ERATOSTHENES'S
MEASUREMENT OF THE CIRCUMFERENCE OF THE EARTH
Eratosthenes of Alexandria (appointed
Director of the Great Library at Alexandria in 236 B.C.) was an innovator
in measurement. Not only did he create a prime number sieve, but
also he figured out how to measure the circumference of the Earth.
To do so, he used Euclidean Geometry and simple measuring tools.
Below, we show the style of measurement that he is said to have made (different
accounts give different details).
Assume the Earth is a sphere.
The circumference of the sphere is measured
along a great circle on the sphere.
Find the circumference of the Earth by finding
the length of intercepted arc of a small central angle.
Find two places on the surface of the Earth
that lie on the same meridian (or close to it): meridians are halves
of great circles.
Eratosthenes chose Alexandria and Syene, near
contemporary Aswan (Figure 1).
Figure 1.
Relative location of Alexandria and Aswan. They are close to lying
on the same meridian (half of a great circle).
Assume that the rays of the Sun are parallel
to each other.
The Sun's rays are directly overhead, on the
Summer Solstice (c. June 21), at 23.5 degrees N. Latitude.
Syene is located at about 23.5 degrees N.
Latitude. Hence, on the Summer Solstice, the reflection of the sun
will appear in a narrow well (and it will not on other days). Eratosthenes
apparently understood this idea.
Alexandria is north of Syene. Thus,
on June 21, objects at Alexandria will cast shadows whereas those at Syene
will not.
Eratosthenes focused on an obelisk or post
located in an open area. He measured the shadow that the obelisk
cast (A'A''), functioning in the manner of a gnomon on a sundial, and then
measured the height of the obelisk (AA') (perhaps using a string anchored
to the tip of the obelisk).
Figure 2.
Eratosthenes's measurement of the circumference of the Earth, based on
a Theorem of Euclid.
According to Euclid, two parallel lines cut
by a transversal have alternate interior angles that are equal. The
Sun's rays are the parallel lines. One ray, at Alexandria, touches
the tip of the obelisk and extends earthward toward the tip of the shadow
of the obelisk, AA''. It is extended to AB in Figure 2. The
other ray, SO, at Syene, goes into the well and extends abstractly to the
center of the Earth, O. The obelisk, AA', also extends abstractly
to the center of the Earth, O; thus, the line, AO, determined by the tip
of the obelisk and the center of the Earth is a transversal cutting the
two parallel rays, SO and AB, of the sun.
Angles (BAO) and (SOA) are thus alternate
interior angles in geometric configuration described above; therefore,
they are equal.
Use the length of the obelisk shadow and the
height of the obelisk to determine angle BAO; triangle AA'A'' is a right
triangle with the right angle at A'. Thus, we would note, tan (A'AA'')
= (length of shadow)/(height of obelisk). Eratosthenes's measurements
of these values led him to conclude that the measure of angle (A'AA") was
7 degrees and 12 minutes.
The value of 7 degrees and 12 minutes is 1/50^{th}
of the degree measure of a circle. Since he assumed that Alexandria
and Syene both lay on a meridian (half a great circle), it followed that
the distance between these two locations was 1/50^{th} of the circumference
of the Earth.
Eratosthenes calculated the distance between
Alexandria and Syene using records involving camel caravans. The
distance he used was 5000 stadia. Thus, the circumference of the
Earth is 250,000 stadia, which translates to somewhat less than 25,000
miles (depending on how ancient units convert to modern units). This
value is remarkably close to current values used.
Many of the assumptions made by Eratosthenes
were not accurate; apparently, however, underfit and overfit of error balanced
out to produce a good result. For example, Syene and Alexandria are
not on the same meridian; Syene is not at exactly 23.5 degrees N. Latitude,
and so forth. See the link to Astronomy Online for more discussion
of historical and astronomical matters.