150 



NJTURE 



[May 20, 1920 



newer Universities, and that they are still con- 

 vinced that Great Britain has only two institutions 

 worthy of the title. Have they any adequate con- 

 ception, for example, of the extent and capacities 

 for teaching- and research of the faculties and 

 departments of metallurgy, engineering, chem- 

 istry, and applied electricity at Sheffield, Leeds, 

 Manchester, and Liverpool, to mention only one 

 group of provincial Universities, and how it might 

 be possible, in connection with a properly organ- 

 ised training corps, to provide instruction for 

 cadets in those branches of specialised military 

 work for which a particular University had special 

 facilities and equipment, involving the application 

 of science to war? 



The Regulations governing the O.T.C. are dated 

 191 2, but we have learnt much since then, and it 

 is essential before these Regulations are revised 

 and re-issued that the Army Council should take 

 the L^niversities into its confidence, and, in con- 

 sultation with their representatives, produce a 

 scheme of training that shall conform to Univer- 

 sity practice and be within the range of University 

 capacity, while at the same time meeting the 

 requirements of the Army Council in its effort 

 to obtain suitably trained men to command the 

 various units of the Army of the future. 



Relativity and Geometry. 



The Foiindations of Einstein's Theory of Gravita- 

 tion. By Erwin Freundlich. Authorised English 

 translation by Henry L. Brose. Preface by 

 Albert Einstein. Introduction by Prof. H. H. 

 Turner. Pp. xvi-f-6i. (Cambridge: At the 

 L'niversity Press, 1920.) Price 55. net. 



PURELY mathematical workers have often found 

 occasion to remark on the prophetic vision 

 of Riemann. He possessed that special genius 

 which catches glimpses of truth, of no special 

 significance to a contemporary, which one day 

 are found to have an importance greater 

 even than the seer himself had dreamed. Certainly 

 this has proved so with much of Riemann 's work. 

 His famous Hahilitationsschrift, " On the Hypo- 

 theses which lie at the Bases of Geometry," was 

 presented to the faculty of philosophv at Got- 

 tingen in 1854, and, in an English translation by 

 Clifford, was brought to the notice of the British 

 public in the columns of Nature (vol. viii., 

 Nos. 183-84, pp. 14-17, 36, 37). It may be per- 

 missible to quote one or two prophetic phrases : 



" It seems that the empirical notions on which 

 the metrical determinations of space are founded, 

 the notion of a solid body and of a ray of light, 

 NO. 2638, VOL. 105] 



cease to be valid for the infinitely small. We are 

 therefore quite at liberty to suppose that the 

 metric relations of space in the infinitely small 

 do not conform to the hypotheses of geometry; 

 and we ought in fact to suppose it, if we can 

 thereby obtain a simpler explanation of 

 phenomena." 



It is worthy of note that Riemann never 

 speaks of space itself as being non-Euclidean. 

 He carefully refers always to the metric or 

 measured relations. The "ground" of these 

 metric relations is to be sought in the nature of 

 the reality underlying space. Is that reality a 

 discrete manifoldness, or is it continuous? If the 

 latter, then the "ground of the metric relations " 

 must be sought in the properties of that reality, 

 or, as he says, " in binding forces which act upon 

 it." Could anything be more prophetic of Ein- 

 stein's conception of gravitation? Then, as if to 

 anticipate the conservative and the scoffer of 

 to-day, he continues : 



" The answer to these questions can only be 

 got by starting from the conception of phenomena 

 which has hitherto been justified by experience, 

 and which Newton assumed as a foundation, and 

 by making in this conception the successive 

 changes required by facts which it cannot ex- 

 plain. Researches starting from general notions, 

 like the investigation we have just made, can, 

 only be useful in preventing this work from being 

 hampered by too narrow views, and progress in 

 knowledge of the interdependence of things from- 

 being checked by traditional prejudices." 



With this open mind, and the work of Gauss. 

 Lobatchevsky, and Bolyai on the geometry of 

 figures on curved surfaces to provoke thought^ 

 Riemann faces the possibility that the geo- 

 metry of three dimensions of actual material 

 bodies may not be so simple as Euclid's 

 system suggests. Geometry in the ordinary sense 

 is, in fact, eliminated ; the metrical relations of 

 bodies are "studied in abstract notions of quan- 

 tity ' ' ; the results of calculation may afterwards 

 be expressed in geometric form. Indeed, what 

 is meant by the "length of a line," or a "line 

 element," becomes far from clear from the geo- 

 metrical point of view. It is merely some quantity 

 which serves to distinguish one point from 

 another. The question is asked : What type of 

 magnitude may be constructed out of the quantities 

 that serve to define two special points in a 

 material body, which may conveniently be taken 

 as a measure of their distinctness one from the 

 other, first from a purely mathematical point of 

 view, but afterwards by an empirical test of its 

 abiding value. Riemann is led to use the general 

 quadratic differential form as the simplest possible 

 expression. 



