CHAPTER XV

.

THE HISTORY OF 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[34] may be interesting 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 instruction given to students was organized on lines similar to those current at Paris and Oxford, and 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 an _opponent_ or _opponents_, the discussion being controlled by a senior 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--though no doubt colleges and private hostels supplemented this instruction in the case of their own students.

The reformation in England was largely 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. The earliest lectureships created by the University seem to have been one in Latin established in or before 1492 and one in mathematics established in or before 1501: they mark the beginning of the system of teaching by experts which has superseded the medieval system of compulsory teaching by all regent masters. The fact that one of these lectureships was in mathematics shows that as early as 1500 the subject was regarded as important. Tunstall, subsequently the most eminent English arithmetician of his time, migrated in 1496 from Oxford to Cambridge, and most of the subsequent English mathematicians of the Tudor period were 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. Under the Elizabethan statutes of 1570, notwithstanding many disadvantages, the mathematical 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 succeeding century.

Newton's mathematical career dates from 1665; his reputation, abilities, and influence attracted general attention to the subject. He created a school of mathematics and mathematical 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.

Until less than a century ago the form of the method of 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 nomen?_ 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. Brass, 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 his questioner--was plucked for want of discrimination 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 philosophical, 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 mental strain involved in keeping a contested act was severe. De Morgan, describing his act kept in 1826, wrote[35]:

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 arose 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 responsibility of abandoning them. This action was singularly high-handed, since a report of 30 May 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 anywhere 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 continued to exist till 1828, though it was not exercised after 1792. Except for the names of these "honorary optimes," this final list was, until 1752, arranged in order of merit into wranglers and senior optimes, junior optimes, and poll-men; after 1752, the wranglers and senior optimes were placed in separate classes. The bachelors on admission to their degrees took seniority according to their 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 οἱ πολλοί.

The final exercises for the bachelor of arts 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. Probably 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[36] these examiners were appointed by the senate with the title of moderator, and with the joint stipend of four shillings for everyone graduating as a bachelor of arts 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 who liked to take part in the examination) to examine them[37], though the opportunity was not always used. Such examinations were conducted in Latin, and originally different candidates attended on different days. Soon after 1710[38] the moderators or proctors began the custom of summoning on one day in January all candidates whom they proposed to question, and conducting the examination in English and in public: the examination did not last more than one day, and was partly on philosophy and partly on mathematics. It was from this examination that the Mathematical Tripos developed.

This introduction of a regular oral examination seems to have been mainly 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, which thus were rendered unavailable for keeping acts. In consequence of this, considerable difficulty was found in arranging for all the candidates to keep the full number of statutable exercises, and obtaining opportunities to compare them one with another: hence the introduction or extension 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 securing an accurate classification, and more confidence felt in the published order. It seems to have been consequent on this that in and after 1748 the final lists were regarded as authoritative and important and that the names of the honorary optimes were definitely indicated: the lists from this time appeared in the _University Calendars_. The lists from 1748 to 1910, with the earlier Ordines Senioritatis from 1499 to 1747, are printed in the _Historical Register of the University_.

Of the detailed history of the examination until the middle of the eighteenth century we know nothing. From 1750 onwards, 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 examination remained entirely oral, and technically was regarded as subsidiary to the discussions which had been previously held in the schools.

Each class contained men of very different abilities, and to meet difficulties thus caused, a custom grew up by which every candidate was liable to be taken aside to be questioned by any master of arts who wished to do so, and this was regarded as an important part of the examination. The examination now continued for two days and a half, the subjects, as before, being mathematics and philosophy. 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. The last half-day was used in revising and rearranging the order of merit.

Richard Cumberland has left an account of the tests to which he was subjected when he took his bachelor 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 continued[39]:

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 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, as above stated, the first class was subdivided into two, called respectively wranglers and senior optimes, a division which is still maintained.

The semi-official examination by masters of arts 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 wrangler in 1761, wrote[40]:

On the following Monday, Tuesday, and Wednesday, we sat in the Senate-house for public examination; during this time I was officially examined by the Proctors and Moderators, and had the honour 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 Waring, 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 part in it constantly gave rise to accusations of partiality.

In 1763 the traditional rules for the conduct of the examination took more definite shape. Henceforth the examiners used the disputations only as a means of classifying the men roughly. On the result of their "acts," and probably partly also of their general reputation, the candidates were divided into eight classes, each arranged in alphabetical order. The subsequent position of the men in the class was determined solely by the senate-house examination. The first two classes comprised all who were expected to be wranglers, the next four classes included the other candidates for honours, and the last two classes consisted of poll-men only. Practically anyone placed in either of the first two classes was allowed, if he wished, to take an aegrotat senior optime, and thus escape all further examination: this was called gulphing it.

All the men from one college were no longer taken together, but each class was examined separately and _vivâ voce_; and hence, since all the students comprised in each class were of about equal attainments, it was possible to make the examination more effective. Richard Watson, of Trinity, claimed that this change was made by him when

## acting as moderator in 1763. He said[41]:

There was more room for partiality ... then [_i.e._ in 1759] than there is now; and I attribute the change, in a great degree, to an alteration which I introduced the first year I was moderator [_i.e._ in 1763], and which has been persevered in ever since. At the time of taking their Bachelor of Arts' degree, the young men are examined in classes, and the classes are now formed according to the abilities shown by individuals in the schools. By this arrangement, persons of nearly equal merits are examined in the presence of each other, and flagrant acts of partiality cannot take place. Before I made this alteration, they were examined in classes, but the classes consisted of members of the same College, and the best and worst were often examined together.

