Page:Carroll - Euclid and His Modern Rivals.djvu/183

Rh Min. That sounds mysterious. Which way along a Line are 'preceding' points to be found?

Nie. Both ways. He adds, directly afterwards, 'a Line has two different directions,' etc.

Min. So your Definition needs a postscript? That is rather clumsy writing. But there is yet another difficulty. How far from a point is the 'next' point?

Nie. At an infinitely small distance, of course. You will find the matter fully discussed in any work on the Infinitesimal Calculus.

Min. A most satisfactory answer for a teacher to make to a pupil just beginning Geometry! I see nothing else to remark on in your treatment of the Line, except that you state, as an Axiom, that 'a straight Line is the shortest way from one point to another.' I have already given, in my review of M. Legendre, my reasons for thinking that this is not a fair Axiom, and ought to be a Theorem (see p. 55).

There is nothing particular to notice in your treatment of angles and right angles. Let us go on to Parallels. How do you prove Euc. I. 32?

P. 9. § 27, Def. 'Parallel Lines are straight Lines which have the same Direction.'

Min. I presume you do not mean to include coincidental Lines?

Nie. Certainly not. We see the omission. Allow us to insert the word 'different.'

Min. Very well. Then your Definition combines the two