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].
9. Some of these inequalities would take place, though the moon, if undisturbed by the sun, would have moved in a circle concentrical to the earth, and in the plane of the earth’s motion; others depend on the elliptical figure, and the oblique situation of the moon’s orbit. One of the first kind is, that the moon is caused so to move, as not to describe equal spaces in equal times, but is continually accelerated, as she passes from the quarter to the new or full, and is retarded again by the like degrees in returning from the new and full to the next quarter. Here we consider not so much the absolute, as the apparent motion of the moon in respect to us.
10. The principles of astronomy teach how to distinguish these two motions. Let S (in fig. 95.) represent the sun, A the earth moving in its orbit B C, D E F G the moon’s orbit, the place of the moon H. Suppose the earth to have moved from A to I. Because it has been shewn, that the moon partakes of all the progressive motion of the earth; and likewise that the sun attracts both the earth and moon equally, when they are at the same distance from it, or that the mean action of the sun upon the moon is equal to its action upon the earth: we must therefore consider the earth as carrying about with it the moon’s orbit; so that when the earth is removed from A to I, the moon’s orbit shall likewise be removed from its former situation into that denoted by K L M N. But now the earth being in I, if the moon were found in O, so that O I should be parallel to H A, though the moon would really have moved from H to O, yet it would not have appeared to a spectator upon the earth to have moved at all, because the earth has moved as much it self; so that the moon would still appear in the same place with respect to the fixed stars. But if the moon be observed in P, it will then appear to have moved, its apparent motion being measured by the angle under O I P. And if the angle under P I S be less than the angle under H A S, the moon will have approached nearer to its conjunction with the sun.
11. To come now to the explication of the mentioned inequality in the moon’s motion: let S (in fig. 96.) represent the sun, A the earth, B C D E the moon’s orbit, C the place of the moon, when in the latter quarter. Here it will be nearly at the same distance from the sun, as the earth is. In this case therefore they will both be equally attracted, the earth in the direction A S, and the moon in the direction C S. Whence as the earth in moving round the sun is continually descending toward it, so the moon in this situation must in any equal portion of time descend as much; and therefore the position of the line A C in respect of A S, and the change, which the moon’s motion produces in the angle under C A S, will not be altered by the sun.
12. But now as soon as ever the moon is advanced from the quarter toward the new or conjunction, suppose to G, the action of the sun upon it will have a different effect. Here, were the sun’s action upon the moon to be applied in the direction G H parallel to A S, if its action on the moon were equal to its action on the earth, no change would be wrought by the sun on the apparent motion of the moon round the earth. But the moon receiving a greater impulse in G than the earth receives in A, were the sun to act in the direction G H, yet it would accelerate the description of the space D A G, and cause the angle under G A D to decrease faster, than otherwise it would. The sun’s action will have this effect upon account of the obliquity of its direction to that, in which the earth attracts the moon. For the moon by this means is drawn by two forces oblique to each other, one drawing from G toward A, the other from G toward H, therefore the moon must necessarily be impelled toward D. Again, because the sun does not act in the direction G H parallel to S A, but in the direction G S oblique to it, the sun’s action on the moon will by reason of this obliquity farther contribute to the moon’s acceleration. Suppose the earth in any short space of time would have moved from A to I, if not attracted by the sun; the point I being in the straight line C E, which touches the earth’s orbit in A. Suppose the moon in the same time would have moved in her orbit from G to K, and besides have partook of all the progressive motion of the earth. Then if K L be drawn parallel to A I, and taken equal to it, the moon, if not attracted by the sun, would be found in L. But the earth by the sun’s action is removed from I. Suppose it were moved down to M in the line I M N parallel to S A, and if the moon were attracted but as much, and in the same direction, as the earth is here supposed to be attracted, so as to have descended during the same time in the line L O, parallel also to A S, down as far as P, till L P were equal to I M; the angle under P M N would be equal to that under L I N, that is, the moon will appear advanced no farther forward, than if neither it nor the earth had been subject to the sun’s action. But this is upon the supposition, that the action of the sun upon the moon and earth were equal; whereas the moon being acted upon more than the earth, did the sun’s action draw the moon in the line L O parallel to A S, it would draw it down so far as to make L P greater than I M; whereby the angle under P M N will be rendred less, than that under L I N. But moreover, as the sun draws the earth in a direction oblique to I N, the earth will be found in its orbit somewhat short of the point M; however the moon is attracted by the sun still more out of the line L O, than the earth is out of the line I N; therefore this obliquity of the sun’s action will yet farther diminish the angle under P M N.
13. Thus the moon at the point G receives an impulse from the sun, whereby her motion is accelerated. And the sun producing this effect in every place between the quarter and the conjunction, the moon will move from the quarter with a motion continually more and more accelerated; and therefore by acquiring from time to time additional degrees of velocity in its orbit, the spaces, which are described in equal times by the line drawn from the earth to the moon, will not be every where equal, but those toward the conjunction will be greater, than those toward the quarter. But now in the moon’s passage from the conjunction D to the next quarter the sun’s action will again retard the moon, till at the next quarter in E it be restored to the first velocity, which it had in C.
