June 16, 1899.] 



SCIENCE. 



845 



consist of three volumes. Concerning the 

 merits of this first part in so far as these may 

 ultimately depend on its relations to the rest of 

 the work, it would be premature to form an 

 opinion. Apart, however, from this coutingeut 

 and inchoate character of the volume, it has 

 a unity and maturity of its own, being avowedly 

 written as an introduction to the calculus, and 

 as such is properly before the public for review. 



The author's aim has been "to penetrate as 

 far as possible, and in as many directions, into 

 the subject — that the student should attain as 

 wide knowledge of the matter, as full compre- 

 hension of the methods, and as clear conscious- 

 ness of the spirit and power of this analysis as 

 the nature of the case would admit." It is not 

 easy to realize so high and composite an ideal. 

 The nature of the case, it is well known, pre- 

 sents some grave difficulties. Of these the most 

 obstinate inheres in the combination of doctrine 

 and applications, of the general and abstract 

 with the particular and concrete, in securing, 

 despite the fragmentariness incident to illus- 

 tration and example, the effect of unity and 

 wholeness in the development of theorJ^ 

 French and German writers, such as Jordan, 

 Harnack, Stolz, escape the difficulty of combin- 

 ing theory and practice by simplj' ignoring the 

 latter. By this easy disregard of the needs of 

 all students except specialists in graduate years, 

 these authors are enabled to attain a coherency 

 and symmetry of development which lend to 

 their work, besides the scientific, something of 

 an artistic character. The Englishman, on the 

 other hand, is prone to lose both of these ad- 

 vantages by sinning in the opposite direction, 

 by a distinct subordination of theory to practice, 

 a collocation, however interesting and useful, 

 of exercises for the ingenuity of students, being 

 neither an sesthetic nor, in strictness, a scien- 

 tific production. 



The problem of overcoming instead of dodg- 

 ing the difficulty in question, of escaping the 

 mentioned vices without losing their peculiar 

 virtues, admits of only approximate solution. 

 The necessary compromise has, as is well known, 

 been skilfully effected in German in the de- 

 servedly much-praised treatise by Kiepert. In 

 the book under review a notably similar suc- 

 cess has been achieved in English. In fact, 



these two works, though differing widely in 

 method and detail, are closely allied in spirit 

 and aim. The motive in both is to guide and 

 inspire ; both are honest, anxious not to de- 

 ceive, faithful in indicating assumptions and lim- 

 itations, and, while seeking first to be intelligi- 

 ble, are in general as rigorous as circumstances 

 will allow. Neither author forgets that in last 

 analysis his science resides in theory, which, 

 therefore, properly receives the greater empha- 

 sis. Nevertheless, both works abound in con- 

 crete examples. These, curiously enough, are 

 nearly all worked out in the German text, 

 while in the English most of them are, as 

 usual, left as exercises for the student. 



In point of matter these works are not coin- 

 cident nor coextensive either with one another 

 or with their rivals, such as the treatises by 

 Edwards, Williamson and Greenhill. For ex- 

 ample, Kiepert gives a concise preliminary 

 treatment of certain algebraic themes, as the 

 binomial theorem, the potential and logarithmic 

 series, convergency and divergency, determi- 

 nants and others, while Smith has, for the sake 

 of brevity, presumed knowledge of some of 

 these, treatment of others being reserved for 

 Vol. II. A like reservation is made in case of 

 the complex variable, and, save for an elegant 

 though very brief account, in case also of differ- 

 ential equations, to each of which topics Kie- 

 pert gives an introduction. On the other hand. 

 Smith, like Williamson, deals with the gamma 

 functions and inserts a helpful chapter on curve 

 tracing, while Kiepert excludes the former sub- 

 ject and considers the latter but incidentally. 

 The omission by the American, as by the Ger- 

 man, of the theory of probability and the calcu- 

 lus of variations is a noticeable departure from 

 British precedent. 



The opening chapter of the volume before us 

 is, in many respects, an admirable presentation 

 of fundamental concepts and operations. The 

 path pursued leads quickly into the heart of 

 the subject. The student meets first things of 

 first importance. The notion of limit is at 

 once lifted into prominence, being carefully un- 

 folded at the very outset, and employed with- 

 out delay in definition and proof. The infini- 

 tesimal is correctly defined, and its subjective 

 character is pointed out, the fact, namely, that 



