Page:Carroll - Euclid and His Modern Rivals.djvu/76

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Min. In the predicate of 2, what right have you to say 'are intersectional'? The true contradictory of 'separational' would be 'have a common point.'

Euc. True: but we may assume as an Axiom 'A Pair of coincidental Lines are equally inclined to any transversal.' This, combined with 1, gives 'A Pair of not-intersectional Lines are equally inclined to any transversal,' whose Contranominal is 2.

Similarly, we may combine, with 6, the Axiom 'A Pair of coincidental Lines are equidistantial from each other,' and thus get a Theorem whose Contranominal is 7.

Min. In classing 15 (a), (b), and (c) under one number, you mean, I suppose, that they are so related that, if any one of them be granted, the others may be deduced ?

Euc. Certainly.

Min. I see that if (a) be given, (b) may be deduced by simply adding 'having a common point' to subject and predicate. And I see that (b) and (c) are Contranominals, so that, if either be given, the other follows. But I don't see how, if (b) only were given, you would prove (a).

Euc. You can prove (c) from it, as you say: and then, from (b) and (c) combined, you can prove (a) thus:—

Any Pair of Lines, which are separational from a third Line, must belong to one or both of the two classes, 'having