Page:Carroll - Euclid and His Modern Rivals.djvu/69

. II. § 4.] Min. I see that 2 (a) is the contranominal of 1. But where does 2 (b) come from?

Euc. It is got from 2 (a) by adding, to each term, the property 'having a common point'—just as if we were to deduce, from 'all men are mortal,' 'all fat men are fat mortals.'

Min. You mean 5 to be contranominal to 4, I suppose. But 'coincidental' is not equivalent to 'non-intersectional.'

Euc. True: but I have added a new condition, viz. 'which have a common point,' to the subject. Non-intersectional Lines, which have a common point, are coincidental, just as, in the next Proposition, non-intersectional Lines, which have a separate point, are separational. Min. 20 is rather a difficult enunciation to grasp.



Euc. A diagram will make it clear. As a matter of fact. No. 2 would be a right Line: but, as we have no right, at present, to assume this, I have drawn it as a wavy line.

Min. I can suggest two Contranominals which you have omitted: one, deducible from 13 (b), 'Two Lines, which are not intersectional and which have different directions, have no common point, i.e. are separational'; the other, deducible from 15, 'Two Lines, which have a