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

Euclid I-29:  A straight line falling on parallel straight lines makes the alternate angles equal to one another […] [9, p.311 ].

Let the angle at the center of the Earth be called angle a.

By hypothesis, the angle formed by the shadow in Alexandria is equal to 1/50th of a circle.  So the measure of this angle is 360°/50 = 7 1/5°.

By Euclid I-29, since the angle in Alexandria and angle a are alternate interior angles, the measure of angle a is also 360°/50 = 7 1/5°.

Euclid III-27:  In equal circles, angles standing on equal circumferences equal one another […] [10, p.58 ].

Some explanation will help to reveal how  Euclid III-27 is used in this argument.

Given two equal circles g and d, with centers p and q respectively, if  arc AB is equal to arc CD,  then angle b is equal to angle a.

Since every circle is equal to itself, by Euclid’s 4th common notion, we can apply this proposition to a single circle.

Given circle g, with center p, if arc AB is equal to arc CD,  then angle b is equal to angle a.                

As real number values, these can be put into ratio form: 

                    arc CD/arc AB = angle a/angle b.

Using this ratio form, Eratosthenes will now use three known values to solve for the unknown fourth value – the circumference of the Earth.