202 



NA TURE 



[Dec. 30, 1886 



its present form by those most interested in pro- 

 moting the higher mathematical studies of the Uni- 

 versity. 



Special reference also should be made to the " three 

 days." It will be remembered that this preliminary por- 

 tion of the examination was the principal feature of the 

 scheme which came into operation in 1848. Both the 

 subjects and the methods of solution that may be em- 

 ployed are defined by a schedule, and only those who 

 satisfy the examiners by their performances in these three 

 days are admitted to the subsequent parts of the 

 examination. It is very singular that an arrangement 

 devised so long ago should not only still continue in 

 force, but even be regarded by some as the most 

 thoroughly satisfactory portion of the whole system. The 

 framers of this scheme and schedule might well have been 

 proud of the lasting character of their work if they could 

 have known that it would outlive two sets of University 

 statutes, and, amidst changes on every siJe, remain 

 unchanged for forty years. 



The early history of the Tripos and its gradual develop- 

 ment into an examination by written answers, and finally 

 by printed papers also, are especially interesting in these 

 days, when the merits of the examination system are so 

 highly appreciated, and its adoption is so universal. The 

 Senate House at Cambridge is the cradle of the modern 

 form of examination in England. 



In connection with the Tripos there is one matter of so 

 much importance that I cannot pass over it entirely with- 

 out mention. I mean the influence of the system of 

 private tuition. I believe that while there is an order of 

 merit it will always be a great assistance to the majority 

 of the candidates to read with a private tutor. JVIathe- 

 matics is a difficult science ; and when a considerable 

 range of subjects has to be traversed in a compara- 

 tively short time, and the knowledge of the candidate has 

 to be finally tested by an examination such as Part I. of 

 the Tripos, it cannot fail to be a great advantage to him 

 to have his difficulties explained, his path smoothed, and 

 his skill in working out problems developed, by an expe- 

 rienced private tutor specially interested in his individual 

 welfare. The system of private tuition has been objected 

 to from two points of view: (i) because it is unsatisfac- 

 tory that the instruttion which is valued most highly by 

 the student should be received from his private tutor 

 instead of from college lecturers or University profes- 

 sors ; and (2) because the student who has followed ini- 

 pHcitly, during his whole undergraduate career, the minute 

 directions of one man with regard to his reading, is placed 

 after his degree, when he is deprived of his guide, in a 

 very unfavourable position for pursuing further his mathe- 

 matical studies. The first objection does not concern us 

 here : my own feeling is, as I have just said, that, when- 

 ever an order of merit exists and the competition for 

 places is keen, the services of private tutors will neces- 

 sarily be called into requisition. The second objection 

 IS one which is of far more importance to our science. 

 There can be no question that, brilliant and eminent as 

 have been the greatest of the private tutors at Cambridge, 

 one result of the system has been that many of the ablest 

 students have been left after graduation not only without 

 any knowledge of the way to follow up the study of the 

 subjects of w^hich they had learned the elements, but even 

 without any taste or inclination to do so. The private 

 tutor's manuscript and verbal instruction had super- 

 seded all need of referring to the original memoirs, and 

 the nascent wrangler knew nothing of the great world of 

 mathematical literature or of the modes of reaching it. 

 On the other hand, it is only fair to say that the amount 

 of mathematical knowledge acquired by the best pupils 

 from their private tutors in the course' of their under- 

 graduate career was really wonderful ; and that till quite 

 recently neither the University nor the colleges offered 



any inducement to the mathematical student to continue 

 his reading after the Tripos. The fact that the student's 

 horizon should have been bounded by the Tripos, and that 

 his training should have been directed with the view to 

 giving him skill in working out questions rather than to 

 developing his taste for the science he was studying, was 

 principally the fault of the system as a whole ; but it was 

 certainly intensified by the complete subjection of the 

 pupil to the course of reading placed before him by the 

 private tutor. A student whose interests and aspirations 

 had been at least held in check, and perhaps entirely 

 stifled, throughout his whole undergraduate career, was 

 generally too subdued or helpless to be able to make 

 use of his freedom when the examination was over. 



Under the new scheme the private tutor still occupies 

 in the main his old position with respect to Part I., 

 although, of course, the higher places in this examination 

 have much less significance than before. With respect to 

 Part II. it is quite different. No attempt is made by 

 private tutors to teach these higher subjects, which, 

 both from their character and extent, are clearly unsuited 

 for private tuition ; and the students are compelled to 

 rely upon the lectures delivered in the colleges and Uni- 

 versity in their preparation for this final examination. 

 Thus, in their fourth year they are brought into contact 

 with the leading mathematicians in Cambridge ; and 

 when the examination sets them free to pursue their own 

 studies or researches, they start on their new career 

 fresh from the best teaching which the University 

 affords. 



Although the subject of my address is the Mathemati- 

 cal Tripos, it may be regarded as still falling within my 

 province to refer to other changes that have taken place 

 in the University for the purpose of encouraging original 

 mathematical work. Fourteen years ago Trinity College 

 invited mathematical candidates for Fellowships to send 

 in, before the examination, dissertations upon any sub- 

 jects of their own selection. It was announced that these 

 dissertations, if possessed of decided merit, would be 

 taken into account in the Fellowship election, together 

 with the results of the Fellowship e.xamination. Not 

 only have these dissertations been of the greatest value 

 in guiding the choice of the electors, but many of them 

 have been important contributions to mathematical litera- 

 ture.' The example of Trinity College has been re- 

 cently followed by St. John's College and King's College. 

 The Smith's Prizes, which for a great number of years had 

 been awarded by a special examination, are now awarded 

 annually for mathematical dissertations, the candidates 

 being free to select their own subjects. This new scheme 

 passed the Senate on October 25, 1883, and the first award 

 of the prizes under it was made in 18S5. Powerful induce- 

 ments are, therefore, now held out by the University and 

 some of the colleges for the best students to devote them- 

 selves to original work. The importance to our science 

 of these direct incentives to research cannot be over- 

 estimated. They come into operation as soon as the 

 stimulus of the examination is removed, and, instead of 

 resting upon their laurels, the ablest mathematicians of 

 the year are induced to concentrate their pow-ers upon a' 

 single subject, just at the time when they are undaunted 

 by any amount of hard work, when their stock of general 

 mathematical knowledge is freshly acquired, and when 

 their minds are flexible, vigorous, and free from care. It 

 is indeed strange to look back upon the changes of the 

 last few years, and to contrast the encouragement now 



^ Among the Trinity dissertations wh'ch have subsequently been printed, 

 1 may mention the late Mr. R. C. Rowe's " Memoir on Abel's Theorem " 

 {Phil. Trans., iSSi), Mr. Forsyth's "Memoir on the Thela Functions" 

 {Phii. Trans.. 1882), Mr. Homersham Cox's ■* Application of Quaternions 

 and Grassmann's Ausdehnungslehre to different kinds of Uniform Space" 

 (Camli Trans., 1882), Mr. Gallop's "Distribution of Electricity on the Cir- 

 cular Disc and Spherical Bowl " {Quart. Math Journ., 1886), and Mr. R. 

 Lachlan's " Systems of Circles and Spheres " {Phil. Trans., 1886). 



