UNIVERSITY OF CALIFORNIA MEDICAL CENTER LIBRARY SAN FRANCISCO 01 Manual F. Morales the library o f R. WiJli arrs HI r A HISTORY OF THE STUDY OF MATHEMATICS AT CAMBRIDGE. C. J. CLAY AND SONS, CAMBEIDGB UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. DEIGHTON, BELL, AND CO. E. A. BROCKHAUS. A HISTOKY OF THE STUDY OF MATHEMATICS AT CAMBEIDGE BY W. W. ROUSE [BALL, FELLOW AND LECTURER OF TRINITY COLLEGE, CAMBRIDGE AUTHOR OF A HISTORY OF MATHEMATICS. AT THE UNIVERSITY PRESS. 7 ,88 9 [All Eights reserved.] 188? 184G1S PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. PEEFACE. 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 of 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 PKEFACE. 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. Poggendorff's 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 names 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. 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 Kobert Kecorde, 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 Eichard Harvey. John Harvey. Simon Forman. ... 24 Edward Wright, died in 1616 25 The earliest treatment of navigation as a science. . . 26 Henry Briggs, 1556 1630 27 His tables of logarithms 28 Introduction of the decimal notation. .... 28 His election to the Savilian chair of geometry at Oxford. 30 William Oughtred, 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 Kooke 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 167684. 58 New results in the theory of equations 58 TABLE OF CONTENTS. xi PAGE The theory of gravitation, 1684. The De motit 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 Kichard Laughton, died in 1726 75 Samuel Clarke, 16751729 76 John Craig, died in 1731 77 John Flamsteed, 16461719 78 Kichard 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 Koger Cotes, 16821716 88 His election to the Plumian chair (founded in 1704). . 89 The second edition of the Principia. .... 89 The Harmonia mensurarum and Opera miscellanea. . . 90 Foundation of the Sadlerian lectureships. 91 Kobert 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, 1680 1770. 105 John Smith. William Lax 105 The lectures of F. J. H. Wollaston and W. Farish 106 Other mathematicians of this time. John Kowning, Francis Wollaston. George Atwood. . . . 107 Francis Maseres. Nevil Maskelyne. 108 Bewick Bridge. William Frend. John Brinckley. . . . 109 Daniel Cresswell. Miles 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. . . . Ill Experimental physicists of this time. Henry Cavendish, 1731 1810 114 Thomas Young, 17731829 115 William Hyde Wollaston, 17761828 116 Chapter VII. The analytical school. Kobert 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, 1791 1858 124 TABLE OF CONTENTS. xiii .PAGE Charles Babbage, 17921871 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. . 12& 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 Kapid 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'** 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 scliolarium 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 mediasval university and colleges 241 Steady development and progress of Cambridge 241 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 Kapid 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, 18001858. . . .252 Prevalent subjects of study at Cambridge, 1600 1858. . . . 253 INDEX 255 ERRATA. Page 14, line 3. After under insert the. Page 34, line 8. For powers read power. Page 38, lines 3 and 5. For Bulialdus read Bullialdus. Page 91, line 12. For seventeenth read eighteenth. Page 92, line 4 from end, and page 95, line 5 from end. For Lahire read La Hire. Page 115, line 12. For His read Cavendish's. Page 183, line 20. For T. Bowstead read Joseph Bowstead. ERRATA ET ADDENDA. Page 7, line 26. Before a parish insert the rolls of a manor or in. Page 7, line 28. For seventeenth read sixteenth. Page 7, footnote. Dele by John Norfolk, and also dele in 1445 and reissued. Page 28, lines 20 and 21. For some 20,000 logarithms read fte logarithms of 70,000 numbers. Page 40, line 26. For Dutch read German. Page 40, line 28. After employed insert ra England; it had been first introduced by Rahn at Zurich in 1659. Page 41, footnote, line 2. For 1833 read 1883. Page 49, line 26. A life of Morland by Halliwell was published at Cambridge in 1838. Page 55, line 25. For 1673 read 1670. Page 57, line 11. For m ( + ) n read (m + ^) n. Page 62, line 22. For on one or two occasions read again in 1701. Page 65, lines 17 and 18. For 1666 read 1668, and for 1667 read 1669. Page 79, line 18. For Halley read Crosthwait and Sharp. An incom- plete edition by Halley was published in 1712. Page 80, footnote, line 1. For W. H. Monk read J. H. Monk. Page 81, line 26. For Stubbs read Stubbe. Page 83, line 24. For 1703 read 1701. Page 90, line 2. For The whole read Most of the. Page 91, line 23. For 1728 read 1738. Page 91, line 26. For 1744 read 1749. Page 93, line 10. For 1704 read 1706. Page 93, line 17. For 1728 read 1738. Page 105, lines 15 and 16. For 1765 1753 read 1758. Page 105, line 23. For 1751 read 1763. Page 107, line 9. The Francis Wollaston here alluded to was not educated at Cambridge. Page 129, line 14. For are read is. Page 135, line 27. For James read John. Page 138, footnote, line 12. For 1848 read 1868. Page 157, footnote, line 3. For expelled read refused the degree. Page 163, line 3. For 1805 read 1796. Page 183, line 20. For T. Bowstead read A. Thurtell. Page 190, line 26. After voce insert These reforms were introduced in 1763 by Richard Watson of Trinity who was moderator in that year. Page 194, line 15. For dialectis read dialecticis. Page 198, line 13. After has insert (except in 1798). Page 209, Ex. 17. For -^ read -i-r- . 1.6 2.6 Page 227, line 4. For By the beginning read Towards the end. Page 229, footnote 1, line 3. For Deinfle read Denifie. Page 249, line 3. Until 1700 the average number of resident under- graduates in any year was apparently much more than four times the number of those who took the B.A. degree in that year. 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 25 MEDIAEVAL 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 vm. 