92 



SCIENCE 



[N. S. Vol. XXXVIII. No. 968 



work the miracle of pleasing at once mathe- 

 maticians who are not engineers and engineers 

 "who are not mathematicians. 



Perhaps the most notable feature of Pro- 

 fessor Hulburt's book is the excellence of its 

 English. No doubt mathematical truth is like 

 other scientific truth in the characteristic re- 

 spect that its significance does not depend pri- 

 marily upon the form in which it is expressed. 

 It ought not to be forgotten, however, that its 

 accessibility does depend upon its form. A 

 loose definition of a mathematical term is not 

 a mathematical definition. A vague statement 

 of a proposition is not a statement of a mathe- 

 matical proposition. Discourse that is not pre- 

 cise, cogent and concatenative is not mathe- 

 matical discourse. For some unexplained 

 cause departments of English fail to give their 

 pupils such facility in English expression as is 

 available for mathematical purposes. And 

 those whose fortune it is to teach undergradu- 

 ate mathematics find it necessary in classroom 

 to devote half their time and energy to trying 

 to secure on the part of their pupils decent, I 

 do not say elegant or imposing or fine, but 

 merely decent expression of ideas. In this im- 

 portant matter, an excellent model is of very 

 great assistance, and such a model Professor 

 Hulbert has furnished. Most excellence is 

 excellence of emphasis. In this respect, 

 too, the book is a model; doctrines are 

 presented in perspective. The nature of 

 the differential and the utility of the dif- 

 ferential notation are made perfectly, un- 

 mistal^ably, intelligible — something that un- 

 fortunately can not be said of some current 

 presentations. As to the order of themes, 

 there may be difference of judgment. Inte- 

 gration is introduced on page 175. Practise 

 in integrating is recommended and afforded 

 before the use of tables, given at page 190. 

 Teachers will value the introduction to an- 

 alytical geometry of three dimensions, page 

 265. Taylor's series is presented as late as 

 page 349. The work closes with an excellent 

 account of simple differential equations, and 

 a list of answers to exercises distributed 

 throughout the volume. Printing and binding 

 aTe well done and the page pleases the eye. 



In the composition of their interesting work, 

 Messrs. Franklin, MacNutt and Charles have 

 been guided by certain convictions. For ex- 

 ample, they believe that " to break the thread 

 of the textual discussion by unnecessary alge- 

 braic developments and by large and frequent 

 groups of purely formal problems," as is com- 

 monly done, is a "really hideous feature"; 

 and they have sought to avoid such a blemish 

 by relegating the majority of the formal prob- 

 lems to an appendix. This plan has not pre- 

 vented them, however, from introducing a 

 plenty of exercises into the body of the text. 

 Again, they are convinced that, very unfortu- 

 nately, nearly all scientific text-books carry the 

 " false suggestion of completeness and final- 

 ity," and, accordingly, in order to guard the 

 reader against gaining such an impression 

 from their book, the authors have very laud- 

 ably given in an appendix " a carefully selected 

 list of treatises on mathematics and on mathe- 

 matical physics." The book is notable for the 

 pains the writers have taken to keep the sci- 

 ence of the calculus attached to reality, and 

 everywhere throughout the work one detects 

 the odor of physical science. On this account, 

 perhaps, theoretical developments seem to have 

 suffered in comparison, sometimes even con- 

 sciously , as in case of the notions of infinitesi- 

 mal, differential, divergence and curl. In- 

 deed the aiithors characterize the articles deal- 

 ing with these ideas as " fallacious," " mere 

 plausibilities," and as being such that " the 

 harder one tries to understand them the more 

 vague and unintelligible they become." We 

 are disposed to think that the authors, if not 

 too modest and frank, have overrated the diffi- 

 culty of presenting the matters in question 

 soundly and clearly. The final chapter, 43 

 pages, is devoted to an elementary exposition 

 of vector analysis, an element of the book that 

 many will gladly welcome. 



Professors Davis, Brenke and Hedrick have 

 produced a very teachable book. It would be 

 more pleasing if the print were larger and the 

 pages less crowded. In an unusual degree one 

 finds here the spirit of the calculus. Designed 

 equally for the college and the engineering 



