August 25, 1911] 



SCIENCE 



231 



after transformation it is doubtful from 

 the standpoint of the engineer if the sub- 

 ject is worth over a page of discussion in 

 the report referred to above, or the many- 

 pages — in one case as many as 18 — of re- 

 cent elementary texts on the calculus. The 

 time given to them in the course on cal- 

 culus could be used to much better ad- 

 vantage elsewhere. 



The relatively few times that the double 

 and triple integrals are used agrees with 

 the results in the undergraduate discussion. 

 Double and triple integrals could be used 

 more, but they aren't; engineers seem to 

 prefer the single integral. From this we 

 would not argue that they should receive 

 attention in a course in the same ratio as 

 they are found used in practise, for both 

 have greater merit and occasions do arise 

 in which they are fundamentally necessary 

 and important. 



The differentiations and integrations, as 

 shown, are, for the most part, limited to a 

 few types and are generally readily ex- 

 ecuted. The algebraic integrations do net 

 at all resemble the heavy forms involving 

 radicals and reduction formulas which 

 were so painfully evident in the college 

 course in calculus. The trigonometric 

 forms, both in differentiation and integra- 

 tion, are limited almost entirely to sines 

 and cosines and their combinations. The 

 heavier integrations, both algebraic and 

 trigonometric, are apt to occur in connec- 

 tion with the solution of the differential 

 equations arising in the discussions. 



Partial differentiation comes in for con- 

 siderable attention partly because of the 

 fact that the journals listed, especially the 

 foreign, always seemed to welcome an 

 article making use of the principles of 

 thermodynamics. The relative importance 

 of this portion of the calculus to the engi- 

 neer seems to be greater than would be 

 indicated by the amount of time given the 



subject in most of the courses in calculus 

 for engineers. 



The principles of angular velocity and 

 acceleration are used with considerable 

 frequency in the foreign journals, thus in- 

 dicating clearly the strong influence of the 

 mechanical engineering side of the tech- 

 nical field. 



The attitude of all the journals in the 

 matter of symbols is the same and agrees 

 with the almost universal custom among 

 mathematicians in this country. The some- 

 what forced efforts made for some time by 

 certain mathematicians in this country to 

 introduce capital letters in connection with 

 derivatives seems to have received no recog- 

 nition whatever among engineers. 



Whenever a differential is used it has a 

 strong resemblance to the infinitesimal of 

 the mathematician, or perhaps a rate, 

 where the time element comes in ; but it will 

 be noticed early that the processes of the 

 engineer, by nature correct to, say, so 

 many decimal places are not always the 

 processes of the mathematician confirmed 

 in his use of the limit. Increments some- 

 times take the place of these differentials 

 and very neat bits of theory are carried out 

 by their use — a "near calculus" as it were. 

 It is at times difScult to distinguish be- 

 tween the use of Arc and dx and yet the 

 idea of the limit is there all the time. 

 Rigor within the limit of allowable error is 

 the key-note throughout. As an illustra- 

 tion I quote: "Let the radii of curvature 

 of the surfaces where the oil film is in- 

 finitely thin be r^ and r^; then, since we 

 are dealing with films of capillary thick- 

 ness, we may treat the dimensions as in- 

 finitesimals as compared with r^ and rj." 

 With which many a pure mathematician 

 will find it difficult to agree. And yet the 

 proofs do not break down at any point. 



More maxima and minima are found 

 directly from the study of algebraic and 



