Page:Carroll - Euclid and His Modern Rivals.djvu/177

. § 1.] such conditions, such and such things follow: but, if one of the conditions were to fail, there would be two Triangles! I must be dreaming! Let me dip my head in cold water, and read it all again. If two Triangles… there would be two Triangles. Oh, my poor brains!'

Nie. You are pleased to be satirical: it is rather obscure writing, we confess.

Min. It is indeed! You do well in calling it the ambiguous case.

At p. 33, I see the heading 'Theorems of equality': but you only give two of them, the second being 'the bisectors of the three angles of a triangle meet in one point,' which, as a specimen of 'Theorems of equality,' is probably unique in the literature of Geometry. I cannot wonder at your not attempting to extend the collection.

At p. 40 I read, 'It is assumed here that if a circle has one point inside another circle, the circumferences will intersect one another.' This I believe to be the boldest assumption yet made in Modern Geometry.

At pp. 40, 42 you assume a length 'greater than half a given Line, without having shewn how to bisect Lines. Two cases of 'Petitio Principii.' (See p. 58.)

P. 69. Here we have a Problem (which you call 'the quadrature of a rectilineal area') occupying three pages and a half. It is 'approached' by four 'stages,' which is a euphemism for saying that this fearful Proposition contains four of Euclid's Problems, viz. I. 42, 44, 45, and II. 14.

P. 73. 2. 'Find a point equally distant from three given