Eratosthenes

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Eratosthenes
An etching of a man's head and neck in profile, looking to the right. The man has a beard and is balding.
Portrait of Eratosthenes
Born 276 BC
Cyrene
Died 194 BC
Alexandria
Ethnicity Greek
Occupation Scholar, librarian, poet, and inventor

Eratosthenes of Cyrene (Ancient Greek: Ἐρατοσθένης, IPA: [eratostʰénɛːs]; English /ɛrəˈtɒsθənz/; c. 276 BC[1] – c. 195/194 BC[2]) was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today.[3]

He is best known for being the first person to calculate the circumference of the Earth, which he did by applying a measuring system using stades, or the length of stadiums during that time period. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable accuracy). Additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day.[4] He created the first map of the world incorporating parallels and meridians, based on the available geographical knowledge of the era.

Eratosthenes was the founder of scientific chronology; he endeavored to revise the dates of the chief literary and political events from the conquest of Troy. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers.

He was a figure of influence who declined to specialize in only one field. According to an entry[5] in the Suda (a 10th-century reference), his critics scorned him, calling him Beta, from the second letter of the Greek alphabet, because he always came in second in all his endeavors.[6] Nonetheless, his devotees nicknamed him Pentathlos, after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world.[7]

Seventeen hundred years after Eratosthenes' death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe that the Earth's circumference was much smaller. Had Columbus set sail knowing that Eratosthenes' larger circumference value was more accurate, he would have known that the place where he made landfall was not Asia, but rather a New World.[8]

Life[edit]

Eratosthenes was born in Cyrene (in modern-day Libya). The son of Aglaos, Eratosthenes was born in 276 BC, in a city ruled by the Ptolemaic dynasty. The city was colonized in the years after Alexander the Great's conquest. It became the capital of Pentapolis (North Africa), a country of five cities: Cyrene, Arsinde, Berenice, Ptolemias, and Apollonia, Cyrenaica. The rule of Cyrene was given to one of Alexander the Great's generals, Ptolemy I Soter. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, a plant used for rich seasoning and medicine.[3] Cyrene became a place of cultivation, where knowledge blossomed. Like any young Greek, Eratosthenes would have studied in the local gymnasium, where he would have learned physical skills and social discourse as well as reading, writing, arithmetic, poetry, and music.[9]

Eratosthenes went to Athens to further his studies. There he was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life.[10] He then studied under Ariston of Chios, who led a more cynical school of philosophy. He also studied under the head of the Platonic Academy, who was Arcesilaus of Pitane. His interest in Plato led him to write his very first work at a scholarly level, Platonikos, inquiring into the mathematical foundation of Plato's philosophies.[11] Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus.[12] He had talent as a most imaginative poet. He wrote poems: one in hexameters called Hermes illustrating the god's life history; and another, in elegiacs, called Erigone, describing the suicide of the Athenian maiden Erigone (daughter of Icarius).[13] He wrote Chronographies, a text that scientifically depicted dates of importance, beginning with the Trojan War. This work was highly esteemed for its accuracy: George Syncellus was later able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes. Eratosthenes also wrote Olympic Victors, a chronology of the winners of the Olympic Games. It is not known when he wrote his works, but they highlighted his abilities.

These works and his great poetic abilities led the pharaoh Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC. Eratosthenes, then thirty years old, accepted Ptolemy's invitation and traveled to Alexandria, where he lived for the rest of his life. Within about five years he became Chief Librarian, a position that the poet Apollonius Rhodius had previously held. As head of the library Eratosthenes tutored the children of Ptolemy, including Ptolemy IV Philopator who became the fourth Ptolemaic pharaoh. He expanded the library's holdings: in Alexandria all books had to be surrendered for duplication. It was said that these were copied so accurately that it was impossible to tell if the library had returned the copy or the original. He sought to maintain the reputation of the Library of Alexandria against competition from the Pergamum. Eratosthenes created a whole section devoted to the examination of Homer, and acquired original works of great tragic dramas of Aeschylus, Sophocles and Euripides[14]

Eratosthenes made several important contributions to mathematics and science, and was a friend of 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 on Elephantine Island near Syene (now Aswan, Egypt).

Eratosthenes believed there was good and bad in every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, and that the Greeks should keep themselves racially pure.[15] As he aged he contracted ophthalmia, becoming blind around 195 BC. Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death. He died in 194 BC at the age of 82 in his beloved Alexandria.[16]

Measurement of the Earth's circumference[edit]

Illustration showing a portion of the globe showing a part of the African continent. The sunbeams shown as two rays hitting the ground at Syene and Alexandria. Angle of sunbeam and the gnomons (vertical sticks) is shown at Alexandria which allowed Eratosthenes' estimates of radius and circumference of Earth.

