Page:Carroll - Euclid and His Modern Rivals.djvu/172

134 taught such proofs as these, would be almost sure to try the plan in cases where the Lines would not really meet, and his assumption would lead him to results more remarkable for novelty than truth.

Let us now take a general survey of your book. And First, as to the Propositions of Euclid which you omit—

Nie. You are alluding to Prop. 7, I suppose. Surely its only use is to prove Prop. 8, which we have done very well without it.

Min. That is quite a venial omission. The others that I miss are 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, and 43: rather a formidable list.

Nie. You are much mistaken! Nearly all of those are in our book, or could be deduced in a moment from theorems in it.

Min. Let us take I. 34 as an instance.

Nie. That we give you, almost in the words of Euclid, at p. 37.

Reads. Th. 22. 'The opposite angles and sides of a Parallelogram will be equal, and the diagonal, or the Line which joins its opposite angles, will bisect it.'

Min. Well, but your Parallelogram is not what Euclid contemplates. He means by the word that the opposite sides are separational—a property whose reality he has demonstrated in I. 27; whereas you mean that they have the same direction—a property whose reality, when asserted of different Lines, has nowhere been satisfactorily proved.

Nie. We have proved it at p. 14. Th. 5. Cor. 2.