July 14, 192 1] 



NATURE 



617 



two bad diagrams are but minor blemishes on 

 this excellent guide for matriculation candidates. 



(2) Mr. Durell's book on the modern geometry 

 of the straight line and circle was intended as a 

 new edition of his "Course of Plane Geometry 

 for Advanced Students, Part I.," published in 

 1909. There have been, however, such consider- 

 able changes that the author has preferred to 

 issue the book under a new name. It contains 

 a pleasant and useful account of the geometry 

 required by scholarship candidates at public and 

 secondary schools, giving the usual work on recti- 

 linear figures, similar figures, harmonic ranges, 

 quadrilaterals and quadrangles, poles and polars, 

 inversion, etc. There is a chapter on vector geo- 

 metry with statical applications, while in dealing 

 with inversion and coaxial circles the author very 

 wisely makes use of analytical methods and nota- 

 tion. The treatment is sound, and the exercises 

 are numerous. 



{3) Many books exist dealing with analytical 

 conies, and presumably every author of such a 

 book aims at making the student interested in 

 this eminently important branch of pure mathe- 

 matics. Nevertheless, new books on the subject 

 will continue to be scanned with anxiety by 

 teachers of mathematics, because there can be no 

 doubt that many students find the subject difficult, 

 and the existing books scarcely afford them the 

 help they need. One must say at once that Dr. 

 Davison's book is no exception to the rule. It is 

 a clear and sound investigation of the ordinary 

 analytical theory of the straight line, circle, and 

 conic sections, carried out on the orthodox prin- 

 ciples and in the orthodox manner. The student 

 who is desirous of learning the subject, and 

 is intellectually and mathematically capable of fol- 

 lowing the argument, will no doubt study the book 

 with profit, for there are very many examples, 

 revision exercises, and a number of problem 

 papers on the subject. The book is well produced 

 and printed in the clear and interesting style that 

 we have learnt to associate with the Cambridge 

 University Press. 



For a possible second edition we would recom- 

 mend a few corrections and slight additions. In 

 dealing with the distance of a point from a 

 straight line, something should be said about the 

 somewhat difficult question of the sign of the 

 distance. There are tiioo tangents to a circle, 

 ellipse or hyperbola, having a given direction. 

 The author assumes that the equation of a circle 

 or conic is of the second degree; this assumption 

 is not good pedagogics in a course of the kind 

 he has produced. Is there any particular reason 

 for putting the equation of an ellipse in the form 

 b2A;2 + aV = «^&^? The classical form with a^, V- 

 NO. 2698, VOL. 107] 



I in the denominators looks simpler and is easier to 

 remember. The director circle of a hyperbola 

 appears to be subject to various vicissitudes, de- 

 pending upon whether the real axis is greater or 

 less than the imaginary axis ; this should be men- 

 tioned. There are several misprints; the worst 

 occurs where the co-ordinates of a point on a circle 

 are called (a cos ^, h sin ^). 



(4) " An Algebra for Engineering Students " 

 aims at giving all the knowledge of algebraic 

 principles and processes that engineers should 

 possess before commencing the calculus as 

 applied to engineering. As a particular class 

 of student is catered for, theoretical proof 

 is in places made to give way to illustration and 

 verification, and no one who has any ex- 

 perience of teaching mathematics to engineers 

 will quarrel with the authors on this ac- 

 count. The subject-matter is the ordinary ele- 

 mentary algebra up to and including quadratic 

 equations, and, in addition, indices, surds, 

 logarithms, arithmetical progressions, ratio and 

 variation are dealt with. Graphs and graphical 

 methods are discussed in a competent manner, 

 and the elementary use of the slide rule is ex- 

 plained. A few nomograms are included, but not 

 in such a way as to afford the reader any real in- 

 sight into their construction or use. The examples 

 are of a practical type, but one cannot help re- 

 marking that the worked example on p. 3 is as 

 artificial as any to be found in the " dry " theo- 

 retical books. 



(5) Dr. Silberstein is an acknowledged exponent 

 of vectorial methods, and anything that he writes 

 on vector algebra bears the stamp of authority. 

 The present book, although intended for optical 

 computers who wish to use vector methods in 

 optical computation, is equally useful to all who 

 wish to read a clear and easy account of the 

 elements of the subject. The ordinary processes 

 of addition and subtraction, and of scalar and 

 vector multiplication, with extensions, are dealt 

 with first ; then follows an account of linear vector 

 operators, leading up to dyads and dyadics. Hints 

 on the differentiation of vectors complete a useful 

 little volume. The division of the book into 

 chapters, and the addition of some examples of 

 a practical nature, would increase its value mani- 

 fold. 



(6) Computation and graphical methods of cal- 

 culation are assuming an increasing importance 

 in mathematical teaching, especially for such 

 students as are preparing to use their mathematics 

 in some industrial or vocational application. 

 Several universities and university colleges have 

 instituted mathematical laboratories, and a book 

 like Dr. Lipka's "Graphical and Mechanical Com- 



