Newton's problems with rigid body dynamics in the light of his treatment of the precession of the equinoxes (Q1266169)

``\dots at last the entire wheelwork became so complicated that only a few could grasp it all, and to those few was attributed an extraordinary degree of learning. This circumstance, together with the religious respect paid in those days to authority, permitted [the book] to appear as undeniable truth \dots'', pp. 37-38 of \textit{O. Dziobek}, ``Mathematical theories of planetary motions''. New York: Dover Publications (1962; Zbl 0113.23801). Newton's \textit{Principia} was heavily scrutinized when it first appeared, and then learned by rote in Cambridge, and now hardly read at all. It is written in a mathematical language now long out of fashion. It contains many sinkholes for the feet of the unwary. It was completed before the principle of angular momentum was fully understood and catechized by Euler, and so contains many expected deficiencies. Contemporaries such as John I and Nicolas Bernoulli pointed some of these out to Newton himself. \textit{G. J. Dobson} here and \textit{R. Weinstock} in ``Newton's \textit{Principia} and inverse-square orbits in a resisting medium: a spiral of twisted logic'', Hist. Math. 25, No. 3, 281-289 (1998), have focussed on two such deficiencies: they are not the first to do so. Dobson attacks, without full mathematical apparatus, an argument of Newton about the precession of the equinoxes. This precession is best expressed by a theory of mechanics including angular momentum, moments of inertia, the torque of forces, moments of momentum, and all the paraphenalia of Euler, d'Alembert and Clairaut of 1745-1755 when a mistake in arithmetic threatened to pull Newton's house down [see \textit{T. L. Hankins}, ``Jean d'Alembert: Science and the Enlightenment''. Oxford: Clarendon Press (1970), pp. 32-33]. Blind acceptance of any master is foolish, and none should be angry at reasoned criticism of the mathematics as is offered by Dobson and Weinstock. Newton uses elementary moments of inertia -- the equatorial ring for the non-spherical part of the earth's shape, a factor 2/5 for the spherical moment of inertia (a fudge factor, not the true moment of inertia of a sphere: see Westfall below). Dobson alleges several sources of error, not always convincingly in the opinion: injudicious use of particle mechanics applied to rigid bodies in his Step 1; use of total (vector) linear momentum instead of angular momentum in his Step 2; and a confusion about moments of inertia in his Step 3; he also points out that Newton uses an unsupported technical hypothesis. \textit{R. S. Westfall} in ``Never at rest. A biography of Isaac Newton''. Cambridge: Cambridge University Press (1983; Zbl 0532.01023) pp. 736-744, details several ``fudge factors'' used by Newton in the various drafts of this article of the \textsl{Principia} [see also his ``Newton and the fudge factor'', Science 179, 751-758 (1973)] also the continuous stream of minor adjustments from the theory and practicalities of the tides which Newton used to measure the ratio of solar and lunar forces which produce the required torque for the precession; and the problem of the motion of a particle in a resisting medium as criticized by Nicolaus Bernoulli in 1712, the problem considered by Weinstock. Regrettably the references in Dobson are few (4), those in Weinstock better and more numerous (21) including John Greenberg's excellent and comprehensive tome ``The problem of the Earth's shape from Newton to Clairaut. The rise of mathematical science in 18th-century Paris and the fall of `normal' science''. Cambridge: Cambridge University Press (1995; Zbl 0833.01012)] where all the mathematics is done.