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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, 14741559 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, 15101558 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, 15271608 19 

Thomas Digges, 15461595 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, 15561630 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, 15741660 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, 16191641 35 

Catalogue of his library 36 

Seth Ward, 16171689 36 

Samuel Foster. Lawrence Eooke 38 

Nicholas Culpepper. Gilbert Clerke 39 

John Pell, 16101685 40 

John Wallis, 16161703 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, 16301677 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 169294 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, 16751729. 76 

John Craig, died in 1731 77 

John Flamsteed, 16461719 78 

Richard Bentley, 16621742 80 

Introduction of examination by written papers. . . 81 

William Whiston, 16671752 83 

Nicholas Saunderson, 16821739 86 

Thomas Byrdall. James Jurin 87 

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

Brook Taylor, 16851731 88 

Roger Cotes, 16821716 

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, 16891768 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, 16801760 100 

Edward Waring, 17361798 101 

Isaac Milner, 17511820 102 

The Plumian professors. 

Anthony Shepherd, 17221795 103 

Samuel Vince, 17541821 103 

Syllabus of his lectures 104 

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

Eoger Long, 16801770 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, 17311810 114 

Thomas Young, 17731829 115 

William Hyde WoUaston, 17761828 116 

Chapter VII. The analytical school. 

Robert Woodhouse, 17731827 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, 17911858. . 124 



TABLE OF CONTENTS. Xlil 

PAGE 

Charles Babbage, 1792 1871. 125 

Sir John Herschel, 17921871 126 

William Whewell, 17941866 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, 16001858. . . .252 

Prevalent political views at Cambridge, 16001858. . . . 252 
Prevalent subjects of study at Cambridge, 16001858. . . .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 below 1 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 Gerbert 1 . 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 stimulus 2 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. 

12 



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 recovered 1 , 
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 arithmetic 1 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 Bacon 2 . 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, 186068 ; see also pp. 480, 487, 52124 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 algorism 1 , 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 1389 2 . 

By the statutes of Prague 3 , 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 
required 4 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 
records 1 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 registers 2 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 Tonstall 1 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. 15351630. 

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 time 1 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 Ramus 1 
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 Eecorde 1 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 he 1 ? 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 Cuningham 1 
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 Dee 1 was born on July 13, 1527, and died in December 
1608. He entered at St John's College 2 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. 

22 



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 used 1 . In one of them known as Pantometria and 
issued in 1571 the theodolite is described: this is the earliest 
known description of the instrument 2 . 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 Brahe 3 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 
earliest 1 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 Forman 1 , 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 Briggs 1 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 Gresham 2 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 Oughtred 1 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 Locke 1 . 

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 Horrox 2 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. 

32 



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 Ward 1 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 Foster 1 , 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 Rooke 1 , 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 Pell 1 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 10 8 . 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 Wallis 1 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, 183388. 
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 x n and that x plq 
stood for the q th root of x p . He next proceeded to find by the 
method of indivisibles the area enclosed between the curve 
y = x m , 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 = ax m , 
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 = Hax m -, 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 + x l + x 2 + . . . its area 
would be x + Jo; 2 + ^x 3 + He then applied this to the quad- 
rature of the curves y = (l-c 2 } , y = (1 - x 2 ) 1 , y = (l-x 2 ) 2 , 
y = (1 cc 2 ) 3 , &c. taken between the limits x = 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 x m 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- 

f l - 
valent to finding the value of I x m dx. 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 = x p ^ 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 = (\- x 2 y', 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 or 2 ) and y = (1 x 2 )\ 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, T 8 y , |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 x 3 = ay 2 . 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 s l : s 2 = v l t l : V 2 t 2 : (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 Barrow 1 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 occupant 1 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 y 1 , 
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 Newton 1 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. 

42 



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 - x 2 )^ 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 - x 2 ) 2 by extracting the 
square root of 1 x 2 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 = ax m (b + cx n ) 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 

x mn-l a.mt+i)*-! 



