N. Kollerstrom

Jnl. for History of Astronomy 1992, 23, 185-192

Newton’s Lunar Density Estimate

Three hundred years ago in 1692, an article by Edmond Halley proposed that the Earth was hollow.(1) Its theory was based on the value of lunar relative density given by Isaac Newton. The first edition of Newton’s Principia (1687) found that “... the mass of the Moon will be to the mass of the Earth as 1 to 26, approximately”, citing the relative densities of Moon to Earth as 9 to 5.(2) This value of lunar relative mass was in excess by a factor of three, as the true mass ratio is 1:81. Arguably the most significant error in the Principia’s Book III, it left an ultra-dense Moon circling our Earth.(3) Edmond Halley simply invoked these figures: “Sir Isaac Newton has demonstrated the Moon to be more solid than our Earth, as 9 to 5; why may we not then suppose four ninths of our globe to be cavity?”(4) It is remarkable that so erroneous a figure, having such unlikely implications, could be thus presented without need for further justification. Halley’s theory appeared as the first significant deduction to be drawn from the Principia.

Newton’s estimate of lunar relative density was derived from the relative tide­raising powers of the Sun and Moon. The Principia had ascertained fairly well the relative density of the Sun, as a quarter that of the Earth (Book III, Prop. 37, Cor. 3), and so by comparing the components of tidal attraction of the two luminaries the lunar relative density was thereby inferred. This was a quite valid method, as shown by the way that French theoretical astronomers used it in the mid-eighteenth century to obtain their estimates of lunar relative mass.(5) However, the Principia’s treatment thereof went greatly astray. It started from the difference between spring and neap tides which occurred twice each month, taking data from Plymouth and in the Bristol channel that gave that ratio as 41 to 23, or 9 to 5.

Newton apprehended (in Proposition 37) that the tide-raising forces of the Sun and Moon varied inversely as the cube of their distances from Earth: only thus would the Moon have a stronger attraction for the tides than the Sun.(6) It was evident to Newton that the solar gravity pull (varying inversely as the square of the distance) was several hundred times stronger than that of the Moon, whatever assumptions about relative densities were made. The Princi­pia’s tidal argument hinged upon this inverse-cube relationship, with little by way of demonstration. Its method of inferring lunar relative density from such considerations would have baffled his contemporaries. Astronomy textbooks by Newtonians such as Whiston and Gregory in the early eighteenth century omitted this argument, for they had no means of following it.

The Principia formulated the equation (S+L)/(S-L) = 9/5, where S and L were tide-raising vectors varying inversely as the cube of distance. Newton apprehended that the ratio involved differentials or gravity field gradients across the Earth and not the forces as such; and that these would sum to maximum values at both the full and new moon positions, but subtract as vectors when the forces were at right angles to each other, that is, at the quadrature positions. This is all quite impressive, and solving the above equation would have given him L:S = 3.5:1, far less disastrous than his final result. For comparison, the astronomically correct ratio of the tide-raising powers of the Moon and Sun is 2.17:1,(7) though we may note that the mean value of this ratio around the shores of Britain is slightly over three to one, as the extent to which the solar (12 hour) and lunar (12.4 hour) diurnal rhythms resonate in the sea varies with local geography.

Instead of solving this equation, Newton inserted various adjustments, of somewhat doubtful astronomical significance,(8) bringing the L:S ratio to 6.3:1. In the Principia’s second edition of 1713, the computation was emended to give an Earth/Moon mass ratio of 39.371 to 1, which became in the third edition 39.788 to 1, thereby reducing its lunar mass estimate to merely 100% in excess of the correct value. From this estimate the second edition obtained a baricentre position (i.e. common centre of gravity of the Earth-Moon system), misplacing it as permanently outside the Earth.(9) Scholars normally refer only to the second and third editions in the context of lunar theory, and I have found no mention of the first edition’s 1:26 Moon/Earth mass ratio estimate in the literature. But developments after the first edition were not used by Halley in his advocacy of a hollow Earth, and we will not refer to them in what follows. Halley viewed Newton’s tidal theory as one of the finest achievements of the Principia’s first edition, as the two reviews he wrote for it make clear (10).

Internal Structure of the Earth

From the study of magnetic compass variations, Halley by 1683 had reached the quite original conclusion that the Earth possessed four magnetic poles (11). He described in the Philosophical transactions of that year how two of these poles were located in the “Southern ocean” and the two northern ones were in the Bering Strait and Spitzbergen (12). However he could not account for the existence of multiple poles, nor their gradual displacement with time, acknowledging that the latter depended on “secrets as yet utterly unknown to Mankind” (13).

