[This article kindly contributed
to the archival project by @ Prof. Michael Fowler @ The
University of Virginia. PJC]
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.
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 orbit.
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.
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.
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.
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 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.
Copyright except where otherwise noted ©1995 Michael Fowler