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Lagrange successfully resolved the first issue and won the prize. The physical cause of the equality between the length of a lunar day and a lunar month had been given by Newton, who argued, by analogy with the tides on the Earth, that the Moon would not be spherical, but a spheroid, the greatest diameter of which, if produced, would pass through the centre of the Earth. The Moon could be in equilibrium only in this position, and any small displacement would result in slow oscillations about this configuration. Despite the efforts of people such as d'Alembert, no one had derived a theory capable of analysing such motions. What Lagrange accomplished was the development of a method by which the period of these so-called librations could be determined.

There are, in fact, two types of libration, the one under discussion being the physical libration. The part of the Moon visible from the Earth varies over the course of a month due simply to the orbit of the Moon being elliptical rather than circular. As the Moon orbits the Earth, its speed changes in accordance with Kepler's second law, but its rotation rate remains fixed, so that the part facing the Earth varies about a mean position. The rotation axis of the Moon is inclined to its orbital plane, and so the result of this oscillation - called the 'optical libration' - is an apparent motion from side to side and up and down during a lunar month.

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