Editing Ancient Greek mathematics

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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}}

==Etymology==
The Greek word {{tlit|grc|mathēmatikē}} ({{lang|grc|μαθηματική}}) derives from {{tlit|grc|máthēma}} ({{lang|grc|μάθημα}} 'lesson'), and ultimately from the verb {{tlit|grc|manthánō}} ({{lang|grc|μανθάνω}} 'I learn'). Strictly speaking, a {{tlit|grc|máthēma}} could be any branch of learning, or anything learnt; however, since antiquity certain {{tlit|grc|mathēmatá}} were granted special status: [[arithmetic]], [[geometry]], [[astronomy]], and [[harmonics]].{{NoteTag|Arithmetic, which dealt with numbers, included not only basic operations of addition, subtraction, multiplication, and division, but also what we would now consider algebra and number theory. Geometry ({{lit.|land mensuration}}) included not only plane and solid geometry and the theory of conic sections, but also optics. Astronomy dealt with phenomena related to the stars and the five planets, and fostered the development of astronomical models and trigonometry. Harmonics dealt primarily with the theory of music scales using means and ratios.}} These four {{tlit|grc|mathēmatá}}, which appear listed together around the time of Archytas and Plato, would later become the medieval [[quadrivium]].<ref>{{cite journal |author=Heath |year=1931 |title=A Manual of Greek Mathematics |journal=Nature |volume=128 |issue=3235 |page=[https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA5 5] |bibcode=1931Natur.128..739T |doi=10.1038/128739a0 }}</ref><ref>{{Cite web |last=Furner |first=J. |date=2020 |title=Classification of the sciences in Greco-Roman antiquity |url=https://www.isko.org/cyclo/greco-roman.htm |access-date=2023-01-09 |website=www.isko.org}}</ref>

==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}}

==Hellenistic and early Roman period==
Ancient Greek mathematics reached its acme during the [[Hellenistic period|Hellenistic]] and early [[Roman Empire|Roman periods]]. [[Alexander the Great|Alexander the Great's]] conquest of the [[Eastern Mediterranean]], [[ancient Egypt|Egypt]], [[Mesopotamia]], the [[Iranian plateau]], [[Central Asia]], and parts of [[India]] led to the spread of the Greek culture and language across these regions. [[Koine Greek]] became the ''[[lingua franca]]'' of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with [[Egyptian mathematics|Egyptian]] and [[Babylonian mathematics]] to give rise to Hellenistic mathematics.<ref>{{Cite book |last=Green |first=P. |title=Alexander to Actium: The Historical Evolution of the Hellenistic Age|date=1990|publisher=University of California Press|isbn=978-0-520-08349-3|edition=1 |jstor=10.1525/j.ctt130jt89 }}</ref><ref>{{Citation|last=Russo|first=L.|title=Hellenistic Mathematics|date=2004 |work=The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn|pages=31–55|place=Berlin, Heidelberg|publisher=Springer |doi=10.1007/978-3-642-18904-3_3 |isbn=978-3-642-18904-3 }}</ref> Several centers of learning also appeared around this time, of which the most important one was the [[Mouseion]] in [[Alexandria]], in [[Ptolemaic Egypt]].{{sfn|Acerbi|2018}} 

Although few in number, Hellenistic mathematicians actively communicated with each other in correspondence; publication consisted of passing and copying someone's work among colleagues.{{sfn|Acerbi|2018}} Much of the work represented by authors such as [[Euclid]], [[Archimedes]], [[Apollonius of Perga|Apollonius]], and [[Ptolemy]] was of a very advanced level and rarely mastered outside a small circle.{{sfn|Høyrup|1990}} 

Euclid collected many previous mathematical results and theorems in the ''[[Euclid's Elements|Elements]]'', a reference work that would become a canon of geometry and elementary number theory for many centuries.{{sfn|Acerbi|2018}} Archimedes used the [[method of exhaustion]] to approximate Pi (''[[Measurement of a Circle]]''), measured the surface area and volume of a sphere (''[[On the Sphere and Cylinder]]''),{{sfn|Acerbi|2018}} devised a mechanical method for developing solutions to mathematical problems using the [[law of the lever]], (''[[Method of Mechanical Theorems]]''),{{sfn|Acerbi|2018}} and developed a way to represent very large numbers (''[[The Sand-Reckoner]]'').<ref>{{Cite journal|last=Reviel Netz|date=2003 |title=The Goal of Archimedes' Sand Reckoner |journal=Apeiron |volume=36 |issue=4 |pages=251–290 |doi=10.1515/APEIRON.2003.36.4.251 }}</ref> Apollonius of Perga, in his extant work ''[[On Conic Sections|Conics]]'', refined and developed the theory of [[conic section]]s that was first outlined by [[Menaechmus]], Euclid, and [[Conon of Samos]].{{sfn|Acerbi|2018}} [[Trigonometry]] was developed around the time of the astronomer [[Hipparchus]],<ref name="10.1111_j.1600-0498.1974.tb00205.x">{{Cite journal|last=Toomer|first=G. J.|date=1974|title=The Chord Table of Hipparchus and the Early History of Greek Trigonometry |journal=Centaurus |volume=18 |issue=1 |pages=6–28 |doi=10.1111/j.1600-0498.1974.tb00205.x }}</ref> and both trigonometry and astronomy were further developed by Ptolemy in his ''[[Almagest]]''.