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 iiow-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 011 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 Batdolt 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, was 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 formula 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 011 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 Holy wood, 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 computo 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 algorithms; and has been reissued in Halliwell's Ear a mathematica, 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 Thomas 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) iheArit/imeticaspeculativa, printed account of his life by J. S. Brewer is prefixed to the Kolls 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 Apergu historique sur... geometric by M. Chasles (first edition). THE FOURTEENTH CENTURY. 7 at Paris in 1502 ; (iii) the Geometries speculative!,, printed at Paris in 1511 ; and (iv) the .De quadrature*, 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. 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, is still extant. It was printed in 1445 and reissued by Halliwell in his Eara mathematica, London, second edition, 1841. 8 MEDIEVAL 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 Campaiius, 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 ^organisation de l'enseignement...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 which 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 14G3 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 IY. 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 Facultat 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 Kalph Churton, Oxford, 1800. 10 MEDIAEVAL 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 bom 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 Record e, the only two English mathematicians of any note of the first half of the sixteenth century, both migrated from Oxford to 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 Eliza- 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 Eamistic 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 Recorde 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 mediaeval 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 iifth as a youth; a third as a man; and spent thirteen years in his dotage : how old was he? Suppose we assume his age to have been 40. Then, by the given conditions, he would have spent 8f (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. Murn. 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 the Whet- stone ofwitte. The title, as is well known, is a play on the old name of algebra as the cossic art : the term 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 j 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 Master. 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 clown 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 English 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 C an tabrig lenses. 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 KENAISSANCE. 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 diiiing-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 is 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 cak es 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 athelida 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. Billings] ey 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 Harveys, and Form an. 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 Grynaeus 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 on 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 spot-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 KENAISSANCE. 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 Jiaven-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 011 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 Ylacq 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 Britannica 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 Constructio published in 1619 it is denned 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 on 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, &c. 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 RENAISSANCE. educated at either university, and his chief work the Artis analyticae 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 in 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 powers 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 HORROX. 35 clid 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 JSTewton 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 Hor rocks 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 Tabulaa Rudolphinae. Alfraganus. Lansbergii Progymn. de Motu Soils. 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. Prutenica3. Cornelii Gemmae Cosmocritice. Comm. in Theor. Purbachii. Herodoti Historia. Theonis Comm. in Ptolom. J. Kepleri Astron. Optica. Tyc. Brahaai 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 wa,s 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 Yenus 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 praevaricator 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 iheDeplenitudine 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 Pell, 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 Apolloriius, 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 Rhonius's algebra, with the addition of considerable new matter: in this work the symbol -f- 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 epliemerides, 1630. On logarithms, 1631. 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. 7 '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, 1833 88. Wallis's mathematical works were published in three volumes at Oxford, 1693-99. 42 THE COMMENCEMENT OF MODERN MATHEMATICS. Mechanical... And among more than two hundred students (at that time) in our college, I do not know of any two (perhaps not any) who had more of Mathematicks than I, (if so much) which was then but little ; and but very few, in that whole university. For the study of Mathematicks was at that time more cultivated in London than in the universities." This pas- sage has been quoted as shewing that no attention was paid to mathematics at that time. I do not think that the facts justify such a conclusion; at any rate Wallis, whether by his own efforts or not, acquired sufficient mathematics at Cambridge to be ranked as the equal of mathematicians such as Descartes, Pascal, and Fermat, Wallis was elected to a fellowship at Queens', commenced M.A. in 1640, and subsequently took orders, but on the whole adhered to the puritan party to whom he rendered great assist- ance in deciphering the royalist despatches. He however joined the moderate presbyterians in signing the remonstrance against the execution of Charles I., by which he incurred the lasting hostility of the Independents a fact which when he subsequently lived at Oxford did something to diminish his unpopularity as a mathematician and a schismatic. There was then no professorship in mathematics and no opening for a mathematician to a career at Cambridge ; and so Wallis reluctantly left the university. In 1649 he was ap- pointed to the Savilian chair of geometry at Oxford, where he lived until his death on Oct. 28, 1703. It was there that all his mathematical works were published. Besides those he wrote on theology, logic, and philosophy ; and was the first to devise a system for teaching deaf-mutes. I do not think it necessary to mention his smaller pamphlets, a full list of which would occupy some four or five pages : but I add a few notes on his more important mathematical writings. The most notable of these was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and WALLIS. 