Actually, there were only seven ``wandering'' heavenly bodies
visible to ancient peoples: the Sun, the Moon, and the five
planets--Mercury, Venus, Mars, Jupiter, and Saturn. The ancients believed that the stars were fixed
to a ``celestial sphere'' which formed the outer boundary of the Universe. However, it was recognized
that the wandering bodies were located *within* this sphere: *e.g.*, because the Moon clearly
passes in front of, and blocks the light from, stars in its path.
It was also recognized that some bodies were closer to the Earth
than others. For instance, ancient astronomers
noted that the Moon occasionally passes in front of the Sun and
each of the planets. Moreover, Mercury and Venus can sometimes be seen to transit in front of the Sun.

The first scientific model of the Solar System was outlined by the Greek philosopher
Eudoxas of Cnidus (409-356BC). According to this model,
the Sun, the Moon, and the planets all execute uniform circular orbits around the Earth--which is
fixed, and non-rotating.
The order of the orbits is as follows: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn--with
the Moon closest to the Earth. For obvious reasons, Eudoxas' model became known as the
*geocentric* model of the Solar System. Note that orbits are circular in this
model for philosophical reasons. The ancients believed the heavens to be the realm of perfection.
Since a circle is the most ``perfect'' imaginable shape, it follows that heavenly objects must execute
circular orbits.

A second Greek philosopher, Aristarchus of Samos (310-230BC), proposed an
alternative model in which the Earth and the planets execute uniform
circular orbits around the Sun--which is fixed. Moreover, the Moon orbits around the Earth, and the Earth
rotates daily about a North-South axis. The
order of the planetary orbits is as follows: Mercury, Venus, Earth, Mars, Jupiter, Saturn--with Mercury
closest to the Sun. This model became known as the *heliocentric* model of the Solar System.

The heliocentric model was generally rejected by the ancient philosophers for three main reasons:

- If the Earth is rotating about its axis, and orbiting around the Sun, then the Earth must be in motion. However, we cannot ``feel'' this motion. Nor does this motion give rise to any obvious observational consequences. Hence, the Earth must be stationary.
- If the Earth is executing a circular orbit around the Sun then the positions of the stars
should be slightly different when the Earth is on opposite sides of the Sun. This effect
is known as
*parallax*. Since no stellar parallax is observable (at least, with the naked eye), the Earth must be stationary. In order to appreciate the force of this argument, it is important to realize that ancient astronomers did not suppose the stars to be significantly further away from the Earth than the planets. The celestial sphere was assumed to lie just beyond the orbit of Saturn. - The geocentric model is far more philosophically attractive than the heliocentric model, since in the former model the Earth occupies a privileged position in the Universe.

The geocentric model was first converted into a proper scientific theory, capable of accurate
predictions, by the Alexandrian philosopher Claudius Ptolemy (85-165AD). The theory that Ptolemy proposed in his famous book, now known as the *Almagest*, remained the dominant scientific picture
of the Solar System for over a millennium. Basically, Ptolemy acquired and extended the extensive
set of planetary observations
of his predecessor Hipparchus, and then constructed a geocentric model capable of accounting
for them. However, in order to fit the observations, Ptolemy was forced to make
some significant modifications to the original model of Eudoxas. Let us
discuss these modifications.

First, we need to introduce some terminology.
As shown in Fig. 100,
*deferants* are large circles centred on the Earth, and *epicyles* are small circles whose
centres move around the circumferences of the deferants. In the Ptolemaic system,
instead of traveling around deferants,
the planets
move around the circumference of epicycles, which, in turn, move around the circumference of
deferants. Ptolemy found, however, that this modification was insufficient to completely account
for
all of his data. Ptolemy's second modification to Eudoxas' model was to displace the Earth slightly from the
common centre of the deferants. Moreover, Ptolemy assumed that the Sun, Moon, and planets rotate
uniformly about an imaginary point, called the *equant*, which is displaced an equal distance in the opposite
direction to the Earth from the centre of the deferants. In other words, Ptolemy assumed that the line
, in Fig. 100,
rotates uniformly, rather than the line .

Figure 101 shows more details of the Ptolemaic model.^{2} Note that this diagram is
not drawn to scale, and the displacement of the Earth from the centre of the deferants has been
omitted for the sake of clarity. It can be seen that the Moon and the Sun do not possess
epicyles.
Moreover, the motions of the *inferior planets* (*i.e.*,
Mercury and Venus) are closely linked to the motion of the Sun. In fact, the centres of the inferior planet
epicycles move on an imaginary line connecting the Earth and the Sun. Furthermore, the
radius vectors connecting the *superior planets* (*i.e.*, Mars, Jupiter, and Saturn)
to the centres of their epicycles are always parallel to the geometric line connecting the
Earth and the Sun. Note that, in addition to the motion indicated in the diagram, all of the
heavenly bodies (including the stars) rotate clockwise (assuming that we are looking down on the
Earth's North pole in Fig. 101) with a period of 1 day. Finally, there are epicycles
within the epicycles shown in the diagram. In fact, some planets need as many as 28
epicycles to account for all the details of their motion. These subsidiary epicycles
are not shown in the diagram, for the sake of clarity.

As is quite apparent, the Ptolemaic model of the Solar System is *extremely complicated*.
However, it successfully accounted for the relatively
crude
naked eye observations made by the ancient Greeks. The Sun-linked epicyles of the
inferior planets are needed to explain why these objects always remain close to the
Sun in the sky. The epicycles of the superior planets are needed to account for
their occasional bouts of *retrograde motion*: *i.e.*, motion in the opposite
direction to their apparent direction of rotation around the Earth. Finally, the
displacement of the Earth from the centre of the deferants, as well as the introduction
of the equant as the centre of uniform rotation, is needed to explain why the planets
speed up slightly when they are close to the Earth (and, hence, appear
brighter in the night sky), and slow down when they are further away.

