It must be remembered that for students pursuing the normal course the senate-house examination still provided the only avenue to a degree. That examination involved a knowledge of the elements of moral philosophy and theology, an acquaintance with the rules of formal logic, and the power of reading and writing scholastic Latin, but mathematics was the predominant subject, and this led to a certain one-sidedness in education. The evil of this was generally recognized, and in 1822 various reforms were introduced in the university curriculum; in particular the Previous Examination was established for students in their second year, the subjects being prescribed Greek and Latin works,[295]a Gospel, and Paley’sEvidences. Set classical books were introduced in the final examination of poll-men; and another honour or tripos examination was established for classical students. These alterations came into effect in 1824; and henceforth the senate-house examination, so far as it related to mathematical students, was known as the Mathematical Tripos.In 1827 the scheme of examination in the mathematical tripos was revised. By regulations68which came into operation in January 1828, four days, exclusive of the day of arranging the brackets, were devoted to the examination; the number of hours of examination was twenty-three, of which seven were assigned to problems. On the first two days all the candidates had the same questions proposed to them, inclusive of the evening problems, and the examination on those days excluded the higher and more difficult parts of mathematics, in order, in the words of the report, “that the candidates for honours may not be induced to pursue the more abstruse and profound mathematics, to the neglect of more elementary knowledge.” Accordingly, only such questions as could be solved without the aid of the differential calculus were set on the first day, and those set on the second day involved only its elementary applications. The classes were reduced[296]to four, determined as before by the exercises in the schools.The regulations of 1827 definitely prescribed that all the papers should be printed. They are also noticeable as being the last which gave the examiners power to askvivâ vocequestions, though such questions “were restricted to asking about propositions contained in the mathematical works commonly in use at the University, or examples and explanations of such propositions.” It was further recommended that no paper should contain more questions than well-prepared students could be expected to answer within the time allowed for it, but that if any candidate, before the end of the time, had answered all the questions in the paper, the examiners might propose additional questionsvivâ voce. The power of granting honorary optime degrees now ceased; it had already fallen into abeyance. Henceforth the examination was conducted under definite rules, and I no longer concern myself with its traditions.In the same year as these changes became effective the examination for the poll degree was separated from the tripos with different sets of papers and a different schedule of subjects69. It was, however, still nominally considered as forming part of the senate-house examination, and until 1858 those[297]who obtained a poll degree were arranged in four classes, described as fourth, fifth, sixth, and seventh, as if in continuation of the junior optimes or third class of the tripos.In the course henceforth ordained for the poll or ordinary degree, the examination, later known as “the General,” represents that part of the old senate-house examination which was intended for the poll-men, but gradually it was moved to an earlier period in the normal course taken by the men. In 1851 admission to the classical tripos70was allowed to others than those who passed the mathematical tripos, and this provided another avenue to a degree entirely independent of the old senate-house examination. In 1852 another set of examinations, at first called “the Professor’s Examinations,” and now somewhat modified and known as “the Specials,” was instituted for all poll-men to take before they could qualify for a degree.In 1858 the fiction that the poll examinations were part of the senate-house examination was abandoned, and subsequently they have been treated as providing an independent method of obtaining the degree: thus now the mathematical tripos is the sole representative of the old senate-house examination. Since 1858 numerous other[298]ways of obtaining a degree in arts have been established, and it is now possible to graduate by showing proficiency in very special, or even technical subjects.Further changes in the mathematical tripos were introduced in 183371. The duration of the examination, before the issue of the brackets, was extended to five days, and the number of hours of examination on each day was fixed at five and a half: seven and a half hours were assigned to problems. The examination on the first day was confined to subjects that did not require the differential calculus, and only the simplest applications of the calculus were permitted on the second and third days. During the first four days of the examination the same papers were set to all the candidates alike, but on the fifth day the examination was conducted according to classes. No reference was made tovivâ vocequestions, though permission was reserved to re-examine candidates if it were found necessary: this right remained in force till 1848, but in fact was never used. In December 1834, a few unimportant details were amended.Mr Earnshaw, the senior moderator in 1836, informed me that he believed that the tripos of that year was the earliest one in which all the papers were marked, and that in previous years the[299]examiners had partly relied on their impression of the answers given.New regulations came into force72in 1839. The examination now lasted for six days, and continued as before for five hours and a half each day: eight and a half hours were assigned to problems. Throughout the whole examination the same papers were set to all candidates, and no reference was made to any preliminary classes. It was no doubt in accordance with the spirit of these changes that the acts in the schools should be abolished, but they were discontinued by the moderators of 1839 without the authority of the senate. The examination was for the future confined73to mathematics.In the same year in which the new scheme came into force a proposal to reopen the subject was rejected on 6 March 1839.The difficulty of bringing professorial lectures into relation with the needs of students has more than once been before the University. The desirability of it was emphasized by a syndicate in February 1843, which recommended conferences at stated intervals between the mathematical professors[300]and examiners. This report, which foreshadowed the creation of a Mathematical Board, was rejected by the senate on 31 March.A few years later the scheme of the examination was again reconstructed by regulations74which came into effect in 1848. The duration of the examination was extended to eight days. The examination lasted in all forty-four and a half hours, twelve of which were devoted to problems. The first three days were assigned to specified elementary subjects; in the papers set on these days riders were to be set as well as bookwork, but the methods of analytical geometry and the calculus were excluded. After the first three days there was a short interval, at the end of which the examiners issued a list of those who had so acquitted themselves as to deserve mathematical honours. Only those whose names were contained in this list were admitted to the last five days of the examination, which was devoted to the higher parts of mathematics. After the conclusion of the examination the examiners, taking into account the whole eight days, brought out the list arranged in order of merit. No provision was made for any rearrangement of this list corresponding to the examination of the brackets. The arrangements of 1848 remained in force till 1873.[301]In the same year as these regulations came into force, a Board of Mathematical Studies (consisting of the mathematical professors, with the moderators and examiners for the current year and the two preceding years) was constituted75by the senate. From that time forward their minutes supply a permanent record of the changes gradually introduced into the tripos. I do not allude to subsequent changes which only concern unimportant details of the examination.In May 1849, the board issued a report in which, after giving a review of the past and existing state of the mathematical studies in the University, they recommended that the mathematical theories of electricity, magnetism, and heat should not be admitted as subjects of examination. In the following year they issued a second report, in which they recommended the omission of elliptic integrals, Laplace’s coefficients, capillary attraction, and the figure of the earth considered as heterogeneous, as well as a definite limitation of the questions in the lunar and planetary theories. In making these recommendations the board were only recognizing what had become the practice in the examination.I may, in passing, mention a curious attempt which was made in 1853 and 1854 to assist candidates to estimate the relative difficulty of the[302]questions asked. This was effected by giving to the candidates, at the same time as the examination paper, a slip of paper on which the marks assigned for the bookworkand rider for each question were printed. I mention the fact merely because these things are rapidly forgotten and not because it is of any intrinsic value. I possess a complete set of slips which came to me from Todhunter.In 1856 there was an amusing difference of opinion between the vice-chancellor and the moderators. The vice-chancellor issued a notice to say that for the convenience of the University he had directed