vary by a factor of about 2, which is clearly not seen in reality. (The apparent angular diameter of the Moon does vary monthly, but only over a much narrower range of about 0.49°–0.55°.) This defect of the Ptolemaic theory led to proposed replacements by Ibn al-Shatir in the 14th century and by Copernicus in the 16th century. Ibn al-Shatir and Copernicus Significant advances in lunar theory were made by the Arab astronomer, Ibn al-Shatir (1304–1375). Drawing on the observation that the distance to the Moon did not change as drastically as required by Ptolemy's lunar model, he produced a new lunar model that replaced Ptolemy's crank mechanism with a double epicycle model that reduced the computed range of distances of the Moon from the Earth. A similar lunar theory, developed some 150 years later by the Renaissance astronomer Nicolaus Copernicus, had the same advantage concerning the lunar distances. Tycho Brahe, Johannes Kepler, and Jeremiah Horrocks Tycho Brahe and Johannes Kepler refined the Ptolemaic lunar theory, but did not overcome its central defect of giving a poor account of the (mainly monthly) variations in the Moon's distance, apparent diameter and parallax. Their work added to the lunar theory three substantial further discoveries. The nodes and the inclination of the lunar orbital plane both appear to librate, with a monthly (according to Tycho) or semi-annual period (according to Kepler). The lunar longitude has a twice-monthly Variation, by which the Moon moves faster than expected at new and full moon, and slower than expected at the quarters. There is also an annual effect, by which the lunar motion slows down a little in January and speeds up a little in July: the annual equation. The refinements of Brahe and Kepler were recognized by their immediate successors as improvements, but their seventeenth-century successors tried numerous alternative geometrical configurations for the lunar motions to improve matters further. A notable success was achieved by Jeremiah Horrocks, who proposed a scheme involving an approximate 6 monthly libration in the position of the lunar apogee and also in the size of the elliptical eccentricity. This scheme had the great merit of giving a more realistic description of the changes in distance, diameter and parallax of the Moon. Newton A first gravitational period for lunar theory started with the work of Newton. He was the first to define the problem of the perturbed motion of the Moon in