Editing History of mathematics (section)

== Greek ==
{{Main|Greek mathematics}}
[[File:Pythagorean.svg|thumb|left|The [[Pythagorean theorem]]. The [[Pythagoreans]] are generally credited with the first proof of the theorem.]]
Greek mathematics refers to the mathematics written in the [[Greek language]] from the time of [[Thales of Miletus]] (~600 BC) to the closure of the [[Platonic Academy|Academy of Athens]] in 529 AD.<ref>Eves, Howard (1990). ''An Introduction to the History of Mathematics'', Saunders, {{ISBN|0-03-029558-0}}</ref> Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following [[Alexander the Great]] is sometimes called [[Hellenistic period|Hellenistic]] mathematics.<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 99}}</ref>

Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of [[inductive reasoning]], that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used [[deductive reasoning]]. The Greeks used logic to derive conclusions from definitions and axioms, and used [[mathematical rigor]] to prove them.<ref>Bernal, Martin (2000). "Animadversions on the Origins of Western Science", pp. 72–83 in Michael H. Shank, ed. ''The Scientific Enterprise in Antiquity and the Middle Ages''. Chicago: University of Chicago Press, p. 75.</ref>

Greek mathematics is thought to have begun with [[Thales of Miletus]] (c. 624–c.546 BC) and [[Pythagoras of Samos]] (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by [[Egyptian mathematics|Egyptian]] and [[Babylonian mathematics]]. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.

Thales used [[geometry]] to solve problems such as calculating the height of [[pyramids]] and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to [[Thales' Theorem]]. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.<ref>{{Harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 43}}</ref> Pythagoras established the [[Pythagoreans|Pythagorean School]], whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".<ref>{{Harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 49}}</ref> It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the [[Pythagorean theorem]],<ref>Eves, Howard (1990). ''An Introduction to the History of Mathematics'', Saunders, {{ISBN|0-03-029558-0}}.</ref> though the statement of the theorem has a long history, and with the proof of the existence of [[irrational numbers]].<ref>{{cite journal|title=The Discovery of Incommensurability by Hippasus of Metapontum|author=Kurt Von Fritz|journal=The Annals of Mathematics|year=1945}}</ref><ref>{{cite journal|title=The Pentagram and the Discovery of an Irrational Number|journal=The Two-Year College Mathematics Journal|author=Choike, James R. |year=1980|volume=11 |issue=5 |pages=312–316 |doi=10.2307/3026893 |jstor=3026893 }}</ref> Although he was preceded by the [[Babylonian mathematics|Babylonians]], [[Indian mathematics|Indians]] and the [[Chinese mathematics|Chinese]],<ref name="Nature"/> the [[Neopythagorean]] mathematician [[Nicomachus]] (60–120 AD) provided one of the earliest [[Greco-Roman]] [[multiplication table]]s, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the [[British Museum]]).<ref>David E. Smith (1958), ''History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics'', New York: Dover Publications (a reprint of the 1951 publication), {{ISBN|0-486-20429-4}}, pp. 58, 129.</ref> The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later [[Middle Ages|Medieval]] name: the ''mensa Pythagorica''.<ref>Smith, David E. (1958). ''History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics'', New York: Dover Publications (a reprint of the 1951 publication), {{ISBN|0-486-20429-4}}, p. 129.</ref>

[[Plato]] (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 86}}</ref> His [[Platonic Academy]], in [[Athens]], became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as [[Eudoxus of Cnidus]] (c. 390 - c. 340 BC), came.<ref name="Boyer 1991 loc=The Age of Plato and Aristotle p. 88">{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 88}}</ref> Plato also discussed the foundations of mathematics,<ref>{{cite web|last=Calian |first=George F. |year=2014 |url=http://www.nec.ro/pdfs/publications/odobleja/2013-2014/FLORIN%20GEORGE%20CALIAN.pdf |title=One, Two, Three… A Discussion on the Generation of Numbers |publisher=New Europe College |url-status=dead |archive-url=https://web.archive.org/web/20151015233836/http://www.nec.ro/pdfs/publications/odobleja/2013-2014/FLORIN%20GEORGE%20CALIAN.pdf |archive-date=2015-10-15 }}</ref> clarified some of the definitions (e.g. that of a line as "breadthless length").