=== Arithmetic ===
Euclid devoted part of his ''[[Euclid's Elements|Elements]]'' (Books VII–IX) to topics that belong to elementary number theory, including [[Prime number|prime numbers]] and [[Divisibility rule|divisibility]]. He gave an algorithm, the [[Euclidean algorithm]], for computing the [[greatest common divisor]] of two numbers (Prop. VII.2) and a [[Euclid's theorem|proof implying the infinitude of primes]] (Prop. IX.20). There is also older material likely based on Pythagorean teachings (Prop. IX.21–34), such as "odd times even is even" and "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it". Ancient Greek mathematicians conventionally separated ''numbers'' (mostly positive integers but occasionally rationals) from ''magnitudes'' or ''lengths'', with only the former being the subject of arithmetic.

The Pythagorean tradition spoke of so-called [[Polygonal number|polygonal]] or [[figurate numbers]]. The study of the sums of triangular and pentagonal numbers would prove fruitful in the [[early modern period]]. Building on the works of the earlier Pythagoreans, [[Nicomachus of Gerasa]] wrote an ''Introduction to Arithmetic'' which would go on to receive later commentary in late antiquity and the Middle Ages. The continuing influence of mathematics in Platonism is shown in [[Theon of Smyrna|Theon of Smyrna's]] ''Mathematics Useful For Understanding Plato'', written around the same time. [[Diophantus]] also wrote on [[polygonal number]]s in addition to a work in pre-modern algebra (''[[Arithmetica]]'').<ref>{{Cite journal |last=Acerbi |first=F. |date=2011 |title=Completing Diophantus, De polygonis numeris, prop. 5 |journal=Historia Mathematica |volume=38 |issue=4 |pages=548–560 |doi=10.1016/j.hm.2011.05.002 |doi-access=free}}</ref><ref>{{Cite journal |last1=Christianidis |first1=J. |last2=Oaks |first2=J. |date=2013 |title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria |journal=Historia Mathematica |volume=40 |issue=2 |pages=127–163 |doi=10.1016/j.hm.2012.09.001 |doi-access=free}}</ref>

An [[epigram]] published by [[Gotthold Ephraim Lessing|Lessing]] in 1773 appears to be a letter sent by [[Archimedes]] to [[Eratosthenes]]. The epigram proposed what has become known as [[Archimedes's cattle problem]]; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed [[Pell's equation]]). As far as it is known, such equations were first successfully treated by Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

=== Geometry ===
{{main|Straightedge and compass construction}}
[[File:Platonic Solids Transparent.svg|thumb|The construction of the [[Platonic solids]], from Book XIII of the ''Elements'', is often credited to Theaetetus, who was active around the time of Plato]]
During the Hellenistic age, three construction problems in geometry became famous: [[doubling the cube]], [[angle trisection|trisecting an angle]], and [[squaring the circle]], all of which are now known to be impossible with a straight edge and compass. Many attempts were made using [[neusis]] constructions including the [[Cissoid of Diocles]], [[Quadratrix]], and the [[Conchoid (mathematics)|Conchoid]] of Nicomedes.{{sfn|Knorr|1986}} Regular polygons and polyhedra had already been known before Euclid's ''Elements'', but Archimedes extended their study to include semiregular polyhedra, also known as [[Archimedean solid]]s. A work transmitted as Book XIV of Euclid's ''Elements'', likely written a few centuries later by [[Hypsicles]], lists other works on the topic, such [[Aristaeus the Elder]]'s ''Comparison of Five Figures'' and Apollonius of Perga's ''Comparison of the Dodecahedron and the Icosahedron''.{{sfn|Acerbi|2018}} Another book, transmitted as Book XV of Euclid's ''Elements'', which was compiled in the 6th century AD, provides further developments.{{sfn|Acerbi|2018}}

Most of the works that became part of a standard mathematical curriculum in late antiquity were composed during the Hellenistic period: ''[[Data (Euclid)|Data]]'' and ''[[Porisms]]'' by Euclid, several works by Apollonius of Perga including ''Cutting off a ratio'', ''Cutting off an area'', ''Determinate section'', ''Tangencies'', and ''Neusis'', and several works dealing with [[Locus (mathematics)|loci]], including ''Plane Loci'' and ''Conics'' by Apollonius, ''Solid Loci'' by [[Aristaeus the Elder]], ''Loci on a Surface'' by Euclid, and ''On Means'' by [[Eratosthenes of Cyrene]]. All of these works other than ''Data'', ''Conics'' Books I–VII, and ''Cutting off a ratio'' are lost but are known from Book 7 of [[Pappus of Alexandria|Pappus]]' ''Collection''.<ref>{{cite book |last1=Pappus |first1=of Alexandria |title=Book 7 of the Collection |date=1986 |location=New York |publisher=Springer |isbn=978-0-387-96257-3 |url=https://archive.org/details/book7ofcollectio0000papp |access-date=4 May 2025}}</ref>