43 shewed that x~ n stood for the reciprocal of x" and that x p ^ q 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 m may be any number positive or negative, and he considered in particular the case of the parabola in which m = 2, and that of the hyperbola in which m = l: 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 oj, its quadrature could be determined. Thus he said that if the equation of a curve was y x + x 1 + x 2 + . . . its area would be x + ^x 2 + ^x 3 + He then applied this to the quad- rature of the curves y = (l-x 2 ) , y = (l-x 2 )\ y = (\x 2 ) 2 y y = (1 x 2 ) 3 , &c. taken between the limits x and x = 1 : and shewed that the areas are respectively 1 2 8 16 X rn L t ^5 TT "5~5' (x) is a function of x and if when x is successively put equal to a^ a a , ... the values of y are known and are b lt b. 2 ... then a parabola whose equation is yp + qx + rx 2 + ... can be drawn through the points (oj, ftj, ( 2 , 5 2 ), . . . and the ordinate of this parabola may be taken as an approximation to the ordinate of the curve. 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 APPEARANCE 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 NEWTON. 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 arid 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 n.), 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\ I should here note the fact that Vince 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 [a:"], or what Leibnitz would have written as l 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 NEWTON. quadratures 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. I 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 Whiston, 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 ] , 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 ] The name was pronounced Laffton : see "Offenbach's account of his visit to Cambridge in 1710 quoted on p. 6 of the Scholae academicae. 76 THE EISE 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 optime 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 Renault'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 Renault 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 analt/tica (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 matliematica 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, Kiccioli'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 K. 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 Brittanica, 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. Rent 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 KISE 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 011 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 K. 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 and 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 arid 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 l 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 EISE OF THE NEWTONIAN SCHOOL. Whiston was succeeded in the Lucasiaii 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 Principia 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. In 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 4- h) =/(*) + hf (x) + jj/" (*) + ..., 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, Laughton, 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 Flainsteed, the astronomer-royal, arid 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, Roger Cotes of Trinity; from 1716 to 1760, Kobert Smith of Trinity (see p. 91); from 1760 to 1796, Anthony Shepherd of Christ's (see p. 103); from 1796 to 1822, Samuel Vinceof 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 EISE 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 $!, Q 2 .. Q n , and 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 W 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 cycloid al 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. 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 Lahire (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 Maclaurin (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 Mario tte 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 THE KISE 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 Eemarks. Baronius's Metaphysicks. March April Euclid's Elem. Locke's Hum. Und. De la Hire Con. Sect. Whiston's Theory with Keill's Ke- 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. WeUs's Trig. Newton's Trig. Eohault'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, Newton's Algebra, and Mil nes'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 Lahire, Sturmius, L'Hospital, Newton, Milnes, or 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 essential. 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 Etiler) 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 Maclauriu 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 fluxiorial 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 Koyal 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 Satmderson'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 bora 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 I + p l 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. 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 1 760 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 Vince, 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 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. 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 believe 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 Cains 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, Koger 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 Scholae 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 1701 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 Handworterbuch. 