The Muse, Urania, looking for all the wrong placesDepartment of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

The Paradox of 1713: Shape of the Lunar Orbit

Figure 1The first edition of the Principia (1687) contained nothing that was of practical use to astronomers  in regard to lunar theory. Its celebrated 'Moon-test' was a one-body problem, dealing merely with uniform circular motion around an immobile force-centre. The chief addition to the Second Edition of the Principia of 1713 was extensive new lunar theory. A dual-track policy was there pursued, of demonstrating how each of the lunar 'equations' were derivable from gravity theory, while also trying to maintain TMM largely intact, e.g. it specified mean motions for Sun, Moon, nodes etc.

The Principia arguments were formulated in terms of absolute sidereal space and so used sidereal periods, whereas TMM was formulated in terms of the tropical reference--i.e., measuring celestial longitude as astronomers did in practice, from the Vernal Point. A more serious problem arose from the Horroxian theory's assumption, that the Moon's motion was elliptical. [figure 1] Using Kepler's first two laws, it viewed the lunar orbit as an ellipse with Earth at one focus. It thus gave the lunar orbit an eccentricity of 0.05505. The ellipse revolved once per nine years in sidereal space (Book III, Propn. 35, Scholium).

Figure 2That would have been fine, except that Newton had also developed an 'explanation' for the inequality know as the Variation, as discovered by Tycho Brahe and Kepler. This put the Moon in an elliptical orbit with the Earth at its centre, not at one of the foci. [figure 2] The short (minor) axis of this ellipse pointed towards the Sun, so that this ellipse revolved annually in space. Its eccentricity was three times larger than the earlier-mentioned ellipse, of the monthly apogee-perigee orbit. This Variation ellipse allowed Newton to account for why lunar motion was slightly faster at the Full and New positions than at the quarters, and his three-body computations credibly explained why solar gravity would deform a circular orbit in this manner (Book III, Propn. 29).

Each of these models would have been O.K. separately, but it was far from evident as to how they could be added together. This mind-wrenching paradox may have been why Newton told Halley that the Moon was the one subject which made his head ache. A discussion of this paradox is contained in a forthcoming article by Curtis Wilson, 'Newton on the Moon's Variation and Apsidal Motion.' Of these two ellipse models Wilson observed, 'The Variational orbit and the Horrocksian ellipse implied two disparate pictures of the lunar orbit.' The Variation was derived from gravity theory in /book III of the Principia several propositions earlier that the TMM-Scholium, as if Newton had more confidence in the former.

The contents of this page remain the copyrighted, intellectual property of Nicholas Kollerstrom.  Details. rev: May 1998