PURE MATHEMATICS 175 



said that Prof. Eddington sees the possibUity of elucidating 

 the forces which bind the component parts of an electron 

 together — a point raised by Sir Oliver Lodge and others in 

 every recent discussion of the ultimate structure of matter, 

 and a point which, as it is becoming clear, rests with the pure 

 mathematician in the last resort — and mathematical readers 

 of Prof. Eddington 's paper cannot but feel that this at least 

 is proved, and that the line of generalisation of Einstein's and 

 Weyl's theories gives more hope than any ad hoc theories 

 developed from the physical side. Prof. Eddington clearly 

 has hopes that the ultimate mathematical meaning of the 

 quantum may emerge from a yet more extensive generalisation. 

 It is possible, but conviction is not immediate, though at least 

 it seems more likely to emerge in this way than in any ad hoc 

 manner. 



The work of Riemann has perhaps never been appreciated 

 at its true value, even by the pure mathematician, and especially 

 on the geometrical side. The debt of Einstein to Riemann is 

 under-estimated by all but Einstein. The small volume of 

 Riemann 's collected works has inspired an amount of later work 

 of which it is difficult to over-estimate the importance. Prof. 

 Eddington 's paper goes far towards redressing this wrong. 



A critical point of the new treatment arises in the intro- 

 duction of the natural gauge of the world, which is determined 

 by the aid of material or optical appliances which measure space 

 and time. As Prof. Eddington points out, any such ap- 

 paratus is part of the world we measure, so that we introduce 

 the assumption that the world is self-gauging. In other words, 

 the tensor of type g^^^ is not really extraneous and arbitrary, 

 but is a tensor already contained in some way in the world 

 geometry. Only one such tensor is found to be available, and 

 it serves to define " natural length," and to clarify and make 

 precise some of the implications of the older theory otherwise 

 difficult to understand. 



Physical space-time can be identified after the gauging 

 equation appears, as what the author calls an alias of the law 

 of gravitation. The next step is to identify " things," which 

 have three types of attributes requiring a geometrical tensor 

 with these properties inherent in it hy virtue of mathematical 

 identities. The author succeeds in identifying readily the 

 energy, momentum, and stress in a region by an energy tensor, 

 electromagnetic potential and force by another, and the electric 

 charge and current vector by a third, showing their correspond- 

 ence with Maxwell's equations and the law of gravitation 

 together. These identifications appear to be unique, but are 

 not definitely proved to be so. 



A vector in general alters length on describing a circuit, 



