more accurately than either factor is. Calculations in celestial mechanics can also be carried out using the units of solar masses, mean solar days and astronomical units rather than standard SI units. For this purpose, the Gaussian gravitational constant was historically in widespread use, , expressing the mean angular velocity of the Sun–Earth system measured in radians per day. The use of this constant, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU since 2012. History of measurement 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 dates to the 1890s), but is not 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. 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: ≈ A measurement was attempted in 1738 by Pierre Bouguer and Charles Marie de La Condamine in their "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 shell, as some thinkers of the day, including Edmond Halley, had suggested. 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 ( times the density of water), about 20% below the modern value. 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's mean radius and the mean gravitational acceleration at Earth's surface, by setting Based on this, Hutton's 1778 result is equivalent to ≈ . The first direct measurement of gravitational attraction