HARVARD PAPER

This paper discusses a hitherto undetected error of 100% in the Earth-Moon mass ratio by Newton, and suggests a reason why it has been overlooked. (Note, 2004: I may have somewhat exaggerated in describing this error as ‘hitherto undetected:’ as Curtis Wilson pointed out to me, it was an issue of major concern to late 18th-century French astronomers. In the UK, however, few ever heard of it.)

An important mistake in Newton’s lunar computa­tions, not hitherto detected, is the overestimation of the Moon’s mass by 100%. The Moon : Earth mass ratio is inferred to be 1 : 40 (Principia Book III Prop. 37, Cor. IV) when in fact it is 1 : 81; and the mean lunar density to be 1-5 times that of Earth (System of the World’, 56) whereas it is 0-6 that of Earth’s mean density.

In what follows, the means by which Newton arrived at the erroneous estimate is first described by way of a comparison of the solar and lunar tide­raising components. Then, the effect of the error upon his estimation of the barycentre of the Earth­Moon system is examined, and it will be seen how his error in positioning the barycentre affected his gravity computation. Lastly, it is asked why such a prominent error has not hitherto been detected.

RELATIVE MASS OF THE MOON FROM TIDAL DATA

Newton arrived at an accurate estimate of the Sun’s mean relative density as one quarter that of Earth (modern value 0.27). This he found using his inverse square law of gravity; he compared the rate at which planets accelerate towards the Sun in their orbits with the rate at which the Moon accelerates towards the Earth in its orbit. However, as nothing is observ­able accelerating towards the Moon, he had no such means of obtaining an estimate of its relative mass.

Newton therefore tried to obtain a comparison of the relative masses of Sun and Moon from their effects on the tides, as he knew their distances from Earth. (Proposition 37, “To find the force of the moon to move the sea” Bk 3 of Principia, 3rd Edn.) Twice a month, at Full and New Moon, the Spring tide occurs, which Newton regarded as being the sum of solar and lunar tidal components, as the two luminaries were then aligned, L+S. The Neap tide occurs a week later, at first and last quarter, and these, the smallest.tides of the month, he regarded as L-S, i.e. the difference between solar and lunar components. (Thus, if the two components were the same there would be no tides twice a month, they would exactly cancel out.)

He effectively assumed that S and L were vectors, and that tidal height could be found by adding or subtracting them at different times of the month. This sounds a very reasonable assumption, and cor­responds to the mathematical Principle of Superpo­sition, namely that two waveforms combine to form a third whose amplitude at any point is the arith­metic sum of the two separate amplitudes. Unfortu­nate’, as we shall see, the actual tides do not operate with such mathematical elegance.

Newton obtained his tidal data from Samuel Sturmy at Bristol. At the mouth of the Avon, the water rose 13.7 metres (45 feet) at the vernal and autumnal equinox syzygies, but at quadrature 7.6 metres (25 feet) only, in the year 1668. At Plymouth, Samuel Colepresse obtained figures of 12.2 metres (41 feet) to 7 metres (23 feet), respec­tively. This ratio Newton found encouragingly simi­lar to Sturmy’s data, and so formulated the equation

(L+S)/(L-S) = 45/25 = 9/55

solving which would have given L = 3 1/2 S. Instead of doing this, however, he introduced an adjustment factor (concerned with the Moon’s declination in quadratures) the relevance of which is, from an astronomical viewpoint, difficult to see. Thereby he obtained L = 5.3 S. (1) A further correction was next inserted because the highest tides arose each month when the Moon was some distance beyond syzygy, about 30°. Thereby he obtained

L = 4.4815 S.

This was his final figure for the tidal ratio. Mathe­matically, deriving a five-figure decimal from rough tidal data is open to criticism, but the figure is much used on the pages following proposition 37 of the Principia.

The modern figure is L = 2.2S, so the value 4.4815 is too large by a factor of two (2). Professor Westfall has described in some detail the process by which Newton arrived at his value, calling it “Newton’s Fudge Factor” (3). However, Newton had no preconceived idea of what the ratio was supposed to be, nothing against which to check it.

How had he gone wrong? Apart from certain correction factors which would have been better omitted, the basic assumption of (L+S) and (L-S) does not very well fit the highly complex and

geographically localized manner in which the oceans resonate to tidal rhythms. The two tidal rhythms of 12 hour (solar) and 12.4 hour (lunar) will resonate to very different extents in a sea, depending on its size and shape. There is no one figure for the Spring/ Neap tidal ratio. Possibly if Newton had ever in his life troubled to visit the seaside, something of this would have dawned on him!

