Page:Carroll - Euclid and His Modern Rivals.djvu/87

. II. § 6.] omitting links in the chain; and some of the new proofs, which at first sight seem to be shorter than mine, are really longer when fully stated. In this particular case I think you will allow that I had good reason for not adopting the method of superposition?

Min. You had indeed.

Euc. Mind, I do not object to that proof, if appended to mine as an alternative. It will do very well for more advanced students. But, for beginners, I think it much clearer to have two non-isosceles Triangles to deal with.

Min. But your objection to laying a Triangle down upon itself does not apply to such a case as I. 24.

Euc. It does not. Let us discuss that case also. The Moderns would, I suppose, take up the Triangle ABC, and apply it to DEF so that AB should coincide with DE?

Min. Yes.

Euc. Well, that would oblige you to say 'and join C, in its new position, to E and F.' The words 'in its new position' would be necessary, because you would now have two points in your diagram, both called C.' And you would also be obliged to give the points B and E additional names, namely A and B.' All which would be very confusing for a beginner. You will allow, I think, that I was right here in constructing a new Triangle instead of transferring the old one?

Min. Cuthbertson evades that difficulty by re-naming the point C, and calling it 'Q.'

Euc. And do the points A and B take their names with them?