=== Applied mathematics ===
Astronomy was considered one of the {{tlit|grc|mathēmatá}}, and accordingly many mathematicians devoted time to astronomy. The ''[[Little Astronomy]]'' is a collection of short works, including [[Theodosius of Bithynia|Theodosius]]'s [[Theodosius' Spherics|''Spherics'']], [[Autolycus of Pitane|Autolycus]]'s ''On the Moving Sphere'', Euclid's ''Optics'' and ''Phaenomena'', [[Aristarchus of Samos|Aristarchus]]'s ''[[On the Sizes and Distances (Aristarchus)|On the Sizes and Distances]]'', that were part of an astronomy curriculum beginning in the 2nd century AD and transmitted as a group. The collection was translated into Arabic with a few additions such as Euclid's ''Data'', [[Menelaus of Alexandria|Menelaus]]'s ''Spherics'' (extant in Arabic only), and various works by Archimedes as the ''Middle Books'', intermediate between Euclid's ''Elements'' and Ptolemy's ''[[Almagest]]''.<ref>{{cite book |last=Evans |first=James |url=https://archive.org/details/historypracticeo0000evan/page/89/mode/1up?q=%22little+astronomy%22 |title=The History & Practice of Ancient Astronomy |publisher=Oxford University Press |year=1998 |isbn=0-19-509539-1 |at="The ''Little Astronomy''", {{pgs|89–91}} |url-access=limited}} {{pb}} {{cite thesis |last=Roughan |first=Christine |title=The Little Astronomy and Middle Books between the 2nd and 13th Centuries CE: Transmissions of Astronomical Curricula |type=PhD thesis |publisher=New York University |url=https://archive.nyu.edu/handle/2451/64391 |year=2023}} </ref>{{sfn|Acerbi|2018|pp=284-285}}

The development of [[trigonometry]] as a synthesis of Babylonian and Greek methods is commonly attributed to [[Hipparchus]], who made extensive astronomical observations and wrote several mathematical treatises, though only his ''Commentary on the Phaenomena of Eudoxus and Aratus'' survives.<ref name="10.1111_j.1600-0498.1974.tb00205.x" /><ref>{{Cite journal |last=Duke |first=D. |date=2011 |title=The very early history of trigonometry. |url=https://people.sc.fsu.edu/~dduke/earlytrig12.pdf |journal=DIO: The International Journal of Scientific History |volume=17 |pages=34–42}}</ref><ref name=":3">{{Cite web |last=Jones |first=A. |date=1994 |title=Greek mathematics to AD 300 |url=https://www.routledge.com/Companion-Encyclopedia-of-the-History-and-Philosophy-of-the-Mathematical/Grattan-Guiness/p/book/9781138688117 |access-date=2021-05-26 |website=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences: Volume One |pages=46–57}}</ref> In the 2nd century AD, Ptolemy wrote a work now called the ''[[Almagest]]'' explaining the motions of the stars and planets according to a geocentric model, and calculated out chord tables to a higher degree of precision than had been done previously, along with an instruction manual, ''[[Handy Tables]]''.<ref>{{Cite web |last=Lambrou |first=M. |date=2003 |title=Theon of Alexandria and Hypatia |url=https://www.historyoftheancientworld.com/2012/01/theon-of-alexandria-and-hypatia/ |access-date=2021-05-26 |website=History of the Ancient World}}</ref><ref>{{Cite book |last=Tihon |url=https://books.google.com/books?id=r1QazgEACAAJ |title=Ptolemaiou Procheiroi Kanones. Ptolemy's Handy Tables. Volume 1b: Tables A1-A2. Transcription and Commentary |date=2011 |publisher=Peeters |isbn=978-2-7584-0117-9}}</ref>

