Eratosthenes

Eratosthenes of Cyrene (Greek Ἐρατοσθένης; c. 276 BC^[1] – c. 195 BC^[2]) was a Greek mathematician, poet, athlete, geographer and astronomer. He made several discoveries and inventions including a system of latitude and longitude. He was the first Greek to calculate the circumference of the Earth (with remarkable accuracy), and the tilt of the earth's axis (also with remarkable accuracy); he may also have accurately calculated the distance from the earth to the sun and invented the leap day.^[3] 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.

According to the entry Ἐρατοσθένης in the Suda (ε 2898), his contemporaries nicknamed him Βῆτα (beta, the second letter of the Greek alphabet) because he supposedly proved himself to be the second best in the world in almost any field.^[4]

Life

19th century reconstruction of Eratosthenes's map of the known world, c.194 BC.

Eratosthenes was born in Cyrene (in modern-day Libya). He was the third chief librarian of the Great Library of Alexandria, the center of science and learning in the ancient world, and died in the capital of Ptolemaic Egypt. He was never married.^{[citation needed]}

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 second librarian, Apollonius of Rhodes, in that post.^[5] He made several important contributions to mathematics and science, and was a good friend to Archimedes. Around 255 BC he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him 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).^{[citation needed]}

Aristotle had argued that humanity was divided into Greeks and everyone else, whom he called barbarians, and that the Greeks should keep themselves racially pure. He thought it was fitting for the Greeks to enslave other peoples. But Eratosthenes criticised Aristotle for his blind chauvinism; he believed there was good and bad in every nation. ^[6]

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, and in the modern day as Aswan) 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 miles or 950 km). 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 stadium 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 stadium"^[7] of about 157.5 m, his measurement turns out to be 39,690 km, an error of less than 1%.^[8]

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 slightly north of 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^{[citation needed]}, especially for travel along the non-linear Nile which was traveled primarily by boat. Assuming enough difficulties in measurements, the accuracy of Eratosthenes' size of the earth is surprising.^{[original research?]}

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.^{[citation needed]}

Eratosthenes and the Alexandria lighthouse; his excessively large earth radius and small AU

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The smaller of the foregoing two readings of Eusebius (4080000 stadia) turns out to be exactly 100 times the terrestrial radius (40800 stadia) implicit in Eratosthenes's Nile map and given in the 1982 paper by Rawlins (p. 212) that analysed this map (see Further Reading). Greek scholars such as Archimedes and Posidonius normally expressed the sun's distance in powers of ten times the earth's radius. The Nile map – Eusebius consistency is developed in a 2008 Rawlins paper. The data would make Eratosthenes's universe the smallest known from the Hellenistic era's height, and made the sun smaller than the earth. His indefensible lunar distance would require the moon to go retrograde among the stars every day for observers in tropical or Mediterranean latitudes, and would predict that half moons occur roughly 10° from quadrature.

The Eusebius-confirmed 1982 paper's empirical Eratosthenes circumference (256000 stadia instead of 250000 or 252000 as previously thought) is 19% too high. But the 2008 paper notes that the theory that atmospheric refraction exaggerated his measurement (a theory originally proposed in the 1982 paper, applied to either mountaintop dip or lighthouse visibility) is thus bolstered as the explanation of Eratosthenes's error. This is because accurately measuring the visibility distance of the Alexandria lighthouse (then world's tallest, built at Eratosthenes's location and during his time) and computing the earth's size from that should have given a result 20% high from refraction, very close to his actual 19% error. The 2008 paper wonders if the 40800 stadia estimate originated with Sostratus of Cnidus (who designed the lighthouse), and offers a reconstructive speculation that the lighthouse was about 93 meters high which is much below previous suppositions.

Prime numbers

Eratosthenes also proposed a simple algorithm for finding prime numbers. This algorithm is known in mathematics as the Sieve of Eratosthenes.

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

• Eratosthenes (crater) on the Moon
• Eratosthenian period in the lunar geologic timescale
• Eratosthenes Seamount in the eastern Mediterranean Sea

See also

Atlas portal

• History of geodesy

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.
• J J O'Connor and E F Robertson (January 1999). "Eratosthenes of Cyrene". MacTutor. School of Mathematics and Statistics University of St Andrews Scotland. http://www-history.mcs.st-andrews.ac.uk/Biographies/Eratosthenes.html. 
• E P Wolfer (1954). Eratosthenes von Kyrene als Mathematiker und Philosoph. Groningen-Djakarta. 
• A V Dorofeeva (1988). "Eratosthenes (ca. 276–194 B.C.)" (in Russian). Mat. v Shkole (4): i. 
• J Dutka (1993). "Eratosthenes' measurement of the Earth reconsidered". Arch. Hist. Exact Sci. 46 (1): 55–66. doi:10.1007/BF00387726. 
• B A El'natanov (1983). "A brief outline of the history of the development of the sieve of Eratosthenes" (in Russian). Istor.-Mat. Issled. 27: 238–259. 
• D H Fowler (1983). "Eratosthenes' ratio for the obliquity of the ecliptic". Isis 74 (274): 556–562. doi:10.1086/353361. 
• B R Goldstein (1984). "Eratosthenes on the "measurement" of the earth". Historia Math. 11 (4): 411–416. doi:10.1016/0315-0860(84)90025-9. 
• E Gulbekian (1987). "The origin and value of the stadion unit used by Eratosthenes in the third century B.C". Arch. Hist. Exact Sci. 37 (4): 359–363. 
• G Knaack (1907). "Eratosthenes". Pauly–Wissowa VI: 358–388. 
• F Manna (1986). "The Pentathlos of ancient science, Eratosthenes, first and only one of the "primes"" (in Italian). Atti Accad. Pontaniana (N.S.) 35: 37–44. 
• A Muwaf and A N Philippou (1981). "An Arabic version of Eratosthenes writing on mean proportionals". J. Hist. Arabic Sci. 5 (1–2): 174–147. 
• D Rawlins (1982). "Eratosthenes' geodesy unraveled : was there a high-accuracy Hellenistic astronomy". Isis 73: 259–265. doi:10.1086/352973. 
• D Rawlins (1982). "The Eratosthenes – Strabo Nile map. Is it the earliest surviving instance of spherical cartography? Did it supply the 5000 stades arc for Eratosthenes' experiment?". Arch. Hist. Exact Sci. 26 (3): 211–219. 
• D Rawlins (2008). "Eratothenes's large earth and tiny universe" (PDF). DIO 14: 3–12. http://www.dioi.org/vols/we0.pdf. 
• C M Taisbak (1984). "Eleven eighty-thirds. Ptolemy's reference to Eratosthenes in Almagest I.12". Centaurus 27 (2): 165–167. doi:10.1111/j.1600-0498.1984.tb00766.x. 

Notes

External links

Wikimedia Commons has media related to: Eratosthenes

• Bernhardy, Gottfried: "Eratosthenica" Berlin 1822 Reprinted Osnabruck 1968 (German text)
• Bernhardy, Gottfried: Eratosthenica Berlin, 1822 (PDF) (Latin/Greek)
• O'Connor, John J.; Robertson, Edmund F., "Eratosthenes", MacTutor History of Mathematics archive .
• 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 [3].

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