A.— MATHEMATICAL AND PHYSICAL SCIENCES. 23 



above treatment there has been no mention either of length or of time : 

 neither measuring-rod nor clock has been introduced in any way. We 

 have left open the question whether the quadratic form does or does not 

 represent anything which can be given directly by measuring-rods and 

 clocks. For my own part I incline to think that the notions of length of 

 material bodies, and time of clocks, are really rather complex notions 

 which do not normally occur in the early chapters of axiomatic physics. 

 The results of the ether-drift experiments of D. C. Miller at Mount 

 Wilson in 1925, if confirmed, would seem to indicate that the geometry 

 which is based on rigid measuring-rods is actually different from the 

 geometry which is based on geodesies and light-rays. 



The actual laws of nature are most naturally derived, it seems to me, 

 from the Minimum Principle enunciated in 1915 by Hilbert, that ' all 

 physical happenings (gravitational, electrical, &c.) in the Universe are 

 determined by a scalar world-function i) being, in fact, such as to annul 

 the variation of the integral 



I II S*> dx u dx l dx 2 dx :i .' 



This principle is the grand culmination of the movement begun 2000 

 years ago by Hero of Alexandria with his discovery that reflected light 

 meets the mirror at a point such that the total path between the source 

 of light and the eye is the shortest possible. In the seventeenth century 

 Hero's theorem was generalised by Fermat into his ' Principle of Least 

 Time ' that ' Nature always acts by the shortest course,' which suffices 

 for the solution of all problems in geometrical optics. A hundred years 

 later this was further extended by Maupertuis, Euler, and Lagrange, into 

 a general principle of ' Least Action ' of dynamical systems, and in 1834 

 Hamilton formulated his famous Principle which was found to be capable 

 of reducing all the known laws of nature — gravitational, dynamical, and 

 electrical — to a representation as minimum-problems. 



Hilbert's minimum principle in general relativity is a direct application 

 of Hamilton's principle, in which the contribution made by gravitation 

 is the integral of the Riemann scalar curvature. Thus gravitation acts 

 so as to make the total amount of the curvature of space-time a minimum : 

 or as we may say, gravitation simply represents a continual effort of the 

 universe to straighten itself out. This is general relativity in a single 

 sentence. 



I have already explained that the curvature of space-time at any 

 point at any instant depends on the physical events that are taking place 

 there : in statical systems, where we can consider space of three dimensions 

 separately from time, the mean curvature (i.e. the sum of the three 

 principal curvatures) of the space at any point is proportional to the 

 energy-density at the point. Since, then, the curvature of space is wholly 

 governed by physical phenomena, the suggestion presents itself that the 

 metric of space-time may be determined wholly by the masses and energy 

 present in the universe, so that space-time cannot exist at all except in 

 so far as it is due to the existence of matter. This doctrine, which is 

 substantially due to Mach, was adopted in 1917 by Einstein, and has led 

 to some interesting developments. The point at issue may be illustrated 



