12. There need nothing more be said of the primary planets; the motions of the secondary shall be next considered.
[Chap. III.]
Of the motion of the MOON and the other SECONDARY PLANETS.
THE excellency of this philosophy sufficiently appears from its extending in the manner, which has been related, to the minutest circumstances of the primary planets motions; which nevertheless bears no proportion to the vast success of it in the motions of the secondary; for it not only accounts for all the irregularities, by which their motions were known to be disturbed, but has discovered others so complicated, that astronomers were never able to distinguish them, and reduce them under proper heads; but these were only to be found out from their causes, which this philosophy has brought to light, and has shewn the dependence of these inequalities upon such causes in so perfect a manner, that we not only learn from thence in general, what those inequalities are, but are able to compute the degree of them. Of this Sir Is. Newton has given several specimens, and has moreover found means to reduce the moon’s motion so completely to rule, that he has framed a theory, from which the place of that planet may at all times be computed, very nearly or altogether as exactly, as the places of the primary planets themselves, which is much beyond what the greatest astronomers could ever effect.
[2.] The first thing demonstrated of these secondary planets is, that they are drawn towards their respective primary in the same manner as the primary planets are attracted by the sun. That each secondary planet is kept in its orbit by a power pointed towards the center of the primary planet, about which the secondary revolves; and that the power, by which the secondaries of the same primary are influenced, bears the same relation to the distance from the primary, as the power, by which the primary planets are guided, does in regard to the distance from the sun[172]. This is proved in the satellites of Jupiter and Saturn, because they move in circles, as far as we can observe, about their respective primary with an equable course, the respective primary being the center of each orbit: and by comparing the times, in which the different satellites of the same primary perform their periods, they are found to observe the same relation to the distances from their primary, as the primary planets observe in respect of their mean distances from the sun[173]. Here these bodies moving in circles with an equable motion, each satellite passes over equal parts of its orbit in equal portions of time; consequently the line drawn from the center of the orbit, that is, from the primary planet, to the satellite, will pass over equal spaces along with the satellite in equal portions of time; which proves the power, by which each satellite is held in its orbit, to be pointed towards the primary as a center[174]. It is also manifest that the centripetal power, which carries a body in a circle concentrical with the power, acts upon the body at all times with the same strength. But Sir Isaac Newton demonstrates that, when bodies are carried in different circles by centripetal powers directed to the centers of those circles, then, the degrees of strength of those powers are to be compared by considering the relation between the times, in which the bodies perform their periods through those circles[175]; and in particular he shews, that if the periodical times bear that relation, which I have just now asserted the satellites of the same primary to observe; then the centripetal powers are reciprocally in the duplicate proportion of the semidiameters of the circles, or in that proportion to the distances of the bodies from the centers[176]. Hence it follows that in the planets Jupiter and Saturn, the centripetal power in each decreases with the increase of distance, in the same proportion as the centripetal power appertaining to the sun decreases with the increase of distance. I do not here mean that this proportion of the centripetal powers holds between the power of Jupiter at any distance compared with the power of Saturn at any other distance; but only in the change of strength of the power belonging to the same planet at different distances from him. Moreover what is here discovered of the planets Jupiter and Saturn by means of the different satellites, which revolve round each of them, appears in the earth by the moon alone; because she is found to move round the earth in an ellipsis after the same manner as the primary planets do about the sun; excepting only some small irregularities in her motion, the cause of which will be particularly explained in what follows, whereby it will appear, that they are no objection against the earth’s acting on the moon in the same manner as the sun acts on the primary planets; that is, as the other primary planets Jupiter and Saturn act upon their satellites. Certainly since these irregularities can be otherwise accounted for, we ought not to depart from that rule of induction so necessary in philosophy, that to like bodies like properties are to be attributed, where no reason to the contrary appears. We cannot therefore but ascribe to the earth the same kind of action upon the moon, as the other primary planets Jupiter and Saturn have upon their satellites; which is known to be very exactly in the proportion assigned by the method of comparing the periodical times and distances of all the satellites which move about the same planet; this abundantly compensating our not being near enough to observe the exact figure of their orbits. For if the little deviation of the moon’s orbit orbit from a true permanent ellipsis arose from the action of the earth upon the moon not being in the exact reciprocal duplicate proportion of the distance, were another moon to revolve about the earth, the proportion between the periodical times of this new moon, and the present, would discover the deviation from the mentioned proportion much more manifestly.
