Chapter XV: The History of the Mathematical Tripos (1)

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in mathematics established in or before 1501: they mark the beginning of the system of teaching by experts which has superseded the medieval system of compulsory teaching by all regent masters.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]: