Page:Carroll - Euclid and His Modern Rivals.djvu/204

166 Min. Do you mean 'separational'?

Nie. Yes.

Min. Have you defined 'parallel' anywhere?

Nie. (after a search) I can't find it.

Min. A careless omission. Moreover, your assertion isn't always true. Suppose your two Lines were drawn from the same point?

Nie. We beg to correct the sentence. 'Two different Lines.'

Min. Very well. Then you assert Table I. 6. (See p. 29.) I grant it.

'Th. 20. Through a point situated outside a straight Line a Parallel, and only one, can be drawn to that Line.'

Min. 'A Parallel,' I grant at once: it is Table I. 9. But only one! That takes us into Table II. What axiom do you assume?

Nie. 'It may be admitted that only one Parallel can be drawn to it.'

Min. That is Table II. 15 (b)—a contranominal of Playfair's Axiom. We need not pursue the subject: all is easy after that. Now hand me the book, if you please: I wish to make a general survey of style, &c.

At p. 4 I read:—'Two Theorems are reciprocal when the hypothesis and the conclusion of one are the conclusion and the hypothesis of the other.' (They are usually called 'converse'—the technical, not the logical, converse, as was mentioned some time ago (p. 47); but let that pass.)