Page:Carroll - Euclid and His Modern Rivals.djvu/151

. § 1.] There is another remark I wish to make, before considering your second assertion. In asserting that there is a real class of non-coincidental Lines that have 'the same direction,' are you not also asserting that there is a real class of Lines that have no common point? For, if they had a common point, they must have 'different directions.'

Nie. I suppose we are.

Min. We will then, if you please, credit you with an Axiom you have not expressed, viz. 'It is possible for two Lines to have no common point.' And here I must express an opinion that this ought to be proved, not assumed. Euclid has proved it in I. 27, which rests on no disputed Axiom; and I think it may be recorded as a distinct defect in your treatise, that you have assumed, as axiomatic, a truth which Euclid has proved.

My conclusion, as to this first assertion of yours, is that it is most decidedly not axiomatic.

Let us now consider your second assertion, that some non-coincidental Lines have 'different directions.' Here I must ask, as before, are you speaking of Lines which have a common point? If so, I am quite ready to grant the assertion.

Nie. Not exactly. It is rather a difficult matter to explain. The Lines we refer to would, as a matter of fact, meet if produced, and yet we do not suppose that fact known in speaking of them. What we ask you to believe is that there is a real class of non-coincidental finite Lines, which we do not yet know to have a common point, but which have 'different directions.' We shall assert presently, in another Axiom, that such Lines will meet if