Lecture Notes by I. R. King: Greek Astronomy and the Ptolemaic System
We're going to go through today the developments that led to the Ptolemaic system of the motion of planets around the Earth, but the development of Greek thought, and in particular Greek science, was far from monolithic and undeviating. Two ``aberrations'' even appeared that are part of our modern view of the world: Heraclides suggested that the daily rising and setting of the sun, moon, and stars could be explained by a rotation of the earth on its axis, and Aristarchus (and probably Heraclides too) thought that the Earth went around the Sun. But these ideas attracted little attention or following. (I don't think Ptolemy even mentions Aristarchus or bothers to refute him.) Now it's very easy today to say, ``Oh, those stupid Greeks! They had people who knew the right answer, but they ignored them.'' But this of course is a ridiculously naive view. They lived in a different world, with views that in many ways are just as intelligent as ours, but different fundamental ideas were dominant. In particular, philosophy dominated over scientific inquiry, and Aristotle's ideas about the nature of the earth and the heavens ruled out both of these motions of the earth. And let's not forget the role of common sense: everybody knows that whenever you accelerate you can feel it, and we certainly don't feel the earth spinning or rushing around the sun.
But I don't want to get off on this side track; let's get back to the development of Greek astronomy. But that brings us right back to Aristarchus.
His heliocentric picture hasn't survived in writing; we know of it
only through mention by others. But one thing he did was definitely
mainstream: measuring the relative distances of the moon and sun.
Magnificent in principle, but flawed in practice, for two reasons.
First, it is hard to measure the time of quarter moon precisely.
(This is usually excited to explain the poor accuracy of his result.)
Second, and rarely noted, is the fact that the eccentricity
(ellipticity) of the moon's orbit makes it move with varying speed,
which vitiates the whole idea, unless you take an average over many
moon orbits. The monthly variations are 5-
larger than the
effect Aristarchus was trying to measure.
One other quantity that was measured in ancient times is the size of
the earth. Around 200 BC, Eratosthenes, who lived in Alexandria, had
heard that in the town of Syene, some distance to the south of him,
the sun was exactly overhead on the summer solstice (when the sun is
farthest north). From the length of a shadow in Alexandria, he
measured that the sun was 
from his zenith at solstice.
This angle is 1/50 of a full circle, so the circumference of the
earth is 
the distance from Syene to Alexandria. The
accuracy of his result is not known with certainty, because the size
of the unit he used is not known. A confusion of units also seems to
have affected Columbus, who thought that the earth's circumference was
smaller than it really is (and used the erroneous figures to justify
his voyage across the ocean).
The greatest achievement of ancient science was the working out of an accurate description of the complicated motions of the planets (for example, retrograde motion). How do we get around this? Appollonius (c. 200 BC) came up with the idea of the epicycle.
Another problem was non-uniform speed, which in the Aristotelian universe was unacceptable. It was well known that the sun doesn't move uniformly throughout the year; the lengths of the four seasons differ by several days. Appollonius had a solution for this one too: the eccentric circle. Motion was not around the cosmic center (earth) but rather around a displaced point. (Ironic that the greatest ancient authority on ellipses seems never to have thought of using them for orbits!)
Hipparchus (c. 140 BC) was the greatest observer of ancient times. He cataloged about 850 stars, and observed the positions of the planets (Hipparchus was to Ptolemy as Tycho later was to Kepler). Magnitudes (a logarithmic measure of brightness) in his catalog established the system of magnitudes that we use today. Hipparchus also applied the ideas of Appollonius (the eccentric and epicycle) to real planetary data.
Ptolemy (c. 150 AD) was the last great natural philosopher of classical
times. He was responsible for putting together a picture of planetary
motion that stood until the beginning of modern times. (He also drew
the best world atlas of ancient times, which later influenced Columbus
toward thinking that China was a reasonably short distance west of
Europe!) He wrote a great compendium of math and astronomy:
,
or Almagest
in its Arabic version.
Ptolemy used observations made by Hipparchus to measure the parallax of the moon. He seems to have gotten the right answer, but made two big mistakes which remarkably canceled out! Parallax exists for the stars too (since the earth moves round the sun) but is very small (since the stars are very far away).
Whereas earlier scientists had been assembling ideas and observations (e.g., epicycle and eccentric of Appollonius; observations of Hipparchus, whose importance Ptolemy acknowledges), it was Ptolemy who finally put it all together to make a system that worked--that is, it would fit observed positions in the past and could be used to predict them in the future. (Ironically, the greatest practical importance for this system was considered to be its use for astrology, which of all ignorant superstitions is the one that modern astronomers most abominate.)
The epicycle is the central device of the Ptolemaic system--it can explain retrograde motion (which is much easier to explain in a heliocentric system!), but it's not enough. Ptolemy had to make things really fit well. He used the eccentric, which can account for varying speed. Today, we know that each planetary orbit has a different eccentricity (that is, degree of ellipticity or non-circularity) of its orbit. So in the Ptolemaic system, each planet has its circular orbit centered at a different place. But even on an eccentric circle, he still couldn't get the speeds right with uniform motion (a philosophical necessity), so he devised a new trick, the equant. This is another point, distinct from both earth and the center of the eccentric circle, from which the planet appears to sweep out equal angles in equal times. When you come to study Copernicus, you'll find that he was quite complacent about epicycles and even used some small ones himself, but he despised the equant.
In summary, a Ptolemaic orbit is defined by the deferent, epicycle, eccentric, and equant. Ptolemy found numerical values for each of the seven ``planets'' (including Sun and Moon), that would fit the observations of his day as well as those of Hipparchus, nearly 300 years earlier. His model provided a good fit (within a fraction of a degree) to all planetary motions for several centuries!
Some medieval Arabic astronomers added subsidiary epicycles, greatly complicating the system and leading to the idea that Ptolemy piled epicycles on top of epicycles. In fact, he didn't, and it was the (relatively) simple system just described that came into western Europe in the thirteenth century and was used there.
Ptolemy had succeeded brilliantly at solving Plato's old challenge to ``save the phenomena''. But his model had some shortcomings. First, it didn't contain any physics--no explanation of why things move the way they do. (But note Copernicus also lacked physics, which was only supplied a century and a half later by Newton.) Secondly, there were some suspicious coincidences unexplained by Ptolemy's model. One was that the centers of the epicycles of Mercury and Venus were fixed on a line between the earth and sun. A second coincidence was that retrograde motion always takes place when the planet is in the opposite direction from the sun (at opposition). But it worked, and it fit many Aristotelian principles, so it was just fine in its day.
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