April io, 1902J 



NATURE 



seems to leave no difficulty unnoticed, the Scottish love 

 of pure logic being prominent throughout. 



It is not a work for the mere smatterer who aims only 

 at learning rules and practical processes, regardless of 

 the logical foundations ; and in its various applications 

 and illustrations it lays under contribution the sciences 

 of electricity, magnetism and heat. The student, how- 

 ever, who has no knowledge of these subjects is not 

 hindered by the introduction of them ; for they can be 

 passed over for a more convenient season, and they are 

 used only as examples which do not belong to the essence 

 of the treatise. 



The first hundred pages (five chapters) of the book are 

 devoted to several subjects which are generally dealt 

 with in other treatises — such as the elements of co- 

 ordinate geometry, the most prominent properties of the 

 conic sections, and the discussion of the limiting value of 



( I -I- - j when w = CO . In these chapters, also, there 



is a good deal about the graphs of functions, algebraic, 

 exponential and trigonometric. It is possible that Prof. 

 Gibson need not have included the portion on conic 

 sections, since students are certain to make a special 

 study of this elsewhere. 



The whole of chapter iv. is devoted to the exposition 

 of rates and limits, and it is minutely logical and 

 illuminating. We cannot imagine Prof. Gibson as 

 accepting that truly wonderful measure or definition of a 

 variable rate which we often find in works on dynamics 

 when their authors treat of a variable velocity : 



"the velocity, when variable, is measured by the distance 

 that would be gone over in a unit of time if the velocity 

 remained constant and equal to that which it is at the 

 particular instant " ! 



It is in chapter vi. that the special subject of the 

 treatise begins formally with the discussion of differenti- 

 ation ; and here Prof. Gibson adopts the good plan of 

 associating in the mind of the student, from the outset, 

 both the " derivative " and the function from which it 

 comes ; in other words, the student is learning his 

 integral as well as his differential calculus, and he is 

 exercised in the art of deducing the function which has 

 a given one for its derivative. There is here a section 

 specially devoted to the properties of hyperbolic func- 

 tions. The names of these functions leave much to be 

 desired, inasmuch as several of them suffer from the 

 fatal defect of being unpronounceable. Thus, how are 

 sinh X and tanh x to be pronounced ? And does not 

 cosh X suggest merriment ? A very simple change would 

 remove the first difficulty. If the prefix hy were put to 

 each of the trigonometrical functions, all the names 

 would be pronounceable and not too long. Thus^i'j/« .r, 

 hytan x, Scz., would at once be pronounceable and 

 indicate the hyperbolic nature of the functions. 



Prof. Gibson's paragraph (p. 157) on the conduction of 

 heat, together with the accompanying page of exercises, 

 is marked by his accustomed love of clearness, but it 

 may not be appreciated by the student of pure calculus 

 unless he happens to have studied previously compara- 

 tively advanced physics. It is, however, a good principle 

 to keep the purely mathematical subject as much as 

 NO. 1693, VOL. 65] 



possible in touch with physics. There is a dangerous 

 tendency visible in some writers to overload the special 

 subject in hand with applications and technicalities in 

 several other subjects, with the result that much of the 

 work is unappreciated by the student. In this respect, 

 however. Prof. Gibson is more judicious than many 

 other authors. The work abounds in warnings to the 

 student against possible (and probable) errors, and the 

 author never hesitates to give a useful collateral piece 

 of information, which it is desirable that the student 

 should have, whenever it can be simply and shortly 

 conveyed — as, for instance, at p. 175, when the different 

 ways in which a battery of a number of cells may be 

 arranged are discussed. This would be regarded by the 

 typical English author as quite outside the bargain which 

 he considers himself to have made with his reader — to 

 give him the truth, the whole truth, and nothing but the 

 truth, with no extra useful information, advice, or 

 warning. 



There is also plenty of the graphic method of illustra- 

 tion in the book in dealing both with the processes of 

 differentiation and with those of integration. 



In dealing with partial differentiation (chapter xi.), the 

 author has a few pages devoted to the elements of co- 

 ordinate geometry of three dimensions, which will, 

 probably, be found by the student who has advanced 

 thus far to be unnecessary. The equations of thermo- 

 dynamics and also Laplace's spherical harmonic equation 

 supply appropriate applications of the subject of the 

 chapter. 



This is followed by a very good chapter on the theory 

 of equations, in which the methods of approximate solu- 

 tion are well discussed, together with the reliance to be 

 placed on successive approximations. 



The successive reductions of the binomial integral 



\ x™~\a-^ bx")idx are dealt with rather too shortly 



at p. 295. It is strange that such an old work as Hymers's 

 " Integral Calculus" should have treated these integrals 

 in such a helpful, complete, and systematic manner, and 

 that Hymers' simple rules for reduction in any specified 

 case should have been quite neglected or overlooked by 

 subsequent writers. 



The mechanical method of integration by Amsler's 

 planimeter, together with the allied geometrical theorems 

 on the displacements of a line, is given in chapter xiv. 

 Near the end of the book there is a thorough discussion 

 of series, their convergency, divergency, &c. The last 

 chapter, xx., is devoted to a short discussion of linear 

 differential equations ; and it is to be hoped that this 

 chapter will be considerably lengthened in the next 

 edition with a good discussion of the symbolical method 

 of integration — a subject on which we should expect 

 Prof. Gibson's acute logic to be very illuminating. 



Finally, the work seems to be exceptionally free from 

 misprints. We notice the extraordinary letter " O " in 

 the centre of fig. 65, p. 315, which looks like a branch of 

 the curve, but is really the origin of coordinates. See a 

 similar defect in fig. 83, p. 362 ; and in line 6 from 

 the end of p. 379 the term p'n should be pTn. 



George M. Minchin. 



