is, as every one knows, that boys may have worked at Euclid for years, and may yet know next to nothing of Geometry." We may readily admit that such may be the case with boys exceptionally stupid or indolent; but if any teacher records this as the average result of his experience, it must I think be singularly to his own discredit.
There is, I see, a notion that the syllogistic form, since it makes the demonstrations a little longer, makes them more difficult; this I cannot admit. The number of words employed is not a test of the difficulty of a demonstration. An examiner, especially if he is examining viva voce, can readily find out where the difficulties of the demonstrations really lie; my own experience leads me to the conclusion that the syllogistic form instead of being an impediment is really a great assistance, especially to early students.
"Unsuggestiveness" has been urged as a fault in Euclid; which is interpreted to mean that it does not produce ability to solve problems. We are told: "Everybody recollects, even if he have not the daily experience, how unavailable for problems a boy's knowledge of Euclid generally is. Yet this is the true test of geometrical knowledge; and problems and original work ought to occupy a much larger share of a boy's time than they do at present." I need not repeat what I have already said, that English mathematicians, hitherto trained in Euclid, are unrivalled for their ingenuity and fertility in the construction and solution of problems. But I will remark that in the important mathematical examinations which are conducted at Cambridge the rapid and correct solution of problems is of paramount value, so that any teacher who can develop that power in his pupils will need no other evidence of the merits of his system.
Euclid's treatment of proportion has been especially marked out for condemnation; indeed, with the boldness which attaches to many assertions on the subject of elementary geometry, it has been pronounced already dead. In my own college it has
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