CHAPTER X.The Law of Gravitation.
We shall proceed to make a few observations on the Law of Gravity, in virtue of which the motions of planets about the sun, and of satellites about their planets take place; and by which also are producedthe fall downwards of all bodies within our reach, and the pressure which they exert upon their supports when at rest. The identification of the latter forces with the former, and the discovery of the single law by which these forces are every where regulated, was the great discovery of Newton: and we wish to make it appear that this law is established by an intelligent and comprehensive selection.
The law of the sun’s attraction upon the planets is, that this attraction variesinverselyas the square of the distance; that is, it decreases as that square increases. If we take three points or planets of the solar system, the distances of which from the sun are in proper proportion one, two, three; the attractive force which the sun at these distances exercises, is as one, one-fourth, and one-ninth respectively. In the smaller variations of distance which occur in the elliptical motion of one planet, the variations of the force follow the same law. Moreover, not only does the sun attract the planets, but they attract each other according to the same law; the tendency to the earth which makes bodies heavy, is one of the effects of this law: and all these effects of the attractions of large masses may be traced to the attractions of the particles of which they are composed; so that the final generalization, including all the derivative laws, is, that every particle of matter in the universe attracts every other, according to the law of the inverse square of the distance.
Such is the law of universal gravitation. Now, the question is, why do either the attractions of masses, or those of their component particles, follow this law of the inverse square of the distance rather than any other? When the distance becomes one, two, and three, why should not the force also become one, two, and three?—or if it must be weaker at points more remote from the attracting body, why should it not be one, a half, a third? or one, an eighth, a twenty-seventh? Such law’s could easily be expressed mathematically, and their consequences calculated.Can any reason be assigned why the law which we find in operationmustobtain? Can any be assigned why itshouldobtain?
The answer to this is, that no reason, at all satisfactory, can be given why such a law must, of necessity, be what it is; but that very strong reasons can be pointed out why, for the beauty and advantage of the system, the present one is better than others. We will point out some of these reasons.
1. In the first place, the system could not have subsisted, if the force had followed adirectinstead of an inverse law, with respect to the distance; that is, if it had increased when the distance increased. It has been sometimes said, that “all direct laws of force are excluded on account of the danger from perturbing forces;”[23]that if the planets had pulled at this earth, the harder the further off they were, they would have dragged it entirely out of its course. This is not an exact statement of what would happen: if the force were to be simply in the direct ratio of the distance, any number of planets might revolve in the most regular and orderly manner. Their mutual effects, which we may call perturbations if we please, would be considerable; but these perturbations would be so combined with the unperturbed motion, as to produce a new motion not less regular than the other. This curious result would follow, that every body in the system would describe, or seem to describe, about every other, an exact elliptical orbit; and that the times of the revolution of every body in its orbit would be all equal. This is proved by Newton, in the sixty-fourth proposition of the Principia. There would be nothing to prevent all the planets, on this supposition, from moving round the sun in orbits exactly circular, or nearly circular, according to the mode in which they were set in motion.
But though the perturbations of the system wouldnot make this law inadmissible, there are other circumstances which would do so. Under this law, the gravity of bodies at the earth’s surface would cease to exist. Nothing would fall or weigh downwards. The greater action of the distant sun and planets would exactly neutralize the gravity of the earth: a ball thrown from the hand, however gently, would immediately become a satellite of the earth, and would for the future accompany it in its course, revolving about it in the space of one year. All terrestrial things would float about with no principle of coherence or stability: they would obey the general law of the system, but would acknowledge no particular relation to the earth. We can hardly pretend to judge of the abstract possibility of such a system of things; but it is clear that it could not exist without an utter subversion of all that we can conceive of the economy and structure of the world which we inhabit.
With any other direct law of force, we should in like manner lose gravity, without gaining the theoretical regularity of the planetary motions which we have described in the case just considered.
2. Amonginverselaws of the distance, (that is, those according to which the force diminishes as the distance from the origin of force increases,) all which diminish the central force faster than thecubeof the distance increases are inadmissible, because they are incompatible with the permanent revolution of a planet. Under such laws it would follow, that a planet would describe a spiral line about the sun, and would either approach nearer and nearer to him perpetually, or perpetually go further and further off: nearly as a stone at the end of a string, when the string is whirled round, and is allowed to wrap round the hand, or to unwrap from it, approaches to or recedes from the hand.