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, Snell, 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 Ilcmdworterbuck. He died on Feb. 9, 1811. Bewick Bridge, a fellow of Peterhouse and mathematical professor at Haileybury College, was born near Cambridge in 17G7, 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 Eiifield 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 arid 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, Vince, 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), Bonny castle'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 fluxional 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 1 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 Farish'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, JepJison. 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. 113 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, Humford, 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. His 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 Gains 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 Plumian 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. u. 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 (JB.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 fluxional 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, dLij in which -j- was denoted by d x y. This was for some years 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 B ABB AGE. 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, 1792, 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, Agnesi'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 a the principles of pure d-ism as opposed to the dot-&ge 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 tScriptores optici for baron Maseres (see p. 108). In 1820 the Astronomical Society was founded mainly through his efforts, and at a later time, 1830 to 1832, he took a prominent part in the foundation of the British Association. In 1828 he succeeded Airy as Lucasian professor and held 1 He left an autobiography under the title Passages from the life of a philosopher. London, 1864. 126 THE ANALYTICAL SCHOOL. the chair till 1839, but by an abuse which was then possible he neither resided nor taught. Babbage will always be famous for his invention of an analytical machine, which could not only perform the ordinary processes of arithmetic, but could tabulate the values of any function and print the results. The machine was never finished, but the drawings of it, now deposited at Kensington, satisfied a scientific commission that it could be constructed. The third of those who helped to establish the new method was Herschel. Sir John Frederick William Herschel was born at Slough on March 7, 1792. His father was Sir William Herschel (1738 1822) who was the most illustrious astronomer of the last half of the last century. Two anec- dotes of his boyish years were frequently told by him as illustrative of his home training, and are sufficiently in- teresting to deserve repetition. One day when playing in the garden he asked his father what was the oldest thing with which he was acquainted. His father replied in Socratic manner by asking what the lad thought " was the oldest of all things." The replies were all open to objection, and finally the astronomer answered the question by picking up a stone and saying that that was the oldest thing of which he had definite knowledge. On another occasion in a conversation he asked the boy what sort of things were most alike. After thinking it over young Herschel replied that the leaves of a tree were most like one another. "Gather then a handful of leaves from that tree," said the philosopher, "and choose two that are alike." Of course it was impossible to do so. Both stories are trivial, but they were typical of the manner in which he was brought up, and these two particular incidents happened to make a deep impression on his mind. Except for one year spent at Eton he was educated at home. In 1809 he entered at St John's College, graduating as senior wrangler and first Smith's prizeman in 1813. His earliest original work was a paper on Cotes's theorem, which he sent when yet an undergraduate to the Royal Society, HERSCHEL. WHEWELL. 127 and immediately after taking his degree it was followed by others on mathematical analysis. He went down from the university in or about 1816, and for a few years read for the bar ; but his natural bent was to chemistry and astronomy, and to those he soon turned his exclusive attention. The desire to complete his father's work led ultimately to his taking up the latter rather than the former subject. He died at Col- lingwood on May 11, 1871. Besides his numerous papers on astronomy, his Outlines of astronomy published in 1849, and his articles on Light and Sound in the Encyclopaedia Metropolitana appear to be the most important of his contributions to science. His addresses to the Astronomical and other societies have been republished, and throw considerable light on the problems of his time. His Lectures on familiar subjects published in 1868 are models of how the mathematical solutions of physical and astronomical problems can be presented in an accurate manner and yet be made intelligible to all readers. Another member of the university who took a prominent part in developing the study of analytical methods was Whewell. William 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 011 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 Macmillari '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 defined 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 B. 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 Venturoli'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 Whe well'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, of Queens' College, who was born in 1798 and died in 1857. King was Lucasian professor from 1839 to 1849 in succession to Babbage. Sir George Biddell Airy, of Trinity College, who was senior wrangler in 1823, was bora 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 Oayley, 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 in 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. xxm. of the Proceedings of the Koyal Society, or in the article contributed by Prof. Tait to the Encyclopaedia Britannica. 