III. Topics of the 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.
  1. Introduction; see THIS LINK.
  2. Mathematicians in Cambridge in the late 1870s; see THIS LINK.
  3. History of the Cambridge Mathematical Tripos; see THIS LINK.
  4. Topics of the Mathematical Tripos;
  5. The Tripos coaches; see THIS LINK.
  6. The Tripos examination; see THIS LINK.
Topics of the Mathematical Tripos.

Just as in the early years of the nineteenth century, there had been an upheaval affecting the character of the methods, so in my time as a student there was the beginning of another upheaval affecting the range of subjects to be studied. The theory of thermodynamics was coming to the front: with the beginning of its activity the name of William Thomson must be associated; and it is not yet exhausted. The theory of gases was claiming attention - the names of Maxwell, Clausius, Tait, Boltzmann, need only be mentioned: the spirits of Clausius and Tait secured a more than ample measure of resounding controversy. More silently, more surely, more steadily, and with results ranging beyond the imagination of that day, Maxwell's mathematical presentation of Faraday's work and his own mathematical developments were revolutionising the systematic treatment of all electrical theory. The initial influence, exercised by these changes upon Tripos study and examination, was small: here a minute fragment, there a tiny snippet, just as beginnings. But all this new work was outside the range and beyond the familiar knowledge of the coaches.

Some of the older subjects remained in the vigour of full demand. Astronomy - then a range of study almost enjoined as a pious duty in the academic home of Newton, and even fortified in its claim by the achievement of Adams in the discovery of Neptune - held a double sway. On one side, there was the mathematics of the geometry of the heavens and the description of cosmical phenomena connected with time, together with all the associated instruments and their correct use. On another side, it was still the matter of active investigation in the motion of the heavenly bodies and the stability of the universe, in the figure and the pulsations of the earth, in the lunar and the planetary theories. Light found devotees (was not Stokes a living authority?): the corpuscular theory had not yet begun to recover from its old dethronement; and the wave-theory reigned unchallenged. Sound was a staple subject: in the Cambridge treatment it lent itself to varied mathematical results, picturesque in form though not contributory to progressive knowledge, and unrelated to the recent physiological work of Helmholz: though, in addition to the examination treatment, Rayleigh's great treatise had just come to our hands if we would but read. Heat (that is, conduction of heat) had pride of place: it lent itself to beautiful conundrums in the guise of mathematics; and there were attractive formulae not yet arithmetised. Sir William Thomson, in one of his frequent fervid moods, had declared Fourier's Treatise to be a poem: we mathematical undergraduates did not pretend to be appreciative critics of poetry in any disguised form, but we read unpoetic condensations in tabloid fragments; especially were we drilled in the use of Fourier series with their ever-new surprises and sudden pitfalls; and Stokes's early papers were our sole source of critical treatment, usually ignored. Rigid Dynamics (including the dynamics of a particle and, strange paradox under such a title, including also the vibrations of strings and bars and plates) claimed a general allegiance. In that range, principles were few and details were multitudinous: Cambridge examiners had an unlimited field for the practice of their ingenuity: bodies could, at will, be made to slip or roll or slide under a friction the amenable quality of which even prodigal Nature could have envied. Particles were made to describe arabesque paths under fantastic laws of force never imagined outside an examination paper. And occasionally, almost like a concession to the high-brows, there were scattered patches of theoretical dynamics which, containing little dynamics, seemed a welter of differential equations.

These were subjects on the grand scale. You may have noted that all of them belong to the domain of applied mathematics, at that epoch in Cambridge still styled Natural Philosophy.

Such pure mathematics as could be discovered in the course was usually made ancillary to this Natural Philosophy, when once beyond the rudimentary stages. Thanks to Salmon, many students knew their analytical conics. Thanks to examiners like Wolstenholme (whose volume of problems, in a more expansive and more formal continental setting, could have been deemed a varied contribution to research), many of us became even skilful in solving the wiliest conundrums. The algebra and the trigonometry (there was no "analysis" in the modern sense) belonged to the old school, the very old school: the proofs, such as were current about the Binomial Theorem, would stir the explosive contempt of the critical mathematicians of today. Nor was the calculus any better in its foundations: I can remember a college question, set years after my student time, "Define a function, and prove that every function has a differential coefficient". The marvel is that our blunders were not greater in number and more atrocious in quality. We were saved, partly by the unconscious benignity that limited the range of application of results; partly by a rough intuitive appeal to physical principles which could expose an error in application but could not construct the correction; partly, even perhaps, because our mentors shared all our blunders.

