How the Greeks Used Geometry to Understand the Stars
Michael Fowler, University of Virginia
Crystal Spheres: Plato, Eudoxus, Aristotle
Plato, with his belief that the world was constructed with geometric simplicity
and elegance, felt certain that the sun, moon and planets, being made of
aither, would have a natural circular motion, since that is the simplest
uniform motion that repeats itself endlessly, as their motion did. However,
although the “fixed stars” did in fact move in simple circles about
the North star, the sun, moon and planets traced out much more complicated
paths across the sky. These paths had been followed closely and recorded
since early Babylonian civilization, so were very well known. Plato
suggested that perhaps these complicated paths were actually combinations of
simple circular motions, and challenged his Athenian colleagues to prove it.
The first real progress on the problem was made by Eudoxus, at Plato’s
academy. Eudoxus placed all the fixed stars on a huge sphere, the earth
itself a much smaller sphere fixed at the center. The huge sphere rotated
about the earth once every twenty-four hours. So far, this is the
standard “starry vault” picture. Then Eudoxus assumed the sun
to be attached to another sphere, concentric with the fixed stars’
sphere, that is, it was also centered on the earth. This new sphere,
lying entirely inside the sphere carrying the fixed stars, had to be
transparent, since the fixed stars are very visible. The new sphere was
attached to the fixed stars’ sphere so that it, too, went around every
twenty-four hours, but in addition it rotated slowly about the two axis
points where it was attached to the big sphere, and this extra rotation was
once a year. This meant that the sun, viewed against the backdrop of the
fixed stars, traced out a big circular path which it covered in a year. This
path is the ecliptic. To get it all right, the ecliptic has to be
tilted at 23½ degrees to the “equator” line of the fixed stars,
taking the North star as the “north pole”.
This gives a pretty accurate representation of the sun’s motion, but
it didn’t quite account for all the known observations at that time.
For one thing, if the sun goes around the ecliptic at an exactly uniform rate,
the time intervals between the solstices and the equinoxes will all be equal.
In fact, they’re not-so the sun moves a little faster around some parts
of its yearly journey through the ecliptic than other parts. This, and
other considerations, led to the introduction of three more spheres to describe
the sun’s motion. Of course, to actually show that the
combination of these motions gave an accurate representation of the sun’s
observed motion required considerable geometric skill! Aristotle wrote a
summary of the “state of the art” in accounting for all the
observed planetary motions, and also those of the sun and the moon. This
required the introduction of fifty-five concentric transparent spheres.
Still, it did account for everything observed in terms of simple
circular motion, the only kind of motion thought to be allowed for aither.
Aristotle himself believed the crystal spheres existed as physical entities,
although Eudoxus may have viewed them as simply a computational device.
It is interesting to note that, despite our earlier claim that the Greeks “discovered
nature”, Plato believed the planets to be animate beings. He argued
that it was not possible that they should accurately describe their orbits year
after year if they didn’t know what they were doing—that is, if
they had no soul attached.
Measuring the Earth, the Moon and the Sun: Eratosthenes and Aristarchus
A little later, Eratosthenes and Aristarchus between them got some idea of
the size of the earth-sun-moon system, as we discussed in an earlier lecture.
And, to quote from Archimedes (see Heath, Greek Astronomy),
“Aristarchus of Samos brought out a book consisting of certain
hypotheses, in which the premises lead to the conclusion that the universe is
many times greater than it is presently thought to be. His hypotheses are
that the fixed stars and the sun remain motionless, that the earth revolves
about the sun in the circumference of a circle, the sun lying in the middle of
the orbit, and that the sphere of the fixed stars, situated about the same
center as the sun, is so great that the circular orbit of the earth is as small
as a point compared with that sphere.”
The tiny size of the earth’s orbit is necessary to understand why the
fixed stars do not move relative to each other as the earth goes around its
Aristarchus’ model was not accepted, nor even was the suggestion that
the earth rotates about its axis every twenty-four hours.
However, the model of the fifty-five crystal spheres was substantially
improved on. It did have some obvious defects. For example, the
sun, moon and planets necessarily each kept a constant distance from the earth,
since each was attached to a sphere centered on the earth. Yet it was
well-known that the apparent size of the moon varied about ten per cent or so,
and the obvious explanation was that its distance from the earth must be
varying. So how could it be attached to a sphere centered on the
earth? The planets, too, especially Mars, varied considerably in brightness
compared with the fixed stars, and again this suggested that the distance from
the earth to Mars must vary in time.