3. By the number of satellites, which move round Jupiter and Saturn, the power of each of these planets is measured in a great diversity of distance; for the distance of the outermost satellite in each of these planets exceeds several times the distance of the innermost. In Jupiter the astronomers have usually placed the innermost satellite at a distance from the center of that planet equal to about 5⅔ of the semidiameters of Jupiter’s body, and this satellite performs its revolution in about 1 day 18½ hours. The next satellite, which revolves round Jupiter in about 3 days 13⅕ hours, they place at the distance from Jupiter of about 9 of that planet’s semidiameters. To the third satellite, which performs its period nearly in 7 days 3¾ hours, they assign the distance of about 14⅖ semidiameters. But the outermost satellite they remove to 25⅓ semidiameters, and this satellite makes its period in about 16 days 16½ hours[177]. In Saturn there is still a greater diversity in the distance of the several satellites. By the observations of the late Cassini, a celebrated astronomer in France, who first discovered all these satellites, except one known before, the innermost is distant about 4½ of Saturn’s semidiameters from his center, and revolves round in about 1 day 21⅓ hours. The next satellite is distant about 5¾ semidiameters, and makes its period in about 2 days 17⅔ hours. The third is removed to the distance of about 8 semidiameters, and performs its revolution in near 4 days 12½ hours. The fourth satellite discovered first by the great Huygens, is near 18⅔ semidiameters, and moves round Saturn in about 15 days 22⅔ hours. The outermost is distant 56 semidiameters, and makes its revolution in about 79 days 7⅘ hours[178]. Besides these satellites, there belongs to the planet Saturn another body of a very singular kind. This is a shining, broad, and flat ring, which encompasses the planet round. The diameter of the outermost verge of this ring is more than double the diameter of Saturn. Huygens, who first described this ring, makes the whole diameter thereof to bear to the diameter of Saturn the proportion of 9 to 4. The late reverend Mr. Pound makes the proportion something greater, viz. that of 7 to 3. The distances of the satellites of this planet Saturn are compared by Cassini to the diameter of the ring. His numbers I have reduced to those above, according to Mr. Pound’s proportion between the diameters of Saturn and of his ring. As this ring appears to adhere no where to Saturn, so the distance of Saturn from the inner edge of the ring seems rather greater than the breadth of the ring. The distances, which have here been given, of the several satellites, both for Jupiter and Saturn, may be more depended on in relation to the proportion, which those belonging to the same primary planet bear one to another, than in respect to the very numbers, that have been here set down, by reason of the difficulty there is in measuring to the greatest exactness the diameters of the primary planets; as will be explained hereafter, when we come to treat of telescopes[179]. By the observations of the forementioned Mr. Pound, in Jupiter the distance of the innermost satellite should rather be about 6 semidiameters, of the second 9-½, of the third 15, and of the outermost 26⅔[180]; and in Saturn the distance of the innermost satellite 4 semidiameters, of the next 6¼, of the third 8¾, of the fourth 20⅓, and of the fifth 59[181]. However the proportion between the distances of the satellites in the same primary is the only thing necessary to the point we are here upon.
4. But moreover the force, wherewith the earth acts in different distances, is confirmed from the following consideration, yet more expresly than by the preceding analogical reasoning. It will appear, that if the power of the earth, by which it retains the moon in her orbit, be supposed to act at all distances between the earth and moon, according to the forementioned rule; this power will be sufficient to produce upon bodies, near the surface of the earth, all the effects ascribed to the principle of gravity. This is discovered by the following method. Let A (in fig. 94.) represent the earth, B the moon, B C D the moon’s orbit, which differs little from a circle, of which A is the center. If the moon in B were left to it self to move with the velocity, it has in the point B, it would leave the orbit, and proceed right forward in the line B E, which touches the orbit in B. Suppose the moon would upon this condition move from B to E in the space of one minute of time. By the action of the earth upon the moon, whereby it is retained in its orbit, the moon will really be found at the end of this minute in the point F, from whence a straight line drawn to A shall make the space B F A in the circle equal to the triangular space B E A; so that the moon in the time wherein it would have moved from B to E, if left to it self, has been impelled towards the earth from E to F. And when the time of the moon’s passing from B to F is small, as here it is only one minute, the distance between E and F scarce differs from the space, through which the moon would descend in the same time, if it were to fall directly down from B toward A without any other motion. A B the distance of the earth and moon is about 60 of the earth’s semidiameters, and the moon completes her revolution round the earth in about 27 days 7 hours and 43 minutes: therefore the space E F will here be found by computation to be about 16⅛ feet. Consequently, if the power, by which the moon is retained in its orbit, be near the surface of the earth greater, than at the distance of the moon in the duplicate proportion of that distance; the number of feet, a body would descend near the surface of the earth by the action of this power upon it in one minute of time, would be equal to 16⅛ multiplied twice into the number 60, that is, equal to 58050. But how fast bodies fall near the surface of the earth may be known by the pendulum[182]; and by the exactest experiments they are found to descend the space of 16⅛ feet in a second of time; and the spaces described by falling bodies being in the duplicate proportion of the times of their fall[183], the number of feet, a body would describe in its fall near the surface of the earth in one minute of time, will be equal to 16⅛ twice multiplied by 60, the same as would be caused by the power which acts upon the moon.
