It could have been a fluke, or he could have "adjusted" the calculation to get the "right" value. He did a not very convincing move like that to "explain" why his calculation predicted only half of the observed value for the motion of lunar apsides. Here is Wilson in Cambridge Companion to Newton:
"The nutation, which Newton had not predicted, required an explanation in terms of inverse-square gravitation, and in mid-1748 Jean le Rond d’Alembert (1717–83) set about deriving it. Nutation is a refinement of the precession of the equinoxes, and d’Alembert soon found that Newton’s explanation of the precession (Proposition 66, Corollary 22, Book 1, and Proposition 39, Book 3 with the preceding lemmas) was deeply flawed.
Newton’s basic error arose from his lack of an appropriate dynamics for the rotational motions of solid bodies, and his attempt to treat problems involving such motions in terms of linear momentum rather than angular momentum. D’Alembert now furnished the elements of the appropriate dynamics, and Leonhard Euler systematized it."
Adams and Leverrier apparently got even more lucky with Neptune. Their Bode's law estimate of its orbital period was off by over half a century, and of its mass by 100-200%. As Kelley writes, "the only actual value they were close to, if one looks at the table, is the location in the sky it would be found", see Was Leverrier-Adams prediction of Neptune a lucky coincidence?