It is probable that before the examination in the senate-house began a candidate, if manifestly placed in too low a class, was allowed the privilege of challenging the class to which he was assigned. Perhaps this began as a matter of favour, and was only granted in exceptional cases, but a few years later it became a right which every candidate could exercise; and I think that it is partly to its development that the ultimate predominance of the tripos over the other exercises for the degree is due.

In the same year, 1763, it was decided that the relative position of the senior and second wranglers, namely, Paley, of Christ's, and Frere, of Caius, was to be decided by the senate-house examination and not by the disputations. Henceforward distinction in that examination was regarded as the most important honour open to undergraduates.

In 1768 Robert Smith, of Trinity College, founded prizes for mathematics and natural philosophy open to two commencing bachelors. The examination followed immediately after the senate-house examination, and the distinction, being much coveted, tended to emphasize the mathematical side of the normal university education of the best men. Since 1883 the prizes have been awarded on the result of dissertations[42]. Additional prizes, awarded at the same time, and associated with the name of Lord Rayleigh[43], were founded in 1909.

Until about 1770, the senate-house examination had been oral, but it began now to be the custom to dictate some or all of the questions and to require answers to be written. Only one question was dictated at a time, and a fresh one was not given out until some student had solved that previously read: a custom which by causing perpetual interruptions to take down new questions must have proved very harassing. We are perhaps apt to think that an examination conducted by written papers is so natural that the custom is of long continuance, but I know no record of any in Europe earlier than the eighteenth century. Until 1830 the questions for the Smith's prizes were dictated.

The following description of the senate-house examination as it existed in 1772 was given by Jebb[44]:

The moderators, some days before the arrival of the time prescribed by the vice-chancellor, meet for the purpose of forming the students into divisions of six, eight, or ten, according to their performance in the schools, with a view to the ensuing examination.

Upon the first of the appointed days, at eight o'clock in the morning, the students enter the senate-house, preceded by a master of arts from each college, who ... is called the "father" of the college....

After the proctors have called over the names, each of the moderators sends for a division of the students: they sit with him round a table, with pens, ink, and paper, before them: he enters upon his task of examination, and does not dismiss the set till the hour is expired. This examination has now for some years been held in the English language.

The examination is varied according to the abilities of the students. The moderator generally begins with proposing some questions from the six books of Euclid, plain (_sic_) trigonometry, and the first rules of algebra. If any person fails in an answer, the question goes to the next. From the elements of mathematics, a transition is made to the four branches of philosophy, viz. mechanics, hydrostatics, apparent astronomy, and optics, as explained in the works of Maclaurin, Cotes, Helsham, Hamilton, Rutherforth, Keill, Long, Ferguson, and Smith. If the moderator finds the set of questionists, under examination, capable of answering him, he proceeds to the eleventh and twelfth books of Euclid, conic sections, spherical trigonometry, the higher parts of Algebra, and sir Isaac Newton's Principia; more particularly those sections, which treat of the motion of bodies in eccentric and revolving orbits; the mutual action of spheres, composed of

## particles attracting each other according to various laws; the

theory of pulses, propagated through elastic mediums; and the stupendous fabric of the world. Having closed the philosophical examination, he sometimes asks a few questions in Locke's Essay on the human understanding, Butler's Analogy, or Clarke's Attributes. But as the highest academical distinctions are invariably given to the best proficients in mathematics and natural philosophy, a very superficial knowledge in morality and metaphysics will suffice.

When the division under examination is one of the highest classes, problems are also proposed, with which the student retires to a distant part of the senate-house, and returns, with his solution upon paper, to the moderator, who, at his leisure, compares it with the solutions of other students, to whom the same problems have been proposed.

The extraction of roots, the arithmetic of surds, the invention of divisers, the resolution of quadratic, cubic, and biquadratic equations; together with the doctrine of fluxions, and its application to the solution of questions "de maximis et minimis," to the finding of areas, to the rectification of curves, the investigation of the centers of gravity and oscillation, and to the circumstances of bodies, agitated, according to various laws, by centripetal forces, as unfolded, and exemplified, in the fluxional treatises of Lyons, Saunderson, Simpson, Emerson, Maclaurin, and Newton, generally form the subject matter of these problems.

When the clock strikes nine, the questionists are dismissed to breakfast: they return at half-past nine, and stay till eleven; they go in again at half-past one, and stay till three; and, lastly, they return at half-past three, and stay till five.

The hours of attendance are the same upon the subsequent day.

On the third day they are finally dismissed at eleven.

During the hours of attendance, every division is twice examined in form, once by each of the moderators, who are engaged for the whole time in this employment.

As the questionists are examined in divisions of only six or eight at a time, but a small portion of the whole number is engaged, at any particular hour, with the moderators; and, therefore, if there were no further examination, much time would remain unemployed.

But the moderator's inquiry into the merits of the candidates forms the least material part of the examination.

The "fathers" of the respective colleges, zealous for the credit of the societies, of which they are the guardians, are incessantly employed in examining those students, who appear most likely to contest the palm of glory with their sons.

This part of the process is as follows:

The father of a college takes a student of a different college aside, and, sometimes for an hour and an half together, strictly examines him in every part of mathematics and philosophy, which he professes to have read.

