Page:Carroll - Euclid and His Modern Rivals.djvu/160

122 Min. Then you must limit your subject, and say 'different Lines.'

Nie. Very well. Reads. 'That different Lines, which are sepcodal with the same Line, are sepcodal with each other. For they each have the same direction as that Line, and therefore the same direction as the other.'

Min. I am willing to grant you, without any proof, that, if such Lines existed, they would have the same direction with regard to each other. The phrase they each have is not remarkably good English. However, you may proceed.

P. 12. Ax. 9. 'An angle may be conceived as transferred from one position to another, the direction of its arms remaining the same.'

Min. Let us first consider the right arm by itself. You assert that it may be transferred to a new position, its direction remaining the same?

Nie. We do.

Min. You might, in fact, have here inserted an Axiom 'A Line may be conceived as transferred from one position to another, its direction remaining the same'?

Nie. That would express our meaning.

Min. And this is virtually identical with your Axiom 'Two different Lines may have the same direction'?

Nie. Certainly. They embody the same truth. But the one contemplates a single Line in two positions, and