Ptolemy's aim in the Almagest is to construct a *kinematic* model of
the solar system, as seen from the earth. In other words, the Almagest outlines a relatively simple geometric model which describes
the apparent motions of the sun, moon, and planets, relative to the earth, but does not attempt to explain
*why* these motions occur (in this respect, the models of Copernicus and Kepler are similar). As such, the fact that the model described
in the Almagest is geocentric in nature is a non-issue, since the earth *is* stationary in its own frame of reference. This is not to say that the
heliocentric hypothesis is without advantages. As we shall see, the
assumption of heliocentricity allowed Copernicus to determine, for the first time, the ratios
of the mean radii of the various planets in the solar system.

We now know, from the work of Kepler, that planetary orbits are actually *ellipses*
which are confocal with the sun. Such orbits possess two main properties.
First, they are *eccentric*: *i.e.*, the sun is displaced from the
geometric center of the orbit. Second, they are *elliptical*: *i.e.*,
the orbit is elongated along a particular axis. Now, Keplerian orbits are characterized by a quantity, , called the *eccentricity*, which
measures their deviation from circularity. It is easily demonstrated
that the eccentricity of a Keplerian
orbit scales as , whereas the corresponding degree of elongation
scales as . Since the orbits of the visible planets in the solar system all possess relatively small values of (*i.e.*, ), it follows that, to an excellent approximation, these orbits can be represented as *eccentric circles*: *i.e.*,
circles which are not quite concentric with the sun. In other words,
we can neglect the ellipticities of planetary orbits compared to
their eccentricities.
This is exactly
what Ptolemy does in the Almagest.
It follows that Ptolemy's
assumption that heavenly bodies move in *circles* is actually
one of the main strengths of his model, rather than
being the main weakness, as is commonly
supposed.

Kepler's second law of planetary motion states that the radius
vector connecting a planet to the sun sweeps out equal areas in equal
time intervals. In the approximation in which planetary orbits
are represented as eccentric circles, this law implies that a typical planet
revolves around the sun at a *non-uniform* rate. However,
it is easily demonstrated that the non-uniform rotation of the radius vector connecting the planet to the sun implies a *uniform* rotation of the radius vector connecting the
planet to the so-called *equant*: *i.e.*, the point directly opposite the sun
relative to the geometric center of the orbit--see Fig. 1.
Ptolemy discovered the equant scheme empirically, and used it
to control the non-uniform rotation of the planets in his model.
In fact, this discovery is one of Ptolemy's main claims to fame.

It follows, from the above discussion, that the geocentric model of Ptolemy is equivalent to a heliocentric
model in which the various planetary orbits are represented as
*eccentric circles*, and in which the radius vector connecting a given planet
to its corresponding equant revolves at a *uniform* rate. In fact, Ptolemy's
model of planetary motion can be thought of as a version of Kepler's model which
is accurate to *first-order* in the planetary eccentricities--see Cha. 4.
According to the Ptolemaic scheme, from the point of view of the earth,
the orbit of the sun is described by a *single* circular motion, whereas
that of a planet is described by a combination of *two* circular motions. In reality, the single circular motion of the sun
represents the (approximately) circular motion of the earth around the sun,
whereas the two circular motions of a typical planet represent a combination of
the planet's (approximately) circular motion around the sun, and the earth's
motion around the sun. Incidentally, the popular myth that Ptolemy's
scheme requires an absurdly large number of circles in order to
fit the observational data to any degree of accuracy has no basis in fact. Actually, Ptolemy's
model of the sun and the planets, which fits the
data very well, only contains 12 circles (*i.e.*, 6 deferents and 6 epicycles).

