Figure of the Earth In 1736, together with
Pierre Louis Maupertuis, he took part in the expedition to
Lapland, which was undertaken for the purpose of estimating a degree of the
meridian arc. The goal of the excursion was to geometrically determine the
figure of the Earth and test
Sir Isaac Newton's prediction presented in his book
Principia that it was an oblate
spheroid. Before the expedition team returned to Paris, Clairaut sent his calculations to the
Royal Society of London. The writing was later published by the society in the 1736–37 volume of
Philosophical Transactions. Initially, Clairaut disagreed with Newton's conclusion. In the article, he outlines several key problems that effectively disprove Newton's calculations, and provides some solutions to the complications. The issues addressed include calculating gravitational attraction, the rotation of an ellipsoid on its axis, and the difference in density of an ellipsoid on its axes. He begins the book by comparing geometric shapes to measurements of land, as it was a subject that most anyone could relate to. He covers topics from lines, shapes, and even some three dimensional objects. Throughout the book, he continuously relates different concepts such as
physics,
astrology, and other branches of
mathematics to geometry. Some of the theories and learning methods outlined in the book are still used by teachers today, in geometry and other topics.
Mathematical astronomy One of the most controversial issues of the 18th century was the
problem of three bodies, or how the Earth, Moon, and Sun are attracted to one another. With the use of the recently founded
Leibnizian calculus, Clairaut was able to solve the problem using four differential equations. He was also able to incorporate Newton's
inverse-square law and law of attraction into his solution, with minor edits to it. However, these equations only offered approximate measurement, and no exact calculations. Another issue still remained with the three body problem; how the Moon rotates on its apsides. Even Newton could account for only half of the motion of the
apsides. The
Théorie de la lune is strictly Newtonian in character. This contains the explanation of the motion of the
apsis. It occurred to him to carry the approximation to the third order, and he thereupon found that the result was in accordance with the observations. This was followed in 1754 by some lunar tables, which he computed using a form of the
discrete Fourier transform. The newfound solution to the problem of three bodies ended up meaning more than proving Newton's laws correct. The unravelling of the problem of three bodies also had practical importance. It allowed sailors to determine the
longitudinal direction of their ships, which was crucial not only in sailing to a location, but finding their way home as well. This held economic implications as well, because sailors were able to more easily find destinations of trade based on the longitudinal measures. Clairaut subsequently wrote various papers on the
orbit of the
Moon, and on the motion of
comets as affected by the perturbation of the planets, particularly on the path of
Halley's comet. He also used applied mathematics to study
Venus, taking accurate measurements of the planet's size and distance from the Earth. This was the first precise reckoning of the planet's size. ==Publications==