THE MATHEMATICAL PRINCIPLES [BOOK I. fore if the force decreases in a yet greater ratio, the body, being now less attracted than before, will ascend to a still greater distance, and so the ec centricity of the orbit will be increased still more. Therefore if the ratio of the increase and decrease of the centripetal force be augmented each revolution, the eccentricity will be augmented also ; and, on the contrary, if that ratio decrease, it will be diminished. Now, therefore, in the system of the bodies T, P, S, when the apsides of the orbit FAB are in the quadratures, the ratio of that increase and de crease is least of all, and becomes greatest when the apsides are in the syzygies. If the apsides are placed in the quadratures, the ratio near the apsides is less, and near the syzygies greater, than the duplicate ratio of the distances : and from that Greater ratio arises a direct motion of the line of 7 o the apsides, as was just now said. But if we consider the ratio of the whole increase or decrease in the progress between the apsides, this is less than the duplicate ratio of the distances. The force in the lower is to the force in the upper apsis in less than a duplicate ratio of the distance of the upper apsis from the focus of the ellipsis to the distance of the lower apsis from the same focus ; and, contrariwise, when the apsides are placed in the syzygies, the force in the lower apsis is to the force in the upper apsis in a greater than a duplicate ratio of the distances. For the forces LM in the quadratures added to the forces of the body T compose forces in a less ra tio ; and the forces KL in the syzygies subducted from the forces of the body T, leave the forces in a greater ratio. Therefore the ratio of the whole increase and decrease in the passage between the apsides is least at the quadratures and greatest at the syzygies ; and therefore in the passage of the apsides from the quadratures to the syzygies it is continually aug mented, and increases the eccentricity of the ellipsis ; and in the passage from the syzygies to the quadratures it is perpetually decreasing, and di minishes the eccentricity. COR. 10. That we may give an account of the errors as to latitude, let us suppose the plane of the orbit EST to remain immovable; and from the cause of the errors above explained, it is manifest, that, of the two forces NM, ML, which are the only and entire cause of them, the force ML acting always in the plane of the orbit PAB never disturbs the mo tions as to latitude ; and that the force NM, when the nodes are in the gyzygies, acting also in the same plane of the orbit, does not at that time affect those motions. But when the nodes are in the quadratures, it dis turbs tliem very much, and, attracting the body P perpetually out of the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and again increases the same in the passage from the syzygies to the quadratures. Hence it comes to pass that when the body is in the syzygies, the inclination is then least of all, and returns to the first magnitude nearly, when the body