REVIEWS 



MATHEMATICS 



Analytic Geometry and Calculus. By Frederic S. Woods and Frederick 

 H. Bailey, Professors of Mathematics in the Massachusetts Institute of 

 Technology. [Pp. xii + 516.] (Boston, U.S.A., and London : Ginn & Co., 

 1917. Price I2.y. 6d. net.) 



This work is a revision and abridgment of the authors' Course in Mathematics for 

 Students of Engineering and Applied Science, and the authors have omitted from 

 the previous work such subjects as determinants, much of the general theory of 

 equations, the general equations of the conic sections, polars and diameters related 

 to conies, centre of curvature, evolutes, certain special method of integration, 

 complex numbers, and some types of differential equations. Further, the material 

 is rearranged so that the first part of the book contains methods for the graphical 

 representation of functions of one variable, both algebraic and transcendental, 

 while analytic geometry of three dimensions is treated later when it is required 

 for the study of functions of two variables. "The transition to the calculus is 

 made early through the discussion of slope and area, the student being thus 

 introduced in the first year of his course to the concepts of a derivative and a 

 definite integral as the limit of a sum" (p. iii). 



The book is undoubtedly useful for engineering students, for whom it will form 

 a sort of compendium of what pure mathematics is necessary for their purpose. 

 But it would seem that an exception is to made in the way differentials are treated 

 (pp. 141-2). They are "not to be thought of as exceedingly minute" (p. 143); 

 yet, in the treatment of slopes and areas, we naturally find that the finite incre- 

 ments of x and y are to proceed indefinitely to zero (pp. 136, 141, 143-4). Also 

 differentials are "infinitesimals," and may, under certain circumstances, be dis- 

 regarded (p. 261). It would, it seems, be much better to introduce the calculus 

 as a method of very close approximation. Other points of criticism, if, indeed, the 

 authors' wish consistently to replace guesswork by logic, are that the comparison 

 test for convergence assumes implicitly a previous definition of irrational 

 numbers (p. 407), and that there are difficulties for a student,— an appeal to the 

 authority of future and more rigorous work,— in the method given for finding 

 e (p. 199), and in the treatment of Fourier's series (p. 429). The examples given 

 to illustrate how differential equations present themselves (for example, pp. 438-41, 

 450-1, 453-4) seem to be very good, and it may be remarked that in many cases 

 they often arise from the actual problems on which mathematicians actually 

 worked in the last years of the seventeenth century. 



Philip E. B. Jourdain. 



Theory of Maxima and Minima. By Harris Hancock, Ph.D. (Berlin), 



Dr.Sc. (Paris), Professor of Mathematics in the University of Cincinnati. 



[Pp. xiv + 193.] (Boston, U.S.A., and London : Ginn & Co., 1917. Price 



io.y. 6d. net.) 



The first who made a distinction between a maximum and a minimum of a 



function seems to be Leibniz (p. 3), and it was Maclaurin who first gave a correct 



•322 



