, 
the  first  four  books  of  the  Conies  of  Apollonius,  and  the  treatise 
of  Theodosius  on  the  sphere.  He  was  appointed  master  of 
Trinity  College  in  1672,  and  died  in  1677. 

He  is  described  as  "low  in  stature,  lean,  and  of  a  pale 
complexion,"  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  prowess 
from  capture  by  pirates.  A  ready  and  caustic  wit  made  him  a 

1  He  actually  denotes  the  coordinates  byp  and  m,  but  I  alter  them  to 
agree  with  the  modern  practice.  For  further  details  of  his  procedure  see 
pp.  269 — 70  of  my  History  of  mathematics,  London,  1888. 


DACRES.   TOOKE.   MORLAND.  49 

favorite  of  Charles  II.,  and  induced  the  courtiers  to  respect 
even  if  they  did  not  appreciate  him.  He  wrote  with  a  sustained 
and  somewhat  stately  eloquence,  and  with  his  blameless  life  and 
scrupulous  conscientiousness  was  one  of  the  most  impressive 
characters  of  the  time. 


Before  proceeding  to  Newton,  who  succeeded  Barrow  in  the 
Lucasian  chair  and  whose  writings  profoundly  modified  the 
subsequent  development  not  only  of  the  Cambridge  school 
of  mathematics  but  of  the  university  system  of  education,  I 
will  mention  three  mathematicians  of  no  great  note  whose 
works  or  teaching  belong  to  the  pre-newtonian  age.  These  are 
Dacres,  Tooke,  and  Morland. 

Arthur  Dacres,  a  fellow  of  Magdalene,  was  born  in  1624, 
and  proceeded  B.A.  in  1645.  He  then  studied  medicine  and 
settled  in  London,  where  he  occupied  a  leading  position.  He 
however  kept  up  his  acquaintance  with  mathematics,  and  in 
1664  was  appointed  professor  of  geometry  at  Gresham  College 
in  succession  to  Barrow.  Dacres  died  in  1678. 

Dacres  was  succeeded  in  his  chair  by  Robert  Hooke,  and 
after  the  death  of  the  latter  in  1704  the  chair  was  offered  to 
Andrew  Tooke.  Tooke  was  born  in  London  in  1673,  took  his 
B.A.  degree  from  Clare  in  1693,  and  died  in  1731.  He  held 
the  professorship  until  1729,  but  with  the  beginning  of  the 
eighteenth  century  an  appointment  at  Gresham  College  ceases 
to  be  a  mark  of  scientific  distinction. 

The  last  of  this  trio  was  Sir  Samuel  Morland.  Morland 
was  born  in  Berkshire  in  1625,  and  was  educated  at  Win- 
chester School  and  Magdalene  College,  but  though  he  resided 
ten  years  at  Cambridge  he  did  not  proceed  to  a  degree.  He 
took  a  prominent  part  in  politics,  and  like  most  of  his 
university  contemporaries  was  a  constitutional  royalist.  On 
the  restoration  he  was  made  master  of  mechanics  to  the  king, 
and  thenceforward  lived  in  or  near  London  till  his  death  on 
Jan.  6,  1696. 

B.  4 


50   THE  COMMENCEMENT  OF  MODERN  MATHEMATICS. 

His  earliest  work  on  the  quadrature  of  curves,  partly- 
printed  in  1666,  was  at  Pell's  request  withdrawn  from  publi- 
cation— why,  I  do  not  know.  In  the  same  year  he  invented 
an  admirable  little  arithmetical  machine,  an  account  of  which 
was  published  in  1673.  Morland  seems  subsequently  to  have 
turned  his  attention  to  the  construction  of  machines.  The 
speaking  tube  is  one  of  his  inventions  :  one  of  the  first  made 
was  presented  in  1671  to  the  library  of  Trinity  College,  and 
is  still  there.  The  form  and  construction  of  capstans,  fire- 
engines,  and  certain  other  pumps  were  greatly  improved  by 
him,  and  the  use  of  the  barometer  as  a  weather-gauge  seems  to 
be  due  to  his  advocacy.  Some  tables  of  interest,  discount,  and 
square  and  cube  roots  were  also  published  by  him  at  different 
dates  after  1679. 


CHAPTER   IV. 
THE   LIFE   AND   WORKS   OF   NEWTON. 

THE  second  occupant  of  the  Lucasian  chair  was  Newton. 
There  is  hardly  a  branch  of  modern  mathematics,  which  cannot 
be  traced  back  to  him,  and  of  which  he  did  not  revolutionize 
the  treatment;  and  in  the  opinion  of  the  greatest  mathema- 
ticians of  subsequent  times — Lagrange,  Laplace,  and  Gauss — 
his  genius  stands  out  without  an  equal  in  the  whole  history 
of  mathematics.  It  will  therefore  be  readily  imagined  how 
powerfully  he  must  have  impressed  his  methods  and  philosophy 
on  the  school  which  he  suddenly  raised  to  be  the  first  in 
Europe ;  and  the  subsequent  history  of  Cambridge  (as  far  as 
this  work  is  concerned  therewith)  is  mainly  that  of  the 
Newtonian  philosophy. 

Isaac  Newton1  was  born  in  Lincolnshire  near  Grantham  on 
Dec.  25,  1642  (O.  S.),  and  died  at  Kensington,  London,  on 
March  20,  1727.  He  went  to  school  at  Grantham,  and  in 
1661  came  up  as  a  subsizar  to  Trinity.  Luckily  he  kept  a 
diary,  and  we  can  thus  form  a  fair  idea  of  the  reading  of  the 
best  men  at  that  time.  He  had  not  read  any  mathematics  before 
€oming  into  residence,  but  was  acquainted  with  Sanderson's 
Logic,  which  was  then  frequently  read  as  preliminary  to 

1  The  account  in  the  text  is  condensed  from  chapter  xvi.  of  my 
History  of  mathematics,  London,  1888,  to  which  I  would  refer  the  reader 
for  authorities  and  fuller  particulars.  An  edition  of  Newton's  works  was 
published  by  S.  Horsley  in  5  volumes,  London,  1779 — 85 :  this  contains 
a  full  bibliography  of  his  writings. 

4—2 


52  THE   LIFE   AND   WORKS   OF   NEWTON. 

mathematics.  At  the  beginning  of  his  first  October  term  he 
happened  to  stroll  down  to  Stourbridge  Fair,  and  there  picked 
up  a  book  on  astrology,  but  could  not  understand  it  on  account 
of  the  geometry  and  trigonometry.  He  therefore  bought  a 
Euclid,  and  was  surprised  to  find  how  obvious  the  propositions 
seemed.  He  thereupon  read  Oughtred's  Clams  and  Descartes' s 
Geometry ',  the  latter  of  which  he  managed  to  master  by  himself 
though  with  some  difficulty.  The  interest  he  felt  in  the  subject 
led  him  to  take  up  mathematics  rather  than  chemistry  as  a 
serious  study.  His  subsequent  mathematical  reading  as  an 
undergraduate  was  founded  on  Kepler's  Optics,  the  works  of 
Vieta,  Schooten's  Miscellanies,  Descartes' s  Geometry,  and 
Wallis's  Arithmetica  infinitorum :  he  also  attended  Barrow's 
lectures.  At  a  later  time  on  reading  Euclid  more  carefully  he 
formed  a  very  high  opinion  of  it  as  an  instrument  of  education, 
and  he  often  expressed  his  regret  that  he  had  not  applied  himself 
to  geometry  before  proceeding  to  algebraic  analysis.  He  made 
some  optical  experiments  and  observations  on  lunar  halos 
while  an  undergraduate.  He  was  elected  to  a  scholarship  in 
1663. 

He  took  his  B.A.  degree  in  1664.  There  is  a  manuscript 
of  his  written  in  the  following  year,  and  dated  May  28,  1665, 
which  is  the  earliest  documentary  proof  of  his  discovery  of 
fluxions.  It  was  about  the  same  time  that  he  discovered  the 
binomial  theorem. 

On  account  of  the  plague  the  college  was  sent  down  in  the 
summer  of  1665,  and  for  the  next  year  and  a  half  Newton 
lived  at  home.  This  period  was  crowded  with  brilliant  dis- 
coveries. He  thought  out  the  fundamental  principles  of  his 
theory  of  gravitation,  namely  that  every  particle  of  matter 
attracts  every  other  particle,  and  he  suspected  that  the  attrac- 
tion varied  as  the  product  of  their  masses  and  inversely  as  the 
square  of  the  distance  between  them.  He  also  worked  out  the 
fluxional  calculus  tolerably  completely:  thus  in  a  manuscript 
dated  Nov.  13  of  the  same  year  he  used  fluxions  to  find  the 
tangent  and  the  radius  of  curvature  at  any  point  on  a  curve, 


NEWTON'S  DISCOVERIES,  1666 — 71.  53 

and  in  October  1666  he  applied  them  to  several  problems  in  the 
theory  of  equations.  Newton  communicated  the  results  to  his 
friends  and  pupils  from  and  after  1669,  but  they  were  not 
published  in  print  till  many  years  later.  It  was  also  while 
staying  at  home  at  this  time  that  he  devised  some  instruments 
for  grinding  lenses  to  particular  forms  other  than  spherical, 
he  perhaps  decomposed  light,  and  he  certainly  devoted  con- 
siderable time  to  astrology  and  alchemy;  indeed  he  never 
abandoned  the  idea  of  transmuting  base  metals  into  gold. 

On  his  return  to  Cambridge  in  1667  Newton  was  elected  to 
a  fellowship,  and  in  1668  took  his  M.A.  degree.  It  is  probable 
that  he  took  pupils.  His  note-books  shew  that  his  attention 
was  now  mostly  occupied  with  chemistry  and  optics,  though 
there  are  a  good  many  problems  in  pure  and  analytical  geometry 
scattered  amongst  them. 

During  the  next  two  years  he  revised  and  edited  Barrow's 
Lectures,  edited  and  added  to  Kinckhuy sen's  Algebra,  and  by 
using  infinite  series  greatly  extended  the  power  of  the  method 
of  quadratures  given  by  Wallis.  These  however  were  only  the 
fruits  of  his  leisure  ;  most  of  his  time  during  these  years  being 
given  up  to  optical  researches. 

In  October  1669  Barrow  had  resigned  the  Lucasian  chair 
in  favour  of  Newton.  Newton  chose  optics  for  the  subject  of 
his  lectures  and  researches,  and  before  the  end  of  the  year  he 
had  worked  out  the  details  of  his  discovery  of  the  decomposition 
of  a  ray  of  white  light  into  rays  of  different  colours,  which 
was  effected  by  means  of  a  prism  bought  at  Stourbridge 
Fair.  The  complete  explanation  of  the  theory  of  the  rainbow 
followed  from  this  discovery.  These  discoveries  formed  the 
subject-matter  of  the  lectures  which  he  delivered  as  Lucasian 
professor  in  the  years  1669,  1670,  and  1671.  The  chief  new 
results  were  embodied  in  papers  published  in  the  Philosophical 
trarisactiotis  from  1671  to  1676.  The  manuscript  of  his  original 
lectures  was  printed  in  1729  under  the  title  Lectiones  opticae. 
This  work  is  divided  into  two  books,  the  first  of  which  contains 
four  sections  and  the  second  five.  The  first  section  of  the  first 


54  THE   LIFE   AND   WORKS   OF   NEWTON. 

book  deals  with  the  decomposition  of  solar  light  by  a  prism  in 
consequence  of  the  unequal  refrangibility  of  the  rays  that  com- 
pose it,  and  gives  a  full  account  of  his  experiments.  The  second 
section  contains  an  account  of  the  method  which  Newton  in- 
vented for  determining  the  coefficients  of  refraction  of  different 
bodies.  This  is  done  by  making  a  ray  pass  through  a  prism  of 
the  material  so  that  the  angle  of  incidence  is  equal  to  the  angle 
of  emergence  :  he  shews  that  if  the  angle  of  the  prism  be  i  and 
the  total  deviation  of  the  ray  be  8  the  refractive  index  will  be 
sin  ^(i  +  8)  cosec  ^i.  The  third  section  is  on  refractions  at  plane 
surfaces.  Most  of  this  section  is  devoted  to  geometrical  solu- 
tions of  different  problems,  many  of  which  are  very  difficult. 
He  here  finds  the  condition  that  a  ray  may  pass  through  a 
prism  with  minimum  deviation.  The  fourth  section  treats  of 
refractions  at  curved  surfaces.  The  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  be  achromatic,  and  instead 
designed  a  reflecting  telescope,  probably  on  the  model  of  a 
small  one  which  he  had  made  in  1668.  The  form  he  invented 
is  that  still  known  by  his  name.  In  1672  he  invented  a  re- 
flecting microscope. 

In  1675  he  set  himself  to  examine  the  problem  as  to  how 
light  was  really  produced.  By  the  close  of  the  year  he  had 
worked  out  the  corpuscular  or  emission  theory.  Only  three 
ways  have  been  suggested  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  perceived  may  send  out  something 
which  hits  or  affects  the  eye  (as  Newton  supposed) ;  or  there 
may  be  some  medium  between  the  eye  and  the  object,  and  the 
object  may  cause  some  change  in  the  form  or  nature  of  this 
intervening  medium  and  thus  affect  the  eye  (as  Huygens  sug- 
gested in  the  wave  or  undulatory  theory).  It  will  be  enough 


NEWTON'S  THEORY  OF  OPTICS.  55 

here  to  say  that  on  either  of  the  two  latter  theories  all  the 
obvious  phenomena  of  geometrical  optics  such  as  reflexion, 
refraction,  &c.  can  be  accounted  for.  Within  the  present 
century  crucial  experiments  have  been  devised  which  give 
different  results  according  as  one  or  the  other  theory  is  adopted; 
all  these  experiments  agree  with  the  results  of  the  undulatory 
theory  and  differ  from  the  results  of  the  Newtonian  theory  : 
the  latter  is  therefore  untenable,  but  whether  the  former  repre- 
sents the  whole  truth  and  nothing  but  the  truth  is  still  an  open 
question.  Until  however  the  theory  of  interference  was  worked 
out  by  Young  the  hypothesis  of  Huygens  failed  to  account  for 
all  the  facts  and  was  open  to  more  objections  than  that  of 
Newton.  Although  Newton  did  not  believe  that  the  wave 
theory  was  the  true  explanation,  he  subsequently  elaborated 
the  fundamental  principles  of  it. 

His  theory  was  embodied  in  two  papers  which  were  com- 
municated to  the  Royal  Society  on  Dec.  9  and  Dec.  16  of  1672. 
In  another  paper  on  physical  optics  which  was  written  in 
1687  he  elaborated  the  theory  of  fits  of  easy  reflexion  and 
transmission,  the  inflexion  of  light  (bk.  n.  part  1),  and  the 
colours  of  thick  plates  (bk.  n.  part  4).  The  three  papers  to- 
gether contain  the  whole  of  his  emission  theory  of  light,  and 
comprise  the  great  bulk  of  his  treatise  on  optics  published  in 
1704,  to  which  the  references  given  immediately  above  refer. 

In  1673  he  had  written  an  account  of  his  method  of 
quadrature  by  means  of  infinite  series  in  letters  to  Collins  or 
Oldenburg;  and  in  1676  in  answer  to  a  request  from  Leibnitz 
he  gave  him  a  very  brief  account  of  his  method  and  added 
the  expansions  of  a  binomial  (i.e.  the  binomial  theorem)  and 
of  sin"1  x ;  from  the  latter  of  which  he  deduced  that  of  sin  x. 
He  also  added  an  expression  for  the  rectification  of  an  elliptic 
arc  in  an  infinite  series. 

Leibnitz  wrote  on  Aug.  27,  1676,  asking  for  fuller  details, 
and  on  Oct.  24  Newton  replied  in  a  long  but  very  interesting 
paper  in  which  he  gives  an  account  of  the  way  in  which  he  had 
been  led  to  some  of  his  results. 


56  THE   LIFE   AND  WORKS  OF  NEWTON. 

He  begins  by  saying  that  altogether  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  the  circle  and  hyperbola. 
Thus,  by  considering  the  series  of  expressions 


he  deduced  by  interpolations  the  law  which  connects  the  suc- 
cessive coefficients  in  the  expansions  of 


He  then  by  analogy  obtained  the  expression  for  the  general 
term  in  the  expansion  of  a  binomial,  i.e.  the  binomial  theorem. 
He  says  that  he  proceeded  to  test  this  by  forming  the  square 

of  the  expansion  of  (1  -  x2)^  which  reduced  to  1  —  or  ;  and  he 
proceeded  in  a  similar  way  with  other  expansions.  He  next 

tested  the  theorem  in  the  case  of  (1  -  x2)2  by  extracting  the 
square  root  of  1  —  x2  more  arithmetico.  He  also  used  the  series 
to  determine  the  areas  of  the  circle  and  hyperbola  in  infinite 
series  and  found  that  they  were  the  same  as  the  results  he  had 
arrived  at  by  other  means. 

Having  established  this  result  he  then  discarded  the  method 
of  interpolation,  and  employed  his  binomial  theorem  as  the 
most  direct  method  of  obtaining  the  areas  and  arcs  of  curves. 
Newton  styled  this  his  second  method  and  it  is  the  basis  of  his 
work  on  analysis  by  infinite  series.  He  states  that  he  had 
discovered  it  before  the  plague  in  1665-66. 

Newton  then  proceeds  to  state  that  he  had  also  a  third 
method;  of  which  (he  says)  he  had  about  1669  sent  an  account 
to  Barrow  and  Collins,  illustrated  by  applications  to  areas, 
rectification,  cubature,  &c.  This  was  the  method  of  fluxions  ; 
but  Newton  gave  no  detailed  description  of  it  in  this  letter, 
probably  because  he  thought  that  Leibnitz  could,  if  he  wished, 
obtain  from  Collins  the  explanation  of  it  alluded  to  above. 
Newton  added  an  anagram  which  described  the  method  but 


THE   LETTER   TO   LEIBNITZ.  57 

which  is  unintelligible  to  any  one  to  whom  the  key  is  not 
given.  He  gives  however  some  illustrations  of  its  use.  The 
first  is  on  the  quadrature  of  the  curves  represented  by 

y  =  axm  (b  +  cxn)p, 

which  he  says  can  be  determined  as  a  sum  of  (m  +  l)/n  terms 
if  (m  +  l)/n  be  a  positive  integer,  and  which  he  thinks  cannot 
otherwise  be  effected  except  by  an  infinite  series.  [This  is  not 
so,  the  integration  is  possible  if  p  +  (m+l)/n  be  an  integer.] 
He  also  gives  a  long  list  of  other  forms  which  are  immediately 
integrable,  of  which  the  chief  are 

xmn-l  a.mt+i)*-! 


a  +  bxn  +  cy?n  '    a  +  bxn  +  ex** 


xmn~l  (a  +  bxn)       (c  + 
and  x(m-i}n-i 


where  m  is  a  positive  integer  and  n  is  any  number  whatever. 

At  the  end  of  his  letter  Newton  alludes  to  the  solution  of 
the  "  inverse  problem  of  tangents,"  a  subject  on  which  Leibnitz 
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 
will  not  describe  except  by  an  anagram  which  being  read  is 
as  follows,  "  Una  methodus  consistit  in  extractione  fluentis 
quantitatis  ex  aequatione  simul  involveute  fluxionem  ejus. 
Altera  tantum  in  assumptione  seriei  pro  quautitate  qualibet 
incognita  ex  qua  caetera  commode  derivari  possunt,  et  in 
collatione  terminorum  homologorum  sequationis  resultantis,  ad 
eruendos  terminos  assumptae  seriei." 

He  adds  in  this  letter  that  he  is  worried  by  the  questions 
he  is  asked  and  the  controversies  raised  about  every  new 
matter  which  he  publishes,  and  he  regrets  that  he  has  allowed 


58  THE   LIFE   AND   WORKS   OF   NEWTON. 

his  repose  to  be  interrupted  by  running  after  shadows ;  and 
he  implies  that  for  the  future  he  will  publish  nothing.  As  a 
matter  of  fact  he  did  refuse  to  allow  any  account  of  his  method 
of  fluxions  to  be  published  till  the  year  1693. 

Leibnitz  did  not  reply  to  this  letter  till  June  21,  1677.  In 
his  answer  he  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  infinitesimal  differences  between  the  co- 
ordinates 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  he  could  integrate. 

I  do  not  know  with  any  certainty  on  what  subjects  Newton 
was  chiefly  occupied  during  the  next  eight  years,  1676 — 1684. 
He  was  partly  engaged  in  chemical  experiments  and  partly  in 
geological  speculations  ;  and  I  believe  that  his  experiments  in 
.electricity  and  magnetism  and  the  law  of  cooling  in  the  theory 
of  heat  are  of  this  date.  A  large  part  of  the  geometry  and 
the  pure  mathematics  subsequently  incorporated  in  the  first 
book  of  the  Principia  should  probably  be  also  referred  to  this 
time ;  and  perhaps  some  parts  of  the  essay  on  cubic  curves. 

It  is  almost  certain  that  the  Universal  arithmetic  which  is 
on  algebra,  theory  of  equations,  and  miscellaneous  problems  con- 
tains the  substance  of  Newton's  lectures  during  these  years. 
His  manuscript  of  it  is  still  extant.  Amongst  several  new  theo- 
rems on  various  points  in  algebra  and  the  theory  of  equations 
the  following  important  results  were  here  first  enunciated.  He 
explained  that  the  equation  whose  roots  are  the  solution  of  a 
given  problem  will  have  as  many  roots  as  there  are  different 
possible  cases ;  and  he  also  considered  how  it  happened  that 
the  equation  to  which  a  problem  led  might  contain  roots  which 
did  not  satisfy  the  original  question.  He  extended  Descartes's 
rule  of  signs  to  give  limits  to  the  number  of  imaginary 
roots.  He  used  the  principle  of  continuity  to  explain  how 
two  real  and  unequal  roots  might  become  imaginary  in  passing 
through  equality,  and  illustrated  this  by  geometrical  considera- 


NEWTON'S  THEORY  OF  GRAVITATION.  59 

tions ;  thence  he  shewed  that  imaginary  roots  must  occur  in 
pairs.  Newton  also  here  gave  rules  to  find  a  superior  limit  to 
the  positive  roots  of  a  numerical  equation,  and  to  determine 
the  approximate  values  of  the  numerical  roots.  He  further 
enunciated  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. 

In  August  1684  Newton  received  a  visit  from  Halley  who 
drew  his  attention  to  the  motion  of  the  moon.  Hooke,  Huygens, 
Halley,  and  Wren  had  all  conjectured  that  the  force  of  the 
attraction  of  the  sun  or  earth  on  an  external  particle  varied 
inversely  as  the  square  of  the  distance.  These  writers  seem  to 
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,  but  they 
could  not  deduce  from  the  law  the  orbits  of  the  planets.  When 
Halley  visited  Cambridge  in  August  1684  he  explained  that 
their  investigations  were  stopped  by  their  inability  to  solve 
this  problem,  and  asked  Newton  if  he  could  find  out  what  the 
orbit  of  a  planet  would  be  if  the  law  of  attraction  were  that  of 
the  inverse  square.  Newton  immediately  replied  that  it  was 
an  ellipse,  and  promised  to  send  or  write  out  afresh  a  demon- 
stration of  it  which  he  had  given  in  1679.  This  was  sent  in 
November  1684. 

Instigated  by  this  question,  Newton  now  attacked  the 
whole  problem  of  gravitation,  and  succeeded  in  shewing  that  if 
the  distances  of  the  members  of  the  solar  system  were  so  great 
that  they  might  for  the  purpose  of  their  mutual  attraction  be 
regarded  as  points  then  their  motions  were  in  accordance  with 
the  law  of  gravitation.  The  elements  of  these  discoveries  were 
put  together  in  the  tract  called  De  motu,  which  contains  the 
substance  of  sections  ii.  and  iii.  of  the  first  book  of  the  Principia, 
and  was  read  by  Newton  for  his  lectures  in  the  Michaelmas 
term  1684. 

Newton  however  had  not  yet  determined  the  attraction  of 


60  THE   LIFE   AND   WORKS   OF   NEWTON. 

a  spherical  body  on  any  external  point,  nor  had  he  calculated 
the  details  of  the  planetary  motions  even  if  the  members  of  the 
solar  system  could  be  regarded  as  points.  The  first  problem 
was  solved  at  the  latest  in  February  1685.  Till  he  had  effected 
this  his  theory  had  been  shewn  to  be  true  only  in  so  far  as  the 
sun  can  be  regarded  as  a  point  compared  with  its  distance  from 
the  planets,  or  the  earth  as  a  point  compared  with  its  distance 
from  the  moon;  but  this  discovery  shewed  that  it  was  mathe- 
matically true,  excepting  only  for  the  slight  deviation  from  a 
perfectly  spherical  form  of  the  sun,  earth  and  planets.  It  was 
thus  now  in  his  power  to  apply  mathematical  analysis  with 
absolute  precision  to  the  explanation  of  the  detailed  phenomena 
of  the  solar  system.  This  he  did  in  the  almost  incredibly  short 
space  of  time  from  March  1686  to  the  end  of  March  1687,  and 
the  result  is  embodied  in  the  Principles1 .  Of  the  three  funda- 
mental principles  there  applied  we  may  say  that  the  idea  that 
every  particle  attracts  every  other  particle  in  the  universe  was 
formed  at  least  as  early  as  1666  ;  the  law  of  equable  description 
of  areas,  its  consequences,  and  the  fact  that  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  were  proved  in  1679 ; 
and  lastly  the  discovery  that  a  sphere,  whose  density  at  any 
point  depends  only  on  the  distance  from  the  centre,  attracts  an 
external  point  as  if  the  whole  mass  were  collected  at  its  centre 
was  made  in  1685.  It  was  this  last  discovery  that  enabled 
him  to  apply  the  first  two  principles  to  the  phenomena  of 
bodies  of  finite  size. 

The  first  book  of  the  Principia  was  finished  on  April  28, 
1686.  This  book  is  given  up  to  the  consideration  of  the 
motion  of  particles  or  bodies  in  free  space  either  in  known 
orbits,  or  under  the  action  of  known  forces,  or  under  their 
mutual  attraction.  In  it  Newton  generalizes  the  law  of  attrac- 
tion into  a  statement  that  every  particle  of  matter  in  the 

1  A  brief  analysis  of  the  subject-matter  of  the  Principia  is  given  on 
pp.  310 — 21  of  my  History  of  mathematics,  London,  1888. 


THE   PRINCIPLE  61 

universe  attracts  every  other  particle  with  a  force  which  varies 
directly  as  the  product  of  their  masses  and  inversely  as  the 
square  of  the  distance  between  them ;  and  he  thence  deduces 
the  law  of  attraction  for  spherical  shells  of  constant  density. 
The  book  is  prefaced  by  an  introduction  on  the  science  of 
dynamics. 

In  another  three  months,  that  is  by  the  summer  of  1686, 
he  had  finished  the  second  book  of  the  Principia.  This  book 
treats  of  motion  in  a  resisting  medium,  and  of  hydrostatics  and 
hydrodynamics,  with  special  applications  to  waves,  tides,  and 
acoustics.  He  concludes  it  by  shewing  that  the  Cartesian 
theory  of  vortices  was  inconsistent  both  with  the  known  facts 
and  with  the  laws  of  motion. 

The  next  nine  or  ten  months  were  devoted  to  the  third 
book.  For  this  he  probably  had  no  materials  ready.  In  it 
the  theorems  obtained  in  the  first  book  are  applied  to  the  chief 
phenomena  of  the  solar  system,  the  masses  and  distances  of  the 
planets  and  (whenever  sufficient  data  existed)  of  their  satellites 
are  determined.  In  particular  the  motion  of  the  moon,  the  various 
inequalities  therein,  and  the  theory  of  the  tides  are  worked 
out  in  great  detail.  He  also  investigates  the  theory  of  comets, 
shews  that  they  belong  to  the  solar  system,  explains  how  from 
three  observations  the  orbit  can  be  determined,  and  illustrates 
his  results  by  considering  certain  special  comets.  The  third 
book  as  we  have  it  is  but  little  more  than  a  sketch  of  what 
Newton  had  proposed  to  himself  to  accomplish.  The  original 
programme  of  the  work  is  extant  and  his  note-books  shew  that 
he  continued  to  work  at  it  for  some  years  after  the  publication 
of  the  first  edition  of  the  Principia. 

The  printing  of  the  work  was  very  slow  and  it  was  not 
finally  published  till  the  summer  of  1687.  The  conciseness, 
absence  of  illustrations,  and  synthetical  character  of  the  book  as 
first  issued  seriously  restricted  the  numbers  of  those  who  were 
able  to  appreciate  its  value ;  and  though  nearly  all  competent 
critics  admitted  the  validity  of  the  conclusions  a  considerable 
time  elapsed  before  it  affected  the  current  beliefs  of  educated 


62  THE   LIFE   AND   WORKS   OF   NEWTON. 

men.  I  should  be  inclined  to  say  (but  on  this  point  opinions 
differ  widely)  that  within  ten  years  of  its  publication  it  was 
generally  accepted  in  Britain  as  giving  a  correct  account  of 
the  laws  of  the  universe  ;  it  was  similarly  accepted  within 
about  twenty  years  on  the  continent,  except  in  France  where 
patriotism  was  urged  in  defence  of  the  Cartesian  theory  until 
Voltaire  in  1738  took  up  the  advocacy  of  the  Newtonian 
theory. 

The  manuscript  of  the  Prindpia  was  finished  by  1686. 
Newton  devoted  the  remainder  of  that  year  to  his  paper  on 
physical  optics,  the  greater  part  of  which  is  given  up  to  the 
subject  of  diffraction  (see  p.  55). 

In  1687  James  II.  having  tried  to  force  the  university  to 
admit  as  a  master  of  arts  a  Roman  Catholic  priest  who  refused 
to  take  the  oaths  of  supremacy  and  allegiance,  Newton  took 
a  prominent  part  in  resisting  the  illegal  interference  of  the 
king,  and  was  one  of  the  deputation  sent  to  London  to  protect 
the  rights  of  the  university.  The  active  part  taken  by 
Newton  in  this  affair  led  to  his  being  in  1689  elected  member 
for  the  university.  This  parliament  only  lasted  thirteen  months, 
and  on  its  dissolution  he  gave  up  his  seat.  At  a  later  date 
he  was  returned  on  one  or  two  occasions,  but  he  never  took 
any  prominent  part  in  politics. 

On  his  coming  back  to  Cambridge  in  1690  he  resumed  his 
mathematical  studies  and  correspondence.  If  he  lectured  at 
this  time  (which  is  doubtful)  it  was  on  the  subject-matter  of 
the  Prindpia.  The  two  letters  to  Wallis  in  which  he  explained 
his  method  of  fluxions  and  fluents  were  written  in  1692,  and 
were  published  in  1693.  Towards  the  close  of  1692  and 
throughout  the  two  following  years  Newton  had  a  long  illness, 
suffering  from  insomnia  and  general  nervous  irritability.  He 
never  quite  regained  his  elasticity  of  mind,  and  though  after 
his  recovery  he  shewed  the  same  power  in  solving  any  question 
propounded  to  him,  he  ceased  thenceforward  to  do  original 
work  on  his  own  initiative,  and  it  was  difficult  to  stir  him 
to  activity. 


THE   LIFE   AND   WORKS   OF   NEWTON.  63 

In  1694  Newton  began  to  collect  data  connected  with 
the  irregularities  of  the  moon's  motion  with  the  view  of  re- 
vising the  part  of  the  Principia  which  dealt  with  that  subject. 
To  render  the  observations  more  accurate  he  forwarded  to 
Flamsteed  a  table  which  he  had  previously  made  of  correc- 
tions for  refraction.  This  was  not  published  till  1721  when 
Halley  communicated  it  to  the  Royal  Society.  The  original 
calculations  of  Newton  and  the  papers  connected  with  it  are 
in  the  Portsmouth  collection  at  Cambridge,  and  shew  that 
Newton  obtained  it  by  finding  the  path  of  a  ray  by  means 
of  quadratures  in  a  manner  equivalent  to  the  solution  of  a 
differential  equation.  As  an  illustration  of  Newton's  genius 
I  may  mention  that  even  as  late  as  1754  Euler  failed  to  solve 
the  same  problem.  In  1782  Laplace  gave  a  rule  for  con- 
structing the  table,  and  his  results  agree  substantially  with 
those  of  Newton. 

I  do  not  suppose  that  Newton  would  in  any  case  have 
produced  much  more  original  work  after  his  illness  ;  but  his 
appointment  in  1695  as  warden,  and  his  promotion  in  1699 
to  the  mastership  of  the  mint  at  a  salary  of  £1500  a  year, 
brought  his  scientific  investigations  to  an  end.  He  now  moved 
to  London.  In  1701  he  resigned  the  Lucasian  chair,  and  in 
1703  he  was  elected  president  of  the  Royal  Society. 

In  1704  he  published  his  Optics,  containing  an  account  of 
his  emission  theory  of  light  (see  p.  55).  To  this  book  two 
appendices  were  added ;  one  on  cubic  curves,  and  the  other  on 
the  quadrature  of  curves  and  his  theory  of  fluxions.  Both  of 
these  were  old  manuscripts  which  had  long  been  known  to  his 
friends  at  Cambridge,  but  had  been  previously  unpublished. 

The  first  of  these  appendices  is  entitled  Enumeratio  linea- 
rum  tertii  ordinis  and  was  apparently  written  before  1676. 
The  object  seems  to  be  to  illustrate  the  use  of  analytical  geo- 
metry, and  as  the  application  to  conies  was  well  known  Newton 
selected  the  theory  of  cubics. 

He  begins  with  some  general  theorems,  and  classifies 
curves  according  as  to  whether  their  equations  are  alge- 


64  THE   LIFE   AND   WORKS   OF   NEWTOX. 

braical  or  transcendental  :  the  former  being  cut  by  a  straight 
line  in  a  number  of  points  (real  or  imaginary)  equal  to  the 
degree  of  the  curve,  the  latter  being  cut  by  a  straight  line  in 
an  infinite  number  of  points.  Newton  then  shews  that  many  of 
the  most  important  properties  of  conies  have  their  analogues 
in  the  theory  of  cubics  ;  of  this  he  gives  numerous  illustrations. 
He  next  proceeds  to  discuss  the  theory  of  asymptotes  and 
curvilinear  diameters  to  curves  of  any  degree. 

After  these  general  theorems  he  commences  his  detailed 
examination  of  cubics  by  pointing  out  that  a  cubic  must  at 
least  have  one  real  asymptotic  direction.  If  the  asymptote 
corresponding  to  this  direction  be  at  a  finite  distance  it  may  be 
taken  for  the  axis  of  y.  This  asymptote  will  cut  the  curve  in 
three  points  altogether,  of  which  at  least  two  are  at  infinity. 
If  the  third  point  be  at  a  finite  distance  then  (by  one  of  his 
general  theorems  on  asymptotes)  the  equation  can  be  written 
in  the  form 

xyz  +  hy  =  ax3  +  bx2  +  cx  +  d, 

while  if  the  third  point  in  which  this  asymptote  cuts  the  curve 
be  also  at  infinity  the  equation  can  be  written  in  the  form 

xy  —  ax3  +  bx2 


Next  he  takes  the  case  where  the  asymptote  corresponding 
to  the  real  asymptotic  direction  is  not  at  a  finite  distance. 
A  line  parallel  to  it  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  one  is  necessarily 
at  a  finite  distance.  He  then  shews  that  if  the  remaining 
point  in  which  this  line  cuts  the  curve  be  at  a  finite  distance 
the  equation  can  be  written  in  the  form 

ys  =  ax3  +  bx2  +  cx  +  d, 

while  if  it  be  at  an  infinite  distance  the  equation  can  be 
written  in  the  form 

y  =  ax3  +  bx2  +  cx  +  d. 


NEWTON'S  CUBIC  CURVES  AND  QUADRATURE  OF  CURVES.  65 

Any  cubic  is  therefore  reducible  to  one  of  four  charac- 
teristic forms.  Each  of  these  forms  is  then  discussed  in  detail, 
and  the  possibility  of  the  existence  of  double  points,  isolated 
ovals,  &c.  is  thoroughly  worked  out.  The  final  result  is  that 
there  are  in  all  seventy-two  possible  forms  which  a  cubic  may 
take.  To  these  Stirling  in  his  Lineae  tertii  ordinis  New- 
tonianae  published  in  1717  added  four;  and  Cramer  and 
Murdoch  in  the  Genesis  curvarum  per  umbras  published  in 
1746  each  added  one  ;  thus  making  in  all  seventy-eight  species. 
In  the  course  of  the  analysis  Newton  states  the  remarkable 
theorem  that  in  the  same  way  as  the  conies  may  be  considered 
as  the  shadows  of  a  circle  (i.e.  plane  sections  of  a  cone  on  a 
circular  base)  so  all  cubics  may  be  considered  as  the  shadows  of 
the  curves  represented  by  the  equation  y2  =  ax*  +  bx2  +  ex  +  d. 

The  second  appendix  to  the  Optics  was  entitled  De  quad- 
ratura  curvarum.  Most  of  it  had  been  communicated  to 
Barrow  in  1666,  and  was  probably  familiar  to  Newton's  pupils 
and  friends  from  about  1667  onwards.  It  consists  of  two 
parts. 

The  bulk  of  the  first  part  had'  been  included  in  the  letter 
to  Leibnitz  of  Oct.  24,  1676.  This  part  contains  the  earliest 
use  of  literal  indices,  and  the  first  printed  statement  of  the 
binomial  theorem  :  these  are  however  introduced  incidentally. 
The  main  object  of  this  part  is  to  give  rules  for  developing  a 
function  of  a?  in  a  series  in  ascending  powers  of  x  ;  so  as  to 
enable  mathematicians  to  effect  the  quadrature  of  any  curve  in 
which  the  ordinate  y  can  be  expressed  as  an  explicit  function 
of  the  abscissa  x.  Wallis  had  shewn  how  this  quadrature 
could  be  found  when  y  was  given  as  a  sum  of  a  number  of 
powers  of  x  (see  p.  43),  and  Newton  here  extends  this  by 
shewing  how  any  function  can  be  expressed  as  an  infinite 
series  in  that  way.  I  should  add  that  Newton  is  generally 
careful  to  state  whether  the  series  are  convergent.  In  this 
way  he  effects  the  quadrature  of  the  curves 


B. 


66  THE   LIFE   AND   WORKS   OF   NEWTON. 

but  the  results  are  of  course  expressed  as  infinite  series.  He 
then  proceeds  to  curves  whose  ordinate  is  given  as  an  implicit 
function  of  the  abscissa  :  and  he  gives  a  method  by  which  y 
can  be  expressed  as  an  infinite  series  in  ascending  powers  of  x, 
but  the  application  of  the  rule  to  any  curve  demands  in  general 
such  complicated  numerical  calculations  as  to  render  it  of  little 
value.  He  concludes  this  part  by  shewing  that  the  recti- 
fication of  a  curve  can  be  effected  in  a  somewhat  similar  way. 
His  process  is  equivalent  to  finding  the  integral  with  regard  to 
x  of  (1  4-  ifY  in  the  form  of  an  infinite  series. 

This  part  should  be  read  in  connection  with  his  Analysis  by 
infinite  series  published  in  1711,  and  his  MetJwdus  dijferentialis 
published  in  1736.  Some  additional  theorems  are  there  given, 
and  in  the  latter  of  these  works  he  discusses  his  method  of 
interpolation.  The  principle  is  this.  If  y  =  <£  (x)  is  a  function 
of  x  and  if  when  x  is  successively  put  equal  to  a^  a2,  ...  the 
values  of  y  are  known  and  are  b^  b2 ...  then  a  parabola 
whose  equation  is  y=p  +  qx  +  rx*  +  ...  can  be  drawn  through 
the  points  (a^  6J,  (a2,  bz),  ...  and  the  ordinate  of  this  parabola 
may  be  taken  as  an  approximation  to  the  ordinate  of  the 
Qurve.  The  degree  of  the  parabola  will  of  course  be  one 
less  than  the  number  of  given  points.  Newton  points  out 
that  in  this  way  the  areas  of  any  curves  can  be  approximately 
determined. 

The  second  part  of  this  second  appendix  contains  a  de- 
scription of  his  method  of  fluxions  and  is  condensed  from  his 
manuscript  to  which  allusion  is  made  a  few  pages  later  (see 
p.  70). 

The  remaining  events  of  Newton's  life  may  be  summed  up 
very  briefly.  In  1705  he  was  knighted.  From  this  time 
onwards  he  devoted  much  of  his  leisure  to  theology,  and  wrote 
at  great  length  on  prophecies  and  predictions  which  had 
always  been  subjects  of  interest  to  him.  His  Universal  arith- 
metic was  published  by  Whiston  in  1707,  and  his  Analysis 
by  infinite  series  in  1711 ;  but  Newton  had  nothing  to  do  with 
preparing  either  of  these  for  the  press.  In  1709  Newton  was 


NEWTON'S  APPEAEANCE  AND  CHARACTER.  67 

persuaded  to  allow  Cotes  to  prepare  the  long-talked-of  second 
edition  of  the  Principia;  it  was  issued  in  March  1713.  A  third 
edition  was  published  in  1726  under  the  direction  of  Henry 
Pemberton.  Newton's  original  manuscript  on  fluxions  was 
published  in  1736,  some  nine  years  after  his  death,  by  John 
Colson.  In  1725  his  health  began  to  fail.  He  died  on  March 
20,  1727,  and  eight  days  later  was  buried  with  great  state  in 
Westminster  Abbey. 

In  appearance  Newton  was  short,  and  towards  the  close  of 
his  life  rather  stout,  but  well  set,,  with  a  square  lower  jaw,  a 
very  broad  forehead,  rather  sharp  features,  and  brown  eyes. 
His  hair  turned  grey  before  he  was  thirty,  and  remained  thick 
and  white  as  silver  till  his  death.  He  dressed  in  a  slovenly 
manner,  was  rather  languid,  arid  was  generally  so  absorbed 
in  his  own  thoughts  as  to  be  anything  but  a  lively  com- 
panion. 

Many  anecdotes  of  his  extreme  absence  of  mind  when 
engaged  in  any  investigation  have  been  preserved.  Thus  once 
when  riding  home  from  Grantham  he  dismounted  to  lead  his 
horse  up  a  steep  hill,  when  he  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  sacrificed  his  time  to  entertain  his  friends,  if  he  left  them  to 
get  more  wine  or  for  any  similar  reason,  he  would  as  often  as 
not  be  found  after  the  lapse  of  some  time  working  out  a  problem, 
oblivious  alike  of  his  expectant  guests  and  of  his  errand.  He 
took  no  exercise,  indulged  in  no  amusements,  and  worked  in- 
cessantly, often  spending  18  or  19  hours  out  of  the  24  in  writing. 
He  modestly  attributed  his  discoveries  largely  to  the  admirable 
work  done  by  his  predecessors-  and  in  answer  to  a  correspondent 
he  explained  that  if  he  had  seen  farther  than  other  men,  it  was 
only  because  he  had  stood  on  the  shoulders  of  giants.  He  was 
morbidly  sensitive  to  being  involved  in  any  discussions.  I 
believe  that  with  the  exception  of  his  two  papers  on  optics 
in  1675,  every  one  of  his  works  was  only  published  under 
pressure  from  his  friends  and  against  his  own  wishes.  There 

5—2 


68  THE   LIFE   AND   WORKS   OF   NEWTOX. 

are  several  instances  of  his  communicating  papers  and  results 
on  condition  that  his  name  should  not  be  published. 

In  character  he  was  perfectly  straightforward  and  honest, 
but  in  his  controversies  with  Leibnitz,  Hooke,  and  others 
though  scrupulously  just  he  was  not  generous.  During  the 
early  half  of  his  life  he  was  parsimonious,  if  not  stingy,  and  he 
was  never  liberal  in  money  matters. 

^"The  above  account,  slight  though  it  is,  will  yet  enable  the 
reader  to  form  an  idea  of  the  immense  extent  of  Newton's  ser- 
vices to  science.  His  achievements  are  the  more  wonderful  if 
we  consider  that  most  of  them  were  effected  within  twenty-five 
years,  1666 — 1692.  Two  branches  of  applied  mathematics 
stand  out  pre-eminent  in  his  work :  first,  his  theories  of  physical 
and  geometrical  optics ;  and  second,  his  theory  of  gravitation 
or  physical  astronomy.  Although  unrivalled  in  his  power  of 
analysis — of  which  his  Universal  arithmetic  and  the  essay  on 
cubic  curves  would  alone  be  sufficient  evidence — he  always  by 
choice  presented  his  proofs  in  a  geometrical  form.  But  it  is 
known  that  for  purposes  of  research  he  generally  used  the 
fluxional  calculus  in  the  first  instance.  Hence  excessive  im- 
portance was  attached  by  the  Newtonian  school  to  these  two 
branches  of  pure  mathematics.  So  completely  did  Newton 
impress  his  individuality  on  English  mathematics  that  during 
the  eighteenth  century  the  subject  at  Cambridge  meant  little 
else  but  a  study  of  the  four  branches  above  mentioned.  I  have 
already  alluded  to  the  subject-matter  of  the  Principia  and 
Optics,  and  I  must  now  say  a  few  words  on  his  method  of 
exposition,  and  his  use  of  geometry  and  fluxions. 

It  is  probable  that  no  mathematician  has  ever  equalled 
Newton  in  his  command  of  the  processes  of  classical  geometry. 
But  his  adoption  of  it  for  purposes  of  demonstration  appears  to 
have  arisen  from  the  fact  that  the  infinitesimal  calculus  was 
then  unknown  to  most  of  his  readers,  and  had  he  used  it  to 
demonstrate  results  which  were  in  themselves  opposed  to  the 
prevalent  philosophy  of  the  time  the  controversy  would  have 
first  turned  on  the  validity  of  the  methods  employed.  Newton 


NEWTON'S  USE  OF  GEOMETRY  AND  FLUXIONS.      69 

therefore  cast  the  demonstrations  of  the  Principia  into  a  geo- 
metrical shape  which,  if  somewhat  longer,  could  at  any  rate  be 
made  intelligible  to  all  mathematical  students  and  of  which  the 
methods  were  above  suspicion.  In  further  explanation  of  this 
I  ought  to  add  that  in  Newton's  time  and  for  nearly  a  century 
afterwards  the  differential  and  fluxional  calculus  were  not  fully 
developed  and  did  not  possess  the  same  superiority  over  the 
method  he  adopted  which  they  do  now.  The  effect  of  his  con- 
fining himself  rigorously  to  classical  geometry  and  elementary 
algebra,  and  of  his  refusal  to  make  any  use  even  of  analytical 
geometry  and  of  trigonometry  is  that  the  Principia  is  written 
in  a  language  which  is  archaic  (even  if  not  unfamiliar)  to 
us.  The  subject  of  optics  lends  itself  more  readily  to  a 
geometrical  treatment,  and  thus  his  demonstrations  of  theo- 
rems in  that  subject  are  not  very  different  to  those  still 
used. 

The  adoption  of  geometrical  methods  in  the  Principia  for 
purposes  of  demonstration  does  not  indicate  a  preference  on 
Newton's  part  for  geometry  over  analysis  as  an  instrument 
of  research,  for  it  is  now  known  that  Newton  used  the  fluxional 
calculus  in  the  first  instance  in  finding  some  of  the  theorems 
(especially  those  towards  the  end  of  book  i.  and  in  book  IL), 
and  then  gave  geometrical  proofs  of  his  results.  This  transla- 
tion of  numerous  theorems  of  great  complexity  into  the  language 
of  the  geometry  of  Archimedes  and  Apollonius  is  I  suppose 
one  of  the  most  wonderful  intellectual  feats  which  was  ever 
performed. 

The  fluxional  calculus  is  one  form  of  the  infinitesimal 
calculus  expressed  in  a  certain  notation  just  as  the  differential 
calculus  is  another  aspect  of  the  same  calculus  expressed  in  a 
different  notation.  Newton  assumed  that  all  geometrical  mag- 
nitudes might  be  conceived  as  generated  by  continuous  motion  : 
thus  a  line  may  be  considered  as  generated  by  the  motion  of  a 
point,  a  surface  by  that  of  a  line,  a  solid  by  that  of  a  surface,  a 
plane  angle  by  the  rotation  of  a  line,  and  so  on.  The  quantity 
thus  generated  was  defined  by  him  as  the  fluent  or  flowing 


70  THE   LIFE   AND   WORKS   OF   NEWTON. 

quantity.  The  velocity  of  the  moving  magnitude  was  defined 
as  the  fluxion  of  the  fluent. 

The  following  is  a  summary  of  Newton's  treatment  of 
fluxions.  There  are  two  kinds  of  problems.  The  object  of  the 
first  is  to  find  the  fluxion  of  a  given  quantity,  or  more  generally 
"the  relation  of  the  fluents  being  given  to  find  the  relation  of 
their  fluxions."  This  is  equivalent  to  differentiation.  The  object 
of  the  second  or  inverse  method  of  fluxions  is  from  the  fluxion 
or  some  relation  involving  it  to  determine  the  fluent,  or  more 
generally  "an  equation  being  proposed  exhibiting  the  relation 
of  the  fluxions  of  quantities  to  find  the  relations  of  those  quan- 
tities or  fluents  to  one  another1."  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  of  tangents.  The  methods 
for  solving  these  problems  are  discussed  at  considerable 
length. 

Newton  then  went  on  to  apply  these  results  to  questions  con- 
nected with  the  maxima  and  minima  of  quantities,  the  method 
of  drawing  tangents  to  curves,  and  the  curvature  of  curves  (viz. 
the  determination  of  the  centre  of  curvature,  the  radius  of  curva- 
ture, and  the  rate  at  which  the  radius  of  curvature  increases). 
He  next  considered  the  quadrature  of  curves  and  the  rectifica- 
tion of  curves2. 

It  has  been  remarked  that  neither  Newton  nor  Leibnitz 
produced  a  calculus,  that  is  a  classified  collection  of  rules ;  and 
that  the  problems  they  discussed  were  treated  from  first  prin- 
ciples. That  no  doubt  is  the  usual  sequence  in  the  history  of 
such  discoveries,  though  the  fact  is  frequently  forgotten  by 
subsequent  writers.  In  this  case  I  think  the  statement,  so  far 
as  Newton  is  concerned,  is  incorrect,  as  the  foregoing  account 
sufficiently  shews. 

If  a  flowing  quantity  or  fluent  were  represented  by  x,  Newton 

1  Colson's  edition  of  Newton's  manuscript,  pp.  xxi.  xxii. 

2  Colson's  edition  of  Newton's  manuscript,  pp.  xxii.  xxiii. 


NEWTON'S  THEORY  OF  FLUXIONS.       71 

denoted  its  fluxion  by  x,  the  fluxion  of  x  or  second  fluxion 
of  x  by  x,  and  so  on.  Similarly  the  fluent  of  x  was  denoted  by 
x'  or  [x~]  or  \x\.  The  infinitely  small  part  by  which  a  fluent 
such  as  x  increased  in  a  small  interval  of  time  measured  by 
o  was  called  the  moment  of  the  fluent;  and  its  value  was  shewn 
to  be  xol.  I  should  here  note  the  fact  that  Yince  and  other 
writers  in  the  eighteenth  century  used  x  to  denote  the  incre- 
ment of  x  and  not  the  velocity  with  which  it  increased ;  that 
is  x  in  their  writings  stands  for  what  Newton  would  have 
expressed  by  xo  and  what  Leibnitz  would  have  written  as  dx. 

They  also  used  the  current  symbol  for  integration.     Thus  I  xn  x 

stands  with  them  for  what  Newton  would  have  usually  ex- 
pressed by  [aj"|,  or  what  Leibnitz  would  have  written  as 

xndx. 

I  need  not  here  concern  myself  with  the  details  as  to  how 
Newton  treated  the  problems  above  mentioned.  I  will  only 
add  that  in  spite  of  the  form  of  his  definition  the  introduction 
in  geometry  of  the  idea  of  time  was  evaded  by  supposing  that 
some  quantity  (e.g.  the  abscissa  of  a  point  on  a  curve)  increased 
equably ;  and  the  required  results  then  depend  on  the  rate  at 
which  other  quantities  (e.g.  the  ordinate  or  radius  of  curvature) 
increase  relatively  to  the  one  so  chosen2.  The  fluent  so  chosen 
is  what  we  now  call  the  independent  variable ;  its  fluxion  was 
termed  the  "principal  fluxion;"  and  of  course  if  it  were 
denoted  by  x  then  x  was  constant,  and  consequently  x  —  0. 

Newton's  manuscript,  from  which  most  of  the  above  sum- 
mary has  been  taken,  is  believed  to  have  been  written  between 
1671  and  1677,  and  to  have  been  in  circulation  at  Cambridge 
from  that  time  onwards.  It  was  unfortunate  that  it  was  not 
published  at  once.  Strangers  at  a  distance  naturally  judged  of 
the  method  by  the  letter  to  Wallis  in  1692  or  the  Tractatus  de 

1  Colson's  edition  of  Newton's  manuscript,  p.  24. 

2  Colson's  edition  of  Newton's  manuscript,  p.  20. 


72  THE   LIFE   AND   WORKS   OF   NEWTOX. 

quadratura  curvarum,  and  were  not  aware  that  it  had  been  so 
completely  developed  at  an  earlier  date.  This  was  the  cause  of 
numerous  misunderstandings. 

The  notation  of  the  fluxional  calculus  is  for  most  purposes 
less  convenient  than  that  of  the  differential  calculus.  The 
latter  was  invented  by  Leibnitz  in  1675,  and  published  in  1684. 
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.  From  what  I  have  read  of  the 
voluminous  literature  on  the  question,  I  think  on  the  whole  it 
points  to  the  fact  that  Leibnitz  obtained  the  idea  of  the  differen- 
tial calculus  from  a  manuscript  of  Newton's  which  he  saw  in 
1673,  but  the  question  is  one  of  considerable  difficulty  and  no 
one  now  is  likely  to  dogmatize  on  it1. 

If  we  must  confine  ourselves  to  one  system  of  notation 
then  there  can  be  no  doubt  that  that  which  was  designed  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.  His  form  of  the  infinitesimal  calculus  was 
adopted  by  all  continental  mathematicians.  In  England  the 
controversy  with  Leibnitz  was  regarded  as  an  attempt  by 
foreigners  to  defraud  Newton  of  the  credit  of  his  invention, 
and  the  question  was  complicated  on  both  sides  by  national 
jealousies.  It  was  therefore  natural  though  it  was  unfortunate 
that  the  geometrical  and  fluxional  methods  (as  used  by  Newton) 
should  be  alone  studied  and  employed  at  Cambridge.  For  more 
than  a  century  the  English  school  was  thus  quite  out  of  touch 
with  continental  mathematicians.  The  consequence  was  that 

1  The  case  in  favour  of  the  independent  invention  by  Leibnitz  is 
stated  in  Biot  and  Lefort's  edition  of  the  Commercium  epistolicum,  Paris, 
1856,  and  in  an  article  in  the  Philosophical  magazine  for  1852.  A  summary 
of  the  arguments  on  the  other  side  is  given  in  Dr  Sloman's  The  claims  of 
Leibnitz  to  the  invention  of  the  differential  calculus  issued  at  Leipzig  in 
1858,  of  which  an  English  translation  was  published  at  Cambridge  in  1860. 


THE  LIFE   AND   WORKS   OF   NEWTON.  73 

in  spite  of  the  brilliant  band  of  scholars  formed  by  Newton  the 
improvements  in  the  methods  of  analysis  gradually  effected  on 
the  continent  were  almost  unknown  in  Cambridge.  It  was 
not  until  about  1820  (as  described  in  chapter  VII.)  that  the 
value  of  analytical  methods  was  fully  recognized  in  England; 
and  that  Newton's  countrymen  again  took  any  large  share  in 
the  developement  of  mathematics. 


CHAPTER  V. 

THE  RISE  OF  THE  NEWTONIAN  SCHOOL. 
CIRC.  1690—1730. 

IN  the  last  chapter  I  enumerated  very  briefly  the  more 
important  discoveries  of  Newton,  and  pointed  out  the  four 
subjects  to  which  he  paid  special  attention.  I  have  now  to 
describe  how  those  discoveries  affected  the  study  of  mathe- 
matics in  the  university,  and  led  to  the  rise  of  the  Newtonian 
school. 

The  mathematical  school  in  the  university  prior  to  Newton's 
time  contained  several  distinguished  men,  but  in  point  of 
numbers  it  was  not  large.  We  need  not  therefore  be  surprised 
to  find  that  it  was  Newton's  theory  of  the  universe  and  not  his 
mathematics  that  excited  most  attention  in  the  university ;  and 
it  was  because  mathematics  supplied  the  key  to  that  theory 
that  it  began  to  be  studied  so  eagerly.  Hence  the  rise  of  the 
Newtonian  school  dates  from  the  publication  of  the  Principia. 

In  considering  the  history  of  this  school,  it  must  be  remem- 
bered that  at  Cambridge  until  recently  professors  only  rarely 
put  themselves  into  contact  with  or  adapted  their  lectures  for 
the  bulk  of  the  students  in  their  own  department.  Accordingly 
if  we  desire  to  find  to  whom  the  spread  of  a  general  study  of 
the  Newtonian  philosophy  was  immediately  due,  we  must  look 
not  to  Newton's  lectures  or  writings,  but  among  those  proc- 
tors, moderators,  or  college  tutors,  who  had  accepted  his 
doctrines.  The  form  in  which  the  Principia  was  cast,  its 
extreme  conciseness,  the  absence  of  all  illustrations,  and  the 


LAUGHTON.  75 

immense  interval  between  the  abilities  of  Newton  and  those 
of  his  contemporaries  combined  to  delay  the  acceptance  of  the 
new  philosophy  ;  and  it  is  a  matter  of  surprise  that  its  truth 
was  so  soon  recognized. 

1  propose  first  to  mention  Richard  Laughton,  Samuel 
Clarke,  John  Craig,  and  John  Flamsteed,  who  were  some  of 
the  earliest  residents  to  accept  the  Newtonian  philosophy. 
I  must  then  devote  a  few  words  to  Bentley,  to  whom  the 
predominance  in  the  university  of  the  Newtonian  school  is 
largely  due:  he  knew  but  little  mathematics  himself,  but  he 
used  his  considerable  influence  to  put  the  study  on  a  satisfactory 
basis.  I  shall  then  briefly  describe  the  works  of  William 
Whist  on,  Nicholas  Saunderson,  Thomas  Byrdall,  James  Jurin, 
Brook  Taylor,  Roger  Cotes,  and  Robert  Smith :  the  three 
mathematicians  last  named  being  among  the  most  powerful  of 
Newton's  immediate  successors.  Lastly  I  propose  to  describe 
the  course  of  reading  in  mathematics  of  a  student  at  Cambridge 
about  the  year  1730,  which  I  take  as  the  limit  of  the  period 
treated  in  this  chapter. 


Among  the  earliest  of  those  who  realized  the  importance  of 
Newton's  discoveries  was  Richard  Laughton1,  a  fellow  of  Clare 
Hall.  I  have  been  unable  to  discover  any  account  of  his  life, 
but  I  find  he  is  referred  to  as  the  most  celebrated  "pupil- 
monger  "  of  his  time,  and  I  gather  from  references  to  him  in 
the  literature  of  the  period  that  he  was  one  of  the  most 
influential  of  those  who  introduced  a  study  of  the  Newtonian 
theory  of  the  universe  into  the  university  curriculum.  In 
1694  he  persuaded  Samuel  Clarke  (who  was  probably  one  of 
his  pupils)  to  defend  in  the  schools  a  question  on  physical 
astronomy  taken  from  the  Principia,  and  in  the  same  year 
the  Cartesian  theory  was  ridiculed  in  the  tripos  verses. 
These  seem  to  be  the  earliest  allusions  in  the  public  exercises 

1  The  name  was  pronounced  Laffton :  see  Uffenbach's  account  of  his 
visit  to  Cambridge  in  1710  quoted  on  p.  6  of  the  Scholae  academicae. 


76  THE   RISE   OF   THE    NEWTONIAN   SCHOOL. 

of  the  university  to  the  Newtonian  philosophy ;  but  so  rapidly 
were  its  merits  appreciated  that  within  twenty  years  it  was  the 
dominant  study  in  the  university.  Later  in  the  same  year 
Laughton  was  made  a  tutor  of  Clare ;  and  thenceforward  he 
took  every  opportunity  of  his  new  position  to  urge  his  pupils 
to  read  Newton. 

In  1710  Laughton  was  proctor,  and  claimed  the  right  to 
preside  in  person  at  the  acts  in  the  schools.  This  was  a  part 
of  the  ancient  duties  of  the  office,  but  since  1680  it  had  been 
customary  for  the  senate  each  year  to  appoint  moderators  who 
performed  it  as  the  deputies  of  the  proctors,  and  even  at  an 
earlier  date  it  was  not  unusual  for  the  latter  officers  to  select 
moderators  (or  posers,  as  they  were  then  generally  designated) 
to  whom  they  delegated  that  part  of  their  work.  Laughton 
presided  in  person,  and  in  summing  up  the  discussions  exposed 
the  assumptions  and  mistakes  in  the  Cartesian  system.  A 
resident1  who  was  no  special  advocate  of  the  new  doctrines 
bears  witness  in  his  diary  to  the  success  of  Laughton's  efforts. 
"It  is  certain,"  says  he,  "that  for  some  years  [before  1710]  he 
had  been  diligently  inculcating  [the  Newtonian]  doctrines,  and 
that  the  credit  and  popularity  of  his  college  had  risen  very 
high  in  consequence  of  his  reputation."  Acting  as  proctor  in 
that  year  Laughton  induced  William  Browne  of  Peterhouse  to 
keep  his  acts  on  mathematical  questions,  and  promised  him  an 
honorary  proctor's  optirne  degree  (see  p.  170)  if  he  would  do  so. 
Laughton  died  in  1726. 

The  earliest  text-book  with  which  I  am  acquainted  written 
to  advocate  the  Newtonian  philosophy  was  by  the  Samuel 
Clarke  to  whom  allusion  has  just  been  made.  Samuel  Clarke2 
was  born  at  Norwich  on  Oct.  11,  1675,  and  took  his  B.A. 
from  Caius  in  1695.  The  text-book  on  physical  astronomy 
then  in  common  use  was  Rohault's  Physics,  which  was 

1  See  the  Diary  of  Ralph  Thorseby  (1677—1724)  edited  by  J.  Hunter, 
2  volumes,  London,  1830. 

2  See  his  life  and  works  by  B.  Hoadly,  4  volumes,  London,  1738;  and 
a  memoir  by  W.  Whiston,  third  edition,  London,  1741. 


CLARKE.      CRAIG.  77 

founded  on  Descartes's  hypothesis  of  vortices.  Clarke  thought 
that  he  could  best  advocate  the  Newtonian  theory  by  issuing 
a  new  edition  of  Rohault  with  notes,  shewing  that  the  con- 
clusions were  necessarily  wrong.  This  curious  mixture  of 
truth  and  falsehood  continued  to  be  read  at  Cambridge  at  least 
as  late  as  1730,  and  went  through  several  editions.  After 
1697  Clarke  devoted  most  of  his  time  to  the  study  of  theology, 
though  in  1706  he  translated  Newton's  Optics  into  "elegant 
Latin,"  with  which  Newton  was  so  pleased  that  he  sent  him  a 
present  of  five  hundred  guineas.  In  1728  Clarke  contributed 
a  paper  to  the  Philosophical  transactions  on  the  controversy 
then  raging  as  to  whether  a  force  ought  to  be  measured  by  the 
momentum  or  by  the  kinetic  energy  produced  in  a  given  mass. 
He  died  in  1729. 

Another  mathematician  of  this  time  who  did  a  good  deal  to 
bring  fluxions  into  general  use  was  Craig.  John  Craig  was 
born  in  Scotland.  He  came  to  Cambridge  about  1680,  but  it  is 
believed  he  never  took  a  degree.  He  went  down  in  1708,  and 
after  holding  various  livings  settled  in  London,  where  he  died 
on  Oct.  11,  1731.  His  chief  works  were  the  Methodus . .  .quad- 
raturas  determinandi  published  in  1685,  the  De  figurarum 
quadraturis  et  locis  geometricis  published  in  1693,  and  the 
De  calculo  fluentium  (2  volumes)  and  De  optica  analytica  (2 
volumes)  which  were  published  in  1718.  In  the  two  works 
first  mentioned  he  argues  in  favour  of  the  ideas  and  notation 
of  the  differential  calculus,  and  in  connection  with  them  he 
had  a  long  controversy  with  Jacob  Bernoulli.  In  the  last 
he  definitely  adopts  the  fluxional  calculus  as  the  correct  way 
of  presenting  the  truths  of  the  infinitesimal  calculus.  These 
works  shew  that  Craig  was  a  good  mathematician. 

Among  his  papers  published  in  the  Philosophical  trans- 
actions I  note  one  in  1698  on  the  quadrature  of  the  logarithmic 
curve,  one  in  1700  on  the  curve  of  quickest  descent,  and 
another  in  the  same  year  on  the  solid  of  least  resistance,  one  in 
1703  on  the  quadrature  of  any  curve,  one  in  1704  containing  a 
solution  of  a  problem  issued  by  John  Bernoulli  as  a  challenge, 


78  THE   RISE    OF    THE    NEWTONIAN    SCHOOL. 

one  in  1708  on  the  rectification  of  any  curve,  and  lastly  one  in 
1710  on  the  construction  of  logarithmic  tables. 

It  is  however  much  easier  to  obtain  a  lasting  reputation  by 
eccentricity  than  by  merit ;  and  hundreds  who  never  heard  of 
Craig's  work  on  fluxions  know  of  him  as  the  author  of  Theologia 
Christianae  principia  mathematica  published  in  1699.  He  here 
starts  with  the  hypothesis1  that  evidence  transmitted  through 
successive  generations  diminishes  in  credibility  as  the  square 
of  the  time.  The  general  idea  was  due  to  the  Mahommedan 
apologists,  who  enunciated  it  as  an  axiom,  and  then  argued  that 
as  the  evidence  for  the  Christian  miracles  daily  grows  weaker 
a  time  must  come  when  they  will  have  no  evidential  value, 
whence  the  necessity  of  another  prophet.  Curiously  enough 
Craig's  formulae  shew  that  the  oral  evidence  would  by  itself 
have  become  worthless  in  the  eighth  century,  which  is  not  so 
very  far  removed  from  the  date  of  Mahommed's  death  (632). 
He  asserts  that  the  gospel  evidence  will  cease  to  have  any  value 
in  the  year  3150.  He  then  quotes  a  text  to  shew  that  at  the 
second  coming  faith  will  not  be  quite  extinct  among  men  : 
and  hence  the  world  must  come  to  an  end  before  3150.  This 
was  reprinted  abroad,  and  seriously  answered  by  many  divines ; 
but  most  of  his  opponents  were  better  theologians  than  mathe- 
maticians, and  would  have  been  wiser  if  they  had  contented 
themselves  with  denying  his  axioms. 

I  must  not  pass  over  this  period  without  mentioning 
Flamsteed.  John  Flamsteed2  was  born  in  Derbyshire  in  1646. 
When  at  school  he  picked  up  a  copy  of  Holywood's  treatise 
on  the  sphere  (see  p.  5)  and  was  so  fascinated  by  it  that  he 
determined  to  study  astronomy.  It  was  intended  to  send  him 
to  Cambridge,  but  for  some  years  he  was  too  delicate  to  leave 
home.  He  however  obtained  copies  of  Street's  Astronomy, 
Riccioli's  Almagestum  novum,  and  Kepler's  Tables,  which  he 
read  by  himself.  By  the  time  he  was  twenty -two  or  three  he 

1  See  pp.  77,  78  of  A  budget  of  paradoxes  by  A.  De  Morgan,  London, 
1872. 

2  See  his  life,  by  E.  F.  Baily,  London,  1835. 


FLAMSTEED.  79 

was  already  one  of  the  best  astronomers  (both  theoretical  and 
practical)  in  Europe.  He  entered  at  Jesus  College  in  1670, 
and  devoted  himself  to  the  study  of  mathematics,  optics,  and 
astronomy.  He  seems  to  have  been  in  constant  communication 
with  Barrow  and  Newton.  He  took  his  B.A.  in  1674,  and  in 
the  following  year  was  appointed  to  take  charge  of  the  national 
observatory  then  being  erected  at  Greenwich.  He  is  thus  the 
earliest  of  the  astronomer-royals.  He  gave  Newton  many  of 
the  data  for  the  numerical  calculations  in  the  third  book  of 
the  Principia,  but  in  consequence  of  a  quarrel,  refused  to  give 
the  additional  ones  required  for  the  second  edition.  He  died 
at  Greenwich  in  1719. 

He  invented  the  system  (published  in  1680)  of  drawing 
maps  by  projecting  the  surface  of  the  sphere  from  the  centre 
on  an  enveloping  cone  which  can  then  be  unwrapped.  He 
wrote  papers  on  various  astronomical  problems,  but  his  great 
work,  which  is  an  enduring  memorial  of  his  skill  and  genius,  is 
his  Historia  coelestis  J3rittanica,  edited  by  Halley  and  published 
posthumously  in  three  volumes  in  1725. 


By  the  beginning  of  the  eighteenth  century  the  immense 
reputation  and  great  powers  of  Newton  were  everywhere 
recognized.  The  adoption  of  his  methods  and  philosophy  at 
Cambridge  was  however  in  no  slight  degree  due  to  other  than 
professed  mathematicians.  Of  these  the  most  eminent  was 
Bentley,  who  invariably  exerted  his  influence  to  make  literature 
and  mathematical  science  the  distinctive  features  of  a  Cambridge 
training.  Philosophy  was  also  still  read  and  was  not  unworthily 
represented  by  Bacon,  Descartes,  and  Locke1.  It  was  from 

1  Francis  Bacon,  born  in  1561,  was  educated  at  Trinity  College, 
Cambridge,  and  died  in  1626 :  the  Novum  organum  was  published  in  1620. 
Rene  Descartes  was  born  in  1596  and  died  in  1650:  his  Discours  was 
published  in  1637,  and  his  Meditations  in  1641.  John  Locke,  born  in 
1632,  was  educated  at  Christ  Church,  Oxford,  and  died  in  1704:  his 
Essay  concerning  human  understanding  was  published  in  1690. 


