186 SCIENCE PROGRESS 



and Sxt in pre-Relativity Theory, an assumption now dropped. 

 This conception enabled Einstein to generaUse dynamical 

 theory, so as to cover the relativity of mass and force to the 

 frame of reference of the observer. Of course, if one chooses to 

 employ a frame of reference in rotational or accelerated motion 

 relative to any of those dealt with, the expression for an element 

 of separation becomes the root of a more general quadratic 

 expression in Bx, viz, S gap dx^ dxp. (In this a and yS are summed 

 independently over i, 2, 3, 4, and there are apparently sixteen 

 terms; these, however, compress to ten by reason of the equalities 

 gap = gpa- The ^-coefficients are, in general, functions of the 

 co-ordinates depending on the relations of the new frame of 

 reference to the old.) This change in the mathematical form of 

 the expression for an element of separation is paralleled physi- 

 cally by the appearance in the new frame of mechanical effects 

 absolutely similar to the effects of introducing a field of gravita- 

 tion. Einstein's Principle of Equivalence assumed that the 

 similarity extended to all physical processes, in particular the 

 propagation of light. Of course such gravitational fields were 

 purely " fictitious " or " geometrical." The mathematical 

 procedure in choosing such a frame is similar to that employed 

 in two-dimensional geometry, when, for one reason or another, we 

 prefer to use polar, elliptical, or some other form of curvilinear 

 co-ordinates instead of Cartesian. It is always possible to revert 

 to the simpler procedure if necessary. Now, in two-dimensional 

 geometry, the possibility of using Cartesian co-ordinates depends 

 on the axioms we start from and the nature of the surface 

 treated. In surfaces which can be developed into a plane (such 

 as the cylinder or cone), co-ordinate systems can (if Euclid's 

 axioms are excepted) be laid down which permit of the distance 

 between two neighbouring points being expressed in the simpler 

 form ; but this is not so on surfaces which cannot be so developed 

 (sphere, ellipsoid, etc.). No co-ordinate system based on a 

 duplicate meshwork of curves on the surface can be found 

 which permits distance to be expressed as ViBxj' + Bx^^) 

 everywhere on the surface. (It is possible in the neighbourhood 

 of an assigned point.) Riemann had already suggested that 

 similar considerations might actually be true of the three- 

 dimensional manifold consisting of the points of an observer's 

 space. Einstein seized this suggestion and applied it to the 

 geometry of the " World," the four-dimensional manifold con- 

 sisting of the " events " in an observer's space-time frame of 

 reference. His geometrical assumption was that the ten g- 

 coefficients could not in general be transformed by any possible 

 change of the co-ordinates to the simple values g^^^ — ^„ — ^sa 

 " ~ i>^44 "" + i> and the rest zero (except for a minute region 

 of space-time neighbouring to an assigned event). His physical 



