Eratosthenes uses these five main assumptions as hypotheses for his famous geometric approximation of the Earth’s circumference. His approximation would not be surpassed for centuries to come. The method devised by Eratosthenes is the basis for the complex “astrogeodetic method” which is used to measure the Earth today [5, p.153 ]. His elegant geometric argument, illustrated below, is sound and simple.
Claim: The circumference of the Earth is approximately 250,000 stades.
Given 1. That Alexandria and Syene lie on the same meridian.
2. That light rays from the Sun which strike the Earth are parallel.
3. That the distance between Alexandria and Syene is 5000 stades.
4. That the angle formed by the shadow and the staff in Alexandria at the
summer solstice is equal to 1/50th of a circle.
5. That the Earth is a sphere.
By construction, the staff in Alexandria is perpendicular to the ground, so in the plane of the meridian it is orthogonal to the cross-sectional circle of the Earth.
By definition of orthogonal, the staff in Alexandria is perpendicular to a line m which is tangent to the Earth at the base of the staff.
Likewise, the staff in Syene is perpendicular to a line n tangent to the Earth at the staff’s base.
Euclid III-19: If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn [10, p.45 ].
Therefore, since the staffs are perpendicular to tangents m and n, if the staffs are extended toward the Earth, their lines intersect at the center of the Earth.
By hypothesis, the light rays striking the Earth are parallel.
Since the staff in Syene casts no shadow, no angle is formed by the intersection of the light rays and the staff, thus the line of the staff is parallel to the light rays.