SINCE the planets move in a void space and are free from resistance; they, like all other bodies, when once in motion, would move on in a straight line without end, if left to themselves. And it is now to be explained what kind of action upon them carries them round the sun. Here I shall treat of the primary planets only, and discourse of the secondary apart in the next chapter. It has been just now declared, that these primary planets move so about the sun, that a line extended from the sun to the planet, will, by accompanying the planet in its motion, pass over equal spaces in equal portions of time[155]. And this one property in the motion of the planets proves, that they are continually acted on by a power directed perpetually to the sun as a center. This therefore is one property of the cause, which keeps the planets in their courses, that it is a centripetal power, whose center is the sun.
2. Again, in the chapter upon centripetal forces[156] it was observ’d, that if the strength of the centripetal power was suitably accommodated every where to the motion of any body round a center, the body might be carried in any bent line whatever, whose concavity should be every where turned towards the center of the force. It was farther remarked, that the strength of the centripetal force, in each place, was to be collected from the nature of the line, wherein the body moved[157]. Now since each planet moves in an ellipsis, and the sun is placed in one focus; Sir Isaac Newton deduces from hence, that the strength of this power is reciprocally in the duplicate proportion of the distance from the sun. This is deduced from the properties, which the geometers have discovered in the ellipsis. The process of the reasoning is not proper to be enlarged upon here; but I shall endeavour to explain what is meant by the reciprocal duplicate proportion. Each of the terms reciprocal proportion, and duplicate proportion, has been already defined[158]. Their sense when thus united is as follows. Suppose the planet moved in the orbit A B C (in fig. 93.) about the sun in S. Then, when it is said, that the centripetal power, which acts on the planet in A, bears to the power acting on it in B a proportion, which is the reciprocal of the duplicate proportion of the distance S A to the distance S B; it is meant that the power in A bears to the power in B the duplicate of the proportion of the distance S B to the distance S A. The reciprocal duplicate proportion may be explained also by numbers as follows. Suppose several distances to bear to each other proportions expressed by the numbers 1, 2, 3, 4, 5; that is, let the second distance be double the first, the third be three times, the fourth four times, and the fifth five times as great as the first. Multiply each of these numbers by it self, and 1 multiplied by 1 produces still 1, 2 multiplied by 2 produces 4, 3 by 3 makes 9, 4 by 4 makes 16, and 5 by 5 gives 25. This being done, the fractions ¼, 1/9, 1/16, 1/25, will respectively express the proportion, which the centripetal power in each of the following distances bears to the power at the first distance: for in the second distance, which is double the first, the centripetal power will be one fourth part only of the power at the first distance; at the third distance the power will be one ninth part only of the first power; at the fourth distance, the power will be but one sixteenth part of the first; and at the fifth distance, one twenty fifth part of the first power.
3. Thus is found the proportion, in which this centripetal power decreases, as the distance from the sun increases, within the compass of one planet’s motion. How it comes to pass, that the planet can be carried about the sun by this centripetal power in a continual round, sometimes rising from the sun, then descending again as low, and from thence be carried up again as far remote as before, alternately rising and falling without end; appears from what has been written above concerning centripetal forces: for the orbits of the planets resemble in shape the curve line proposed in § 17 of the chapter on these forces[159].
4. But farther, in order to know whether this centripetal force extends in the same proportion throughout, and consequently whether all the planets are influenced by the very same power, our author proceeds thus. He inquires what relation there ought to be between the periods of the different planets, provided they were acted upon by the same power decreasing throughout in the forementioned proportion; and he finds, that the period of each in this case would have that very relation to the greater axis of its orbit, as I have declared above[160] to be found in the planets by the observations of astronomers. And this puts it beyond question, that the different planets are pressed towards the sun, in the same proportion to their distances, as one planet is in its several distances. And thence in the last place it is justly concluded, that there is such a power acting towards the sun in the foresaid proportion at all distances from it.