Ancient Greeks often considered the study of optics to be a part of applied geometry.{{sfn|Acerbi|2018|pp=281–282}} An extant work on [[catoptrics]] is dubiously attributed to Euclid. Archimedes is known to have written a now lost work on catoptrics, while [[Diocles (mathematician)|Diocles]]' ''On Burning Mirrors'' is extant in an Arabic translation.{{sfn|Acerbi|2018}} Other examples of [[applied mathematics]] around this time include the construction of analogue computers like the [[Antikythera mechanism]], the accurate measurement of the [[circumference of the Earth]] by [[Eratosthenes]], and the mathematical and mechanical works of [[Heron of Alexandria|Heron]].<ref>{{Cite journal |last=Edmunds |first=M. G. |date=2014-10-02 |title=The Antikythera mechanism and the mechanical universe |journal=Contemporary Physics |volume=55 |issue=4 |pages=263–285 |doi=10.1080/00107514.2014.927280}}</ref><ref>Russo, Lucio (2004). ''The Forgotten Revolution''. Berlin: Springer. pp. 273–277.</ref><ref>{{Cite journal |last=Tybjerg |first=Karin |date=2004-12-01 |title=Hero of Alexandria's Mechanical Geometry |journal=Apeiron |volume=37 |issue=4 |pages=29–56 |doi=10.1515/APEIRON.2004.37.4.29}}</ref>
==Mathematics in late antiquity==
The mathematicians in the later Roman era from the 4th century onward generally had few notable original works, however, they are distinguished for their commentaries and expositions on the works of earlier mathematicians. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity.{{sfn|Mansfeld|2016}}

=== Pappus' ''Collection''===
[[Pappus of Alexandria]] compiled a canon of results of earlier mathematics in the ''Collection'' in eight books, of which part of book II and books III–VII are extant in Greek and book VIII is extant in Arabic. The collection attempts to sum up the whole of Ancient Greek mathematics up to that time as interpreted by Pappus: Book III is framed as a letter to [[Pandrosion]], a mathematician in Athens, and discusses three construction problems and attempts to solve them: [[doubling the cube]], [[angle trisection]], and [[squaring the circle]]. Book IV discusses classical geometry, which Pappus divides into plane geometry, Line geometry, and Solid geometry, and includes a discussion of Archimedes' construction of the [[Arbelos]], otherwise only known via a pseudo-Archimedean work, [[Book of Lemmas]]. Book V discusses isoperimetric figures, summarizing otherwise lost works by [[Zenodotus]] and [[Archimedes]] on isoperimetric plane figures and solid figures, respectively. Book VI deals with astronomy, providing commentary on some of the works of the Little Astronomy corpus. Book VII deals with analysis, providing epitomes and lemmas from otherwise lost works. Book VIII deals with mechanics. The Greek version breaks off in the middle of a sentence discussing [[Hero of Alexandria]], but a complete edition of the book survives in Arabic.<ref>{{cite book |last1=Pappus |first1=of Alexandria |title=Book 7 of the Collection |date=1986 |publisher=Springer-Verlag |isbn=978-0-387-96257-3 |url=https://archive.org/details/book7ofcollectio0000papp |access-date=4 May 2025}}</ref>

=== Commentaries ===
The commentary tradition, which had begun during the Hellenistic period, continued into late antiquity. The first known commentary on the ''Elements'' was written by [[Hero of Alexandria]], who likely set the format for future commentaries. [[Serenus of Antinoöpolis]] wrote a lost commentary on the ''Conics'' of Apollonius, along with two works that survive, ''Section of a Cylinder'' and ''Section of a Cone'', expanding on specific subjects in the ''Conics''.{{sfn|Acerbi|2018|p=274}} Pappus wrote a commentary on Book X of the elements, dealing with incommensurable magnitudes. [[Heliodorus of Larissa]] wrote a summary of the Optics.{{sfn|Netz|2022}}

Many of the late antique commentators were associated with Neoplatonist philosophy; [[Porphyry of Tyre]], a student of Plotinus, the founder of [[Neoplatonism]], wrote a commentary on Ptolemy's ''Harmonics''. [[Iamblichus]], who was himself a student of Porphyry, wrote a commentary on Nicomachus' Introduction to Arithmetic. In Alexandria in the 4th century, [[Theon of Alexandria]] wrote commentaries on the writings of [[Ptolemy]], including a commentary on the ''Almagest'' and two commentaries on the ''Handy Tables'', one of which is more of an instruction manual ("Little Commentary"), and another with a much more detailed exposition and derivations ("Great Commentary"). [[Hypatia]], Theon's daughter, also wrote a commentary on Diophantus' ''Arithmetica'' and a commentary on the ''Conics'' of Apollonius, which have not survived.{{sfn|Cameron|1990}} 

In the 5th century, in Athens, [[Proclus]] wrote a commentary on Euclid's elements, which the first book survives. Proclus' contemporary, [[Domninus of Larissa]], wrote a summary of Nicomachus' Introduction to Arithmetic, while [[Marinus of Neapolis]], Proclus' successor, wrote an ''Introduction to Euclid's Data''. Meanwhile in Alexandria, [[Ammonius Hermiae]], [[John Philoponus]] and [[Simplicius of Cilicia]] wrote commentaries on the works of Aristotle that preserve information on earlier mathematicians and philosophers. [[Eutocius of Ascalon]] (c.&nbsp;480–540), another student of Ammonius, wrote commentaries that are extant on Apollonius' ''Conics'' along with some treatises of Archimedes: ''On the Sphere and Cylinder'', ''Measurement of a Circle'', and ''On Balancing Planes'' (though the authorship of the last one is disputed).{{sfn|Netz|2022|pp=427-428}} In Rome, Boethius, seeking to preserve Ancient Greek philosophical, translated works on the [[quadrivium]] into Latin, deriving much of his work on Arithmetic and Harmonics from Nicomachus.{{sfn|Netz|2022|pp=429-430}}

