May 31, 1918] 



, SCIENCE 



541 



claims of the theoretician, on the one hand, 

 and the practician, on the other, furnishes the 

 mathematical test-book writer with a perpetu- 

 ally fascinating problem; for the theoretician 

 and the practician have nothing in common 

 save their insistency; in other respects they are 

 as wide apart as the poles; the theoretician is 

 a rigorist, a logician, a lover of the abstract, 

 demands a minimum of assumption and a 

 maximum of proof; the practician hates the ab- 

 stract, loves the concrete, and mainly depends 

 for his happiness on getting results by the use 

 of rules and formulas that he neither under- 

 stands nor cares to understand. To win the 

 unqualified approval of both these types of 

 critic is impossible, a contradiction in terms; 

 to incur the unqualified condemnation of both 

 is not impossible; the target to be aimed at is 

 somewhere between; to locate it and to hit it 

 squarely — two very different things — require a 

 rare combination of sanity, skill and good 

 luck. 



It: may be added that in these times the 

 test-book writer receives additional stimula- 

 tion from the keen competition of other sub- 

 jects and from the challenge of certain cun- 

 ning educators who have shrewdly discovered 

 that the educational value of mathematics has 

 always been greatly overestimated. 



The aim of Professors March and Wolff has 

 been to present " the calculus in such a way 

 that it will appeal to the average student rather 

 as a means of studying scientific problems 

 than as a collection of proofs and formulas." 

 The aim is commendable but in saj-ing so we 

 do not intend to imply, and the authors would 

 probably not contend, that the calculus must 

 appear either as such a " means " or as such a 

 " collection " for it has other aspects, aspects 

 both attractive and worthy. Integration is in- 

 troduced at an early stage. In connection 

 with the employment of infinitesimals, Du- 

 hamel's theorem is used but without too much 

 finesse. There are numerous applications to 

 elementary classical problems of geometry, 

 physics and mechanics. A brief introduction 

 to analytical geometry of three dimensions is 

 inserted for such readers as may require it. 

 The phrasing is in general so careful and so 



good that its very excellence operates as a chal- 

 lenge, and one is tempted to ask whether it 

 would not be a trifle better to say that the 

 phrase, division by zero, is meaningless than to 

 say (p. 30) that " division by zero is an impos- 

 sible operation"; to say that the fraction, 

 (V — 4) : (x — 2), has no value for x = 2 than 

 to say " its value is not determined at this 

 point"; to say in such a case that there is no 

 quotient than to say that " the quotient has no 

 meaning." The volume closes with a very brief 

 chapter dealing with simple types of differen- 

 tial equations. 



To differential equations Professor Love de- 

 votes three chapters amounting to more than 

 fifty pages. Integration is not presented 

 earlier than page 116. This is preceded by a 

 chapter on curve tracing. The reader is im- 

 pressed with the possibility of calculating the 

 most important mass-moments of first and sec- 

 ond order by means of simple integration. 

 Applications are drawn exclusively from geom- 

 etry and mechanics with unusual emphasis on 

 the latter. The importance of checking re- 

 sults, particularly in integration problems, is 

 stressed. An excellent feature is the presence 

 of " worked examples " to assist the reader in 

 making transition from theory to practise. In 

 Professor Love's book as in that of Professors 

 March and Wolff the fundamental theorems 

 respecting limits are set down without proof. 



In Mr. Barker's book we have a pretty plain 

 specimen of plane trigonometrj-. Trigonomet- 

 ric series are not present. Of the wider bear- 

 ings and higher attachments of the subject the 

 reader is not made aware. Much attention is 

 rightly given to simple applications. The 

 large page and open type please the eye. The 

 punctuation is unusual and not consistent 

 with itself. The radian is defined as if it must 

 be conceived as always having its vertex at the 

 center of a circle. The words " these " and 

 " this " (pp. 2, 3) are assigned to duties that 

 they are unable to perform. In article 4 one is 

 at a loss to determine the significance of the 

 repeated phrase, " said to be." The author has 

 sometimes allowed himself the freedom of 

 such colloquial expressions as " Expand the 

 left members and we have" (p. 86). 



