GRAVITATION. - I. General considerations.
In the study of natural phenomena, over the centuries, the opportunity has repeatedly arisen to admit the hypothesis that material bodies may, or indeed must, mutually attract (and sometimes possibly also repel) each other even when not in contact or even when significantly distant. In assertions to this effect, which Empedocles, Anaxagoras, and later Leucippus and Democritus posited as the foundation of their cosmogonies, are recalled in the works of Plato and Aristotle, and in these also interestingly linked to the phenomena of falling bodies near the Earth’s surface. In this field, there still remained, at least latently, the admission of a mysterious cohesion or attraction until the time of Galileo, who, with his simultaneous discoveries of the fundamental law of dynamics and the law of falling bodies, was finally able to clarify its fundamental character in a much clearer manner. It was then proven that the Earth, considered globally as a whole, near its surface, exerts on all material bodies a force capable of imparting to them an equal acceleration (i.e., an equal increase in their velocity, approximately equal to 980 centimeters per second, an acceleration now conventionally denoted by the letter g). It follows immediately from Galileo’s law of dynamics that the force of attraction on each body (i.e., its weight) must necessarily be proportional to its mass.
Yet despite the extraordinary importance of this Galilean result, which, far more than his celebrated dialogues on the *Maxims of the Systems*, was what “to the Englishman who so much saw and first cleared the paths of the firmament,” one does not yet encounter the notion and the universal law of gravitation. The precise formulation of that notion and the complete statement of its law were chiefly the work of Newton.
II. NEWTON’S DEDUCTION.
First, Newton’s third law of dynamics, by affirming the existence, by reaction, of an equal attraction of the considered body upon the Earth, established a complete reciprocity between them. One could therefore immediately extend Galileo’s result by affirming that “between any two bodies of masses m and m’ there always occurs an attraction proportional to the product of those masses.” Yet the decisive step toward the conception of the universality of the phenomenon of gravitation was taken by Newton only when, to his mind occupied with the search for the cause by which the Moon accompanies the Earth in its motion through space (in 1666), the idea flashed upon him that it must consist in an attractive force mutually binding Earth and Moon, of the same origin and nature as that which, near the Earth’s surface, binds Earth and any material body, determining its weight and, if free, its fall with acceleration g.
On the basis of the Galilean experimental criterion, the plausibility of these hypotheses could be demonstrated only by the full agreement of all their consequences with reality. The Moon, by the mere fact of moving around the Earth in an approximately circular orbit, is necessarily subjected to a centripetal acceleration, which kinematics, already sufficiently developed at that time, allowed one to calculate without difficulty from its distance D from the Earth’s center and its period of revolution T—two quantities already known with sufficient accuracy. This centripetal acceleration turned out to be equal to 0.272 centimeters per second, i.e., 3600 times weaker than that on material bodies at the Earth’s surface. For the plausibility of Newton’s hypothesis, the Earth’s attractive force on any body situated at distance D from its center would have to have that same value.
At this point a notable difficulty arose: the fundamental principles of Galileo and Newton had allowed, given the existence of an attractive force between two masses, to affirm that it must be proportional to their product; yet they said nothing about the dependence of such a force on their distance. But here Newton generally supplied two further hypotheses, which would later have to be fully consistent a posteriori with the original hypothesis. At that time it was a widely held opinion that the intensity of phenomena that could be considered as symmetrically emanating from a center (such as, e.g., light from a point source) must diminish in proportion to the spherical surface it reached, and thus in proportion to the square of the distance from that center. In that case, even if the emanation did not occur solely from the center but from spherical layers symmetrically around it, the effect at an external point had to be equal to what would have occurred if all the emanation came from that center. Thus, assuming the Earth’s masses to be symmetrically distributed around the center, Newton was able to frame his calculations as if the entire mass of the Earth were condensed at its central point. Then, denoting by R the Earth’s radius, and given the accepted law regarding the effect of distance, the forces and hence the accelerations at distances R and D must satisfy the condition F at R : F at D = D² : R². But since the acceleration at the Earth’s surface a at R = 980 and that at the distance of the Moon a at D = 0.272 centimeters per second, and thus a at R : a at D = 3600, it follows that the ratio D/R ought to be equal to 60, i.e., that the distance of the Moon should be equal to 60 R. But at that time, owing to the inexact knowledge of R, it was believed to be only 56 R. Newton’s hypothesis therefore did not accord sufficiently with apparent reality.
But in 1683, the new and more rigorous measurement of a meridian arc carried out by Picard was made known. On the basis of this, the Earth’s radius turned out to be appreciably smaller than had hitherto been believed, precisely about 1/60 of the distance of the Moon. Naturally Newton was delighted to be able to reconsider his hypothesis of 1666, and in 1687, after four years of reflection and work, he published his immortal work, in which, having systematized the principles of Galileo’s and his own dynamics, he broadly applied the classical formula of gravitational forces
\[
F = G \cdot \frac{M_1 M_2}{D^2},
\]
now valid for any mass in the universe, at any distance. The universal constant G was determined through successive delicate experiments, increasingly precise, as G = 6.66 × 10⁻⁸ cm³/sec²·gr⁻¹.
Yet the unlimited universality of the notion of gravitation and of its law can only be adequately grasped when it is considered elementarily, i.e., as referring to every particle of matter in the universe conceived as attracting and as attracted. The phenomena that manifest themselves are nothing other than the resultants of these innumerable elementary phenomena.
III. MERITS AND DEFECTS OF THE NEWTONIAN THEORY.
As regards the merits of the Newtonian conception, there is certainly no need to insist. Everyone knows that in the *Principia* there was already contained, far more than in embryo, celestial mechanics, which for centuries has passed from triumph to triumph by applying the conception and the law of Newton, valid even at inconceivable distances. And not only valid with mediocre approximations, but with approximations that are almost inconceivably close. It suffices to recall that two minute discrepancies, which in other natural sciences would have been considered negligible, led to the discovery of the two most distant planets of our solar system and the strange companions of some double stars.
But there are also some difficulties. The admission of actions between distant bodies without the intervention of a transmitting medium, or, as it was immediately called, action at a distance, was repugnant even to its own author, who insisted on declaring that he was not at all affirming that such actions were truly at a distance, but only that subsequent phenomena unfolded as if they were indeed so. It is noteworthy that, while they disturbed people for two centuries, successive thinkers sought to develop numerous interpretations and theories of the phenomenon of gravity, Newton rigorously abstained from doing so, as if he had intuitively grasped what only became evident after the mid-19th century: “that at the deepest foundations of the knowledge of nature, one cannot arrive through interpretations of a mechanistic type.”
Another significant difficulty arises from the fact that, if one considers the universe as infinitely extended, as was practically the norm in previous centuries, and with a destructive notion of matter analogous to that of the part accessible to observation, Newton’s theory could no longer be valid. For this and other difficulties, see RELATIVITÀ.