Science & Technology Studies

Nicholas Kollerstrom's

Newton's 1702 Lunar Theory

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.

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.

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).

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*rev: May 1998*