Page:Carroll - Euclid and His Modern Rivals.djvu/86

48 Book: but the modern fancy is to use it on all possible occasions. The Syllabus indicates (to use the words of the Committee) 'the free use of this principle as desirable in many cases where Euclid prefers to keep it out of sight.'

Euc. Give me an instance of this modern method.

Min. It is proposed to prove I. 5 by taking up the isosceles Triangle, turning it over, and then laying it down again upon itself.

Euc. Surely that has too much of the Irish Bull about it, and reminds one a little too vividly of the man who walked down his own throat, to deserve a place in a strictly philosophical treatise?

Min. I suppose its defenders would say that it is conceived to leave a trace of itself behind, and that the reversed Triangle is laid down upon the trace so left.

Euc. That is, in fact, the same thing as conceiving that there are two coincident Triangles, and that one of them is taken up, turned over, and laid down upon the other. And what does their subsequent coincidence prove? Merely this: that the right-hand angle of the first is equal to the left-hand angle of the second, and vice versâ. To make the proof complete, it is necessary to point out that, owing to the original coincidence of the Triangles, this same 'left-hand angle of the second' is also equal to the left-hand angle of the first: and then, and not till then, we may conclude that the base-angles of the first Triangle are equal. This is the full argument, strictly drawn out. The Modern books on Geometry often attain their much-vaunted brevity by the dangerous process of