Clairaut's theorem is a general mathematical law applying to spheroids of revolution. Published in 1743 by Alexis Claude Clairaut in his Théorie de la figure de la terre, tirée des principes de l'hydrostatique, which synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid, it was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes.
In the 18th century, there was much controversy over the shape of the Earth. In Issac Newton's Principia, he outlines his theory and calculations on the shape of the Earth. Newton theorized that the Earth had an ellipsoid shape, with flattening at the poles. Using geometric calculations, he gave a concrete argument as to the hypothetical Ellipsoid shape of the Earth.
The goal of Principia was not to provide exact answer for natural phenomena, but to theorize potential solutions to these unresolved factors in science. He pushed for scientists to look further into the unexplained variables. Two prominent researchers that he inspired were Alexis Clairaut and Pierre Louis Maupertuis. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to Lapland in an attempt to accurately measure the meridian arc. By doing so, they could hypothetically gauge the shape of the Earth. Clairaut found that Newton's calculations were incorrect, and wrote a letter to the Royal Society of London with his findings. The society published an article in Philosophical Transactions the following year in 1737 that revealed his discovery. Clairaut showed how Newton's equations were incorrect, and did not prove an ellipsoid shape to the Earth. However, he corrected problems with the theory, that in effect would prove Newton's theory correct. Clairaut believed that Newton had reasons for choosing the shape that he did, but he did not support it in Principia. Clairaut's article did not provide an valid equation to back up his argument as well. This created much controversy in the scientific community.
It was not until Clairaut wrote Théorie de la figure de la terre in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorem.
(where a = semimajor axis, b = semiminor axis).
Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density. This work was subsequently pursued by Laplace, who relaxed the initial assumption that surfaces of equal density were spheroids. Stokes showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium. A history of the subject, and more detailed equations for g can be found in Khan.
The above expression for g has been supplanted by the Somigliana equation (after Carlo Somigliana):
- is the spheroid's eccentricity, squared;
- is the defined gravity at the equator and poles, respectively;
- (formula constant);
The spheroidal shape of the Earth is the result of the interplay between gravity and centrifugal force caused by the Earth's rotation about its axis. In his Principia, Newton proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening f given by 1/230. As a result, gravity increases from the equator to the poles. By applying Clairaut's theorem, Laplace was able to deduce from 15 gravity values that f = 1/330. A modern estimate is 1/298.25642. See Figure of the Earth for more detail.
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- Table 1.1 IERS Numerical Standards (2003))
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