14. Again as the moon moves from E to the full or opposition to the sun in B, it is again accelerated, the deficiency of the sun’s action upon the moon, from what it has upon the earth, producing here the same effect as before the excess of its action. Consider the moon in Q, moving from E towards B. Here if the moon were attracted by the sun in a direction parallel to A S, yet being acted on less than the earth, as the earth descends toward the sun, the moon will in some measure be left behind. Therefore Q F being drawn parallel to S B, a spectator on the earth would see the moon move, as if attracted from the point Q in the direction Q F with a degree of force equal to that, whereby the sun’s action on the moon falls short of its action on the earth. But the obliquity of the sun’s action has also here an effect. In the time the earth would have moved from A to I without the influence of the sun, let the moon have moved in its orbit from Q to R. Drawing therefore R T parallel to A I, and equal to the same, for the like reason as before, the moon by the motion of its orbit, if not at all attracted by the sun, must be found in T; and therefore, if attracted in a direction parallel to S A, would be in the line T V parallel to A S; suppose in W. But the moon in Q being farther off the sun than the earth, it will be less attracted, that is, T W will be less than I M, and if the line S M be prolonged toward X, the angle under X M W will be less than that under X I T. Thus by the sun’s action the moon’s passage from the quarter to the full would be accelerated, if the sun were to act on the earth and moon in a direction parallel to A S: and the obliquity of the sun’s action will still more increase this acceleration. For the action of the sun on the moon is oblique to the line S A the whole time of the moon’s passage from Q to T, and will carry the moon out of the line T V toward the earth. Here I suppose the time of the moon’s passage from Q to T so short, that it shall not pass beyond the line S A. The earth also will come a little short of the line I N, as was said before. From these causes the angle under X M W will be still farther lessened.
15. The moon in passing from the opposition B to the next quarter will be retarded again by the same degrees, as it is accelerated before its appulse to the opposition. Because this action of the sun, which in the moon’s passage from the quarter to the opposition causes it to be extraordinarily accelerated, and diminishes the angle, which measures its distance from the opposition; will make the moon slacken its pace afterwards, and retard the augmentation of the same angle in its passage from the opposition to the following quarter; that is, will prevent that angle from increasing so fast, as otherwise it would. And thus the moon, by the sun’s action upon it, is twice accelerated and twice restored to its first velocity, every circuit it makes round the earth. This inequality of the moon’s motion about the earth is called by astronomers its variation.
16. The next effect of the sun upon the moon is, that it gives the orbit of the moon in the quarters a greater degree of curvature, than it would receive from the action of the earth alone; and on the contrary in the conjunction and opposition the orbit is less inflected.
17. When the moon is in conjunction with the sun in the point D, the sun attracting the moon more forcibly than it does the earth, the moon by that means is impelled less toward the earth, than otherwise it would be, and so the orbit is less incurvated; for the power, by which the moon is impelled toward the earth, being that, by which it is inflected from a rectilinear course, the less that power is, the less it will be inflected. Again, when the moon is in the opposition in B, farther removed from the sun than the earth is; it follows then, though the earth and moon are both continually descending to the sun, that is, are drawn by the sun toward it self out of the place they would otherwise move into, yet the moon descends with less velocity than the earth; insomuch that the moon in any given space of time from its passing the point of opposition will have less approached the earth, than otherwise it would have done, that is, its orbit in respect of the earth will approach nearer to a straight line. In the last place, when the moon is in the quarter in F, and equally distant from the sun as the earth, we observed before, that the earth and moon would descend with equal pace toward the sun, so as to make no change by that descent in the angle under F A S; but the length of the line F A must of necessity be shortned. Therefore the moon in moving from F toward the conjunction with the sun will be impelled more toward the earth by the sun’s action, than it would have been by the earth alone, if neither the earth nor moon had been acted on by the sun; so that by this additional impulse the orbit is rendred more curve, than it would otherwise be. The same effect will also be produced in the other quarter.
18. Another effect of the sun’s action, consequent upon this we have now explained, is, that though the moon undisturbed by the sun might move in a circle having the earth for its center; by the sun’s action, if the earth were to be in the very middle or center of the moon’s orbit, yet the moon would be nearer the earth at the new and full, than in the quarters. In this probably will at first appear some difficulty, that the moon should come nearest to the earth, where it is least attracted to it, and be farthest off when most attracted. Which yet will appear evidently to follow from that very cause, by considering what was last shewn, that the orbit of the moon in the conjunction and opposition is rendred less curve; for the less curve the orbit of the moon is, the less will the moon have descended from the place it would move into, without the action of the earth. Now if the moon were to move from any place without farther disturbance from that action, since it would proceed in the line, which would touch its orbit in that place, it would recede continually from the earth; and therefore if the power of the earth upon the moon, be sufficient to retain it at the same distance, this diminution of that power will cause the distance to increase, though in a less degree. But on the other hand in the quarters, the moon, being pressed more towards the earth than by the earth’s single action, will be made to approach it; so that in passing from the conjunction or opposition to the quarters the moon ascends from the earth, and in passing from the quarters to the conjunction and opposition it descends again, becoming nearer in these last mentioned places than in the other.
19. All these forementioned inequalities are of different degrees, according as the sun is more or less distant from the earth; greater when the earth is nearest the sun, and less when it is farthest off. For in the quarters, the nearer the moon is to the sun, the greater is the addition to the earth’s action upon it by the power of the sun; and in the conjunction and opposition, the difference between the sun’s action upon the earth and upon the moon is likewise so much the greater.
20. This difference in the distance between the earth and the sun produces a farther effect upon the moon’s motion; causing the orbit to dilate when less remote from the sun, and become greater, than when at a farther distance. For it is proved by Sir Isaac Newton, that the action of the sun, by which it diminishes the earth’s power over the moon, in the conjunction or opposition, is about twice as great, as the addition to the earth’s action by the sun in the quarters[188]; so that upon the whole, the power of the earth upon the moon is diminished by the sun, and therefore is most diminished, when the action of the sun is strongest: but as the earth by its approach to the sun has its influence lessened, the moon being less attracted will gradually recede from the earth; and as the earth in its recess from the sun recovers by degrees its former power, the orbit of the moon must again contract. Two consequences follow from hence: the moon will be most remote from the earth, when the earth is nearest the sun; and also will take up a longer time in performing its revolution through the dilated orbit, than through the more contracted.
