Search Loci: Convergence:
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.