He is best remembered as the first known person to calculate theEarth's circumference, which he did by using the extensive survey results he could access in his role at the Library. His calculation was remarkably accurate (his error margin turned out to be less than 1%).[2][3] He was the first to calculateEarth's axial tilt, which similarly proved to have remarkable accuracy.[4][5] He created thefirst global projection of the world incorporatingparallels andmeridians based on the available geographic knowledge of his era.[4] Eratosthenes was the founder of scientificchronology;[6] he used Egyptian and Persian records to estimate the dates of the main events of theTrojan War, dating the sack ofTroy to 1184 BC. Innumber theory, he introduced thesieve of Eratosthenes, an efficient method of identifyingprime numbers and composite numbers.
He was a figure of influence in many fields who tried to understand the complexities of the known world.[7] His devotees nicknamed himPentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Yet, according to an entry[8] in theSuda (a 10th-century encyclopedia), some critics scorned him, calling himNumber 2 because he always came in second in all his endeavours.[9]
The son of Aglaos, Eratosthenes was born in276 BC inCyrene. Now part of modern-dayLibya, Cyrene had been founded by Greeks centuries earlier and became the capital ofPentapolis (North Africa), a country of five cities: Cyrene,Arsinoe,Berenice,Ptolemais, andApollonia.Alexander the Great conquered Cyrene in332 BC, and following his death in323 BC, its rule was given to one of his generals,Ptolemy I Soter, the founder of thePtolemaic Kingdom. Under Ptolemaic rule the economy prospered, based largely on the export of horses andsilphium, a plant used for rich seasoning and medicine. Cyrene became a place of cultivation, where knowledge blossomed. According to Roller, the rarity of both Eratosthenes' and his father's names are indicative of his humble origins, though due to the possibilities of upward mobility in theHellenistic world he was not limited by them.[10] However, Matthew suggests that his name, meaning "lovely strength" suggests noble upbringing,[11] as does his education from a young age, which could imply his belonging to the aristocracy of Cyrene.[12] Like any young Greek at the time, Eratosthenes would have studied in the localgymnasium, where he would have learned physical skills and social discourse as well as reading, writing, arithmetic, poetry, and music.[citation needed]
By the late260s BCE, Eratosthenes went toAthens to further his studies.[10] According toStrabo, he was taughtStoicism there by the school's founder,Zeno of Citium. Zeno taught philosophical lectures on living a virtuous life, though their interaction would have been minimal since Zeno died shortly after Eratosthenes arrived.[13] Strabo lists the little-known Apelles ofChios among his teachers.[13] Eratosthenes later studied under theperipateticAristo of Chios,[14] who led acynical school of philosophy, and the eclectic-viewedBion of Borysthenes.[13] He studied under the recently appointed head of thePlatonic Academy,Arcesilaus of Pitane.[13] Eratosthenes' later mathematical work implies that he received mathematical training there.[13] According to theSuda, Eratosthenes studied under Lysanias of Cyrene, a philologist and grammarian who focused onHomer. Poet, scholar, and librarianCallimachus likely crossed paths with Eratosthenes in local debates and scholarly discourse,[13] even though he was likely never his formal teacher.[15]
Strabo criticized Eratosthenes for his association with such varied schools of thought, believing that he was unwilling to commit to philosophy and had learned only enough to appear as a philosopher, while seeing it as nothing more than a distraction from his regular work.[16][13] Later authors may have shared this view to some extent: Eratosthenes was referred to asBeta (Second), as he was not seen as the leading expert in any given field[13] (though this may have been because he was the second chief librarian in Alexandria).[17] Others dubbed himPentathlos (Πένταθλος - All-Rounded), given his various skills and areas of knowledge;[17]Pentathlos is the title of an athlete who competes in many events but comes in second in all of them.[18]Strabo described Eratosthenes as a mathematician among geographers and a geographer among mathematicians,[19] and complained that he did not pay enough respect to Zeno. This comment by Strabo reflects Eratosthenes' independence in thought and practice.[13]
