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Divide it then, and in its place let gravity act, and the body will circulate as before ; its tendency to the center, or its weight, being just balanced by its centrifugal force. Knowing the radius of the earth, we can calculate by the principles of mechanics the periodical time in which a body so balanced must circulate to keep it up; and this appears to be lh 23m 22.
(443.) If we make the same calculation for a body at the distance of the moon, supposing its weight or gravity the same as at the earth's surface, we shall find the period required to be 106 45m 30%. The actual period of the moon's revolution, however, is 27d 7h 43m; and hence it is clear that the moon's velocity is not nearly sufficient to sustain it against such a power, supposing it to revolve in a circle, or neglecting (for the present) the slight ellipticity of its orbit. In order that a body at the distance of the moon (or the moon itself) should be capable of keeping its distance from the earth by the outward effort of its centrifugal force, while yet its time of revolution should be what the moon's actually is, it will appear* that gravity, instead of being as intense as at the surface, would require to be very nearly 3600 times less energetic; or, in other words, that its intensity is so enfeebled by the remoteness of the body on which it acts, as to be capable of producing in it, in the same time, only stooth part of the motion which it would impart to the same mass of matter at the earth's surface.
(444.) The distance of the moon from the earth's center is a very little less than sixty times the distance from the center to the surface, and 3600 : 1 :: 602 : 1°; so that the proportion in which we must admit the earth's gravity to be enfeebled at the moon's distance, if it be really the force which retains the moon in her orbit, must be (at least in this particular instance) that of the squares of the distances at which it is compared. Now, in such a diminution of energy with increase of distance, there is nothing prima facie inadmissible. Emanations from a center, such as light and heat,
• Newton, Princip. b. i., Prop. 4., Cor. 2.
do really diminish in intensity by increase of distance, and in this identical proportion; and though we cannot certainly argue much from this analogy, yet we do see that the power of magnetic and electric attractions and repulsions is actually enfeebled by distance, and much more rapidly than in the simple proportion of the increased distances. The argument therefore, stands thus :— On the one hand, Gravity is a real power, of whose agency we have daily experience. We know that it extends to the greatest accessible heights, and far beyond ; and we see no reason for drawing a line at any particular height, and there asserting that it must cease entirely; though we have analogies to lead us to suppose its energy may diminish as we ascend to great heights from the surface, such as that of the moon. On the other hand we are sure the moon is urged towards the earth by some power which retains her in her orbit, and that the intensity of this power is such as would correspond to a gravity, diminished in the proportion — otherwise not improbable- of the squares of the distances. If gravity be not that power, there must exist some other; and, besides this, gravity must cease at some inferior level, or the nature of the moon must be different from that of ponderable matter; — for if not, it would be urged by both powers, and therefore too much urged and forced inwards from her path.
(445.) It is on such an argument that Newton is understood to have rested, in the first instance, and provisionally, his law of universal gravitation, which may be thus abstractly stated :— “Every particle of matter in the universe attracts every other particle, with a force directly proportioned to the mass of the attracting particle, and inversely to the square of the distance between them.” In this abstract and general form, however, the proposition is not applicable to the case before us. The earth and moon are not mere particles, but great spherical bodies, and to such the general law does not immediately apply; and, before we can make it applicable, it becomes necessary to enquire what will be the force with which a congeries of particles, constituting a solid mass of any assigned figure, will attract another such collection of material atoms. This problem is one purely dynamical, and, in this its general form, is of extreme difficulty. Fortunately however, for human knowledge when the attracting and attracted bodies are spheres, it admits of an easy and direct solution. Newton himself has shown (Princip. b. i. prop. 75.) that, in that case, the attraction is precisely the same as if the whole matter of each sphere were collected into its center, and the spheres were single particles there placed; so that, in this case, the general law applies in its strict wording. The effect of the trifling deviation of the earth from a spherical form is of too minute an order to need attention at present, It is, however, perceptible, and may be hereafter noticed.
