Ptolemy's complete lunar model in Almagest V produces a large variation of distance of the centre of the epicycle, between its maximum at mean conjunction and opposition to the mean position of the Sun and its minimum at mean quadrature, so that the resulting path of the centre of the epicycle about the Earth is an oval figure. In Figure 1, with the Earth at T, the centre of the epicycle L moves uniformly about T on an eccentric of radius R with centre E and apogee A, through the mean elongation from the mean position of the Sun S in the direction of increasing longitude, while the apsidal line of the eccentric rotates in the opposite direction through the same mean elongation. Consequently, L reaches the apogee A and perigee B of the eccentric twice in each mean synodic month, which produces the oval path, farthest from the Earth at mean conjunction and opposition, closest at mean quadrature. The Moon M moves on the epicycle of radius r in the direction opposite to the motion of L through the mean anomaly, completed in an anomalistic month, uniformly with respect to the mean apogee F, which has an 'inclination' (pmsneusis) toward a point P, opposite to the direction of F from the Earth and with the same eccentricity, e = PT = TE.1 The true apogee G lies on the line TEG from the Earth. Our concern here is the path described by the variable distance of the lunar epicycle from the Earth p = TL. At mean conjunction and opposition, p = EA + ET=R + e and at mean quadrature p = EA - ET = R - e. The figure has been drawn using historical values due to Muhyi al-Din al-Maghrib! (d. 1283) of the Maragha Observatory in north-western Iran, where R + e = 60, R = 51,e = 9, and r = 5;12.2 Thus at conjunction and opposition p = R + e = 60 and at quadrature p = R - e = 42.Ptolemy, as Pedersen remarks, always conceives the motion of the lunar epicycle centre as a circular motion around the moving centre of the deferent. He never asks for the orbit described by the lunar epicycle centre relative to the centre of the Earth.3 Abu al-Rayhan al-Biruni (973-1048) in his al-Qanun al-mas'udi VII.7.14 considers this problem in the form of question and answer. The translation of the relevant passage is as follows:Q: What [shape] does its [i.e., the Moon's] epicycle centre describe by this motion [i.e., according to Ptolemy's model of a movable eccentric]?A: If it is assumed that the Sun is at rest and if the lunar epicycle centre is at its orb's apogee in its [mean] conjunction or opposition [to the Sun] and is at the perigee in its [mean] quadrature, it will describe a rounded rectangular shape by its motion. It might be thought that it [i.e., the path of the epicycle centre] is an ellipse of the [right circular] conic or cylindrical sections. [But,] it is not so.Take the Moon's orb