Page:Carroll - Euclid and His Modern Rivals.djvu/228

190 Nos. (widly) Say no more! My brain reels!

Min. I spare you. Let us go on to p. 5, where I find the following:—

'Rule of Conversion. If of the hypotheses of a group of demonstrated Theorems it can be said that one must be true, and of the conclusions that no two can be true at the same time, then the converse of every Theorem of the group will necessarily be true.'

Let us take an instance:—

Those will do for 'demonstrated Theorems,' I suppose?

Nos. I suppose so.

Min. And the 'hypothesis' of the first 'must be true,' simply because it is true.

Nos. It would seem so.

Min. And it is quite clear that 'of the conclusions no two can be true at the same time,' for they contradict each other.

Nos. Clearly.

Min. Then it ought to follow that 'the converse of every Theorem of the group will necessarily be true.' Take the converse of the second, i.e.

Is this 'necessarily true'? Is every thing which is less than 3 necessarily less than 2?

Nos. Certainly not. I think you have misinterpreted the phrase 'it can be said that one must be true,' when used of the hypotheses. It does not mean 'it can be said, from a knowledge of the subject-matter of some one