The majority of Eratosthenes' studies focused on philosophy; mathematics was less prominent, and philology even less so.[13] Despite his later contributions to the field, evidence for his study of geography is completely absent, though this is not surprising as such a discipline did not exist in Athens at the time.[13] Eratosthenes was exposed to extensive geographic literature, such as the works of Homer, who was considered the first geographer in his eyes,Hecataeus of Miletus (Circuit of the Earth),Aeschylus,Herodotus and others.[13] Eratosthenes was born forty years after the death ofAlexander the Great, whose travel companions,Androsthenes,Nearchos,Onesikratos,Ptolemy I and others, wrote about their journeys with him, and whose conquests cleared the path for Hellenistic explorers.[20]
Eratosthenes remained in Athens for twenty years, studying and writing.[20] During this period he wrotePlatonikos, inquiring into the mathematical foundation of Plato's philosophy, as well as the poetic works ofHermes andErigone. HisChronographies focused on the important dates of theTrojan War, and hisOlympic Victors compiled a list of the winners of the Olympic games.[21] Little more is known about this period of his life.[20]
In246 BCE,Ptolemy III succeeded his father,Ptolemy II. Over the next twenty-five years, thePtolemaic empire reached its greatest extent andAlexandria attained its zenith as an intellectual center.[22] The post oflibrarian, which included the position of royal tutor toPtolemy IV Philopator,[18] became the most prestigious academic appointment.[22] The reigning librarian,Apollonius of Rhodes, was forced into retirement by the new king (possibly through the influence of Callimachus), and Eratosthenes, who by this time was gaining fame as a scholar and a poet in the tradition of Callimachus, was summoned from Athens to replace him.[23] Roller suggests that Eratosthenes' roots in Cyrene, the native city of Callimachus, and more importantly QueenBerenike, contributed favorably to his appointment.[23]
The beginning of Eratosthenes' career in Alexandria was focused on mathematics. He was closely affiliated withArchimedes, who sent him material for comment and praised him enthusiastically for his contributions;[23] hisMethod of Mechanical Theorems was written as a letter to Eratosthenes.[24] Eratosthenes subsequently wrote compositions ongeography,philosophy,rhetoric,literary criticism,grammar,poetry andastronomy,[25][17] though some suggest that his astronomical contributions were hardly notable.[26] It was said that his poetry strangely contained the very didactic elements which he condemned.[27] Toward the end of his days, he served as an advisor and companion toArsinoe, sister and wife of Ptolemy IV.[28]
According to theSuda, as he aged his eyesight began to fail.[29] Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death.[29] He died at the age of 80 in Alexandria[29] around the year196 BCE.[30] Roller notes that Dionysios of Kyzikos recorded the genuine epitaph of Eratosthenes bemoaning the fact that he was buried in a foreign land, with reference to the "shore of Proteus", a Homeric allusion to the land of Egypt:[31]
A softening old age with no darkening through disease quenched you and put you to deserved sleep pondering great things, Eratosthenes. Mother Kyrene did not receive you into the paternal tombs, son of Aglaos, but you are buried as a friend in a foreign land, here on the edge of the shore of Proteus.[31]
TheSuda records four students of Eratosthenes:Aristophanes of Byzantium, his successor as Librarian of Alexandria, the geographerMnaseus ofPatara inLycia, the historian Menander, probably of Ephesos, and Aristis, who was otherwise unknown.[32]
It appears that, outside of the geographical context, Eratosthenes did not contribute any original work in the field of astronomy.[26] His name was not associated with any astronomical observations, nor was he cited as an authority in Ptolemy's works on astronomical calendars andparapegmata.[26] Additionally, doubt has been cast on the attribution of the measurement of the sun to him by Eusebius and Macrobius, and the one astronomical title associated with his name,Catasterismi, is considered to be incorrectly attributed, and the lost work upon which it was possibly based can hardly be considered astronomical.[26] However, in the field of astronomical geography his contributions were substantial.[citation needed]
Measure of Earth's circumference according to Cleomedes's simplified version, based on the approximation thatSyene is on theTropic of Cancer and on the same meridian asAlexandria.