Eratosthenes calculated the circumference of the Earth without leaving Egypt. Eratosthenes knew that at local noon on the summer solstice in the Ancient Egyptian city of Swenet (known in ancient Greek as Syene, and now as Aswan) on the Tropic of Cancer, the Sun would appear at the zenith, directly overhead. He knew this because he had been told that the shadow of someone looking down a deep well in Syene would block the reflection of the Sun at noon off the water at the bottom of the well. Using a gnomon, he measured the Sun's angle of elevation at noon on the solstice in Alexandria, and found it to be 1/50th of a circle (7°12') south of the zenith. He may have used a compass to measure the angle of the shadow cast by the Sun.[17] Assuming that the Earth was spherical (360°), and that Alexandria was due north of Syene, he concluded that the meridian arc distance from Alexandria to Syene must therefore be 1/50th of a circle's circumference, or 7°12'/360°.

His knowledge of the size of Egypt after many generations of surveying trips for the Pharaonic bookkeepers gave a distance between the cities of 5,000 stadia. This distance was corroborated by inquiring about the time that it took to travel from Syene to Alexandria by camel. He rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. Some claim Erathostenes used the Egyptian stade of 157.5 meters, which would imply a circumference of 39,690 km, an error of 1.6%, but the 185 meter Attic stade is the most commonly accepted value for the length of the stade used by Eratosthenes in his measurements of the Earth, [18] which imply a circumference of 46,620 km, an error of 16.3%. It is unlikely, however, that Eratosthenes got an accurate measurement of the circumference of the Earth, given three errors in the assumptions he made:[17]

  1. That Alexandria and Syene lie on the same meridian.
  2. That the distance between Alexandria and Syene is 5000 stades.
  3. That the Earth is a sphere.

If we repeat Eratosthenes' calculation with more accurate data, the result is 40,074 km, which is 66 km different (0.16%) from the currently accepted circumference of the Earth.[17]

"Father of Geography"[edit]

Eratosthenes' map of the world (194 B.C.)
19th-century reconstruction of Eratosthenes' map of the known world, c. 194 BC

See History of geodesy

Eratosthenes continued from his knowledge about the Earth, his discoveries of its size and shape, and began to sketch it. He wanted a way to view his world simplistically, but to a higher degree.[clarification needed] In the Library of Alexandria he had access to various travel books, which contained various items of information and representations of the world that needed to be pieced together in some organized format.[19] In his three-volume work Geographika, he described and mapped his entire known world, even dividing the Earth into five climate zones:[20] two freezing zones around the pole, two temperate zones, and a zone encompassing the equator and the tropics.[21] He had invented geography. He created terminology that is still used today. He placed grids of overlapping lines over the surface of the Earth. He used parallels and meridians to link together every place in the world. It was now possible to estimate one's distance from remote locations with this network over the surface of the Earth. In Geographika the names of over 400 cities and their locations were shown: this had never been achieved before.[3] Unfortunately Geographika has been lost to history, but fragments of the work can be pieced together from other great historians like Pliny, Polybius, Strabo, and Marcianus.

  • The first book was something of an introduction and gave a review of his predecessors, recognizing their contributions that he compiled in the library. In this book Eratosthenes denounced Homer as not providing any insight into what he now described as geography. His disapproval of Homer's topography angered many who believed the world depicted in the Odyssey to be legitimate.[7][22] He also commented on the ideas of the nature and origin of the Earth: he had thought of Earth as an immovable globe; while on its surface was a place that was changing. He had hypothesized that at one time the Mediterranean was a vast lake that covered the countries that surrounded it; and had only become connected to the ocean to the west when a passage had opened up sometime in its history.
  • In the second book is his discovery about the circumference of the Earth. This is where, according to Pliny, "The world was grasped." Eratosthenes described his famous story of the well in Syene, described above. This book would now be considered a text on mathematical geography.
  • His third book of Geographika contained political geography. He cited countries and used parallel lines to divide the map into sections, to give accurate descriptions of the realms. This was a breakthrough, and can be considered the beginning of geography.[23]

Other achievements[edit]

