September 22, 1923] 



NA TURE 



429 



symbolism was introduced to accompany the geo- 

 metrical reasoning ; in the third chapter of the present 

 volume this symbohsm is applied to the matters in 

 hand. The symbols employed consist of the iterative 

 ^symbols, and those derived from them as the irrational 

 lumbers of arithmetic are derived from the rational 

 lumbers, together with combinations of such symbols 

 [of the form x + iy, i denoting a new symbol such 

 that ?^= -I. In Chapter IV. it is shown that if we 

 ^introduce certain laws of order of succession, the 

 symbols are, in manipulation, indistinguishable from 

 16 complex numbers of ordinary analysis. The 

 listinction between real and imaginary elements is 

 then discussed. The last chapter deals with the notion 

 )f the interval of two points of a line, and the angular 

 iterval of two lines through a point, defined pro- 

 tectively in regard to an absolute conic, and leads up 

 Ito a discussion of non-Euclidean geometry. There 

 iioUow two important appendices dealing with certain 

 [configurations of points and lines, and, in particular, 

 irith the complete figure of Pascal's theorem, which 

 is best considered from four dimensions. 



Much of the matter contained in this work is, of 

 Icourse, familiar enough, though often presented from 

 |a new point of view ; in places, especially in Chapter 

 illl., extreme condensation of treatment makes difficult 

 [reading, but one can browse with pleasure and profit 

 |from almost anywhere in its pages, and surely that 

 lis a test of a good book. The printing and diagrams 

 fare excellent, as one would expect from the Cambridge 

 [University Press ; we would like to single out for 

 special mention the frontispiece, the Hexagrammum 

 |Mysticum, which any one who has tried to draw the 

 igure will recognise as simply marvellous. 

 (2) Going on from Prof. Baker's book to Prof. 

 |Woods', one feels a little confused. Prof. Woods is 

 jncerned with " advanced work in algebraic geo- 

 letry " and so does not worry about the foundations. 

 It it is rather difficult to determine what his founda- 

 ions are. One's first impression is that he defines 

 point (in a plane) by means of three numbers, real 

 )r complex, and then the line joining two points 

 i, yi as the set of points aCt + Ajyf (: = i, 2, 3), which 

 ps quite logical, though in Prof. Baker's opinion it 

 r* appears to beg one of the main, and most interesting, 

 Iquestions arising in the foundations of geometry," 

 [but then, on p. 28, Prof. Woods refers for the proof 

 of the theorem that any linear equation represents a 

 straight line " to any text-book on analytic geometry." 

 This criticism may appear pedantic, but the under- 

 lying idea of the book is, very properly, the group 

 concept, and the logical attitude is, surely, to begin 

 with the projective group and afterwards to consider 

 its sub-groups, the metrical group, and so on. Also 



NO. 2812, VOL. 1 12] 



discussions of non-Euclidean geometry (Chapter VII.) 

 seem a httle unsatisfactory if the idea of distance 

 has been present from the beginning. 



Prof. Woods' book, however, contains a very great 

 deal of interesting and valuable matter not elsewhere 

 accessible in any one volume. His plan is to study 

 different co-ordinate systems, based upon various 

 geometric elements and classified according to the 

 number of dimensions involved. Thus in three- 

 dimensional geometry he considers first the circles 

 of a plane and then point and plane co-ordinates ; 

 in four-dimensional geometry the lines of three- 

 dimensional space, spheres and four-dimensional point 

 space, in each case studying the meaning of the linear 

 and quadratic equations. Contact transformations, 

 tetracyclical and pentaspherical co-ordinates are also 

 dealt with. There are numerous exercises. The 

 author is to be congratulated on his determination 

 to " preserve the English idiom " by not using such 

 a phrase as "a line on a point," although this has 

 considerable authority behind it now and was intro- 

 duced, we believe, by an Englishman. The word 

 " nonminimum " would have looked better, surely, 

 with a hyphen ; the extra expense involved in printing 

 could have been saved by omitting the diaeresis in the 

 much more frequently occurring word " coordinate." 



(3) There is little to say about the third work under 

 review. It is a clearly set out, elementary school- 

 book on projective geometry on the ordinary lines, 

 built up upon a metric foundation and excluding any 

 consideration of imaginary elements. A desire to be 

 simple has led to some doubtful statements, ^.g. " the 

 greatest number of points of a figure that lie on a line 

 which is not entirely in the figure is called the order 

 of the figure." But the book may be recommended 

 as a good example of its class ; and there is an attractive 

 Greek alphabet on p. vi. The historical note at the 

 end is not so good as one would have expected in a 

 book with which Prof. D. E. Smith is associated. 



F. P. W. 



The Distribution of Mental Products. 



A Short History of the International Language Movement. 

 By Albert Leon Guerard. Pp. 268. (London : 

 T. Fisher Unwin, Ltd., 1922.) 215. net. 



PROBABLY no subject is more distasteful to the 

 average educated Englishman than the question 

 of an " artificial auxihary language." If he be a 

 literary scholar, he feels insulted ; if a man of business 

 and affairs, he is coldly indifferent and incredulous. 

 A few men of science may, perhaps, be mildly curious 

 and politely tolerant. If anything can awaken interest 

 and overcome prejudice, it will be this book written 



iM I 



