66 



NATURE 



[March i8, 1920 



Vector Algebra" we have a very lucid exposition 

 of a subject somewhat removed from the ordinary 

 interests of the mathematical teacher or re- 

 searcher. Vectorial representation is a common 

 feature of many branches of physical science, and 

 the author's share in the encouragement of the 

 use of vectorial methods amply justifies his further 

 contributions to the discussion of the nature and 

 properties of vectors, whether as means of cal- 

 culation and research, or as illustrative of funda- 

 mental geometrical properties of space. The 

 present book aims at the construction of an 

 algebra of vectors, based solely on the axioms of 

 connection and of order. Only addition and sub- 

 traction of unlocalised vectors are dealt with in 

 the book itself ; in a subsequent paper in the 

 Philosophical Magazine the treatment is extended 

 so as to include multiplication and division. 



Opinions may differ as to the utility of the 

 system thus constructed ; there does not seem to 

 be any obvious application of the ideas to the 

 <liscovery of new results in pure mathematics or 

 in investigations of a physical character. But 

 the methods are elegant, and the exposition 

 is admirable. The proofs afforded of theorems 

 on the prbjective geometry of rectilinear figures 

 and conies amply repay the few pleasant hours 

 spent in reading the book and its continuation in 

 the above-mentioned paper. One may perhaps 

 question whether the book is really adapted for 

 *' beginners in geometry." 



It would have added to the value of the in- 

 vestigation if the book had been divided into 

 chapters and a reasonable number of examples 

 inserted for exercise in the methods developed. 

 The construction for scalar multiples of a vector 

 admits of some simplification. 



(3) This is a useful account of the application 

 of graphical methods to dynamical problems, 

 especially such as are of an engineering character. 

 The process of graphical integration is applied to 

 work, to space, to velocity and acceleration, and 

 to action, the auxiliary parabola being used for 

 the last. Polar diagrams are used with special 

 application to simple harmonic motion and to 

 combinations of simple harmonic motions, to cams, 

 etc. Velocities changing in direction are then 

 considered with applications to rotating bodies 

 and the turbine. Linkages and static forces in 

 mechanisms are followed by the elements of fly- 

 wheel design and the theory of the balancing of 

 rotating parts. There are many diagrams and 

 exercises. 



Though primarily intended for the engineer, the 



book contains much that should be incorporated 



into ordinary elementary courses on dynamics. 



Actual live problems with their practical solutions 



NO. 2629, VOL. 105] 



are far more valuable, pedagogically, than the 



numerous artificial exercises that are given in so 



many of the books written "for schools and 



colleges." 



Mr. Andrews should take more pains with his 



notation ; the needless use of x for ordinates must 



surely annoy the student. The statement 



„ area below curve 



mean effort =-, , j 



length of curve 



(p. 14) needs obvious correction. The definition 



of work on p. 34 applies only to a force constant 



in -magnitude and direction. On p. 47 simple 



harmonic motion is defined in the usual manner, 



but with the addition that the force acts in a 



direction opposite to the direction of motion of 



the body. This is not an oversight, for it is 



repeated on p. 64 ! 



(4) This is not a book for beginners, although 

 Dr. Davison follows the usual practice of indi- 

 cating what might be omitted on a first reading. 

 The whole book should be put aside on a first 

 reading of the subject, and a more suitable pre- 

 sentation selected for the purpose. 



But the student who has already mastered the 

 elements of the calculus, and understands the 

 meaning of a limit and the notion of differentia- 

 tion and integration, is ready for Dr. Davison's 

 book. It is brief, yet full. Part i. contains first 

 principles — i.e. differentiation, successive differ- 

 entiation, expansions, and indeterminate forms. 

 Part ii. deals with the applications to maxima 

 and minima, and to the theory of curves, including 

 curvature, asymptotes, singular points, curve- 

 tracing in Cartesian and in polar co-ordinates, 

 envelopes, evolutes, and pedals. There are 

 numerous examples, including sets of revision 

 exercises. Two excellent features are the problem 

 papers and the suggestions for a number of 

 mathematical essays. The form of the book is 

 pleasant, and the diagrams are well drawn and 

 reproduced. 



A few improvements are possible. Thus §§28 

 and 34 are ambiguously worded. There is a trap in 

 the formulae of § 61. In the chapter on polar 

 co-ordinates nothing is said about the ambiguity 

 inherent in polar equations, as mentioned in these 

 columns in a recent review of another book. 

 These are but a few blemishes in what is an 

 excellent production on well-known traditional 

 lines. 



(5) There is much excellent matter in Mr. 

 Milne's discussion of the analytical geometry of 

 the ^straight line and circle. The treatment is 

 lucid and such as will appeal to the beginner ; the 

 subject-matter is very well chosen, and pre- 

 sented in abundant detail with numerous illus- 

 trative exercises, both worked and unworked. 



