A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

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.