masses, so differently defined, is a fact which is confirmed by experiments of very high accuracy (experiments of Eötvös), and classical mechanics offers no explanation for this equality. It is, however, clear that science is fully justified in assigning such a numerical equality only after this numerical equality is reduced to an equality of the real nature of the two concepts. That this object may actually be attained by an extension of the principle of relativity, follows from the following consideration. A little reflection will show that the theorem of the [Pg 60] equality of the inert and the gravitational mass is equivalent to the theorem that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, is It is only when there is numerical equality between the inert and gravitational mass that the acceleration is independent of the nature of the body. Let now be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to , free from acceleration. We shall also refer these masses to a system of co-ordinates ' uniformly accelerated with respect to . Relatively to ' all the masses have equal and parallel accelerations; with respect to ' they behave just as if a gravitational field were present and ' were unaccelerated. Overlooking for the present the question as to the "cause" of such a gravitational field, which will occupy us later, there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that ' is "at rest" and a gravitational field is present we may consider as equivalent to the conception that only is an "allowable" system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of co-ordinates, and ', we call the "principle of equivalence;" this principle is evidently intimately connected with the theorem of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to co-ordinate systems which are in [Pg 61] non-uniform motion relatively to each other. In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. For according to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to ) or under the combined action of inertia and gravitation (with respect to '). The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered in development must be considered as small in comparison.