Page:Carroll - Euclid and His Modern Rivals.djvu/141

. § 1.] the same direction,' we must begin by discussing direction.

Min. Undoubtedly. How do you define direction?

Nie. Well, we have not attempted that. The idea seemed to us to be too elementary for definition. But let me read you what we have said about it.

P. 2. Def. 2. 'A geometrical Line has position, and length, and at every point of it it has direction…'

P. 3. Def. 4. ' A straight Line is a Line which has the same direction at all parts of its length. It has also the opposite direction… A straight Line may be conceived as generated by a point moving always in the same direction.'

I will next quote what we have said about two Lines having 'the same direction' and 'different directions.'

Min. We will take that presently: I have a good deal to say first as to what you have read. I gather that you consider direction to be a property of a geometrical entity, but not itself an entity?

Nie. Just so.

Min. And you ascribe this property to a Line, and also to the motion of a point?

Nie. We do.

Min. For simplicity's sake, we will omit all notice of curved Lines, etc., and will confine ourselves to straight Lines and rectilinear motion, so that in future, when I use the word 'Line,' I shall mean 'straight Line.' Now may we not give a notion of 'direction' by saying—that a