5. This power, when referred to the planets, our author calls centripetal, when to the sun attractive; he gives it likewise the name of gravity, because he finds it to be of the same nature with that power of gravity, which is observed in our earth, as will appear hereafter[161]. By all these names he designs only to signify a power endued with the properties before mentioned; but by no means would he have it understood, as if these names referred any way to the cause of it. In particular in one place where he uses the name of attraction, he cautions us expressly against implying any thing but a power directing a body to a center without any reference to the cause of it, whether residing in that center, or arising from any external impulse[162].
6. But now, in these demonstrations some very minute inequalities in the motion of the planets are neglected; which is done with a great deal of judgment; for whatever be their cause, the effects are very inconsiderable, they being so exceeding small, that some astronomers have thought fit wholly to pass them by[163]. However the excellency of this philosophy, when in the hands of so great a geometer as our author, is such, that it is able to trace the least variations of things up to their causes. The only inequalities, which have been observed common to all the planets, are the motion of the aphelion and the nodes. The transverse axis of each orbit does not always remain fixed, but moves about the sun with a very slow progressive motion: nor do the planets keep constantly the same plane, but change them, and the lines in which those planes intersect each other by insensible degrees. The first of these inequalities, which is the motion of the aphelion, may be accounted for, by supposing the gravitation of the planets towards the sun to differ a little from the forementioned reciprocal duplicate proportion of the distances; but the second, which is the motion of the nodes, cannot be accounted for by any power directed towards the sun; for no such can give the planet any lateral impulse to divert it from the plane of its motion into any new plane, but of necessity must be derived from some other center. Where that power is lodged, remains to be discovered. Now it is proved, as shall be explained in the following chapter, that the three primary planets Saturn, Jupiter, and the earth, which have satellites revolving about them, are endued with a power of causing bodies, in particular those satellites, to gravitate towards them with a force, which is reciprocally in the duplicate proportion of their distances; and the planets are in all respects, in which they come under our examination, so similar and alike, that there is no reason to question, but they have all the same property. Though it be sufficient for the present purpose to have it proved of Jupiter and Saturn only; for these planets contain much greater quantities of matter than the rest, and proportionally exceed the others in power[164]. But the influence of these two planets being allowed, it is evident how the planets come to shift continually their planes: for each of the planets moving in a different plane, the action of Jupiter and Saturn upon the rest will be oblique to the planes of their motion; and therefore will gradually draw them into new ones. The same action of these two planets upon the rest will cause likewise a progressive motion of the aphelion; so that there will be no necessity of having recourse to the other cause for this motion, which was before hinted at[165]; viz, the gravitation of the planets towards the sun differing from the exact reciprocal duplicate proportion of the distances. And in the last place, the action of Jupiter and Saturn upon each other will produce in their motions the same inequalities, as their joint action produces in the rest. All this is effected in the same manner, as the sun produces the same kind of inequalities and many others in the motion of the moon and the other secondary planets; and therefore will be best apprehended by what shall be said in the next chapter. Those other irregularities in the motion of the secondary planets have place likewise here; but are too minute to be observable: because they are produced and rectified alternately, for the most part in the time of a single revolution; whereas the motion of the aphelion and nodes, which continually increase, become sensible in a long series of years. Yet some of these other inequalities are discernible in Jupiter and Saturn, in Saturn chiefly; for when Jupiter, who moves faster than Saturn, approaches near to a conjunction with him, his action upon Saturn will a little retard the motion of that planet, and by the reciprocal action of Saturn he will himself be accelerated. After conjunction, Jupiter will again accelerate Saturn, and be likewise retarded in the same degree, as before the first was retarded and the latter accelerated. Whatever inequalities besides are produced in the motion of Saturn by the action of Jupiter upon that planet, will be sufficiently rectified, by placing the focus of Saturn’s ellipsis, which should otherwise be in the sun, in the common center of gravity of the sun and Jupiter. And all the inequalities in the motion of Jupiter, caused by Saturn’s action upon him, are much less considerable than the irregularities of Saturn’s motion[166].
