Page:Carroll - Euclid and His Modern Rivals.djvu/139

. § 1.] table, and each player tries to have a hand at the top of the pile?

Nie. I know the game.

Min. Well, did you ever see both players succeed at once?

Nie. No.

Min. Whenever that feat is achieved, you may then expect to be able to place two angles 'on one another'! You have hardly, I think, grasped the physical fact that, when one of two things is on the other, the second is underneath the first. But perhaps I am hypercritical. Let us try an example of your Axiom: let us place an angle of 90° on one of 270°. I think I could get the vertices and arms to coincide in the way you describe.

Nie. But the one angle would not be on the other; one would extend round one-fourth of the circle, and the other round the remaining three-fourths.

Min. Then, after all, the angle is a mysterious entity, which extends from one of the Lines to the other? That is much the same as Euclid's Definition. Let us now take your definition of a Right Angle.

Nie. We first define 'one revolution,' which is the angle described by a Line revolving, about one extremity, round into its original position.

Min. That is clear enough.

Nie. We then say (p. 7. Def. 9) 'When it coincides with what was initially its continuation, it has described half a revolution, and the angle it has then described is called a straight angle.'