Eratosthenes was described by the Suda Lexicon as a Πένταθλος (Pentathlos) which can be translated as "All-Rounder", for he was skilled in a variety of things: He was a true polymath. He was nicknamed Beta, because he was great at many things and tried to get his hands on every bit of information, but never achieved the highest rank in anything, so much so that Strabo accounts Eratosthenes as a mathematician among geographers, and a geographer among mathematicians.[24]

  • 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 καὶ ὀκτωκισμυρίας" (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. With a stade of 185 meters, 804,000,000 stadia is 149,000,000 kilometers, approximately the distance from the Earth to the Sun.
  • Eratosthenes also calculated the Sun's diameter. According to Macrobious, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth.[23]
  • During his time at the Library of Alexandria, Eratosthenes devised a calendar using his predictions about the ecliptic of the Earth. He calculated that there are 365 days in a year and that every fourth year there would be 366 days.[25]
  • He was also very proud of his solution for Doubling the Cube. His motivation was that he wanted to produce catapults. Eratosthenes constructed a mechanical line drawing device to calculate the cube, called the mesolabio. He dedicated his solution to King Ptolemy, presenting a model in bronze with it a letter and an epigram.[26] Archimedes was Eratosthenes' friend and he, too, worked on the war instrument with mathematics. Archimedes dedicated his book The Method to Eratosthenes, knowing his love for learning and mathematics.[27]

Prime numbers[edit]

Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square).

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

In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite, i.e., not prime, the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.

Works[edit]

Eratosthenes was one of the most pre-eminent scholarly figures of his time, and produced works covering a vast area of knowledge before and during his time at the Library. He wrote on many topics — geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry, and even old comedies. Unfortunately, there are only fragments left of his works after the Destruction of the Library of Alexandria.[28]

The Burning of the Library at Alexandria in 391 AD, an illustration from "Hutchinsons History of the Nations", c. 1910

Titles[edit]

  • Platonikos
  • Hermes
  • Erigone
  • Chronographies
  • Olympic Victors
  • Περὶ τῆς ἀναμετρήσεως τῆς γῆς (On the Measurement of the Earth)[29] (lost, summarized by Cleomedes)
  • Гεωγραϕικά (Geographika) [30] (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.

Things named after Eratosthenes[edit]

Eratosthenes
Eratosthenes crater 4114 h2.jpg
Coordinates 14°30′N 11°18′W / 14.5°N 11.3°W / 14.5; -11.3Coordinates: 14°30′N 11°18′W / 14.5°N 11.3°W / 14.5; -11.3
Diameter 58 km
Depth 3.6 km
Colongitude 12° at sunrise
Eponym Eratosthenes

References[edit]

