Dec. 2, 1886] 



NATURE 



103 



can approach them. Of necessity the literature of mathe- 

 matics must always be in the main a journal literature, for 

 the audience addressed is small ; but I cannot help feeling 

 that the disproportion between the amount of exploration 

 effected and the attempts made to render accessible the 

 territories explored and conquered might be greater than 

 it is. No one can realise more vividly than I do how 

 vastly more difficult it is to write a book than a collection 

 of memoirs, and how beset with anxieties, for any one 

 who is at all fastidious, is the task of arranging the funda- 

 mental properties of any comparatively new subject in 

 clear and logical form. The sustained struggle to attain 

 clearness, exactitude, and thoroughness in the orderly 

 development of a complicated and mutually-connected 

 system of propositions wears out the worker more than 

 thrice the same amount of labour devoted to new investi- 

 gations with all the fascinating excitement of successes 

 and failures, rewards and disappointments. In writing a 

 memoir, the mathematician begins where he pleases, and 

 confines himself to what has interested him and what he 

 knows he has done well. In composing a book, the author 

 has not only to marshal into order an array of theorems 

 of various kinds, assigning to each its due place and im- 

 portance, but he has— hardest task of all, perhaps — to 

 confine his treatise within bounds, to keep it from growing 

 to gigantic proportions as his increased study of the subject 

 opens up 10 him fresh vistas. On the other hand, how- 

 ever, is to be considered the great service he can thus 

 render to his favourite study : an introductory treatise on 

 a subject not otherwise approachable by any direct route, 

 even if it be not of the highest class, may have done far 

 more for its advance than could have been effected by the 

 most brilliant memoir. Time, care, and thought are 

 essential for the preparation of any valuable treatise, and 

 full references to the original memoirs should be always 

 given ; if these conditions have been fulfilled, the writer 

 has deserved well of mathematical science. 



I have not been able to forbear from making the few 

 preceding remarks upon two subjects on which I have 

 long felt strongly ; but I pass now without further delay 

 to the main subject of my address— the Mathematical 

 Tripos. I have thought that, in view of the importance of 

 this examination to our science, and the frequent changes 

 that have ta'ien place recently, this might be a subject of 

 no ordinary interest to our members as w-ell as to my- 

 self. Since 1S72 change has succeeded change with great 

 rapidity, and there are probably not many outside the 

 mathematical portion of the resident body at Cambridge 

 who are fully aware of the present mode of conducting 

 the examination or of the further changes already sanc- 

 tioned by the Senate and which take effect next June. It 

 is, indeed, generally known that the list of wranglers, 

 senior optimes, and junior optimes is published in June, 

 at about the same time as many other Tripos lists, instead 

 of by itself in January, and that the senior wrangler is 

 displaced from his throne, and no longer owes his position 

 to the results of the whole examination, so that he is not 

 necessaiily — even from an examination standpoint — the 

 first mathematician of his year. So much only is generally 

 known ; and it has seemed to me that it might be of 

 interest, considering the influence for good that it is 

 hoped the examination in its new form will have upon the 

 progress of mathematics, to give some account of the 

 successive developments that have taken place in this 

 time-honoured examination, and the causes and efforts 

 that have led to them. The difficulties connected with 

 the placing of all the mathematical candidates of the year 

 in one order of merit, the extension or limitation of the sub- 

 jects of examination, and various other questions connected 

 with the Tripos, are matters that have been continually 

 discussed and re-discussed in the light of fresh experience 

 by tho_^e concerned with the mathematical course of studies 

 at Cambridge, but I may, nevertheless, perhaps be per- 

 mitted to-night very briefly to refer to some of the familiar 



arguments in the presence of a more extended audience 

 of mathematicians. 



It is convenient to preface the principal remarks I have 

 to make by an outline of the history of the Tripos. In 

 doing so, I must pass very lightly over its origin and early 

 development, as anything approaching to a complete 

 history of its origin and rise in the last century would 

 amount almost to a history of the studies of the University. 



At the beginning of the last century, besides certain 

 merely formal disputations, the onlyexercises requiredfrom 

 candidates for degrees were the keeping of acts and oppon- 

 encies. Each candidate for honours in the course of his 

 third year had to maintain publicly a thesis, the subject 

 of which was chosen by himself, against three opponents, 

 in the presence of one of the Moderators, who acted as 

 umpire. The subjects selected were philosophical or 

 mathematical ; the discussion took place in Latin and in 

 logical form. After hearing the discussions, the Proctors 

 and Moderators prepared a final list of candidates qualified 

 to receive degrees. This can scarcely be considered to 

 have been an order of merit, for each of the Proctors and 

 Moderators, and also the Vice-Chancellor, had the right 

 to introduce the name of one candidate into the list 

 whenever he pleased ; still, except in the case of the 

 recipients of these honorary degrees, it is probable that 

 the list in the main fairly represented the merits of the 

 candidates. It was divided into three classes, consisting 

 of (i) the wranglers and senior optimes ; (2) the senior 

 optimes who had done fairly well but had not distinguished 

 themselves ; and (3) 01 ttoXXoj, or the poll-men. The first 

 class received their degrees on Ash Wednesday, taking 

 seniority according to their order on the list, and the two 

 other classes received their degrees later. 



With regard to the origin of the Tripos, Mr. W. W. 

 Rouse Ball, in his interesting sketch of its history,' 

 writes : — 



" The impressions gathered from these disputations in 

 the schools were necessarily rather vague, and when they 

 became the sole University exercise for a degree they 

 hardly afforded a sufficient basis for an accurate arrange- 

 ment of the men in order of merit. It was, I believe, to 

 correct this fault that the Senate Houss examination was 

 introduced, and I am inclined to think that it had its origin 

 about the year 1730. At first it probably consisted only 

 of a feiv vivA voce questions addressed by the Proctors 

 and Moderators in the week after the schools to those 

 candidates about whose abilities and position some doubt 

 was felt ; but its advantages were so patent that within 

 ten or twelve years it had become systematised into a 

 regular examination to which all questionists were liable, 

 although technically it was still regarded as only supple- 

 mentary to theexercises in the schools. From the beginning 

 it was conducted in English, and accurate lists were made 

 of the order of merit of the candidates ; two advantages 

 to which, I think, its final and definite establishment must 

 be largely attributed." 



Mr. Ball divides the time during which the exercises in 

 the schools and the Tripos were concurrent into five 

 periods : (l) from 1730 to about 1750, during which time 

 it was probably unauthorised and regarded as an experi- 

 ment ; (2) from 1750 to 1763, during which it was gradu- 

 ally establishing itself,— in the last year of this period it wa: 

 officially decided that when a candidate's position in the 

 class-list was doubtful the Senate House examination and 

 not the disputation was to be taken as the final test ; (3) 

 from 176310 1779, during which definite rules were framed 

 and laid down for conducting it ; (4) from 1779 to 1827, 

 durino- which it practically superseded the disputations ; 

 (5) from 1S27 to 1S41, the year in which the disputations 

 were abolished. 



The lists published in the Cambridge University 

 Calendars begin with the year 1747, because in that year 



^ '* The Origin and History of the Mathematical Tripos," Cambridge, 

 i38o. (Reprinted from the Cambridge Review.) 



