20 SECTIONAL ADDRESSES. 



to be an imperfection, and it was felt that sooner or later everything, 

 including electro-magnetism, would be re-interpreted and represented in 

 some way as consequences of the pure geometry of space and time. In 

 1918 Weyl proposed to effect this by rebuilding geometry once more 

 on a new foundation, which we must now examine. 



Weyl fixed attention in the first place on the ' light-cone,' or aggregate 

 of directions issuing from a world-point P, in which light-signals can go 

 out from it. The light-cone separates those world-points which can be 

 affected by happenings at P, from those points whose happenings can 

 affect P ; it, so to speak, separates past from future, and therefore lies at 

 the basis of physics. Now the light-cone is represented by the equation 

 ds 2 =0, where ds is the element of proper time, and Weyl argued that 

 this equation, rather than the quantity ds 2 itself, must be taken as the 

 starting-point of the subject; in other words, it is the ratios of the ten co- 

 efficients g, m in ds 2 , and not the actual values of these coefficients, which 

 are to be taken as determined by our most fundamental physical ex- 

 periences. Following up this principle, he devised a geometry more 

 general than the Riemannian geometry which had been adopted by 

 Einstein : instead of being specified, like the Riemannian geometry, by a 

 single quadratic differential form 



^ 9 m dx p dx a 



it is specified by a quadratic differential form 



■*■* g pq dx p ax q 



and a linear differential form 2<p p dx p together. The coefficients g m 



of the quadratic form can be interpreted, as in Einstein's theory, as the 

 potentials of gravitation, while the four coefficients (p p of the linear form 

 can be interpreted as the scalar-potential and the three components of the 

 vector-potential in Maxwell's electromagnetic theory. Thus Weyl suc- 

 ceeded in exhibiting both gravitation and electricity as effects of the 

 metric of the world. 



The enlargement of geometrical ideas thus achieved was soon followed 

 by still wider extensions of the same character, due to Eddington, Schou- 

 ten, Wirtinger, and others. From the point of view of the geometer, 

 they constituted striking and valuable advances in his subject, and they 

 seemed to offer an attractive prospect to the physicist of combining the 

 whole of our knowledge of the material universe into a single unified 

 theory. The working out of the various possible alternative schemes for 

 identifying these more general geometries with physics has been the chief 

 occupation of relativists during the last nine years. Many ingenious 

 proposals and adaptations have been published, and more than one author 

 has triumphantly announced that at last the problem has been solved. 

 But I do not think that any of the theories can be regarded as satisfactory, 

 and within the last year or two a note of doubt has been perceptible ; 

 were we after all on the right track 1 At last Einstein himself 3 has made 



8 Math Ann. 97 (1926), p. 99. 



