Page:Carroll - Euclid and His Modern Rivals.djvu/173

. § 1.] Min. Which, if traced back, will be seen to depend ultimately on your 6th Axiom, where you assume the reality of such Lines. But, even if your Theorem had been shown to refer to a real figure, how would that prove Euc. I. 34?

Nie. You only need the link that separational Lines have the same direction.

Min. Have you supplied that link?

Nie. No: but the reader can easily make it for himself. It is the 'Contranominal' (as you call it) of our 8th Axiom, 'two straight Lines which have different directions would meet if prolonged indefinitely.'

Min. Your pupils must be remarkably clever at drawing deductions and filling up gaps in an argument, if they usually supply that link, as well as the proof that separational Lines exist at all, for themselves. But, as you do not supply these things, it seems fair to say that your book omits all the Propositions which I have enumerated.

I will now take a general survey of your book, and select a few points which seem to call for remark.

P. 14. Th. 5. Cor. 1. 'Hence if two straight Lines which are not parallel are intersected by a third, the alternate angles will be not equal, and the interior angles on the same side of the intersecting Line will be not supplementary.' Excuse the apparent incivility of the remark, but this Corollary is false.

Nie. You amaze me!