Editing Eratosthenes (section)

==Scholarly career==
===Measurement of Earth's circumference{{anchor|Earth's circumference|Arc measurement}}===
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{{main|Earth's circumference#Eratosthenes}}

[[File:Eratosthenes_measure_of_Earth_circumference.svg|thumb|upright=1.4|Measure of Earth's circumference according to Cleomedes' simplified version, based on the approximation that [[Syene]] is on the [[Tropic of Cancer]] and on the same meridian as [[Alexandria]].]]

The measurement of [[Earth's circumference]] is the most famous among the results obtained by Eratosthenes,<ref>{{cite book
 |last=Russo
 |first=Lucio
 |author-link=Lucio Russo
 |url=https://books.google.com/books?id=MOTpnfz7ZuYC
 |title=The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn
 |date=2004
 |publisher=Springer
 |isbn=3-540-20396-6
 |location=Berlin
 |page=68
 |oclc=52945835
 |access-date=2024-08-28
 |url-status=live
 |archive-url=https://web.archive.org/web/20240828024309/https://www.google.com/books/edition/The_Forgotten_Revolution/MOTpnfz7ZuYC
 |archive-date=2024-08-28
}}</ref> who estimated that the meridian has a length of 252,000 [[Stadion (unit)|stadia]] ({{convert|39060|to|40320|km|mi}}), with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between {{convert|155|and|160|m|ft}}).<ref name="russo273277" /> Eratosthenes described his [[arc measurement]] technique,<ref name="Torge Müller 2012 p. 5">{{cite book | last1=Torge | first1=W. | last2=Müller | first2=J. | title=Geodesy | publisher=De Gruyter | series=De Gruyter Textbook | year=2012 | isbn=978-3-11-025000-8 | url=https://books.google.com/books?id=RcfmBQAAQBAJ&pg=PA6 | access-date=2021-05-02 | page=5}}</ref> in a book entitled ''{{visible anchor|On the Measure of the Earth}}'', which has not been preserved. However, a simplified version of the method has been preserved, as described by [[Cleomedes]].<ref>Cleomedes, ''Caelestia'', i.7.49–52.</ref>  Modern day measurements of the actual circumference around the [[equator]] is {{cvt|40075.017|km|mi}}, and passing through the [[geographical pole|poles]] the circumference is {{cvt|40007.863|km|mi}}.<ref>{{cite web |first1=Sigurd |last1=Humerfelt |date=26 October 2010 |title=How WGS 84 defines Earth |url=http://home.online.no/~sigurdhu/WGS84_Eng.html |access-date=27 February 2025 |url-status=dead |archive-url=https://web.archive.org/web/20110424104419/http://home.online.no/~sigurdhu/WGS84_Eng.html |archive-date=24 April 2011 }}</ref>  

The simplified method works by considering two cities along the same [[meridian (geography)|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 [[bematist]]s.<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 [[turn (angle)|one turn]].

{{anchor|Geography}}<!--linked-->

===Geography===
[[File:Mappa di Eratostene.jpg|thumb|right|alt=Eratosthenes' map of the world (194 BC)|19th-century reconstruction of [[Eratosthenes' Map of the World|Eratosthenes' map of the (for the Greeks) known world]], {{nowrap|{{circa}} 194 BC}}]]
{{see also|History of geodesy|History of longitude}}

Eratosthenes now continued from his knowledge about the Earth. Using his discoveries and knowledge of its size and shape, he began to sketch it. In the Library of Alexandria he had access to various travel books, which contained various items of information and representations of the world that needed to be pieced together in some organized format.<ref name="Smith2005" /> In his three-volume work ''Geography'' ({{langx|grc-Latn|Geographika}}), 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" /> This book is the first recorded instance of many terms still in use today, including the name of the discipline [[geography]].<ref name='intro1'>{{cite book |last1=Dahlman |first1=Carl |last2=Renwick |first2=William |title=Introduction to Geography: People, Places & Environment |date=2014 |publisher=Pearson |isbn=978-0-13-750451-0 |edition=6 |url=https://www.pearson.com/store/en-us/pearsonplus/p/9780137504510.html?creative=545445680380&keyword=&matchtype=&network=g&device=c&gclid=CjwKCAjwpKyYBhB7EiwAU2Hn2QPXxmu7Nqnx04A__xcaDqM3GuPh2cbR2wI7G7ihOs2cQpV7CUFAxxoCzLEQAvD_BwE&gclsrc=aw.ds |access-date=28 August 2022}}</ref> He placed grids of overlapping lines over the surface of the Earth. He used parallels and meridians to link together every place in the world. It was now possible to estimate one's distance from remote locations with this network over the surface of the Earth. In the ''Geography'' the names of over 400 cities and their locations were shown, which had never been achieved before.<ref name="roller"/> However, his ''Geography'' has been lost to history, although fragments of the work can be pieced together from other great historians like [[Pliny the Elder|Pliny]], [[Polybius]], [[Strabo]], and [[Marcianus]]. While this work is the earliest we can 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 now 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>{{Cite web|url=https://www.khanacademy.org/partner-content/big-history-project/solar-system-and-earth/knowing-solar-system-earth/a/eratosthenes-of-cyrene|title=Eratosthenes of Cyrene|website=Khan Academy|language=en|access-date=2019-11-19}}</ref> This book would now 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 and can be considered the beginning of geography. For this, Eratosthenes was named the "Father of Modern Geography."<ref name="Smith2005" />

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.<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 give us confirmation of his sources.</ref> 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".<ref>Isaac, Benjamin. Invention of Racism in Classical Antiquity. Princeton University Press, 2013.</ref>

===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.<ref name="Dicks1971" />
* [[Eusebius of Caesarea]] in his ''[[Preparatio Evangelica]]'' includes a brief chapter of three sentences on celestial distances ([http://www.tertullian.org/fathers/eusebius_pe_15_book15.htm Book XV], Chapter 53). He states simply that Eratosthenes found the distance to the Sun to be "{{lang|grc|σταδίων μυριάδας τετρακοσίας καὶ ὀκτωκισμυρίας}}" (literally "of [[Stadia (length)|stadia]] [[myriad]]s 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 {{cvt|185|m|||}}, 804,000,000 stadia is {{cvt|149000000|km|||}}, 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" />
* During his time at the Library of Alexandria, Eratosthenes devised a calendar using his predictions about the [[ecliptic]] of the Earth. He calculated that there are 365 days in a year and that every fourth year there would be 366 days.<ref name="manawatu" />
* 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.<ref name="zhumud" /> Archimedes was Eratosthenes' friend and he, too, worked on the war instrument with mathematics. Archimedes dedicated his book ''The Method'' to Eratosthenes, knowing his love for learning and mathematics.<ref name="chondros" />

===Number theory===
[[File:Sieve of Eratosthenes animation.gif|right|frame|Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from the prime's square).]]
{{Main|Sieve of Eratosthenes|Primality test}}

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

In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of [[prime number sieve]]s, 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.