Various attempts had been made to account for the gradual motion of the lines of magnetic declination by a few minutes of arc each year, as this had great relevance to navigation.(14) Descartes for example had suggested that such motion was due to physical accretion and movement of iron ore deposits (15), but Halley realized during his 1676 voyage to St Helena that no theory depending on surface deposits of iron could explain the compass variations. Henry Bond claimed to be able to predict secular changes in terrestrial magnetism, and in 1674 the King set up a committee to investigate this claim (16). Robert Hooke, who was a member of the committee, proposed in 1674 that the magnetic poles were moving in a circular path at 10° from the geographic poles, possibly rotating once in 370 years (17).

Halley had gathered much data on the subject from his voyage to St Helena, as he also had from the unpublished manuscripts of Peter Perkins. Perkins had been researching this topic and addressed the Royal Society on the subject in 1680 some months before his death. Halley purchased Perkins’s papers on the variation of geomagnetism immediately following the latter’s demise (18).While it is true that Halley never acknowledged this, it is equally true that nothing has substantiated Flamsteed’s allegation that components of Halley’s theory de­rived from Perkins. Perkins was developing a theory of his own to account for the variation, as was indicated in his address to the Royal Society, but it is hard to discern any resemblance to that developed by Halley (19). The first part of Halley’s 1692 essay commented on the inadequacy of existing theories, conclud­ing that something deep below Earth’s surface must be causing the phenomenon.

Halley had been much involved with the production of the Principia, and it now seemed to provide him with a key. Its estimate of the Earth/Moon mass ratio suggested to him that the Earth was hollow. How else could that ratio be explained? The germ of the idea may have dawned upon him while reading Burnet’s Sacred theory of the Earth which had appeared (in Latin) in 1681. This assigned hollow cavities to the Earth, catacombs and subterranean grottoes, but did so in a traditional mode in accord with classic myth and lore (20). The tenor of Burnet’s vision was, as Schaffer has observed (21), in stark contrast to that of Halley.

With this assumption, Halley could gain a physical explanation for the existence of his four magnetic poles, and also their motions. Within the Earth, concentric to its hollow shell and rotating coaxially, he discerned another sphere, whose existence had hitherto been unsuspected. Possibly there were further spheres hidden within that one. Both spheres had magnetic poles embedded within them at a distance from their common axis of rotation. A very gradual differential rotation between the two spheres accounted for the drift of the magnetic poles with time. One pair of magnetic poles was stationary, being embedded in the outer shell, while the other pair drifted westwards, because the inner sphere was revolving at a slower rate. Of the two northern poles, Halley opined that the pole due north of Land’s End was the moving one, while the other lying on a meridian through California, at some 15° from the pole, was static. Of the two southern poles, that south of America was the moving one.

Halley’s Earth was composed of an outer shell 500 miles thick, with an air gap of the same distance between it and the inner sphere. To the objection that the latter might collide with the outer shell, and thereby damage it, he explained that it was held at the centre by the force of gravity. Halley was confident his readers would perceive the necessity of this: “should these globes be adjusted once to the same common centre, the Gravity of the parts of the Concave would press towards the centre of the inner ball ... it follows that the Nucleus being once fixt in the common centre, must always here remain (22). Halley pointed out that “the Ring environing the Globe of Saturn”, which remained coaxial to the planet, was held there by gravity. (No-one then knew that Saturn’s rings were rotating. The Principia had not discussed the matter.) By analogy, could not gravity also hold a globe concentric inside the hollow Earth?

Happily, Halley’s argument was in accord with “Almighty Wisdom”, which would not have arranged all the matter of Earth “barely to support its surface”. Rather, the matter had been distributed “to yield as great a Surface for the use of living Creatures as can consist with the conveniency and security of the whole”. Halley acknowledged that there might be objections to his new theory: for example, might not the oceans leak away if an earthquake opened up a crack beneath them, leading to flooding of the lower regions? His answer was that within the ground there existed “saline and Vitriolick Particles as may contri­bute to petrefaction”. Water flowing down would soon find its path blocked by these petrifying particles.

