IN preparing this second edition for press I have altered as slightly as possible those portions of the work which were written entirely by Prof. Kelland. The mode of presentation which he employed must always be of great interest, if only from the fact that he was an exceptionally able teacher; but the success of the work, as an introduction to a method which is now rapidly advancing in general estimation, would of itself have been a sufficient motive for my refraining from any serious alteration. A third reason, had such been necessary, would have pre sented itself in the fact that I have never considered with the necessary care those metaphysical questions connected with the growth and development of mathematical ideas, to which my late venerated teacher paid such particular attention. My own part of the book (including mainly Chap. X. And worked out Examples 10 — 24 in Chap. IX.) was written hurriedly, and while I was deeply engaged with work of a very different kind so that I had no hesitation in determining to re-cast it where I fancied I could improve it.