Notices respecting New Books. 131 



direct paths, which intersect, are each intersected by a third 

 direct path in two separate points, any unterminated direct path 

 extending in the same direction as the last one, which intersects 

 one of the two former, will also intersect the other." There are 

 besides three axioms : on the possible transference of geometrical 

 figures, the possible extension of a straight line and the three-way 

 extension of space, but the implicit definition we have quoted 

 above in eoctenso is the crucial point in the theory. Our readers 

 are now furnished with the key to Mr. Dixon's method, and, 

 whether they agree with him or not, they will find much of 

 interest in this elaboration of his argument. In Part i. he writes 

 on the Logical Status of the Science of Geometry. In Part ii. we 

 have a subjective Theory of Geometry deduced from the two 

 fundamental Concepts, Position and Direction. Herein the 

 student is carried through the equivalent of great part of Euclid's 

 books i. and xi. with the exceptions of those propositions in book i. 

 which deal with plane areas. Part iii. dilates on the applicability 

 of the foregoing subjective geometry to the geometry of material 

 space. This contains some very interesting remarks upon what 

 Mr. Dixon prefers to call " Geometry of four independent direc- 

 tions " in place of the common " Geometry of four dimensions," 

 and an account of a method of forming diagrams in material space 

 which are " orthogonal projections of the true figures" (p. 101). 

 The printing is very well done and the figures are good and clear. 

 We have detected only three typographical errors, one being 

 " Eaucoult's " pendula (p. 22) ; and on p. 53 occurs the funny 

 expression M a cornery path." 



Explication de V epoch Quaternaire sans Hypotheses. 

 Par H. Hermite. Neuchatel, Attinger Ereres, Editeurs. 1891. 



To anyone who has speculated on the causes of geological epochs 

 the title of this work is calculated to raise an incredulous smile. 

 That a period the characteristics of which are themselves more or 

 less matters of surmise can be explained without recourse to hypo- 

 theses seems impossible. A careful perusal has not shown us that 

 M. Hermite performs his promise, for a greater collection of 

 unsupported assumptions it would be difficult to find put together. 



The author commences by attempting to show that the figure 

 of the earth is not an ellipsoid of revolution on the assumption of 

 homogeneity, and that Newton's calculations give a larger equatorial 

 diameter compared to the polar diameter than measurement has 

 proved to exist. 



By what he calls " the principle of the Equilibrium of the Seas," 

 he thinks that this difference between theory and observation can 

 be explained. We confess our inability to follow the reasoning by 

 which he arrives at this conclusion ; but it is connected with the 

 greater salinity, and, as he thinks, consequent density of the seas 

 at the Equator than at the Poles. But he takes no account of the 



K2 



