  Propositions 25 through 35 derive quantitative results for three lunar inequalities — the systematic departure from the area rule called the “the variation,” the 18 year motion of the line of nodes, and the fluctuating inclination of the orbit — from the perturbing action of the Sun. For all three Newton starts with a circular orbit, so these too involve departures from an idealization. The values he obtained for the different components of the solar perturbing force in Proposition 25 and subsequently, as needed, were accurate to several significant figures. All three derivations, which are mathematically demanding, were successful in obtaining agreement with the values of the inequalities obtained from observation, especially so the derivation for the recession of the lunar nodes, for which he achieves agreement with the known value to better than 98 percent. (Newton must have been mystified by the failure of his seemingly parallel derivation of the 9 year precession of the line of apsides to achieve better than 50 percent agreement.)