May 13, 1898.] 



SCIENCE. 



679 



conceptions and methods of the science, and to 

 provide him with the simplest means linown at 

 the present time for dealing with them. 



Professor Lamb has produced a text-book the 

 distinctive feature of which, to our mind, is that 

 a serious and successful attempt has been made 

 to meet these latter demands. The author says : 

 "Considerable attention has been paid to the 

 logic of the subject. Writers of text-books, 

 however elementary, cannot remain perma- 

 nently indifferent to the investigations of the 

 modern Theory of Functions (of a real variable), 

 although opinions may differ widely as to the 

 character and extent of the influence which 

 these should exert. It is not claimed that the 

 proofs of fundamental propositions which are 

 here offered have the formal precision of state- 

 ment which is de rigiieur in the theory referred 

 to ; but it is hoped that in substance they will 

 be found to be correct. Occasionally, where 

 a rigorous proof of a theorem in its full gen- 

 erality would be too long or intricate, it has 

 been found possible by introducing some addi- 

 tional condition into the statement, to simplify 

 the argument, without really impairing the 

 practical value of the theorem." In this im- 

 portant respect the book is the iirst of its kind 

 on the subject of Calculus to appear in the 

 English language. May future writers on Cal- 

 culus emulate the example of Mr. Lamb in 

 trying to make their presentation rigorous 

 according to the highest standards of their day, 

 and at the same time not beyond the compre- 

 hension of the students whom they would in- 

 struct ! 



The choice of material is varied and compre- 

 hensive. Both the indefinite and the definite in- 

 tegral are introduced at an early stage. There 

 is a chapter of 44 pp. on Physical Applications 

 and one of 62 pp. on Special Curves, besides an 

 earlier chapter of 47 pp. on Geometrical Appli- 

 cations and a later one of 62 pp. entitled Curva- 

 ture and containing, among other things, a treat- 

 ment of the instantaneous center and of the 

 space and body centrodes, including an appli- 

 cation to teeth of wheels. Then follow chapters 

 on Differential Equations of the first and second 

 orders (34 + 51 pp.). Iq order, however, to 

 deal with some of the most important differen- 

 tial equations that arise in practice, some 



knowledge of the properties of power series is 

 indispensable ; and so a chapter on Infinite 

 Series (25 pp.) is introduced. This is one of 

 the first elementary treatments in English of 

 the continuity of infinite series and of the con- 

 ditions under which they can be integrated and 

 differentiated term by term. It is decidedly 

 well done, and the collection of examples at the 

 end is a valuable contribution to the presenta- 

 tion of this important subject. The book ends- 

 with a chapter on Taylor's Theorem. 



This is not the place for detailed criticism. 

 We cannot refrain, however, from deploring, 

 especially in a book characterized in the main 

 by rigor, the utter inadequacy of the treatment 

 of the important subject of infinitesimals. As- 

 one of the consequences of this neglect, a sat- 

 isfactory definition of the differential is impos- 

 sible. Again, some of the applications of the 

 calculus to geometry might have been dispensed 

 with to make place for a somewhat fuller treat- 

 ment of multiple integration. An unfortunate 

 lapse occurs in the foot-note on p. 544. The 

 power series has not been proved ' uniformly 

 convergent for values of x ranging up to a, 

 exclusively.' The text to which this note is 

 appended is, however, clear and accurate. 



The author tells us that ' ' this book attempts 

 to teach those portions of the Calculus which 

 are of primary importance in the application 

 to such subjects as Physics and Engineering." 

 For the vast majority of the students of the 

 calculus their interest is quickened and their 

 insight into the nature of the calculus is deep- 

 ened if they are shown the applications of 

 analysis to the problems of every-day life. We 

 could wish that the author had laid more stress 

 on such problems, had not a most excellent 

 book representing this side of the calculus 

 recently appeared from the pen of Professor 

 John Perry.* Mr. Lamb's plan, however, is 

 a different one. He says himself: "It is to 

 be clearly understood, indeed, that the object 

 aimed at in this book is not to teach Dynamics 

 or Physics or Engineering, but to exercise the 

 reader in the kind of Mathematics which he 

 will find most useful for the study of those sub- 

 jects." 



* The Calculus for Engineers, Edward Arnold, Lon- 

 don, 1897. 



