Page:Carroll - Euclid and His Modern Rivals.djvu/163

. § 1.] the same; the magnitude of the included angle will remain the same.

Have I represented your meaning correctly?

Nie. We have no objection to make.

Min. We will return to this subject directly. I must now ask you to read the enunciation of Th. 4, omitting, for simplicity's sake, all about supplementary angles, and assuming the Lines to be taken 'the same way.'

P. 12. Th. 4. 'If two Lines are respectively sepcodal with two other Lines, the angle made by the first Pair will be equal to the angle made by the second Pair.'

Min. The 'sep' is of course superfluous, for if the Lines are 'compuncto-codirectional,' it is equally true. May I re-word it thus?—

'If two Pairs of Lines, each terminated at a point, be such that the directions of one Pair are respectively the same as those of the other; the included angles are equal.'

Nie. Yes, if you like.

Min. But surely the only difference, between Ax. 9 and this, is that in the Axiom we contemplated a single Pair of Lines transferred, while here we contemplate two Pairs?

Nie. That is the only difference, we admit.

Min. Then I must say that it is anything but good logic to take two Propositions, distinguished only by a trivial difference in form, and to call one an Axiom and the other a Theorem deduced from it! A very gross case of 'Petitio Principii,' I fear!