(446.) The next step in the Newtonian argument is one which divests the law of gravitation of its provisional character, as derived from a loose and superficial consideration of the lunar orbit as a circle described with an average or mean velocity, and elevates it to the rank of a general and primordial relation by proving its applicability to the state of existing nature in all its circumstances. This step consists in demonstrating, as he has done* (Princip. i. 17. i. 75.), that, under the influence of such an attractive force mutually urging two spherical gravitating bodies towards each other, they will each, when moving in each other's neighbourhood, be deflected into an orbit concave towards the other, and describe, one about the other regarded as fixed, or both round their common centre of gravity, curves whose forms are limited to those figures known in geometry by the general name of conic sections. It will depend, he shows, in any assigned case, upon the particular circumstances or velocity, distance, and direction, which of these curves shall be described, - whether an ellipse, a circle, a parabola, or
. We refer for these fundamental propositions, as a point of duty, to the immortal work in which they were first propounded. It is impossible for us, in this volume, to go into these investigations : even did our limits permit, it would be utterly inconsistent with our plan; a general idea, however, of their conduct will be given in the next chapter. We trust that the careful and attentive study of the Principia in its original form will never be laid aside, whatever be the improvements of the modern analysis as respects facility of calculation and expression. From no other quarter can a thorough and complete comprehension of the mechanism of our system, (so far as the immediate scope of that work extends,) be anything like so well, and we may add, so easily obtained
an hyperbola; but one or other it must be; and any one of any degree of eccentricity it may be, according to the circumstances of the case; and, in all cases, the point to which the motion is referred, whether it be the centre of one of the spheres, or their common centre of gravity, will of necessity be the focus of the conic section described. He shows, furthermore (Princip. i. 1.), that, in every case, the angulur velocity with which the line joining their centres moves, must be inversely proportional to the square of their mutual distance, and that equal areas of the curves described will be swept over by their line of junction in equal times.
(447.) All this is in conformity with what we have stated of the solar and lunar movements. Their orbits are ellipses, but of different degrees of eccentricity; and this circumstance already indicates the general applicability of the principles in question.
(448.) But here we have already, by a natural and ready implication (such is always the progress of generalization), taken a further and most important step, almost unperceived. We have extended the action of gravity to the case of the earth and sun, to a distance immensely greater than that of the moon, and to a body apparently quite of a different nature than either. Are we justified in this ? or, at all events, are there no modifications introduced by the change of data, if not into the general expression, at least into the particular interpretation, of the law of gravitation ? Now, the moment we come to numbers, an obvious incongruity strikes us. When we calculate, as above, from the known distance of the sun (art. 357.), and from the period in which the earth circulates about it (art. 305.), what must be the centrifugal force of the latter by which the sun's attraction is balanced, (and which, therefore, becomes an exact measure of the sun's attractive energy as exerted on the earth,) we find it to be immensely greater than would suffice to counteract the earth's attraction on an equal body at that distance-greater in the high proportion of 354936 to 1. It is clear, then, that if the earth be retained in its orbit about the sun by solar attraction, conformable in its rate of diminution with the general law, this force must be no less than 354936 times more intense than what the earth would be capable of exerting, cæteris paribus, at an equal distance.
(449.) What, then, are we to understand from this result ? Simply this, — that the sun attracts as a collection of 354936 earths occupying its place would do, or, in other words, that the sun contains 354936 times the mass or quantity of ponderable matter that the earth consists of. Nor let this conclusion startle us. We have only to recall what has been already shown in (art. 358.) of the gigantic dimensions of this magnificent body, to perceive that, in assigning to it so vast a mass, we are not outstepping a reasonable proportion. In fact, when we come to compare its mass with its bulk, we find its density * to be less than that of the earth, being no more than 0.2543. So that it must consist, in reality, of far lighter materials, especially when we consider the force under which its central parts must be condensed. This consideration renders it highly probable that an intense heat prevails in its interior by which its elasticity is reinforced, and rendered capable of resisting this almost inconceivable pressure without collapsing into smaller dimensions.
(450.) This will be more distinctly appretiated, if we estimate, as we are now prepared to do, the intensity of gravity at the sun's surface.
The attraction of a sphere being the same (art. 445.) as if its whole mass were collected in its centre, will, of course, be proportional to the mass directly, and the square of the distance inversely; and, in this case, the distance is the radius of the sphere. Hence we concludes, that the intensities of solar and terrestrial gravity at the surfaces of the two globes are in the proportions of 27:9 to 1. A pound of terrestrial matter at the sun's surface, then, would exert a pressure equal to what 27.9 such pounds would do at the
* The density of a material body is as the mass directly, and the volume inversely: hence density of O: density of :: 1924479 : 1 : 0·2543 : 1.
† Solar gravity : terrestrial :: 1420000 : 2000;2::27.9 : 1; the respective radii of the sun and earth being 440000, and 4000 miles.