## Planetary theory

Books IX-XIII of the Almagest are devoted to the motion of the planets, with longitudes and latitudes being considered separately. For the longitude theories, there are two anomalies to model. The first is manifested by the varying speed of the planet as it travels round the ecliptic and is thus similar to the anomaly in the Sun's motion. This suggests an eccentric deferent as a suitable geometrical scheme to account for the irregularity. The second anomaly is the phenomenon of retrograde motion and this ultimately is linked to the motion of the Sun. The superior planets always reach the centre of their retrograde arcs when they are

See p. 54. Ptolemy's method is described in detail in van Helden (1985). Unlike the procedure adopted by Hipparchus, Ptolemy's approach is extremely sensitive to errors in the measured parameters.

in opposition to the Sun, whereas this happens at conjunction for the inferior planets. As Apollonius had shown, retrograde motion can be modelled by an epicyclic theory, and so some combination of eccentric deferent and epicycle suggests itself.

However, it turns out that this is insufficient by itself accurately to predict the correct positions at which the retrograde motions begin and the correct angular widths of the retrograde arcs.34 To solve the problem, Ptolemy introduced a further modification to Apollonius' scheme and separated the centre of the deferent circle from the centre of uniform rotation. Thus, he introduced a new point - the equant - about which the centre of the deferent rotated with uniform angular speed. Nowhere does Ptolemy state how he devised this construction, but it works astonishingly well - a Ptolemaic equant produces planetary longitudes differing from modern theory by less than 10' of arc, even for the comparatively large eccentricity of Mars. The discovery of the equant mechanism thus

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