After he hath, from this examination, formed an accurate idea of the student's abilities and acquired knowledge, he makes a report of his absolute or comparative merit to the moderators, and to every other father who shall ask him the question.

Besides the fathers, all masters of arts, and doctors, of whatever faculty they be, have the liberty of examining whom they please; and they also report the event of each trial, to every person who shall make the inquiry.

The moderators and fathers meet at breakfast, and at dinner. From the variety of reports, taken in connection with their own examination, the former are enabled, about the close of the second day, so far to settle the comparative merits of the candidates, as to agree upon the names of four-and-twenty, who to them appear most deserving of being distinguished by marks of academical approbation.

These four-and-twenty [wranglers and senior optimes] are recommended to the proctors for their private examination; and, if approved by them, and no reason appears against such placing of them from any subsequent inquiry, their names are set down in two divisions, according to that order, in which they deserve to stand; are afterwards printed; and read over upon a solemn day, in the presence of the vice-chancellor, and of the assembled university.

The names of the twelve [junior optimes], who, in the course of the examination, appear next in desert, are also printed, and are read over, in the presence of the vice-chancellor, and of the assembled university, upon a day subsequent to the former....

The students, who appear to have merited neither praise nor censure [the poll-men], pass unnoticed: while those, who have taken no pains to prepare themselves for the examination, and have appeared with discredit in the schools, are distinguished by particular tokens of disgrace.

Jebb's statement about the number of wranglers and senior optimes is only approximate.

It may be added that it was now frankly recognized that the examination was competitive[45]. Also that though it was open to any member of the senate to take part in it, yet the determination of the relative merit of the students was entirely in the hands of the moderators[46]. Although the examination did not occupy more than three days it must have been a severe physical trial to anyone who was delicate. It was held in winter and in the senate-house: that building was then noted for its draughts, and was not warmed in any way; and, according to tradition, on one occasion the candidates on entering in the morning found the ink frozen in the pots on their desks.

The University was not altogether satisfied[47] with the regulations, and in 1779[48] the scheme of examination was amended in various respects. In particular the examination was extended to four days, a third day being given up entirely to natural religion, moral philosophy, and Locke's _Essay_. It was further announced[49] that a candidate would not receive credit for advanced subjects unless he had satisfied the examiners in Euclid's _Elements_ and elementary natural philosophy.

A system of brackets or "classes quam minimae" was now introduced. Under this system the examiners issued on the morning of the fourth day a provisional list of men who had obtained honours, with the names of those of about equal merit bracketed, and that day was devoted to arranging the names in each bracket in order of merit: the examiners being given explicit authority to invite the assistance of others in this work. Whether at this time a candidate could request to be re-examined with the view of being moved from one bracket to another is uncertain, but later this also was allowed.

The number of examiners was also increased to four, the moderators of one year becoming, as a matter of course, the examiners of the next. Thus of the four examiners in each year, two had taken part in the examination of the previous year, and the continuity of the system of examination was maintained. The names of the moderators appear on the tripos lists, but the names of the examiners were not printed on the lists till some years later.

The right of any master of arts to take part in the examination was not affected, though henceforth it was exercised more sparingly, and I believe was not insisted on after 1785. But it became a regular custom for the moderators to invite particular residents to examine and compare specified candidates: Milner, of Queens', was constantly asked to assist in this way.

It was not long before it became an established custom that a candidate, who was dissatisfied with the class in which he had been placed as the result of his disputations, might challenge it before the examination began. This power seems to have been used but rarely; it was, however, a recognition of the fact that a place in the tripos list was to be determined by the senate-house examination alone, and the examiners soon acquired the habit of settling the preliminary classes without exclusive reference to the previous disputations.

The earliest extant paper actually set in the senate-house, to which we can with certainty refer, is a problem paper set in 1785 or 1786 by W. Hodson, of Trinity, then a proctor. The autograph copy from which he gave out the questions was luckily preserved, and is in the library[50] of Trinity College. It must be almost the last problem paper which was dictated, instead of being printed and given as a whole to the candidates. The paper is as follows:

1. To determine the velocity with which a Body must be thrown, in a direction parallel to the Horizon, so as to become a secondary planet to the Earth; as also to describe a parabola, and never return.

2. To demonstrate, supposing the force to vary as _1/D²_ how far a body must fall both within and without the Circle to acquire the Velocity with which a body revolves in a Circle.

3. Suppose a body to be turned (_sic_) upwards with the Velocity with which it revolves in an Ellipse, how high will it ascend? The same is asked supposing it to move in a parabola.

4. Suppose a force varying first as _1/D³_, secondly in a greater ratio than _1/D²_ but less than _1/D³_, and thirdly in a less ratio than _1/D²_, in each of these Cases to determine whether at all, and where the body parting from the higher Apsid will come to the lower.

5. To determine in what situation of the moon's Apsid they go most forwards, and in what situation of her Nodes the Nodes go most backwards, and why?

6. In the cubic equation _x³ + qx + r = 0_ which wants the second term; supposing _x = a + b_ and _3ab = -q_, to determine the value of _x_. (_sic._)

7. To find the fluxion of _x^r × (y^n + z^m)^{1/q}_.

8. To find the fluent of _aẋ / (a + x)_.

9. To find the fluxion of the _m_^th power of the Logarithm of _x_.

10. Of right-angled Triangles containing a given Area to find that whereof the sum of the two legs _AB + BC_ shall be the least possible. [This and the two following questions are illustrated by diagrams. The angle at _B_ is the right angle.]

