GRAVITAZIONE. - I. GENERALITIES. - In the study of natural phenomena, the opportunity repeatedly arose over the centuries to admit the hypothesis that material bodies might, or indeed might necessarily, mutually attract one another (and sometimes possibly also repel one another), even without being in contact or even when substantially distant from one another. Clear affirmations to this effect, which Empedocles, Anaxagoras, and later Leucippus and Democritus made the basis of their cosmogonies, are recalled in the works of Plato and Aristotle, and in these are also interestingly brought into connection with the phenomena of the falling of heavy bodies near the earth’s surface. In this field, the admission of one of these mysterious attractions remained at least latent until the time of Galileo, who, through his simultaneous discoveries of the fundamental law of dynamics and the law of the fall of heavy bodies, was finally able to establish its fundamental character much more clearly. It was then proved that the earth, still considered globally, as a whole, attracted all material bodies near its surface with a force capable of imparting to them an equal acceleration (that is, an equal increase in their velocity, approximately equal to 980 centimetres per second, an acceleration now conventionally indicated by the letter g). It follows immediately from Galileo’s dynamic law that the force of attraction acting on each body (that is, its weight) is necessarily proportional to its mass.
But despite the extraordinary importance of this Galilean result which, much more than his celebrated dialogues on The Great Systems, was what “to the Englishman who so greatly desired to see it—first cleared the paths of the firmament,” we are not yet confronted with the notion and universal law of g. The clarification of that notion and the complete formulation of its law were predominantly the work of Newton.
II. THE NEWTONIAN DEDUCTION. – In the first place, Newton’s third principle of dynamics, by affirming the existence, by reaction, of an equal attraction exerted by the body under consideration upon the earth, established between them a complete reciprocity. It was therefore possible immediately to extend Galileo’s result by asserting that “between any two bodies of masses m and m′ there always occurs an attraction proportional to the product m m′ of those masses.” However, the decisive step toward conceiving the universality of the phenomenon of g. was taken by Newton only when, as his mind was occupied with seeking the cause whereby the moon accompanies the earth in its motion through space (in 1666), the idea flashed upon him that this cause must consist in an attractive force mutually binding earth and moon, of the same origin and the same nature as that which, near the earth’s surface, binds the earth and every material body, determining its weight and, if free, its fall with the acceleration g.
According to the Galilean experimental criterion, the reliability of these hypotheses could be demonstrated only by the complete agreement of all their consequences with reality. Merely by moving around the earth in an approximately circular orbit, the moon is necessarily subject to a centripetal acceleration, which kinematics, already sufficiently developed at that time, made it possible to calculate without difficulty on the basis of its distance D from the centre of the earth and its period of revolution T—two quantities already known with sufficient accuracy at that time. This centripetal acceleration proved to be equal to 0.272 centimetres per second, that is, 3,600 times weaker than that acting on material bodies at the earth’s surface. For the Newtonian hypothesis to be reliable, the attractive force of the earth upon any body situated at distance D from its centre ought to have had that same value.
At this point a considerable difficulty arose: the fundamental principles of Galileo and Newton had made it possible, given the existence of an attractive force between two masses, to assert that it must be proportional to their product; however, they said nothing about the dependence of this force upon their distance. But here too Newton ingeniously supplied the deficiency with two further hypotheses, which later would have to prove a posteriori to be in complete agreement with the original hypothesis. At that time it was widely held that the intensity of phenomena that could be regarded as emanating symmetrically from a centre—such as, for example, light from a point source—ought to diminish in proportion to the spherical surface that it reached, and hence in proportion to the square of the distance from that centre. In that case, even if the emanation did not proceed from the centre alone but from spherical layers symmetrical around it, the effect at an external point had to be equal to what would have been obtained if the entire emanation had proceeded from that centre. Thus, assuming the earth’s masses to be distributed symmetrically around the centre, Newton was able to set up his calculations as though the entire terrestrial mass were condensed at its central point. Then, denoting the terrestrial radius by R, and given the accepted law concerning the effect of distance, the forces, and therefore the accelerations, at distances R and D ought to satisfy the condition F at R : F at D = D² : R². But since the acceleration at the earth’s surface was a at R = 980 and that at the moon’s distance was a at D = 0.272 centimetres per second, and therefore a at R : a at D = 3,600, it follows that the ratio D/R ought to be equal to 60, that is, the moon’s distance ought to be equal to 60 R. But at that time, because of the inaccurate knowledge
of R, it was believed to be only 56 R. Newton’s hypothesis therefore did not appear to be in sufficient agreement with apparent reality.
But in 1683 the new and more rigorous measurement of the meridian arc carried out by Picard became known. On its basis, the radius of the earth proved to be appreciably smaller than had hitherto been believed, and specifically approximately 1/60 of the moon’s distance. Newton was naturally delighted to be able to reconsider his 1666 hypothesis, and in 1687, after four years of meditation and work, he published his immortal work, in which, having systematized the principles of the dynamics of Galileo and of himself, he presented and applied extensively the classic formula of gravitational forces F
F = G · frac{M₁ M₂}{D²}
now valid for any mass in the universe, at any distance. The universal constant G was determined through subsequent delicate experiments of increasing accuracy as G = 6.66 · 10^{-8} cm³/sec² · gr^{-1}.
But an adequate idea of the unlimited universality of the notion of g. and of its law can be formed only when it is considered in elementary terms, that is, by referring it to every particle of matter in the universe, conceived as both attracting and attracted. The phenomena that manifest themselves are nothing other than the resultants of these countless elementary phenomena.
III. MERITS AND DEFECTS OF THE NEWTONIAN THEORY
As regards the merits of the Newtonian conception, there is certainly no need to dwell on them. Everyone knows that the Principia already contained, far more than in germ, the celestial mechanics that for centuries has gone from triumph to triumph by applying Newton’s conception and law, valid even at inconceivably great distances. And not merely valid with mediocre approximations, but with approximations that are almost equally inconceivable. It is enough to recall that two minimal divergences, which in other natural sciences would have been considered negligible, led to the discovery of the two most distant planets in our solar system and of the strange companions of certain double stars.But there is also no lack of some difficulty. The admission of actions between distant bodies without the intervention of a transmitting medium—or, as it was immediately called, actions at a distance—was repugnant to its own author, who insisted on declaring that he did not at all assert that those actions were really at a distance, but only that the subsequent phenomena took place as if they actually were so. And it is noteworthy that, while contemporaries and, for two centuries, successors sought to develop numerous interpretations and theories of the phenomenon of g., Newton rigorously abstained from doing so, as though he had had a brilliant intuition of what finally became evident only after the middle of the nineteenth century: “that the deepest foundations of the knowledge of nature cannot be reached through interpretations of a mechanistic type.”
Another significant difficulty derives from the fact that, if the universe is considered infinitely extended, as was practically customary in previous centuries, and with an average distribution of matter analogous to that of the portion of it accessible to observation, Newtonian theory could no longer be valid. For this and other difficulties V. RELATIVITÀ.