Page:Carroll - Euclid and His Modern Rivals.djvu/256

218 And I do not know that any change has been suggested in your test of a right Line in Prop. 14.

The next subject is 'Angles.'

Your definition would perhaps be improved, if for 'inclination to' we were to read 'declination from,' for, the greater the angle the greater the declination, and the less (as it seems to me) the inclination.

Euc. I agree with you.

Min. The next point is that you limit the size of an angle to something less than the sum of two right angles.

Euc. What advantage is claimed for the extension of the Definition?

Min. It is a prospective rather than an immediate one. It must be granted you that the larger angles are not needed in the first four Books—

Euc. In the first six Books.

Min. Nay, surely you need them in the Sixth Book?

Euc. Where?

Min. In Prop. 33, where you treat of 'any equimultiples whatever' of an angle, of an arc, and of a sector. You cannot possibly assume the multiple angle to be always less than two right angles.

Euc. You think, then, that a multiple of an angle must itself be an angle?

Min. Surely.

Euc. Then a multiple of a man must itself be a man. If I contemplate a man as multiplied by the number ten thousand, I must realise the idea of a man ten thousand times the size of the first?

Min. No, you need not do that.