the tripos lists to be published at 8.0 a.m. as well as at 9.0 a.m., but when members of the senate arrived at 8.0 the moderators said that the list should not be read until 9.0.Considerable changes in the scheme of examination were introduced in 1873. On 5 December 1865, the board had recommended the addition of Laplace’s coefficients and the figure of the earth considered as heterogeneous as subjects of the examination; the report does not seem to have been brought before the senate, but attention was called to the fact that certain departments of mathematics and mathematical physics found no place in the tripos schedules, and were neglected by most students. Accordingly, a syndicate was appointed on 6 June 1867, to consider the matter, and a scheme drawn[303]up by them was approved in 186876and came into effect in 1873.The new scheme of examination was framed on the same lines as that of 1848. The subjects in the first three days were left unchanged, but an extra day was added, devoted to the elements of mathematical physics. The essence of the modification was the greatly extended range of subjects introduced into the schedule of subjects for the last five days, and their arrangement in divisions; the total marks awarded to the questions in each of the five divisions being approximately in a proportion to the total marks assigned to the questions in the first three days as 2, 1, 1, 1, 2/3 to 1 respectively. Under these regulations the number of examiners was increased from four to five.The assignment of marks to groups of subjects was made under the impression that the best candidates would concentrate their abilities on a selection of subjects from the various divisions. But it was found that, unless the questions were made extremely difficult, more marks could be obtained by reading superficially all the subjects in the five divisions than by attaining real proficiency in a few of the higher ones: while the wide range of subjects rendered it practically impossible to[304]cover all the ground thoroughly in the time allowed. The failure was so pronounced that in 1877 another syndicate was appointed to consider the mathematical studies and examinations of the University. They presented an elaborate scheme, but on 13 May 1878, some of the most important parts of it were rejected; their subsequent proposals, accepted on 21 November 1878 (by 62 to 49), represented a compromise which pleased few members of the senate77.Under the new scheme which came into force in 1882 the tripos was divided into two portions: the first portion was taken at the end of the third year of residence, the range of subjects being practically the same as in the regulations of 1848, and the result brought out in the customary order of merit. The second portion was held in the following January, and was open only to those who had been wranglers in the preceding June. This portion was confined to higher mathematics and appealed chiefly to specialists: the result was brought out in three classes, each arranged in alphabetical order. The moderators and examiners conducted the whole examination without any extraneous aid.In the next year or two further amendments[305]were made78, the second part of the examination being moved to the June of the fourth year, and thrown open to all men who had graduated in the tripos of the previous June. At the same time the conduct of the examination in partIIwas transferred to four examiners nominated by the board: this put it largely under the control of the professors. The range of subjects of partIIwas also greatly extended, and candidates were encouraged to select only a few of them. It was further arranged that partImight be taken at the end of a man’s second year of residence, though in that case it would not qualify for a degree. A student who availed himself of this leave could take partIIat the end either of his third or of his fourth year as he pleased.The general effect of these changes was to destroy the homogeneity of the tripos. Objections to the new scheme were soon raised. Especially, it was said—whether rightly or wrongly—that partIcontained too many technical subjects to serve as a general educational training for any save mathematicians; that the distinction of a high place in the historic list produced on its results tended to prevent the best men taking it in their second year, though by this time they had read enough to be able to do so; and that partIIwas so constructed as[306]to appeal only to professional mathematicians, and thus the higher branches of mathematics were neglected in the University by all save a few specialists.Whatever value be attached to these opinions, the number of students studying mathematics fell rapidly under the scheme of 1886. In 1899 the board proposed79further changes. These seemed to some members of the senate to be likely still further to decrease the number of men who took up the subject as one of general education; and the two main proposals were rejected, 15 February 1900 by votes of 151 to 130 and 161 to 129.A few years later, in 190780, the board brought forward another scheme, proposing changes so sweeping as almost to destroy the identity of the tripos. Under this the examination in partIIwas abolished—a change on which all parties were agreed. There was introduced an examination, called partI, confined to elementary mathematics, which could be taken as early as the second term of residence, and for which in certain cases of failure a student could present himself again, but this, although an examination for honours, did not qualify for a degree.[307]In the new partII, taken normally at the end of the third year of residence and qualifying for a degree, candidates were given some option in the subjects of their examination, and order of merit was abolished. The first examination under this scheme was held in 1908.A remarkable feature in the history of the Cambridge mathematical school is the fact that for nearly two hundred years most students were accustomed to rely for preparation for it on work done with a private tutor or “Coach.” Towards the close of the seventeenth century we first read of these “pupil-mongers” (among whom Laughton of Clare was the most famous) who made it their business to prepare men for their “acts.”With the rise of the senate-house examination the importance of this class of teachers increased, for success in that examination was regarded as the crown of the academic course, and brought with it, in the shape of a fellowship, an immediate competence with a reasonable prospect of an assured career. It was the business of private tutors to prepare their pupils for the examination, and among those who in this way came to the front shortly after the middle of the eighteenth century were Richard Watson, John Wilson whose name is still known by its association with a proposition in the theory of numbers, and Robert Thorp. The last named[308]teacher was described, about 1761, as being “of eminent use to young men in preparing them for the Senate-House Examinations and peculiarly successful”; and it was added that “one young man of no shining reputation with the assistance of Mr Thorp’s tuition had stood at the head of wranglers.”In a grace of the senate, passed in 1781, it is stated that almost all sophs then resorted to private tuition, and for more than a century subsequently, the practice was well established. These were the men who really directed the reading of the students. Even non-residents, if reputed to be successful coaches, drew pupils. Thus John Dawson, a medical practitioner at Sedbergh, regularly prepared pupils in the vacations for the senate-house examination, and at least eleven of the senior wranglers between 1781 and 1800 are known to have studied under him.During the nineteenth century the system developed under two remarkable teachers, William Hopkins, 1793–1866, and Edward John Routh, 1831–1907, to whom the vast majority of the better known Cambridge mathematicians of this century owed most of what they learnt in their undergraduate days. Hopkins in the twenty-two years from 1828–49, had among his pupils one hundred and seventy-five wranglers, of whom seventeen were[309]senior, forty-four in one of the first three places, and one hundred and eight in one of the first ten places. So too Routh, in the thirty-one years from 1858–88, had between six hundred and seven hundred pupils, most of whom became wranglers, twenty-seven being senior in the tripos and forty-one Smith’s prizemen. To organize teaching on this scale demanded rare gifts.Perhaps it may be of interest to describe, by way of example, the general features of Routh’s system. He gave catechetical lectures three times a week to classes of eight or ten men of approximately equal knowledge and ability. The work to be done between two lectures was heavy, and included the solution of some eight or nine fairly hard examples on the subject of the lectures. Examination papers were also constantly set on tripos lines (bookwork and riders), while there was a weekly paper of problems set to all pupils alike. All papers sent up were marked in public, the comments on them in class were generally brief, and, to save time, solutions of the questions were circulated in manuscript. Teaching also was supplemented by manuscripts on the subjects. Finally to the more able students he was accustomed, shortly before their tripos, to give memoirs or books for analyses and commentaries. The course for the first three years and the two earlier long vacations covered all the subjects of the[310]examination—the last long vacation and the first term of the fourth year were devoted to a thorough revision.Under Hopkins and Routh there was no trace of what is called cramming; they might say that a particular demonstration was so long that it could not be required in the tripos, but none the less they expected their pupils to master it. The system had faults, but it had the merit of providing a