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II. History of the Cambridge Mathematical Tripos.
Part of a version of an address to the South-West Wales Branch of the Mathematical Association, Swansea, on 2 March, 1935, by Andrew R Forsyth. It was published in The Mathematical Gazette 19 (234) (1935), 162-179.
History of the Cambridge Mathematical Tripos.
Let me summarise, briefly, [the position of the Tripos] in the University. In early days, there was only a single test for the bachelor's degree at Cambridge. By the beginning of the eighteenth century, the test had been systematised into what was called the Senate-House Examination, largely mathematical in range, with more than doubtful Latin as the medium of expression. In 1824, a new examination in what we call classical learning (the language and literature of Greece and Rome) had been instituted; and from that date the old Senate-House Examination was called the Mathematical Tripos (the word Tripos itself being of ancient usage at Cambridge). But every candidate for admission to the new Classical Tripos must have qualified in the Mathematical Tripos: such was the ordinance. Macaulay's omniscient schoolboy (or does nobody read Macaulay's Essays now!) will remember how Macaulay himself failed to qualify: the Trevelyan biography contains a sympathy-begging letter to his mother in which, on the eve of some examination, he complains that Milton's descriptions of Heaven and Hell alike have been driven out of his head by two abominable trigonometrical formulae, which he quotes (one of them wrongly) as evidence of their foul character. Macaulay failed in his mathematics and had to be content with a poll degree - an instance, not a solitary instance, of the occasional ironies of academic records.
By that date the Tripos had become a venerable and venerated institution: it had acquired the characteristics of an ancient establishment: in spite of modifications, it retained some of those characteristics even in my own day. Bear in mind that the years of the eighteenth century, when the examination was settling into shape, were not far removed from the Newton-Leibniz controversy about the differential calculus. Partisans in science can be as fierce as partisans in theology, though happily in our day they wield no spiritual weapon of excommunication. In patriotic duty bound, the Cambridge of Newton adhered to Newton's fluxions, to Newton's geometry, to the very text of Newton's Principia: in my own Tripos in 1881 we were expected to know any lemma in that great work by its number alone, as if it were one of the commandments or the 100th Psalm. Thus English mathematics were isolated: Cambridge became a school that was self-satisfied, self-supporting, self-content, almost marooned in its limitations; and gradually it developed a special organisation the like of which has never appeared in other countries. The professors of the University did not examine: the lecturers of the colleges never examined, as teachers: the University did not, in any way, prepare students for its own examinations. But a new profession arose - not officially recognised: open to all, for there was neither a test of admission nor means of exclusion: that of the private coach. He made it the business of his life to prepare candidates for the examination. The University, correctly impartial, chose the examiners on the successive nominations of the colleges in a statutory automatic cycle, established for the appointment of the proctors who were best known to us as the active disciplinary police of the University. Thus the examiners changed from year to year. But the private coach was continuous. He accumulated experience and skill: he sifted all examination papers, recent and old alike: he codified mathematical knowledge into small tracts or pamphlets, kept in manuscript as his own private prescription for his own set of students. Thus it came about that there were relatively few books: Euclid of course: textbooks on algebra, or trigonometry, in a limited range, with a morbid devotion to the fascinating tortuosities of fiendish problems. It is true that Griffin, who was Senior Wrangler when Sylvester was second, produced a book on geometrical optics: and that Parkinson, who was Senior Wrangler when William Thomson was second, produced an edition of that same book (it was in use in my time) with pictures of telescopes which had simplicity of coffee-canisters. But it was upon the coaches that teaching depended: in their profession, they condensed available knowledge into potted abstracts; and so there were tabloid manuscripts on heat, on light, on sound, on lunar theory, on planetary theory, on practically every subject included in the Tripos schedule. The examination questions fell out of the unknown upon candidates: the private teaching devoted its powers to a preparation of students to face new conundrums by solving all that were old. Thus the continuous coaching gradually established a learned profession; and the Tripos became practically a game preserve for the coaches.
Yet do not imagine that the result was a mere fossilisation of eighteenth-century ideas and methods. Such a tendency had shown itself at the beginning of the nineteenth century. The dot-notation and the notions of fluxions had full sway; and, in Cambridge the differentials of the type dx, pervading continental mathematics, were avoided if not implicitly banned. One group of men - the leading spirits among whom were Whewell, Peacock, Babbage, and Herschel - fought the tendency by protest, by argument, by work; and one outcome of their action was to admit the differential calculus or, in the phrase of Babbage, to substitute pure d-ism for the dot-age of the University. How complete was the change may be gathered from even the single fact that Pollock, the Senior Wrangler in 1806, later a Fellow of the Royal Society, never acquired the differential calculus.
Gradually, however, the coaching system was firmly established: in the teaching, the coaches were all-dominant, almost the sole agents, because nearly all the college lectures were concerned with preliminary subjects. Happily for Cambridge and for mathematics, there were superb teachers engaged in the systematised round; and among all their names two survive in historic prominence, William Hopkins and Edward John Routh, both of Peterhouse. Hopkins was the leading coach through the period in which Sylvester, Green, Stokes, Cayley, Adams, William Thomson, Todhunter, Routh, Maxwell, graduated: a marvellous roll of pupils. As Hopkins retired, Routh took his place, apparently by natural merit. He had begun by taking the pupils of W J Steele - once described to me by Tait as the greatest mathematical teacher who ever lived - when Steele was ill: and Steele died all too soon. Routh produced a Senior Wrangler, if production be the proper word: he went on producing Senior Wranglers, some twenty-seven or twenty-eight in succession.
When I was an undergraduate, Routh was supreme: though supreme, he was not alone. There were other coaches of high repute, his contemporaries. There was Percival Frost: some of you may know his Solid Geometry, a few may know his Curve Tracing, probably none will have known his Newton, I, II III. There was Isaac Todhunter, whose textbooks blazed a new development in the teaching of mathematics in the country, and whose more ambitious works (such as his History of the Theory of Probability) endure as authorities of reference. There was William Besant (the name of his novelist-brother, Walter Besant, was better known to the large world outside Cambridge), the last of the old school devoted to fluxions, whose re-edited book on hydrodynamics still has its vogue with students and teachers. All these had been elected Fellows of the Royal Society. There was one, much junior in standing, superb teacher, a man (as I believe) whose powers would have taken him far as a pioneer into the domain of new knowledge had they been devoted to research rather than coaching - R R Webb, the earliest of the old Senior Wranglers now alive. Let me mention one other name in this hierarchy - my old friend, R A Herman, who died a few years ago, a man of unusual manipulative skill, a great and stimulating teacher, the last of the great coaches. Their names remain in the history of Cambridge mathematics.
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