80  THE   RISE   OF   THE   NEWTONIAN    SCHOOL. 

Newton  aided  by  Bentley  that  the  Cambridge  of  the  eighteenth 
century  drew  its  inspiration,  and  it  was  their  influence  that 
made  the  intellectual  life  of  the  university  during  that  time  so 
much  more  active  than  that  of  Oxford. 

Richard  Bentley1  was  born  in  Yorkshire  on  Jan.  27,  1662, 
and  died  at  Cambridge  on  July  14,  1742.  He  took  his  B.A. 
from  St  John's  College  in  1680  as  third  wrangler,  but  in 
consequence  of  the  power  of  conferring  honorary  optime  degrees 
(see  p.  170)  his  name  appears  as  sixth  in  the  list.  He  was  not 
eligible  for  a  fellowship,  and  in  1682  went  down. 

In  1692  he  was  selected  to  deliver  the  first  course  of  the 
Boyle  lectures  on  theology,  which  had  been  founded  by  the 
will  of  Robert  Boyle,  who  died  in  1691.  In  the  sixth,  seventh, 
and  eighth  sermons  he  gave  a  sketch  of  the  Newtonian  dis- 
coveries :  this  was  expressed  in  non-technical  language  and 
excited  considerable  interest  among  those  members  of  the 
general  public  who  had  been  unable  to  follow  the  mathematical 
form  in  which  Newton's  arguments  and  investigations  had  been 
previously  expressed. 

In  1699  Bentley  was  appointed  master  of  Trinity  College, 
and  from  that  time  to  his  death  an  account  of  his  life  is  the 
history  of  Cambridge.  It  is  almost  impossible  to  overrate  his 
services  to  literature  and  scientific  criticism,  and  his  influence 
on  the  intellectual  life  of  the  university  was  of  the  best.  It  is 
however  indisputable  that  many  of  his  acts  were  illegal,  and 
the  fact  that  he  wished  to  promote  the  interests  of  learning  is 
no  excuse  for  the  arrogance,  injustice,  and  tyranny  which 
characterized  his  rule. 

One  reform  of  undoubted  wisdom  which  he  introduced  may 

1  See  the  Life  of  Bentley  by  W.  H.  Monk,  2  vols.,  London,  1833 :  see 
also  the  volume  by  R.  C.  Jebb  in  the  series  of  English  men  of  letters, 
London,  1882;  the  latter  on  the  whole  is  eulogistic,  and  it  must  be 
remembered  that  most  of  Bentley's  Cambridge  contemporaries  would  not 
have  taken  so  favourable  a  view  of  his  character.  Another  brilliant 
monograph  on  Bentley  from  the  pen  of  Hartley  Coleridge  will  be  found  in 
the  Worthies  of  Yorkshire  and  Lancashire,  London,  1836. 


BENTLEY.  81 

be  here  mentioned.  Elections  to  scholarships  and  fellowships 
at  that  time  took  place  on  the  result  of  a  viva  voce  examination 
by  the  master  and  seniors  in  the  chapel.  To  give  an  oppor- 
tunity for  written  exercises  and  time  for  discussion  by  the 
electors  of  the  merits  of  the  candidates,  Bentley  arranged 
that  every  candidate  should  be  first  examined  by  each  elector. 
In  practice  part  of  the  examination  was  always  oral  and  part 
written.  He  also  made  the  award  of  scholarships  annual 
instead  of  biennial,  and  admitted  freshmen  to  compete  for 
them.  In  1789  the  examination  was  made  the  same  for  all 
candidates  and  conducted  openly.  A  survival  of  the  old 
practice — after  nearly  two  hundred  years — exists  in  the  fact 
that  the  electors  to  fellowships  and  scholarships  still  always 
adjourn  to  the  chapel  to  make  the  technical  election  and 
declaration. 

The  following  account  of  the  scholarship  examination  for 
1709  taken  from  a  letter1  of  one  of  the  candidates  (John 
Byrom)  to  his  father  may  interest  the  reader,  as  it  is  the 
earliest  account  of  such  an  examination  which  I  have  seen.  In 
that  year  there  were  apparently  ten  vacancies,  and  nineteen 
students  "  sat "  for  them.  At  the  end  of  April  every  candidate 
sent  a  letter  in  Latin  to  the  master  and  each  of  the  seniors 
announcing  that  he  should  present  himself  for  the  examination. 
On  May  7  Byrom  was  examined  by  the  vice-master,  on  the 
following  Monday  and  Tuesday  he  was  examined  by  Bentley, 
Stubbs,  and  Smith  in  their  respective  rooms,  and  on  Wednesday 
he  went  to  the  lodge  and  while  there  wrote  an  essay:  the 
other  seniors  seem  to  have  shirked  taking  part  in  the  examina- 
tion. "  On  Thursday,"  writes  Byrom,  "  the  master  and  seniors 
met  in  the  chapel  for  the  election ;  Dr  Smith  had  the  gout  and 
was  not  there.  They  stayed  consulting  about  an  hour  and  a 
half,  and  then  the  master  wrote  the  names  of  the  elect,  who 
(sic)  shewed  me  mine  in  the  list.  Fifteen  were  chosen.  [The 

1  See  p.  6  of  the  Remains  of  John  Byrom,  Chetham  Society  Publica- 
tions, Manchester,  1854. 

B.  6 


82  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

five  lowest  being  pre-elected  to  the  next  vacancies]. ...  Friday 
noon  we  went  to  the  master's  lodge,  where  we  were  sworn  in 
in  great  solemnity,  the  senior  Westminster  reading  the  oath  in 
Latin,  all  of  us  kissing  the  Greek  Testament.  Then  we 
kneeled  down  before  the  master,  who  took  our  hands  in  his 
and  admitted  us  scholars  in  the  name  of  the  Father,  Son,  &c. 
Then  we  went  and  wrote  our  names  in  the  book  and  came 
away,  and  to-day  gave  in  our  epistle  of  thanks  to  the  master. 
We  took  our  places  at  the  scholars'  table  last  night.  To-day 
the  new  scholars  began  to  read  the  lessons  in  chapel  and  wait 
[i.e.  to  read  grace]  in  the  hall,  which  offices  will  come  to  me 
presently." 

In  appearance  Bentley  was  tall  and  powerful,  the  forehead 
was  high  and  not  very  broad,  but  the  great  development  and 
rather  coarse  lines  of  the  lower  part  of  the  face  and  cheeks 
seem  to  me  the  most  prominent  features  and  always  strike  me 
as  indicative  of  cruelty  and  selfishness.  The  hair  was  brown 
and  the  hands  small.  Of  his  appearance  Prof.  Jebb  says,  "  The 
pose  of  the  head  is  haughty,  almost  defiant ;  the  eyes,  which 
are  large,  prominent,  and  full  of  bold  vivacity,  have  a  light  in 
them  as  if  Bentley  were  looking  straight  at  an  impostor  whom 
he  detected,  but  who  still  amused  him;  the  nose,  strong  and 
slightly  tip-tilted,  is  moulded  as  if  nature  had  wished  to  shew 
what  a  nose  can  do  for  the  combined  expression  of  scorn  and 
sagacity ;  and  the  general  effect  of  the  countenance,  at  a  first 
glance,  is  one  which  suggests  power — frank,  self-assured, 
sarcastic,  and  I  fear  we  must  add  insolent." 

In  character  he  was  warm-hearted,  impulsive,  and  no  doubt 
well-intentioned ;  and  separated  from  him  by  a  century  and  a 
half  we  may  give  him  credit  for  the  reforms  he  made — in 
spite  of  the  illegal  manner  in  which  they  were  introduced, 
and  of  his  injustice  and  petty  meanness  against  those  who 
opposed  him.  Even  his  apologists  admit  that  he  was  grasping, 
arrogant,  arbitrary,  intolerant,  and  at  any  rate  in  manner  not 
a  gentleman,  while  in  the  latter  part  of  his  life  he  neglected 
the  duties  of  his  office.  But  his  abilities  immeasurably  ex- 


WHISTON.  83 

ceeded  those  of  his  contemporaries,  and  such  as  he  was  he  has 
left  a  permanent  impress  on  the  history  of  Cambridge. 

The  interest  that  Bentley  felt  in  the  Newtonian  philosophy 
arose  from  the  nature  of  the  conclusions  and  of  the  irrefutable 
logic  by  which  they  were  proved.  He  was  not  however  capable 
of  appreciating  the  mathematical  analysis  by  which  they  had 
been  attained.  Of  those  who  were  urged  by  him  to  take  up 
the  study  of  mathematics,  one  of  the  earliest  was  Whiston. 
William  Whiston1  was  born  in  Leicestershire  on  Dec.  9,  1667. 
He  entered  in  1685  at  Clare,  and  mentions  in  his  biography 
that  he  attended  Newton's  lectures.  He  took  his  B.A.  in  the 
Lent  term  of  1690,  in  the  same  year  was  elected  a  fellow,  and 
for  some  time  subsequently  took  pupils.  In  1696  he  published 
his  celebrated  Theory  of  the  earth.  The  fanciful  manner  in 
which  he  accounted  for  the  deluge  by  means  of  the  tail  of  a 
comet  is  well  known ;  but  Bentley's  criticism  that  Whiston  had 
forgotten  to  provide  any  means  for  getting  rid  of  the  water  with 
which  he  had  covered  the  earth,  and  that  it  was  of  little  use  to 
explain  the  origin  of  the  deluge  by  natural  means  if  it  were 
necessary  to  invoke  the  aid  of  the  Almighty  to  finish  the  opera- 
tion, is  a  sound  one. 

When  in  1699  Newton  was  appointed  master  of  the  mint  he 
asked  Whiston  to  act  as  his  deputy  in  the  Lucasian  chair.  As 
such  Whiston  lectured  on  the  Principia.  In  1703  Newton  re- 
signed his  professorship  and  Whiston  was  chosen  as  his  successor. 

In   1702   Whiston  brought  out  an   edition  of   Tacquet's2 

1  Whiston  wrote  an  autobiography,  published  at  London  in  1749,  but 
many  of  the  events  related  are  not  described  accurately :  see  Monk's  Life 
of  Bentley,  vol.  i.  pp.  133,  151,  215,  290,  and  vol.  ii.  p.  18.     An  account 
of  his  life  is  given  in  the  Biographia  Britannica,  first  edition,  6  vols., 
London,  1747—66. 

2  Andrew  Tacquet,  who  was  born  at  Antwerp  in  1611  and  died  in  1660, 
was  one  of  the  best  known  Jesuit  mathematicians  and  teachers  of  the 
seventeenth  century.    His  translation  of  Euclid's  Elements  was  published 
in  1655,  and  remained  a  standard  text-book  on  the  continent  until  super- 
seded by  Legendre's   Geometrie.      Tacquet   also  wrote  on  optics  and 
astronomy.     His  collected  works  were  republished  in  two  volumes  at 
Antwerp  in  1669. 

6—2 


84  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

Euclid  which  remained  the  standard  English  text-book  on  ele- 
mentary geometry  until  displaced  by  the  edition  of  Robert 
Simson  issued  in  1756.  A  year  or  so  later  Whiston  asked 
Newton  to  be  allowed  to  print  the  Universal  arithmetic, 
manuscript  copies  of  which  were  circulating  in  the  university 
in  much  the  same  way  as  manuscripts  containing  matter  which 
has  not  yet  got  incorporated  into  text-books  do  at  the  present 
time.  Newton  gave  a  reluctant  consent,  and  it  was  published 
by  Whiston  in  1707. 

Whiston  seems  to  have  been  an  honest  and  well-meaning 
man  but  narrow,  dogmatic,  and  intolerant ;  and  having  adopted 
certain  religious  opinions  he  not  only  preached  them  on  all 
occasions,  but  he  questioned  the  honesty  of  those  who  differed 
from  him.  The  following  account  of  the  beginning  of  the  con- 
troversy is  taken  from  a  letter  of  William  Reneu  of  Jesus,  an 
undergraduate  of  the  time. 

I  have  a  peice  of  very  ill  news  to  send  you  i.e.  viz.  y*  one  Whiston  our 
Mathematicall  Professor,  a  very  learned  (and  as  we  thought  pious)  man 
has  written  a  Book  concerning  ye  Trinity  and  designs  to  print  it,  wherein 
he  sides  wth  ye  Arrians ;  he  has  showed  it  to  severall  of  his  freinds,  who 
tell  him  it  is  a  damnable,  heretical  Book  and  that,  if  he  prints  it,  he'll 
Lose  his  Professorship,  be  suspended  ab  officio  et  beneficio,  but  all  won't 
do,  he  sales,  he  can't  satisfy  his  Conscience,  unless  he  informs  ye  world 
better  as  he  thinks  than  it  is  at  present,  concerning  ye  Trinity. 

It  is  characteristic  of  the  tolerancy  of  the  Cambridge  of  the 
time  that,  although  Whiston's  opinions  were  contrary  to  the 
oath  he  had  taken  on  commencing  his  M.A.,  yet  no  public 
notice  was  taken  of  them  until  he  began  to  attack  individuals 
who  did  not  agree  with  him.  It  was  impossible  to  allow  the 
scandal  thus  occasioned  to  continue  indefinitely.  Whiston  was 
warned  and  as  he  persisted  in  going  on  he  was  in  1711  expelled 
from  his  chair.  The  details  of  his  opinions  are  now  of  no 
interest. 

After  leaving  the  university  Whiston  wrote  several  books 
on  astronomy  and  theology,  but  they  are  not  material  to  my 
purpose.  A  list  of  them  will  be  found  in  his  life.  His  trans- 


WHISTON.  85 

lation  of  Josephus  is  still  in  common  use.  He  and  Desaguliers 
gave  lectures  on  experimental  physics  illustrated  by  experi- 
ments in  or  about  1714:  these  are  said  to  have  been  the  earliest 
of  the  kind  delivered  in  London. 

An  attempt  to  prosecute  him  was  made  in  London  by  some 
clergymen ;  but  the  courts  deemed  it  vindictive,  and  strained 
the  law  to  delay  the  sentence  till  1715,  when  all  past  heresy 
was  pardoned  by  an  act  of  grace.  Whiston  rather  cleverly  made 
use  of  these  proceedings  to  push  his  opinions  aud  in  particular 
his  theory  of  the  deluge  into  general  notice :  on  one  occasion  he 
put  an  account  of  the  latter  instead  of  a  petition  into  the  legal 
pleadings  and  the  judges  discussed  it  with  great  gravity  and 
bewilderment  until  they  found  it  had  nothing  to  do  with  the 
suit.  As  so  often  happened  in  similar  cases  the  prosecution 
only  served  to  disseminate  his  opinions  and  excite  sympathy  for 
his  undoubted  honesty  aud  candour.  Queen  Caroline  who  liked 
to  see  celebrated  heretics  ordered  him  to  preach  before  her,  and 
after  the  sermon  in  talking  to  him  said  she  wished  he  would 
tell  her  of  any  faults  in  her  character,  to  which  he  replied  that 
talking  in  public  worship  was  certainly  a  prominent  one,  and 
on  her  asking  whether  there  were  any  others  he  refused  to  tell 
her  till  she  had  amended  that  one.  He  died  in  London  on 
Aug.  22,  1752. 

Intolerant,  narrow,  vain,  and  with  no  idea  of  social  pro- 
prieties :  he  was  yet  honest  and  courageous ;  and  though  not  a 
specially  distinguished  mathematician  himself,  his  services  in 
disseminating  the  discoveries  of  others  were  considerable.  His 
tenure  of  the  professorship  was  marked  by  the  publication  of 
Newton's  writings  on  algebra  and  theory  of  equations  (the 
Universal  arithmetic),  analytical  geometry  (cubic  curves),  the 
fluxional  calculus,  and  optics.  Copies  of  lectures  and  papers  in 
the  transactions  of  learned  societies  are  and  always  will  be 
inaccessible  to  many  students.  Henceforth  Newton's  mathe- 
matical works  were  open  to  all  readers,  and  the  credit  of  that  is 
partly  due  to  Whiston. 

1  See  e.g.  p.  183  of  his  memoirs. 


86  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

Whiston  was  succeeded  in  the  Lucasian  chair  by  Saunderson. 
Nicholas  Saunderson1  was  born  in  Yorkshire  in  1682,  and  be- 
came blind  a  few  months  after  his  birth.  Nevertheless  he 
acquired  considerable  proficiency  in  mathematics,  and  was  also 
a  good  classical  scholar.  When  he  grew  up  he  determined  to 
make  an  effort  to  support  himself  by  teaching,  and  attracted  by 
the  growing  reputation  of  the  Cambridge  school  he  moved  to 
Cambridge,  residing  in  Christ's  College.  There  with  the  per- 
mission of  Whiston  he  gave  lectures  on  the  Universal  arith- 
metic, Optics,  and  Princij)ia  of  Newton,  and  drew  considerable 
audiences.  His  blindness,  poverty,  and  zeal  for  the  study  of 
mathematics  procured  him  many  friends  and  pupils ;  and 
among  the  former  are  to  be  reckoned  Newton  and  Whiston. 

When  in  1711  Whiston  was  expelled  from  the  Lucasian 
chair,  queen  Anne  conferred  the  degree  of  M.A.  by  special 
patent  on  Saunderson  so  as  to  qualify  him  to  hold  that  pro- 
fessorship, and  he  continued  to  occupy  it  till  his  death  on  April 
19,  1739. 

His  lectures  on  algebra  and  fluxions  were  embodied  in 
text-books  published  posthumously  in  1740  and  1756.  The 
algebra  contains  a  description  of  the  board  and  pegs  by  the  use 
of  which  he  was  enabled  to  represent  numbers  and  perform 
numerical  calculations.  The  work  on  fluxions  contains  his 
illustrations  of  the  Principia  and  of  Cotes's  Logometria ;  and 
probably  gives  a  fair  idea  of  how  the  subject  was  treated  in  the 
Cambridge  lecture-rooms  of  the  time. 

He  is  described  by  one  of  his  pupils  as  "justly  famous  not 
only  for  the  display  he  made  of  the  several  methods  of  reason- 
ing, for  the  improvement  of  the  mind,  and  the  application  of 
mathematics  to  natural  philosophy  ;  but  by  the  reverential 
regard  for  Truth  as  the  great  law  of  the  God  of  truth,  with 
which  he  endeavoured  to  inspire  his  scholars,  and  that  peculiar 
felicity  in  teaching  whereby  he  made  his  subject  familiar  to 

1  An  account  of  his  life  is  prefixed  to  his  Algebra  published  in  two 
volumes  at  Cambridge  in  1740. 


BYRDALL.      JURIN.  87 

their  minds."  He  was  passionate,  outspoken,  and  truthful,  and 
seems  to  be  fairly  described  as  "better  qualified  to  inspire 
admiration  than  to  make  or  preserve  friends." 

I  notice  references  to  two  other  mathematicians  of  this 
time  as  having  taken  a  prominent  part  in  the  introduction 
of  the  Newtonian  philosophy,  but  I  can  find  no  particulars  of 
their  lives  or  works.  The  first  of  these  is  Thomas  Byrdall,  of 
King's  College,  who  died  in  1721,  and  is  said  to  have  not  only 
assisted  Newton  in  preparing  the  Principia  for  the  press,  but 
to  have  checked  most  of  the  numerical  calculations.  Contem- 
porary rumour  is  not  to  be  lightly  rejected,  but  I  have  never 
seen  any  evidence  for  the  statement.  The  second  of  these 
writers  is  James  Jurin,  a  fellow  of  Trinity  College,  who  was 
born  in  1684,  graduated  as  B.A.  in  1705,  and  died  in  1750. 
He  wrote  in  1732  on  the  theory  of  vision,  and  was  one  of 
the  earliest  philosophers  who  tried  to  apply  mathematics  to 
physiology.  He  took  a  prominent  part  in  the  controversies 
between  the  followers  of  Newton  and  Leibnitz,  and  in  par- 
ticular engaged  in  a  long  dispute l  with  Michelotti  on  a  question 
connected  with  the  momentum  of  running  water. 


During  this  time  the  Newtonian  philosophy  had  become 
dominant  in  the  mathematical  schools  at  Oxford :  the  Savilian 
professors  of  astronomy  being  David  Gregory  from  1691  to 
1708,  and  John  Keill  from  1708  to  1721;  and  the  Savilian 
professors  of  geometry  being  Wallis  (see  p.  42)  till  1703, 
and  thence  till  1720  Edmund  Halley;  but  mathematics  was 
still  an  exotic  study  there,  and  the  majority  of  the  residents 
regarded  mathematics  and  puritanism  as  allied  and  equally 
unholy  subjects.  Jri  London  the  Newtonian  philosophy  was 
worthily  represented  by  Abraham  de  Moivre  and  by  Brook 
Taylor,  while  Newton  himself  regularly  presided  at  the  meet- 
ings of  the  Royal  Society. 

1  See  Philosophical  transactions  vols.  LX.  to  LXVI. 


88  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

The  only  one  of  those  immediately  above  mentioned  who 
came  from  Cambridge  was  Brook  Taylor1,  who  was  born  at 
Edmonton  on  Aug.  18,  1685,  and  died  in  London  on  Dec.  29, 
1731.  He  entered  at  St  John's  College  in  1705,  and  graduated 
as  LL.B.  in  1709.  After  taking  his  degree  he  went  to  live  in 
London,  and  from  the  year  1708  onwards  he  wrote  numerous 
papers  in  the  Philosophical  transactions,  in  which  among  other 
things  he  discussed  the  motion  of  projectiles,  the  centre  of 
oscillation,  and  the  forms  of  liquids  raised  by  capillarity.  He 
wrote  on  linear  perspective,  two  volumes,  1715  and  1719.  But 
the  work  by  which  he  is  generally  known  is  his  Methodus 
incrementorum  directa  et  inversa  published  in  1715.  This  con- 
tained the  enunciation  and  a  proof  of  the  well-known  theorem 

f(x  +  h)  =/(*)  +  hf  (x)  +  j^/"  (*)  +  ..., 

by  which  any  function  of  a  single  variable  can  be  expanded. 
He  did  not  consider  the  convergency  of  the  series,  and  the 
proof,  which  contains  numerous  assumptions,  is  not  worth  re- 
producing. In  this  treatise  he  also  applied  the  calculus  to  various 
physical  problems,  and  in  particular  to  the  theory  of  the  trans- 
verse vibrations  of  strings. 

Regarded  as  mathematicians,  Whiston,  Laugh  ton,  and 
Saunderson  barely  escape  mediocrity,  but  their  contemporary 
Cotes,  of  whom  I  have  next  to  speak,  was  a  mathematician  of 
exceptional  power,  and  his  early  death  was  a  serious  blow  to 
the  Cambridge  school.  The  remark  of  Newton  that  if  only 
Cotes  had  lived  "we  should  have  learnt  something"  indicates 
the  opinion  of  his  abilities  generally  held  by  his  contempora- 
ries. 

Roger  Cotes2  was  born  near  Leicester  on  July  10,  1682. 
He  entered  at  Trinity  in  1699,  took  his  B.A.  in  1703,  and  in 

1  An  account  of  his  life  by  Sir  William  Young  is  prefixed  to  the 
Contemplatio  philosophica,  London,  1793. 

2  See  the  Biographia  Britannica,  second  edition,  London,  1778 — 93, 
and  also  the  Dictionary  of  national  biography. 


COTES.  89 

1705  was  elected  to  a  fellowship.  In  1704  Dr  Plume,  the  arch- 
deacon of  Rochester  and  formerly  of  Christ's  College  (bachelor 
of  theology,  1661),  founded  a  chair  of  astronomy  and  experi- 
mental philosophy.  The  first  appointment  was  made  in  1707, 
and  Cotes  was  elected1.  Whiston  was  one  of  the  electors,  and 
he  writes,  "I  was  the  only  professor  of  mathematics  directly 
concerned  in  the  choice,  so  my  determination  naturally  had  its 
weight  among  the  rest  of  the  electors.  I  said  that  I  pretended 
myself  to  be  not  much  inferior  in  mathematics  to  the  other  can- 
didate's master,  Dr  Harris,  but  confessed  that  I  was  but  a  child 
to  Mr  Cotes  :  so  the  votes  were  unanimous  for  him2."  Newton, 
to  whom  Bentley  had  introduced  Cotes,  also  wrote  a  very  strong 
testimonial  in  his  favour. 

Bentley  at  once  urged  the  new  professor  to  establish  an 
astronomical  observatory  in  the  university.  The  university 
gave  no  assistance,  but  Trinity  College  consented  to  have  one 
erected  on  the  top  of  the  Great  Gate,  and  to  allow  the  Plumian 
professor  to  occupy  the  rooms  in  connection  with  it ;  consider- 
able subscriptions  were  also  raised  in  the  college  to  provide 
apparatus.  The  observatory  was  pulled  down  in  1797. 

In  1709  Newton  was  persuaded  to  allow  Cotes  to  prepare 
the  long-talked-of  second  edition  of  the  Principia.  The  first 
edition  had  been  out  of  print  by  1690;  but  though  Newton  had 
collected  some  materials  for  a  second  and  enlarged  edition,  he 
could  not  at  first  obtain  the  requisite  data  from  Flamsteed,  the 
astronomer-royal,  and  subsequently  he  was  unable  or  unwill- 
ing to  find  the  time  for  the  necessary  revision.  The  second 
edition  was  issued  in  March  1713,  but  a  considerable  part  of  the 

1  The   successive  professors  were  as  follows.     From  1707  to  1716, 
Koger  Cotes  of  Trinity;  from  1716  to  1760,  Eobert  Smith  of  Trinity  (see 
p.. 91);  from  1760  to  1796,  Anthony  Shepherd  of  Christ's  (see  p.  103); 
from  1796  to  1822,  Samuel  Vince  of  Caius  (see  p.  103) ;  from  1822  to  1828, 
Robert  Woodhouse  of  Caius  (see  p.  118) ;  from  1828  to  1836,  Sir  George  B. 
Airy  of  Trinity  (see  p.  132) ;  from  1836  to  1883,  James  Challis  of  Trinity 
(see  p.  132) ;  who  in  1883  was  succeeded  by  G.  H.  Darwin  of  Trinity,  the 
present  professor. 

2  See  p.  133  of  Whiston's  Memoirs. 


90  THE   RISE   OF  THE  NEWTONIAN  SCHOOL. 

new  work  contained  in  it  was  due  to  Cotes  and  not  to  Newton. 
The  whole  correspondence  between  Newton  and  Cotes  on  the 
various  alterations  made  in  this  edition  is  preserved  in  the 
library  of  Trinity  College.  Cambridge  :  it  was  edited  by  Edle- 
ston  for  the  college  in  1850.  This  edition  was  sold  out  within 
a  few  months,  but  a  reproduction  published  at  Amsterdam 
supplied  the  demand.  Cotes  himself  died  on  June  5,  1716, 
shortly  after  the  completion  of  this  work. 

He  is  described  as  possessing  an  amiable  disposition,  an 
imperturbable  temper,  and  a  striking  presence;  and  he  was  cer- 
tainly loved  and  regretted  by  all  who  knew  him. 

His  writings  were  collected  and  published  in  1722  under 
the  titles  Harmonia  mensurarum  and  Opera  miscellanea.  His 
professorial  lectures  on  hydrostatics  were  published  in  1738. 
A  large  part  of  the  Harmonia  mensurarum  is  given  up  to  the 
decomposition  and  integration  of  rational  algebraical  expres- 
sions ;  that  part  which  deals  with  the  theory  of  partial 
fractions  was  left  unfinished,  but  was  completed  by  de  Moivre. 
Cotes's  theorem  in  trigonometry  which  depends  on  forming  the 
quadratic  factors  of  xn  -  1  is  well  known.  The  proposition 
that  "  if  from  a  fixed  point  0  a  line  be  drawn  cutting  a  curve 
in  $i»  Qz-  Qn>  and  a  point  P  be  taken  on  it  so  that  the 
reciprocal  of  OP  is  the  arithmetic  mean  of  the  reciprocals  of 
OQi,  OQ2,...OQn,  then  the  locus  of  P  will  be  a  straight  line"  is 
also  due  to  Cotes.  The  title  of  the  book  was  derived  from  the 
latter  theorem.  The  Opera  miscellanea  contains  a  paper  on 
the  method  for  determining  the  most  probable  result  from  a 
number  of  observations :  this  was  the  earliest  attempt  to 
frame  a  theory  of  errors.  It  also  contains  essays  on  Newton's 
Methodus  differ entialis,  on  the  construction  of  tables  by  the 
method  of  differences,  on  the  descent  of  a  body  under  gravity, 
on  the  cycloidal  pendulum,  and  on  projectiles. 

It  was  unfortunate  for  Cotes's  reputation  that  his  friend 
Brook  Taylor  stated  the  property  of  the  circle  which  Cotes  had 
discovered  as  a  challenge  to  foreign  mathematicians  in  a 
manner  which  was  somewhat  offensive.  John  Bernoulli  solved 


SMITH.  91 

the  question  proposed  in  1719,  and  his  friends  seized  on  his 
triumph  as  a  convenient  opportunity  for  shewing  their  dislike 
of  Newton  by  depreciating  Cotes. 

,  The  study  of  mathematics  in  the  different  colleges  received 
at  this  time  a  considerable  stimulus  by  the  establishment  in 
1710  of  certain  lectureships  by  Lady  Sadler.  On  the  advice  of 
William  Croone  (born  about  1629  and  died  in  1684),  a  fellow 
of  Emmanuel  and  professor  of  rhetoric  at  Gresham  College,  she 
gave  to  the  university  an  estate  of  which  the  income  was  to  be 
divided  amongst  the  lecturers  on  algebra  at  certain  colleges. 
This  no  doubt  helped  to  promote  the  interest  in  that  subject 
/£•  *  during  the  seventeenth  century.  With  the  advance  in  the 
standard  of  education  it  ceased  to  be  productive  of  much 
benefit,  and  in  1860  it  was  changed  into  a  professorship  of 
pure  mathematics  ;  in  1863  Arthur  Cayley  of  Trinity  was 
appointed  professor. 

Cotes  was  succeeded  as  Plumian  professor  by  his  cousin 
Robert  Smith.  Robert  Smith  was  born  in  1689,  entered  at 
Trinity  in  1707,  took  his  B.A.  in  1711,  and  was  elected  to  a 
fellowship  in  the  following  year.  He  held  the  office  of  master 
of  mechanics  to  the  king.  As  Plumian  professor  he  lectured 
on  optics  and  hydrostatics,  and  subsequently  he  wrote  text- 
books on  both  those  subjects.  His  Opticks  published  in  1728 
is  one  of  the  best  text-books  on  the  subject  that  has  yet 
appeared,  and  with  a  few  additions  might  be  usefully  reprinted 
now.  He  also  published  in  1744  a  work  on  sound,  entitled 
Harmonics,  which  contains  the  substance  of  lectures  he  had  for 
many  years  been  giving.  He  edited  Cotes's  works.  He  was 
made  master  of  Trinity  in  1742,  and  died  at  Cambridge  on 
Feb.  2,  1768.  He  founded  by  his  will  two  annual  prizes  for 
proficiency  in  mathematics  and  natural  philosophy,  to  be  held 
by  commencing  bachelors  and  known  by  his  name.  They 
proved  productive  of  the  best  results,  and  at  a  later  time  they 
enabled  the  university  to  encourage  some  of  the  higher 
branches  of  mathematics  which  did  not  directly  come  into  the 
university  examinations  for  degrees. 


92  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

The  labours  of  Laughton,  Bentley,  Whiston,  Saunderson, 
Cotes,  and  Smith  were  rewarded  by  the  definite  establishment 
about  the  year  1730  of  the  Newtonian  philosophy  in  the 
schools  of  the  university.  The  earliest  appearance  of  that 
philosophy  in  the  scholastic  exercises  is  the  act  kept  by 
Samuel  Clarke  in  1694  and  above  alluded  to.  Ten  years  later 
it  was  not  unusual  to  keep  one  act  from  Newton's  writings ;  but 
from  1730  onwards  it  was  customary  to  require  at  least  one  dis- 
putation to  be  on  a  mathematical  subject — usually  on  Newton — 
and  in  general  to  expect  one  to  be  on  a  philosophical  thesis, 
although  after  1750  it  was  possible  to  propose  mathematical 
questions  only.  The  decade  from  1725  to  1735  is  an  important 
one  in  a  history  of  mathematics  at  Cambridge,  not  only  for  the 
reasons  given  above,  but  because  the  mathematical  tripos,  which 
profoundly  affected  the  subsequent  development  of  mathe- 
matics in  the  university,  originated  then.  The  history  of  the 
origin  and  growth  of  that  examination  may  be  left  for  the 
present.  The  death  of  Newton  and  the  retirement  or  death  of 
nearly  all  those  who  had  been  brought  under  his  direct  in- 
fluence also  fall  within  this  decade,  and  it  thus  naturally  marks 
the  conclusion  of  this  chapter. 


The  effect  of  the  teaching  of  the  above-mentioned  mathema- 
ticians in  extending  the  range  of  reading  is  shewn  by  the  fol- 
lowing list  of  mathematical  text-books  which  were  in  common 
use  by  the  year  1730.  The  dates  given  are  those  of  the  first 
editions,  but  in  most  cases  later  editions  had  been  issued  incor- 
porating the  discoveries  of  subsequent  writers. 

First,  for  the  subjects  of  pure  mathematics.  The  usual 
text-books  on  pure  geometry  were  the  Elements  of  Euclid  (edi- 
tions of  Barrow,  Gregory,  or  Whiston),  the  Conies  of  Apollonius 
(Halley's  edition,  1710),  or  of  de  LahirG  (1685),  to  which  we 
may  perhaps  add  the  fourth  and  fifth  sections  of  the  first  book 
of  the  Principia.  [Simson's  Conies  was  published  in  1735, 
and  became  the  recognized  text-book  for  that  subject  for  the 


MATHEMATICAL   TEXT-BOOKS.  93 

remainder  of  the  eighteenth  century.]  The  usual  text-book  on 
arithmetic  was  Oughtred's  Clavis,  or  E.  Wingate's  Arithmetic 
(1630).  The  usual  text-books  on  algebra  were  those  by  Harriot, 
Oughtred,  Wallis,  and  Newton  (Universal  arithmetic).  The 
usual  text-books  on  trigonometry  were  those  by  Oughtred 
(the  Clavis),  Seth  Ward  (1654),  Caswell  (1685),  and  E. 
Wells  (1714).  The  usual  text-books  on  analytical  geometry 
were  those  by  Wallis  (1665),  and  Maclauriu  (1720).  The  usual 
text-books  on  the  infinitesimal  calculus  were  those  by  Humphry 
Ditton  (1704),  W.  Jones  (1711),  and  Brook  Taylor  (1715). 

Next  for  the  subjects  of  applied  mathematics.  I  know  of 
no  work  on  mechanics  of  this  time  suitable  for  students  other 
than  the  treatises  -by  Stevinus,  Huygens,  and  Wallis,  and  the 
introduction  to  the  Principia :  no  one  of  these  is  what  we 
should  call  a  text-book. 

Geometrical  optics  was  generally  studied  in  the  pages  of 
Newton,  Gregory  (1695),  or  Robert  Smith  (1728).  In  elementary 
hydrostatics  a  translation  of  a  text-book  by  Mariotte  was  used, 
but  copies  or  notes  of  the  lectures  of  Cotes  and  Whiston  were 
probably  accessible.  The  elements  of  both  the  last-named  and 
other  physical  subjects  were  also  read  in  W.  J.  'sGravesande's 
work  (published  in  1720  and  translated  by  Desaguliers  in 
1738).  The  mathematical  treatment  of  the  higher  parts  of  the 
subject,  if  studied  at  all,  was  read  in  the  edition  of  Newton's 
lectures. 

There  were  numerous  works  on  astronomy  in  common  use. 
Selected  portions  of  the  Principia,  Clarke's  translation  and 
commentary  on  Rohault,  and  Kepler's  writings  were  read  by 
the  more  advanced  students,  but  I  suspect  that  most  men  con- 
tented themselves  with  one  or  more  of  the  popular  summaries  of 
which  several  were  then  in  circulation — one  of  the  best  being 
that  by  David  Gregory  (1702). 

Of  course  a  much  longer  list  of  text-books  then  obtainable 
might  be  drawn  up,  but  I  think  the  above  includes  all,  or  nearly 
all,  the  books  then  in  common  use.  I  believe  the  writings  of 
Leibnitz,  the  Bernoullis,  and  their  immediate  followers  were 


94 


THE   RISE   OF   THE    NEWTONIAN    SCHOOL. 


but  rarely  consulted,  though  they  probably  were  included  in 
the  more  important  mathematical  libraries  of  the  time.  I  may 
here  add  that  the  libraries  of  Cotes  and  Robert  Smith  are  both 
preserved  in  Trinity. 

Two  tutors  of  a  somewhat  earlier  date  drew  out  time  tables 
shewing  the  order  in  which  the  subjects  should  be  read,  accom- 
panied by  a  list  of  the  books  in  common  use.  They  are  pub- 
lished in  the  third  and  fourth  appendices  to  the  Scholae  aca- 
demicae,  from  which  the  following  account  is  condensed. 

In  the  Student's  guide  written  about  1706  by  Daniel 
Waterland,  a  fellow  and  subsequently  master  of  Magdalene 
College,  the  following  course  of  reading  in  "philosophical 
studies"  is  recommended  :  Waterland  adds  that  by  January 
and  February  he  means  the  two  first  months  of  residence  and 
not  necessarily  the  calendar  months  named.  It  will  be  noticed 


First  year 

Second  year 

Third  year 

Fourth  year 

Jan. 
Feb. 

Wells's  Arithm. 

Wells's  Astron. 
Locke. 

Burnet's  Theo- 
ry with  Keill's 
Remarks. 

Baronius's 
Metaphysicks. 

March 
April 

Euclid's  Elem. 

Locke's    Hum. 
Und. 
De  la  Hire  Con. 
Sect. 

Whiston's 
Theory    with 
Keill's       Re- 
marks. 

Newton's 
Opticks. 

May 
June 

Euclid's  Elem. 
Burgersdicius's 
Logick. 

Whiston's 
Astron. 

Wells's  Chron. 
Beveridge's 
Chronology. 

Whiston's 
Praelect. 
Phys.  Math. 

July 

Aug. 

Euclid's  Elem. 
Burgersdicius. 

Keil's      Intro- 
duction. 

Whitby's  Eth. 
Puffendorfs 
Law  of  Nat. 

Gregory's 
Astronomy. 

Sept. 
Oct. 

Wells's  Geogr. 

Cheyne's  Phil. 
Principles. 

Puffendorf. 
Grotius  de  Jure 
Belli. 

Nov. 
Dec. 

Wells's  Trig. 
Newton's  Trig. 

Renault's 
Physics. 

Puffendorf. 
Grotius. 

MATHEMATICAL    TEXT-BOOKS.  95 

that  a  mathematician  was  expected  to  read  the  elements  of 
various  sciences,  and  the  curriculum  was  not  a  narrow  one. 

Waterland  remarks  on  this  course  that  Hammond's  Algebra, 
Wells's  Mechanics,  and  Wells's  Optics  should  also  be  added  at 
some  time  in  the  first  three  years.  Further,  a  bachelor  if  he  did 
not  intend  to  take  orders  should  before  proceeding  to  the  M.A. 
degree  read  Newton's  Principia,  Ozanam's  Cursus,  Sturmius's 
Works,  Huygens's  Works,  New  ton's  Algebra,  and  Milnes's  Conic 
sections. 

In  a  third  edition  issued  in  1740  the  Arithmetic,  Trigono- 
metry, and  Astronomy  of  Wells  are  respectively  replaced  by 
Wingate's  Arithmetic,  Keill's  Trigonometry,  and  Harris's  Astro- 
nomy •  Simpson's  Conies  is  substituted  for  that  by  de  la  Hire  ; 
Bartholin's  Physics  is  to  be  read  as  well  as  Renault's ;  finally 
Winston's  Astronomy  is  struck  out  and  Milnes's  Conic  sections 
recommended  to  be  then  read.  Besides  these  the  attention 
of  the  student  is  directed  to  Maclaurin's  Algebra,  Simpson's 
Algebra,  and  Huygens's  Planetary  worlds. 

A  somewhat  similar  course  was  sketched  out  in  1707  by 
Robert  Green,  a  fellow  and  tutor  of  Clare,  who  took  his  B.A. 
in  1699  and  died  in  1730.  Green  was  almost  the  last  Cantab 
of  any  position  who  rejected  the  Newtonian  theory  of  physical 
astronomy.  He  recommended  his  pupils  to  spend  the  first 
year  on  the  study  of  classics :  the  second  on  logic,  ethics,  geo- 
metry (Euclid,  Sturmius,  Pardies,  or  Jones),  arithmetic  (Wells, 
Tacquet,  or  Jones),  algebra  (Pell,  Wallis,  Harriot,  Kersey, 
Newton,  Descartes,  Harris,  Oughtred,  Ward,  or  Jones),  and 
corpuscular  philosophy  (Descartes,  Rohault,  Yarenius,  Le  Clerk, 
or  Boyle):  the  third  on  natural  science,  optics  (Gregory, 
Rohault,  Dechales,  Barrow,  NEWTON,  Descartes,  Huygens, 
Kepler,  or  Molyneux),  and  conic  sections  and  other  curves  (De 
Witt,  De  Lahk»o,  Sturmius,  L'Hospital,  Newton,  Milnes,  or  •C0"  ^ 
Wallis):  the  fourth  year  on  mechanics  of  solids  and  fluids 
(Marriotte,  Keill,  Huygens,  Sturmius,  Boyle,  Newton,  Ditton, 
Wallis,  Borellus,  or  Halley),  fluxions  and  infinite  series  (Wallis, 
Newton,  Raphson,  Hays,  DITTON,  Jones,  Nieuwentius,  or 


96  THE   RISE   OF   THE   NEWTONIAN   SCHOOL. 

L'Hospital),  astronomy  (Gassendi,  Mercator,  BULLIALDUS, 
Horrocks,  Flamsteed,  Newton,  Gregory,  Whiston,  or  Kepler), 
and  logarithms  and  trigonometry  (Sturmius,  Briggs,  Vlacq, 
Gellibrand,  Harris,  Mercator,  Jones,  Newton,  or  Caswell). 
The  authors  whose  names  are  printed  in  small  capitals  are 
those  specially  recommended.  The  order  in  which  the  subjects 
are  to  be  taken  is  curious. 


CHAPTER  VI. 
THE  LATER  NEWTONIAN  SCHOOL. 

CIRC.  1730—1820. 

I  HAVE  already  explained  that  the  results  of  the  infinite- 
simal calculus  may  be  expressed  in  either  of  two  notations. 
In  most  modern  books  both  are  used,  but  if  we  must  confine 
ourselves  to  one  then  that  adopted  by  Leibnitz  is  superior  to 
that  used  by  Newton,  and  for  some  applications — such  as  the 
calculus  of  variations — is  almost  essentia].  The  question  as 
to  the  relative  merits  of  the  two  methods  was  unfortunately 
mixed  up  with  the  question  as  to  whether  Leibnitz  had  dis- 
covered the  fundamental  ideas  of  the  calculus  for  himself,  or 
whether  he  had  acquired  them  from  Newton's  papers,  some  of 
which  date  back  to  1666.  Personal  feelings  and  even  national 
jealousies  were  appealed  to  by  both  sides.  Finally  Newton's 
notation  was  generally  adopted  in  England,  while  that  invented 
by  Leibnitz  was  employed  by  most  continental  mathematicians. 
The  latter  result  was  largely  due  to  the  influence  of  John 
Bernoulli,  the  most  famous  and  successful  mathematical 
teacher  of  his  age,  who  through  his  pupils  (especially  Euler) 
determined  the  lines  on  which  mathematics  was  developed  on 
the  continent  during  the  larger  part  of  the  eighteenth  century. 

A  common  language  and  facility  of  intercommunication  of 

ideas  are  of  the  utmost  importance  in  science,  and  even  if  the 

Cambridge  school  had  enjoyed  the  use  of  a  better  notation  than 

their  continental  contemporaries  they  would  have  lost  a  great 

B.  7 


98          THE  LATER  NEWTONIAN  SCHOOL. 

deal  by  their  isolation.  So  little  however  did  they  realize  this 
truth  that  they  made  no  serious  efforts  to  keep  themselves 
acquainted  with  the  development  of  analysis  by  their  neigh- 
bours. On  the  continent  on  the  other  hand  the  results  arrived 
at  by  Newton,  Taylor,  Maclaurin,  and  others  were  translated 
from  the  fluxional  into  the  differential  notation  almost  as  soon 
as  they  were  published ;  to  this  I  should  add  that  the  journals 
and  transactions  in  which  continental  mathematicians  embodied 
their  discoveries  were  circulated  over  a  very  wide  area  and 
large  numbers  of  them  were  distributed  gratuitously. 

The  use  of  the  differential  notation  may  be  taken  as  defi- 
nitely adopted  on  the  continent  about  the  year  1730.  The 
separation  of  the  Newtonian  school  from  the  general  stream 
of  European  thought  begins  to  be  observable  about  that  time, 
and  explains  why  I  closed  the  last  chapter  at  that  date. 

Modern  analysis  is  derived  from  the  writings  of  Leibnitz 
and  John  Bernoulli  as  interpreted  by  d'Alembert,  Euler,  La- 
grange,  and  Laplace.  Even  to  the  end  the  English  school  of 
the  latter  half  of  the  eighteenth  century  never  brought  itself 
into  touch  with  these  writers.  Its  history  therefore  leads  no- 
where, and  hence  it  is  not  necessary  to  discuss  it  at  any  great 
length. 

The  isolation  of  the  later  Newtonian  school  would  suffi- 
ciently account  for  the  rapid  falling  off  in  the  quality  of  the 
work  produced,  but  the  effect  was  intensified  by  the  manner  in 
which  its  members  confined  themselves  to  geometrical  demon- 
strations. If  Newton  had  given  geometrical  proofs  of  most  of 
the  theorems  in  the  Principia  it  was  because  their  validity 
was  unimpeachable,  and  as  his  results  were  opposed  to  the 
views  then  prevalent  he  did  not  wish  the  discussion  as  to  their 
truth  to  turn  on  the  correctness  of  the  methods  used  to  demon- 
strate them.  But  his  followers,  long  after  the  principles  of 
the  infinitesimal  calculus  had  been  universally  recognized  as 
valid,  continued  to  employ  geometrical  proofs  wherever  it  was 
possible.  These  proofs  are  elegant  and  ingenious,  but  it  is 
necessary  to  find  a  separate  kind  of  demonstration  for  every 


THE   LATER   NEWTONIAN    SCHOOL.  99 

distinct  class  of  problems  so  that  the  processes  are  not  nearly 
so  general  as  those  of  analysis. 

During  the  whole  of  the  period  treated  in  this  chapter  only 
two  mathematicians  of  the  first  rank  can  be  claimed  for  the 
Newtonian  school.  These  were  Maclaurin  in  Scotland  and 
Clairaut  in  France  :  the  latter  being  the  sole  distinguished 
foreigner  who  by  choice  used  the  Newtonian  geometrical 
methods.  Neither  of  them  had  any  special  connection  with 
Cambridge.  Waring  might  perhaps  under  more  favourable 
circumstances  have  taken  equal  rank  with  them,  but  except 
for  him  I  can  recall  the  names  of  no  Cambridge  men  whose 
writings  at  this  distance  of  time  are  worth  more  than  a  passing 
notice. 

Although  the  quality  of  the  mathematical  work  produced 
in  this  period  was  so  mediocre  yet  the  number  of  eminent 
lawyers  educated  in  the  mathematical  schools  of  Cambridge 
was  extraordinarily  large.  Many  careful  observers  have  as- 
serted that  in  the  majority  of  cases  a  mathematical  training 
affords  the  ideal  general  education  which  a  lawyer  should  have 
before  he  begins  to  read  law  itself.  A  study  of  analytical 
mathematics  is  among  the  best  instruments  for  training  the 
reasoning  faculties,  and  for  many  students  it  provides  the  best 
available  preliminary  education  for  a  scientific  lawyer;  but  I 
doubt  if  it  has  that  special  fitness  which  geometry  and  the  use 
of  geometrical  methods  seem  to  possess  for  the  purpose. 

Throughout  the  time  considered  in  this  chapter  the  New- 
tonian philosophy  was  dominant  in  the  schools  of  the  university, 
but  the  senate-house  examination  gradually  took  the  place  of 
the  scholastic  exercises  as  the  real  test  of  a  man's  abilities.  An 
account  of  those  exercises  and  of  the  origin  and  development 
of  the  mathematical  tripos  is  given  in  chapters  ix.  and  x. 
I  will  merely  here  remark  that  the  tripos  (then  known  as 
the  senate-house  examination)  became  by  the  middle  of  the 
eighteenth  century  the  only  avenue  to  a  degree,  and  that  all 
undergraduates  from  that  time  forward  had  to  read  at  least 
the  elements  of  mathematics. 

7—2 


100  THE   LATER   NEWTONIAN   SCHOOL. 

Of  course  geometry,  algebra,  and  the  fluxional  calculus 
were  read  by  all  mathematical  students  ;  but  the  subjects  which 
attracted  most  attention  during  this  time  were  astronomy  and 
optics.  The  papers  in  the  transactions  of  the  Royal  Society 
and  the  problems  published  in  the  form  of  challenges  in  the 
pages  of  the  Ladies'  diary  (1707 — 1817)  and  other  similar 
publications  will  give  a  fair  idea  of  the  kind  of  questions  that 
excited  most  interest  in  England.  If  any  one  will  compare 
these  with  the  papers  then  being  published  on  the  continent 
by  d'Alembert,  Euler,  Lagrange,  Laplace,  Legendre,  Gauss, 
and  others  he  will  not  I  think  blame  me  for  making  my 
account  of  the  Cambridge  mathematical  school  of  this  time 
little  else  than  a  list  of  names. 

I  shall  first  consider  very  briefly  the  mathematical  pro- 
fessors of  this  time,  and  shall  then  similarly  enumerate  a  few 
other  contemporary  mathematicians  and  physicists. 


I  begin  then  by  mentioning  the  professors. 

The  occupants  of  the  Lucasiau  chair  were  successively 
John  Colson,  Edward  Waring,  and  Isaac  Milner.  Saunderson 
died  in  1739,  and  was  succeeded  by  Colson.  John  Colson1  was 
born  at  Lichfield  in  1680.  In  1707  he  communicated  a  paper 
to  the  Royal  Society  on  the  solution  of  cubic  and  biquadratic 
equations.  He  was  then  a  schoolmaster,  and  having  acquired 
some  reputation  as  a  successful  teacher  was  recommended  by 
Robert  Smith  the  master  of  Trinity  to  come  to  Cambridge  and 
lecture  there.  He  had  rooms  in  Sidney,  but  apparently  was 
not  a  member  of  that  college :  subsequently  he  moved  to 
Emmanuel,  whence  he  took  his  M.A.  degree  in  1728.  While 
residing  there  he  contributed  a  paper  on  the  principles  of 
algebra  to  the  Philosophical  transactions,  1726. 

He   then   accepted   a    mastership  at  Rochester  grammar- 

1  No  contemporary  biography  of  Colson  is  extant ;  but  nearly  all  the 
known  references  to  him  have  been  collected  in  the  Dictionary  of 
national  biography. 


COLSON.     WARING.  101 

school.  In  1735  he  wrote  a  paper  on  spherical  maps1;  and  in 
1736  he  published  the  original  manuscript  of  Newton  on 
fluxions,  together  with  a  commentary  (see  pp.  70,  71). 

*  When  a  candidate  for  the  Lucasian  chair  in  1739  he  was 
opposed  by  Abraham  de  Moivre,  who  was  admitted  a  member 
of  Trinity  College  and  created  M.A.  to  qualify  him  for  the 
office.  Smith  really  decided  the  election,  and  as  de  Moivre 
was  very  old  and  almost  in  his  dotage  he  pressed  the  claims  of 
Colson.  The  appointment  was  admitted  to  be  a  mistake,  and 
even  Cole,  who  was  a  warm  friend  of  Colson,  remarks  that  the 
latter  merely  turned  out  to  be  "a  plain  honest  man  of  great 
industry  and  assiduity,  but  the  university  was  much  disap- 
pointed in  its  expectations  of  a  professor  that  was  to  give  credit 
to  it  by  his  lectures."  Colson  died  at  Cambridge  on  Jan.  20, 
1760. 

Besides  the  papers  sent  to  the  Royal  Society  enumerated 
above  and  his  edition  of  Newton's  Fluxions,  Colson  wrote  an 
introductory  essay  to  Saunderson's  Algebra,  1740,  and  made  a 
translation  of  Agnesi's  treatise  on  analysis:  he  completed  the 
latter  just  before  his  death,  and  it  was  published  by  baron 
Maseres  in  1801. 

Colson  was  succeeded  in  1760  by  Waring,  a  fellow  of  Mag- 
dalene. Edward  Waring  was  born  near  Shrewsbury  in  1736, 
took  his  B.A.  as  senior  wrangler  in  1757,  and  died  on  Aug. 
15,  1798.  He  is  described  as  being  a  man  of  unimpeach- 
able honour  and  uprightness  but  painfully  shy  and  diffident. 
The  rival  candidate  for  the  Lucasian  chair  was  Maseres;  and 
as  Waring  was  not  then  of  standing  to  take  the  M.A.  degree 
he  had  to  get  a  special  license  from  the  crown  to  hold  the 
professorship. 

Waring  wrote  Miscellanea  analytica,  issued  in  1762,  Medi- 
tationes  algebraicae,  issued  in  1770,  Proprietates  algebraicarum 
curvarum,  issued  in  1772;  and  Meditationes  analyticae,  issued 
in  1776.  The  first  of  these  is  on  algebra  and  analytical  geometry, 

1  Philosophical  transactions  1735. 


102  THE   LATER  NEWTONIAN   SCHOOL. 

and  includes  some  papers  published  when  he  was  a  candidate 
for  the  Lucasian  chair  as  a  proof  of  his  fitness  for  the  post. 
The  third  of  these  works  is  that  which  is  most  celebrated  :  it 
contains  several  results  that  were  previously  unknown.  From 
a  cursory  inspection  of  these  writings  I  think  they  shew  con- 
siderable power,  but  the  classification  and  arrangement  of 
them  are  imperfect. 

Waring  contributed  numerous  papers  to  the  Philosophical 
transactions.  Most  of  these  are  on  the  summation  of  series, 
but  in  one  of  them,  read  in  1778,  he  enunciated  a  general 
method  for  the  solution  of  an  algebraical  equation  which  is 
still  sometimes  inserted  in  text-books ;  his  rule  is  correct  in 
principle  but  involves  the  solution  of  a  subsidiary  equation 
which  is  sometimes  of  a  higher  order  than  the  equation  origi- 
nally proposed.  Papers  by  him  on  various  algebraical  problems 
will  be  found  in  the  Philosophical  transactions  for  1763,  1764, 
1779,  1784,  1786,  1787,  1788,  1789,  and  1791. 

In  a  reply  to  some  criticisms  which  had  been  made  on  the 
first  of  the  above-mentioned  works  he  enunciated  the  celebrated 
theorem  that  if  p  be  a  prime  then  1  +  p  —  1  is  a  multiple  of  p\ 
for  this  result  he  was  indebted  to  one  of  his  pupils,  John 
Wilson,  who  was  then  an  undergraduate  at  Peterhouse.  Wilson 
was  born  in  Cumberland  on  Aug.  6,  1741,  graduated  as 
senior  wrangler  in  1761,  and  subsequently  took  pupils.  He 
was  a  good  teacher  and  made  his  pupils  work  hard,  but  some- 
times when  they  came  for  their  lessons  they  found  the  door 
sported  and  'gone  a  fishing'  written  on  the  outside,  which 
Paley  (who  was  one  of  them)  deemed  the  addition  of  insult 
to  injury,  for  he  was  himself  very  fond  of  that  sport.  Wilson 
later  went  to  the  bar,  and  was  appointed  a  justice  in  the 
Common  Pleas.  He  died  at  Kendal  on  Oct.  18,  1793. 

Waring  was  succeeded  in  1798  by  Milner,  who  was  then 
professor  of  natural  philosophy,  master  of  Queens'  College, 
and  dean  of  Carlisle.  Isaac  Milner1  was  born  at  Leeds  in 

1  His  life  has  been  written  by  Mary  Milner,  London,  1842. 


104  THE   LATER   NEWTONIAN   SCHOOL. 

English  observers  :    this  was  preceded  by  a  work  on  practical 
astronomy  issued  in  1790. 

He  also  wrote  text-books  on  conic  sections,  algebra,  tri- 
gonometry, fluxions,  the  lever,  hydrostatics,  and  gravitation, 
which  form  part  of  a  general  course  of  mathematics :  these 
were  all  published  or  reissued  in  1805  or  1806,  and  for  a  short 
time  were  recognised  as  standard  text-books  for  the  tripos  ; 
but  they  are  badly  arranged  and  were  superseded  by  the  works 
of  Wood.  His  treatise  on  fluxions  first  published  in  1805 
went  through  numerous  editions,  and  is  one  of  the  best  ex- 
positions of  that  method.  In  it,  however,  as  in  all  the 
Cambridge  works  of  that  time,  he  used  x  to  denote,  not  the 
fluxion  of  x,  but  the  increment  of  x  generated  in  a  small  time ; 
that  is  what  Newton  would  have  written  as  xo.  He  asserts 
that  "this  is  agreeable  to  Sir  I.  Newton's  ideas  on  the 
subject,"  and  "as  the  velocities  are  in  proportion  to  the  in- 
crements or  decrements  which  would  be  generated  in  a  given 
time,  if  at  any  instant  the  velocities  were  to  become  uniform, 
such  increments  or  decrements  will  represent  the  fluxions  at 
that  instant1."  He  also  used  the  symbol  of  integration  (see 

P-  71). 

A  public  advertisement  of  his  lectures  for  1802  is  as 
follows. 

The  lectures  are  experimental,  comprising  mechanics,  hydrostatics, 
optics,  astronomy,  magnetism,  and  electricity;  and  are  adapted  to  the 
plan  usually  followed  by  the  tutors  in  the  university.  All  the  funda- 
mental propositions  in  the  first  four  branches,  are  proved  by  experiments, 
and  accompanied  with  such  explanations  as  may  be  useful  to  the 
theoretical  student.  Various  machines  and  philosophical  instruments 
are  exhibited  in  the  course  of  the  lectures,  and  their  construction  and 
use  explained.  And  in  the  two  latter  branches  a  set  of  experiments  are 
instituted  to  shew  all  the  various  phenomena,  and  such  as  tend  to 
illustrate  the  different  theories  which  have  been  invented  to  account  for 
them.  The  lectures  are  always  given  in  the  first  half  of  the  midsummer 
term  at  4  o'clock  in  the  afternoon,  in  the  public  Lecture-room  under  the 
front  of  the  Public  Library.  Terms  are  3  guineas  for  the  first  course, 
2  guineas  for  the  second,  and  afterwards  gratis. 

1  Vince's  Fluxions,  p.  1. 


MILNER.     SHEPHERD.     VINCE.  103 

1751,  took  his  B.A.  in  1774  as  senior  wrangler,  and  died 
in  London  on  April  1,  1820.  He  wrote  several  works  on 
theology.  A  contemporary  says  that  he  had  "extensive  learning 

always  at  his  command great  talents  for  conversation  and 

a  dignified  simplicity  of  manner,"  but  he  does  not  seem  to 
have  possessed  any  special  qualifications  for  the  Lucasian  chair. 
At  an  earlier  time  he  had  frequently  taken  part  in  the  exami- 
nations in  the  senate-house,  but  I  believe  I  am  right  in  saying 
that  after  his  election  to  the  professorship  he  never  lectured, 
or  taught,  or  examined  in  the  tripos,  or  presided  in  the  schools. 


The  occupants  of  the  Plumian  chair  during  the  period 
treated  in  this  chapter  were  Robert  Smith  (see  p.  91),  Anthony 
Shepherd,  and  Samuel  Vince. 

In  1760  Robert  Smith  was  succeeded  by  Shepherd.  Anthony 
Shepherd  was  born  in  Westmoreland  in  1722,  took  his  B.A. 
from  St  John's  in  1743,  was  subsequently  elected  a  fellow  of 
Christ's,  and  died  in  London  on  June  15,  1795.  Of  him  I 
know  nothing  save  that  in  1772  he  published  some  refraction 
and  parallax  tables,  and  that  in  1776  he  printed  a  list  of 
some  experiments  on  natural  philosophy  which  he  had  used 
to  illustrate  a  course  of  lectures  he  had  given  in  Trinity 
College. 

Shepherd  was  followed  in  1796  by  Yince,  a  fellow  of  Caius. 
Samuel  Vince  was  born  in  Suffolk  about  1754,  took  his 
B.A.  as  senior  wrangler  in  1775,  and  died  in  December,  1821. 
His  original  researches  consisted  chiefly  of  numerous  obser- 
vations on  the  laws  of  friction  and  the  motion  of  fluids,  and  he 
contributed  papers  on  these  subjects  to  the  Philosophical  trans- 
actions for  1785,  1795,  and  1798.  His  results  are  substantially 
correct.  A  list  of  all  his  papers  sent  to  various  societies  is 
given  in  Poggendorff.  His  most  important  work  is  an  astronomy 
published  in  three  volumes  at  Cambridge,  1797 — 1808;  the 
first  volume  is  descriptive,  the  second  an  account  of  physical 
astronomy,  and  the  third  a  collection  of  tables  arranged  for 


LONG.      SMITH.      LAX.  105 

A  "plan"  of  his  lectures  with  a  detailed  account  of  his 
experiments  was  published  in  1793,  and  another  one  was  issued 
in  1797.  His  lectures  are  said  to  have  been  good,  and  I 
beiieve  he  was  always  willing  to  assist  students  in  their  reading. 
His  successors  will  be  mentioned  in  the  next  chapter. 


In  1749  Thomas  Lowndes  of  Overton  founded  another  pro- 
fessorship1 of  astronomy  and  geometry.  The  first  occupant  of 
the  chair  was  Roger  Long,  a  fellow  and  subsequently  master  of 
Pembroke  College,  and  the  friend  of  the  poet  Gray.  Long  was 
born  in  Norfolk  on  Feb.  2,  1680,  graduated  as  B.A.  in  1701, 
and  died  on  Dec.  16,  1770.  His  chief  work  is  one  on 
astronomy  in  two  quarto  volumes  published  in  1742  :  fresh 
editions  were  issued  in  1764  and  1784,  and  it  became  a 
standard  text-book  at  Cambridge;  the  descriptive  parts  are 
said  to  be  well  written.  In  1765,  or  according  to  some 
accounts  1753,  he  constructed  a  zodiack  or  large  sphere  capable 
of  containing  several  people  and  on  the  inside  of  which  the 
constellations  visible  from  Cambridge  were  marked.  This 
famous  globe  stood  in  the  grounds  of  Pembroke  College,  and 
was  only  destroyed  in  1871. 

Long  was  succeeded  in  1771  by  John  Smith,  the  master 
of  Caius  College,  who  in  his  turn  was  followed  in  1795  by 
William  Lax,  a  fellow  of  Trinity,  who  was  born  in  1751  and 
held  the  chair  till  his  death  on  Oct.  29,  1836.  Both  of  these 
professors  seem  to  have  neither  lectured  nor  taught.  Lax 
wrote  a  pamphlet  on  Euclid,  1808  :  and  in  1821  issued  some 
tables  for  use  with  the  Nautical  almanack.  He  also  con- 
tributed papers  to  the  Philosophical  transactions  for  1799  and 
1809. 

1  The  successive  professors  were  as  follows.  From  1749  to  1771, 
Eoger  Long  of  Pembroke;  from  1771  to  1795,  John  Smith  of  Caius; 
from  1795  to  1836,  William  Lax  of  Trinity ;  from  1836  to  1858,  George 
Peacock  of  Trinity  (see  p.  124) ;  who  in  1858  was  succeeded  by  J.  C. 
Adams  of  Pembroke,  the  present  professor. 


106  THE   LATER  NEWTONIAN   SCHOOL. 

To  meet  the  want  of  the  lectures  they  should  have  given 
Francis  John  Hyde  Wollaston  (born  about  1761,  took  his  B.A. 
in  1783,  and  died  in  1823),  a  fellow  of  Trinity  Hall  and  Jack- 
sonian  professor,  lectured  on  astronomy  from  1785  to  1795,  and 
William  Parish  (born  in  1759  and  died  in  1837),  a  fellow  of 
Magdalene,  who  was  professor  of  chemistry  from  1794  to  1813 
and  of  natural  experimental  philosophy  from  1813  to  1837, 
lectured  on  mechanics.  A  paper  by  Farish  on  isometrical 
perspective  appears  in  the  Cambridge  philosophical  transactions 
for  1822. 

Farish  was  also  vicar  of  St  Giles's,  Cambridge,  and  many 
stories  of  the  complications  produced  by  his  extraordinary 
absence  of  mind  are  still  current.  He  is  celebrated  in  the 
domestic  history  of  the  university  for  having  reduced  the 
practice  of  using  Latin  as  the  official  language  of  the  schools 
and  the  university  to  a  complete  farce.  On  one  occasion, 
when  the  audience  in  the  schools  was  unexpectedly  increased 
by  the  presence  of  a  dog,  he  stopped  the  discussion  to  give  the 
peremptory  order  Verte  canem  ex.  At  another  time  one  of  the 
candidates  had  forgotten  to  put  on  the  bands  which  are  still 
worn  on  certain  ceremonial  occasions.  Farish,  who  was  presiding, 
said,  Domine  opponentium  tertie,  non  habes  quod  debes.  Ubi 
sunt  tui...(with  a  long  pause)  Anglice  bands?  To  whom  with 
commendable  promptness  the  undergraduate  replied,  Dignissime 
domine  moderator,  sunt  in  meo  (Anglice)  pocket.  Another  piece 
of  scholastic  Latin  quoted  by  Wordsworth  is,  Domine  opponens 
non  video  vim  tuum  argumentum1. 


The  only  other  mathematicians  of  this  time  whom  I  deem, 
it  necessary  to  mention  here  are  George  Atwood,  Miles  Bland, 
Bewick  Bridge,  John  Brinkley,  Daniel  Cresswell,  William 
Frend,  Francis  Maseres,  Nevil  Maskelyne,  John  Rowning, 
Francis  Wollaston,  and  James  Wood.  I  confine  myself  to  a 

1  See  p.  41  of  the  Scholac  academicae;  and  Nichol's  Literary 
anecdotes,  vm.  541. 


ROWNING.     WOLLASTON.     ATWOOD.  107 

short  note  on  each,  and  I  have  arranged  these  notes  roughly  in 
chronological  order. 

John  Rowning,  a  fellow  of  Magdalene  College,  was  born  in 
17Q1  and  died  in  London  in  1771.  He  wrote  A  compendious 
system  of  natural  philosophy,  published  in  two  volumes  in 
1738 ;  a  treatise  on  the  method  of  fluxions,  published  in  1756  ; 
and  a  description  of  a  machine  for  solving  equations,  published 
in  the  Philosophical  transactions  for  1770. 

Francis  Wollaston,  a  fellow  of  Sidney  College,  who  was 
born  on  Nov.  23,  1731,  and  took  his  B.A.  as  second  wrangler 
in  1758,  wrote  several  papers  and  works  on  practical  astronomy; 
a  list  of  these  is  given  in  Poggendorff's  Handwdrterbuch.  He 
died  at  Chiselhurst  on  Oct.  31,  1815. 

George  Atwood  was  born  in  1746,  was  educated  at  West- 
minster School,  took  his  B.A.  as  third  wrangler  and  first 
Smith's  prizeman  in  1769,  and  subsequently  was  elected  a 
fellow  and  tutor  of  Trinity  College.  The  inefficiency  of  the 
professorial  body  served  as  a  foil  to  his  lectures,  which  attracted 
all  the  mathematical  talent  of  the  university.  They  were  not 
only  accurate  and  clear,  but  delivered  fluently  and  illustrated 
with  great  ingenuity.  The  apparatus  for  calculating  the 
numerical  value  of  the  acceleration  produced  by  gravity  which 
is  still  known  by  his  name  was  invented  by  him  and  used  in 
his  Trinity  lectures  in  1782  and  1783.  Analyses  of  the  courses 
delivered  in  1776  and  in  1784  were  issued  by  him,  and  are 
still  extant.  Pitt  attended  Atwood's  lectures,  and  was  so  much 
interested  in  them  that  he  gave  him  a  post  in  London ;  and 
for  the  last  twenty  years  of  his  life  Atwood  was  the  financial 
adviser  of  every  successive  government.  Atwood  died  in  London 
on  July  11,  1807. 

His  most  important  work  was  one  on  dynamics,  published 
at  Cambridge  in  1784.  He  also  wrote  a  treatise  on  the  theory 
of  arches  published  in  1804.  Besides  these  he  contributed 
several  papers  to  the  Philosophical  transactions :  these  include 
one  in  1781  on  the  theory  of  the  sextant;  one  in  1794  on  the 
mathematical  theory  of  the  watch,  especially  the  times  of  vibra- 


108  THE    LATER   NEWTONIAN    SCHOOL. 

tion  of  balances;  one  in  1796,  to  which  the  Copley  medal  was 
awarded,  on  the  positions  of  equilibrium  of  floating  bodies;  and 
lastly  one  in  1798  on  the  stability  of  ships. 

Waring's  rival  for  the  Lucasian  chair  was  Francis  Maseres1, 
a  fellow  of  Clare  Hall.  Maseres  was  descended  from  a  family 
of  French  Huguenots  who  had  settled  in  England  :  he  was  born 
in  London  on  Dec.  15,  1731,  and  took  his  B.A.  as  senior 
wrangler  in  1752.  After  failing  to  be  elected  to  the  profes- 
sorship he  went  to  the  bar,  and  subsequently  as  attorney- 
general  to  the  province  of  Canada;  on  his  return  in  1773  he 
was  made  a  cursitor  baron  of  the  Exchequer,  and  held  that 
office  till  his  death  on  May  19,  1824.  In  1750  he  published  a 
trigonometry,  and  at  a  later  time  several  tracts  on  algebra  and 
the  theory  of  equations  :  these  are  of  no  value,  as  he  refused  to 
allow  the  use  of  negative  or  impossible  quantities.  In  1783 
he  wrote  a  treatise  in  two  volumes  on  the  theory  of  life  assur- 
ance, which  is  a  creditable  attempt  to  put  the  subject  on  a 
scientific  basis.  He  has  however  acquired  considerable  cele- 
brity from  the  reprints  of  most  of  the  works  either  on  loga- 
rithms or  on  optics  by  mathematicians  of  the  seventeenth 
century,  including  those  by  Napier,  Siiell,  Descartes,  Schooten, 
Huygens,  Barrow,  and  Halley.  These  were  published  in  six 
volumes,  1791 — 1807,  at  his  expense  after  a  careful  revision 
of  the  text  under  the  titles  Scriptores  logarithmici  and  Scrip- 
tores  optici. 

