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60 + 4; 37 + 2; 30 inagreementwithIbnal-Shatir's, albeit erroneous, value. The whole mechanism then is attached to a larger geocentric sphere, the encompassing orb, which has the observed steady motion of the solar apogee. Ibn al-Shatir stated explicitly that he believed the theory of trepidation to be unsound because it was not supported by observational evidence.

For the Moon, Ibn al-Shatir again noted that one reason for modifying Ptolemy's scheme was better to predict the changes in its apparent diameter:

[the model of Ptolemy also] requires that the diameter of the moon should be twice as large at quadrature than at the beginning, which is impossible, because it was not seen as such.32

The modified lunartheory is illustrated inFigure 4.7 and canbe seen to be similar in structure to the solar theory, with an epicycle on an epicycle. The mean moon is M and this rotates around the Earth once per synodic month (relative to ES), the radius of this deferent circle being taken as 60. The point B, which is the centre of the epicycle that carries the Moon M rotates in the opposite sense around M once in each anomalistic month. Finally, the Moon rotates around its epicycle in such a way that the angle MBM is twice the angle MES. Ibn

Quoted from Saliba (1987b).

Fig. 4.8. Ibn al-Shatir's theory for the superior planets.

al-Shatir took the radii of the two epicycles to be ri = 6; 35 and r2 = 1; 25, and then this mechanism implies that at the syzygies the distance of the Moon from the Earth varies between r ± (r1 - r2) (i.e. between 54; 50 and 65; 10) and at the quadratures between r ± (r1 + r2), i.e. between 52; 0 and 68; 0.

Ibn al-Shatir also devised a method for incorporating similar mechanisms into the models for planetary longitudes. Here, the motivation would appear to have been entirely aesthetic, since, as far as accuracy was concerned, his models did not improve on those of Ptolemy, but simply removed the unpleasant equants. Indeed, Ibn al-Shatir chose his parameters so as to make his models equivalent mathematically to those of Ptolemy. However, he was very pleased with the result, as in the introduction to his zij (The New Astronomical Handbook), he wrote:

I therefore asked Almighty God to give me inspiration and help me to invent models that would achieve what was required, and God—may He be praised and exalted, all praise and gratitude to Him—did enable me to devise universal models for the planetary motions in longitude and latitude and all other observable features of their motions, models that were free—thank God—from the doubts surrounding previous models.

The mechanism that Ibn al-Shatir used to model the superior planets is shown in Figure 4.8. As in the solar and lunar schemes, the Earth is at the centre, but

Quoted from King (1975a).

now there are three, rather than two, epicycles. The point D rotates around a circle, radius 60, and B rotates around D in the opposite sense and at the same rate, so that DB is always parallel to EA, A being the apogee of the orbit. This is equivalent to B moving anticlockwise around an eccentric circle (the dashed circle in the figure, centre G) with eccentricity equal to |EG|, which is the same as | BD |. The centre of the third epicycle, C, rotates around B so that ZDBC is twice ZDEA and, just as in the solar theory, this ensures that C rotates uniformly about a fixed point Q. Finally, the planet, P, rotates around C on this third epicycle. The uniform rotation of P is measured with respect to the line QCF and is such that CP always is parallel to ES. In order to make this geometrical arrangement match closely Ptolemy's system, it is necessary to choose the radii of the first and second epicycles, | BD | = ri and | BC | = r2, so that they satisfy r1 — r2 = e and r1 + r2 = 2e, where e is the eccentricity in Ptolemy's model (i.e. e = |EO| in Figure 3.13, p. 77) and this is precisely what Ibn al-Shatir did. Ibn al-Shatir's models for the inferior planets were similar though, not surprisingly, that for the motion of Mercury was more complicated, involving the use of yet another epicycle.

The work of Ibn al-Shatir represents the culmination of a programme of research that was driven by the desire to correct certain deficiencies in Ptolemaic astronomy. In the main, the perceived defects were at a philosophical level - astronomers objected to some of Ptolemy's geometrical models because they could not physically be realized without violating some well-established principle. Most Islamic astronomers were quite happy to take Ptolemy's observations as they were and, apart from correcting his values for things like precession and obliquity, wanted to produce theories that reproduced exactly the same phenomena as Ptolemy's did. Thus, for example, Mu'ayyad al-Din al-'Urdi, an astronomer from Damascus who developed alternative lunar and planetary theories, explicitly stated that he had made no observations of his own and stressed that his criticisms of Ptolemy were at a higher conceptual level.

The new models that were devised by the Maragha School of astronomers were a great success in that they produced longitude predictions as accurate as those from the Almagest, but without the use of the physically impossible devices introduced by Ptolemy. The majority of practising astronomers, on the

Discussions of various aspects of al-'Urdi's work can be found in a number of articles that have been reprinted in Saliba (1994) and his ideas on the size of the Cosmos, which differed greatly from those of Ptolemy, are described in van Helden (1985), pp. 32-3.

other hand, stuck with the tried and tested procedures that had been in existence for centuries.

The Maragha Observatory was not the only significant establishment of its type in the history of Islamic astronomy. There was another observatory, built by Ulugh Beg in Samarkand. Ulugh Beg (this is not his original name, but means 'great prince') was, from the age of 15, a provincial governor in Samarkand but, unlike his grandfather, the Mongol conqueror Tamerlane, he was not interested in conquest but in science. He succeeded his father as ruler of the Mongol empire in 1447, but was murdered by his son 2 years later. During the quarter century leading up to Ulugh Beg's assassination in 1449, Samarkand was the most important scientific centre in the East and in 1420 he founded an institute of higher learning there (in which astronomy was the most important subject). In 1424, he built an observatory, which was destroyed in the sixteenth century, its location being rediscovered in 1908 by the archaeologist V. L. Vyatkin. Ulugh Beg's observatory was responsible for many accurate determinations of solar and planetary parameters, as well as the construction of extremely accurate trigonometric tables. However, it is most famous for the production of the first star catalogue not based on that of Hipparchus and Ptolemy. The influence of this catalogue on later developments in the West was minimal, though, since it only became well known in Europe in the mid seventeenth century, after the publication of the more accurate observations of Tycho Brahe.

One of the chief scientists at Ulugh Beg's observatory was al-Kashi, who wrote a number of works, the most famous of which was an encyclopedia of elementary mathematics that included large sections on methods of calcula-

tion for astronomers. Al-Kashi also achieved a significant computational feat when he computed 2n to sixteen decimal places, greatly exceeding all previous results. His stated motivation for doing this was so that he could calculate the circumference of the Universe to within the thickness of a horse's hair! Rather more useful was his computation of sin 1° to ten sexagesimal places.

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