21. These irregularities the sun would produce in the moon, if the moon, without being acted on unequally by the sun, would describe a perfect circle about the earth, and in the plane of the earth’s motion; but though neither of these suppositions obtain in the motion of the moon, yet the forementioned inequalities will take place, only with some difference in respect to the degree of them; but the moon by not moving in this manner is subject to some other inequalities also. For as the moon describes, instead of a circle concentrical to the earth, an ellipsis, with the earth in one focus, that ellipsis will be subjected to various changes. It can neither preserve constantly the same position, nor yet the same figure; and because the plane of this ellipsis is not the same with that of the earth’s orbit, the situation of the plane, wherein the moon moves, will continually change; neither the line in which it intersects the plane of the earth’s orbit, nor the inclination of the planes to each other, will remain for any time the same. All these alterations offer themselves now to be explained.
22. I shall first consider the changes which are made in the plane of the moon’s orbit. The moon not moving in the same plane with the earth, the sun is seldom in the plane of the moon’s orbit, viz. only when the line made by the common intersection of the two planes, if produced, will pass through the sun, as is represented in fig. 97. where S denotes the sun; T the earth; A T B the earth’s orbit described upon the plane of this scheme; C D E F the moon’s orbit, the part C D E being raised above, and the part C F E depressed under the plane of this scheme. Here the line C E, in which the plane of this scheme, that is, the plane of the earth’s orbit and the plane of the moon’s orbit intersect each other, being continued passes through the sun in S. When this happens, the action of the sun is directed in the plane of the moon’s orbit, and cannot draw the moon out of this plane, as will evidently appear to any one that shall consider the present scheme: for suppose the moon in G, and let a straight line be drawn from G to S, the sun draws the moon in the direction of this line from G toward S: but this line lies in the plane of the orbit; and if it be prolonged from S beyond G, the continuation of it will lie on the plane C D E; for the plane itself, if sufficiently extended, will pass through the sun. But in other cases the obliquity of the sun’s action to the plane of the orbit will cause this plane continually to change.
23. Suppose in the first place, the line, in which the two planes intersect each other, to be perpendicular to the line which joins the earth and sun. Let T (in fig. 98, 99, 100, 101.) represent the earth; S the sun; the plane of this scheme the plane of the earth’s motion, in which both the sun and earth are placed. Let A C be perpendicular to S T, which joins the earth and sun; and let the line A C be that, in which the plane of the moon’s orbit intersects the plane of the earth’s motion. To the center T describe in the plane of the earth’s motion the circle A B C D. And in the plane of the moon’s orbit describe the circle A E C F, one half of which A E C will be elevated above the plane of this scheme, the other half A F C as much depressed below it.
24. Now suppose the moon to set forth from the point A (in fig. 98.) in the direction of the plane A E C. Here she will be continually drawn out of this plane by the action of the sun: for this plane A E C, if extended, will not pass through the sun, but above it; so that the sun, by drawing the moon directly toward it self, will force it continually more and more from that plane towards the plane of the earth’s motion, in which it self is; causing it to describe the line A K G H I, which will be convex to the plane A E C, and concave to the plane of the earth’s motion. But here this power of the sun, which is said to draw the moon toward the plane of the earth’s motion, must be understood principally of so much only of the sun’s action upon the moon, as it exceeds the action of the same upon the earth. For suppose the preceding figure to be viewed by the eye, placed in the plane of that scheme, and in the line C T A on the side of A, the plane A B C D will appear as the straight line D T B, (in fig. 102.) and the plane A E C F as another straight line F E; and the curve line A K G H I under the form of the line T K G H I.
Now it is plain, that the earth and moon being both attracted by the sun, if the sun’s action upon both was equally strong, the earth T, and with it the plane A E C F or line F T E in this scheme, would be carried toward the sun with as great a pace as the moon, and therefore the moon not drawn out of it by the sun’s action, excepting only from the small obliquity of the direction of this action upon the moon to that of the sun’s action upon the earth, which arises from the moon’s being out of the plane of the earth’s motion, and is not very considerable; but the action of the sun upon the moon being greater than upon the earth, all the time the moon is nearer to the sun than the earth is, it will be drawn from the plane A E C or the line T E by that excess, and made to describe the curve line A G I or T G I. But it is the custom of astronomers, instead of considering the moon as moving in such a curve line, to refer its motion continually to the plane, which touches the true line wherein it moves, at the point where at any time the moon is. Thus when the moon is in the point A, its motion is considered as being in the plane A E C, in whose direction it then essaies to move; and when in the point K (in fig. 99.) its motion is referred to the plane, which passes through the earth, and touches the line A K G H I in the point K. Thus the moon in passing from A to I will continually change the plane of her motion. In what manner this change proceeds, I shall now particularly explain.
25. Let the plane, which touches the line A K I in the point K (in fig. 99.) intersect the plane of the earth’s orbit in the line L T M. Then, because the line A K I is concave to the plane A B C, it falls wholly between that plane, and the plane which touches it in K; so that the plane M K L will cut the plane A E C, before it meets with the plane of the earth’s motion; suppose in the line Y T, and the point A will fall between K and L. With a semidiameter equal to T Y or T L describe the semicircle L Y M. Now to a spectator on the earth the moon, when in A, will appear to move in the circle A E C F, and, when in K, will appear to be moving in the semicircle L Y M. The earth’s motion is performed in the plane of this scheme, and to a spectator on the earth the sun will appear always moving in that plane. We may therefore refer the apparent motion of the sun to the circle A B C D, described in this plane about the earth. But the points where this circle, in which the sun seems to move, intersects the circle in which the moon is seen at any time to move, are called the nodes of the moon’s orbit at that time. When the moon is seen moving in the circle A E C D, the points A and C are the nodes of the orbit; when she appears in the semicircle L Y M, then L and M are the nodes. Now here it appears, from what has been said, that while the moon has moved from A to K, one of the nodes has been carried from A to L, and the other as much from C to M. But the motion from A to L, and from C to M, is backward in regard to the motion of the moon, which is the other way from A to K, and from thence toward C.
