  Book 3 ends with a revolutionary analysis of comet trajectories that occupies roughly one-third of the total length of the Book in all three editions. This analysis was slow in coming. As late as June 1686, Newton wrote: “the third [book] wants the Theory of Comets” [C, II, 437]. What made the problem difficult, as compared to planet trajectories, was the need to work from a small number of imprecise one-shot observations made from a moving Earth. The method presented in the Principia fits a parabola iteratively to the observations, employing novel finite-difference methods that Newton later expanded into a full tract in mathematics, “Methodis Differentialis.” The method presupposes the theory of gravity first in opting for a parabola and second in assuming that the inverse-square centripetal forces known from the planets act on comets along their entire trajectory. The text notes that the trajectories may well be ellipses, but the period of return in that case would be the best way of determining the ellipse. (The parabola approximates the high-curvature end of ellipses with high eccentricity.) The proposal that comets may return was novel, but even more revolutionary at the time was the claim that they button-hook around the Sun, implying that what had sometimes in the past been taken for two distinct comets were really one comet before and after perihelion.