After the closure of the Neoplatonic schools by the emperor [[Justinian]] in 529 AD, the institution of mathematics as a formal enterprise entered a decline. However, two mathematicians connected to the Neoplatonic tradition were commissioned to build the [[Hagia Sophia]]: [[Anthemius of Tralles]] and [[Isidore of Miletus]]. Anthemius constructed many advanced mechanisms and wrote a work ''On Surprising Mechanisms'' which treats "burning mirrors" and skeptically attempts to explain the function of [[Archimedes' heat ray]]. Isidore, who continued the project of the Hagia Sophia after Anthemius' death, also supervised the revision of Eutocius' commentaries of Archimedes. From someone in Isidore's circle we also have a work on polyhedra that is transmitted pseudepigraphically as ''Book XV'' of Euclid's ''Elements.''{{sfn|Netz|2022|pp=432-433}} 

==Reception and legacy==
[[File:P. Oxy. I 29.jpg|thumb|upright|A papyrus fragment ([[Papyrus Oxyrhynchus 29|P. Oxy. 29]]) from [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' Book II, dated to approximately 100 AD.]]
The majority of mathematical treatises written in Ancient Greek, along with the discoveries made within them, have been lost; around 30% of the works known from references to them are extant.<ref>{{cite journal |last1=Acerbi |first1=Fabio |last2=Masià |first2=Ramon |title=The Greek Mathematical Corpus: a Quantitative Appraisal |journal=Histoire & mesure |date=30 June 2022 |volume=XXXVII |issue=1 |pages=15–36 |doi=10.4000/histoiremesure.15779}}</ref> Authors whose works survive in Greek manuscripts include: [[Euclid]], [[Autolycus of Pitane]], [[Archimedes]], [[Aristarchus of Samos]], [[Philo of Byzantium]], [[Biton of Pergamon]], [[Apollonius of Perga]], [[Hipparchus]], [[Theodosius of Bithynia]], [[Hypsicles]], [[Athenaeus Mechanicus]], [[Geminus]], [[Hero of Alexandria]], [[Apollodorus of Damascus]], [[Theon of Smyrna]], [[Cleomedes]], [[Nicomachus]], [[Ptolemy]], [[Cleonides]], [[Gaudentius (music theorist)|Gaudentius]], [[Anatolius of Laodicea]], [[Aristides Quintilian]], [[Porphyry (philosopher)|Porphyry]], [[Diophantus]], [[Alypius of Alexandria|Alypius]], [[Heliodorus of Larissa]], [[Pappus of Alexandria]], [[Serenus of Antinoöpolis]], [[Theon of Alexandria]], [[Proclus]], [[Marinus of Neapolis]], [[Domninus of Larissa]], [[Anthemius of Tralles]], and [[Eutocius]]. 

The earliest surviving papyrus to record a Greek mathematical text is P. Hib. i 27, which contains a parapegma of Eudoxus' astronomical calendar, along with several [[Elephantine papyri and ostraca|ostraca]] from the 3rd century BC that deal with propositions XIII.10 and XIII.16 of Euclid's ''Elements''.{{sfn|Fowler|1999|pp=209}} A papyrus recovered from [[Herculaneum]]<ref>P. Herc. 1061</ref> contains an essay by the Epicurean philosopher [[Demetrius Lacon]] on Euclid's Elements.{{sfn|Fowler|1999|pp=210}} 

Most of the oldest extant manuscripts for each text date from the 9th century onward, copies of works written during and before the Hellenistic period.<ref>{{cite web|url= http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Greek_sources_1.html|title= How do we know about Greek mathematics?|author= J J O'Connor and E F Robertson|date= October 1999|work= The MacTutor History of Mathematics archive|publisher= University of St. Andrews|access-date= 18 April 2011|archive-date= 30 January 2000|archive-url= https://web.archive.org/web/20000130113411/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Greek_sources_1.html|url-status= dead}}</ref> The two major sources of manuscripts are Byzantine-era codices, copied some 500 to 1500 years after their originals, and [[Graeco-Arabic translation movement|Arabic]] translations of Greek works; what has survived reflects the preferences of readers in late antiquity along with the interests of mathematicians in the Byzantine empire and the medieval Islamic world who preserved and copied them.{{sfn|Høyrup|1990}}

Despite the lack of original manuscripts, the dates for some Greek mathematicians are more certain than the dates of surviving Babylonian or Egyptian sources because a number of overlapping chronologies exist, though many dates remain uncertain.

