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in agreement with the value given by the Roman writer Pliny in AD first century. The effect of a non-zero value for the eccentricity on this simple argument is, however, quite complex.

Fig. 3.14. Ptolemy's theory for Venus.

When it came to Mercury, however, Ptolemy found that he could not use the same geometrical scheme. Of the planets known to Ptolemy, Mercury is by far the hardest to observe, and its orbit deviates significantly from a circle. The observations Ptolemy used for his theory of Mercury - some of which were very inaccurate - led him to believe that Mercury's orbit had two perigees. Thus, he came up with a geometrical device that modelled this phenomenon, and this is shown in Figure 3.15. The line EQO points to the apogee of Mercury's orbit, the longitude of which is fixed with respect to the stars, and the equant Q is the midpoint of EO. The centre of the epicycle C rotates uniformly around the equant, following the mean sun, in such a way that it lies always on a deferent circle, the centre D of which is rotating in the opposite sense around a small circle centred on O. The rates of rotation are chosen so that the angles DO A and AQC always are equal. This construction causes C to move in an oval orbit that has two points of closest approach to the Earth, Px and P2. For a deferent radius of 60, Ptolemy calculated the radius of Mercury's epicycle as 22; 30 and |EQ | as 6; 0. Based on the simple argument described above for Venus, an epicycle radius of 22; 30 corresponds to a maximum elongation of 22°, which is again the value quoted by Pliny.37 In his later work, the Planetary Hypotheses,

Detailed discussions of the empirical basis for Ptolemy's theories of the inferior planets can be found in Wilson (1973), Swerdlow (1989), and the accuracy of the Mercury model is investigated in Nevalainen (1996).

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