Page:Carroll - Euclid and His Modern Rivals.djvu/95

. II.] parts of Geometry all he really needs is to grasp the fact that it is shorter than any broken Line made up of straight Lines.

Nie. That is true.

Min. And all cases of broken Lines may be deduced from their simplest case, which is Euclid's I. 20.

Nie. Well, we will abate our claim and simply ask to have I. 20 granted us as an Axiom.

Min. But it can be proved from your own Axioms and it is a generally admitted principle that, at least in dealing with beginners, we ought not to take as axiomatic any Theorem which can be proved by the Axioms we already possess.

Nie. For beginners we must admit that Euclid's method of treating this point is the best. But you will allow ours to be a legitimate and elegant method for the advanced student

Min. Most certainly. The whole of your beautiful treatise is admirably fitted for advanced students it is only from the beginner's point of view that I venture to criticise it at all.

Your treatment of angles and right angles does not, I think, differ much from Euclid's?

Nie. Not much. We prove, instead of assuming, that all right angles are equal, deducing it from the Axiom that two right Lines cannot enclose a space.

Min. I think some such proof a desirable interpolation.

I will now ask you how you prove Euc. I. 29.

Nie. What preliminary Propositions will you grant us as proved?