Eratosthenes

Eratosthenes (Greek Ἐρατοσθένης) (276 BC - 194 BC) was a Hellenistic mathematician, geographer and astronomer. His contemporaries nicknamed him beta β because he supposedly proved himself to be the second in the mediterranean world in many fields. He is noted for devising a system of latitude and longitude, and for being the first known to have calculated the circumference of the Earth.

Life

Eratosthenes was born in Cyrene (in modern-day Libya), but worked and died in Alexandria capital of Ptolemaic Egypt. He never married. He was reputedly known for his haughty character. Eratosthenes studied at Alexandria and for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library, succeeding the first librarian, Zenodotos, in that post. He made several important contributions to mathematics and science, and was a good friend to Archimedes. Around 255 BC he invented the armillary sphere, which was widely used until the invention of the orrery in the 18th century. In 195 BC he became blind and a year later he supposedly starved himself to death.

He is credited by Cleomedes in On the Circular Motions of the Celestial Bodies with having calculated the Earth's circumference around 240 BC, using trigonometry and knowledge of the angle of elevation of the Sun at noon in Alexandria and Syene (now Aswan, Egypt).

Measurement of the Earth

Eratosthenes knew that on the summer solstice at local noon in the town of Syene on the Tropic of Cancer, the sun would appear at the zenith, directly overhead. He also knew, from measurement, that in his hometown of Alexandria, the angle of elevation of the Sun would be 7.2° south of the zenith at the same time. Assuming that Alexandria was due north of Syene he concluded that the distance from Alexandria to Syene must be 7.2/360 of the total circumference of the Earth. The distance between the cities was known from caravan travellings to be about 5000 stadia: approximately 800 km. He established a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion he used is no longer known (the common Attic stadion was about 185 m), but it is generally believed that the circumference calculated by Eratosthenes corresponds to 39,690 km ^{[citation needed]}.

Although Eratosthenes' method was well founded, the accuracy of his calculation was inherently limited. The accuracy of Eratosthenes' measurement would have been reduced by the fact that Syene is not precisely on the Tropic of Cancer, is not directly south of Alexandria, and the Sun appears as a disk located at a finite distance from the Earth instead of as a point source of light at an infinite distance. There are other sources of experimental error: the greatest limitation to Eratosthenes' method was that, in antiquity, angles could only be measured to within about a quarter of a degree, and overland distance measurements were even less reliable. So the accuracy of the result of Eratosthenes' calculation is surprising: the circumference of the Earth around the poles is now measured at 40,008 km.

Eratosthenes' experiment was highly regarded at the time, and his estimate of the Earth’s size was accepted for hundreds of years afterwards. His method was used by Posidonius about 150 years later.

Other work

Eratosthenes' other contributions include:

The Sieve of Eratosthenes as a way of finding prime numbers.
Possibly, the measurement of the Sun-Earth distance, now called the astronomical unit and of the distance to the Moon (see below).
The measurement of the inclination of the ecliptic with an angle error 7'.
He compiled a star catalogue containing 675 stars, which was not preserved.
A map of the Nile's route as far as Khartoum.
A map of the entire known world, from the British Isles to Ceylon, and from the Caspian Sea to Ethiopia. Only Hipparchus, Strabo, and Ptolemy were able to make more accurate maps in the classical and post-classical world.
A number of works on theatre and ethics
A calendar with leap years, in which he attempted to work out the precise dates and relations of various events in politics and literature from his day back to the Trojan War.

The fragmentary collection of Hellenistic sky-myths called Catasterismi (Katasterismoi) was attributed to Eratosthenes; it is believed that it was so attributed to add to its credibility.

The mysterious astronomical distances

Eusebius of Caesarea in his Praeparatio Evangelica includes a brief chapter of three sentences on celestial distances (Book XV, Chapter 53). He states simply that Eratosthenes found the distance to the sun to be "σταδίων μυριάδας τετρακοσίας και οκτωκισμυρίας" (literally "of stadia myriads 400 and 80000") and the distance to the moon to be 780,000 stadia. The expression for the distance to the sun has been translated either as 4,080,000 stadia (1903 translation by E. H. Gifford), or as 804,000,000 stadia (edition of Edouard des Places, dated 1974-1991). The meaning depends on whether Eusebius meant 400 myriad plus 80000 or "400 and 80000" myriad.

This testimony of Eusebius is dismissed by the scholarly Dictionary of Scientific Biography. It is true that the distance Eusebius quotes for the moon is much too low (about 144,000 km) and Eratosthenes should have been able to do much better than this since he knew the size of the earth and Aristarchos of Samos had already found the ratio of the moon's distance to the size of the earth. But if what Eusebius wrote was pure fiction, then it is difficult to explain the fact that, using the Greek stadium of 185 metres, the figure of 804 million stadia that he quotes for the distance to the sun comes to 149 million kilometres. The difference between this and the modern accepted value is less than 1%.

Named after Eratosthenes

Sieve of Eratosthenes
Eratosthenes crater on the Moon
Eratosthenian period in the lunar geologic timescale
Eratosthenes Seamount in the eastern Mediterranean Ocean

External links

Bernhardy, Gottfried: "Eratosthenica" Berlin 1822 Reprinted Osnabruck 1968 (German text)
O'Connor, John J.; Robertson, Edmund F., "Eratosthenes", MacTutor History of Mathematics Archive, University of St Andrews
Eratosthenes' sieve in Javascript
Eratosthenes' sieve as a simple algorithm
About Eratosthenes' methods, including a Java applet
How to measure the earth with Eratosthenes' method
How the Greeks estimated the distances to the moon and sun
Eratosthenes on PBS.org
Inter-collegiate project for measuring the earth with Eratosthenes' method
Measuring the earth with Eratosthenes' method
List of ancient Greek mathematecians and contemporaries of Eratosthenes
New Advent Encyclopedia article on the Library of Alexandria
Eratosthenes' sieve explored and visualised in Flash

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