Page:Carroll - Euclid and His Modern Rivals.djvu/169

. § 1.], and to place oneself in the mental attitude of one who is totally ignorant of the subject.

Min. Very difficult, no doubt, but absolutely essential, if you mean to write a book adapted to the use of beginners. My own belief as to the course of thought needed to grasp the theory of 'direction' is this:—first you grasp the idea of 'the same direction' as regards Lines which have a common point; next, you convince yourself, by some other means, that different Lines exist which make equal angles with all transversals; thirdly, you go back, armed with this new piece of knowledge, and use it unconsciously, in forming an idea of 'the same direction' as regards different Lines. And I believe that the course of thought in the mind of a beginner is simply this:—he grasps, easily enough, the idea of 'the same direction' as regards Lines which have a common point; but when you put before him the idea of different Lines, and ask him to realise the meaning of the phrase, when applied to such Lines, he, finding that the former geometrical conception of 'coincidence' is not applicable in this case, and knowing nothing of the idea, which is latent in your mind, of Lines which make equal angles with all transversals, simply fails to attach any idea at all to the phrase, and accepts it blindly, from faith in his teacher, and is from that moment, until he reaches the Theorem about transversals, walking in the dark.

Nie. If this be true, of course the theory of 'direction,' however beautiful in itself, is not adapted for purposes of teaching.

Min. That is my own firm conviction. But I fear I may have wearied you by discussing this matter at such