428 



NAl UKh 



[September 22, 192. 



it tnie, asks Berj^son, that the disk constitutes one 

 system ? It is only a system if we suppose it immol)ile ; 

 but in that case we are placing a real physicist on it, 

 and then on whatever point of the disk we immobilise 

 this real physicist with his real clock, we have the time 

 which is one and lived. In short, we have to choose. 

 Kit her the disk is thought of as rotating, and then 

 gravitation is resolved into inertia. This is how the 

 physicist represents it, and not as it is for him living 

 and conscious ; l)ut then the times measured by the 

 retarded clocks are represented times, and of these 

 there is infinity ; the disk will be a multiplicity of 

 systems. Or else this same rotating disk is thought of 

 as immobile. Then inertia at once becomes gravita- 

 tion. The real physicist now lives its time, and so 

 considered time is one and the same everywhere. 



The importance of the book from the point of view 

 of philosophy can scarcely be exaggerated. It accepts 

 frankly the paradox of relativity, goes behind it, and 

 exposes it. The retarding of clocks in systems acceler- 

 ated relatively to the observer's immobilised system 

 is shown to be a case in point of the relativity of 

 magnitudes. Just as the real dimensions of an object 

 are its spatial magnitudes for an observer immobilised 

 at that point of the universe at which the object is, 

 so the time t belonging to any system is the time 

 lived by an observer immobilised in that system. 

 For every immobilised observer the times and spaces 

 of other systems are infinitely variable, but these 

 variations are perspectives, represented not lived. 



H. WiLDON Carr, 



Projective Geometry. 



(i) Principles of Geometry. By Prof. H. F. Baker. 

 Vol. 2 : Plane Geometry, Conies, Circles, Non- 

 Euclidean Geometr}'. Pp. xv + 243. (Cambridge : 

 At the University Press, 1922.) 155. net. 



(2) Higher Geometry : An Introduction to Advanced 

 Methods in Analytic Geometry. By Prof. F. S. 

 Woods. Pp. x-i-423. (Boston and London : Ginn 

 and Co., 1922.) 22s. 6d. net. 



(3) Elements of Projective Geometry. By G. H. Ling, 

 G. Wentworth, and D. E. Smith. (Wentworth- 

 Smith Mathematical Series.) Pp. vi + i86. (Boston 

 and London : Ginn and Co., 1922.) 12s. 6d. net. 



(i) ^HRISTIAN VON STAUDT'S " Beitrage zur 

 V_x Geometric der Lage " was published so 

 long ago as 1857 ; about the year 187 1 Felix Klein 

 wrote a series of papers emphasising the fact that it 

 is possible to build up, on von Staudt's lines, the whole 

 of projective geometry, independently not only of 

 axioms of parallelism but also of the notions of dis- 

 NO. 2812, VOL. 112] 



tance and congruence. Yet it is astonishing how little 

 effect this di.scovery has liad upon English treatises 

 on projective geometry, which still, with very few- 

 exceptions, base iheir subject upon metrical geometr)', 

 and are content to prove purely projective properties 

 of conies by " projecting into a circle." There are, 

 it is true, Whitehead's two tracts on the " Axioms of 

 Projective Geometry " and *' Axioms of Descriptive 

 Geometry," but these, as their titles imply, deal only 

 with the logical preliminaries. There is also G. B. 

 Mathews' " Projective Geometry," which suffers 

 rather from undue compression and somewhat con* 

 fuses the i.ssue by talking about infinity so early as 

 Chapter II. ; and there is the important two- volume 

 treatise by Veblen and Young, which is certainly not 

 for the ordinary man. 



. There was obviously room for a lucid and logical 

 account of the whole of the more elementary parts 

 of geometry, conies, and quadrics and cubic surfaces, 

 developed from the projective point of view, and that 

 is what Prof. Baker's series on the " Principles of 

 Geometry," of which this is the second volume, aims 

 at supplying. Its publication, then, is an event of 

 the greatest importance. Prof. Baker believes that 

 much time " could be saved by following, from the 

 beginning, after an extensive study of diagrams and 

 models, the order of development adopted in this hook ; 

 and such a plan would make much less demand upon 

 the memory " than does the traditional treatment. 

 Is it not about time that some such course were adopted 

 for University students of scholarship standard in 

 their first year ? The ideas involved are, perhaps, 

 difficult, but not more so than those which the Cam- 

 bridge freshman is expected to assimilate from lectures 

 on analysis. 



In the first chapter a conic is defined in the usual 

 way as the locus of the intersection of corresponding 

 rays of two related pencils of lines in the same plane ; 

 next, Pascal's theorem and the theon.- of polarity are 

 developed ; and then there are forty most interesting 

 pages of examples of the application of the foregoing 

 theorems in various directions. The theor}- of out- 

 polar conies, Poncelet's theorem and Hamilton's 

 extension of Feuerbach's theorem may be mentioned. 

 Chapter II. summarises prof>erties of conies relative 

 to two points of reference, and gives a number of 

 results containing those usually develof>ed as con- 

 sequences of the notion of distance. The terms 

 current in metrical geometry, perpendicular, circle, 

 rectangular hyperbola, and so on, are used for the 

 sake of clearness, but have here, of course, a much 

 more general meaning, depending upon the choice 

 of the absolute points of reference. 



In the first volume of the series an algebraic 



