Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 35

• 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 ( AA ′ ) or post located in an
open area ( Figure 2.1 ) . The base of the post is at A ′ in Alexandria
and its tip is at A (assuming the obelisk is a straight extension of
OA ′ —that is, it is not a “leaning” tower). He measured the shadow
( A ′ A ″ ) that the obelisk cast, functioning in the manner of a gno-
mon on a sundial, and then measured the height of the obelisk
( AA ′ ) (perhaps using a string anchored to the tip of the obelisk).
• According to Euclid, two parallel lines cut by a transversal have
alternate interior angles that are equal. The Sun's rays are the par-
allel lines. One ray, at Alexandria, touches the tip of the obelisk
and extends earthward toward the tip of the shadow of the obe-
lisk, AA ″ . It is extended to AB in Figure 2.1 . 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 the
geometric configuration described above; therefore, they are equal.
• Trigonometry permits angle measurement (see review in subsequent
section). Use the length of the obelisk shadow and the height of the
obelisk to determine angle ( BAO ); triangle ( A ′ AA ″ ) is a right triangle
with the right angle at A ′ . Thus, tan ( A ′ AA ″ ) = (length of shadow, A ′ A ″ )/
(height of obelisk, AA ′ ). Eratosthenes's measurements of these values
led him to conclude that the measure of angle ( A ′ AA ″ ) was 7 degrees
and 12 minutes and so therefore it was also of angle ( BAO ).
• The value of 7 degrees and 12 minutes is approximately 1/50th of
the degree measure of a circle. Since he assumed that Alexandria
and Syene both lay on a meridian (half of a great circle), it fol-
lowed that the distance between these two locations was 1/50th
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 more (depending on
how ancient units convert to modern units) than current accepted
values although it is remarkably close.
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, and Syene
is not at exactly 23.5 degrees north latitude. To gain a deeper understanding
of the principles involved in this measurement, you have the opportunity to
participate in the activity or “practice” in the following section. The activity