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the matter, must have caused the author much thought : he tells us that he acted 

 upon two principles, (1) that of including everything that is utilisable^ even 

 though it is used but once, and assembling about this nucleus all that was 

 necessary or even desirable for the demonstration of the indispensable theorems ; 

 (2) that of rejecting systematically everything else. Unfortunately these two 

 principles do not establish everlasting canons which will guide future writers. 

 They constitute at best an involuntary confession of the impossibility of laying 

 down a cast-iron programme. It is the function of authors to guide the footsteps 

 of those who direct our studies ; but the utilisable will never prove a trustworthy 

 lodestar. As M. Appell tells us in an introductory preface, a solid mathematical 

 basis is indispensable for every serious scientific study, theoretical or practical ; 

 and the value of this book is that it attempts to provide this basis for all, not being 

 addressed to specialists. 



The range of subjects covered by the book is necessarily wide : an English 

 student finds here information that he would have to seek in a series of four or five 

 text-books in his own language. The variety of the contents naturally suggests 

 the question whether a student derives more good from isolated treatises or from 

 the encyclopaedic treatment. Every student will have his own answer to this 

 question, but indirectly, by the sections into which he divides his book, the author 

 seems to indicate his sympathy for the English method of treatment. These are : 

 I. Geometry and Analytical Geometry ; II. Algebra, Theory of Functions, Deriva- 

 tives and Applications ; III. Integral Calculus and Applications. In the first 

 part, Geometry is mainly the geometry of vectors, and includes the composition of 

 a system of localised vectors, while Analytical Geometry is almost restricted to the 

 study of the forms of the equations of planes, lines, conies, and quadrics, and 

 does not include tangents or tangent planes, which are reserved for Part II. In 

 the third part we have almost an embarrassment of riches ; for besides the usual 

 applications, we find chapters on elliptic functions, Fourier series, and a short 

 treatise on ordinary and partial differential equations. But it is perhaps in 

 the second part that the author must have had his chief struggle to select 

 the really utilisable. The following summary of topics selected in their order 

 gives some notion of the contents — complex numbers, binomial theorem, 

 determinants, the infinitely small, the infinitely great, series, functions, deriva- 

 tives, variation of functions, construction of curves, expansion in series, appli- 

 cations of the differential calculus to curves, to surfaces, kinematics, solution 

 of equations, numerical calculation and graphics. The impression left is some- 

 thing that reminds one of the collector into whose soul, side by side with the 

 virtue of order, had crept the spirit of acquisitiveness ; he could not bear to 

 part with anything, and on his death left many well-packed drawers which were 

 carefully labelled " useless." M. Zoretti's second part might have been more simply 

 labelled " utilisable." We cannot help regretting that the general plan of the 

 book has suffered from the desire to exhibit in it anything which could be of use, 

 ne serait-ce qtfune fois. Every part of the book exhibits great skill in treatment, 

 and in particular we would select the last chapter of the second part, which every 

 teacher should read and study ; it is real practical mathematics, the very best 

 statement of this important branch of the subject. 



The object of a mathematical treatise should be to treat mathematics from the 

 mathematical standpoint, and to allow illustrations to group themselves in their 

 natural place and in their fitting proportions, but above all the first object is to 

 teach mathematics. It is idle to suppose that there is one mathematics for 

 engineers and physicists and another for mathematicians. It is only by learning 



