He is best known for being the first person known to calculate the Earth'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 calculate Earth's axial tilt, which similarly proved to have remarkable accuracy.[4][5] He created the first global projection of the world incorporating parallels and meridians based on the available geographic knowledge of his era.[4]
Eratosthenes was the founder of scientific chronology;[6] he used Egyptian and Persian records to estimate the dates of the main events of the Trojan War, dating the sack of Troy to 1184 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers and composite numbers.
He was a figure of influence in many fields who yearned to understand the complexities of the entire world.[7] 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. Yet, according to an entry[8] in the Suda (a 10th-century encyclopedia), some critics scorned him, calling him Number 2 because he always came in second in all his endeavours.[9]
Life
The son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Now part of modern-day Libya, Cyrene had been founded by Greeks centuries earlier and became the capital of Pentapolis (North Africa), a country of five cities: Cyrene, Arsinoe, Berenice, Ptolemais, and Apollonia. Alexander the Great conquered Cyrene in 332 BC, and following his death in 323 BC, its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, a plant used for rich seasoning and medicine.[10] Cyrene became a place of cultivation, where knowledge blossomed. Like any young Greek at the time, 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.[11]
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.[12] He then studied under Aristo 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 first work at a scholarly level, Platonikos, inquiring into the mathematical foundation of Plato's philosophies.[7] Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus.[11] 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).[7] 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 king 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 original or the copy. He sought to maintain the reputation of the Library of Alexandria against competition from the Library of 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.[7]
Eratosthenes believed there was both good and bad in every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, as well as for arguing that the Greeks should keep themselves racially pure.[14] 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 194BC at the age of 82 in Alexandria.[11]
Scholarly career
Astronomy
Measurement of Earth's circumference
Measure of Earth's circumference according to Cleomedes's simplified version, based on the approximation that Syene is on the Tropic of Cancer and on the same meridian as Alexandria.
The Earth's circumference is the most famous measurement obtained by Eratosthenes,[2] who estimated that the meridian has a length of 252,000 stadia (39,060 to 40,320 kilometres (24,270 to 25,050mi)), 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 525ft)).[2] Eratosthenes described his arc measurement technique,[15] in a book entitled On the Measure of the Earth, which has not been preserved. However, a simplified version of the method was preserved, as described by Cleomedes.[16] Modern day measurements of the actual circumference around the equator is 40,075.017km (24,901.461mi), and passing through the poles the circumference is 40,007.863km (24,859.734mi).[17]
The simplified method works by considering two cities along the same meridian and measuring both the distance between them and the difference in angles of the shadows cast by the sun on a vertical rod (a gnomon) in each city at noon on the summer solstice. The two cities used were Alexandria and Syene (modern Aswan), and the distance between the cities was measured by professional bematists.[18] 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 of one turn.
Obliquity of the ecliptic
Eratosthenes determined the obliquity of the ecliptic.[19] The ecliptic 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.[19] 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.[19] He determined the angular distance between the two tropics asof the full circle (360°), i.e., 47° 42′ 40″, which, when halved, yields a value of 23° 51′ 20″ for ε.[19] How he arrived at this result is unknown; the hypotheses considered in research are speculative.
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.[20] In his three-volume work Geography (Ancient Greek: Geographika), he described and mapped his entire known world, even dividing the Earth into five climate zones:[21] two freezing zones around the poles, two temperate zones, and a zone encompassing the equator and the tropics.[22] This book is the first recorded instance of many terms still in use, including the name of the science geography.[23] 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 the Geography the names of over 400 cities and their locations were shown, which had never been achieved before.[10] However, his Geography has been lost to history, although fragments of the work can be pieced together from other great historians like Pliny, Polybius, Strabo, and Marcianus. While this work is the earliest to trace certain ideas, words, and concepts in the historical record, earlier contributions may have been lost to history.
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 described as geography. His disapproval of Homer's topography angered many who believed the world depicted in the Odyssey to be legitimate.[7][24] He also 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 the Mediterranean 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.[25] This book would later be considered a text on mathematical geography.
His third book of the Geography 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 that can be considered the beginning of geography. For this, Eratosthenes was named the "Father of Modern Geography."[20]
According to Strabo, Eratosthenes argued against the Greek-Barbarian dichotomy. He says Alexander ignored his advisers by his regard for all people with law and government.[26] Strabo says that Eratosthenes was wrong to claim that Alexander had disregarded the counsel of his advisers. Strabo argues 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".[27]
The philosopher and mathematician Theon of Smyrna quotes two passages from a work by Eratosthenes entitled Platōnikós, which has not survived. The literary genre to which the Platonikos belonged is disputed. Some researchers have considered it a commentary on Plato's dialogue Timaeus, but Eratosthenes does not seem to have limited himself to a discussion of just one of Plato's works. It has often been assumed that it was a dialogue in which Plato was the main speaker, but in that case, according to ancient custom, the work would have to be called Plato and not Platonikos. Platonikos is probably to be understood in the sense of Platonikos logos (work on Plato). It was probably a handbook intended to make Plato's works easier for a wider audience to access by clarifying terms and explaining difficult passages.[28]
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 to Platonic philosophy. For him, mathematical knowledge also meant philosophical knowledge. The tool of the ratio equation ("a is to b as c is to d"), which he called "analogy", was also intended to help in gaining non-mathematical knowledge. He generally strove to solve problems by looking for analogies in the sense of ratio equations.[29] 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.