TheEarth's circumference is the most famous measurement obtained by Eratosthenes,[2] who estimated that the meridian has a length of 252,000stadia, or 39,060 to 40,320 kilometres (24,270 to 25,050 mi), with an error on the real value between −2.4% and +0.8%, assuming a value for the stadion between 155 and 160 metres (509 and 525 ft).[2] Eratosthenes described hisarc measurement technique[33] in his bookOn the Measure of the Earth, which has not been preserved. However, a simplified version of the method as described byCleomedes was preserved.[34] Modern day measurements of the actual circumference around theequator is 40,075.017 km (24,901.461 mi); passing through thepoles the circumference is 40,007.863 km (24,859.734 mi).[35]
The simplified method works by considering two cities along the samemeridian and measuring both the distance between them and the difference in angles of the shadows cast by the sun on a vertical rod (agnomon) in each city at noon on the summersolstice. The two cities used by Eratosthenes wereAlexandria andSyene (modern Aswan), with the distance between the cities measured by professionalbematists.[36] A geometric calculation reveals that the circumference of the Earth is the distance between the two cities divided by the difference in shadow angles expressed as a fraction ofone turn, or expressed algebraically as
Eusebius of Caesarea in hisPreparatio 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 "ofstadiamyriads 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 m (607 ft), 804,000,000 stadia is 149,000,000 km (93,000,000 mi), approximately the distance from the Earth to the Sun.
Eratosthenes also calculated the Sun's diameter. According toMacrobius, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth.[37] The actual figure is approximately 109 times.[38]
Eratosthenes determined theobliquity of the ecliptic.[39] Theecliptic is the apparent circular orbit of the sun projected onto the imaginary celestial sphere over the course of a year; its obliquity is the inclination of its plane relative to the plane of the equator.[39] The value of this angle (ε) is not constant; at the time of Eratosthenes, it was 23° 43′ 40″. As early as the 5th century BC,Oenopides of Chios had determined 24°; Eratosthenes improved the accuracy of the measurement.[39] He determined the angular distance between the two tropics as of the full circle (360°), i.e., 47° 42′ 40″, which, when halved, yields a value of 23° 51′ 20″ for ε.[39] How he arrived at this result is unknown; the hypotheses considered in research are speculative. While at the Library of Alexandria, Eratosthenes devised a calendar using his predictions about theecliptic of the Earth. He calculated that there are 365 days in a year and that every fourth year there would be 366 days.[40] TheGreek astronomerHipparchus (c. 190 – c. 120 BC) credited Eratosthenes (276 – 194 BC) as the inventor of the armillary sphere,[41][42][43][44][45] a model ofobjects in the sky (on thecelestial sphere), consisting of a spherical framework ofrings, centered onEarth or theSun, that represent lines ofcelestial longitude and latitude and other astronomically important features, such as theecliptic.[46]
Eratosthenes continued using his knowledge about the Earth. With his discoveries and knowledge of its size and shape, he began to sketch it. In the Library of Alexandria he had access to travel books, which contained information and representations of the world that needed to be pieced together in some organized format.[37] In his three-volume workGeography (Ancient Greek:Geographika), he described and mapped his entire known world, even dividing the Earth into five climate zones:[47] two freezing zones around the poles, two temperate zones, and a zone encompassing the equator and the tropics.[48] 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 then possible to estimate the distance from remote locations with this network over the surface of the Earth. In theGeography he recorded the names of over 400 cities and their locations were shown, a feat without precedent.[49]
According to Strabo, Eratosthenes argued against the Greek-Barbarian dichotomy and said Alexander ignored his advisers by his regard for all people with law and government.[50] Though he argued that Eratosthenes was wrong to claim that Alexander had disregarded the counsel of his advisers asserting that it was Alexander's interpretation of their "real intent" in recognizing that "in some people there prevail the law-abiding and the political instinct, and the qualities associated with education and powers of speech".[51]
InPlatonikos, primarily mathematical questions were dealt with; the concepts discussed included distance, ratio, continuous and discontinuous proportion, mathematical mean, prime number and point. The focus was on the theory of proportions, in which Eratosthenes saw the key toPlatonic philosophy. For him, mathematical knowledge meant philosophical knowledge. The tool of the ratio equation ("a is to b as c is to d"), which he called "analogy", was intended to help in gaining non-mathematical knowledge. He generally strove to solve problems by looking for analogies in the sense of ratio equations.[52] In proportion, he believed he had found the unifying bond of the "mathematical" sciences (arithmetic,geometry,astronomy,music theory), since all statements of these sciences could ultimately be traced back to statements about proportions.[citation needed]
Just as one is the starting point(archḗ) and the primary element(stoicheíon) of numbers and thus of quantity, and just as the point is the insoluble, irreducible element of length, for Eratosthenes equality (as the primary ratio 1:1) is the element and origin of all relationships and proportions. Numbers arise through addition, and the various ratios through the enlargement of the terms of the initial ratio; the line, on the other hand, cannot be produced by the combination of individual points, since the individual point has no extension, but rather it arises from the continuous movement of a point.[53] This view was later criticized by the skepticSextus Empiricus.[53]
Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from the prime's square).
