Page:Carroll - Euclid and His Modern Rivals.djvu/137

. § 1.] Min. That I have already fully discussed in reviewing M. Legendre's book (see p. 55).

Nie. We have also 'A Line may be conceived as transferred from any position to any other position, its magnitude being unaltered.' (p. 5. Ax. 3.)

Min. True of any geometrical magnitude: but hardly worth stating, I think. I have now to ask you how you define an Angle?

Nie. 'Two straight Lines that meet one another form an angle at the point where they meet.' (p. 5.)

Min. Do you mean that they form it 'at the point' and nowhere else?

Nie. I suppose so.

Min. I fear you allow your angle no magnitude, if you limit its existence to so small a locality!

Nie. Well, we don't mean 'nowhere else.'

Min. (meditatively) You mean 'at the point—and somewhere else.' Where else, if you please?

Nie. We mean—we don't quite know why we put in the words at all. Let us say 'Two straight Lines that meet one another form an angle.'

Min. Very well. It hardly tells us what an angle is, and, so far, it is inferior to Euclid's Definition: but it may pass. Do you put any limit to the size of an angle?

Nie. We have not named any, but the largest here treated of is what we call 'one revolution.'

Min. You admit reëntrant angles then?

Nie. Yes.

Min. Then your Definition only states half the truth: you should have said 'form two angles.'