Page:Carroll - Euclid and His Modern Rivals.djvu/203

. § 4.] Nie. We abandon the Axiom.

Min. Better luck next time! Try another Definition.

Nie. 'A broken Line is a Line composed of straight Lines.'

Min. But a straight Line also is 'a Line composed of straight Lines,' isn't it?

Nie. Well, we abandon the Definition.

Min. This is quite a new process in our navigation. Instead of heaving the lead, we seem to be throwing overboard the whole of our cargo! Let us hear something about Angles.

Nie. 'The figure formed by two Lines that intersect is called an Angle.'

Min. What do you mean by 'figure'? Do you define it anywhere?

Nie. Yes. 'The name of figure is given to volumes, surfaces, and lines.'

Min. Under which category do you put 'Angle'?

Nie. I don't know.

Min. Anything new about the Definition, or equality, of right angles?

Nie. No, except that we prove that all right angles are equal.

Min. That we have discussed already (see p. 57). Let us go on to Pairs of Lines, and your proof of Euc. I. 29, 32.

'Th. 19. Two Lines perpendicular to the same Line are parallel.'