Nevil  Maskelyne  was  born  in  London  on  Oct.  6,  1732,  was 
educated  at  Westminster  School,  and  took  his  B.A.  as  seventh 
wrangler  in  1754,  and  was  subsequently  elected  to  a  fellowship 
at  Trinity.  In  1765  he  succeeded  Bliss  at  Greenwich  as 
astronomer-royal :  the  rest  of  his  life  was  given  up  to  practical 
astronomy.  The  issue  of  the  Nautical  almanack  was  wholly 
due  to  him,  and  began  in  1767;  in  1772  he  made  the 
Schehallien  observations  from  which  he  calculated  (then  for 

1  An  account  of  his  life  is  given  in  the  Gentleman's  magazine  for 
June,  1824 :  see  also  pp.  121 — 3  of  the  Budget  of  paradoxes  by  A.  De 
Morgan,  London,  1872. 


BRIDGE.  FREND.  BRINKLEY.          109 

the  first  time)  the  mean  density  of  the  earth;  lastly  in  1790 
he  published  the  earliest  standard  catalogue  of  stars,  and 
Delambre  for  that  reason  considers  modern  observational  astro- 
nomy to  date  from  that  year.  A  list  of  his  numerous  papers 
contributed  to  the  Philosophical  transactions  will  be  found 
in  Poggendorff's  Handworterbuch.  He  died  on  Feb.  9,  1811. 

Bewick  Bridge,  a  fellow  of  Peterhouse  and  mathematical 
professor  at  Haileybury  College,  was  born  near  Cambridge  in 
1767,  graduated  B.A.  as  senior  wrangler  in  1790,  and  died  at 
Cherryhinton,  of  which  he  was  vicar,  on  May  15,  1833.  He 
wrote  text-books  on  geometrical  conies  (two  volumes,  1810), 
algebra  (1810,  1815,  and  1821),  trigonometry  (1810  and  1818), 
and  mechanics  (1813). 

William  Frend  was  born  at  Canterbury  on  Nov.  22,  1757, 
took  his  B.A.  from  Christ's  College  as  second  wrangler  in  1780, 
and  was  subsequently  elected  to  a  fellowship  in  Jesus  College. 
He  published  in  1796  a  work  entitled  Principles  of  algebra,  in 
which  he  rejected  negative  quantities  as  nonsensical.  He  is 
probably  better  known  in  connection  with  his  banishment  in 
1793  from  the  university  on  account  of  his  publication  of  a 
certain  pamphlet  called  Peace  and  Union.  I  should  add  that 
he  was  only  refused  leave  to  reside,  and  was  not  deprived  of  his 
fellowship.  Any  sympathy  for  the  harsh  treatment  which  he 
seems  to  have  experienced  will  probably  be  dissipated  by  read- 
ing his  own  account  of  the  proceedings  which  he  published  at 
Cambridge  in  1793.  He  died  in  London  on  Feb.  21,  1841. 

John  Brinkley,  a  fellow  of  Caius,  and  subsequently  bishop 
of  Cloyne,  who  was  born  in  Suffolk  in  1763  and  graduated  as 
senior  wrangler  and  first  Smith's  prizeman  in  1788,  acquired 
considerable  reputation  as  professor  of  astronomy  at  Dublin. 
He  contributed  numerous  papers  either  to  the  Royal  Society 
or  to  the  corresponding  society  in  Ireland  on  various  problems 
in  astronomy,  also  a  few  on  different  questions  connected  with 
the  use  of  series.  A  complete  list  of  these  will  be  found  in 
the  Catalogue  of  scientific  papers  from  the  year  1800  issued 
by  the  Royal  Society.  He  died  in  Dublin  on  Sept.  14,  1835. 


110  THE   LATER   NEWTONIAN   SCHOOL. 

Daniel  Cresswell,  a  fellow  of  Trinity,  who  was  born  at 
Wakefield  in  1776  and  graduated  as  seventh  wrangler  in  1797, 
was  a  well-known  "  coach "  of  his  day.  In  1822  he  took  a 
college  living,  and  died  at  Enfield  on  March  21,  1844.  His 
most  important  works  are  the  Elements  of  linear  perspective, 
Cambridge,  1811;  a  translation  of  Venturoli's  Mechanics,  Cam- 
bridge, 1822;  and  a  work  on  the  geometrical  treatment  of 
problems  of  maxima  and  minima. 

Miles  Bland,  a  fellow  and  tutor  of  St  John's  College,  who 
was  born  in  1786  and  graduated  as  second  wrangler  in  1808, 
was  one  of  the  best  known  writers  of  elementary  books  at  the 
beginning  of  the  century:  he  went  down  from  the  university  in 
1823  and  died  in  1868.  In  1812  he  published  a  collection  of 
algebraical  problems,  and  in  1819  another  of  geometrical 
problems:  these  became  well-known  school  books.  In  1824 
he  issued  an  elementary  work  on  hydrostatics;  and  this  was 
followed  in  1830  by  a  collection  of  mechanical  problems. 

James  Wood,  a  fellow  and  subsequently  the  master  of  St 
John's  College  and  dean  of  Ely,  was  born  in  Lancashire  about 
1760,  graduated  as  senior  wrangler  in  1782,  and  died  at 
Cambridge  on  April  23,  1839.  His  algebra  was  long  a 
standard  work,  it  formed  originally  a  part  of  his  Principles  of 
mathematics  and  natural  philosophy  in  four  volumes,  Cam- 
bridge, 1795 — 99  ;  the  section  on  astronomy  (vol.  iv.  part  ii.) 
was  contributed  by  Vince.  Wood  also  wrote  a  paper  On 
the  roots  of  equations  which  will  be  found  in  the  Philosophical 
transactions  for  1798. 

It  was  with  difficulty  that  I  made  out  a  list  of  some  thirty 
or  forty  writers  on  mathematics  of  this  time  who  were  educated 
at  Cambridge ;  and  the  above  names  comprise  every  one  of  them 
whose  works  can  as  far  as  I  know  be  said  to  have  influenced 
the  development  of  the  study  at  Cambridge  or  elsewhere. 


It  is  not  easy  to  make  out  exactly  what  books  were  usually 
read  at  this  time,  but  Whewell  says  that  they  certainly  included 


THE   LATER   NEWTONIAN   SCHOOL.  Ill 

considerable  parts  of  the  Principia,  the  works  of  Cotes,  Atwood, 
Yince,  and  Wood  :  the  treatises  by  the  two  last-named  mathe- 
maticians were  probably  read  by  all  mathematical  students. 

Sir  Frederick  Pollock  of  Trinity,  who  was  senior  wrangler 
in  1806,  in  the  account  printed  in  the  next  paragraph,  asserts 
that  in  his  freshman's  year  he  read  Wood's  Algebra  (to  quad- 
ratic equations),  Bonnycastle's  Algebra,  and  Simpson's  Euclid: 
in  his  second  year  he  read  algebra  beyond  quadratic  equations 
in  Wood's  work,  and  the  theory  of  equations  in  the  works  by 
Wood  and  Yince  :  in  his  third  year  he  read  the  Jesuit  edition 
of  Newton's  Principia,  Yince's  Fluxions,  and  copied  numerous 
manuscripts  or  analyses  supplied  by  his  coach.  There  is  no 
doubt  that  he  is  right  in  saying  that  this  was  less  than  was  usual. 

The  letter  to  which  I  have  just  referred  was  sent  by  Sir 
Frederick  Pollock  in  July,  1869,  to  Prof.  De  Morgan  in 
answer  to  a  request  for  a  trustworthy  account,  which  would 
be  of  historical  value,  about  the  mathematical  reading  of  men 
at  the  beginning  of  this  century.  It  is  so  interesting  that  no 
excuse  is  necessary  for  reproducing  it. 

I  shall  write  in  answer  to  your  inquiry  all  about  my  books,  my 
studies,  and  my  degree,  and  leave  you  to  settle  all  about  the  proprieties 
which  my  letter  may  give  rise  to,  as  to  egotism,  modesty,  &c.  The  only 
books  I  read  the  first  year  were  Wood's  Algebra  (as  far  as  quadratic 
equations),  Bonnycastle's  ditto,  and  Euclid  (Simpson's).  In  the  second 
year  I  read  Wood  (beyond  quadratic  equations),  and  Wood  and  Vince 
for  what  they  called  the  branches.  In  the  third  year  I  read  the  Jesuit's 
Newton  and  Vince's  Fluxions ;  these  were  all  the  books,  but  there  were 
certain  MSS.  floating  about  which  I  copied — which  belonged  to  Dealtry, 
second  wrangler  in  Kempthorne's  year.  I  have  no  doubt  that  I  had  read 
less  and  seen  fewer  books  than  any  senior  wrangler  of  about  my  time,  or 
any  period  since ;  but  what  I  knew  I  knew  thoroughly,  and  it  was  com- 
pletely at  my  fingers'  ends.  I  consider  that  I  was  the  last  geometrical 
and  Jluxional  senior  wrangler  :  I  was  not  up  to  the  differential  calculus, 
and  never  acquired  it.  I  went  up  to  college  with  a  knowledge  of  Euclid 
and  algebra  to  quadratic  equations,  nothing  more ;  and  I  never  read  any 
second  year's  lore  during  my  first  year,  nor  any  third  year's  lore  during 
my  second  ;  my  forte  was,  that  what  I  did  know  I  could  produce  at  any 
moment  with  PERFECT  accuracy.  I  could  repeat  the  first  book  of  Euclid 
word  by  word  and  letter  by  letter.  During  my  first  year  I  was  not  a 


112  THE  LATER   NEWTONIAN   SCHOOL. 

'reading'  man  (so  called) ;  I  had  no  expectation  of  honours  or  a  fellowship, 
and  I  attended  all  the  lectures  on  all  subjects — Harwood's  anatomical, 
Wollaston's  chemical,  and  Parish's  mechanical  lectures — but  the  exami- 
nation at  the  end  of  the  first  year  revealed  to  me  my  powers.  I  was  not 
only  in  the  first  class,  but  it  was  generally  understood  I  was  first  in  the 
first  class ;  neither  I  nor  any  one  for  me  expected  I  should  get  in  at  all. 
Now,  as  I  had  taken  no  pains  to  prepare  (taking,  however,  marvellous 
pains  while  the  examination  was  going  on),  I  knew  better  than  any  one 
else  the  value  of  my  examination  qualities  (great  rapidity  and  perfect 
accuracy) ;  and  I  said  to  myself,  '  If  you're  not  an  ass,  you'll  be  senior 
wrangler;'  and  I  took  to  'reading'  accordingly.  A  curious  circumstance 
occurred  when  the  brackets1  came  out  in  the  senate-house  declaring  the 
result  of  the  examination :  I  saw  at  the  top  the  name  of  Walter  bracketed 
alone  (as  he  was) ;  in  the  bracket  below  were  Fiott,  Hustler,  Jephson.  I 
looked  down  and  could  not  find  my  own  name  till  I  got  to  Bolland,  when 
my  pride  took  fire,  and  I  said,  '  I  must  have  beaten  that  man,  so  I  will 
look  up  again ; '  and  on  looking  up  carefully  I  found  the  nail  had  been 
passed  through  my  name,  and  I  was  at  the  top  bracketed  alone,  even 
above  "Walter.  You  may  judge  what  my  feelings  were  at  this  discovery; 
it  is  the  only  instance  of  two  such  brackets,  and  it  made  my  fortune — 
that  is,  made  me  independent,  and  gave  me  an  immense  college  reputa- 
tion. It  was  said  I  was  more  than  half  of  the  examination  before  any 
one  else.  The  two  moderators  were  Hornbuckle,  of  St  John's,  and  Brown 
(Saint  Brown),  of  Trinity.  The  Johnian  congratulated  me.  I  said 
perhaps  I  might  be  challenged ;  he  said,  '  Well,  if  you  are  you're  quite 
safe — you  may  sit  down  and  do  nothing,  and  no  one  would  get  up  to  you 

in  a  whole  day.' 

My  experience  has  led  me  to  doubt  the  value  of  competitive  exami- 
nation. I  believe  the  most  valuable  qualities  for  practical  life  cannot  be 
got  at  by  any  examination — such  as  steadiness  and  perseverance.  It 
may  be  well  to  make  an  examination  part  of  the  mode  of  judging  of  a 
man's  fitness ;  but  to  put  him  into  an  office  with  public  duties  to  perform 
merely  on  his  passing  a  good  examination  is,  I  think,  a  bad  mode  of 
preventing  mere  patronage.  My  brother  is  one  of  the  best  generals  that 


1  The  '  brackets '  were  a  preliminary  classification  in  order  of  merit. 
They  were  issued  on  the  morning  of  the  last  day  of  the  tripos  examina- 
tion. The  names  in  each  bracket  were  arranged  in  alphabetical  order. 
A  candidate  who  considered  that  he  was  placed  too  low  in  the  list  could 
challenge  any  one  whose  name  appeared  in  the  bracket  next  above  that 
in  which  his  own  was  placed,  and  if  on  re-examination  he  proved  himself 
the  equal  of  the  man  so  challenged  his  name  was  transferred  to  the 
higher  bracket  (see  p.  200). 


THE  LATER  NEWTONIAN   SCHOOL.  133 

ever  commanded  an  army,  but  the  qualities  that  make  him  so  are  quite 
beyond  the  reach  of  any  examination.  Latterly  the  Cambridge  exami- 
nations seem  to  turn  upon  very  different  matters  from  what  prevailed  in 
my  time.  I  think  a  Cambridge  education  has  for  its  object  to  make  good 
members  of  society — not  to  extend  science  and  make  profound  mathema- 
ticians. The  tripos  questions  in  the  senate-house  ought  not  to  go  beyond 
certain  limits,  and  geometry  ought  to  be  cultivated  and  encouraged  much 
more  than  it  is. 

To  this  De  Morgan  replied  : 

Your  letter  suggests  much,  because  it  gives  possibility  of  answer. 
The  branches  of  algebra  of  course  mainly  refer  to  the  second  part  of 
Wood,  now  called  the  theory  of  equations.  Waring  was  his  guide. 
Turner — whom  you  must  remember  as  head  of  Pembroke,  senior  wrangler 
of  1767 — told  a  young  man  in  the  hearing  of  my  informant  to  be  sure 
and  attend  to  quadratic  equations.  '  It  was  a  quadratic,'  said  he,  '  made 
me  senior  wrangler.'  It  seems  to  me  that  the  Cambridge  revivers  were 
Waring,  Paley,  Vince,  Milner. 

You  had  Dealtry's  MSS.  He  afterwards  published  a  very  good  book  on 
fluxions.  He  merged  his  mathematical  fame  in  that  of  a  Claphamite 
Christian.  It  is  something  to  know  that  the  tutor's  MS.  was  in  vogue  in 
1800-1806. 

Now — how  did  you  get  your  conic  sections  ?  How  much  of  Newton 
did  you  read?  From  Newton  direct,  or  from  tutor's  manuscript? 

Surely  Fiott  was  our  old  friend  Dr  Lee.  I  missed  being  a  pupil  of 
Hustler  by  a  few  weeks.  He  retired  just  before  I  went  up  in  February 
1823.  The  echo  of  Hornbuckle's  answer  to  you  about  the  challenge 
has  lighted  on  Whewell,  who,  it  is  said,  wanted  to  challenge  Jacob,  and 
was  answered  that  he  could  not  beat  [him]  if  he  were  to  write  the 
whole  day  and  the  other  wrote  nothing.  I  do  not  believe  that  Whewell 
would  have  listened  to  any  such  dissuasion. 

I  doubt  your  being  the  last  fluxional  senior  wrangler.  So  far  as  I 
know,  Gipps,  Langdale,  Alderson,  Dicey,  Neale,  may  contest  this  point 
with  you. 

The  answer  of  Sir  Frederick  Pollock  to  these  questions  is 
dated  August  7,  1869,  and  is  as  follows. 

You  have  put  together  as  revivers  five  very  different  men.  Woodhouse 
was  better  than  Waring,  who  could  not  prove  Wilson's  (Judge  of  C.  P.) 
guess  about  the  property  of  prime  numbers;  but  Woodhouse  (I  think) 
did  prove  it,  and  a  beautiful  proof  it  is.  Vince  was  a  bungler,  and  I 
think  utterly  insensible  of  mathematical  beauty. 

B.  8 


114  THE  LATER  NEWTONIAN   SCHOOL. 

Now  for  your  questions.  I  did  not  get  my  conic  sections  from  Vince. 
I  copied  a  MS.  of  Dealtry's.  I  fell  in  love  with  the  cone  and  its  sections, 
and  everything  about  it.  I  have  never  forsaken  my  favourite  pursuit ; 
I  delighted  in  such  problems  as  two  spheres  touching  each  other  and  also 
the  inside  of  a  hollow  cone,  &c.  As  to  Newton,  I  read  a  good  deal  (men 
now  read  nothing),  but  I  read  much  of  the  notes.  I  detected  a  blunder 
which  nobody  seemed  to  be  aware  of.  Tavel,  tutor  of  Trinity,  was  not ; 
and  he  augured  very  favourably  of  me  in  consequence.  The  application 
of  the  Principia  I  got  from  MSS.  The  blunder  was  this :  in  calculating 
the  resistance  of  a  globe  at  the  end  of  a  cylinder  oscillating  in  a  resisting 
medium  they  had  forgotten  to  notice  that  there  is  a  difference  between 
the  resistance  to  a  globe  and  a  circle  of  the  same  diameter. 

The  story  of  Whewell  and  Jacob  cannot  be  true.  Whewell  was  a  very, 
very  considerable  man,  I  think  not  a  great  man.  I  have  no  doubt  Jacob 
beat  him  in  accuracy,  but  the  supposed  answer  cannot  be  true ;  it  is  a 
mere  echo  of  what  actually  passed  between  me  and  Hornbuckle  on  the 
day  the  Tripos  came  out — for  the  truth  of  which  I  vouch.  I  think  the 
examiners  are  taking  too  practical  a  turn ;  it  is  a  waste  of  time  to  calculate 
actually  a  longitude  by  the  help  of  logarithmic  tables  and  lunar  observa- 
tions. It  would  be  a  fault  not  to  know  how,  but  a  greater  to  be  handy 
at  it1. 


I  may  mention  in  passing  that  experimental  physics  began 
about  this  time  to  attract  considerable  attention.  This  was 
largely  due  to  the  influence  of  Cavendish,  Young,  W.  H. 
Wollaston,  Rumford,  and  Dalton  in  England,  and  of  Lavoisier 
and  Laplace  in  France.  The  first  three  of  these  writers  came 
from  Cambridge ;  and  I  add  a  few  lines  on  the  subject-matter 
of  their  works. 

The  honourable  Henry  Cavendish2  was  born  at  Nice  on 
Oct.  10,  1731.  His  tastes  for  scientific  research  and  mathe- 
matics seem  to  have  been  formed  at  Cambridge,  where  he 
resided  from  1749  to  1753.  He  was  a  member  of  Peterhouse, 

1  Memoir  of  A.  De  Morgan  (pp.  387—392),   by  S.  E.  De  Morgan, 
London,  1882. 

2  An  account  of  his  life  by  G.  Wilson  will  be  found  in  the  first 
volume  of  the  publications  of  the  Cavendish  Society,  London,  1851.     His 
Electrical  researches  were  edited  by  J.  C.  Maxwell,  and  published  at 
Cambridge  in  1879. 


CAVENDISH.     YOUNG.     WOLL  ASTON.  115 

but  like  all  fellow-commoners  of  the  time  did  not  present  him- 
self for  the  senate-house  examination,  and  in  fact  he  did  not 
actually  take  a  degree.  He  created  experimental  electricity, 
and -was  one  of  the  earliest  writers  to  treat  chemistry  as  an 
exact  science.  In  1798  he  determined  the  density  of  the 
earth  by  estimating  its  attraction  as  compared  with  that  of 
two  given  lead  balls :  the  result  is  that  the  mean  density  of  the 
earth  is  about  five  and  a  half  times  that  of  water.  This  ex- 
periment was  carried  out  in  accordance  with  a  suggestion  which 
had  been  first  made  by  John  Michell,  a  fellow  of  Queens' 
[B.A.  1748],  who  had  died  before  he  was  able  to  carry  it  into 
effect.  Si&  note-books  prove  him  to  have  been  much  inte- 
rested in  mathematical  questions  but  I  believe  he  did  not  publish 
any  of  his  results.  He  died  in  London  on  Feb.  24,  1810. 

Thomas  Young1,  born  at  Milverton  on  June  13,  1773,  and 
died  in  London  on  May  10,  1829,  was  among  the  most  eminent 
physicists  of  his  time.  He  seems  as  a  boy  to  have  been  some- 
what of  a  prodigy,  being  well  read  in  modern  languages  and 
literature  as  well  as  in  science;  he  always  kept  up  his  literary 
tastes  and  it  was  he  who  first  furnished  the  key  to  decipher 
the  Egyptian  hieroglyphics.  He  was  destined  to  be  a  doctor, 
and  after  attending  lectures  at  Edinburgh  and  Gottingen 
•entered  at  Emmanuel  College,  Cambridge,  from  which  he  took 
his  degree  in  1803  ;  and  to  his  stay  at  the  university  he 
attributed  much  of  his  future  distinction.  His  medical  career 
was  not  particularly  successful,  and  his  favorite  maxim  that  a 
medical  diagnosis  is  only  a  balance  of  probabilities  was  not 
appreciated  by  his  patients,  who  looked  for  certainty  in  return 
for  their  fee.  Fortunately  his  private  means  were  ample. 
Several  papers  contributed  to  various  learned  societies  from 
1798  onwards  prove  him  to  have  been  a  mathematician  of 
considerable  power;  but  the  researches  which  have  immortalized 
his  name  are  those  by  which  he  laid  down  the  laws  of  inter- 
ference of  waves  and  of  light,  and  was  thus  able  to  overcome 

1  For  further  details  see  his  life  and  works  by  G.  Peacock,  4  vols. 
1855. 

8—2 


116  THE   LATER   NEWTONIAN   SCHOOL. 

the   chief   difficulties   in   the   way   of  the  acceptance    of  the 
undulatory  theory  of  light. 

Another  experimental  physicist  of  the  same  time  and 
school  was  William  Hyde  Wollaston,  who  was  born  at  Dereham 
on  Aug.  6,  1766,  and  died  in  London  on  Dec.  22,  1828.  He 
was  educated  at  Caius  College  (M.B.  1788),  of  which  society  he 
was  a  fellow.  Besides  his  well-known  chemical  discoveries,  he 
is  celebrated  for  his  researches  on  experimental  optics,  and  for 
the  improvements  he  effected  in  astronomical  instruments. 


One  characteristic  of  this  period  to  which  I  have  not  yet 
alluded  is  the  rise  of  a  class  of  teachers  in  the  university  who 
are  generally  known  as  coaches  or  private  tutors,  but  I  may 
conveniently  defer  any  remarks  on  this  subject  until  I  consider 
the  general  question  of  the  organization  of  education  in  the 
university  (see  pp.  160 — 163). 


CHAPTER  VII. 
THE   ANALYTICAL   SCHOOL1. 

THE  isolation  of  English  mathematicians  from  their  conti- 
nental contemporaries  is  the  distinctive  feature  of  the  history 
of  the  latter  half  of  the  eighteenth  century.  Towards  the 
close  of  that  century  the  more  thoughtful  members  of  the  uni- 
versity recognized  that  this  was  a  serious  evil,  and  it  would 
seem  that  the  chief  obstacle  to  the  adoption  of  analytical 
methods  and  the  notation  of  the  differential  calculus  arose  from 
the  professorial  body  and  the  senior  members  of  the  senate, 
who  regarded  any  attempt  at  innovation  as  a  sin  against  the 
memory  of  Newton. 

I  propose  in  this  chapter  to  give  a  sketch  of  the  rise  of  the 
analytical  school,  and  shall  briefly  mention  the  chief  works  of 
Robert  Woodhouse,  George  Peacock,  Charles  Babbage,  and 
Sir  John  Herschel.  The  later  history  of  that  school  is  too 
near  our  own  times  to  render  it  possible  or  desirable  to  discuss 
it  in  similar  detail :  and  I  shall  make  no  attempt  to  do  so. 

The  earliest  attempt  in  this  country  to  explain  and  ad- 
vocate the  notation  and  methods  of  the  calculus  as  used  on  the 
continent  was  due  to  Woodhouse,  who  stands  out  as  the  apostle 
of  the  new  movement. 

1  For  the  few  biographical  notes  given  in  this  chapter  I  am  generally 
indebted  to  the  obituary  notices  which  are  printed  in  the  transactions  of 
the  Eoyal  and  other  similar  learned  societies. 


118  THE  ANALYTICAL   SCHOOL. 

Robert  Woodhouse1  was  born  at  Norwich  on  April  28, 
1773,  took  his  B.A.  as  senior  wrangler  and  first  Smith's  prize- 
man in  1795  from  Caius  College,  was  elected  to  a  fellowship 
in  due  course,  and  continued  to  live  at  Cambridge  till  his  death 
on  Dec.  23,  1827. 

His  earliest  work,  entitled  the  Principles  of  analytical 
calculation,  was  published  at  Cambridge  in  1803.  In  this  he 
explained  the  differential  notation  and  strongly  pressed  the 
employment  of  it,  but  he  severely  criticized  the  methods  used 
by  continental  writers,  and  their  constant  assumption  of  non- 
evident  principles.  Woodhouse  was  a  brilliant  logician,  but, 
perhaps  partly  for  that  reason,  the  style  of  the  book  is  very 
crabbed ;  and  it  is  difficult  to  read,  on  account  of  the  extra- 
ordinary complications  of  grammatical  construction  in  which 
he  revels.  This  was  followed  in  1809  by  a  trigonometry 
(plane  and  spherical),  and  in  1810  by  a  historical  treatise  on 
the  calculus  of  variations  and  isoperimetrical  problems.  He 
next  produced  an  astronomy  :  the  first  volume  (usually  bound 
in  two)  on  practical  and  descriptive  astronomy  being  issued  in 
1812,  the  second  volume,  containing  an  account  of  the  treat- 
ment of  physical  astronomy  by  Laplace  and  other  continental 
writers,  being  issued  in  1818.  All  these  works  deal  critically 
with  the  scientific  foundation  of  the  subjects  considered — a 
point  which  is  not  unfrequently  neglected  in  modern  text- 
books. 

In  1820  Woodhouse  succeeded  Milner  as  Lucasian  pro- 
fessor, but  in  18222  he  resigned  it  in  exchange  for  the  Plunrian 
chair.  The  observatory  at  Cambridge  was  finished  in  1824, 
and  Woodhouse  was  appointed  superintendent,  but  his  health 
was  then  rapidly  failing,  though  he  lingered  on  till  1827. 

1  See  the  Penny  Cyclopaedia,  vol.  xxvn. 

2  It  will  be  convenient  to  state  here  that  Woodhouse's  successor  in  the 
Lucasian  chair  was  Thomas  Turton,  of  St  Catharine's  College.     Turton 
was  born  in  1780  and  graduated  as  senior  wrangler  in  1805.     I  am  not 
aware  that  he  ever  lectured.     In  1826  he  exchanged  the  chair  for  one 
of  divinity;  in  1842  he  was  made  dean  of  Westminster;   and  in  1845 
bishop  of  Ely.     He  died  in  1864. 


WOODHOUSE.  119 

A  man  like  Woodhouse,  of  scrupulous  honour,  universally- 
respected,  a  trained  logician,  and  with  a  caustic  wit,  was  well 
fitted  to  introduce  a  new  system.  "The  character,"  says  De 
Morgan,  "which  must  be  given  of  the  several  writings  of 
Woodhouse  entitles  us  to  suppose  that  the  revolution  in  our 
mathematical  studies,  of  which  he  was  the  first  promoter, 
would  not  have  been  brought  about  so  easily  if  its  earliest 
advocacy  had  fallen  into  less  judicious  hands.  For  instance, 
had  he  not,  when  he  first  called  attention  to  the  continental 
analysis,  exposed  the  unsoundness  of  some  of  the  usual  methods 
of  establishing  it  more  like  an  opponent  than  a  partizan,  those 
who  were  averse  from  the  change  would  probably  have  made  a 
successful  stand  against  the  whole  upon  the  ground  which,  as 
it  was,  Woodhouse  had  already  made  his  own.  From  the 
nature  of  his  subjects,  his  reputation  can  never  equal  that  of 
the  first  seer  of  a  comet  with  the  world  at  large :  but  the  few 
who  can  appreciate  what  he  did  will  always  regard  him  as  one 
of  the  most  philosophical  thinkers  and  useful  guides  of  his 
time." 

Woodhouse's  writings  were  of  no  use  for  the  public  ex- 
aminations and  were  scouted  by  the  professors,  but  apparently 
they  were  eagerly  studied  by  a  minority  of  students.  Her- 
schel1,  with  perhaps  a  pardonable  exaggeration,  describes  the 
general  feeling  of  the  younger  members  of  the  university  thus. 
"Students  at  our  universities,  fettered  by  no  prejudices,  en- 
tangled by  no  habits  and  excited  by  the  ardour  and  emulation 
of  youth,  had  heard  of  the  existence  of  masses  of  knowledge 
from  which  they  were  debarred  by  the  mere  accident  of  posi- 
tion. They  required  no  more.  The  prestige  which  magnifies 
what  is  unknown,  and  the  attractions  inherent  in  what  is  for- 
bidden, coincided  in  their  impulse.  The  books  were  procured 
and  read,  and  produced  their  natural  effects.  The  brows  of 
many  a  Cambridge  moderator  were  elevated,  half  in  ire,  half 
in  admiration,  at  the  unusual  answers  which  began  to  appear 

1  The  reader  will  find  another  account  by  Whewell  of  the  same  move- 
ment in  Todhunter's  edition  of  his  life  (vol.  n.  pp.  16,  29,  30). 


120  THE   ANALYTICAL   SCHOOL. 

in  examination  papers.  Even  moderators  are  not  made  of  im- 
penetrable stuff:  their  souls  were  touched,  though  fenced  with 
seven-fold  Jacquier,  and  tough  bull-hide  of  Vince  and  Wood." 

But  while  giving  Woodhouse  all  the  credit  due  to  his 
initiation,  I  doubt  whether  he  exercised  much  influence  on  the 
majority  of  his  contemporaries,  and  I  think  the  movement 
might  have  died  away  for  the  time  being,  if  the  advocacy  of 
Peacock  had  not  given  it  permanence.  I  allude  hereafter  very 
briefly  to  him  and  others  of  those  who  worked  with  him.  I 
will  only  say  here  that  in  1812  three  undergraduates — Peacock, 
Herschel,  and  Babbage — who  were  impressed  by  the  force  of 
Woodhouse's  remarks  and  were  in  the  habit  of  breakfasting 
together  every  Sunday  morning,  agreed  to  form  an  Analytical 
Society,  with  the  object  of  advocating  the  general  use  in  the 
university  of  analytical  methods  and  of  the  differential  notation, 
and  thus  as  Herschel  said  "do  their  best  to  leave  the  world 
wiser  than  they  found  it."  The  other  original  members  were 
William  Henry  Maule  of  Trinity,  senior  wrangler  in  1810  and 
subsequently  a  justice  of  the  common  pleas,  Thomas  Robinson 
of  Trinity,  thirteenth  wrangler  in  1813,  Edward  Ryan  of 
Trinity,  who  took  his  B.A.  in  1814,  and  Alexander  Charles 
Louis  d'Arblay  of  Christ's,  tenth  wrangler  in  1818.  In  1816 
the  Society  published  a  translation  of  Lacroix's  Elementary 
differential  calculus. 

In  1817  Peacock,  who  was  moderator  for  that  year,  in- 
troduced the  symbols  of  differentiation  into  the  papers  set  in 
the  senate-house  examination.  But  his  colleague,  John  White 
of  Caius  (B.A.  1808),  continued  to  use  the  fluxional  notation. 
Peacock  himself  wrote  on  March  17  of  1817  (i.e.  just  after 
the  examination)  on  the  subject  as  follows  :  "  I  assure  you 
that  I  shall  never  cease  to  exert  myself  to  the  utmost  in  the 
cause  of  reform,  and  that  I  will  never  decline  any  office  which 
may  increase  my  power  to  effect  it.  I  am  nearly  certain  of 
being  nominated  to  the  office  of  moderator  in  the  year  1818—19, 
and  as  I  am  an  examiner  in  virtue  of  my  office,  for  the  next 
year  I  shall  pursue  a  course  even  more  decided  than  hitherto, 


THE  ANALYTICAL  SCHOOL.  121 

since  I  feel  that  men  have  been  prepared  for  the  change,  and 
will  then  be  enabled  to  have  acquired  a  better  system  by  the 
publication  of  improved  elementary  books.  I  have  consider- 
able influence  as  a  lecturer,  and  I  will  not  neglect  it.  It  is 
by  silent  perseverance  only  that  we  can  hope  to  reduce  the 
many-headed  monster  of  prejudice,  and  make  the  university 
answer  her  character  as  the  loving  mother  of  good  learning 
and  science." 

The  action  of  G.  Peacock  and  the  translation  of  Lacroix's 
treatise  were  severely  criticised  by  D.  M.  Peacock  in  a  work 
which  was  published  at  the  expense  of  the  university  in  1819. 
The  reformers  were  however  encouraged  by  the  support  of 
most  of  the  younger  members  of  the  university;  and  in  1819 
G.  Peacock,  who  was  again  moderator,  induced  his  colleague 
Richard  Gwatkin  of  St  John's  (B.A.  1814)  to  adopt  the  new 
notation.  It  was  employed  in  the  next  year  by  Whewell1, 
and  in  the  following  year  by  Peacock  again,  by  which  time  the 
notation  was  well-established2 :  and  subsequently  the  language 
of  the  fluxional  calculus  only  appeared  at  rare  intervals  in  the 
examination.  It  should  however  be  noted  in  passing  that  it 
was  only  the  exclusive  use  of  the  fluxional  notation  that  was  so 
hampering,  and  in  fact  the  majority  of  modern  writers  use  both 
systems.  It  was  rather  as  the  sign  of  their  isolation  and  of 
the  practice  of  treating  all  questions  by  geometry  that  the 
fluxional  notation  offended  the  reformers,  than  on  account  of 
any  inherent  defects  of  its  own. 

The  Analytical  Society  followed  up  this  rapid  victory  by 

1  Whewell  gave  but  a  wavering  support  to  Peacock's  action  so  long  as 
its   success  was  doubtful :    see  vol.  n.  p.  16,   of  Todhunter's  Life   of 
Whewell,  London,  1876. 

2  A  letter  by  Sir  George  Airy  describing  his  recollections  of  the 
senate-house  examination  of  1823  and  the  introduction  of  analysis  into 
the  university  examinations  is  printed  in  the  number  of  Nature  for  Feb. 
24,  1887.     I  think  the  contemporary  statements  of  Herschel,  Peacock, 
Whewell,  and  the  criticisms  of  De  Morgan,    shew  that  the  analytical 
movement  was  somewhat  earlier  than  the  time  mentioned  by  Sir  George 
Airy. 


122  THE   ANALYTICAL   SCHOOL. 

the  issue  in  1820  of  two  volumes  of  examples  illustrative  of  the 
new  method :  one  by  Peacock  on  the  differential  and  integral 
calculus,  and  the  other  by  Herschel  on  the  calculus  of  finite 
differences.  Since  then  all  elementary  works  on  the  subject 
have  abandoned  the  exclusive  use  of  the  fluxiona!  notation. 
But  of  course  for  a  few  years  the  old  processes  continued  to  be 
employed  in  college  lecture-rooms  and  examination  papers  by 
some  of  the  senior  members  of  the  university. 

Amongst  those  who  materially  assisted  in  extending  the 
use  of  the  new  analysis  were  Whewell  and  Airy.  The  former 
issued  in  1819  a  work  on  mechanics,  and  the  latter,  who  was  a 
pupil  of  Peacock,  published  in  1826  his  Tracts,  in  which  the 
new  method  was  applied  with  great  success  to  various  physical 
problems.  Finally,  the  efforts  of  the  society  were  supplemented 
by  the  publication  by  Parr  Hamilton  in  1826  of  an  analytical 
geometry,  which  was  an  improvement  on  anything  then  ac- 
cessible to  English  readers. 

The  new  notation  had  barely  been  established  when  a  most 
ill-advised  attempt1  was  made  to  introduce  another  system, 

in  which  -^-  was  denoted  by  dyOj.     This  was  for  some  years 

CLOC 

adopted  in  the  Johnian  lecture-rooms  and  examination  papers, 
but  fortunately  the  strong  opposition  of  Peacock  and  De  Mor- 
gan prevented  its  further  spread  in  the  university.  In  fact 
uniformity  of  notation  is  essential  to  freedom  of  communi- 
cation, and  one  would  have  supposed  that  those  who  admitted 
the  evil  of  the  isolation  to  which  Cambridge  and  England  had 
for  a  century  been  condemned  would  have  known  better  than 
to  at  once  attempt  to  construct  a  fresh  language  for  the  whole 
mathematical  world. 

1  See  On  the  notation  of  the  differential  calculus,  Cambridge,  1832: 
and  also  the  article  by  A.  De  Morgan  in  the  Quarterly  journal  of  educa- 
tion for  1834.  De  Morgan  says  it  was  first  used  in  Trinity,  but  I  can 
find  no  trace  of  it  in  the  examination  papers  of  that  college.  It  occurs  in 
the  papers  set  in  the  annual  examination  at  St  John's  in  the  years  1830, 
1831,  and  1832.  I  suspect  that  it  was  invented  by  Whewell,  but  I  have 
no  definite  evidence  of  the  fact. 


THE  ANALYTICAL  SCHOOL.  123 

The  use  of  analytical  methods  spread  from  Cambridge  over 
the  rest  of  the  country,  and  by  1830  they  had  almost  entirely 
superseded  the  fluxional  and  geometrical  methods.  It  is 
possible  that  the  complete  success  of  the  new  school  and  the 
brilliant  results  that  followed  from  their  teaching  led  at  first 
to  a  somewhat  too  exclusive  employment  of  analysis ;  and 
there  has  of  late  been  a  tendency  to  revert  to  graphical  and 
geometrical  processes.  That  these  are  useful  as  auxiliaries 
to  analysis,  that  they  afford  elegant  demonstrations  of  results 
which  are  already  known,  and  that  they  enable  one  to  grasp 
the  connection  between  different  parts  of  the  same  subject  is 
universally  admitted,  but  it  has  yet  to  be  proved  that  they  are 
equally  potent  as  instruments  of  research.  To  that  I  may  add, 
that  in  my  opinion  the  analytical  methods  are  peculiarly 
suited  to  the  national  genius. 

I  have  often  thought  that  an  interesting  essay  might  be 
written  on  the  influence  of  race  in  the  selection  of  mathematical 
methods.  The  Semitic  races  had  a  special  genius  for  arithmetic 
and  algebra,  but  as  far  as  I  know  have  never  produced  a  single 
geometrician  of  any  eminence.  The  Greeks  on  the  other  hand 
adopted  a  geometrical  procedure  wherever  it  was  possible,  and 
they  even  treated  arithmetic  as  a  branch  of  geometry  by  means 
of  the  device  of  representing  numbers  by  lines.  In  the  modern 
and  mixed  races  of  Europe  the  effects  are  more  complex,  but  I 
think  until  Newton's  time  English  mathematics  might  be 
characterized  as  analytical.  Some  admirable  text-books  on 
arithmetic  and  algebra  were  produced,  and  the  only  three 
writers  previous  to  Newton  who  shewed  marked  original 
power  in  pure  mathematics — Briggs,  Harriot,  and  Wallis — 
generally  attacked  geometrical  problems  by  the  aid  of  algebra 
or  analysis.  For  more  than  a  century  the  tide  then  ran  the 
other  way ;  and  the  methods  of  classical  geometry  were  every- 
where used.  This  was  wholly  due  to  Newton's  influence,  and 
as  with  the  lapse  of  time  that  died  away  the  analytical  methods 
again  came  into  favour. 


124  THE  ANALYTICAL   SCHOOL. 

I  add  a  few  notes  on  the  writers  above-mentioned  and 
their  immediate  successors,  but  with  the  establishment  of  the 
analytical  school  I  consider  my  task  is  finished. 

George  Peacock,  who  was  the  most  influential  of  the  early 
members  of  the  new  school,  was  born  at  Denton  on  April  9, 
1791,  and  took  his  B.A.  from  Trinity  as  second  wrangler  and 
second  Smith's  prizeman  in  1813.  He  was  elected  to  a  fellow- 
ship in  1814,  and  subsequently  was  made  a  tutor  of  the  college. 
I  have  already  alluded  to  the  prominent  part  which  he  took 
in  introducing  analysis  into  the  senate-house  examination. 

Of  his  work  as  a  tutor  there  seems  to  be  but  one  opinion. 
An  old  pupil,  himself  a  man  of  great  eminence,  says,  "  While 
his  extensive  knowledge  and  perspicuity  as  a  lecturer  main- 
tained the  high  reputation  of  his  college,  and  commanded  the 
attention  and  admiration  of  his  pupils,  he  succeeded  to  an 
extraordinary  degree  in  winning  their  personal  attachment  by 
the  uniform  kindliness  of  his  temper  and  disposition,  the  prac- 
tical good  sense  of  his  advice  and  admonitions,  and  the  absence 
of  all  moroseness,  austerity,  or  needless  interference  with  their 
conduct."  "His  inspection  of  his  pupils,"  says  another  of 
them,  "  was  not  minute,  far  less  vexatious;  but  it  was  always 
effectual,  and  at  all  critical  points  of  their  career,  keen  and 
searching.  His  insight  into  character  was  remarkable." 

The  establishment  of  the  university  observatory  was  mainly 
due  to  his  efforts.  In  1836  he  was  appointed  to  the  Lown- 
dean  professorship  in  succession  to  W.  Lax  (see  p.  105).  The 
rival  candidate  was  Whewell.  In  1839  Peacock  was  made 
dean  of  Ely,  and  resided  there  till  his  death  on  Nov.  8,  1858. 

Although  Peacock's  influence  on  the  mathematicians  of 
his  time  and  his  pupils  was  very  considerable  he  has  left  few 
remains.  The  chief  are  his  Examples  illustrative  of  the  use  of 
the  differential  calculus,  1820;  his  article  on  Arithmetic  in  the 
Encyclopaedia  Metropolitans,  1825,  which  contains  the  best 
historical  account  of  the  subject  yet  written,  though  the 
arrangement  is  bad;  his  Algebra,  1830  and  1842;  and  his 
Report  on  recent  progress  in  analysis,  1833,  which  commenced 


BABBAGE.  125 

those  valuable  summaries  of  scientific  progress  which  enrich 
many  of  the  annual  volumes  of  the  British  Association. 

The  next  most  important  member  of  the  Analytical  Society 
was  Charles  Babbage1,  who  was  born  at  Totnes  on  Dec.  26, 
17^,  and  died  in  London  on  Oct.  18,  1871.  He  entered  at 
Trinity  College  in  April,  1810,  as  a  bye-term  student  and  was 
thus  practically  in  the  same  year  as  Herschel  and  Peacock. 
Before  coming  into  residence  Babbage  was  already  a  fair 
mathematician,  having  mastered  the  works  on  fluxions  by 
Humphry  Ditton,  Maclaurin,  and  Simpson,  Aguesi's  Analysis 
(in  the  English  translation  of  which  by  the  way  the  fluxional 
notation  is  used),  Woodhouse's  Principles  of  analytical  calcu- 
lation, and  Lagrange's  Theorie  des  fonctions. 

It  was  he  who  gave  the  name  to  the  Analytical  Society, 
which  he  stated  was  formed  to  advocate  "the  principles  of 
pure  d-ism  as  opposed  to  the  dot-age  of  the  university."  The 
society  published  a  volume  of  memoirs,  Cambridge,  1813;  the 
preface  and  the  first  paper  (on  continued  products)  are  due  to 
Babbage  :  this  work  is  now  very  scarce. 

Finding  that  he  was  certain  to  be  beaten  in  the  tripos  by 
Herschel  and  Peacock,  Babbage  migrated  in  1813  to  Peterhouse 
and  entered  for  a  poll  degree,  in  order  that  he  might  be  first  both 
in  his  college  and  his  examination  in  the  senate-house.  After 
taking  his  B.A.  he  moved  to  London,  and  an  inspection  of  the 
catalogue  of  scientific  papers  issued  by  the  Royal  Society  shews 
how  active  and  many-sided  he  was.  The  most  important  of 
his  contributions  to  the  Philosophical  transactions  seem  to  be 
those  on  the  calculus  of  functions,  1815  to  1817,  and  the  mag- 
netisation of  rotating  plates,  1825.  In  1823  he  edited  the 
Scriptores  optici  for  baron  Maseres  (see  p.  108).  In  1820  the 
Astronomical  Society  was  founded  mainly  through  his  efforts, 
and  at  a  later  time,  1830  to  1832,  he  took  a  prominent  part  in 
the  foundation  of  the  British  Association. 

In  1828  he  succeeded  Airy  as  Lucasian  professor  and  held 

1  He  left  an  autobiography  under  the  title  Passages  from  the  life  of  a 
philosopher.     London,  1864. 


126  THE   ANALYTICAL   SCHOOL. 

the  chair  till  1839,  but  by  an  abuse  which  was  then  possible  he 
neither  resided  nor  taught. 

Babbage  will  always  be  famous  for  his  invention  of  an 
analytical  machine,  which  could  not  only  perform  the  ordinary 
processes  of  arithmetic,  but  could  tabulate  the  values  of  any 
function  and  print  the  results.  The  machine  was  never  finished, 
but  the  drawings  of  it,  now  deposited  at  Kensington,  satisfied 
a  scientific  commission  that  it  could  be  constructed. 

The  third  of  those  who  helped  to  establish  the  new  method 
was  Herschel.  Sir  John  Frederick  William  Herschel  was 
born  at  Slough  on  March  7,  1792.  His  father  was  Sir 
William  Herschel  (1738—1822)  who  was  the  most  illustrious 
astronomer  of  the  last  half  of  the  last  century.  Two  anec- 
dotes of  his  boyish  years  were  frequently  told  by  him  as 
illustrative  of  his  home  training,  and  are  sufficiently  in- 
teresting to  deserve  repetition.  One  day  when  playing  in 
the  garden  he  asked  his  father  what  was  the  oldest  thing 
with  which  he  was  acquainted.  His  father  replied  in  Socratic 
manner  by  asking  what  the  lad  thought  "  was  the  oldest  of  all 
things."  The  replies  were  all  open  to  objection,  and  finally  the 
astronomer  answered  the  question  by  picking  up  a  stone  and 
saying  that  that  was  the  oldest  thing  of  which  he  had  definite 
knowledge.  On  another  occasion  in  a  conversation  he  asked 
the  boy  what  sort  of  things  were  most  alike.  After  thinking  it 
over  young  Herschel  replied  that  the  leaves  of  a  tree  were  most 
like  one  another.  "Gather  then  a  handful  of  leaves  from  that 
tree,"  said  the  philosopher,  "and  choose  two  that  are  alike." 
Of  course  it  was  impossible  to  do  so.  Both  stories  are  trivial, 
but  they  were  typical  of  the  manner  in  which  he  was  brought 
up,  and  these  two  particular  incidents  happened  to  make  a 
deep  impression  on  his  mind. 

Except  for  one  year  spent  at  Eton  he  was  educated  at 
home.  In  1809  he  entered  at  St  John's  College,  graduating 
as  senior  wrangler  and  first  Smith's  prizeman  in  1813. 

His  earliest  original  work  was  a  paper  on  Cotes's  theorem, 
which  he  sent  when  yet  an  undergraduate  to  the  Royal  Society, 


HERSCHEL.     WHEWELL.  127 

and  immediately  after  taking  his  degree  it  was  followed  by- 
others  on  mathematical  analysis.  He  went  down  from  the 
university  in  or  about  1816,  and  for  a  few  years  read  for  the 
bar;  but  his  natural  bent  was  to  chemistry  and  astronomy, 
and  to  those  he  soon  turned  his  exclusive  attention.  The 
desire  to  complete  his  father's  work  led  ultimately  to  his  taking 
up  the  latter  rather  than  the  former  subject.  He  died  at  Col- 
lingwood  on  May  11,  1871. 

Besides  his  numerous  papers  on  astronomy,  his  Outlines  of 
astronomy  published  in  1849,  and  his  articles  on  Light  and 
Sound  in  the  Encyclopaedia  Metropolitana  appear  to  be  the 
most  important  of  his  contributions  to  science.  His  addresses 
to  the  Astronomical  and  other  societies  have  been  republished, 
and  throw  considerable  light  on  the  problems  of  his  time.  His 
Lectures  on  familiar  subjects  published  in  1868  are  models  of 
how  the  mathematical  solutions  of  physical  and  astronomical 
problems  can  be  presented  in  an  accurate  manner  and  yet  be 
made  intelligible  to  all  readers. 

Another  member  of  the  university  who  took  a  prominent 
part  in  developing  the  study  of  analytical  methods  was  Whewell. 
William  Whewell1,  of  Trinity  College,  was  born  at  Lancaster  on 
May  24,  1794,  graduated  as  second  wrangler  and  second  Smith's 
prizeman  in  1816,  and  was  in  due  course  elected  to  a  fellowship. 
His  life  was  spent  in  the  work  of  his  college  and  university. 
He  was  tutor  of  Trinity  from  1823  to  1839,  and  master  from 
1841  to  his  death  in  1866 ;  while  at  different  times  he  held  in 
the  university  the  chairs  of  mineralogy  and  moral  philosophy. 

His  chief  original  works  were  his  History  of  the  inductive 
sciences  and  his  papers  on  the  tides,  for  the  latter  of  which  he 
received  a  medal  of  the  Royal  Society ;  but  for  my  purpose  he 
is  chiefly  noticeable  for  the  great  influence  he  exerted  on  his 
contemporaries. 

1  Two  accounts  of  his  life  have  been  written  :  one  by  I.  Todhunter  in 
two  volumes,  London,  1876 ;  and  the  other  by  Stair  Douglas,  London, 
1881.  The  more  important  facts  form  the  subject  of  an  appreciative  and 
graceful  article  by  W.  G.  Clark  in  Macmillan's  magazine  for  April,  1866. 


128  THE  ANALYTICAL  SCHOOL. 

Whewell  occupied  to  his  generation  somewhat  the  same 
position  that  Bentley  had  done  to  the  Cambridge  of  his  day. 
But  though  Whewell  was  almost  as  masterful  and  combative 
as  Bentley  he  was  honest,  generous,  and  straightforward.  He 
lived  to  see  his  unpopularity  pass  away,  his  wonderful  attain- 
ments universally  recognized,  and  to  enjoy  the  hearty  respect 
of  all  and  the  love  of  many.  His  contemporaries  seem  to  have 
regarded  him  as  the  most  striking  figure  of  the  present  century, 
but  his  range  of  knowledge  was  so  wide  and  discursive  that  it 
could  not  be  very  deep,  and  his  reputation  has  faded  with 
great  rapidity.  Perhaps  a  future  generation  will  rate  him 
more  highly  than  that  of  to-day,  though  he  will  always  occupy 
a  prominent  position  in  the  history  of  the  university  and  his 
college. 

With  a  view  of  stimulating  still  further  the  interest  in 
mathematical  and  scientific  subjects  and  the  new  methods  of 
treating  them,  a  permanent  association  known  as  the  Cambridge 
Philosophical  Society  was  established  in  1819.  It  proved  very 
useful,  and  noticeably  so  during  the  first  twenty  or  thirty 
years  after  its  formation.  It  was  incorporated  in  1832. 


The  character  of  the  instruction  in  mathematics  at  the 
university  has  at  all  times  largely  depended  on  the  text-books 
then  in  use.  The  importance  of  good  books  of  this  class  has 
been  emphasized  by  a  traditional  rule  that  questions  should 
not  be  set  on  a  new  subject  in  the  tripos  unless  it  had  been 
discussed  in  some  treatise  suitable  and  available  for  Cambridge 
students.  Hence  the  importance  attached  to  the  publication 
of  the  work  on  analytical  trigonometry  by  Woodhouse  in  1809, 
and  of  the  works  on  the  differential  calculus  by  the  Analytical 
Society  in  1816  and  1820.  It  will  therefore  be  advisable  to 
enumerate  here  some  of  the  mathematical  text-books  brought 
out  by  members  of  the  new  school.  I  generally  confine  myself 
to  those  published  before  1840,  and  thus  exclude  the  majority 
of  those  known  to  undergraduates  of  the  present  day. 


MATHEMATICAL  TEXT-BOOKS.  129 

Wallis  had  published  a  treatise  on  analytical  conic  sections 
in  1665,  but  it  had  fallen  out  of  use;  and  the  only  work  on 
the  subject  commonly  read  at  Cambridge  at  the  beginning  of 
the  century  was  an  appendix  of  about  thirty  pages  at  the  end 
of  Wood's  Algebra.  This  was  headed  On  the  application  of 
algebra  to  geometry,  and  it  contained  the  equations  of  the 
straight  line,  ellipse,  and  a  few  other  curves,  -rules  for  the 
construction  of  equations,  and  similar  problems. 

The  senate-house  papers  from  1800  to  1820  shew  that  at 
the  beginning  of  the  century  analytical  geometry  was  always 
represented  to  some  extent,  though  scarcely  as  an  independent 
subject.  Most  of  the  questions  relate  to  areas  and  loci,  in 
which  little  more  than  the  mode  of  representation  by  means  of 
abscissae  and  ordinates  are  involved.  Even  as  late  as  1830 
the  editor  of  the  ninth  edition  of  Wood's  Algebra  deemed  that 
the  chapter  above  mentioned  afforded  a  sufficient  account  of 
the  subject. 

The  need  of  a  text-book  on  analytical  geometry  was  first 
supplied  by  the  work  by  Henry  Parr  Hamilton  issued  in  1826, 
and  above  alluded  to.  Hamilton  was  born  at  Edinburgh  on 
April  3,  1794,  and  graduated  from  Trinity  College  as  ninth 
wrangler  in  1816;  he  was  elected  in  due  course  to  a  fellowship, 
and  held  various  college  offices.  He  went  down  in  1830.  In 
1850  he  was  appointed  dean  of  Salisbury,  and  lived  there  till 
his  death  on  Feb.  7,  1880.  In  1826  Hamilton  published  his 
Principles  of  analytical  geometry,  in.  which  he  denned  the  conic 
sections  by  means  of  the  general  equation  of  the  second  degree, 
and  discussed  the  elements  of  solid  geometry.  Two  years  later, 
in  1828,  he  supplemented  this  by  another  and  more  elementary 
work,  termed  An  analytical  system  of  conic  sections,  in  which  he 
defined  the  curves  by  the  focus  and  directrix  property,  as  had 
been  first  suggested  by  Boscovich  :  the  latter  of  these  books 
went  through  numerous  editions,  and  was  translated  into 
German. 

In  1830  John  Hymers  (of  St  John's,  second  wrangler  in 
1826,  died  in  1887)  published  his  Analytical  geometry  of  three 
B.  9 


130  •  THE   ANALYTICAL   SCHOOL. 

dimensions.  In  1833  Peacock  issued  (anonymously)  a  Syllabus 
of  trigonometry,  and  the  application  of  algebra  to  geometry, 
seventy  pages  of  which  are  devoted  to  analytical  geometry ; 
there  was  a  second  edition  in  1836.  Hymers's  Conic  sections 
appeared  in  1837;  it  superseded  Hamilton's  in  the  university, 
and  remained  the  standard  work  until  the  publication  of  the 
text-books  still  in  use. 

Among  works  on  the  calculus  subsequent  to  those  of 
Peacock  and  Herschel  I  should  mention  one  by  Thomas 
Grainger  Hall  (of  Magdalene  College,  fifth  wrangler  in  1824, 
and  subsequently  professor  of  mathematics  at  King's  College, 
London),  issued  in  1834,  and  the  work  by  De  Morgan  pub- 
lished in  1842.  Henry  Kuhff,  of  St  Catharine's  (B.A.  1830, 
died  in  1842),  issued  a  work  on  finite  differences  in  1831 ;  but 
I  have  never  seen  a  copy  of  it.  In  1841  a  Collection  of  ex- 
amples illustrative  of  the  use  of  the  calculus  was  published  by 
Duncan  Farquharson  Gregory,  a  fellow  of  Trinity  College  :  this 
was  a  work  of  great  ability  and  was  one  of  the  earliest  attempts 
to  bring  the  calculus  of  operations  into  common  use.  Gregory 
was  born  at  Edinburgh  in  April,  1813,  graduated  as  fifth 
wrangler  in  1837,  and  died  on  Feb.  23,  1844.  His  writings, 
edited  by  W.  Walton,  accompanied  by  a  biographical  memoir 
by  R.  L.  Ellis1,  were  published  at  Cambridge  in  1865. 

There  was  not  the  same  need  in  applied  mathematics  for  a 
new  series  of  text-books,  since  optics,  hydrostatics,  and  astro- 
nomy were  already  fairly  represented,  and  Woodhouse's  work 
on  the  latter  involved  the  analytical  discussion  of  dynamics. 
There  was  however  no  good  work  on  elementary  mechanics, 
and  one  was  urgently  required  :  this  was  supplied  by  the  pub- 
lication in  1819  of  WhewelPs  Mechanics,  and  in  1823  of  the 
same  author's  Dynamics.  Another  text-book  on  the  subject 
was  the  translation  of  Yenturoli's  Mechanics  by  D.  Cresswell, 

1  Robert  Leslie  Ellis,  of  Trinity  College,  who  was  born  at  Bath  in 
1817  and  died  at  Cambridge  in  1859,  was  senior  wrangler  in  1840.  His 
memoirs  were  collected  and  published  in  1863,  and  a  life  by  H.  Goodwin, 
the  present  bishop  of  Carlisle,  is  prefixed  to  them. 


MATHEMATICAL   TEXT-BOOKS.  131 

issued  in  1822  (see  p.  110).  In  1832-34  Whewell  re-issued  his 
Dynamics  in  a  greatly  enlarged  form  and  in  three  parts,  and  in 
1837  published  the  Mechanical  Euclid.  Most  of  the  older 
text-books  in  hydrostatics  were  superseded  by  Eland's  Ele- 
ments of  hydrostatics,  published  in  1824. 

In  1823  Henry  Coddington,  of  Trinity  College  (who  was 
senior  wrangler  in  1820  and  died  at  Rome  on  March  3,  1845), 
issued  a  text-book  on  geometrical  optics,  which  was  practically 
a  transcript  of  Whewell's  lectures  in  Trinity  on  the  subject. 
In  1838  William  Nathaniel  Griffin  (senior  wrangler  in  1837) 
published  his  Optics,  and  this  remained  for  many  years  a 
standard  work.  In  1829  Coddington  issued  a  treatise  on 
physical  optics,  which  was  followed  by  papers  on  various 
problems  in  that  subject. 

The  publication  by  Sir  George  Airy  of  his  Tracts  in  1826 
exercised  a  far  greater  influence  on  the  study  of  mathematical 
physics  in  the  university  than  the  works  just  mentioned.  A 
second  edition  of  the  Tracts,  which  appeared  in  1831,  con- 
tained a  chapter  on  the  Undulatory  theory  of  light,  a  subject 
which  was  thenceforth  freely  represented  in  the  tripos. 

I  should  add  to  the  above  remarks  that  between  1823  and 
1830  Dionysius  Lardner  (born  in  1793  and  died  in  1859) 
brought  out  a  series  of  treatises  on  the  greater  number  of  the 
subjects  above  mentioned. 

From  1840  onwards  an  immense  number  of  text-books 
were  issued.  I  do  not  propose  to  enumerate  them,  but  I  may 
in  passing  just  allude  to  the  works  on  most  of  the  subjects  of 
elementary  mathematics  brought  out  at  a  somewhat  later  date 
by  Isaac  Todhunter,  of  St  John's  College,  who  was  born  at 
Rye  in  1820,  graduated  as  senior  wrangler  in  1848,  and  died 
at  Cambridge  in  1884.  His  text-books,  if  somewhat  long, 
were  always  reliable,  and  for  some  years  they  were  in  general 
use.  Besides  these  Todhunter  wrote  histories  of  the  calculus 
of  variations,  of  the  theory  of  probabilities,  and  of  the  theory 
of  attractions. 

It  would  be  an  invidious  task  to  select  a  few  out  of  the 

9—2 


132  THE  ANALYTICAL   SCHOOL. 

roll  of  eminent  mathematicians  who  have  been  educated  at 
Cambridge  under  the  analytical  school.  But  the  names  of 
those  who  have  held  important  mathematical  chairs  will 
serve  to  shew  how  powerful  that  school  has  been,  and  con- 
fining myself  strictly  to  the  above,  and  omitting  any  reference 
to  others — no  matter  how  influential — I  may  just  mention  the 
following  names  as  a  sort  of  appendix  to  this  chapter.  The 
order  in  which  they  are  arranged  is  determined  by  the  dates- 
of  the  tripos  lists.  I  add  a  few  remarks  on  the  works  of 
Augustus  De  Morgan,  George  Green,  and  James  Clerk  Max- 
well, but  in  general  I  confine  myself  to  giving  the  name  of 
the  professor  and  mentioning  the  chair  that  he  held  or  holds. 

The  senior  wrangler  in  the  tripos  of  1819  was  Joshua  Kingy 
of  Queens'  College,  who  was  born  in  1798  and  died  in  1857. 
King  was  Lucasian  profestor  from  1839  to  1849  in  succession 
to  Babbage. 

Sir  George  BiddeU  Airy,  of  Trinity  College,  who  was  senior 
wrangler  in  1823,  was  born  in  Northumberland  on  July  27, 
1801.  In  1826  he  succeeded  Thomas  Turton  in  the  Lucasian 
chair,  which  in  1828  he  exchanged  for  the  Plumian  professor- 
ship, where  he  followed  Woodhouse  :  he  held  this  professorship 
until  his  appointment  as  astronomer-royal  in  1836,  in  succession 
to  John  Pond. 

The  senior  wrangler  of  1825  was  James  Challis,  of  Trinity,, 
who  was  born  in  1803  and  died  on  Dec.  3,  1882:  Challis  was 
Plumian  professor  in  succession  to  Sir  George  Airy  from  1836 
to  1882. 

The  year  1827  is  marked  by  the  name  of  Augustus  De 
Morgan1,  who  graduated  from  Trinity  as  fourth  wrangler.  He 
was  born  in  Madura  (Madras)  in  June  1806.  In  the  then 
state  of  the  law  he  was  (as  a  Unitarian)  unable  to  stand  for 
a  fellowship,  and  accordingly  in  1828  he  accepted  the  chair  of 
mathematics  at  the  newly-established  university  of  London, 
which  is  the  same  institution  as  that  now  known  as  Uni- 

1  His  life  has  been  written  by  his  widow  S.  E.  De  Morgan.  London,. 
1882. 


THE  ANALYTICAL  SCHOOL.  133 

versity  College.  There  (except  for  five  years  from  1831  to 
1835)  he  taught  continuously  till  1867,  and  through  his 
works  and  pupils  exercised  a  wide  influence  on  English 
mathematics.  The  London  Mathematical  Society  was  largely 
his  creation,  and  he  took  a  prominent  part  in  the  proceedings 
-of  the  Royal  Astronomical  Society.  He  died  in  London  on 
March  18,  1871. 

He  was  perhaps  more  deeply  read  in  the  philosophy  and 
history  of  mathematics  than  any  of  his  contemporaries,  but  the 
results  are  given  in  scattered  articles  which  well  deserve  col- 
lection and  republication.  A  list  of  these  is  given  in  his  life, 
and  I  have  made  considerable  use  of  some  of  them  in  this  book. 
The  best  known  of  his  works  are  the  memoirs  on  the  founda- 
tion of  algebra,  Cambridge  philosophical  transactions,  vols.  vin. 
and  ix. ;  his  great  treatise  on  the  differential  calculus  published 
in  1842,  which  is  a  work  of  the  highest  ability;  and  his  articles 
on  the  calculus  of  functions  and  on  the  theory  of  probabilities 
in  the  Encyclopaedia  Metropolitana.  The  article  on  the  cal- 
culus of  functions  contains  an  investigation  of  the  principles 
of  symbolic  reasoning,  but  the  applications  deal  with  the  solu- 
tion of  functional  equations  rather  than  with  the  general  theory 
of  functions.  The  article  on  probabilities  gives  a  very  clear 
analysis  of  the  mathematics  of  the  subject  to  the  time  at  which 
it  was  written. 

In  1830  we  have  the  names  of  Charles  Thomas  Whitley, 
subsequently  professor  of  mathematics  at  the  university  of 
Durham ;  James  William  Lucas  Heaviside,  subsequently  pro- 
fessor of  mathematics  at  the  East  India  College,  Haileybury ; 
and  Charles  Pritchard,  now  Savilian  professor  of  astronomy 
at  the  university  of  Oxford. 

In  1837  the  second  wrangler  was  James  Joseph  Sylvester, 
who  is  now  Savilian  professor  of  geometry  at  the  university  of 
Oxford.  Among  the  numerous  memoirs  he  has  contributed  to 
learned  societies  I  may  in  particular  single  out  those  on 
canonical  forms,  the  theory  of  contravariants,  reciprocants,  the 
theory  of  equations,  and  lastly  that  on  Newton's  rule.  He 


134  THE  ANALYTICAL   SCHOOL. 

has   also  created   the  language  and    notation  of   considerable 
parts  of  the  various  subjects  on  which  he  has  written. 

In  the  same  list  appears  the  name  of  George  Green,  who 
was  one  of  the  most  remarkable  geniuses  of  this  century. 
Green  was  born  near  Nottingham  in  1793.  Although  self- 
educated  he  contrived  to  obtain  copies  of  the  chief  mathe- 
matical works  of  his  time.  In  a  paper  of  his,  written  in  1827 
and  published  by  subscription  in  the  following  year,  the  term 
potential  was  first  introduced,  its  leading  properties  proved, 
and  the  results  applied  to  magnetism  and  electricity.  In  1832 
and  1833  papers  on  the  equilibrium  of  fluids  and  on  attractions 
in  space  of  n  dimensions  were  presented  to  the  Cambridge 
Philosophical  Society,  and  in  the  latter  year  one  on  the  motion 
of  a  fluid  agitated  by  the  vibrations  of  a  solid  ellipsoid  was 
read  before  the  Royal  Society  of  Edinburgh.  In  1833  he 
entered  at  Caius  College,  graduated  as  fourth  wrangler  in 
1837,  and  in  1839  was  elected  to  a  fellowship.  Directly  after 
taking  his  degree  he  threw  himself  into  original  work,  and 
produced  in  1837  his  paper  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  hypothesis,  the  phe- 
nomenon of  total  reflexion  is  explained  physically,  and  certain 
properties  of  the  vibrating  medium  are  deduced.  In  1839,  he 
read  a  paper  on  the  propagation  of  light  in  any  crystalline 
medium.  All  the  papers  last  named  are  printed  in  the 
Cambridge  philosophical  transactions  for  1839.  He  died  at 
Cambridge  in  1841.  A  collected  edition  of  his  works  was 
published  in  1871. 

The  senior  wrangler  in  1841  was  George  Gabriel  Stokes,  of 
Pembroke  College,  who  was  born  in  Sligo  on  Aug.  13,  1819, 
and  in  1849  succeeded  Joshua  King  as  Lucasian  professor. 
In  the  following  year  Arthur  Cayley,  of  Trinity  College,  was 
senior  wrangler :  he  was  born  at  Richmond,  Surrey,  on  Aug. 
16,  1821,  and  in  1863  was  appointed  Sadlerian  professor. 
In  the  tripos  of  the  next  year  John  Couch  Adams,  of  St 


THE  ANALYTICAL   SCHOOL.  135 

John's  College,  and  now  of  Pembroke  College,  was  senior 
wrangler:  he  was  born  in  Cornwall  on  June  5,  1819,  and 
in  1858  succeeded  Peacock  as  Lowndean  professor. 

-The  second  wrangler  in  1843  was  Francis  Bashforth,  who 
was  subsequently  appointed  professor  at  Woolwich.  His  re- 
searches, especially  those  on  the  motion  of  a  projectile  in  a 
resisting  medium  (London,  1873),  have  been  and  are  in  con- 
stant use  among  artillerymen  and  engineers  of  all  nations. 

The  second  wrangler  iu  1845  was  Sir  William  Thomson,  of 
Peterhouse,  who  was  born  at  Belfast  in  June,  1824,  and  is 
now  professor  of  natural  philosophy  at  the  university  of  Glasgow. 
I  need  hardly  say  here  that  Sir  William  Thomson  has  enriched 
every  department  of  mathematical  physics  by  his  writings. 
His  collected  papers  are  now  being  published  by  the  university 
of  Cambridge.  Among  other  names  in  the  same  tripos  are 
those  of  Hugh  Blackburn,  of  Trinity  College,  who  was  sub- 
sequently professor  of  mathematics  at  the  university  of  Glasgow, 
and  of  George  Robarts  Smalley,  the  astronomer-royal  of  New 
South  Wales. 

The  senior  wrangler  of  1852  was  Peter  Guthrie  Tait,  now 
professor  of  natural  philosophy  at  the  university  of  Edinburgh, 
who  besides  other  well-known  works  was  joint  author  with 
Sir  William  Thomson  of  the  epoch-marking  Treatise  on  natural 
philosophy,  of  which  the  first  edition  was  published  in  1867. 