5. In this computation the earth is supposed to be at rest, whereas it would have been more exact to have supposed it to move, as well as the moon, about their common center of gravity; as will easily be understood, by what has been said in the preceding chapter, where it was shewn, that the sun is subjected to the like motion about the common center of gravity of it self and the planets. The action of the sun upon the moon, which is to be explain’d in what follows, is likewise here neglected: and Sir Isaac Newton shews, if you take in both these considerations, the present computation will best agree to a somewhat greater distance of the moon and earth, viz. to 60½ semidiameters of the earth, which distance is more conformable to astronomical observations.
[6.] These computations afford an additional proof, that the action of the earth observes the same proportion to the distance, which is here contended for. Before I said, it was reasonable to conclude so by induction from the planets Jupiter and Saturn; because they act in that manner. But now the same thing will be evident by drawing no other consequence from what is seen in those planets, than that the power, by which the primary planets act on their secondary, is extended from the primary through the whole interval between, so that it would act in every part of the intermediate space. In Jupiter and Saturn this power is so far from being confined to a small extent of distance, that it not only reaches to several satellites at very different distances, but also from one planet to the other, nay even through the whole planetary system[184]. Consequently there is no appearance of reason, why this power should not act at all distances, even at the very surfaces of these planets as well as farther off. But from hence it follows, that the power, which retains the moon in her orbit, is the same, as causes bodies near the surface of the earth to gravitate. For since the power, by which the earth acts on the moon, will cause bodies near the surface of the earth to descend with all the velocity they are found to do, it is certain no other power can act upon them besides; because if it did, they must of necessity descend swifter. Now from all this it is at length very evident, that the power in the earth, which we call gravity, extends up to the moon, and decreases in the duplicate proportion of the increase of the distance from the earth.
[7.] This finishes the discoveries made in the action of the primary planets upon their secondary. The next thing to be shewn is, that the sun acts upon them likewise: for this purpose it is to be observed, that if to the motion of the satellite, whereby it would be carried round its primary at rest, be superadded the same motion both in regard to velocity and direction, as the primary it self has, it will describe about the primary the same orbit, with as great regularity, as if the primary was indeed at rest. The cause of this is that law of motion, which makes a body near the surface of the earth, when let fall, to descend perpendicularly, though the earth be in so swift a motion, that if the falling body did not partake of it, its descent would be remarkably oblique; and that a body projected describes in the most regular manner the same parabola, whether projected in the direction, in which the earth moves, or in the opposite direction, if the projecting force be the same[185]. From this we learn, that if the satellite moved about its primary with perfect regularity, besides its motion about the primary, it would participate of all the motion of its primary; have the same progressive velocity, with which the primary is carried about the sun; and be impelled with the same velocity as the primary towards the sun, in a direction parallel to that impulse of its primary. And on the contrary, the want of either of these, in particular of the impulse towards the sun, will occasion great inequalities in the motion of the secondary planet. The inequalities, which would arise from the absence of this impulse towards the sun are so great, that by the regularity, which appears in the motion of the secondary planets, it is proved, that the sun communicates, the same velocity to them by its action, as it gives to their primary at the same distance. For Sir Isaac Newton informs us, that upon examination he found, that if any of the satellites of Jupiter were attracted by the sun more or less, than Jupiter himself at the same distance, the orbit of that satellite, instead of being concentrical to Jupiter, must have its center at a greater or less distance, than the center of Jupiter from the sun, nearly in the subduplicate proportion of the difference between the sun’s action upon the satellite, and upon Jupiter; and therefore if any satellite were attracted by the sun but 1/1000 part more or less, than Jupiter is at the same distance, the center of the orbit of that satellite would be distant from the center of Jupiter no less than a fifth part of the distance of the outermost satellite from Jupiter[186]; which is almost the whole distance of the innermost satellite. By the like argument the satellites of Saturn gravitate towards the sun, as much as Saturn it self at the same distance; and the moon as much as the earth.
8. Thus is proved, that the sun acts upon the secondary planets, as much as upon the primary at the same distance: but it was found in the last chapter, that the action of the sun upon bodies is reciprocally in the duplicate proportion of the distance; therefore the secondary planets being sometimes nearer to the sun than the primary, and sometimes more remote, they are not alway acted upon in the same degree with their primary, but when nearer to the sun, are attracted more, and when farther distant, are attracted less. Hence arise various inequalities in the motion of the secondary planets[187].