642 IX. ON EMPIRICAL AND VERBAL STRUCTURES
but which do not contain members that express a change in 'time', and hence do not take into account the transmission of electricity with finite velocity.1
The modern theories, as for instance, the Maxwell theory of electromag-netism, and the Einstein theory, are based on action by contact. These theories not only use the differential method, but they also introduce explicitly the finite velocity of propagation.
The invention of differential geometry with the recent contribution of Weyl, which we have already mentioned, transforms the geometry of Euclid from a language of action at a distance into contact geometry, or a language of indefinitely near action.
It should be mentioned perhaps that the riemannian differential geometry is more general than all the E geometries which preceded it, and includes them, as well as the E geometry, as special cases. Perhaps, as Weyl points out,2 the investigation of the famous fifth postulate, which was the beginning of E geometry, was accidental in importance and the main structural value of the E geometries lies precisely in the application of the differential methods to geometry which was originated by the great work of Riemann. This work, we see, has carried us from metaphysical action at a distance to a physical action by contact. In passing from the older mechanics to electromagnetic events a very striking analogy appears, which explains the finite velocity of propagation.
In mechanics, when we have waves in an elastic medium, the finite velocity of propagation is due to the delay which occurs due to the inertia of materials.
(d?s/dP), which represents the rate
Now inertia is determined by acceleration
(v=ds/dt), velocity itself being a rate of change of
of change of the velocity
displacement. We see that this retardation, or negative acceleration, is represented by a double differentiation.
Something analogous occurs in electromagnetic events. The rate of change
(de/dt) determines the magnetic field; and then the rate of
of the electric field
(dk/dt) of the latter determines the electric field at a neighbouring
point. The advance of the electric field from point to point is thus conditioned by two differentiations with respect to 'time', which is quite analogous to acceleration.
It is due to this double differentiation with respect to 'time' that the formulation of electromagnetic waves are structurally possible. If the partial effects were to occur without loss of 'time', no propagation of the electric waves would occur. The maxwellian 'field equations' not only express the above-mentioned relations, but introduce structurally the finite velocity of propagation which makes the Maxwell's electromagnetic theory structurally a contact theory.
The Einstein theory is also structurally a contact theory, and it may be said that it was originated by this contact tendency, and has carried it to the limit, as we shall see later. The gaussian theory of surfaces, whose extension to any number of dimensions was made by Riemann, also represents action by contact. This theory does not state the laws of surfaces on a large scale,