Ptolemy's model of the Solar System was rescued from the wreck of ancient European civilization
by the Roman Catholic Church, which, unfortunately, converted it into a minor article
of faith, on the basis of a few references in the Bible which seemed to imply that the Earth is
stationary and the Sun is moving (*e.g.*, Joshua 10:12-13, Habakkuk 3:11).
Consequently, this model was not subject to proper scientific criticism for over a
millennium. Having said this, few medieval or renaissance philosophers were entirely satisfied with
Ptolemy's model. Their dissatisfaction focused, not on the many epicycles (which to the
modern eye seem rather absurd), but on the displacement of the Earth from the
centre of the deferants, and the introduction of the equant as the centre of uniform
rotation. Recall, that the only reason planetary orbits are constructed from circles
in Ptolemy's model is to preserve the assumed ideal symmetry of the heavens. Unfortunately,
this symmetry is severely compromised when the Earth is displaced from the
apparent centre of the Universe. This problem so perplexed the Polish priest-astronomer
Nicolaus Copernicus (1473-1543) that he eventually decided to reject the geocentric
model, and revive the heliocentric model of Aristarchus. After many years of mathematical
calculations, Copernicus published a book entitled *De revolutionibus orbium coelestium*
(On the revolutions of the celestial spheres) in 1543 which outlined his new heliocentric
theory.

Copernicus' model is illustrated in Fig. 102. Again, this diagram is not to scale. The planets execute uniform circular orbits about the Sun, and the Moon orbits about the Earth. Finally, the Earth revolves about its axis daily. Note that there is no displacement of the Sun from the centres of the planetary orbits, and there is no equant. Moreover, in this model, the inferior planets remain close to the Sun in the sky without any special synchronization of their orbits. Furthermore, the occasional retrograde motion of the superior planets has a more natural explanation than in Ptolemy's model. Since the Earth orbits more rapidly than the superior planets, it occasionally ``overtakes'' them, and they appear to move backward in the night sky, in much the same manner that slow moving cars on a freeway appears to move backward to a driver overtaking them. Copernicus accounted for the lack of stellar parallax, due to the Earth's motion, by postulating that the stars were a lot further away than had previously been supposed, rendering any parallax undetectably small. Unfortunately, Copernicus insisted on retaining uniform circular motion in his model (after all, he was trying to construct a more symmetric model than that of Ptolemy). Consequently, Copernicus also had to resort to epicycles to fit the data. In fact, Copernicus' model ended up with more epicycles than Ptolemy's!

The real breakthrough in the understanding of planetary motion occurred--as most breakthroughs in physics occur--when better data became available. The data in question was produced by the Dane Tycho Brahe (1546-1601), who devoted his life to making naked eye astronomical observations of unprecedented accuracy and detail. This data was eventually inherited by Brahe's pupil and assistant, the German scientist Johannes Kepler (1571-1630). Kepler fully accepted Copernicus' heliocentric theory of the Solar System. Moreover, he was just as firm a believer as Copernicus in the perfection of the heavens, and the consequent need for circular motion of planetary bodies. The main difference was that Kepler's observational data was considerably better than Copernicus'. After years of fruitless effort, Kepler eventually concluded that no combination of circular deferants and epicycles could completely account for his data. At this stage, he started to think the unthinkable. Maybe, planetary motion was not circular after all? After more calculations, Kepler was eventually able to formulate three extraordinarily simple laws which completely accounted for Brahe's observations. These laws are as follows:

- The planets move in elliptical orbits with the Sun at one focus.
- A line from the Sun to any given planet sweeps out equal areas in equal time intervals.
- The square of a planet's period is proportional to the cube of the planet's mean distance from the Sun.

Figure 103 illustrates Kepler's second law. Here, the ellipse represents a planetary orbit, and represents the Sun, which is located at one of the focii of the ellipse. Suppose that the planet moves from point to point in the same time it takes to move from point to point . According the Kepler's second law, the areas of the elliptic segments and are equal. Note that this law basically mandates that planets speed up when they move closer to the Sun.

Table 5 illustrates Kepler's third law. The mean distance, , and orbital period, , as well as the ratio , are listed for each of the first six planets in the Solar System. It can be seen that the ratio is indeed constant from planet to planet.

Since we have now definitely adopted a heliocentric model of the Solar System, let us discuss the ancient Greek objections to such a model, listed earlier. We have already dealt with the second objection (the absence of stellar parallax) by stating that the stars are a lot further away from the Earth than the ancient Greeks supposed. The third objection (that it is philosophically more attractive to have the Earth at the centre of the Universe) is not a valid scientific criticism. What about the first objection? If the Earth is rotating about its axis, and also orbiting the Sun, why do we not ``feel'' this motion? At first sight, this objection appears to have some force. After all, the rotation velocity of the Earth's surface is about . Moreover, the Earth's orbital velocity is approximately . Surely, we would notice if we were moving this rapidly? Of course, this reasoning is faulty because we know, from Newton's laws of motion, that we only ``feel'' the acceleration associated with motion, not the motion itself. It turns out that the acceleration at the Earth's surface due to its axial rotation is only about . Moreover, the Earth's acceleration due to its orbital motion is only . Nowadays, we can detect such small accelerations, but the ancient Greeks certainly could not.

Kepler correctly formulated the
three laws of planetary motion in 1619. Almost seventy years later, in 1687,
Isaac Newton published his *Principia*, in which he presented, for the
first time, a universal theory of motion. Newton then went on to
illustrate his
theory by using it to deriving Kepler's laws from first principles. Let us now discuss Newton's
monumental achievement in more detail.