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 1/50th 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 1/50th 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 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.

J. E. Diggins, The Whole Round Earth, http://www.anselm.edu/homepage/dbanach/erat.htm
Astronomy Online:  http://www.algonet.se/~sirius/eaae/aol/market/collabor/erathost/

Institute of Mathematical Geography.  Copyright, 2005, held by authors.
Spatial Synthesis:  Centrality and Hierarchy, Volume I, Book 1.
Sandra Lach Arlinghaus and William Charles Arlinghaus