which act upon them in the same gravitational field (gravitational mass). The equality of these two 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 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