2 \6 



NA TURE 



[July i i, 1907 



the kinetic energy of the system is necessarily posi- 

 tive; still more, if the potential energy is a true 

 minimum. 



Prof. Bromwich has given an excellent account of 

 the analytical theory, with various geometrical and 

 dynamical illustrations, and he has added a very 

 useful bibliography. As he has pointed out himself, 

 he has selected Kronecker's method of proof of the 

 invariance of the factors of the discriminant of 

 S + ^T; and he has made no reference to the specially 

 arithmetical form of the problem, where the coeffici- 

 ents of the forms are integers, and the equations of 

 transformation have to be unitary and integral. For 

 this, the student will consult Frobenius, Hensel, and 

 H. J. Smith, whose memoirs, of course. Prof. Brom- 

 wich includes in his list of references. 



Dr. Whitehead's chapters deal with a subject 

 which, on the one hand, is more ancient than that 

 of Prof. Bromwich, but, on the other, has changed 

 its aspect recently in a much rnore remarkable way. 

 Dr. Whitehead is one of the company of sappers who 

 are reducing all the mathematical part of geometry 

 to a system of abstract logic applied to a minimum 

 number of undefinable entities, connected by a mini- 

 mum number of undefinable relations. Put in this 

 bald way, their work seems purely destructive and 

 hateful, but in reality it is not so. In the tract on 

 projective geometry it is shown how, with the help 

 of Dedekind's axiom, and those of order, it is possible 

 to make rigorous von Staudt's proof that all the 

 points on a line are either reached by harmonic con- 

 structions starting from three given points, or de- 

 finable as limiting points of a set of such points. 

 This leads to definitions of numerical cross-ratios 

 and of numerical homogeneous coordinates which arc 

 independent of any theory of distance or measure- 

 ment, a very remarkable and far-reaching result. It 

 is very encouraging to find that the magnificent 

 genius of von Staudt is gradually gaining the recog- 

 nition that it deserves ; the interval between him and 

 his predecessors is at least as great as that between 

 Apollonius and Steiner. 



By " descriptive " geometry Dr. Whitehead means 

 "any geometry in which two straight lines in a 

 plane do npt necessarily intersect." Besides the dis- 

 cussion of preliminary axioms and definitions, his 

 second tract falls into two principal parts ; the first 

 deals with the problem of enlarging a descriptive 

 space into a projective space (the simplest example 

 is that of adjoining the plane at infinity to Euclidean 

 space), the second with the theory of displacements 

 and measurement. The latter is based upon what is, 

 perhaps, the only satisfactory method — ^that of Sophus 

 Lie. The last chapter gives the formulae of metrical 

 geometry in the shape given to them by Cayley and 

 Laguerre, so that, neglecting a constant numerical 

 factor, a distance and an angle are each measured 

 by the logarithm of a cross-ratio. The cross-ratio, 

 of course, must be projectivcly defined, otherwise we 

 should be in a vicious circle, and it is in the avoid- 

 ance of this circle that the latest perfection of the 

 theorv consists. 



G. B. M. 



NO. 1967, VOL. 76] 



OUR BOOK SHELF. 

 Practical Agricultural Chemistry. By F. D. S. 

 Robertson. Pp. x + 210. (London: Bailli^re, Tin- 

 dall and Cox, 1907.) Price 7s. 6</. net. 

 In his introduction the author tells us that his book 

 is intended as a course of laboratory practice for the 

 use of students in agricultural chemistry. How much 

 time does he suppose such students can give to agri- 

 cultural chemistry to allow them to deal with subjects 

 like the valuation of tea and coffee, or the deter- 

 mination of hop resin and glycerin in beer? To what 

 class of readers is a paragraph like the follow- 

 ing of use? " The Bitter Used. This is neces- 

 sarilv a tedious operation, and for full particulars the 

 reader is referred to such books upon poisons as de- 

 scribe the processes of Dragendorff and others. The pre- 

 pared and concentrated beer is subjected to a series of 

 extractions with petroleum ether, benzene, chloroform, 

 and amvl alcohol, each of which is examined in turn." 

 Even in the more properly agricultural parts of the 

 book there is little evidence that the author possesses 

 any working acquaintance with his subject, e.g. the 

 chapter on the analvsis of soils is the merest skeleton, 

 possessing small reference to the methods in regular 

 use, and containing actual errors, such as the attempt 

 to estimate humus by solution in ammonia without a 

 preliminary treatment of the soil with acid. 



.^gain, in his description of the Reichert-Wollny 

 process for estimating volatile acids in butter, the 

 author savs nothing of the ofticial standardisation ot 

 the dimensions of the apparatus and other details, 

 which, however, must be followed if figures are to be 

 obtained comparable with those of other analysts, and 

 indeed are absolutely essential if the analyst is doing 

 public work. We cannot recommend Mr. Robertson's 

 book. 



.in Episode of Flatland, or Hoiv a Plane Folk dis- 

 covered the Third Dimension, to which is added An 

 Outline of the History of Unaea. By C. H. Hinton. 

 Pp 181. (London : Swan Sonnenschein and Co., Ltd. 

 These plane people live on the edge of a disc which 

 is their world. A third dimension exists only in their 

 mathematics. Their astronomers find that a catas- 

 trophe will certainly happen. One cranky philosopher 

 believes that there is a third dimension, and shows a 

 scared people how their world may be tilted and the 

 catastrophe averted. The author's characters act and 

 make love much like three-dimensional people, and 

 they talk of a higher dimension just as Mr. Hinton 

 would himself talk of a fourth dimension. There 

 never was an allegory, not even that of Bunyan, which 

 was consistent with itself for one chapter, but Mr. 

 Hinton 's is more inconsistent with itself than any 

 other allegory we have seen. J. P- 



The Bernese Oherland. Vol. iii. Dent de Morclcs to 

 the Gemmi. By H. Diibi. Pp. xxiv+136. 

 (London : T. Fisher Unwin, 1907.) Price los. 

 This conveniently arranged pocket-book is the most 

 recent addition to the " Climbers' Guides " edited by 

 .Sir Martin Conway and the Rev. W. .\. B. Coolidge. 

 The southern limit of the region described is the 

 Rhone \'allcy from Martigny to Leuk ; and the 

 northern is marked by the low passes leading from 

 the Ormonts-Lessus glen to Kandersteg by the heads 

 of the Grande Eau, the Sarine, the Simme, and the | 

 Kander valleys, which mark it off from the foot- ' 

 hills. The preface of the book directs attention to the 

 fact that the present is the jubilee year of the forma- 

 tion of the .'\lpine Club and of the ascents of the 

 Oldenhorn and the Wildstrubel, and we arc confident 

 that the increase in the number of climbers during 

 the last fiftv vears will ensure a wide popularity for 

 this workmanlike volume of " marching orders." 