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



x mn ~ l (a + bx n ) (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 Principles 1 . 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 

xy z + hy = ax 3 + bx 2 + 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 ax 3 + bx 2 



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 

y s = ax 3 + bx 2 + cx + d, 

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

y = ax 3 + bx 2 + 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 y 2 = ax* + bx 2 + 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^ a 2 , ... the 
values of y are known and are b^ b 2 ... then a parabola 
whose equation is y=p + qx + rx* + ... can be drawn through 
the points (a^ 6J, (a 2 , b z ), ... 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 

52 



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 another 1 ." 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 curves 2 . 

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 xo l . 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 x n x 

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

x n dx. 

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 chosen 2 . 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 it 1 . 

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. 16901730. 

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 Laughton 1 , 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 
resident 1 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 Clarke 2 
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 (16771724) 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 hypothesis 1 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 Flamsteed 2 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 Locke 1 . 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 Bentley 1 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 letter 1 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 Whiston 1 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's 2 

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, 174766. 

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. 

62 



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 y e Trinity and designs to print it, wherein 
he sides w th y e 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 y e world 
better as he thinks than it is at present, concerning y e 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 Saunderson 1 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 Taylor 1 , 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 Cotes 2 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 elected 1 . 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 him 2 ." 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 x n - 1 is well known. The proposition 
that " if from a fixed point a line be drawn cutting a curve 
in $i Qz- Q n > an d a point P be taken on it so that the 
reciprocal of OP is the arithmetic mean of the reciprocals of 
OQi, OQ 2 ,...OQ n , 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 Lahko, Sturmius, L'Hospital, Newton, Milnes, or C " ^ 
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. 17301820. 

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. 

72 



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 Colson 1 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 maps 1 ; 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 Milner 1 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 instant 1 ." 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- 
fessorship 1 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 argumentum 1 . 



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 Maseres 1 , 
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 brackets 1 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 it 1 . 



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 Cavendish 2 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. 387392), 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 Young 1 , 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. 

82 



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 SCHOOL 1 . 

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 Woodhouse 1 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 1822 2 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- 
schel 1 , 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 181819, 
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 Whewell 1 , 
and in the following year by Peacock again, by which time the 
notation was well-established 2 : 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 attempt 1 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 Babbage 1 , 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 (17381822) 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 Whewell 1 , 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. Ellis 1 , 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 

92 



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 King y 
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 
Morgan 1 , 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. Maxwell 1 
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 EDUCATION 1 . 

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 enactments 1 . 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 glomeriae 1 . This master of glomery had as such 
no special right over the other students of the university 2 , 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 faculty 3 ." 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. 

statutes 1 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 
acquainted 2 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 instruction 2 . 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 schools 3 , and the uni- 
versity derived a considerable part of its scanty income from 
the rents taken from the lecturers. Gratuitous lectures were 
forbidden 4 . 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 philosophy 2 ; 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 year 1 . 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 introduced 2 . 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- 
ment 1 . 

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 
derived 2 ), "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 commendations 1 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 them 2 ." 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. 

102 



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 permissible 1 . 

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 schools 1 . 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 Peacock 1 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 students 2 . Noblemen at Oxford and Cam- 
bridge were exempted from this restrictive rule 3 . 

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 statute 1 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 1549 2 . 
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 expressed 3 was the very reasonable one that masters 

1 Mullinger, i. 5963. 

2 Mullinger, n. 109115. 

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 
1502 1 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 
utterly 1 ; 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 recast 1 . 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 follows 2 . An undergraduate was obliged to be a member 
of a college. After he had resided for three years 3 , 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 
fourth 3 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 
thenceforth 1 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 century 1 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. 48. 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 eight 1 ", 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 pupils 1 ; 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 life 1 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 graces 2 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 

every 1 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. 



112 



CHAPTER IX. 
THE EXERCISES IN THE SCHOOLS 1 . 

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 Cartesian 2 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 students 1 . 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 Trinity 1 , 
it is apparently the original manuscript handed to the modera- 
tors at the close of the disputation. The manuscript begins 

Q[uestiones] 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. l mug . 
Cragg, S.S. Trinitatis, Opp. 2 US . 
Newcome, coll. Begin., Opp. 3 US . 

Finally at the bottom is the signature of the presiding mo- 
derator Littlehales Mod r . 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 congregation 1 . 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 u 4 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 1 st Opponent's 2 nd 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 Arg s 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 2 nd 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 Nov r 11 th under 
Newton against Wingfield & a second opponency for Nov r 19 th 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. 

