Eratosthenes

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Template:Short description Template:About Template:Infobox person Eratosthenes of Cyrene (Template:IPAc-en; Template:Langx Template:IPA; Template:Nowrap – Template:Nowrap) was an Ancient Greek polymath and scholar. He was known as a mathematician, geographer, poet, astronomer, and music theorist. Eratosthenes later became the chief librarian at the Library of Alexandria. His work is comparable to the modern discipline of geography, and he introduced some of its terminology, coining the terms geography and geographer.<ref>Template:Cite book</ref>

He is best remembered as the first known person 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%).<ref name="russo273277" /><ref>Template:Cite web</ref> He was the first to calculate Earth's axial tilt, which similarly proved to have remarkable accuracy.<ref name=":02">Template:Cite book</ref><ref name="cornell" /> He created the first global projection of the world incorporating parallels and meridians based on the available geographic knowledge of his era.<ref name=":02" /> Eratosthenes was also the founder of scientific chronology;<ref>Template:Cite book</ref> 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 tried to understand the complexities of the known world.<ref name="chambers" /> 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<ref name="epsilon" /> 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.<ref name="asimov" />

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. 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 the Hellenistic world he was not limited by them.<ref name=":03">Template:Cite book</ref> However, Matthew suggests that his name, meaning "lovely strength" suggests noble upbringing,<ref>Template:Cite book</ref> as does his education from a young age, which could imply his belonging to the aristocracy of Cyrene.<ref>Template:Cite book</ref> 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.Template:Citation needed

By the late 260s BCE, Eratosthenes went to Athens to further his studies.<ref name=":03" /> According to Strabo, he was taught Stoicism 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.<ref name=":13">Template:Cite book</ref> Strabo also lists the little-known Apelles of Chios among his teachers.<ref name=":13" /> Eratosthenes later studied under the peripatetic Aristo of Chios,<ref>Template:Cite book</ref> who led a cynical school of philosophy, and the eclectic-viewed Bion of Borysthenes.<ref name=":13" /> He also studied under the recently appointed head of the Platonic Academy, Arcesilaus of Pitane.<ref name=":13" /> Eratosthenes' later mathematical work implies that he received mathematical training there.<ref name=":13" /> According to the Suda, Eratosthenes also studied under Lysanias of Cyrene, a philologist and grammarian who focused on Homer. Poet, scholar, and librarian Callimachus likely crossed paths with Eratosthenes in local debates and scholarly discourse,<ref name=":13" /> even though he was likely never his formal teacher.<ref>Template:Cite book</ref>

File:Eratosthenes Teaching in Alexandria (Bernardo Strozzi, Montreal).jpg
Eratosthenes teaching in Alexandria by Bernardo Strozzi (1635)

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.<ref>Template:Cite book</ref><ref name=":13" /> Later authors may have shared this view to some extent: Eratosthenes was referred to as Beta (Second), as he was not seen as the leading expert in any given field<ref name=":13" /> (though this may have been because he was the second chief librarian in Alexandria).<ref name=":62">Template:Cite book</ref> Others dubbed him Pentathlos (Πένταθλος - All-Rounded), given his various skills and areas of knowledge;<ref name=":62" /> Pentathlos also refers to the title of an athlete who competes in many events but comes in second in all of them.<ref name=":4">Template:Cite book</ref> Strabo also described Eratosthenes as a mathematician among geographers and a geographer among mathematicians,<ref name="Dicks1971" /> and complained that he did not pay enough respect to Zeno. This comment by Strabo reflects Eratosthenes' independence in thought and practice.<ref name=":13" />

The majority of Eratosthenes' studies focused on philosophy; mathematics was less prominent, and philology even less so.<ref name=":23">Template:Cite book</ref> 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.<ref name=":23" /> 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.<ref name=":23" /> Eratosthenes was born forty years after the death of Alexander 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.<ref>Template:Cite book</ref>

Eratosthenes remained in Athens for twenty years, studying and writing.<ref name=":7">Template:Cite book</ref> During this period he wrote Platonikos, inquiring into the mathematical foundation of Plato's philosophy, as well as the poetic works of Hermes and Erigone. His Chronographies focused on the important dates of the Trojan War, and his Olympic Victors compiled a list of the winners of the Olympic games.<ref>Template:Cite book</ref> Little more is known about this period of his life.<ref name=":7" />

