REVIEWS 343 



astonishing flair for methods of inquiry long since discredited, and known to be 

 barren or deceptive. 



Hugh Elliot. 



MATHEMATICS 



Differential Calculus. By H. B. Phillips, Ph.D., Assistant Professor of 

 Mathematics in the Massachusetts Institute of Technology. [Pp. vi + 162.] 

 (New York : John Wiley & Sons ; London : Chapman & Hall, 1916. Price 

 $s. bd. net.) 



The author says in his Preface that in this book " a few central methods are 

 expounded and applied to a large variety of examples to the end that the student 

 may learn principles and gain power. In this way the differential calculus makes 

 only a brief text suitable for a term's work, and leaves for the integral calculus, 

 which in many respects is far more important, a greater proportion of time than is 

 ordinarily devoted to it." After an Introduction, in which there are explanations 

 of function, limit, and indeterminate forms, which are not quite satisfactory from 

 a logical point of view, the concepts of derivative and differential are described. 

 The explanation of the idea of a differential is good (pp. 15-16), though the way 

 in which higher differentials are treated in certain cases (p. 30) is not satisfactory. 

 The usual topics treated in courses on the differential calculus are gone through. 

 Rates of change of position and velocity on a straight line and a curved path are 

 not introduced, I think, as early as they might be with advantage. The enuncia- 

 tion and exemplification — it is not more than this — of Rolle's theorem (p. 94) has 

 a bad mistake ; the examples are not examples of discontinuity of the derivative, 

 as the author states, but of its non-existence. There are certainly some advantages 

 in the way in which the subject of the differential calculus is exposed in this book : 

 for example, I may notice the — to a student — illuminating definition given (p. 10) 

 of a " continuous " function. But I hardly think that such advantages compensate 

 fully for the decided disadvantages of carelessness and lack of logic. Even ele- 

 mentary students in mathematics are far more puzzled by a teacher's logical 

 obscurities than teachers seem inclined to admit, and, at any rate for the training 

 of future mathematicians, even what may seem to some to be excessive care in 

 logic may not be out of place. It is quite true that even more important in teach- 

 ing than strict logic is suggestiveness of ideas and stimulation of the student's 

 imagination ; but this book seems to fall short in this requirement also. 



Philip E. B. Jourdain. 



Elliptic Integrals. By Harris Hancock, Professor of Mathematics in the 

 University of Cincinnati. [Pp. 104.] (New York : John Wiley & Sons, 

 Inc. ; London : Chapman and Hall, Ltd., 1917. Price 6s. net.) 

 This volume is the eighteenth of the excellent series of " Mathematical Mono- 

 graphs " edited by Mansfield Merriman and Robert S. Woodward, and is a 

 treatise almost entirely on the well-known elliptic integrals of Legendre. It may 

 thus be regarded as a somewhat more advanced part of a course on the integral 

 calculus, in which are considered integrals of rational expressions involving square 

 roots of cubics and quartics in one variable. The integrals of the three kinds 

 and the Legendre transformations are treated in the first chapter, but on the 

 whole, in order to keep within a limit of about one hundred pages, the author has 

 had to confine the discussion almost entirely to the elliptic integrals of the first 

 and second kinds. In the second chapter the functions which arise from the 

 inversion of elliptic integrals are shortly described, and their doubly periodic 

 properties are emphasised. The third chapter is concerned with the reduction of 