  1. ^ The Suda states that he was born in the 126th Olympiad, (276–272 BC). Strabo (Geography, i.2.2), though, states that he was a "pupil" (γνωριμος) of Zeno of Citium (who died 262 BC), which would imply an earlier year-of-birth (c. 285 BC) since he is unlikely to have studied under him at the young age of 14. However, γνωριμος can also mean "acquaintance", and the year of Zeno's death is by no means definite. Cf. Eratosthenes entry in the Dictionary of Scientific Biography (1971)
  2. ^ The Suda states he died at the age of 80, Censorinus (De die natali, 15) at the age of 81, and Pseudo-Lucian (Makrobioi, 27) at the age of 82.
  3. ^ a b c Roller, Duane W. Eratosthenes' Geography. New Jersey: Princeton University Press, 2010.
  4. ^ Alfred, Randy (June 19, 2008). "June 19, 240 B.C.E: The Earth Is Round, and It's This Big". Wired. Retrieved 2013-06-22. 
  5. ^ Entry ε 2898
  6. ^ See also Asimov, Isaac. Asimov's Biographical Encyclopedia of Science and Technology, new revised edition. 1975. Entry #42, "Eratosthenes", Page 29. Pan Books Ltd, London. ISBN 0-330-24323-3. It was also asserted by Carl Sagan, 31 minutes into his Cosmos episode The Shores of the Cosmic Ocean
  7. ^ a b Chambers, James T. "Eratosthenes of Cyrene." Dictionary Of World Biography: The Ancient World (January 1998): 1–3.
  8. ^ Gow, Mary. "Measuring the Earth: Eratosthenes and His Celestial Geometry, p. 6 (Berkeley Heights, NJ: Enslow, 2010).
  9. ^ Bailey, Ellen. 2006. "Eratosthenes of Cyrene." Eratosthenes Of Cyrene 1–3. Book Collection Nonfiction: High School Edition.
  10. ^ Rist, J.M. "Zeno and Stoic Consistency," in Phronesis. Vol. 22, No. 2, 1977.
  11. ^ Chambers, James T. "Eratosthenes of Cyrene." in Dictionary Of World Biography: The Ancient World January 1998: 1–3.
  12. ^ Bailey, Ellen. "Eratosthenes of Cyrene," in Eratosthenes of Cyrene 1–3. Book Collection Nonfiction: High School Edition, 2006.
  13. ^ Chambers, James T. "Eratosthenes of Cyrene" in Dictionary Of World Biography: The Ancient World (January 1998): 1–3.
  14. ^ Chambers, James T. "Eratosthenes of Cyrene" Dictionary Of World Biography: The Ancient World, January 1998.
  15. ^ * p439 Vol. 1 William Woodthorpe Tarn Alexander the Great. Vol. I, Narrative; Vol. II, Sources and Studies0. Cambridge: Cambridge University Press, 1948. (New ed., 2002 (paperback, ISBN 0-521-53137-3)).
  16. ^ Bailey, Ellen. "Eratosthenes of Cyrene." Eratosthenes Of Cyrene (January 2006): 1–3.
  17. ^ a b c How did Eratosthenes measure the circumference of the Earth?
  18. ^ Eratosthenes and the Mystery of the Stades – How Long Is a Stade?
  19. ^ Smith, Sir William. "Eratosthenes", in A Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
  20. ^ Morris, Terry R. "Eratosthenes of Cyrene." in Encyclopedia Of The Ancient World. November 2001.
  21. ^ 2011. "Eratosthenes." Hutchinson's Biography Database 1.
  22. ^ Eckerman, Chris. Review of (D.W.) Roller 'Eratosthenes' Geography. Fragments Collected and Translated, with Commentary and Additional Material. The Classical Review. 2011.
  23. ^ a b Smith, Sir William. "Eratosthenes," in A Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
  24. ^ Dicks, D.R.. "Eratosthenes," in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
  25. ^ Greek Scholar's Work Shows Usefulness of Measurement." Manawatu Standard, June 19, 2012., 07, Newspaper Source Plus
  26. ^ Zhumud, Leonid. Plato as "Architect of Science". in Phonesis. Vol. 43 (3) 1998. 211–244.
  27. ^ Chondros, Thomas G. Archimedes Life Works and Machines. in Mechanism and Machine Theory. Vol.45(11) 2010. 1766–1775.
  28. ^ Dicks, D.R.. "Eratosthenes", in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
  29. ^ Mentioned by Hero of Alexandria in his Dioptra. See p. 272, vol. 2, Selections Illustrating the History of Greek Mathematics, tr. Ivor Thomas, London: William Heinemann Ltd.; Cambridge, MA: Harvard University Press, 1957.
  30. ^ Dicks, D.R.. "Eratosthenes," in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.

Further reading[edit]

