Page:Carroll - Euclid and His Modern Rivals.djvu/229

. § 1.] hypothesis, that it is, and therefore must be, true,' but 'it can be said, from a knowledge of the mutual logical relation of all the hypotheses, as a question of form alone, and without any knowledge of their subject-matter, that one must be true, though we do not know which it is.'

Min. Your power of uttering long sentences is one that does equal honour to your head and your—lungs. And most sincerely do I pity the unfortunate learner who has to make out all that for himself! Let us proceed.

P. 9. Def. 13. 'The bisector of an angle is the straight Line that divides it into two equal angles.'

This assumes that 'an angle has one and only one bisector,' which appears as Ax. 4, at the foot of p. 10.

P. 10. Def. 21. 'The opposite angles made by two straight Lines that intersect &c.'

This seems to imply that 'two Lines that intersect' always do make 'opposite angles.'

Nos. Surely they do?

Min. By no means. Look at p. 12, Def. 32, where, in speaking of a Triangle, you say 'the intersection of the other two sides is called the vertex.'

Nos. A slip, I confess.

Min. One of many.

P. 12, Def. 31. 'All other Triangles are called acute-angled Triangles.' What? If a Triangle had two right angles, for instance?

Nos. But there is no such Triangle.

Min. That is a point you do not prove till we come to Th. 18, Cor. 1, two pages further on. The same remark applies to your Def. 33, in the same page. 'The side…