measure the meridian arc. From such measurements they could calculate the eccentricity of the Earth, its degree of departure from a perfect sphere. Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but his calculations were in error, 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 a 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. Formula Clairaut's formula for the acceleration due to gravity g on the surface of a spheroid at latitude , was: where is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as: (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). Geodesy 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