Page:Carroll - Euclid and His Modern Rivals.djvu/82

44 are (by my Prop. 13) equal to four right angles. But this is absurd, since they were made equal to two right angles.

Hence D is nearer to CE than B is; i.e. BD approaches CE, and so will meet it if produced.

Min. You certainly have made your Axiom a little more axiomatic. It is, I presume, an afterthought of yours: otherwise you would have made your Axiom deal with approaching Lines, and would then have proved your present Axiom as a Theorem.

Euc. Excuse me. Whatever the habits of modern geometricians may be, in our day we always investigated a subject down to the very roots. No 'afterthought' was possible. You of the nineteenth century may 'look before and after,' if it so please you, so long as we have liberty to look at what is at our feet: you may 'sigh for what is not,' and welcome, so long as we may chuckle at what is!

Min. Flippancy will not serve your turn. If you have no better reason than that—

Euc. I have a better reason. How could I have dealt with approaching Lines without first strictly defining 'the distance of a point from a Line'?

Min. Nohow, I grant you.

Euc. Which would have entailed a definition of 'the distance of a point from a point,' i.e. the length of the shortest path by which the one can pass to the other—which again would have entailed the comparison of all possible paths—which again would have entailed the estimation of the lengths of curved Lines—which again—

Min. This is uncanny! It is whichcraft!

Euc. (preserves a disgusted silence).