system to gain the highest places in the Cambridge Mathematical Tripos, and then other teachers will readily follow in the path thus opened to distinction. But it may naturally be said that as long as Euclid is prescribed for the text-book, the conditions of competition are unfair towards those who adopt some modern substitute; I will examine this point. In the Cambridge Examination for Mathematical Honours there are at present sixteen papers; a quarter of the first paper is devoted to book-work questions on Euclid. Now suppose that 1000 marks are assigned to the whole examination, and that about five of these fall to the book-work in Euclid. A student of any modern system would surely be able to secure some of these five marks, even from a stern Euclidean partisan. But to take the worst case, suppose the candidate deliberately rejects all chance of these five marks, and turns to the other matter on the paper, especially to the problems; here the advantage will be irresistibly on his side owing to the "superiority of the modern to the ancient methods of geometry" which is confidently asserted. It must be remembered that in spite of all warning and commands to the contrary, examiners will persist in making their papers longer than can be treated fully in the assigned time, so that the sacrifice of the book-work will be in itself trifling and will be abundantly compensated by the greater facility at the solution of problems which is claimed for the modern teaching, as compared with the "unsuggestiveness" of Euclid, and by the greater accuracy of reasoning, since we are told that "the logical training to be got from Euclid is very imperfect and in some respects bad." Thus on the whole the disciple of the modern school will even in the first paper of the Cambridge Tripos Examination be more favourably situated than the student of Euclid; and of course in the other papers the advantages in his favour become largely increased. For we must remember that we are expressly told that Euclid is "an unsuitable preparation for the higher mathematical training of the present day;" and that "those who continue their mathematical reading with a view of obtaining