The  year  1854  is  distinguished  by  the  name  of  James  Clerk 
Maxwell,  of  Trinity  College,  who  was  second  wrangler ;  Edward 
James  Routh,  of  Peterhouse,  being  senior  wrangler.  Maxwell1 
was  born  in  Edinburgh  on  June  13,  1831.  His  earliest  paper 
was  written  when  only  fourteen  on  a  mechanical  method  of 
tracing  cartesian  ovals,  and  was  sent  to  the  Royal  Society  of 

1  A  tolerably  full  account  of  his  life  and  a  list  of  his  writings  will  be 
found  either  in  vol.  xxin.  of  the  Proceedings  of  the  Koyal  Society,  or  in 
the  article  contributed  by  Prof.  Tait  to  the  Encyclopaedia  Eritannica. 
For  fuller  details,  his  life  by  L.  Campbell  and  W.  Garnett,  London,  1882, 
may  be  consulted.  His  collected  works  are  being  edited  by  Prof.  Niven, 
and  will  shortly  be  published  by  the  university  of  Cambridge. 


136  THE   ANALYTICAL   SCHOOL. 

Edinburgh.  His  next  paper  written  three  years  later  was  on 
the  theory  of  rolling  curves,  and  was  immediately  followed  by 
another  on  the  equilibrium  of  elastic  solids.  At  Cambridge  in 
1854  after  taking  his  degree  he  read  papers  on  the  transfor- 
mation of  surfaces  by  bending,  and  on  Faraday's  lines  of  force. 
These  were  followed  in  1859  by  the  essay  on  the  stability  of 
Saturn's  rings,  and  various  articles  on  colour.  He  held  a  chair 
of  mathematics  at  Aberdeen  from  1856  to  1860;  and  at  King's 
College,  London,  from  1860  to  1868;  in  1871  he  was  ap- 
pointed to  the  Cavendish  chair  of  physics  at  Cambridge.  His 
most  important  subsequent  works  were  his  Electricity  and 
magnetism  issued  in  1873,  his  Theory  of  heat  published  in 
1871,  and  his  elementary  text-book  on  Matter  and  motion. 
To  these  works  I  may  add  his  papers  on  the  molecular  theory 
of  gases  and  the  articles  on  cognate  subjects  which  he  con- 
tributed to  the  ninth  edition  of  the  Encyclopaedia  Britannica. 
He  died  at  Cambridge  on  Nov.  5,  1879. 

His  Electricity  and  magnetism,  in  which  the  results  of 
various  papers  are  embodied,  has  revolutionized  the  treatment 
of  the  subject.  Poisson  and  Gauss  had  shewn  how  electro- 
statics might  be  treated  as  the  effects  of  attractions  and  re- 
pulsions between  imponderable  particles ;  while  Sir  William 
Thomson  in  1846  had  shewn  that  the  effects  might  also  and 
with  more  probability  be  supposed  analogous  to  a  flow  of  heat 
from  various  sources  of  electricity  properly  distributed.  In 
electro-dynamics  the  only  hypothesis  then  current  was  the 
exceedingly  complicated  one  proposed  by  Weber,  in  which  the 
attraction  between  electric  particles  depends  on  their  relative 
motion  and  position.  Maxwell  rejected  all  these  hypotheses 
and  proposed  to  regard  all  electric  and  magnetic  phenomena  as 
stresses  and  motions  of  a  material  medium ;  and  these,  by  the 
aid  of  generalized  coordinates,  he  was  able  to  express  in 
mathematical  language.  He  concluded  by  shewing  that  if  the 
medium  were  the  same  as  the  so-called  luminiferous  ether,  the 
velocity  of  light  would  be  equal  to  the  ratio  of  the  electro- 
magnetic and  electrostatic  units.  This  appears  to  be  the  case, 


THE   ANALYTICAL   SCHOOL.  137 

though  these  units  have  not  yet  been  determined  with  sufficient 
precision  to  enable  us  to  speak  definitely  on  the  subject. 

Hardly  less  eventful,  though  less  complete,  was  his  work 
on  the  kinetic  theory  of  gases.  The  theory  had  been  es- 
tablished by  the  labours  of  Joule  in  England  and  Clausius 
in  Germany ;  but  Maxwell  reduced  it  to  a  branch  of  mathe- 
matics. He  was  engaged  on  this  subject  at  the  time  of  his 
death,  and  his  two  last  papers  were  on  it.  It  has  been  the 
subject  of  some  recent  papers  by  Boltzmann. 

In  the  tripos  list  of  1859  appear  the  names  of  William  Jack, 
professor  of  mathematics  at  the  university  of  Glasgow ;  of 
Edward  James  Stone,  the  Radcliffe  observer  at  the  university 
of  Oxford ;  and  of  Robert  Bellamy  Clifton,  the  professor  of 
physics  at  the  university  of  Oxford. 

I  repeat  again  that  the  above  list  is  in  no  way  intended  to 
be  exhaustive,  but  is  rather  to  be  taken  as  one  illustration  of 
the  growing  numbers  and  reputation  of  the  Cambridge  school 
of  mathematics. 

The  year  at  which  I  stop  is  the  first  of  the  Victorian 
statutes;  and  is  a  well-defined  date  at  which  I  may  close  this 
history. 

We  live  in  an  age  somewhat  analogous  to  that  of  the  com- 
mencement of  the  renaissance.  The  system  of  education  under 
the  Elizabethan  statutes — narrow  in  its  range  of  studies  and 
based  on  theological  tests — has  given  way  to  one  where  subjects 
of  all  kinds  are  eagerly  studied.  The  rise  of  the  analytical 
school  in  mathematics  and  the  establishment  of  the  classical 
tripos  in  1824  are  the  first  outward  and  visible  signs  of  the 
new  intellectual  activity  which  was  quickening  the  whole  life 
of  the  university.  The  mathematicians  have  taken  their  full 
share  in  that  life,  and  that  they  have  again  raised  Cambridge 
to  the  position  of  one  of  the  chief  mathematical  schools  of 
Europe  will  I  think  be  admitted  by  the  historian  of  the  subse- 
quent history  of  mathematics  in  Cambridge. 


CHAPTER    VIII. 
THE  ORGANIZATION  AND  SUBJECTS  OF  EDUCATION1. 

SECTION  1.     The  mediaeval  system  of  education. 

SECTION  2.     The  period  of  transition. 

SECTION  3.     The  system  of  education  under  the  Elizabethan  statutes. 

IN  the  preceding  chapters  I  have  enumerated  most  of  the 
eminent  mathematicians  educated  at  Cambridge,  and  have  in- 
dicated the  lines  on  which  the  study  of  mathematics  developed. 
I  propose  now  to  consider  very  briefly  the  kind  of  instruction 
provided  by  the  university,  and  the  means  adopted  for  testing 
the  proficiency  of  students. 

Until  1858  the  chief  statutable  exercises  for  a  degree  were 
the  public  maintenance  of  a  thesis  or  proposition  in  the  schools 

1  In  writing  this  chapter  I  have  mainly  relied  on  Observations  on 
the  statutes  of  the  university  of  Cambridge  by  G.  Peacock,  London, 
1841,  and  on  the  University  of  Cambridge  by  J.  Bass  Mullinger,  2 
volumes,  Cambridge,  1873  and  1884.  The  most  complete  collection  of 
documents  referring  to  Cambridge  is  that  contained  in  the  Annals  of 
Cambridge  by  C.  H.  Cooper,  5  volumes,  Cambridge,  1842 — 52;  but  the 
collection  of  Documents  relating  to  the  university  and  colleges  of  Cam- 
bridge, issued  by  the  Eoyal  Commissioners  in  1852,  is  for  many  purposes 
more  useful.  The  Statuta  antiqua  are  printed  at  the  beginning  of  the 
edition  of  the  statutes  issued  at  Cambridge  in  1785,  and  are  reprinted  in 
the  Documents.  It  would  seem  from  the  Munimenta  academica  by  Henry 
Anstey  in  the  Kolls  Series,  London,  1848,  that  the  customs  at  Oxford 
only  differed  in  small  details  from  those  at  Cambridge,  and  the  regula- 
tions of  either  university  may  be  used  to  illustrate  contemporary  student 
life  at  the  other :  but  migration  between  them  was  so  common  that  it 
would  have  been  strange  if  it  had  been  otherwise. 


THE   MEDIAEVAL   SYSTEM   OF   EDUCATION.  139 

against  certain  opponents,  and  the  opposition  of  a  proposition 
laid  down  by  some  other  student.  Every  candidate  for  a 
degree  had  to  take  part  in  a  certain  number  of  these  discus- 
sions. 

The  subject-matter  of  these  "acts"  varied  at  different 
times.  In  the  course  of  the  eighteenth  century  it  became  the 
custom  at  Cambridge  to  "keep"  some  or  all  of  them  on  mathe- 
matical questions,  and  I  had  at  first  intended  to  con6ne  myself 
to  reproducing  one  of  the  disputations  kept  in  that  century. 
But  as  the  whole  mediaeval  system  of  education — teaching  and 
examining — rested  on  the  performance  of  similar  exercises, 
and  as  our  existing  system  is  derived  from  that  without  any 
break  of  continuity,  I  thought  it  might  be  interesting  to  some 
of  my  readers  if  I  gave  in  this  chapter  a  sketch  of  the  course 
of  studies,  the  means  of  instruction,  and  the  tests  imposed  on 
students  in  earlier  times ;  leaving  the  special  details  of  a 
mathematical  act  to  another  chapter.  It  will  therefore  be 
understood  that  I  am  here  only  indirectly  concerned  with  the 
history  of  the  development  of  mathematical  studies. 

I  also  defer  to  a  subsequent  chapter  the  description  of  the 
origin  and  history  of  the  mathematical  tripos.  I  will  only 
here  remark  that  the  university  was  not  obliged  to  grant  a 
degree  to  any  one  who  performed  the  statutable  exercises,  and 
after  the  middle  of  the  eighteenth  century  the  university  in 
general  refused  to  pass  a  supplicat  for  the  B.A.  degree  unless 
the  candidate  had  also  presented  himself  for  the  senate-house 
examination.  That  examination  had  its  origin  somewhere 
about  1725  or  1730,  and  though  not  recognized  in  the  statutes 
or  constitution  of  the  university  it  gradually  superseded  the 
discussions  as  the  actual  test  of  the  ability  of  students. 

The  mediaeval  system  of  education. 

The  rules  of  some  of  the  early  colleges,  especially  those  of 
Michael-house  (founded  in  1324,  which  now  forms  part  of 
Trinity  College),  regulated  every  detail  of  the  daily  life  of 


140  THE   MEDIAEVAL   SYSTEM   OF   EDUCATION. 

their  members,  and  together  with  the  ancient  statutes  of  the 
university  enable  us  to  picture  the  ordinary  routine  of  the 
career  of  a  mediaeval  student. 

In  the  thirteenth  or  fourteenth  century  then  a  boy  came 
up  to  the  university  at  some  age  between  ten  and  thirteen 
under  the  care  of  a  "  fetcher,"  whose  business  it  was  to  collect 
from  some  district  about  twenty  or  thirty  lads  and  bring  them 
up  in  one  party.  These  "bringers  of  scholars"  were  pro- 
tected by  special  enactments1.  On  his  arrival  the  boy  was 
generally  entered  under  some  master  of  arts  who  kept  a  hostel 
(i.e.  a  private  boarding-house  licensed  by  the  university)  or  if 
very  lucky  got  a  scholarship  at  a  college.  The  university  in 
its  corporate  capacity  did  not  concern  itself  much  about  the 
discipline  or  instruction  of  its  younger  members :  times  were 
rough  and  life  was  hard,  and  if  one  student  more  or  less  died 
or  otherwise  came  to  grief  no  one  cared  about  it,  so  that  a 
student  who  relied  on  the  university  alone  or  got  into  a  bad 
hostel  was  in  sorry  straits. 

If  we  follow  the  course  of  a  student  who  was  at  one  of  the 
colleges  or  better  hostels  we  may  say  that  in  general  he  spent 
the  first  four  years  of  his  residence  in  studying  the  subjects 
of  the  trivium,  that  is,  Latin  grammar,  logic,  and  rhetoric. 
During  that  time  he  was  to  all  intents  a  schoolboy,  and  was 
treated  exactly  like  one.  It  is  noticeable  that  the  technical 
term  for  a  student  on  presentation  for  the  bachelor's  degree  is 
still  juvenis,  and  the  word  vir  is  reserved  for  those  who  are  at 
least  full  bachelors. 

Few  of  those  who  thus  came  up  knew  anything  beyond  the 
merest  elements  of  Latin,  and  the  first  thing  a  student  had  to 
learn  wa.s  to  speak,  read,  and  write  that  language.  It  is  proba- 
ble that  to  the  end  of  the  fourteenth  century  the  bulk  of  those 
who  came  to  the  university  did  not  progress  beyond  this,  and 
were  merely  students  in  grammar  attending  the  glomerel 
schools.  There  would  seem  to  have  been  nearly  a  dozen  such 

1  Munimenta  academica,  346  ;  Lyte,  198. 


PKOCEEDINGS  IN   GRAMMAR.  141 

schools  in  the  thirteenth  century,  each  under  one  master,  and 
all  under  the  supervision  of  a  member  of  the  university,  known 
as  the  magister  glomeriae1.  This  master  of  glomery  had  as  such 
no  special  right  over  the  other  students  of  the  university2,  but 
the  "  glomerels  "  were  of  course  subject  to  his  authority;  and 
to  enhance  his  dignity  he  had  a  bedell  to  attend  him.  To 
these  glomerels  the  university  gave  the  degree  of  "  master  in 
grammar,"  which  served  as  a  license  to  teach  Latin,  gave  the 
coveted  prefix  of  dominus  or  magister  (which  in  common  lan- 
guage was  generally  rendered  dan,  don,  or  sir),  and  distinguished 
the  clerk  from  a  mere  "hedge-priest."  To  get  this  degree  the 
glomerel  had  first  to  shew  that  he  had  studied  Priscian  in  the 
original,  and  then  to  give  a  practical  demonstration  of  pro- 
ficiency in  the  mechanical  part  of  his  art.  The  regulations 
were  that  on  the  glomerel  proceeding  to  his  degree  "  then  shall 
the  bedell  purvay  for  every  master  in  grammar  a  shrewd  boy, 
whom  the  master  in  grammar  shall  beat  openly  in  the  grammar 
schools,  and  the  master  in  grammar  shall  give  the  boy  a  groat 
tor  his  labour,  and  another  groat  to  him  that  provideth  the  rod 
and  the  palmer,  etcetera,  de  singidis.  And  thus  endeth  the 
act  in  that  faculty3."  The  university  presented  the  new 
master  in  grammar  with  a  palmer,  that  is  a  ferule ;  he  took  a, 
solemn  oath  that  he  would  never  teach  Latin  out  of  any  inde- 
cent book ;  and  he  was  then  free  of  the  exercise  of  his  pro- 
fession. The  last  degree  in  grammar  was  given  in  1542. 
A  student  in  grammar  in  general  went  down  as  soon  as  he  got 
his  degree.  The  resident  masters  in  grammar  occupied  a  very 
subordinate  position  in  the  university  hierarchy.  They  not 
only  yielded  precedence  to  bachelors,  but  there  were  express 

1  Mullinger,  i.  340. 

2  These  rules  were  laid  down  in  1275  by  Hugh  Balsham,  the  bishop 
of  Ely. 

3  The  account  of  this  and  other  ceremonies  of  the  mediaeval  univer- 
sity is  taken  from  the  bedell's  book  compiled  in  the  sixteenth  century  by 
Matthew  Stokes,  a  fellow  of  King's  and  registrary  of  the  university.    It 
is  printed  at  length  in  an  appendix  to  Peacock's  Observations. 


142  THE  MEDLEVAL   SYSTEM   OF   EDUCATION. 

statutes1  that  the  university  should  not  attend  the  funeral  of 
one  of  them. 

The  corresponding  degree  of  master  of  rhetoric  was  occa- 
sionally given.  The  last  degree  in  this  faculty  was  conferred 
in  1493. 

Ambitious  students  or  the  scholars  of  a  college  were  ex- 
pected to  know  something  of  Latin  before  they  came  up ;  but 
the  knowledge  was  generally  of  a  very  elementary  character, 
and  not  more  than  could  be  picked  up  at  a  monastic  or 
cathedral  school.  These  lads  formed  the  honour  students,  and 
took  their  degrees  in  arts. 

To  obtain  the  degree  of  master  of  arts  in  the  thirteenth 
century  it  was  necessary  first  to  obtain  a  licentia  docendi,  anr. 
secondly  to  be  "incepted,"  that  is,  admitted  by  the  whole  body 
of  teachers  or  regents  as  one  of  themselves.  The  licentia 
docendi  was  originally  obtained  on  proof  of  good  moral  charac- 
ter from  the  chancellor  of  the  chapter  of  the  church  with 
which  the  university  was  in  close  connection.  For  inception 
the  student  was  then  recommended  by  a  master  of  the  univer- 
sity under  whom  he  had  studied,  and  the  student  had  to  keep 
an  act  or  give  a  lecture  before  the  whole  university.  On  his 
inception  he  gave  a  dinner  or  presents  to  his  new  colleagues. 
Possibly  the  procedure  was  as  elaborate  as  that  described 
immediately  hereafter,  but  we  do  not  know  any  details  beyond 
the  above. 

At  a  later  time,  as  education  became  more  general,  the  lads 
were  somewhat  older  when  they  came  up,  and  were  already 
acquainted2  with  Latin  grammar.  The  students  in  grammar 
thus  gradually  declined  in  numbers,  and  finally  were  hardly 
regarded  as  being  members  of  the  university.  By  the  fifteenth 
century  the  average"  age  at  entrance  was  thirteen  or  four- 

1  Statuta  antiqua,  178;  Documents,  i.  404.     Similar  regulations  ex- 
isted at  Oxford,  Munimenta  academica,  264,  443. 

2  In  founding  King's  College  Henry  VI.  seems  to  have  assumed  that 
the  scholars  would  have  already  mastered  all  the  subjects  of  the  trivium 
at  Eton.     The  statute  is  quoted  in  Mullinger,  i.  308. 


THE  LECTURES  IN  THE  FACULTY  OF  ARTS.    143 

teen l,  and  most  of  the  students  proceeded  in  arts.  From  this 
time  forward  the  statuta  antiqua  of  the  university  enable  us  to 
sketch  the  course  of  a  student  in  far  greater  detail,  but  there 
is  no  reason  to  suppose  that  it  was  substantially  different  from 
that  of  a  student  in  arts  in  the  two  preceding  centuries. 

A  student  in  arts  spent  the  first  year  of  his  course  in  learn- 
ing Latin.  This  at  first  meant  Priscian  and  grammar  only, 
but  in  the  fifteenth  century  Terence,  Virgil,  and  Ovid  were 
added  as  text-books  which  should  be  used,  and  versification  is 
mentioned  as  a  possible  subject  of  instruction2.  The  next  two 
years  were  devoted  to  logic;  the  text-books  being  the  Sum- 
mulae  and  the  commentary  of  Duns  Scotus.  The  fourth  year 
was  given  up  to  rhetoric  :  this  meant  certain  parts  of  Aris- 
totelian philosophy,  as  derived  from  Arabic  sources. 

Instruction  in  these  subjects  was  given  by  the  cursory 
lectures  of  students  in  their  fifth,  sixth,  or  seventh  years  of  resi- 
dence (which  had  to  be  delivered  before  nine  in  the  morning  or 
after  noon) ;  and  by  the  ordinary  lectures  which  every  (regent) 
master  was  obliged  to  give  for  at  least  one  year  after  taking 
his  degree.  All  other  lectures  were  termed  extraordinary. 
Every  lecture  had  to  be  given  in  the  schools3,  and  the  uni- 
versity derived  a  considerable  part  of  its  scanty  income  from 
the  rents  taken  from  the  lecturers.  Gratuitous  lectures  were 
forbidden4.  A  statute  of  Urban  Y.  in  1366  addressed  to  the 
university  of  Paris  expressly  forbad  to  students  the  use  of 
benches  or  seats  in  lecture-rooms ;  this  was  probably  held 
binding  at  Cambridge,  and  all  students  attending  lectures  were 
expected  to  sit  or  lie  on  straw  scattered  on  the  floor,  as  we 
know  was  the  case  in  Paris.  Only  extraordinary  lectures 
were  permissible  in  the  Long  Vacation. 

1  See  the  regulations  of  King's  Hall,  quoted  in  Mullinger,  i.  253. 

2  See  MuUinger,  i.  350. 

3  A  list  of  pictures  of  lectures  in  illuminated  manuscripts  is  given  in 
Lyte,  228. 

4  Cambridge  documents,  i.  391;  similar  regulations  existed  at  Oxford, 
Munimenta  academica,  110,  129,  256,  279. 


144  THE   MEDIAEVAL   SYSTEM  OF  EDUCATION. 

The  lectures  were  either  dictatory,  or  analytical,  or  dialec- 
tical l.  The  first  or  nominatio  ad  pennam  consisted  in  dictating 
text-books,  for  few  students  possessed  copies  of  any  works 
except  the  Summulae  and  the  Sententiae :  the  former  being 
the  standard  work  on  logic,  and  the  latter  011  theology.  The 
second  or  analytical  lecture  was  purely  formal,  and  tradition- 
ally was  never  allowed  to  vary  in  any  detail — an  illustration 
of  it  is  extant  in  the  commentary  by  Aquinas  on  Aristotle's 
Ethics.  The  lecturer  commenced  with  a  general  question;  men- 
tioned the  principal  divisions;  took  one  of  them  and  subdivided 
it ;  repeated  this  process  over  and  over  again  till  he  got  to  the 
first  sentence  in  that  part  of  the  work  on  which  he  was 
lecturing;  he  then  expressed  the  result  in  several  ways. 
Having  finished  this  he  started  again  from  the  beginning  to 
get  to  his  second  sentence.  No  explanatory  notes  or  allusions 
to  other  parts  of  the  same  work  or  to  other  authorities  were 
permitted.  These  lectures  were  the  resource  of  those  masters 
who  wished  to  get  through  their  regency  with  as  little  trouble 
as  possible,  but  for  the  credit  of  the  mediaeval  students  I  am 
glad  to  say  that  they  were  not  popular.  Thirdly,  there  was  the 
dialectical  lecture,  where  each  sentence,  or  some  interpretation 
of  it,  was  propounded  as  a  question  and  defended  against  all 
objections,  the  arguments  being  thrown  into  the  syllogistic 
form  and  of  course  expressed  in  Latin.  Any  student  might 
be  called  on  to  take  part  in  the  discussion,  and  it  thus  prepared 
him  for  the  ordeal  through  which  he  had  subsequently  to  pass 
to  obtain  a  degree.  An  illustration  of  this  is  extant  in  the 
Quaestiones  of  Buridanus. 

To  supplement  the  instruction  given  by  the  regents,  three 
teachers  (known  as  the  Barnaby  lecturers)  were  annually  ap- 
pointed by  the  university,  at  stipends  of  £3.  Gs.  Sd.  a  year, 
to  lecture  on  Terence,  logic,  and  philosophy2;  and  subsequently 
a  fourth  lectureship  on  the  subjects  of  the  quadrivium  was 


1  See  Mullinger,  i.  359  et  seq.;  and  Peacock,  appendix  A. 

2  See  Peacock,  appendix  A,  v. 


THE   EXERCISES   REQUIRED   FROM   A   SOPHISTER.     145 

created  with  a  stipend  of  £4  a  year1.  These  officers  were  re- 
gularly appointed  till  1858,  though  for  nearly  three  centuries 
they  had  given  no  lectures. 

By  the  Lent  term  of  his  third  year  of  residence  a  student 
was  supposed  to  have  read  the  subjects  of  the  trivium,  and  he 
was  then  known  as  a  general  sophister.  As  such  he  had  to 
dispute  publicly  in  the  schools  four  times ;  twice  as  a  respond- 
ent to  defend  some  thesis  which  he  asserted,  and  twice  as  an 
opponent  to  attack  those  asserted  by  others.  A  bachelor  pre- 
sided over  these  discussions.  The  subject-matter  of  these  acts 
in  mediaeval  times  was  some  scholastic  question  or  a  pro- 
position taken  from  the  Sentences.  About  the  end  of  the 
fifteenth  century  religious  questions,  such  as  the  interpreta- 
tion of  biblical  texts,  began  to  be  introduced2.  Some  fifty 
or  sixty  years  later  the  favorite  subjects  were  drawn  either 
from  dogmatic  theology  (or  possibly  from  philosophy).  In  the 
seventeenth  century  the  questions  were  usually  philosophical, 
but  in  the  eighteenth  century  most  of  them  were  mathematical. 
Some  of  these  are  printed  later.  A  complete  list  of  the  acts  of 
any  year  would  give  a  very  fair  idea  of  the  prevalent  studies. 

After  keeping  his  acts  the  sophister  was  examined  by  the 
university  as  to  his  character  and  academical  standing,  and  if 
nothing  was  reported  against  him,  presented  himself  as  a  ques- 
tionist  to  be  examined  by  the  proctors  and  regents  in  the  arts 
school.  In  general  he  had  then  to  defend  some  question 
against  the  most  practised  logicians  in  the  university — a  some- 
what severe  ordeal.  Stupid  men  propounded  some  irrefu- 
table truism,  but  the  ambitious  student  courted  attack  by 
affirming  some  paradox. 

The  influence  of  these  acts,  especially  those  for  the  higher 
degrees,  was  very  considerable.  Thus  the  brilliant  declama- 
tion of  Peter  Ramus  for  his  master's  degree  at  Paris  on  the 
subject  QiMiecumque  ab  Aristotele  dicta  essent  commenticia  esse 
drew  a  crowded  and  critical  audience,  and  the  subsequent 

1  See  Statuta  antiqua,  136. 

2  Mullinger,  i.  568. 

B.  10 


146  THE   MEDIAEVAL   SYSTEM   OF    EDUCATION. 

discussion  really  affected  the  whole  subsequent  development 
of  philosophy  in  Europe. 

A  candidate  was  never  rejected,  but  reputation  or  contempt 
followed  the  popular  verdict  as  to  how  he  acquitted  himself. 
The  desirability  of  having  on  these  occasions  a  numerous 
and  friendly  audience  was  so  great  that  a  man's  friends  not 
only  came  themselves,  but  used  forcible  means  to  bring  in 
all  passers-by.  So  considerable  a  nuisance  did  the  practice 
become  that  a  statute  at  Oxford  is  extant  in  which  it  is  con- 
demned under  the  penalty  of  excommunication  and  imprison- 
ment1. 

This  test  having  been  passed  the  student  obtained  a  sup- 
plicat  to  the  senate  from  his  hostel  or  college.  He  was  then 
admitted  as  an  incepting  bachelor.  This  was  not  a  degree,  but 
it  marked  the  transition  to  the  studies  and  life  of  an  under- 
graduate. The  official  account  of  the  ceremony  is  sufficientlv 
quaint  to  be  worth  quoting.  On  a  day  shortly  before  Ash- 
Wednesday  about  nine  o'clock  in  the  morning  the  bedells, 
each  carrying  his  silver  staff  of  office  or  bacillarius  (from  which, 
it  has  been  suggested,  the  title  of  bachelor  may  possibly  be 
derived2),  "shall  go  to  the  College,  House,  Hall,  or  Hostel 
where  the  said  Questionists  be,  and  at  their  entry  into  the  said 
House  shall  call  and  give  warning  in  the  midst  of  the  Court 
with  these  words,  Alons,  Alons,  goe,  Masters,  goe,  goe ;  and 
then  toll,  or  cause  to  be  tolled  the  bell  of  the  House  to 
gather  the  Masters,  Bachelors,  Scholars,  and  Questionists 
together.  And  all  the  company  in  their  habits  and  hoods 
being  assembled,  the  Bedells  shall  go  before  the  junior  Ques- 
tionist,  and  so  all  the  rest  in  their  order  shall  follow  bare- 
headed, and  then  the  Father,  and  after  all,  the  Graduates  and 

1  Munimenta  academica,  i.  247. 

2  See  p.  208  of  University  society  in  the  eighteenth  century,  by  C. 
Wordsworth,  Cambridge,  1874.     The  derivation  usually  given  is  from  the 
Celtic  bach,  little,  from  which  comes  the  old  French  baceller,  to  make 
love :  but  Prof.  Skeat  in  his  dictionary  says  that  this  is  a  bad  guess,  and 
in  the  supplement  he  repeats  that  the  derivation  is  uncertain. 


THE    INCEPTION   OF   A   BACHELOR.  147 

company  of  the  said  House,  unto  the  common  schools  in  due 
order.  And  when  they  do  enter  into  the  schools,  one  of  the 
Bedells  shall  say,  noter  mater  [academia],  bona  nova,  bona 
nova;  and  then  the  Father  being  placed  in  the  responsall's 
seat,  and  his  children  standing  over  against  him  in  order, 
and  the  eldest  standing  in  the  hier  hand  and  the  rest  in 
their  order  accordingly,  the  Bedell  shall  proclaim,  if  he 
have  any  thing  to  be  proclaimed,  and  further  say,  Reverende 
Pater,  licebit  tibi  incipere,  sedere,  et  cooperiri  si  placet.  That 
done,  the  Father  shall  enter  his  commendations1  of  his  chil- 
dren, and  propounding  of  his  questions  unto  them,  which  the 
eldest  shall  first  answer,  and  the  rest  in  order.  And  when  the 
Father  has  added  his  conclusion  unto  the  questions,  the  Bedell 
shall  bring  them  home  in  the  same  order  as  they  went... and  at 
the  uttermost  school  door  the  Questionists  shall  turn  them  to  the 
Father  and  the  company  and  give  them  thanks  for  their  coming 
with  them2."  But  the  regulations  add  that  if  the  Father  shall 
ask  too  hard  questions  or  entrap  his  children  into  an  argument 
"the  Bedell  shall  knock  him  out,"  by  which  was  meant  knock- 
ing the  door  so  loudly  that  nothing  else  could  be  heard. 

At  a  later  time  the  incepting  bachelors  were  divided  into 
classes,  the  higher  classes  being  admitted  to  the  title  of 
bachelor  a  few  weeks  before  the  lower  ones.  The  former 
correspond  to  the  honour  students  of  the  present  time,  the 
latter  to  the  poll  men. 

During  the  remainder  of  the  Lent  term  the  newly  incepted 
bachelor  was  expected  to  spend  every  afternoon  in  the  schools. 
In  addition  to  the  necessity  of  "disputing"  with  any  regent 
who  cared  to  come  and  test  his  abilities,  he  was  required  to 
preside  at  least  nine  times  over  the  disputations  which  those 
who  were  studying  the  trivium  were  keeping,  criticize  the 
arguments  used,  and  sum  up  or  determine  the  whole  discussion. 

1  At  this  point  of  the  ceremony  the  candidates  knelt,  and  the  bedells 
are  directed  to  pluck  the  hoods  of  the  candidates  over  their  faces,  so  that 
the  blushes  raised  by  their  modesty  may  not  be  seen. 

2  Peacock,  Appendix  A,  iv — vi. 

10—2 


148  THE  MEDIAEVAL   SYSTEM   OF   EDUCATION. 

Heiice  he  was  usually  known  as  a  determiner,  and  was  said  to 
stand  in  quadragesima. 

There  was  a  master  of  the  schools  whose  business  it  was  to 
keep  order.  But  his  task  must  have  been  very  difficult,  and 
apparently  was  generally  beyond  his  powers ;  for  we  read  that 
drinking,  wrestling,  cockfighting,  and  such  like  amusements 
were  common.  These  "determinations"  were  regarded  as  a 
great  opportunity  for  distinction,  but  the  school  was  a  rough 
one,  and  many  students  preferred  to  determine  by  proxy  which 
was  permissible1. 

It  will  be  noticed  that  the  quadragesimal  disputations  took 
place  after  Ash- Wednesday,  and  therefore  after  the  admission 
of  some  or  all  the  students  to  the  title  of  bachelor.  In  early 
times  it  is  believed  that  the  inception  took  place  even  before 
the  examination  by  the  proctors. 

The  bachelor  was  supposed  to  devote  the  next  three  years 
to  the  study  of  the  quadrivium ;  namely,  arithmetic,  geometry 
(including  geography),  music,  and  astronomy ;  and  before  he 
could  proceed  to  the  degree  of  master  he  had  to  make  a 
declaration  that  he  had  studied  these  subjects.  There  was. 
however  no  public  test  of  his  knowledge,  and  practically,  unless 
he  had  a  marked  interest  in  them,  he  continued  to  devote  his 
time  to  logic,  metaphysics,  or  theology,  which  then  afforded 
the  only  avenues  to  distinction. 

I  have  already  pointed  out  that  a  bachelor  was  expected  to 
give  cursory  lectures,  by  which  it  may  be  added  he  earned 
some  pocket-money.  He  was  also  required  to  be  present  at 
all  public  disputations  of  masters  of  arts  unless  expressly 
excused  by  the  proctors,  to  keep  three  acts  against  a  regent 
master,  two  acts  against  bachelors,  and  give  one  declamation. 

It  is  usually  said  that  most  bachelors  resided  and  in  due 
course  commenced  master.  That  is  true  of  scholars  at  the 
colleges  who  were  obliged  by  statute  to  do  so,  but  I  suspect 
that  most  students  at  the  hostels  went  down  after  their  ad- 
mission to  the  title  of  bachelor. 

1  See  Statuta  antiqua,  141. 


THE  CREATION  OF  A  MASTER.          149 

At  the  end  of  the  seventh  year  from  his  entry  the  student 
who  had  performed  all  these  exercises  could  become  a  master. 
The  degree  itself  or  the  formal  ceremony  of  creation  was  given 
on  the  second  Tuesday  in  July,  called  the  day  of  commence- 
ment. On  the  previous  evening  certain  exercises  of  inception, 
known  as  the  vespers,  were  performed  in  the  schools1.  On  the 
Tuesday  morning  the  whole  university  met  in  Great  St  Mary's 
(which  was  fitted  up  for  the  occasion  something  like  a  theatre) 
at  7  A.M.  to  hear  high  mass.  The  supplicat  for  the  degree  was 
then  presented.  If  this  were  passed  the  youngest  regent 
present  (or  his  proxy),  known  as  the  praevaricator,  opened 
the  proceedings  with  a  speech  in  which  any  questions  then 
affecting  the  university  were  discussed  with  considerable 
license.  Next  a  doctor  of  divinity,  acting  as  the  "father," 
placed  the  pileum  or  cap  (symbolical  of  a  master's  degree)  on 
the  head  of  the  incepting  master.  The  latter  then  defended  a 
proposition  taken  from  Aristotle,  first  against  the  prsevaricator, 
and  then  against  the  youngest  non-regent;  finally  the  youngest 
doctor  of  divinity  summed  up  the  conclusion.  Each  successive 
inceptor  went  through  a  similar  exercise. 

Anthony  Wood  discovered  a  manuscript  containing  a  few 
questions  proposed  at  the  similar  congregation  at  Oxford. 
They  apparently  owe  their  preservation  to  the  fact  that  the 
inceptor  put  the  proposition  into  metrical  form,  which  struck 
the  audience  as  an  ingenious  conceit.  I  give  one  as  a  specimen 
of  the  kind  of  questions  propounded.  "  Questio  quinta  ad 
quam  respondebit  quintus  noster  inceptor  dominus  Robertus 
Gloucestrise,  quse  de  licentia  duorum  procuratorum  et  cum 
supportatione  hujus  venerabilis  auditorii  est  diutius  pertrac- 
tanda,  est  in  hac  forma.  Utrum  potentiarum  imperatrix  |  celsa 
morum  gubernatrix,  \  vis  libera  rationalis,  \  sit  laureata  digni- 
tate  |  electionis  consiliatae  \  ut  Domina  principalis." 

1  The  students  by  immemorial  custom  were  permitted  to  seize  the 
new  inceptor  as  he  came  out,  and  whether  he  liked  it  or  no  (and  the 
extant  references  shew  that  he  usually  didn't)  shave  him  in  preparation 
for  the  morrow. 


150  THE   MEDIEVAL   SYSTEM   OF    EDUCATION. 

The  subsequent  ceremonies  of  inception  are  described  at 
length  in  Peacock1  and  were  chiefly  formal.  The  incepting 
master  was  expected  to  make  a  present  of  either  a  gown  or 
gloves  to  every  officer  of  the  university,  and  to  give  a  dinner 
to  all  the  regents,  to  which  however  he  was  allowed  to  ask  his 
own  friends.  The  cost  of  this  must  have  been  considerable. 
lu  the  fourteenth  century  the  universities  of  Paris,  Oxford, 
and  Cambridge  passed  identical  stroutes  that  no  one  should 
spend  on  his  inception  more  than  .£41.  13s.  4c/.,  a  sum  which  is 
equivalent  to  about  ,£500  now,  and  must  have  been  far  above 
the  means  of  most  students2.  Noblemen  at  Oxford  and  Cam- 
bridge were  exempted  from  this  restrictive  rule3. 

A  student  could  apparently  plead  poverty  as  an  excuse  for 
not  fulfilling  these  duties,  or  could  incept  by  proxy — the  proxy 
receiving  a  degree  too.  The  conditions  under  which  this  was 
allowed  are  not  fully  known. 

These  presents  and  the  cost  of  the  dinner  were  ultimately 
changed  into  a  fee  to  the  university  chest.  The  difficulty  of 

1  See  Appendix  A  to  Peacock's  Observations. 

2  Statuta  antiqua,  127.     Mullinger,  i.  357. 

3  I  can  quote  the  menu  of  one  feast  given  by  a  wealthy  inceptor,  the 
cost  of  which  must  have  far  exceeded  the  statutable  limit ;  but  it  owes 
its  preservation  to  the  fact  that  it  was  an  exceptional  case.     The  wealth 
of  the  host  was  fabulously  large,  and  no  conclusion  can  be  drawn  as  to 
the  usual  practice.     The  "dinner"  to  which  I  refer  was  that  given  by 
George  Nevill,  the  brother  of  the  Earl  of  Warwick,  on  taking  his  master's 
degree  in  1452.    It  lasted  two  days ;  on  the  first  of  which  sixty,  and  on 
the  second,  two  hundred  dishes  were  served.     The  following  is  the  bill  of 
fare  for  the  chief  table,  which  in  my  ignorance  of  matters  culinary  I 
transcribe  verbatim :  a  suttletee,  the  bore  head  and  the  bull ;  frumenty 
and  venyson ;  fesant  in  brase ;  swan  with  chowdre ;  capon  of  grece ;  hern- 
shew  ;  poplar ;  custard  royall ;  grant  flanport  desserted ;  leshe  damask ; 
frutor  lumbent ;  a  suttletee.     The  dishes  served  at  the  second  table  were 
viant  in  brase ;  crane  in  sawce ;  yong  pocock ;  cony ;  pygeons ;  bytter ; 
curlew ;  carcall ;  partrych ;  venyson  baked  ;    fryed  meat   in   port ;   lesh 
lumbent ;   a  frutor ;   a  suttletee.      At  the  third  table  were  gely  royall 
desserted ;  hanch  of  venson  rested ;  wodecoke  ;  plover  ;  knottys ;  styntis ; 
quayles ;   larkys ;   quyuces   baked ;   viaunt   in   port ;    a  frutor ;   lesh ;    a 
suttletee. 


THE   CAREER  OF  A  MASTER.  151 

raising  the  money  for  these  expenses  was  to  some  extent  met 
by  the  university  allowing  the  proctors  to  take  jewels,  manu- 
scripts, or  even  clothes,  as  pledges.  It  would  seem  that  the 
university  sometimes  made  a  bad  bargain,  for  by  a  statute1  of 
unknown  date  the  proctors  are  forbidden  to  advance  money  on 
any  books  or  manuscripts  which  are  written  on  paper,  but  they 
are  expressly  allowed  to  continue  to  take  vellum  manuscripts 
as  a  security  for  fees.  The  new  master  was  not  permitted  to 
exercise  his  functions  until  the  term  after  that  in  which  he 
incepted — a  custom  which  still  exists  at  Cambridge — but  sub- 
ject to  that  restriction  he  was  obliged  to  reside  and  teach  for 
at  least  one  year,  and  was  both  entitled  and  obliged  to  charge 
a  fee  to  those  who  attended  his  lectures.  His  duties  were  then 
at  end,  and  if  he  went  down  he  was  tolerably  sure  of  getting 
his  livelihood,  while  his  degree  served  as  a  license  to  lecture  on 
the  trivium  and  quadrivium  in  any  university  in  Europe. 

The  genuine  student,  or  the  man  who  aimed  at  worldly 
success,  generally  proceeded  to  the  doctor's  degree  in  civil  law, 
canon  law,  or  theology;  and  in  most,  colleges  it  was  obligatory 
on  a  fellow  to  do  so.  A  similar  degree  was  also  obtainable  in 
medicine  or  music.  No  one  could  obtain  the  doctorate  in  any 
subject  who  did  not  really  know  it  as  it  was  then  understood. 
These  courses  took  from  eight  to  ten  years,  and  are  too  elabo- 
rate for  me  to  describe  here. 

It  was  not  uncommon  for  the  new  master  to  migrate  to 
another  university  and  take  his  doctorate  there.  Paris  was 
especially  thus  favoured,  and  a  mediaeval  scholar  was  rarely 
content  if  he  had  not  spent  a  few  years  in  the  famous  rue  du 
fouarre.  This  migration  facilitated  the  propagation  of  ideas, 
and  served  somewhat  the  same  purpose  as  the  multiplication 
of  a  book  by  printing  at  a  later  time. 

If  we  were  to  judge  solely  by  the  statutes  and  ordinances 

of  the  university,  this  curriculum  would  seem  to  have  been  well 

designed  as  a  general  and  elastic  system  of  education.     The 

scientific  subjects  of  the  quadrivium  were  however  frequently 

1  See  Statuta  antiqua,  182. 


152  THE   MEDLEVAL    SYSTEM   OF   EDUCATION. 

neglected.  This  was  partly  due  to  the  fact  that  they  had 
practical  applications,  for  the  universities  of  Paris,  Oxford,  and 
Cambridge  systematically  discouraged  all  technical  instruction, 
holding  that  a  university  education  should  be  general  and  not 
technical.  The  chief  reason  for  the  neglect  was  however  that 
no  distinction  could  be  obtained  except  in  philosophy  and 
transcendental  theology.  Thjse  subjects  are  interesting  in 
themselves,  and  valuable  as  a  branch  of  higher  education,  but 
experience  seems  to  shew  that  only  those  who  have  already 
mastered  some  exact  science  are  likely  to  derive  benefit  from 
their  study.  Be  this  as  it  may,  it  was  not  the  belief  of  the 
schoolmen.  They  captured  the  mediaeval  universities,  and 
there  is  a  general  consensus  of  opinion  that  the  absence  of 
fruitful  work  was  mainly  due  to  the  fact  that  they  controlled 
its  studies  and  induced  men  to  read  philosophy  before  their 
opinions  were  sufficiently  mature. 

I  should  add  that  the  popular  idea  that  the  schoolmen  did 
nothing  but  dispute  about  questions  such  as  how  many  angels 
could  simultaneously  dance  on  the  point  of  a  needle  is  grossly 
unjust.  Besides  discussing  various  questions  which  are  still 
debated,  they  created  the  science  of  formal  logic,  and  it  is  to 
them  that  the  precision  and  flexibility  of  the  Romance  tongues 
is  mainly  due.  No  doubt  some  of  their  more  foolish  members 
said  some  foolish  things,  but  to  judge  them  by  the  propositions 
which  Erasmus  selected  when  he  was  attacking  them  and  ridi- 
culing their  pretensions  is  manifestly  unfair.  It  is  said  that 
in  philosophy  they  settled  nothing,  but  that  was  hardly  their 
fault,  for  it  is  characteristic  of  the  subject  that  no  question  is 
ever  definitely  settled.  It  must  also  be  remembered  that  the 
schoolmen  held  that  the  value  of  a  general  education  was  to  be 
tested  by  the  methods  used  rather  than  the  results  attained. 

The  only  subject  that  rivalled  philosophy  as  a  popular 
study  was  theology.  It  did  not  enter  directly  into  the  cur- 
riculum for  the  master's  degree,  but  it  involved  the  most 
burning  questions  of  the  day,  and  could  not  fail  to  excite 
general  interest.  The  standard  text-book  for  this  was  the 


THE   EDWARDIAN   STATUTES   (1549).  153 

work  known  as  the  Sentences^.  This  was  a  collection  made 
by  Peter  Lombard,  in  1150,  of  the  opinions  (sententiae)  of  the 
Fathers  and  other  theologians  on  the  most  difficult  points  in 
the  Christian  belief.  The  logicians  adopted  it  as  a  magazine 
of  indisputable  major  premises,  and  created  a  large  literature 
of  deductions  therefrom. 


The  period  of  transition. 

The  mediseval  system  of  education  was  terminated  by  the 
royal  injunctions  of  1535,  which  forbad  the  teaching  of  the 
logic  and  metaphysics  of  the  schoolmen,  and  in  place  thereof 
commanded  the  study  of  classical  and  biblical  literature  and  of 
science.  The  subsequent  rearrangements  of  the  studies  of  the 
university  were  briefly  as  follows. 

The  first  serious  attempt  to  reorganize  the  studies  of  the 
university  was  embodied  in  the  Edwardian  code  of  15492. 
To  check  the  presence  of  those  who  were  merely  schoolboys,  it 
directed  that  for  the  future  students  (except  those  at  Jesus 
College)  should  be  required  to  have  learnt  the  elements  of 
Latin  before  coming  into  residence.  The  curriculum  laid  down 
was  as  follows.  The  freshman  was  to  be  first  taught  mathe- 
matics, as  giving  the  best  general  training  :  this  was  to  be 
followed  by  dialectics,  and  if  desirable  by  philosophy :  the 
whole  forming  the  course  for  the  bachelor's  degree.  The 
bachelor  in  his  turn  was  expected  to  read  perspective,  astro- 
nomy, Greek,  and  the  elements  of  philosophy  before  taking  the 
master's  degree.  Finally,  a  resident  master,  after  acting  as 
regent  for  three  years  was  expected  to  study  law,  medicine,  or 
theology.  These  reforms  represented  the  views  of  the  mo- 
derate conservative  party  in  the  university,  and  the  only 
objection  expressed3  was  the  very  reasonable  one  that  masters 

1  Mullinger,  i.  59—63. 

2  Mullinger,  n.  109—115. 

3  By  Ascham:  see  p.  16  of  Original  letters  of  eminent  literary  men 
edited  by  Sir  Henry  Ellis,  Camden  Society,  London,  1843. 


154  THE   PERIOD   OF   TRANSITION. 

should  be  at  liberty  to  take  the  doctorate  in  any  branch  of 
literature  or  science  that  they  pleased. 

These  statutes  were  replaced  in  1557  by  others,  known 
as  Cardinal  Pole's ;  but  the  latter  were  repealed  and  the 
Edwardian  (with  a  few  minor  alterations)  re-enacted  in  1559. 

The  period  of  transition  was  marked  by  the  commencement 
of  the  professorial  system  of  instruction.  The  mediaeval  plan 
of  making  every  master  lecture  for  at  least  one  year  was 
essentially  bad ;  and  in  practice  it  had  to  be  supplemented  by 
the  hostels  and  colleges.  By  the  beginning  of  the  sixteenth 
century  it  was  generally  admitted  that  this  method  was  not 
adapted  to  the  requirements  of  the  university;  and  it  was  then 
proposed  to  endow  professorships  whereby  it  was  hoped  that 
the  university  would  obtain  for  its  students  the  best  available 
teaching.  The  new  system  originated  with  the  foundation  in 
15021  by  the  Lady  Margaret  of  a  chair  of  divinity;  and 
in  1540  her  grandson,  Henry  VIII. ,  endowed  the  five 
regius  professorships  of  divinity,  law,  physic,  Hebrew,  and 
Greek. 

The  age  of  transition  was  also  contemporaneous  with  the 
establishment  of  the  college  system,  as  we  know  it.  The  early 
colleges  were  at  first  founded  for  a  few  fellows  and  scholars 
only.  When  however  the  insignificant  little  hall  of  God's 
House  (which  had  been  founded  in  1439  and  whose  members 
never  read  beyond  the  trivium)  was  in  1505  enlarged  and  re- 
incorporated  by  Lady  Margaret  as  Christ's  College,  a  power 
was  taken  to  admit  pensioners,  then  called  convivae,  and  at 
the  same  time  the  government  was  vested  in  the  fellows  as 
well  as  the  master.  These  changes  were  introduced  on  the 
advice  of  Bishop  Fisher,  the  confessor  of  Lady  Margaret,  to 
whom  Cambridge  is  perhaps  more  indebted  than  to  any  other 
of  its  numerous  and  illustrious  benefactors.  A  similar  provi- 
sion was  inserted  in  the  statutes  of  the  other  colleges  which 

1  The  earliest  professorships  founded  at  Oxford  were  those  endowed 
by  Henry  VIII.  in  1546.  I  believe  professorships  were  established  at 
Paris  in  the  fifteenth  centurv. 


THE   PERIOD   OF   TRANSITION.  155 

were  shortly  afterwards  founded,  viz.  St  John's,  Buckingham 
(now  known  as  Magdalene),  Trinity,  Emmanuel,  and  Sidney. 

The  colleges  concerned  themselves  with  the  health,  morals, 
and  discipline  of  their  students,  as  well  as  with  their  educa- 
tion. As  soon  as  the  college  and  university  systems  of  tuition 
and  discipline  came  into  competition  the  latter  broke  down 
utterly1;  and  twenty  years  sufficed  to  change  the  university 
from  one  where  nearly  all  the  students  were  directly  under  the 
authority  of  the  university  to  one  where  they  were  grouped  in 
colleges,  each  college  supervising  the  education  and  discipline 
of  its  students,  subject  of  course  to  the  general  rules  of  the 
whole  body  of  graduates  by  whom  the  final  test  of  a  proper 
education  was  applied  before  a  degree  was  granted.  The 
university  imposed  no  exercises  until  a  student's  third  year  of 
residence  and  abandoned  the  duty  of  providing  instruction  for 
undergraduates  to  the  colleges.  It  is  easy  to  criticize  the 
theory  of  the  college  system,  but  there  can  be  no  doubt  that 
it  at  once  met  and  still  meets  the  general  requirements  of  the 
nation  at  large. 


The  system  of  education  under  the  Elizabethan  statutes. 

The  period  of  transition  in  the  studies  of  the  university 
was  brought  to  a  close  by  the  promulgation  of  the  Elizabethan 
code  of  1570,  which  remained  almost  intact  till  1858.  These 
statutes  are  memorable  for  the  complete  revolution  which  they 
effected  in  the  constitution  of  the  university,  making  it  directly 
amenable  to  the  influence  of  the  crown  and  distinctly  ecclesi- 
astical in  character.  The  manner  in  which  these  changes  were 

1  Dr  Caius  had  been  educated  under  the  old  system,  but  when  he 
returned  in  1558  (to  refound  Gonville  Hall)  he  found  the  collegiate 
system  was  firmly  established.  The  history  of  the  university  which  he 
wrote  is  thus  particularly  valuable,  for  he  describes  in  detail  exactly  how 
the  older  system  differed  from  that  under  which  he  then  found  himself 
living. 


156          THE  ELIZABETHAN  STATUTES. 

introduced  is  described  later  (see  pp.  245-247).  The  curriculum 
was  also  recast1.  Mathematics  was  again  excluded  from  the 
trivium,  and  in  lieu  thereof  undergraduates  were  directed 
to  read  rhetoric  and  logic;  but  the  commissioners  made  no 
material  alterations  in  thr  course  for  the  master's  degree.  The 
power  to  interpret  these  statutes,  and  to  arrange  the  times  and 
details  of  all  lectures  and  necessary  exercises,  was  vested  in  the 
heads  of  colleges  alone. 

Although  the  subjects  of  education  were  changed  the  ex- 
ercises for  degrees,  the  manner  of  taking  them,  and  the  intervals 
between  them  were  left  substantially  unaltered,  save  only  that 
the  conditions  under  which  the  exercises  had  to  be  performed 
were  rigorously  defined  by  statute,  and  no  longer  left  to  the 
discretion  of  the  governing  body  of  the  university. 

The  statutable  course  for  the  degree  of  bachelor  of  arts  was 
as  follows2.  An  undergraduate  was  obliged  to  be  a  member 
of  a  college.  After  he  had  resided  for  three  years3,  and  had 
studied  Greek,  arithmetic,  rhetoric,  and  logic,  he  was  created  a 
general  sophister  by  his  college.  He  then  attended  the  in- 
cepting bachelors,  comprising  students  one  year  senior  to  him- 
self who  were  standing  in  quadragesima ;  and  besides  this  read 
two  theses,  and  kept  at  least  two  responsions  and  two  op- 
ponencies  under  the  regency  of  a  master.  At  the  end  of  his 
fourth3  year  he  was  examined  by  his  college,  and  if  approved 
presented  as  a  questionist.  In  the  week  preceding  Ash- Wed- 
nesday (or  earlier  in  the  same  term)  he  was  examined  by  the 
proctors  (or  by  their  deputies,  the  posers,  subsequently  termed 
moderators)  and  any  other  regents  who  wished  to  do  so.  A 
supplicat  from  the  student's  college  was  then  presented,  and  if 
granted  the  undergraduate  was  admitted  ad  respondendum 
quaestioni.  "  I  admit  you,"  said  the  vice-chancellor,  "  to  be 
bachelor  of  arts  upon  condition  that  you  answer  to  your 

1  Mullinger,  n.  232  et  seq. 

2  Peacock,  8 — 10  et  seq. 

3  The  requisite  residence  was  in  practice  shortened  by  reckoning  the 
time  from  the  term  in  which  the  name  was  put  on  the  college  boards. 


THE  STATUTABLE   COUKSE  IN  ARTS.  157 

questions:  rise  and  give  God  thanks."  The  student  then 
rose,  crossed  the  senate-house,  and  knelt  down  to  say  "  his 
private  prayers."  The  ceremony  of  "  entering  the  questions  " 
took  place  immediately  afterwards  in  the  schools,  the  father  or 
proctor  asking  a  question  from  Aristotle's  analytics.  It  was 
purely  formal,  and  the  bedells  attended  to  "  knock  out "  any 
one  who  began  to  argue.  The  questionist  was  admitted  as  a 
bachelor  designate  on  Ash-Wednesday  (or  if  not  worthy  of  this 
was  admitted  a  few  weeks  later).  He  then  became  a  de- 
terminer, and  after  standing  in  quadragesima  until  the  Thursday 
before  Palm  Sunday,  the  complete  degree  of  bachelor  was  con- 
ferred by  the  proctors. 

A  candidate  for  the  degree  of  master  of  arts  was  required 
to  reside,  to  attend  lectures,  and  to  be  present  at  all  public 
acts  kept  by  masters.  Besides  these  he  had  to  deliver  one 
declamation,  and  to  keep  three  respondencies  against  M.A. 
opponents,  two  respondencies  against  B.A.  opponents,  and  six 
opponencies  against  B.A.  respondents.  The  caput  however  in 
1608  decided  that  residence  should  no  longer  be  necessary  for 
taking  the  master's  degree.  The  decision  was  contrary  to  the 
statutes,  but  it  only  sanctioned  a  practice  which  had  already 
become  prevalent.  The  exercises  and  acts  for  that  degree  were 
thenceforth1  reduced  to  a  mere  formality,  so  that  the  only  real 
tests  subsequently  imposed  by  the  university  on  its  students 
were  those  immediately  preceding  and  attending  the  admission 
to  the  bachelor's  degree. 

Like  all  immutable  codes,  which  deal  minutely  with  every 
detail  of  administration,  the  new  statutes  proved  unworkable 
in  some  parts.  It  is  doubtful  if  the  performance  of  all  the 
exercises  and  acts  was  ever  enforced,  and  it  «was  not  long 
before  some  of  the  most  important  provisions  of  the  new  code 
were  habitually  and  systematically  neglected. 

1  I  should  add  that  in  1748  William  Ridlington  of  Trinity  Hall  (B.A. 
1739)  who  was  then  proctor,  required  the  strict  performance  of  the 
statutable  exercises,  and  Christopher  Anstey  of  King's  was  expelled  for 
resisting  the  claim. 


158          THE  ELIZABETHAN  STATUTES. 

I  come  next  to  the  method  of  giving  instruction,  which  was 
usual  during  most  of  this  period. 

The  professorial  system  was  already  well  established.  The 
regius  chairs  and  others  founded  at  a  later  time,  brought 
eminent  men  to  the  university,  and  it  would  be  difficult  to 
overrate  the  influence  thus  exerted ;  but  as  a  means  of  getting 
the  best  teaching  suitable  for  the  bulk  of  the  students  the 
scheme  failed.  In  fact,  the  power  of  advancing  the  bounds  of 
knowledge  in  any  particular  study  and  the  art  of  expounding 
and  teaching  results  that  are  already  known  are  rarely  united 
in  the  same  person.  The  professors  were  generally  selected  for 
the  first  qualification.  On  the  whole  I  think  they  were,  in 
nearly  all  cases,  the  most  eminent  members  of  the  university  in 
their  own  departments ;  and  if  in  the  eighteenth  century  some 
of  them  not  only  did  not  teach  but  did  very  little  to  encourage 
advanced  work,  the  fault  is  rather  to  be  attributed  to  the  age 
than  to  the  system. 

We  must  however  recognize  as  a  historical  fact  that  till  the 
end  of  the  eighteenth  century  the  professors  did  not — with  a 
few  exceptions,  and  notably  of  Newton — influence  the  in- 
tellectual life  of  the  university  as  much  as  might  have  been 
reasonably  expected,  and  they  were  generally  glad  to  abandon 
nearly  all  teaching  to  the  colleges. 

Throughout  the  period  in  which  the  Elizabethan  statutes 
were  in  force  the  college  and  tutorial  systems  of  education  were 
much  as  we  now  know  them.  I  add  in  the  following  para- 
graphs a  brief  account  of  what  the  colleges  expected  from  their 
students. 

In  the  sixteenth  century1  an  undergraduate  was  expected 
to  rise  at  4.30,  after  his  private  prayers  (in  a  stated  form)  lie 
went  to  chapel  at  5.0.  After  service  (and  possibly  breakfast) 
he  adjourned  to  the  hall,  where  he  did  exercises  and  attended 
lectures  from  six  to  nine.  At  nine  the  college  lectures  gene- 

1  This  account  is  taken  from  the  statutes  of  Trinity  College:  see 
Peacock,  pp.  4—8.  The  statutes  of  1552  and  1560  are  printed  as  an 
appendix  to  the  second  volume  of  Mullinger's  work. 


THE   COLLEGE    SCHEME   OF   EDUCATION.  159 

rally  ceased,  and  the  great  body  of  the  students  proceeded 
to  the  public  schools,  either  to  hear  lectures,  or  to  listen  to, 
or  take  part  in  the  public  disputations  which  were  requisite 
for  the  degree  of  bachelor  or  master.  Dinner  was  served  at 
eleven,  and  at  one  o'clock  the  students  returned  to  their 
attendance  on  the  declamations  and  exercises  in  the  schools. 
From  three  until  six  in  the  afternoon  they  were  at  liberty 
to  pursue  their  amusements  or  their  private  studies  :  at  six 
o'clock  they  supped  in  the  college-hall  and  immediately  after- 
wards retired  to  their  chambers.  There  was  no  evening 
service  in  the  college  chapels  on  ordinary  days  until  the  reign 
of  James  I.  Whether  most  students  lived  up  to  this  ideal  is 
doubtful :  some  certainly  did  not. 

As  time  went  on  the  average  age  at  entrance  rose  from 
about  sixteen  in  the  sixteenth  century  to  seventeen  or  eighteen 
in  the  seventeenth,  and  to  eighteen  or  nineteen  in  the  eighteenth 
century.  The  hours  also  gradually  got  later,  and  the  strictness 
of  the  regulations  was  somewhat  relaxed.  At  the  beginning  of 
the  eighteenth  century  the  "  college  day  began  with  morning 
chapel,  usually  at  six.  Breakfast  was  not  a  regular  meal,  but 
it  was  often  taken  at  a  coffee-house  where  the  London  news- 
papers could  be  read.  Morning  lectures  began  at  seven  or 
eight  in  the  college-hall.  Tables  were  set  apart  for  different 
subjects.  At  'the  logick  table'  one  lecturer  is  expounding 
Duncan's  treatise,  while  another,  at  'the  ethick  table'  is  in- 
terpreting Puffendorf  on  the  duty  of  a  man  and  a  citizen  ; 
classics  and  mathematics  engage  other  groups.  The  usual 
college  dinner-hour  which  had  long  been  11  a.m.,  had  ad- 
vanced before  1720  to  noon.  The  afternoon  disputations  in 
the  schools  often  drew  large  audiences  to  hear  respondent  and 
opponent  discuss  such  themes  as  'natural  philosophy  does  not 
tend  to  atheism,'  or  'matter  cannot  think.'  Evening  chapel 
was  usually  at  five;  a  slight  supper  was  provided  in  hall  at 
seven  or  eight1",  or  in  summer  even  later.  Sometimes  after 
supper  acts  (preparatory  to  those  in  the  schools)  were  kept : 
1  See  Jebb's  Life  of  Bentley,  p.  88. 


160  THE   ELIZABETHAN   STATUTES. 

the  origin  of  the  college  fees  for  those  degrees  is  the  re- 
muneration paid  to  the  M.A.'s  who  presided  at  these  intra- 
mural exercises.  At  other  times  plays  were  then  performed 
in  hall,  and  once  a  week  a  viva  voce  examination  (of  course  in 
Latin)  was  held.  Some  of  the  tutors  also  gave  evening  lectures 
in  their  rooms. 

In  the  sixteenth  and  seventeenth  centuries  the  educational 
work  of  the  university  was  mainly  performed  by  the  college 
tutors.  It  was  at  first  usual  to  allow  men  to  choose  each  his 
own  tutor  according  to  the  subject  he  wished  to  read,  and  to 
allow  any  fellow  or  the  master  to  take  pupils1;  but  the  ad- 
ministrative and  disciplinary  difficulties  connected  with  such 
a  scheme  proved  insuperable,  while  it  was  found  to  be  almost 
impossible  for  a  corporation  to  prevent  an  inefficient  fellow 
from  taking  pupils.  The  number  of  tutors  was  therefore 
limited,  but  it  was  still  assumed  that  a  tutor  was  able  to 
give  to  every  man  all  the  instruction  he  required.  Of  course 
this  universal  knowledge  was  not  generally  possessed,  and 
towards  the  beginning  of  the  eighteenth  century  we  hear  of 
other  teachers  who  were  ready  to  give  instruction  in  all  the 
mathematical  subjects  required  by  the  university. 

There  can  be  no  question  that  some  members  of  the  uni- 
versity had  given  such  private  instruction  in  earlier  times. 
I  should  however  say  that  the  difference  between  the  mediaeval 
system  of  coaching  and  that  which  sprang  up  in  the  eighteenth 
century  was  that  the  former  was  resorted  to  either  by  students 
who  were  backward  and  wanted  special  assistance,  or  by  those 
who  wished  to  specialize  and  went  to  specialists,  while  the 
latter  was  used  by  those  who  desired  to  master  the  maximum 
number  of  subjects  in  the  minimum  time  with  a  view  to  taking 
as  high  a  place  in  the  tripos  as  possible.  As  soon  as  that  ex- 
amination, with  its  strictly  denned  order  of  merit,  became  the 
sole  avenue  to  a  degree  coaching  became  usual  and  perhaps 

1  On  the  former  tutorial  system  see  e.g.  the  Scholae  academicae,  259  et 
seq.;  and  also  vol.  ii.,  pp.  438 — 9  of  Todhunters  Life  of  Whewell,  London, 
1876. 


PRIVATE  TUTORS.  161 

inevitable,  for  a  high  place  in  the  tripos  was  not  only  the 
chief  university  distinction,  but  had  a  considerable  pecuniary 
value. 

There  is  no  doubt  that  mathematics  is  most  efficiently 
taught  either  by  private  instruction,  or  by  lectures  supple- 
mented by  private  instruction.  Every  part  of  it  has  to  be  read 
in  a  tolerably  well-defined  sequence,  and  with  the  varying 
abilities  and  knowledge  of  men  this  requires  a  certain  amount 
of  individual  assistance  which  cannot  be  given  in  a  large 
lecture.  Most  of  the  tutors  and  professors  of  the  eighteenth 
century  neglected  this  fact.  Indeed  the  professors,  taken  as 
a  whole,  made  no  effort  to  influence  the  teaching  of  the 
university,  while  the  majority  of  the  college  tutors  of  that 
time  were  not  sorry  to  be  relieved  of  the  most  laborious  part 
of  their  work.  On  the  other  hand,  the  instruction  given  by 
the  coaches  was  both  thorough  and  individual;  while  as  men 
were  free  to  choose  their  own  private  tutor,  inefficient  teachers 
were  rare.  Of  course  where  the  examination  included  a  very 
large  subject,  such  as  a  book  of  the  Principia,  that  subject  had 
to  be  taught  by  means  of  an  analysis,  and  such  analyses  and 
manuscripts  containing  matter  not  incorporated  into  text-books 
were  and  are  in  constant  circulation  in  the  university. 

The  result  of  the  movement  was  that  the  whole  instruction 
of  the  bulk  of  the  more  advanced  students  (in  mathematics) 
passed  into  the  hands  of  a  few  men  who  were  independent  both 
of  the  university  and  of  the  colleges — a  fact  which  seems  to  be 
as  puzzling  as  it  is  inexplicable  to  foreign  observers. 

I  am  satisfied  that  the  system  originated  in  the  eighteenth 
century,  but  I  have  found  it  very  difficult  to  arrive  at  any 
definite  facts  or  dates.  In  particular  I  am  not  clear  how 
far  the  "pupil-mongers"  of  the  beginning  of  that  century, 
such  as  Laughton,  are  to  be  regarded  as  private  tutors  or 
not.  I  suspect  that  they  were  college  lecturers  who  threw 
their  lectures  open  to  the  university,  but  supplemented  them 
by  additional  assistance  for  which  they  were  paid  a  private 
fee. 

B.  11 


162  THE   ELIZABETHAN   STATUTES. 

The  earliest  indisputable  reference  to  a  coach,  across  which 
I  have  come  is  in  the  life1  of  William  Paley  of  Christ's.  His 
"  private  tutor  "  was  Wilson  of  Peterhouse  (see  p.  102),  by  whom 
"  he  was  recommended  to  Mr  Thorp  [Robert  Thorp,  of  Peter  - 
house,  B.A.  1758,  and  afterwards  archdeacon  of  Northumber- 
land] who  was  at  that  time  of  eminent  use  to  young  men 
in  preparing  them  for  the  senate-house  examination  and 
peculiarly  successful.  One  young  man  of  no  shining  reputation 
with  the  assistance  of  Mr  Thorp's  tuition  had  stood  at  the 
head  of  wranglers."  Thorp — to  cut  a  long  story  short — con- 
sented to  coach  Paley,  and  brought  him  out  as  senior  in  1763. 
A  grace  passed  by  the  senate  in  1781  commences  with  a  pre- 
amble in  which  it  is  stated  that  almost  all  sophs  then  resorted 
to  private  tuition. 

At  that  time  the  moderators  in  the  tripos  often  prepared 
pupils  for  the  examination  they  were  about  to  conduct. 
Various  graces2  of  the  senate  were  passed  from  1777  onwards 
to  stop  this  custom.  At  a  later  period  different  attempts  were 
made  to  prevent  private  tutors  from  acting  as  examiners,  but 
all  such  legislation  broke  down  in  practice. 

Even  non-residents  acquired  a  reputation  as  successful 
coaches.  Thus  John  Dawson,  a  medical  practitioner  at  Sed- 
bergh  (born  in  January,  1734,  and  died  in  September,  1820), 
regularly  prepared  pupils  for  Cambridge,  and  read  with  them 
in  the  long  vacation.  At  least  eleven  of  the  senior  wranglers 
between  1781  and  1800  are  known  to  have  studied  under  him, 
but  the  names  of  his  pupils  cannot  in  general  be  now  deter- 
mined. 

During  the  first  three-quarters  of  the  present  century 
(i.e.  beyond  the  point  to  which  my  history  extends)  nearly 

1  See  p.  29  of  his  life  by  E.  Paley,  London,  1838.    William  Paley 
was  the  author  of  the  well  known  View  of  the  evidences  of  Christianity, 
first  published  in  1794 :  he  was  born  in  1743,  and  died  in  1803. 

2  A  list  of  them  is  given  in  chap.  in.  section  3  of  Whewell's  Of  a 
liberal  education,  second  edition,  London,  1850.     See  also  the  Scholae 
academicae  pp.  260 — 261. 


PRIVATE  TUTORS.  163 

every1  mathematical  student  read  with  a  private  tutor.  So 
universal  was  the  practice  that  William  Hopkins  (who  was  born 
in  1805,  graduated  as  seventh  wrangler  in  1827,  and  died  in 
1866)  was  able,  in  1849,  to  say  that  since  his  degree  he 
had  had  among  his  pupils  nearly  two  hundred  wranglers,  of 
whom  17  had  been  senior  and  44  in  one  of  the  first  three 
places.  So  again  at  the  recent  presentation  of  his  portrait  to 
Dr  Routh  by  his  old  pupils  it  was  remarked  that  he  had 
directed  the  undergraduate  mathematical  education  of  nearly 
all  the  younger  Cambridge  mathematicians  of  the  present  time. 
Thus  in  the  thirty-one  years  from  1858  to  1888  he  had  had  no 
less  than  631  pupils,  most  of  whom  had  been  wranglers,  and 
27  of  whom  had  been  senior  wranglers. 

Private  tuition  in  other  subjects  became  for  a  short  time 
usual,  but  with  the  recent  developments  and  improvements  in 
college  teaching  by  the  aid  of  a  large  staff  of  teachers  in  addi- 
tion to  the  tutors,  the  necessity  for  coaching  has  gradually  dis- 
appeared— at  any  rate  in  subjects  other  than  mathematics. 
Whether  in  that  subject  it  is  possible  to  give  all  the  requisite 
teaching  by  college  lectures  without  sacrificing  the  advantages 
of  order  of  merit  in  the  tripos  is  one  of  the  problems  of  the 
present  time. 

1  There  were  exceptions ;  thus  G.  Pryme,  who  was  sixth  wrangler  in 
1803,  writes  in  his  Reminiscences  (p.  48)  that  coaching  was  not  really 
necessary,  and  that  he  found  college  lectures  sufficient. 


11—2 


CHAPTER  IX. 
THE   EXERCISES   IN   THE   SCHOOLS1. 

I  PURPOSE  now  to  give  an  account  of  the  scholastic  acts  to 
which  so  many  references  were  made  in  the  last  chapter,  and 
to  illustrate  their  form  by  reproducing  one  on  a  mathematical 
subject. 