If we endeavour to compare the law of the inverse square of the distance, which really regulates the central force, with other laws, not obviously inadmissible,as for instance, the inverse simple ratio of the distance, a considerable quantity of calculation is found to be necessary in order to trace the results, and especially the perturbations in the two cases. The perturbations in the supposed case have not been calculated; such a calculation being a process so long and laborious that it is never gone through, except for the purpose of comparing the results of theory with those of observation, as we can do with regard to the law of inverse square. We can only say, therefore, that the stability of the system, and the moderate limits of the perturbations, which we know to be secured by the existing law, would not, so far as we know, be obtained by any different law.
Without going into further examination of the subject, we may observe that there are some circumstances in which the present system has a manifest superiority in its simplicity over the condition which would have belonged to it if the force had followed any other law. Thus, with the present law of gravitation the planets revolve, returning perpetually on the same track, very nearly. The earth describes an oval, in consequence of which motion she is nearer to the sun in our winter than in our summer by about one-thirtieth part of the whole distance. And, as the matter now is, the nearest approach to the sun, and the farthest recess from him, occur always at the same points of the orbit. There is indeed a slight alteration in these points arising from disturbing forces, but this is hardly sensible in the course of several ages. Now if the force had followed any other law, we should have had the earth running perpetually on a new track. The greatest and least distances would have occurred at different parts in every successive revolution. The orbit would have perpetually intersected and been interlaced with the path described in former revolutions; and the simplicity and regularity which characterizes the present motion would have been quite wanting.
3. Another peculiar point of simplicity in the present law of mutual attraction is this: that it makes the law of attraction for spherical masses the same as for single particles. If particles attract with forces which are inversely as the square of the distance, spheres composed of such particles, will exert a force which follows the same law. In this character the present law is singular, among all possible laws, excepting that of the direct distance which we have already discussed. If the law of the gravitation of particles had been that of the inverse simple distance, the attraction of a sphere would have been expressed by a complex series of mathematical expressions, each representing a simple law. It is truly remarkable that the law of the inverse square of the distance, which appears to be selected as that of themassesof the system, and of which the mechanism is, that it arises from the action of theparticlesof the system, should lead us to the same law for the action of these particles: there is a strikingprerogativeof simplicity in the law thus adopted.
The law of gravitation actually prevailing in the solar system has thus great and clear advantages over any law widely different from it; and has moreover, in many of its consequences, a simplicity which belongs to this precise law alone. It is in many such respects auniquelaw; and when we consider that it possesses severalpropertieswhich arepeculiarto it, and severaladvantageswhich may be peculiar to it, and which are certainly nearly so; we have some ground, it would appear, to look upon its peculiarities and its advantages as connected. For the reasons mentioned in the last chapter, we can hardly expect to see fully the way in which the system is benefited by the simplicity of this law, and by the mathematical elegance of its consequences: but when we see that it has some such beauties, and some manifest benefits, we may easily suppose that our ignorance and limited capacity alone prevent our seeing that there are, for the selection of this law of force,reasons of a far more refined and comprehensive kind than we can distinctly apprehend.
4. But before quitting this subject we may offer a few further observations on the question, whether gravitation and the law of gravitation benecessaryattributes of matter. We have spoken of the selection of this law, but is it selected? Could it have been otherwise? Is not the force of attraction a necessary consequence of the fundamental properties of matter?
This is a question which has been much agitated among the followers of Newton. Some have maintained, as Cotes, that gravity is an inherent property of all matter; others, with Newton himself, have considered it as an appendage to the essential qualities of matter, and have proposed hypotheses to account for the mode in which its effects are produced.
The result of all that can be said on the subject appears to be this: that no one can demonstrate the possibility of deducing gravity from the acknowledged fundamental properties of matter: and that no philosopher asserts, that matter has been found to exist, which was destitute of gravity. It is a property which we have no right to callnecessaryto matter, but every reason to supposeuniversal.
