Page:Carroll - Euclid and His Modern Rivals.djvu/260

222 Min. That is true.

Euc. Let us now consider the new methods of proof suggested by my Rivals.

Min. Prop. 5 has been much attacked—I may say trampled on—by your Modern Rivals.

Euc. Good. So that is why you call it 'The Asses' Bridge'? Well, how many new methods do they suggest for crossing it?

Min. One is 'hypothetical construction,' M. Legendre bisecting the base, and Mr. Pierce the vertical angle, but without any proof that the thing can be done.

Euc. So long as we agree that beginners in Geometry shall be limited to the use of Lines and Circles, so long will it be unsafe to assume a point as found, or a Line as drawn, merely because we are sure it exists. For example, it is axiomatic, of course, that every angle has a bisector: but it is equally obvious that it has two trisectors: and if I may assume the one as drawn, why not the others also? However we have discussed this matter already (p. 20).

Min. A second method is 'superposition,' adopted by Mr. Wilson and Mr. Cuthbertson—a method which here involves the reversing of the triangle, before applying it to its former position.

Euc. That also we have discussed (p. 47). What is the method adopted in the new Manual founded on the Syllabus of the Association?

Min. The same as Mr. Pierce's. Mr. Reynolds has a curious method: he treats the sides as obliques 'equally remote from the perpendicular.'

Euc. Curious, indeed.