The extensive sets of observations of gravity provided the basis of calculations of the ellipticity of the earth. Col. A. R. Clarke in his _Geodesy_ (London, 1880) calculated the ellipticity from the results of gravity surveys to be 1/(292.2 ± 1.5). Of interest is the calculation by Charles S. Peirce, who used only determinations made with Kater invariable pendulums and corrected for elevation, atmospheric effect, and expansion of the pendulum through temperature.[110] He calculated the ellipticity of the earth to be 1/(291.5 ± 0.9). The 19th century witnessed the culmination of the ellipsoidal era of geodesy, but the rapid accumulation of data made possible a better approximation to the figure of the earth by the geoid. The geoid is defined as the average level of the sea, which is thought of as extended through the continents. The basis of geodetic calculations, however, is an ellipsoid of reference for which a gravity formula expresses the value of normal gravity at a point on the ellipsoid as a function of gravity at sea level at the equator, and of latitude. The general assembly of the International Union of Geodesy and Geophysics, which was founded after World War I to continue the work of _Die Internationale Erdmessung_, adopted in 1924 an international reference ellipsoid,[111] of which the ellipticity, or flattening, is Hayford's value 1/297. In 1930, the general assembly adopted a correlated International Gravity Formula of the form [gamma] = [gamma]_{E}(1 + [beta]sin^{2} [phi] + [epsilon]sin^{2} 2[phi]) where [gamma] is normal gravity at latitude [phi], [gamma]_{E} is the value of gravity at sea level at the equator, [beta] is a parameter which is computed on the basis of Clairaut's theorem from the flattening value of the meridian, and [epsilon] is a constant which is derived theoretically. The plumb line is perpendicular to the geoid, and the components of angle between the perpendiculars to geoid and reference ellipsoid are deflections of the vertical. The geoid is above the ellipsoid of reference under mountains and it is below the ellipsoid on the oceans, where the geoid coincides with mean sea level. In physical geodesy, gravimetric data are used for the determination of the geoid and components of deflections of the vertical. For this purpose, one must reduce observed values of gravity to sea level by various reductions, such as free-air, Bouguer, isostatic reductions. If g_{0} is observed gravity reduced to sea level and [gamma] is normal gravity obtained from the International Gravity Formula, then [Delta]g = g_{0} - [gamma] is the gravity anomaly.[112] In 1849, Stokes derived a theorem whereby the distance N of the geoid from the ellipsoid of reference can be obtained from an integration of gravity anomalies over the surface of the earth. Vening Meinesz further derived formulae for the calculation of components of the deflection of the vertical.