Writings of Charles S. Peirce: A Chronological Edition, Volume 8: 1890–1892

9. Review of Muir’s The Theory of Determinants

36 9 Review of Muir’s The Theory of Determinants 28 August 1890 The Nation The Theory of Determinants in the Historical Order of Its Development . Part I. “Determinants in General: Leibnitz (1693) to Cayley (1841).” By Thomas Muir, M.A., LL.D., F.R.S.E. Macmillan & Co. 1890. The only history of much interest is that of the human mind. Tales of great achievements are interesting, but belong to biography (which still remains in a prescientific stage) and do not make history, because they tell little of the general development of man and his creations. The history of mathematics, although it relates only to a narrow department of the soul’s activity, has some particularly attractive features. In the first place, the different steps are perfectly definite; neither writer nor reader need be in the least uncertain as to what are the things that have to be set forth and explained. Then, the record is, as compared with that of practical matters, nearly perfect. Some writings of the ancients are lost, some early matters of arithmetic and geometry lie hidden in the mists of time, but almost everything of any consequence to the modern development is in print. Besides, this history is a chronicle of uninterrupted success, a steady succession of triumphs of intelligence over primitive stupidity, little marred by passionate or brutal opposition. Dr. Muir, already well known by many investigations into determinants and continued fractions, and by a charming little introduction to determinants, has thoroughly studied the history of this subject, and has arranged his account of it with remarkable clearness. Each writer’s results are stated in his own language, followed by a luminous commentary . An ingenious table shows the history of forty-four theorems, and at the same time serves as an index to the first half of this volume, which, it is to be presumed, is one-half of the first part, and not more than one-fourth of the whole work. 9. Muir’s Theory of Determinants, 1890 37 Perhaps Dr. Muir attaches a little too much importance to theorems, as contradistinguished from methods and ideas. Thus, he speaks rather unfavorably of Bézout’s work (1779), although it contains the idea of polar multiplication; but because this is not made a theorem, Dr. Muir hardly notices it. The first paper analyzed in the book is by Leibnitz, and contains the umbral notation, which is the quintessential idea of the theories of determinants as well as that of matrices, to which the theory of determinants is but an appendage. We have already mentioned that the last number of the American Journal of Mathematics contains an admirable memoir upon matrices by Dr. Henry Taber of Clark University. ...