CHAPTER XL

THE MATHEMATICAL TRIPOS.
The Mathematical Tripos has played so prominent a part in the history of education at Cambridge and of mathematics in England, that a sketch of its development* may be inter- esting to general readers.
So far as mathematics is concerned the history of the University before Newton may be summed up very briefly. The University was founded towards the end of the twelfth century. Throughout the middle ages the studies were or- ganized on lines similar to those at Paris and Oxford. To qualify for a degree it was necessary to perform various exercises, and especially to keep a number of acts or to oppose acts kept by other students. An act consisted in effect of a debate in Latin, thrown, at any rate in later times, into syllogistic form. It was commenced by one student, the respondent, stating some proposition, often propounded in the form of a thesis, which was attacked by one or more oppone^its, the discussion being controlled by a graduate. The teaching was largely in the hands of young graduates — every master of arts being compelled to reside and teach for at least one year —
* The following pages are mostly summarized from my Histoi-y of the Study of Mathematics at Cambridge, Cambridge, 1889. The subject is also treated in Whewell's Liberal Education, Cambridge, three parts, 1845, 1850, 1853; Wordsworth's Scholae Academicae, Cambridge, 1877 ; my own Origin and History of the Mathematical Tripos, Cambridge, 1880 ; and Dr Glaisher's Presidential Address to the London Mathematical Society, Transactions, vol. xvm, 1836, pp. 4—38.
CH. Xl] THE MATHEMATICAL Till PCS 249
though no doubt Colleges and private hostels supplemented this instruction in the case of their own students.
The Reformation in England was mainly the work of Cambridge divines, and in the University the Renaissance was warmly welcomed. In spite of the disorder and confusion of the Tudor period, new studies and a system of professional instruction were introduced. Probably the science (as distinct from the art) of mathematics, save so far as involved in the quadrivium, was still an exotic study, but it was not wholly neglected. Tonstall, subsequently the most eminent English arithmetician of his time, migrated, perhaps about 1495, from Balliol College, Oxford, to King's Hall, Cambridge, and in 1530 the University appointed a mathematical lecturer in the person of Paynell of Pembroke Hall. Most of the subsequent English mathematicians of the Tudor period seem to have been educated at Cambridge ; of these I may mention Record, who migrated, probably about 1535, from Oxford, Dee, Digges, Blundeville, Buckley, Billingsley, Hill, Bedwell, Hood, Richard and John Harvey, Edward Wright, Briggs, and Oughtred. The Elizabethan statutes restricted liberty of thought and action in many ways, but, in spite of the civil and religious disturbances of the early half of the 17th century, the mathe- matical school continued to grow. Horrox, Seth Ward, Foster, Rooke, Gilbert Clerke, Pell, Wallis, Barrow, Dacres, and Morland may be cited as prominent Cambridge mathematicians of the time.
Newton's mathematical career dates from 1665 ; his repu- tation, abilities, and influence attracted general attention to the subject. He created a school of mathematics and mathe- matical physics, among the earliest members of which I note the names of Laughton, Samuel Clarke, Craig, Flamsteed, Whiston, Saunderson, Jurin, Taylor, Cotes, and Robert Smith. Since then Cambridge has been regarded as, in a special sense, the home of English mathematicians, and from 1706 onwards we have fairly complete accounts of the course of reading and work of mathematical students there.
Until less than a century ago the form of the method of
250 THE MATHEMATICAL TRIPOS [CH. XI
qualifying for a degree remained substantially unaltered, but the subject-matter of the discussions varied from time to time with the prevalent studies of the place.
