Miscellaneous Intelligence. 



235 



Problem I : The sums of distinct joth powers of which the 

 greatest is x p , equal to a pih power. 



Problem II: The sums of distinct £>th powers equal to a given 

 pt\\ power. 



Problem III: The sums of consecutive pth powers equal to 

 a joth power. 



Several other topics closely allied to the above are also consid- 

 ered. The work as a whole is divided into four parts, preceded 

 by a short introduction. 



In Part I the case p = l is considered, the problem here reduc- 

 ing to a treatment of sums of integers. The latter half of this 

 section contains a comprehensive discussion of prime and com- 

 posite numbers, a number of methods being given for determin- 

 ing that a given integer is or is not a prime, together with the 

 means, in the latter case, of resolving it into factors. 



Part II treats the case of squares : three necessary and suffi- 

 cient conditions that a sum of squares shall be a square are ob- 

 tained. There are also several special topics, such as methods of 

 solving the equation y* + z 2 = x*, and identities relating to sums 

 of squares. 



The case, p> 3, is contained in Part III, where a generalization 

 of the conditions of Part II are given. These are worked out in 

 detail for p = 3, 4, 5, 6, 7, and 8. The work closes with a short 

 chapter on polygonal numbers. 



The whole treatment is most freely illustrated by examples 

 making clear the methods of application of the various principles. 

 In this connection it may not be amiss to mention the tables, 

 which comprise the first 5,000 triangular numbers, the primes up 

 to 10,000 and the powers and sums of powers up to the sixth of 

 the numbers 1 — 30. Finally it should be remarked that the work 

 is complete in itself, a feature which ought to commend it to 

 readers. w. A. w. 



2. The Grammar of Science ; by Karl Pearson. Part I. — 

 Physical. Third Edition, revised and enlarged. Pp. xx, 394. 

 London, 1911 (Adam and Charles Black). — The first edition of 

 the present work appeared in 1892 and was followed by a second 

 in 1899. The time which has elapsed since the last date has seen 

 so rapid a development of science as to call for an extension of 

 the field discussed, by the author, and while the present volume 

 deals with physical ideas and phenomena, a second is to follow 

 having to do with living forms. The latter is expected to appear 

 in the course of the present year. The present volume is thought- 

 ful and suggestive and the every-day physicist accustomed to 

 assume the reality of matter and force, of the ether, the law of 

 cause and effect and the like, will find his ideas challenged and 

 subjected to a critical philosophical scrutiny of the most search- 

 ing kind. Thus, for example, the author remarks in his preface : 

 "Beyond such discarded fundamentals as 'matter' and 'force' 

 lies another fetish amidst the unscrutable arena of even modern 

 science, namely, the category of cause and effect." 