11. To find the Surface of the Cone _ABC_. [The cone is a right one on a circular base.]

12. To rectify the arc _DB_ of the semicircle _DBV_.

In cases of equality in the senate-house examination, the acts were still taken into account in settling the tripos order: and in 1786, when the second, third, and fourth wranglers came out equal in the examination, a memorandum was published that the second place was given to that candidate who _dialectis magis est versatus_, and the third place to that one who _in scholis sophistarum melius disputavit_.

At this time there were various intervals in the examination by the moderators, and the examinations by the extraneous examiners took place in these intervals. Those candidates who at any time were not being examined occupied themselves with amusements, provided they were not too boisterous and obvious: probably dice and cards played a large

## part in them. Gunning in an amusing account of his examination in 1788

talks of playing with a teetotum[51] on the Wednesday (when specified works by Locke and Paley formed the subjects of examination), and says this game "was carried on with great spirit ... by considerable numbers during the whole of the examination."

About this period, 1790, the custom of printing the problem papers was introduced, but until 1828 the other papers continued to be dictated. Since then all the papers have been printed.

I insert here the following letter[52] from William Gooch, of Caius, in which he described his examination in the senate-house in 1791. It must be remembered that it is the letter of an undergraduate addressed to his father and mother, and was not intended either for preservation or publication: a fact which certainly does not detract from its value.

_Monday_ ¼ aft. 12.

We have been examin'd this Morning in pure Mathematics & I've hitherto kept just about even with Peacock which is much more than I expected. We are going at 1 o'clock to be examin'd till 3 in Philosophy.

From 1 till 7 I did more than Peacock; But who did most at Moderator's Rooms this Evening from 7 till 9, I don't know yet;--but I did above three times as much as the Sen^r Wrangler last year, yet I'm afraid not so much as Peacock.

Between One & three o'Clock I wrote up 9 sheets of Scribbling Paper so you may suppose I was pretty fully employ'd.

_Tuesday Night._

I've been shamefully us'd by Lax to-day;--Tho' his anxiety for Peacock must (of course) be very great, I never suspected that his

## Partially (_sic_) w^d get the better of his Justice. I had

entertain'd too high an opinion of him to suppose it.--he gave Peacock a long private Examination & then came to me (I hop'd) on the same subject, but 'twas only to _Bully_ me as much as he could,--whatever I said (tho' right) he tried to convert into Nonsense by seeming to misunderstand me. However I don't entirely dispair of being first, tho' you see Lax seems determin'd that I shall not.--I had no Idea (before I went into the Senate-House) of being able to contend at all with Peacock.

_Wednesday evening._

Peacock & I are still in perfect Equilibrio & the Examiners themselves can give no guess yet who is likely to be first;--a New Examiner (Wood of St. John's, who is reckon'd the first Mathematician in the University, for Waring doesn't reside) was call'd solely to examine Peacock & me only.--but by this new Plan nothing is yet determin'd.--So Wood is to examine us again to-morrow morning.

_Thursday evening._

Peacock is declar'd first & I second,--Smith of this Coll. is either 8^th or 9^th & Lucas is either 10^th or 11^th.--Poor Quiz Carver is one of the οἱ πολλοί;--I'm perfectly _satisfied_ that the Senior Wranglership is Peacock's due, but _certainly_ not so very undisputably as Lax pleases to represent it--I understand that _he_ asserts 'twas 5 to 4 in Peacock's favor. Now Peacock & I have explain'd to each other how we went on, & can _prove indisputably_ that it wasn't 20 to 19 in his favor;--I _cannot_ therefore be displeas'd for being plac'd second, tho' I'm provov'd (_sic_) with Lax for his false report (so much beneath the Character of a Gentleman.)--

N.B. it is my very _particular Request_ that you dont mention Lax's behaviour to me to any one.

Such was the form ultimately taken by the senate-house examination, a form which it retained substantially without alteration for nearly half-a-century. It soon became the sole test by which candidates were judged. The University was not obliged to grant a degree to anyone who performed the statutable exercises, and it was open to the senate to refuse to pass a supplicat for a bachelor's degree in arts unless the candidate had presented himself for the senate-house examination. In 1790 James Blackburn, of Trinity, a questionist of exceptional abilities, was informed that in spite of his good disputations he would not be allowed a degree unless he also satisfied the examiners in the tripos. He accordingly solved one "very hard problem," though in consequence of a dispute with the authorities he refused to attempt any more[53].

Henceforth the examination was compulsory on all candidates pursuing the normal course for the B.A. degree. In 1791 the University laid down rules[54] for its conduct, so far as it concerned poll-men, decreeing that those who passed were to be classified in four divisions or classes, the names in each class to be arranged alphabetically, but not to be printed on the official tripos lists. The classes in the final lists must be distinguished from the eight preliminary classes issued before the commencement of the examination. The men in the first six preliminary classes were expected to take honours; those in the seventh and eighth preliminary classes were _primâ facie_ poll-men.