In 246 BCE, Ptolemy III succeeded his father, Ptolemy II. Over the next twenty-five years, the Ptolemaic empire reached its greatest extent and Alexandria attained its zenith as an intellectual center.<ref name=":33">Template:Cite book</ref> The post of librarian, which also included the position of royal tutor to Ptolemy IV Philopator,<ref name=":4" /> became the most prestigious academic appointment.<ref name=":33" /> 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.<ref name=":43">Template:Cite book</ref> Roller suggests that Eratosthenes' roots in Cyrene, the native city of Callimachus, and more importantly Queen Berenike, also contributed favorably to his appointment.<ref name=":43" />

The beginning of Eratosthenes' career in Alexandria was focused on mathematics. He was closely affiliated with Archimedes, who sent him material for comment and praised him enthusiastically for his contributions;<ref name=":43" /> his Method of Mechanical Theorems was written as a letter to Eratosthenes.<ref>Template:Cite book</ref> Eratosthenes subsequently wrote compositions on geography, philosophy, rhetoric, literary criticism, grammar, poetry and astronomy,<ref>Template:Cite book</ref><ref name=":0">Template:Cite book</ref> though some suggest that his astronomical contributions were hardly notable.<ref name=":1">Template:Cite book</ref> It was also said that his poetry strangely contained the very didactic elements which he condemned.<ref name=":6">Template:Cite book</ref> Toward the end of his days, he served as an advisor and companion to Arsinoe, sister and wife of Ptolemy IV.<ref name=":9">Template:Cite book</ref>

According to the Suda, as he aged his eyesight began to fail.<ref name=":8">Template:Cite book</ref> Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death.<ref name=":8" /> He died at the age of 80 in Alexandria<ref name=":8" /> around the year 196 BCE.<ref>Template:Cite book</ref> 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:<ref name=":10">Template:Cite book</ref>

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.<ref name=":10" />

The Suda records four students of Eratosthenes: Aristophanes of Byzantium, his successor as Librarian of Alexandria, the geographer Mnaseus of Patara in Lycia, the historian Menander, probably of Ephesos, and Aristis, who was otherwise unknown. <ref>Template:Cite book</ref>

Achievements

Astronomy

Measurement of Earth's circumferenceTemplate:Anchor

Template:Main It appears that, outside of the geographical context, Eratosthenes did not contribute any original work in the field of astronomy.<ref name=":1" /> His name was not associated with any astronomical observations, nor was he cited as an authority in Ptolemy's works on astronomical calendars and parapegmata.<ref name=":1" /> 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.<ref name=":1" /> However, in the field of astronomical geography his contributions were substantial.Template:Citation needed

File:Eratosthenes measure of Earth circumference.svg
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,<ref name="russo273277" /> who estimated that the meridian has a length of 252,000 stadia, or Template:Convert, with an error on the real value between −2.4% and +0.8%, assuming a value for the stadion between Template:Convert.<ref name="russo273277" /> Eratosthenes described his arc measurement technique<ref name="Torge Müller 2012 p. 52">Template:Cite book</ref> in his book Template:Visible anchor, which has not been preserved. However, a simplified version of the method as described by Cleomedes was preserved.<ref>Cleomedes, Caelestia, i.7.49–52.</ref> Modern day measurements of the actual circumference around the equator is Template:Cvt; passing through the poles the circumference is Template:Cvt.<ref>Template:Cite web</ref>

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 by Eratosthenes were Alexandria and Syene (modern Aswan), with the distance between the cities measured by professional bematists.<ref>Martianus Capella, De nuptiis Philologiae et Mercurii, VI.598.</ref> 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, or expressed algebraically as <math>C = \frac{360^\circ d}{\Delta\theta}</math>

Sun measurements

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 "Template:Lang" (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 Template:Cvt, 804,000,000 stadia is Template:Cvt, 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.<ref name="Smith2005" /> The actual figure is approximately 109 times.<ref name="caltech" />