  • Aujac, G. (2001). Eratosthène de Cyrène, le pionier de la geographie. Paris: Édition du CTHS. 224p.
  • Bulmer-Thomas, Ivor (1939–1940). Selections Illustlating the History of Greek Mathematics. Cambridge, MA: Harvard University Press. 
  • Cameron McPhail (2011). Reconstructing Eratosthenes' Map of the World: a Study in Source Analysis. A Thesis Submitted for the Degree of Master of Arts at the University of Otago. Dunedin, New Zealand.
  • Diller, A. (1934). "Geographical Latitudes in Eratosthenes, Hipparchus and Posidonius". Klio 27(3): 258–269.
  • Dorofeeva, A. V. (1988). "Eratosthenes (ca. 276–194 B.C.)". Mat. V Shkole (in Russian) (4): i. 
  • Dutka, J. (1993). "Eratosthenes' measurement of the Earth reconsidered". Arch. Hist. Exact Sci. 46 (1): 55–66. doi:10.1007/BF00387726. 
  • El'natanov, B. A. (1983). "A brief outline of the history of the development of the sieve of Eratosthenes". Istor.-Mat. Issled. (in Russian) 27: 238–259. 
  • Fischer, I. (1975). "Another look at Eratosthenes' and Posidonius' determinations of the Earth's circumference". Quarterly Journal of the Royal Astronomical Society 16: 152—167.
  • Fowler, D. H.; Rawlins, Dennis (1983). "Eratosthenes' ratio for the obliquity of the ecliptic". Isis 74 (274): 556–562. doi:10.1086/353361. 
  • Fraser, P. M. (1970). "Eratosthenes of Cyrene". Proceedings of the British Academy 56: 175–207. 
  • Fraser, P. M. (1972). Ptolemaic Alexandria. Oxford: Clarendon Press. 
  • Fuentes González, P. P., "Ératosthène de Cyrène", in R. Goulet (ed.), Dictionnaire des Philosophes Antiques, vol. III, Paris, Centre National de la Recherche Scientifique, 2000, pp. 188–236.
  • Geus K. (2002). Eratosthenes von Kyrene. Studien zur hellenistischen Kultur- und Wissenschaftgeschichte. München: Verlag C.H. Beck. (Münchener Beiträge zur Papyrusforschung und antiken Rechtsgeschichte. Bd. 92) X, 412 S.
  • Goldstein, B. R. (1984). "Eratosthenes on the "measurement" of the Earth". Historia Math. 11 (4): 411–416. doi:10.1016/0315-0860(84)90025-9. 
  • Gulbekian, E. (1987). "The origin and value of the stadion unit used by Eratosthenes in the third century B.C". Archive for History of Exact Sciences 37 (4): 359–363. doi:10.1007/BF00417008. 
  • Honigmann, E. (1929). Die sieben Klimata und die πολεις επισημοι. Eine Untersuchung zur Geschichte der Geographie und Astrologie in Altertum und Mittelalter. Heidelberg: Carl Winter's Universitätsbuchhandlung. 247 S.
  • Knaack, G. (1907). "Eratosthenes". Pauly–Wissowa VI: 358–388. 
  • Manna, F. (1986). "The Pentathlos of ancient science, Eratosthenes, first and only one of the "primes"". Atti Accad. Pontaniana (N.S.) (in Italian) 35: 37–44. 
  • Muwaf, A.; Philippou, A. N. (1981). "An Arabic version of Eratosthenes writing on mean proportionals". J. Hist. Arabic Sci. 5 (1–2): 174–147. 
  • Nicastro, Nicholas (2008). Circumference: Eratosthenes and the ancient quest to measure the globe. New York: St. Martin's Press. ISBN 0-312-37247-7. 
  • O'Connor, John J.; Robertson, Edmund F., "Eratosthenes", MacTutor History of Mathematics archive, University of St Andrews .
  • Marcotte, D. (1998). "La climatologie d'Ératosthène à Poséidonios: genèse d'une science humaine". G. Argoud, J.Y. Guillaumin (eds.). Sciences exactes et sciences appliquées à Alexandrie (IIIe siècle av J.C. – Ier ap J.C.). Saint Etienne: Publications de l'Université de Saint Etienne: 263–277.
  • Pfeiffer, Rudolf (1968). History of Classical Scholarship From the Beginnings to the End of the Hellenistic Age. Oxford: Clarendon Press. 
  • Rawlins, D. (1982). "Eratosthenes' geodesy unraveled : was there a high-accuracy Hellenistic astronomy". Isis 73 (2): 259–265. doi:10.1086/352973. 
  • Rawlins, D. (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. 
  • Rawlins, D. (2008). "Eratosthenes's large Earth and tiny universe" (PDF). DIO 14: 3–12. 
  • Roller, Duane W. (2010). Eratosthenes' Geography: Fragments collected and translated, with commentary and additional material. Princeton: Princeton University Press. ISBN 978-0-691-14267-8. 
  • Shcheglov, D.A. (2004/2006). "Ptolemy's System of Seven Climata and Eratosthenes' Geography". Geographia Antiqua 13: 21–37.
  • Shcheglov, D.A. (2006). "Eratosthenes' Parallel of Rhodes and the History of the System of Climata". Klio 88: 351–359.
  • Strabo (1917). The Geography of Strabo. Horace Leonard Jones, trans. New York: Putnam. 
  • Taisbak, C. M. (1984). "Eleven eighty-thirds. Ptolemy's reference to Eratosthenes in Almagest I.12". Centaurus 27 (2): 165–167. Bibcode:1984Cent...27..165T. doi:10.1111/j.1600-0498.1984.tb00766.x. 
  • Thalamas, A. (1921). La géographe d'Ératosthène. Versailles.
  • Wolfer, E. P. (1954). Eratosthenes von Kyrene als Mathematiker und Philosoph. Groningen-Djakarta. 

External links[edit]

Preceded by
Apollonius of Rhodes
Head of the Library of Alexandria Succeeded by
Aristophanes of Byzantium