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 the theorem enunciated by Newton, that every cubic is a projection of one of five divergent parabolas. Clairaut formed the acquaintance of Maupertius, whom he accompanied on an expedition to Lapland to measure the length of a degree of the meridian. At that time the shape of the earth was a subject of serious disagreement. Newton and Huygens had concluded from theory that the earth was flattened at the poles. About 1713 Dominico Cassini measured an arc extending from Dunkirk to Perpignan and arrived at the startling result that the earth is elongated at the poles. To decide between the conflicting opinions, measurements were renewed. Maupertius earned by his work in Lapland the title of "earth flattener" by disproving the Cassinian tenet that the earth was elongated at the poles, and showing that Newton was right. On his return, in 1743, Clairaut published a work, Théorie de la figure de la Terre, which was based on the results of Maclaurin on homogeneous ellipsoids. It contains a remarkable theorem, named after Clairaut, that the sum of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to $$\scriptstyle{2\frac{1}{2}}$$ times the fraction expressing the centrifugal force at the equator, the unit of force being represented by the force of gravity at the equator. This theorem is independent of any hypothesis with respect to the law of densities of the successive strata of the earth. It embodies most of Clairaut's researches. Todhunter says that "in the figure of the earth no other person has accomplished so much as Clairaut, and the subject remains at present substantially as he left it, though the form is different. The splendid analysis which Laplace supplied, adorned but did not really alter the theory which started from the creative hands of Clairaut."

In 1752 he gained a prize of the St. Petersburg Academy for his paper on Théorie de la Lune, in which for the first time modern analysis is applied to lunar motion. This contained