systematic grounding in a wide field of subjects. The effectiveness of teaching of this kind was dependent on intimate constant personal intercourse, and the importance of this cannot be overrated. The scandal of the system consisted in the fact that a man was compelled to pay heavy fees to the University and his College for instruction, and yet found it advantageous at his own expense to go elsewhere to get it.During the last quarter of the nineteenth century college lecturers began to share with the coaches the general direction of studies. Post-graduate work was also to some extent brought under the influence of professors and university lecturers—these not uncommonly suggesting subjects for dissertations for fellowships, Smith’s prizes, etc. But the students thus influenced were not numerous, and it still remains true that the majority of mathematical undergraduates are so out of touch[311]with the professors in the subject as to be unacquainted even with their personal appearance.Such was the mathematical tripos and its history. Whatever its demerits, it dominated the situation, and Cambridge mathematics and mathematicians of the nineteenth century were the direct product of the system it embodied. Judged by the output, I do not think it can be said to have resulted in failure; and perhaps Cayley, Sylvester, Adams, Green, Stokes, Kelvin, and Maxwell—to mention no others—were none the worse for having been compelled to go through the course.The reconstitution in 1907 of the tripos, and the destruction of many of its distinctive features must profoundly modify the future history of mathematics at Cambridge, but forecasts on such a theme would be useless.The curious origin of the term tripos has been repeatedly told, and an account of it may fitly close this chapter. Formerly there were three principal occasions on which questionists were admitted to the title or degree of bachelor. The first of these was at the comitia priora, held on Ash-Wednesday, for the best men in the year. The next was at the comitia posteriora, which was held a few weeks later, and at which any student who had distinguished himself in the quadragesimal exercises subsequent to Ash-Wednesday had his seniority reserved to him.[312]Lastly, there was the comitia minora, for students who had in no special way distinguished themselves.In the fifteenth century an important part in the ceremony on each of these occasions was taken by a certain “ould bachilour,” who sat upon a three-legged stool or tripos before the proctors and tested the abilities of the would-be graduates by arguing some question with the “eldest son,” who was selected from them as their representative. To assist the latter in what might be an unequal contest his “father,” that is, the officer of his college who was to present him for his degree, was allowed to come to his assistance.The discussion took place in Great St Mary’s Church, and marked the admission of the student to a position with new responsibilities, while the season of Lent was chosen with a view to bring this into prominence. The puritan party objected to the semi-ecclesiastical character of the proceedings, and in the course of the sixteenth century set themselves to bring the ceremony into disrepute. The part played by the questionist now became purely formal, though a serious debate still sometimes took place between the father of the senior questionist and a regent master who represented the University: this, however, came to be prefaced by a speech by the bachelor, who was now called Mr Tripos, just as we speak of a judge as the bench, or of a rower[313]as an oar. Ultimately public opinion permitted Mr Tripos to say pretty much what he pleased, so long as it was not dull and was scandalous. The speeches he delivered or the verses he recited were generally printed and preserved by the registrary, and were known as the tripos verses: originally they referred to the subjects of the disputations then propounded. The earliest copies now extant are those for 1575.The university officials, to whom the personal criticisms in which Mr Tripos indulged were by no means pleasing, repeatedly exhorted him to remember “while exercising his privilege of humour, to be modest withal.” In 1710, says Mullinger81, “the authorities after condemning the excessive license of the tripos announced that the comitia at Lent would in future be conducted in the Senate-House; and all members of the University, of whatever order or degree, were forbidden to assail or mock the disputants with scurrilous jokes or unseemly witticisms. About the year 1747–8, the moderators initiated the practice of printing the honour lists on the back of the sheets containing the tripos verses, and after the year 1755 this became the invariable practice. By virtue of this purely arbitrary connection these lists[314]themselves became known as the tripos; and eventually the examination itself, of which they represented the results, also became known by the same designation.”Mr Tripos ceased to deliver his speech about 1750, but the issue of tripos verses continued for nearly 150 years longer. During the latter part of this time they consisted of four sets of verses, usually in Latin, but occasionally in Greek, in which current topics in the University were treated lightly or seriously as the writer thought fit. They were written for the proctors and moderators by undergraduates or commencing bachelors, each of whom was supposed to receive a pair of white kid gloves in recognition of his labours. Thus gradually the word tripos changed its meaning “from a thing of wood to a man, from a man to a speech, from a speech to sets of verses, from verses to a sheet of coarse foolscap paper, from a paper to a list of names, and from a list of names to a system of examination82.”In 1895 the proctors and moderators, without consulting the senate, sent in no verses, and thus, in spite of widespread regret, an interesting custom of many centuries standing was destroyed. In defence of this action, it was said that the custom had never been embodied in statute or ordinance,[315]and thus was not obligatory, and further that its continuance was not of material benefit to anybody. Such arguments are not conclusive, and we may well regret the disappearance of historic ties unless it can be shown that they cause inconvenience, which of course in this case could not be asserted.By way of supplement to the foregoing account, I append a list of those who have held or hold the various university mathematical chairs and lectureships.TheLucasian Professorship of Mathematicswas founded in 1663 by Henry Lucas. The successive occupants of the chair have been: Isaac Barrow, 1664–1669; Isaac Newton, 1669–1702; William Whiston, 1702–1711; Nicholas Saunderson (Sanderson), 1711–1739; John Colson, 1739–1760; Edward Waring, 1760–1798; Isaac Milner, 1798–1820; Robert Woodhouse, 1820–1822; Thomas Turton, 1822–1826; George Biddell Airy, 1826–1828; Charles Babbage, 1828–1839; Joshua King, 1839–1849; George Gabriel Stokes, 1849–1903; Joseph Larmor, 1903et seq.ThePlumian Professorship of Astronomy and Experimental Philosophywas founded in 1704 by Thomas Plume. The successive occupants of the chair have been: Roger Cotes, 1707–1716; Robert Smith, 1716–1760; Anthony Shepherd, 1760–1796; Samuel Vince, 1796–1822; Robert Woodhouse, 1822–1828; George Biddell Airy, 1828–1836; James Challis, 1836–1883; George Howard Darwin, 1883–1912; Arthur Stanley Eddington, 1913et seq.TheLowndean Professorship of Astronomy and Geometrywas founded in 1749 by Thomas Lowndes. The successive occupants of the chair have been: Roger Long, 1750–1771; John Smith, 1771–1795; William Lax, 1795–1836; George Peacock, 1836–1858; John Couch Adams, 1858–1892; Robert Stawell Ball, 1892–1913; Henry Frederick Baker, 1914et seq.TheSadleirian Professorship of Pure Mathematicswas founded, in 1863 from a benefaction given in 1710 by Lady Sadleir. The successive occupants of the chair have been: Arthur Cayley, 1863–1895; Andrew Russell Forsyth, 1895–1910; Ernest William Hobson, 1910et seq.[316]TheCavendish Professorship of Experimental Physicswas founded in 1871 by the University; the laboratory attached being built at the expense of the then Chancellor, the Duke of Devonshire. The successive occupants of the chair have been: James Clerk Maxwell, 1871–1879; John William, Baron Rayleigh, 1879–1884; Joseph John Thomson, 1884et seq.TheProfessorship of Mechanism and Applied Mechanics, with laboratories and shops attached, was founded by the University in 1875. The successive occupants of the chair have been: James Stuart, 1875–1890; James Alfred Ewing, 1890–1903; Bertram Hopkinson, 1903et seq.FiveLectureships in Mathematicswere created in 1882 under the directions of Royal Commissioners, and subsequently two others (now reduced to one other) tenable, if desired, with one of the above, were founded. The successive holders have been: Joseph John Thomson, 1884; Andrew Russell Forsyth, 1884–1895; William Herrick Macaulay, 1884–1887; Richard Tetley Glazebrook, 1884–1898; Ernest William Hobson, 1884–1910; Joseph Larmor, 1885–1903; Richard Pendlebury, 1888–1901; Henry Frederick Baker, 1895–1914; Augustus Edward Hough Love, 1898–1899; Hector Munro Macdonald, 1899–1904; Herbert William Richmond, 1901et seq.; George Ballard Mathews, 1903–1905; James Hopwood Jeans, 1904–1906, 1910–1912; John Gaston Leathem, 1905–1909; Robert Alfred Herman, 1906et seq.; Edmund Taylor Whittaker, 1905–1906; Thomas James I’Anson Bromwich, 1909et seq.; John Hilton Grace, 1901et seq.; Godfrey Harold Hardy, 1914et seq.; Arthur Berry, 1914et seq.34The greater part of this chapter formerly appeared in myMathematical Recreations and Essays, but a few paragraphs on “coaching” have been taken from a paper which I wrote for distribution to those who attended the International Congress of Mathematicians held in England in 1912. The subject is treated in Whewell’sLiberal Education, Cambridge, three parts, 1845, 1850, 