result upon the Questionist's place in the list of Honours. The Moderator must take the vivá voce performance into his account, in arranging the classes. I do not conceive that this would be attended with any difficulty, if the Moderators were freely and carefully to use their judgment. At least, the oral examination might be taken into account in determining the Junior Optimes, with regard to whom I should mainly propose to employ it; as I shall have to explain.

248 With regard to the nature of the vivá voce Examination thus proposed, it may consist, as a College Examination in Euclid generally consists, in requiring the oral proof of a proposition, and afterwards propounding questions respecting the connexion of the steps of the proof: or it may consist in the examiner propounding difficulties which he requires the examinee to solve; either such difficulties as the Opponents in former times used to bring against the Respondent, or such difficulties as the history of science supplies; those, for instance, which at first prevented the acceptance of Galileo's or Newton's discoveries. Or the viva voce examination may be combined with a paper examination (their combination, indeed, is on all accounts to be recommended); and this being done, the examinee may be required to explain, vivá voce, steps of proofs which he has already written down. All these kinds of examination would be, I believe, very decisive as to the talents and attainments of the candidates, and very instructive to the bystanders who were looking forwards to a similar trial. Nor do I think they would be at all difficult to conduct, by Moderators who were good mathematicians, and who had reconsidered their mathematical knowledge, with a view to such examinations.

249 I will only observe further, that the vivá voce examination would be worse than useless if it were not conducted with vigour and spirit. If a viva voce examination were to be established, and were to be

allowed to be, or to become, a mere matter of form, like the disputations at the Schools in their later days, it would answer no purpose; and would probably be soon abolished on account of the weariness and disgust it would excite in those who were compelled to undergo it.

250 I may remark, as some compensation for the trouble and time which the vivá voce examination would demand, that it might be made to abridge very materially the paper part of the Examination. Questions from books might be answered more expeditiously by each candidate, orally than on paper; and the answers would, as we have seen, be, or at least might be made, more decisive as to his merit. Hence the Questions from Books to be answered on paper might be greatly reduced in amount. The Problems, which must of course be proposed and answered on paper, would thus bear a larger proportion to the Questions; as they did before the practical discontinuance of the Disputations.

251 I should recommend, then, that the University should institute, for the Candidates for Mathematical Honours, an Examination in a Standard Course of Elementary Mathematics, consisting of a combination of paper Examinations with public vivá voce Examinations, to be continued through the Michaelmas term. But in order to attain another object, of which I have spoken, namely, an attention to the Elementary Subjects in those who are Candidates for the highest honours, I should recommend that this Examination take place previously to the Examination for the highest honours, and that its result should affect the access to the higher honours. In order to make my meaning clear, I may say, that I would make this an Examination to determine who should be Junior Optimes; and that none but those who had been pronounced Junior Optimes, after such an Examination, should be allowed

to compete for the honours of Senior Optimes and Wranglers.

252 I conceive that by this arrangement, and only by some such arrangement as this, should we effectually guard against the danger of candidates trying to make lucky hits in the higher subjects, without an adequate knowledge of the lower. If the two kinds of subjects are mingled together in the same examination, many candidates will always prefer to try their sagacity and good fortune in picking out parts of the higher, rather than encounter any labour of thought in mastering the lower, so as to be able to explain and apply them. If the two portions of the examination are separated, this game can no longer be played. And when this is prevented, we may hope to get rid of the ignorance and confusion of mind which may sometimes be found in those who obtain high honours in an examination, indiscriminate in its subjects, and conducted entirely upon paper.

253 When this Junior Optime Examination has thus been detached from the Examination for the Higher Honours, there will be much less difficulty in conducting the latter in a satisfactory manner. I have already said that I conceive it would be an advantage, if those Treatises on the various branches of Mathematics which are the subjects of study of the candidates for these Honours, were determined by authority. I shall not attempt to point out, further than I have already done, the books which ought to be selected to constitute the University Course of the Higher Mathematics. I conceive that such a selection might with advantage be committed to a body appointed by the University for this purpose ;-a Board of Studies. This Body should be so constituted as to give permanency and fixity to the University Course, and at the same time, to admit into it gradually those changes which the real progress of Mathematical Science might

require. The Board of Studies should therefore consist partly of permanent and partly of changing elements; it should represent both the past Teaching and the present Examinations. I should conceive that such a Board might be very fitly composed of the Mathematical Professors of the University, along with about an equal number of recent Moderators. A Course of the Higher Mathematics, drawn up by a Board, and modified from time to time, would be a fit guide both for Studies and for Examinations; and might, I think, without much difficulty, have its due influence secured to it.

254 In the plan which I have proposed, the Examination for the Higher Honours might perhaps be conducted entirely on paper; although I do not see why verbal explanations of written papers should not be admitted. In order that the examination might be instructive to those who had afterwards to undergo the like, it ought to be public; and to be effectually so, as I conceive, not only should the Questions be made public (as at present they are) but also the Answers which are made to the questions, or at least some sufficient specimens of such answers. I do not mean that such answers should be printed; but that the answers in general, or at least those of the candidates who obtained the highest places, should be made accessible to the students and the University in general. Such a publication would make the examinations really public, and would make the performances of each race of successful students useful in the education of their successors. It would also, probably, improve the form which our most active mathematical students would give to their answers; for with the prospect of this publicity to be given to their performances, they would hardly be content to present their knowledge in a rude, obscure, conventional form; but would, it may be supposed, aim at the clearness, fulness and symmetry [PT. I.]

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which would commend their papers to the judgments of mathematicians in general.

255 I will make one more remark on the subject of our Examinations. If the prevalence of mere paper examinations be, as I have endeavoured to show that it is, a source of great evil in an educational system, the University of Cambridge is called upon to seek some remedy for the evil, not only by her duty to herself, but also by her duty to several other institutions. For the practice of paper examinations, as one of the main modes of action of educational bodies, has become very general in this country and its dependencies, mainly, I think, through the influence of Cambridge. The impartiality and purity which have of late years been universally acknowledged as the constant attributes of our examinations, have produced a general confidence in examinations constituted on their plan. Cambridge men have gone into every part of the empire as professors, teachers, and officers of various kinds in educational bodies; and they have carried with them their conviction in favour of paper examinations, and their habit of conducting such examinations. In this manner paper examinations have been very extensively adopted in other Universities, Colleges, and Schools; and appear to be still extending themselves, and threatening to supersede all other modes of educational action. If there be, as I have endeavoured to show, a great evil in the preponderance of this one mode of dealing with young persons in their education, it becomes us, who have set the example of the errour, to lead the way to the remedy. If we have gone wrong in allowing oral examinations to fall into decline, we should, for the sake of others as well as for our own sake, attempt to restore oral examinations. We should aim at familiarizing our students with vivá voce examinations, and public lectures, for the sake of those whom they may hereafter have to teach or to