A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

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.