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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.

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# 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.