Editing Ancient Greek mathematics (section)

==Origins==
[[File:Cropped image of Pythagoras from Raphael's School of Athens.jpg|thumb|upright|[[Pythagoras]] with a tablet of ratios, detail from ''[[The School of Athens]]'' by [[Raphael]] (1509). Modern historians question whether Pythagoras made any mathematical discoveries such as the [[Pythagorean theorem]].]] 
The origins of Greek mathematics are not well understood.<ref name="LH">{{cite book|first=Luke|last=Hodgkin|title=A History of Mathematics: From Mesopotamia to Modernity|url=https://archive.org/details/historyofmathema0000hodg|url-access=registration|publisher=Oxford University Press|year=2005|isbn=978-0-19-852937-8|chapter=Greeks and origins}}</ref><ref>{{Cite book|last=Knorr|first=W.|title=On the early history of axiomatics: The interaction of mathematics and philosophy in Greek Antiquity.|publisher=D. Reidel Publishing Co.|year=1981|pages=145–186}} Theory Change, Ancient Axiomatics, and Galileo's Methodology, Vol. 1</ref> The earliest advanced civilizations in Greece were the [[Minoan]] and later [[Mycenaean Greece|Mycenaean]] civilizations, both of which flourished in the second half of the [[Bronze Age]]. While these civilizations possessed writing, and many [[Linear B]] tablets and similar objects have been deciphered, no mathematical writings have yet been discovered.{{sfn|Netz|2022|p=13}} The mathematics from the preceding Babylonian and Egyptian civilizations were primarily focused on land mensuration and accounting. Although some problems were contrived to be challenging beyond any obvious practical application, there are no signs of explicit theoretical concerns as found in Ancient Greek mathematics. It is generally thought that [[Babylonian mathematics|Babylonian]] and [[Ancient Egyptian mathematics|Egyptian mathematics]] had an influence on the younger Greek culture, possibly through an oral tradition of mathematical problems over the course of centuries, though no direct evidence of transmission is available.{{sfn|Høyrup|1990}}{{sfn|Netz|2022|pp=25-26}}

When Greek writing re-emerged in the 7th century BC, following the [[Late Bronze Age collapse]], it was based on an entirely new system derived from the [[Phoenician alphabet]], with Egyptian [[papyrus]] being the preferred medium.{{sfn|Netz|2022|pp=14-15}} Because the earliest known mathematical treatises in Greek, starting with [[Hippocrates of Chios]] in the 5th century&nbsp;BC, have been lost, the early history of Greek mathematics must be reconstructed from information passed down through later authors, beginning in the mid-4th century BC.{{sfn|Netz|2022}}{{sfn|Boyer|1991|pp=40–89}} Much of the knowledge about early Greek mathematics is thanks to references by Plato, Aristotle, and from quotations of [[Eudemus of Rhodes]]' histories of mathematics by later authors. These references provide near-contemporary accounts for many mathematicians active in the 4th century&nbsp;BC.{{sfn|Boyer|1991|pp=43-61}}{{sfn|Netz|2022|pp=89-90}} Euclid's ''Elements'' is also believed to contain many theorems that are attributed to mathematicians in the preceding centuries.{{sfn|Netz|2022|pp=120-121}}

=== Archaic period ===
Ancient Greek tradition attributes the origin of Greek mathematics to either [[Thales of Miletus]] (7th century BC), one of the legendary [[Seven Sages of Greece]], or to [[Pythagoras|Pythagoras of Samos]] (6th century BC), both of whom supposedly visited Egypt and Babylon and learned mathematics there.{{sfn|Boyer|1991|pp=43-61}} However, modern scholarship tends to be skeptical of such claims as neither Thales or Pythagoras left any writings that were available in the Classical period. Additionally, widespread literacy and the [[scribe|scribal culture]] that would have supported the transmission of mathematical treatises did not emerge fully until the 5th century; the [[oral literature]] of their time was primarily focused on public speeches and recitations of poetry.{{sfn|Netz|2022|pp=16-19}} The standard view among historians is that the discoveries Thales and Pythagoras are credited with, such as [[Thales's theorem|Thales' Theorem]], the [[Pythagorean theorem]], and the [[Platonic solids]], are the product of attributions by much later authors.{{sfn|Netz|2022|pp=16-17}} 

=== 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}}