Cycles and Epicycles: Hipparchus and Ptolemy
A new way of combining circular motions to account for the movements of the
sun, moon and planets was introduced by Hipparchus (second century BC) and
realized fully by Ptolemy (around AD 150). Hipparchus was aware the
seasons weren’t quite the same length, so he suggested that the sun went
around a circular path at uniform speed, but that the earth wasn’t in the
center of the circle. Now the solstices and equinoxes are determined by
how the tilt of the earth’s axis lines up with the sun, so the directions
of these places from the earth are at right angles. If the circle is off
center, though, some of these seasons will be shorter than others. We
know the shortest season is fall (in our hemisphere).
Another way of using circular motions was provided by Hipparchus’
theory of the moon. This introduced the idea of the “epicycle”,
a small circular motion riding around a big circular motion. (See below
for pictures of epicycles in the discussion of Ptolemy.) The moon’s
position in the sky could be well represented by such a model. In fact,
so could all the planets. One problem was that to figure out the planet’s
position in the sky, that is, the line of sight from the earth, given its
position on the cycle and on the epicycle, needs trigonometry. Hipparchus
developed trigonometry to make these calculations possible.
Ptolemy wrote the “bible” of Greek (and other ancient)
astronomical observations in his immense book, the “Almagest”.
This did for astronomy at the time what Euclid’s Elements did for
geometry. It gave huge numbers of tables by which the positions of
planets, sun and moon could be accurately calculated for centuries to come.
We cannot here do justice to this magnificent work, but I just want to mention
one or two significant points which give the general picture.
To illustrate the mechanism, we present here a slightly simplified version
of his account of how the planets moved. The main idea was that each
planet (and also, of course, the sun and moon) went around the earth in a
cycle, a large circle centered at the center of the earth, but at the same time
the planets were describing smaller circles, or epicycles, about the point that
was describing the cycle. Mercury and Venus, as shown in the figure, had
epicycles centered on the line from the earth to the sun. This picture
does indeed represent fairly accurately their apparent motion in the sky—note
that they always appear fairly close to the sun, and are not visible in the
middle of the night.
For an animation, click here!
The planets Mars, Jupiter and Saturn, on the other hand, can be seen through
the night in some years. Their motion is analyzed in terms of cycles
greater than the sun’s, but with epicycles exactly equal to the sun’s
cycle, and with the planets at positions in their epicycles which correspond to
the sun’s position in its cycle—see the figure below.
For an animation, click here!
This system of cycles and epicycles was built up to give an accurate account
of the observed motion of the planets. Actually, we have significantly
simplified Ptolemy’s picture. He caused some of the epicycles to be
not quite centered on the cycles, they were termed eccentric. This
departure from apparent perfection was necessary for full agreement with
observations, and we shall return to it later. Ptolemy’s book was
called the Almagest in the Middle Ages, the Arabic prefix al with
the Greek for “the greatest” the same as our prefix mega.
Ptolemy’s View of the Earth
It should perhaps be added that Ptolemy, centuries after
Aristarchus, certainly did not think the earth rotated. (Heath, Greek
Astronomy, page 48). His point was that the aither was lighter than
any of the earthly elements, even fire, so it would be easy for it to move
rapidly, motion that would be difficult and unnatural for earth, the heaviest
material. And if the earth did rotate, Athens would be moving at
several hundred miles per hour. How could the air keep up? And even if
somehow it did, since it was light, what about heavy objects falling through
the air? If somehow the air was carrying them along, they must be very firmly
attached to the air, making it difficult to see how they could ever move
relative to the air at all! Yet they can be, since they can fall, so the whole
idea must be wrong.
Ptolemy did, however, know that the earth was spherical. He pointed
out that people living to the east saw the sun rise earlier, and how much
earlier was proportional to how far east they were located. He also noted
that, though all must see a lunar eclipse simultaneously, those to the east
will see it as later, e.g. at 1 a.m., say, instead of midnight, local
time. He also observed that on traveling to the north, Polaris rises in
the sky, so this suggests the earth is curved in that direction too. Finally,
on approaching a hilly island from far away on a calm sea, he noted that the
island seemed to rise out of the sea. He attributed this phenomenon
(correctly) to the curvature of the earth.