Search

The American mathematician Robert Jackson Adcock (1826–1895) is an obscure figure, hitherto associated with the history of regression analysis and least-squares, whose identity and life are described in Part II of this work. In 1872, he self-published a pamphlet, 'Gravitation to the sphere and the two ellipsoids of revolution: ratio of the axes of a rotating fluid mass', which seems to have been largely ignored at the time. It effectively became lost thereafter, until a copy was recently discovered in the Library of the Royal Society. In it, he determined the degree of flattening assumed by a rotating homogeneous ellipsoidal fluid Earth with a uniform density, subject to gravitational attraction and in hydrostatic equilibrium—a problem previously considered, but not solved, by Newton, Laplace, Gauss and Dirichlet. Adcock successfully obtained an explicit solution for the potential of a homogeneous ellipsoid and correctly calculated the flattening of such a model. Until now, this result was believed to have first been obtained by a German engineer, Otto Heymann, in 1935. Adcock's pamphlet is transcribed here with a commentary on its contents. It is hoped that recognition of his remarkable achievement will enhance his reputation as a mathematician. [ABSTRACT FROM AUTHOR]

The sphericity of the form of the Earth was questioned around the year 1687, primarily, by Isaac Newton who deduced from his theory of universal gravitation that the Earth has the form of a spheroid flattened at the poles and elongated at the equator. In France, some preeminent geographers were not convinced by Newton’s arguments, and about the same period, based on empirical measurements, they emitted another theory, claiming that on the contrary, the Earth has the form of a spheroid flattened at the equator and elongated at the poles. To find the real figure of the Earth became one of the major questions that were investigated by geographers, astronomers, mathematicians and other scientists in the 18th century, and the work done around this question had an impact on the development of all these fields. In this paper, we review the work of the 18th-century French mathematician, astronomer and geographer Alexis-Claude Clairaut related to the question of the figure of the Earth. We report on the relation between this work and that of Leonhard Euler. At the same time, we comment on the impact of the question of the figure of the Earth on mathematics, astronomy and hydrostatics. Finally, we review some later mathematical developments that are due to various authors that were motivated by this question. It is interesting to see how a question on geography had such an impact on the theoretical sciences. [ABSTRACT FROM AUTHOR]

A generalized theory of the figures of the Earth's interior to a third-order precision of ellipticity is proposed in accompanying paper in which all the odd degree and nonzero order spherical harmonic terms are included. As both the direct and indirect contributions of the asymmetric crust are included, this theory makes a significant improvement for calculating the asymmetric equilibrium figures of the real Earth comparing with the traditional theories which can only deal with the ideal symmetric Earth. The principal moments of inertia (PMOI: A, B, C) and global dynamical flattening (H) are important quantities in studying the rotating Earth. Precession and gravity observations give observation value of H (Hobs≈1/305.4559) with very high precision, while its theoretical calculated value (Htheory≈1/308.5) from traditional theories and a starting symmetric Earth model (like PREM model) is about 1% less than Hobs. Using the new theory in accompanying paper and replacing the homogeneous outermost crust and oceanic layers in PREM with CRUST1.0 model, we recalculate the equilibrium figures of the Earth's interior and finally get new values of PMOI and Htheory (≈1/304.7167) whose consistency with Hobs are significantly improved to 0.24%. Furthermore, the asymmetric figures of some interesting boundaries, like inner core boundary, core-mantle boundary, are also given as by-products of this work as these boundaries' figures are key input for studies of their topographic effect on global rotation and geodynamics, like nutation, normal modes, especially like free core nutation. [ABSTRACT FROM AUTHOR]

Traditionally a laterally homogeneous and spherical base Earth model (e.g., the PREM model) is considered as input when computing the Earth's equipotential surfaces, which are then resulted to be in symmetric shape. However, the Earth, known with a complex distribution of interior material and density, especially in the upper mantle and the crust, cannot be treated as a symmetric sphere. Recently, a CRUST1.0 model of crust layer is published and well accepted. But the effect caused by the asymmetric crust (and mantle) on equilibrium figures of the Earth cannot be analyzed by the traditional theories. A generalized theory of the figure of the Earth to third-order precision is firstly proposed in this paper, as well as the iterative calculation strategy to solve the complex equation system. In order to validate this generalized theory, the degeneration of this generalized theory with the PREM model is made and is compared with traditional theories, and it is shown that the result of this generalized theory, after degeneration, is consistent very well with traditional theory. Meanwhile, the effect (including both the direct and indirect effects) of the crust layer, from the CRUST1.0 model, on the figures of equipotential surfaces of the Earth's interior, as well as their effects on the global dynamics flattening, will be presented as an application of this theory in accompanying paper. [ABSTRACT FROM AUTHOR]

Geodesy, by definition, requires international collaboration on a global scale. An organized cooperation started in 1862, and has become today’s International Association of Geodesy (IAG). The roots of modern geodesy in the 18th century, with arc measurements in several parts of the world, and national geodetic surveys in France and Great Britain, are explained. The manifold local enterprises in central Europe, which happened in the first half of the 19th century, are described in some detail as they prepare the foundation for the following regional project. Simultaneously, Gauss, Bessel and others developed a more sophisticated definition of the Earth’s figure, which includes the effect of the gravity field. In 1861, the retired Prussian general J.J. Baeyer took up earlier ideas from Schumacher, Gauss, Struve and others, to propose a Central European Arc Measurement in order to study the figure of the Earth in that region. This led to a scientific organization, which soon extended from Central Europe to the whole continent and later to the globe, and changed its name in 1886 to ‘Internationale Erdmessung’ (International Geodetic Association). The scientific programme also widened remarkably from more local studies based on geometric data to regional and global investigations, with gravity measurements as an important source of information. The Central Bureau of the Internationale Erdmessung was hosted at the Prussian Geodetic Institute in Potsdam, and with Baeyer as Director, developed as an efficient tool of the Association. The scientific research extended and deepened after 1886, when F.R. Helmert became Director of the Central Bureau. A stronger international participation then took place, while the influence of the German states reduced. Of great practical importance were questions of standardization and reference systems, but first attempts to interpret gravity field variations and to monitor geodynamic phenomena by geodetic methods indicated future tendencies. With the First World War and the expiry of the last international convention in 1916, the international cooperation within the frame of the Association practically came to an end, which ended the first epoch of the Association. Nevertheless, due to the strong commitment of two scientists from neutral countries, the International Latitude Service continued to observe polar motion and to deliver the data to the Berlin Central Bureau for evaluation. After the First World War, geodesy became one of the founding members of the International Union for Geodesy and Geophysics (IUGG), and formed one of its Sections (respectively Associations). It has been officially named the International Association of Geodesy (IAG) since 1932. [ABSTRACT FROM AUTHOR]