H Kollerstrom's Newton's Lunar Theory in STS@UCL Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

The Lunar Mass Error

Newton never visited the seaside. If he had, it might have dawned upon him that rough tidal data from the Bristol channel did not provide the most reliable basis for estimating lunar mass. The first edition of the Principia overestimated this by a factor of three, finding that: '... the mass of the Moon will be to the mass of the Earth as 1 to 26, approximately', citing the relative densities as 9 to 5 (Bk.3, Propn. 37, Cors 3 and 4). The actual ratio is 1:81, as the lunar density is merely 0.6 that of Earth. This was the biggest error in the Principia. How had the calculation gone so horribly wrong?

It started (Propn 37) from rough tidal height data, of fortnightly spring and neap tides, at Plymouth and Exeter, which were similar, giving average maxima and minima of 41 to 23, or 9 to 5. It then used the equation (S+L)/(S-L) = 9/5, where S and L were the solar and lunar tide-raising vectors, which varied inversely as the cube of the distance; solving which would have given L/S = 3.5, as would not have been too bad. The astronomically-correct value is 2.2, though the mean value around the shores of Britain is slightly over three, as the extent to which solar (12 hour) and lunar (12.4 hour) diurnal rhythms resonate in any given part of the sea depends upon local geography.

Instead of solving this equation however, Newton inserted various adjustments, of somewhat doubtful astronomical significance, bringing the L/S ratio to 6.3. This led him to the bizarre conclusion that 'the body of the moon is more dense and more earthy than the earth itself.' (Propn. 37) His contemporaries seemed to accept this conclusion, eg William Whiston in his 'Astronomical Lectures' of 1710 gave the lunar relative density as 7.00 and Earth's as 3.87, compared to the Sun as 1.00 (Whiston, 1710, Frontespiece). See Halley's work on the Hollow Earth.

The error was reduced in the second edition of the Principia of 1713, where the Earth/Moon mass ratio became 1:39.371. A greatly increased level of bogus accuracy is here evident, with a five-figure value carrying a hundred percent error. The tidal-pull ratio appeared as L/S = 4.4815, a figure much used on the pages following Proposition 37. There is a diverting account of Newton and Cotes adjusting the figures in this context, in the Westfall biography: 'As he completed Book III, Newton doctored still another computation in his effort to create an illusion of great accuracy.' (1980, p.736). The figure became 1:39.788 in the Third Edition of 1726.

Tides and the inverse cube law

Newton was the first person who ever explained why there were two tides a day, which Halley viewed this as one of the finest achievements of the Principia, as the review he wrote for it makes clear (PT xvi 1687 291-7; xix 1696 445-57). Newton's tidal computation hinged upon an inverse cube relation, of tide-raising power of the luminaries with distance, whose derivation he did not explain, so that his contemporaries were at a loss to understand it. It was evident that solar gravity pulled the Earth about two hundred times more strongly than did that of the Moon, but it was equally evident that the Moon exerted a stronger pull upon the tides than the Sun. Plainly, the lunar day was more important for tides than the solar day.

The inverse-cube relation determines the tidal vector because, in modern terms, it is the differential or gravity field gradient that affects the height of tides: 'but the force of the moon to move the sea varies inversely as the cube of its distance from the Earth' (Propn 37). The Principia's tidal argument hinged upon this statement, with little by way of demonstration. The early eighteenth-century astronomy textbooks by Newtonians such as Whiston and Gregory omitted this argument, for they had no means of following it.

Brougham in his 1855 analysis of Newton's lunar theory referred, in passing, to Newton's inverse-cube tide-raising law, giving no account of how he used or derived it, and I have not come across any other account of its place in Newton's lunar studies. Also, Brougham stated concerning the lunar mass: 'it is now known that its true value is about 1/49th that of Earth' (1855, p.294), indicating that the true value was still unknown a little more than a century ago. As Curtis Wilson has shown, more exact values had been obtained by French astronomers in the previous century (1987, p.252). A correct value for this ratio was obtained by British astronomers in the 1870s, using Earth's motion around its baricentre.

A Baricentre outside the Earth

Newton in 1713 found that the Earth-Moon system revolved around a point outside the Earth, their baricentre or common centre of gravity. (In fact, their common centre of gravity remains always several thousand miles below the Earth's surface.) The notion of a baricentre had been discussed in general terms by Wren, Hooke and Wallis in their studies of momentum conservation in the 1670s, but this was the first calculation. As a consequence of his overestimation of lunar mass, the common centre of gravity was moved right outside the Earth.The Principia positions the baricentre outside the Earth (1713).

This misplacement of the baricentre produced an undue contraction in the lunar orbit radius, resulting in the 'Moon-test' of 1713 being less accurate than his first one of 1687. Although carried through in nine-figure computations, and claiming to achieve an accuracy of one part in 4000 (as Westfall observed in his 'Newton and the Fudge Factor' 1973), it used a lunar orbit radius that was 1% too small. Thereby celestial and earthly mechanics were linked together to around 1% accuracy - though his nine-figure computations indicate Newton had a greater accuracy in mind.

Also see the Moon Test page.

Sources: Kollerstrom, 1985, 1991,1992 in bibliography  (NB, The above material is not contained in my Doctoral thesis).

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