===Byzantine mathematics===
With the closure of the Neoplatonist schools in the 6th century, Greek mathematics declined in the medieval Byzantine period, although many works were preserved in medieval manuscript transmission and translated into first [[Syriac language|Syriac]] and [[Arabic]], and later into Latin.{{sfn|Netz|2022}}  The transition to [[miniscule]] manuscript in the 9th century, however, many works that were not copied during this time period were lost, although a few uncial manuscripts do survive. Many surviving works are derived from only a single manuscript; such as  Pappus' ''Collection'' and Books I–IV of the Conics.{{sfn|Acerbi|2018}} Many of the surviving manuscripts originate from two scholars in this period in the circle of [[Photios I]], [[Leo the Mathematician]] and [[Arethas of Caesarea]]. [[Scholia]] written in the margins of Euclid's elements that have been copied throughout multiple extant manuscripts that were also written by Arethas, derived from Proclus' commentary along with many commentaries on Euclid which are now lost. The works of Archimedes survived in three different recensions in manuscripts from the 9th and 10th centuries; two of which are now lost after being copied, the third of which, the [[Archimedes Palimpsest]], was only rediscovered in 1906.{{sfn|Netz|2022}}

In the later Byzantine period, [[George Pachymeres]] wrote a summary of the quadrivium, and [[Maximus Planudes]] wrote scholia on the first two books of ''Diophantus.''{{sfn|Netz|2022}}

===Medieval Islamic mathematics===
Numerous mathematical treatises were translated into Arabic in the 9th century; many works that are only extent today in Arabic translation, and there is evidence for several more that have since been lost.<ref>{{cite journal |last1=Lorch |first1=Richard |title=Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages |journal=Science in Context |date=June 2001 |volume=14 |issue=1–2 |pages=313–331 |doi=10.1017/S0269889701000114 |url=https://epub.ub.uni-muenchen.de/15929/ }}</ref><ref>{{cite journal |last1=Toomer |first1=G. J. |title=Lost greek mathematical works in arabic translation |journal=The Mathematical Intelligencer |date=January 1984 |volume=6 |issue=2 |pages=32–38 |doi=10.1007/BF03024153}}</ref>

Medieval Islamic scientists such as [[Alhazen]] developed the ideas of the Ancient Greek geometry into advanced theories in optics and astronomy, and Diophantus' ''Arithmetica'' was synthezied with the works of [[Al-Khwarizmi]] and works from [[Indian mathematics]] to develop a theory of [[algebra]].{{sfn|Netz|2022}}

The following works are extant only in Arabic translations:{{sfn|Høyrup|1990|pp=1-2}}

* Apollonius, ''Conics'' books V–VII, ''Cutting Off of a Ratio''
* Archimedes, ''[[Book of Lemmas]]''
* [[Diocles (mathematician)|Diocles]], ''On Burning Mirrors''
* Diophantus, ''[[Arithmetica]]'' books IV–VII
* Euclid, ''On Divisions of Figures'', ''On Weights''
* [[Menelaus of Alexandria|Menelaus]], ''Sphaerica''
* Hero, ''Catoptrica'', ''Mechanica''
* Pappus, ''Commentary on Euclid's Elements book X'', ''Collection'' Book VIII
* Ptolemy, ''[[Planisphaerium]]'', 

Additionally, the work ''[[Optics (Ptolemy)|Optics]]'' by Ptolemy only survives in a [[Latin translations of the 12th century|Latin translations]] of the Arabic translation of a Greek original.

=== In Latin Medieval Europe ===
[[File:Diophantus-cover.png|thumb|upright|Cover of Diophantus' ''[[Arithmetica]]'' in Latin]]
The works derived from Ancient Greek mathematical writings that had been written in late antiquity by [[Boethius]] and [[Martianus Capella]] had formed the basis of early medieval quadrivium of arithmetic, geometry, astronomy, and music. In the 12th century the original works of Ancient Greek mathematics were translated into Latin first from Arabic by [[Gerard of Cremona]], and then from the original Greek a century later by [[William of Moerbeke]].{{sfn|Netz|2022}}

===Renaissance===
The publication of Greek mathematical works increased their audience; Pappus's collection was published in 1588, Diophantus in 1621.  Diophantus would go on to influence [[Pierre de Fermat]]'s work on number theory; Fermat scribbled his famous note about [[Fermat's Last theorem]] in his copy of ''Arithmetica''. Descartes, working through the [[Problem of Apollonius]] from his edition of Pappus, proved what is now called [[Descartes' theorem]] and laid the foundations for [[Analytic geometry]].{{sfn|Netz|2022}}

