The Measurements Of Eratosthenes

Eratosthenes of Cyrene (c. 276 B.C.–c. 195 B.C.) is justly famous for his ingenious method of calculating the circumference of the earth.  What is less widely known is the fact that he made other contributions to the history of mathematics, some of which we will discuss here.

He must have had considerable merits, or else Archimedes would not have addressed his work The Method to him.  We know that when Eratosthenes was about forty he was invited to tutor Philopator, the son of the Ptolemaic king of Egypt, Ptolemy Euergetes.  Around this time he was appointed to be rector of the library at Alexandria.  One of his more interesting discoveries in arithmetic was a primitive device, perhaps still useful today, for finding prime numbers.  The method was called a sieve (κόσκινον)and worked this way.  A series of odd numbers is laid out beginning with the number 3:

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25…

The number 3 is a prime number, but multiples of it are not.  A series of multiples of 3 is formed by passing over two numbers at a time beginning with 3.  Then certain multiples are “stricken out” which are not primes.  From the number 5 (which is prime), we pass over four numbers at a time beginning from 5, and get a string of multiples of 5 (15, 25, etc.).  This process is repeated, which is rather laborious, until we are left with numbers that are primes.  The method may not be extremely efficient or practical for large numbers, but it at least shows a keen awareness of numeric quantities.  In geometry he solved the problem of finding any number of mean proportionals between two given straight lines.

With regard to his famous scheme for measuring the earth’s circumference, we should note that Eratosthenes stood on the shoulders of others that came before him.  It had been observed by others that Lysimachia on the Hellespont and the city of Syene in Egypt were on the same meridian.  Two stars which were at an apogee at the two places were at an angular distance of 1/15th of a circle; from this fact it was inferred that the distance between Lysimachia and Syene was about 20,000 stades.  And if this were the case, then the entire circumference of the earth must be about 300,000 stades.  Eratosthenes’s contribution was to improve greatly on the accuracy of this measurement; but he did not invent the technique.  Once Eratosthenes made his famous measurement using the cities of Alexandria and Syene as reference points, he arrived at he final number.

What was this final number?  One writer, Cleomedes, says the figure arrived at was 250,000 stades, but the geographer Strabo and the writer Theon of Smyrna say the figure was 252,000 stades.  We do not know for certain which figure is correct.  Pliny says that Eratosthenes equated his “stades” unit of measurement with the Egyptian skoinos (about .525 meters).  In any case, the final measurement he arrived at was very accurate.

The circumference of the earth was not the only measurement Eratosthenes made.  He is said to have calculated the distance of the moon from the earth to be 780,000 stades.  His measurement of the distance of the sun from the earth was about 804,000,000 stades.  These calculations were made using data gained from lunar eclipses, but I have not been able to discover the exact method he used to arrive at these figures.  Another measurement he made was that of the size of the sun.  He noted that the sun cast no shadows at noon at Syene in Egypt for a radius of 300 stades.  That is, in a circle of radius 300 stades at Syene, no shadows were cast at noon.  He then reasoned that a circle of the sun was about 10,000 times the size of a circular section of the earth.  Finally, he roughly concluded that the sun’s diameter must be about 3,000,000 stades.  This figure may not be very accurate, but it was at least an attempt at celestial measurement in the era before telescopes.  I have read that he also calculated the height of the earth’s highest mountain to be about 10 stades (or 1/8000th of the earth’s diameter), but it is not clear how he arrived at this conclusion.


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