Page:Carroll - Euclid and His Modern Rivals.djvu/214

176 P. 61. Th. (of Book III) it is stated that Parallelograms, on equal bases and between the same Parallels, 'may always be placed so that their equal bases coincide,' and it is clearly assumed that they will still be 'between the same Parallels.' And again, in p. 63, the altitude of a Parallelogram is defined as 'the perpendicular distance of the opposite side from the base,' clearly assuming that there is only one such distance. In both these passages the Theorem is assumed 'Parallels are equidistant from each other,' of which no proof has been given, though of course it might have been easily deduced from Th. (p. 19).

The Theorems in Euc. II are here proved algebraically, which I hold to be emphatically a change for the worse, chiefly because it brings in the difficult subject of incommensurable magnitudes, which should certainly be avoided in a book meant for beginners.

I have little else to remark on in this book. Several of the new Theorems in it seem to me to be premature, e. g. Th. , &c. on 'Loci': but the sins of omission are more serious. He actually leaves out Euc. I. 7, 17, 21 (2nd part), 24, 25, 26 (2nd part), 48, and II. 1, 2, 3, 8, 9, 10, 12, 13. Moreover he separates Problems and Theorems, which I hold to be a mistake. I will not trouble you with any further remarks.