26. Farther the angle, which the plane, wherein the moon at any time appears, makes with the plane of the earth’s motion, is called the inclination of the moon’s orbit at that time. And I shall now proceed to shew, that this inclination of the orbit, when the moon is in K, is less than when she was in A; or, that the plane L Y M, which touches the line of the moon’s motion in K, makes a less angle with the plane of the earth’s motion or with the circle A B C D, than the plane A E C makes with the same. The semicircle L Y M intersects the semicircle A E C in Y; and the arch A Y is less than L Y, and both together less than half a circle. But it is demonstrated by the writers on that part of astronomy, which is called the doctrine of the sphere, that when a triangle is made, as here, by three arches of circles A L, A Y, and Y L, the angle under Y A B without the triangle is greater than the angle under Y L A within, if the two arches A Y, Y L taken together do not amount to a semicircle; if the two arches make a complete semicircle, the two angles will be equal; but if the two arches taken together exceed a semicircle, the inner angle under Y L A is greater than the other[189]. Here therefore the two arches A Y and L Y together being less than a semicircle, the angle under A L Y is less, than the angle under B A E. But from the doctrine of the sphere it is also evident, that the angle under A L Y is equal to that, in which the plane of the circle L Y K M, that is, the plane which touches the line A K G H I in K, is inclined to the plane of the earth’s motion A B C; and the angle under B A E is equal to that, in which the plane A E C is inclined to the same plane. Therefore the inclination of the former plane is less than the inclination of the latter.
27. Suppose now the moon to be advanced to the point G (in fig. 100.) and in this point to be distant from its node a quarter part of the whole circle; or in other words, to be in the midway between its two nodes. And in this case the nodes will have receded yet more, and the inclination of the orbit be still more diminished: for suppose the line A K G H I to be touched in the point G by a plane passing through the earth T: let the intersection of this plane with the plane of the earth’s motion be the line W T O, and the line T P its intersection with the plane L K M. In this plane let the circle N G O be described with the semidiameter T P or N T cutting the other circle L K M in P. Now the line A K G I is convex to the plane L K M, which touches it in K; and therefore the plane N G O, which touches it in G, will intersect the other touching plane between G and K; that is, the point P will fall between those two points, and the plane continued to the plane of the earth’s motion will pass beyond L; so that the points N and O, or the places of the nodes, when the moon is in G, will be farther from A and C than L and M, that is, will have moved farther backward. Besides, the inclination of the plane N G O to the plane of the earth’s motion A B C is less, than the inclination of the plane L K M to the same; for here also the two arches L P and N P taken together are less than a semicircle, each of these arches being less than a quarter of a circle; as appears, because G N, the distance of the moon in G from its node N, is here supposed to be a quarter part of a circle.
28. After the moon is passed beyond G, the case is altered; for then these arches will be greater than quarters of the circle, by which means the inclination will be again increased, tho’ the nodes still go on to move the same way. Suppose the moon in H, (in fig. 101.) and that the plane, which touches the line A K G I in H, intersects the plane of the earth’s motion in the line Q T R, and the plane N G O in the line T V, and besides that the circle Q H R be described in that plane; then, for the same reason as before, the point V will fall between H and G, and the plane R V Q will pass beyond the last plane O V N, causing the points Q and R to fall farther from A and C than N and O. But the arches N V, V Q are each greater than a quarter of a circle, N V the least of them being greater than G N, which is a quarter of a circle; and therefore the two arches N V and V Q together exceed a semicircle; consequently the angle under B Q V will be greater, than that under B N V.
29. In the last place, when the moon is by this attraction of the sun, drawn at length into the plane of the earth’s motion, the node will have receded yet more, and the inclination be so much increased, as to become somewhat more than at first: for the line A K G H I being convex to all the planes, which touch it, the part H I will wholly fall between the plane Q V R and the plane A B C; so that the point I will fall between B and R; and drawing I T W, the point W will be farther remov’d from A than Q. But it is evident, that the plane, which passes through the earth T, and touches the line A G I in the point I, will cut the plane of the earth’s motion A B C D in the line I T W, and be inclined to the same in the angle under H I B; so that the node, which was first in A, after having passed into L, N and Q, comes at last into the point W; as the node which was at first in C has passed successively from thence through the points M, O and R to I: but the angle under H I B, which is now the inclination of the orbit to the plane of the ecliptic, is manifestly not less than the angle under E C B or E A B, but rather something greater.
30. Thus the moon in the case before us, while it passes from the plane of the earth’s motion in the quarter, till it comes again into the same plane, has the nodes of its orbit continually moved backward, and the inclination of its orbit is at first diminished, viz. till it comes to G in fig. 100, which is near to its conjunction with the sun, but afterwards is increased again almost by the same degrees, till upon the moon’s arrival again to the plane of the earth’s motion, the inclination of the orbit is restored to something more than its first magnitude, though the difference is not very great, because the points I and C are not far distant from each other[190].