Ptolemy is often accused of slavish adherence to the tenants of Aristotelian philosophy,
to the overall detriment of his model. However, despite Ptolemy's conventional geocentrism, his model of the solar system deviates from orthodox Aristotelism in a number of crucially important respects. First of all,
Aristotle argued--from a purely philosophical standpoint--that
heavenly bodies should move in *single uniform circles*.
However, in the Ptolemaic system, the motion of the planets is
a combination of *two* circular motions. Moreover, at least one
of these motions is *non-uniform*. Secondly, Aristotle also argued--again
from purely philosophical grounds--that the earth is located at the
*exact center* of the universe, about which all
heavenly bodies orbit in concentric circles. However, in the Ptolemaic system, the earth is *slightly displaced* from the center of the universe.
Indeed, there is no unique center of the universe, since the circular orbit
of the sun and
the circular planetary deferents all have slightly different geometric centers,
none of which coincide with the earth. As described in the Almagest, the non-orthodox
(from the point of view of Aristolelian philosophy)
aspects of Ptolemy's model were ultimately dictated by *observations*.
This suggests that, although Ptolemy's world-view was based on
Aristolelian philosophy, he did not hesitate to deviate from this standpoint
when required to by observational data.

From our heliocentric point of view, it is easily appreciated that the *epicycles*
of the *superior* planets (*i.e.*, the planets further from the sun than
the earth) in Ptolemy's model actually represent the *earth's* orbit around the sun, whereas the
*deferents* represent the *planets'* orbits around the sun--see Fig. 29.
It follows that the *epicycles* of the superior planets should all be the *same size* (*i.e.*,
the size of the earth's orbit), and that the radius vectors connecting
the centers of the epicycles to the planets should always all point
in the *same direction* as the vector connecting the earth to the sun.

We can also appreciate that the *deferents* of the *inferior*
planets (*i.e.*, the planets closer to the sun than
the earth) in Ptolemy's model actually represent the *earth's* orbit around the sun, whereas the
*epicycles* represent the *planets'* orbits around the sun--see Fig. 33.
It follows that the *deferents* of the inferior planets should all be the
*same size* (*i.e.*, the size of the earth's orbit), and that the
centers of the epicycles (relative to the earth) should all correspond to the position of the sun (relative to the earth).

The geocentric model of the solar system outlined above represents a perfected version of Ptolemy's model, constructed with a knowledge of the true motions of the planets around the sun. Not surprisingly, the model actually described in the Almagest deviates somewhat from this ideal form. In the following, we shall refer to these deviations as ``errors'', but this should not be understood in a perjorative sense.

Ptolemy's first error lies in his model of the sun's apparent motion around the earth, which he inherited
from Hipparchus. Figure 1 compares what Ptolemy actually
did, in this respect, compared to
what he should have done in order to be completely consistent
with the rest of his model. Let us normalize the mean radius of the sun's
apparent orbit to unity, for the sake of clarity.
Ptolemy should
have adopted the model shown on the left in Fig. 1, in which the earth
is displaced from the center of the sun's orbit a distance (the eccentricity of the earth's orbit around the sun) towards the perigee (the point of the sun's closest approach
to the earth), and the equant is displaced the same distance in the opposite
direction. The instantaneous angular position of the sun is then obtained
by allowing the radius vector connecting the equant to the sun to rotate uniformly at the sun's mean orbital angular velocity. Of course, this implies that the
sun rotates *non-uniformly* about the earth. Ptolemy actually adopted the Hipparchian model
shown on the right in Fig. 1. In this model, the earth is displaced
a distance from the center of the sun's orbit in the direction of the perigee,
and the sun rotates at a *uniform* rate (*i.e.*, the radius vector rotates uniformly). It turns out that, to first-order in , these two
models are equivalent in terms of their ability to predict the *angular position* of the sun relative to the earth--see Cha. 4. Nevertheless, the Hippachian
model is incorrect, since it predicts too large (by a factor of ) a variation in the *radial distance* of the sun from the earth (and, hence, the angular size of the sun) during the
course of a year (see Cha. 4). Ptolemy probabaly adopted
the Hipparchian model because his Aristotelian leanings prejudiced him in favor of uniform circular motion whenever this was consistent with observations.
(It should be noted that Ptolemy was not interested in explaining the relatively small variations in the angular size of the sun during the year--presumably, because this effect was difficult for him to accurately measure.)