“I have adventured to make these Subterranean orbs capable of being inhabited”, Halley added, in words for which science fiction writers of futurity would be grateful (23). His reason for this was, that all nature teemed with life. The argument was teleological. His under-world would be occupied, as the planets too were doubtless inhabited, and so some illumination was therefore needed: “The concave arches may in several places shine with such a substance as invests the surface of the Sun.” Halley conceded that he was here using a “final cause” in asserting that the luminosity of the upper atmospheres of his under-world was provided for its denizens.

It has been argued by David Kubrin that Halley took over Robert Hooke’s view of the Earth as “a series of concentric shells, one of which contained the magnetic poles”, and that Halley’s model of internal terrestrial structure was “rather similar to Hooke’s (24). David Oldroyd has likewise claimed that “... Hooke’s theory was thought sufficiently adequate to furnish a model for the structure of the Earth by Halley in 1692, in his explanation of the Earth’s magnetic poles (25). There are however objections to this view.

Hooke’s model of the Earth’s interior was layered like an onion, and hardly resembled that developed by Halley (26). There is indeed no single feature which the two views shared in common. Hooke’s model lacked any air-gaps between its strata, or subterranean spheres, or principle of differential rotation, or multiple magnetic poles; nor had it any mechanism for a rotation of magnetic poles around the geographic poles. Only, in Hooke’s view, in the Earth’s pristine condition may it once have had spherical concentric shells of matter, but earthquakes had long since disrupted that ideal scheme. Hooke did not locate the Earth’s magnetic power in a specific shell as Kubrin claimed, but rather left it diffusely spread through the “magnetical core or Magnetical Globe of the Earth (27). These views appeared in Hooke’s posthumously published essay “On earthquakes”. In 1674 Hooke gave his opinion to the Royal Society that the Earth’s magnetic poles were rotating around the Earth’s axis, but this was accompanied by no suggestion as to how such an effect could be produced: he never theorized, as did Halley, as to what might produce a periodic rotation of the magnetic poles. For Hooke, earthquakes were an instrument for the transformative processes that he envisaged through the Earth’s geological history, whereas on Halley’s model they presented a difficulty which required accounting for. It therefore appears that not one of the highly distinctive characteristics of Halley’s model came from Hooke. Halley’s theory was original, and derived not from the lectures of Hooke but rather from the error in lunar mass of Newton’s Principia.

Grit in the Clockwork

Halley presented his discovery of the ‘secular acceleration’ of the Moon, whereby its orbit gradually alters over centuries, to the Royal Society in 1695 (28). Schaffer has pointed out that his embryonic ideas on this matter had already appeared in the 1692 paper, in the last argument there given for a hollow Earth (29). This argument concerned the viscosity of interplanetary space.

The Moon had been designed as denser than the Earth, Halley explained, not least so that it could keep up with the Earth in its journey through space. Had the two spheres been of similar density, the Moon would soon have been left behind: “the cavity I assign to the Earth may well serve to adjust its weight to the Moon. For otherwise the Earth would leave the Moon behind it and she become another primary planet.” His argument was subtly mathematical: the deceleration experienced by a planet due to the viscous drag of space would be in proportion to its cross-sectional area (radius’) but inversely as its mass (radius’); which meant that, if Earth and Moon were both of the same density, but of very different sizes, they would be differentially retarded. Earth had been designed as hollow, to avoid losing its Moon.

Book II of the Principia had demonstrated that the virtually perfect vacuum of outer space offered negligible resistance to a moving body. This was the sole issue on which Halley ever ventured to disagree with the Principia. Halley had failed in 1691 to be elected to the Savilian chair at Oxford on grounds of atheism, then associated with any belief in “the eternity of the world”. If Halley could show that the solar system was running down, from the friction of interplanetary space, then he would have evidence for a limited duration of the world. Schaffer has argued that he aimed thereby to evade the charge of atheism (30).

If there was such an ether-drag from space, it was not at once evident to Halley what effect this would have. If it caused the Moon to fall nearer to the Earth, then this would have to be accompanied by an acceleration in its orbit, not a retardation. In 1693, Halley published a paper suggesting that mediaeval eclipses were occurring slightly later in time than would have been expected (31). Halley saw in these temporal displacements evidence for his theory that the Moon was accelerating in its monthly orbit and was gradually drawing nearer to the Earth. His discovery came to be called ‘secular acceleration’. In 1694, Newton told him that the Earth’s mass was increasing due to such things as the tails of comets falling upon it, from which Halley found support for his view of the Moon as being pulled ever closer to the Earth, and thereby accelerated (32).

