Ptolemy’s
solar model is equivalent to that of Hipparchus’. In Heilbron’s book,
Ptolemy’s method for obtaining the solar eccentricity and the angle between
the line of apsides and the line joining the solstices *y*
is illustrated and explained; the calculations involved are provided in the
section entitled "Ptolemy's Solar Eccentricity" of this paper. When modern values for time differences between solstice and
equinox are applied to Ptolemy’s model, the value found for the angle *y*
is fairly close to that found in the past. However the same could not be said
for the value of eccentricity. In fact, the old value for solar eccentricity *e*
exceeds the modern value by a factor of two. More discussion on this issue is
provided later in this section.

To determine planetary positions accurately and conveniently, Ptolemy made use of a model that involved epicycles, deferents and equant points. I shall proceed to elaborate more on this model with reference to the same paper quoted earlier in the section "Frames of Reference" by James Evans, and another of his, entitled “On the function and the probable origin of Ptolemy’s equant”.

As indicated in the section named "Hipparchus' Solar Model", when we shift the point of view of a planet from the Sun to the Earth, the planet moves on an epicycle, with its centre carried on a deferent about the Earth. When seen from Earth, the planet appears to display periodic retrograde motion, that is, it seems to move backwards sometimes. Apollonios of Perge used the deferent-and-epicycle theory to provide an explanation for retrograde motion in the zero-eccentricity planetary model, or the model in which the centre of the deferent coincides with Earth.

Figure
11

In Figure 11, the planet Z moves uniformly on an epicycle with centre K. K also moves uniformly around a deferent centred at Earth E. The fixed reference line O¡ points to the vernal equinoctial point. Thus angles and increase uniformly with time. If increases at a much higher rate than , then the planet would appear to display retrograde motion. In other words, in the ancient planetary theory, the epicycle is the mechanism that produces retrograde motion.

However, even though the model accounts for the retrogradations qualitatively, it fails to do so quantitatively. By the model, the retrograde loops produced would have the same size and shape, and be uniformly spaced around the ecliptic, as shown in Figure 12.

Figure 12

Yet this is overly simplified and does not occur in reality. The precise distance between one retrogradation and the next is quite variable, and thus the retrograde arcs are not equally spaced around the zodiac. There is no way for the uniformly spaced retrograde loops of the model to reproduce the unevenly spaced retrograde arcs of the planet itself.

A step taken to improve this zero-eccentricity model was to displace the Earth E from the centre of the deferent, C, slightly.

Figure 13

In
Figure 13, planet Z moves uniformly on the epicycle and the centre of the
epicycle, K, moves around the deferent at uniform speed. However, C no longer
coincides with E, as in the “zero-eccentricity model”. This is known as the
“intermediate model”. The intermediate model provided just the kind of
variable spacing between two retrogradations of the planet yet it was still
unable to fit the width of the retrograde loops perfectly. To tackle the
problem, Ptolemy added a third device called the **equant point**, defined as
a point about which the angular velocity of a body on its orbit is constant.
Figure 14 shows Ptolemy’s planetary model.

Figure 14

Here, Z, K and E have their usual meanings. However K is now assumed not to move uniform either with respect to the Earth E or the centre of the deferent, C. Instead, uniform motion of K is observed at X which is the mirror image of the Earth E, and is what Ptolemy named as the “equant point”.

As
a result of these two points X and E, there are now two eccentricities to be
defined. Take the radius of the deferent to be *a*, the eccentricity of the
Earth E with respect to centre C to be
*e*_{1}
and the eccentricity of the equant
point with respect to centre C to be
*e*_{2}. Then,

The
rule of equant motion – that K moves at constant angular speed as viewed from
the equant – produces a physical variation in the speed of K. This variation
in speed is determined by
*e*_{2}
and if
*e*_{2}
goes to zero, the motion of K
reduces to uniform circular motion. Even if
*e*_{2}
were zero, the eccentricity
*e*_{1}
would cause the motion of K to
appear non-uniform from the Earth E due to the same optical illusion mentioned
in "Hipparchus' Solar Model". The sum
*e*_{1}
+
*e*_{2} is
called the “total eccentricity”. Ptolemy always puts
*e*_{1}
=
*e*_{2} and
such a situation has come to be referred to as the “bisection of
eccentricity”. This notion would once again be brought up in a later section where
we make comparisons across Hipparchus’, Ptolemy’s and Kepler’s solar and
planetary models.