7. This one principle therefore of the planets having a power, as well as the sun, to cause bodies to gravitate towards them, which is proved by the motion of the secondary planets to obtain in fact, explains all the irregularities relating to the planets ever observed by astronomers.
8. Sir Isaac Newton after this proceeds to make an improvement in astronomy by applying this theory to the farther correction of their motions. For as we have here observed the planets to possess a principle of gravitation, as well as the sun; so it will be explained at large hereafter, that the third law of motion, which makes action and reaction equal, is to be applied in this case[167]; and that the sun does not only attract each planet, but is it self also attracted by them; the force, wherewith the planet is acted on, bearing to the force, wherewith the sun it self is acted on at the same time, the proportion, which the quantity of matter in the sun bears to the quantity of matter in the planet. From the action between the sun and planet being thus mutual Sir Isaac Newton proves that the sun and planet will describe about their common center of gravity similar ellipsis’s; and then that the transverse axis of the ellipsis described thus about the moveable sun, will bear to the transverse axis of the ellipsis, which would be described about the sun at rest in the same time, the same proportion as the quantity of solid matter in the sun and planet together bears to the first of two mean proportionals between this quantity and the quantity of matter in the sun only[168].
9. Above, where I shewed how to find a cube, that should bear any proportion to another cube[169], the lines F T and T S are two mean proportionals between E F and F G; and counting from E F, F T is called the first, and F S the second of those means. In numbers these mean proportionals are thus found.
Suppose A and B two numbers, and it be required to find C the first, and D the second of the two mean proportionals between them. First multiply A by it self, and the product multiply by B; then C will be the number which in arithmetic is called the cubic root of this last product; that is, the number C being multiplied by it self, and the product again multiplied by the same number C, will produce the product above mentioned. In like manner D is the cubic root of the product of B multiplied by it self, and the produce of that multiplication multiplied again by A.
10. It will be asked, perhaps, how this correction can be admitted, when the cause of the motions of the planets was before found by supposing the sun the center of the power, which acted upon them: for according to the present correction this power appears rather to be directed to their common center of gravity. But whereas the sun was at first concluded to be the center, to which the power acting on the planets was directed, because the spaces described round the sun in equal times were found to be equal; so Sir Isaac Newton proves, that if the sun and planet move round their common center of gravity, yet to an eye placed in the planet, the spaces, which will appear to be described about the sun, will have the same relation to the times of their description, as the real spaces would have, if the sun were at rest[170]. I farther asserted, that, supposing the planets to move round the sun at rest, and to be attracted by a power, which every where should act with degrees of strength reciprocally in the duplicate proportion of the distances; then the periods of the planets must observe the same relation to their distances, as astronomers find them to do. But here it must not be supposed, that the observations of astronomers absolutely agree without any the least difference; and the present correction will not cause a deviation from any one astronomer’s observations, so much as they differ from one another. For in Jupiter, where this correction is greatest, it hardly amounts to the 3000th part of the whole axis.
11. Upon this head I think it not improper to mention a reflection made by our excellent author upon these small inequalities in the planets motions; which contains under it a very strong philosophical argument against the eternity of the world. It is this, that these inequalities must continually increase by slow degrees, till they render at length the present frame of nature unfit for the purposes, it now serves[171]. And a more convincing proof cannot be desired against the present constitution’s having existed from eternity than this, that a certain period of years will bring it to an end. I am aware this thought of our author has been represented even as impious, and as no less than casting a reflection upon the wisdom of the author of nature, for framing a perishable work. But I think so bold an assertion ought to have been made with singular caution. For if this remark upon the increasing irregularities of the heavenly motions be true in fact, as it really is, the imputation must return upon the asserter, that this does detract from the divine wisdom. Certainly we cannot pretend to know all the omniscient Creator’s purposes in making this world, and therefore cannot undertake to determine how long he designed it should last. And it is sufficient, if it endure the time intended by the author. The body of every animal shews the unlimited wisdom of its author no less, nay in many respects more, than the larger frame of nature; and yet we see, they are all designed to last but a small space of time.
12. There need nothing more be said of the primary planets; the motions of the secondary shall be next considered.