REVIEWS 679 



the three subjects which have to be treated in elementary lectures at Univer- 

 sities into one coherent whole, while allowing the important differences of 

 method between analysis and geometry to become apparent. This aim is 

 indeed a worthy one, and a successful account of how unity of treatment 

 could be attained in the teaching of mathematics suitable for pass students 

 or students reading for honours in' physics or economics or other sciences 

 would be of very great value. It will be remarked that the calculus is 

 untouched in this course. And this, indeed, appears to be the cause of some 

 of the defects of the book. The authors in the preface remark that " no 

 attempt has been made to introduce the terminology of the calculus, as it 

 is found that there is ample material in the more elementary field which 

 should be covered before the student embarks upon what may properly 

 be called higher mathematics." In spite of this infinite series are introduced, 

 and indeed are introduced without adequate explanation. The reason for 

 this can doubtless be traced to the view that the notion of an infinite series 

 presents serious difficulties to the elementary student. But the modern 

 theory of limits is involved in so fundamental a manner in the theory of 

 convergence and can be explained so simply that it seems a great mistake 

 to arrange elementary courses so as to avoid its introduction. The remarks 

 made about the geometrical series 



1 + r + r 2 . . . + r n + . . r < 1 



are all entirely correct, but do not seem to be full enough to meet the case 

 of the student who is unfamiliar with the notion of a limit. 



The treatment of complex numbers also seems to be unsatisfactory ; 

 i, defined as a number which multiplied by itself equals — 1, is introduced 

 at the very beginning. It is stated that complex numbers are combined 

 according to the laws of numbers, and that the combination of any two or 

 more complex numbers . . . gives a complex number. It is not clear whether 

 these statements are postulated or derived from other postulates implicitly 

 introduced in the (very slight) discussion of the geometrical interpretation. 

 But if they are assumed, no hint is given why such results should be intro- 

 duced. It is never made clear that i is introduced in order that the laws 

 for complex numbers or ordered pairs of numbers may be formally the same 

 as the laws for real numbers. For teaching purposes it seems desirable to 

 discuss first the introduction of ordered pairs of real numbers and the way 

 in which the four fundamental algebraic operations are to be defined ; and 

 only subsequently to consider the possibility of there being a meaning for 

 i which will bring the laws of combination of ordered pairs of real numbers 

 (x, y) (x f , y'), (x" , y") under the same formal treatment as numbers such 

 as (x + iy), [x' + iy'), (x" + iy") .... To introduce i ab initio as a number 

 which, multiplied by itself, gives — . 1, is to run the danger of losing the atten- 

 tion of the student, and of letting him assume from the beginning the exis- 

 tence of a number i which will make a uniform treatment of real and complex 

 numbers possible. 



Throughout the book there is a wealth of examples and illustrations. 

 These, together with the clear treatment of simple trigonometrical problems, 

 and of the elementary theory of two- and three-dimensional geometry, make 

 the book as a whole (apart from the two important respects already men- 

 tioned) useful as an introduction of very simple mathematics to the student. 



Dorothy Wrinch. 



A Non-Euclidean Theory of Matter and Electricity. By P. A. Campbell. 

 [Pp. 44.] (Cambridge, Mass. : University Press, 1907, Price 35 

 cents.) 



The prominent position which Einstein's theory of space and time has 

 recently taken in the realm of scientific thought has resulted in the re- 



