Page:Carroll - Euclid and His Modern Rivals.djvu/147

. § 1.] any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. This relationship of direction, which you call 'having the same direction,' and I 'having identical directions,' we may express by the word 'co-directional.'

Nie. All very true. My only puzzle is, why you have explained it at such enormous length: my meerschaum has gone out while I have been listening to you!

Min. Allow me to hand you a light. As to the 'enormous length' of my explanation, we are in troubled waters, my friend! There are breakers ahead, and we cannot 'heave the lead' too often.

Nie. It is 'lead' indeed!

Min. Let us now return to our fixed Line: and this time we will take a point not on it, and through this point we will draw a second Line. You say that we can, if we choose, draw it in 'the same direction' as that of the first Line?

Nie. We do.

Min. In that case let me remind you of the warning I gave you a few minutes ago, that we have no geometrical meaning for the phrase 'the same direction,' unless when used of Lines having a common point. What geometrical meaning do you attach to the phrase when used of other Lines?

Nie. (after a pause) I fear we cannot give you a geometrical definition of it at present.

Min. No? Can you construct such Lines?

Nie. No, but really that is not necessary. We allow of 'hypothetical constructions' now-a-days.