REVIEWS 667 



of London, and does not di£Eer materially from previous editions. A new 

 chapter has been added on Point-Reciprocation, which is the special case of 

 polar reciprocation with regard to a conic arising when the conic is a circle. 

 Its importance is due to the fact that by it metrical properties may be obtained ; 

 also it is considered that the student is most easily familiarised with the 

 general notion of reciprocation by examining this particular case in some detail. 



Another novelty is introduced in chapter ii, where Menelaus' and Ceva's 

 Theorems are deduced from the proposition that the continued product of 

 ratios of segments of the sides of a triangle is unaltered by projection. 



A useful set of Miscellaneous Examples has been added at the end. 



In comparison with such a book as Enriques' Geometria Proiettiva the 

 general treatment may seem lacking in breadth, and the book rather over- 

 burdened with detail ; the author has in mind, of course, the elementary 

 student, and is somewhat circumscribed by the peculiarities of the syllabuses 

 of the University of London, by which involution is postponed to a relatively 

 late stage ; but his book remains one of the best introductions to the subject 

 that there is in English, and may be safely recommended to the serious student. 



F. P. W. 



Elements d' Analyse Mathlmatique. Par Paul Appell. Quatri^me Edition. 



[Pp. X + 715.] (Paris : Gauthier-Villars et Cie., 1921. Price 65 fr.) 

 M. Paul Appell, in 1898, founded the first edition of this book, a general 

 course of mathematics for engineers and physicists, on his lectures at the 

 ficole centrale des Arts et Manufactures. The fourth edition, which has 

 just appeared, contains many additions and is practically a new book. 



The first impression that one gets on glancing through its pages is how very 

 much better it is than corresponding books in English. Its scope is, of 

 course, a great deal larger than that of most of our practical introductions 

 to the calculus ; in fact, one would expect a strike among science students in 

 this country who were required to know about the curvature and torsion 

 of twisted curves and properties of asymptotic lines on a surface. But in 

 the more elementary parts the superiority is obvious. Take, for instance, his 

 treatment of series in chapter vi. Use is made in plenty of our intuitional 

 notion of a curve (e.g. in the proof of the mean value theorem), but when he 

 gets to the question of differentiation and integration of a power series he 

 does not give a shoddy half-proof of his proposition ; he assumes it and says 

 so. " C'est la un thdordme trds important que nous admettrons " (italics in 

 the original). 



The contents include, as has been said, a fair amount of the elements of 

 differential geometry, in the plane and in three dimensions ; there is a chapter 

 on Stoke's Theorem and on Green's Theorem, and five on differential equa- 

 tions. Many examples are worked out in detail in the text, which has the 

 effect of increasing the size of the book and making it rather difficult to read 

 through. There is no doubt, however, that not only the mathematical 

 physicist but also the student of pure mathematics, in his early stages, would 

 benefit by reading it. *F. P, W. 



A Study of Mathematical Education, including the Teaching of Arithmetic. 



By Benchara Branford. 2nd Edition. [Pp. xii + 420.] (Oxford : 



At the Clarendon Press, 1921. Price 75. 6d. net.) 

 This book is a study, from the point of view of general educational principles, 

 of the methods which should be employed in mathematical education. The 

 author has had a long experience of school and college education, and has 

 combined mathematical knowledge with educational ideas. The importance 

 of mathematical laboratories well stocked with clay, cardboard, wire and 

 wooden models in mathematical education, both at schooland at the University, 

 is only now being realised. It is of course evident that a teacher of mathe- 



