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[PDF page 194] 178 THE MATHEMATICAL TRIPOS. [CH. VII examination followed immediately after the Senate-House examination, andthedistinction, beingmuchcoveted, tendedtoemphasizethemath- ematical side of the normal University education of the best men. Since 1883 the prizes have been awarded on the result of dissertations*. Until now the Senate-House examination had been oral, but about this time, circ. 1770, it began to be the custom to dictate some or all of the questions and to require answers to be written. Only one question was dictated at a time, and a fresh one was not given out until some student had solved that previously read: a custom which by causing perpetual interruptions to take down new questions must have proved very harassing. We are perhaps apt to think that an ex- amination 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 Prize were dictated. The following description of the Senate-House examination as it existed in 1772 is given by Jebb†. The moderators, some days before the arrival of the time prescribed by the vice-chancellor, meet for the purpose of forming the students into divisionsofsix, eight, orten, accordingtotheirperformanceintheschools, 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 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 op- tics, as explained in the works of Maclaurin, Cotes, Helsham, Hamil- ton, Rutherforth, Keill, Long, Ferguson, and Smith. If the moderator finds the set of questionists, under examination, capable of answering * See Grace of October 25, 1885; and theCambridge University Reporter, October 23 and 30, 1883. † The Works of J. Jebb, London, 1787, vol. ii, pp. 290–297.

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[PDF page 195] CH. VII] THE MATHEMATICAL TRIPOS. 179 him, he proceeds to the eleventh and twelfth books of Euclid, conic sec- tions, 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 examina- tion, he sometimes asks a few questions in Locke’s Essay on the human understanding, Butler’s Analogy, or Clarke’s Attributes. But as the high- est 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 divis- ers, the resolution of quadratic, cubic, and biquadratic equations; together with the doctrine of fluxions, and its application to the solution of ques- tions “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 cen- tripetal 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 break- fast: 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 partic- ular 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:

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[PDF page 196] 180 THE MATHEMATICAL TRIPOS. [CH. VII 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 ab- solute or comparative merit to the moderators, and to every other father who shall ask him the question. Besidesthefathers, allmastersofarts, anddoctors, ofwhateverfaculty 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 recom- mended 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 exam- ination was competitive*. 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†. Al- though 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 * “Emulation, which is the principle upon which the plan is constructed.”The Works of J. Jebb, London, 1787, vol. iii, p. 261. † The Works of J. Jebb, London, 1787, vol. iii, p. 272.

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[PDF page 197] CH. VII] THE MATHEMATICAL TRIPOS. 181 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 in the pots on their desks frozen. The University was not altogether satisfied* with the scheme in force, and in 1779†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. It was further announced‡ that a candidate would not re- ceive credit for advanced subjects unless he had satisfied the examiners in Euclid and elementary Natural Philosophy. A system of brackets or “classes quam minimae” was now intro- duced. 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 oth- ers 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. Under the scheme of 1779 also the number of examiners was in- creased 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 M.A. to take part in the examination was not affected, though henceforth it was exercised more sparingly, and I be- lieve was not insisted on after 1785. But it became a regular custom for the moderators to invite particular M.A.s 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 * See Graces of July 5, 1773, and of February 17, 1774. † See Graces of March 19, 20, 1779. ‡ Notice issued by the Vice-Chancellor, dated May 19, 1779.

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[PDF page 198] 182 THE MATHEMATICAL TRIPOS. [CH. VII 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 papers actually set in the Senate-House, and now ex- tant, are two problem papers set in 1785 and 1786 by W. Hodson, of Trinity, then a proctor. The autograph copies from which he gave out the questions were luckily preserved, and are in the library* of Trinity College. They must be almost the last problem papers which were dic- tated, instead of being printed and given as a whole to the candidates. The problem paper in 1786 was as follows: 1. To determine the velocity with which a Body must be thrown, in a direc- tion 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 than1 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. Todetermineinwhatsituationofthemoon’sApsidtheygomostforwards, 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 ofx. 7. To find the fluxion of xr × (yn + zm)1/q. 8. To find the fluent of a ˙x a + x. 9. To find the fluxion of the mth power of the Logarithm ofx. 10. Of right-angled Triangles containing a given Area to find that whereof the sum of the two legsAB + BC shall be the least possible. [This and the two following questions are illustrated by diagrams. The angle atB 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 . * The Challis Manuscripts, iii, 61.

