Eratosthenes and the Mystery of the Stades - Eratosthenes' Argument (III)

Newlyn Walkup

Let the arc of the Earth between Alexandria and Syene be called arc AS, and let the full circumference of the Earth be called arc EC.

Let the angle at the center of the Earth be called angle a, and let the full 360° of the circle be called angle b.

By Euclid III-27, we have  arc EC/arc AS = angle b/angle a.

By hypothesis, the length of arc AS is 5000 stades, and angle a is equal to 1/50th of a circle.

Since angle b is the angle measure of a complete circle, angle b= 1 circle.

Substituting these real number values into the previous ratio, we get


                      arc EC/arc AS = angle b/angle a              

                      arc EC /5000 stades = 1/(1/50)

                      arc EC    =  5000 stades/(1/50)

                      arc EC    =  250,000 stades.

Therefore, since the length of arc EC is equal to the circumference of the Earth, we get that the circumference of the Earth is approximately 250,000 stades.