M re Hankinson & Miss Paget of Lynn are now at Cambridge, I drank 
tea & supp'd with them on Thursday at M r Smithson's (the Cook's of 
S l . 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 M r 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 M r 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. M re 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 

122 



180 THE EXERCISES IN THE SCHOOLS. 

next thursday if possible to think nothing of my 2 nd (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 Dec 1 ". 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 
men 1 ." 

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- 
fiable 2 . 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 quarters 1 ." 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- commoners 1 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. Jebb 1 
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 TRIPOS 1 . 

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 
thousand 1 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 English 2 , 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 extant 1 . 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 192204 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 partiality 1 . 

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 said 1 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 Trinity 2 . 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 = to a cubic 
one. 

8. To find the fluent of x x Ja* - x 2 . 

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 

132 



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 x z + qx + r = 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 x r x (y n + z m ). 

8. To find the fluent of -^- . 

a + x 

9. To find the fluxion of the m th 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 Sen r 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 (tho 5 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 8 th 
or 9 th & Lucas is either 10 th or 11 th . 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 one 1 .' 

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, 32223. 



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 . 9 X 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 = a x . 

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 X s -px* + qx-r = Q be a, 6, and c, 
to transform it into another, whose roots are a 2 , b 2 , c 2 . 

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? - 5x 2 + Sx - 4 = 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 x 2 - 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 , 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 x n - px n-1 + qx n ~ 2 - &c. = 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 -x z y-a s = 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 2 2 2 . 3 . 4 



26. Given the fluent (a + cz n ) m x P+-I 2 
to find the fluent (a + cz n ) 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 - px z + 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* + a 2 ). 

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. 

a 3 

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 x 3 - 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 a 3 - x 3 . 

10. Solve the equation 3x 2 - 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* - Ja 2 - 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 z 4 - 17x 3 + 88x 2 - 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-house 1 . 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 follows 2 . 

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 regulations 1 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. 

142 



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 marked 1 . 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 paper 1 . 

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 Lent 1 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 
Mullinger 1 , ''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 examination 1 ." 

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 fair 1 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 allusion 2 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 England 1 ." 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 reside 2 , 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 (11991216) 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 established 1 , 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 regents 2 . Finally, every measure had to be approved 
by the chancellor. 

The executive of the non-regent house was vested in the 
two scrutators 3 . But the proctors (sometimes also called 
rectors) were the two great officers of the university : they 

- 1 See Buleus, vol. v. p. 377. 

2 Statuta antiqua, 2, 21, 50, 71, 163. 

3 Peacock, 21 et seq. 

152 



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 excommunication 1 , 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 consent 2 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 others 1 . 

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 
schools 2 . 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 master 1 : 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 university 2 ." 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 pupils 1 . There was no formal matriculation of students 
until the year 1543 2 . 

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 statutes 3 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 out 4 . 

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. 218220. 

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 name 1 . 

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 
caput 2 (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 university 3 . They were 
accordingly expelled; but in 1306 4 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 year 1 , 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. 
Thurot 1 , 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 says 2 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 one 3 . 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 uncommon 4 . 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 students 1 . 
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 hostels 2 . 
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 year 3 . 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. 252254. 

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 20s 1 . 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 life 1 . 

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 cost 2 . 

The amusements 3 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. 366372. 

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 assault 1 . 

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 organized 1 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- 
cellor 2 . 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. 500663. 



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 Cambridge 1 ; 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 poor 2 . 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 

162 



244 OUTLINES OF THE HISTORY OF THE UNIVERSITY. 

by Cramner from the work of Wied and Bucer 1 . 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 old 2 . 

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 manner 1 . 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 changes 2 . 
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. 22234. 



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 
colleges 1 . 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. 109120). 

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 incorrect 1 . 



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 played 1 . 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 clubs 2 , 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 later 1 . 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, 119129. 



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 
disappeared 1 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 attitude 1 . 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. 



INDEX 1 . 



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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Q.A Ball, Walter William Rouse 
17 History of the study of 
C3P2 mathematics at Cambridge 



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