6i8 



NATURE 



[February 3, 19 16 



different publishers will announce their pro- 

 gramme as soon as they can. Sir George Green- 

 hill, as a veteran tractarian, ought to be person- 

 ally interested in these successors of his tract on 

 the calculus. 



The present series, so far, deals mainly with 

 practical affairs — fortunately not without regard 

 to theory ; four of the six tracts are said to be 

 "for the mathematical laboratory." It is to be 

 hoped that this term will not be hackneyed : it is 

 not very happy, in any case, for " a mathematical 

 laboratory " ought to include any place where 

 mathematical experiments are carried on, and 

 experiments are more important than methods of 

 computation. " Number-room " would be shorter 

 and more descriptive; but let that pass, and let 

 us go on to summarise the contents of this parcel. 



(i) This has the merit of appreciating the work 

 of Monge, in many ways the greatest master of 

 orthogonal descriptive geometry. The introduc- 

 tion is interesting, though rather amorphous ; for 

 instance, the article on " laboratory methods " 

 would have found a better place at the beginning 

 of chapter ii. This chapter deals with a good 

 number of the really fundamental problems on 

 lines and planes; chap. iii. on curved surfaces 

 and space-curves is of a more familiar and less 

 valuable type; chap. iv. is on perspective, and, 

 though very brief, gives some useful hints ; 

 chapter v. is on photogrammetry, and the most 

 novel of all. The weakest point in the tract is 

 that nothing is said about dealing with lines that 

 do not meet on the paper ; theoretically this is 

 unimportant, but in a drawing-office it is another 

 matter, and auxiliary vanishing points have to be 

 used frequently. 



{2) So far as we can judge, this seems to be 

 very well done. There is an outline of the theory 

 of finite differences, the classical formulae of 

 Simpson, Lagrange, Gauss, etc., and a number 

 of worked and unworked examples. In another 

 editioi-. it might be worth while to give the con- 

 stants required for Gauss's method (p. 88) in the 

 form of decimals as well as in that of surds. 



(3) It is difficult to say what will be the ultim- 

 ate physical way of stating the principle of re- 

 lativity, but, thanks chiefly to Minkowski, the 

 mathematical theory is assuming a simple in- 

 variant form. The four chapters of this tract 

 deal respectively with Einstein's formulae, trans- 

 formation of electromagnetic equations, applica- 

 tions to electron theory, and Minkowski's trans- 

 formation. Criticism of a physical nature must 

 be left to others; the purely mathematical treat- 

 ment seems to be all that can be desired. Un- 

 fortunately there are very few references^ — not 

 even one to Minkowski's papers. Without re- 

 NO. 2414, VOL. 96] 



ferences, a tract cannot perform its proper ser- 

 vice as an introduction to an important subject. 



(4) This is perhaps the most interesting of the 

 "laboratory" tetrad. The subject being limited, 

 the author is able to give a detailed account, with 

 good examples, of the way in which an irregular 

 periodic graph can be reduced to its harmonic 

 constituents. For work on electric oscillations 

 this tract ought to be very useful. There is also 

 a chapter on spherical harmonics. 



(5) This will appeal to astronomers and 

 ordnance survey people, and sailors. The main 

 novelty is the last chapter, on graphical methods, 

 which gives an account of the ingenious inven- 

 tions of D'Ocagne, Chauvenet, and others. There 

 are numerous examples, worked out to seven 

 places ; and others unsolved for the practice of 

 the reader. 



(6) This is an excellent tract on what is now 

 an extensive subject. The main points are very 

 clearly put ; room has even been found for an out- 

 line of non-Euclidean geometry, and the expres- 

 sion of co-ordinates of points on an algebraic curve 

 as one-valued functions. There is a bibliography 

 which seems to include most of the books and 

 papers of really first-rate importance ; and there 

 is a sufficient number of diagrams. English- 

 speaking students ought now, at any rate, to 

 appreciate Poincare's wonderful discoveries in 

 this field. 



Mathematicians owe a special debt to Prof. 

 Whittaker for the work he is doing in connection 

 with this series ; his encouragement and help are 

 acknowledged in handsome terms by several of 

 the authors. G. B. M. 



OVR BOOKSHELF. 

 Evolution. By J. A. S. Watson. Pp. vii+153. 



(London : T. C. and E. C. Jack, 191 5.} Price 



55. net. 

 The object of this work is apparently to provide, 

 in small compass and with copious illustrations, 

 an account of evolution from the lowest forms of 

 life to man. "The Evidence for Evolution " forms 

 the subject-matter of the first chapter, in which 

 well-chosen lines of argument are briefly laid down 

 and illustrated. The three succeeding chapters 

 treat respectively of " Unicellular and Multicellular 

 Animals," "The Worms and some of their Pos- 

 terity," and "The Early Vertebrates and the 

 Fishes." The fifth, "The Conquest of the Land," 

 leads on to the sixth and concluding chapter, "The 

 Mammals and Man." The illustrations, 146 in 

 number, are largely borrowed from German 

 sources, although the author is wrong in attri- 

 buting the bust of Homo primigenius (Fig. 145) 

 to a German sculptor. It was modelled by an 

 American lady, daughter of Alphaeus Hyatt, who 

 will be affectionately remembered by many British 