After the Renaissance some of the statutable exercises were " huddled," that is, were reduced to a mere form. To huddle an act, the proctor generally asked some question such as Quid est noinen? to which the answer usually expected was Nescio. In these exercises considerable license was allowed, particularly if there were any play on the words involved. For example, J. Brasse, of Trinity, was accosted with the question. Quid est aes ? to which he answered, Nescio nisi finis examinationis. 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 discrimina- tion in time and place. In earlier years even the farce of huddling seems to have been unnecessary, for it was said 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.**
In medieval times acts had been usually kept on some scholastic question or on a proposition taken from the Sentences. About the end of the fifteenth century religious questions, such as the interpretation of Biblical texts, began to be introduced. Some fifty or sixty years later the favourite subjects were drawn either from dogmatic theology or from philosophy. In the seventeenth century the questions were usually philo- sophical, but in the eighteenth century, under the influence of the Newtonian school, a large proportion of them were mathematical.
Further details about these exercises and specimens of acts kept in the eighteenth century are given in my History of Mathematics at Cambridge. Here I will only say that they provided an admirable training in the art of presenting an argument, and in dialectical skill in attack and defence. The
CH. Xl] THE MATHEMATICAL TRIPOS 251
mental strain in a contested act was severe. De Morgan,
describing his act kept in 1826, wrote*:
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 honore, 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.
Had the language of the discussions been changed to English, as was repeatedly urged from 1774 onwards, these exercises might have been retained with advantage, but the barbarous Latin and the syllogistic form in which they were carried on prejudiced their retention.
About 1830 a custom grew up for the respondent and opponents to meet previously and arrange their arguments together. The discussions then became an elaborate farce, and were a mere public performance of what had been already rehearsed. Accordingly the moderators of 1839 took the re- sponsibility of abandoning them. This action was singularly high-handed, since a report of May 30, 1838, had recommended that they should be continued, and there was no reason why they should not have been reformed and retained as a useful feature in the scheme of study.
On the result of the acts a list of those qualified to receive degrees was drawn up. This list was not arranged strictly in order of merit, because the proctors could insert names any- where in it, but by the beginning of the eighteenth century this power had 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 cjntinued to exist till 1828, though it was not exercised after 1792. Subject to the granting of these honorary degrees, this final list was arranged in order ot merit into wranglers and senior optimes, junior optimes, and poll-men. The bachelors on receiving their degrees took seniority according to their
* Biuhjct of Paradoxes, by A. De Morgan, London, 1872, p. 305.
252 THE MATHEMATICAL TRIPOS [CU. XI
order on this list. The title wrangler is derived from these contentious discussions; the title optime from the customary- compliment given by the moderator to a successful disputant, Domine..., optime disputasti, or even optime quidem disputasti, and the title of poll-man from the description of this class as ol
The final exercises for the B.A. degree were never huddled, and until 1839 were carried out strictly. University officials were responsible for approving the subject-matter of these acts. Stupid men offered some irrefutable truism, but the ambitious student courted reputation by affirming some paradox. Prob- ably all honour men kept acts, but poll-men were deemed to comply with the regulations by keeping opponencies. The proctors were responsible for presiding at these acts, or seeing that competent graduates did so. In and after 1649 two examiners were specially appointed for this purpose. In 1680* these examiners were appointed by the Senate with the title of moderator, and with the joint stipend of four shillings for everyone graduating as B.A. during their year of office. In 1688 the joint stipend of the moderators was fixed at £40 a year. The moderators, like the proctors, were nominated by the Colleges in rotation.
From the earliest times the proctors had the power of questioning a candidate at the end of a disputation, and probably all candidates for a degree attended the public schools on certain days to give an opportunity to the proctors, or any master that liked, to examine themf, though the opportunity was not always used. Different candidates attended on different days. Probably such examinations were conducted in Latin. But soon after 1710 1 the moderators or proctors began the custom of summoning on one day in January all candidates whom they proposed to question. The examination was held
* See Grace of October 25, 1680.
t Ex. gr. see De la Pryme's account of his graduation in 1694, Surtees Society, vol. liv, 1870, p. 32.
X W. Eeneu, in bis letters of 1708-1710 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.
CH. XlJ THE MATHEMATICAL TRIPOS 253
in public, and from it the Senate-House Examination arose. The examination at this time did not last more than one day, and was, there can be no doubt, partly on philosophy and partly on mathematics. It is believed that it was always conducted in English, and it is likely that its rapid development was largely due to this.
