Schiehallion experiment

56°40′4″N 4°5′52″W / 56.66778°N 4.09778°W / 56.66778; -4.09778

The Schiehallion experiment was an 18th-century

The experiment had previously been considered, but rejected, by

An experiment to test Newton's idea would both provide supporting evidence for his law of universal gravitation, and estimates of the mass and density of the Earth. Since the masses of astronomical objects were known only in terms of relative ratios, the mass of the Earth would provide reasonable values to the other planets, their moons, and the Sun. The data were also capable of determining the value of the Newtonian constant of gravitation G, though this was not a goal of the experimenters; references to a value for G would not appear in the scientific literature until almost a hundred years later.^[4]

Finding the mountain

Chimborazo, 1738

A pair of French astronomers,

Between 1763 and 1767, during operations to survey the Mason–Dixon line between the states of Pennsylvania and Maryland, British astronomers found many more systematic and non-random errors than might have been expected, extending the work longer than planned.^[7] When this information reached the members of the Royal Society, Henry Cavendish realized that the phenomenon might have been due to the gravitational pull of the nearby Allegheny Mountains, which had probably diverted the plumb lines of the theodolites and the liquids inside spirit levels.^[8]

Prompted by this news, a further attempt on the experiment was proposed to the Royal Society in 1772 by Nevil Maskelyne, Astronomer Royal.^[9] He suggested that the experiment would "do honour to the nation where it was made"^[3] and proposed Whernside in Yorkshire, or the Blencathra-Skiddaw massif in Cumberland as suitable targets. The Royal Society formed the Committee of Attraction to consider the matter, appointing Maskelyne, Joseph Banks and Benjamin Franklin amongst its members.^[10] The Committee dispatched the astronomer and surveyor Charles Mason to find a suitable mountain.^[1]

After a lengthy search over the summer of 1773, Mason reported that the best candidate was

Mason declined to conduct the work himself for the offered commission of one

Maskelyne published his initial results in the Philosophical Transactions of the Royal Society in 1775,^[14] using preliminary data on the mountain's shape and hence the position of its center of gravity. This led him to expect a deflection of 20.9″ if the mean densities of Schiehallion and the Earth were equal.^[3][15] Since the deflection was about half this, he was able to make a preliminary announcement that the mean density of the Earth was approximately double that of Schiehallion. A more accurate value would have to await completion of the surveying process.^[14]

Maskelyne took the opportunity to note that Schiehallion exhibited a gravitational attraction, and thus all mountains did; and that Newton's inverse square law of gravitation had been confirmed.^[14][16] An appreciative Royal Society presented Maskelyne with the 1775 Copley Medal; the biographer Chalmers later noting that "If any doubts yet remained with respect to the truth of the Newtonian system, they were now totally removed".^[17]

Surveying

The work of the surveying team was greatly hampered by the inclemency of the weather, and it took until 1776 to complete the task.

That the mean density of the Earth should so greatly exceed that of its surface rocks naturally meant that there must be more dense material lying deeper. Hutton correctly surmised that the core material was likely metallic, and might have a density of 10,000 kg·m⁻³.[18] He estimated this metallic portion to occupy some 65% of the diameter of the Earth.^[19] With a value for the mean density of the Earth, Hutton was able to set some values to Jérôme Lalande's planetary tables, which had previously only been able to express the densities of the major solar system objects in relative terms.^[19]

Repeat experiments

A more accurate measurement of the mean density of the Earth was made 24 years after Schiehallion, when in 1798

The Scottish scientist John Playfair carried out a second survey of Schiehallion in 1811; on the basis of a rethink of its rock strata, he suggested a density of 4,560 to 4,870 kg·m⁻³,^[22] though the then elderly Hutton vigorously defended the original value in an 1821 paper to the Society.^[3][23] Playfair's calculations had raised the density closer towards its modern value, but was still too low and significantly poorer than Cavendish's computation of some years earlier.

The Schiehallion experiment was repeated in 1856 by

A modern re-examination of the geophysical data was able to take account of factors the 1774 team could not. With the benefit of a 120-km radius digital elevation model, greatly improved knowledge of the geology of Schiehallion, and the help of a computer, a 2007 report produced a mean Earth density of 5,480 ± 250 kg·m⁻³.^[26] When compared to the modern figure of 5,515 kg·m⁻³, it stood as a testament to the accuracy of Maskelyne's astronomical observations.^[26]

Mathematical procedure

Consider the

where G is the Newtonian constant of gravitation. G and m can be eliminated by taking the ratio of F to W:

${\displaystyle {\frac {F}{W}}={\frac {GmM_{M}/d^{2}}{GmM_{E}/r_{E}^{2}}}={\frac {M_{M}}{M_{E}}}\left({\frac {r_{E}}{d}}\right)^{2}={\frac {\rho _{M}}{\rho _{E}}}{\frac {V_{M}}{V_{E}}}\left({\frac {r_{E}}{d}}\right)^{2}}$

where V_M and V_E are the volumes of the mountain and the Earth. Under

${\displaystyle W=T\cos \theta ,\quad F=T\sin \theta }$

Substituting for T:

${\displaystyle \tan \theta ={\frac {F}{W}}={\frac {\rho _{M}}{\rho _{E}}}{\frac {V_{M}}{V_{E}}}\left({\frac {r_{E}}{d}}\right)^{2}}$

Since V_E, V_M and r_E are all known, θ has been measured and d has been computed, then a value for the ratio ρ_E : ρ_M can be obtained:^[22]

${\displaystyle {\frac {\rho _{E}}{\rho _{M}}}={\frac {V_{M}}{V_{E}}}\left({\frac {r_{E}}{d}}\right)^{2}{\frac {1}{\tan \theta }}}$

Notes

References