Eratosthenes proposed a mathematical approximate solution to the problem ofdoubling the cube, the "Delian problem," which was unsolvable with compass and ruler. In order to solve this problem, Eratosthenes constructed a mechanical line drawing device to calculate the cube, called theMesolabio.[54] He dedicated his solution to King Ptolemy, presenting a model in bronze with it a letter and an epigram.[55]
For prime number research, he used analgorithm that allows one to separate allprime numbers from the set of all odd natural numbers that are less than or equal to a given number. This method is known as theSieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους). However, according toHans-Joachim Waschkies he did not invent it - as was previously believed; rather, it was already known, and he only coined the term "sieve."[56]
Eratosthenes' sieve is one of a number ofprime number sieves, and is a simple, ancient algorithm for finding allprime 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.
A secondary subject ofPlatonikos wasmusic theory, in which Eratosthenes applied the theory of proportions to music,[57] In this regard he is considered one of the oldest authorities in the field of music in antiquity.[57] The scholarPtolemy preserved Eratosthenes' calculations for the tetrachord,[58] which show that he used the "Pythagorean" tuning, which he then refined.[58] Eratosthenes knew and considered the system of the music theoristAristoxenus.[59] However, Ptolemy does not disclose how he proceeded with his calculations.
Eratosthenes addressed metaphysics such as the doctrine of the soul in thePlatonikos. Like the PlatonistCrantor, by whom he was probably influenced, he held the view that the soul could not be purely immaterial, but must have something corporeal about it, for it exists in the world of sensible things; moreover, it is always in a body.[60] This is based on the idea that the soul can only grasp sensible objects if it has a corresponding disposition in its own structure. Accordingly, it is a mixture of two components, one incorporeal and one corporeal.[61]
Eratosthenes was one of the most eminent scholars 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. There are no documents left of his work after thedestruction of the Library of Alexandria.
Platonikos - Most probably Eratosthenes' main mathematical treatise, of which only few extracts remain, found in theExpositio rerum mathematicarum ad legendum Platonem utilium, byTheon of Smyrna.[26] It is unclear whether the work was a commentary onPlato'sTimaeus or a dialogue with Plato as the principal speaker, but its central theme was the fundamental mathematics underlying Plato's philosophy.[18] It is suggested that it served as a handbook intended to make Plato's works easier for a wider audience to access by clarifying terms and explaining difficult passages.[62] The topics covered included proportion and progression, and as a derivative, the theory of musical scales, and the solution of "the Delian problem" in response to the godly demand of doubling the cube-shaped altar inDelos to stop a plague[18] (preserved together with Eratosthenes' epigram and letter to Ptolemy III by Eutocius in hisEutocii commentarii in libros de sphaera et cylindro, II, 1, inArchimedes opera omnia).[26]
On the Old Comedy - A work of literary criticism consisting of twelve books, which attempted to derive the authorship of plays from the dates they were performed, included discussions of textual criticism and contained a section on the meaning and usage of words.[26] The latter was highly praised and often cited by ancient authors.[26]
Anterinys/Hesiod - A poetic work, now lost, the contents of which are unknown.[63]
