548 



SCIENCE. 



[N. S. Vol. XI. No. 275. 



spirit prevails, a theory is presented, it is the 

 understanding which is addressed throughout , 

 and the student, if he be fit, will not easily es- 

 cape the conviction, which not every elemen- 

 tary presentation of the calculus is good enough 

 to produce, that he is dealing with a logically 

 coherent body of doctrine, whose applications, 

 moreover, yield absolutely valid results. 



A good book points the way to its own im- 

 provement. We may, therefore, venture in 

 course of the following remarks to indicate 

 some respects in which what is well done may, 

 in a second edition, be done perhaps even better. 



Being confined to the differential calculus, 

 the work possesses, on that account, some 

 peculiar merits but fails, on the same account, 

 to gain what many in recent years have come 

 to regard as the very considerable advantages 

 of presenting differentiation and integration 

 simultaneously. Knowledge of some algebraic 

 matters treated in the brief introductory chap- 

 ter might, of course, have been defensibly pre- 

 supposed ; but as a precaution such prelimi- 

 nary review seemsjustified and would, perhaps, 

 be even more acceptable were it more compre- 

 hensive. However, a certain fragmentariness 

 and discontinuity of thought could have been 

 avoided if the discussion of continuity, here 

 begun, had been reserved for the next chapter 

 where the discussion is resumed after an in- 

 terval of twenty pages. Prom a statement of 

 the properties which a function must possess if 

 it is to be continuous in a given interval, the 

 reader is left to infer what is meant by con- 

 tinuity ' at' and in the ' vicinity' of a point or 

 a value. If the notion of continuity is very 

 important, it is equally elusive, and as the be- 

 ginner best learns what the idea is bj' learning 

 what it is not, the authors' discussion, which is 

 good, would have been enhanced, we believe, 

 by a somewhat minute examination of at least 

 several examples of discontinuity. The just 

 observation, p. 11, that the ' essence' (of the 

 infinitesimal) "lies in its power of decreasing 

 numerically, having zero for its limit, and not 

 in the smallness of any of the constant values 

 it may pass through," seems to impose a restric- 

 tion on the statement of theorem 2 on the fol- 

 lowing page and to invalidate the proof there 

 given ; for, of course, the sum of a finite num- 



ber of infinitesimals may be constant, zero, 

 while to speak of the ' largest' of the infinites- 

 imals does not go to the ' essence' of the mat- 

 ter. The necessity of the word finite in the 

 theorem is happily shown by examples, though 

 definition of the term finite has not at this stage 

 been attempted. The definition, later given, 

 of finite number as being one which ' is neither 

 infinite nor zero,' is, like that given by G. Can- 

 tor, not only negative (which is but a trifling 

 objection) but also unavailable so long as the 

 •notion of the infinite is not formed. The in- 

 finite has, it is true, been defined as a variable 

 but not as a constant. As constants, neverthe- 

 less, capable, moreover, of being 'given,' some 

 infinites must be regarded, if, as in comparison 

 of variables, the phrase ' infinite limit,' is to be 

 recognized as legitimate, unless indeed one be 

 prepared to reconstruct the idea of limit. 



The notion of derivative, being attached by 

 definition to that of continuous function, while 

 it assumes the cardinal theorem that every 

 function having a derivative is continuous, is, 

 besides, not unlikely to prove a little bewilder- 

 ing to the student when a few pages later he is 

 informed, without explanation, that some con- 

 tinuous functions do not possess derivatives. 

 And being directed to " show that a function is 

 not differentiable at a discontinuity," the stu- 

 dent has only to reply that the function being 

 discontinuous at a point is not continuous there, 

 which is scarcely the answer expected. How- 

 ever, the imperfections noted relate mainly to 

 minutise, they are histological imperfections 

 and do not greatly mar the presentation as a 

 whole, which, designed for the novice, is pri- 

 marily concerned with the more obvious anat- 

 omy of the subject. 



The chapter on fundamental principles is per- 

 haps the best in the book, though some others, 

 as those on expansion of functions and inde- 

 terminate forms, are specially worthy of praise. 

 While no list of ' higher plane curves' has been 

 inserted, there is a chapter on curve tracing 

 and still another, unusually long but luminous, 

 devoted to asymptotes. In dealing with the 

 vexed and vexing question of the differential 

 and the differential notation, the convenient 

 though logically extraneous notion of 'rates,' 

 is employed as sole medium of explanation. It 