In 1799 the moderators announced[55] that for the future they would require every candidate to show a competent knowledge of the first book of Euclid's _Elements_, arithmetic, vulgar and decimal fractions, simple and quadratic equations, and selected books by Locke and Paley. Paley's works seem to be held in esteem by modern divines, and his _Evidences_, though not his _Philosophy_, still remains (1917) one of the subjects of the Previous Examination, but his contemporaries thought less highly of his writings, or at any rate of his philosophy. Thus Best is quoted by Wordsworth[56] as saying of Paley's _Philosophy_, "The tutors of Cambridge no doubt neutralize by their judicious remarks, when they read it to their pupils, all that is pernicious in its principles": so also Richard Watson, bishop of Llandaff, in his anecdotal autobiography[57], says, in describing the senate-house examination in which Paley was senior wrangler, that Paley was afterwards known to the world by many excellent productions, "though there are some ... principles in his philosophy which I by no means approve."

In 1800 the moderators extended to all men in the first four preliminary classes the privilege of being allowed to attempt the problem papers: hitherto this privilege had been confined to candidates placed in the first two classes. Until 1828 the problem papers were set in the evenings, and in the rooms of the moderator, but many of the so-called problems were really pieces of bookwork or easy riders. No problems were ever set to the men in the seventh and eighth preliminary classes, which contained the poll-men.

The _University Calendars_ date from 1796, and from 1802 to 1882 inclusive contain the printed tripos papers of the previous January. The papers from 1801 to 1820 and from 1838 to 1849 inclusive were also published in separate volumes, which are to be found in most public libraries. None of the bookwork papers of this time are now extant, but it is believed that they contained few, if any, riders. In looking at these papers to form an opinion of the knowledge current at the time it is necessary to bear in mind that the text-books then in circulation were far from satisfactory.

The _Calendar_ of 1802 contains a diffuse account of the examination. It commences as follows:

On the Monday morning, a little before eight o'clock, the students, generally about a hundred, enter the Senate-House, preceded by a master of arts, who on this occasion is styled the father of the College to which he belongs. On two pillars at the entrance of the Senate-House are hung the classes and a paper denoting the hours of examination of those who are thought most competent to contend for honours. Immediately after the University clock has struck eight, the names are called over, and the absentees, being marked, are subject to certain fines. The classes to be examined are called out, and proceed to their appointed tables, where they find pens, ink, and paper provided in great abundance. In this manner, with the utmost order and regularity, two-thirds of the young men are set to work within less than five minutes after the clock has struck eight. There are three chief tables, at which six examiners preside. At the first, the senior moderator of the present year and the junior moderator of the preceding year. At the second, the junior moderator of the present, and the senior moderator of the preceding year. At the third, two moderators of the year previous to the two last, or two examiners appointed by the Senate. The two first tables are chiefly allotted to the six first classes; the third, or largest, to the οἱ πολλοί.

The young men hear the propositions or questions delivered by the examiners; they instantly apply themselves; demonstrate, prove, work out and write down, fairly and legibly (otherwise their labour is of little avail) the answers required. All is silence; nothing heard save the voice of the examiners; or the gentle request of some one, who may wish a repetition of the enunciation. It requires every person to use the utmost dispatch; for as soon as ever the examiners perceive anyone to have finished his paper and subscribed his name to it another question is immediately given....

The examiners are not seated, but keep moving round the tables, both to judge how matters proceed and to deliver their questions at proper intervals. The examination, which embraces arithmetic, algebra, fluxions, the doctrine of infinitesimals and increments, geometry, trigonometry, mechanics, hydrostatics, optics, and astronomy, in all their various gradations, is varied according to circumstances: no one can anticipate a question, for in the course of five minutes he may be dragged from Euclid to Newton, from the humble arithmetic of Bonnycastle to the abstruse analytics of Waring. While this examination is proceeding at the three tables between the hours of eight and nine, printed problems are delivered to each person of the first and second classes; these he takes with him to any window he pleases, where there are pens, ink, and paper prepared for his operations.

The examination began at eight o'clock in the morning. At nine the papers had to be given up, and half-an-hour was allowed for breakfast. At half-past nine the candidates came back, and were examined in the way described above till eleven, when the senate-house was again cleared. An interval of two hours then took place. At one o'clock all returned to be again examined. At three the senate-house was cleared for half-an-hour, and, on the return of the candidates, the examination was continued till five. At seven in the evening the first four classes went to the senior moderator's rooms to solve problems. They were finally dismissed for the day at nine, after eight hours of examination. The work of Tuesday was similar to that of Monday: Wednesday was partly devoted to logic and moral philosophy.

At eight o'clock on Thursday morning a first list was published with all candidates of about equal merits bracketed. Until nine o'clock a candidate had the right to challenge anyone above him to an examination to see which was the better. At nine a second list came out, and a candidate's right of challenge was then confined to the bracket immediately above his own. If he proved himself the equal of or better than the man so challenged his name was transferred to the upper bracket. To challenge and then to fail to substantiate the claim to removal to a higher bracket was considered rather ridiculous. Revised lists were published at eleven, three, and five, according to the results of the examination during that day. At five the whole examination ended. The proctors, moderators, and examiners then retired to a room under the public library to prepare the list of honours, which was sometimes settled in a few hours, but sometimes not before two or three the next morning. The name of the senior wrangler was generally announced at midnight, and the rest of the list the next morning. In 1802 there were eighty-six candidates for honours, and they were divided into fifteen brackets, the first and second brackets containing each one name only, and the third bracket four names.