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 Hadcliffe 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, 184252; but the collection of Documents relating to the university and colleges of Cam- bridge, issued by the Royal 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 Rolls 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 MEDIEVAL 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 confine 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 MEDIEVAL 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 " f etcher," 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 was 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. PROCEEDINGS 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 " glornerels " 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 lor his labour, and another groat to him that provideth the rod and the palmer, etcetera, de singulis. 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 MEDIAEVAL 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, and 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 1 , 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 V. 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 Mullinger, 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 MEDLEVAL SYSTEM OF EDUCATION. The lectures were either dietatory, or analytical, or dialec- tical \ 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 on 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. 6s. 8d. 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 sopkister. 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 Quaecumque ah Aristotele dicta essent commenticia esse drew a crowded and critical audience, and the subsequent 1 See Statuta antiqua, 136. - 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 sufficiently 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, seder e, 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. Hence he was usually known as a determiner, and was said to stand in quaclragesima. 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 domiims 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 MEDIAEVAL SYSTEM OF EDUCATION. The subsequent ceremonies of inception are described at length in Peacock 1 arid 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. In the fourteenth century the universities of Paris, Oxford, and Cambridge passed identical statutes 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 MEDIEVAL 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. These 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 Homance 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 1 . 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. ] 09 115. 3 By Ascham: see p. 16 of Original letters of eminent literary men edited by Sir Henry Ellis, Camden Society, London, 1843. 154 THE PEEIOD 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 century. 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 the 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, 810 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 COURSE 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 respond encies 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) he went to chapel at 5.0. After service (and possibly breakfast) he adjourned to the hall, where he did exercises and attended lectures from six to nine. At nine the college lectures gene- 1 This account is taken from the statutes of Trinity College : see Peacock, pp. 4 8. The statutes of 1552 and 1560 are printed as an appendix to the second volume of Mullinger's work. THE COLLEGE SCHEME OF EDUCATION. 159 rally ceased, and the great body of the students proceeded to the public schools, either to hear lectures, or to listen to, or take part in the public disputations which were requisite for the degree of bachelor or master. Dinner was served at eleven, and at one o'clock the students returned to their attendance on the declamations and exercises in the schools. From three until six in the afternoon they were at liberty to pursue their amusements or their private studies : at six o'clock they supped in the college-hall and immediately after- wards retired to their chambers. There was no evening service in the college chapels on ordinary days until the reign of James I. Whether most students lived up to this ideal is doubtful : some certainly did not. As time went on the average age at entrance rose from about sixteen in the sixteenth century to seventeen or eighteen in the seventeenth, and to eighteen or nineteen in the eighteenth century. The hours also gradually got later, and the strictness of the regulations was somewhat relaxed. At the beginning of the eighteenth century the " college day began with morning chapel, usually at six. Breakfast was not a regular meal, but it was often taken at a coffee-house where the London news- papers could be read. Morning lectures began at seven or eight in the college-hall. Tables were set apart for different subjects. At 'the logick table' one lecturer is expounding Duncan's treatise, while another, at 'the ethick table' is in- terpreting Puffendorf on the duty of a man and a citizen ; classics and mathematics engage other groups. The usual college dinner-hour which had long been 11 a.m., had ad- vanced before 1720 to noon. The afternoon disputations in the schools often drew large audiences to hear respondent and opponent discuss such themes as 'natural philosophy does not tend to atheism,' or 'matter cannot think.' Evening chapel was usually at five; a slight supper was provided in hall at seven or 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 defined 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 vaiying 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 bis life by E. Paley, London, 1838. William Paley was the author of tbe 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 Whe well'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 E/outh 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- carum formulae by R. F., published in 1638. A list of questions 011 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 dominus 1 Scholae academicae, pp. 363, 364. 168 THE EXERCISES IN THE SCHOOLS. respondens, 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 consequential). 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 , 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 officio functus es, or sometimes THE EXERCISES IN THE SCHOOLS. 