Eudoxus developed the [[method of exhaustion]], a precursor of modern [[Integral|integration]]<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 92}}</ref> and a theory of ratios that avoided the problem of [[incommensurable magnitudes]].<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 93}}</ref> The former allowed the calculations of areas and volumes of curvilinear figures,<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 91}}</ref> while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, [[Aristotle]] (384–{{circa|322 BC}}) contributed significantly to the development of mathematics by laying the foundations of [[logic]].<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 98}}</ref>

[[File:P. Oxy. I 29.jpg|right|thumb|One of the oldest surviving fragments of Euclid's ''Elements'', found at [[Oxyrhynchus]] and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.<ref>{{cite web |url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html |title=One of the Oldest Extant Diagrams from Euclid |author=Bill Casselman |author-link=Bill Casselman (mathematician) |publisher=University of British Columbia |access-date=2008-09-26}}</ref>]]
In the 3rd century BC, the premier center of mathematical education and research was the [[Musaeum]] of [[Alexandria]].<ref>{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 100}}</ref> It was there that [[Euclid]] ({{circa|300 BC}}) taught, and wrote the ''[[Euclid's Elements|Elements]]'', widely considered the most successful and influential textbook of all time.<ref name="Boyer 1991 loc=Euclid of Alexandria p. 119"/> The ''Elements'' introduced [[mathematical rigor]] through the [[axiomatic method]] and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the ''Elements'' were already known, Euclid arranged them into a single, coherent logical framework.<ref name="Boyer 1991 loc=Euclid of Alexandria p. 104">{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 104}}</ref> The ''Elements'' was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.<ref>Eves, Howard (1990). ''An Introduction to the History of Mathematics'', Saunders. {{ISBN|0-03-029558-0}} p. 141: "No work, except [[The Bible]], has been more widely used..."</ref> In addition to the familiar theorems of [[Euclidean geometry]], the ''Elements'' was meant as an introductory textbook to all mathematical subjects of the time, such as [[number theory]], [[algebra]] and [[solid geometry]],<ref name="Boyer 1991 loc=Euclid of Alexandria p. 104"/> including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also [[Euclid#Other works|wrote extensively]] on other subjects, such as [[conic sections]], [[optics]], [[spherical geometry]], and mechanics, but only half of his writings survive.<ref>{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 102}}</ref>

[[File:Archimedes pi.svg|thumb|left|upright=1.2|Archimedes used the [[method of exhaustion]] to approximate the value of [[pi]].]]
[[Archimedes]] ({{circa|287}}–212 BC) of [[Syracuse, Italy|Syracuse]], widely considered the greatest mathematician of antiquity,<ref>{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 120}}</ref> used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], in a manner not too dissimilar from modern calculus.<ref name="Boyer1991">{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 130}}</ref> He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, {{nowrap|3+{{sfrac|10|71}} < π < 3+{{sfrac|10|70}}}}.<ref>{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 126}}</ref> He also studied the [[Archimedes spiral|spiral]] bearing his name, obtained formulas for the [[volume]]s of [[surface of revolution|surfaces of revolution]] (paraboloid, ellipsoid, hyperboloid),<ref name="Boyer1991" /> and an ingenious method of [[exponentiation]] for expressing very large numbers.<ref>{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 125}}</ref> While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles.<ref>{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 121}}</ref> He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.<ref>{{Harv|Boyer|1991|loc="Archimedes of Syracuse" p. 137}}</ref>

[[File:Conic sections 2.png|thumb|right|upright=1.25|[[Apollonius of Perga]] made significant advances in the study of [[conic sections]].]]
[[Apollonius of Perga]] ({{circa|262}}–190 BC) made significant advances to the study of [[conic sections]], showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.<ref>{{Harv|Boyer|1991|loc="Apollonius of Perga" p. 145}}</ref> He also coined the terminology in use today for conic sections, namely [[parabola]] ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").<ref>{{Harv|Boyer|1991|loc="Apollonius of Perga" p. 146}}</ref> His work ''Conics'' is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.<ref>{{Harv|Boyer|1991|loc="Apollonius of Perga" p. 152}}</ref> While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.<ref>{{Harv|Boyer|1991|loc="Apollonius of Perga" p. 156}}</ref>

