NATURE 



THURSDAY, APRIL 23, ic 



.1 .V7J11' C-.4LCL'LLS. 



.1 first Course i)i the Differential and Integral Cal- 

 culus. By Dr. W. F. Osgood. Pp. xv + 423; with 

 125 figures. (New York : The Macmillan Com- 

 pany; London : .Macmillan and Co., Ltd., igoy.) 

 Price JOS. 6d. 



THE introduction of the Calculus at an early stage 

 in a course of elementary mathematics has ren- 

 -dered necessary the substitution of simplified methods 

 of treatment for those occurring in the earlier text- 

 books ; for example, an abandonment of the lavish 

 and unnecessary use of infinite series, the convergency 

 of which was generally ill-understood, in the differen- 

 tiation of simple expressions. 



.\ number of good books have recently appeared 

 written more or less with this object in view, but we 

 have seen none in which the survival of old and 

 clumsy methods has been reduced to the vanishing 

 point in the same way that has been done in this 

 book. 



The present reviewer has been in the habit of con- 

 •ducting a class in the calculus on simplified lines 

 identical in nearly every respect with those adopted 

 independently by Prof. Osgood; indeed, this book 

 represents almost word for word what he would have 

 wished to write had he undertaken to write a Calculus. 

 The reviewer is thus greatly indebted to the author 

 for having saved him this troublesome and thankless 

 task, or the alternative of continuing the elaborate 

 lecture notes which he has found necessary to dictate 

 to his pupils on the bookwork of the subject. The 

 following is a brief summary of some of the salient 

 features of the book to which the reviewer attaches 

 especial importance. 



The methods of the calculus are discussed and 

 exemplified in the first instance by the study of the 

 differentiation of series of positive integral powers 

 nnly. The reviewer would prefer to see the binomial 

 theorem omitted from the proof for the derivative 

 of X" and a proof based on the product rule substi- 

 Juted, but this is a minor detail which any teacher can 

 si'.pply for himself and his class. 



The very important method of " differentiating an 

 equation as it stands " is explicitly used as such in 

 finding the tangents to algebraic curves as well as in 

 the differentiation of fractional powers, inverse func- 

 ■lions, and the like. The introduction of this subject 

 binder the title of " differentiation of an implicit 

 I miction " is quite unnecessary, and we arc glad to 

 see that the perfectly simple method really required 

 /or these cases is viewed in its right light. 



In the chapter on transcendental functions the 

 author clearly points out that the reason for measur- 

 ing angles in radians is essentially explained by the 

 < alculus, and he also gives the differentiation formula 

 4iir angles measured in degrees. The e.xplanation is 

 necessary in order to dispel any doubts the beginner 

 m.'iy have previously formed as to the mental sanity 

 <i| those mathematicians who deliberately chose :in 



NO. 200S, VOL. 77] 



incommensurable unit for the measurement of com- 

 mensurable angles. 



The author's introduction of the incommensurable 

 base e, though only one of a number of different 

 possible methods, is even more satisfactory, no 

 previous knowledge being assumed regarding this 

 base, which is shown to make its existence felt as 

 soon as we attempt to differentiate a power of any 

 constant a witfi respect to its index, or to differentiate 

 the logarithm of the variable to any assumed base a. 



When integration is explained the author does not 

 waste too much time in discussing the methods of 

 integrating long and complicated expressions, but 

 proceeds very soon to the consideration of definite 

 integrals and of geometrical and mechanical illustra- 

 tions. 



" Volumes of revolution " only constitute a par- 

 ticular application of a general method of finding the 

 volume of a solid the sections of which parallel to a 

 fixed plane are circles, squares, triangles, or other 

 simple figures. The examples on pp. 159-161 should 

 make this point clear. 



Curvature, evolutes, properties of the cycloid, 

 moments of inertia, and attractions are discussed at 

 an early stage. So also are harmonic motion, resisted 

 motion, and damped oscillations. 



When infinite series arc introduced the student 

 should be ready for the satisfactory and sufficiently 

 rigorous treatment given, especially in connection 

 with convergence. 



In dealing with Taylor's theorem, the remainder is 

 carefully attended to, and specially mentioned in con- 

 nection with the binomial theorem. We should like 

 to have seen the remainder given as a definite integral, 

 but this can readily be supplied in lecture notes. 



Partial differentiation is fully discussed, and we 

 notice among the examples the familiar thermo- 

 dynamic application 



(/^ (fr rf^ ^ _ J 

 dzj dt dp 



There is a useful chapter on solid geometry which 

 introduces the notion of direction cosines, the ortho- 

 gonal propertv of confocal conicoids, and the oscu- 

 lating plane of twisted curves. 



Double and triple integrals are well discussed. The 

 artifices so commonly used in the older treatment for 

 calculating volumes and centres of gravity by means 

 of single integrations in particular cases had the 

 disadvantage, from which Prof. Osgood's treatment 

 is exempt, of failing to familiarise the student with 

 notions which he necessarily encounters in the study 

 of electricity and other branches of physics. 



Hyperbolic functions are not introduced until the end. 

 In the opinion of the reviewer they have figured far 

 too prominently in previous treatments of the calculus, 

 with the result that the student has been encouraged 

 to waste time in working out integrals in compli- 

 cated forms involving " sneezes " and " coughs," and 

 "sneeze and cough minus ones," which he cannot 

 interpret. It is to be presumed that the above words 

 represent the most natural equivalent in speaking of 

 th° new-fashioned hvperbolic notation, for to say 



C C 



