June 19, 1908] 



SCIENCE 



955 



ing organic and fundamental, can be neither 

 composed nor annulled. Fortunately for an 

 age that glories in possessing and is bent on 

 advancing a material civilization that theorists 

 never would and that practicians never could 

 produce, there is a third class among those 

 who have to do with mathematics, a class com- 

 posed of two groups of men : a group in- 

 terested primarily in theory, in mathematics 

 as a science, yet having a strong secondary 

 interest in applications, in practise, in mathe- 

 matics as a tool; and another group chiefly 

 interested in practise that involves applica- 

 tions of mathematics, but having at the same 

 time a potent secondary interest in the sub- 

 ject as a science, as a body of consistencies, an 

 ensemble of coherent doctrines. The latter 

 group,' the engineers, tend to keep mathe- 

 matics sane, serviceable, attached to reality, 

 adapted to the needs of the surveyor, the 

 miner, the excavator, the bridge-builder, and 

 the rest; the former group, comprising most 

 of the professional mathematicians and the 

 teachers of mathematics, serve to save the 

 science from degenerating into a mere drudge, 

 and by extending its structures far above the 

 conscious needs of man, make it an everlast- 

 ing monument to his dignity and an honor to 

 his spirit. Thus the interests of these two 

 groups, unlike those of the theorist and the 

 practician, intersect; and as there is the need 

 of better cooperation between the groups, there 

 is also, by virtue of their community of 

 temperament, the possibility of securing it. 

 The engineer says to the teacher of mathe- 

 matics : " Make your science more serviceable, 

 lay bare its instrumental significance, teach 

 us how to use it." The teacher replies : " Your 

 demand is just and reasonable, but you should 

 understand that the application of a difficult 

 doctrine to a difficult concrete problem pre- 

 supposes an understanding of the doctrine as 

 a doctrine and that such understanding re- 

 quires native ability and prolonged study." 

 As a reasonable man, the engineer must admit 

 that the teacher, too, is right. What, then, is 

 to be done? The answer is: compromise. 



How to effect the compromise to the best 

 advantage of all the interests involved — the 

 integrity of the science itself, the insistent 



claims of the technologist, the indubitable 

 rights of those who pursue the study of mathe- 

 matics solely as a discipline and especially of 

 those rarer spirits who hope to make it the 

 object of a life's devotion — that is the ques- 

 tion that presses upon the teachers of mathe- 

 matics in our day and that, owing to the 

 familiar rapid multiplication of technological 

 schools, presses especially hard upon teachers 

 of the calculus. In Professor Osgood's book 

 culminate the eilorts of nearly a generation of 

 mathematicians to produce a beginner's cal- 

 culus that shall be both rigorous and under- 

 standable, theoretic enough to be scientific and 

 sufficiently practical for the student of engi- 

 neering, not too spiritless for those whose aim 

 is liberality of culture and yet adequate as a 

 preparation for the intending student of still 

 higher disciplines. For directness and sim- 

 plicity of presentation, clearness and correct- 

 ness of statement, judicious accentuation and 

 ordering of topics, and for the happy mingling 

 of the concrete and particular with the ab- 

 stract and general, this work attains a level 

 of excellence not likely to be soon surpassed. 

 The author estimates that the time required 

 for covering the matter of the book corre- 

 sponds roughly to a five-hour course through- 

 out one year. The estimate is based, however, 

 upon the lecture method of presentation. In 

 the case of lectures adapted to undergraduates, 

 this method, whatever be its compensatory ad- 

 vantages, is undoubtedly less rapid than that 

 of assigning definite lessons and requiring 

 recitations upon them. By employing the lat- 

 ter method of instruction and by omitting the 

 chapter of about forty pages devoted to me- 

 chanics — an omission entirely practicable in 

 a considerable number of institutions that 

 provide a separate course in elementary me- 

 chanics to follow the calculus — it would seem 

 to be possible to cover the remainder of the 

 matter fairly well in a three-hour year course 

 or even in a five-hour half-year course. In- 

 deed, if one make the mentioned omission, the 

 remainder of the book, owing to wider margins 

 and other physical features, only appears to 

 contain more reading matter than such a book, 

 for example, as Osborne's revised " Calculus," 

 and this last, as experience has shown, can be 



