## Eratosthenes' Calculation of the Circumference of the Earth:

It was near the summer solstice of 240 BC that Eratosthenes, curator of the
famed Library of Alexandria and renowned mathematician and geographer, performed
his famous experiment in Egypt to calculate the size of the Earth. The
bottom of a deep well in the city of Syene, Egypt (near the present day Aswan
Dam and very near the Tropic of Cancer) was known to be illuminated by the sun
directly at mid-day on the longest day of the year (the solstice). But on the
same day, a vertical pole in Alexandria, some 800 km to the north, cast a
distinct shadow. By measuring the shadow and applying the geometry of a sphere,
Eratosthenes calculated the Earth's diameter with remarkable accuracy. Sadly,
the concept of a spherical Earth was lost from common thought for over a
thousand years until Christopher Columbus and others proved the fact by sailing
west to go east. The image below of Egypt and the Nile River is
provided by the NASA MODIS instrument. -- *From Martin Ruzek and the Earth Science Picture of the Day (see below) at http://epod.usra.edu/.*

Eratosthenes' deduction of the circumference of the Earth is based on the
following argument.

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Postulate 1**: The Earth is a sphere.

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Postulate 2:** The sun is far enough that rays of light intersect the Earth as parallel lines.

**Given:** The length of arc AC, the distance between two given cities, is known. On a particluar day,
the sun casts no shadow in the city at point C. On the same day, a pole with length AD casts a shadow of
length AE.

**Method of Calculation:** (1) Calculate (or measure) angle ADE, (2) Deduce measure of angle ABC using "alternate interior angles are congruent",
(3) Calculate 360/(measure of angle ABC), (4) Multiply length of arc AC by the result of step 3 to deduce the circumference of the Earth.

A modern estimate for the circumference of the Earth is around 24,875 miles.

### In the figure below, try moving the points A, B, C and D.

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Eratosthenes knew the distance from Alexandria to Syene (now Aswan), he knew on
a certain day there was no shadow in Syene while, on that same day, there was a
shadow in Alexandria. And he knew some geometry...

*After putting a stick in the ground, and using some
principles of geometry, he deduced the size of the world!*