===Modern mathematics===
Ancient Greek mathematics constitutes an important period in the [[history of mathematics]]: fundamental in respect of [[geometry]] and for the idea of [[formal proof]].<ref>{{Citation|last1=Grant|first1=H.|title=Axiomatics—Euclid's and Hilbert's: From Material to Formal|date=2015 |work=Turning Points in the History of Mathematics|pages=1–8|publisher=Springer|doi=10.1007/978-1-4939-3264-1_1|isbn=978-1-4939-3264-1|last2=Kleiner|first2=I.|series=Compact Textbooks in Mathematics }}</ref>  Greek mathematicians also contributed to [[number theory]], [[Theoretical astronomy|mathematical astronomy]], [[combinatorics]], [[mathematical physics]], and, at times, approached ideas close to the [[integral calculus]].{{sfn|Knorr|1996|pages=67-88}}<ref>Powers, J. (2020). Did Archimedes do calculus? ''History of Mathematics Special Interest Group of the MAA'' [https://homsigmaa.net/wp-content/uploads/2020/05/Jeffery-Powers-1.pdf]</ref>

[[Richard Dedekind]] acknowledged Eudoxus's theory of proportion as an inspiration for the [[Dedekind cut]], a method of contructing the [[real number]]s.<ref>{{Cite journal|last=Stein|first=Howard| date=1990 |title=Eudoxos and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics |journal=Synthese |volume=84 |issue=2 |pages=163–211 |doi=10.1007/BF00485377 }}</ref>

== See also ==
{{portal|Greece|Mathematics}}
* {{annotated link|Timeline of ancient Greek mathematicians}}
* {{annotated link|List of Greek mathematicians}}
* {{annotated link|Music of ancient Greece}}

==Notes==
=== Footnotes ===
{{NoteFoot}}
=== Citations ===
{{reflist}}

==References==
*{{Cite encyclopedia|last=Acerbi|first=Fabio |editor1-first=Paul T|editor1-last=Keyser|editor2-first=John| editor2-last=Scarborough| date=2018| title=Hellenistic Mathematics | encyclopedia = Oxford Handbook of Science and Medicine in the Classical World | url=https://www.academia.edu/36286615/Hellenistic_Mathematics |access-date=2021-05-26| pages=268–292| doi=10.1093/oxfordhb/9780199734146.013.69| isbn=978-0-19-973414-6}}
*{{Cite book |first1=Carl B. |last1=Boyer |author-link=Carl Benjamin Boyer | title=A History of Mathematics |edition=3rd |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=978-0-471-54397-8}}
*{{Cite journal |last=Cameron |first=A. |date=1990 |title=Isidore of Miletus and Hypatia: On the Editing of Mathematical Texts |url=https://grbs.library.duke.edu/article/view/4171 |journal=Greek, Roman, and Byzantine Studies|volume=31|issue=1|pages=103–127}}
*{{cite book|last1= Fowler| first1= D. H. | title = The Mathematics of Plato's Academy |publisher = Clarendon Press |date=1999| edition = 2nd}}
*{{Citation |last=Høyrup |first=J. |year=1990 |title=''"Sub-scientific mathematics: Undercurrents and missing links in the mathematical technology of the Hellenistic and Roman world"'' |type=Unpublished manuscript, written for ''[[Aufstieg und Niedergang der römischen Welt]]'' |url=http://webhotel4.ruc.dk/~jensh/Publications/1990%7bg%7d_Undercurrents.PDF }}
*{{cite book|last1=Knorr |first1 = Wilbur R. |date=1986 |title = The Ancient Tradition of Geometric Problems| url= https://books.google.com/books?id=_poUuMFvA3oC}}
*{{Cite encyclopedia|last=Knorr|first=Wilbur R.|title=The method of indivisibles in Ancient Geometry|publisher=MAA Press|year=1996|encyclopedia=Vita Mathematica|pages=67–86}}
* {{Cite book|author1-link=Jaap Mansfeld|last=Mansfeld|first=J.|url=https://brill.com/view/title/6580|title=Prolegomena Mathematica: From Apollonius of Perga to the Late Neoplatonism. With an Appendix on Pappus and the History of Platonism|date=2016|publisher=Brill|isbn=978-90-04-32105-2}}
*{{cite book |last1=Netz |first1=Reviel |title=A New History of Greek Mathematics |date=2022 |publisher=Cambridge University Press |isbn=978-1-108-83384-4}}
*{{Cite encyclopedia |last=Netz |first=Reviel |title=The problem of Pythagorean mathematics| date=2014| doi=10.1017/CBO9781139028172.009 |encyclopedia=A History of Pythagoreanism |pages=167–184 |editor-last=Huffman |editor-first=Carl A. |publisher=Cambridge University Press |isbn=978-1-107-01439-8 }}
*{{Cite encyclopedia |last=Schofield |first=Malcolm |title=Archytas| date=2014| doi=10.1017/CBO9781139028172.009 |encyclopedia=A History of Pythagoreanism |pages=69–87 |editor-last=Huffman |editor-first=Carl A. |publisher=Cambridge University Press |isbn=978-1-107-01439-8 }}

