Editing Ancient Greek mathematics (section)

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[[File:Pythagoras Euclid.svg|thumb|upright=0.8|An illustration of [[Euclid]]'s proof of the [[Pythagorean theorem]]]]
'''Ancient Greek mathematics''' refers to the history of mathematical ideas and texts in [[Ancient Greece]] during [[Classical antiquity|classical]] and [[late antiquity]], mostly from the 5th century BC to the 6th century AD.<ref>{{Cite journal |last=Sidoli |first=Nathan |date=2020 |editor-last=Taub |editor-first=Liba |title=Ancient Greek Mathematics |url=http://individual.utoronto.ca/acephalous/Sidoli_2020_Ancient_Greek_Mathematics.pdf |journal=The Cambridge Companion to Ancient Greek and Roman Science |pages=190–191 |doi=10.1017/9781316136096.010 |isbn=978-1-316-13609-6}}</ref><ref>{{Cite book |last=Netz |first=Reviel |date=2002 |chapter=Greek mathematics: A group picture. |doi=10.1093/acprof:oso/9780198152484.003.0011 |title=Science and Mathematics in Ancient Greek Culture |pages=196–216 |isbn=978-0-19-815248-4}}</ref> Greek mathematicians lived in cities spread around the shores of the ancient [[Mediterranean]], from [[Anatolia]] to [[Italy]] and [[North Africa]], but were united by [[Greek culture]] and the [[Ancient Greek|Greek language]].{{sfn|Boyer|1991|p=48}} The development of mathematics as a theoretical discipline and the use of [[deductive reasoning]] in [[Mathematical proof|proofs]] is an important difference between Greek mathematics and those of preceding civilizations.<ref>{{Cite book |last=Knorr |first=W. |title=Mathematics |publisher=Harvard University Press |year=2000 |location=Greek Thought: A Guide to Classical Knowledge |pages=386–413}}</ref><ref>{{Citation |last=Schiefsky |first=Mark |title=The Creation of Second-Order Knowledge in Ancient Greek Science as a Process in the Globalization of Knowledge |date=2012-07-20 |url=https://mprl-series.mpg.de/studies/1/12/index.html |work=The Globalization of Knowledge in History |series=MPRL – Studies |place=Berlin |publisher=Max-Planck-Gesellschaft zur Förderung der Wissenschaften |isbn=978-3-945561-23-2}}</ref>

The early history of Greek mathematics is obscure, and traditional narratives of [[Theorem|mathematical theorems]] found before the fifth century&nbsp;BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subject is the ''[[Elements (Euclid)|Elements]]'', written during the [[Hellenistic period]]. The works of renown mathematicians [[Archimedes]] and [[Apollonius of Perga|Apollonius]], as well as of the astronomer [[Hipparchus]], also belong to this period. In the [[Imperial Roman]] era, [[Ptolemy]] used trigonometry to determine the positions of stars in the sky, while [[Nicomachus]] and other ancient philosophers revived ancient [[number theory]] and harmonics. During [[late antiquity]], [[Pappus of Alexandria]] wrote his ''Collection'', summarizing the work of his predecessors, while [[Diophantus]]' ''[[Arithmetica]]'' dealt with the solution of arithmetic problems by way of pre-modern algebra. Later authors such as [[Theon of Alexandria]], his daughter [[Hypatia]], and [[Eutocius of Ascalon]] wrote commentaries on the authors making up the ancient Greek mathematical corpus.

The works of ancient Greek mathematicians were copied in the medieval Byzantine period and translated into Arabic and Latin, where they exerted influence on mathematics in the Islamic world and in Medieval Europe. During the [[Renaissance]], the texts of Euclid, Archimedes, Apollonius, and Pappus in particular went on to influence the development of [[early modern]] mathematics. Some problems in Ancient Greek mathematics were solved only in the modern era by mathematicians such as [[Carl Friedrich Gauss]], and attempts to prove or disprove Euclid's parallel line postulate spurred the development of [[non-Euclidean geometry]]. Ancient Greek mathematics was not limited to theoretical works but was also used in other activities, such as business transactions and in land mensuration, as evidenced by extant texts where [[Numerical analysis|computational procedures]] and practical considerations took more of a central role.{{sfn|Høyrup|1990}}