Page:Carroll - Euclid and His Modern Rivals.djvu/170

132 great length. Let us turn to another subject. What is your practical test for knowing whether two finite Lines will meet if produced?

Nie. You have already heard our 8th Axiom (p. 11). 'Two straight lines which have different directions would meet if produced.'

Min. But, even if that were axiomatic (which I deny), it would be no practical test, for you have admitted that you have no means of knowing whether two Lines, not known to have a common point, have or have not different directions.

Nie. We must refer you to p. 14. Th. 5. Cor. 2, where we prove that Lines, which make equal angles with a certain transversal, have the same direction.

Min. Which you had already asserted, if you remember, in Ax. 9.

Nie. Well then, we refer you to Ax. 9 as containing the same truth.

Min. And having got that truth, whether lawfully or not, what do you do with it?

Nie. Why, surely it is almost the same as saying that, if they make unequal angles, they have different directions.

Min. And what then?

Nie. Then, combining this with the Axiom you refused to grant, namely, that Lines having different directions will meet, we get a practical test, such as you were asking for.

Min. (dreamily) I see! You get rid of the 'different directions' altogether, and the result is that 'Lines, which make unequal angles with a certain transversal, will meet if produced,' which is Tab. II. 2 (see p. 34). And this you assert as axiomatic truth?