1.^,6 



NATURE 



[August 4, 1910 



MATHEMATICAL TEXT-BOOKS. 



(1) Elements of the Differential and Integral Calculus. 

 By Prof. A. E. H. Love, F.R.S. Pp. xiv+208. 

 (Cambridj^e : University Press, 1909.) Price 55. 



(2) Plane Trigonometry. An Elementary Text-book 

 for the Higher Classes of Secondary Scliools and for 

 Colleges. By Prof. H. S. Carslaw. Pp. .\viii + 293 

 + xi. (London : Macmillan and Co., Ltd., 1909.) 

 Price 4s. bd. 



(3) Elementary Projective Geometry. By A. G. 

 Pickford. Pp. xii + 256. (Cambridge : University 

 Press, 1909.) Price 4.5. 



(4) .4 First Course in .Analytical Geometry. Plane and 

 Solid, leitli Numerous Examples. Bv C. N. Sc'nmall. 

 Pp. viii + ;,i,S. (London: Blacliie and .Son, Ltd., 

 1909.) Price 65. net. 



(l)' I 'HIS booli is founded on lectures delivered by 

 -»■ Prof. Love at Oxford to students of applied 

 sciences. The object, both of the book and of the 

 lectures, is to encourage the study of the Calculus 

 among-st a wider circle than has been commonlv the 

 case hitherto. To quote from Prof. Love's preface : — 



"The principles of the Differential and Integral 

 Calculus ought to be counted as a part of the heritage 

 of every educated man or woman in the twentieth 

 century, no less than the Copernican system or the 

 Darwinian theory. In order to make a beginning 

 no previous knowledge of mathematics is needed be- 

 yond the most elementary notions of geometry, a little 

 algebra, including the law of indices, and the defini- 

 tions of the trigonometric functions." 



The more difficult theorems on limits which are 

 needed have been proved with considerable detail, but 

 the proofs are placed in appendices so as to avoid 

 discouraging the beginner. The most novel of these 

 is the very complete and satisfactory discussion of the 

 length of an arc of a circle (App. v.); but it seems 

 a pity that the same method was not carried on (in 

 App. vi.) to obtain the limits of sin a/a and tan a/a 

 (as a tends to zero) instead of following the more 

 intuitive method which is given in most text-books. 



With the aim and methods of Prof. Love's book 

 we are in hearty sympathy; our sole criticism would 

 refer to the difificult problem of treating the logarithmic 

 function, about which opinions will probably always 

 differ. In conclusion, we may refer to the welcome 

 practice of reducing definite integrals to numbers, in 

 suitable cases, instead of stopping at analytical for- 

 mulae; it is very instructive to beginners to compare 

 the result of such a calculation with the Talue found 

 from a graph, by estimating its area roughly. 



(2) The earlier part of this book is based upon 

 lectures delivered by Prof. Carslaw to first-year pass 

 classes, first at Glasgow and afterwards at Sydney. 

 Very good graphs of the trigonometrical functions, 

 direct (pp. 60-2) and inverse (pp. 205-8), including the 

 less familiar functions, such as sec x and sec '.-v, will 

 be found in the book; and a set of four-figure tables 

 is given at the end of the book. While welcoming 

 the spread of these tables, it seems a pity that more 

 modern tables (in tenths and hundredths of degrees) 

 were not used instead of Bottomley's forms. 



The second part of the book, on analytical trigono- 

 metry, is less completely handled, proofs of the more 

 NO. 2127, VOL. 84] 



difficult theorems (such as the power-series for sin x 

 and cos x, the product for sin x) being outlined only, 

 without any attempt at rigorous investigations. 



.\ few notes of difificulties may be added, in view of 

 later editions. In discussing the limit of sin 9/9 as » 

 tends to zero (S 92) it is assumed that the length of 

 a circular arc is less than the sum of the tangents 

 drawn at its extremities. .'Although this assumption 

 is natural enough in a jiurely intuitive discussion, yet, 

 after having given an arithmetic definition of length in 

 §S 89, 90, it would be only reasonable to deduce the 

 theorem in question from the definition. 



When discussing the solution of trigonometrical 

 equations, it seems strange that no use is made of the 

 substitution / = cos9 + i sin " ; and the more so since 

 all the examples solved in the book (§ Ij6) can be more 

 easily treated by this substitution than by any other. 

 In § 14S the convergence of the series !"«„ cos «^, 

 l^ih, sin n6 is treated graphically by the aid of a 

 spiral polvgon. This method is interesting on account 

 of its applications in physical optics, leading up to the 

 graphical treatment of diffraction-integrals by a smooth 

 spiral curve (such as Cornu's spiral) ; but it is not 

 quite obvious where the geometrical discussion intro- 

 duces the condition ««>««+]. It would be helpful to 

 give an algebraical treatment as well, following the 

 classical methods of Abel and Dirichlet, from which 

 the essential character of the condition <7«>a« + i is 

 at once evident. 



Prof. Carslaw 's book may be heartily recommended 

 to anyone wishing for a good knowledge of ele- 

 mentary trigonometry, together with a first introduc- 

 tion to more advanced methods. 



(3) It is not easy to estimate the effect which 

 a geometrical text-book will produce on a beginner ; 

 and we have had no opportunity of testing this particu- 

 lar book in actual teaching. But on a first reading the 

 arrangement adopted seems less satisfactory than in 

 several existing books : in a course on projective 

 geometry, the method of projection should take a 

 prominent part, and not be left until the last chapter 

 in the book. There is a tendency also to give a 

 variety of proofs of theorems which are really all 

 special cases of one general theorem (such as Pascal's 

 or Brianchon's), and this helps to make the book 

 longer, without making it any easier to read. 



Two details may perhaps be criticised : the idea of 

 involution is introduced very early, before defining 

 projective ranges on the same line ; but in actual teach- 

 ing it is generallv found easier to define involution 

 as a special type of homography. Also the pole-locus 

 of a line with respect to a system of four-point conies 

 is called the nine-point conic, instead of the eleven- 

 point conic; the latter term is now generally adopted, 

 and the reason for the change is not obvious. 



It seems to us that there is some need for a book on 

 projective geometry which makes occasional use of 

 analytical methods — in fact, a book written more on 

 the lines of the second half of Salmon's "Conies" — 

 and a really useful addition would be some plates of 

 drawings, on a fairly large scale, showing the actual 

 construction of conies by means of pencil and ruler, in 

 various ways. 



(4) There is but little to distinguish the present 



