Page:Carroll - Euclid and His Modern Rivals.djvu/145

. § 1.] Nie. We accept it.

Min. And, since a finite Line has the same direction as the infinite Line of which it is a portion, we may generalise thus:—'Coincidental Lines have the same direction. Non-coincidental Lines, which have a common point, have different directions.'

But it must be carefully borne in mind that we have as yet no geometrical meaning for these phrases, unless when applied to two Lines which have a common point.

Nie. Allow me to remark that what you call 'coincidental Lines' we call 'the same Line' or 'parts of the same Line,' and that what you call 'non-coincidental Lines' we call 'different Lines.'

Min. I understand you: but I cannot employ these terms, for two reasons: first, that your phrase 'the same Line' loses sight of a fact I wish to keep in view, that we are considering a Pair of Lines; secondly, that your phrase 'different Lines' might be used, with strict truth, of two different portions of the same infinite Line, so that it is not definite enough for my purpose.

Let us now proceed 'to consider the relations of two or more straight Lines in one Plane in respect of direction.'

And first let me ask which of the propositions of Table II you wish me to grant you as an axiom?

Nie. (proudly) Not one of them! We have got a new patent process, the 'direction' theory, which will dispense with them all.

Min. I am very curious to hear how you do it.