Eratosthenes

Eratosthenes of Cyrene (Greek Ἐρατοσθένης; 276 BC - 194 BC) 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 (again with remarkable accuracy); he may also have accurately calculated the distance from the earth to the sun and invented the leap day. [1] 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.

His contemporaries nicknamed him "beta" (the second letter in the Greek alphabet) because he supposedly proved himself to be the second best in many fields.

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 chief librarian of the Great Library of Alexandria and died in the capital of Ptolemaic Egypt. He was 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 [2]. 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 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"^[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 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, 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 meters, 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] Scribal errors in copying the numbers, either of Eusebius' text or of the text that Eusebius read, are probably responsible.

Eratosthenes and the Alexandria lighthouse; his overlarge earth radius and oversmall AU

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 years back by the 1982 Rawlins paper (p.212) which analysed this map, a paper which has been listed for years in Further Readings below. 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, now also listed below. The data would make Eratosthenes's universe the smallest known from the Hellenistic era's height, and would have boosted the Alexandrian stock of geocentricity by making 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 (who designed the lighthouse), and offers a reconstructive speculation that the lighthouse was about 93 meters high which is much below previous suppositions.

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)

See also

• 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. 
• 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". 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. 
• Pa\mias, Jordi, and Klaus Geus (trans., comm.), Eratosthenes. Sternsagen (Catasterismi). Griechisch / Deutsch. Bibliotheca Classicorum, 2. Oberhaid: Utopica, 2007. Pp. 258. EUR 29.95. ISBN 978-3-938083-05-5.

Notes

External links

• 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].