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[PDF page 199] CH. VII] THE MATHEMATICAL TRIPOS. 183 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 exami- nation a memorandum was published that the second place was given to that candidate whodialectis magis est versatus, and the third place to that one whoin scholis sophistarum melius disputavit. There seem to have been considerable intervals in the examination by the moderators, and the examinations by the extraneous examin- ers took place in these intervals. Those candidates who at any time were not being examined occupied themselves with amusements, pro- vided 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 games with a teetotum* in which he took part on the Wednesday (when Locke and Paley formed the sub- jects of examination), but “which was carried on with great spirit...by considerable numbers during the whole of the examination.” About this time, 1790, the custom of printing the problem papers was introduced, but until 1828 the other papers continued to be dic- tated. Since 1827 all the papers have been printed. I insert here the following letter† from William Gooch, of Caius, in which he describes his examination in the Senate-House in 1791. It must be remembered that it is the letter of an undergraduate addressed to his father and mother, and was not intended either for preservation or publication: a fact which certainly does not detract from its value. Monday 1 4 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 Modera- tor’s Rooms this Evening from 7 till 9, I don’t know yet;—but I did above three times as much as the Senr 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’vebeenshamefullyus’dbyLaxto-day;—Tho’hisanxietyforPeacock must (of course) be very great, I never suspected that his Partially (sic) wd get the better of his Justice. I had entertain’d too high an opinion of * H. Gunning, Reminiscences, second edition, London, 1855, vol. i, p. 82. † C. Wordsworth,Scholae Academicae, Cambridge, 1877, pp. 322–23.

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[PDF page 200] 184 THE MATHEMATICAL TRIPOS. [CH. VII 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 toBully 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 dispairofbeingfirst, tho’youseeLaxseemsdetermin’dthatIshallnot.—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 8th or 9th & Lucas is either 10th or 11th.—Poor Quiz Carver is one of theοÉ πολλοÐ;—I’m perfectlysatisfied that the Senior Wranglership is Peacock’s due, but certainly not so very indisputably 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, & canprove indisputably that it wasn’t 20 to 19 in his favor;—Icannot 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 veryparticular Request that you don’t mention Lax’s behaviour to me to any one. Such was the form ultimately taken by the Senate-House exami- nation, a form which it substantially retained 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 any- one who performed the statutable exercises, and it was open to the University to refuse to pass a supplicat for the B.A. degree 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.* It will be recollected that the examination was now compulsory on all candidates pursuing the normal course for the B.A. degree. In 1791 * Gunning, Reminiscences, second edition, London, 1855, vol. i, p. 182.

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[PDF page 201] CH. VII] THE MATHEMATICAL TRIPOS. 185 the University laid down rules* 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 alphabet- ically, 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 wereprimâ facie poll-men. In 1799 the moderators announced†that for the future they would require every candidate to show a competent knowledge of the first book of Euclid, arithmetic, vulgar and decimal fractions, simple and quadratic equations, and Locke and Paley. Paley’s works seem to be held in esteem by modern divines, and hisEvidences, though not his Philosophy, still remains (1905) one of the subjects of the Previous Ex- amination, but his contemporaries thought less highly of his writings, or at any rate of his Philosophy. Thus Best is quoted by Wordsworth‡ as saying of Paley’sPhilosophy, “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§, says, in describing the Senate-House examination in which Paley was senior wrangler, that Paley was afterwards known to the world by many excellent produc- tions, “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 pre- liminary 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. 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. No problems were ever set to the men in the seventh and * See Grace of April 8, 1791. † Communicated by the moderators to fathers of Colleges on January 18, 1799, and agreed to by the latter. ‡ C. Wordsworth,Scholae Academicae, Cambridge, 1877, p. 123. § Anecdotes of the Life of R. Watson, London, 1817, p. 19.

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[PDF page 202] 186 THE MATHEMATICAL TRIPOS. [CH. VII eighth preliminary classes, which contained the poll-men. None of the bookworkpapersofthistimearenowextant, butitisbelievedthatthey contained but few riders. Many of the so-called problems were really pieces of bookwork or easy riders: it must however be remembered that the text-books then in circulation were inferior and incomplete as compared with modern ones. TheCalendar of1802containsadiffuseaccountoftheexamination. 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 examinedarecalledout, andproceedtotheirappointedtables, wherethey 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 theοÉ πολλοÐ. 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, flux- ions, the doctrine of infinitesimals and increments, geometry, trigonom- etry, 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