Obliquity of the ecliptic

Eratosthenes determined the obliquity of the ecliptic.<ref name=":42">Template:Cite journal</ref> 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.<ref name=":42" /> 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.<ref name=":42" /> 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.<ref name=":42" /> 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 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.<ref name="manawatu" /> The Greek astronomer Hipparchus (Template:Circa) credited Eratosthenes (276Template:Snd194 BC) as the inventor of the armillary sphere,<ref>Williams, p. 131</ref><ref>Walter William Bryant: A History of Astronomy, 1907, p. 18</ref><ref>John Ferguson: Callimachus, 1980, Template:ISBN, p. 18</ref><ref>Henry C. King: The History of the Telescope, 2003, Template:ISBN, p. 7</ref><ref>Dirk L. Couprie, Robert Hahn, Gerard Naddaf: Anaximander in Context: New Studies in the Origins of Greek Philosophy, 2003, Template:ISBN, p. 179</ref> a model of objects in the sky (on the celestial sphere), consisting of a spherical framework of rings, centered on Earth or the Sun, that represent lines of celestial longitude and latitude and other astronomically important features, such as the ecliptic.<ref>Template:Cite web</ref>

Template:Anchor

Geography

Eratosthenes's map of the world (194 BC)
19th-century reconstruction of Eratosthenes's map of the (for the Greeks) known world, Template:Nowrap

Template:See also 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.<ref name="Smith2005" /> In his three-volume work Geography (Template:Langx), he described and mapped his entire known world, even dividing the Earth into five climate zones:<ref name="Morris" /> two freezing zones around the poles, two temperate zones, and a zone encompassing the equator and the tropics.<ref name="Hutchinson" /> 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 he recorded the names of over 400 cities and their locations were shown, a feat without precedent.<ref name="roller" />

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.<ref>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.</ref> 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".<ref>Isaac, Benjamin. Invention of Racism in Classical Antiquity. Princeton University Press, 2013.</ref>

Mathematics, music theory and metaphysics

Template:See alsoIn Platonikos, 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.<ref>Heinrich Dörrie (Hrsg.): Der Platonismus in der Antike, Bd. 1, Stuttgart-Bad Cannstatt 1987, S. 351, 355, 361f., 367–386.</ref> 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.Template: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.<ref name=":12">Template:Cite book</ref> This view was later criticized by the skeptic Sextus Empiricus.<ref name=":12" />

File:Sieve of Eratosthenes animation.gif
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. In order to solve this problem, Eratosthenes constructed a mechanical line drawing device to calculate the cube, called the Mesolabio.<ref>Template:Cite web</ref> He dedicated his solution to King Ptolemy, presenting a model in bronze with it a letter and an epigram.<ref name="zhumud" />

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."<ref>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.</ref>

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 Platonikos was music theory, in which Eratosthenes applied the theory of proportions to music,<ref name=":22">Template:Citation</ref> In this regard he is considered one of the oldest authorities in the field of music in antiquity.<ref name=":22" /> The scholar Ptolemy preserved Eratosthenes' calculations for the tetrachord,<ref name=":32">Template:Cite book</ref> which show that he used the "Pythagorean" tuning, which he then refined.<ref name=":32" /> Eratosthenes also knew and considered the system of the music theorist Aristoxenus.<ref>Template:Cite book</ref> However, Ptolemy does not disclose how he proceeded with his calculations.

Eratosthenes also addressed metaphysics such as the doctrine of the soul in the Platonikos. 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.<ref>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.</ref> 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.<ref>Klaus Geus: Eratosthenes von Kyrene, München 2002, S. 185f.</ref>

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 topicsTemplate:Sndgeography, 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.

Athenian period

  • Platonikos - Most probably Eratosthenes' main mathematical treatise, of which only few extracts remain, found in the Expositio rerum mathematicarum ad legendum Platonem utilium, by Theon of Smyrna.<ref name=":1" /> It is unclear whether the work was a commentary on Plato's Timaeus or a dialogue with Plato as the principal speaker, but its central theme was the fundamental mathematics underlying Plato’s philosophy.<ref name=":4" /> 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.<ref>Klaus Geus: Eratosthenes von Kyrene, München 2002, S. 142, 192–194.</ref> 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 in Delos to stop a plague<ref name=":4" /> (preserved together with Eratosthenes' epigram and letter to Ptolemy III by Eutocius in his Eutocii commentarii in libros de sphaera et cylindro, II, 1, in Archimedes opera omnia).<ref name=":1" />
  • 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.<ref name=":1" /> The latter was highly praised and often cited by ancient authors.<ref name=":1" />
  • Anterinys/Hesiod - A poetic work, now lost, the contents of which are unknown.<ref name=":3" />
  • Erigone - A poetic work depicting the star legend of Icarius, his daughter Erigone and her dog,<ref name=":3" /> according to which Erigone committed suicide upon hearing about the death of her father.<ref name=":6" /> The work contained astronomical elements, as the characters were translated as the heavenly bodies of Boötes, Virgo, and Sirius.<ref name=":3" />
  • Hermes - A poetic work, of which some sixteen lines have survived.<ref>Template:Cite book</ref> It paralleled the beginning of the Homeric hymn, but added to it the heavenward ascent of Hermes which included a vivid description of the different climate zones of the inhabited world,<ref name=":6" /><ref name=":3">Template:Cite book</ref> and contained "a good deal of descriptive astronomy" in the words of Thomas Heath.<ref>Template:Cite book</ref>