Around the same time, [[Eratosthenes of Cyrene]] ({{circa|276}}–194 BC) devised the [[Sieve of Eratosthenes]] for finding [[prime numbers]].<ref>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 161}}</ref> The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.<ref name=autogenerated3>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 175}}</ref> Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably [[trigonometry]], largely to address the needs of astronomers.<ref name=autogenerated3 /> [[Hipparchus of Nicaea]] ({{circa|190}}–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.<ref>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 162}}</ref> [[Heron of Alexandria]] ({{circa|10}}–70 AD) is credited with [[Heron's formula]] for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots.<ref>S.C. Roy. ''Complex numbers: lattice simulation and zeta function applications'', p. 1 [https://books.google.com/books?id=J-2BRbFa5IkC&dq=Heron+imaginary+numbers&pg=PA1]. Harwood Publishing, 2007, 131 pages. {{ISBN|1-904275-25-7}}</ref> [[Menelaus of Alexandria]] ({{circa|100 AD}}) pioneered [[spherical trigonometry]] through [[Menelaus' theorem]].<ref>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 163}}</ref> The most complete and influential trigonometric work of antiquity is the ''[[Almagest]]'' of [[Claudius Ptolemy|Ptolemy]] ({{circa|AD 90}}–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.<ref>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 164}}</ref> Ptolemy is also credited with [[Ptolemy's theorem]] for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.<ref>{{Harv|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 168}}</ref>

[[File:Diophantus-cover.png|thumb|right|upright|Title page of the 1621 edition of Diophantus' ''Arithmetica'', translated into [[Latin]] by [[Claude Gaspard Bachet de Méziriac]].]]
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 178}}</ref> During this period, [[Diophantus]] made significant advances in algebra, particularly [[indeterminate equation|indeterminate analysis]], which is also known as "Diophantine analysis".<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 180}}</ref> The study of [[Diophantine equations]] and [[Diophantine approximations]] is a significant area of research to this day. His main work was the ''Arithmetica'', a collection of 150 algebraic problems dealing with exact solutions to determinate and [[indeterminate equation]]s.<ref name=autogenerated1>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 181}}</ref> The ''Arithmetica'' had a significant influence on later mathematicians, such as [[Pierre de Fermat]], who arrived at his famous [[Fermat's Last Theorem|Last Theorem]] after trying to generalize a problem he had read in the ''Arithmetica'' (that of dividing a square into two squares).<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 183}}</ref> Diophantus also made significant advances in notation, the ''Arithmetica'' being the first instance of algebraic symbolism and syncopation.<ref name=autogenerated1 />

[[File:Hagia Sophia Mars 2013.jpg|thumb|left|The [[Hagia Sophia]] was designed by mathematicians [[Anthemius of Tralles]] and [[Isidore of Miletus]].]]
Among the last great Greek mathematicians is [[Pappus of Alexandria]] (4th century AD). He is known for his [[Pappus's hexagon theorem|hexagon theorem]] and [[Pappus's centroid theorem|centroid theorem]], as well as the [[Pappus configuration]] and [[Pappus graph]]. His ''Collection'' is a major source of knowledge on Greek mathematics as most of it has survived.<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" pp. 183–90}}</ref> Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.

The first woman mathematician recorded by history was [[Hypatia]] of Alexandria (AD 350–415). She succeeded her father ([[Theon of Alexandria]]) as Librarian at the Great Library{{citation needed|date=December 2018}} and wrote many works on applied mathematics. Because of a political dispute, the [[Christianity in the Roman Empire|Christian community]] in Alexandria had her stripped publicly and executed.<ref>{{Cite web|url=https://sourcebooks.fordham.edu/source/hypatia.asp|title=Internet History Sourcebooks Project|website=sourcebooks.fordham.edu}}</ref> Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as [[Proclus]], [[Simplicius of Cilicia|Simplicius]] and [[Eutocius]].<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" pp. 190–94}}</ref> Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic [[Platonic Academy|Academy of Athens]] by the emperor [[Justinian]] in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the [[Byzantine empire]] with mathematicians such as [[Anthemius of Tralles]] and [[Isidore of Miletus]], the architects of the [[Hagia Sophia]].<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 193}}</ref> Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 194}}</ref>