There was a special devotion to an ancient and limited geometry: to Euclid, not the old man, but to the Simson-Todhunter presentation: to Geometrical Conics, a tessellated pavement of scrappily short proofs of properties of the ellipse and the parabola, with the hyperbola rather clumsily lurking in the background: and to what was called Newton. This was not the great Principia which, indeed, few of us saw: certainly the first time I ever saw that work was when a copy of the Glasgow edition, duly bound and stamped, came to me as the then usual bonus additional to the Smith's Prize. What we had was an English translation, doubtless accurate, certainly overloaded with a bewildering heap of peddling minutiae, not unworthy of ancient grammarians as they dwelt on the letter of sacred texts. Even so, we never attained to the applied mathematics of that work: we were restricted to the rudiments of a geometry of infinitesimals: and we did not know that Newton had specially devised this geometry to avoid the use of the analytical calculus which had been his original weapon of research. The Anglicised form of these Newtonian Lemmas and their proofs had to be set out by us with almost the same meticulous verbal accuracy as was exacted in so-called Euclid proofs. Finally, in the earlier section of the Tripos Examination (officially described as "qualifying for honours", commonly known as "the three days"), there was a rigid rule against the explicit use of a differential coefficient and of an integration-process: we might substitute x + h for x and subtract, dodging onwards to the satisfaction of the examiner: we might use a Newton curve, if we could devise it, to effect a quadrature: but never might we use d/dx or the ∫-sign of integration which were taboo.

A few words more in amplification of a remark already made, in connection with the position of pure mathematics in the Cambridge studies in our day - that (except as occasional exercises in intellectual gymnastic) they were ancillary to applications in Natural Philosophy. Not long before my time there was a curious instance of this requirement of servitude. Airy, who had been a professor at Cambridge and had written useful Tracts, made it a public reproach against Cayley that he had used partial differential equations of the third order; and by way of driving the reproach home, declared that such equations of the second order marked the limit of the subject worthy of consideration, because they were the utmost that were required in natural philosophy. Cayley's reply - gentle and courteous of course, but definite equally of course - was that, in such a matter and in kindred matters, he was performing the duty enjoined by the statute on the holder of his professorship, "to explain the principles of pure mathematics". Airy had been thinking of the natural philosophy of his own earlier day: his riper years devoted to astronomy, were spent in the administration of Greenwich Observatory: and Maxwell's published papers (to cite nothing else) might have revealed to the obstinate conservative the antiquated character of his scientific dogma.

Let me return to the course of a mathematical student. In that course, pure mathematics, as a systematic range of ordered knowledge in many subjects of diverse content, received scant attention. The theory of numbers was represented by one chapter of snippets in Todhunter's large book on algebra; it was deemed to be covered by a single lecture; and we never even heard of a congruence, the very sign of which was appropriated for another purpose, let alone any work by Gauss. The theory of invariance was ignored beyond a sidling admission of some geometrical properties of conics and (for a few select adventurous minds) similar properties of quadrics. Differential equations were either catchy numerical examples of a few standard forms - mere trivialities - or they were the equations of mathematical physics, each treated by an isolated method to its own issue. The theory of functions, in any form, was absolutely unknown even in title. The imaginary i was suspiciously regarded as an untrustworthy intruder. The complex variable (a phrase that had not then penetrated to Cambridge) was described either as imaginary or as impossible; it compelled recognition as a root of a quadratic equation, or a cubic, or a quartic; and when I went to Cayley's lectures for one term in my third year, at the beginning the very word plunged me into complete bewilderment. A scrap of elliptic functions had been introduced to the Tripos - the scrap was too much for the coaches; we were fortunate at Trinity with Glaisher, as was St John's with Pendlebury: but from the Tripos standpoint the subject was a cross, between a trigonometry with two cosines and Legendre exercises on elliptic integrals regarded as beyond evaluation: while double periodicity was established as a thing all by itself, and was left in the air as isolated as the coffin of Mahomet in the legend. Cayley and Salmon had made vital contributions to analytical geometry beyond conics and quadrics, and Salmon's books reposed on the shelves of college libraries. But only mathematical wanderers, the Esaus and the Ishmaels of the Tripos knew even a fragment of such novelties; and differential geometry, save as an exercise in differential calculus, was out of bounds because Tripos questions were never set! Occasional attention, by individual students who had been pupils of Tait at Edinburgh or pupils of Barker at Manchester, was paid to quaternions: never as a non-commutative algebra: always with an introductory testimonial from Natural Philosophy. The theory of groups was unknown, except of course to Cayley; its title would have conveyed no meaning to any Cambridge student: yet nearly ten years earlier, Jordan at Paris had attracted to his lectures on that subject a German student, Felix Klein, and a Norwegian student, Sophus Lie. Perhaps the easiest indication of my estimate of the position may come from a personal confession. Something of differential equations beyond mere examples, the elements of Jacobian elliptic functions, and the mathematics of Gauss's method of least squares, I had learnt from Glaisher's lectures; and by working at the matter of the one course by Cayley which had been attended, I began to understand that pure mathematics was more than a collection of random tools mainly fashioned for use in the Cambridge treatment of natural philosophy. Otherwise very nearly the whole of such knowledge of pure mathematics as is mine began to be acquired only after my Tripos degree. In that Cambridge atmosphere we all were reared to graduation on applied mathematics. There is no regret: on the contrary, I am glad to have had the varied training in that range of knowledge, as glad as of the classical training in schooldays: my reason for seeming to dwell on the topic is that it may indicate the main trend of Cambridge studies fifty to sixty years ago.

The next section is at THIS LINK

Last Updated April 2020