Having obtained his soli-lunar tidal ratio, Newton proceeds to a very remarkable astronomical inference, never yet properly appreciated. It would have been evident to him that the Sun’s pull on Earth was several hundred times stronger than the Moon’s pull, simply by applying the inverse square law. Whatever value for lunar density is taken this must be so. And yet it was also evident that the Moon had a stronger effect on the tides than the Sun: high tides tended to occur when the Moon was overhead not when the Sun was overhead.

The tide-raising power of the two luminaries varies as the inverse cube of their distance: “Although the gravitational force diminishes as the inverse square of distance, the height of the tidal bulge varies as the inverse third power of the separa­tion between the satellite and the planet, reflecting the fact that the tide is due to the difference in the satellite’s attraction at opposite sides of the planet” (4).

It was somehow evident to Newton that his soli­lunar tidal ratio reflected the inverse cube ratio of the distances involved, solar and lunar, although he merely stated this without any sign of having derived it:

    “But the force of the moon to move the sea varies inversely as the cube of its distance from the earth.” (Prop. 37)

and, of the Sun

    “.. its force to raise the sea is ... inversely as the cube of its distance from the earth.” (Prop. 36)

Brougham (5) in 1855 referred, in passing, to Newton’s use of the inverse cube tide-raising law, but gave 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.

The inverse cube tidal law can be derived in a fairly staightforward manner from the inverse square law of gravity. I have the impression that Newton’s realization of this law-for he was the first to use it-was more intuitive than deductive. It reminds one of De Morgan’s comment that Newton was “so happy in his conjectures as to seem to know more than he could possibly have any means of proving.”

If this inverse cube tide-raising law is assumed, and if it is further assumed that the Sun and Moon are of the same angular size, as they are quite closely, then the soli-lunar tidal ratio turns out to be equal to the ratio of the mean densities of the Sun and Moon (see Appendix). In Corollary III of Proposition 37, Newton applies this principle, which follows from the inverse cube law, but making a slight adjustment for a supposed difference in angular diameter of the two luminaries:

    “Cor. III. Because the force of the moon for moving the sea is to the like force of the sun as 4.4815 to 1, and those forces (by Cor. XIV, Prop. 66, Book 1) are as the densities of the bodies of the sun and moon and the cubes of their apparent diameters conjointly, the density of the moon will be to the density of the sun directly as 4.4815 to 1, and inversely as the cube of the moon’s diameter to the cube of the sun’s diameter; that is (seeing the mean apparent diameters of the moon and sun are 31’ 16’/z”, and 32’ 12”), as 4891 to 1000. But the density of the sun was to the density of the earth as 1000 to 4000; and therefore the density of the moon is to the density of the earth as 4891 is to 4000, or as 11 to 9. Therefore the body of the moon is more dense and more earthy than the earth itself.”

Thereby he derives the Earth/Moon mass ratio as 39.788 to 1. Thus a 100% error arrived in making the first estimate of lunar mass. Newton’s derivation thereof is chiefly remarkable for his anticipation of the inverse cube tide-raising law.

NEWTON’S BARYCENTRE COMPUTATION

Newton seldom gets the credit he deserves for being the first to conceive, estimate and apply the common centre of gravity for the Earth-Moon system. His estimate of the mean radius of the orbit of the Moon around this barycentre was in error by 1-2% owing to the error in lunar mass.

He had arrived at a remarkably accurate estimate of the mean distance of the Moon from Earth, as 60 2/5 Earth-radii (Prop. 37, Cor. VII). The true figure is 60.27, so he was within 0.2%. It was probably the most accurate figure used by an astronomer. up to that time.

He had discussed in Part I how two orbs circling each other will rotate about their common centre of gravity. Thus the distances of Moon and Earth from it will be, he estimates, in the ratio of 40:1 approxi­mately, i.e. the Earth orbits once a month around a point outside itself! The barycentre is actually located 1000 km beneath the Earth’s surface. Pro­fessor Westfall writes of this computation (Prop. 37, Cors. VI, VII) that “considerations of the tides led to the determination of the common centre of gravity,” (6) missing the point that a barycentre was computed from the mass ratio of the two orbs involved, not from tidal data.

Through decreasing the 60 2/5 figure in the ratio of 40.788 to 39.788 Newton obtained a mean radius for the lunar orbit as 1,158,268,534 feet (353,272 km). (or, mean lunar radius about the barycentre is given by d = 60.4x39.8/40.8 = 58.9 Earth radii.) This occurs in the well-known section where gravita­tional force at the Moon is related to that at the Earth’s surface (Prop. 37, Cor. VII). Unfortunately he shifted over to 9-figure calculations, whereas previously he had been working to 5- or 6-figure accuracy.

