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tion.

it out of the plane AFC towards the sun, while the Theory of moon passed from A to I; so during its passage from Universal C to W, the moon being all that time farther from the Gravita. sun than the earth, it will be attracted less; and the earth, together with the plane AECF, will as it were be drawn from the moon, in such a manner, that the path the moon describes shall appear from the earth as it did in the former case by the moon being drawn away.

Theory of the two arches LP and NP, taken together, are less Universal than a semicircle, each of them being leɛs than a quaGravita. drant, as appears, because GN, the distance of the moon in G from its node N, is here supposed to be a quarter part of a circle. After the moon is passed beyond G, the case is altered; for then these arches will be greater than quarters of a circle; by which means the inclination will be again increased, though the nodes still go on to move the same way. Suppose the moon in H (fig. 143.), and that the plane which touches the line AKGI in H intersects the plane of the earth's motion in the line QTR, and the plane NGO in the line TV, and besides, that the circle QHR be described in that plane; then, for the same reason as before, the point V will fall between H and G, and the plane RVQ will pass beyond the last plane OVN, causing the points Q and K to fall farther from A and C than N and O. But the arches NV, VQ are each greater than the quarter of a circle: consequently the angle under BQV will be greater than that under BNV. Lastly, when the moon is by this attraction of the sun drawn at length into the plane of the earth's orbit, 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 AKGHI being convex to all the planes which touch it, the part HI will wholly fall between the plane QVR and the plane ABC; so that the point I will fall between B and R; and, drawing ITW, the point W will be farther removed from A than Q. But it is evident, that the plane which passes through the earth T, and touches the line AGI in the point I, will cut the plane of the earth's motion ABCD in the line ITW, and be inclined to the same in the angle under HIB; so that the node which was first in A, after having passed into L, N, and Q, comes at last in the point W, as the node which was at first in C has passed from thence successively through the points M, O, and R, to I. But the angle HIB, which is now the inclination of the orbit to the plane of the ecliptic, is manifestly not less than the angle under ECB or EAB, but rather something greater. Thus the moon, 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 it at first diminished till it comes to G in fig. 128. 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.

In like manner, if the moon had departed from the quarter at C, it should have described the curve line CXW in fig. 140. between the planes AH and ADC, which would be convex to the former planes and concave to the latter; so that here also the nodes would 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 become 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

354 Such are the changes which the nodes and inclina- Motion of tion of the moon's orbit undergo when the nodes are the nodes in the quarters; but when the nodes by their motion, explained, 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.-Let AGH, in fig. 145. be a circle described in the plane of the earth's motion, having the earth in T for its centre, A the point opposite to the sun, and G a fourth part of the circle distant from A. Let the nodes of the moon's orbit be situated in the line BTD, and B the node falling between A, the place where the moon would be in the full, and G the place where she would be in the quarter. Suppose BEDF to be the plane in which the moon attempts to move when it proceeds from the point B: then, because the moon in B is more distant from the sun than the earth, it will be less attracted by the sun, and will not descend towards the sun so fast as the earth, consequently it will quit the plane BEDF, which is supposed to accompany the earth, and describe the line BIK convex to it, 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 will change its course, and the following part of the path it describes will be concave towards the plane BED or BGD, and continue concave to the plane BGD till it crosses that plane in L just as in the preceding case. Now, to show that the nodes, while the moon is passing from B to K, will proceed forward, or move the same way with the moon, and at the same time the inclination of the orbit will increase when the moon is in the point I, let the line MIN 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 MTN, and the line BED, in TO. Because the line BIK is convex to the plane BED, which touches it in B, the plane NIM must cross the plane DEB before it meets the plane CGB; and therefore the point M will fall from G towards B; and the node of the moon's orbit being translated from B towards M is moved forward.