Erigone - A poetic work depicting the star legend ofIcarius, his daughterErigone and her dog,[63] according to which Erigone committed suicide upon hearing about the death of her father.[27] The work contained astronomical elements, as the characters were translated as the heavenly bodies ofBoötes,Virgo, andSirius.[63]
Hermes - A poetic work, of which some sixteen lines have survived.[63] It paralleled the beginning of theHomeric hymn, but added to it the heavenward ascent of Hermes which included a vivid description of the different climate zones of the inhabited world,[27][63] and contained "a good deal of descriptive astronomy" in the words of Thomas Heath.[64]
On Intermediate Terms (Peri mesotḗtōn) - A work attributed to Eratosthenes byPappus, of the latethird century CE.[65] Its contents were lost, but it can be said that it consisted of two books, and was of enough importance to be included in what Pappus called the "Treasury of Analysis" together with the writings ofEuclid,Apollonius, andAristaeus, thus implying that it was a systematic geometrical composition.[65] In another passage, Pappus refers to "loci with reference to means" which were discussed by Eratosthenes, supposedly in the work mentioned, the nature of these loci in unknown.[65] Since this work is not mentioned anywhere else in ancient sources, some have suggested that it is identical withPlatonikos.[66] In 1981, a medieval Arabic translation of a text by "Aristanes" (Eratosthenes) on mean proportionals was published. However, this is not the lost workOn Intermediate Terms mentioned by Pappus, but an alleged letter from Eratosthenes toKing Ptolemy III about the doubling of a cube, which is preserved in the original Greek text. The authenticity of the letter is disputed.[67]
The Catasterismi, ("Placings among stars"), cited in the Suda under the titleAstronomy.[68] The extant work by this name in its current form cannot be attributed to Eratosthenes, however it is rooted in a genuine work by him with the same name.[69] TheCatasterismi contained astar catalogue, which references the writings of Aratus, but as opposed to the largely technical descriptions of Aratus, it includes a collection of legends relating to individual stars and constellations.[68] The catalogue contains 42 entries covering all the constellations, one entry on the planets and one entry on the milky way; it includes a list of stars belonging to each constellation, with their locations within the constellation, all together number 736,[68] (though Hipparchus has approximately 1000).[70] It has been pointed out, that Eratosthenes did not invent the myths, which had been transmitted over centuries through Greek traditions, rather he connected these tales to the constellations and attributed the different mythical characters to them.[68]
Arsinoe (a memoir of queenArsinoe; lost; quoted byAthenaeus in theDeipnosophistae) - A biography or eulogy of Arsinoe III, wife and sister of Ptolemy IV, who was murdered at the age of 30 after her husband's death.[28] Eratosthenes had been her advisor and companion in public events.[28] The writing of the work is the last datable event in the life of Eratosthenes, and the work itself is likely the last that he wrote, as Arsinoe's death occurred in 204 BCE, Eratosthenes was about eighty years old at the time, and he did not live for much longer.[28]
On the Measurement of the Earth (Περὶ τῆς ἀναμετρήσεως τῆς γῆς) - Described as a separate work by Heron in his Dioptra, and according to Galen it dealt with astronomical or mathematical geography.[71] Among the topics discussed were the size of the equator, the distance of the tropic and polar circles, the size of the polar area, the sizes of the sun and the moon and the distances from them and their total and partial eclipses and the changes in the length of the day according to location and date.
