Febbuabt 22, 1907] 



SCIENCE 



299 



SCIENTIFIC BOOKS 

 Lectures on the Theory of Functions of Real 



Variables. Volume I. By James Pier- 



PONT. Boston, Ginn and Company. Pp. 



xii + 560. 



A considerable part of the present volume 

 is in very close touch with problems which 

 confront the students of elementary mathe- 

 matics, dealing with such questions as the dif- 

 ference between rational and irrational num- 

 bers, the theory of limits and the concepts of 

 continuity and discontinuity. Bright and 

 thoughtful students frequently seek more light 

 on these subjects than they can find in the 

 elementary text-books and many teachers will 

 doubtless rejoice to find that a large amount 

 of most interesting information along these 

 lines has been made accessible by a scholar in 

 whom they can have the utmost confidence. 



A little more than a hundred pages are de- 

 voted to the fundamental matters which are to 

 serve as a basis for the notion of function in 

 general. This notion is illustrated by means 

 of the trigonometric functions with which the 

 reader is supposed to be familiar and a very 

 brief proof is given of the interesting fact that 

 these functions are transcendental. The de- 

 scriptive introduction to functions is followed 

 by a similar introduction to point aggregates 

 in which several fundamental theorems re- 

 lating to limiting points are proved and a 

 number of the common terms are defined and 

 illustrated. The theory of point aggregates 

 furnishes some of the most interesting in- 

 stances of the distinction between finite and 

 infinite multitudes, and the importance of this 

 theory is partially illustrated by the fact that 

 one of its terms (dense) is needed as early 

 as page 20 to describe the system of rational 

 numbers. 



The greater part of the present volume deals 

 with questions which the student approaches 

 in the elementary calculus. The processes of 

 differentiation and integration are treated with 

 a completeness which seems impracticable in 

 a first course, yet this completeness is essential 

 for a thorough comprehension of the subject. 

 A very helpful feature is furnished by the 

 ' numerous examples of incorrect forms of 

 reasoning currently found in standard works 



on calculus.' It has been " the author's ex- 

 perience that nothing stimulates the student's 

 critical sense so powerfully as to ask him to 

 detect the flaws in a piece of reasoning which 

 at an earlier stage of his training he con- 

 sidered correct." 



The vast extent of the applications of the 

 processes of calculus have frequently led 

 writers to overlook the regions where these 

 processes do not always lead to correct results. 

 Even some of the most useful formulas, 

 such as 



dy/dt ^= dy/dx ■ dx/dt, 



appear in nearly all, if not in all, of the other 

 English texts with an incomplete demonstra- 

 tion. Arts. 378-80, which are devoted to a 

 satisfactory demonstration of this formula, 

 exhibit also the missing elements in the com- 

 mon demonstrations and suggest a method for 

 an elementary demonstration in case a func- 

 tion has only a finite number of oscillations. 



The last three chapters are devoted to im- 

 proper integrals and to multiple proper inte- 

 grals. These naturally contain much more 

 original matter than those which precede. 

 This is especially true of the last chapter, 

 which is practically an original contribution. 

 The definition of an integral is taken in the 

 most general fashion and includes all the pos- 

 sible fields, whereas until then the most gen- 

 eral was Jordan's and this is restricted to the 

 inner points of a field. No other work con- 

 tains such a complete treatment of the sub- 

 ject of uniform convergence as is found in 

 these chapters. 



The present volume, which is to be followed 

 by another along the same lines, seems 

 especially timely in view of the movement to 

 employ the notion of function much more 

 generally in the elementary courses in algebra 

 and geometry. Teachers of secondary mathe- 

 matics should have a clear understanding of 

 the concept of function and we know of no 

 other work where an accurate knowledge of 

 this concept can be acquired as readily as 

 from the earlier chapters and the criticisms 

 of the present treatise. The fact that Ginn & 

 Company should undertake the publication of 

 such works as this and Goursat's ' Course in 



