Page:Carroll - Euclid and His Modern Rivals.djvu/104

66 if a Line revolves about its extremity, it all moves at once.

Min. Which, I take the liberty to think, is quite as great an assumption as Euclid's. I think the Axiom quite plain enough without any proof.

Your treatment of angles, and right angles, is the same as Euclid's, I think?

Nie. Yes, except that we prove that 'all right angles are equal.'

Min. Well, it is capable of proof, and therefore had better not be retained as an Axiom.

I must now ask you to give me your proof of Euc. I. 32.

Nie. We prove as far as I. 28 as in Euclid. In order to prove I. 29, we first prove, as a Corollary to Euc. I. 20, that 'the shortest distance between two points is a straight Line.'

Min. What is your next step?

Nie. A Problem (Pr. F. p. 52) in which we prove the Theorem that, of all right Lines drawn from a point to a Line, the perpendicular is the least.

Min. We will take that as proved.

Nie. We then deduce that the perpendicular is the shortest path from a point to a Line.

Next comes a Definition. 'By the distance of a point from a straight Line is meant the shortest path from the point to the Line.'

Min. Have you anywhere defined the distance of one point from another?

Nie. No.