Page:Carroll - Euclid and His Modern Rivals.djvu/113

.] magnitude, I should like to know? What kinds of magnitude is a Line capable of possessing?

Nie. Length only, of course.

Min. Two Lines?

Nie. (unasealy) Length only.

Min. A million?

Nie. (more unasealy) Length only.

Min. A pencil?

Nie. (faintly) Spare me!

Min. So much for the quality of your angular magnitude! Now for its quantity. What is the length of one of these half-rays?

Nie. Infinite, of course.

Min. And the aggregate length of all the half-rays in your 'angle' cannot well be less. Thus we may deduce a truly delightful definition of angular magnitude. 'As to quality, it is linear. As to quantity, it is infinite'!

Nie. (writhes, but says nothing).

Min. Will you not throw up your brief?

Nie. Not yet: I must fight it out.

Min. Then we must review this marvellous book 'to the bitter end.' What have you to say about 'right angles'?

Nie. We have 'angles of rotation' and 'angles of continuation' (p. 48); and the axiom 'all angles of rotation are equal' (p. 49) as a substitute for 'all right angles are equal.'

Min. It is a practicable method, but not so suitable for beginners as Euclid's. This matter I have already