Euclid and His Modern Rivals/Act III. Scene I. § 5.

by Charles L. Dodgson

 

 

ACT III.

 

Scene I.

 

§ 5. Reynolds.

 

'Though this be madness, yet there's method in it.'

Nie. I lay before you 'Modern Methods in Elementary Geometry,' by E. M. Reynolds, M.A., Mathematical Master in Clifton College, Modern Side; published in 1868.

Min. The first remark I have to make on it is, that the Definitions and Axioms are scattered through the book, instead of being placed together at the beginning, and that there is no index to them, so that the reader only comes on them by chance: it is quite impossible to refer to them.

Nie. I cannot defend the innovation.

Min. In Th. i (p. 3), I read 'the angles CDA, CDB are together equal to two right angles. For they fill exactly the same space.' Do you mean finite or infinite space? If 'finite,' we increase the angle by lengthening its sides: if 'infinite,' the idea is unsuited for elementary teaching. You had better abandon the idea of an angle 'filling space,' which is no improvement on Euclid's method.

P. 61. Th. ii (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. xvi (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. xix, &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.