Page:Carroll - Euclid and His Modern Rivals.djvu/162

124 Min. In that case you assert that, when a pair of Lines, terminated at a point, is transferred so that its vertex has a new position, these three conditions can be simultaneously fulfilled:—
 * (1) the right arm has 'the same direction' as before;
 * (2) the left arm has 'the same direction' as before;
 * (3) the magnitude of the angle is unchanged.

Nie. We do not dispute it.

Min. But any two of these conditions are sufficient, without the third, to determine the new state of things. For instance, taking (1) and (3), if we fix the position of the right arm, by giving it 'the same direction' as before, and also keep the magnitude of the angle unchanged, is not that enough to fix the position of the left arm, without mentioning (2)?

Nie. It certainly is.

Min. Your Axiom asserts, then, that any two of these conditions lead to the third as a necessary result?

Nie. It does.

Min. Your Axiom then contains two distinct assertions: the data of the first being (1) and (3) [or (2) and (3), which lead to a similar result], the data of the second being (1) and (2). These I will state as two separate Axioms:—

9 . If a Pair of Lines, terminated at a point, be transferred to a new position, so that the direction of one of the Lines, and the magnitude of the included angle, remain the same; the direction of the other Line will remain the same.

9 . If a Pair of Lines, terminated at a point, be transferred to a new position, so that their directions remain