I  have  already  enumerated  the  subjects  of  instruction 
enjoined  by  the  Elizabethan  statutes,  and  it  is  certain  that  it 
was  intended  that  the  scholastic  disputations  should  be  kept  on 
philosophical  questions  drawn  from  that  curriculum. 

The  statutes  however  had  hardly  received  the  royal  assent 
before  the  philosophy  of  Ramus  (see  p.  14)  became  dominant 
in  the  university;  and  the  discussions  were  tinged  by  his  views. 
About  1650  the  tenets  of  the  Baconian  and  Cartesian2  systems 
of  philosophy  became  the  favourite  subjects  in  the  schools  of 
the  university.  Some  fifty  years  later  they  were  displaced  by 
subjects  drawn  from  the  Newtonian  philosophy,  and  thenceforth 
it  was  usual  to  keep  some  of  the  disputations  on  mathematical 
subjects;  though  it  always  remained  the  general  custom  to 

1  The  substance  of  this  chapter  is  reprinted  from  my  Origin  and 
history  of  the  mathematical  tripos,  Cambridge,  1880.     The  materials  for 
that  were  mainly  taken  from  Of  a  liberal  education,  by  W.  Whewell, 
Cambridge,  1848,  and  the  Scholae  academicae,  by  C.  Wordsworth,  Cam- 
bridge, 1877. 

2  I  think  there  can  be  no  doubt  that  the  Cartesian  philosophy  was 
read:   Whewell,  however,  always  maintained  the  contrary,  but  in  this 
opinion  he  was  singular. 


THE   EXERCISES  IN   THE   SCHOOLS.  165 

propound  at  least  one  philosophical  question,  which  was  fre- 
quently taken  from  Locke's  Essay.  In  1750  it  was  decided  in 
Cumberland's  case  that  it  was  not  necessary  for  a  candidate 
to  offer  any  except  mathematical  subjects. 

The  earliest  list  with  which  I  am  acquainted  of  questions 
kept  in  the  schools  is  contained  in  the  Disputationum  academi- 
carurti  formulae  by  R.  F.,  published  in  1638.  A  list  of 
questions  on  philosophy  in  common  use  during  the  early  years 
of  the  eighteenth  century  was  published  in  1735  by  Thomas 
Johnson,  who  was  a  fellow  of  Magdalene  College  and  master  at 
Eton. 

The  procedure  seems  to  have  remained  substantially  un- 
altered from  the  thirteenth  to  the  nineteenth  centuries,  and  it 
is  probable  that  the  following  account  taken  from  the  records 
of  the  eighteenth  century  would  only  differ  in  details  from  the 
description  of  a  similar  exercise  kept  in  the  middle  ages. 

The  disputation  commenced  by  the  candidate  known  as  the 
act  or  respondent  proposing  three  propositions  [in  the  middle 
ages  he  only  proposed  one]  on  one  of  which  he  read  a  thesis. 
Against  this  other  students  known  as  opponents  had  then  to 
argue.  The  discussions  were  presided  over  by  the  moderators 
[or  before  1680  by  the  proctors,  or  their  deputies  the  posers], 
who  moderated  the  discussion  and  awarded  praise  or  blame  as 
the  case  might  require.  The  discussions  were  always  carried 
on  in  Latin  and  in  syllogistic  form. 

In  the  eighteenth  century,  when  the  system  had  crys- 
tallized into  a  rigid  form,  it  was  the  invariable  custom  to  have 
in  the  sophs's  schools  three  opponents  to  each  respondent.  Of 
these  the  first,  who  took  the  lead  in  the  discussion,  was  expected 
to  urge  five  objections  against  the  first  of  the  propositions  laid 
down  by  the  respondent,  three  against  the  second,  and  one 
against  the  third.  The  respondent  replied  to  each  in  turn, 
and  when  an  argument  had  been  disposed  of,  the  moderator 
called  for  the  next  by  saying  Probes  aliter.  When  the  dispu- 
tation had  continued  long  enough  the  opponent  was  dismissed 
with  some  such  phrase  as  Bene  disputasti.  The  second  op- 


166  THE  EXERCISES  IN   THE   SCHOOLS. 

ponent  followed,  and  urged  three  objections  against  the  first 
proposition  and  one  against  each  of  the  others.  His  place  was 
then  taken  by  the  third  opponent,  of  whom  but  one  argument 
against  each  question  was  required.  If  a  candidate  failed 
utterly  he  was  dismissed  with  the  order  Descendas,  which  was 
equivalent  to  a  modern  pluck.  Such  cases  were  extremely 
rare.  Finally,  the  respondent  was  examined  by  the  moderator, 
and  according  as  he  acquitted  himself  was  released  with  some 
suitable  remark. 

The  following  is  a  more  detailed  account  of  the  procedure 
in  the  eighteenth  century.  By  that  time  all  the  exercises 
subsequent  to  the  admission  to  the  degree  of  bachelor  had 
become  reduced  to  a  mere  formality ;  but  every  student  (un- 
less he  intended  to  proceed  in  civil  law,  or  was  a  fellow-com- 
moner) had  in  the  course  of  his  third  year  of  residence  to 
keep  one  or  more  disputations  in  the  sophs's  schools. 

At  the  beginning  of  the  Lent  term  the  moderators  (or, 
before  1680,  the  proctors)  applied  to  the  tutors  of  the  dif- 
ferent colleges  for  lists  of  the  candidates  for  the  next  year. 
An  undergraduate  had  no  right  to  present  himself,  and  several 
cases  are  mentioned  in  which  permission  to  keep  exercises  in 
the  schools  was  refused  to  students  who  were  not  likely  to  do 
credit  to  the  college.  To  see  if  this  were  the  case  it  was  usual 
for  the  college  authorities  to  examine  their  students  before  the 
latter  were  allowed  to  keep  an  act  in  public,  and  to  prepare 
them  for  it  by  mock  exercises  in  the  college  hall.  The  college 
fee  for  students  taking  a  bachelor's  or  master's  degree  was,  as 
I  have  already  said,  originally  imposed  to  cover  the  cost  of 
this  preliminary  examination  and  preparation. 

The  lists  sent  by  the  college  tutors  were  supplemented  by 
memoranda  such  as  'reading  man,'  'non-reading  man,'  &c.,  and 
guided  by  these  remarks  and  the  general  reputation  of  the 
students  the  moderators  fixed  on  those  who  should  keep  the 
acts  and  opponencies.  The  expectant  wranglers  were  generally 
chosen  to  be  the  respondents,  they  and  the  senior  optimes  were 
reserved  for  the  first  and  second  opponencies  (on  whom  the 


THE  EXERCISES  IN  THE  SCHOOLS.  167 

brunt  of  the  discussion  fell),  and  the  third  opponencies  were 
given  to  those  who  were  expected  to  take  a  poll  degree,  the 
appearance  of  the  latter  in  the  schools  being  often  little  more 
than  nominal. 

By  a  happy  accident  the  private  list  of  Moore  Meredyth, 
of  Trinity  (B.A.  1736),  who  was  one  of  the  proctors  for  1752 
has  been  preserved,  and  is  now  in  the  university  registry.  It 
contains  altogether  the  names  of  seventy-seven  students1.  Of 
these  twelve  are  placed  first  in  a  class  by  themselves  headed  by 
the  letter  R,  which  means  that  they  were  selected  to  be  respon- 
dents. Fourteen  are  put  next  by  themselves  in  another  division 
marked  0,  and  these  men  were  most  likely  chosen  to  keep  first 
opponencies.  The  names  of  those  who  were  not  expected  to 
take  honours  form  a  third  list.  The  names  in  each  set  begin 
with  the  Trinity  men,  and  those  from  the  other  colleges  follow. 
From  the  list  which  the  moderators  had  thus  drawn  up  of 
the  candidates,  and  some  three  weeks  before  any  particular 
respondent  had  to  keep  an  act,  he  received  a  notice  from  the 
proctors  calling  on  him  to  propose  for  their  approval  three  sub- 
jects for  discussion.  In  practice  he  was  allowed  to  choose  any 
questions  taken  from  the  traditional  subjects  of  examination, 
and  to  select  the  one  in  support  of  which  he  should  read  his 
thesis.  So  important  was  the  work  of  preparation  that  even 
a  college  dean  relented  somewhat  of  his  sternness,  and  the 
student  was  permitted  to  take  out  a  dormiat,  and  thus  excused 
from  morning  chapels  was  able  to  concentrate  all  his  attention 
on  the  approaching  contest.  One  of  his  first  duties  was  to 
make  the  acquaintance  of  his  opponents,  inform  them  on 
which  of  the  three  subjects  he  intended  to  read  his  thesis,  and 
arrange  other  details  of  the  contest.  In  earlier  times  the 
opponents  had  no  such  assistance.  The  opponents  in  a  similar 
way  arranged  amongst  themselves  the  order  and  plan  of  their 
arguments. 

The  disputation  began  about  three  o'clock.     As  soon  as  the 
moderator   had   taken   his  seat    he    said    Ascendat    dominiis 
1  Scholae  academicae,  pp.  363,  364. 


168  THE  EXERCISES   IN   THE  SCHOOLS. 

respondent,  and  thereupon  the  respondent  walked  up  into  a 
sort  of  desk  facing  the  moderator.  The  exercise  commenced 
by  his  reading  a  Latin  thesis,  which  lasted  about  ten  minutes, 
in  support  of  one  of  his  propositions  :  this  essay  was  after- 
wards given  to  the  moderators.  As  soon  as  it  was  finished  the 
moderator,  turning  to  the  first  opponent,  said  Ascendat  oppo- 
nentium  primus.  The  latter  then  entered  a  box  below  or  by 
the  side  of  the  moderator  and  facing  the  respondent.  He 
opposed  the  proposition  laid  down  in  the  thesis  in  five  argu- 
ments, the  second  question  in  three,  and  the  third  in  one. 
Every  argument  was  put  into  the  form  of  a  hypothetical 
syllogism  and  ran  as  follows.  Major  premise :  If  A  is  B  (the 
antecedentia),  C  is  D  (the  consequens,  or  more  generally  but 
inaccurately  spoken  of  as  the  consequentia).  Minor  premise  : 
But  A  is  B.  Conclusion  :  Therefore  C  is  D  (the  consequentia). 
The  respondent  denied  any  step  in  this  that  was  not  clear, 
generally  admitting  that  A  was  B,  but  alleging  that  it  did  not 
follow  that  C  was  D.  The  opponent  then  explained  how  he 
maintained  his  objection,  and  this  process  was  continually 
repeated  until  he  had  fairly  stated  his  case,  when  the  respond- 
ent replied ;  and  the  discussion  was  then  carried  on  until  the 
moderator  stopped  it  by  saying  to  the  opponent  Probes  aliter. 
After  the  eighth  argument  the  first  opponent  was  sent  down 
with  some  compliment  such  as  Domine  opponens,  bene  disputasti, 
or  optime  disputasti,  or  even  optime  quidem  disputasti.  It  is 
from  this  use  of  the  word  that  the  terms  senior  optime  and 
junior  optime  are  derived.  As  soon  as  the  first  opponent  had 
finished,  the  second  opponent  followed  and  urged  three  ob- 
jections against  the  first  proposition  and  one  against  each  of 
the  others.  His  place  was  then  taken  by  the  third  opponent, 
of  whom  but  one  argument  against  each  question  was  required. 
Finally,  the  respondent  was  examined  by  the  presiding  mode- 
rator, and  according  as  he  did  badly  or  well  was  released  with 
the  remark  Tu  autem,  domine  respondens,  bene  (or  satis,  or 
satis  et  bene)  disputasti,  or  even  satis  et  optime  quidem  et  in 
thesi  et  in  disputationibus  tuo  qfficio  functus  es,  or  sometimes 


THE   EXERCISES  IN  THE   SCHOOLS.  169 

with  the  highest  compliment  of  all,  summo  ingenii  acumine 
disputasti. 

In  general  optime  guidem  was  the  highest  praise  expected, 
but  towards  the  close  of  the  eighteenth  century  Lax  introduced 
the  custom  of  giving  elaborate  compliments,  much  to  the  dis- 
gust of  some  of  the  older  members  of  the  university.  An 
order  to  quit  the  desk  was  equivalent  to  rejection,  but  the 
power  was  very  rarely  used. 

A  copy  of  the  thesis  read  on  Feb.  25,  1782,  by  John 
Addison  Carr  of  Jesus  for  his  act  is  in  the  library  of  Trinity1, 
it  is  apparently  the  original  manuscript  handed  to  the  modera- 
tors at  the  close  of  the  disputation.  The  manuscript  begins 

Q[u«estiones]  S[unt] 

Eecte  statuit  Newtonus  in  tertia  sua  sectione. 
Eecte  statuit  Emersonus  de  motu  projectiiium. 
Origo  mail  moral  is  solvi  potest  salvis  Dei  attributis. 
De  postrema. 

Then  follows  an  essay  on  the  third  question  ;  and  on  the  last 
page  of  the  manuscript  there  is  a  memorandum 

Carr,  coll.  Jes.  Eesp.  Feb.  25,  1782. 
Bere,  Sid.  coll.,  Opp.  lmug. 
Cragg,  S.S.  Trinitatis,  Opp.  2US. 
Newcome,  coll.  Begin.,  Opp.  3US. 

Finally  at  the  bottom  is  the  signature  of  the  presiding  mo- 
derator Littlehales  Modr.  Coll.  Johann.  which  he  affixed  at 
the  conclusion  of  the  act.  The  essay  covers  some  eight  and 
a  half  cl(  >sely  written  pages  of  a  foolscap  quarto  note-book,  and 
is  not  worth  quoting.  In  the  tripos  list  of  1783,  Carr  came 
out  as  eleventh  senior  optime,  Bere  as  ninth  senior  optime, 
Cragg  as  sixth  junior  optime  (i.e.  last  but  two),  and  Newcome 
as  twelfth  wrangler. 

On  the  results  of  these  discussions  the  final  list  of  those 
qualified  to  receive  degrees  was  prepared.  The  order  of  this 
list  in  early  times  had  been  settled  according  to  the  discretion 

1  The  Challis  manuscripts,  in.  59. 


170  THE   EXERCISES  IN  THE  SCHOOLS. 

of  the  proctors  and  moderators,  and  every  candidate  before 
presenting  himself  took  an  oath  that  he  would  abide  by  their 
decision.  The  list  was  not  arranged  strictly  in  order  of  merit, 
because  the  proctors  could  insert  names  anywhere  in  it;  but 
except  for  these  honorary  distinctions,  the  recipients  of  which 
were  called  proctors's  or  honorary  optimes,  it  probably  fairly 
represented  the  merits  of  the  candidates.  The  names  of  those 
who  received  these  honorary  degrees  subsequent  to  1747  are 
struck  out  from  the  lists  given  in  all  the  calendars  issued 
subsequent  to  1799.  It  is  only  in  exceptional  cases  that  we 
are  acquainted  with  the  true  order  for  the  earlier  tripos  lists, 
but  in  a  few  cases  the  relative  positions  of  the  candidates  are 
known;  for  example,  in  1680  Bentley  came  out  third  though 
he  was  put  down  as  sixth  in  the  list  of  wranglers.  By 
the  beginning  of  the  eighteenth  century  this  power  had  ap- 
parently become  restricted  to  the  right  reserved  to  the  vice- 
chancellor,  the  senior  regent,  and  each  proctor  to  place  in  the 
list  one  candidate  anywhere  he  liked — a  right  which  continued 
to  exist  till  1827,  though  it  was  not  exercised  after  1797. 

Subject  to  the  granting  of  these  honorary  degrees,  this  final 
list  was  arranged  in  order  of  merit  into  three  classes,  con- 
sisting of  (i)  the  wranglers  and  senior  optimes ;  (ii)  the  junior 
optimes  who  had  passed  respectably  but  had  not  distinguished 
themselves;  and  (iii)  ot  TroAAot,  or  the  poll  men.  The  first 
class  included  those  bachelors  quibus  sua  reservatur  seniorita-s 
comitiis  prioribus :  they  received  their  degrees  on  Ash-Wed- 
nesday, taking  seniority  according  to  their  order  on  the  list. 
The  two  other  classes  received  their  degrees  a  few  weeks  later. 

The  order  as  determined  by  the  performance  of  these  acts 
seems  to  have  been  accurately  foreshadowed  by  the  preliminary 
lists  framed  by  the  moderators.  Thus  the  tripos  list  for  1753 
shews  that  all  the  undergraduates  selected  to  be  respondents 
became  wranglers.  Of  the  first  opponents,  three  (probably 
personal  friends  of  the  moderators)  received  honorary  optime 
degrees  as  second,  third,  and  fourth  wranglers  respectively ;  four 
obtained  a  place  in  the  first  class  by  their  own  merits ;  and  the 


THE   EXERCISES   IX   THE   SCHOOLS.  171 

rest  appear  as  senior  optimes — one,  who  was  ill,  receiving  it  as 
an  honorary  degree.  The  book  lay  before  the  moderators  during 
the  discussions,  and  if  any  third  opponent  shewed  unexpected 
skill  in  the  acts  his  name  was  marked,  and  transferred  from 
the  seventh  or  eighth  class  comprising  the  poll  men  to  the  fifth 
or  sixth  which  contained  the  expectant  junior  optimes.  In 
the  list  of  1752  sixteen  names  are  thus  crossed  out,  and  these 
form  the  third  class  of  that  tripos.  The  rest  of  the  candidates, 
thirty-five  in  number,  together  with  seven  others  who  kept  no 
acts  (at  any  rate  before  the  moderators)  form  the  poll  list  for 
that  year. 

At  a  later  time,  as  we  shall  see  in  the  next  chapter,  the 
acts  were  only  used  as  a  means  of  arranging  the  men  into  four 
groups,  namely,  those  expected  to  be  wranglers,  senior  optimes, 
junior  optimes,  and  poll  men  respectively ;  and  the  order  in 
each  group  was  determined  by  the  senate-house  examination, 
in  which  a  different  set  of  papers  was  given  to  each  group. 
Finally,  a  means  of  passing  from  one  group  to  another  by 
means  of  the  senate-house  examination  was  devised.  Thence- 
forth the  acts  ceased  to  be  of  the  same  importance,  though 
they  still  afforded  a  test  by  which  public  opinion  as  to  the 
abilities  of  men  was  largely  influenced. 

The  moderators's  book  for  1778  has  been  preserved  and  is 
in  the  library  of  Trinity.  It  may  be  interesting  if  I  describe 
briefly  the  way  in  which  it  is  arranged.  Each  page  is  dated, 
and  contains  a  list  of  the  three  subjects  proposed  for  that  day 
together  with  the  names  of  the  respondent  and  the  three  oppo- 
nents. Of  the  three  questions  proposed  by  each  respondent 
the  first  was  invariably  on  a  mathematical  subject,  and  with 
one  exception  was  always  taken  from  Newton.  In  all  but  ten 
cases  the  second  was  also  on  some  mathematical  question.  The 
last  was  on  some  point  in  moral  philosophy. 

According  as  the  acts  were  well  kept  or  not  the  moderators 
marked  the  names  of  the  candidates.  Very  good  performances 
were  rewarded  with  the  mark  A  +,  A,  or  A  —  ;  good  perform- 
ances with  E  -f ,  JZ,  or  E  — ;  fair  performances  with  a  +,  a,  or 


172  THE   EXERCISES   IN  THE   SCHOOLS. 

a  - ;  and  indifferent  ones  with  e  4-  or  e.  It  was  on  these 
marks  that  the  subsequent  "classes"  were  drawn  up. 

Between  Feb.  3  and  July  2  sixty-six  exercises  in  all  were 
kept,  each  of  course  involving  four  candidates:  between  Oct. 
26  and  Dec.  11  thirty  were  kept.  Three  acts  were  stopped 
when  only  half  finished  because  the  book  of  statutes  (without 
the  presence  of  which  a  moderator  had  no  power)  was  sent  for 
by  the  proctors  to  consult  at  a  congregation1.  Two  or  three 
others  are  included  in  the  book  but  are  cancelled  ;  most  of 
them  I  gather  because  of  some  irregularity,  but  one  because 
the  selected  respondent  had  died. 

Altogether  112  students  of  that  year  presented  themselves 
for  the  bachelor's  degree,  but  they  did  not  all  appear  in  the 
schools.  Of  the  honour  candidates,  forty-seven  in  number, 
one  kept  two  acts,  another  kept  three,  and  three  kept  four ;  all 
the  rest  kept  five,  six,  or  seven  acts.  Five  honorary  optime 
degrees  were  also  given.  There  were  sixty  poll  men  :  of  these 
thirty -seven  presented  themselves  at  the  proper  time  and 
formed  the  first  list,  all  save  eight  of  these  having  kept  one 
or  more  acts.  Eight  bye-term  men  received  their  degrees 
as  baccalaurei  ad  baptistam  in  the  following  Michaelmas  term, 
and  eight  more  as  baccalaurei  ad  diem  cinerum  on  Ash- 
Wednesday  or  "duiices's  day."  It  was  not  usual  for  the 

1  Thus  W.  Chafin  of  Emmanuel,  describing  his  act  kept  in  1752,  says 
that  he  had  got  off  tolerably  well  against  W.  Disney  of  Trinity,  who  was 
his  first  opponent,  but  that  W.  Craven  of  St  John's  "  brought  an  argu- 
ment against  me  fraught  with  fluxions ;  of  which  I  knew  very  little  and 
was  therefore  at  a  nonplus,  and  should  in  one  minute  have  been  exposed, 
had  not  at  that  instant  the  esquire  bedell  entered  the  schools  and  de- 
manded the  book  which  the  moderator  carries  with  him,  and  is  the  badge 
of  his  office.  A  convocation  was  that  afternoon  held  in  the  senate-house, 
and  on  some  demur  that  happened,  it  was  found  requisite  to  inspect  this 
book,  which  was  immediately  delivered,  and  the  moderator's  authority 
stopped  for  that  day,  and  we  were  all  dismissed  ;  and  it  was  the  happiest 
and  most  grateful  moment  of  my  life,  for  I  was  saved  from  imminent 
disgrace,  and  it  was  the  last  exercise  that  I  had  to  keep  in  the  schools." 
(From  the  Gentleman's  magazine  for  January,  1818  ;  quoted  on  pp.  29,  30 
of  the  Scholae  academicae.) 


THE   EXERCISES  IN  THE   SCHOOLS.  173 

moderators  to  preside  over  the  acts  of  bye-term  men,  and  the 
exercises  of  these  sixteen  men  do  not  therefore  appear  in  this 
book.  Of  the  remaining  candidates  two  were  "  plucked  "  out- 
right, four  took  a  poll  degree  in  the  following  year,  and  one 
candidate  died  during  his  questionist's  year. 

The  senior  wrangler  of  the  year  was  Thomas  Jones  of 
Trinity,  whose  reputation,  if  we  may  believe  tradition,  was  so 
well  established  that  his  attendance  at  the  senate-house  exami- 
nation was  excused  by  the  moderators.  Of  course  this  did  not 
prevent  his  position  as  senior  being  challenged  (in  the  manner 
described  on  p.  200)  if  any  candidate  thought  himself  badly  used. 
Jones  had  "coached"  the  second  wrangler  in  his  own  year.  He 
was  afterwards  tutor  of  Trinity,  and  one  of  the  most  influential 
members  of  the  university  at  the  end  of  the  last  century. 

No  detailed  records  of  these  disputations  prior  to  the 
eighteenth  century  now  exist.  The  official  accounts  by  the 
proctors  and  moderators  were  usually  destroyed  as  soon  as 
the  men  were  admitted  to  their  degrees,  and  it  is  only  by 
accident  that  the  two  from  which  I  have  made  quotations 
above  have  been  preserved. 

The  only  verbatim  reports  (with  which  I  am  acquainted) 
of  any  disputations  actually  kept  are  of  some  which  took 
place  between  1780  and  1790.  These  are  contained  in  a 
small  manuscript  now  in  the  library  of  Caius  College.  One 
of  them,  by  the  kindness  of  that  society,  I  was  able  to  insert 
in  my  Origin  and  history  of  the  mathematical  tripos ,  published 
at  Cambridge  in  1880,  and  I  here  reproduce  it.  The  manu- 
script consists  of  rough  notes  of  exercises  performed  in  the 
schools,  with  the  addition  of  suggested  objections  to  the 
questions  most  usually  chosen  by  the  respondents.  Many  of 
the  arguments  are  crossed  out  as  being  obviously  untenable, 
while  several  of  the  pages  are  torn  and  defaced,  presenting 
much  the  same  appearance  as  a  copy  book  of  an  ordinary 
schoolboy  would  if  it  were  preserved  in  some  library  as  the 
sole  specimen  of  its  kind.  Altogether  the  manuscript  contains 
the  whole  or  portions  of  twenty-three  distinct  disputations. 


174  THE   EXERCISES   IN   THE    SCHOOLS. 

The  conversational  parts  (i.  e.  the  real  discussions)  are  omitted 
throughout — indeed  it  was  useless  to  take  notes  of  these,  since 
the  debate  was  not  likely  to  take  exactly  the  same  turn  on 
any  subsequent  occasion — and  the  collection  should  therefore 
be  regarded  as  an  analysis  of  the  arguments  brought  forward 
rather  than  as  giving  the  actual  disputations. 

The  discussion  to  which  I  alluded  and  which  I  here  quote 
as  an  illustration  of  the  form  of  these  scholastic  exercises  was 
kept  on  Feb.  20,  1784,  by  Joshua  Watson  of  Sidney,  as  first 
opponent,  against  the  questions  proposed  by  William  Sewell  of 
Christ's.  The  report  of  it  is  one  of  the  fullest  of  those  pre- 
served in  the  book,  and  it  seems  also  a  good  example  both  of 
the  nature  of  the  objections  raised,  and  the  form  in  which  they 
were  urged.  In  reference  to  the  former,  it  is  only  fair  to 
remember  that  the  opponent  had  in  general  to  deny  a  proposi- 
tion which  he  knew  perfectly  well  was  true,  and  which  the 
respondent  had  usually  chosen  because  it  was  very  difficult  to 
controvert.  In  reference  to  the  latter,  the  minor  premise  has 
been  omitted  from  the  manuscript  in  all  save  one  of  the  dispu- 
tations, but  I  have  ventured  to  replace  it  and  to  add  such  other 
technical  phrases  as  were  always  used.  I  have  only  to  add 
that  those  portions  which  are  not  in  the  original  are  printed  in 
square  brackets :  and  that  wherever  the  mark  f  is  placed,  there 
are  pencil  notes  explaining  how  the  conclusion  is  deduced;  but 
time  has  rendered  these  so  illegible  that  it  is  impossible  to 
decipher  them  with  certainty.  The  Latin  is  that  of  the  schools, 
and  I  reprint  it  as  it  stands  in  the  original. 

The  propositions  were  (i)  Solis  parallaxis  ope  Veneris  intra 
solem  conspiciendse  a  method  o  Halleiirecte  determinari  potest; 
(ii)  E/ecte  statuit  Newtonus  in  tertia  sua  sectioue  libri  primi ; 
(iii)  Diversis  sensibus  non  ingrediuntur  ideae  communes. 

After  Sewell  had  read  an  essay  on  the  first  of  these  ques- 
tions, the  discussion  began  as  follows. 

Moderator.    [Ascendat  dominus  opponentium  primus.] 
Opponent.     [Probo]  contra  primam  [quaestionem].     Si  asserat  Hal- 
leius  Venerem  cum  Soli  sit  proxima  Londini  visam  a  centro  Solis  qua- 


SPECIMEN   OF   A   DISPUTATION.  175 

tuor  minutis  primis  distare,  cadit  quaestio.  [Sed  asserit  Halleius  Vene- 
rem  cum  Soli  sit  proxima  Londini  visam  a  centre  Solis  quatuor  minutis 
primis  distare.  Ergo  cadit  quaestio.] 

Respondent.     [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.     [Probo  consequentiam.]    Si  in  schemate  posuit  semitam  Vene- 
ris  ad  os  Gangeticum  quatuor  etiam  minutis  primis  distare,  valet  conse- 
quentia  [Sed  in  schemate  posuit  semitam  Veneris  ad  os   Gangeticum 
quatuor  etiam  minutis  primis  distare.     Ergo  valet  consequentia.] 
Eesp.    [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.    [Iterum    probo  consequentiam.]     Si  spectatoribus  positis  in 
diversis  parallelis  latitudinis  non  eadem  appareat  distantia  atque  igitur 
non  licet  eandem  visibilem  sumere   distantiam  in  hisce  duobus  locis 
valent  consequentia  et  argumentum.    [Sed  spectatoribus  positis  in  diver- 
sis  parallelis  latitudinis  non  eadem  apparet  distantia  atque  non  licet 
eandem  visibilem  sumere  distantiam  in  hisce  duobus  locis.     Ergo  valent 
consequentia  et  argumentum.] 

The  conclusion  valet  argumentum  meant  that  the  opponent 
considered  that  he  had  fairly  stated  his  case,  and  here  therefore 
ought  to  follow  first  the  respondent's  exposition  of  the  fallacy 
in  the  opponent's  argument,  and  then  the  opponent's  answer 
sustaining  his  objection  to  the  original  proposition  given  above. 
As  soon  as  each  had  fairly  stated  and  illustrated  his  case  or 
the  discussion  began  to  degenerate  into  an  interchange  of  per- 
sonalities, the  moderator  turning  to  the  opponent  said  Probes 
aliter,  and  a  fresh  argument  was  accordingly  begun.  All  these 
steps  are  missing  in  the  manuscript. 

The  remaining  seven  arguments  of  the  opponent  were  as 
follows. 

Opp.  [Probo]  aliter  [contra  primam].  Si  in  figura  Halleiana  cen- 
trum Solis  correspondeat  cum  loco  spectatoris  in  Tellure,  cadit  quaestio. 
[Sed  in  figura  Halleiana  centrum  Solis  correspondet  cum  loco  spectatoris 
in  Tellure.  Ergo  cadit  quaestio.  ]t 

Eesp.     [Concedo  antecedentiam  et  nego  consequentiam.] 
Opp.     [Probo  consequentiam.]     Si  locus  centri  Solis  a  vero  centre 
amoti  ob  motum  spectatoris  fit  curva  linea,  valet  consequentia.     [Sed 
locus  centri  Solis  a  vero  centre  amoti  ob  motum  spectatoris  fit  curva 
linea.     Ergo  valet  consequentia.] 

Eesp.     [Concedo  antecedentiam  et  nego  consequentiam.] 

Opp.    [Iterum  probo   consequentiam.]     Si  composite  motu  Veneris 


176  THE   EXERCISES   IN  THE   SCHOOLS. 

uniformi  in  recta  linea  cum  motu  Solari  in  curva  linea  fit  semita  Veneris 
in  disco  Solis  curva  linea,  valet  consequentia.     [Sed  composito  motu 
Veneris  uniformi  in  recta  linea  cum  motu  Solari  in  curva  linea  fit  semita 
Veneris  in  disco  Solis  curva  linea.     Ergo  valet  consequentia.] 
Resp.    [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.     [Iterum  probo  consequentiam.]     Si  longitudo  hujusce  lineae 
non  recte  determinari  potest,  valent  consequentia  et  argumentum.     [Sed 
longitudo  hujusce  lineae  non  recte  determinari  potest.     Ergo  valent  con- 
sequentia et  argumentum.] 

The  next  argument  against  the  first  proposition  ran  as 
follows. 

Opp.  [Probo]  aliter  [contra  primam].  Si  spectator!  ad  os  Gangeti- 
cum  posito  ob  terraa  motum  rnotui  Veneris  contrarium  contrahatur 
transitus  tempus  integrum,  cadit  quaestio.  [Sed  spectatori  ad  os  Gan- 
geticum  posito  ob  terras  motum  motui  Veneris  contrarium  contrahitur 
transitus  tempus  integrum.  Ergo  cadit  quaestio.] 

Resp.  [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.  [Iterum  probo  consequentiam.]  Si  assumat  Halleius  contrac- 
tionem  hanc  duodecim  minutis  primis  temporis  sequalem,  et  deinde  huie 
hypothesi  insistendo  eidem  tempori  asqualem  probat,  valent  consequentia 
et  argumentum.  [Sed  assumat  Halleius  contractionem  hanc  duodecim 
minutis  primis  temporis  asqualem,  et  deinde  huic  hypothesi  insistendo 
eidem  tempori  sequalem  probat.  Ergo  valent  consequentia  et  argu- 
mentum.] 

The  fourth  objection  to  the  first  proposition  was  as  follows. 

Opp.  [Probo]  aliter  [contra  primam].  Si  posuit  Halleius  eandem 
visibilem  semitam  Veneris  per  discum  Solarem  ad  os  Gangeticum  et 
porturn  Nelsoni,  et  hanc  semitam  dividat  in  aequalia  horaria  spatia,  cadit 
quaestio.  [Sed  Halleius  posuit  eandem  visibilem  semitam  Veneris  per 
discum  Solarem  ad  os  Gangeticum  et  portum  Nelsoni,  et  hanc  semitam 
dividit  in  aequalia  horaria  spatia.  Ergo  cadit  quaestio.] 
Resp.  [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.  [Probo  consequentiam.]  Si  motus  horarius  Veneris  accele- 
ratur  vel  retardatur  per  motum  totum  spectatoris  in  medio  transitu,  quo 
magis  autem  distat,  minus  acceleratur  vel  retardatur,  valet  consequentia. 
[Sed  motus  horarius  Veneris  acceleratur  vel  retardatur  per  motum  totum 
spectatoris  in  medio  transitu,  quo  magis  autem  distat,  minus  acceleratur 
vel  retardatur.  Ergo  valet  consequentia.] 

Resp.    [Concede  antecedentiam,  et  nego  consequentiam.] 

Opp.    [Iterum  probo  consequentiam.]     Si  igitur  ob  motum  Veneris 


SPECIMEN  OF  A  DISPUTATION.  177 

acceleratum  ad  os  Gangeticum  et  retardatum  ad  portum  Nelsoni  bi 
motus  non  debent  repraesentari  per  idem  spatium,  valent  consequentia  et 
argumentum.  [Sed  ob  motum  Veneris  acceleratum  ad  os  Gangeticum  et 
retardatum  ad  portum  Nelsoni  hi  motus  non  debent  repraasentari  per 
idem  spatium.  Ergo  valent  consequentia  et  argumentum.] 

The  last  argument  against  the  first  question  was  as  follows. 

Opp.  [Probo]  aliter  [contra  primam].  Si  secundum  constructionem 
Halleianam  spectator!  ad  portum  Nelsoni,  posito  tempore  extensionis 
majore,  major  etiam  fit  transitus  duratio,  cadit  quaestio.  [Sed  secun- 
dum constructionem  Halleianam  spectator!  ad  portum  Nelsoni,  posito 
tempore  extensionis  majore,  major  fit  transitus  duratio.  Ergo  cadit 
quasstio.Jt 

Eesp.  [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.  [Probo  consequentiam.]  Si  secundum  eandem  constructionem 
posito  quod  spectatori  ad  os  Gangeticum  tempus  contractionis  majus  sit 
duodecim  minutis  primis,  evadat  tempus  durationis  majus  etiam,  valet 
consequentia.  [Sed  secundum  eandem  constructionem  posito  quod  spec- 
tatori ad  os  Gangeticum  tempus  contractionis  majus  est  duodecim  minu- 
tis primis,  et  evadit  tempus  durationis  majus  etiam.  Ergo  valet  conse- 
quentia.]t 

Eesp.     [Concede  antecedentiam  et  nego  consequentiam.] 
Opp.     [Iterum  probo  consequentiam.]     Si  hae  duae  conclusiones  inter 
se  pugnent,  valent  consequentia  et  argumentum.     [Sed  has  duae  conclu- 
siones inter  se  pugnant.    Ergo  valent  consequentia  et  argumentum.] 

The  opponent  then  proceeded  to  attack  the  second  proposi- 
tion, and  his  first  objection  to  it  was  as  follows. 

Opp,  [Probo]  contra  secundam  [quaestionem].  Si  vis  in  parabola  ad 
infinitam  distantiam  sit  infinitesimalis  secundi  ordinis,  cadit  quasstio. 
[Sed  ad  infinitam  distantiam  vis  in  parabola  est  infiuitesimalis  secundi 
ordinis.  Ergo  cadit  quaestio.] 

Eesp.  [Conceclo  antecedentiam  et  nego  consequentiam.] 
Opp.  [Probo  consequentiam.]  Si  vis  sit  u4  igiturque  ad  infinitam  dis- 
tantiam sit  infinitesimalis  quarti  ordinis,  valent  consequentia  et  argu- 
mentum. (The  manuscript  here  is  almost  unintelligible.)  [Sed  vis  est  w* 
igiturque  ad  infinitam  distantiam  est  infinitesimalis  quarti  ordinis.  Ergo 
valent  consequentia  et  argumentum.] 

The  second  objection  to  this  question  was  as  follows. 

Mod.    [Probes  aliter.] 

Opp.  [Probo]  aliter  [contra  secundam].  Si  velocitates  ad  extremitates 
axium  minorum  diversarum  ellipsium  quarum  latera  recta  aequantur  siut 

B.  12 


178  THE   EXERCISES  IN  THE   SCHOOLS. 

inter  se  inverse  ut  axes  minores,   cadit  quasstio.      [Sed  velocitates  ad 
extremitates  axium  minorum  diversarum  ellipsium  quarum  latera  recta 
aequantur  sunt  inter  se  inverse  ut  axes  minores.     Ergo  cadit  quaestio.] 
Resp.     [Concedo  antecedentiam  et  nego  consequentiam.] 
Opp.     [Probo  consequentiam.]  Si  locus  extremitatum  omnium  axium 
minorum  sit  parabola,  valet  consequentia.    [Sed  locus  extremitatum  om- 
nium axium  minorum  est  parabola.    Ergo  valet  consequentiam.] 
Resp.     [Concedo  antecedentiam  et  nego  consequentiam.] 
Opp.    [Iterum  probo  consequentiam.]  Si  velocitas  corporis  revolventis 
in  ista  parabola  sit  ad  velocitatem  ad  mediam  distantiam  correspondentis 
ellipseos  ut  ^2  :  1,  valet  consequentia.     [Sed  velocitas  corporis  revolven- 
tis in  ista  parabola  est  ad  velocitatem  ad  mediam  distantiam  correspon- 
dentis ellipseos  ut  J%  :  1.     Ergo  valet  consequentia.] 

Resp.     [Concedo  antecedentiam  et  nego  consequentiam.] 
Opp.     [Iterum  probo  consequentiam.]  Si  velocitas  in  parabola  sit  in- 
verse ut  ordinata,  valent  consequentia  et  argumentum.     [Sed  velocitas 
in  parabola  est  inverse  ut  ordinata.     Ergo  valent  consequentia  et  argu- 
mentum. ] 

The  argument  against  the  third  proposition  was  as  follows. 

Hod.    [Probes  aliter.] 

Opp.  [Probo]  contra  tertiam  [quaestionem].  Aut  cadit  tua  quaestio 
aut  non  possibile  est  hominem  ab  ineunte  aetate  caecum  et  jam  adultum 
visum  recipientem  visu  dignoscere  posse  id  quod  tangendo  prius  solum- 
modo  dignoscebat.  Sed  possibile  [est  hominem  ab  ineunte  aetate  cascum 
et  jam  adultum  visum  recipientem  visu  dignoscere  posse  id  quod  tangendo 
prius  solummodo  dignoscebat.  Ergo  cadit  quaestio]. 

Resp.    [Concedo  majorem  sed  nego  minorem.] 

Opp.  [Probo  minorem.]  Si  eadem  ratio  quae  prius  eum  docebat  dig- 
noscere tangendo  inter  cubum  et  globum  eum  etiam  docebit  intuendo 
recte  dignoscere,  valent  minor  et  argumentum.  [Sed  eadem  ratio  quae 
prius  eum  docebat  dignoscere  tangendo  inter  cubum  et  globum  eum 
etiam  docebit  intuendo  recte  dignoscere.  Ergo  valent  minor  et  argu- 
mentum.] 

Watson  was  subsequently  followed  on  the  same  side  by 
W.  Lax  of  Trinity  as  second  opponent,  and  Richard  Biley  of 
St  John's  as  third  opponent ;  and  it  would  seem  from  the 
tripos  list  of  1785  that  Sewell  was  altogether  overmatched  by 
his  antagonists. 

The  following  account  of  some  disputations  in  1790  is 
taken  from  a  letter  by  William  Gooch  of  Caius,  who  was 


THE   EXERCISES   IN   THE   SCHOOLS.  179 

second  wrangler  in  1791.  It  is  especially  valuable  as  giving 
us  an  undergraduate's  view  of  these  exercises.  Another  letter 
by  him  descriptive  of  the  senate-house  examination  in  1791  is 
printed  in  the  next  chapter.  The  letter  in  question  is  dated 
Nov.  6,  1790,  and  after  some  gossip  about  himself  he  goes  on 

Peacock  kept  a  very  capital  Act  indeed  and  had  a  very  splendid  Honor 
of  which  I  can't  remember  a  Quarter,  however  among  a  great  many  other 
things,  Lax  told  him  that  "Abstruse  and  difficult  as  his  Questions  were, 
no  Argument  (however  well  constructed)  could  be  brought  against  any 
Part  of  them,  so  as  to  baffle  his  inimitable  Discerning  &  keen  Penetration" 
&c.  &c.  &c. — However  the  Truth  was  that  he  confuted  all  the  Arguments 
but  one  which  was  the  1st  Opponent's  2nd  Argument, — Lax  lent  him  his 
assistance  too,  yet  still  he  didn't  see  it,  which  I  was  much  surpris'd  at  as 
it  seem'd  easier  than  the  Majority  of  the  rest  of  the  Args — Peacock  with 
the  Opponents  return'd  from  the  Schools  to  my  Eoom  to  tea,  when  (agree- 
able to  his  usual  ingenuous  Manner)  he  mention 'd  his  being  in  the  Mud 
about  Wingfield's  2nd  argument,  &  requested  Wingfield  to  read  it  to  him 
again  &  then  upon  a  little  consideration  he  gave  a  very  ample  answer  to 
it. — I  was  third  opponent  only  and  came  off  with  ' '  optime  quidem  dispu- 
tasti"  i.e.  "you've  disputed  excellently  indeed"  (quite  as  much  as  is  ever 
given  to  a  third  opponency) — I've  a  first  opponency  for  Novr  11th  under 
Newton  against  Wingfield  &  a  second  opponency  for  Novr  19th  under 
Lax  against  Gray  of  Peter-House.  Peacock  is  Gray's  first  opponent  & 
"Wingfield  his  third,  so  master  Gray  is  likely  to  be  pretty  well  baited. 
His  third  Question  (of  all  things  in  the  world)  is  to  defend  Berkley's  im- 
material System. 

Mre  Hankinson  &  Miss  Paget  of  Lynn  are  now  at  Cambridge,  I  drank 
tea  &  supp'd  with  them  on  Thursday  at  Mr  Smithson's  (the  Cook's  of 
Sl.  Johns  Coll.)  &  yesterday  I  din'd  drank  tea  and  supp'd  there  again  with 
the  same  Party,  and  to  day  I'm  going  to  meet  them  at  Dinner  at  Mr  Hall's 
of  Camb.  Hankinson  of  Trin.  (as  you  may  suppose)  have  (sic)  been  there 
too  always  when  I  have  been  there ;  as  also  Smithson  of  Emmanuel  Coll. 
{son  of  this  Mr  Smithson).  Miss  Smithson  is  a  very  accomplished  girl, 
&  a  great  deal  of  unaffected  Modesty  connected  with  as  much  Delicacy 
makes  her  very  engaging. — She  talks  French,  and  plays  well  on  the 
Harpsichord.  Mre  H.  will  continue  in  Camb.  but  for  a  day  or  two  longer 
or  I  should  reckon  this  a  considerable  Breach  upon  my  Time ; — However 
I  never  can  settle  well  to  any  thing  but  my  Exercises  when  I  have  any 
upon  my  Hands,  and  I'm  sure  I  don't  know  what  purpose  'twould  answer 
to  fagg  much  at  my  Opponencies,  as  I  doubt  whether  I  should  keep  at  all 
the  better  or  the  worse  they  being  upon  subjects  I've  long  been  pretty  well 
acquainted  with. — Yet  I'm  resolv'd  when  I've  kept  my  first  Opponency 

12—2 


180  THE    EXERCISES    IN   THE   SCHOOLS. 

next  thursday  if  possible  to  think  nothing  of  my  2nd  (for  friday  se'nnight) 
till  within  a  day  or  two  of  the  time — One  good  thing  is  I  can  now  have  no 
more,  so  I've  the  luck  to  be  free  from  the  schools  betimes,  for  the  term 
doesn't  end  till  the  middle  of  Dec1".1 

My  readers  may  be  interested  to  know  that  Gooch  was  quite 
captivated  by  Miss  Smithson,  and  he  intended  to  propose  to  her 
on  his  return  from  the  astronomical  expedition  sent  out  by  the 
government  in  1791 — 3,  in  which  he  took  part.  He  was  cap- 
tured by  the  South  Sea  islanders  in  May,  1792,  and  murdered 
before  assistance  could  reach  him. 

The  following  list  of  subjects  of  acts  known  to  have  been 
kept  between  1772  and  1792  is  taken  from  Wordsworth. 
Some  were  chosen  more  than  once.  The  questions  on  mathe- 
matics were  as  follows. 

Newton's  Principia,  book  i,  section  i;  book  i,  sections  ii  and  iii;  book 
i,  section  iii;  book  i,  section  vii;  book  i,  section  viii;  book  i,  section  xii, 
props.  1 — 5;  book  i,  section  xii,  props.  39  and  40;  book  i,  section  xii, 
prop.  66  and  one  or  more  corollaries.  Cotes's  Harmonia  mensurarum, 
prop.  1.  Cotes's  theorem  on  centripetal  force.  Cotes's  proposition  on 
the  five  trajectories.  The  path  of  a  projectile  is  a  parabola.  Halley's 
determination  of  the  solar  parallax.  Correction  of  the  aberration  of 
rays  by  conic  sections.  The  method  of  fluxions.  Smith  de  focalibus 
distantibus.  Maclaurin,  chapter  in,  sections  1 — 8  and  11 — 22.  Morgan 
on  mechanical  forces.  Morgan  on  the  inclined  plane.  Hamilton  on 
vapour. 

The  questions  on  philosophy  were  as  follows. 

Berkeley  on  sight  and  touch.  Montesquieu  Laws,  chapter  i,  section  i. 
Locke  on  faith  and  reason.  Can  matter  think?  The  signification  of 
words.  Wollaston  on  happiness.  From  Paley,  On  penalties;  On 
happiness;  On  promises.  Free  press.  Imprisonment  for  debt.  Duel- 
ling. The  slave  trade.  Common  ideas  do  not  enter  by  different  senses. 
Composite  ideas  have  no  absolute  existence.  The  immortality  of  the 
soul  may  be  inferred  by  the  light  of  nature.  The  immortality  of  the  soul 
may  be  inferred  by  the  light  of  nature,  but  no  more  than  that  of  other 
animals.  The  soul  is  immaterial.  Omnia  nostra  de  causa  facimus. 

A  candidate  was  not  however  allowed  to  offer  any  question. 
Thus  a  proposition  taken  out  of  Euclid's  Elements  was  gene- 

1  Scholae  academicae,  321 — 22. 


THE   EXERCISES   IN  THE   SCHOOLS.  181 

rally  rejected  by  the  moderators,  probably  because  of  the  diffi- 
culty of  arguing  against  its  correctness.  In  1818  as  a  great 
concession  a  questionist  was  allowed  to  "keep"  in  the  eleventh 
book  of  Euclid.  The  moderators  also  refused  to  allow  the  main- 
tenance of  any  doctrine  which  they  regarded  as  immoral  or 
heretical.  Thus  when  Paley  of  Christ's,  in  1762,  proposed  for  his 
theses  the  subjects  that  punishment  in  hell  did  not  last  through- 
out eternity,  and  that  a  judicial  sentence  of  death  for  any  crime 
was  unjustifiable  they  were  rejected ;  whereupon  he  upheld 
the  opposite  views  in  the  schools,  leaving  to  his  opponents 
the  duty  of  sustaining  his  original  propositions. 

Of  the  disputations  in  1819  Whewell,  who  was  then 
moderator,  writes  as  follows.  "  They  are  held  between  under- 
graduates in  pulpits  on  opposite  sides  of  the  room,  in  Latin 
and  in  a  syllogistic  form.  As  we  are  no  longer  here  in  the 
way  either  of  talking  Latin  habitually  or  of  reading  logic, 
neither  the  one  nor  the  other  is  very  scientifically  exhibited. 
The  syllogisms  are  such  as  would  make  Aristotle  stare,  and 
the  Latin  would  make  every  classical  hair  in  your  head  stand 
on  end.  Still  it  is  an  exercise  well  adapted  to  try  the  clear- 
ness and  soundness  of  the  mathematical  ideas  of  the  men, 
though  they  are  of  course  embarrassed  by  talking  in  an  un- 
known tongue It  does  not,  at  least  immediately,  produce 

any  effect  on  a  man's  place  in  the  tripos,  and  is  therefore  con- 
siderably less  attended  to  than  used  to  be  the  case,  and  in 
most  years  is  not  very  interesting  after  the  five  or  six  best 
men1." 

Even  to  the  last  they  sometimes  led  to  a  brilliant 
passage  of  arms.  Thus  Richard  Shilleto  of  Trinity  College 
(B.A.  1832,  and  subsequently  a  fellow  of  Peterhouse),  kept  an 
act  on  the  well-worn  subject  as  to  whether  suicide  was  justi- 
fiable2. Quid  est  suicidium,  said  he,  ut  Latine  nos  loquamur 
nisi  suum  caesio  ?  and  then  he  went  on  to  defend  it  on  the 

1  See  vol.  n.  pp.  35,  36  of  Todhunter's  Life  of  Whewell,  London,  1876. 

2  The  story  is  told  differently  by  Wordsworth,  but  I  give  it  as  I  have 
heard  it.     Suicidium  was  the  scholastic  translation  of  suicide. 


18*2  THE   EXERCISES   IN    THE   SCHOOLS. 

ground  that  roast  pig  and  boiled  ham  were  delicacies  appre- 
ciated by  all.  His  opponent,  a  Johnian  and  good  mathe- 
matician but  ignorant  of  classics,  could  not  understand  a 
word  of  this,  but  the  moderator,  Francis  Martin  of  Trinity, 
entered  into  the  spirit  of  the  fun  and  himself  carried  on  the 
discussion.  In  earlier  times  (and  even  a  few  years  previously) 
the  acts  were  a  serious  matter,  and  a  joke  such  as  this  would 
not  have  been  tolerated. 

The  form  in  which  they  were  carried  out  required  a 
knowledge  of  formal  logic,  and  (at  least)  a  smattering  of  con- 
versational Latin;  and  till  within  a  few  years  of  their  abolition 
in  1839,  the  publicity  of  the  discussion  ensured  the  most 
thorough  preparation.  This  previous  preparation  was  the  more 
necessary  as  the  respondent  had  to  answer  off-hand  any 
objection  from  any  source,  or  any  apparent  argument  however 
fallacious,  which  the  opponent  (in  general  previously  prompted 
by  his  tutor)  might  bring  against  his  thesis. 

Thus  De  Morgan  writing  about  his  act  kept  in  1826  says, 
"  I  was  badgered  for  two  hours  with  arguments  given  and 
answered  in  Latin, — or  what  we  call  Latin — against  Newton's 
first  section,  Lagrange's  derived  functions,  and  Locke  on 
innate  principles.  And  though  I  took  off  everything,  and 
was  pronounced  by  the  moderator  to  have  disputed  magno 
honor 6,  I  never  had  such  a  strain  of  thought  in  my  life. 
For  the  inferior  opponents  were  made  as  sharp  as  their  betters 
by  their  tutors,  who  kept  lists  of  queer  objections  drawn  from 
all  quarters1."  James  Devereux  Hustler,  the  third  wrangler 
of  1806  and  subsequently  a  tutor  of  Trinity,  had  a  special 
reputation  for  prompting  men  with  such  objections  (seep.  113). 

I  believe  that  so  long  as  the  discussion  was  a  real  one  and 
carried  on  in  the  language  of  formal  logic  (which  prevented  the 
argument  wandering  from  the  point),  it  was  an  admirable 
training,  though  to  be  productive  of  the  best  effects  it  required 
a  skilled  moderator.  It  not  only  gave  considerable  dialectical 

1  See  p.  305  of  the  Budget  of  paradoxes  by  A.  De  Morgan,  London,  1872. 


THE   EXERCISES   IN   THE   SCHOOLS.  183 

practice  but  was  a  corrective  to  the  thorough  but  somewhat 
narrow  training  of  the  tripos. 

Had  the  language  of  the  discussions  been  changed  to 
English,  as  was  repeatedly  urged  from  1774  onwards,  these 
exercises  might  have  been  kept  with  great  advantage,  but  the 
barbarous  Latin  and  the  syllogistic  form  in  which  they  were 
carried  on  prejudiced  their  retention.  I  do  not  know  whether 
disputations  are  now  used  in  any  university,  except  as  a  more 
or  less  formal  ceremony,  after  a  man's  ability  has  been  tested 
in  other  ways;  but  I  am  told  that  they  still  form  a  part  of 
the  training  in  some  of  the  Jesuit  colleges  where  the  students 
have  to  maintain  heresies  against  the  professors,  and  that  the 
directors  of  those  institutions  have  a  high  opinion  of  their 
value. 

About  1830  a  custom  grew  up  for  the  respondent  and  oppo- 
nents to  meet  previously  and  arrange  their  arguments  together. 
The  whole  ceremony  then  became  an  elaborate  farce  and  was 
a  mere  public  performance  of  what  had  been  already  re- 
hearsed. Accordingly  the  moderators  of  1840,  T.  Gaskin  and 

Bowstead,  took  the  responsibility  of  discontinuing  them. 
Their  action  was  singularly  high-handed,  as  a  report  of  May  30, 
1838,  had  recommended  that  the  moderators  should  continue  to 
be  guided  by  these  exercises. 

No  one,  however  distinguished,  appeared  more  than  twice 
as  a  respondent  and  twice  in  each  grade  of  opponency,  that  is, 
eight  times  altogether — some  of  the  exercises  being  performed 
in  the  Lent  and  Easter  terms  of  the  third  year  of  residence, 
and  the  remainder  in  the  October  term  of  the  fourth  year. 
The  non-reading  men  were  perhaps  only  summoned  once  or 
twice,  and  before  1790  fellow- commoners1  seemed  to  have  been 
excused  all  attendance. 

1  The  earliest  certain  instance  of  a  fellow-commoner  presenting  him- 
self for  the  senate-house  examination  is  that  of  T.  Gisborne  of  St  John's, 
who  was  sixth  wrangler  in  1780.  The  first  known  case  of  a  fellow-com- 
moner appearing  in  the  schools  is  that  of  James  Scarlett  (Lord  Abinger) 
of  Trinity,  who  took  a  poll  B.A.  degree  in  1790.  Before  that  time  their 


184  THE  EXERCISES  IN  THE  SCHOOLS. 

By  the  Elizabethan  code  every  student  before  being  ad- 
mitted to  a  degree  had  to  swear  that  he  had  performed  all  the 
statutable  exercises.  The  additional  number  thus  required 
to  be  performed  were  kept  by  what  was  called  huddling.  To 
do  this  a  regent  took  the  moderator's  seat,  one  candidate  then 
occupied  the  respondent's  rostrum,  and  another  the  opponent's. 
Recte  statuit  Newtonus,  said  the  respondent.  Recte  non  statuit 
Newtonus,  replied  the  opponent.  This  was  a  disputation,  and 
it  was  repeated  a  sufficient  number  of  times  to  count  for  as 
many  disputations.  The  men  then  changed  places,  and  the. 
same  process  was  repeated,  each  maintaining  the  contrary  of 
his  first  assertion — an  admirable  practice,  as  De  Morgan  ob- 
served, for  those  who  were  going  to  enter  political  life.  Jebb1 
asserts  that  in  his  time  (1772)  a  candidate  in  this  way  could 
as  a  respondent  read  two  theses,  propound  six  questions,  and 
answer  sixteen  arguments  against  them,  all  in  five  minutes. 

Throughout  the  eighteenth  century  the  ceremony  of  enter- 
ing the  questions  (see  pp.  147, 155)  was  purely  formal.  So  also 
were  the  quadragesimal  exercises,  which  it  will  be  remembered 
were  held  after  Ash- Wednesday,  and  therefore  after  the  degree 
of  bachelor  had  been  conferred.  All  of  these  were  huddled. 
The  proctor  generally  asked  some  question  such  as  Quid  est 
nomen  ?  to  which  the  answer  usually  expected  was  Nescio.  In 
these  exercises  more  license  was  allowable,  and  if  the  proctor 
could  think  of  any  remark  which  he  was  pleased  to  consider 
witty,  particularly  if  there  was  any  play  on  words  in  it,  he 
was  at  liberty  to  give  free  scope  to  his  fancy.  Some  of  the 
repartees  to  which  these  personal  remarks  gave  rise  have  been 
preserved.  For  example,  J.  Brasse,  of  Trinity,  who  was  sixth 
wrangler  in  1811,  was  accosted  with  the  question,  Quid  est 
ces?  to  which  he  answered,  Nescio  nisi  finis  examinationis. 

appearance  was  optional,  but  Thomas  Jones  of  Trinity,  the  senior 
wrangler  of  1779,  when  moderator  in  1786 — 7,  introduced  a  grace  by 
which  fellow-commoners  were  subjected  to  the  same  exercises  as  other 
students. 

1  Jebb's  Works,  vol.  n.  p.  298. 


HUDDLING.  185 

So  again  Joshua  King  of  Queens'  was  asked  Quid  est  rex? 
to  which  he  promptly  replied,  Socius  reginalis,  as  ultimately 
turned  out  to  be  the  case. 

A  diligent  reader  of  the  literature  connected  with  the 
university  of  the  eighteenth  century  may  find  numbers  of  these 
mock  disputations ;  but  I  will  content  myself  with  one  more 
specimen.  Domine  respondens,  says  the  moderator,  quidfecisti 
in  academia  triennium  commorans  ?  Anne  circulum  quadrasti  ? 
To  which  the  student  shewing  his  cap  with  the  board  broken 
and  the  top  as  much  like  a  circle  as  anything  else,  replied  : 
Minime  domine  eruditissime :  sed  quadratum  omnino  circulavi. 

It  should  be  added  that  retorts  such  as  these  were  only 
allowed  in  the  pretence  exercises,  and  a  candidate  who  in  the 
actual  examination  was  asked  to  give  a  definition  of  happiness 
and  replied  an  exemption  from  Payne — that  being  the  name  of 
the  moderator  then  presiding — was  plucked  "for  want  of  dis- 
crimination in  time  and  place." 

In  earlier  times  even  the  farce  of  huddling  seems  to  have 
been  unnecessary,  for  the  Heads  reported  to  a  royal  commission 
in  1675  that  it  was  not  uncommon  for  the  proctors  to  take 
"cautions  for  the  performance  of  the  statutable  exercises,  and 
accept  the  forfeit  of  the  money  so  deposited  in  lieu  of  their 
performance." 

The  exercises  for  the  higher  degrees  (if  kept  at  all)  were 
universally  performed  by  huddling.  The  statutable  exercises  for 
the  M.A.  degree  were  three  respondencies,  each  against  a 
master  as  opponent,  two  respondencies  against  bachelor  oppo- 
nents, and  one  declamation.  In  the  eighteenth  century  these 
had  become  reduced  to  a  mere  form  and  were  all  huddled. 
The  usual  procedure  was  to  "declaim"  two  lines  of  the  ^Eneid 
or  of  Virgil's  first  Eclogue;  and  then  to  keep  three  acts  with 
the  formula,  Recte  statuit  N'ewtonus,  Woodius,  et  Paleius.  To 
this  the  opponent  replied  (thus  keeping  three  opponencies),  Si 
non  recte  statuerunt  JVewtonus,  Woodius,  et  Paleius  cadunt 
quaestiones:  sed  non  recte  statuerunt  Newtonus,  Woodius,  et 
Paleius :  ergo  cadunt  quaestiones. 


186  THE   EXERCISES   IN   THE   SCHOOLS. 

At  some  time  early  in  the  present  century  (I  suspect  about 
1820)  the  practice  of  huddling,  at  any  rate  for  the  master's 
degree,  almost  ceased.  It  was  generally  felt  that  it  was  better 
to  openly  violate  an  antiquated  statute  than  to  keep  the  letter 
and  not  the  spirit  of  it.  This  was  largely  due  to  Farish  and 
Peacock. 

I  may  here  add  that  though  the  standards  of  education 
and  examination  for  the  bachelor's  degree  at  Oxford  during 
the  seventeenth  and  eighteenth  centuries  were  very  far  below 
those  at  Cambridge,  yet  the  performance  of  certain  exercises 
for  the  master's  degree  was  always  there  enforced,  and  these 
to  some  extent  counteracted  the  evil  effects  of  the  absence  of 
any  honour  examination  and  of  any  real  disputations  for  those 
who  took  the  bachelor's  degree. 


CHAPTER   X. 
THE  MATHEMATICAL  TRIPOS1. 

I  TRACED  in  chapter  V.  the  steps  by  which  mathematics 
became  in  the  eighteenth  century  the  dominant  study  in  the 
university.  I  purpose  in  this  chapter  to  give  a  sketch  of  the 
rise  of  the  mathematical  tripos,  that  is,  of  the  instrument  by 
which  the  proficiency  of  students  in  mathematics  came  ulti- 
mately to  be  tested. 

The  proctors  had  from  the  earliest  time  had  the  power  of 
questioning  a  candidate  when  a  disputation  was  closed.  I  be- 
lieve that  it  was  about  1725  that  the  moderators  began  the 
custom  of  regularly  summoning  those  candidates  in  regard  to 
whose  abilities  and  position  some  doubt  was  felt.  In  earlier 
times  each  candidate  had  been  examined  when  his  act  was 
finished,  but  now  all  the  candidates  to  be  questioned  were 
present  at  the  same  time,  and  this  enabled  the  moderators  to 
compare  one  man  with  another. 

An  additional  reason  why  it  was  then  desirable  to  use  this 
latent  power  was  the  fact  that  at  that  time  it  had  become 
impossible  to  get  rooms  in  which  all  the  statutable  exercises 

1  The  substance  of  this  chapter  is  taken  from  my  Origin  and  history 
of  the  mathematical  tripos,  Cambridge,  1880.  The  history  of  the  tripos 
is  also  treated  in  Of  a  liberal  education,  by  W.  Whewell,  Cambridge, 
1848,  and  in  the  Scholae  academicae  by  C.  Wordsworth,  Cambridge,  1877. 
In  1888  Dr  Glaisher  chose  the  subject  for  his  inaugural  address  to  the 
London  Mathematical  Society :  all  the  more  important  facts  are  there 
brought  together  in  a  convenient  form,  and  in  some  places  in  the  latter 
part  of  the  chapter  I  have  utilized  his  summary  of  the  later  regulations 
for  the  conduct  of  the  examination. 


188  THE   MATHEMATICAL   TRIPOS. 

could  be  properly  performed,  and  many,  even  of  the  best  men, 
had  no  opportunity  to  shew  their  dialectical  skill  by  means 
of  the  exercises  in  the  schools.  This  arose  from  the  fact  that 
when  George  I.  in  1710  presented  the  university  with  thirty 
thousand1  books  and  manuscripts,  there  was  no  suitable  place 
in  which  they  could  be  arranged.  It  was  accordingly  decided 
to  build  a  new  senate-house,  and  use  the  old  one  as  part  of  the 
library,  and  meanwhile  the  books  were  stored  in  the  schools 
and  the  old  senate-house.  The  new  building  was  more  than 
twenty  years  in  course  of  construction,  and  during  that  in- 
terval the  authorities  found  it  impossible  to  compel  the  perform- 
ance of  all  the  exercises  required  from  candidates  for  degrees. 

During  the  confusion  so  caused,  the  discipline  and  studies 
of  the  university  suffered  seriously.  The  new  senate-house 
was  opened  in  1730,  and  Matthias  Mawson,  the  master  of 
Corpus,  who  was  vice-chancellor  in  1730  and  1731,  made  a 
determined  effort  to  restore  order.  It  was  however  found 
almost  impossible  to  enforce  all  the  statutable  exercises,  and 
there  was  the  less  necessity  as  the  examination,  which  had 
begun  to  grow  up,  supplied  a  practical  means  of  testing  the 
abilities  of  the  candidates.  The  advantages  of  the  latter 
system  were  so  patent  that  within  ten  or  twelve  years  it 
had  become  systematized  into  an  organized  test  to  which  all 
questionists  were  liable,  although  it  was  still  regarded  as  only 
supplementary  to  the  exercises  in  the  schools.  From  the  be- 
ginning it  was  conducted  in  English2,  and  accurate  lists  were 
made  of  the  order  of  merit  of  the  candidates ;  two  advantages 
to  which  I  think  its  final  and  definite  establishment  must  be 
largely  attributed. 

I  therefore  place  the  origin  of  the  senate-house  exami- 
nation about  the  year  1725;  but  there  are  no  materials  for 

1  The  library  had  been  shamefully  neglected.     It  contained  at  that 
time  less  than  fifteen  thousand  volumes :  many  thousands  having  been 
lost  or  stolen  in  the  two  preceding  centuries. 

2  I  have  no  doubt  that  this  was  the  case;  but  Jebb's  statement  (made 
in  1772),  if  taken  by  itself,  rather  implies  the  contrary. 


THE   MATHEMATICAL   TRIPOS.  189 

forming  an  accurate  opinion  as  to  how  it  was  then  conducted. 
It  is  however  probable  that  for  about  twenty  years  or  so  after 
its  commencement  it  was  looked  upon  as  a  tentative  and 
unauthorized  experiment.  Two  changes  which  were  then  made 
caused  greater  attention  to  be  paid  to  the  order  of  the  tripos 
list,  and  thus  served  to  give  it  more  prominence.  In  the  first 
place,  from  1747  onwards  the  final  lists  were  printed  and 
distributed ;  from  that  time  also  the  names  of  the  honorary 
or  proctor's  optimes  (see  p.  170)  were  specially  marked,  and 
it  was  thus  possible,  by  erasing  them,  to  obtain  the  correct 
order  of  the  other  candidates.  The  lists  published  in  the 
calendars  begin  therefore  with  that  date,  and  in  the  issues  for 
all  years  subsequent  to  1799  the  names  of  those  who  received 
these  honorary  degrees  have  been  omitted.  In  the  second 
place,  it  was  found  possible  by  means  of  the  new  examination 
to  differentiate  the  better  men  more  accurately  than  before ; 
and  accordingly,  in  1753,  the  first  class  was  subdivided  into 
two,  called  respectively  wranglers  and  senior  optimes,  a  division 
which  is  still  maintained. 

From  1750  onwards  the  examination  was  definitely  re- 
cognized by  the  university,  and  we  have  now  more  materials 
to  enable  us  to  judge  how  it  was  conducted.  It  would  seem 
from  these  that  it  was  presided  over  by  the  proctors  and 
moderators,  who  took  all  the  men  from  each  college  together 
as  a  class,  and  passed  questions  down  till  they  were  answered ; 
but  it  still  remained  entirely  oral,  and  technically  was  regarded 
as  subsidiary  to  the  discussions  in  the  schools.  As  each  class 
thus  contained  men  of  very  different  abilities,  a  custom  grew 
up  by  which  every  candidate  was  liable  to  be  taken  aside  to  be 
questioned  by  any  M.A.  who  wished  to  do  so,  and  this  was 
regarded  as  the  more  important  part  of  the  examination.  The 
subjects  were  mathematics  and  a  smattering  of  philosophy.  At 
first  the  examination  lasted  only  one  day,  but  at  the  end  of  this 
period  it  continued  for  two  days  and  a  half.  At  the  conclusion 
of  the  second  day  the  moderators  received  the  reports  of  those 
masters  of  arts  who  had  voluntarily  taken  part  in  the  exami- 


190  THE   MATHEMATICAL   TRIPOS. 

nation,  and  provisionally  settled  the  final  list ;  while  the  last 
half-day  was  used  in  revising  and  rearranging  the  order  of 
merit.  In  1763  it  was  decided  that  the  position  of  Paley  of 
Christ's  as  senior  in  the  tripos  list  to  Frere  of  Cains  was  to  be 
decided  by  the  senate-house  examination  and  not  by  the  dis- 
putations. 

During  the  following  years,  that  is  from  1763  to  1779,  the 
traditionary  rules  which  had  previously  guided  the  examiners 
in  each  year  took  definite  shape,  and  the  senate-house  exami- 
nation and  not  the  disputations  became  the  recognized  test  by 
which  a  man's  final  place  in  the  list  was  determined.  This  was 
chiefly  due  to  the  fact  that  henceforth  the  examiners  used  the 
disputations  only  as  a  means  of  classifying  the  men  roughly. 
On  the  result  of  their  'acts'  (and  probably  partly  also  of 
their  general  reputation)  the  candidates  were  divided  into 
eight  classes,  each  being  arranged  in  alphabetical  order.  Their 
subsequent  position  in  the  class  was  determined  solely  by  the 
senate-house  examination.  The  first  two  classes  comprised  all 
who  were  expected  to  be  wranglers,  the  next  four  classes 
included  the  other  candidates  for  honours,  and  the  last  two 
classes  consisted  of  poll  men  only.  Practically  any  one  placed 
in  either  of  the  first  two  classes  was  allowed,  if  he  wished,  to 
take  an  aegrotat  senior  optime,  and  thus  escape  all  further 
examination :  this  was  called  gulphing  it.  All  the  men  from 
one  college  were  no  longer  taken  together,  but  each  class  was 
examined  separately  and  vivd  voce.  As  henceforth  all  the 
students  comprised  in  each  class  were  of  about  equal  attain- 
ments, it  was  possible  to  make  the  examination  more  efficient. 