If we could show gravity to be a necessary consequence of those properties which we adopt as essential to our notion of matter, (extension, solidity, mobility, inertia) we might then call it also one of the essential properties. But no one probably will assert that this is the case. Its universality is a fact of observation merely. How then can a property,—in its existence so needful for the support of the universe, in its laws so well adapted to the purposes of creation,—how came it to be thus universal? Its being found every where is necessary for its uses; but this is so far from being a sufficient explanation of its existence, that it is an additional fact to be explained. We have here, then, an agency mostsimple in its rule, most comprehensive in its influence, most effectual and admirable in its operation. What evidence could be afforded of design, by laws of mechanical action, which this law thus existing and thus operating does not afford us?
5. It is not necessary for our purpose to consider the theories which have been proposed to account for the action of gravity. They have proceeded on the plan of reducing this action to the result of pressure or impulse. Even if such theories could be established, they could not much, or at all, affect our argument; for the arrangements by which pressure or impact could produce the effects which gravity produces, must be at least as clearly results of contrivance, as gravity itself can be.
In fact, however, none of these attempts can be considered as at all successful. That of Newton is very remarkable: it is found among the Queries in the second edition of his Optics. “To show,” he says, “that I do not take gravity for an essential property of bodies, I have added one question concerning its cause, choosing to propose it by way of question, because I am not yet satisfied about it for want of experiments.” The hypothesis which he thus suggests is, that there is an elastic medium pervading all space, and increasing in elasticity as we proceed from dense bodies outwards: that this “causes the gravity of such dense bodies to each other: every body endeavouring to go from the denser parts of the medium towards the rarer.” Of this hypothesis we may venture to say, that it is in the first place quite gratuitous; we cannot trace in any other phenomena a medium possessing these properties: and in the next place, that the hypothesis contains several suppositions which are more complex than the fact to be explained, and none which are less so. Can we, on Newton’s principles, conceive an elastic medium otherwise than as a collection of particles, repelling each other? and is the repulsion of such particles a simpler fact than theattraction of those which gravitate? And when we suppose that the medium becomes more elastic as we proceed from each attracting body, what cause can we conceive capable of keeping it in such a condition, except a repulsive force emanating from the body itself: a supposition at least as much requiring to be accounted for, as the attraction of the body. It does not appear, then, that this hypothesis will bear examination; although, for our purpose, the argument would be rather strengthened than weakened, if it could be established.
6. Another theory of the cause of gravity, which at one time excited considerable notice, was that originally proposed by M. Le Sage, ina memoir entitled“Lucrece Newtonien,” and further illustrated by M. Prevost; according to which all space is occupied by currents of matter, moving perpetually in straight lines, in all directions, with a vast velocity, and penetrating all bodies. When two bodies are near each other, they intercept the current which would flow in the intermediate space if they were not there, and thus receive a tendency towards each other from the pressure of the currents on their farther sides. Without examining further this curious and ingenious hypothesis, we may make upon it the same kind of observations as before;—that it is perfectly gratuitous, except as a means of explaining the phenomena; and that, if it were proved, it would still remain to be shown what necessity has caused the existence of thesetwo kindsof matter; the first kind being that which is commonly called matter, and which alone affects our senses, while it is inert as to any tendency to motion; the second kind being something imperceptible to our senses, except by the effects it produces on matter of the former kind; yet exerting an impulse on every material body, permeating every portion of common matter, flowing with inconceivablevelocity, ininexhaustible abundance, from every part of the abyss of infinity on one side, to the opposite part of the sameabyss; and so constituted that through all eternity it can never bend its path, or return, or tarry in its course.
If we were to accept this theory, it would little or nothing diminish our wonder at the structure of the universe. We might well continue to admire the evidence of contrivance, if such a machinery should be found to produce all the effects which flow from the law of gravitation.
7. The arguments for and against the necessity of the law of the inverse square of the distance in the force of gravity, were discussed with great animation about the middle of the last century. Clairault, an eminent mathematician, who did more than almost any other person for the establishment and development of the Newtonian doctrines, maintained, at one period of his researches, not only that the inverse square was not thenecessarylaw, but also that it was not thetruelaw. The occasion of this controversy was somewhat curious.