It was a century-and-a-half before astronomers realized that the converse was the case: the Moon is in fact decelerating in its monthly orbit as it gradually recedes from Earth (33). Halley’s final argument over the Earth/Moon density needs to be seen within this context: as part of his quest for the grit in the cosmic clockwork whereby he could avoid the charge of asserting “the eternity of the world”.

Halley’s essay was popular, and frequently reprinted, appearing several times in abridged editions of the Philosophical transactions. The fourth edition of his essay appeared in 1732, the fifth posthumously in 1749 and a later edition in 1819 (34). It was reprinted in vol. i of Miscellanea curiosa, consisting of the “most valuable” discourses read and delivered to the Royal Society, which passed through several editions between 1705 and 1726.

In later life, when Savilian Professor of Geometry at Oxford, Halley together with much of north-west Europe witnessed an extraordinary aurora borealis display, on 6 March 1716, visible even in the daytime. Halley was requested to compose a commentary upon this strange phenomenon for the Royal Society’s journal. Halley had never before witnessed an aurora, in his sixty years as an astronomer (35). After discussing certain magnetic aspects to this phenomenon, he suggested that it might be due to inherently luminous matter inside the Earth emerging through fissures in the North Pole. This substance, he explained, would “transude through and penetrate the Cortex of our Earth” and then spread through the atmosphere as the aurora. He could explain why the luminous vapour would only “transude” in the polar region (36). It was because, as Newton had shown, the Earth was a flattened sphere, and so its shell would therefore be thinnest at the poles. Halley’s portrait at 80 years of age as Astronomer Royal in 1736 shows him holding a diagram of the hollow Earth.

Whiston’s Astronomical lectures of 1715 endorsed Newton’s ultra-dense Moon, citing its mean density as 1.8 times that of the Earth (37). This was the value Newton had published in 1687, suggesting that Whiston had not perused that part of the Principia’s second edition in which this value was revised down­wards. Whiston then left his readers mystified as to how the Moon could have come by such a high density, but his Astronomical principles of religion of 1717 supplied an answer (38). Whiston there explained that the “total Cavity of the Central Regions” of the Earth could well account for the density ratio, and referred to Halley’s paper of the previous year. Whiston found scriptural corroboration (as Halley had not) for these lower regions being inhabited.


1. E. Halley, “An account of the cause of the change of the variation of the magnetical needle with an hypothesis of the structure of the internal parts of the Earth”, Philosophical transac­tions, xvi (1692), 563-87. The paper was read to the Royal Society on 25 November 1691.

2. 1. Newton, Philosophiae naturalis principia mathematica (London, 1687), 466 (Book 3, Prop. 37, Cors. 3 and 4).

3. N. Kollerstrom, “Newton’s lunar mass error”, Journal of the British Astronomical Association, xcv (1985), 151-3.

4. Halley, op. cit. (ref 1), 568.

5. C. Wilson, “D’Alembert versus Euler on the precession of the equinoxes”, Archive for history of exact sciences, xxxvii (1987), 233-73, p. 252.

6. E. J. Aiton, “The contributions of Newton, Bernoulli and Euler to the theory of the tides”, Annals of science, xi (1955), 206-23, p. 211.

7. C. A. Young, A text book of general astronomy (London, 1889), 282: "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."

8. R. S. Westfall, "Newton and the fudge factor", Science, cli (1973), 751-8, p. 756.

9. Newton, op. cit. (ref. 2), 2nd edn (London, 1713), 430 (Book 3, Prop. 37, Cor. 3). See also N. Kollerstrom, "Newton's two moon-tests", The British journal for the history of science, xxiv (1991),369-72.

10. Halley's review of the Principia appeared in Philosophical transactions, xvi (1687), 291-7; further comments of his appeared in his "True theory of the tides", ibid., xix (1696), 445-57.

11. E. Halley, "A theory of the variation of the magnetic compass", ibid., xiii (1683), 208-21.

12. The nonexistence of two of Halley's four magnetic poles was demonstrated in 1817, when the first complete chart of magnetic meridians appeared: Sydney Chapman, "Edmond Halley and geomagnetism", Nature, clii (1943), 231-7.

13. Halley, op. cit. (ref. 11), 221.

14. J. A. Bennett, "Cosmology and the magnetical philosophy, 1640-1680", Journal for the history of astronomy, xii (1981), 165-77.