Ptolemy's next error was to neglect the non-uniform rotation of the
superior planets on their epicycles. This is equivalent to neglecting
the orbital eccentricity of the earth (recall that the epicycles of the superior
planets actually represent the earth's orbit) compared to those of
the superior planets. It turns out that this is a fairly good approximation,
since the superior planets all have significantly greater orbital eccentricities
than the earth. Nevertheless,
neglecting the non-uniform rotation of the
superior planets on their epicycles has the unfortunate effect of obscuring the tight coupling between
the apparent motions of these planets, and that of the sun. The radius vectors connecting the epicycle centers of the
superior planets to the planets themselves should always all point *exactly* in the
same direction as that of the sun relative to the earth.
When the aforementioned non-uniform rotation is neglected, the radius
vectors instead point in the direction of the *mean sun* relative to
the earth. The mean sun is a fictitious body which has the
same apparent orbit around the earth as the real sun, but which circles
the earth at a *uniform* rate. The mean sun only coincides with the
real sun twice a year.

Ptolemy's third error is associated with his treatment of the
inferior planets. As we have seen, in going from the superior to the
inferior planets, deferents and epicycles effectively swap roles.
For instance, it is the *deferents* of the inferior planets, rather than the epicycles, which represent
the earth's orbit. Hence, for the sake of consistency with his treatment of
the superior planets, Ptolemy should have neglected the non-uniform
rotation of the epicycle centers around the deferents of the inferior
planets, and retained the non-uniform rotation of the planets themselves around the
epicycle centers. Instead, he did exactly the opposite. This
is equivalent to neglecting the inferior planets' orbital eccentricities relative to
that of the earth. It follows that this approximation only works when an inferior
planet has a *significantly smaller* orbital eccentricity than that of the earth.
It turns out that this is indeed the case for Venus, which has the smallest eccentricity
of any planet in the solar system. Thus, Ptolemy was able to
successfully account for the apparent motion of Venus. Mercury, on the
other hand, has a much larger orbital eccentricity than the earth. Moreover, it
is particularly difficult to obtain good naked-eye positional data for Mercury, since this planet always appears
very close to the sun in the sky. Consequently,
Ptolemy's Mercury data was highly inaccurate.
Not surprisingly, then, Ptolemy was not able to account for the apparent motion of Mercury
using his standard deferent-epicycle approach. Instead, in order to fit the data, he was forced
to introduce an additional, and quite spurious, epicycle into his model of
Mercury's orbit.

Ptolemy's fourth, and possibly largest, error is associated with his treatment of the moon. It should be noted that the moon's motion around the earth is extremely complicated in nature, and was not fully understood until the early 20th century CE. Ptolemy constructed an ingenious geometric model of the moon's orbit which was capable of predicting the lunar ecliptic longitude to reasonable accuracy. Unfortunately, this model necessitates a monthly variation in the earth-moon distance by a factor of about two, which implies a similarly large variation in the moon's angular diameter. However, the observed variation in the moon's diameter is much smaller than this. Hence, Ptolemy's model is not even approximately correct.

Ptolemy's fifth error is associated with his treatment of planetary
ecliptic latitudes. Given that the deferents and epicycles of
the superior planets represent the orbits of the planets themselves around the
sun, and the sun's apparent orbit around the earth, respectively, it follows
that one should take the slight inclination of planetary orbits to
the ecliptic plane (*i.e.*, the plane of the sun's apparent orbit)
into account by tilting the deferents of superior planets, whilst keeping their epicycles
parallel to the ecliptic. Similarly, given that the epicycles and deferents of
inferior planets represent the orbits of the planets themselves around the
sun, and the sun's apparent orbit around the earth, respectively, one should tilt the epicycles of
inferior planets, whilst keeping their deferents parallel to the ecliptic.
Finally, since the inclination of planetary orbits are all essentially constant in time, the inclinations
of the epicycles and deferents should also be constant.
Unfortunately, when Ptolemy constructed his theory of
planetary latitudes he tilted the both deferents and epicyles of
all the planets. Even worse, he allowed the inclinations of the epicycles to
the ecliptic plane to vary in time. The net result is a theory which is far more
complicated than is necessary.

The final failing in Ptolemy's model of the solar system lies in its *scale invariance*.
Using angular position data alone, Ptolemy was able to determine the *ratio*
of the epicycle radius to that of the deferent for each planet, but was
not able to determine the *relative sizes* of the deferents of different planets.
In order to break this scale invariance it is necessary to make an
additional assumption--*i.e.*, that the earth orbits the sun. This
brings us to Copernicus.