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continue to be useful to students of actuarial work ; but the mere fact that such 

 a book has been called for suggests that there is something wrong in our teaching 

 of mathematics. If the whole subject of series were taken together, as it ought 

 to be, finite differences would naturally precede the study of the infinitesimal 

 calculus. As it is, series are generally dealt with by inadequate and ungeneralised 

 algebraic methods, and the student is then plunged headlong into the infinitesimal 

 calculus — so that the whole of the finite calculus, so useful in every branch 

 of science, is apt to be passed entirely. This is the result of a scheme of teaching 

 which is academical rather than practical. The book itself may be briefly defined 

 as being very largely Boole with numerous examples taken from actuarial 

 examination papers, together with very full solutions of many of the problems — all 

 useful and instructive work for students of many domains of science. The 

 principal additions in the second edition consist of a chapter on functions of two 

 variables and one on Stirling's formula of interpolation. Symbolic methods are 

 introduced early, and the writing is generally clear and easy. 



The excellent translation of Mr. Arne Fisher's book lies on the opposite pole 

 of the theoretical and even the metaphysical side of the subject, and has the 

 advantage of giving much continental work which is not well known in England — 

 especially Scandinavian and Russian work. After a lucid philosophical intro- 

 duction and a short historical chapter, the author proceeds with his theme in a 

 manner which will be particularly helpful to those who have not read in this line. 

 Laplace's work is taken as the basis and the treatment adopted gives new points 

 of view. The examples are well put, and the book continues with Bayes' Rule, 

 Tchebycheff's Theorem, the Theory of Dispersion, and Homograde Statistical 

 Series, etc., a knowledge of both the finite and the infinitesimal calculus being 

 rightly demanded. The second volume will be looked forward to, and should 

 provide something like the required text-book. We can find no mention of 

 Todhunter's admirable history. 



The primer of Elderton and Fippard attempts to explain " each step verbally 

 with the help of arithmetical examples, in order to avoid the introduction of 

 algebraical formulas." It appears to us that one might as well try to teach rifle 

 shooting without live cartridges, or swimming without water. Surely everyone 

 should be well enough educated nowadays to understand " algebraical formula?," 

 and the absence of them seems to us merely to add to the difficulties. But as far 

 as it goes the book is very well done. 



Mr. Burn's new work is prepared on a similar basis, but contains more 

 details helpful, say, to Medical Officers of Health — who are often notoriously 

 non-mathematical. We doubt the wisdom of writing down to ignorance— a thing 

 which is too frequently done. If a person does not possess enough energy to 

 learn a little mathematics, he will not possess enough to deal profitably with 

 statistics, and had better be left alone, without any attempts at spoon-feeding. 

 The work contains, however, some useful information apart from any special study 

 of vital statistics. 



The Declining Birth-Rate : its Causes and Effects. Being the Report of and 

 the chief Evidence taken by the National Birth-Rate Commission, instituted 

 with official recognition by the National Council of Public Morals, for the 

 Promotion of Race Regeneration — Spiritual, Moral, and Physical. [Pp. xiv 

 + 450.] (London : Chapman & Hall, 1916. Price 10s. bd. net.) 



The seriousness of the population question in this country must have been 

 impressed upon all thinking people for some time past by the almost monotonous 



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