Page:Carroll - Euclid and His Modern Rivals.djvu/175

. § 1.] enough to say 'if two Triangles have two angles of the one equal to two angles of the other &c.' Nie. Well, it is at worst a superfluity: the enunciation is really identical with Euclid's.

Min. By no means. The logical effect of a superfluous datum is to limit the extent of a Proposition: and, if the Proposition be 'universal,' it reduces it to 'particular'; i.e. it changes 'all A is B into 'some A is B.' For suppose we take the Proposition 'all A is B,' and substitute for it 'all that is both A and X is B,' we may be accidentally making an assertion of the same extent as before, for it may happen that the whole class A possesses the property X; but, so far as logical form is concerned, we have reduced the Proposition to some things that are A (viz. those which are also X) are B.'

I turn now to p. 27, where I observe a new proof for Euc. I. 24.

Nie. New and, we hope, neat and short.

Min. Charmingly neat and short, as it stands: but this method really requires the discussion of five cases, each with its own figure.

Nie. How do you make that out?

Min. The five cases are:—

(1) Vertical angles together less than two right angles, and adjacent base angles acute (the case you give).

(2) Adjacent base-angles right.

(3) Adjacent base-angles obtuse.

These two cases are proved along with the first.

(4) Vertical angles together equal to two right angles.

This requires a new proof, as we must substitute for the