Editing Gravitational constant (section)

=== Early history ===

The existence of the constant is implied in [[Newton's law of universal gravitation]] as published in the 1680s (although its notation as {{math|''G''}} dates to the 1890s),<ref name=BoysG/> but is not [[Algebra#Algebra as a branch of mathematics|calculated]] in his ''[[Philosophiæ Naturalis Principia Mathematica]]'' where it postulates the [[inverse-square law]] of gravitation. In the ''Principia'', Newton considered the possibility of measuring gravity's strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable.<ref name="Davies">{{cite journal|last=Davies|first=R.D.|title=A Commemoration of Maskelyne at Schiehallion|journal=Quarterly Journal of the Royal Astronomical Society|volume=26|issue=3|pages=289–294|bibcode=1985QJRAS..26..289D|date=1985}}</ref> Nevertheless, he had the opportunity to estimate the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order:<ref>"Sir Isaac Newton thought it probable, that the mean density of the earth might be five or six times as great as the density of water; and we have now found, by experiment, that it is very little less than what he had thought it to be: so much justness was even in the surmises of this wonderful man!" Hutton (1778), p. 783</ref>
: {{math|''G''}} ≈ {{val|6.7|0.6|e=-11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}

A measurement was attempted in 1738 by [[Pierre Bouguer]] and [[Charles Marie de La Condamine]] in their "[[French Geodesic Mission|Peruvian expedition]]". Bouguer downplayed the significance of their results in 1740, suggesting that the experiment had at least proved that the Earth could not be a [[Hollow Earth|hollow shell]], as some thinkers of the day, including [[Edmond Halley]], had suggested.<ref name="Poynting_p50-56">{{cite book|last=Poynting|first=J.H.|title=The Earth: its shape, size, weight and spin|publisher=Cambridge|date=1913 |pages=50–56 |url=https://books.google.com/books?id=whA9AAAAIAAJ&pg=PA50}}</ref>

The [[Schiehallion experiment]], proposed in 1772 and completed in 1776, was the first successful measurement of the mean density of the Earth, and thus indirectly of the gravitational constant. The result reported by [[Charles Hutton]] (1778) suggested a density of {{val|4.5|u=g/cm3}} ({{sfrac|4|1|2}} times the density of water), about 20% below the modern value.<ref name="Hutton">{{cite journal|last=Hutton|first=C. |date=1778 |title=An Account of the Calculations Made from the Survey and Measures Taken at Schehallien |journal=Philosophical Transactions of the Royal Society |volume=68 |pages=689–788 |doi=10.1098/rstl.1778.0034|doi-access=free }}</ref> This immediately led to estimates on the densities and masses of the [[Sun]], [[Moon]] and [[planets]], sent by Hutton to [[Jérôme Lalande]] for inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, given [[Earth radius#Mean radius|Earth's mean radius]] and the [[little g|mean gravitational acceleration]] at Earth's surface, by setting<ref name=BoysG>[https://books.google.com/books?id=ZrloHemOmUEC&pg=PA353 Boys 1894], p.330 In this lecture before the Royal Society, Boys introduces ''G'' and argues for its acceptance. See:
[https://archive.org/details/meandensityeart00poyngoog/page/n26 <!-- pg=4 --> Poynting 1894], p.&nbsp;4, [https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA1 MacKenzie 1900], p.vi</ref>
<!--modern values: g=9.80665 ms^-2, Re=  6.3781e+6 m
3*g/(4*pi*Re)=3.6706e-7
3.6706e-7/5.448e3=6.7375e-11
the "correct" value (for G=6.674e-11) would be 5.500 gcm^-3.
-->
<math display="block">G = g\frac{R_\oplus^2}{M_\oplus} = \frac{3g}{4\pi R_\oplus\rho_\oplus}.</math>
Based on this, Hutton's 1778 result is equivalent to {{nowrap|{{math|''G''}} ≈ {{val|8|e=-11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}}}.

[[File:Cavendish Torsion Balance Diagram.svg|thumb|Diagram of torsion balance used in the [[Cavendish experiment]] performed by [[Henry Cavendish]] in 1798, to measure G, with the help of a pulley, large balls hung from a frame were rotated into position next to the small balls.]]
The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, by Henry Cavendish.<ref>Published in ''[[Philosophical Transactions of the Royal Society]]'' (1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S., ''Scientific Memoirs'' Vol. 9: ''The Laws of Gravitation''. American Book Co. (1900), pp. 59–105.</ref> He determined a value for {{math|''G''}} implicitly, using a [[Torsion spring#Torsion balance|torsion balance]] invented by the geologist Rev. [[John Michell]] (1753). He used a horizontal [[torsion beam]] with lead balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. In spite of the experimental design being due to Michell, the experiment is now known as the Cavendish experiment for its first successful execution by Cavendish.

Cavendish's stated aim was the "weighing of Earth", that is, determining the average density of Earth and the [[Earth's mass]]. His result, {{nowrap|1={{math|1=''ρ''<sub>🜨</sub>}} = {{val|5.448|(33)|u=g.cm-3}}}}, corresponds to value of {{nowrap|1={{math|1=''G''}} = {{val|6.74|(4)|e=-11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}}}. It is remarkably accurate, being about 1% above the modern [[CODATA]] recommended value {{physconst|G|round=3|ref=no}}, consistent with the claimed relative standard uncertainty of 0.6%.