Newton and other astronomers had found that the line of the moon’sapsides(that is of her greatest and least distances from the earth) moves round to different parts of the heavens with a velocity twice as great as that which the calculation from the law of gravitation seems at first to give. According to the theory, it appeared that this line ought to move round once in eighteen years; according to observation, it moves round once in nine years. This difference, the only obvious failure of the theory of gravitation, embarrassed mathematicians exceedingly. It is true, it was afterwards discovered that the apparent discrepancy arose from a mistake; the calculation, which is long and laborious, was supposed to have been carried far enough to get close to the truth; but it appeared afterwards that the residue which had been left out as insignificant, produced, by an unexpected turn in the reckoning, an effect as large as that which had been taken for the whole. But this discovery was not made till afterwards; and inthe mean time the law of the inverse square appeared to be at fault. Clairault tried to remedy the defect by supposing that the force of the earth’s gravity consisted of a large force varying as the square of the distance, and a very small force varying as the fourth power (the square of the square.) By such a supposition, observation and theory could be reconciled; but on the suggestion of it, Buffon came forward with the assertion that the forcecouldnot vary according to any other law than the inverse square. His arguments are rather metaphysical than physical or mathematical. Gravity, he urges, is a quality, an emanation; and all emanations are inversely as the square of the distance, as light, odours. To this Clairault replies by asking, how we know that light and odours have their intensity inversely as the square of the distance from their origin: not, he observes, by measuring the intensity, but bysupposingthese effects to be material emanations. But who, he asks, supposes gravity to be a material emanationfromthe attracting body.
Buffon again pleads that so many facts prove the law of the inverse square, that a single one, which occurs to interfere with this agreement, must be in some manner capable of being explained away. Clairault replies, that the facts donotprove this law to obtain exactly; that small effects, of the same order as the one under discussion, have been neglected; and that therefore the law is only known to be true,as faras such an approximation goes, and no farther.
Buffon then argues, that there can be no such additional fraction of the force, following a different law, as Clairault supposes: for what, he asks, is there to determine the magnitude of the fraction to one amount rather than another? why should nature select for it any particular magnitude? To this it is replied, that, whether we can explain the fact or not, nature does select certain magnitudes in preference to others: that where we ascertain she does this, we are not to deny the fact because we cannot assign thegrounds of her preference. What is there, it is asked, to determine the magnitude of the whole force at any fixed distance? We cannot tell; yet the force is of a certain definite intensity and no other.
Finally, Clairault observes, that we have, in cohesion, capillary attraction, and various other cases, examples of forces varying according to other laws than the inverse square; and that therefore this cannot be the only possible law.
The discrepancy between observation and theory which gave rise to this controversy was removed, as has been already stated, by a more exact calculation: and thus, as Laplace observes, in this case the metaphysician turned out to be right and the mathematician to be wrong. But most persons, probably, who are familiar with such trains of speculation, will allow, that Clairault had the best of the argument, and that the attempts to show the law of gravitation to be necessarily what it is, are fallacious and unsound.
8. We may observe, however, that the law of gravitation according to the inverse square of the distance, which thus regulates the motions of the solar system, is not confined to that province of the universe, as has been shown by recent researches. It appears by the observations and calculations of Sir John Herschel, that several of the stars, calleddouble stars, consist of a pair of luminous bodies which revolve above each other in ellipses, in such a manner as to show that the force, by which they are attracted to each other, varies according to the law of the inverse square. We thus learn a remarkable fact concerning bodies which seemed so far removed that no effort of our science could reach them; and we find that the same law of mutual attraction which we have before traced to the farthest bounds of the solar system, prevails also in spaces at a distance compared with which the orbit of Saturn shrinks into a point. The establishment of such a truth certainly suggests, as highly probable, the prevalence of this law amongall the bodies of the universe. And we may therefore suppose, that the same ordinance which gave to the parts of our system that rule by which they fulfil the purposes of their creation, impressed the same rule on the other portions of matter which are scattered in the most remote parts of the universe; and thus gave to their movements the same grounds of simplicity and harmony which we find reason to admire, as far as we can acquire any knowledge of our own more immediate neighbourhood.