Geographica (ГεωγραΦικά) - The work was the first attempt at providing a mathematical foundation for geographical studies, as well as the first recorded instance of many terms still in use, including the name of the sciencegeography.[72] It is now lost, but 155 fragments survive, 105 in the writings of Strabo, 16 in the writings of Pliny the elder, and the rest scattered in Byzantine sources.[28] According to Strabo, who is the primary source for its form and content, it consisted of three parts.[73] For a long time it was the main authority on geographical matters, and was referred to by Julius Caesar inDe Bello Gallico, when he mentioned that Eratosthenes knew of the Hercynian forest.[73] Even the critical Strabo admitted that Eratosthenes was the leading authority on the southeastern quarter of the inhabited world.[73] The work described the global landmass as a whole, discussed its division into regions, estimated distances, landscape alterations, the location of the inhabited world, and included limited descriptions of lands and peoples.[73] The work was criticized by Strabo, who complained that Eratosthenes' approach was too mathematical, and by Hipparchus, who argued that it was not mathematical enough, as Eratosthenes did not make sufficient use of astronomical data in establishing the reference lines of his map.[74] It is possible that the circumference of the Earth was written as part of theGeographica, though if it wasn't, it was most likely mentioned in it.[74] Its detailed description is now known only throughDe Motu Circulari by Cleomedes.[74] 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 denouncedHomer as not providing any insight into what he described as geography. His disapproval of Homer's topography angered many who believed the world depicted in theOdyssey to be legitimate.[7][75] He commented on the ideas of the nature and origin of the Earth: he thought of Earth as an immovable globe while its surface was changing. He hypothesized that at one time theMediterranean had been a vast lake that covered the countries that surrounded it and that it only became connected to the ocean to the west when a passage opened up sometime in its history. The second book contains his calculation of the circumference of the Earth. This is where, according to Pliny, "The world was grasped." Here Eratosthenes described his famous story of the well in Syene, wherein at noon each summer solstice, the Sun's rays shone straight down into the city-center well.[76] This book would later be considered a text onmathematical geography. His third book of theGeography containedpolitical 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 that can be considered the beginning of geography. For this, Eratosthenes was named the "Father of Modern Geography."[37]
Chronographies and Olympic Victors - Two works that represent the first systematic, scientific treatment of chronological questions by a Greek author[26] and that established a dating system based on the Olympiads.[77]Olympic Victors was likely a popularizing work and included numerous anecdotes, some preserved by Plutarch.[26] For several events, the fall of Troy (1184/1183 BCE), the Dorian migration (1104/1103 BCE), the first Olympiad (777/776 BCE), Xerxes’ invasion (480/479 BCE), and the outbreak of the Peloponnesian War (432/431 BCE), Eratosthenes’ dates are still considered authoritative.[26]
^TheSuda states that he was born in the 126thOlympiad, (276–272 BC).Strabo (Geography, i.2.2), though, states that he was a "pupil" (γνωριμος) ofZeno of Citium (who died in 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.[79]
^TheSuda states he died at the age of 80,Censorinus (De die natali, 15) at the age of 81, andPseudo-Lucian (Makrobioi, 27) at the age of 82.
^See also Asimov, Isaac.Asimov's Biographical Encyclopedia of Science and Technology, new revised edition. 1975. Entry #42, "Eratosthenes", p. 29. Pan Books Ltd, London.ISBN0-330-24323-3. This was also asserted by Carl Sagan 31 minutes into his Cosmos episodeThe Shores of the Cosmic Ocean
^abEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 8.ISBN978-0-691-14267-8.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 bc). Oxford: Oxford university press. p. 10.ISBN978-0-19-887429-4.
^Eratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 11.ISBN978-0-691-14267-8.
^abcdefghijklEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 9.ISBN978-0-691-14267-8.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 BC). Oxford: Oxford University Press. p. 13.ISBN978-0-19-887429-4.
^Dicks, D.R. "Eratosthenes", inComplete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
^abcEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 10.ISBN978-0-691-14267-8.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 BC). Oxford: Oxford University Press. p. 12.ISBN978-0-19-887429-4.
^abEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. pp. 10–11.ISBN978-0-691-14267-8.
^abcEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 12.ISBN978-0-691-14267-8.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 bc). Oxford: Oxford university press. p. 12.ISBN978-0-19-887429-4.
^abcÉratosthène; Roller, Duane W. (2010).Eratosthenes' geography: fragments collected and translated. Princeton: Princeton university press. p. 115.ISBN978-0-691-14267-8.
^abcdeMatthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 BC). Oxford: Oxford University Press. p. 15.ISBN978-0-19-887429-4.
^abcMatthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 bc). Oxford: Oxford university press. p. 301.ISBN978-0-19-887429-4.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 BC). Oxford: Oxford University Press. p. 74.ISBN978-0-19-887429-4.
^abEratosthenes; Roller, Duane W.; Strabo (2010).Eratosthenes' Geography. Princeton, N.J: Princeton University Press. p. 270.ISBN978-0-691-14267-8.
^Matthew, Christopher Anthony (2023).Eratosthenes and the measurement of the Earth's circumference (c. 230 BC). Oxford: Oxford University Press. p. 14.ISBN978-0-19-887429-4.