1853; Wordsworth’sScholae Academicae, Cambridge, 1877; my ownOrigin and History of the Mathematical Tripos, Cambridge, 1880; Glaisher’s Presidential Address to the London Mathematical Society,Transactions, vol. XVIII, 1886, pp. 4–38; and myHistory of the Study of Mathematics at Cambridge, Cambridge, 1889.35Budget of Paradoxes, by A. De Morgan, London, 1872, p. 305.36See grace of 25 October 1680.37Ex. gr.see De la Pryme’s account of his graduation in 1694,Surtees Society, vol.LIV, 1870, p. 32.38W. Reneu, in his letters of 1708–10 describing the course for the B.A. degree, makes no mention of the senate-house examination, and I think it is a reasonable inference that it had not then been established.39Memoirs of Richard Cumberland, London, 1806, pp. 78–79.40Quoted by C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 30–31.41Anecdotes of the Life of Richard Watson, London, 1817, pp. 18–19.42See grace of 25 October 1883; and theCambridge University Reporter, 23 October 1883.43See grace of 11 February 1909, and theCambridge University Reporter, 8 December 1908.44The Works of J. Jebb, London, 1787, vol.II, pp. 290–297.45“Emulation, which is the principle upon which the plan is constructed.”The Works of J. Jebb, London, 1787, vol.III, p. 261.46The Works of J. Jebb, London, 1787, vol.III, p. 272.47See graces of 5 July 1773, and of 17 February 1774.48See graces of 19, 20 March 1779.49Notice issued by the vice-chancellor, dated 19 May 1779.50TheChallis Manuscripts,III, 61. There are two copies almost identical, one dated 1785, the other 1786. Probably the paper printed in the text was set in 1786.51H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 82.52C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 322–323.53H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 182.54See grace of 8 April 1791.55Communicated by the moderators to fathers of colleges on 18 January 1799, and agreed to by the latter.56C. Wordsworth,Scholae Academicae, Cambridge, 1817, p. 123.57Anecdotes of the Life of Richard Watson, London, 1817, p. 19.58Memoir of A. De Morgan, London, 1882, pp. 387–392.59See graces, 15 December 1808.60S. Douglas,Life of W. Whewell, London, 1881, p. 20.61For a contemporary account of this, see C. A. Bristed,Five Years in an English University, New York, 1852, pp. 233–239.62Seeex. gr.the grace of 14 November 1827, referred to below.63Proceedings of the Royal Society, London, 1859, vol.IX, pp. 538–539.64Whewell’s Writings and Correspondence, ed. Todhunter, London, 1876, vol.II, p. 36.65S. Douglas,Life of Whewell, London, 1881, p. 56.66Alma Mater, London, 1827, vol.II, pp. 58–98.67SeeNature, vol.XXXV, 24 February 1887, pp. 397–399. See also hisAutobiography, Cambridge, 1896, chapter ii.68See grace, 14 November 1827.69See grace, 21 May 1828, confirming a report of 27 March 1828.70See grace of 31 October 1849.71See grace of 6 April 1832.72See grace of 30 May 1838.73Under a badly-worded grace passed on 11 May 1842, on the recommendation of a syndicate on theological studies, candidates for mathematical honours were, after 1846, required to attend the poll examination on Paley’sMoral Philosophy, the new testament and ecclesiastical history. This had not been the intention of the senate, and on 14 March 1855, a grace was passed making this clear.74See grace of 13 May 1846, confirming a report of 23 March 1846.75See grace of 31 October 1848.76See grace of 2 June 1868. It was carried by a majority of only five in a house of 75.77See graces of 17 May 1877; 29 May 1878; and 21 November 1878; and theCambridge University Reporter, 2 April, 14 May, 4 June, 29 October, 12 November, and 26 November 1878.78See graces of 13 December 1883; 12 June 1884; 10 February 1885; 29 October 1885; and 1 June 1886.79See reports dated 7 November 1899, and 20 January 1900.80See the reports of the special board,Cambridge University Reporter, 29 May and 20 November 1906, and the graces of 2 February 1907. The voting on the first grace was 776 placet and 644 non-placet.81J. B. Mullinger,The University of Cambridge, Cambridge, vol.I, 1873, pp. 175–176.82C. Wordsworth,Scholae Academicae, Cambridge, 1877, p. 21.
It must be remembered that for students pursuing the normal course the senate-house examination still provided the only avenue to a degree. That examination involved a knowledge of the elements of moral philosophy and theology, an acquaintance with the rules of formal logic, and the power of reading and writing scholastic Latin, but mathematics was the predominant subject, and this led to a certain one-sidedness in education. The evil of this was generally recognized, and in 1822 various reforms were introduced in the university curriculum; in particular the Previous Examination was established for students in their second year, the subjects being prescribed Greek and Latin works,[295]a Gospel, and Paley’sEvidences. Set classical books were introduced in the final examination of poll-men; and another honour or tripos examination was established for classical students. These alterations came into effect in 1824; and henceforth the senate-house examination, so far as it related to mathematical students, was known as the Mathematical Tripos.
In 1827 the scheme of examination in the mathematical tripos was revised. By regulations68which came into operation in January 1828, four days, exclusive of the day of arranging the brackets, were devoted to the examination; the number of hours of examination was twenty-three, of which seven were assigned to problems. On the first two days all the candidates had the same questions proposed to them, inclusive of the evening problems, and the examination on those days excluded the higher and more difficult parts of mathematics, in order, in the words of the report, “that the candidates for honours may not be induced to pursue the more abstruse and profound mathematics, to the neglect of more elementary knowledge.” Accordingly, only such questions as could be solved without the aid of the differential calculus were set on the first day, and those set on the second day involved only its elementary applications. The classes were reduced[296]to four, determined as before by the exercises in the schools.
The regulations of 1827 definitely prescribed that all the papers should be printed. They are also noticeable as being the last which gave the examiners power to askvivâ vocequestions, though such questions “were restricted to asking about propositions contained in the mathematical works commonly in use at the University, or examples and explanations of such propositions.” It was further recommended that no paper should contain more questions than well-prepared students could be expected to answer within the time allowed for it, but that if any candidate, before the end of the time, had answered all the questions in the paper, the examiners might propose additional questionsvivâ voce. The power of granting honorary optime degrees now ceased; it had already fallen into abeyance. Henceforth the examination was conducted under definite rules, and I no longer concern myself with its traditions.
In the same year as these changes became effective the examination for the poll degree was separated from the tripos with different sets of papers and a different schedule of subjects69. It was, however, still nominally considered as forming part of the senate-house examination, and until 1858 those[297]who obtained a poll degree were arranged in four classes, described as fourth, fifth, sixth, and seventh, as if in continuation of the junior optimes or third class of the tripos.
In the course henceforth ordained for the poll or ordinary degree, the examination, later known as “the General,” represents that part of the old senate-house examination which was intended for the poll-men, but gradually it was moved to an earlier period in the normal course taken by the men. In 1851 admission to the classical tripos70was allowed to others than those who passed the mathematical tripos, and this provided another avenue to a degree entirely independent of the old senate-house examination. In 1852 another set of examinations, at first called “the Professor’s Examinations,” and now somewhat modified and known as “the Specials,” was instituted for all poll-men to take before they could qualify for a degree.
In 1858 the fiction that the poll examinations were part of the senate-house examination was abandoned, and subsequently they have been treated as providing an independent method of obtaining the degree: thus now the mathematical tripos is the sole representative of the old senate-house examination. Since 1858 numerous other[298]ways of obtaining a degree in arts have been established, and it is now possible to graduate by showing proficiency in very special, or even technical subjects.
Further changes in the mathematical tripos were introduced in 183371. The duration of the examination, before the issue of the brackets, was extended to five days, and the number of hours of examination on each day was fixed at five and a half: seven and a half hours were assigned to problems. The examination on the first day was confined to subjects that did not require the differential calculus, and only the simplest applications of the calculus were permitted on the second and third days. During the first four days of the examination the same papers were set to all the candidates alike, but on the fifth day the examination was conducted according to classes. No reference was made tovivâ vocequestions, though permission was reserved to re-examine candidates if it were found necessary: this right remained in force till 1848, but in fact was never used. In December 1834, a few unimportant details were amended.
Mr Earnshaw, the senior moderator in 1836, informed me that he believed that the tripos of that year was the earliest one in which all the papers were marked, and that in previous years the[299]examiners had partly relied on their impression of the answers given.
