SCIENCE PROGRESS 



RECENT ADVANCES IN SCIENCE 



PURE MATHEMATICS. By Dorothy M. Wrinch, D.Sc, Fellow of 

 Girton College, Cambridge, and Member of the Research Staff, University 

 College, London. 



In the current number of the Proceedings of the Royal Society 

 (A. 697, issued May 2), Prof. Eddmgton makes an important 

 extension of Einstein and Weyl's theories of the gravitational 

 field. The quantities g^,„ which occur in the Riemann geometry 

 become, according to Einstein, potentials in the gravitational 

 field. The extension of this interpretation, due to H. Weyl, 

 is less well known in this country, and is, in fact, very recent. 

 We may describe the magnitude which is represented by the 

 quantities g^^ as the " metric," and what Weyl showed was 

 that the metric can be made to include the four potentials of 

 the electromagnetic field, if a certain restriction made in the 

 Riemannian geometry — not in itself a natural type of restriction 

 — is removed. 



Prof. Eddington starts from the position that Weyl's 

 geometry is still restricted in an unnecessary way, and that 

 something very much more comprehensive can be found, and 

 begins with a question. What more general can be expected, 

 since gravitation enters the scheme of equations with Einstein, 

 when we discard the geometry of Euclid, and then, when we 

 adopt Weyl's, electromagnetic forces enter the scheme also ? 

 Prof. Eddington contends that the forces of non-Maxwellian 

 type which hold an electron together must be, in all probability, 

 the next to enter into a suitable further generalisation ; but he 

 does not contend that the present paper has necessarily done 

 this already. But at least an important restriction, of a purely 

 mathematical or geometrical type, now disappears from the 

 aspect of the theory of relativity which most concerns the 

 pure mathematician. 



Weyl's theory deals with gauge-systems, or systems of unit 

 standards of interval length, or distance in space-time, and is 

 based on the view that any particular system can only be used 

 for a special time and place. The gauge system of the universe 

 is, in fact, arbitrary, much in the same sense as the axes of co- 

 ordinates. But Weyl admitted a specific gauge-system of 

 fundamental importance in the geometrical relations of the 

 universe, about which more will be said later. 

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