Page:Carroll - Euclid and His Modern Rivals.djvu/112

74 Min. That will serve very well to give a notion of 'straight.' For a working definition we require of course some practical test, such as 'two straight Lines cannot enclose a space.'

Nie. We have that. At p. 20 we give you 'Axiom IV. Through two points always one, and only one, Line can be drawn.' And at p. 18 we at last distinguish 'Line' and 'curve.' 'A straight Line will in future be called a Line simply. All other Lines will be called curved Lines, or curves.'

Min. Better late than never: though it makes wild work of your former theory—in which you got the notion of 'Line' from a bent wire, and of 'curve' from the path of a moving point. Now for the Definition of 'angle.'

Nie. (after turning the leaves backwards and forwards for some time, begins to read in an unsteady voice) 'The part of a pencil of half-rays, described by a half-ray on turning about its end point from one position to another, is called an angle.' (p. 47.)

Min. So you reject the notion of 'inclination' (or rather 'declination')? Well! This is an innovation! We must investigate it thoroughly. You mean by 'half-ray,' I presume, what Euclid calls 'a Line terminated in one direction but not in the other'?

Nie. Certainly.

Min. Now what is a 'pencil'?

Nie. 'The aggregate of all Lines in a plane which pass through a given point.' (p. 38.)

Min. Aha! And where will you get your angular