Alexandrian period

  • On Intermediate Terms (Peri mesotḗtōn) - A work attributed to Eratosthenes by Pappus, of the late third century CE.<ref name=":5">Template:Cite book</ref> 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 of Euclid, Apollonius, and Aristaeus, thus implying that it was a systematic geometrical composition.<ref name=":5" /> 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.<ref name=":5" /> Since this work is not mentioned anywhere else in ancient sources, some have suggested that it is identical with Platonikos.<ref>Klaus Geus: Eratosthenes von Kyrene, München 2002, S. 190f.</ref> 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.<ref>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.</ref>
  • The Catasterismi, ("Placings among stars"), also cited in the Suda under the title Astronomy.<ref name=":53">Template:Cite web</ref> 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.<ref>Template:Cite book</ref> 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.<ref name=":53" /> 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,<ref name=":53" /> (though Hipparchus has approximately 1000).<ref>Template:Cite book</ref> 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.<ref name=":53" />
  • Arsinoe (a memoir of queen Arsinoe; lost; quoted by Athenaeus in the Deipnosophistae) - 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.<ref name=":9" /> Eratosthenes had been her advisor and companion in public events.<ref name=":9" /> 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.<ref name=":9" />
  • 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.<ref>Template:Cite book</ref> 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 science geography.<ref name="intro12">Template:Cite book</ref> 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.<ref name=":9" /> According to Strabo, who is the primary source for its form and content, it consisted of three parts.<ref name=":0" /> For a long time it was the main authority on geographical matters, and was referred to by Julius Caesar in De Bello Gallico, when he mentioned that Eratosthenes knew of the Hercynian forest.<ref name=":0" /> Even the critical Strabo admitted that Eratosthenes was the leading authority on the southeastern quarter of the inhabited world.<ref name=":0" /> 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.<ref name=":0" /> 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.<ref name=":2">Template:Cite book</ref> It is possible that the circumference of the Earth was written as part of the Geographica, though if it wasn't, it was most likely mentioned in it.<ref name=":2" /> Its detailed description is now known only through De Motu Circulari by Cleomedes.<ref name=":2" /> 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.<ref name="chambers" /><ref name="Eckerman" /> 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.<ref>Template:Cite web</ref> 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."<ref name="Smith2005" />
  • Chronographies and Olympic Victors - Two works that represent the first systematic, scientific treatment of chronological questions by a Greek author<ref name=":1" /> and that established a dating system based on the Olympiads.<ref>Template:Cite book</ref> Olympic Victors was likely a popularizing work and included numerous anecdotes, some preserved by Plutarch.<ref name=":1" /> 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.<ref name=":1" />

Additional works

  • A means of determining prime numbers (the Sieve of Eratosthenes)<ref name="ad-bib"/>
  • A work on instrumentation<ref name="ad-bib"/>
  • The calculation of harmonics<ref name="ad-bib"/>
  • A treatise on philosophy (On Good and Bad)<ref name="ad-bib"/>
  • A work on rhetoric (On Declamation)<ref name="ad-bib"/>
  • A literary critique of the works of the poet Homer<ref name="ad-bib"/>
  • An extensive discussion of the nature of old comedy<ref name="ad-bib"/>
  • A correction of the calendar (On the 8-Year Cycle)<ref name="ad-bib"/>
  • An examination of planetary orbits<ref name="ad-bib"/>
  • An examination of the winds<ref name="ad-bib"/>
  • Philosophical analyses (On the Philosophical Sects and On Freedom from Pain)<ref name="ad-bib"/>
  • Dialogues and grammatical works<ref name="ad-bib"/>
  • A discussion of wealth and poverty<ref name="ad-bib"/>
  • A history of the campaigns of Alexander the Great (uncertain)<ref name="ad-bib"/>

See also

Notes

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References

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Further reading

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