## The motion of the Sun and precession

Perhaps Hipparchus' greatest contributions concerned the motion of the Sun. In On the Length of the Year he claimed that the length of the tropical year (the time between identical equinoxes or solstices) was constant, and he measured it at 365 + 4 - 3I0 days (365 days 5 h 55 min 12 s) which exceeds the modern value by about 61 minutes but, nevertheless, represents a significant improvement over the previous value of 3651 days. This probably was done by taking the Babylonian value for the synodic month (i.e. 29; 31, 50, 8, 20 days) and using the approximate equality given by the Metonic cycle (i.e. 19 years = 235 months) and then checking the result with observations.7

Hipparchus was the first to attempt to calculate the parameters needed for the eccentric circle theory of Apollonius to agree with observations of the Sun's position, and his model of the solar motion, and the basic principles by which the parameters for the model were deduced, remained standard until the seventeenth century. The success of Hipparchus' solar model was due to being mathematically simple and yet very accurate. Provided the parameters in the model are calculated accurately, the errors in the predicted solar longitudes will not be detectable from naked-eye observations. Hipparchus' method for determining the parameters is illustrated in Figure 3.3, in which E is the Earth and the Sun S rotates around an eccentric circle centre O with an angular speed of m = 1 revolution per year & 59' 8" per day. The points P\, P2, P3, and P4 represent the position of the Sun at the vernal equinox, the summer solstice, the autumnal equinox and the winter solstice, respectively. In order to be able to use the model to compute the position of the Sun, we need to calculate the longitude X of the apogee A of the orbit of the Sun (the apogee is the point on the orbit furthest from the Earth), which we choose to measure from the vernal equinox, and the ratio of the eccentricity | EO | to the radius | OS| of the orbit of the Sun.

7 See Swerdlow (1979).

SS Ol

Fig. 3.3. Hipparchus'solar theory.

The technique Hipparchus used to do this can be described using modern trigonometry as follows. First, consider the triangles EOP1 and EOP2. The sine rule gives

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