^Torge, W.; Müller, J. (2012).Geodesy. De Gruyter Textbook. De Gruyter. p. 5.ISBN978-3-11-025000-8. Retrieved2021-05-02.
^Martianus Capella,De nuptiis Philologiae et Mercurii, VI.598.
^abcSmith, Sir William. "Eratosthenes", inA Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
^Dirk L. Couprie, Robert Hahn, Gerard Naddaf:Anaximander in Context: New Studies in the Origins of Greek Philosophy, 2003,ISBN978-0-7914-5537-1, p. 179
^Roller, Duane W. Eratosthenes' Geography. New Jersey: Princeton University Press, 2010.
^Plutarch's similar discussion claiming that Alexander ignoredAristotle's advice in this matter may have been influenced by Eratosthenes, but Plutarch does not confirm his sources.
^Isaac, Benjamin. Invention of Racism in Classical Antiquity. Princeton University Press, 2013.
^Heinrich Dörrie (Hrsg.):Der Platonismus in der Antike, Bd. 1, Stuttgart-Bad Cannstatt 1987, S. 351, 355, 361f., 367–386.
^abSextus Empiricus; Bett, Richard (2018).Against those in the disciplines. Oxford: Oxford university press. pp. 161–163.ISBN978-0-19-871270-1.
^Zhumud, Leonid. Plato as "Architect of Science". inPhonesis. Vol. 43 (3) 1998. 211–244.
^Hans-Joachim Waschkies:Anfänge der Arithmetik im Alten Orient und bei den Griechen, Amsterdam 1989, S. 280–288; Klaus Geus:Eratosthenes von Kyrene, München 2002, S. 189.
^abChalmers, John H.; Polansky, Larry (1993).The divisions of the tetrachord: = Peri tōn toy tetrachordoy katatomōn = Sectiones tetrachordi ; a prolegomenon to the construction of musical scales. Hanover, NH: Frog Peak Music. p. 10.ISBN978-0-945996-04-0.
^Chalmers, John H.; Polansky, Larry (1993).The divisions of the tetrachord: = Peri tōn toy tetrachordoy katatomōn = Sectiones tetrachordi ; a prolegomenon to the construction of musical scales. Hanover, NH: Frog Peak Music. p. 48.ISBN978-0-945996-04-0.
^Hans Krämer:Eratosthenes. In:Grundriss der Geschichte der Philosophie. Die Philosophie der Antike, Bd. 3:Ältere Akademie – Aristoteles – Peripatos, hrsg.Hellmut Flashar. 2. Auflage, Basel 2004, S. 126. Zur Seelenlehre des Eratosthenes siehe auch Friedrich Solmsen:Eratosthenes as Platonist and Poet. In: Solmsen,Kleine Schriften, Bd. 1, Hildesheim 1968, S. 212–216.
^Klaus Geus:Eratosthenes von Kyrene, München 2002, S. 185f.
^Klaus Geus:Eratosthenes von Kyrene, München 2002, S. 142, 192–194.
^Klaus Geus:Eratosthenes von Kyrene, München 2002, S. 190f.
^Klaus Geus:Eratosthenes von Kyrene, München 2002, S. 133–135, 195–205 plädiert für Echtheit des Briefs, der meist als Fälschung betrachtet wird, und bietet S. 196–200 eine deutsche Übersetzung.
^Eckerman, Chris. Review of (D.W.) Roller 'Eratosthenes' Geography. Fragments Collected and Translated, with Commentary and Additional Material. The Classical Review. 2011.
^abcdefghijklmnÉratosthène; Roller, Duane W. (2010).Eratosthenes' geography: fragments collected and translated. Princeton: Princeton university press. pp. 12–13.ISBN978-0-691-14267-8.
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.
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Fowler, D. H.; Rawlins, Dennis (1983). "Eratosthenes' ratio for the obliquity of the ecliptic".Isis.74 (274):556–562.doi:10.1086/353361.S2CID144617495.
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.
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. New Series (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):147–175.
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.
Rawlins, D. (1982). "Eratosthenes' geodesy unraveled: was there a high-accuracy Hellenistic astronomy".Isis.73 (2):259–265.doi:10.1086/352973.S2CID120730515.
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.doi:10.1007/BF00348500.S2CID118004246.
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