New regulations came into force72in 1839. The examination now lasted for six days, and continued as before for five hours and a half each day: eight and a half hours were assigned to problems. Throughout the whole examination the same papers were set to all candidates, and no reference was made to any preliminary classes. It was no doubt in accordance with the spirit of these changes that the acts in the schools should be abolished, but they were discontinued by the moderators of 1839 without the authority of the senate. The examination was for the future confined73to mathematics.
In the same year in which the new scheme came into force a proposal to reopen the subject was rejected on 6 March 1839.
The difficulty of bringing professorial lectures into relation with the needs of students has more than once been before the University. The desirability of it was emphasized by a syndicate in February 1843, which recommended conferences at stated intervals between the mathematical professors[300]and examiners. This report, which foreshadowed the creation of a Mathematical Board, was rejected by the senate on 31 March.
A few years later the scheme of the examination was again reconstructed by regulations74which came into effect in 1848. The duration of the examination was extended to eight days. The examination lasted in all forty-four and a half hours, twelve of which were devoted to problems. The first three days were assigned to specified elementary subjects; in the papers set on these days riders were to be set as well as bookwork, but the methods of analytical geometry and the calculus were excluded. After the first three days there was a short interval, at the end of which the examiners issued a list of those who had so acquitted themselves as to deserve mathematical honours. Only those whose names were contained in this list were admitted to the last five days of the examination, which was devoted to the higher parts of mathematics. After the conclusion of the examination the examiners, taking into account the whole eight days, brought out the list arranged in order of merit. No provision was made for any rearrangement of this list corresponding to the examination of the brackets. The arrangements of 1848 remained in force till 1873.
[301]In the same year as these regulations came into force, a Board of Mathematical Studies (consisting of the mathematical professors, with the moderators and examiners for the current year and the two preceding years) was constituted75by the senate. From that time forward their minutes supply a permanent record of the changes gradually introduced into the tripos. I do not allude to subsequent changes which only concern unimportant details of the examination.
In May 1849, the board issued a report in which, after giving a review of the past and existing state of the mathematical studies in the University, they recommended that the mathematical theories of electricity, magnetism, and heat should not be admitted as subjects of examination. In the following year they issued a second report, in which they recommended the omission of elliptic integrals, Laplace’s coefficients, capillary attraction, and the figure of the earth considered as heterogeneous, as well as a definite limitation of the questions in the lunar and planetary theories. In making these recommendations the board were only recognizing what had become the practice in the examination.
I may, in passing, mention a curious attempt which was made in 1853 and 1854 to assist candidates to estimate the relative difficulty of the[302]questions asked. This was effected by giving to the candidates, at the same time as the examination paper, a slip of paper on which the marks assigned for the bookworkand rider for each question were printed. I mention the fact merely because these things are rapidly forgotten and not because it is of any intrinsic value. I possess a complete set of slips which came to me from Todhunter.
In 1856 there was an amusing difference of opinion between the vice-chancellor and the moderators. The vice-chancellor issued a notice to say that for the convenience of the University he had directed the tripos lists to be published at 8.0 a.m. as well as at 9.0 a.m., but when members of the senate arrived at 8.0 the moderators said that the list should not be read until 9.0.
Considerable changes in the scheme of examination were introduced in 1873. On 5 December 1865, the board had recommended the addition of Laplace’s coefficients and the figure of the earth considered as heterogeneous as subjects of the examination; the report does not seem to have been brought before the senate, but attention was called to the fact that certain departments of mathematics and mathematical physics found no place in the tripos schedules, and were neglected by most students. Accordingly, a syndicate was appointed on 6 June 1867, to consider the matter, and a scheme drawn[303]up by them was approved in 186876and came into effect in 1873.
The new scheme of examination was framed on the same lines as that of 1848. The subjects in the first three days were left unchanged, but an extra day was added, devoted to the elements of mathematical physics. The essence of the modification was the greatly extended range of subjects introduced into the schedule of subjects for the last five days, and their arrangement in divisions; the total marks awarded to the questions in each of the five divisions being approximately in a proportion to the total marks assigned to the questions in the first three days as 2, 1, 1, 1, 2/3 to 1 respectively. Under these regulations the number of examiners was increased from four to five.
The assignment of marks to groups of subjects was made under the impression that the best candidates would concentrate their abilities on a selection of subjects from the various divisions. But it was found that, unless the questions were made extremely difficult, more marks could be obtained by reading superficially all the subjects in the five divisions than by attaining real proficiency in a few of the higher ones: while the wide range of subjects rendered it practically impossible to[304]cover all the ground thoroughly in the time allowed. The failure was so pronounced that in 1877 another syndicate was appointed to consider the mathematical studies and examinations of the University. They presented an elaborate scheme, but on 13 May 1878, some of the most important parts of it were rejected; their subsequent proposals, accepted on 21 November 1878 (by 62 to 49), represented a compromise which pleased few members of the senate77.
Under the new scheme which came into force in 1882 the tripos was divided into two portions: the first portion was taken at the end of the third year of residence, the range of subjects being practically the same as in the regulations of 1848, and the result brought out in the customary order of merit. The second portion was held in the following January, and was open only to those who had been wranglers in the preceding June. This portion was confined to higher mathematics and appealed chiefly to specialists: the result was brought out in three classes, each arranged in alphabetical order. The moderators and examiners conducted the whole examination without any extraneous aid.
In the next year or two further amendments[305]were made78, the second part of the examination being moved to the June of the fourth year, and thrown open to all men who had graduated in the tripos of the previous June. At the same time the conduct of the examination in partIIwas transferred to four examiners nominated by the board: this put it largely under the control of the professors. The range of subjects of partIIwas also greatly extended, and candidates were encouraged to select only a few of them. It was further arranged that partImight be taken at the end of a man’s second year of residence, though in that case it would not qualify for a degree. A student who availed himself of this leave could take partIIat the end either of his third or of his fourth year as he pleased.
The general effect of these changes was to destroy the homogeneity of the tripos. Objections to the new scheme were soon raised. Especially, it was said—whether rightly or wrongly—that partIcontained too many technical subjects to serve as a general educational training for any save mathematicians; that the distinction of a high place in the historic list produced on its results tended to prevent the best men taking it in their second year, though by this time they had read enough to be able to do so; and that partIIwas so constructed as[306]to appeal only to professional mathematicians, and thus the higher branches of mathematics were neglected in the University by all save a few specialists.
Whatever value be attached to these opinions, the number of students studying mathematics fell rapidly under the scheme of 1886. In 1899 the board proposed79further changes. These seemed to some members of the senate to be likely still further to decrease the number of men who took up the subject as one of general education; and the two main proposals were rejected, 15 February 1900 by votes of 151 to 130 and 161 to 129.
A few years later, in 190780, the board brought forward another scheme, proposing changes so sweeping as almost to destroy the identity of the tripos. Under this the examination in partIIwas abolished—a change on which all parties were agreed. There was introduced an examination, called partI, confined to elementary mathematics, which could be taken as early as the second term of residence, and for which in certain cases of failure a student could present himself again, but this, although an examination for honours, did not qualify for a degree.[307]In the new partII, taken normally at the end of the third year of residence and qualifying for a degree, candidates were given some option in the subjects of their examination, and order of merit was abolished. The first examination under this scheme was held in 1908.
A remarkable feature in the history of the Cambridge mathematical school is the fact that for nearly two hundred years most students were accustomed to rely for preparation for it on work done with a private tutor or “Coach.” Towards the close of the seventeenth century we first read of these “pupil-mongers” (among whom Laughton of Clare was the most famous) who made it their business to prepare men for their “acts.”
With the rise of the senate-house examination the importance of this class of teachers increased, for success in that examination was regarded as the crown of the academic course, and brought with it, in the shape of a fellowship, an immediate competence with a reasonable prospect of an assured career. It was the business of private tutors to prepare their pupils for the examination, and among those who in this way came to the front shortly after the middle of the eighteenth century were Richard Watson, John Wilson whose name is still known by its association with a proposition in the theory of numbers, and Robert Thorp. The last named[308]teacher was described, about 1761, as being “of eminent use to young men in preparing them for the Senate-House Examinations and peculiarly successful”; and it was added that “one young man of no shining reputation with the assistance of Mr Thorp’s tuition had stood at the head of wranglers.”
