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See Roberts (1957), Kennedy and Roberts (1959), Abbud (1962), Roberts (1966).

31 Quoted from Saliba (1987b).

always is parallel to the line EA which points toward the solar apogee. If BC is drawn parallel to DE, then EDBC is a parallelogram. If B were taken to represent the Sun, then the epicycle model just described would be equivalent to an eccentric circular motion centred at C (see Figure 2.11, p. 46) and, hence, equivalent to Ptolemy's solar theory.

However, Ibn al-Shatir introduced a third circle, centred at B, called the 'director', that carries the Sun S. This rotates at twice the angular speed of the deferent and parecliptic and in the opposite sense to that with which B moves around the deferent. This geometrical arrangement ensures that if SQ is drawn parallel to DE, the point Q remains fixed, with |EC| equal to the radius of the deferent and | QC| the radius of the director, and the Sun rotates uniformly around Q. This is demonstrated in the diagram accompanying the main figure, which shows an enlarged version of the parallelogram EDBC in which all the angles marked are equal. Ibn al-Shatir thus managed, in effect, to incorporate an equant into the solar theory using only uniform circular motions. The parameters he found to be appropriate for his model were |EC| = 4; 37, | QC| = 2; 30 and so |EQ| = 2; 7 which is close to Ptolemy's value of the eccentricity of 60/24 = 2; 30, so that Ibn al-Shatir's model predicts solar longitudes that are

close to those predicted by Ptolemy. The variation in apparent diameter is now

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