# Eratosthenes

## 276-194 B.C.

Eratosthenes of Cyrene was a Greek mathematician, astronomer, geographer, philosopher, and poet, whose varied talents earned him the epithet "pentathlete." Educated in Athens, he was called to Alexandria (about 245 B.C.) by Ptolemy III to serve as tutor to Ptolemy's son, Philopator, and as librarian at the Museum. He died in Alexandria.

Although he was a prolific writer, his work survives only in fragments and in reports by other writers. He is best known for his ingenious measurement of the earth's circumference. Having determined that the noon sun at the summer solstice stands almost directly overhead at the town of Syene (now Aswan) , he measured the angle between the sun's rays and a perpendicular line at the same time in Alexandria. Assuming that Syene and Alexandria lie on the same meridian and that the sun's rays are parallel, he reasoned that the angle, which measured approximately 1/50 of a full circle (7° 12'), corresponded to the angular distance (difference in latitude) between the two towns. Hence, the linear distance between them, established by direct measurement to be 5,000 stadia, represented 1150 of the earth's circumference. To the resulting value of 250,000 stadia he added another 2,000 to account for various inaccuracies, giving a total circumference of 252,000 stadia (probably about 24,500 miles or 40,000 kilometers). Eratosthenes' method is theoretically highly accurate, but uncertainty regarding the modem equivalent of the stadion makes it difficult to judge his actual accuracy.

Eratosthenes also attempted to measure the distances of the sun and moon from the earth and wrote a Geography that served as a model for subsequent authors. In mathematics, he wrote a long treatise on Platonic mathematics and devised an instrument, the mesolabon, for determining two mean proportionals between two given magnitudes. The numbers x and y are mean proportionals between a and b if and only if a:x = x:y = y:b. He also established a procedure for systematically determining all prime numbers less than a given value. Called the "sieve of Eratosthenes," the procedure essentially eliminates all multiples of the successive primes beginning with 2.

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