In a grace of the senate, passed in 1781, it is stated that almost all sophs then resorted to private tuition, and for more than a century subsequently, the practice was well established. These were the men who really directed the reading of the students. Even non-residents, if reputed to be successful coaches, drew pupils. Thus John Dawson, a medical practitioner at Sedbergh, regularly prepared pupils in the vacations for the senate-house examination, and at least eleven of the senior wranglers between 1781 and 1800 are known to have studied under him.
During the nineteenth century the system developed under two remarkable teachers, William Hopkins, 1793–1866, and Edward John Routh, 1831–1907, to whom the vast majority of the better known Cambridge mathematicians of this century owed most of what they learnt in their undergraduate days. Hopkins in the twenty-two years from 1828–49, had among his pupils one hundred and seventy-five wranglers, of whom seventeen were[309]senior, forty-four in one of the first three places, and one hundred and eight in one of the first ten places. So too Routh, in the thirty-one years from 1858–88, had between six hundred and seven hundred pupils, most of whom became wranglers, twenty-seven being senior in the tripos and forty-one Smith’s prizemen. To organize teaching on this scale demanded rare gifts.
Perhaps it may be of interest to describe, by way of example, the general features of Routh’s system. He gave catechetical lectures three times a week to classes of eight or ten men of approximately equal knowledge and ability. The work to be done between two lectures was heavy, and included the solution of some eight or nine fairly hard examples on the subject of the lectures. Examination papers were also constantly set on tripos lines (bookwork and riders), while there was a weekly paper of problems set to all pupils alike. All papers sent up were marked in public, the comments on them in class were generally brief, and, to save time, solutions of the questions were circulated in manuscript. Teaching also was supplemented by manuscripts on the subjects. Finally to the more able students he was accustomed, shortly before their tripos, to give memoirs or books for analyses and commentaries. The course for the first three years and the two earlier long vacations covered all the subjects of the[310]examination—the last long vacation and the first term of the fourth year were devoted to a thorough revision.
Under Hopkins and Routh there was no trace of what is called cramming; they might say that a particular demonstration was so long that it could not be required in the tripos, but none the less they expected their pupils to master it. The system had faults, but it had the merit of providing a systematic grounding in a wide field of subjects. The effectiveness of teaching of this kind was dependent on intimate constant personal intercourse, and the importance of this cannot be overrated. The scandal of the system consisted in the fact that a man was compelled to pay heavy fees to the University and his College for instruction, and yet found it advantageous at his own expense to go elsewhere to get it.
During the last quarter of the nineteenth century college lecturers began to share with the coaches the general direction of studies. Post-graduate work was also to some extent brought under the influence of professors and university lecturers—these not uncommonly suggesting subjects for dissertations for fellowships, Smith’s prizes, etc. But the students thus influenced were not numerous, and it still remains true that the majority of mathematical undergraduates are so out of touch[311]with the professors in the subject as to be unacquainted even with their personal appearance.
Such was the mathematical tripos and its history. Whatever its demerits, it dominated the situation, and Cambridge mathematics and mathematicians of the nineteenth century were the direct product of the system it embodied. Judged by the output, I do not think it can be said to have resulted in failure; and perhaps Cayley, Sylvester, Adams, Green, Stokes, Kelvin, and Maxwell—to mention no others—were none the worse for having been compelled to go through the course.
The reconstitution in 1907 of the tripos, and the destruction of many of its distinctive features must profoundly modify the future history of mathematics at Cambridge, but forecasts on such a theme would be useless.
The curious origin of the term tripos has been repeatedly told, and an account of it may fitly close this chapter. Formerly there were three principal occasions on which questionists were admitted to the title or degree of bachelor. The first of these was at the comitia priora, held on Ash-Wednesday, for the best men in the year. The next was at the comitia posteriora, which was held a few weeks later, and at which any student who had distinguished himself in the quadragesimal exercises subsequent to Ash-Wednesday had his seniority reserved to him.[312]Lastly, there was the comitia minora, for students who had in no special way distinguished themselves.
In the fifteenth century an important part in the ceremony on each of these occasions was taken by a certain “ould bachilour,” who sat upon a three-legged stool or tripos before the proctors and tested the abilities of the would-be graduates by arguing some question with the “eldest son,” who was selected from them as their representative. To assist the latter in what might be an unequal contest his “father,” that is, the officer of his college who was to present him for his degree, was allowed to come to his assistance.
The discussion took place in Great St Mary’s Church, and marked the admission of the student to a position with new responsibilities, while the season of Lent was chosen with a view to bring this into prominence. The puritan party objected to the semi-ecclesiastical character of the proceedings, and in the course of the sixteenth century set themselves to bring the ceremony into disrepute. The part played by the questionist now became purely formal, though a serious debate still sometimes took place between the father of the senior questionist and a regent master who represented the University: this, however, came to be prefaced by a speech by the bachelor, who was now called Mr Tripos, just as we speak of a judge as the bench, or of a rower[313]as an oar. Ultimately public opinion permitted Mr Tripos to say pretty much what he pleased, so long as it was not dull and was scandalous. The speeches he delivered or the verses he recited were generally printed and preserved by the registrary, and were known as the tripos verses: originally they referred to the subjects of the disputations then propounded. The earliest copies now extant are those for 1575.
The university officials, to whom the personal criticisms in which Mr Tripos indulged were by no means pleasing, repeatedly exhorted him to remember “while exercising his privilege of humour, to be modest withal.” In 1710, says Mullinger81, “the authorities after condemning the excessive license of the tripos announced that the comitia at Lent would in future be conducted in the Senate-House; and all members of the University, of whatever order or degree, were forbidden to assail or mock the disputants with scurrilous jokes or unseemly witticisms. About the year 1747–8, the moderators initiated the practice of printing the honour lists on the back of the sheets containing the tripos verses, and after the year 1755 this became the invariable practice. By virtue of this purely arbitrary connection these lists[314]themselves became known as the tripos; and eventually the examination itself, of which they represented the results, also became known by the same designation.”
Mr Tripos ceased to deliver his speech about 1750, but the issue of tripos verses continued for nearly 150 years longer. During the latter part of this time they consisted of four sets of verses, usually in Latin, but occasionally in Greek, in which current topics in the University were treated lightly or seriously as the writer thought fit. They were written for the proctors and moderators by undergraduates or commencing bachelors, each of whom was supposed to receive a pair of white kid gloves in recognition of his labours. Thus gradually the word tripos changed its meaning “from a thing of wood to a man, from a man to a speech, from a speech to sets of verses, from verses to a sheet of coarse foolscap paper, from a paper to a list of names, and from a list of names to a system of examination82.”
In 1895 the proctors and moderators, without consulting the senate, sent in no verses, and thus, in spite of widespread regret, an interesting custom of many centuries standing was destroyed. In defence of this action, it was said that the custom had never been embodied in statute or ordinance,[315]and thus was not obligatory, and further that its continuance was not of material benefit to anybody. Such arguments are not conclusive, and we may well regret the disappearance of historic ties unless it can be shown that they cause inconvenience, which of course in this case could not be asserted.
By way of supplement to the foregoing account, I append a list of those who have held or hold the various university mathematical chairs and lectureships.
