Editing Ancient Greek mathematics (section)

=== Classical Greece ===
[[File:Lune.svg|thumb|One of the earliest documented results in Ancient Greek mathematics is the [[Lune of Hippocrates]], from the late 5th century BC. The shaded portion in the upper left is the same area as the shaded part of the triangle]]
The earliest traces of Greek mathematical treatises appear in the second half of the fifth century BC.{{sfn|Netz|2022}} According to Eudemus,<ref>s.v. Proclus, Commentary on Euclid's Elements</ref> [[Hippocrates of Chios]] was the first to write a book of ''Elements'' in the tradition later continued by Euclid.{{sfn|Fowler|1999|pp=382-383}} Fragments from another treatise written by Hippocrates on [[Lune of Hippocrates|lunes]] also survives, possibly as an attempt to [[square the circle]].<ref>s.v. [[Simplicius of Cilicia]], Commentary on Aristotle's Physics</ref> Eudemus' states that Hippocrates studied with an astronomer named [[Oenopides of Chios]]. Other mathematicians associated with Chios include Andron and Zenodotus, who may be associated with a "school of Oenopides" mentioned by Proclus.{{sfn|Netz|2022}}

Although many stories of the early Pythagoreans are likely apocryphal, including stories about people being drowned or exiled for sharing mathematical discoveries, some fifth-century Pythagoreans may have contributed to mathematics.{{sfn|Netz|2014}} Beginning with [[Philolaus of Croton]], a contemporary of [[Socrates]], studies in arithmetic, geometry, astronomy, and harmonics became increasingly associated with [[Pythagoreanism]]. Fragments of Philolaus' work are preserved in quotations from later authors.{{sfn|Netz|2014}} Aristotle is one of the earliest authors to associate Pythagoreanism with mathematics, though he never attributed anything specifically to Pythagoras.<ref>{{cite book |last1=Tredennick |first1=Hugh |url=https://archive.org/details/in.ernet.dli.2015.185284/page/n65/mode/2up |title=Aristotle The Metaphysics |date=1923 |publisher=Heinemann |page=66 |access-date=27 April 2025}}</ref><ref>{{Cite journal |last=Cornelli |first=Gabriele |date=2016-05-20 |title=A review of Aristotle's claim regarding Pythagoreans fundamental Beliefs: All is number? |url=http://revistas.unisinos.br/index.php/filosofia/article/view/fsu.2016.171.06 |journal=Filosofia Unisinos |volume=17 |issue=1 |pages=50–57 |doi=10.4013/fsu.2016.171.06 |doi-access=free}}</ref><ref>Hans-Joachim Waschkies, "Introduction" to "Part 1: The Beginning of Greek Mathematics" in ''Classics in the History of Greek Mathematics'', pp. 11–12</ref>

Other extant evidence shows fifth-century philosophers' acquaintance with mathematics: [[Antiphon (orator)|Antiphon]] claimed to be able to construct a rectilinear figure with the same area as a given circle, while [[Hippias]] is credited with [[Quadratrix of Hippias|a method]] for squaring a circle with a neusis construction. [[Protagoras]] and [[Democritus]] debated the possibility for [[Tangent|a line to intersect a circle at a single point]]. According to Archimedes, Democritus also asserted, apparently without proof, that the area of a cone was 1/3 the area of a cylinder with the same base, a result which was later proved by [[Eudoxus of Cnidus]].{{sfn|Netz|2022}}

==== Mathematics in the time of Plato ====
While Plato was not a mathematician, numerous early mathematicians were associated with [[Plato]] or with his [[Platonic Academy|Academy]]. Familiarity with mathematicians' work is also reflected in several Platonic dialogues were mathematics are mentioned, including the ''[[Meno]]'', the ''[[Theaetetus (dialogue)|Theaetetus]]'', the ''[[Republic]]'', and the ''[[Timaeus (dialogue)|Timaeus]]''.{{sfn|Fowler|1999}}

[[Archytas]], a Pythagorean philosopher from Tarentum, was a friend of Plato who made several contributions to mathematics, including solving the problem of [[doubling the cube]], now known to be impossible with only a compass and a straightedge, using an alternative method. He also systematized the [[Pythagorean means|study of means]], and possibly worked on optics and mechanics.<ref>{{Cite journal |last=Burnyeat |first=M. F. |date=2005 |title=Archytas and Optics |url=https://www.cambridge.org/core/journals/science-in-context/article/abs/archytas-and-optics/BDBF3868CEF7004C16547836D66A4F24 |journal=Science in Context |volume=18 |issue=1 |pages=35–53 |doi=10.1017/S0269889705000347 |doi-broken-date=16 December 2024}}</ref> Archytas has been credited with early material found in Books VII–IX of the ''Elements'', which deal with [[elementary number theory]].{{sfn|Netz|2014}}

[[Theaetetus (mathematician)|Theaetetus]] is one of the main characters in the Platonic [[Theaetetus (dialogue)|dialogue named after him]], where he works on a problem given to him by [[Theodorus of Cyrene]] to demonstrate that the square roots of several numbers from 3 to 17 are irrational, leading to the construction now known as the [[Spiral of Theodorus]]. Theaetetus is traditionally credited with much of the work contained in Book X of the ''Elements'', concerned with [[incommensurable magnitudes]], and Book XIII, which outlines the construction of the [[regular polyhedra]]. Although some of the regular polyhedra were certainly known previously, he is credited with their systematic study and the proof that only five of them exist.<ref>Elements Book XIII, Proposition 18</ref>{{sfn|Acerbi|2018|pp=277-278}}

Another mathematician who might have visited Plato's Academy is [[Eudoxus of Cnidus]], associated with the theory of proportion found in Book V of the ''Elements''. [[Archimedes]] credits Eudoxus with a proof that the volume of a cone is one-third the volume of a cylinder with the same base, which appears in two propositions in Book XII of the ''Elements''.{{sfn|Acerbi|2018|p=279}} He also developed an astronomical calendar, now lost, that remains partially preserved in [[Aratus]]' poem ''[[Phaenomena]].''{{sfn|Netz|2022}} Eudoxus seems to have founded a school of mathematics in [[Cyzicus]], where one of Eudoxus' students, [[Menaechmus]], went on to develop a theory of conic sections.{{sfn|Netz|2022}}