31. After the same manner, if the moon had departed from the quarter in C, it should have described the curve line C X W (in fig. 98.) between the planes A F C and A D C, which would be convex to the former of those planes, and concave to the latter; so that, here also, the nodes should continually recede, and the inclination of the orbit gradually diminish more and more, till the moon arrived near its opposition to the sun in X; but from that time the inclination should again increase, till it became a little greater than at first. This will easily appear, by considering, that as the action of the sun upon the moon, by exceeding its action upon the earth, drew it out of the plane A E C towards the sun, while the moon passed from A to I; so, during its passage from C to W, the moon being all that time farther from the sun than the earth, it will be attracted less; and the earth, together with the plane A E C F, will as it were be drawn from the moon, in such sort, that the path the moon describes shall appear from the earth, as it did in the former case by the moon’s being drawn away.
32. These are the changes, which the nodes and the inclination of the moon’s orbit undergo, when the nodes are in the quarters; but when the nodes by their motion, and the motion of the sun together, come to be situated between the quarter and conjunction or opposition, their motion and the change made in the inclination of the orbit are somewhat different.
33. Let A G C H (in fig. 103.) be a circle described in the plane of the earth’s motion, having the earth in T for its center. Let the point opposite to the sun be A, and the point G a fourth part of the circle distant from A. Let the nodes of the moon’s orbit be situated in the line B T D, and B the node, falling between A, the place where the moon would be in the full, and G the place where the moon would be in the quarter. Suppose B E D F to be the plane, in which the moon essays to move, when it proceeds from the point B. Because the moon in B is more distant from the sun than the earth, it shall be less attracted by the sun, and shall not descend towards the sun so fast as the earth: consequently it shall quit the plane B E D F, which we suppose to accompany the earth, and describe the line B I K convex thereto, till such time as it comes to the point K, where it will be in the quarter: but from thenceforth being more attracted than the earth, the moon shall change its course, and the following part of the path it describes shall be concave to the plane B E D or B G D, and shall continue concave to the plane B G D, till it crosses that plane in L, just as in the preceding case. Now I say, while the moon is passing from B to K, the nodes, contrary to what was found in the foregoing case, will proceed forward, or move the same way with the moon[191]; and at the same time the inclination of the orbit will increase[192].
34. When the moon is in the point I, let the plane M I N pass through the earth T, and touch the path of the moon in I, cutting the plane of the earth’s motion, in the line M T N, and the plane B E D in the line T O. Because the line B I K is convex to the plane B E D, which touches it in B, the plane N I M must cross the plane D E B, before it meets the plane C G B; and therefore the point M will fall from B towards G, and the node of the moon’s orbit being translated from B to M is moved forward.
35. I say farther, the angle under O M G, which the plane M O N makes with the plane B G C, is greater than the angle under O B G, which the plane B O D makes with the same. This appears from what has been already explained; because the arches B O, O M are each less than the quarter of a circle, and therefore taken both together are less than a semicircle.
36. Again, when the moon is come to the point K in its quarter, the nodes will be advanced yet farther forward, and the inclination of the orbit also more augmented. Hitherto the moon’s motion has been referred to the plane, which passing through the earth touches the path of the moon in the point, where the moon is, according to what was asserted at the beginning of this discourse upon the nodes, that it is the custom of astronomers so to do. But here in the point K no such plane can be found; on the contrary, seeing the line of the moon’s motion on one side the point K is convex to the plane B E D, and on the other side concave to the same, no plane can pass through the points T and K but will cut the line B K L in that point. Therefore instead of such a touching plane, we must here make use of what is equivalent, the plane P K Q, with which the line B K L shall make a less angle than with any other plane; for this plane does as it were touch the line B K in the point K, since it so cuts it, that no other plane can be drawn so, as to pass between the line B K and the plane P K Q. But now it is evident, that the point P, or the node, is removed from M towards G, that is, has moved yet farther forward; and it is likewise as manifest, that the angle under K P G, or the inclination of the moon’s orbit in the point K, is greater than the angle under I M G, for the reason so often assigned.
37. After the moon has passed the quarter, the path of the moon being concave to the plane A G C H, the nodes, as in the preceding case, shall recede, till the moon arrives at the point L; which shews, that considering the whole time of the moon’s passing from B to L, at the end of that time the nodes shall be found to have receded, or to be placed backwarder, when the moon is in L, than when it was in B. For the moon takes a longer time in passing from K to L, than in passing from B to K; and therefore the nodes continue to recede a longer time, than they moved forwards; so that their recess must surmount their advance.
38. In the same manner, while the moon is in its passage from K to L, the inclination of the orbit shall diminish, till the moon comes to the point, in which it is one quarter part of a circle distant from its node; suppose in the point R; and from that time the inclination shall again increase. Since therefore the inclination of the orbit increases, while the moon is passing from B to K, and diminishes itself again only, while the moon is passing from K to R, and then augments again, till the moon arrive in L; while the moon is passing from B to L, the inclination of the orbit is much more increased than diminished, and will be distinguishably greater, when the moon is come to L, than when it set out from B.
39. In like manner, while the moon is passing from L on the other side the plane A G C H, the node shall advance forward, as long as the moon is between the point L and the next quarter; but afterwards it shall recede, till the moon come to pass the plane A G C H again in the point V, between B and A: and because the time between the moon’s passing from L to the next quarter is less, than the time between that quarter and the moon’s coming to the point V, the node shall have more receded than advanced; so that the point V will be nearer to A, than L is to C. So also the inclination of the orbit, when the moon is in V, will be greater, than when the moon was at L; for this inclination increases all the time the moon is between L and the next quarter; it decreases only while the moon is passing from this quarter to the mid way between the two nodes, and from thence increases again during the whole passage through the other half of the way to the next node.
