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ones, confirmed the Babylonian lunar periods as:

251 synodic months = 269 anomalistic months,

5458 synodic months = 5923 draconitic months.

Hipparchus' lunar model is illustrated in Figure 3.4. It consists of an epicycle carrying the Moon, M, the centre C of which rotates around a deferent circle, the deferent-epicycle system being inclined at an angle of 5° to the ecliptic and intersecting that circle in the two nodes A and B . The nodal line AOB was made to rotate in order to account for the fact that the longitude of the position of maximum latitude of the Moon changes gradually. The motion on the epicycle ensures that the Moon's speed is variable and by making the period of revolution of M around the epicycle different from the period of C around the deferent, we ensure that the longitude of the position of maximal speed (which occurs when M is at its closest to O) varies over time.

Hipparchus made the simplifying assumption that the motions in latitude caused by the 5° angle of the orbit could be treated separately from the motion in longitude due to the epicyclic system. This introduces only very minor errors. The rotation of the nodal line thus becomes irrelevant for the longitude theory, which is reduced to a simple two-dimensional deferent-epicycle system. In order to use the theory, various parameters have to be computed. Thus, for the longitude calculations, we require the rates of rotation of C around the deferent and M around the epicycle as well as the ratio of the radii of the epicycle and

See Swerdlow and Neugebauer (1984), I, p. 198.

Fig. 3.4. Hipparchus' lunar theory.

deferent, whereas for the latitude calculations we require simply the rate of rotation of the nodal line. This latter parameter can be determined from direct observations of eclipses. It turns out that in order to reproduce the observed behaviour of the Moon, the nodal line must rotate approximately once every 18 years (the so-called 'Saros period').

The rates of rotation of the deferent and the epicycle also are easily obtained. The former is simply the rate required for C to complete 1 revolution in 1 sidereal month which turns out to be about 13° 10' 35" per day, and the latter is chosen to ensure that the epicycle rotates once in each anomalistic month, which implies a rate of 13° 3' 54 ' per day. The final parameter, the ratio of the radii of the epicycle and deferent, presents a far harder problem. Hipparchus developed a geometrical method that enabled this ratio to be obtained from

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