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.