His final estimation, deduced from lunar orbital data, of the distance that an object dropped on Earth would fall in 1 second, is 15.11175 feet (4.6 metres); the modern figure for free fall in vacuo is 16.09 feet (6.41 metres) (1/2 g), which corresponds to 15.07 Paris feet, which Newton was using (1 Paris foot = 1.068 imperial feet). Thus his supposedly deduced figure agrees with the modern figure by 0.2%. It is an academic question, as to how from a lunar orbital radius value which was 1.0% too small, he deduced a free-fall value at the Earth’s surface which was 0.2% too high. Professor Westfall may be right that the value of the lunar radius had really been deduced from the free-fall value, i.e. that Newton worked backwards. Whichever way the computation was done, celestial and earthly mechanics were linked together at around 1 % accuracy. Admittedly 9-figure computations indicate that Newton had a greater accuracy in mind.

There is one other area in which the lunar mass error is relevant, and that is Newton’s calculation of the precession. of the equinoxes. Some have regarded the computation as suspicious; for example William Whewell wrote in 1857 that, of precession, “in all, his means of calculation were insufficient” (7). He used again the 1:4.4817 S/L ratio, i.e. he assumed precession was mainly due to the pull of the Moon. It is normally claimed that he computed the magnitude of precession, as he indeed claimed to have done, and his calculations showed an annual precession of the equinoxes of 50” 00”’ 12iv

    “which motion agrees with the phenomena for the precession of the equinoxes by astronomical observa­tions is about 50” yearly.” (Prop. 39)

Doubt is unfortunately cast on this close agree­ment between theory and observation, if his lunar mass is in fact out by 100%. His estimate of the bulge around Earth’s equator was out by nearly 50% (54.7 km compared with 37 km), so the precession section is more an indication of the way in which lunar gravity produces the gyration of Earth’s axis than a real computation of its magnitude.

As I. Bernard Cohen wrote in 1975, there is a need for “a coherent presentation of Newton’s theory of the moon-historical and analytical and critical-indicating both its real achievements and its failures” (8).

In 1855 Brougham in his analysis of Newton’s lunar theory stated that “It is now known that the true mass is about 1/49th that of Earth” (9). So the true ratio was still unknown a century ago. It is nowadays determined from an estimation of the position of the barycentre-the opposite process to that used by Newton. The Earth’s motion around the barycentre is accurately estimated from stellar parallax observa­tions, and then from this an Earth-Moon mass ratio of 81.3: 1 is inferred.

In 1873 the Earth/Moon mass ratio was taken as 81.4: 1 by Proctor, this being a mean value of esti­mates obtained by the astronomers Stone, Leverrier and Newcomb (10). They were derived not only from parallax observations, but also from the magnitude of nutation (the 18.6-year motion of Earth’s axis due to rotation of the lunar nodes).

By the time the Earth/Moon mass ratio became accurately known, physicists and astronomers had ceased reading the Principia, so this key error lay undetected.

APPENDIX

If an orb lies at a distance d from the Earth, and has a radius r, a mean density p, and subtends a small angle 2A, then its mass ? pr3.

But r = d sin A, so the mass ? pd3 sin3A.

If the tidal pull a mass/d3, then the pull ? psin3A.

If S and L have the same value for A, the tidal pull

S/L = pS/pL.

Figure 2. Tidal pull and ratio of mean densities
of Sun and Moon.

REFERENCES

1 System of the World, 1803 edition, p. 47. Newton’s steps in deriving the L/S tidal ratio are described in Westfall, R., Never at Rest: a biography of Isaac Newton, 736, Cambridge, 1980.

2 Young, C. A., A Text Book of General Astronomy, 282, Lon­don 1889. He wrote: “Since the tide-raising power varies as the cube of the distance inversely, while the attracting force varies only with the inverse square, it turns out that although the sun’s attraction on the Earth is nearly 200 times as great as that of the moon, its tide-raising power is only about two fifths as much.”

3 Westfall, R., ‘Newton and the Fudge Factor’, Science, 179, 751 (1973).

4 Goldreich, P., ‘Tides and the Earth-Moon System’, Scientific American, 226, 43 (1972).

5 Brougham, H. P. and Routh, E. J., Analytical View of Sir Isaac Newton’s Principia, 289, London, 1855.

6 Westfall, R., op. cit., 733.

7 Whewell, W., History of the Inductive Sciences, 135, London (3rd Edn.), 1857.

8 Isaac Newton’s Theory of the Moon’s Motion, with introduc­tion by I. Bernard Cohen, 67, Folkestone, 1975.

9 Brougham, H. P. and Routh, E. J., op. cit., 294.

10 Proctor, R. A., The Moon, London, 1873.

Note: Curtis Wilson had discussed this lunar mass error, in: 'Perturbations and Solar Tables from Lacaille to Delambre', Archive for History of Exact Sciences, 1980, 22, 53-188, 88 and in 'D'Alembert versus Euler on the Precession of the
Equinoxes' AHES 1987, 37, 233-273, 242.