# Eratosthenes

**Eratosthenes of Cyrene** (Greek Ερατοσθένης 276 BC - 194 BC) was a Greek mathematician, poet, athlete, geographer and astronomer. His contemporaries nicknamed him "beta" (Greek for "number two") because he supposedly proved himself to be the second in the ancient Mediterranean region in many fields. He is noted for devising a system of latitude and longitude, and for being the first known person to have calculated the circumference of the Earth. He also created a map of the world based on the available geographical knowledge of the era. Eratosthenes was also the founder of scientific chronology; he endeavored to fix the dates of the chief literary and political events from the conquest of Troy.

## Contents

## Life

Eratosthenes was born in Cyrene (in modern-day Libya). He was the chief librarian of the Great Library of Alexandria and died in the capital of Ptolemaic Egypt. He never married.

Eratosthenes studied in Alexandria and claimed to have also studied 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, Apollonius of Rhodes, in that post[1]. 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 194 BC Eratosthenes became blind and, according to legends, a year later, he 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 knowledge of the angle of elevation of the Sun at noon on the summer solstice in Alexandria and in the Elephantine Island near Syene (now Aswan, Egypt).

## Eratosthenes' measurement of the Earth's circumference

Eratosthenes knew that on the summer solstice at local noon in the Ancient Egyptian city of Swenet (known in Greek as 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 1/50 of a full circle (7°12') 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 1/50 of the total circumference of the Earth. His estimated distance between the cities was 5000 stadia (about 500 geographical or nautical miles). He rounded the result to 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 frequently argued. The common Attic stadion was about 185 m, which would imply a circumference of 46,620 km, i.e. 16.3% too large. However, if we assume that Eratosthenes used the "Egyptian stadion"^{[1]} of about 157.5 m, his measurement turns out to be 39,690 km, an error of less than 1%.^{[2]}

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, overland distance measurements were not reliable, especially for travel along the non-linear Nile which was traveled primarily by boat. So the accuracy of Eratosthenes' size of the earth is surprising.

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.

## The mysterious astronomical distances

Eusebius of Caesarea in his *Preparatio 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 80,000") 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 80,000 or "400 and 80,000" 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 Aristarchus 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, or Olympic, 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%.^{[3]}

## Works

*On the Measurement of the Earth*(lost, summarized by Cleomedes)*Geographica*(lost, criticized by Strabo)*Arsinoe*(a memoir of queen Arsinoe; lost; quoted by Athenaeus in the*Deipnosophistae*)- A fragmentary collection of Hellenistic myths about the constellations, called
*Catasterismi*(*Katasterismoi*), was attributed to Eratosthenes, perhaps to add to its credibility.

## Named after Eratosthenes

- Sieve of Eratosthenes
- Eratosthenes crater on the Moon
- Eratosthenian period in the lunar geologic timescale
- Eratosthenes Seamount in the eastern Mediterranean Sea
- Jules Eratosthenes Brown (fictional character from the
*Back to the Future*franchise)

## Further reading

- Kathryn Lasky.
*The Librarian Who Measured the Earth*. New York: Little, Brown and Company, 1994. ISBN 0-316-51526-4. An illustrated biography for children focusing on the measurement of the earth. Kevin Hawkes, illustrator.

## External links

- Bernhardy, Gottfried: "Eratosthenica" Berlin 1822 Reprinted Osnabruck 1968 (German text)
- 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
- Eratosthenes' sieve in classic BASIC all-web based interactive programming environment
- Following in the footsteps of Eratosthenes : project [2].

A portion of content for this article is credited to Wikipedia. *Content under GNU Free Documentation License(GFDL)*

- ↑ traianus.rediris.es/topo01/surveying.pdf
- ↑ There is a huge Eratosthenes-got-it-right literature based upon attacking the applicability of the standard 185m stadium to his experiment. Among advocates: F. Hultsch,
*Griechische und Römische Metrologie*, Berlin, 1882; E. Lehmann-Haupt, Stadion entry in*Paulys Real-Encyclopädie*, Stuttgart, 1929; I. Fischer,*Q. Jl. R. astr. Soc. 16.2*:152-167, 1975; Gulbekian (1987); Dutka (1993). The means employed include worrying various ratios of the stadium to the unstably defined "schoenus", or using a truncated passage from Pliny. (Gulbekian just computes the stadium from Eratosthenes's experiment instead of the reverse.) A disproportionality of literature exists because professional scholars of ancient science have generally regarded such speculation as special pleading and so have not bothered to write extensively on the issue. Skeptical works include E. Bunbury's classic*History of Ancient Geography*, 1883; D. Dicks,*Geographical Fragments of Hipparchus*, University of London, 1960; O. Neugebauer,*History of Ancient Mathematical Astronomy*, Springer, 1975; J. Berggren and A. Jones,*Ptolemy's Geography*, Princeton, 2000. Some difficulties with the several arguments for Eratosthenes's exact correctness are discussed by Rawlins in 1982b page 218 and in his Contributions and Distillate. - ↑ Other than the distance to the moon, no celestial distance is unambiguously established as known in antiquity even to within a factor of two. As late as a century ago, the earth's distance to the sun (the A. U.) was known less accurately than 1%.