Again the angle under OMG, which the plane MON makes with the plane BGC, is greater than the angle OBG, which the plane BOD makes with the same. This appears from what bas been already demonstrated, because the arches BO and OM are each of them less than the quarter of a circle; and therefore, taken both together, are less than a semicircle. But further, 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 angmented. Hitherto we have referred the moon's motion to that plane, which, passing through the earth, touches the path of the moon in the point where the moon is, as we have already said that the custom of R 2

astronomers

Theory of astronomers is. But in the point K no such plane can Universal be found on the contrary, seeing the line of the Gravita- moon's motion on one side the point K is convex to tion. the plane BED, and on the other side concave to the same, so that no plane can pass through the points T and K, but will cut the line BKL in that point; therefore instead of such a touching plane, we must make use of PKQ, which is equivalent, and with which the line BKL shall make a less angle than with any other plane; for this does as it were touch the line BK in the point K, since it cuts it in such a manner that no other plane can be drawn so as to pass between the line BK and the plane PKQ. But now it is evident, that the point P, or the node, is removed from M towards G, that is, has moved yet further forward; and it is likewise as manifest, that the angle under KPG, or the inclination of the moon's orbit in the point K, is greater than the angle under IMG, for the reason already given.

After the moon has passed the quarter, her plane being concave to the plane AGCH, the nodes will recede as before till she arrives at the point L; which shows, 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 more backward, 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 there fore the nodes continue to recede a longer time than they moved forwards; so that their recess must surmount their advance. 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 come 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 will 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, then augments again while the moon passes from B to L; it thence comes to be much more increased than diminished, and thus will be distinguishably greater when the moon comes to L than when it sets out from B. In like manner, when the moon is passing from L on the other side the plane AGCH, the node will advance forward as long as the moon is between the point L and the next quarter; but afterwards it will recede till the moon come to pass the plane AGCH 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 will have receded more than it has 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 she was in L; for this inclination increases all the time the moon is betwixt L and the next quarter, decreasing only when she 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.

In this manner we see, that at every period of the moon the nodes will have receded, and thereby have approached towards a conjunction with the sun: but

this will be much forwarded by the motion of the Theard earth, or the apparent motion of the sun himself. In Universal the last scheme the sun will appear to have moved from Gravita S towards W. Let us suppose it had appeared to have moved from S to W while the moon's node has receded from B to V; then drawing the line WTX, the arch VX will represent the distance of the line drawn between the nodes from the sun when the moon is in V; whereas the arch BA 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 in one year, while the node is near nineteen in making its revolution. We have also seen that when the moon was in the quadrature, the inclination of her orbit decreased till she came to the conjunction or opposition, according to the 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 she 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 nodes come into conjunction with the sun at which time it has been shown that the latter has no power to change the plane of her orbit. As soon, however, as the nodes are got out of conjunction towards the other quarters, they begin 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. 146. represent one of the moon's nodes placed between the point of opposition B and the quarter C. Let the plane ADE pass through the earth T, and touch the path of the moon in A. Let the line AFGH be the path of the moon in her passage from A to H, where she crosses again the plane of the earth's moon. This line will be convex towards the plane ADE, 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 towards this plane. All the time this line is convex towards the plane ADE, the nodes will recede; and, on the contrary, move forward when the line is concave towards that plane. But the moon is longer in passing from A to G, and therefore the nodes go backward farther than they proceed; and therefore, on the whole, when the moon has 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 at 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. Similar effects, both with 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 ABEC from H, till it comes over that plane again in I.

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

degrees, that the inequalities of the motion of the apo- Theory? geon, arising from this last consideration, are much Univers greater than what arise from the other.

Gravita

tion.

This unsteady motion of the apogeon gives rise to another inequality in the motion of the moon herself, so that it cannot at all times be explained by the same Occacia. ellipsis. For whenever the apogeon moves in conse- another in quence, the motion of the luminary must be referred equality to an orbit more eccentric than what the moon would tricity di describe, if the whole power by which the moon was the mom acted upon in its passing from the apogeon changed orbit according to the reciprocal duplicate proportion of its 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 former of the two figures last referred to, the true place of the moon L falls without the orbit AMB, to which its motion is referred whence the orbit ALE truly described by the moon, is less incurvated in the point A than is the orbit AMB: therefore this orbit is more oblong, and differs farther from a circle than the ellipsis would, whose curvature in A were equal to that of the line ALB: that is, the proportion of the distance of the earth T from the centre of the ellipsis to its axis, will be greater in AMB 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; 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 towards the earth, and will not descend so low. Now, suppose, in the former of these figures, that the apogeon A is in the situation where it is approaching towards the conjunction or opposition of the sun; in this case its progressive motion will be more and more accelerated. Here suppose the moon, after having descended from A through the orbit AE as far as F, where it is come to its nearest distance from the earth, ascends again up the line FG. As the motion of the apogeon is here more and more accelerated, it is plain that the cause of its motion must also be on the increase that is, the power by which the moon is drawn to the earth, will decrease with the increase of the moon's distance in her ascent from F, in a greater proportion than that wherewith it is increased with the decrease of distance in the moon's distance to it. Consequently the moon will ascend to a greater distance than AT from whence it is descended; therefore the proportion of the greatest distance of the moon to the least is increased. But farther, when the moon again descends, the power will increase yet farther with the decrease of distance than in the last ascent it increased 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 be yet more increased. Thus, as long as the apogeon is advancing to 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

tion.