TheLucasian Professorship of Mathematicswas founded in 1663 by Henry Lucas. The successive occupants of the chair have been: Isaac Barrow, 1664–1669; Isaac Newton, 1669–1702; William Whiston, 1702–1711; Nicholas Saunderson (Sanderson), 1711–1739; John Colson, 1739–1760; Edward Waring, 1760–1798; Isaac Milner, 1798–1820; Robert Woodhouse, 1820–1822; Thomas Turton, 1822–1826; George Biddell Airy, 1826–1828; Charles Babbage, 1828–1839; Joshua King, 1839–1849; George Gabriel Stokes, 1849–1903; Joseph Larmor, 1903et seq.ThePlumian Professorship of Astronomy and Experimental Philosophywas founded in 1704 by Thomas Plume. The successive occupants of the chair have been: Roger Cotes, 1707–1716; Robert Smith, 1716–1760; Anthony Shepherd, 1760–1796; Samuel Vince, 1796–1822; Robert Woodhouse, 1822–1828; George Biddell Airy, 1828–1836; James Challis, 1836–1883; George Howard Darwin, 1883–1912; Arthur Stanley Eddington, 1913et seq.TheLowndean Professorship of Astronomy and Geometrywas founded in 1749 by Thomas Lowndes. The successive occupants of the chair have been: Roger Long, 1750–1771; John Smith, 1771–1795; William Lax, 1795–1836; George Peacock, 1836–1858; John Couch Adams, 1858–1892; Robert Stawell Ball, 1892–1913; Henry Frederick Baker, 1914et seq.TheSadleirian Professorship of Pure Mathematicswas founded, in 1863 from a benefaction given in 1710 by Lady Sadleir. The successive occupants of the chair have been: Arthur Cayley, 1863–1895; Andrew Russell Forsyth, 1895–1910; Ernest William Hobson, 1910et seq.[316]TheCavendish Professorship of Experimental Physicswas founded in 1871 by the University; the laboratory attached being built at the expense of the then Chancellor, the Duke of Devonshire. The successive occupants of the chair have been: James Clerk Maxwell, 1871–1879; John William, Baron Rayleigh, 1879–1884; Joseph John Thomson, 1884et seq.TheProfessorship of Mechanism and Applied Mechanics, with laboratories and shops attached, was founded by the University in 1875. The successive occupants of the chair have been: James Stuart, 1875–1890; James Alfred Ewing, 1890–1903; Bertram Hopkinson, 1903et seq.FiveLectureships in Mathematicswere created in 1882 under the directions of Royal Commissioners, and subsequently two others (now reduced to one other) tenable, if desired, with one of the above, were founded. The successive holders have been: Joseph John Thomson, 1884; Andrew Russell Forsyth, 1884–1895; William Herrick Macaulay, 1884–1887; Richard Tetley Glazebrook, 1884–1898; Ernest William Hobson, 1884–1910; Joseph Larmor, 1885–1903; Richard Pendlebury, 1888–1901; Henry Frederick Baker, 1895–1914; Augustus Edward Hough Love, 1898–1899; Hector Munro Macdonald, 1899–1904; Herbert William Richmond, 1901et seq.; George Ballard Mathews, 1903–1905; James Hopwood Jeans, 1904–1906, 1910–1912; John Gaston Leathem, 1905–1909; Robert Alfred Herman, 1906et seq.; Edmund Taylor Whittaker, 1905–1906; Thomas James I’Anson Bromwich, 1909et seq.; John Hilton Grace, 1901et seq.; Godfrey Harold Hardy, 1914et seq.; Arthur Berry, 1914et seq.
TheLucasian Professorship of Mathematicswas founded in 1663 by Henry Lucas. The successive occupants of the chair have been: Isaac Barrow, 1664–1669; Isaac Newton, 1669–1702; William Whiston, 1702–1711; Nicholas Saunderson (Sanderson), 1711–1739; John Colson, 1739–1760; Edward Waring, 1760–1798; Isaac Milner, 1798–1820; Robert Woodhouse, 1820–1822; Thomas Turton, 1822–1826; George Biddell Airy, 1826–1828; Charles Babbage, 1828–1839; Joshua King, 1839–1849; George Gabriel Stokes, 1849–1903; Joseph Larmor, 1903et seq.
ThePlumian Professorship of Astronomy and Experimental Philosophywas founded in 1704 by Thomas Plume. The successive occupants of the chair have been: Roger Cotes, 1707–1716; Robert Smith, 1716–1760; Anthony Shepherd, 1760–1796; Samuel Vince, 1796–1822; Robert Woodhouse, 1822–1828; George Biddell Airy, 1828–1836; James Challis, 1836–1883; George Howard Darwin, 1883–1912; Arthur Stanley Eddington, 1913et seq.
TheLowndean Professorship of Astronomy and Geometrywas founded in 1749 by Thomas Lowndes. The successive occupants of the chair have been: Roger Long, 1750–1771; John Smith, 1771–1795; William Lax, 1795–1836; George Peacock, 1836–1858; John Couch Adams, 1858–1892; Robert Stawell Ball, 1892–1913; Henry Frederick Baker, 1914et seq.
TheSadleirian Professorship of Pure Mathematicswas founded, in 1863 from a benefaction given in 1710 by Lady Sadleir. The successive occupants of the chair have been: Arthur Cayley, 1863–1895; Andrew Russell Forsyth, 1895–1910; Ernest William Hobson, 1910et seq.
[316]TheCavendish Professorship of Experimental Physicswas founded in 1871 by the University; the laboratory attached being built at the expense of the then Chancellor, the Duke of Devonshire. The successive occupants of the chair have been: James Clerk Maxwell, 1871–1879; John William, Baron Rayleigh, 1879–1884; Joseph John Thomson, 1884et seq.
TheProfessorship of Mechanism and Applied Mechanics, with laboratories and shops attached, was founded by the University in 1875. The successive occupants of the chair have been: James Stuart, 1875–1890; James Alfred Ewing, 1890–1903; Bertram Hopkinson, 1903et seq.
FiveLectureships in Mathematicswere created in 1882 under the directions of Royal Commissioners, and subsequently two others (now reduced to one other) tenable, if desired, with one of the above, were founded. The successive holders have been: Joseph John Thomson, 1884; Andrew Russell Forsyth, 1884–1895; William Herrick Macaulay, 1884–1887; Richard Tetley Glazebrook, 1884–1898; Ernest William Hobson, 1884–1910; Joseph Larmor, 1885–1903; Richard Pendlebury, 1888–1901; Henry Frederick Baker, 1895–1914; Augustus Edward Hough Love, 1898–1899; Hector Munro Macdonald, 1899–1904; Herbert William Richmond, 1901et seq.; George Ballard Mathews, 1903–1905; James Hopwood Jeans, 1904–1906, 1910–1912; John Gaston Leathem, 1905–1909; Robert Alfred Herman, 1906et seq.; Edmund Taylor Whittaker, 1905–1906; Thomas James I’Anson Bromwich, 1909et seq.; John Hilton Grace, 1901et seq.; Godfrey Harold Hardy, 1914et seq.; Arthur Berry, 1914et seq.
