to the plane of the earth’s equator. Th e Pythagorean sect of
southern Italy saw in nature a symmetrical structure, with
all things having an underlying unity of form, based on
numbers, ratios, and proportions.
Observed nature is not always symmetrical, however,
and later Greek thinkers had to account for many strange
anomalies. For the astronomer, the most serious was retro-
grade motion: Th e earth’s orbit relative to that of the planets
occasionally makes it appear that the planets are moving in
reverse. Eudoxus of Cnidus (ca. 400–ca. 347 b.c.e.) tackled
this problem while proposing a geometrical model for the
motion of the heavenly bodies. He believed in simple circu-
lar motion, as did the Pythagoreans, but added two spheres
that were concentric—sharing the same center point—and
nested within each other. Th e planets moved along the edg-
es of these spheres, which rotated in opposite motion. Th e
smaller, planetary sphere was inclined to the larger sphere,
which lay on a diff erent axis. Th is arrangement explained
the apparent motion of the planet backward and forward
in the sky.
Aristotle (384–322 b.c.e.) refi ned the system of Eudoxus,
proposing that the motion of each of the heavenly spheres
has an eff ect on the others. Aristotle confi rmed the belief
in a spherical earth by observing that earth’s shadow on the
moon during lunar eclipses is curved. Of all the Greek phi-
losophers, Aristotle had the strongest infl uence on medieval
astronomers, who imitated his elaborate schema to explain
all observable phenomena—including the earth and stars,
planets, sun and moon, and the four basic elements of water,
air, fi re, and earth.
Th e varying distances of the stars, planets, sun, and
moon—shown by their changing brightness—could not be
explained by spheres and perfect circular motion. More so-
phisticated geometric models of the universe were required.
Using the Pythagorean theorem and the geometry of Euclid
(fl. ca. 300 b.c.e.), Aristarchus (ca. 310–230 b.c.e.) measured
the ratio of distances to the moon and sun, showing that the
sun was farther from the earth than was the moon. He also
proposed a heliocentric, or sun-centered, universe, an idea
counter to the prevailing wisdom that placed the earth at the
center of the universe. Aristarchus explained the apparently
fi xed position of the stars as a result of their great distance
from the earth. Th e planets in his system moved around the
earth in an eccentric orbit, with the earth located off the
orbit’s center. Th is arrangement accounted for the varying
speeds and brightness of the planets.
By comparing the length of shadows at two distant loca-
tions in Egypt, Eratosthenes (276–194 b.c.e.) of Alexandria
came up with a fairly accurate measure of the earth’s cir-
cumference. He also calculated the distance of the sun and
the moon and the 23.5-degree tilt of the earth toward the
plane of its orbit around the sun. Apollonius of Perga (ca. 262
b.c.e.–ca. 190 b.c.e.) introduced a theory of epicycles. Th e
planets moved on a circle, or epicycle, whose center moved
about a greater circle known as the deferent. In his book Th e
Sand Reckoner, the mathematician Archimedes of Syracuse
(287–212 b.c.e.) calculated the diameter of earth’s orbit and
that of the sun and used this ratio to determine the diameter
of the universe.
Hipparchus (ca. 190 b.c.e.–ca. 120 b.c.e.) was an as-
tronomer and mathematician of Nicaea, in Asia Minor. He
proposed mathematical theorems for the motion of the sun
and moon; created a system of brightness magnitudes for
the stars; compiled a star catalogue; built a celestial globe;
and discovered precession, the movement of the axis of a ro-
tating body. Hipparchus was the fi rst to divide the circle into
360 degrees, which allows astronomers to map the heavens
and geographers the earth. Many consider Hipparchus to
be the greatest of all the Greek astronomers, but only one of
his books survives. By contrast, Claudius Ptolemaeus (Ptol-
emy) of Alexandria (ca. 90–ca. 168 c.e.) earned the highest
renown of all Greek astronomers. Ptolemy wrote an impor-
tant work of observations, theories, and calculations known
in Greek as Mathematike Syntaxis and more familiarly by
the Arabic-derived name Almagest (Th e Great Book). Th is
13-book treatise drew on Greek and Babylonian astronomy
as well as observations that Ptolemy himself made over a
span of 40 years.
Th e Almagest was fi rst translated into Latin in the 12th
century by Gerard of Cremona. Th e book gives a mathemati-
cal theory of the motion of the sun, moon, and planets. Th e
sphere theory propounded by Aristotle was used by Ptolemy
as the basis for his description of the epicycles of the spheres.
Ptolemy compares observations of the solar equinoxes and
solstices with those of Meton and Hipparchus and gives a
precise measurement of the length of the seasons. Ptolemy
postulated a spherical universe, at the center of which lay the
spherical, fi xed, and unmoving earth. Th e planetary spheres
contained, in ascending order from the earth, the moon, Mer-
cury, Venus, the sun, Mars, Jupiter, Saturn, and the stars.
Ptolomy’s Almagest also contains a star catalogue up-
dated from that of Hipparchus and listing 1,022 stars and 48
constellations. Th e star-mapping techniques described the
stars as points on a grid, a system that he later extended to
earthly locations in other books. Ptolemy’s system of a 360-
degree globe, further divided into minutes and seconds,
could also be applied to earthly locations—what is known
at present as latitude and longitude. Th is system of location
was later used in his book Geographia. Th e Almagest was the
standard scholarly text on astronomy until the 16th century
in Europe. Ptolemy’s geocentric universe was accepted by
European and Arab astronomers for more than 1,000 years
aft er his death, until it was replaced by the heliocentric sys-
tem devised by Copernicus.
Alexandria remained the center of Greek astronomy,
philosophy, and mathematics aft er the time of Ptolemy. Writ-
ers created commentaries on the works of past astronomers
and helped preserve their theories. Th eon of Alexandria (ca.astronomy: Greece 131