To the critical discussion of Relativity which he so brilliantly presented at the Academy of Sciences M. Paul Painlevé added a valuable mathematical contribution of which the chief result is the following: It is possible to excogitate other laws of gravitation than that offered by Einstein, and all of them will fulfil the Einsteinian conditions. The learned French geometrician indicated several of these, especially one of which the formula differs considerably from that of Einstein, yet equally and precisely explains the motions of the planets, the displacement of the perihelion of Mercury, and the deviation of rays of light near the sun. This new formula corresponds to a space that is independent of time, and it does not involve the consequence that Einstein’s formula does in regard to the shifting toward the red of all the lines in the solar spectrum. The verification or non-verification of this consequence of Einstein’s equation, of which we pointed out the difficulties (perhaps insurmountable) in a previous chapter, thus acquires a new importance. It is a remarkable thing that many of the formulæ of gravitation given by M. Painlevé lead to the conclusion, differently from that of Einstein, that space remains Euclidean even near the sun, in the sense that measures are not necessarily contracted. All this light on the astronomical horizon seems like the dawn of a new era in which observations of unprecedented delicacy will provide tests that are calculated to give a more precise and less ambiguous form to the law of gravitation. There are great days—or, rather, great nights—in store for the astronomer. * * * * * As far as the principles are concerned, the controversy will go on. It must end in something like the following dialogue: _The Newtonian_: Do you admit that at a point in the universe that is far away from all material masses a moving object left to itself must follow a straight line? If so, you recognise the existence of privileged observers—those for whom the line is straight. For another observer the line is a parabola. Therefore his point of view is wrong. _The Relativist_: Yes, I grant it; but in point of fact there is no point in the universe where there is no influence of distant material masses. Therefore your moving object left to itself is a mere fiction, and I am not going to base science upon an unverifiable piece of imagination. The whole aim of the Relativist is to rid science of everything that has no experimental significance. As to the observer who sees the moving object in question describe a parabola, he will interpret his observation to mean that the object is in a gravitational field.