The earth rotates about the sun in an orbit which is almost a circle. If the earth is thought of as stationary, then it is the sun which rotates about us, carrying with its satellite planets. For an inferior planet, one whose orbit is smaller than the earth’s (or of the sun’s), its resultant motion with respect to the earth is easy to visualize. It is the motion of a point on a wheel rotating independently at the same time that the wheel’s centre rotates upon the rim of a larger wheel. To adopt ancient terminology, the larger wheel (or orbit) is the deferent, the smaller the epicycle.
The case of a superior planet is somewhat more involved, since its orbit is larger than the sun’s. But the two orbits can be interchanged without affecting the position of the planet, to retain the deferent as the larger with centre in the vicinity of the earth, and the epicycle as the smaller, outer wheel carrying the planet. Under all circumstances, then, the planet moves with respect to the earth in a series of loops congruent, is insufficiently precise for anything but a good approximation. With an actual planet, the size and character of the retrogradations vary, depending upon the region of the sky in which they take place.
Having set up the problem, it is useful to note three ways by which, in various places and times, it was solved:
- a) Purely numerical techniques may be used, with no appeal at any stage to a geometric model. This highly sophisticated approach was developed by “Babylonian” astronomers in the late Achaemenian and Parthian Period; it then disappeared until the clay tablets on which it was recorded were excavated and deciphered in recent times. Its eschewal of geometry gives it a singularly modern aspect.
- b) The simple deferent-epicycle model may be accepted, and the necessary variation in the retrogradations introduced by means of computational schemes which have no immediate geometric motivation or rationale. Unlike (a) above, these procedures involve trigonometric, rather than algebraic transformations because of the implicit presence of the epicycle. They were characteristic of Indian and Iranian (Sasanian) astronomy.
- c) The deferent-epicycle configuration may be modified geometrically, by making the earth eccentric with respect to the deferent and by introducing a periodic variation in the speed of the epicycle centre, in order to improve the correspondence between the model and the facts. This is the method of Ptolemy, and his solution is about as good a job as can be hoped for without abandonment of circular orbits. As with (a) and (b) the ultimate result is numerical, a set of true longitudes corresponding to given instants. But here computation is postponed until the end, and the figure determines the direction the computation shall take.
In the Sasanian period there existed an astronomical literature in the Royal Tables (zîg î shahryârân or “zîg î shah”), which are known to us only from citations in Arabic treatises. These documents are scarce and corrupt. The job is made both frustrating and fascinating by the fact that no astronomical documents in Middle Persic (Pahlavi) have survived; often the student can judge only from internal evidence whether a particular element comes from a Sasanian source, or whether it originated in a later period. We can distinguish three sets of Royal Tables.
The first set of Royal Tables was composed in the early Sasanian period. The first two Sasanian rulers (Ardashêr and Shâbuhr) sponsored Persic translations of Greek and Sanskrit works on astronomy and astrology. Among the texts so translated were the Greek astrological treatises of Dorotheus of Sidon and Vettius Valens, and the astronomical Syntaxis mathematike (Almagest) of Ptolemy, as well as a Sanskrit astrological work by one Faramâsb (parameshvara?).
The second was composed in 556, under Khusrô Anushervân. This version of the Royal Tables was used by Mâshâ’allâh in his Ketâb fi’l-qerânât wa’l-adyân wa’l-melal, written about 810; from this we can see that it rejected the Indian method of finding the mean longitudes of the planets by means of their integer rotations in a yuga (world period). This zîg was with four kardajas.
The last set of Royal Tables was written under Yazdegird III in the 630’s or 640’s and was translated into Arabic by Tamîmî; we have only fragments of this translation. According to Al-Hâshemî: “Yazdegird brought out a zîg and he named it after the example of the Shâh [Khusrô]. He made it in three kardajas and called it The Triple. Its explanatory text and apogees and nodes and mean motions and equations correspond to those of the Arkand as to midnight epoch. People still work with it, except that they regard the heavy ones [the superior planets] more correct in the Arkand by observation, and the light ones more correct in the Shâh”. This zîg and The Indian astronomy have the following common elements:
- There must exist a way of determining the “equation of the anomaly” say
e1, the modification in the planet’s position due to its place on the epicycle;
- There must exist a way of determining the “equation of the centre”, say
e2, the modification in the planet’s position due to its situation relative to a point in the sky fixed for each planet, its apogee.
- There must exist a rule causing
e1, since the two equations are not independent of each other.
e2 were defined by means of numerical tables. In the Indian sources, the earlier versions of the Zîg î Shâh,
e1 tables are computed by the use of the epicycle configuration, while the
e2 functions are of the more crude “sinusoidal” form
k depends upon the planet, and t is the argument. In the latest version of the Zîg î Shâh
e1 also seems to be sinusoidal; and this “method of sines” is widespread, as is also the “method of declinations”, which substitutes
k•d(t) is the declination of a point on the ecliptic having longitude