June 14, 19 17] 



NATURE 



303 



■" straight lines " circles which cut a fixed ordinary 

 sphere orthogonally, and to regard all points out- 

 side this sphere either as non'^xistent or as 

 "images" of accessible points within the sphere. 

 The plane version of this is given bv Prof. 

 Carslaw (pp. 153—75) in the clearest manner con- 

 ceivable ; but he does not seem (in this book) to 

 have considered the analogous theory in solido. 

 There is no satisfactory theory of three-dimen- 

 sional non-Euclidean geometry, from an intui- 

 tional point of view, unless it gives us a clear 

 three-dimensional image in our ordinary space, 

 assuming, of course, that our powers of "intui- 

 tion " are confined to ordinary space. 



One of the great merits of Prof. Carslaw 's book 

 is that he gives a good account of the history of 

 the subject. In a certain sense Saccheri is the 

 great pioneer, and as much justice seems to be 

 done to him as the scope of the work permits. 

 The next is presumably Gauss, but, as usual, he 

 lost his claim by delay in publication. 



It should be noticed that theories of parallels 

 and theories of distance are, or mav be made, 

 essentially distinct. Thus, if we define, parallel 

 lines as those which cut the fundamental sphere 

 orthogonally in the same point, they may or may 

 not be continually at the same distance from 

 each other, according as we define the measure 

 of the distance of two parallel lines. 



Altogether, we think Prof. Carslaw 's book is 

 one of the best introductions to the subject that 

 we have seen. He ought to have given a refer- 

 ence to Mr. Somerville's bibliography. 



(3) Let Fj, Fo, . . . Fn be n assigned poly- 

 nomials in m variables ; then [Fj, Fg, . . . F„] 

 is defined to be the set of polynomials 

 X1F1 + X2F2+ . ._ . +X„F„, where X,, Xo, 

 . . . Xn are arbitrary polynomials in the same 

 variables. We also speak of [F,, Fo, . . . F„] 

 as a "modulus" or "module," this term being 

 due to Kronecker, who first emphasised the im- 

 portance of algebraical moduli. The importance 

 of arithmetical moduli, in the wider sense, was 

 discovered by Dedekind, and the whole theory of 

 algebraic integers in a given field may be reduced 

 to that of moduli contained in that field. The 

 aljrebraic theory is analogous, but much more 

 difficult, and Dr. Macaulay has done a real service 

 to mathematics by his original and critical tract. 

 Even men such as Kronecker and Lasker seem 

 to have made mistakes (in detail) in this peculiarly 

 difficult field of research. 



The originality and conscientiousness of this 

 tract are so great that the reader must forgive 

 the author for occasional obscurities. For in- 

 stance, the " array " on p. 7 is fundamental, but 

 we fear that many readers may fail to see 

 precisely what it means, and the " reverse " nota- 

 tion fp. 4) for Fj, F<, is not justified bv any remark 

 in the text. 



The main result, illustrated by well-chosen ex- 

 amples, is that whereas, in the arithmetical 

 theory, a modulus is uniquely expressible as a 

 product of prime moduli, and all moduli are, so 

 to speak, homogeneous in the sense that numbers 

 NO. 2485, VOL. 99] 



of the natural scale are homogeneous, the same 

 is not true of algebraic moduli in general, and 

 we have to introduce technical epithets to distin- 

 guish one kind of modulus from another. In fact, 

 it seems clear that the problem of classifying 

 algebraic moduli according to their essential pro- 

 perties is at least as complicated as the corre- 

 sponding problem in group-theory ; and if we 

 attend to the arithmetical nature of the coefficients 

 {e.g. if, instead of taking them as umbrae, we 

 take them as integers in a given finite field), addi- 

 tional difficulties present themselves. We hope 

 that Dr. Macaulay will continue his researches; 

 meanwhile this tract ought to be welcomed as 

 one of the most valuable in the series to which 

 it belongs. G. B. M. 



SOME ASPECTS OF TEXTILE 

 MANUFACTURE. 



Dyeing in Germany and America, with Notes on 

 Colour Production. By S. H. Higgins. 

 Second edition, rewritten and enlarged. Pp. 

 viii-fi43. (Manchester: At the. University 

 Press ;■ London : Longmans, Green, and Co., 

 1916.) Price 5s. net. 



THE first edition of this book was reviewed in 

 Nature for November 7, 1907. Since the 

 completion of h's work as a Travelling Scholar 

 under the Gartside Foundation scheme, the 

 author has gained much practical experience in 

 dye and bleach works, the results of which are 

 embodied in the new volume. This has added 

 considerably to its value, particularly in the 

 sections dealing with mercerisation and bleaching. 



In a new chapter the German and English 

 methods of manufacturing flannelettes are con- 

 trasted. This is of interest as raising the general 

 question of the relative efficiency of the British 

 and German methods of textile manufacture. 

 Generalisation on such a topic is, of course, open 

 to many pitfalls, and an adequate discussion of 

 the matter would be impossible in this review ; 

 but, broadly speaking, the British textile in- 

 dustry has developed along the lines of 

 specialisation of processes, whilst the German 

 specialises in products. This contrast is seen 

 very acutely in the worsted industry, in which 

 it is quite usual here for at least five distinct 

 firms to be concerned in the production of a piece 

 of cloth — the comber, the spinner, the weaver, the 

 dyer and finisher, and the merchant. Each of 

 these carries out its section of the work with the 

 maximum amount of skill and at the minimum cost, 

 but there is an obvious, and very real, danger 

 that the various processes are not sufficiently co- 

 ordinated. On the other hand, the usual German 

 practice is to carry out all processes in one works 

 and under one general control, when it is much 

 easier to correlate the various stages of manufac- 

 ture and subordinate each process to the final 

 result desired. The ultimate aim should be to 

 combine the advantages of both systems. 



The author of the book has also added a new 

 chapter on "Instruction in Dyeing," and gives it 