==Further reading==
* A. Barker, Porphyry’s Commentary on Ptolemy’s Harmonics
* A. Barker, Greek Musical Writings, Vol. 2: Harmonic and Acoustic Theory
* A. Bernard, “Ancient Rhetoric and Greek Mathematics: A Response to a Modern Historiographical Dilemma,” 
* I. Bodnár, Oenopides of Chius: A Survey of the Modern Literature with a Collection of the Ancient Testimonia
* {{Citation
 | first=David M.
 | last=Burton
 | title=The History of Mathematics: An Introduction
 | edition=3rd
 | publisher=The McGraw-Hill Companies, Inc.
 | year=1997
 | isbn=978-0-07-009465-9
 }}
* M. F. Burnyeat, “Plato on Why Mathematics Is Good for the Soul,” Proceedings of the British Academy 2000
* M. F. Burnyeat, “The Philosophical Sense of Theaetetus’ Mathematics,” 1978
* L. Corry, A Brief History of Number
* S. Cuomo, Pappus of Alexandria and the Mathematics of Late Antiquity
* {{cite book |editor1-last=Christianidis |editor1-first=Jean |title=Classics in the History of Greek Mathematics |date=2004 |publisher=Kluwer |location=Dordrecht |isbn=978-1-4020-0081-2}}
* {{Citation
 | first=Roger
 | last=Cooke
 | title=The History of Mathematics: A Brief Course
 | publisher=Wiley-Interscience
 | year=1997
 | isbn=978-0-471-18082-1
 | url=https://archive.org/details/historyofmathema0000cook
 }}
* {{Citation
 | first=John
 | last=Derbyshire
 | author-link=John Derbyshire
 | title=Unknown Quantity: A Real And Imaginary History of Algebra
 | publisher=Joseph Henry Press
 | year=2006
 | isbn=978-0-309-09657-7
 | url=https://archive.org/details/isbn_9780309096577
 }}
* E. J. Dijksterhuis, Archimedes
* M. N. Fried, and S. Unguru, Apollonius of Perga’s Conica: Text, Context, Subtext
* {{Citation
 | first=Thomas Little
 | last=Heath
 | author-link= T. L. Heath
 | title=[[A History of Greek Mathematics]]
 | publisher=Dover publications
 | year=1981
| orig-year=First published 1921
 | isbn=978-0-486-24073-2
 }}
* {{Citation
 | first=Thomas Little
 | last=Heath
 | author-link= T. L. Heath
 | title=A Manual of Greek Mathematics
 | publisher=Dover publications
 | year=2003
 | orig-year=First published 1931
 | isbn=978-0-486-43231-1
 }}
* Huffman, Archytas
* Huffman, Philolaus
* A. Jones, A Portable Cosmos
* R. W. Knorr, The Evolution of the Euclidean Elements, 1975
* H. Mendell, “Reflections on Eudoxus, Callippus and Their Curves: Hippopedes and Callippopedes,”
* I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements
* Netz, “Eudemus of Rhodes, Hippocrates of Chios and the Earliest Form of a Greek Mathematical Text,” 
* R. Netz, Ludic Proof: Greek Mathematics and the Alexandrian Aesthetics 
* R. Netz, The Shaping of Deduction in Greek Mathematics 
* O. Pedersen, A Survey of the Almagest: With Annotation and New Commentary by Alexander Jones
* D. N. Sedley, “Epicurus and the Mathematicians of Cyzicus,” 
* M. Sialaros, J. Christianidis, and A. Megremi (eds.), “On Mathemata: Commenting on Greek and Arabic Mathematical Texts,” 
*{{cite book |last1=Sing |first1=Robert |last2=Berkel |first2=Tazuko Angela van |last3=Osborne |first3=Robin |title=Numbers and numeracy in the Greek polis |date=2022 |publisher=Brill |isbn=978-90-04-46721-7}}
* {{Citation
 | first=John
 | last=Stillwell
 | author-link=John Stillwell
 | title=Mathematics and its History
 | edition=2nd
 | publisher=Springer Science + Business Media Inc.
 | year=2004
 | isbn=978-0-387-95336-6
 }}
*{{cite book |last1=Szabó |first1=Árpád |last2=Szabó |first2=Árpád |title=The Beginnings of Greek Mathematics |date=1978 |publisher=Akadémiai Kiadó |location=Budapest |isbn=978-963-05-1416-3}}
* S. Unguru, “On the Need to Rewrite the History of Greek Mathematics,” Archive for History of Exact Sciences 15 (1975): 67-114
* G. Vlastos, “Elenchus and Mathematics: A Turning-Point in Plato’s Philosophical Development,”
* I. Yavetz, “On the Homocentric Spheres of Eudoxus,” Archive for History of Exact Sciences

==External links==
{{Wikiquote|Ancient Greek mathematics}}
* [http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Mathematics.html Vatican Exhibit]
* [http://aleph0.clarku.edu/~djoyce/mathhist/greece.html History of Mathematics]
* [https://mathshistory.st-andrews.ac.uk/ MacTutor archive of History of Mathematics]
{{History of mathematics}}
{{Ancient Greek mathematics|state=uncollapsed}}
{{Ancient Greece topics}}
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