REVIEWS 



MATHEMATICS 



A Course of Pure Mathematics. By G. H. Hardy, M.A., F.R.S. Second 

 edition. [Pp. xii + 442.] (Cambridge: at the University Press. Price 

 12s. net.) 

 A comparison between the treatises on the Differential and Integral Calculus of 

 Dr. Todhunter and the book before us would convince even a superficial observer 

 of the change which has revolutionised mathematical instruction in this country in 

 the last quarter of a century. The writers of text-books at the beginning of that 

 period believed that the subject, as they expounded it, was as firmly established 

 and as logically deduced from its premises as geometry. Their practice had the 

 authority of great names, and the form of their books reflected their confidence. 

 Proofs which had been devised by Euler and Leibnitz seemed above challenge. 

 If cracks were visible in the armour in which these authors did their work, they 

 were not pointed out ; the brilliant results achieved sufficed to give sanction to the 

 theory. Under such circumstances it was unnecessary to examine foundations 

 too closely ; the theory expounded fitted certain classes of functions, and by 

 established custom the student's attention was confined to such functions. A good 

 deal of comfort was then, as it is to-day in another branch of philosophy, 

 administered to doubting disciples by the use of the mysterious word continuous. 

 Even in universities it was no uncommon thing to hear a lecturer declare that a 

 function was continuous, and therefore could be differentiated. English mathe- 

 maticians did not concern themselves with functions which declined to obey their 

 rules, indeed, such functions were regarded as freaks. The work of Weierstrass, 

 communicated in the lecture room to his students, made its way slowly across the 

 Channel, while Cantor and Dedekind found few disciples in our midst. But the 

 work of these great thinkers has as certainly changed the light in which every 

 mathematician regards the Infinitesimal Calculus, as Newton's discoveries affected 

 the outlook of the natural philosopher upon the material universe. 



Now Mr. Hardy's book is almost the first attempt to bring before junior 

 English mathematical students at the outset of their career the rigorous methods 

 by which alone this subject can be safely established. Mr. Hardy's task is one of 

 no common difficulty ; he can at the best select. It is impossible for him to 

 present the theory in its entirety even to clever undergraduates. No reader can, 

 however, finish the course of Pure Mathematics without knowing a good deal 

 about limits, infinity, functions, continuity, and without wishing to know more of 

 these subjects. Such a reader will have no excuse for thinking that functions 

 which are continuous are necessarily differentiable, and he may even have learnt 

 that continuous functions can be integrated. It is an excellent thing for 

 mathematics and its progress in this country that the book was written, and it 

 is a sign, full of promise, that a new edition has been demanded so soon. In 

 the second edition the author has made certain excisions and some additions, 

 the result of which, as he warns us, is to make the book a little more difficult. 



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