169 with the highest compliment of all, summo ingenii acumine disputasti. In general optime quidem 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[uaestiones] S[unt] Kecte statuit Newtonus in tertia sua sectione. Eecte statuit Emersonus de motu projectilium. Origo mali moralis 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. Resp. Feb. 25, 1782. Bere, Sid. coll., Opp. l mus . Cragg, S.S. Trinitatis, Opp. 2 US . Newcome, coll. Regin., 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 closely 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 TroAAoi, or the poll men. The first class included those bachelors quibus sua reservatur senioritas 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 IN 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 +, E, or E ; fair performances with a -f , a, or 172 THE EXERCISES IN THE SCHOOLS. a - ; and indifferent ones with e + 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 "durices'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 methodo Halleii recte determinari potest; (ii) Recte statuit Newtonus in tertia sua sectione 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 [qusestionem]. Si asserat Hal- leius Venerem cum Soli sit proxima Londini visam a centre Solis qua- SPECIMEN OF A DISPUTATION. 175 tuor minutis primis distare, cadit qusestio. [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 quffistio. [Sed in figura Halleiana centrum Solis correspondet cum loco spectatoris in Tellure. Ergo cadit qusestio.]t Eesp. [Concede 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. [Concede antecedentiam et nego consequentiam.] Opp. [Iterum probo consequentiam.] Si composite motu Veneris 176 THE EXERCISES IN THE SCHOOLS. uniform! in recta linea cum motu Solari in curva linea fit semita Veneris in disco Solis curva linea, valet consequentia. [Sed composite motu Veneris uniformi in recta linea cum motu Solari in curva linea fit semita Veneris in disco Solis curva linea. Ergo valet consequentia.] Eesp. [Concedo antecedentiam et nego consequentiam.] Opp. [Iterum probo consequentiam.] Si longitude hujusce line* non recte determinari potest, valent consequentia et argumentum. [Sed longitudo hujusce linese 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 terras motum motui Veneris contrarium contrahatur transitus tempus integrum, cadit quaestio. [Sed spectator! ad os Gan- geticum posito ob terras motum motui Veneris contrarium contrahitur transitus tempus integrum. Ergo cadit quasstio.] Eesp. [Concedo antecedentiam et nego consequentiam. ] Opp. [Iterum probo consequentiam.] Si assumat Halleius contrac- tionem hanc duodecim minutis primis temporis aaqualem, et deinde huic hypothesi insistendo eidem tempori aequalem probat, valent consequentia et argumentum. [Sed assumat Halleius contractionem hanc duodecim minutis primis temporis aequalem, et deinde huic hypothesi insistendo eidem tempori aequalem 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 portum Nelsoni, et hanc semitam dividat in aequalia horaria spatia, cadit quasstio. [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.] Eesp. [Concedo 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.] Eesp. [Concedo antecedentiam, et nego consequentiam.] Opp. [Iterum probo consequentiam.] Si igitur ob motum Veneris SPECIMEN OF A DISPUTATION. 177 acceleratum ad os Gangeticum et retardatura ad portum Nelsoni hi 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 repraesentari 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 spectatori ad portum Nelsoni, posito tempore extensionis majore, major fit transitus duratio. Ergo cadit quaestio. ]f Resp. [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 Resp. [Concede antecedentiam et nego consequentiam.] Opp. [Iterum probo consequentiam.] Si hae duae conclusiones inter se pugnent, valent consequentia et argumentum. [Sed hae 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 quaestio. [Sed ad infinitam distantiam vis in parabola est infiuitesimalis secundi ordinis. Ergo cadit quaestio.] Resp. [Concede antecedentiam et nego consequentiam.] Opp. [Probo consequentiam.] Si vis sit w 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 sint B. 12 178 THE EXERCISES IN THE SCHOOLS. inter se inverse ut axes minores, cadit quaestio. [Sed velocitates ad extremitates axium minorum diversarum ellipsium quarum latera recta aequantur sunt inter se inverse ut axes minores. Ergo cadit quaestio.] Eesp. [Concedo antecedentiam efc 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 ^/2 : 1. Ergo valet consequentia.] Eesp. [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. Mod. [Probes aliter.] Opp. [Probo] contra tertiam [quasstionem]. Aut cadit tua quaestio aut non possibile est hominem ab ineunte astate caecum et jam adultum visum recipientem visu dignoscere posse id quod tangendo prius solum- modo dignoscebat. Sed possibile [est hominem ab ineunte setate cascum et jam adultum visum recipientem visu dignoscere posse id quod tangendo prius solummodo dignoscebat. Ergo cadit quaestio]. Eesp. [Concedo majorem sed nego minorem.] Opp. [Probo minorem.] Si eadem ratio quas 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 " &G. &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 Boom 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 opponeiicy 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'. 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 rs 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 Opponeucies, 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 18 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. ii. 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. 182 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 imderstand 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 e, 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 T. 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 qiiadrasti ? 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 IVewtonus, Woodius, et Paleius. To this the opponent replied (thus keeping three opponencies), Si non recte statuerunt Newtonus, 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 Y. 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 TKIPOS. 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 segrotat senior optime, and thus escape all further examination : this was called gutyhing 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 WhewelTs 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 finding 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 mariner 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 8. To find the fluent of x x Ja?^x z . 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 DBRS. [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 s , secondly in a greater ratio than 2 but less than A , 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 3 + qx + r=0 which wants the second term; supposing x a + b and 3a&= -q, to determine the value of x. i 7. To find the fluxion of x r x (y n + z m )v. 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 d 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' 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 js. 6d. 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