Eratosthenes was called "Beta" because he was never first place in anything, but the third librarian at Alexandria, who studied under the Stoic philosopher Zeno, Ariston, Lysanias, and the poet-philosopher Callimachus, is more famous than his "Alpha" teachers because his discoveries are still used in science.

Eratosthenes, born in 276 B.C., in Cyrene (Syrene) in what is Libya today, is known for many observations and calculations. Chief among these are the calculation of the circumference of the earth (note: the Greeks did know the earth was spherical) and the development of a mathematical sieve named after him.

Much of what he wrote is now lost, including a geometrical treatise, On Means, and one on the mathematics behind Plato's philosophy, Platonicus. He also wrote the fundamentals of astronomy in a poem called Hermes. His most famous calculation, in the now lost treatise On the measurement of the Earth, explains how he compared the shadow of the sun at Summer Solstice noon in two places, Alexandria and Syrene. Based on the distance between the two, he calculated the circumference of the earth to be 250,000 stadia (about 24,662 miles). stadia. He calculated the distance from the sun at 804,000,000 stadia and the distance from the moon at 780,000 stadia.

He also devised a calendar with leap years, a 675-star catalogue, and maps. He recognized a lake as the source for the Nile and that rains in the lake region caused the Nile to flood.

He is reported to have starved himself to death at Alexandria in 194 B.C.

## Measuring the earth's circumference

The two cities Eratosthenes used to measure the circumference of the earth were Alexandria and Syene, not Cyrene. Syene is modern-day Aswan, Egypt. Cyrene was Eratosthenes' birthplace; it corresponds to modern-day Shahhat, Libya.

The method used by Eratosthenes works like this:

• Assumption: the sun is infinitely far from the earth. This is borne out by the fact that the first quarter and last quarter phases of the moon each occur 90 degrees away from the sun.
• Eratosthenes observes that on the summer solstice the sun is directly overhead at Aswan. On that same day the sun makes an angle of just over 7 degrees with the zenith at Alexandria. This gives him the angular separation of the two cities on the curve of the earth's surface. Call it a.
• Eratosthenes hires someone to measure the distance between Alexandria and Syene. Call it d
• The ratio of a to a full circle (360 degrees) must equal the ratio of d to the circumference of the earth (C). Since a and d have been measured, it is easy to calculate C = 360/a * d
• Eratosthenes did his calculations using stadia as his unit of distance. The exact length of a stadium is uncertain, so it's not clear how good this measurement is; however, even the worst estimates suggest it was good to about 4% of the currently accepted value.

## Eratosthenes' other achievements

In astronomy Eratosthenes also measured distances to the sun and moon, using observations of lunar eclipses, and he calculated a value of about 24 degrees for the tilt of the earth's rotational axis.

In geography Eratosthenes sketched the route of the Nile river as far as Khartoum. He also suggested that the river had its source in lakes to the south and that heavy rains in those regions were the reason for the river's yearly floods.

Eratosthenes also wrote history and commentary about several branches of mathematics, and he contributed to that field a method of finding prime numbers and a method for constructing mean proportionals between two line segments, a construction necessary in duplicating the cube. Finally, as if all that weren't enough, he wrote a number of works of poetry, literature, and philosophy.

## References

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html

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