784 



NATURE 



[FEBRUARY 17, 192 1 



tively to systems of co-ordinates possessing arbi- 

 trary motion with respect to an inertial system. 

 With the aid of these special fields we should be 

 able to study the law which is satisfied in general 

 by gravitational fields. In this connection we shall 

 have to take account of the fact that the ponder- 

 able masses will be the determining factor in pro- 

 ducing the field, or, according to the fundamental 

 result of the special theory of relativity, the energy 

 density — a magnitude having the transformational 

 character of a tensor. 



On the other hand, considerations based on the ■ 

 metrical results of the special theory of relativity I 

 led to the result that Euclidean metrics can no | 

 longer be valid with respect to accelerated sys- 

 tems of co-ordinates. Although it retarded the 

 progress of the theory several years, this enor- 

 mous difficulty was mitigated by our knowledge 

 that Euclidean metrics holds for small domains. 

 As a consequence, the magnitude ds, which was , 

 physically defined in the special theory of rela- \ 

 tjvity hitherto, retained its significance also in the 

 general theory of relativity. But the co-ordinates 

 themselves lost their direct significance, and 

 degenerated simply into numbers with no physical 

 meaning, the sole purpose of which was the num- 

 bering of the space-time points. Thus in the '\ 

 general theory of relativity the co-ordinates per- i 

 form the same function as the Gaussian co-ordi- ■ 

 nates in the theory of surfaces. A necessary con- 

 sequence of the preceding is that in such general 

 co-ordinates the measurable magnitude ds must 

 be capable of representation in the form 



where the symbols gm are functions of the space- I 



time co-ordinates. From the above it also follows | 



that the nature of the space-time variation of the | 

 factors gm determines, on one hand the space- 



time metrics, and on the other the gravita- 

 tional field which governs the mechanical 

 behaviour of material points. 



The law of the gravitational field is determined 

 mainly by the following conditions : First, it shall 

 be valid for an arbitrary choice of the system of 

 co-ordinates ; secondly, it shall be determined by 

 the energy tensor of matter; and thirdly, it shall 

 contain no higher differential coeflScients of the 

 factors gm than the second, and must be linear in 

 these. In this way a law was obtained which, 

 although fundamentally different from Newton's 

 law, corresponded so exactly to the latter in the 

 deductions derivable from it that only very few 

 criteria were to be found on which the theory 

 could be decisively tested by experiment. 



The following are some of the important ques- 

 tions which are awaiting solution at the present 

 time. Are electrical and gravitational fields really 

 so different in character that there is no formal 

 unit to which they can be reduced? Do gravita- 

 tional fields play a part in the constitution of 

 matter, and is the continuum within the atomic 

 nucleus to be regarded as appreciably non- 

 Euclidean? A final question has reference to the 

 cosmological problem. Is inertia to be traced to 

 mutual action with distant masses? And con- 

 nected with the latter: Is the spatial extent of the 

 universe finite? It is here that my opinion differs 

 from that of Eddington. With Mach, I feel that 

 an affirmative answer is imperative, but for the 

 time being nothing can be proved. Not until a 

 dynamical investigation of the large systems of 

 fixed stars has bejsn performed from the point of 

 view of the limits of validity of the Newtonian 

 law of gravitation for immense regions of space 

 will it perhaps be possible to obtain eventually an 

 exact basis for the solution of this fascinating 

 question. 



Relativity: The Growth of an Idea. 



By E. Cunningham. 



SACCHERI, in his "Logica Demonstrativa," 

 published in 1697, ten years after Newton's 

 "Principia Mathematica," lays down a distinction 

 between real and nominal definitions which should 

 be kept in mind if we are to do justice to Newton. 

 Euclid defines a square as a four-sided figure the 

 sides of which are all equal, and the angles of 

 which are all right-angles. That is what he means 

 by the name "square." It is a nominal definition. 

 It remains to be shown that such a figure exists. 

 This is done in Book I., Prop. 46. The definition 

 then becomes real. Euclid is not guilty of the 

 error of presupposing the existence of the figure. 

 Newton prefixes to his principles of natural 

 philosophy certain definitions of absolute, true, 

 and mathematical space and time. The former 

 remains fixed and immovable ; the latter flows 

 uniformly on, without regard to material bodies. 

 He strives here against the imperfections of lan- 



NO. 2677, VOL. 106] 



guage to give words to the thought in the back 

 of his mind. The philosopher attacks him on 

 these definitions ; he has no right to presuppose 

 that these words correspond to any reality. What 

 then? Suppose these offending definitions re- 

 moved, or recognised as purely nominal. Then 

 the definitions of velocity, acceleration, mass, and 

 force are nominal, too, and the whole of Newton's 

 structure of dynamics is a paper scheme of words 

 and relations which may or may not correspond 

 to the world of sense. 



But that is exactly what it is. That is what all 

 scientific theory is, until experiment demonstrates 

 that the correspondence exists. The justification 

 of Newton's theory comes, not in the discovery 

 of a time that flows uniformly on, but in the fact 

 that the observed phenomena of the tides, of 

 planetary motion, and of mechanics in general do 

 fit on to his scheme. But the fit does not consist 



