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.