Page:Carroll - Euclid and His Modern Rivals.djvu/161

. § 1.] the other contemplates two Lines: the difference is very slight.

Min. Exactly so. Now let me ask you, do you mean, by the word 'angle,' a constant or a variable angle?

Nie. I do not quite understand your question.

Min. I will put it more fully. Do you mean that the arms of the angle are rigidly connected, so that it cannot change its magnitude, or that they are merely hinged loosely together, as it were, so that it depends entirely on the relative motions of the two arms whether the angle changes its magnitude or not?

Nie. Why are we bound to settle the question at all?

Min. I will tell you why. Suppose we say that the arms are merely hinged together: in that case all you assert is that each arm may be transferred, its direction remaining the same; that is, you merely assert your 6th Axiom twice over, once for the right arm and once for the left arm; and you do not assert that the angle will retain its magnitude. But in the Theorem which follows, you clearly regard it as a constant angle, for you say 'the angle AOD would coincide with the angle EKH. Therefore the angle AOD = EKH.' But the 'therefore' would have no force if AOD could change its magnitude. Thus you would be deducing, from an Axiom where 'angle' is used in a peculiar sense, a conclusion in which it bears its ordinary sense. You have heard of the fallacy A dicto secundum Quid ad dictum Simpliciter?

Nie. (hastily) We are not going to commit ourselves to that. You may assume that we mean, by 'angle,' a rigid angle, which cannot change its magnitude.