It is clear from the above account that the competition fostered by the examination had developed so much as to threaten to impair its usefulness as guiding the studies of the men. On the other hand, there can be no doubt that the carefully devised arrangements for obtaining an accurate order of merit stimulated the best men to throw all their energies into the work for the examination. It is easy to point out the double-edged result of a strict order of merit. The problem before the University was to retain its advantages while checking any abuses to which it might lead.

It was the privilege of the moderators to entertain the proctors and some of the leading resident mathematicians the night before the issue of the final list, and to communicate that list in confidence to their guests. This pleasant custom survived till 1884. I revived the practice in 1890 when acting as senior moderator, but it seems to have now ceased.

In 1806 Sir Frederick Pollock was senior wrangler, and in 1869 in answer to an appeal from De Morgan for an account of the mathematical study of men at the beginning of the century he wrote a letter[58] which is sufficiently interesting to bear reproduction:

I shall write in answer to your inquiry, _all_ about my books, my study, and my degree, and leave you to settle all about the proprieties which my letter may give rise to, as to egotism, modesty, &c. The only books I read the first year were Wood's _Algebra_ (as far as quadratic equations), Bonnycastle's ditto, and _Euclid_ (Simpson's). In the second year I read Wood (beyond quadratic equations), and Wood and Vince, for what they called the _branches_. In the third year I read the _Jesuit's_ Newton and Vince's _Fluxions_; these were all the _books_, but there were certain MSS. floating about which I copied--which belonged to Dealtry, second wrangler in Kempthorne's year. I have no doubt that I had read less and seen fewer books than any senior wrangler of about my time, or any period since; but what I knew I knew thoroughly, and it was completely at my fingers' ends. I consider that I was the last _geometrical_ and _fluxional_ senior wrangler; I was not up to the _differential_ calculus, and never acquired it. I went up to college with a knowledge of Euclid and algebra to quadratic equations, nothing more; and I never read any second year's lore during my first year, nor any third year's lore during my second; my _forte_ was, that what I _did_ know I _could produce at any moment with PERFECT accuracy_. I could repeat the first book of Euclid word by word and letter by letter. During my first year I was not a "_reading_" man (so called); I had no expectation of honours or a fellowship, and I attended all the lectures on all subjects--Harwood's anatomical, Wollaston's chemical, and Farish's mechanical lectures--but the examination at the end of the first year revealed to me my powers. I was not only in the first class, but it was generally understood I was _first_ in the first class; neither I nor anyone for me expected I should get in at all. Now, as I had taken no pains to prepare (taking, however, marvellous pains while the examination was going on), I knew better than anyone else the value of my _examination qualities_ (great rapidity and perfect accuracy); and I said to myself, "If you're not an ass, you'll be senior wrangler"; and _I took to "reading" accordingly_. A curious circumstance occurred when the Brackets came out in the Senate-house declaring the result of the examination: I saw at the top the name of Walter _bracketed alone_ (as he was); in the bracket below were _Fiott_, _Hustler_, _Jephson_. I looked down and could not find my own name till I got to Bolland, when my pride took fire, and I said, "I must have beaten _that man_, so I will look up again"; and on looking up carefully I found the nail had been passed through my name, and I was at the top bracketed _alone_, even above Walter. You may judge what my feelings were at this discovery; it is the only instance of two such brackets, and it made my fortune--that is, made me independent, and gave me an immense college reputation. It was said I was more than half of the examination before anyone else. The two moderators were Hornbuckle, of St John's, and Brown (Saint Brown), of Trinity. The Johnian congratulated me. I said perhaps I might be challenged; he said, "Well, if you are you're quite safe--you may sit down and do nothing, and no one would get up to you in a whole day." ...

Latterly the Cambridge examinations seem to turn upon very different matters from what prevailed in my time. I think a Cambridge education has for its object to make good members of society--not to extend science and make profound mathematicians. The tripos questions in the Senate-house ought not to go beyond certain limits, and geometry ought to be cultivated and encouraged much more than it is.

To this De Morgan replied:

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

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

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

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

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

The answer, dated 7 August 1869, of Sir Frederick Pollock to these questions was as follows:

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

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

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

A few minor changes in the senate-house examination were made in 1808[59]. A fifth day was added to the examination. Of the five days thus given up to it three were devoted to mathematics, one to logic, philosophy, and religion, and one to the arrangement of the brackets. Apart from the evening paper the examination on each of the first three days lasted six hours: of these eighteen hours, eleven were assigned to bookwork and seven to problems. The problem papers were set from six to ten in the evening.

A letter from Whewell, dated 19 January 1816, thus describes his examination in the senate-house[60]:

Jacob. Whewell. Such is the order in which we are fixed after a week's examination.... I had before been given to understand that a great deal depended upon being able to write the greatest possible quantity in the smallest time, but of the rapidity which was actually necessary I had formed the most distant idea. I am upon no occasion a quick writer, and upon subjects where I could not go on without sometimes thinking a little I soon found myself considerably behind. I was therefore surprised, and even astonished, to find myself bracketed off, as it is called, in the second place; that is, on the day when a new division of the classes is made for the purpose of having a closer examination of the respective merits of men who come pretty near to each other, I was not classed with anybody, but placed alone in the second bracket. The man who is at the head of the list is of Caius College, and was always expected to be very high, though I do not know that anybody expected to see him so decidedly superior as to be bracketed off by himself.

The tendency to cultivate mechanical rapidity was a grave evil, and lasted long after Whewell's time. According to rumour the highest honours in 1845 were obtained by assiduous practice in writing[61].

