NATURE 



i.37 



THURSDAY, FEBRUARY 8, 1906. 



A NEW INTEGRAL CALCULUS. 



Integral Calculus for Beginners. By Alfred Lodge, 

 M.A. Pp. xiii + 203. (London: George Bell and 

 Sons, 1905.) Price 4s. 6d. 



•" "T^HIS is a sequel to the author's ' Differential 

 J- Calculus,' and is intended for students of 

 physics and mechanics who require a good working 

 knowledge of integration and its more simple appli- 

 cations." Such is the claim put forward by Prof. 

 Lodge in his preface. We naturally expect a book 

 in which simple useful applications figure more 

 prominently than lengthy theoretical investigations, 

 and in this we are not disappointed. Moreover, a 

 number of interesting features strike us as being 

 particularly good, although a few others are capable 

 of improvement. 



The first of the good features is the insertion of 

 the integration constant C in the elementary worked- 

 out examples on integration. Its omission frequently 

 leads a beginner astray. Another feature possessing 

 many advantages is that the chapter on rational 

 fractions is reduced to a minimum. The process of 

 integrating a rational fraction with a denominator 

 of high degree is not often required in actual prac- 

 tice. Moreover, the graph of such a fraction has 

 infinite branches corresponding to the real factors of 

 the denominator, so that unless the areas of these 

 infinite branches are carefully discussed, by intro- 

 ducing the notion of the " principal value " of a 

 definite integral, the result only enables us to find 

 the areas of limited portions of the curves, for which 

 approximate methods of quadrature would in many 

 cases suffice. On the other hand, the mode of intro- 

 ducing the connection between integration and 

 summation — a point on which Prof. Lodge rightly 

 lays special stress (§§ 2, 43, 48) — will probably be re- 

 garded by most readers as not so satisfactory as it 

 might be. Thus, to go no further, a relation on 

 p. 4 is stated to be true to the first order which on 

 p. 62 is shown to contain an error of the first order. 

 This is the greater pity as the investigation of § 48 

 would, with the addition of a couple of lines, contain 

 all that is necessary for a rigorous graphical proof, 

 far shorter than that given in § 2 ; we hope this 

 point will receive attention in future editions. 



In reading the sections dealing with Simpson's rule 

 and its modifications, one is surprised at the conserv- 

 atism that prevails in the retention of a formula 

 in which odd and even ordinates have unequal weight 

 — a conservatism quite independent of the present 

 book. When it is recollected that cutting off the 

 first and last strips of a curvilinear area reverses the 

 weights of the ordinates it will be seen easily that 

 the trapezoidal rule, modified by suitable corrections 

 for the two ends, may be made to give results quite 

 as accurate as those of Simpson's rule. 

 NO. 1893, v O L - 73] 



Under applications of the calculus, we find areas, 

 centres of mass, volumes and surfaces of revolu- 

 tion, and moments of inertia with especial reference 

 to plane areas and their centres of pressure. The 

 sections on differential equations contain what has 

 for some time past been regarded as a standard 

 elementary course on the subject, namely, the simpler 

 equations of the first order and linear equations with 

 constant coefficients. The study of the first integral 

 of the equation d 2 y/dx 2 = f(y) in connection with its 

 kinematic interpretation, and the discussion of small 

 oscillations in connection with the equation of 

 harmonic motion, are good features. Finally, the 

 chapter on the Gamma function has given Prof. 

 Lodge an opportunity of saying something he wanted 

 to say, and of saying it in his own way instead of 

 cutting or drying it down to the requirements of a 

 syllabus. It contains an interesting discussion 

 of the extension of the conception of a fac- 

 torial to negative and fractional arguments. It 

 is much to be hoped that this chapter will en- 

 courage other writers of text-books to launch out 

 into something new and original. This might to 

 some extent help to save English mathematical teach- 

 ing from sinking down to a uniform dead level of 

 mediocrity, reminding one of an open, barren veldt, 

 in which all the smaller hills have been levelled down 

 by the steam-roller of the examination and the 

 syllabus, and all high eminences have crumbled to 

 the ground as a result of the starvation salaries paid 

 to really competent mathematical teachers. 



G. H. Bryan. 



[Since the above w«s written I have had some corre- 

 spondence with Proi. Lodge quite independently of 

 1 h<_' present review. The treatment of the summation 

 of infinitesimals contemplated by him in the articles 

 criticised above may be stated more clearly somewhat 

 as follows: — Let y = f(x), and let x increase from a 

 to b by a series of increments dx. Then if dy denote 

 the corresponding increment of y, the sum of the 

 increments dy is exactly equal to f(b) — f(a). More- 

 over, the " exact differential " dy becomes equal to 

 the " differential product " f(x)dx when dx is in- 

 finitesimal, and under this condition we may put 

 /(&) — /( a )> equal also to the sum of the differential 

 products f(x)dx. Also in all practical applications 

 Prof. Lodge contends that what we really want is 

 the sum of the exact differentials dy rather than the 

 sum of the corresponding differential products 

 f'(x)dx. This contention I believe to be correct, and 

 if Prof. Lodge can re-write the articles once more 

 — for he says that he has already repeatedly re-written 

 them — and make it more clear that he is not merely 

 giving an inaccurate reproduction of Todhunter's 

 rigorous proof, but something quite different, his 

 treatment may be made one of the many valuable 

 features of his book. The method interests me 

 greatly, and appears to be of sufficient general 

 interest to justify the present explanatory note. — 

 G. H. B.] 



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