A  full  description  of  the  senate-house  examination  as  it 
existed  in  1772  is  extant1.  It  was  written  by  John  Jebb, 
who  had  been  second  wrangler  in  1757.  From  this  account  we 
find  that  it  had  then  become  usual  for  the  junior  moderator 
of  the  year  and  the  senior  moderator  of  the  preceding  year  to 
take  the  first  two  or  three  classes  together  by  themselves  at 

1  It  is  reprinted  in  §§  192—204  of  Whewell's  Of  a  liberal  education, 
second  edition,  London,  1850. 


THE   MATHEMATICAL   TRIPOS.  191 

one  table.  In  a  similar  way  the  next  four  or  three  classes 
sat  at  another  table,  presided  over  by  the  senior  moderator  of 
that  year  and  the  junior  moderator  of  the  preceding  one  \  while 
the  last  two  classes  containing  the  poll  men  were  examined  by 
themselves.  Thus,  in  all,  three  distinct  sets  of  papers  were 
set.  It  is  probable  that  before  the  examination  in  the  senate- 
house  began  a  candidate,  if  manifestly  placed  in  too  low  a 
class,  was  allowed  the  privilege  of  challenging  the  class  to 
which  he  was  assigned.  Perhaps  this  began  as  a  matter  of 
favour,  and  was  only  granted  in  exceptional  cases,  but  a  few 
years  later  it  became  a  right  which  every  candidate  could 
exercise;  and  I  think  that  it  is  partly  to  its  development 
that  the  ultimate  predominance  of  the  tripos  over  all  the  other 
exercises  for  degrees  is  due. 

The  examination  took  place  in  January  and  lasted  three 
days.  The  range  of  subjects  for  the  first  or  highest  class  is 
described  by  Jebb  as  follows. 

The  moderator  generally  begins  with  proposing  some  questions  from 
the  six  books  of  Euclid,  plane  trigonometry,  and  the  first  rules  of  algebra. 
If  any  person  fails  in  an  answer,  the  question  goes  to  the  next.  From 
the  elements  of  mathematics,  a  transition  is  made  to  the  four  branches 
of  philosophy,  viz.  mechanics,  hydrostatics,  apparent  astronomy,  and 
optics,  as  explained  in  the  works  of  Maclaurin,  Cotes,  Helsham,  Hamilton, 
Kutherforth,  Keill,  Long,  Ferguson,  and  Smith.  If  the  moderator  finds 
the  set  of  questionists,  under  examination,  capable  of  answering  him,  he 
proceeds  to  the  eleventh  and  twelfth  books  of  Euclid,  conic  sections, 
spherical  trigonometry,  the  higher  parts  of  algebra,  and  Sir  Isaac  Newton's 
Principia;  more  particularly  those  sections  which  treat  of  the  motion 
of  bodies  in  eccentric  and  revolving  orbits  ;  the  mutual  action  of  spheres, 
composed  of  particles  attracting  each  other  according  to  various  laws ; 
the  theory  of  pulses,  propagated  through  elastic  mediums;  and  the 
stupendous  fabric  of  the  world.  Having  closed  the  philosophical  exami- 
nation, he  sometimes  asks  a  few  questions  in  Locke's  Essay  on  the 
human  understanding,  Butler's  Analogy,  or  Clarke's  Attributes.  But 
as  the  highest  academical  distinctions  are  invariably  given  to  the  best 
proficients  in  mathematics  and  natural  philosophy,  a  very  superficial 
knowledge  in  morality  and  metaphysics  will  suffice. 

When  the  division  under  examination  is  one  of  the  higher  classes, 
problems  are  also  proposed,  with  which  the  student  retires  to  a  distant 


192  THE   MATHEMATICAL   TRIPOS. 

part  of  the  senate-house,  and  returns,  with  his  solution  upon  paper,  to 
the  moderator,  who,  at  his  leisure,  compares  it  with  the  solutions  of 
other  students,  to  whom  the  same  problems  have  been  proposed. 

The  extraction  of  roots,  the  arithmetic  of  surds,  the  invention  of 
divisors,  the  resolution  of  quadratic,  cubic,  and  biquadratic  equations ; 
together  with  the  doctrine  of  fluxions,  and  its  application  to  the  solution 
of  questions  'de  maximis  et  minimis,'  to  the  rinding  of  areas,  to  the 
rectification  of  curves,  the  investigation  of  the  centers  of  gravity  and 
oscillation,  and  to  the  circumstances  of  bodies,  agitated,  according  to 
various  laws,  by  centripetal  forces,  as  unfolded,  and  exemplified,  in  the 
fluxional  treatises  of  Lyons,  Saunderson,  Simpson,  Emerson,  Maclaurin, 
and  Newton,  generally  form  the  subject-matter  of  these  problems. 

As  the  questionists  in  each  class  were  examined  in  divisions 
of  six  or  eight  at  a  time,  a  considerable  number  were  dis- 
engaged at  any  particular  hour.  Any  master  or  doctor  could 
then  call  a  man  aside  and  examine  him.  This  separate  ex- 
amination or  scrutiny  was  the  test  by  which  the  best  men  were 
differentiated.  Any  one  who  thus  voluntarily  took  part  in  the 
examination  had  to  report  his  impressions  to  the  proper  officers. 
This  right  of  examination  was  a  survival  of  the  part  taken 
by  every  regent  in  the  exercises  of  the  university ;  but  it 
constantly  gave  rise  to  accusations  of  partiality1. 

Although  the  examination  lasted  but  a  few  days  it  must 
have  been  a  severe  physical  trial  to  any  one  who  was  delicate. 
It  was  held  in  winter  and  in  the  senate-house.  That  building 
was  then  noted  for  its  draughts  and  was  not  warmed  in  any 
way;  and  we  are  told  that  upon  one  occasion  the  candidates 
on  entering  in  the  morning  found  the  ink  frozen  at  their  desks. 
The  duration  of  the  examination  must  have  been  even  more 
trying  than  the  circumstances  under  which  it  was  conducted. 
The  hours  on  Monday  and  Tuesday  were  from  8  to  9,  9.30  to 
11,  1  to  3,  3.30  to  5,  and  7  to  9.  The  evening  paper  was  set 
in  the  rooms  of  the  moderator,  and  wine  or  tea  was  provided. 
The  examination  on  Wednesday  ended  at  11.  On  Thursday, 
morning  at  eight  a  first  list  was  published  with  all  candidates 

1  See  for  example  Gooch's  letter  reprinted  later  on  p.  196 :  see  also 
Bligh's  pamphlets  of  1780  and  1781. 


THE  MATHEMATICAL   TRIPOS.  193 

of  about  equal  merit  bracketed,  and  that  day  was  devoted  to 
arranging  the  men  whose  names  appeared  in  the  same  bracket 
in  their  proper  order.  A  man  rarely  rose,  above  or  sunk  below 
his  bracket,  but  during  the  first  hour  he  had  the  right,  if  dis- 
satisfied with  his  position,  to  challenge  any  one  above  him  to  a 
fresh  examination  in  order  to  see  which  was  the  better.  At 
nine  a  second  list  came  out,  and  a  candidate's  power  of  chal- 
lenging was  then  confined  to  the  bracket  immediately  above 
his  own.  Fresh  lists  revised  and  corrected  came  out  at  11  a.  m., 
3  p.m.,  and  5  p.m.  The  final  list  was  then  prepared.  The  name 
of  the  senior  wrangler  was  announced  at  midnight,  and  the 
rest  of  the  list  the  next  morning.  The  publication  of  the  list 
was  attended  with  great  excitement. 

About  this  time,  circ.  1772,  it  began  to  be  the  custom  to 
dictate  some  or  all  of  the  questions  and  to  require  answers  to 
be  written.  Only  one  question  was  dictated  at  a  time,  and  a 
fresh  one  was  not  given  out  until  some  student  had  solved  that 
previously  read — a  custom  which  by  causing  perpetual  inter- 
ruptions to  take  down  new  questions  must  have  proved  very 
harassing.  We  are  perhaps  apt  to  think  that  an  examination 
conducted  by  written  papers  is  so  natural  that  the  custom 
is  of  long  continuance.  But  I  can  find  no  record  of  any  (in 
Europe)  earlier  than  those  introduced  by  Bentley  at  Trinity 
in  1702  (see  p.  81):  though  in  them  it  will  be  observed  that 
every  candidate  had  a  different  set  of  questions  to  answer, 
so  that  a  strict  comparison  must  have  been  very  difficult.  The 
questions  for  the  Smith's  prizes  continued  until  1830  to  be 
dictated  in  the  manner  described  above.  Even  at  the  present 
time  it  is  usual  to  dictate  the  mathematical  papers  for  the 
baccalaureate  degree  in  the  university  of  France,  but  all  the 
questions  are  read  out  at  once. 

In  1779  the  senate-house  examination  was  extended  to  four 
days,  the  third  day  being  given  up  entirely  to  moral  philosophy ; 
at  the  same  time  the  number  of  examiners  was  increased, 
and  the  system  of  brackets  recognized  as  a  formal  part  of  the 
procedure.  The  right  of  any  M.A.  to  take  part  in  it,  though 
B.  13 


194  THE   MATHEMATICAL   TRIPOS. 

continuing  to  exist,  was  much  more  sparingly  exercised,  and 
I  believe  was  not  insisted  on  after  1785.  A  candidate  who 
was  dissatisfied  with  the  class  in  which  he  had  been  placed  as 
the  result  of  his  disputations  was  henceforth  allowed  to 
challenge  it  before  the  examination  began.  This  power  seems 
to  have  been  used  but  rarely;  it  was  however  a  recognition  of 
the  fact  that  a  place  in  the  tripos  list  was  to  be  determined  by 
the  senate-house  examination  alone,  and  the  examiners  soon 
acquired  the  habit  of  settling  the  preliminary  classes  without 
much  reference  to  the  previous  disputations. 

In  cases  of  equality  the  acts  were  still  taken  into  account 
in  settling  the  tripos  order;  and  in  1786  when  the  second, 
third,  and  fourth  wranglers  came  out  equal  in  the  examination 
a  memorandum  was  published  that  the  second  place  was  given 
to  that  candidate  who  in  dialectis  magis  est  versatus,  and  the 
third  place  to  that  one  who  in  scholis  sophistarum  melius  dis- 
putavit. 

In  1786  a  question  set  to  the  expectant  wranglers  which 
required  the  extraction  of  the  square  root  of  a  number  to  three 
places  of  decimals  is  said1  to  have  been  considered  unreasonably 
hard. 

The  only  papers  of  this  date  which  as  far  as  I  know  are 
now  extant  are  one  of  the  problem  papers  set  in  1785  and 
one  of  those  set  in  1786.  These  were  composed  by  William 
Hodson,  of  Trinity  (seventh  wrangler  in  1764,  and  vice-master 
of  the  college  from  1789  to  1793),  who  was  then  proctor.  The 
autograph  copies  from  which  he  gave  out  the  questions  were 
luckily  preserved,  and  have  recently  been  placed  in  the  library 
of  Trinity2.  They  must  be  almost  the  last  problem  papers 
which  were  dictated,  instead  of  being  printed  and  given  as 
a  whole  to  the  candidates. 

1  See  Gunning's  Reminiscences,  vol.  i.  chap.  in.     Note  however  that 
the  Reminiscences  were  not  written  till  60  or  70  years  later ;  and  this 
statement  only  represents  the  author's  recollections  of  the  rumours  of  the 
time.     There  are  reasons  for  thinking  that  the  statement  is  exaggerated. 

2  The  Challis  Manuscripts,  in.  61.  ** 


PROBLEM    PAPERS   SET   IN    1785    AND    1786.          195 

The  paper  for  1785  is  headed  by  a  memorandum  to  warn 
candidates  to  write  distinctly  and  to  observe  that  "  at  least  as 
much  will  depend  upon  the  clearness  and  precision  of  the  answers 
as  upon  the  quantity  of  them."  The  questions  are  as  follows. 

1.  To  prove  how  many  regular  Solids  there  are,  what  are  those 
Solids  called,  and  why  there  are  no  more. 

2.  To  prove  the  Asymptotes  of  an  Hyperbola  always  external  to  the 
Curve. 

3.  Suppose  a  body  thrown  from  an  Eminence  upon  the  Earth,  what 
must  be  the  Velocity  of  Projection,  to  make  it  become  a  secondary  planet 
to  the  Earth  ? 

4.  To  prove  in  all  the  conic  sections  generally  that  the  force  tending 
to  the  focus  varies  inversely  as  the  square  of  the  Distance. 

5.  Supposing  the  periodical  times  in  different  Ellipses  round  the 
same  center  of  force,  to  vary  in  the  sesquiplicate  ratio  of  the  mean 
distances,  to  prove  the  forces  in  those  mean  distances  to  be  inversely  as 
the  square  of  the  distance. 

6.  What  is  the  relation  between  the  3rd  and  7th  Sections  of  Newton, 
and  how  are  the  principles  of  the  3rd  applied  to  the  7th? 

7.  To  reduce  the  biquadratic  equation  x*  +  qx'2  +  rx  +  s  =  0  to  a  cubic 
one. 

8.  To  find  the  fluent  of  x  x  Ja*  -  x2. 

9.  To  find  a  number  from  which  if  you  take  its  square,  there  shall 
remain  the  greatest  difference  possible. 

10.  To  rectify  the  arc  DB  of  the  circle  DBES.    [A  figure  in  the 
margin  shews  that  an  arc  of  any  length  is  meant.] 

The  problem  paper  for  1786  is  as  follows. 

1.  To  determine  the  velocity  with  which  a  Body  must  be  thrown,  in 
a  direction  parallel  to  the  Horizon,  so  as  to  become  a  secondary  planet 
to  the  Earth;  as  also  to  describe  a  parabola,  and  never  return. 

2.  To  demonstrate,  supposing  the  force  to  vary  as  — ,  how  far  a 

body  must  fall  both  within  and  without  the  Circle  to  acquire  the  Velocity 
with  which  a  body  revolves  in  a  Circle. 

3.  Suppose  a  body  to  be  turned  (sic)  upwards  with  the  Velocity  with 
which  it  revolves  in  an  Ellipse,  how  high  will  it  ascend?    The  same  is 
asked  supposing  it  to  move  in  a  parabola. 

4.  Suppose  a  force  varying  first  as  — ^ ,  secondly  in  a  greater  ratio 
than  —  but  less  than  -=-^ ,  and  thirdly  in  a  less  ratio  than  -2 ,  in  each 

13—2 


196  THE  MATHEMATICAL   TRIPOS. 

of  these  Cases  to  determine  whether  at  all,  and  where  the  body  parting 
from  the  higher  Apsid  will  come  to  the  lower. 

5.  To  determine  in  what  situation  of  the  moon's  Apsids  they  go  most 
forwards,  and  in  what  situation  of  her  Nodes  the  Nodes  go  most  back- 
wards, and  why  ? 

6.  In  the  cubic  equation  xz  +  qx  +  r  =  0  which  wants  the  second  term ;. 
supposing  x  =  a  +  b  and  3a&=  -  q,  to  determine  the  value  of  x. 

i 

7.  To  find  the  fluxion  of  xr  x  (yn  +  zm)«. 

8.  To  find  the  fluent  of  -^-  . 

a  +  x 

9.  To  find  the  fluxion  of  the  mth  power  of  the  Logarithm  of  x. 

10.  Of  right-angled  Triangles  containing  a  given  Area  to  find  that 
whereof  the  sum  of  the  two  legs  AB  +  BC  shall  be  the  least  possible. 
[This  and  the  two  following  questions  are  illustrated  by  diagrams.     The 
angle  at  B  is  the  right  angle.] 

11.  To  find  the  Surface  of  the  Cone  ABC.     [The  cone  is  a  right  one 
on  a  circular  base.] 

12.  To  rectify  the  arc  DB  of  the  semicircle  DBV. 

I  insert  here  the  following  letter  from  William  Gooch,  of 
Caius,  in  which  he  describes  his  examination  in  the  senate- 
house  in  1791.  It  must  be  remembered  that  it  is  the  letter 
of  an  undergraduate  addressed  to  his  father  and  mother,  and 
was  not  intended  either  for  preservation  or  publication — a  fact 
which  certainly  does  not  detract  from  its  value.  His  account 
of  his  acts  in  1790  was  printed  in  the  last  chapter.  This 
letter  is  dated  January,  1791,  and  is  written  almost  like  a 
diary. 

'Monday  \  aft.  12. 

We  have  been  examin'd  this  Morning  in  pure  Mathematics  &  I've 
hitherto  kept  just  about  even  with  Peacock  which  is  much  more  than  I 
expected.  We  are  going  at  1  o'clock  to  be  examin'd  till  3  in  Philosophy. 

From  1  till  7  I  did  more  than  Peacock ;  But  who  did  most  at  Mode- 
rator's Booms  this  Evening  from  7  till  9,  I  don't  know  yet ; — but  I  did 
above  three  times  as  much  as  the  Senr  Wrangler  last  year,  yet  I'm  afraid 
not  so  much  as  Peacock. 

Between  One  &  three  o'Clock  I  wrote  up  9  sheets  of  Scribbling  Paper 
so  you  may  suppose  I  was  pretty  fully  employ'd. 

Tuesday  Night. 

I've  been  shamefully  us'd  by  Lax  to-day; — Tho'  his  anxiety  for 
Peacock  must  (of  course)  be  very  great,  I  never  suspected  that  his  Par- 


THE   MATHEMATICAL  TRIPOS.  197 

tially  (sic)  w*1  get  the  better  of  his  Justice.  I  had  entertain'd  too  high  an 
opinion  of  him  to  suppose  it. — he  gave  Peacock  a  long  private  Examina- 
tion &  then  came  to  me  (I  hop'd)  on  the  same  subject,  but  'twas  only  to 
Bully  me  as  much  as  he  could, — whatever  I  said  (tho5  right)  he  tried  to 
convert  into  Nonsense  by  seeming  to  misunderstand  me.  However  I 
don't  entirely  dispair  of  being  first,  tho'  you  see  Lax  seems  determin'd 
that  I  shall  not. — I  had  no  Idea  (before  I  went  into  the  Senate-House)  of 
being  able  to  contend  at  all  with  Peacock. 

Wednesday  evening. 

Peacock  &  I  are  still  in  perfect  Equilibrio  &  the  Examiners  them- 
selves can  give  no  guess  yet  who  is  likely  to  be  first ; — a  New  Examiner 
(Wood  of  St.  John's,  who  is  reckon'd  the  first  Mathematician  in  the  Uni- 
versity, for  Waring  doesn't  reside)  was  call'd  solely  to  examine  Peacock 
<fe  me  only. — but  by  this  new  Plan  nothing  is  yet  determin'd. — So  Wood 
is  to  examine  us  again  to-morrow  morning. 

Thursday  evening. 

Peacock  is  declar'd  first  &  I  second, — Smith  of  this  Coll.  is  either  8th 
or  9th  &  Lucas  is  either  10th  or  11th. — Poor  Quiz  Carver  is  one  of  the  ol 
TroXXot; — I'm  perfectly  satisfied  that  the  Senior  Wranglership  is  Peacock's 
due,  but  certainly  not  so  very  indisputably  as  Lax  pleases  to  represent  it 
— I  understand  that  he  asserts  'twas  5  to  4  in  Peacock's  favor.  Now 
Peacock  &  I  have  explain'd  to  each  other  how  we  went  on,  &  can  prove 
indisputably  that  it  wasn't  20  to  19  in  his  favor; — I  cannot  therefore  be 
displeas'd  for  being  plac'd  second,  tho'  I'm  provov'd  (sic)  with  Lax  for 
his  false  report  (so  much  beneath  the  Character  of  a  Gentleman.) — 

N.B.  it  is  my  very  particular  Request  that  you  don't  mention  Lax's 
behaviour  to  me  to  any  one1.' 

It  was  about  this  time  that  the  custom  of  printing  the 
problem  (but  not  the  other)  papers  was  introduced. 

Such  was  the  form  ultimately  taken  by  the  senate-house 
examination,  a  form  which  it  substantially  retained  without 
alteration  for  nearly  half  a  century,  and  which  may  fairly  be 
considered  as  the  archetype  of  the  numerous  competitive  ex- 
aminations now  existing  in  England.  It  soon  became  the 
sole  test  by  which  candidates  were  judged.  In  1790  James 
Blackburn  of  Trinity,  a  questionist  of  exceptional  abilities, 
was  informed  that  in  spite  of  his  good  disputations  he  would 
not  be  allowed  a  degree  unless  he  also  satisfied  the  examiners 
1  Scholae  academicae,  322—23. 


198  THE   MATHEMATICAL   TKIPOS. 

in  the  tripos.  He  accordingly  solved  one  'very  hard  problem/ 
though  in  consequence  of  a  dispute  with  the  authorities  he 
refused  to  attempt  any  more.  In  1799  a  further  step  in  the 
same  direction  was  taken,  and  it  was  determined  to  require 
from  every  candidate  a  knowledge  of  the  first  book  of  Euclid, 
arithmetic  (to  fractions),  elementary  algebra,  Locke's  Essay, 
and  Paley's  Evidences.  A  knowledge  of  the  first  two  books 
of  Euclid,  algebra  to  simple  and  quadratic  equations,  and 
the  early  chapters  of  Paley's  Evidences  of  Christianity  was 
still  considered  sufficient  to  secure  a  position  in  the  senior 
optimes. 

Since  1796  a  calendar  containing  an  account  of  the  uni- 
versity constitution  and  customs  has  been  annually  published. 
The  following  garrulous  account  of  the  examination  in  1802  is 
taken  from  the  calendar  of  that  year. 

On  the  Monday  morning,  a  little  before  eight  o'clock,  the  students, 
generally  about  a  hundred,  enter  the  senate-house,  preceded  by  a  master 
of  arts,  who  on  this  occasion  is  styled  the  father  of  the  college  to  which 
he  belongs.  On  two  pillars  at  the  entrance  of  the  senate-house  are  hung 
the  classes  and  a  paper  denoting  the  hours  of  examination  of  those  who 
are  thought  most  competent  to  contend  for  honours.  Immediately  after 
the  university  clock  has  struck  eight,  the  names  are  called  over,  and  the 
absentees,  being  marked,  are  subject  to  certain  fines.  The  classes  to  be 
examined  are  called  out,  and  proceed  to  their  appointed  tables,  where 
they  find  pens,  ink,  and  paper  provided  in  great  abundance.  In  this 
manner,  with  the  utmost  order  and  regularity,  two-thirds  of  the  young 
men  are  set  to  work  within  less  than  five  minutes  after  the  clock  has 
struck  eight.  There  are  three  chief  tables,  at  which  six  examiners  preside. 
At  the  first,  the  senior  moderator  of  the  present  year  and  the  junior 
moderator  of  the  preceding  year.  At  the  second,  the  junior  moderator 
of  the  present  and  the  senior  moderator  of  the  preceding  year.  At  the 
third,  two  moderators  of  the  year  previous  to  the  two  last,  or  two  ex- 
aminers appointed  by  the  senate.  The  two  first  tables  are  chiefly  allotted 
to  the  six  first  classes ;  the  third,  or  largest,  to  the  oi  iro\\ol. 

The  young  men  hear  the  propositions  or  questions  delivered  by  the 
examiners;  they  instantly  apply  themselves;  demonstrate,  prove,  work 
out  and  write  down,  fairly  and  legibly  (otherwise  their  labour  is  of  little 
avail)  the  answers  required.  All  is  silence ;  nothing  heard  save  the  voice 
of  the  examiners;  or  the  gentle  request  of  some  one,  who  may  wish  a 


THE   MATHEMATICAL   TRIPOS.  199 

repetition  of  the  enunciation.  It  requires  every  person  to  use  the  utmost 
dispatch ;  for  as  soon  as  ever  the  examiners  perceive  any  one  to  have 
finished  his  paper  and  subscribed  his  name  to  it  another  question  is 
immediately  given.  A  smattering  demonstration  will  weigh  little  in  the 
scale  of  merit;  everything  must  be  fully,  clearly,  and  scientifically 
brought  to  a  true  conclusion.  And  though  a  person  may  compose  his 
paper  amidst  hurry  and  embarrassment,  he  ought  ever  to  recollect  that 
his  papers  are  all  inspected  by  the  united  abilities  of  six  examiners  with 
coolness,  impartiality,  and  circumspection. 

The  examiners  are  not  seated,  but  keep  moving  round  the  tables,  both 
to  judge  how  matters  proceed  and  to  deliver  their  questions  at  proper 
intervals.  The  examination,  which  embraces  arithmetic,  algebra, 
fluxions,  the  doctrine  of  infinitesimals  and  increments,  geometry,  trigo- 
nometry, mechanics,  hydrostatics,  optics,  and  astronomy,  in  all  their 
various  gradations,  is  varied  according  to  circumstances:  no  one  can 
anticipate  a  question,  for  in  the  course  of  five  minutes  he  may  be  dragged 
from  Euclid  to  Newton,  from  the  humble  arithmetic  of  Bonnycastle  to 
the  abstruse  analytics  of  Waring.  While  this  examination  is  proceeding 
at  the  three  tables  between  the  hours  of  eight  and  nine,  printed  problems 
are  delivered  to  each  person  of  the  first  and  second  classes ;  these  he  takes 
with  him  to  any  window  he  pleases,  where  there  are  pens,  ink,  and  paper 
prepared  for  his  operations. 


At  nine  o'clock  the  papers  had  to  be  given  up,  and  half-an- 
hour  was  allowed  for  breakfast.  At  half-past  nine  the  can- 
didates came  back,  and  were  examined  in  the  way  described 
above  till  eleven,  when  the  senate-house  was  again  cleared. 
An  interval  of  two  hours  then  took  place.  At  one  o'clock  all 
returned  to  be  again  examined.  At  three  the  senate-house 
was  cleared  for  half-an-hour,  and,  on  the  return  of  the  can- 
didates, the  examination  was  continued  till  five.  At  seven  in 
the  evening  the  first  four  classes  went  to  the  senior  moderator's 
rooms  to  solve  problems.  They  were  finally  dismissed  for  the 
day  at  nine,  after  eight  hours  of  examination.  The  work  on 
Tuesday  was  similar  to  that  of  Monday;  Wednesday  was  partly 
devoted  to  logic  and  moral  philosophy.  At  eight  o'clock  on 
Thursday  morning  the  brackets  or  preliminary  classifications 
in  order  of  merit,  each  containing  the  names  of  the  candidates 
placed  alphabetically,  were  hung  upon  the  pillars.  The  exa- 


200  THE   MATHEMATICAL  TRIPOS. 

initiation  that  day  was  devoted  to  arranging  the  men  in  each 
bracket  in  their  proper  order :  but  every  candidate  had  the 
right  to  challenge  any  one  whose  name  appeared  in  the  bracket 
immediately  above  his  own.  If  he  proved  himself  the  equal 
of  the  man  so  challenged  his  name  was  transferred  to  the 
upper  bracket.  To  challenge  and  then  to  fail  to  substantiate 
the  claim  to  removal  to  a  higher  bracket  was  considered 
rather  ridiculous.  Fresh  editions  and  revisions  of  the  brackets 
were  published  at  9  a.m.,  11  a.m.,  3  p.m.,  and  5  p.m.,  accord- 
ing to  the  results  of  the  examination  during  that  day.  At 
five  the  whole  examination  ended.  The  proctors,  moderators, 
and  examiners  then  retired  to  a  room  under  the  public  library 
to  prepare  the  list  of  honours,  which  was  sometimes  settled 
without  much  difficulty  in  a  few  hours,  but  sometimes  not  be- 
fore two  or  three  the  next  morning.  The  name  of  the  senior 
wrangler  was  generally  published  at  midnight. 

In  1802,  there  were  eighty-six  candidates  for  honours,  and 
they  were  divided  into  fifteen  brackets,  the  first  and  second 
brackets  containing  each  one  name  only,  and  the  third  bracket 
four  names. 

Until  1883  the  tripos  papers  of  the  current  year  were 
printed  in  the  calendar.  The  papers  from  1801  to  1820  were 
also  published  separately  under  the  title  Cambridge  problems  ; 
being  a  collection  of  the  printed  questions  proposed  to  the  candi- 
dates...^ the  general  examinations  from  1801  to  1820  inclusive. 
As  complete  sets  of  all  the  problems  set  to  each  of  the  classes 
are  now  rare,  I  propose  to  print  here  the  whole  of  the  problem 
papers  set  in  1802. 


MONDAY  MORNING  PROBLEMS.— MR.  PALMER. 

First  and  second  classes  (i.e.  the  expectant  wr 'anglers). 

1.  GIVEN  the  three  angles  of  a  plane  triangle,  and  the  radius  of  its 
inscribed  circle,  to  determine  its  sides. 

2.  The  specific  gravities  of  two  fluids,  which  will  not  mix,  are  to 
each  other  as  n  :  1,  compare  the  quantities  which  must  be  poured  into  a 


THE  PROBLEM  PAPERS  SET  IN  1802.       201 

cylindrical  tube,  whose  length  is  (a)  inches,  that  the  pressures  on  the 
concave  surfaces  of  the  tube,  which  are  in  contact  with  the  fluids,  may  be 
equal. 

3.  Determine  that  point  in  the  arc  of  a  quadrant  from  which  two 
lines  being  drawn,  one  to  the  centre  and  the  other  bisecting  the  radius, 
the  included  angle  shall  be  the  greatest  possible. 

4.  Required  the  linear  aperture  of  a  concave  spherical  reflector  of 
glass,  that  the  brightness  of  the  sun's  image  may  be  the  same  when 
viewed  in  the  reflector  and  in  a  given  glass  lens  of  the  same  radius. 

5.  Determine  the  evolute  to  the  logarithmic  spiral. 

6.  Prove  that  the  periodic  times  in  all  ellipses  about  the  same  center 
are  equal. 

7.  The  distance  of  a  small  rectilinear  object  from  the  eye  being  given, 
compare  its  apparent  magnitude  when  viewed  through  a  cylindrical  body 
of  water  with  that  perceived  by  the  naked  eye. 

8.  Find  the  fluents  of  the  quantities      .  9X    9t ,  and ^— -  . 

*(«•-«*)  y(a  +  y)* 

9.  Through  what  space  must  a  body  fall  internally,  towards  the 
centre  of  an  ellipse,  to  acquire  the  velocity  in  the  curve? 

10.  Find  the  principal  focus  of  a  globule  of  water  placed  in  air. 

11.  Determine,  after  Newton's  manner,  the  law  of  the  force  acting 
perpendicular  to  the  base,  by  which  a  body  may  describe  a  common 
cycloid. 

12.  Find  the  area  of  the  curve  whose  equation  is  xy  =  ax. 

13.  What  is  the  value  of  q  that  force  x  (period)2  =  q  x  radius  of  circle? 

14.  Two  places,  A  and  B,  are  so  situated  that  when  the  sun  is  in  the 
northern  tropic  it  rises  an  hour  sooner  at  A  than  at  B  ;  and  when  the  sun 
is  in  the  southern  tropic  it  rises  an  hour  later  at  A  than  at  B.     Required 
the  latitudes  of  the  places. 

15.  From  what  point  in  the  periphery  of  an  ellipse  may  an  elastic 
body  be  so  projected  as  to  return  to  the  same  point,  after  three  successive 
reflections  at  the  curve,  having  in  its  course  described  a  parallelogram? 

MONDAY  AFTERNOON  PROBLEMS.— MR.  DEALTRY. 

Third  and  fourth  classes  (i.e.  the  expectant  senior  optimes). 

1.  Inscribe  the  greatest  cylinder  in  a  given  sphere. 

2.  Rays,  which  pass  through  a  globe  at  equal  distances  from  the 
centre,  are  turned  equally  out  of  their  course. — Required  a  proof. 

3.  Given  a  declination  of  the  sun  and  the  latitude  of  the  place,   to 
find  the  duration  of  twilight. 

4.  A  cylindrical  vessel,  16  feet  high,  empties  itself  in  four  hours  by 
a  hole  in  the  bottom. — What  space  does  the  surface  describe  in  each 
hour? 

5.  Prove  that  if  two  circles  touch  each  other  externally,  and  parallel 
diameters  be  drawn,  the  straight  lines,  which  join  the  opposite  extremities 
of  these  diameters,  will  pass  through  the  point  of  contact. 

6.  A  ball,  whose  elasticity  :  perfect  elasticity  ::  n  :  1,  falls  from  a 
given  height  upon  a  hard  plane,  and  rebounds  continually  till  its  whole 
motion  is  lost. — Find  the  space  passed  over. 


202  THE   MATHEMATICAL   TRIPOS. 

7.  If  a  body  revolves  in  any  curve,  compare  the  angular  velocity  of 
the  perpendicular  with  that  of  the  distance. 

8.  How  far  must  a  body  fall  externally  to  acquire  the  velocity  in  a 
circle,  the  force  varying  as  the  distance? 

9.  Given  the  right  ascensions  and  declinations  of  two  stars,  to  find 
their  distance. 

10.  Find   the   velocity  with   which    air   rushes   into   an   exhausted 
receiver. 

11.  Let  the  roots  of  the  equation  Xs -px*  +  qx-r  =  Q  be  a,  6,  and  c, 
to  transform  it  into  another,  whose  roots  are  a2,  b2,  c2. 

12.  Find  the  fluent  - — ^,  a  being  less  than  1 ;  and  of       ^         . 


13.  Find  that  point  in  the  ellipse,  where  the  velocity  is  a  geometric 
mean  between  the  greatest  and  least  velocities,  the  force  varying  —  . 

14.  Determine  the  position  of  a  line  drawn  from  a  given  point  to  a 
given  inclined  plane,  through  which  the  body  will  fall  in  the  same  time 
as  through  the  given  plane. 

15.  The  equation  y?  -  5x2  +  Sx  -  4  =  0  has  two  equal  roots. — Find  them. 

16.  Find  the  sum   of  the   cube   numbers    1  +  8  +  27  +  &c.    by   the 
differential   method;    and  sum   the  following  series  by  the  method  of 
increments : 

1.2  +  2. 3+3. 4+ <&c.  n  terms. 

=— s  +  ^— ~  +  75— j  &c.  n  terms  and  ad  infinitum. 

1  .  J       &  .  o       o  .  4 

17.  If  half  of  the  earth  were  taken  off  by  the  impulse  of  a  comet, 
what  change  would  be  produced  in  the  moon's  orbit? 

18.  Prove  that  if  the  eye  be  placed  in  the  principal  focus  of  a  lens, 
the  image  of  a  given  object  would  always  appear  the  same. 

19.  Find  the  time  of  emptying  a  given  paraboloid  by  a  hole  made  in 
the  vertex. 

20.  Find   the  proportion  between  the  centripetal  and  centrifugal 
forces  in  a  curve ;  and  apply  the  expression  to  the  reciprocal  spiral. 

MONDAY  AFTERNOON  PROBLEMS.— MR.  DEAL/TRY. 

Fifth  and  sixth  classes  (i.e.  the  eocpectant  junior  optimes). 

1.  Prove  that  an  arithmetic  mean  is  greater  than  a  geometric. 

2.  Every  section  of  a  sphere  is  a  circle. — Kequired  a  proof. 

3.  If  |  of  an  ell  of  Holland  cost  ££.  what  will  12|  ells  cost? 

4.  Prove    the  method  of   completing  the   square  in    a    quadratic 
equation. 

5.  Take  away  the  second  term  of  the  equation  x2-  12#  +  o  =  0. 

6.  Inscribe  the  greatest  rectangle  in  a  given  circle. 

7.  Sum  the  following  series : 

1  +  3  +  5  +  7  +  &c.  to  n  terms. 

.  ad  inf. 


THE  PROBLEM  PAPERS  SET  IN  1802.       203 


172T3     27374    37T75-- 
8-,   Find  the  value  of  x  in  the  following  equations  : 
42a        35z_ 
a;  -  2  ~  x^3 


9.  In  a  given  circle  to  inscribe  an  equilateral  triangle. 

10.  Two   equal  bodies  move  at  the  same  instant  from  the  same 
extremity  of  the  diameter  of  a  circle  with  equal  velocities  in  opposite 
semi-circles.     Required  the  path  described  by  the  centre  of  gravity  ;  find 
the  path  also  when  the  bodies  are  unequal. 

11.  Through  what  chord  of  a  circle  must  a  body  fall  to  acquire  half 
the  velocity  gained  by  falling  through  the  diameter? 

12.  Given  the  latitude  of  the  place  and  the  sun's  meridian  altitude, 
to  find  the  declination. 

13.  Given  the  sun's  altitude  and  azimuth  and  the  latitude  of  the 
place,  to  find  the  declination  and  the  hour  of  the  day. 

14.  Prove  that  the  velocity  in  a  parabola  :  velocity  in  a  circle  at  the 
same  distance  ::  fj2  :  1. 

15.  How  far  must  a  body  fall  internally  to  acquire  the  velocity  in  a 

circle,  the  force  varying  -     ? 


MONDAY  EVENING  PROBLEMS.—  MR.  DEALTRY. 

First,  second,  third,  and  fourth  classes. 

1.  Find  four  geometric  means  between  1  and  32,  and  three  arithmetic 
means  between  1  and  11. 

2.  Suppose   a  straight  lever  has  some  weight,  and  at  one  end  a 
weight  is  suspended  equal  to  that  of  the  lever;  where  must  the  fulcrum 
be  placed,  that  there  may  be  an  equilibrium? 

3.  Determine  the  latitude  of  the  place,  where  the  sun's  meridian 
altitude  is  73°.  24'.  13",  its  declination  south  being  16°.  36'.  47". 

4.  If  Q  represent  the  length  of  a  quadrant,  whose  radius  is  R,  and 

the  force  vary  —  0  ,  the  time  of  descent  half  way  to  the  centre  of  force  : 

the  time  through  the  remaining  half  ::  Q  +  R  :  Q  -  R.     Required  a  proof. 

5.  P  and  W  represent  two  weights  hung  over  a  fixed  pulley  ;  supposing 
P  to  descend,  what  space  will  it  describe  in  t",  the  inertia  of  the  pulley 
being  taken  into  the  account? 

6.  If  a  pendulum,  whose  length  is  40  inches  would  oscillate  in  1"  at 
the  pole  of  a  sphere,  the  radius  of  which  is  4000  miles  ;  what  must  be  the 
time  of  rotation  round  its  axis,  that  the  same  pendulum  at  the  equator 
may  oscillate  twice  in  3"  ? 

7.  A  given  cone  is  immersed  in  water  with  its  vertex  downward  ; 


204 


THE   MATHEMATICAL   TRIPOS. 


what  part  of  the  axis  will  be  immersed,  if  the  specific  gravity  of  the  fluid  : 
that  of  the  cone  ::  8  :  1  ? 

8.  The  axis  of  a  wheel  and  axle  is  placed  in  a  horizontal  position, 
and  a  weight  y,  which  is  applied  to  the  circumference  of  the  axle,  is  raised 
by  the  application  of  a  given  moving  force  p  applied  to  the  circumference 
of  the  wheel;  given  the  radii  of  the  wheel  and  axle,  it  is  required  to 
assign  the  quantity  y,  when  the  moment  generated  in  it  in  a  given  time 
is  a  maximum,  the  inertia  of  the  wheel  and  axle  not  being  considered. 

9.  Would  Venus  ever  appear  retrograde  according  to  the  Tychonic 
system? 

10.  A  perfectly  elastic  ball  begins  to  fall 
from  a  given  distance  SA  in  a  right  line 
towards  the  centre  of  force  S,  the  force  vary- 
ing =-^;  in  its  descent,  it  impinges  upon  a 


S 


hard  plane  OP  inclined  to  SA  at  a  given 
angle,  and  after  describing  a  certain  curve 
comes  to  the  plane  on  the  other  side,  and  is 
then  reflected  to  the  center  ;  find  the  nature 
of  this  curve;  and  determine  the  whole  time 
of  descent  to  the  center  S  in  terms  of  the 
periodic  time  of  a  body  revolving  in  a  circle 
at  the  distance  SA. 

11.  Let  parallel  rays  be  refracted  through 
two  contiguous  double  convex  lenses;  find 
the  focal  length  on  the  supposition  that  the 

radii  of  all  the  surfaces  are  equal,  and  the  sine  of  incidence  :  sine  of 

refraction  ::  5  :  4. 

12.  Given  the  latitude  of  the  place  and  the  declination  of  the  sun, 
the  former  being  less  than  the  latter  ;  to  find  at  what  time  of  the  day  the 
shadow  of  a  stick  would  be  stationary,  and  how  far  it  would  afterwards 
recede  on  the  horizontal  plane. 

13.  Transform  the  equation  xn  -  pxn-1  +  qxn~2  -  &c.  =  0    into    one, 
whose  roots  are  the  reciprocals  of  the  sum  of  every  n  -  1  roots  of  the 
original  equation. 

14.  A  body  descends  down  the  cycloidal  arc  AM,  the  base  AL  being 
parallel  to  the  horizon  and  M  the  lowest  point  of  the  cycloid;  determine 
that  point  where  its  velocity  in  a  direction  perpendicular  to  the  horizon 
is  a  maximum. 

15.  Construct  the  equation  a?y  -xzy-as  =  0. 

16.  Compare  the  time  of  descent  to  the  center  in  the  logarithmic 
spiral  with  the  periodic  time  in  a  circle,  whose  radius  is  equal  to  the 
distance  from  which  the  body  is  projected  downward. 

17.  Given  the  difference  of  altitudes  of  two  stars,  which  are  upon 
the  meridian  at  the  same  time,  and  their  difference  of  altitudes  and 
difference  of  azimuths  an  hour  afterwards,  to  find  the  latitude  of  the 
place. 

18.  A  person's  face  in  a  reflecting  concave  decreases  to  the  principal 
focus,  and  then  increases  in  going  from  it.  —  Kequired  a  demonstration. 

19.  Prove  that  the  mean  quantity  of  the  disturbing  force  of  S  upon 
P,  in  the  66th  proposition  of  Newton,  during  one  revolution  of  P  round  T, 
is  ablatitious,  and  equal  to  half  the  mean  addititious  force. 


THE  PROBLEM  PAPERS  SET  IN  1802.       205 

20.  The  time  of  the  sun's  rising  is  the  tune  which  elapses  between 
the  appulse  of  the  upper  and  under  limb  of  the  sun's  disc  to  the  horizon ; 
given  the  sun's  apparent  diameter  and  the  latitude  of  the  place,  it  iB 
required  to  determine  the  declination,  when  this  time  is  a  minimum. 

21.  Through  a  given  point  situate  between  two  right  lines  given  in 
position,  to  draw  a  third  line  cutting  them  in  such  a  manner,  that  the 
rectangle  under  the  parts  intercepted  between  the  point  and  the  two  lines 
may  be  a  minimum. 

22.  Let  a  spherical  body  descend  in  a  fluid  from  rest ;  having  given 
the  diameter  of  the  sphere,  and  its  specific  gravity  with  reference  to  that 
of  the  fluid,  it  is  required  to  assign  the  velocity  of  the  sphere  at  any 
given  point  of  the  space  described. 

23.  The  distance  of  the  centre  of  gravity  from  the  vertex  of  a  solid 
formed  by  the  revolution  of  a  curved  surface  is  f  of  its  axis. — Determine 
the  nature  of  the  generating  curve. 

24.  Suppose  a  given  cylindrical  vessel  filled  with  water  to  revolve 
with  a  given  angular  velocity  round  its  axis. — Required  the  quantity 
contained  in  the  cylinder,  when  the  water  and  cylinder  are  relatively  at 
rest. 

25.  Sum  the  following  series  : 

+  — — — : — „  +- — -t — =- -n  +  &c.  to  n  terms  and  ad  inf. 


1.2.3.4  '  2.3.4.5     3.4.5. 


1  ,  .  , 

x  =s  +^— ~ — A  x  ^  +  K-   — *  x  TTA  +  &o.  ad  mf. 


1  .  2  .  3      22     2  .  3  .  4 


26.     Given  the  fluent      (a  +  czn)m  x  «P»+»-I  2 
to  find  the  fluent  (a  +  czn)m+1  x  2*™-1  2. 


,  22 

Required  also  fluent  -  *—  r  -  ;    and  of  -  --  ,   6   being  a  whole 

X"  1  +7712 

positive  number. 

TUESDAY   MORNING   PROBLEMS.—  MR.   DEALTRY. 

First  and  second  classes. 

1.  Inscribe  the  greatest  cone  in  a  given  spheroid. 

2.  A  parabolic  surface  is  immersed  vertically  in  a  fluid,  whose  density 
increases  as  the  depth,  with  its  base  contiguous  to  the  surface  of  the 
fluid  ;  find  upon  which  of  the  ordinates  to  the  axis  there  is  the  greatest 
pressure. 

3.  Solve  the  equation  a:3  -  pxz  +  qx  -  r  =  0,  whose  roots  are  in  geometric 
progression. 

4.  Suppose  the  reflecting  curve  to  be  a  circular  arc,  and  the  focus  of 
incident  rays  in  the  circumference  of  the  circle,  to  find  the  nature  of  the 
caustic. 

5.  If  the  sine  of  incidence  :  sine  of  refraction  ::  m  :  n,  required  the 
focal  length  of  a  hemisphere,  the  rays  falling  first  on  the  convex  side. 

6.  If  the  subtangent  of  a  logarithmic  curve  be  equal  to  the  sub- 


206  THE   MATHEMATICAL   TRIPOS. 

tangent  of  the  reciprocal  spiral,  prove  that  the  arc  intercepted  between 
any  two  rays  in  the  spiral  is  equal  to  the  arc  intercepted  between  any  two 
ordinates  of  the  curve  respectively  equal  to  the  former. 

7.  In  what  direction  must  a  body  be  projected  from  the  top  of  a  given 
tower  with  a  given  velocity,  so  that  it  may  fall  upon  the  horizontal  plane 
at  the  greatest  distance  possible  from  the  bottom  of  the  tower? 

8.  Draw  an  asymptote  to  the  elliptic  spiral. 

9.  If  water  or  any  fluid  ascends   and  descends  with   a  reciprocal 
motion  in  the  legs  of  a  cylindrical  canal  inclined  at  any  angle,  to  find 
the  length  of  a  pendulum  which  will  vibrate  in  the  same  time  with  the 
fluid.  

10.  Find  the  fluent  vxx,  where  v=hyp.  log.  (x  -i  Jx*  •+  a2). 

11.  The  centrifugal  force  at  the  equator  arising  from  the  rotation  of 
the  earth  round  its  axis  :  the  centrifugal  force  in  any  parallel  of  latitude  :: 
(rad.)2  :  (sine.)2  of  the  co-latitude. — Eequired  a  proof. 

12.  Given  the  latitudes  of  two  places  together  with  their  difference 
of  longitudes,  to  find  the  declination  of  the  sun,  when  it  sets  to  the  two 
places  at  the  same  time. 

13.  Required  the  equation  to  a  curve,  whose  subtangent  is  equal  to  n 
times  its  abscissa. 

14.  If  the  force  vary       +1 ,  how  far  must  a  body  fall  externally  to 

acquire  the  velocity  in  any  curve,  whose  chord  of  curvature  at  the  point 
of  projection  is  c?  and  apply  the  expression  to  the  parabola  and  logarith- 
mic spiral. 

TUESDAY  AFTERNOON  PROBLEMS.— MR.  PALMER. 

Third  and  fourth  classes. 

1.  Find  the  value  of  £123333,  &c.  (sic) 

2.  Determine  geometrically  a  mean  proportional  between  the  sum 
and  difference  of  two  given  straight  lines. 

3.  What  is  the  general  form  of  parallelograms,  whose  diameters  cut 
each  other  at  right  angles? 

4.  Investigate  the  area  of  a  circle,  whose  diameter  is  unity;    and 
prove  that  the  areas  of  different  circles  are  in  a  duplicate  ratio  of  their 
diameters. 

5.  Divide   a   given   line   into   two   parts,   such    that   their  product 
multiplied  by  their  difference  may  be  a  maximum. 

6.  Prove  that  in  any  curve  the  velocity  :  velocity  in  a  circle  at  the 
same  distance  (SP)  ::  ^/ chord  of  curvature  :  <J%SP 

7.  A  body  projected  from  one  extremity  of  the  diameter  of  a  circle,  at 
an  angle  of  45°,  strikes  a  marked  place  in  the  center.     Eequired  the 
velocity  of  projection  and  greatest  altitude. 

a3 

8.  Find  the  area  of  a  curve  whose  equation  is  y  =  -^ -z . 

9.  In  how  many  years  will  the  interest  due  upon  £100  be  equal  to 
the  principal,  allowing  compound  interest  ? 

10.  Admitting  the  periods  of  the  different  planets  to  be  in  a  sesqui- 


THE  PROBLEM  PAPERS  SET  IN  1802.       207 

plicate  ratio  of  the  principal  axes  of  their  orbits,  shew  that  they  are 
attracted  towards  the-  sun  by  forces  reciprocally  proportional  to  the 
squares  of  their  several  distances  from  it. 

11.  Prove  that  in  the  course  of  the  year  the  sun  is  as  long  above  the 
horizon  of  any  place  as  he  is  below  it. 

12.  Determine  the  limits  within  which  an  eclipse  of  the  sun  or  moon 
may  be  expected  ;  and  shew  what  is  the  greatest  number  of  both  which 
can  happen  in  one  year. 

13.  Prove  that  the  time  in  which  any  regular  vessel  will  freely  empty 
itself  :  time  in  which  a  body  will  freely  fall  down  twice  its  height  ::  area 
of  base  :  area  of  orifice. 

14.  Find  the  fluents  of     XX          XX 


15.  Find  the  principal  focus  of  a  lens  ;  and  shew  how  an  object  may 
be  placed  before  a  double  convex  lens,  that  its  image  may  be  inverted  and 
magnified  so  as  to  be  twice  as  great  as  the  object. 

16.  Prove  that  Cardan's  rule  fails  unless  two  roots  of  the  proposed 
cubic  be  impossible  ;  and  determine  whether  that  rule  be  applicable  to  the 
equation  x3  -  237x  -  884  =  0. 

17.  Deduce  Newton's  general  expression  in  Sect.  9,  for  the  force  in 
the  moveable  orbit. 

18.  Define  logarithms,  and  explain  their  use;  also,  prove  that 


19.  Explain  the  different  kinds  of  parallax  ;  and  shew  from  the  want 
of  parallax  in  the  fixed  stars,  that  their  distance  from  the  earth  bears  no 
finite  ratio  to  that  of  the  sun. 


TUESDAY  AFTERNOON  PROBLEMS.— MR.  PALMER. 

Fifth  and  sixth  classes. 

1.  How  many  yards  of  cloth,  worth  3s.  7%d.  per  yard,  must  be  given 
in  exchange  for  935£  yards,  worth  18s.  l^d.  per  yard? 

2.  Find  the  interest  of  £873.  15s.  Od.  for  2£  years  at  4£  per  cent. 

3.  Prove  that  the  diameters  of  a  square  bisect  each  other  at  right 


4.  Prove  the  opposite  angles  of  a  quadrilateral  figure  inscribed  in  a 
circle  equal  to  two  right  angles. 

5.  Prove  that  if  A  oc  B  when  C  is  given,  and  A  oc  C  when  B  is  given, 
when  neither  B  nor  C  is  given,  A  x  BC. 

6.  Prove  radius  a  mean  proportional  between  tangent  and  cotangent; 
and  that  sine  x  cosine  oc  (sine)2  of  twice  the  angle. 

7.  Given  the  sine  of  an  angle,  to  find  the  sine  of  twice  that  angle. 

8.  Prove  that  in  the  parabola  (ordinate)2  =  abscissa  x  parameter. 

9.  Extract  the  square  root  of  a3  -  x3. 

10.  Solve  the  equation  3x2  -  19x  + 16  =  0. 

11.  Prove  that  motion  when  estimated  in  a  given  direction  is  not 
increased  by  resolution. 

12.  Find  the  ratio  of  P  :  W  when  every  string  in  a  system  of  pullies 
is  fastened  to  the  weight. 


208  THE  MATHEMATICAL   TRIPOS. 

13.  Prove  that  time  of  oscillation  a  *  Gngt    . 

^/force 

14.  Prove  that  when  a  fluid  passes  through  pipes  kept  constantly 
full,  velocity  <x  inversely  as  area  of  section. 

15.  Define  the  centre  of  a  lens;  and  find  the  centre  of  a  meniscus. 

16.  Find  the  fluxion  of  Jo?  +  x*  -  Ja2  -  x\ 

17.  Prove  elevation  of  the  equator  above  the  horizon  =  co-latitude. 

18.  Prove  that  sagita  a  (arc)2. 

19.  Prove  that  in  the  same  orbit  velocity  oc  inversely  as  perp. 

TUESDAY  EVENING  PROBLEMS.—  MR.  PALMER. 

First,  second,  third,  and  fourth  classes. 


1.  When  £100  stock  may  be  purchased  in  the  3  per  cents,  for 

at  what  rate  may  the  same  quantity  of  stock  be  purchased  in  the  5  per 
cents,  with  equal  advantage  ? 

2.  A  ball  of  wood  being  balanced  in  air  by  the  same  weight  of  iron, 
how  will  the  equilibrium  be   affected  when  the  bodies  are  weighed  in 
vacuo  ?  and  by  what  weight  of  wood,  properly  disposed,  may  the  equi- 
librium be  restored  ? 

3.  Investigate  the  value  of  the  circumference  of  a  circle  whose  radius 
is  unity. 

4.  Compare  the   areas  of  the  parabolas  described  by  two  bodies 
projected   together  from  the   same  point,  and  with  the  same  velocity, 
towards  a  mark  situated  in  an  horizontal  plane,  the  angles  of  elevation 
being  to  each  other  ::  2  :  1. 

5.  Prove  the  rule  for  finding  the  quadratic  divisors  of  any  equation  ; 
and  apply  it  to  the  equation  z4-  17x3  +  88x2-  172#  +  112  =  0. 

6.  On  what  point  of  the  compass  does  the  sun  rise  to  those  who  live 
under  the  equinoctial,  when  he  is  in  the  northern  tropic? 

7.  How  many  equal  circles  may  be  placed  around  another  circle  of 
the  same  diameter,  touching  each  other  and  the  interior  circle? 

8.  Determine  the  resistance  of  the  medium  in  which  a  body  by  an 
uniform  gravity  may  describe  a  parabolic  orbit  ? 

9.  Prove  that  a  body  moving  in  the  reciprocal  spiral,  approaches  or 
leaves  the  centre  uniformly. 

10.  Find  the  velocity  and  time  of  flight  of  a  body  projected  from  one 
extremity  of  the  base  of  an  equilateral  triangle,  and  in  the  direction  of  the 
side  adjacent  to  that  extremity  towards  an  object  placed  in  the  other 
extremity  of  the  base. 

11.  Define  similar  curves  ;  and  prove  that  conterminous  arcs  of  such 
curves  have  their  chords  of  curvature  at  the  point  of  contact  in  a  given 
ratio. 

12.  Compare  the  time  of  a  revolution  about  the  center  of  a  given 
ellipse,  with  that  about  its  focus. 

13.  Find   the   attraction    of  a   corpuscle   placed   in   the   axis   of  a 
cylindrical   superficies,  whose   particles  attract  in  an  inverse  duplicate 
ratio  of  the  distance. 

14.  Prove  that  if  the  center  of  oscillation  of  a  pendulum  be  made 


THE   MATHEMATICAL   TRIPOS.  209 

the  point  of  suspension,   the  former  point  of  suspension  becomes  the 
center  of  oscillation. 

15.    Determine  the  content  of  the  solid  generated  by  a  semicircle 
revolving  about  a  tangent  parallel  to  its  base. 


16.     Find  the  fluents  of 


17.  Sum  the  series  1  -  -3  +  -5  -  &?  +  &c.  ad  inf.  and  also  to  n  terms. 

i— 5 +  o  +  377+&c- to  n  terms-  rs +  377 +  67u+&c- ad  inf- 

18.  Required  the  sun's  place  in  the  ecliptic,  when  the  increment  of 
his  declination  is  equal  to  that  of  his  right  ascension. 

19.  Prove  that  the  force  by  which  a  body  may  describe  a  curve, 
whose  ordinates  are  parallel,  is  proportioned  to  ±y;  and  determine  the 
quantity  q  such  that  force  =  q  x  ±j/. 

20.  Compare  the  times  in  which  a  cylinder,  whose  axis  is  parallel  to 
the  horizon,  will  discharge  the  first  and  last  half  of  its  content  through 
an  orifice  in  its  lowest  section. 

21.  Prove  that   the  image  of  a   straight  line  immersed  in  water 
appears  concave  to  an  eye  placed  anywhere  between  the  extremities  of 
the  line. 

22.  At  what  distance  from  the  earth  would  the  apparent  brightness 
of  the  moon  be  equal  to  that  of  Saturn  and  his  ring  together,  supposing 
the  apparent  brightness  of  Saturn  to  that  of  his  ring  ::  2  :  1? 

No  problems  were  ever  set  to  the  seventh  and  eighth 
classes,  which  contained  the  poll  men.  None  of  the  book- 
work  papers  of  this  time  are  now  extant,  but  it  is  believed 
that  they  contained  no  riders.  It  will  be  seen  from  the  above 
specimens  that  many  of  the  so-called  problems  were  really 
pieces  of  book-work  or  easy  riders :  it  must  however  be  re- 
membered that  the  text-books  then  in  circulation  were  inferior 
and  incomplete  as  compared  with  modern  ones. 

A  few  minor  changes  in  the  senate-house  examinations 
were  made  in  the  following  years.  In  1808  a  fifth  day  was 
added  to  the  examination.  Of  the  five  days  thus  given  up  to 
it,  three  were  devoted  to  mathematics,  one  to  logic,  philosophy, 
and  religion,  and  one  to  the  arrangement  of  the  brackets. 
Apart  from  the  evening  paper,  the  examination  on  each  of  the 
first  three  days  lasted  six  hours.  Of  these  eighteen  hours 
eleven  were  assigned  to  book-work  and  seven  to  problems. 
In  1800  the  first  four  classes  had  been  allowed  to  take  the 
B.  14 


210  THE   MATHEMATICAL   TRIPOS. 

problem  papers,  and  in  1818  they  were  opened  to  all  the  candi- 
dates for  honours,  i.e.  the  first  six  classes,  and  set  from  6  to 
10  in  the  evening :  the  hours  of  examination  being  thus 
extended  to  ten  a  day. 

Some  observations  on  the  tripos  examination  of  1806  will 
be  found  in  the  letter  by  Sir  Frederick  Pollock  to  which  refer- 
ence has  been  already  made  (see  p.  112).  A  letter  from 
Whewell,  dated  January  19,  1816,  describes  his  examination 
in  the  senate-house1.  It  was  at  this  time  that  the  character 
of  the  examination  was  changing  and  that  the  differential 
notation  and  analysis  were  being  introduced  in  the  place  of 
fluxions  and  geometry.  The  remarks  of  Peacock  and  others  on 
this  subject  have  been  already  quoted  (see  chapter  vn.).  Whewell 
was  moderator  in  1820,  and  in  a  letter  to  his  sister  dated 
Jan.  20,  1820,  he  describes  the  examination.  There  is  nothing 
of  any  historical  interest  in  his  account,  save  that  it  shews 
that  many  of  the  questions  were  still  dictated.  The  letter  is 
as  follows2. 

The  examination  in  the  senate-house  begins  to-morrow,  and  is  rather 
close  work  while  it  lasts.  We  are  employed  from  seven  in  the  morning 
till  five  in  the  evening  in  giving  out  questions  and  receiving  written 
answers  to  them ;  and  when  that  is  over,  we  have  to  read  over  all  the 
papers  which  we  have  received  in  the  course  of  the  day,  to  determine  who 
have  done  best,  which  is  a  business  that  in  numerous  years  has  often 
kept  the  examiners  up  the  half  of  every  night ;  but  this  year  is  not  par- 
ticularly numerous.  In  addition  to  all  this,  the  examination  is  conducted 
in  a  building  which  happens  to  be  a  very  beautiful  one,  with  a  marble 
floor  and  a  highly  ornamented  ceiling ;  and  as  it  is  on  the  model  of  a 
Grecian  temple,  and  as  temples  had  no  chimneys,  and  as  a  stove  or  a  fire 
of  any  kind  might  disfigure  the  building,  we  are  obliged  to  take  the 
weather  as  it  happens  to  be,  and  when  it  is  cold  we  have  the  full  benefit 
of  it — which  is  likely  to  be  the  case  this  year.  However,  it  is  only  a  few 
days,  and  we  have  done  with  it. 

In  the  decade  from  1820  to  1830  a  powerful  party  arose  in 
the  university,  as  in  the  country,  which  desired  to  overhaul  all 

1  See  p.  20  of  Douglas's  Life  of  Whewell,  London,  1881. 

2  See  p.  56  of  Douglas's  Life  of  Whewell,  London,  1881. 


THE   MATHEMATICAL   TRIPOS.  211 

existing  methods  and  regulations.  Among  other  changes  the 
Previous  Examination,  or  Little-Go,  was  established  in  1824, 
for  students  in  their  second  year ;  a  reform  which  was  urgently 
needed,  as  till  then  the  university  required  nothing  from  its 
undergraduate  members  until  they  had  entered  their  third 
year  of  residence.  The  power  of  granting  honorary  op  time 
degrees,  which  had  already  fallen  into  abeyance,  was  abolished. 
At  the  same  time  the  classical  tripos  was  founded  for  those 
who  had  already  taken  honours  in  mathematics,  and  the  plan 
of  the  senate-house  examination  was  re-arranged.  Henceforth 
it  is  known  as  the  mathematical  tripos. 

From  this  time  onwards  the  examination  was  conducted  in 
each  year  by  four  examiners,  namely,  the  two  moderators  and 
the  two  examiners,  the  moderators  of  one  year  becoming  as  a 
matter  of  course  the  examiners  of  the  next.  Thus  of  the  four 
examiners  in  each  year,  two  had  taken  part  in  the  examination 
of  the  previous  year.  The  continuity  of  the  examination  was 
well  kept  up  by  this  arrangement ;  but  it  had  the  effect  of 
causing  its  traditions  to  be  somewhat  punctiliously  observed, 
the  papers  of  each  year  being,  as  regards  the  subjects  included, 
exact  counterparts  of  the  corresponding  papers  of  the  previous 
year. 

By  regulations1  which  were  confirmed  by  the  senate  on 
November  13,  1827,  and  came  into  operation  in  January  1828, 
another  day  was  added,  so  that  the  examination  in  mathe- 
matics extended  over  four  days,  exclusive  of  the  day  of  arrang- 
ing the  brackets ;  the  number  of  hours  of  examination  was 
twenty-three,  of  which  seven  were  assigned  to  problems.  On 
the  first  two  days  all  the  candidates  had  the  same  questions 
proposed  to  them,  inclusive  of  the  evening  problems,  and  the 
examination  on  those  days  excluded  the  higher  and  more 
difficult  parts  of  mathematics,  in  order,  in  the  words  of  the 
report,  "that  the  candidates  for  honours  may  not  be  induced 

1  Most  of  the  analysis  here  given  of  the  regulations  of  1827,  1832, 
and  1848  is  taken  from  Dr  Glaisher's  inaugural  address  to  the  London 
Mathematical  Society  in  1888. 

14—2 


212  THE  MATHEMATICAL   TKIPOS. 

to  pursue  the  more  abstruse  and  profound  mathematics,  to 
the  neglect  of  more  elementary  knowledge."  Accordingly, 
only  such  questions  as  could  be  solved  without  the  aid  of 
the  differential  calculus  were  set  on  the  first  day,  and  those  set 
on  the  second  day  involved  only  its  elementary  applications. 
The  classes  were  reduced  to  four,  determined  as  before  by  the 
exercises  in  the  schools.  The  regulations  of  1827  are  especially 
important  because  they  first  prescribed  that  all  the  papers, 
should  be  printed.  They  are  also  noticeable  as  being  the  last 
which  gave  the  examiners  power  to  ask  vivd  voce  questions. 
After  recommending  that  there  be  not  contained  in  any  paper 
more  questions  than  well-prepared  students  have  generally 
been  found  able  to  answer  within  the  time  allowed  for  the 
paper,  the  report  proceeds  "but  if  any  candidate  shall,  before 
the  end  of  the  time,  have  answered  all  the  questions  in  the 
paper,  the  examiners  may  at  their  discretion  propose  addi- 
tional questions  vivd  voce." 

At  the  same  time  as  these  changes  were  made  (i.e.  in  1828) 
the  examination  for  the  poll  degree  was  separated  from  the 
tripos  and  placed  in  the  following  week,  with  different  sets  of 
papers  and  a  different  schedule  of  subjects.  It  was,  however, 
still  nominally  considered  as  forming  part  of  the  senate-house 
examination.  It  is  perhaps  worthy  of  remark  that  this  fiction 
was  maintained  till  1858,  and  those  who  obtained  a  poll  degree 
were  arranged  according  to  merit  into  four  classes,  viz.,  a 
fourth,  fifth,  sixth,  and  seventh,  as  if  in  continuation  of  the 
junior  optimes  or  third  class  of  the  tripos.  Till  1850  all 
members  of  the  university  who  took  the  degree  of  bachelor 
of  arts  were  expected  to  pass  what  we  now  call  the  mathe- 
matical tripos,  but  which  was  then  the  only  examination  held 
for  that  degree.  The  year  1828  therefore  shews  us  the 
examination  dividing  into  two  distinct  parts.  In  1850  the 
classical  tripos  was  made  independent  of  the  mathematical 
tripos,  and  thus  provided  another  and  separate  avenue  to  a 
degree.  In  1858  the  poll-examination  was  finally  separated 
from  the  other  part  of  the  mathematical  tripos,  and  provided 


THE  MATHEMATICAL  TRIPOS.  213 

a  third  way  of  obtaining  the  degree.  Since  then  numerous 
other  ways  of  obtaining  the  degree  have  been  established,  and 
it  is  now  possible  to  get  it  by  shewing  proficiency  in  very 
special  or  even  technical  subjects.  I  may  just  add  in  pass- 
ing that  the  examination  usually  termed  "the  general"  is 
historically  the  survival  of  the  old  senate-house  examination 
for  the  poll  men;  and  that  in  1852  a  third  examination,  at 
first  called  "the  professors's  examinations,"  and  now  known  as 
"the  specials,"  was  instituted  for  all  poll  men  to  take  at  the 
end  of  their  third  year. 

New  regulations  concerning  the  mathematical  tripos  were 
confirmed  by  the  senate  on  April  6,  1832,  and  took  effect  in 
1833.  The  commencement  of  the  examination  was  placed  a 
day  earlier,  the  duration  was  extended  to  five  days,  and  the 
number  of  hours  of  examination  on  each  day  was  fixed  at  five 
and  a-half.  Twenty  hours  were  assigned  to  book-work,  and 
seven  and  a-half  to  problems.  The  examination  on  the  first  day 
was  confined  to  subjects  that  did  not  require  the  differential 
calculus,  and  only  the  simplest  applications  of  the  calculus 
were  permitted  on  the  second  and  third  days.  During  the 
first  four  days  of  the  examination  the  same  papers  were  set  to 
.all  the  candidates  alike,  but  on  the  fifth  day  the  examination 
was  conducted  according  to  classes.  No  reference  was  made 
to  vivd  voce  questions,  and  the  preliminary  classification  of  the 
brackets  only  survived  in  a  permission  to  use  it  if  it  were 
found  necessary. 

The  tripos  of  1836  is  said  to  have  been  the  earliest  one  in 
which  all  the  papers  were  marked1.  In  previous  years  the 
examiners  had  partly  relied  on  their  impression  of  the  answers 
given. 

The  regulations  of  1832  were  superseded  by  a  new  system, 
which  passed  the  senate  on  June  2,  1838,  and  came  into 
•operation  in  January  1839.  By  these  new  rules  the  examina- 
tion lasted  for  a  week.  It  began  on  the  Wednesday  week 

1  This  comes  to  me  on  the  authority  of  the  late  Samuel  Earnshaw, 
the  senior  moderator  of  that  year. 


214  THE  MATHEMATICAL  TRIPOS. 

preceding  the  first  Monday  in  the  Lent  term,  and  ended  on 
the  following  Tuesday  night;  and  continued  every  day  from 
nine  to  half-past  eleven  in  the  morning,  and  from  one  to  four 
in  the  afternoon.  The  list  was  published  on  the  Friday  week 
following.  Of  the  thirty-three  hours  of  examination,  eight 
and  a- half  were  assigned  to  problems.  Throughout  the  whole 
examination  the  same  papers  were  set  to  all  the  candidates. 
The  permissive  rule  relating  to  the  re-examination  of  the 
candidates  (a  relic  of  the  brackets)  was  retained  in  these 
regulations  in  the  same  form  as  in  those  of  1832.  The 
examination  was  for  the  future  confined  to  mathematics, 
and  "religion"  and  "philosophy"  henceforth  disappear  from  the 
schedule  of  subjects.  The  former  of  these  was,  it  is  true, 
temporarily  reintroduced  in  1846  in  the  form  of  papers  on  the 
New  Testament,  Paley,  and  Ecclesiastical  history,  but  as  in 
settling  the  final  list  no  account  was  taken  of  the  marks  ob- 
tained in  these  papers  they  were  generally  neglected.  They 
were  accordingly  again  struck  out  by  a  grace  of  the  senate 
in  1855,  and  have  never  been  reinstated. 

These  regulations  contain  no  allusion  to  the  classes,  and  it 
was  no  doubt  in  accordance  with  the  spirit  of  these  changes 
that  the  acts  in  the  schools  should  be  abolished,  but  they  seem 
to  have  been  discontinued  by  the  moderators  of  1839  on  their 
own  authority  (see  p.  183). 

A  few  years  later  the  scheme  of  the  examination  was  again 
reconstructed  by  regulations  which  came  into  effect  in  1848. 
The  examination,  as  thus  constituted,  underwent  no  further 
alteration  till  1873,  and  the  first  three  days  remain  practically 
unchanged  at  the  present  time.  The  duration  of  the  exami- 
nation was  extended  from  six  to  eight  days,  the  first  three 
days  being  assigned  to  the  elementary  and  the  last  five  to  the 
higher  parts  of  mathematics.  After  the  first  three  days  there 
was  an  interval  of  a  few  days  at  the  end  of  which  the  moderators 
and  examiners  issued  a  list  of  those  who  had  so  acquitted  them- 
selves as  to  deserve  mathematical  honours.  Only  those  whose 
names  were  contained  in  this  list  were  admitted  to  the  last 


THE  MATHEMATICAL   TRIPOS.  215 

five  days  of  the  examination.  After  the  conclusion  of  the 
examination  the  moderators  and  examiners,  taking  into  account 
the  whole  eight  days,  brought  out  the  list  arranged  in  order  of 
merit.  No  provision  was  made  for  any  re-arrangement  of  this 
list  corresponding  to  the  examination  of  the  brackets,  which, 
though  forming  part  of  the  previous  scheme,  had  been  dis- 
continued for  some  time.  An  important  part  of  the  new 
regulations  was  the  limitation,  by  a  schedule,  of  the  subjects 
of  examination  in  the  first  three  days,  and  of  the  manner  in 
which  the  questions  were  to  be  answered;  the  methods  of 
analytical  geometry  and  differential  calculus  being  excluded. 
In  all  the  subjects  contained  in  this  schedule  examples  and 
questions  arising  directly  out  of  the  propositions  were  to  be 
introduced  into  the  papers,  in  addition  to  the  propositions 
themselves.  Taking  the  whole  eight  days,  the  examination 
lasted  forty-fouV  and  a  half  hours,  twelve  hours  of  which  were 
devoted  to  problems. 

