While most
of the mathematics we have studied to this point is mathematics for its
own sake, it was seen by most Greeks to be a window into the true nature
of things in the real world. Plato
of Athens (427 - 327), one of the great lights of Greek philosophy,
developed a very influential theory
of forms, which asserts that objects in the real world have an ideal
existence accessible to us only through contemplation and thought, and
that what we perceive through our senses is just a rough approximation,
a "shadow", of their ideal essences. In this respect, geometrical
objects, like "line" and "circle", are ideal, distinguishable from any
representation we might given them in our real world drawings of "line"
or "circle". Therefore, the mathematician engaged in geometrical
thought would be regarded as accomplishing much the same task as a modern-day
physicist in his lab would be regarded today, since the geometer would
be seen to be in touch with the world in its ideal form. Knowledge
of mathematical objects therefore informs one about physical objects in
their ideal form. This is the subtext with which Euclid's *Elements*,
for instance, should be read. Euclid seems to believe that he is
laying down in rigorously logical form a physics of the ideal world rather
than just a theory of abstract mathematics.

There were amongst
the Greek philosophers those who studied natural philosophy, what we would
today call physics, astronomy, biology, chemistry, etc. Pre-eminent
amongst these was Aristotle, whose writings
in natural philosophy would be taken as definitive works in the subject
for many centuries after his death.

Many of these
thinkers made significant advances in astronomy, as we have seen in the
mathematical applications of Eratosthenes and Aristarchus. From Babylonian
astronomers, the Greeks inherited the catalog of constellations, especially
the constellations of the zodiac, the twelve regions of the sky through
which the sun, moon, and planets travelled in their circuits. The
sun makes a complete circuit of the heavens in one year and the twelve
signs of the zodiac correspond roughly to the regions in which the sun
lies in successive months. They also adopted the sexagesimal system
of notation for doing astronomy.

In addition
to advances in measurement, other Greek astronomers, convinced that they
could understand the underlying forces at work in the cosmos by proposing
mathematical models that would describe these motions (instead of attributing
them to the action of the gods!), devised explanations for the astronomical
phenomena they observed. Eudoxus, for instance, was as accomplished
as an astronomer as he was as a mathematician. According to Aristotle,
he taught (in the fourth century, BC) that the rotating celestial sphere
of the fixed stars enclosed a nested sequence of many smaller rotating
spheres on which the planetary lights rested; on the innermost sphere rested
the moon, then the other planets: the Sun, Mercury, Venus, Mars, Jupiter
and Saturn (this is found in book
12 of Artistotle's *Metaphysics*). The three outer planets,
which are sometimes observed to reverse direction in their motion across
the sky, required a series of spheres each, rotating on different axes,
to explain this strange behavior. Significantly, Eudoxus was convinced
that the heavenly bodies had to be explained solely by means of spheres
and spherical motion, for the sphere was the most perfect three-dimensional
shape.

Apollonius (third
century, BC) noticed that this did not explain why the planets wax and
wane in brightness, suggesting that their distances from earth vary over
time. He solved this problem by supposing that instead of a *homocentric*
system of spheres which were all centered on the earth, the planets moved
along *eccentric* spheres, in which the centers of the planetary spheres
varied to allow them to be sometimes nearer and sometimes farther from
earth.

Hipparchus
of Nicaea (180? - 125?) was interested in calculating various important
astronomical constants, like the length of the year, which he obtained
accurately to within 7 minutes, and the amount of the *precession of
the equinoxes*, that very small amount of drift that the axis of the
earth is subject to. (In Babylonian times, the vernal equinox, or
position of the sun at the beginning of spring when day and night are of
equal length, was found in the constellation Aries, but by the time of
Hipparchus, it had drifted into Pisces; today, it is entering "the age
of Aquarius". This position moves at a rate of 50 seconds of arc--that's
one sixtieth of one sixtieth of a degree--each year. This phenomenon
is called the precession of the equinoxes.) He also published a detailed
series of star charts, which required careful calculations, especially
the need to measure sides of triangles. To facilitate these calculations,
he also drew up a **table of chords**. For angles measured at
the center of a unit circle, in steps of 7^{1}/_{2}°,
he calculated geometrically the lengths of the chords that subtend those
arcs.

Two centuries later, around the year 147 AD, Claudius Ptolemy (85? - 165?), an Egyptian-born Greek with Roman citizenship, compiled a systematic summary of mathematical astronomy entitled

While no mention of familiar trigonometric functions like sine or cosine appear in Greek geometry (the modern trigonometric functions of sine, cosine, etc., were first used by Hindu mathematicians as early as the fifth century AD and were transmitted through Arab scholars int he tenth century and later into the West), the ideas are equivalent. We learn from modern trigonometry that the sine of a central angle in a unit circle is the height of its opposite side. It follows that the formula

Read the text

Return to the calendar

last modified 10/8/02

Copyright (c) Daniel E. Otero