342 SCIENCE PROGRESS 



Differential Calculus for Colleges and Secondary Schools. By Charles 

 Davison, Sc.D., Mathematical Master at King Edward's High School, 

 Birmingham. [Pp. viii + 309.] (London : G. Bell & Sons, 1919. Price 6^-.) 



This good but somewhat conventional addition to the " Cambridge Mathematical 

 Series " is divided into two parts, the first dealing with the principles, and the 

 second with the applications, of the differential calculus. But the author wisely 

 points out (p. v) that he does not suggest that this order should be adhered 

 to in reading the book. The first part consists of chapters on the differential 

 coefficient, differentiation of certain functions, successive differentiation, the ex- 

 pression of functions in power-series, and the evaluation of indeterminate forms. 

 The second part deals with maxima and minima, and the usual geometrical 

 applications, including singular points and curve-tracing in rectangular and polar 

 co-ordinates. A very welcome feature is the suggestion (pp. 246 sqq.) of essays 

 on rather more general aspects of the subject than are included in ordinary 

 examination questions. 



In the treatment of Rolle's theorem (p. 61), the conclusion that a continuous 

 function which is first negative and then positive "must pass through the value 

 zero for some value of x " relies implicitly on a crude geometrical notion of the 

 continuity of a function. It is, of course, important that a student should be 

 shown that all concepts in mathematics start from crude notions, but he should 

 also be shown that concepts must be distinguished from the crude notions. 



Philip E. B. Jourdain. 



Projective Vector Algebra: an Algebra of Vectors independent of the 

 Axioms of Congruence and of Parallels. By L. Silberstein, Ph.D., 

 Lecturer in Mathematical Physics at the University of Rome. [Pp. 

 viii + 78.] (London : G. Bell & Sons, 1919. Price 7s. 6d. net.) 



This very original little book was at first intended as a paper, and its purpose 

 is to construct, by means of the axioms of connexion and of order alone, a very 

 simple algebra of vectors that is to embrace only the equality, the addition, and 

 hence also the subtraction, of vectors (pp. 1-2). " There is no essential difficulty 

 in introducing also an appropriate vector multiplication of two vectors. This, 

 however, besides being superfluous, does not share the remarkable simplicity 

 which will be seen to belong to the proposed vector addition, and did not, 

 therefore, seem sufficiently interesting to be given in this little book" (p. 2). 

 In fact, the whole of projective geometry falls within the scope of an algebra 

 in which multiplication is not defined for vectors (p. 38). 



Starting from the postulates given by F. Schur in his Grundlagen der Geometrie 

 of 1909, and the mostj essential Desargues' theorem (pp. 2, 7-8), — indeed, the 

 reader requires hardly anything more than the knowledge of this theorem as a 

 consequence of the axioms of order and of connexion (p. 4), — the addition of 

 two coinitial vectors is defined without the concept of " equality " of vectors being 

 introduced (pp. 4-5). With this definition, which depends on two arbitrary 

 points on the lines representing vectors, a most valuable extension of the usual 

 definition of addition is obtained ; indeed, Dr. Silberstein's definition of vector 

 sum " is manifestly a generalisation of the Euclidean one, and is valid also for 

 such spaces in which there are no Euclidean parallels 'nor even Lobatschevskyan 

 parallels or asymptotic lines " (p. 5). The associative law is then proved (pp. 

 6-9), and the sum of collinear vectors defined (pp. 9-18). The concept of 