In  the  same  year  as  these  regulations  came  into  force,  the 
Board  of  mathematical  studies  (consisting  of  the  mathematical 
professors,  and  the  moderators  and  examiners  for  the  current 
and  two  preceding  years)  was  constituted  by  the  senate.  In 
May  1849  they  issued  a  report  in  which,  after  giving  a 
short  review  of  the  past  and  existing  state  of  mathematical 
studies  in  the  university,  they  recommended  that,  consider- 
ing the  great  number  of  subjects  occupying  the  attention 
of  the  candidates,  and  the  doubt  existing  as  to  the  range 
of  subjects  from  which  questions  might  be  proposed,  the 
mathematical  theories  of  electricity,  magnetism,  and  heat 
should  not  be  admitted  as  subjects  of  examination.  In  the 
following  year  they  issued  a  second  report,  in  which  they 
recommended  the  omission  of  elliptic  integrals,  Laplace's  co- 
efficients, capillary  attraction,  and  the  figure  of  the  earth  con- 
sidered as  heterogeneous,  as  well  as  a  definite  limitation  of  the 
questions  in  lunar  and  planetary  theory.  In  making  these 
recommendations,  the  Board  stated  that  they  were  only  giving 
expression  to  what  had  become  the  practice  in  the  examina- 


216  THE  MATHEMATICAL  TRIPOS. 

tion,  and  were  merely  putting  before  the  candidates  such 
results  as  might  have  been  deduced  by  any  one  from  a  study  of 
the  senate-house  papers  of  the  preceding  years.  The  Board 
also  recommended  that  the  papers  containing  book-work  and 
riders  should  be  shortened. 

From  that  time  forward  their  minutes  supply  a  permanent 
record  of  the  changes  gradually  introduced  into  the  tripos. 
Those  changes  lie  beyond  the  limits  of  this  book. 

I  may  just,  in  passing,  mention  a  curious  attempt  which 
was  made  in  1854  to  assist  candidates  in  judging  of  the  relative 
difficulty  of  the  questions  asked,  by  informing  them  of  the 
marks  assigned  to  each  question.  The  marks  for  the  book-work 
and  rider  of  each  question  were  printed  on  a  little  slip  of 
paper  which  was  given  to  the  candidates  at  the  same  time  as 
the  examination  paper1. 

It  is  not  unusual  to  hear  the  remark  that  the  scheme  of 
the  tripos  from  1839  to  1873  was  framed  so  as  to  discourage 
those  who  wished  to  apply  mathematics  to  physical  questions ; 
but  that  opinion  is,  I  think,  framed  on  a  misunderstanding. 
The  university  insisted  that  her  mathematical  graduates  should 
have  a  thorough  knowledge  of  all  the  elementary  subjects,  and 
left  to  them  the  particular  sciences  to  which  they  might  (if 
they  felt  inclined)  apply  it.  It  only  needs  a  glance  at  the 
tripos  lists  to  see  that  this  course  was  in  no  way  prejudicial  to 
any  branch  of  mathematical  science.  Indeed  I  believe  that  if 
the  senate  had  not  been  so  anxious  to  define  exactly  what 
might  and  what  might  not  be  asked,  but  had  allowed  the 
subjects  of  the  examination  to  grow  by  the  gradual  introduction 
of  questions  from  the  more  recent  applications  of  mathematics, 
there  is  no  reason  why  the  regulations  of  1841  or  of  1848 
should  not  meet  all  the  requirements  of  the  present  time. 
Under  those  regulations  the  Cambridge  graduate  who  devoted 
himself  to  mathematical  research  possessed  a  great  advantage 

1  I  mention  the  fact  rather  because  these  things  are  rapidly  forgotten 
than  because  it  is  of  any  intrinsic  value.  I  possess  a  complete  set  of 
slips  which  came  to  me  from  Dr  Todhunter. 


ORIGIN   OF   THE   TERM  TRIPOS.  217 

over  his  continental  colleagues  in  the  wider  range  of  his 
general  mathematical  knowledge.  That  advantage  has  recently 
been  abandoned,  but  on  the  other  hand  a  man  on  taking  his 
degree  is  now  a  specialist  in  some  small  part  of  one  branch  of 
the  subject.  Time  alone  can  shew  which,  is  the  better  system. 
I  myself  have  no  doubt  that  it  is  in  general  wiser  to  defer 
specialization  until  after  a  man  has  taken  his  first  degree,  but 
the  drift  of  recent  legislation  has  been  in  the  other  direction. 

The  curious  origin  of  the  term  tripos  has  been  repeatedly 
told,  and  an  account  of  it  may  fitly  close  this  chapter.  There 
were  three  principal  occasions  on  which  questionists  were 
admitted  to  the  degree  of  bachelor.  The  first  of  these  was  the 
comitia  prior  a  held  on  Ash -Wednesday  for  the  best  men  in 
the  year.  The  next  was  the  comitia  posteriora  which  was  held 
a  few  weeks  later,  and  at  which  any  student  who  had  dis- 
tinguished himself  in  the  quadragesimal  exercises  subsequent 
to  Ash- Wednesday  had  his  seniority  reserved  to  him.  Lastly, 
there  was  the  comitia  minor  a,  or  the  general  bachelor's  com- 
mencement, for  students  who  had  in  no  special  way  dis- 
tinguished themselves.  In  the  fifteenth  century  an  important 
part  in  the  ceremony  on  each  of  these  occasions  was  taken  by 
a  certain  "ould  bachilour,"  who  as  the  representative  of  the 
university  had  to  sit  upon  a  three-legged  stool  or  tripos  "before 
Mr  Proctours"  and  test  the  abilities  of  the  would-be  graduates 
by  arguing  some  question  with  the  "eldest  son,"  who  was  the 
senior  and  representative  of  them.  To  assist  the  latter  in 
what  was  generally  an  unequal  contest,  his  "father,"  that  is, 
the  officer  of  his  college  who  was  to  present  him  for  his  degree, 
was  allowed  to  come  to  his  assistance. 

The  ceremony  was  a  serious  one,  and  had  a  certain  religious 
character.  It  took  place  in  Great  St  Mary's  Church,  and 
marked  the  admission  of  the  student  to  a  position  with  new 
responsibilities,  while  the  season  of  Lent1  was  chosen  with  a 
view  to  bring  this  into  prominence.  The  puritan  party  ob- 

1  Grave  scandal  was  caused  at  Oxford  by  a  custom  of  giving  suppers 
after  the  quadragesimal  exercises  for  the  day  were  over,  and  this  even  in 


218  THE   MATHEMATICAL   TRIPOS. 

jected  to  the  observance  of  such  ecclesiastical  ceremonies,  and 
in  the  course  of  the  sixteenth  century  they  converted  the 
proceedings  into  a  sort  of  licensed  buffoonery.  The  part 
played  by  the  questionist  became  purely  formal.  A  serious 
debate  still  sometimes  took  place  between  the  father  of  the 
senior  questionist  and  a  regent  master,  who  represented  the 
university;  but  the  discussion  always  began  with  an  intro- 
ductory speech  by  the  bachelor,  who  came  to  be  called  Mi- 
Tripos  just  as  we  speak  of  a  judge  as  the  bench  or  of  a  rower 
as  an  oar.  Ultimately  the  tripos  was  allowed  to  say  pretty 
much  what  he  pleased,  so  long  as  it  was  not  dull  and  was 
scandalous.  The  speeches  he  delivered  or  the  verses  he 
recited  were  generally  preserved  by  the  registrary,  and  were 
known  as  the  tripos  verses :  originally  they  referred  to  the 
subjects  of  the  disputations  then  propounded.  The  earliest 
copies  now  extant  are  those  for  1575. 

The  university  officials,  to  whom  the  personal  criticisms 
in  which  the  tripos  indulged  were  by  no  means  pleasing, 
repeatedly  exhorted  him  to  remember  "while  exercising  his 
privilege  of  humour,  to  be  modest  withal."  In  1740,  says  Mr 
Mullinger1,  ''the  authorities  after  condemning  the  excessive 
license  of  the  tripos  announced  that  the  cornitia  at  Lent  would 
in  future  be  conducted  in  the  senate-house  ;  and  all  members 
of  the  university,  of  whatever  order  or  degree,  were  forbidden 
to  assail  or  mock  the  disputants  with  scurrilous  jokes  or  un- 
seemly witticisms.  About  the  year  1747-8,  the  moderators 
initiated  the  practice  of  printing  the  honour  lists  on  the  back 
of  the  sheets  containing  the  tripos-verses,  and  after  the  year 
1755  this  became  the  invariable  practice.  By  virtue  of  this 

"  the  holy  season  of  Lent."  Bachelors  detected  in  so  acting  were  liable 
to  immediate  expulsion:  but  as  a  concession  to  juvenile  weakness  the 
sophister  was  allowed  to  give  an  entertainment  in  the  previous  term 
provided  the  expenditure  did  not  exceed  sixteen-pence.  See  vol.  n. 
p.  453  of  Munimenta  academica,  by  Henry  Anstey,  in  the  Kolls  Series, 
London,  1868. 

1  Mullinger's  Cambridge,  pp.  175,  176. 


ORIGIN   OF   THE   TERM  TRIPOS.  219 

purely  arbitrary  connection  these  lists  themselves  became 
known  as  the  tripos;  and  eventually  the  examination  itself, 
of  which  they  represented  the  results,  also  became  known  by 
the  same  designation." 

A  somewhat  similar  position  at  the  comitia  majora  (or 
congregation  held  on  Commencement-day)  to  that  of  the  tripos 
on  Ash- Wednesday  was  filled  by  the  prsevaricator  or  varier, 
who  was  the  junior  M.A.  regent  of  the  previous  year,  or  his 
proxy.  But  he  never  indulged  in  as  much  license  as  the  "  ould 
bachilor,"  and  no  determined  effort  to  turn  that  ceremony  into 
a  farce  was  ever  made. 

The  tripos  and  prsevaricator  ceased  to  recite  their  speeches 
about  1750,  but  the  issue  of  the  verses  by  the  former  has  never 
been  discontinued.  At  present  these  verses  are  published  011 
the  last  day  of  the  Michaelmas  term,  and  consist  of  four  odes, 
usually  in  Latin  but  occasionally  in  Greek,  in  which  current 
events  or  topics  of  conversation  in  the  university  are  treated 
satirically  or  seriously.  They  are  written  for  the  two  proctors 
and  two  moderators  by  undergraduates  or  commencing  bachelors, 
who  are  supposed  each  to  receive  a  pair  of  white  kid  gloves  in 
recognition  of  their  labours.  Since  1859  the  two  sets,  corre- 
sponding to  the  two  days  of  admission,  have  been  printed 
together  on  the  first  three  pages  of  a  sheet  of  foolscap  paper. 
On  the  fourth  page  the  order  of  seniority  of  the  honour  men 
of  the  year  is  printed  crosswise  in  columns,  the  sheet  being 
folded  into  four  parts,  so  that  all  the  names  can  be  read  with- 
out opening  the  page  to  more  than  half  its  extent. 

Thus  gradually  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  verses  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  examination1." 

1  Wordsworth,  p.  21. 


CHAPTER  XI. 
OUTLINES  OF  THE  HISTORY  OF  THE  UNIVERSITY.1 


SECTION  1.     The  mediaeval 

SECTION  2.     The  university  from  1525  to  1858. 

MY  object  in  writing  the  foregoing  pages  was  to  trace  the 
development  of  the  study  of  mathematics  at  Cambridge  from 
the  foundation  of  the  university  to  the  year  1858.  Some 
knowledge  of  the  history,  constitution,  and  organization  of  the 
university  is  however  (in  my  opinion)  essential  to  any  who 
would  understand  the  manner  in  which  mathematics  was  intro- 
duced into  the  university  curriculum  and  the  way  in  which  it 
developed.  To  a  sketch  of  these  subjects  this  chapter  is  accord- 
ingly devoted.  I  have  made  it  somewhat  fuller  than  is  abso- 
lutely essential  for  my  purpose,  in  the  hope  that  I  may  enable 
the  reader  to  realize  the  life  of  a  student  in  former  times. 

1  The  materials  for  this  chapter  are  mainly  taken  from  the  University 
of  Cambridge  by  J.  Bass  Mullinger,  Cambridge,  (vol.  i.  to  1535),  1873, 
(vol.  ii.  to  1625),  1884;  the  Annals  of  Cambridge  by  C.  H.  Cooper,  5 
vols.,  Cambridge,  1842 — 1852;  Observations  on  the  statutes  by  George 
Peacock,  London,  1841 ;  the  collection  of  Documents  relating  to  the  uni- 
versity and  colleges  of  Cambridge,  issued  by  the  Royal  Commissioners 
in  1852 ;  and  lastly  the  Scholae  academicae  by  C.  Wordsworth,  Cambridge, 
1877.  For  the  corresponding  references  to  Oxford  I  am  mainly  indebted 
to  the  Munimenta  academica,  by  H.  Anstey,  Bolls  Series,  London,  1868, 
and  to  a  History  of  Oxford  to  1530,  by  H.  C.  M.  Lyte,  London,  1886. 
The  works  of  Peacock,  Mullinger  and  Lyte  contain  references  to  all  the 
more  important  facts. 


THE  MEDIAEVAL   UNIVERSITY.  221 

The  history  of  the  university  is  divisible  into  three  toler- 
ably distinct  periods.  The  first  commences  with  its  founda- 
tion towards  the  close  of  the  twelfth  century,  and  terminates 
with  the  royal  injunctions  of  1535.  This  was  followed  by  some 
thirty  or  forty  years  of  confusion,  but  about  the  end  of  the 
sixteenth  century  the  university  assumed  that  form  and 
character  which  continued  with  but  few  material  changes  to 
the  middle  of  this  century.  Most  of  its  members  would,  I 
think  agree  that  a  fresh  departure  in  its  development  then 
began,  the  outcome  of  which  cannot  yet  be  predicted. 

The  mediaeval  university. 

Cambridge,  like  all  the  early  mediaeval  universities,  arose 
from  a  voluntary  association  of  teachers  who  were  exercising 
their  profession  in  the  same  place.  Of  the  exact  details  of  its 
early  history  we  know  nothing ;  but  the  general  outlines  are 
as  follows. 

A  university  of  the  twelfth  or  thirteenth  century  usually 
began  in  connection  with  some  monastic  or  cathedral  school  in 
the  vicinity  of  which  lecturers  had  settled.  As  soon  as  a  few 
teachers  and  scholars  had  thus  taken  up  their  permanent 
residence  in  the  neighbourhood  they  organized  themselves  (but 
in  all  cases  quite  distinct  from  the  monastic  schools)  as  a  sort 
of  trades  union  or  guild,  partly  to  protect  themselves  from  the 
extortionate  charges  of  tradesmen  and  landlords,  partly  be- 
cause all  men  with  a  common  pursuit  were  then  accustomed  to 
form  such  unions.  Such  an  association  was  known  as  a  uni- 
versitas  magistrorum  et  scholarium.  A  universitas  scholarium, 
if  successful  in  attracting  students  and  acquiring  permanency, 
always  sought  special  legal  privileges,  such  as  the  right  of 
fixing  the  price  of  provisions  and  the  power  of  trying  legal 
actions  in  which  their  members  were  concerned.  These  pri- 
vileges generally  led  to  a  recognition,  explicit  or  implicit,  of 
the  guild  by  the  crown  as  a  studium  generale,  i.e.  a  body  with 
power  to  grant  degrees  which  conferred  a  right  of  teaching 


222     OUTLINES  OF  THE   HISTORY   OF   THE   UNIVERSITY. 

anywhere  within  the  kingdom.  The  university  was  frequently 
incorporated  at  or  about  the  same  time.  It  was  still  only  a 
local  corporation,  but  it  entered  on  its  third  and  final  stage  of 
development  when  it  obtained  recognition,  explicit  or  implicit, 
from  the  pope  (or  emperor).  This  gave  its  degrees  currency 
throughout  Christendom,  and  it  thenceforward  became  a  re- 
cognized member  of  a  body  of  closely  connected  corporations. 
Such  is  the  general  outline  of  the  history  of  a  mediaeval 
university.  In  later  times  the  title  of  university  was  confined 
to  degree-granting  bodies,  and  any  other  place  of  higher 
education  was  termed  a  studium  generale. 

The  records  and  charters  of  the  university  of  Cambridge 
were  burnt  in  1261,  in  1322,  and  again  in  1381.  We  must 
therefore  refer  to  the  analogy  of  other  universities,  and  parti- 
cularly of  Paris  (which  was  the  typical  mediaeval  university, 
and  was  taken  as  a  model  by  those  who  first  organized  Oxford 
and  Cambridge),  to  obtain  an  idea  of  its  early  history,  filling  in 
the  dates  of  the  various  steps  in  its  development  by  means  of 
allusions  thereto  in  trustworthy  authorities. 

It  seems  almost  certain  that  there  was  no  university  at 
Cambridge  in  1112,  when  the  canons  of  St  Giles's  moved  from 
the  church  of  that  name  to  their  new  priory  at  Barnwell.  It  is 
also  known  that  the  university  existed  in  its  first  stage,  (i.e. 
as  a  self-constituted  and  self-governing  community),  in  the  year 
1209,  since  several  students  from  Oxford  migrated  in  that  year 
to  the  university  of  Cambridge.  At  some  time  before  the 
latter  date,  and  probably  subsequent  to  1112,  one  or  more 
grammar-schools  were  opened  in  Cambridge,  either  under  the 
care  of  the  monks  at  Barnwell  priory,  or  of  the  conventual 
church  at  Ely,  or  possibly  of  both  authorities.  The  connection 
between  these  schools  and  the  beginning  of  the  university  has 
always  appeared  to  me  to  be  a  singularly  interesting  historical 
problem,  though  it  has  hitherto  attracted  but  little  attention. 

Most  critics  consider  that  the  university  of  Paris  arose  from 
the  audiences  that  came  together  to  hear  William  of  Cham- 
peaux  lecture  on  logic  in  1109,  or  his  pupil  Abelard  on 


THE   MEDIEVAL   UNIVERSITY.  223 

theology  some  thirty  years  later;  and  that  these  lectures  were 
delivered  with  the  sanction  of  the  chapter  of  Ste.  Genevieve. 
It  is  generally  believed  that  the  university  of  Oxford  arose  in 
a  similar  way  from  the  students  who  were  attracted  there  to 
hear  the  lectures  of  Robert  Pullen  on  theology  in  1133,  and  of 
Vacarius  on  civil  law  in  1149;  and  that  as  the  monks  of 
St  Friedeswyde's  were  probably  French,  the  lectures  were  given 
in  their  house  and  by  their  invitation.  Paris  and  Oxford  were 
important  towns,  and  not  unnaturally  became  universities. 
Cambridge,  however,  was  a  small  village.  In  1086  it  only  con- 
tained 373  hovels  grouped  round  St  Peter's  church,  while 
about  half  a  mile  off  were  a  few  cottages  clustered  round 
St  Benet's  Church;  and  in  1174,  after  being  burnt  to  the 
ground,  it  was  only  partially  rebuilt.  It  is  thus  at  first  sight 
difficult  to  see  why  lecturers  should  have  settled  there,  and 
the  analogies  of  other  universities  throw  but  little  light  on  it. 
I  suspect  the  explanation  is  that  students  were  attracted  in 
the  first  instance  by  the  great  fair  held  once  every  year  at 
Stourbridge,  which  is  an  open  common  lying  within  the  boun- 
daries of  the  borough. 

The  village  of  Cambridge  was  situated  at  the  end  of  a  pro- 
montory which  projected  into  the  fens,  and  commanded  the 
northernmost  ford  by  which  the  eastern  counties  could  commu- 
nicate with  the  midlands.  Away  to  the  Wash  stretched  a  vast 
succession  of  watery  fens,  across  which  a  stranger  could  scarcely 
hope  to  pass  in  safety  save  at  the  end  of  a  dry  summer  or  after 
a  long  frost.  The  position  was  thus  an  important  one,  both 
strategically  and  commercially  ;  and  the  annual  fair  at  Stour- 
bridge became  one  of  the  two  great  centres  of  trade  for  northern 
and  central  Europe l.  Thither  the  merchants  from  Germany  and 
the  Low  countries  came  by  boat  from  Bishop's  Lynn  up  the  Ouse 
and  Cam  to  exchange  their  goods  for  the  wool  and  horses  from 
the  western  counties  and  midland  shires;  and  miles  of  tents 

1  The  other  great  mediaeval  fairs  were  Leipzig  and  Nijnii  Novgorod. 
Stourbridge,  though  now  a  mere  shadow  of  its  former  self  and  yearly 
diminishing  in  importance,  is  -still  one  of  the  largest  fairs  in  England. 


224     OUTLINES   OF  THE  HISTORY   OF   THE  UNIVERSITY. 

and  booths  were  put  up  in  streets  according  to  elaborate  rules, 
which  at  a  later  time  were  regulated  by  act  of  parliament. 

Thus  for  a  month  in  the  year  many  thousands  of  travellers 
were  brought  to  Cambridge,  and  led,  I  conjecture,  to  the  estab- 
lishment of  a  universitas  scholarium,  for  which  the  monks  and 
more  advanced  students  of  the  grammar-schools  supplied  part 
of  the  audiences.  It  is  noticeable  that  until  a  few  years  ago 
doctors  were  required  to  wear  scarlet  when  the  fair1  was 
proclaimed — thus  putting  that  ceremony  on  a  level  for  univer- 
sity purposes  with  the  five  or  six  great  feasts  of  the  church. 
Even  as  late  as  Newton's  time  it  was  apparently  an  important 
mart  for  scientific  books  and  instruments  (see  pp.  52,  53). 

Whatever  was  the  cause  of  its  location  at  Cambridge  the 
university  existed  in  1209;  and  from  an  allusion2  in  some 
legal  proceedings  in  1225  to  the  chancellor  of  the  university, 
and  from  the  fact  that  when  in  1229  Heniy  III.  invited 
French  students  to  leave  Paris  and  settle  in  England  the 
majority  preferred  to  come  to  Cambridge,  it  is  clear  that  it  was 
then  an  organized  and  well-known  university. 

In  1231  Henry  III.  gave  to  the  university  jurisdiction  over 
certain  classes  of  townsmen;  in  1251  he  extended  it  so  as  to 
give  exclusive  legal  jurisdiction  in  all  matters  concerning 
scholars,  and  finally  confirmed  all  its  rights  in  1260.  These 
powers  were  granted  by  letters  and  enactments,  and  the 
first  charter  of  which  we  now  know  anything  was  that  given 
by  Edward  I.  in  1291.  It  was,  however,  the  custom  at 
both  universities  to  solicit  a  renewal  of  their  privileges  at  the 
beginning  of  each  reign  (an  opportunity  of  which  they  often 
took  advantage  to  get  them  extended),  and  it  is  possible  that  the 
dates  here  given  may  be  those  of  the  renewals  of  the  original 
charters  which,  as  stated  above,  were  burnt  in  the  fourteenth 
century. 

1  A  collection  of  references  to  the  fair  will  be  found  in  pp.  153 — 165 
of  the  Life  of  Ambrose  BonwicJce  edited  by  J.  E.  B.  Mayor,  Cambridge, 
1870. 

2  Record  office,  Coram  Eege  Rolls,  Hen.  III.  nos.  20  and  21. 


THE   MEDIAEVAL   UNIVERSITY.  225 

The  university  was  recognized  by  letters  from  the  pope  in 
1233,  but  in  1318  John  xxn.  gave  it  all  the  rights  which  were 
or  could  be  enjoyed  by  any  university  in  Christendom.  Under 
these  sweeping  terms  it  obtained,  as  settled  iu  the  Barnwell 
process  1430,  exemption  from  the  jurisdiction  both  of  the 
bishop  of  Ely  and  the  archbishop  of  Canterbury.  A  survival 
of  this  papal  recognition,  which  involved  a  right  of  migration, 
still  exists  in  the  customary  admission  of  a  graduate  of  Oxford 
or  Cambridge  to  an  ad  eundem  degree  at  the  other  university. 
The  singular  privilege  of  conferring  degrees  possessed  by  the 
archbishop  of  Canterbury  is  also  derived  from  the  position  of 
the  pope  as  the  head  of  every  university  in  Christendom. 

It  may  be  interesting  if  I  add  the  corresponding  dates  for 
Paris  and  Oxford,  since  the  mediaeval  histories  of  the  three 
universities  are  closely  connected.  The  university  of  Paris 
was  formed  at  some  time  between  1100  and  1169;  legal 
privileges  were  conferred  by  the  state  in  1200;  and  its  degrees 
were  recognized  as  conferring  a  right  to  teach  throughout 
Christendom  in  1283.  The  university  of  Oxford  was  formed 
at  some  time  between  1149  and  1180;  legal  privileges  were 
conferred  by  the  state  in  1214;  and  its  degrees  were  recognized 
by  the  pope  in  1296.  The  university  of  Cambridge,  as  I 
have  just  explained,  was  formed  at  some  time  between  1112  and 
1209;  legal  privileges  were  conferred  by  the  state  in  1231; 
and  its  degrees  were  recognized  by  the  pope  in  1318.  Two 
other  mediaeval  universities  rival  Paris  in  antiquity:  the^e 
were  the  legal  school  at  Bologna  and  the  medical  school  at 
Salerno,  but  at  these  the  education  was  technical  rather  than 
general. 

The  characteristic  feature  of  these  five  mediaeval  univer- 
sities— Paris,  Bologna,  Salerno,  Oxford,  and  Cambridge '—is 
that  they  thus  grew  into  the  form  they  ultimately  took.  They 
were  recognized  by  the  state  and  church,  but  they  were  not, 
like  the  later  universities,  created  by  a  definite  act  or  charter. 

A  mediaeval  university  was  at  first  formed  of  a  collection 

1  They  are  probably  the  five  oldest  universities  in  Europe. 
B.  15 


226     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

of  teachers  and  pupils  with  hardly  any  pretence  of  organization. 
So  loose  was  the  connection  of  its  members  with  one  another 
that  there  was  a  constant  series  of  secessions.  These  secessions 
play  a  much  smaller  part  in  the  history  of  Oxford  and  Cam- 
bridge than  in  that  of  the  continental  universities,  as  after 
1334  the  English  universities  imposed  an  oath  on  their 
graduates  never  to  teach  as  in  a  university  anywhere  in 
England  except  at  Cambridge  and  Oxford,  "nor  to  acknow- 
ledge as  legitimate  regents  those  who  had  commenced  in  any 
other  town  in  England1."  It  must  be  remembered  that  the  two 
universities  were  very  closely  connected,  and  that  till  1535  a 
certain  proportion  of  the  students  divided  their  time  between 
the  two. 

It  is  probable  that  at  the  beginning  of  the  thirteenth 
century  there  was  no  code  of  rules  at  Cambridge  for  the 
guidance  of  its  members.  The  ancient  statutes  are  undated, 
but  there  is  every  reason  to  believe  that  the  constitution  of  the 
university  in  the  fourteenth  century,  which  is  described  in  the 
following  pages,  only  differed  in  details  from  that  which  was 
in  practical  force  during  most  of  the  preceding  century. 

The  governing  body  of  the  university  was  termed  the 
regent-house,  and  it  was  at  first  strictly  confined  to  those 
graduates  who  were  actively  engaged  in  teaching.  In  the 
twelfth  and  thirteenth  centuries  the  final  degree  of  master 
was  merely  a  license  to  teach  :  no  one  sought  it  who  did  not 
intend  to  use  it  for  that  purpose  and  to  reside2,  and  only  those 
who  had  a  natural  aptitude  for  such  work  were  likely  to  enter 

1  Peacock,  Appendix  A,  xxviii ;  Munimenta  academica,  375.   At  Oxford 
until  1827  every  newly-created  master  had  also  to  swear  that  he  would 
never  consent  to  the  "reconciliation  of  Henry  Symeon."    Henry  Symeon 
is  said  to  have  been  a  master  of  arts  who  obtained  an  office  in  the  reign 
of  King  John  (1199—1216)  by  representing  that  he  was  only  a  bachelor. 
For  this  offence  the  implacable  university  held  him  up  for  over  600  years 
to  the  obloquy  of  every  successive  generation.      Peacock,   A.,    xxiii ; 
Munimenta  academica,  432,  473 ;  Lyte,  214. 

2  A  survival  of  this  idea  exists  in  the  technical  description  of  a  doctor 
of  divinity  at  Oxford  and  Cambridge  as  sacrae  theologiae  professor. 


THE   MEDIEVAL    UNIVERSITY.  227 

so  ill-paid  a  profession.  It  was  thus  obtainable  by  any  student 
who  had  gone  through  the  recognized  course  of  study  and 
shewn  he  was  of  good  moral  character.  Outsiders  were  also 
admitted,  but  not  as  a  matter  of  course.  By  the  beginning  of 
the  fourteenth  century  students  began  to  seek  for  degrees 
without  any  intention  of  teaching;  and  in  1426  the  university 
of  Paris  took  on  itself  to  refuse  a  degree  to  a  student — a 
Slavonian,  one  Paul  Nicolas — who  had  performed  the  necessary 
exercises  in  a  very  indifferent  manner.  He  took  legal  pro- 
ceedings against  the  university  to  compel  them  to  grant  the 
degree,  but  their  right  to  withhold  it  was  established1,  and 
other  universities  then  assumed  a  similar  power.  He  was,  I 
believe,  the  first  student  who  was  "  plucked." 

The  degree  gave  the  right  to  teach,  but  after  about  1400 
the  university  only  granted  it  on  condition  that  the  new 
master  should  lecture  in  the  schools  of  the  university  for  at 
least  one  year.  Many  of  those  who  had  ceased  to  do  so  were 
however  still  resident  and  engaged  in  the  work  of  the  univer- 
sity; and  in  course  of  time  heads  of  hostels,  various  executive 
officers,  and  finally  all  graduates  who  had  ceased  to  teach, 
formed  a  second  assembly  called  the  non-regent  house,  whose 
consent  was  necessary  to  the  more  important  graces.  The  two 
houses  taken  together  formed  the  senate  of  the  university. 

The  constitution  was  thus  rendered  singularly  complex. 
Some  matters  were  decided  by  the  regents  alone,  others  by  the 
concurrence  of  both  houses  voting  separately,  others  by  both 
houses  sitting  and  voting  together,  and  lastly,  others  by  both 
houses  sitting  together  but  with  the  right  of  voting  confined 
to  the  regents2.  Finally,  every  measure  had  to  be  approved 
by  the  chancellor. 

The  executive  of  the  non-regent  house  was  vested  in  the 
two  scrutators3.  But  the  proctors  (sometimes  also  called 
rectors)  were  the  two  great  officers  of  the  university :  they 

-1  See  Bul«eus,  vol.  v.  p.  377. 

2  Statuta  antiqua,  2,  21,  50,  71,  163. 

3  Peacock,  21  et  seq. 

15—2 


228     OUTLINES  OF  THE   HISTOKY   OF  THE   UNIVERSITY. 

acted  as  the  executive  both  of  the  regent-house  and  of  the 
whole  university,  and  together  were  competent  to  perform  the 
duties  of  the  chancellor  in  case  of  an  emergency.  Even  the 
power  of  veto  possessed  by  him  could  be  challenged  if  they 
thought  fit;  and  on  their  initiative  the  whole  university  as- 
sembled in  Great  St  Mary's  could  override  the  chancellor's 
veto,  or  even  expel  him  from  his  office.  It  was  the  proctors- 
as  representing  the  regents  (and  not  the  chancellor)  who 
conferred  degrees. 

The  chancellor  was  chosen  biennially  by  the  regents,  and 
acted  as  head  of  the  university  during  his  tenure  of  the  office. 
He  was  always  a  resident,  and  it  was  not  until  the  election  for 
life  of  Fisher  in  1514  that  the  office  became  honorary.  It 
is  possible  that  at  first  the  chancellor  represented  the  bishop 
of  Ely,  with  whose  sanction  or  under  whose  protection  the 
university  had  originated,  and  from  whom  was  derived  the 
power  of  excommunication1,  which  was  freely  used  against 
troublesome  students.  The  chancellor  was  however  quite  in- 
dependent of  the  bishop;  and  so  jealous  was  the  university 
of  any  possibility  of  episcopal  interference  that  any  official 
or  nominee  of  the  bishop  was  absolutely  ineligible  for  the 
office. 

The  other  officers  of  the  university  were  the  taxors,  who 
fixed  the  rent  of  hostels  and  lodgings,  and  in  conjunction  with 
two  burgesses  determined  the  price  of  eatables  sold  in  open 
market,  and  four  or  five  beadles  who  attended  on  the  officers 
of  the  university  :  of  the  latter  two  are  still  retained  as  the 
esquire  bedells. 

It  may  be  added  that  so  soon  as  a  master  of  arts  became  a 
non-regent  he  was  unable  to  become  a  regent  again  except 
with  the  consent2  of  the  chancellor  and  the  regent-house,  a 
consent  which  was  by  no  means  always  given. 

Besides  these  houses  the  teachers  in  arts,  law,  divinity,  &c. 
were  constituted  into  separate  faculties,  but  probably  without 

1  Peacock,  B.,  LXV. 

2  Statuta  antiqua,  11,  144. 


THE   MEDIEVAL    UNIVERSITY.  229 

legislative   powers :    the  faculty  of  arts  is  considerably  older 
than  the  others1. 

It  is  probable  that  at  first  the  university  possessed  no 
buildings  or  appurtenances.  Lectures  were  given  in  barns, 
private  rooms,  or  in  any  place  where  shelter  could  be  obtained; 
while  congregations  of  the  university  and  formal  meetings 
were  generally  held  in  Great  St  Mary's  Church.  At  some 
time  before  1346  the  university  obtained  a  room  or  rooms  in 
which  exercises  could  be  performed  :  these  were  situated  in 
Free-school  lane,  and  were  possibly  identical  with  the  glomerel 
schools2.  The  divinity  school  was  commenced  in  1347  and 
opened  in  1398;  and  the  art  and  law  schools  were  added  in 
1458.  The  former  is  now  included  in  the  library,  and  is 
underneath  the  present  catalogue  room  (which  is  itself  the  old 
senate-house  of  the  university).  The  quadrangle  was  finished 
in  1475 3.  Most  of  the  colleges  and  monasteries  had  libraries 

1  Almost  all  the  above  remarks  are  applicable  to  Paris  and  Oxford. 
The  early  history  of  the  former  has  been  investigated  with  great  care  in 
Die    Universitfiten  des  Hittelalters  bis  1400,  by  P.  H.  Deinfle,  Berlin, 
1885 ;    and  the   chief  facts   connected  with   it    are   given    in   Bulasus. 
Materials   for  the   history  of  the   university   of  Oxford  exist   in   great 
abundance,  but  I  know  of  no  work  on  it  of  the  same  character  as  that  of 
Deinfle  on  Paris,  or  Mullinger  on  Cambridge. 

2  Mullinger,    i.  299,  300.      The  earliest  buildings  at  Oxford  were 
erected  in  1320.     (Lyte,  68,  99.) 

3  The  following  account  of  the  buildings  surrounding  the  eastern 
quadrangle  of  the  library  is  taken  from  the  Cambridge  university  reporter 
of  Oct.  20,  1881  (pp.  62,  63).     "The  northern  building,  which  had  the 
school  of  theology  on  the  ground-floor,  and  the  'capella  nova  universi- 
tatis,'  or,  as  it  would  now  be  called,  the  senate-house,  on  the  first  floor, 
was  finished  about  1400.     The  west  side,  which  had  the  school  of  canon 
law  on  the  ground-floor,  and  the  'libraria  nova'  on  the  first  floor,  had 
heen  commenced  in  1440,  but  was  not  completed  until  1458.     The  south 
side,  which  had  the  schools  of  philosophy  and  civil  law  on  the  ground- 
floor,  and  some  other  schools,  together  with  a  library,  on  the  first  floor, 
was  erected  between  1458  and  1467.     The  narrow  building  that  joined 
the  north  and  south  sides  together,  and  formed  a  west  front,  continuous 
with  the  eastern  gables  of  the  north  and  south  sides,  was  erected  between 
1470  and   1475.     The  ancient  aspect  of  this  quadrangle  is  shewn  in 


230     OUTLINES   OF  THE  HISTORY   OF   THE  UNIVERSITY. 

of  their  own,  but  the  university  or  common  library  was  not 
built  till  1424. 

The  question  of  how  suitable  provision  should  be  made  for 
the  board  and  lodging  of  the  students  was  however  far  more 
pressing  than  that  of  providing  accommodation  for  the  cor- 
porate life  of  the  university.  The  town  was  originally  only  a 
collection  of  unhealthy  cottages,  and  unlike  Paris  and  Oxford 
had  no  importance  except  that  derived  from  the  presence  of 
the  university.  The  character  of  the  accommodation  offered 
did  not  however  prevent  the  townsmen  from  utilizing  their 
monopoly  to  make  extortionate  charges;  and  almost  the  first 
act  of  the  university  of  which  we  know  anything  was  to  at- 
tempt to  find  a  remedy  for  the  evils  and  dangers  to  which  the 
lads  who  nocked  to  the  university  were  thus  exposed.  In  1231 
a  rule  was  made  that  every  scholar  must  place  himself  under 
the  tuition  of  some  master1:  and  in  1276  the  university,  in 
virtue  of  powers  conferred  by  the  crown,  passed  a  grace  that 
no  lodging-house  keeper  or  teacher  was  to  receive  a  scholar 
unless  the  latter  "had  a  fixed  master  within  fifteen  days  after 
his  entry  into  the  university2."  No  record  of  this  tutorial 
relation  was  kept  by  the  university,  but  at  stated  periods  the 
masters  attended  in  the  schools  and  read  out  the  roll  of  their 

Loggan's  print,  taken  about  1688.  The  porch  and  staircase  at  the  N.W. 
angle,  together  with  the  west  wall  as  far  as  the  northernmost  buttress  on 
that  side,  was  taken  down  in  1714,  in  order  to  make  a  room  on  the  first 
floor  large  enough  to  receive  Bishop  Moore's  library.  At  the  same  time 
the  windows,  of  which  there  was  originally  only  one,  of  three  lights, 
between  each  pair  of  buttresses,  were  replaced  by  the  existing  round- 
headed  ones.  Shortly  after,  in  1727,  the  present  senate-house  being 
completed,  the  old  'capella  universitatis '  was  absorbed  into  the  library. 
The  classical  building,  which  now  replaces  the  central  block  on  the  east 
side,  was  begun  in  1754,  the  style  being  selected  in  order  to  make  it 
harmonize  with  the  senate-house.  The  old  divinity  school  on  the 
groundfloor  of  the  north  side  was  taken  into  the  library  in  1856.  These 
various  changes  have  utterly  destroyed  the  ancient  character  of  the 
quadrangle. " 

1  Cooper,  i.  42. 

2  Statuta  antiqua,  42. 


THE   MEDIAEVAL   UNIVERSITY.  231 

own  pupils1.  There  was  no  formal  matriculation  of  students 
until  the  year  15432. 

The  university  also  took  steps  to  encourage  the  resident 
masters  to  open  hostels  or  boarding-houses,  and  until  the 
sixteenth  century  the  majority  of  the  students  lived  in  these 
houses.  One  of  the  earliest  of  the  extant  statutes3  of  the 
university  gives  the  detailed  rules  which  the  university  laid 
down  about  the  year  1300  for  regulating  the  hiring  of  these 
hostels.  It  illustrates  how  completely  the  university  was  then 
the  dominant  power  in  the  town,  that  if  a  master  of  arts  wished 
to  take  any  particular  house  for  a  hostel  and  could  give  security 
for  the  rent  the  university  turned  the  owner  out4. 

Another  way  of  meeting  the  difficulty  was  by  the  establish- 
ment of  colleges,  the  idea  of  which  was  borrowed  from  Paris 
and  Oxford.  The  earliest  to  be  established  was  that  which  is  now 
known  as  Peterhouse  in  or  before  1280.  At  first  this  and  other 
similar  foundations  were  designed  to  house  and  support  a  master 
with  certain  fellows  and  scholars  (to  give  them  their  modern 
designations)  only,  but  not  pensioners  or  ordinary  students. 

Another  danger  of  a  different  kind  existed  in  the  constant 
efforts  at  proselytizing  by  the  religious  orders.  In  the  course 
of  the  thirteenth  century  all  the  great  monastic  orders  esta- 
blished houses  in  Cambridge  where  food,  shelter,  the  use  of  a 
library,  and  assistance  were  offered  to  all  who  would  join  the 
order.  The  number  of  these  houses  shew  that  the  reputation 
of  the  university  must  have  been  considerable.  The  Augus- 
tinian  canons  were  already  established  at  Barnwell,  but  they 
enlarged  their  abbey  till  it  became  one  of  the  wealthiest  in  the 
kingdom.  The  Franciscans  built  a  house  in  1224,  and  shortly 

1  Cambridge  documents,  i.  332.     Lyte,  198. 

2  Mullinger,  n.  63. 

3  It  is  printed  at  length  in  Mullinger,  i.  639,  and  a  translation  is 
given  on  pp.  218—220. 

4  See  vol.  i.  p.  65  of  Cooper's  Annals  on  a  case  which  happened  in 
1292:  it  is  evident  from  the  references  that  the  university  was  legally 
entitled  to  exercise  the  power. 


232     OUTLINES   OF  THE   HISTORY   OF   THE   UNIVERSITY. 

afterwards  erected  the  magnificent  church  and  monastery  which 
formerly  stood  on  the  site  of  Sidney.  By  the  middle  of  the 
thirteenth  century  representatives  of  nearly  all  the  orders  were 
living  in  Cambridge.  Among  others  the  Carmelites  had  oc- 
cupied the  site  of  Queens';  the  Dominicans  the  site  of  Em- 
man  u  el ;  the  Augustinian  friars  the  site  of  the  new  museums; 
the  Benedictines  the  site  of  Magdalene;  the  White  canons  the 
site  of  Peterhouse  Lodge ;  and  the  brethren  of  St  John  the  site 
of  the  college  of  that  name1. 

Now  the  university,  though  it  was  closely  connected  with 
the  regular  clergy  and  though  the  majority  of  its  members  were 
even  in  orders,  was  still  essentially  a  secular  institution.  It 
was  natural,  therefore,  that  this  crowd  of  monks,  who  merely 
became  masters  of  the  university  in  order  to  recruit  among  its 
junior  members,  should  be  regarded  with  great  suspicion.  The 
successful  ruse  by  which  in  1228  the  Dominicans  had 
temporarily  obtained  the  entire  control  of  the  university  of 
Paris  gave  warning  of  what  was  designed,  but  with  that  tolera- 
tion which  has  always  been  a  marked  feature  in  Cambridge  life 
an  open  rupture  was  avoided — the  monks  were  admitted  to 
degrees  so  long  as  they  conformed  to  the  regulations  of  the 
university,  and  by  courtesy  one  was  always  elected  on  the 
caput2  (see  p.  245). 

The  university,  however,  never  ceased  to  be  on  its  guard 
against  these  "foreigners  who,"  so  ran  the  phrase,  "  cajoled  lads 
before  they  could  well  distinguish  betwixt  a  cap  and  a  cowl." 
In  1303  two  of  them,  Nicholas  de  Dale  and  Adam  de  Haddon, 
insisted  that  the  rights  of  their  respective  monasteries  were 
paramount  to  all  privileges  of  the  university3.  They  were 
accordingly  expelled;  but  in  13064  the  university  allowed 
monks  to  proceed  to  degrees  in  divinity  without  having 
previously  incepted  in  arts.  Instead  of  accepting  this  decision 


1  Mullinger,  i.  138,  139,  564. 

2  Statuta  antiqua,  4;  Peacock,  21. 

3  Peacock,  26. 

4  Peacock,  33. 


THE   MEDIAEVAL   UNIVERSITY.  233 

as  a  favor  and  concession  the  monks  treated  it  as  a  sign  of 
their  triumph,  and  in  1336  a  grace  had  to  be  passed  forbidding 
the  friars  to  receive  into  their  orders  any  scholar  under  the 
age  of  eighteen.  Oxford  passed  a  similar  statute  in  1358. 
Under  pressure  from  Rome  these  statutes  were  subsequently 
repealed,  but  in  1359  the  university  passed  a  grace  by  which 
only  two  friars  from  each  house  were  allowed  to  incept  in  the 
same  year1,  which  sufficiently  served  to  protect  the  university 
from  excessive  proselytizing. 

The  establishment  of  these  numerous  and  powerful  bodies 
had  however  another  and  more  lasting  effect.  Although  the 
monks  and  friars  were  nominally  members  of  the  univer- 
sity, they  were  divided  from  the  rest  of  the  masters  on  nearly 
every  question  of  policy,  and  thus  acted  as  a  counterpoise 
to  the  overwhelming  power  of  the  university  in  local  matters. 
They  were  also  wealthy,  and  materially  increased  the  pro- 
sperity of  the  town,  so  that  by  1300  the  mayor  and  burgesses 
formed  a  well-organized  corporate  body.  In  that  year  the  total 
population  of  the  university  and  town  was  about  4000 2,  but 
except  at  the  time  of  the  annual  Stourbridge  fair  there  does  not 
seem  to  have  been  any  considerable  trade,  save  that  arising  from 
the  supply  of  the  needs  of  the  university  and  the  monasteries. 

The  statements  about  the  number  of  students  at  the  medi- 
aeval universities  must  be  received  with  considerable  caution. 
They  represent  vague  impressions  rather  than  the  result  of  an 
accurate  census.  It  must  also  be  recollected  that  it  was 
customary  to  reckon  as  members  of  the  university  all  servants 
and  tradesmen  whose  chief  employment  was  in  connection  with 
students,  while  the  fact  that  the  average  student  spent  at 
least  seven  years  at  the  university  before  he  became  a  master, 
and  generally  twenty  years  or  more  if  he  aspired  to  become  a 
doctor  (after  which  he  probably  still  resided  for  some  years), 
caused  the  university  to  be  largely  composed  of  permanent 
residents  of  every  age  from  1 2  to  40. 

1  Statuta  antiqua,  163,  164.     Peacock,  xliii;  Mullinger,  i.  263. 

2  Cooper,  i.  58. 


234     OUTLINES   OF  THE  HISTOKY   OF  THE  UNIVERSITY. 

The  question  has  been  very  carefully  considered  by  M. 
Thurot1,  who  comes  to  the  conclusion  that  the  total  number  of 
students  at  Paris  never  rose  much  above  1500  nor  of  regents 
above  200.  I  think  I  should  probably  not  be  far  wrong  if  I 
estimated  the  total  number  of  masters  and  students  (exclusive 
of  monks)  at  Cambridge  during  the  thirteenth,  fourteenth, 
and  fifteenth  centuries  as  varying  between  500  and  1000.  The 
numbers  at  Oxford  in  the  thirteenth  century  were  perhaps 
about  700;  in  the  fourteenth  century  probably  nearly  2000  ; 
in  the  fifteenth  century  the  university  is  described  as  "wholly 
deserted,"  perhaps  the  total  number  then  did  not  exceed  200 
or  300.  I  ought  to  add  that  all  these  numbers  are  considerably 
less  than  those  usually  given,  but  the  latter  probably  include 
servants  and  tradespeople.  Peacock  says2  that  the  number  of 
regent-masters  created  at  Cambridge  in  each  year  [I  presume 
in  the  fifteenth  century]  averaged  about  40 ;  and  that  of 
bachelors  in  law  about  15.  This,  as  far  as  I  can  judge,  will 
give  a  result  not  very  different  from  that  which  I  had  in- 
dependently arrived  at. 

The  question  as  to  the  social  position  of  the  students  in 
mediaeval  times  is  a  difficult  one3.  The  balance  of  opinion  is 
that  a  large  majority  were  poor,  and  it  is  certain  from  several 
of  the  ancient  statutes  that  poverty  was  not  uncommon4.  On 
the  other  hand,  a  considerable  minority  must  have  been  wealthy. 
The  grace,  to  which  allusion  was  made  in  chapter  VIII.,  by 
which  any  incepting  master  was  forbidden  to  spend  in  presents 
and  dinners,  on  the  occasion  of  taking  his  degree,  what  would 
now  be  equivalent  to  .£500,  would  have  been  absurd  if  there 
were  no  wealthy  men  at  the  university.  Moreover  it  is  clear 
from  internal  evidence,  that  Richard  II.  in  framing  the 
statutes  of  King's  Hall  (which  had  been  founded  by  Edward  II. 

1  See  pp.  32,  42  of  De  V organisation  de  Venseignement  au  moyen  age? 
by  C.  Thurot,  Paris,  1850.     See  also  Munimenta  academica,  p.  xlviii. 

2  Observations,  33. 

3  Mullinger,  i.  345,  note. 

4  See  Cooper,  i.  245,  343. 


THE   MEDIEVAL   UNIVEKSITY.  235 

and  Edward  III.,  and  is  now  a  part  of  Trinity  College), 
expressly  designed  it  for  wealthy  and  aristocratic  students1. 
All  regulations  about  poverty  were  erased  from  its  rules,  while 
in  place  of  them  various  sumptuary  and  disciplinary  regu- 
lations were  inserted.  Among  these  I  notice  that  the  daily 
expenditure  of  food  for  each  student  was  not  to  exceed  Is.  2d. 
a  week,  which  would  be  worth  now  say  about  14s.  or  15s.  and, 
was  nearly  half  as  much  again  as  at  Gonville  Hall.  Other 
rules  were  that  students  should  not  keep  dogs  in  college,  or 
play  the  flute  to  the  annoyance  of  their  neighbours.  The 
additional  provision  that  no  one  should  practise  with  the 
cross-bow  in  the  courts  or  walks  of  the  college  must  com- 
mend itself  to  every  one  of  mature  age.  A  tradition  that 
the  society  laid  down  a  rule  that  no  student  should  strike  a 
fellow,  or.  under  any  circumstances  the  master,  is  suggestive 
that  its  members  were  not  wholly  devoted  to  study.  In  the 
fifteenth  century  no  one  was  admitted  who  was  not  bene 
natus. 

I  think  therefore  we  may  safely  say  that  the  students  were 
drawn  from  all  classes  and  ranks  in  the  kingdom,  but  that  a 
large  proportion  were  poor. 

I  may  perhaps  be  pardoned  for  adding  a  few  words  on  the 
social  side  of  the  life  of  a  mediseval  student.  The  majority  of 
the  students  and  all  the  wealthier  ones  resided  in  hostels2. 
Some  of  these  houses  no  doubt  contained  all  the  comforts 
which  were  then  customary,  but  no  account  of  life  at  a  hostel 
is  now  extant.  It  would  seem,  however,  that  there  was  usually 
a  common  sitting-room  or  hall ;  and  at  the  better  hostels  a  lad 
could  hire  a  bedroom  for  his  sole  use,  the  rent  of  which  varied 
from  7s.  Qd.  to  13s.  4d.  a  year3.  The  total  expenditure  of  the 
son  of  a  well-to-do  tradesman  at  Oxford  in  the  reign  of  Edward 
III.  came  to  £9.  10s.  8d.;  board  was  charged  at  the  rate  of  2s. 

1  Mullinger,  i.  252—254. 

2  See  Lever's  sermon  at  St  Paul's  Cross,  preached  in  1550:  Arber's 
edition,  p.  121. 

3  Munimenta  academica,  556,  655. 


236     OUTLINES   OF   THE    HISTORY   OF   THE    UNIVERSITY. 

a  week,  tuition  at  26s.  Sd.  a  year,  and  clothes  cost  20s1.  In 
1289  the  allowance  to  two  brothers  de  la  Fyte  was  half-a-mark 
each  per  week,  which  was  raised  in  their  second  year  of 
residence  to  35  marks  a  year:  besides  this  bills  for  certain 
necessary  expenses,  which  seem  to  have  averaged  nearly  £5  a 
year  for  each  of  them,  were  paid  by  the  king.  This  scale  of 
allowance  was  exceptionally  high,  as  the  boys  were  well  con- 
nected, and  protected  by  the  king :  they  had  a  manservant  to 
themselves.  At  the  other  end  of  the  social  scale  two  poor  lads 
named  Bongs  wood  were  sent  by  bishop  Swinfield  to  Oxford  in 
1288,  and  the  bills  for  both  of  them  for  forty  weeks'  residence 
came  to  £13.  19s.  2c?.2  From  these  and  similar  facts  it  would 
seem  that  a  student  could  hardly  support  himself  on  less  than 
<£9  a  year,  and  that  anything  beyond  .£15  a  year  was  a  hand- 
some allowance.  If  these  totals  be  multiplied  by  12  or  13  they 
will  represent  about  their  equivalents  in  modern  value. 

The  colleges,  except  King's  Hall,  were  intended  for  poor 
students,  but  compared  with  those  of  Paris  seem  to  have  been 
fairly  comfortable,  and  indeed  for  that  age  luxurious.  Every 
student  swore  obedience  to  the  college  authorities,  and  it  was 
rigidly  enforced  with  birch  and  rod.  The  younger  students 
slept  three  or  four  in  a  room,  which  also  served  as  study, 
but  was  more  often  than  not  unwarmed.  There  was  a  dining 
hall,  in  which  on  great  occasions  a  fire  was  lit.  Here  meals 
were  served,  namely,  dinner  about  10  a.m.  and  supper  about 
5  p.m. ;  meat  being  apparently  provided  on  each  occasion,  ex- 
cept in  Lent.  The  colleges  generally  required  their  members 
to  speak  nothing  but  Latin  (or  in  a  few  cases  French)  in  hall 
and  on  all  formal  occasions  except  the  great  festivals  of  the 
church.  In  the  evening  mock  contests  were  held  in  the  hall, 
by  which  students  were  practised  for  the  acts  they  had  to  keep 
in  the  schools.  There  was  usually  an  attic  fitted  up  as  a 
library  where  students  could  find  the  text-books  of  the  day,  and 

1  The  accounts  of  the  guardian  of  Hugh  atte  Boure,  quoted  in  Eiley's 
London,  p.  379. 

2  The  authorities  are  quoted  in  Lyte,  93. 


THE   MEDIAEVAL    UNIVEKSITY.  237 

from  which  a  fellow  could  borrow  books  :  this  use  of  a  library 
was  one  of  the  most  highly  valued  privileges  of  college  life1. 

The  disciplinary  rules  of  the  colleges  were  naturally 
stricter  than  those  in  force  in  the  hostels.  Until  a  student 
of  a  college  became  a  bachelor  he  was  not  allowed  to  go 
out  of  college  bounds  unless  accompanied  by  a  master  of 
arts.  A  bachelor  had  much  the  same  freedom  as  an  under- 
graduate now-a-days,  except  that  he  generally  had  but  one 
room,  which  he  had  to  share  with  another  man,  and  only  a 
fellow  of  considerable  standing  had  a  room  to  himself. 
Allowances  were  conditional  on  residence,  but  were  generally 
sufficient  to  supply  all  the  necessaries  of  a  student's  life.  The 
master  was  absolute  within  the  college :  a  fatal  defect  in 
organization,  for  a  single  incompetent  master  could  destroy  the 
progress  of  centuries,  as  every  mediaeval  college  in  succession 
found  to  its  cost2. 

The  amusements3  of  the  students  were  much  what  we 
should  expect  from  English  lads.  Contests  with  the  cross- 
bow were  common,  and  cock-fighting — at  any  rate  in  the 
hostels — was  a  usual  amusement.  To  the  more  adventurous 
student  the  opportunity  of  a  fight  with  the  townsmen  was 
always  open.  As  far  as  we  can  judge  at  this  distance  of  time 
the  university  authorities  in  their  dealings  with  the  town  were 
arrogant  and  exasperating,  but  always  kept  within  the  law; 
and  technically  in  all  the  serious  riots  the  townsmen  were  in  the 
wrong.  The  riots  of  1261,  1322,  and  1381  were  particularly 
violent,  and  the  townsmen  not  only  committed  outrages  of 
every  kind,  but  burnt  some  of  the  hostels,  and  all  the  charters 
and  documents  of  the  university  as  well  as  of  such  colleges  as 
they  were  able  to  sack.  After  the  last  of  these  riots  the 
government  confiscated  the  liberties  of  the  town,  and  bestowed 
them  on  the  chancellor,  in  whom  they  remained  vested  till  the 
reign  of  Henry  VIII.  To  this  stringent  measure  the  subse- 

1  Mullinger,  i.  366—372. 

2  See  for  example  Mullinger,  i.  424. 

3  Mullinger,  i.  373,  374. 


238     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

quent  prosperity  of  the  university  (and  so  indirectly  of  the 
town)  was  largely  due.  The  ill  feeling  which  existed  at  every 
mediaeval  university  between  town  and  gown  was  intensified  at 
Cambridge  by  the  fact  that  the  fishing  in  the  river  was  unusually 
good,  and  belonged  absolutely  to  the  mayor  and  corporation, 
who  refused  to  allow  university  men  to  fish  in  it  under  any 
circumstances.  Such  a  right  could  not  be  enforced  without 
considerable  friction,  and  as  the  university  claimed  and  exer- 
cised exclusive  jurisdiction  to  try  cases  where  their  own 
members  were  concerned,  the  dispute  was  complicated  by 
differences  of  opinion  on  the  evidence  requisite  to  prove  a 
trespass  or  assault1. 

Besides  these  amusements  there  was  rarely  a  year  in  which 
some  tournament  or  form  of  sport  was  not  held  in  the  im- 
mediate neighbourhood,  and  like  the  fair  at  Stourbridge  gave 
opportunity  for  plenty  of  adventures,  as  well  as  the  interesting 
spectacle  of  bear  and  bull  baiting.  The  prohibitions  in  the 
statutes  of  New  College,  Oxford,  of  dice  and  chess  as  in- 
struments of  gaming  imply  that  they  were  constantly  used. 
Among  the  more  wealthy  members  of  the  university  tennis, 
cock-fighting,  and  riding  seem  to  have  been  especially  popular; 
but  many  of  the  college  statutes  enjoin  that  a  daily  walk  with 
a  companion,  and  conversation  "on  scholarship  or  some  proper 
and  pleasant  topic"  should  if  possible  be  enforced. 

Lastly,  it  should  be  added  that  local  ties  and  prejudices 
were  very  strongly  maintained.  Students  born  anywhere 
south  of  the  Trent  formed  one  "  nation,"  while  those  born  to  the 
north  of  it  formed  another.  These  nations  took  opposite  sides 
on  every  question  ;  thus  when  Occam,  who  was  a  southerner, 
advocated  nominalism,  the  northerners  at  once  adopted  the 
1  Finally,  in  despair  of  obtaining  their  rights  otherwise,  the  corpora- 
tion farmed  their  powers  piscatorial  to  certain  poor  men,  who  it  was 
thought  "needing  all  the  money  they  could  obtain  would  not  fail  in  well 
guarding  that  which  they  had  purchased."  This  ingenious  scheme 
failed,  for  the  poor  men  shortly  petitioned  the  corporation  to  cancel  the 
agreement,  since  "many  times  had  they  been  driven  out  of  their  boats 
with  stones  and  other  like  things,  to  the  danger  of  their  bodies. " 


THE   MEDIEVAL   UNIVERSITY.  239 

realistic  views  of  Scotus.  They  were  organized1  almost  like  regi- 
ments, and  the  smouldering  hostility  between  them  was  always 
ready  to  break  into  open  riot,  which  not  unfrequently  ended  in 
loss  of  life.  So  high  did  local  feeling  run  that  most  of  the  college 
statutes  expressly  guarded  against  the  favoritism  that  arose  from 
it  by  a  provision  that  not  more  than  two  or  three  scholars  or 
fellows  born  in  the  same  county  could  be  on  the  foundation  at 
the  same  time. 

The  students  dressed  much  like  other  Englishmen  of  the 
same  period.  Efforts  to  enforce  the  tonsure  and  ecclesiastical 
robe  were  not  unfrequently  made,  but  seem  to  have  been  always 
evaded.  Perhaps  knee-breeches,  a  coat  (the  cut  of  which 
varied  at  different  times)  bound  round  the  waist  with  a  belt, 
stockings,  and  shoes  (not  boots)  fairly  represent  the  visible 
part  of  the  dress  of  an  average  student  at  an  average  time. 
The  dress  of  a  blue-coat  boy  may  be  compared  with  this.  To 
this  most  students  seem  to  have  added  a  cloak  edged  or  lined 
with  fur,  which  often  found  its  way  into  the  university  chest 
as  a  pledge  for  loans  advanced.  Girdles,  shoes,  rings,  &c.  varied 
with  the  fashion  of  the  day. 

The  earliest  inventory  of  the  possessions  of  a  Cambridge 
student  that  I  can  quote  is  one  of  the  belongings  of  Leonard 
Metcalfe,  a  scholar  of  St  John's  College,  who  was  executed  in 
1541  for  the  murder  of  a  townsman.  All  his  goods  were  con- 
fiscated to  the  crown,  and  therefore  scheduled  by  the  vice-chan- 
cellor2. His  wardrobe  consisted  of  a  gown  faced  with  satin,  an 
old  jacket  of  tawny  chamblet  (i.e.  silk  and  hair  woven  cross- 
wise), an  old  doublet  of  tawny  silk,  a  jacket  of  black  serge,  a 
doublet  of  canvass,  one  pair  of  hose,  an  old  sheet  or  shirt,  a 
cloak,  and  an  old  hat.  I  suppose  these  were  in  addition  to  the 
clothes  he  wore  when  being  executed,  as  the  latter  were  the 

1  See  Statuta  antiqua,  44. 

2  See  vol.  i.  pp.  109,  110  of  the  Privileges  of  the  university  of  Cam- 
bridge, by  George  Dyer,  London,  1824.     For  corresponding  inventories 
of  Oxonians,   see  Munimenta  academica,  numerous  references  between 
pp.  500—663. 


240     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

perquisite  of  the  hangman.  He  had  besides  a  coverlet,  two 
blankets  (one  being  very  old),  and  a  pair  of  sheets — but  most 
of  these  are  stated  to  have  been  pawned  before  he  went  to 
prison.  His  furniture  consisted  of  a  wardrobe-chest  with  a 
hanging  lock  and  key,  a  piece  of  carpet,  a  chair,  a  knife, 
and  a  lute.  The  table  and  bedstead  were  fixtures,  and  be- 
longed to  the  college.  His  books  with  their  respective  values 
were  as  follows.  A  Latin  dictionary,  Is.  8d.;  Vocabularius 
juris  et  Gesta  Romanorum,  4d.;  Introductions  Fabri,  3c?.; 
Horatius  sine  commenti,  4d. ;  Tartaretus  super  Summulas,  2d. ; 
The  shepheard's  kalender,  2d. ;  Moria  Erasmi,  Qd. ;  and  Compen- 
dium quatuor  librorum  institutionum,  3d. ;  the  total  value  being 
three  shillings  and  eight-pence,  equivalent  to  rather  more  than 
two  pounds  now-a-days.  He  had  not  taken  his  bachelor's 
degree,  and  it  is  therefore  not  surprising  that  he  possessed 
no  mathematical  works.  His  total  assets  were  valued  at 
£4.  Is.  8c?.,  equivalentto  .£50  or  £60  at  the  present  time.  The 
above  list  seems  fairly  to  represent  the  belongings  of  a  mediaeval 
student,  except  that  Metcalfe's  library  was  unusually  large. 

A  gown  or  some  similar  distinctive  dress  has  always  been 
worn  at  Cambridge1;  but  the  cut  and  material  varied  at  dif- 
ferent times.  Masters  wore  a  square  cap,  and  doctors  a  biretta, 
but  it  is  not  clear  whether  any  cap  was  worn  by  undergrad- 
uates. From  the  original  statutes  of  New  College,  Oxford, 
and  Winchester  School,  it  seems  probable  that  at  that  time 
the  students  went  bareheaded,  as  they  still  do  at  Christ's 
Hospital.  The  earliest  reference  to  caps  being  worn  by 
students  as  a  part  of  their  academical  dress  occurs  in  the 
sixteenth  century.  The  cap  then  worn  was  circular  in  shape 
and  flabby,  lined  with  black  silk,  with  a  brim  of  black  velvet 
for  pensioners  or  black  silk  for  sizars.  The  square  cap  for 
undergraduates  was  not  generally  introduced  till  1769  :  the 
puritan  party  having  objected  to  it  in  the  sixteenth  and 
seventeenth  centuries  as  a  symbol  of  popery. 

The  cut  of  the  B.A.  hood  has  not  varied  from  the  thir- 
1  See  Cooper's  Annals,  vol.  i.  pp.  156,  157,  182,  215,  355. 


THE   MEDIEVAL    UNIVERSITY.  241 

teenth  century,  except  that  the  two  ends  were  formerly  sewn 
together  instead  of  being  connected  by  a  string  as  they  are 
now/  In  the  middle  ages  it  was  lined  with  wool  and  not 
rabbit-skin.  The  shape  is  different  to  that  of  all  other  univer- 
sities, as  it  includes  what  is  called  a  tippet.  The  M.A.  hood 
for  regents  was  the  same  as  at  present.  The  hoods  of  non- 
regents  were  of  the  same  shape,  but  lined  with  black.  The 
proctors  invariably  wore  the  hood  squared,  as  they  do  now  : 
and  the  scrutators  and  taxors  had  the  same  privilege 1. 

It  must  be  remembered  that  the  mediaeval  university  and 
colleges  were  very  poor2.  The  members  of  the  latter  often 
found  themselves  unable  to  obtain  money,  even  for  their  daily 
food,  except  by  selling  books  or  pledging  their  house.  The 
former  had  a  few  scholarships,  the  earliest  of  which  was 
founded  in  1255,  and  possessed  a  few  funds  for  the  purpose 
of  loans.  Every  separate  bequest  or  gift  was  for  simplicity  of 
accounts  kept  in  a  separate  chest,  and  some  of  these  coffers  are 
still  preserved  in  the  registry.  The  name  has  also  been  re- 
tained as  a  synonym  for  the  university  treasury. 

The  development  of  the  university  throughout  the  middle 
ages  seems  to  have  been  one  of  steady,  uniform  progress.  This 
was  partly  due  to  its  own  merits,  but  partly  to  the  gradual 
deterioration  of  the  monastic  schools.  There  was  no  sudden  out- 
burst of  prosperity,  such  as  that  which  in  the  fourteenth  century 
made  Oxford  the  most  celebrated  seat  of  learning  in  Europe, 
but  neither  was  there  any  collapse  such  as  that  which  in 
the  fifteenth  century  left  Oxford  almost  deserted  \  though  the 
numbers  at  Cambridge  do  not  seem  to  have  increased  during 
that  century. 

1  The  above  account  is  summarized  from  pp.  454 — 543  of  University 
life  in  the  eighteenth  century,  by  C.  Wordsworth,  Cambridge,  1874. 

2  Even  now  the  corporate  revenue  of  the  university  proper  (as  distin- 
guished from  the  colleges)  is  less  than  £2,500  a  year.     I  suppose  very 
few  people  realize  how  pressed  for  means  is  the  university,  and  that  it  is 
only  by  contributions  from  the  colleges  (out  of  property  which  was  really 
left  for  other  purposes)  that  the  university  contrives  to  balance  its  ac- 
counts.    The  much  greater  wealth  of  the  sister  university  has  largely 
contributed  to  the  idea  that  the  university  of  Cambridge  is  also  wealthy. 

B.  16 


242     OUTLINES   OF  THE  HISTORY  OF  THE   UNIVERSITY. 


The  university  from  1525  to  1858. 

The  close  of  the  fifteenth  century  was  marked  by  the 
commencement  of  schools  of  science  and  divinity.  A  similar 
development  was  general  throughout  Europe,  but  it  was  some 
years  before  the  English  universities  felt  the  full  force  of  the 
movement.  The  intellectual  life  at  Oxford  during  the  middle 
ages  had  been  far  more  vigorous  and  active  than  that  at 
Cambridge,  and  in  literature  (though  probably  not  in  science 
and  divinity)  the  renaissance  in  England  had  commenced  about 
the  year  1440  at  Oxford.  The  logicians  there  bitterly  opposed 
the  new  movement,  and  succeeded  in  temporarily  stopping  it. 
The  consequence  was  that  the  revival  of  the  study  of  literature 
in  England  was  mainly  effected  at  Cambridge.  The  effects 
of  this  preeminence  in  the  sixteenth  century  lasted  long  after 
the  immediate  causes  had  ceased  to  act,  and  until  the  close  of 
the  eighteenth  century  the  literary  and  scientific  schools  of 
Cambridge  were  superior  to  those  of  Oxford. 

It  was  to  Fisher,  and  subsequently  to  Erasmus,  that  Cam- 
bridge owed  the  creation  of  its  literary  schools,  which  originated 
about  the  year  1510.  I  think,  however,  that  during  the  pre- 
ceding century — in  fact  since  the  suppression  of  the  Lollard 
movement  by  Archbishop  Arundel  on  his  visit  in  1401 — the 
drift  of  opinion  in  Cambridge  had  steadily  set  towards 
moderate  puritanism  and  the  study  of  science.  I  suspect  that 
the  divergence  in  the  opinions  prevalent  at  Oxford  and  Cam- 
bridge which  here  first  shews  itself  was  due  to  the  fact  that 
the  residents  at  Cambridge  were  every  year  brought  into  con- 
tact at  the  Stourbridge  fair  with  merchants  and  scholars  from 
Germany,  and  apparently  through  them  with  the  Italian 
universities  (especially  Padua),  while  Oxford  was  a  much  more 
self-contained  society.  It  is  noteworthy  that  almost  all  the 
Cambridge  reformers  came  from  Norfolk,  which  was  in  close 
commercial  connection  with  the  Netherlands,  and  that  the 
literary  party  in  the  university  were  nicknamed  Germans. 