The devotion of the Cambridge school to geometrical and fluxional methods had led to its isolation from contemporary continental mathematicians. Early in the nineteenth century the evil consequence of this began to be recognized; and it was felt to be little less than a scandal that the researches of Lagrange, Laplace, and Legendre were unknown to many Cambridge mathematicians save by repute. An attempt to explain the notation and methods of the calculus as used on the continent was made by Woodhouse, later professor in the University, who stands out as the apostle of the new movement.

It is doubtful if Woodhouse could have brought analytical methods into vogue by himself; but his views were enthusiastically adopted by three students, Peacock, Babbage, and Herschel, who succeeded in carrying out the reforms he had suggested. They created an Analytical Society which Babbage explained was formed to advocate "the principles of pure _d_-ism as opposed to the _dot_-age of the University." The character of the instruction in mathematics at the University has at all times largely depended on the text-books in use, and the importance of good books of this class was emphasized by a traditional rule that questions should not be set on a new subject in the tripos unless it had been discussed in some treatise suitable and available for Cambridge students[62]. Hence the importance attached to the publication of the work on analytical trigonometry by Woodhouse in 1809, and of the works on the differential calculus issued by members of the Analytical Society in 1816 and 1820.

In 1817 Peacock, who was moderator, introduced the symbols for differentiation into the papers set in the senate-house examination; his colleague, however, continued to use the fluxional notation. Peacock himself wrote on 17 March 1817 (_i.e._ shortly after the examination) on the subject as follows[63]:

I assure you ... that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-19, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only that we can hope to reduce the many-headed monster of prejudice, and make the University answer her character as the loving mother of good learning and science.

In 1818 all candidates for honours, that is, all men in the first six preliminary classes, were allowed to attempt the problems: this change was made by the moderators.

In 1819 Peacock, who was again moderator, induced his colleague to adopt the new notation. It was employed in the next year by Whewell, and in the following year by Peacock again. Henceforth the calculus in its modern language and analytical methods were freely used, new subjects were introduced, and for many years the examination provided a mathematical training fairly abreast of the times.

By this time the disputations had ceased to have any immediate effect on a man's place in the tripos. Thus Whewell[64], writing about his duties as moderator in 1820, said:

You would get very exaggerated ideas of the importance attached to it [an Act] if you were to trust Cumberland; I believe it was formerly more thought of than it is now. It does not, at least immediately, produce any effect on a man's place in the tripos, and is therefore considerably less attended to than used to be the case, and in most years is not very interesting after the five or six best men: so that I look for a considerable exercise of, or rather demand for, patience on my part. The other part of my duty in the Senate House consists in manufacturing wranglers, senior optimes, etc. and is, while it lasts, very laborious.

Of the examination itself in this year he wrote as follows[65]:

The examination in the Senate House begins to-morrow, and is rather close work while it lasts. We are employed from seven in the morning till five in the evening in giving out questions and receiving written answers to them; and when that is over, we have to read over all the papers which we have received in the course of the day, to determine who have done best, which is a business that in numerous years has often kept the examiners up the half of every night; but this year is not particularly numerous. In addition to all this, the examination is conducted in a building which happens to be a very beautiful one, with a marble floor and a highly ornamented ceiling; and as it is on the model of a Grecian temple, and as temples had no chimneys, and as a stove or a fire of any kind might disfigure the building, we are obliged to take the weather as it happens to be, and when it is cold we have the full benefit of it--which is likely to be the case this year. However, it is only a few days, and we have done with it.

A sketch of the examination in the previous year from the point of view of an examinee was given by J.M.F. Wright[66], but there is nothing of special interest in it.

Sir George B. Airy[67] gave the following sketch of his recollections of the reading and studies of undergraduates of his time and of the tripos of 1823, in which he had been senior wrangler:

At length arrived the Monday morning on which the examination for the B.A. degree was to begin.... We were all marched in a body to the Senate-House and placed in the hands of the Moderators. How the "candidates for honours" were separated from the οἱ πολλοί I do not know, I presume that the Acts and the Opponencies had something to do with it. The honour candidates were divided into six groups: and of these Nos. 1 and 2 (united), Nos. 3 and 4 (united), and Nos. 5 and 6 (united), received the questions of one Moderator. No. 1, Nos. 2 and 3 (united), Nos. 4 and 5 (united), and No. 6, received those of the other Moderator. The Moderators were reversed on alternate days. There were no printed question-papers: each examiner had his bound manuscript of questions, and he read out his first question; each of the examinees who thought himself able proceeded to write out his answer, and then orally called out "Done." The Moderator, as soon as he thought proper, proceeded with another question. I think there was only one course of questions on each day (terminating before 3 o'clock, for the Hall dinner). The examination continued to Friday mid-day. On Saturday morning, about 8 o'clock, the list of honours (manuscript) was nailed on the door of the Senate House.

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, a Gospel, and Paley's _Evidences_. 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 regulations[68] which 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 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 ask _vivâ voce_ questions, 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 questions _vivâ 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 subjects[69]. It was, however, still nominally considered as forming part of the senate-house examination, and until 1858 those 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 tripos[70] was 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 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 1833[71]. 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 to _vivâ voce_ questions, 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 examiners had partly relied on their impression of the answers given.

New regulations came into force[72] in 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 confined[73] to 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 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 regulations[74] which 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.

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 constituted[75] by 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 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 bookwork and 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 up by them was approved in 1868[76] and 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 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 senate[77].