40. Thus we have traced the moon from her node in the quarter, and shewn, that at every period of the moon the nodes will have receded, and thereby will have approached toward a conjunction with the sun. But this conjunction will be much forwarded by the visible motion of the sun itself. In the last scheme the sun will appear to move from S toward W. Suppose it appeared to have moved from S to W, while the moon’s node has receded from B to V, then drawing the line W T X, the arch V X will represent the distance of the line drawn between the nodes from the sun, when the moon is in V; whereas the arch B A represented that distance, when the moon was in B. This visible motion of the sun is much greater, than that of the node; for the sun appears to revolve quite round each year, and the node is near 19 years in making one revolution. We have also seen, that when the node was in the quadrature, the inclination of the moon’s orbit decreased, till the moon came to the conjunction, or opposition, according to which node it set out from; but that afterwards it again increased, till it became at the next node rather greater than at the former. When the node is once removed from the quarter nearer to a conjunction with the sun, the inclination of the moon’s orbit, when the moon comes into the node, is more sensibly greater, than it was in the node preceding; the inclination of the orbit by this means more and more increasing till the node comes into conjunction with the sun; at which time it has been shewn above, that the sun has no power to change the plane of the moon’s motion; and consequently has no effect either on the nodes, or on the inclination of the orbit.
41. As soon as the nodes, by the action of the sun, are got out of conjunction toward the other quarters, they begin again to recede as before; but the inclination of the orbit in the appulse of the moon to each succeeding node is less than at the preceding, till the nodes come again into the quarters. This will appear as follows. Let A (in fig. 104.) represent one of the moon’s nodes placed between the point of opposition B and the quarter C. Let the plane A D E pass through the earth T, and touch the path of the moon in A. Let the line A F G H be the path of the moon in her passage from A to H, where she crosses again the plane of the earth’s motion. This line will be convex toward the plane A D E, till the moon comes to G, where she is in the quarter; and after this, between G and H, the same line will be concave toward this plane. All the time this line is convex toward the plane A D E, the nodes will recede; and on the contrary proceed, while it is concave to that plane. All this will easily be conceived from what has been before so largely explained. But the moon is longer in passing from A to G, than from G to H; therefore the nodes recede a longer time, than they proceed; consequently upon the whole, when the moon is arrived at H, the nodes will have receded, that is, the point H will fall between B and E. The inclination of the orbit will decrease, till the moon is arrived to the point F, in the middle between A and H. Through the passage between F and G the inclination will increase, but decrease again in the remaining part of the passage from G to H, and consequently at H must be less than at A. The like effects, both in respect to the nodes and inclination of the orbit, will take place in the following passage of the moon on the other side of the plane A B E C, from H, till it comes over that plane again in I.
42. Thus the inclination of the orbit is greatest, when the line drawn between the moon’s nodes will pass through the sun; and least, when this line lies in the quarters, especially if the moon at the same time be in conjunction with the sun, or in the opposition. In the first of these cases the nodes have no motion, in all others, the nodes will each month have receded: and this regressive motion will be greatest, when the nodes are in the quarters; for in that case the nodes have no progressive motion during the whole month, but in all other cases the nodes do at some times proceed forward, viz. whenever the moon is between either quarter, and the node which is less distant from that quarter than a fourth part of a circle.
43. It now remains only to explain the irregularities in the moon’s motion, which follow from the elliptical figure of the orbit. By what has been said at the beginning of this chapter it appears, that the power of the earth on the moon acts in the reciprocal duplicate proportion of the distance: therefore the moon, if undisturbed by the sun, would move round the earth in a true ellipsis, and the line drawn from the earth to the moon would pass over equal spaces in equal portions of time. That this description of the spaces is altered by the sun, has been already declared. It has also been shown, that the figure of the orbit is changed each month; that the moon is nearer the earth at the new and full, and more remote in the quarters, than it would be without the sun. Now we must pass by these monthly changes, and consider the effect, which the sun will have in the different situations of the axis of the orbit in respect of that luminary.
44. The action of the sun varies the force, wherewith the moon is drawn toward the earth; in the quarters the force of the earth is directly increased by the sun; at the new and full the same is diminished; and in the intermediate places the influence of the earth is sometimes aided, and sometimes lessened by the sun. In these intermediate places between the quarters and the conjunction or opposition, the sun’s action is so oblique to the action of the earth on the moon, as to produce that alternate acceleration and retardment of the moon’s motion, which I observed above to be stiled the variation. But besides this effect, the power, by which the earth attracts the moon toward itself, will not be at full liberty to act with the same force, as if the sun acted not at all on the moon. And this effect of the sun’s action, whereby it corroborates or weakens the action of the earth, is here only to be considered. And by this influence of the sun it comes to pass, that the power, by which the moon is impelled toward the earth, is not perfectly in the reciprocal duplicate proportion of the distance. Consequently the moon will not describe a perfect ellipsis. One particular, wherein the moon’s orbit will differ from an ellipsis, consists in the places, where the motion of the moon is perpendicular to the line drawn from itself to the earth. In an ellipsis, after the moon should have set out in the direction perpendicular to this line drawn from itself to the earth, and at its greatest distance from the earth, its motion would again become perpendicular to this line drawn between itself and the earth, and the moon be at its nearest distance from the earth, when it should have performed half its period; after performing the other half of its period its motion would again become perpendicular to the forementioned line, and the moon return into the place whence it set out, and have recovered again its greatest distance. But the moon in its real motion, after setting out as before, sometimes makes more than half a revolution, before its motion comes again to be perpendicular to the line drawn from itself to the earth, and the moon is at its nearest distance; and then performs more than another half of an intire revolution before its motion can a second time recover its perpendicular direction to the line drawn from the moon to the earth, and the moon arrive again to its greatest distance from the earth. At other times the moon will descend to its nearest distance, before it has made half a revolution, and recover again its greatest distance, before it has made an intire revolution. The place, where the moon is at its greatest distance from the earth, is called the moon’s apogeon, and the place of the least distance the perigeon. This change of the place, where the moon successively comes to its greatest distance from the earth, is called the motion of the apogeon. In what manner the sun causes the apogeon to move, I shall now endeavour to explain.
