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... lic, ix. 587-8, viii. 546) by a composite magic square of the order 27. He says frankly that this explanation is conjectural, but it is plausible, and may be commended to Platonic students. Trinity College, Cambridge. W. W. ROUSE BALL. Les Nombres positifs. Expose des theories modernes de l'arithmetique el6mentaire. Par M. STUYVAERT. Gand [no date; preface dated June 22nd, 1906J. Pp. xii + 133. M. Stuyvaert's doctrine is that arithmetic borrows a few fundamental concepts (like 'number' and 'addition') and axioms (like the commutativity of addition) from experience, and builds on these its direct operations. The inverse operations implied by these direct ones are not always possible, and " give rise to symbols, such as negative numbers, etc.; and the mechanism of the calculus of these symbols is pure convention, subject only to the condition of not contradicting the preceding theories, in the case where these symbols represent possible operations. As for the sciences of application, they are subject to this calculus by means of conventions also, and postulates particular to each of them." There is nothing in the exposition which would show that it was not written thirty years ago, and consequently, although it is more carefully done than the part devoted to principles in some modtern text-books, it can hardly be called modern,-neglecting as it does the facts, now fairly well-known, of the definability of all arithmetical concepts, the provability of all arithmetical axioms, the independence of arithmetic of experience, and the untenability of the view that the 'numbers,' in the extended sense of arithmetic, are ever mere symbols. PHILIP E. B. JOrRDAIN. Five Figure Mathematical Tables. By A. DtJ PRE DENNING. (Longmans, Green & Co. 2s.) These are intended for school and laboratory work. By the use of a large page (ll"x 7") and a small but clear type the author contrives to bring the difficult tables included in his collection (Logarithms, Antilogarithms, Trigonometrical Functions, Squares, Cubes, etc.) into a small number of pages (21). Thus Circular Measures, Sines, Tangents and their Logarithms, are all comprised within 6 pages. In the Tables of Logarithms and Antilogarithms the weak parts of both are avoided. Thus the Logs. of the numbers from 4'0000 to 9'9999 are given, and, on the opposite page, the Antilogs. for those from 1I0000 to 4'0644, all the differences being consequently small. Useful tables of constants and formulae are added. It seems a timely publication and can be recommended for Laboratory use in cases where 5-fig. work is necessary. We should have been glad to see the superfluous 10 rejected from the Logarithmic Trigonometrical function. Its retention throws unnecessary difficulties in the way of the computer. This book contains 568 examples in applied mechanics, dealing with shearing force, bending moments, the compression of struts, and simple cases of torsion and of stresses in cylinders, pipes, and plates exposed to fluid pressure. The examples are nearly all numerical, but answers are not given "so as to emphasize that the goal is a proper solution." The policy of this omission, in a book containing, very properly, several cases differing only in numerical data, appears doubtful. There is a short but pithy note on the design of riveted joints at p. 12, and the book concludes with a table of constants. C. S. J. Integration by Trigonometric and Imaginary Substitution. By C. 0.