This introduction of a regular oral examination seems to have been largely due to the fact that when, in 1710, George I gave the Ely library to the University, it was decided to assign for its reception the old Senate-House — now the Catalogue Room in the Library — and to build a new room for the meetings of the Senate. Pending the building of the new Senate-House the books were stored in the Schools. As the Schools were thus rendered unavailable for keeping acts, considerable difficulty was found in arranging for all the candi- dates to keep the full number of statutable exercises, and thus obtaining opportunities to compare them one with another: hence the introduction of a supplementary oral examination. The advantages of this examination as providing a ready means of testing the knowledge and abilities of the candidates were so patent that it was retained when the necessity for some system of the kind had passed away, and finally it became systematized into an organized test to which all questionists were subjected.
In 1731 the University raised the joint stipend of the moderators to £60 " in consideration of their additional trouble in the Lent Term." This would seem to indicate that the Senate-House Examination had then taken formal shape, and perhaps that a definite scheme for its conduct had become customary.
As long as the order of the list of those approved for degrees was settled on the result of impressions derived from acts kept by the different candidates at different times and on different subjects, it was impossible to arrange the men in strict order of merit, nor was much importance attached to the order. But, with the introduction of an examination of all the candidates on one day, much closer attention was paid to
254 THE MATHEMATICAL TRIPOS [CH. XI
securing a strict order of merit, and more confidence was felt in the published order. It seems to have been consequent on this that in and after 1748 (N.S.) the final lists were circulated, and it was further arranged that the names of the honorary optimes should be indicated. The lists from this time appear in the University calendars : the earlier lists from 1499, edited with biographical notes by C. M. Neale, were published in 1909.
Of the detailed history of the examination until the middle of the eighteenth century we know nothing. From 1750 on- wards, however, we have more definite accounts of it. At this time, it would seem that all the men from each College were taken together as a class, and questions passed down by the proctors or moderators till they were answered : but the ex- amination remained entirely oral, and technically was regarded as subsidiary to the discussions which had been previously held in the schools. As each class 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 an important part of the examination. The subjects were mathematics and philosophy. The examination now 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 examination, and provisionally settled the final list; while the last half-day was used in revising and re-arranging the order of merit.
Richard Cumberland has left an account of the tests to which he was subjected when he took his B.A. degree in 1751. Clearly the disputations still played an important part, and it is difficult to say what weight was attached to the subsequent Senate-House Examination ; his reference to it is only of a general character. After saying that he kept two acts and two opponencies he continues*:
The last time I was called upon to keep an act in the schools I sent in three questions to the Moderator, which he withstood as being all mathematical, and
* Memoirs of Eichard Cumberland, London, 1806, pp. 78 — 79.
CH. Xl] THE MATHEMATICAL TIUPOS 255
required me to conform to the usage of proposing one metaphysical question in the place of that, which I should think fit to withdraw. This was ground I never liked to take, and I appealed against his requisition: the act was accordingly put by till the matter of right should be ascertained by the statutes of the university, and in the result of that enquiry it was given for me, and my
question stood I yielded now to advice, and paid attention to my health, till
we were cited to the senate house to be examined for our Bachelor's degree. It was hardly ever my lot during that examination to enjoy any respite. I seemed an object singled out as every man's mark, and was kept perpetually at the table under the process of question and answer.
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.
The semi-official examination by M.A.s was regarded as the more important part of the test, and the most eminent residents in the University took part in it. Thus John Fenn, of Caius, 5th Avrangier in 1761, writes* :
On the following Monday, Tuesday, and Wednesday, we sat in the Senate- house for public exainination ; during this time I was officially examined by the Proctors and Moderators, and had the honor of being taken out for examination by Mr Abbot, the celebrated mathematical tutor of St John's College, by the eminent professor of mathematics Mr Waving, of Magdalene, and by Mr Jebb of Peterhouse, a man thoroughly versed in the academical studies.
This irregular examination by any master who chose to take