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.[30] This view was later criticized by the skeptic Sextus Empiricus.[30]
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 of doubling the cube, the "Delian problem," which was unsolvable with compass and ruler. For prime number research, he used an algorithm that allows one to separate all prime numbers from the set of all odd natural numbers that are less than or equal to a given number. This method is known as the Sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους). However, according to Hans-Joachim Waschkies he did not invent it - as was previously believed; rather, it was already known, and he only coined the term "sieve."[31]
Eratosthenes' sieve is one of a number of prime number sieves, and 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.
A secondary subject of Platonicus was music theory, in which Eratosthenes applied the theory of proportions to music,[32] In this regard he is considered one of the oldest authorities in the field of music in antiquity.[32] The scholar Ptolemy preserved Eratosthenes' calculations for the tetrachord,[33] which show that he used the "Pythagorean" tuning, which he then refined.[33] Eratosthenes also knew and considered the system of the music theorist Aristoxenus.[34] However, Ptolemy does not disclose how he proceeded with his calculations.
Furthermore, Eratosthenes also addressed metaphysics such as the doctrine of the soul in the Platonicus. Like the Platonist Crantor, by whom he was probably influenced, he held the view that the soul could not be purely immaterial, but must also have something corporeal about it, for it exists in the world of sensible things; moreover, it is always in a body.[35] 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.[36]
The late antique mathematician Pappus mentions a mathematical work by Eratosthenes entitled On Intermediate Terms (Peri mesotḗtōn). Since this work is not mentioned anywhere else in ancient sources, it can be assumed that it is identical with Platonikos.[37] In 1981, a medieval Arabic translation of a text by "Aristanes" (Eratosthenes) on mean proportionals was published. However, this is not the lost work On Intermediate Terms mentioned by Pappus, but an alleged letter from Eratosthenes to King Ptolemy III about the doubling of a cube, which is also preserved in the original Greek text. The authenticity of the letter is disputed.[38]
Achievements
Eratosthenes was described by the Suda Lexicon as a Πένταθλος (Pentathlos) which can be translated as "All-Rounded", for he was skilled in a variety of things; he was a true polymath. His opponents nicknamed him "Number 2" 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; Strabo accounts Eratosthenes as a mathematician among geographers and a geographer among mathematicians.[39]
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 stadiamyriads 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 185m (607ft), 804,000,000 stadia is 149,000,000km (93,000,000mi), approximately the distance from the Earth to the Sun.
Eratosthenes also calculated the Sun's diameter. According to Macrobius, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth.[20] The actual figure is approximately 109 times.[40]
While 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.[41]
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.[42] Archimedes was Eratosthenes's 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.[43]
Works
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 the destruction of the Library of Alexandria.[39]
Astronomical writings
Three astronomical writings by Eratosthenes are known, but only fragmentary:
The Catasterismi,("Placings among stars") which is also cited in the Suda under the title Astronomy.[44] The Catasterismi contained a star catalogue, which references the writings of Aratus, but as opposed to the largely technical descriptions of Aratus, it also includes a collection of legends relating to individual stars and constellations.[44] The catalogue contains 42 entries covering all the constellations, one entry on the planets and one entry on the milky way; it also includes a list of stars belonging to each constellation, with their locations within the constellation, all together number 736.[44] 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.[44]
Hipparchus (c.190– c.120BC), a Greek mathematician who measured the radii of the Sun and the Moon as well as their distances from the Earth.
Posidonius (c.135– c.51BC), a Greek astronomer and mathematician who calculated the circumference of the Earth.
Notes
↑ The Suda states that he was born in the 126th Olympiad, (276–272BC). Strabo (Geography, i.2.2), though, states that he was a "pupil" (γνωριμος) of Zeno of Citium (who died in 262BC), 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.[47]
↑ 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.
↑ 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 episode The Shores of the Cosmic Ocean
1 2 Roller, Duane W. Eratosthenes' Geography. New Jersey: Princeton University Press, 2010.
1 2 3 Bailey, Ellen. 2006. "Eratosthenes of Cyrene." Eratosthenes Of Cyrene 1–3. Book Collection Nonfiction: High School Edition.
↑ Rist, J.M. "Zeno and Stoic Consistency," in Phronesis. Vol. 22, No. 2, 1977.
↑ p.439 Vol. 1 William Woodthorpe Tarn Alexander the Great. Vol. I, Narrative; Vol. II, Sources and Studies. Cambridge: Cambridge University Press, 1948. (New ed., 2002 (paperback, ISBN0-521-53137-3)).
↑ Torge, W.; Müller, J. (2012). Geodesy. De Gruyter Textbook. De Gruyter. p.5. ISBN978-3-11-025000-8. Retrieved 2021-05-02.
1 2 3 Smith, Sir William. "Eratosthenes", in A Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
↑ Morris, Terry R. "Eratosthenes of Cyrene." in Encyclopedia Of The Ancient World. November 2001.
↑ Eckerman, Chris. Review of (D.W.) Roller 'Eratosthenes' Geography. Fragments Collected and Translated, with Commentary and Additional Material. The Classical Review. 2011.
↑ Plutarch's similar discussion claiming that Alexander ignored Aristotle'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.
↑ Klaus Geus: Eratosthenes von Kyrene, München 2002, S. 142, 192–194.
↑ Heinrich Dörrie (Hrsg.): Der Platonismus in der Antike, Bd. 1, Stuttgart-Bad Cannstatt 1987, S. 351, 355, 361f., 367–386.
1 2 Sextus Empiricus; Bett, Richard (2018). Against those in the disciplines. Oxford: Oxford university press. pp.161–163. ISBN978-0-19-871270-1.
↑ 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.
1 2 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.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. 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.
1 2 3 Dicks, D.R. "Eratosthenes", in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
↑ 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, Massachusetts: Harvard University Press, 1957.
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