45. Our author shews, that if the moon were attracted toward the earth by a composition of two powers, one of which were reciprocally in the duplicate proportion of the distance from the earth, and the other reciprocally in the triplicate proportion of the same distance; then, though the line described by the moon would not be in reality an ellipsis, yet the moon’s motion might be perfectly explained by an ellipsis, whose axis should be made to move round the earth; this motion being in consequence, as astronomers express themselves, that is, the same way as the moon itself moves, if the moon be attracted by the sum of the two powers; but the axis must move in antecedence, or the contrary way, if the moon be acted on by the difference of these powers. What is meant by duplicate proportion has been often explained; namely, that if three magnitudes, as A, B, and C, are so related, that the second B bears the same proportion to the third C, as the first A bears to the second B, then the proportion of the first A to the third C, is the duplicate of the proportion of the first A to the second B. Now if a fourth magnitude, as D, be assumed, to which C shall bear the same proportion as A bears to B, and B to C, then the proportion of A to D is the triplicate of the proportion of A to B.
46. The way of representing the moon’s motion in this case is thus. T denoting the earth (in fig. 105, 106.) suppose the moon in the point A, its apogeon, or greatest distance from the earth, moving in the direction A F perpendicular to A B, and acted upon from the earth by two such forces as have been named. By that power alone, which is reciprocally in the duplicate proportion of the distance, if the moon let out from the point A with a proper degree of velocity, the ellipsis A M B may be described. But if the moon be acted upon by the sum of the forementioned powers, and the velocity of the moon in the point A be augmented in a certain proportion[193]; or if that velocity be diminished in a certain proportion, and the moon be acted upon by the difference of those powers; in both these cases the line A E, which shall be described by the moon, is thus to be determined. Let the point M be that, into which the moon would have arrived in any given space of time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D in such sort, that the angle under A T M shall bear the same proportion to the angle under A T C, as the velocity, with which the ellipsis A M B must have been described, bears to the difference between this velocity, and the velocity, with which the moon must set out from the point A in order to describe the path A E. Let the angle A T C be taken toward the moon (as in fig. 105.) if the moon be attracted by the sum of the powers; but the contrary way (as in fig. 106.) if by their difference. Then let the line A B be moved into the position C D, and the ellipsis A M B into the situation C N D, so that the point M be translated to L: then the point L shall fall upon the path of the moon A E.
47. The angular motion of the line A T, wereby it is removed into the situation C T, represents the motion of the apogeon; by the means of which the motion of the moon might be fully explicated by the ellipsis A M B, if the action of the sun upon it was directed to the center of the earth, and reciprocally in the triplicate proportion of the moon’s distance from it. But that not being so, the apogeon will not move in the regular manner now described. However, it is to be observed here, that in the first of the two preceding cases, where the apogeon moves forward, the whole centripetal power increases faster, with the decrease of distance, than if the intire power were reciprocally in the duplicate proportion of the distance; because one part only is in that proportion, and the other part, which is added to this to make up the whole power, increases faster with the decrease of distance. On the other hand, when the centripetal power is the difference between these two, it increases less with the decrease of the distance, than if it were simply in the reciprocal duplicate proportion of the distance. Therefore if we chuse to explain the moon’s motion by an ellipsis (as is most convenient for astronomical uses to be done, and by reason of the small effect of the sun’s power, the doing so will not be attended with any sensible error;) we may collect in general, that when the power, by which the moon is attracted to the earth, by varying the distance, increases in a greater than in the duplicate proportion of the distance diminished, a motion in consequence must be ascribed to the apogeon; but that when the attraction increases in a less proportion than that named, the apogeon must have given to it a motion in antecedence[194]. It is then observed by Sir Is. Newton, that the first of these cases obtains, when the moon is in the conjunction and opposition; and the latter, when the moon is in the quarters: so that in the first the apogeon moves according to the order of the signs; in the other, the contrary way[195]. But, as was said before, the disturbance given to the action of the earth by the sun in the conjunction and opposition being near twice as great as in the quarters[196], the apogeon will advance with a greater velocity than recede, and in the compass of a whole revolution of the moon will be carried in consequence[197].
48. It is shewn in the next place by our author, that when the line A B coincides with that, which joins the earth and the sun, the progressive motion of the apogeon, when the moon is in the conjunction or opposition, exceeds the regressive in the quadratures more than in any other situation of the line A B[198]. On the contrary, when the line A B makes right angles with that, which joins the earth and sun, the retrograde motion will be more considerable[199], nay is found so great as to exceed the progressive; so that in this case the apogeon in the compass of an intire revolution of the moon is carried in antecedence. Yet from the considerations in the last paragraph the progressive motion exceeds the other; so that in the whole the mean motion of the apogeon is in consequence, according as astronomers find. Moreover, the line A B changes its situation with that, which joins the earth and sun, by such slow degrees, that the inequalities in the motion of the apogeon arising from this last consideration, are much greater than what arises from the other[200].
