Page:Carroll - Euclid and His Modern Rivals.djvu/136

98 Nie. 'I am of opinion that the time is come for making an effort to supplant Euclid in our schools and universities.' (Pref. p. xiv.)

Min. It will be necessary, considering how great a change you are advocating, to examine your book very minutely and critically.

Nie. With all my heart. I hope you will show, in your review, 'the spirit without the prejudices of a geometrician.' (Pref. p. xv.)

Min. We will begin with the Right Line. And first, let me ask, how do you define it?

Nie. As 'a Line which has the same direction at all parts of its length.' (p. 3)

Min. You do not, I think, make any practical use of that as a test, any more than Euclid does of the property of lying evenly as to points on it?

Nie. No, we do not.

Min. You construct and test it as in Euclid, I believe? And you have his Axiom that 'two straight Lines cannot enclose a space?'

Nie. Yes, but we extend it. Euclid asserts, in effect, that two Lines, which coincide in two points, coincide between those points: we say they 'coincide wholly,' which includes coincidence beyond those points.

Min. Euclid tacitly assumes that.

Nie. Yes, but he has not expressed it.

Min. I think the addition a good one. Have you any other Axioms about it?

Nie. Yes, 'that a straight Line marks the shortest distance between any two of its points.' (p. 5. Ax. 1.)