Theory of will become more and more eccentric. As soon, howUniversal ever, as the apogeon is past the conjunction or oppoGravita sition with the sun, its progressive motion abates, and with it the proportion of the greatest distance of the moon from the earth to the least will also diminish: and when the apogeon becomes retrograde, the diminution of this proportion will be still farther continued, until 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. These changes in the nodes, 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 the sun. Sir Isaac Computa. Newton has computed the very quantity of many of lunar ine- the moon's inequalities. That acceleration of the qualities. moon's motion which is called the variation, when

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greatest, removes the luminary out of the place in which it would otherwise be found, somewhat more than half a degree. If the moon, without disturbance from the sun, would bave described a circle concentrical to the earth, his action will cause her approach nearer in the conjunction and opposition than in the quarters, nearly in the proportion of 69 to 70. It has already been mentioned, that the nodes perform their period in almost 19 years. This has been found by observation; and the computations of Sir Isaac assigned to them the same period. The inclination of the moon's orbit, when least, is an angle about one. eighteenth of that which constitutes a right angle; and the difference between the greatest and least inclination, is about one-eighteenth of the least inclination, according to our author's computation: which is also agreeable to the general observations of astronomers.

There is one empirical equation of the moon's motion which the comparison of ancient and modern eclipses obliges the astronomers to employ, without being able to deduce it, like the rest, à priori, from the theory of an universal force inversely proportional to the square of the distance. It has therefore been considered as a The secu- stumbling block in the Newtonian philosophy. This lar equa is what is called the secular equation of the moon's mean tion of the motion. The mean motion is deduced from a comparison of distant observations. The time between them, being divided by the number of intervening revolutions, gives the average time of one revolution, or the mean lunar period. When the ancient Chaldean observations are compared with those of Hipparchus, we obtain a certain period; when those of Hipparchus are compared with some in the 9th century, we obtain a period somewhat shorter; when the last are compared with those of Tycho Brahe, we obtain one still shorter; and when Brahe's are compared with those of our day, we obtain the shortest period of all-and thus the moon's mean motion appears to accelerate continually; and the accelerations appear to be in the duplicate ratio of the times. The acceleration for the century which ended in 1700 is about 9 seconds of a degree; that is to say, the whole motion of the moon during the 17th centu

tion.

ry must be increased 9 seconds, in order to obtain its Theory of motion during the 18th; and as much must be taken Universal from it, or added to the computed longitude, to obtain Gravitaits motion during the 16th: and the double of this must be taken from the motion during the 16th, to obtain its motion during the 15th, &c. Or it will be sufficient to calculate the moon's mean longitude for any time past or to come by the secular motion which obtains in the present century, and then to add to this longitude the product of 9 seconds, multiplied by the square of the number of centuries which intervene. Thus having found the mean longitude for the year 1200, add 9 seconds, multiplied by 36, for six centuries. By this method we shall make our calculation agree with the most ancient and all intermediate observations. If we neglect this correction, we shall differ more than a degree from the Chaldean observation of the moon's place in the heavens.

The mathematicians having succeeded so completely in deducing all the observed inequalities of the planetary motions, from the single principle, that the deflecting forces diminished in the inverse duplicate ratio of the distances, were fretted by this exception, the reality of which they could not contest. Many opinions were formed about its cause. Some have attempted to deduce it from the action of the planets on the moon; others have deduced it from the oblate form of the earth, and the translation of the ocean by the tides; others have supposed it owing to the resistance of the ether in the celestial spaces; and others have imagined that the action of the deflecting force requires time for its propagation to a distance: But their deductions have been proved unsatisfactory, and have by no means the precision and evidence that have been attained in the other questions of physical astronomy. At last M. de la Place, of the Royal Academy of Sciences at Paris, has happily succeeded, and deduced the secular equation of the moon from the Newtonian law of planetary deflection. It is produced in the following manner.

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