Was Newton's successful calculation of precession of equinoxes a fluke?

Question

I have looked at several sources, and Newton was right about the fact that the Earth is not a perfect sphere, but an ellipsoid caused the precession of equinoxes, as the Moon's gravitational attraction of the Earth causes torque, if the Earth is not a perfect sphere.

However, I looked at the English translation of Principia

https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2

[408] "And since the mean semi-diameter of the earth, according to Picart's mensuration, is 19615800 Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will be higher at the equator than at the poles by 85472 feet, or 17 1⁄10 miles. And its height at the equator will be about 19658600 feet, and at the poles 19573000 feet."

I also looked at the Latin edition, even though I don't know Latin: http://www.gutenberg.org/files/28233/28233-h/28233-h.htm

The value was slightly different, but not that much

"Ideoque cùm Terræ semidiameter mediocris, juxta nuperam Gallorum mensuram, sit pedum Parisiensium 19615800 seu milliarium 3923 (posito quod milliare sit mensura pedum 5000;) Terra altior erit ad æquatorem quàm ad polos, excessu pedum 85200 seu milliarium 17."

I searched the Internet and found out that the actual difference is neither 17.1 miles nor 17 miles but 13.3 miles. So, Newton got the radius difference quite wrong. I thought about the possibility that the miles then and now were different, but this was not the case. Internet says that the radius of the earth is 3950~3963 miles, which almostly agrees with Newton's value 3923.16 miles or 3923 miles.

So, if Newton got wrong about the radius difference, how did he end up getting the right value? (50 arcseconds per a year)

https://en.wikipedia.org/wiki/Axial_precession Wikipedia says

"Over a century later precession was explained in Isaac Newton's Philosophiae Naturalis Principia Mathematica (1687), to be a consequence of gravitation (Evans 1998, p. 246). Newton's original precession equations did not work, however, and were revised considerably by Jean le Rond d'Alembert and subsequent scientists."

Does anyone know what actually happened?

Answer 1

Numerical agreement was (at best) a fluke; quoth e.g. A. Berry, A short history of astronomy (1898, p. 235):

The amount of the precession as calculated by Newton did as a matter of fact agree pretty closely with the observed amount, but this was due to the accidental compensation of two errors, arising from his imperfect knowledge of the form and construction of the earth, as well as from erroneous estimates of the distance of the sun and of the mass of the moon

The first correct computation is attributed to D’Alembert (1749, “disorderly”) and then Euler (1751, “a model of clarity”) — appreciations taken from Curtis Wilson’s good summary, The precession of the equinoxes from Newton to d’Alembert and Euler (1995, pp. 47–54).

For detailed analyses of Newton’s “errors”: D’Alembert (ibid.) or Wilson’s exposition of it (1987, §3); Laplace (1825, pp. 275–278); and Westfall (1973; 1980, pp. 736–739) who convincingly argues that their “accidental compensation” happened quite by design. For instance,

the correction of a faulty lemma in edition one imposed the necessity of an adjustment of more than 50 percent in the remaining numbers. Without even pretending that he had new data, Newton brazenly manipulated the old figures on precession so that he not only covered the apparent discrepancy but carried the demonstration to a higher plane of accuracy.

... whence the changes you noted between the first (Latin) and third (translated) edition. (The correspondence with Cotes, in which they double the alleged accuracy to reach 1:3000, is a riot.)

Answer 2

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?