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 were made[78], 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

## part II was transferred to four examiners nominated by the board: this

put it largely under the control of the professors. The range of subjects of part II was also greatly extended, and candidates were encouraged to select only a few of them. It was further arranged that

## part I might 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 part II at 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 part I contained 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 part II was so constructed as 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 proposed[79] further 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 1907[80], the board brought forward another scheme, proposing changes so sweeping as almost to destroy the identity of the tripos. Under this the examination in part II was abolished--a change on which all parties were agreed. There was introduced an examination, called part I, 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. In the new part II, 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 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 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 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 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. 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 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 Mullinger[81], "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 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 examination[82]."

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, 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.

The _Lucasian Professorship of Mathematics_ was 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, 1903 _et seq._

The _Plumian Professorship of Astronomy and Experimental Philosophy_ was 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, 1913 _et seq._

The _Lowndean Professorship of Astronomy and Geometry_ was 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, 1914 _et seq._

The _Sadleirian Professorship of Pure Mathematics_ was 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, 1910 _et seq._

The _Cavendish Professorship of Experimental Physics_ was 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, 1884 _et seq._

The _Professorship 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, 1903 _et seq._

Five _Lectureships in Mathematics_ were 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, 1901 _et seq._; George Ballard Mathews, 1903-1905; James Hopwood Jeans, 1904-1906, 1910-1912; John Gaston Leathem, 1905-1909; Robert Alfred Herman, 1906 _et seq._; Edmund Taylor Whittaker, 1905-1906; Thomas James I'Anson Bromwich, 1909 _et seq._; John Hilton Grace, 1901 _et seq._; Godfrey Harold Hardy, 1914 _et seq._; Arthur Berry, 1914 _et seq._

[Footnote 34: The greater part of this chapter formerly appeared in my _Mathematical 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'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; Glaisher's Presidential Address to the London Mathematical Society, _Transactions_, vol. XVIII, 1886, pp. 4-38; and my _History of the Study of Mathematics at Cambridge_, Cambridge, 1889.]

[Footnote 35: _Budget of Paradoxes_, by A. De Morgan, London, 1872, p. 305.]

[Footnote 36: See grace of 25 October 1680.]

[Footnote 37: _Ex. gr._ see De la Pryme's account of his graduation in 1694, _Surtees Society_, vol. LIV, 1870, p. 32.]

[Footnote 38: W. 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.]

[Footnote 39: _Memoirs of Richard Cumberland_, London, 1806, pp. 78-79.]

[Footnote 40: Quoted by C. Wordsworth, _Scholae Academicae_, Cambridge, 1877, pp. 30-31.]

[Footnote 41: _Anecdotes of the Life of Richard Watson_, London, 1817, pp. 18-19.]

[Footnote 42: See grace of 25 October 1883; and the _Cambridge University Reporter_, 23 October 1883.]

[Footnote 43: See grace of 11 February 1909, and the _Cambridge University Reporter_, 8 December 1908.]

[Footnote 44: _The Works of J. Jebb_, London, 1787, vol. II, pp. 290-297.]

[Footnote 45: "Emulation, which is the principle upon which the plan is constructed." _The Works of J. Jebb_, London, 1787, vol. III, p. 261.]

[Footnote 46: _The Works of J. Jebb_, London, 1787, vol. III, p. 272.]

[Footnote 47: See graces of 5 July 1773, and of 17 February 1774.]

[Footnote 48: See graces of 19, 20 March 1779.]

[Footnote 49: Notice issued by the vice-chancellor, dated 19 May 1779.]

[Footnote 50: The _Challis 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.]

[Footnote 51: H. Gunning, _Reminiscences_, second edition, London, 1855, vol. I, p. 82.]

[Footnote 52: C. Wordsworth, _Scholae Academicae_, Cambridge, 1877, pp. 322-323.]

[Footnote 53: H. Gunning, _Reminiscences_, second edition, London, 1855, vol. I, p. 182.]

[Footnote 54: See grace of 8 April 1791.]

[Footnote 55: Communicated by the moderators to fathers of colleges on 18 January 1799, and agreed to by the latter.]

[Footnote 56: C. Wordsworth, _Scholae Academicae_, Cambridge, 1817, p. 123.]

[Footnote 57: _Anecdotes of the Life of Richard Watson_, London, 1817, p. 19.]

[Footnote 58: _Memoir of A. De Morgan_, London, 1882, pp. 387-392.]

[Footnote 59: See graces, 15 December 1808.]

[Footnote 60: S. Douglas, _Life of W. Whewell_, London, 1881, p. 20.]

[Footnote 61: For a contemporary account of this, see C.A. Bristed, _Five Years in an English University_, New York, 1852, pp. 233-239.]

[Footnote 62: See _ex. gr._ the grace of 14 November 1827, referred to below.]

[Footnote 63: _Proceedings of the Royal Society_, London, 1859, vol. IX, pp. 538-539.]

[Footnote 64: _Whewell's Writings and Correspondence_, ed. Todhunter, London, 1876, vol. II, p. 36.]

[Footnote 65: S. Douglas, _Life of Whewell_, London, 1881, p. 56.]

[Footnote 66: _Alma Mater_, London, 1827, vol. II, pp. 58-98.]

[Footnote 67: See _Nature_, vol. XXXV, 24 February 1887, pp. 397-399. See also his _Autobiography_, Cambridge, 1896,