Page:Carroll - Euclid and His Modern Rivals.djvu/140

102 Min. How do you know that it has described half a revolution?

Nie. Well, it is not difficult to prove. Let that portion of the Plane, through which it has revolved, be rolled over, using as an axis the arm (in its initial position) and its continuation, until it falls upon the other portion of the Plane. The two angular magnitudes will now together make up 'one revolution': therefore each is 'half a revolution.'

Min. A proof, I grant: but you are very sanguine if you expect beginners in the subject to supply it for themselves.

Nie. It is an omission, we admit.

Min. And then 'a straight angle'! 'Straight' is necessarily unbending: while 'angle' is from, 'a bend or hook': so that your phrase is exactly equivalent to 'an unbending bend'! In 'the Bairnslea Foaks' Almanack' I once read of 'a mad chap' who spent six weeks 'a-trying to maäk a straät hook': but he failed. He ought to have studied your book. Have you Euclid's Axiom 'all right angles are equal'?

Nie. We deduce it from 'all straight angles are equal': and that we prove by applying one straight angle to another.

Min. That is all very well, though I cannot think 'straight angles' a valuable contribution to the subject. I will now ask you to state your method of treating Pairs of Lines, as far as your proof of Euc. I. 32.

Nie. To do that we shall of course require parallel Lines: and, as our definition of them is 'Lines having