49. Farther, this unsteady motion in the apogeon is attended with another inequality in the motion of the moon, that it cannot be explained at all times by the same ellipsis. The ellipsis in general is called by astronomers an eccentric orbit. The point, in which the two axis’s cross, is called the center of the figure; because all lines drawn through this point within the ellipsis, from side to side, are divided in the middle by this point. But the center, about which the heavenly bodies revolve, lying out of this center of the figure in one focus, these orbits are said to be eccentric; and where the distance of the focus from this center bears the greatest proportion to the whole axis, that orbit is called the most eccentric: and in such an orbit the distance from the focus to the remoter extremity of the axis bears the greatest proportion to the distance of the nearer extremity. Now whenever the apogeon of the moon moves in consequence, the moon’s motion must be referred to an orbit more eccentric, than what the moon would describe, if the whole power, by which the moon was acted on in its passing from the apogeon, changed according to the reciprocal duplicate proportion of the distance from the earth, and by that means the moon did describe an immoveable ellipsis; and when the apogeon moves in antecedence, the moon’s motion must be referred to an orbit less eccentric. In the first of the two figures last referred to, the true place of the moon L falls without the orbit A M B, to which its motion is referred: whence the orbit A L E, truly described by the moon, is less incurvated in the point A, than is the orbit A M B; therefore the orbit A M B is more oblong, and differs farther from a circle, than the ellipsis would, whose curvature in A were equal to that of the line A L B, that is, the proportion of the distance of the earth T from the center of the ellipsis to its axis will be greater in the ellipsis A M B, than in the other; but that other is the ellipsis, which the moon would describe, if the power acting upon it in the point A were altered in the reciprocal duplicate proportion of the distance. In the second figure, when the apogeon recedes, the place of the moon L falls within the orbit A M B, and therefore that orbit is less eccentric, than the immoveable orbit which the moon should describe. The truth of this is evident; for, when the apogeon moves forward, the power, by which the moon is influenced in its descent from the apogeon, increases faster with the decrease of distance, than in the duplicate proportion of the distance; and consequently the moon being drawn more forcibly toward the earth, it will descend nearer to it. On the other hand, when the apogeon recedes, the power acting on the moon increases with the decrease of distance in less than the duplicate proportion of the distance; and therefore the moon is less impelled toward the earth, and will not descend so low.
50. Now suppose in the first of these figures, that the apogeon A is in the situation, where it is approaching toward the conjunction or opposition of the sun. In this case the progressive motion of the apogeon is more and more accelerated. Here suppose that the moon, after having descended from A through the orbit A E as far as F, where it is come to its nearest distance from the earth, ascends again up the line F G. Because the motion of the apogeon is here continually more and more accelerating, the cause of its motion is constantly upon the increase; that is, the power, whereby the moon is drawn to the earth, will decrease with the increase of distance, in the moon’s ascent from F, in a greater proportion than that wherewith it increased with the decrease of distance in the moon’s descent to F. Consequently the moon will ascend higher than to the distance A T, from whence it descended; therefore the proportion of the greatest distance of the moon to the least is increased. And when the moon descends again, the power will yet more increase with the decrease of distance, than in the last ascent it decreased with the augmentation of distance; the moon therefore must descend nearer to the earth than it did before, and the proportion of the greatest distance to the least yet be more increased. Thus as long as the apogeon is advancing toward the conjunction or opposition, the proportion of the greatest distance of the moon from the earth to the least will continually increase; and the elliptical orbit, to which the moon’s motion is referred, will be rendered more and more eccentric.
51. As soon as the apogeon is passed the conjunction with the sun or the opposition, the progressive motion thereof abates, and with it the proportion of the greatest distance of the moon from the earth to the least distance will also diminish; and when the apogeon becomes regressive, the diminution of this proportion will be still farther continued on, till the apogeon comes into the quarter; from thence this proportion, and the eccentricity of the orbit will increase again. Thus the orbit of the moon is most eccentric, when the apogeon is in conjunction with the sun, or in opposition to it, and least of all when the apogeon is in the quarters.
52. These changes in the nodes, in the inclination of the orbit to the plane of the earth’s motion, in the apogeon, and in the eccentricity, are varied like the other inequalities in the motion of the moon, by the different distance of the earth from the sun; being greatest, when their cause is greatest, that is, when the earth is nearest to the sun.
53. I said at the beginning of this chapter, that Sir Isaac Newton has computed the very quantity of many of the moon’s inequalities. That acceleration of the moon’s motion, which is called the variation, when greatest, removes the moon out of the place, in which it would otherwise be found, something more than half a degree[201]. In the phrase of astronomers, a degree is 1/360 part of the whole circuit of the moon or any planet. If the moon, without disturbance from the sun, would have described a circle concentrical to the earth, the sun will cause the moon to approach nearer to the earth in the conjunction and opposition, than in the quarters, nearly in the proportion of 69 to 70[202]. We had occasion to mention above, that the nodes perform their period in almost 19 years. This the astronomers found by observation; and our author’s computations assign to them the same period[203]. The inclination of the moon’s orbit when least, is an angle about 1/18 part of that angle, which constitutes a perpendicular; and the difference between the greatest and least inclination of the orbit is determined by our author’s computation to be about 1/18 of the least inclination[204]. And this also is agreeable to the observations of astronomers. The motion of the apogeon, and the changes in the eccentricity, Sir Isaac Newton has not computed. The apogeon performs its revolution in about eight years and ten months. When the moon’s orbit is most eccentric, the greatest distance of the moon from the earth bears to the least distance nearly the proportion of 8 to 7; when the orbit is least eccentric, this proportion is hardly so great as that of 12 to 11.
54. Sir Isaac Newton shews farther, how, by comparing the periods of the motion of the satellites, which revolve round Jupiter and Saturn, with the period of our moon round the earth, and the periods of those planets round the sun with the period of our earth’s motion, the inequalities in the motion of those satellites may be derived from the inequalities in the moon’s motion; excepting only in regard to that motion of the axis of the orbit, which in the moon makes the motion of the apogeon; for the orbits of those satellites, as far as can be discerned by us at this distance, appearing little or nothing eccentric, this motion, as deduced from the moon, must be diminished.