15. Chapman, op. cit. (ref. 12), 234.

16. E. G. Forbes, History of Greenwich Observatory, i (London, 1975), 15-17.

17. T. Birch, History of the Royal Society (London, 1756), iii, 131.

18. Forbes, op. cit. (ref. 16), 74; F. Baily, An account of the Revd. John Flamsteed (London, 1835), 194. A letter to Flamsteed by Thomas Perkins of 12 December 1700 recalled the purchase of his late brother's papers in 1680 by Edmond Halley.

19. For Perkins's theory see Birch, op. cit. (ref. 17), iv, 18-19; no-one has ever taken Flamsteed's claim seriously, see E. F. McPike, Hevelius, Flamsteed and Halley (London, 1937), 92.

20. See Athanasius Kircher, Mundus subterraneus (Amsterdam, 1665). For discussion of Kirchner and magnetic cosmology, see M. Baldwin, "Magnetism and the anti-Copernican polemic", Journal for the history of astronomy, xvi (1985), 155-74.

21. S. Schaffer, "Halley's atheism and the end of the world", Notes and records of the Royal Society, xxxii (1977), 17-40, p. 18.

22. Halley, op. cit. (ref. 1), 573.

23. C. Zircle, "The theory of concentric spheres: Halley, Mather and Symmes", Isis, xxxvii (1947), 155-9.

24. D. Kubrin, "Such an impertinently litiginous lady", in Standing on the shoulders of giants, ed. by N. J. Thrower (Oxford, 1990), 55-87, pp. 59, 64.

25. D. R. Oldroyd, "Geological controversy in the seventeenth century: Hooke vs Wallis and its aftermath", in Robert Hooke: New studies, ed. by Michael Hunter and Simon Schaffer (Woodbridge, Suffolk, 1989), 207-33, p. 230.

26. For Hooke's geological theories in the 1690s, see Y. Ito, "Hooke's cyclic theory of the Earth", The British journal for the history of science, xxi (1988), 295-314.

27. R. Hooke, "A discourse of earthquakes", Posthumous works, ed. by R. Waller (London, 1707), 279-450, p. 349.

28. E. Halley, "Some account of the ancient state of the city of Palmyra", Philosophical transactions, xix (1695), 160-75, p. 174.

29. Schaffer, op. cit. (ref. 21), 22. 30. Ibid., 18.

31. E. Halley, "Emendationes ac notae in vetustas Albatenii observationes astronomicas", Philoso-phical transactions, xvii (1693), 913-21.

32. Royal Society, Journal Book, 31 October 1694. See D. Kubrin, "Newton and the cyclical cosmos", Journal of the history of ideas, xxviii (1967), 325-46, p. 337.

33. In the mid-nineteenth century the true character of the phenomenon called `secular acceleration' of the Moon came to be understood, as due to the slowing of the Earth's rotation rate. The increasing day-length causes the lunar month to appear shorter over the centuries. The apparent effect has been regarded as sufficiently similar to credit Halley with the discovery of the Moon's `secular acceleration'. See D. Kusher, "Secular acceleration of the Moon's mean motion", Archive for history of exact sciences, xxxix (1989), 291-316.

34. E. Halley, "An account of the cause of the change of the variation of the magnetic needle",Philosophical transactions abridged, 1705 edn, ii, 610-20; 1809 edn (5th edn), iii, 470-8; Miscellanea curiosa, 3rd edn (1726), i, 43-59.

35. The aurora of 1716 marked the conclusion of the sixty-year Maunder Minimum (J. Eddy, “The case of the missing sunspots”, Scientific American, May 1977, 80-92, p. 82), whereby the Sun largely bereft of sunspots had refrained from causing magnetic storms or aurorae, as it has done ever since.

36. E. Halley, “An account of the late surprizing appearance of the lights seen in the air, on the sixth of March last, with an attempt to explain the principal phaenomena thereof”, Philosophical transactions, xxix (1716), 406-28, p. 427.

37. W. Whiston’s Astronomical lectures (London, 1715, 1728) gave the relative densities as: Sun 1.00, Earth 3.87, Moon 7.00 (Frontispiece). This Frontispiece did not appear in the original Latin edition of this work, Praelectiones astronomicae (Cambridge, 1707).

38. W. Whiston, Astronomical principles of religion (London, 1717), 95-96.

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