Page:Carroll - Euclid and His Modern Rivals.djvu/235

. § 2.] At p. 27, Theorem 14, is a new proof of Euc. I. 24, apparently an amended version of Mr. Wilson's five-case proof, which I discussed at p. 137. He has now reduced it to three cases, but I still think the bisector of the angle a superfluity.

At p. 37 we have those curious specimens of 'Theorems of Equality,' which I discussed at p. 139.

At p. 53 is the Theorem which asserts, in its conclusion, part of its own data (see p. 192).

At p. 54 we are told that parallel Lines, which have equal projections on another Line, are equal (see p. 193).

At p. 55 we have the inconceivable triad of equal intercepts made by a Line cutting three Parallels (see p. 193).

At p. 161 I am surprised to see him fall into a trap in which I have often seen unwary students caught, while trying to say Euc. III. 30 ('To bisect a given arc') After proving two chords equal, they at once conclude that certain arcs, cut off by them, are equal; forgetting to prove that the arcs in question are both minor arcs.

But I must go no further: I have already wandered beyond the limits of Euc. I, II. The one great merit of this book

Nie. You have mentioned all the faults, then?

Min. By no means. You are too impatient. The one great merit, as I was saying, of Mr. Wilson's new book (and a most blessed change it is!) is that it ignores the whole theory of 'direction.' That he has finally abandoned that night-mare of Elementary Geometry, I dare not hope: so all I have said about it had better stand, lest