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IN the first of these tracts Prof. Dedekind gives a theory of irrational numbers and of the arithmetical continuum which is logically perfect, and in form, perhaps, more simple and direct than any other which has been or could be suggested; in the second he proceeds, by a marvellous chain of subtle inferences, from the idea of a manifold (or system of distinguishable objects in the widest sense) to the series of natural numbers and the elementary operations of arithmetic. It is to be hoped that the translation will make the essays better known to English mathematicians; they are of the very first importance, and rank with the work of Weierstrass, Kronecker and Cantor in the same field. The translation is rather painfully literal, and does not convey much idea of the graceful style of the original; but it is, on the whole, correct. On p. 46, l. 15, “hereafter” is a wrong rendering of hierauf; on p. 52, l. 18, ψ(s′) and s should be ψ(S′) and S; p. 61, last line but one, “such” is superfluous. On p. 34 there is an amusing complication of errors. What the author means is, “In this sense” (or “in the light of this fact”), “which I wish to express by the words αει ο ανθρωπος ανιθμητίζει, formed after a well-known saying, I hope,“ &c. The reference is to the motto on the title-page of the German edition, which was coined by the author in imitation of the Platonic dictum, αει ο θεος γεωμετρίζει.Essays on the Theory of Numbers. I. Continuity and Irrational Numbers. II. The Nature and Meaning of Numbers.By Richard Dedekind. Authorised translation by W. W. Beman. Pp. 116. (Chicago: The Open Court Publishing Co.; London: Kegan Paul and Co., Ltd., 1901).

Essays on the Theory of Numbers I. Continuity and Irrational Numbers II. The Nature and Meaning of Numbers