PG26839.B00204
[PDF page 203] CH. VII] THE MATHEMATICAL TRIPOS. 187 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. At nine o’clock 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 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 11 a.m., 3 p.m., and 5 p.m., 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 without much difficulty in a few hours, but sometimes not before 2 a.m. or 3 a.m. 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

PG26839.B00205
[PDF page 204] 188 THE MATHEMATICAL TRIPOS. [CH. VII energies into the work for the examination. It is easy to point out the usual 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 mathemat- ical study of men at the beginning of the century he wrote a letter* which is sufficiently interesting to bear reproduction: I shall write in answer to your inquiry,all about my books, my studies, 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’sAlgebra (as far as quadratic equations), Bon- nycastle’s ditto, andEuclid (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 theJesuit’sNewton and Vince’s Fluxions; these were all thebooks, 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 lastgeometrical and fluxional senior wran- gler; I was not up to thedifferential 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; myforte was, that what Idid know Icould produce at any moment withperfect ac- curacy. 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 lec- tures on all subjects—Harwood’s anatomical, Woollaston’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 wasfirst in the first class; neither I nor any one for me expected I should get in at all. Now, as I had taken no pains to pre- pare (taking, however, marvellous pains while the examination was going * Memoir of A. de Morgan, London, 1882, pp. 387–392.

PG26839.B00206
[PDF page 205] CH. VII] THE MATHEMATICAL TRIPOS. 189 on), I knew better than any one else the value of myexamination qualities (great rapidity and perfect accuracy); and I said to myself, ‘If you’re not an ass, you’ll be senior wrangler;’ andI 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 Walterbracketed alone (as he was); in the bracket below wereFiott, 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 any one 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 Cambridgerevivers were Waring, Paley, Vince, Milner. You had Dealtry’smss. 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’sms. 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

PG26839.B00207
[PDF page 206] 190 THE MATHEMATICAL TRIPOS. [CH. VII 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 of Sir Frederick Pollock to these questions is dated August 7, 1869, and is as follows. You have put together asrevivers 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 agreat man. I have no doubt Jacob beat him in accuracy, but the supposed answercannot 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 toopractical a turn; it is a waste of time to calculateactually a longitude by the help of logarithmic tables and lunar observations. It would be a fault not to knowhow, but a greater to be handy at it. A few minor changes in the Senate-House examinations were made in 1808*. 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 book-work and seven to problems. The problem papers were set from 6 to 10 in the evening. * See Graces, December 15, 1808.

PG26839.B00208
[PDF page 207] CH. VII] THE MATHEMATICAL TRIPOS. 191 A letter from Whewell dated January 19, 1816, describes his ex- amination in the Senate-House*. 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, to the general regret of the University, by assiduous practice in writing†. The devotion of the Cambridge school to geometrical and fluxional methods has led to its isolation from contemporary continental mathe- maticians. 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 Conti- nent was made by R. Woodhouse, who stands out as the apostle of the new movement. It is doubtful if he could have brought analytical meth- ods 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 princi- ples of pure d-ism as opposed to thedot-age of the University.” The character of the instruction in mathematics at the University has at all * S. Douglas, Life of W. Whewell, London, 1881, p. 20. † For a contemporary account of this see C.A. Bristed,Five Years in an English University, New York, 1852, pp. 233–239.

PG26839.B00209
[PDF page 208] 192 THE MATHEMATICAL TRIPOS. [CH. VII times largely depended on the text-books then in use, and the impor- tance 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 Cam- bridge students*. 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. But his colleague continued to use the fluxional notation. Peacock himself wrote on March 17 of 1817 (i.e. shortly after the examination) on the subject as follows†: 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 G. Peacock, who was again moderator, induced his col- league 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 examina- tion 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‡, writing about * See ex. gr., the Grace of November 14, 1827, referred to below. † Proceedings of the Royal Society, London, 1859, vol. ix, pp. 538–9. ‡ Whewell’s Writings and Correspondence, ed. Todhunter, London, 1876, vol. ii, p. 36.

PG26839.B00210
[PDF page 209] CH. VII] THE MATHEMATICAL TRIPOS. 193 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*: 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†, but there is nothing of special interest in it. Sir George Airy‡ 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 theοÉ πολλοÐ 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), * S. Douglas, Life of Whewell, London, 1881, p. 56. † Alma Mater, London, 1827, vol. ii, pp. 58–98. ‡ See Nature, vol. 35, Feb. 24, 1887, pp. 397–399.

PG26839.B00211
[PDF page 210] 194 THE MATHEMATICAL TRIPOS. [CH. VII 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 for- mal 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 cur- riculum; 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’sEvidences. Set classical books were introduced in the final examination of poll-men; and another honour or tripos examination was established for classical students. These al- terations 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* which came into operation in January, 1828, another day was added, so that the examination extended over four days, exclusive of the day of arranging the brackets; 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 can- didates for honours may not be induced to pursue the more abstruse and profound mathematics, to the neglect of more elementary knowl- * See the Grace, November 14, 1827.

PG26839.B00212
[PDF page 211] CH. VII] THE MATHEMATICAL TRIPOS. 195 edge.” 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 askvivâ voce questions, though such questions were restricted to “propositions contained in the math- ematical works commonly in use in 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 ques- tions 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 the traditions of the examination. In the same year as these changes became effective the examina- tion for the poll degree was separated from the tripos with different sets of papers and a different schedule of subjects*. 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 continu- ation of the junior optimes or third class of the tripos. The year 1828 therefore shews us the Senate-House examination dividing into two dis- tinct parts; one known as the mathematical tripos, the other as the poll examination. In 18511 the classical tripos was made independent of the mathematical tripos, and thus provided a separate avenue to a degree. Historically, the examination usually known as “the General” represents the old Senate-House examination for the poll-men, but gradually it has been moved to an earlier period in the normal course taken by the men. In 1852 another set of examinations, at first called “the professor’s ex- aminations,” and now somewhat modified and known as “the Specials,” was instituted for all poll-men to take before they could qualify for * See Grace, May 21, 1828, confirming a Report of March 27, 1828. 1. ‘1850’ corrected to ‘1851’ as per errata sheet