34The greater part of this chapter formerly appeared in myMathematical Recreations and Essays, but a few paragraphs on “coaching” have been taken from a paper which I wrote for distribution to those who attended the International Congress of Mathematicians held in England in 1912. The subject is treated in Whewell’sLiberal Education, Cambridge, three parts, 1845, 1850, 1853; Wordsworth’sScholae Academicae, Cambridge, 1877; my ownOrigin and History of the Mathematical Tripos, Cambridge, 1880; Glaisher’s Presidential Address to the London Mathematical Society,Transactions, vol. XVIII, 1886, pp. 4–38; and myHistory of the Study of Mathematics at Cambridge, Cambridge, 1889.35Budget of Paradoxes, by A. De Morgan, London, 1872, p. 305.36See grace of 25 October 1680.37Ex. gr.see De la Pryme’s account of his graduation in 1694,Surtees Society, vol.LIV, 1870, p. 32.38W. Reneu, in his letters of 1708–10 describing the course for the B.A. degree, makes no mention of the senate-house examination, and I think it is a reasonable inference that it had not then been established.39Memoirs of Richard Cumberland, London, 1806, pp. 78–79.40Quoted by C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 30–31.41Anecdotes of the Life of Richard Watson, London, 1817, pp. 18–19.42See grace of 25 October 1883; and theCambridge University Reporter, 23 October 1883.43See grace of 11 February 1909, and theCambridge University Reporter, 8 December 1908.44The Works of J. Jebb, London, 1787, vol.II, pp. 290–297.45“Emulation, which is the principle upon which the plan is constructed.”The Works of J. Jebb, London, 1787, vol.III, p. 261.46The Works of J. Jebb, London, 1787, vol.III, p. 272.47See graces of 5 July 1773, and of 17 February 1774.48See graces of 19, 20 March 1779.49Notice issued by the vice-chancellor, dated 19 May 1779.50TheChallis Manuscripts,III, 61. There are two copies almost identical, one dated 1785, the other 1786. Probably the paper printed in the text was set in 1786.51H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 82.52C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 322–323.53H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 182.54See grace of 8 April 1791.55Communicated by the moderators to fathers of colleges on 18 January 1799, and agreed to by the latter.56C. Wordsworth,Scholae Academicae, Cambridge, 1817, p. 123.57Anecdotes of the Life of Richard Watson, London, 1817, p. 19.58Memoir of A. De Morgan, London, 1882, pp. 387–392.59See graces, 15 December 1808.60S. Douglas,Life of W. Whewell, London, 1881, p. 20.61For a contemporary account of this, see C. A. Bristed,Five Years in an English University, New York, 1852, pp. 233–239.62Seeex. gr.the grace of 14 November 1827, referred to below.63Proceedings of the Royal Society, London, 1859, vol.IX, pp. 538–539.64Whewell’s Writings and Correspondence, ed. Todhunter, London, 1876, vol.II, p. 36.65S. Douglas,Life of Whewell, London, 1881, p. 56.66Alma Mater, London, 1827, vol.II, pp. 58–98.67SeeNature, vol.XXXV, 24 February 1887, pp. 397–399. See also hisAutobiography, Cambridge, 1896, chapter ii.68See grace, 14 November 1827.69See grace, 21 May 1828, confirming a report of 27 March 1828.70See grace of 31 October 1849.71See grace of 6 April 1832.72See grace of 30 May 1838.73Under a badly-worded grace passed on 11 May 1842, on the recommendation of a syndicate on theological studies, candidates for mathematical honours were, after 1846, required to attend the poll examination on Paley’sMoral Philosophy, the new testament and ecclesiastical history. This had not been the intention of the senate, and on 14 March 1855, a grace was passed making this clear.74See grace of 13 May 1846, confirming a report of 23 March 1846.75See grace of 31 October 1848.76See grace of 2 June 1868. It was carried by a majority of only five in a house of 75.77See graces of 17 May 1877; 29 May 1878; and 21 November 1878; and theCambridge University Reporter, 2 April, 14 May, 4 June, 29 October, 12 November, and 26 November 1878.78See graces of 13 December 1883; 12 June 1884; 10 February 1885; 29 October 1885; and 1 June 1886.79See reports dated 7 November 1899, and 20 January 1900.80See the reports of the special board,Cambridge University Reporter, 29 May and 20 November 1906, and the graces of 2 February 1907. The voting on the first grace was 776 placet and 644 non-placet.81J. B. Mullinger,The University of Cambridge, Cambridge, vol.I, 1873, pp. 175–176.82C. Wordsworth,Scholae Academicae, Cambridge, 1877, p. 21.
34The greater part of this chapter formerly appeared in myMathematical Recreations and Essays, but a few paragraphs on “coaching” have been taken from a paper which I wrote for distribution to those who attended the International Congress of Mathematicians held in England in 1912. The subject is treated in Whewell’sLiberal Education, Cambridge, three parts, 1845, 1850, 1853; Wordsworth’sScholae Academicae, Cambridge, 1877; my ownOrigin and History of the Mathematical Tripos, Cambridge, 1880; Glaisher’s Presidential Address to the London Mathematical Society,Transactions, vol. XVIII, 1886, pp. 4–38; and myHistory of the Study of Mathematics at Cambridge, Cambridge, 1889.
35Budget of Paradoxes, by A. De Morgan, London, 1872, p. 305.
36See grace of 25 October 1680.
37Ex. gr.see De la Pryme’s account of his graduation in 1694,Surtees Society, vol.LIV, 1870, p. 32.
38W. Reneu, in his letters of 1708–10 describing the course for the B.A. degree, makes no mention of the senate-house examination, and I think it is a reasonable inference that it had not then been established.
39Memoirs of Richard Cumberland, London, 1806, pp. 78–79.
40Quoted by C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 30–31.
41Anecdotes of the Life of Richard Watson, London, 1817, pp. 18–19.
42See grace of 25 October 1883; and theCambridge University Reporter, 23 October 1883.
43See grace of 11 February 1909, and theCambridge University Reporter, 8 December 1908.
44The Works of J. Jebb, London, 1787, vol.II, pp. 290–297.
45“Emulation, which is the principle upon which the plan is constructed.”The Works of J. Jebb, London, 1787, vol.III, p. 261.
46The Works of J. Jebb, London, 1787, vol.III, p. 272.
47See graces of 5 July 1773, and of 17 February 1774.
48See graces of 19, 20 March 1779.
49Notice issued by the vice-chancellor, dated 19 May 1779.
50TheChallis Manuscripts,III, 61. There are two copies almost identical, one dated 1785, the other 1786. Probably the paper printed in the text was set in 1786.
51H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 82.
52C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 322–323.
53H. Gunning,Reminiscences, second edition, London, 1855, vol.I, p. 182.
54See grace of 8 April 1791.
55Communicated by the moderators to fathers of colleges on 18 January 1799, and agreed to by the latter.
56C. Wordsworth,Scholae Academicae, Cambridge, 1817, p. 123.
57Anecdotes of the Life of Richard Watson, London, 1817, p. 19.
58Memoir of A. De Morgan, London, 1882, pp. 387–392.
59See graces, 15 December 1808.
60S. Douglas,Life of W. Whewell, London, 1881, p. 20.
61For a contemporary account of this, see C. A. Bristed,Five Years in an English University, New York, 1852, pp. 233–239.
62Seeex. gr.the grace of 14 November 1827, referred to below.
63Proceedings of the Royal Society, London, 1859, vol.IX, pp. 538–539.
64Whewell’s Writings and Correspondence, ed. Todhunter, London, 1876, vol.II, p. 36.
65S. Douglas,Life of Whewell, London, 1881, p. 56.
66Alma Mater, London, 1827, vol.II, pp. 58–98.
67SeeNature, vol.XXXV, 24 February 1887, pp. 397–399. See also hisAutobiography, Cambridge, 1896, chapter ii.
68See grace, 14 November 1827.
69See grace, 21 May 1828, confirming a report of 27 March 1828.
70See grace of 31 October 1849.
71See grace of 6 April 1832.
72See grace of 30 May 1838.
73Under a badly-worded grace passed on 11 May 1842, on the recommendation of a syndicate on theological studies, candidates for mathematical honours were, after 1846, required to attend the poll examination on Paley’sMoral Philosophy, the new testament and ecclesiastical history. This had not been the intention of the senate, and on 14 March 1855, a grace was passed making this clear.
74See grace of 13 May 1846, confirming a report of 23 March 1846.
75See grace of 31 October 1848.
76See grace of 2 June 1868. It was carried by a majority of only five in a house of 75.
77See graces of 17 May 1877; 29 May 1878; and 21 November 1878; and theCambridge University Reporter, 2 April, 14 May, 4 June, 29 October, 12 November, and 26 November 1878.
78See graces of 13 December 1883; 12 June 1884; 10 February 1885; 29 October 1885; and 1 June 1886.
79See reports dated 7 November 1899, and 20 January 1900.
80See the reports of the special board,Cambridge University Reporter, 29 May and 20 November 1906, and the graces of 2 February 1907. The voting on the first grace was 776 placet and 644 non-placet.
81J. B. Mullinger,The University of Cambridge, Cambridge, vol.I, 1873, pp. 175–176.
82C. Wordsworth,Scholae Academicae, Cambridge, 1877, p. 21.