THE   PERIOD   OF   TRANSITION   (1535 — 1570).         243 

On  the  other  hand  it  should  be  noted  that  some  of  the 
most  influential  leaders  of  the  renaissance  (such  as  Tonstal, 
Tyndale,  Recorde,  and  Erasmus)  came  from  Oxford,  bringing 
with  them  the  best  traditions  of  that  university;  and  the 
rapidly  rising  reputation  of  Cambridge  was  greatly  stimulated 
by  those  new-comers.  So  completely  successful  were  the 
philosophers  at  Oxford  in  destroying  the  study  of  literature 
there,  that  Wolsey  was  obliged  to  come  to  Cambridge,  much 
though  he  disliked  it,  to  get  scholars  acquainted  with  the 
subject  to  put  on  the  foundation  of  his  new  Cardinal  College. 
The  same  reason  probably  explains  why  some  fifty  years  later 
the  society  of  Trinity  College,  Dublin,  was  at  first  almost 
wholly  recruited  from  the  members  of  Trinity  College,  Cam- 
bridge. 

The  triumph  of  the  Oxford  logicians  was  synonymous  with 
the  ascendancy  there  of  the  narrow  orthodox  theological  party. 
Hence  the  reformation  was  mainly  the  work  of  Cambridge 
divines.  The  preliminary  meetings  in  which  the  general  lines 
of  the  movement  were  laid  down  were  all  held  at  Cambridge  at 
the  White  Horse  Inn,  where  the  house  of  the  tutor  of  King's 
now  stands.  The  most  prominent  of  these  proto-reformers  were 
Barnes,  Bilney,  Coverdale,  Tyndale,  and  Parker.  The  preva- 
lent feeling  of  the  university  is  shewn  by  the  fact  that  when  in 
1525  Wolsey  ordered  the  arrest  of  Barnes  the  students  broke 
into  the  room  in  which  the  court  before  which  he  had  been 
summoned  was  sitting,  and  Wolsey  had  to  adjourn  the  trial 
to  London  before  he  could  secure  a  hostile  verdict.  Many 
of  the  most  eminent  members  of  the  university,  such  as 
Cranmer,  Ridley,  Latimer,  Ascham,  and  Cheke,  did  not 
conceal  their  sympathy  with  the  reformers.  The  fall  of 
Wolsey  and  the  rise  of  Cranmer  (who  had  suggested  Henry's 
divorce)  threw  the  control  of  the  movement  entirely  into  the 
hands  of  graduates  of  Cambridge,  and  perhaps  no  more  strik- 
ing evidence  of  that  can  be  given  than  the  fact  that  out  of 
the  thirteen  compilers  of  the  new  prayer-book  issued  in  1549 
twelve  came  from  Cambridge,  while  the  litany  was  prepared 

16—2 


244     OUTLINES   OF   THE   HISTORY   OF   THE    UNIVERSITY. 

by  Cramner  from  the  work  of  Wied  and  Bucer1.  On  the 
other  hand,  all  the  leaders  of  the  Roman  party  (save  Fisher, 
who  belonged  to  an  older  generation)  were  Oxonians. 


The  development  of  the  study  of  classical  and  biblical 
literature  and  of  science,  and  the  rise  of  a  critical  spirit 
evoked  by  the  renaissance  mark  the  approaching  end  of  the 
reign  of  the  schoolmen,  and  the  mediaeval  curriculum  was  defi- 
nitely terminated  by  the  royal  injunctions  of  1535.  In 
these  the  king  ordered  that  henceforth  no  lectures  should  be 
given  on  the  sentences  or  on  canon  law ;  but  that  Greek, 
Latin,  and  divinity  should  be  taught  in  addition  to  the  tri- 
vium  and  quadrivium,  and  that  the  scriptures  should  be  read. 
The  university  system  of  teaching  by  means  of  the  lectures  of 
the  regents  was  essentially  bad.  To  remedy  this  it  was  ordered 
that  permanent  lecturers  should  be  appointed.  At  the  same 
time  the  large  number  of  clergy  and  others  who  were  living  at 
Cambridge  to  enjoy  the  social  advantages  of  the  place,  without 
any  intention  of  studying,  were  ordered  to  quit  it  at  once  if 
over  forty  years  old2. 

This  break-up  of  the  mediaeval  system  of  education  was 
followed  by  a  serious  fall  in  the  number  of  students,  until  in 
1545  the  entries  barely  exceeded  30,  while  at  Oxford  they 
sank  to  20.  So  serious  did  the  situation  become  that  the 
university  directed  all  "useless  books"  in  the  university  library 
to  be  sold ;  and  abolished  some  of  the  annual  offices  in  the 
university,  directing  that  their  duties  should  be  performed  by 
the  proctors  as  best  they  might.  In  1535  and  1537  the 
university  even  suspended  the  Barnaby  lecturer  on  mathe- 
matics, so  that  they  might  appropriate  his  salary  of  <£4  a 
year  for  the  benefit  of  the  lecturers  on  Hebrew  and  Greek. 

After   the   dissolution    of  the  monasteries,   Henry  VIII. 

1  Bucer  was  regius  professor  of  theology  at  Cambridge,  and  worked 
in  collaboration  with  Wied. 

2  Mullinger,  i.  630. 


THE   ELIZABETHAN   STATUTES.  245 

personally  investigated  the  position  of  the  universities,  and 
decided  that  they  were  doing  admirable  work  in  an  economical 
and  'efficient  manner1.  To  promote  study  he  endowed  at 
Cambridge  in  1540  five  regius  professorships  (see  p.  154). 

It  was  at  this  time  that  the  colleges  began  to  admit  pen- 
sioners as  well  as  scholars  (see  p.  154).  The  effect  on  the 
members  of  the  university  was  immediate  and  striking.  In 
1564  the  number  of  residents  had  risen  to  1267,  and  in  1569 
it  was  1630.  The  corresponding  numbers  at  Oxford  were 
rather  less  than  two-thirds  those  of  Cambridge. 

The  Edwardian  statutes  of  1549  were  an  honest  attempt  to 
reorganize  the  university  in  a  manner  suited  to  the  changed 
conditions  of  education  (see  p.  153),  but  no  serious  alterations 
were  made  in  the  constitution. 

The  Elizabethan  code  of  1570  made  numerous  changes2. 
That  code  was  mainly  designed  to  effect  three  things :  first,  on 
the  advice  of  Cecil,  to  make  the  university  directly  amenable  to 
the  influence  of  the  crown;  secondly,  on  the  advice  of  the 
bishops,  to  make  it  a  distinctly  ecclesiastical  organization,  with 
a  view  to  provide  a  supply  of  educated  clergy  for  the  realm; 
and  thirdly,  probably  by  command  of  the  queen,  to  ensure  that 
the  best  general  education  for  laymen  as  well  as  clergy  should 
be  obtainable;  finally,  the  better  to  secure  these  objects  it  was 
decided  to  offer  no  direct  encouragement  to  any  other  work. 
The  university  strenuously  opposed  this  limitation  of  its  powers 
and  studies,  but  without  success. 

The  subjection  of  the  university  to  the  power  of  the  crown 
was  effected  by  an  ingenious  artifice  suggested,  it  is  believed, 
by  Cecil.  From  time  immemorial  the  first  grace  at  a  congre- 
gation was  to  appoint  a  committee  of  five,  termed  the  caput,  to 
assist  the  chairman  at  that  meeting.  To  prevent  objectionable  or 
surprise  motions  a  grace  could  not  be  submitted  if  any  member 
of  the  caput  objected  to  it.  By  the  new  statutes  the  caput  was 
constituted  as  a  permanent  committee,  to  be  elected  by  the 

1  Mullinger,  i.  461. 

-  Mullinger,  ii.  222—34. 


246     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

heads  of  colleges,  doctors,  and  the  two  scrutators,  and  to  hold 
office  for  a  year.  Without  going  into  further  details- it  may  be 
said  that  this  gave  an  absolute  veto,  and  also  the  whole  power 
of  initiating  legislation,  to  an  irresponsible  committee  appointed 
by  the  heads :  and  even  then,  the  vice-chancellor  could  frus- 
trate all  legislation  by  refusing  to  summon  the  committee,  as 
happened  in  1751 — 52.  The  heads  were  also  directed  to 
nominate  two  names  for  the  vice-chancellorship,  one  of  whom 
must  be  chosen ;  and  consequently  since  1586  no  one  but  a 
head  has  been  elected  to  that  office.  Finally,  the  heads  were 
to  act  as  a  council  to  advise  the  chancellor  on  all  matters 
affecting  the  conduct  of  students,  and  were  to  fix  the  times  and 
subjects  of  all  exercises  and  lectures.  Besides  this  each  head 
was  given  a  power  of  veto  on  any  public  act  or  election  in  his 
own  college.  The  rights  of  the  regent  and  non-regent  houses 
were  not  directly  touched,  but  practically  the  heads  were  made 
supreme ;  and  as  there  were  but  fourteen  of  them,  nearly  all  of 
whom  were  hoping  for  preferment  at  the  hands  of  the  crown, 
there  was  little  difficulty  in  getting  their  sanction  to  anything 
the  government  wished.  The  proctors,  who  were  entitled  if 
they  wished  to  set  aside  both  chancellor  and  caput  and  to 
appeal  directly  to  the  university,  were  deprived  of  most  of 
their  powers,  and  expressly  declared  to  be  like  all  other  officers 
subordinate  to  the  chancellor.  Henceforth  they  were  nomi- 
nated by  the  colleges  according  to  a  certain  cycle,  and  the 
nomination  was  conditional  on  the  approval  of  the  heads. 

That  the  old  democratic  construction  was  open  to  grave- 
abuses  is  evident  from  the  unscrupulous  tactics  of  the  puritans 
at  some  of  the  congregations  in  the  spring  of  1570.  That 
party  were  not  then  strong  enough  to  control  the  policy  of 
the  university,  but  they  were  able  to  block  all  business  and 
legislation.  Several  congregations  broke  up  in  great  disorder, 
and  it  was  necessary  to  make  the  executive  efficient,  which- 
ever party  controlled  it.  The  new  oligarchic  constitution  erred 
on  the  other  side  and  almost  stifled  the  independent  criticism 
of  the  senate.  At  the  same  time  I  should  observe  that  any 


THE   ELIZABETHAN    STATUTES.  247 

member  of  the  senate  could  propose  a  grace,  and,  except  in 
times  of  great  excitement,  it  was  usual  to  allow  it  to  be  put  to 
the  vote.  It  will  be  noticed  that  by  the  statutes  of  1858  many 
of  the  powers  of  the  caput  were  transferred  to  a  council 
elected  by  the  resident  graduates,  which  is  so  far  perhaps  a 
reasonable  compromise,  but  against  this  must  be  set  the  fact 
that  the  members  of  the  senate  have  practically  been  deprived 
of  the  power  of  initiating  a  grace. 

To  secure  the  ecclesiastical  character  of  the  university  a 
decree  of  1553  was  confirmed,  by  which  the  subscription  of  the 
forty-two  articles  was  required  from  all  those  proceeding  to 
the  degree  of  M.A.,  B.D.,  and  D.D.;  and  in  1616  this  was 
extended  to  all  degrees. 

The  commissioners  who  drafted  the  Elizabethan  statutes  of 
1570  not  only  reorganized  the  constitution  of  the  university 
but  recast  the  curriculum.  Mathematics  was  excluded  from 
the  trivium,  and  undergraduates  were  directed  to  read  rhetoric 
and  logic,  but  the  course  for  the  master's  degree  was  left 
almost  unaltered  (see  p.  156).  The  necessary  exercises  for 
degrees  and  intervals  between  them  were  left  as  before,  except 
that  they  were  defined  rigorously  by  statute,  and  no  resident 
could  be  excused  from  any  of  them.  The  regency  of  masters 
was  extended  to  five  years,  after  which  a  master  became 
necessarily  a  non-regent.  Generally  the  discipline  of  the 
university  was  made  more  precise  and  rigid. 

The  new  statutes  recognized  the  change  which  had  taken 
place  in  the  system  of  education  by  assigning  to  a  regent  the 
duty  of  presiding  over  or  taking  part  in  the  public  disputa- 
tions, and  not  as  formerly  that  of  teaching  and  reading  in  the 
schools.  Finally,  new  statutes  could  only  be  made  if  they  in 
no  way  interfered  with  these. 

The  commissioners  saw  that  the  mediaeval  university  had 
failed  to  provide  teaching  suitable  for  most  of  its  members,  and 
had  made  no  proper  provisions  for  the  safety  and  discipline  of 
the  students;  and  they  realized  that  for  the  future  the  efficiency 
of  the  university  must  largely  depend  on  that  of  the  colleges. 


248     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

They  accordingly  spent  two  months  in  visiting  the  separate 
colleges1.  The  chief  object  of  the  changes  introduced  was  to 
secure  good  discipline  and  teaching,  and  decency  in  public 
worship 2.  The  commissioners  entered  into  such  detail  as  to 
settle  the  dress  of  members  of  the  university  for  all  time  to 
come,  and  even  the  private  prayers  they  should  use  when 
they  got  out  of  bed  in  the  morning. 

Some  of  the  provisions  of  these  statutes,  such  as  the  regency 
of  five  years,  the  power  of  veto  in  all  college  matters  by  its 
master,  and  possibly  the  residence  of  bachelors,  were  never 
enforced,  and  others  were  constantly  broken ;  but  taken  as  a 
whole  they  were  accepted  by  the  university  and  acted  on. 

Shortly  after  the  Elizabethan  statutes  came  into  effect  the 
incomes  of  the  colleges  began  to  rise,  partly  through  their  good 
management  of  their  estates,  partly  by  gifts  of  their  members. 
It  became  not  uncommon  to  have  a  surplus  after  meeting  the 
expenses  of  the  house,  and  as  the  surplus,  if  any,  was  divisible 
among  the  fellows,  a  fellowship  began  to  be  regarded  as  a  money 
prize  which  might  serve  as  a  provision  for  life — an  idea  which  no 
doubt  materially  retarded  the  intellectual  life  of  the  university. 

The  following  table,  which  is  as  complete  as  the  material 
at  my  command  permits,  will  enable  the  reader  to  judge  of  the 
progress  of  the  university.  It  gives  for  the  various  periods 
mentioned  the  average  yearly  number  of  matriculations,  and  the 
average  yearly  number  of  bachelor  degrees  (exclusive  of  those  of 

1  See   the   contemporary   account   published  in  Lamb's  Documents, 
London,  1838  (pp.  109—120). 

2  I  think  few  people  realize  how  intolerant  were  the  extreme  puritan 
party  at  this  time,  and  how  anxious  they  were  to  display  their  principles 
in  such  a  way  as  to  hurt  what  they  regarded  as  the  prejudices  of  their 
contemporaries.     As  an  illustration  of  the  length  to  which  they  were  pre- 
pared to  go,  I  may  mention  that  at  Emmanuel  (their  head-quarters  in  the 

university)  they  took  the  communion  "sittinge  upon  forms  about & 

did  pull  the  loafe  one  from  the  other and  soe  the  cupp,  one  drinking 

as  it  were  to  another  like  good  fellows. "     (Baker  vi.  85 — 86,  quoted  by 
Mullinger.)     Had  they  been  more  tolerant  and  courteous  I  believe  they 
would  have  triumphed ;   but  their  excessive  zeal  provoked  a  continual 
reaction  against  them  and  their  doctrines. 


THE   NUMBER  OF   STUDENTS. 


249 


medicine  and  theology)  which  were  conferred.  The  number  of 
undergraduates  resident  in  any  year  after  1600  may  be  taken 
roughly  as  being  four  times  the  number  of  those  who  took  the 
B.A.  degree  in  that  year.  I  have  added  the  corresponding 
numbers  for  Oxford  wherever  I  could  obtain  sufficient  data, 
but  I  have  no  doubt  that  the  statements  about  the  numbers 
of  matriculations  there  in  the  sixteenth  and  seventeenth  cen- 
turies (although  founded  on  official  data)  are  incorrect1. 


Period 

Cambridge 
matric  illations 

Oxford 
matriculations 

Cambridge 
bachelors 

Oxford 
bachelors 

From  1501  to  1516 

48 

1518  „  1570 

50   . 

...     43 

1571  „  1599 
1600      1633 



...    258  (?) 
..  312  (?) 

...178... 
229  .. 

...  110... 
...  191 

1634       1666 

...193... 

1667       1699 

.    326  (?) 

...185... 

...  174    . 

1700       1733 

297 

151 

1734       1766 

214     .. 

...106.  . 

1767       1799 

153 

241 

114 

1800       1833 

...     342  ... 

...    332 

.  .230  .. 

1834       1866 

447 

423 

346 

,      1867       1886 

743  .  . 

693      . 

...565... 

In  1887 

...  1012  ... 

...    766    ... 

...786... 

...612... 

There  is  but  little  difficulty  in  describing  the  life,  studies, 
and   amusements  of  the  students  of  this  period.     From  the 

1  The  numbers  given  for  different  years  are  extraordinarily  various 
and  bear  no  relation  to  the  number  of  B.A.  degrees  conferred  four  years 
later.  Thus  the  matriculations  for  1573  and  1575  are  returned  as  35 
and  467  respectively,  while  the  number  of  B.A.  degrees  taken  sixteen 
terms  (four  years)  later  are  given  as  97  and  115 :  the  latter  are  pro- 
bably correct.  In  some  years  the  entry  is  stated  as  having  been  larger 
than  is  the  case  now  (e.g.  the  return  for  1581  is  829),  and  it  is  certain 
that  there  was  then  no  accommodation  in  the  colleges  for  such  numbers. 
We  have  also  good  reason  for  saying  that  from  1570  to  1620  the  number 
of  residents  at  Oxford  was  about  two-thirds  of  the  corresponding  numbers 
at  Cambridge,  and  thus  must  have  been  much  smaller  than  the  alleged 
number  of  matriculations.  I  have  therefore  no  doubt  that  the  data  are 
untrustworthy. 


250     OUTLINES  OF  THE   HISTORY   OF  THE   UNIVERSITY. 

close  of  the  sixteenth  century  there  is  a  constant  succession  of 
diaries,  and  a  great  mass  of  correspondence  by  resident  mem- 
bers of  the  university.  The  social  life  of  the  seventeenth 
century  is  described  at  length  by  Mullinger  (vol.  n.  chap,  v.), 
and  that  of  the  eighteenth  century  by  Wordsworth.  It  was 
rougher  and  coarser  than  that  to  which  we  are  accustomed, 
but  it  was  more  civilized  and  courteous  than  that  of  the 
middle  ages. 

The  most  popular  amusements  of  the  undergraduates  of  the 
upper  classes  in  the  seventeenth  century  seem  to  have  been 
tennis,  cock-fighting,  fishing,  hawking,  hunting,  fencing,  and 
quoits  (at  one  time  or  another).  Football  also  was  apparently 
occasionally  played1.  Students  of  the  lower  classes  seem  to 
have  indulged  in  a  good  deal  of  rough  horse-play.  The  long 
winter  evenings  were  relieved  by  plays  performed  in  hall  after 
supper  on  Saturday  and  Simday  evenings ;  and  at  Christmas 
every  one,  young  and  old,  played  cards.  But  with  compulsory 
morning  chapel  at  6  a.m.,  and  deans  who  would  take  no  excuse 
for  absence,  the  hour  for  bed  was  earlier  than  at  present. 

The  usual  amusements  of  the  undergraduates  of  the 
eighteenth  century  were  tennis,  racquets,  and  bowls :  fives 
and  billiards  were  also  occasionally  played.  There  were  no 
athletic  clubs2,  and  the  only  organized  societies  (other  than 
dining  clubs)  that  I  know  of  were  those  for  ringing  peals  on 
church-bells  and  giving  concerts.  The  annual  fair  at  Stour- 
bridge  was  the  meeting-place  of  nearly  every  conjurer,  mounte- 
bank, and  company  of  strolling  actors  in  the  kingdom,  and  for 
a  fortnight  provided  a  perfect  surfeit  of  amusements. 

Discipline  was  stern.  The  birch  rod,  which  during  the 
seventeenth  century  and  the  early  half  of  the  eighteenth  cen- 
tury hung  up  at  the  butteries,  was  in  regular  use ;  and  once  a 

1  D'Ewes  mentions  a  match  in  1620  between  Trinity  and  St  John's. 

2  Boat-racing  on  the  river  was  apparently  introduced  about  1820,  and 
cricket  some  twenty  or  thirty  years  earlier :  it  is  said  that  the  first  public 
match  of  cricket  in  its  present  form  ever  played  was  that  of  Kent  against 
England  in  1746. 


COLLEGE   LIFE.  251 

week  the  college  dean  attended  in  hall — usually  on  Thursday 
evenings — to  see  that  the  butler  applied  it  to  such  youths 
under  the  age  of  eighteen  years  as  had  infringed  any  college 
rules,  or  sometimes  to  any  lad  who  was  beginning  to  shew 
himself  "too  forward,  pragmatic,  and  conceited". 

At  sunset  the  college  gates  were  locked.  All  the  students 
however  lived  in  college,  and  the  more  popular  colleges  were 
so  overcrowded  that  usually  three  or  four  men  had  to  share  a 
room.  Except  at  Trinity,  where  most  of  the  students  were 
sons  of  county  squires  or  parsons,  the  bulk  of  the  students 
came  from  what  is  called  the  lower  middle  class,  but  there  was 
a  fair  sprinkling  of  members  of  the  aristocracy  who  lived  apart 
from  the  rest  of  the  community.  The  expense  to  the  son  of 
a  county  squire  seems  to  have  been  equivalent  to  from  £180  to 
£220  a  year ;  to  a  fellow-commoner  about  £330  a  year.  The 
servants  of  the  college,  porters,  cooks,  &c.  were  mostly  sizars, 
who  received  education,  board,  and  lodging  in  return  for  their 
services. 

The  hour  of  dining  gradually  grew  later1.  In  1570  it  was 
at  9-0,  or  at  Trinity  at  lO'O.  By  1755  it  had  got  shifted  to 
noon.  In  1800  it  was  at  2 -15  at  Trinity,  and  at  1*30  at  most 
of  the  other  colleges;  and  the  senior  members  of  the  university 
began  to  complain  that  the  afternoon  attendance  at  the  schools 
was  in  consequence  much  diminished.  A  few  years  later 
dinner  was  usually  served  at  3-0,  but  until  1850  the  hour  did 
not,  I  think,  get  later  than  5.0.  Since  then  the  same  movement 
has  gone  on,  and  now  (1889)  dinner  at  Trinity  is  at  7.30. 

The  main  outlines  of  the  history  of  the  university  under 
the  Elizabethan  code  are  probably  well  known  to  most  of  my 
readers.  The  leading  features  are  connected  with  the  history 
of  the  theological  school,  the  rise  of  the  mathematical  and 
Newtonian  schools,  and  finally  the  outburst  of  activity  in  all 
departments  of  knowledge  which  preceded  the  grant  of  the  first 
Victorian  statutes. 

The  supremacy  of  the  Cambridge  school  of  theologians 
i  Wordsworth,  119—129. 


252     OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

remained  unbroken  till  the  death  of  James  I. ;  and  it  may  be 
illustrated  by  the  fact  that  110  less  than  four  out  of  the  five 
delegates  from  Britain  to  the  synod  of  Dort  in  1618  came  from 
Cambridge.  Its  influence  in  the  country  was  then  destroyed 
by  the  rise  of  the  high  church  party  under  Laud.  It  still 
however  remained  the  intellectual  centre  of  the  puritan  party ; 
and  of  the  numerous  university  graduates  who  emigrated  to 
America  between  1620  and  1647  over  three-fourths  came  from 
Cambridge. 

The  moderate  puritanism  which  had  been  predominant 
among  the  junior  members  of  the  university  for  a  century  and 
a  half,  and  the  moderate  anglicanism  which  the  majority  of  the 
senior  members  had  professed  for  the  same  time,  alike  almost 
disappeared1  with  the  excesses  and  violence  in  which  the 
Independents  indulged  in  the  middle  of  the  seventeenth 
century. 

With  the  accession  of  Charles  II.  the  same  difference  of 
opinion  which  had  marked  the  Oxford  and  Cambridge  of  the 
reigns  of  Henry  and  Elizabeth  again  shewed  themselves.  Oxford 
adopted  the  anglicanism  of  Laud,  and  the  politics  of  the 
extreme  tories.  Cambridge,  on  the  other  hand,  gave  rise  to 
the  school  now  known  as  that  of  the  Cambridge  Platonists,  and 
was  the  centre  of  the  whig  party.  I  gather  from  Mullinger's 
work  that  the  leading  members  of  the  Platonic  school  were 
Whichcote,  Cudworth,  Henry  More,  Culver  well,  Rust,  Glanvil, 
and  Norris:  they  form  the  successors  to  the  puritan  divines 
of  an  earlier  generation.  The  Platonists  were  succeeded  in 
natural  sequence  by  the  school  of  Sherlock,  Law,  and  Paley. 
They  in  their  turn  gave  place  on  the  one  side  to  the  evan- 
gelical school  of  Berridge,  Milner,  and  Simeon;  and  on  the 
other  side,  but  somewhat  later,  to  the  school  of  Maurice, 
Trench,  and  Hallam. 

External  politics  did  not  play  so  large  a  part  in  the 
internal  history  of  the  university  as  was  the  case  at  Oxford. 
Cambridge  was  the  centre  of  the  constitutional  royalists  at  the 
1  See  for  example  Pepys's  diary  for  February  1660. 


PREVALENT   STUDIES.  253 

beginning  of  the  sixteenth  century,  and  of  the  whig  party  at 
the  close  of  that  century.  The  revolution  of  1688  was  the 
triumph  of  the  latter.  Towards  the  latter  half  of  the  eighteenth 
century  the  politics  of  the  majority  of  the  residents  became 
tory  rather  than  whig,  but  the  toryism  was  of  a  moderate 
and  progressive  type. 

In  fact,  both  in  religion  and  politics,  the  dominant  tone  of 
the  university  was  what  its  friends  would  call  moderation, 
tolerance,  and  a  respect  for  the  rights  of  others,  and  what  its 
opponents  would,  I  suppose,  describe  as  lukewarm  ness,  and  a 
failure  to  carry  principles  to  all  their  logical  consequences. 

The  studies  prevalent  at  the  two  universities  mark  the 
same  difference  of  attitude1.  At  Oxford  dogmatic  theology, 
classical  philosophy,  and  political  history  occupied  most  atten- 
tiou.  At  Cambridge  the  negative  and  critical  philosophy  and 
logic  of  Ramus  was  followed  by  the  philosophy  of  Bacon  (and 
possibly  of  Descartes),  which  in  turn  was  displaced  by  that 
of  Locke.  The  modern  school  of  classical  literature  was 
worthily  represented  by  Bentley,  Porson,  and  others. 

But  it  was  the  mathematical  school  which  displayed  the 
most  marked  originality  and  power.  The  writings  of  Briggs, 
Horrox,  Wallis,  Barrow,  Newton,  Cotes,  and  Taylor  had 
placed  Cambridge  in  the  first  rank  of  European  schools. 
Under  the  influence  of  the  Newtonian  philosophy  mathematics 
gradually  became  the  dominant  study  of  the  place,  and  for  the 
latter  half  of  this  time  the  mathematicians  controlled  the 
studies  of  the  university  almost  as  absolutely  as  the  logicians 

1  It  is  interesting  to  observe  how  persistently  particular  studies  have 
been  prevalent  at  each  of  the  two  universities.  Leaving  aside  literature 
and  theology  (to  which  much  attention  was  paid  at  both  universities),  we 
may  say  that  interest  at  Oxford  has  always  been  specially  centred  in  philo- 
sophy in  its  wider  sense,  and  history  (constitutional  and  political);  while 
at  Cambridge  the  study  of  mathematical,  physical,  and  natural  science, 
and  the  applications  thereof,  have  generally  attracted  more  attention.  Of 
course  it  is  easy  to  cite  particular  instances  to  the  contrary,  but  I  believe 
the  assertion  above  made  is  substantially  true,  and  has  been  so  for  the 
last  four  hundred  years. 


2o4    OUTLINES   OF   THE   HISTORY   OF   THE   UNIVERSITY. 

had  controlled  those  of  the  mediaeval  university.  There  can 
be  no  doubt  that  this  was  a  real  misfortune,  and  that  it  led  to 
a  certain  one-sidedness  in  education.  At  the  same  time  it 
must  be  remembered  that  a  knowledge  of  the  elements  of 
moral  philosophy  and  theology,  an  acquaintance  with  the  rules 
of  formal  logic,  and  the  power  of  reading  and  writing  scholastic 
Latin  were  required  from  all  students. 

The  mathematicians,  to  do  them  justice,  threw  no  obstacle 
in  the  way  of  the  introduction  of  other  branches  of  learning; 
and  the  predominance  of  mathematical  studies  was  mainly  due 
to  the  fact  that  they  were  the  only  ones  in  which  any  con- 
tinuous and  conspicuous  intellectual  activity  was  displayed. 

The  isolation  of  the  Cambridge  mathematical  school  and 
the  falling-off  in  the  quality  of  the  work  produced  are  the  most 
striking  points  in  its  position  at  the  end  of  the  last  century. 
The  adoption  of  the  continental  notation,  the  development  of 
analytical  methods,  and  the  removal  of  the  barriers  which 
separated  Cambridge  mathematicians  from  their  contemporaries 
of  other  schools  distinguish  the  opening  years  of  this  century. 
Those  reforms  may  be  taken  as  effected  by  1825.  The  achieve- 
ments of  the  mathematical  school  for  the  years  subsequent  to 
that  will  form  a  brilliant  chapter  in  the  intellectual  history 
of  the  university,  but  those  who  created  the  new  school  are  too 
near  our  own  time  to  render  it  possible  or  desirable  to  analyse 
the  general  characteristics  of  their  work. 

It  was  not  however  only  in  mathematics  that  this  new 
renaissance  was  visible.  In  all  branches  of  learning  there  was 
an  awakening,  and  the  last  few  years  in  which  the  Elizabethan 
statutes  were  in  force  are  distinguished  by  the  opening  out  of 
fresh  studies,  no  less  than  by  the  development  of  old  ones.  Thus 
the  year  1858  is  the  close  of  a  well-defined  period  in  the  history 
of  the  university,  and  the  new  constitution  then  given  to  the 
university  marks  the  beginning  of  another  era,  which  I  prefer 
to  treat  as  wholly  outside  the  limits  of  this  work. 


INDEX1. 


Abacus,  2. 

Abelard,  223. 

Abinger,  Lord,  183. 

Acts,  chap,  ix,  also  145.  194.  214. 

Adams,  J.  C.,  134.     ref.  to,  105. 

Addition,  symbol  for,  15 

Adelhard  of  Bath,  4. 

Agnesi,  125. 

Airy,  Sir  George  Biddell,  132. 

—  ref.  to,  89.  121.  122.  131. 
Alembert,  d',  98. 

Algebra,  works  on,  17.  40.  45.  86. 

95.  102.  104.  108.  109.  111.  124. 

129. 

Algebraic  curves,  64. 
Algorism,  5.  7. 
Alkarismi,  5. 
Almagest,  8.  23. 
Almanack,  Nautical,  105.  108. 
American  journ.  of  math.,  23. 
Amusements  of  students,  237-8, 250. 
Analytical  conic  sections,  44.  129. 

—  geometry,   introduction  of,  44. 
129. 

—  works  on,  63.  102.  132. 
Analytical  school,  The,  chap.  vn. 

—  society,  120,  121.  125,  128. 

—  machine,  126. 

Annals  of  Cambridge,  ref.  to,  138. 

220.  231.  240. 
Anne,  Queen,  86. 
Anstey,  Christopher,  157. 
Anstey,  Henry,  ref.  to,  138.  220. 
Apollonius,  3.  4.  40.  48.  69.  92. 


Aquinas,  144. 
Arab  science,  3.  4. 
Arabic  numerals,  4.  5.  7. 
Archimedes,  3.  4.  48.  69. 
Aristotle,  45.  143.  149. 
Aristotelian  philosophy,  35.  143. 
Arithmetic,  see  quadrivium. 
Arithmetic,  works  on,  4.  5.  6.  15. 
93.  95.  124. 

—  mediaeval,  2. 

—  Newton's  Universal,  58.  66.  68. 
84.  86.  93. 

Arithmetica  infinitorum,  of  Wallis, 

42.  44. 

Arithmetica  logarithmica,  28. 
Arts,  Bachelor  of,  title  of,  2.  148. 

—  degree  of,  139.  145-8.  156-7. 

—  Master  of,  2.  13.  142.  157. 
Arundel,  Abp,  242. 
Ascham,  ref.  to,  153. 
Assumption,  rule  of  false,  16. 
Astrolabe,  work  on,  5.  21. 
Astrology,  8.  9,  see  also  quadrivium. 
Astronomical  society,  125. 127. 133. 
Astronomy,  see  quadrivium. 

—  works  on,  36.  95.  96.  103.  104. 
105.  109.  118.  127. 

—  Ptolemaic,  23. 

Athense  Cantabrigienses,  ref.  to,  10. 

15.  19.  234. 
Attraction,  law  of,  60. 

—  capillary,  215. 

Atwood,  George,  107,  ref.  to,  106. 
Aubrey,  J.,  ref.  to,  36.  37. 


i  The  Index  has  been  prepared  at  the  University  Press.    I  have  revised  and 
added  to  it  and  hope  there  are  no  omissions  of  importance.    W.  W.  R.  B. 


256 


INDEX. 


Babbage,  Charles,  125,  ref.  to,  117. 
Bachelor  of  arts/title  of,  2.  148. 

—  degree  of,  139.  145-8.  156-7. 
Bacon,  Francis,  79.  253. 
Bacon,  Eoger,  5-6.     ref.  to,  3. 
Ball's  Hist,  of  math.,  ref.  to,  3.  51. 
Balsham,  Hugh,  141. 

Barnaby  lecturers,  144.  244.  • 

Barn  well  Priory,  222. 

Barrow,  Isaac,  46-49. 

-  ref.  to,  14.  34.  53.  56.  79.92.  95. 

108.  253. 

Bashforth,  Francis,  135. 
Bedells,  146.  147.  228. 
Bedwell,  Thomas,  23.  ref.  to,  15. 27. 
Bentley,  Richard,  80-81. 

—  ref.  to,  75.  79.  81.  89.  92.  128. 
170.  193.  253. 

Bernoulli,  Jacob,  77.  93. 
Bernoulli,  John,  77.  90.  93.  97.  98. 
Billingsley,  Sir  Henry,  22-23. 

—  ref.  to,  15. 

Binomial  theorem,  52.  65.  66. 
Biographia  Britannica,  ref.  to,  41. 

83.  89. 

Biot,  ref.  to,  72. 
Biretta,  240. 
Blackburn,  Hugh,  135. 
Bland,  Miles,  110. 
Bligh,  Reginald,  ref.  to,  192. 
Blundeville,  Thomas,  21-22. 

—  ref.  to,  13.  26. 
Boethius,  2.  3. 
Bologna,  University  of,  9. 
Boltzmann,  137. 
Bowstead,  Joseph,  183. 
Boyle,  Robert,  95. 
Boyle  lectures,  80. 
Bradwardine,  6. 
Brasse,  John,  184. 
Breda,  College  at,  40. 
Brewer,  ref.  to,  6. 
Bridge,  Bewick,  109. 
Bridges,  Noah,  ref.  to,  31. 
Briggs,  Henry,  27-30. 

—  ref.  to,  15.  253. 
Brinkley,  John,  109. 
British  association,  125. 
Bucer,  Martin,  244. 

Buckley,  William,  22.    ref.  to,  13. 
Bulseus,  ref.  to,  227. 
Bullialdus,  hypothesis  of,  38.  96. 


Burgon,  J.  W.,  ref.  to,  27. 
Buridanus,  144. 

Byrdall,  Thomas,  87.     ref.  to,  75. 
Byrom,  John,  81. 

Caius  College,  103.  109.  116.  118. 

120.  134.  173. 
Caius,  Dr.,  155. 
Calculus,  The,  34.  71.  72.  77.  88. 

100,  111.  122.  124.  212.  213, 
Cambridge,  University  of,  7.  8.  10. 

11.  14.  221.  225. 
—  Mathematics  at,  42,  46.  72.  73. 

92. 97.  99. 119. 134. 137.  220.  253. 

254. 

—  Observatory  at,  89.  118.  124. 

—  Annals  of,  138.  220.  231.  240. 

—  university  reporter,  ref.  to,  229. 
Campanus,  4.  7.  8. 

Campbell,  L.,  ref.  to,  135. 

Capillary  attraction,  215. 

Caps,  College,  240. 

Cardan,  13. 

Caroline,  Queen,  85. 

Carr,  J.  A.,  169. 

Cartesian  theory,  48.  61.   62.  75. 

164. 

Cartesian  philosophy,  see  Descartes. 
Cassiodorus,  2.  3. 
Caswell,  96. 
Cavalieri,  33. 

Cavendish,  Hon.  H.,  114-5. 
Cavendish  professorship,  136. 
Cayley,  Arthur,  134.     ref.  to,  91. 
Chafin,  W.,  172. 
Challis,  James,  132.     ref.  to,  89. 
Challis  MSS.,  ref.  to,  169.  194. 
Champeaux,  William  of,  222. 
Chancellor,  The,  228. 
Charles  II.,  King,  49. 
Charterhouse,  46. 
Charts,  Mercator's,  26. 
Chasles,  5.     ref.  to,  6. 
Christ  Church,  Oxford,  79. 
Christ's  College,  103.  120.  154. 
Churton,  Ralph,  ref.  to,  9. 
Clairaut,  99. 

Clare  College,  49.  75.  83.  95.  108. 
Clark,  W.  G.,  ref.  to,  129. 
Clarke,  Samuel,  76-77. 

—  ref.  to,  75.  92.  93. 
Clausius,  137. 


INDEX. 


257 


Clavis  mathematica,  30.  93. 
Clerke,  Gilbert,  39. 
Clifton,  Eobert  Bellamy,  137. 
Coaching,  116.  160-3. 
Coddington,  Henry,  131. 
Co-efficients,  Laplace's,  215. 
Coleridge,  Hartley,  ref.  to,  80. 
Collins,  55.  56. 
Colson,  John,  70.  100. 
Comets,  theory  of,  61. 
Comitia  majora,  219. 

—  minora,  217. 

—  priora,  217. 

Commencement-day,  149.  219. 
Conic  sections,  44. 

—  works  on,  92.  95.  104.  109.  129. 
130. 

Constructio,  Napier's,  29. 

Convivas,  154. 

Cooper,  ref.  to,  138.  220.  231.  240. 

Copernican  hypothesis,  14.  18.  20. 

Copernicus,  14.  18.  20. 

Copley  medal,  108. 

Corpus  Christi  College,  17. 

Cossic  art,  The,  17. 

Cotes,  Eoger,  88. 

—  ref.   to,  67.  75.  86.  90.  91.  94. 
111.  126.  180.  191.  253. 

Craig,  John,  77-78.     ref.  to,  75. 
Cramer,  Gabriel,  ref.  to,  65. 
Craven,  W.,  172. 
Cremona,  Gerard  of,  4. 
Creswell,  Daniel,  110. 
Cricket,  introduction  of,  250. 
Croone,  William,  91. 
Cubics,  Newton's  classification  of, 

63-5. 

Culpepper,  Nicholas,  39. 
Cumberland,  Earl  of,  26. 
Cunningham,  William,  17. 
Curriculum  for  M.A.  degree,  2.  14. 

157.  247. 
Curves,  quadrature  of,  43.  50.  63. 

65.  70.  77. 

—  rectification  of,  44.  66. 
Cycloidal  pendulum,  90. 
Cycloids,  44. 

Dacres,  Arthur,  49. 
D'Alembert,  98. 
Dalton,  John,  114. 
D'Arblay,  A.  C.  L.,  120. 

B. 


Darwin,  G.  H.,  89. 

Dawson,  John,  ref.  to,  162. 

Dealtry,  111.  113.  114. 

Dechales,  95. 

Decimal  notation,  introduction  of, 

28. 

Dee,  John,  19-21.     ref.  to,  13.  22. 
Degrees,  B.A.,  2.  139.  145-8.  156. 

—  M.A.,  2.  13.  142. 

—  in  medicine,  151. 

—  in  music,  151. 
Deinfle,  P.  H.,  ref.  to,  229. 
Deluge,  Whiston's  theory  of  the,  83. 
De  Moivre,  Abraham,  87.  90.  101. 
De  Morgan,  Augustus,  132. 

—  ref.  to,  4.  5.  10.  21.  22.  78.  108. 
111.  113.  119.  121.  122. 130.  132. 
180.  182.  184. 

Desaguliers,  93. 

Descartes,  ref.  to,  33.  42.  44.  52. 

77.  79.  95.  108.  253. 
Determinations,  148.  157. 
D'Ewes,  ref.  to,  250. 
Differential  calculus,  72.   77.  111. 

122. 124.  212.  213. 
Diffraction,  Theory  of,  55.  62. 
Digges,  Leonard,  21. 
Digges,  Thos.,  21.     ref.  to,  13. 
Diophantus,  40. 
Disney,  W.,  172. 
Disputations,  chapter  ix. 
Ditton,  Humphry,  93.  95.  125. 
Dormiat,  167. 
Dort,  Synod  of,  252. 
Dress  of  students,  239. 
Dublin,  Trinity  College,  243. 
Duns  Scotus,  143.  239. 
Durham,  University  of,  133. 
Dyer,  George,  ref.  to,  239. 
Dynamics,  works  on,  45.  107.  130. 

Earnshaw,  Samuel,  213. 
Edinburgh,  University  of,  135. 
Education,    Systems    of,    chapter 

VIII. 

Edward  I.,  King,  224. 
Edward  HI.,  King,  235. 
Edward  IV.,  King,  9. 
Edward  VI.,  King,  15.  22. 
Edwardian  statutes,  13.  153.  154. 

245. 
Egyptian  hieroglyphics,  115. 

17 


258 


INDEX. 


Elastic  bodies,  45. 
Electricity,  works  on,  104.  136. 
Elliptic  integrals,  215. 
Ellis,  Kobert  Leslie,  130. 
Ellis,  Sir  Henry,  ref.  to,  153. 
Elizabeth,  Queen,  20.  26. 
Elizabethan  statutes,  13.  35.  139. 

155.  158. 164.  184.  245.  247.  251. 
Emmanuel  College,  35.  38.  41.  91. 

100.  155.  172. 
Encyclopaedia  Britannica,  25.  135. 

136. 
Encyclopaedia  Metropolitana,  124. 

133. 

Equality,  Symbol  for,  16. 
Equations,  Theory  of,  58.  59. 
Erasmus,  152.  242.  243. 
Esquire  bedells,  228. 
Euclid's  Elements,  3.  4.  7.  8.  9.  13. 

14.  18.  22.  23.  46.  83.  92.  180. 
Euclid's  works,  ref.  to,  3.  29.52.95. 

105.  111.  131. 
Euler,  63,  97,  98. 
Examination    papers    (problems), 

195-197.  200-208. 
Expenses  of  students,  236.  251. 
Experimental  physics,  114.  115. 

Fairs,  Stourbridge,  223.  233.  242. 
250. 

—  Leipzic,  223. 

—  Nijnii  Novgorod,  223. 
False  assumption,  rule  of,  16. 
Faraday,  Michael,  136. 
Farish,  William,  106. 

—  ref.  to,  112.  186. 

Father  of  a  college,  The,  147.  149. 

217. 

Fellow-commoners,  183. 
Felstead  School,  46. 
Fermat,  42.  44. 
Fisher,  Bp,  154.  228.  242. 
Flamsteed,  John,  78-79. 

—  ref.  to,  63.  75.  89.  96. 
Fluids,  motion  of,  103. 
Fluxional  calculus,  72.  100.  121. 
Fluxions,  works  on,  52.  58.  63.  66. 

70.  71.  78.  95.  104.  111.  121. 
Forman,  Simon,  24-25.   ref.  to,  15. 
Foster,  Samuel,  38. 
Frederick  II.,  Emperor,  4. 
Frend,  William,  109. 


Frere,  190. 

Friction,  laws  of,  103. 

Galileo,  18. 
Garnett,  Wm.,  135. 
Gaskin,  Thomas,  183. 
Gassendi,  96. 
Gauss,  51.  136. 
General  examination,  213. 
Gentleman's  Magazine,  108.  172. 
Geometrical  optics,  works  on,  93. 
Geometry,  works  on,  7.  109.  110. 
122. 129. 130.  see  also  quadrivium. 

—  Savilian  professorship  of,  37.  42. 

—  analytical,  44.  122.  129. 
George  I.,  King,  188. 
Gerard  of  Cremona,  4. 
Gherardi,  9. 

Gisborne,  Thomas,  183. 
Glaisher,  J.  W.  L.,  187.  211. 
Glasgow,  University  of,  137. 
Glomerel,  141. 

Gooch,  Wm.,  178.  179.  180.  192. 
Gowns,  Academical,  240. 
Grammar,  degrees  in,  141. 
Gravesande,  W.  J.  's,  93. 
Gravitation,  theory  of,  52.  59.  60. 
Greek,  professorship  of,  47.  154. 
Green,  George,  134. 
Green,  Kobert,  95.  132. 
Gregory,  David,  87.  92.  93.  96. 
Gregory,  Duncan  Farquharson,  130. 
Gresham  College,  38.  47.  49. 
Gresham,  Sir  Thos.,  27. 
Griffin,  Wm.  Nathaniel,  131. 
Grosseteste,  6. 
Grynseus,  23. 
Gunning,  ref.  to,  194. 
Giinther,  work  by,  1. 
Gwatkin,  Kichard,  121. 

Haileybury  College,  109.  133. 

Hall,  Thos.  G.,  130. 

Halley,  Edmund,  59.   63.   79.   87. 

108. 

Halliwell,  ref.  to,  5.  7.  21. 
Halsted,  ref.  to,  23. 
Hamilton,  Parr,  122.  129. 
Hammond,  95. 
Hankel,  ref.  to,  8. 
Harmonics,  Smith's,  91. 
Harriot,  Thomas,  31.  32.  93.  95. 


INDEX. 


259 


Harvey,  Gabriel,  24. 

Harvey,  John,  24.    ref.  to,  15. 

Harvey,  Eichard,  24.    ref.  to,  15. 

Heaviside,  J.  W.  L.,  133. 

Hebrew,  professorship  of,  154. 

Henry  III.,  King,  224. 

Henry  VI.,  King,  142. 

Henry  VIII.,  King,  154.  244-5. 

Henry,  Charles,  16. 

Herbert,  Lord,  13. 

Herschel,  Sir  John,  126. 

—  ref.  to,  117.  119.  121.  125.  130. 
Hervagius,  23. 

HeveKus,  36. 

Hieroglyphics,  Egyptian,  115. 

Hill,  Thos.,  23.    ref.  to,  15. 

Hist,  of  mathematics,  ref.  to,  3.  51. 

Hoadly,  B.,  ref.  to,  76. 

Hodgkins,  John,  9. 

Hodson,  William,  194. 

Holbroke,  John,  9. 

Holywood,  5.    ref.  to,  8.  78. 

Hood,  Thos.  23-24.    ref.  to,  15. 

Hoods  for  graduates,  240.  241. 

Hook,  W.  F.,  ref.  to,  6. 

Hooke,  Robert,  49.  59.  68. 

Hopkins,  Wm.,  163. 

Horrox,  Jeremiah,  35. 

—  ref.  to,  33.  253. 
Hostels,  231.  235. 
Huddling,  184-6. 
Hustler,  J.  D.,  113.  182. 
Huygens,  54.  55.  59.  93.  108. 
Hydrodynamics,  works  on,  61. 
Hydrostatics,  works  on,  61.  90. 104. 

110. 
Hymers,  John,  129. 

Inception,  149.  150. 
Indices,  law  of,  42.  43. 
Indivisible  college,  The,  38. 
Infinitesimal  calculus,  34.  72. 
Injunctions  of  1535,  12.  153.  221. 

244. 

Integral  calculus,  122. 
Integrals,  elliptic,  215. 
Interpolation,  principle  of,  43.  44. 

56. 

Inverse  problem  of  tangents,  57. 
Isidorus,  2.  3. 

Isometrical  perspective,  106. 
Isoperimetrical  problems,  118. 


Jack,  William,  137. 

James  I.,  King,  159.  252. 

James  II. ,  King,  62. 

Jebb,  John,  ref.  to,  184.  188.  190. 

191. 

Jebb,  E.  C.,  ref.  to,  80.  82.  159. 
Jesus  College,  79.  109.  153.  169. 
Johnson,  J.,  165. 
Jones,  Thomas,  173.  184. 
Jones,  William,  93.  95.  96. 
Joule,  137. 

Journal  of  math. ,  American,  23. 
Julian  calendar,  20. 
Junior  optimes,  168.  171. 
Jupiter  and  Saturn,  conjunction  of, 

Jurin,  James,  87.    ref.  to,  75. 

Keill,  John,  87. 
Keningham,  William,  17. 
Kepler,  52.  59.  78.  93.  96. 
Kersey,  95. 
Kinckhuysen,  53. 
King,  Joshua,  132. 

—  ref.  to,  134.  185. 
King's  College,  9.  10.  38. 
King's  Hall,  10.  234.  236. 
Kollar,  V.,  ref.  to,  8. 
Kuhff,  Henry,  130. 
Kurtze,  M.,  ref.  to,  -9. 

Lacroix,  120. 

Ladies's  diary,  100. 

Lady  Margaret  professorship,  154. 

Lagrange,  51.  98.  182. 

Lamb's  Documents,  ref.  to,  248. 

Laplace,  51.  98.  114.  118. 

Laplace's  coefficients,  215. 

Lardner,  Dionysius,  131. 

Latin  grammar  and  language.  10G. 

140-3.   153.  160.  165.  182.  254. 

see  also  trivium. 
Laughton,  Eichard,  75. 

—  ref.  to,  75.  88.  92. 
Law,  degrees  in,  151. 

—  of  attraction,  60. 

—  of  indices,  42.  43. 
Laws  of  motion,  61. 

—  of  friction,  103. 
Lax,  William,  105. 

—  ref.  to,  105.  125.  169,  178.  179. 
Le  Clerk,  95. 


260 


INDEX. 


Lectiones  mathematics,  47. 

—  opticse  et  geometric®,  47. 
Lecturers,  244. 

-  Barnaby,  144.  244. 
Lectures,  Boyle,  80. 

-  times  of,  143.  144. 

—  places  for,  228. 
Lefort,  72. 
Legendre,  83. 

Leibnitz,  54.  56.  57.  58.  65.  68.  71. 

72.  87.  93.  97. 
Leipzic,  University  of,  8.  9. 

-  Fair  of,  223. 
Leonardo  of  Pisa,  4. 
Leslie,  J.,  ref.  to,  10. 
Liber  abbaci,  The,  4. 
Libraries,  229.  230. 

Light,  reflexion  and  refraction  of, 

48. 

Lilly,  William,  24. 
Linear  perspective,  88.  110. 
Lists,  publication   of  tripos,  193. 

214. 

Little-G-o,  211. 

Locke,  John,  35.  79.  164.  191.  253. 
Lodgings,  230. 
Logarithms,  invention  of,  28. 

—  works  on,  28.  96. 
Logic,  see  trivium. 

London,  University  of,  132.  133. 

—  mathematical  society,  133.  187. 
211. 

Long,  Koger,  105.     ref.  to,  105. 
Lowndean  professorship,  105.  135. 
Lucas,  Henry,  47. 
Lucasian  professorship,   47.    100. 

101.  118.  125.  132. 
Lux  Mercatoria,  31. 
Lyte,  ref.  to,  140.  143.  220.  226. 

229. 

Machine,  Analytical,  146. 
Maclaurin,   93.   98.  99.   125.    180. 

191.  192. 
Magdalene   College,  49.  101.  106. 

165. 

Magnetism,  works  on,  104.  136. 
Maps  on  Mercator's  scale,  26. 
Marie,  Maximilian,  ref.  to,  41. 
Marks  in  tripos,  216. 
Marshall,  Koger,  9. 
Martin,  Francis,  182. 


Mary,  Queen,  15. 

Maseres,    Francis,    108.     ref.    to, 

101.  125. 

Maskelyne,  Nevil,  108. 
Master,  Richard,  10. 
Master  of  arts,  2.  13.  14.  142.  157. 

247. 

Master  of  grammar,  141. 
Master  of  rhetoric,  141. 
Mathematics  at  Cambridge,  42.  46. 

72.  73.  92.  97.  99.  119.  134.  137. 

215.  220.  253-4. 

—  at  Oxford,  46.  87. 
Mathematical  studies,  Board  of,  215. 

-  tables,  5.  28.  41. 

—  tripos,  chapter  x. 

Mathesis  universalis,  of  Wallis,  44. 
Maule,  W.  H.,  120. 
Mawson,  Matt.,  188. 
Maxwell,  James  Clerk,  135. 

-  ref.  to,  114.  132.  137. 
Mayor,  J.  E.  B.,  ref.  to,  224. 
Mechanics,  works  on,  95.  104.  109. 

130. 

Medal,  Copley,  108. 
Medicine,  degrees  in,  151. 
Mediaeval  mathematics,  chapter  i. 

—  education,  138-152. 
Melanchthon,  13. 
Mercator,  Gerard,  25.  96. 

—  charts  of,  26. 
Meredyth,  Moore,  167. 
Merton  College,  Oxford,  6.  29. 
Michael-house,  139. 
Michell,  John,  115. 
Michelotti,  87. 
Microscope,  54. 

Milner,  Isaac,  102.  ref.  to,  100. 113. 

Milnes,  95. 

Moderators,  166.  167.  170.  190. 
191.  210.  215.  219. 

Modern  mathematics,  commence- 
ment of,  chapter  in. 

Molyneux,  95. 

Monasteries,  231-3. 

Moors,  mathematics  of  the,  3.  4. 

Monk,  W.  H.,  80.  83. 

Morland,  Sir  Samuel,  49. 

Motion,  laws  of,  61. 

—  of  fluids,  103. 

MuUinger,  ref.  to,  8.  14.  138.  140. 
141-5.  150.  153.  156.  158.  220. 


INDEX. 


261 


229.  231-4.  235.  237.  239.  244. 

245.  248.  250.  252. 
Multiplication,  symbol  for,  30. 
Munimenta  academica,   138.  140. 

142.  143.  218.  220.  226.  234.  235. 
Murdoch,  Patrick,  ref.  to,  65. 
Music,  see  quadrivium. 
Music,  degrees  in,  151. 

Napier  of  Mercbiston,  27.  28.  30. 

108. 

Napier,  A.,  ref.  to,  46. 
Nash,  Thos. ,  24. 
Natural    philosophy,     works    on, 

107.  110.  135. 

—  science,  works  on,  95.  107.  110. 
135. 

Nautical  almanack,  105.  108. 
Navigation,  earliest  scientific  treat- 
ment of,  26. 
Neil,  William,  44. 
New  Eiver  Company,  23.  27. 
•  Newton,  Isaac,  chapter  iv. 

—  ref.  to,  14.  34.  36.  45.  48.  79. 
83.  84.  85.  87.  90.  91.  93.  95,  96. 
97.  101.  104.  113.  123.  124.  133. 
158.  171.  174.  180.  182. 184. 191. 
192.  253. 

Newton  MSS.,  Portsmouth  collec- 
tion of,  63. 

Nij nii-Novgorod,  Fair  of,  223. 

Niven,  W.  D.,  135. 

Non-regent  house,  227.  246. 

Norfolk,  John,  7. 

Notation,  introduction  of  decimal, 
28. 

Numbers,  square,  40. 

Numerals,  Arabic,  4.  5.  7. 

—  Roman,  7. 

Observatory  at  Cambridge,  89. 118. 

124. 

01  TroAXof,  170.  171. 
Oldenburg,  55. 
Opponent,  165.  167. 
Optics,  works  on,  62.  63.  65.  68.  77, 

86.  91.  95.  104.  131. 
Optime,  168.  171.  189. 
Oughtred,  William,  30-31. 
-  ref.  to,  15.  37.  38.  39.  52.  93. 
Ovid,  143. 


Oxford,  mathematics  at,  46.  87. 
Oxford,  University  of,  3.   5.   7.  9. 

10.   11.  29.  133.    137.   143.  150. 

152.  154.  225.  253. 
Ozanam,  95. 

Pacioli,  10. 

Padua,  University  of,  9.  10.  242. 
Paley,  Win.,  113. 162. 180. 190.  252. 
Paris,  University  of,  5.  7.  9.  143. 

150.  152.  154.  193.  222.  225.  227. 

236. 

Pascal,  42. 

Paynell,  Nicholas,  10. 
Peace  and  Union,  Frere  on,  109. 
Peacock,  D.  M.,  121. 
Peacock,  George,  124. 

—  ref.  to,  105.  115.  117.  120.  121. 
125. 138.  141.  144.  147.  150.  156. 
158.  179.  186.  210.  220.  226.  227. 
232-3. 

Pell,  John,  40.  41. 

—  ref.  to,  31.  33.  95. 
Pemberton,  Henry,  67. 
Pembroke  College,  9.  10.  105.  134. 

135. 

Pendulum,  cycloidal,  90. 
Penny  Cyclopaedia,  25.  40.  118. 
Pensioners,  245. 
Pepys's  Diary,  ref.  to,  252. 
Perspective,  isometrical,  106. 

—  linear,  88.  110. 
Peterhouse,  9.  109.  114.  125.  135. 

162.  181.  231. 
Philosophical  Society,  Cambridge, 

128. 
Philosophical  transactions,  77.  87. 

88.  100.  101.  102.  103.  105.  107. 

109.  110.  125.  133.  134. 
Philosophy,  Aristotelian,  35,  143. 
Physics,  works  on,  95. 

—  experimental,  114. 
Pileum,  149. 

Pisa,  University  of,  9. 
Platonists,  the  Cambridge,  252. 
Plume,  Thomas,  89. 
Plumian  professorship,  89.  91. 103. 

132. 

Poggendorff,  ref.  to,  103.  107.  109. 
Poisson,  136. 

Pole,  Cardinal,  Statutes  of,  154. 
Pollock,  Sir  Frederick,  111.  210. 


262 


INDEX. 


Pond,  John,  132. 

Pope,  Walter,  ref.  to,  36. 

Portsmouth  collection  of  Newton 

MSS.,  63. 

Prague,  University  of,  8.  9. 
Previous  examination,  211. 
Principia  of  Newton,  ref.  to,  36.  45. 

58.59.  61.  62.  63.  67.  68.  74.75.79. 

83.  86.  89.  93.  98.  111.  161.  181. 
Priory  of  Barn  well,  222. 
Priscian,  141.  143. 
Prisms,  53.  54. 
Pritchard,  Charles,  133. 
Private  tutors,  160-3. 
Problem  papers  in  tripos,  195-197. 

200-9. 
Proctors,  166.  167.  170.  217.  219. 

227.  241.  246. 

Professorships,  Cavendish,  136. 
—  Lady  Margaret,  154. 

—  Lowndean,  105.  135. 

—  Lucasian,   47.    100.    101.    118. 
125.  132. 

—  Plumian,  89.  91.  103.  132. 

—  Kegius,  154.  245. 

—  Sadlerian,  91.  134. 

—  Savilian  (at  Oxford),  37.  42.  87. 
133. 

Proportion,  rules  of,  6.  7. 

—  symbol  for,  31. 
Pryme,  G.,  163. 

Ptolemaic  astronomy,  work  on,  23. 

—  ref.  to,  31.  33.  95. 
Ptolemy's  works,  3.  4.  8.  9.  13. 
Puffendorf,  159. 

Quadragesimal  exercises,  148.  157. 
Quadrature  of  curves,  50.  63.  65. 

70.  77. 
Quadrivium,  the,  2.  3.  6.  7.  9.  13. 

148.  244. 

Queens'  College,  42.  102.  115.  132. 
Questionists,  145.  146.  192. 

Eaces,  Semitic,  123. 
Eainbow,  theory  of,  53. 
Eamus,  Peter,  14. 

—  ref.  to,  23.  35.  145.  164.  253. 
Eatdolt,  4. 

Eay,  John,  46. 

Eecord  Office,  ref.  to,  224. 

Eecorde,  Eobert,  15-19. 


Eecorde,  ref.  to,  11.  12. 18.  19.  243. 
Eeflexion,  laws  of,  48. 
Eeformation,  the,  243. 
Eefraction,  laws  of,  48.  54. 
Eegent-house,  the,  226.  228.  246. 
Eegiomontanus,  10. 
Eegius  professorships,  154.  245. 
Eenaissance,  the,  12.  137.  242. 
Eeneu,  William,  ref.  to,  84. 
Eespondent,  165.  167. 
Eheims,  College  of,  19. 
Ehetoric,  see  trivium. 

—  Master  of,  141. 
Ehonius,  algebra  of,  40. 
Eiccioli's  Almagest,  78. 
Eichard  II.,  King,  234. 
Eidlington,  Wm.,  157. 
Eiley,  E.,  178. 
Eobinson,  T.,  120. 
Eohault,  works  of,  76.  93.  95. 
Eomau  numerals,  use  of,  7. 
Eooke,  Laurence,  38. 
Eouth,  E.  J.,  135.  163. 
Eowning,  John,  107.      ref.  to,  106. 
Eoyal  astronomical  society,  133. 
Eoyal  society,  37.  63.  87.  100.  109. 

125.  126. 

-  of  Edinburgh,  134. 136. 
Eule,  of  proportion,  6.  7. 

—  of  false  assumption,  16. 
Eumford,  Count,  114. 
Eyan,  E.,  120. 

Sacrobosco,  5.     ref.  to,  8.  78. 
Sadlerian  professorship,  91.  134. 
St  Catharine's  College,  118. 
St  John's  College,  47.  80.  88.  110. 

121.  126.  135.  155. 
Salerno,  University  of,  225. 
Sanderson's  Logic,  ref.  to,  51. 
Saturn  and  Jupiter,  conjunction  of, 

24. 
Saunderson,  Nicholas,  86. 

—  ref.  to,  75.  88.  92.  101. 
Savile,  Sir  Henry,  29. 

Savilian  professorships,  37.  42.  87. 

133. 

Scarborough,  Charles,  37. 
Schneider,  ref.  to,  5. 
Schola?  academic®,  ref.  to,  75.  94. 

106.  160. 162.  164.  167.  187. 
Schooten,  ref.  to,  52.  108. 


INDEX. 


263 


Scott,  Sir  Walter,  ref.  to,  17. 
Scotus,  Duns,  143.  239. 
Scrutators,  227.  241.  246. 
Semitic  races,  123. 
Senate-house,  the  old,  229. 

—  erection  of  existing,  188. 

—  examination,  chapter  x. 
Senior  optimes,  168.  171.  189. 
Sentences,  the,  145.  153. 
Sextant,  107. 

Shepherd,  Anthony,  103.  ref.  to,  89. 

Shilleto,  Richard,  181. 

Sidney  Sussex  College,  36. 100. 155. 

Simpson,  125. 

Simson,  Eobert,  84.  92. 

Sloman,  H.,  72. 

Smalley,  G.  E.,  135. 

Smith,  John,  105. 

Smith,  Eobert,  91. 

—  ref.  to,  75.  89.  94.  103. 
Smith,  Thos.,  19.  24. 
Smith's  Prizes,  91,  124.  193. 
Snell,  108. 

Social  life  of  students,  235.  250. 
Solar  system,  Newton's  theory  of, 

61. 

Sophister,  145.  162. 
Speaking  tube,  50. 
Square  numbers,  40. 
Stair  Douglas,  ref.  to,  127.  210. 
Statuta  antiqua,  ref.  to,  142.  145. 

148.  150.  151.  227.  230.  232.  233. 

239. 
Statutes,  Edwardian,  13.  153.  154. 

245. 

—  Elizabethan,  13.  35.  139.  155. 
158.  164.  184.  245.  247.  251. 

—  Victorian,  137.  247.  251. 

—  of  Cardinal  Pole,  154. 

—  of  Trinity  College,  158. 
Stevinus,  28.  93. 
Stirling,  James,  ref.  to,  65. 
Stokes,  G.  G. ,  134. 
Stokes,  Matt.,  ref.  to,  141. 
Stone,  Edward  James,  137. 
StourbridgeFair,  223.  233.  242.  250. 
Street's  Astronomy,  78. 
Students,   amusements  of,  237-8. 

250. 

—  dress  of,  239. 

—  expenses  of,  236.  251. 

—  numbers  of,  233-4. 


Students,  social  life  of,  235.  250. 
Studium  generale,  221. 
Sturmius,  95.  96. 
Subtraction,  symbol  for,  16. 
Supplicats,  146.  149.  156. 
Suter,  H.,  work  by,  1. 
Sylvester,  James  Joseph,  133. 
Symbol  for  addition,  15. 

—  for  multiplication,  30. 

—  for  proportion,  31. 

—  for  subtraction,  16. 
Symeon,  Henry,  226. 
Synod  of  Dort,  252. 

Tables,  mathematical,  5.  28.  41. 
Tacquet,  Andrew,  83.  95. 
Tait,  Peter  Guthrie,  135. 
Tangents,  inverse  problem  of,  57. 
Taxors,  228,  241. 
Taylor,  Brook,  88. 

—  ref.  to,  75.  87.  90.  93.  253. 
Terence,  143.  144. 

Text  books  in  use  circ.  1200,  2.  3. 

—  1549,  13. 

—  1660,  52. 

—  1730,  92-96. 

—  1800,  111. 

—  1830,  128-131. 
Theodolite,  derivation  of,  21. 
Theodosius,  works  of,  48. 
Thompson,  see  Eumford. 
Thomson,  Sir  Wm.,  135. 

—  ref.  to,  136. 
Thoresby,  Ealph,  ref.  to,  76. 
Thorp,  Eobert,  162. 
Thurot,  ref.  to,  8.  234. 
Todhunter,  Isaac,  131. 

—  ref.  to,  121.  127.  160.  181.  216. 
Tonstall,  Cuthbert,  10. 

—  ref.  to,  12.  13.  243. 
Tooke,  Andrew,  49. 
Torricelli,  39. 

Transactions,  Philosophical,  77. 87. 
88.  100-103.  105.  107.  109.  110. 
125.  133.  134. 

Trigonometrica  Britannica,  28. 

Trigonometry,  plane,  earliest  Eng- 
lish use  of,  22. 

—  spherical,  earliest  English  use 
of,  21. 

—  works  on,  96.  96.  104.  108.  109. 
118.  128. 


264 


INDEX. 


Trinity  College,  40.  46.  51.  79.  80. 
100.  105.  110.  120.  124.  127. 129. 
131.  132.  134.  135.  139. 140. 155. 
173.  182.  193. 194.  235.  243.  251. 

—  Statutes  of,  158. 
Trinity  College,  Dublin,  243. 
Trinity  Hall,  157. 

Tripos,  Mathematical,  chapter  x. 

—  origin  of  the  term,  217-219. 
Tripos  verses,  218.  219. 
Trivium,  the,  2.  140.  142. 147.  156. 

244.  247. 

Tuition,  private,  116.  160-3. 
Turton,  Thomas,  118.  132. 
Tycho  Brahe,  ref.  to,  21. 

Uffenbach,  ref.  to,  75. 

Universal   arithmetic,  of  Newton, 

58.  66.  68.  84.  85.  86.  93. 
Universitas  scholarium,  221.  224. 
University,  of  Bologna,  9.  225. 

—  of  Cambridge,  7.  8.  10.  11.  14. 
also  chapters  vm.  and  xi. 

—  of  Durham,  133. 

—  of  Leipzic,  8.  9. 

-  of  Oxford,  3.  5.  7.  9-11.  29.  225. 

—  of  Padua,  9.  10.  242. 

-  of  Paris,  5.  7.  9.  143.  150.  152. 
154.  193.  222.  225.  227.  236. 

—  of  Pisa,  9. 

—  of  Prague,  8.  9. 

—  of  Salerno,  225. 

—  of  Vienna,  8. 

Urban  V.,  Statutes  of,  143. 
Urstitius,  23. 

Varenius,  95. 
Venturoli,  110.  130. 
Verses,  Tripos,  218.  219. 
Vice-Chancellorship,  246. 
Victorian  Statutes,  137.  247.  251. 
Vienna,  University  of,  8. 
Vieta,  52. 
Vince,  Samuel,  103. 

—  ref.  to,  89.  103.  104.  111.  113. 
120. 

Virgil,  143.  185. 
Vlacq,  28.  96. 


Wallis,  John,  41. 

—  ref.  to,  14.  33.  35.  52.  53.  71. 
93.  95.  129. 

Walton,  W. ,  130. 
Ward,  John,  27.  38.  46. 
Ward,  Seth,  33.  36-38.  93. 
Waring,  Edward,  101. 

—  ref.  to,  99.  100.  113. 
Waterland,  Daniel,  94. 
Weber,  136. 

Weissenborn,  ref.  to,  1.  3. 
Wells,  E.,  93.  95. 
Westminster  School,  82.  107.  108. 
Whatton,  A.  B.,  35. 

Whewell,  William,  127-8. 

—  ref.  to,  46.  110.  114.  119.  121. 
122.  130.  160.  162.  164. 181. 187. 
190.  210. 

Whiston,  William,  83. 

-  ref.  to,  75.  76.  88.  89.  92.  96. 

White,  John,  120. 

WTaitley,  Charles  Thomas,  133. 

Whytehead,  22.  23. 

Wilson,  John,  102. 

Winchester  School,  49. 

Wingate,  E.,  93. 

Wollaston,  Francis,  107. 

Wollaston,  F.  J.  H.,  106. 

WoUaston,  W.  H.,  116.  ref.  to,  114. 

Wolsey,  Cardinal,  243. 

Wood,  Anthony,  149. 

Wood,  James,  110.  ref.  to,  111.  120. 

Woodhouse,  Eobert,  118. 

—  ref.  to,  89.  117.  128.  132. 
Wordsworth,   Chris.,  ref.    to,    146. 

164.  180.  187.  219.  220.  241.  250. 

251. 

Wranglers,  170.  171.  189. 
Wren,  Sir  Christopher,  ref.  to,  59. 
Wright,  Edward,  25-27. 

—  ref.  to,  15.  28. 

Young,  Sir  Wm.,  ref.  to,  88. 
Young,  Thomas,  115.     ref.  to,  114. 

Zamberti,  23. 
Zodiack,  Long's,  105. 


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