October 20, 1923] 



NATURE 



583 



he persist in acquainting himself with the first chapters, 

 he would be delighted with the precision of language 

 and thought and with the homeliness of the contents ; 

 indeed, it may be said that the number of readers of 

 this beautifully executed work will be a fair measure 

 of the Greek spirit among our geometers of the present 

 day. To barbarians it will seem to cut right across the 

 course of modern geometry with an independence 

 which shows itself in nomenclature and notation, in 

 absence of references, and most of all in the limitations 

 which the author has placed upon himself in the 

 selection of his material. This is partly accounted for 

 by the fact that Prof. Neville is avowedly a disciple of 

 Mr. Russell, whose well-known aphorisms are scattered 

 over the book, and it is scarcely to be expected that 

 a subject written in the form which modem logic 

 demands should develop itself along lines which appear 

 fundamental in discovery. 



The earlier part of the book is an introduction to 

 vector analysis followed by an excellent discussion of 

 Cartesian axes and vector frames. Perhaps it should 

 be mentioned that " anisotropic " space does not imply 

 any " medium " theory — Prof. Neville's words have no 

 implications but are equivalents of the symbols of the 

 Principia. Anisotropic space is flat space of three 

 dimensions which does not touch the absolute in four 

 dimensions. The second half of the book is devoted 

 to the construction of algebraic space out of those 

 properties of vectors and points which were suggested 

 as significant in the earlier chapters. This is a most 

 valuable contribution, and we confine our attention 

 to it. 



Geometers say that a circle is cut by a line of its 

 plane at two points, real or imaginary. There are 

 great advantages in doing so, but if asked for reasons 

 they content themselves with explaining that this is 

 a conventional way of talking and that imaginary 

 " points " merely stand for certain pairs of imaginary 

 numbers. How they stand for them is not clear. To 

 find a logical basis one of two methods may be adopted. 

 The first, that of von Staudt, consists of replacing the 

 imaginary points by an equivalent real elliptic involu- 

 tion : any construction which has been algebraically 

 thought out by the use of imaginaries at intermediate 

 steps can be replaced by a more elaborate real con- 

 struction which can be actually carried out by pencil 

 on paper. This method has the beauty of being 

 geometrically relevant. 



The second plan, which is that adopted in this book, 

 has the logical advantage of allowing the real points 

 no special privilege. Algebraic complex space is built 

 up from such fundamental relations as hold between 

 vectors and vectors and between vectors and points 

 in ordinary geometry ; in other words, we remove the 



NO. 2816, VOL. I 12] 



loose convention or postulate used by the " teacher 

 in a hurry," and carefully devise a unique construct 

 within which all the required operations can be carried 

 out. This however has obvious geometrical dis- 

 advantages, as it involves an embarrassing array of 

 relations in which we have no reason to be interested. 



It may be doubted if there can be any true inter- 

 pretation of a space in the modern sense which does not 

 deal with the group of transformations for which it is 

 the accepted field. The ordinary geometry, as intro- 

 duced by Prof. Neville, involves lines, directions, 

 distances, all accepted from experience ; no such 

 geometry can dispense with the idea of motion unless 

 it has first laid down a series of postulates such as he 

 dislikes. This geometry, which he repeatedly refers- 

 to as " kinematical," cannot be any more logical, and 

 is far less vivid when all reference to motion is excluded. 

 His original space is the field of such transformations, 

 and as such is really trivial in the complex domain. 

 " To use geometrical language," writes Russell, "... 

 is only a convenient help to the imagination." Prof. 

 Neville's geometry reminds us of the notorious 

 Euclidean point when it has moved, for what help to 

 the imagination can come from a discussion of lines 

 perpendicular to themselves or the bizarre metrical 

 geometry of the isotropic plane ? Just as the logician 

 objects to Staudt's method as a search for complex 

 space within real space, we fear most geometers will 

 not pleasantly accept the task of picking out projective 

 properties from the mass of metrical relations which. 

 Prof. Neville's method imposes on them. 



George Westinghouse. 



A Life of George Westinghouse. By Dr. Henry G. 

 Prout. (For a Committee of the American Society 

 of Mechanical Engineers.) Pp. xiii -t- 375. (London : 

 Benn Bros. Ltd., 1922.) 185. net. 



THE American Society of Mechanical Engineers has 

 undertaken to issue volumes devoted to the lives 

 of some of its great men; and the supervision of the work 

 has been entrusted to a committee of the Society. 

 The first book of the series was a special edition of the 

 autobiography of John Fritz, honorary member and 

 past president. The present volume is the second of 

 the series. 



In the almost complete absence of personal records, 

 letters, notes, and other material from which a biography 

 could be prepared, the committee has had to draw upon 

 the memories and impressions of those men still living 

 who were nearest to Westinghouse, and the editor's 

 duty has been to co-ordinate their contributions. 

 This method of preparing a biography has both its 



