Editing History of mathematics

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[[File:Euclid-proof.jpg|thumb|right|upright=1.5|A proof from [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' ({{circa|300 BC}}), widely considered the most influential textbook of all time.<ref name="Boyer 1991 loc=Euclid of Alexandria p. 119">{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 119}}</ref>]]
{{Math topics TOC}}
The '''history of mathematics''' deals with the origin of discoveries in [[mathematics]] and the [[History of mathematical notation|mathematical methods and notation of the past]]. Before the [[modern age]] and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the [[Mesopotamian]] states of [[Sumer]], [[Akkad (region)|Akkad]] and [[Assyria]], followed closely by [[Ancient Egypt]] and the Levantine state of [[Ebla]] began using [[arithmetic]], [[algebra]] and [[geometry]] for purposes of [[taxation]], [[commerce]], trade and also in the field of [[astronomy]] to record time and formulate [[calendars]].

The earliest mathematical texts available are from [[Mesopotamia]] and [[Ancient Egypt|Egypt]] – ''[[Plimpton 322]]'' ([[Babylonian mathematics|Babylonian]] {{circa|2000}} – 1900 BC),<ref name=":0">Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", ''Historia Mathematica'', 8, pp. 277–318.</ref> the ''[[Rhind Mathematical Papyrus]]'' ([[Egyptian mathematics|Egyptian]] c. 1800 BC)<ref name=":1">{{Cite book | edition = 2 | publisher = [[Dover Publications]] | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = The Exact Sciences in Antiquity | series = Acta Historica Scientiarum Naturalium et Medicinalium | orig-year = 1957 | year = 1969 | volume = 9 | pages = 1–191 | pmid = 14884919 | isbn = 978-0-486-22332-2 | url = https://books.google.com/books?id=JVhTtVA2zr8C}} Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.</ref> and the ''[[Moscow Mathematical Papyrus]]'' (Egyptian c. 1890 BC). All of these texts mention the so-called [[Pythagorean triple]]s, so, by inference, the [[Pythagorean theorem]] seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the [[Pythagoreans]], who coined the term "mathematics" from the ancient [[Greek language|Greek]] ''μάθημα'' (''mathema''), meaning "subject of instruction".<ref>{{cite journal|author=Turnbull|title=A Manual of Greek Mathematics|journal=Nature|volume=128|issue=3235|page=5|bibcode=1931Natur.128..739T|year=1931|doi=10.1038/128739a0|s2cid=3994109}}</ref> [[Greek mathematics]] greatly refined the methods (especially through the introduction of deductive reasoning and [[mathematical rigor]] in [[mathematical proof|proofs]]) and expanded the subject matter of mathematics.<ref>Heath, Thomas L. (1963). ''A Manual of Greek Mathematics'', Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."</ref> The [[ancient Romans]] used [[applied mathematics]] in [[surveying]], [[structural engineering]], [[mechanical engineering]], [[bookkeeping]], creation of [[Lunar calendar|lunar]] and [[solar calendar]]s, and even [[Roman art|arts and crafts]]. [[Chinese mathematics]] made early contributions, including a [[place value system]] and the first use of [[negative numbers]].<ref name=":2">Joseph, George Gheverghese (1991). ''The Crest of the Peacock: Non-European Roots of Mathematics''. Penguin Books, London, pp. 140–48.</ref><ref>Ifrah, Georges (1986). ''Universalgeschichte der Zahlen''. Campus, Frankfurt/New York, pp. 428–37.</ref> The [[Hindu–Arabic numeral system]] and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in [[Indian mathematics|India]] and were transmitted to the [[Western world]] via [[Islamic mathematics]] through the work of [[Muḥammad ibn Mūsā al-Khwārizmī]].<ref>Kaplan, Robert (1999). ''The Nothing That Is: A Natural History of Zero''. Allen Lane/The Penguin Press, London.</ref><ref>"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html</ref> Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.<ref>[[Adolf Yushkevich|Juschkewitsch, A. P.]] (1964). ''Geschichte der Mathematik im Mittelalter''. Teubner, Leipzig.</ref> Contemporaneous with but independent of these traditions were the mathematics developed by the [[Maya civilization]] of [[Mexico]] and [[Central America]], where the concept of [[zero]] was given a standard symbol in [[Maya numerals]].

Many Greek and Arabic texts on mathematics were [[Latin translations of the 12th century|translated into Latin]] from the 12th century onward, leading to further development of mathematics in [[Middle Ages|Medieval Europe]]. From ancient times through the [[Postclassical age|Middle Ages]], periods of mathematical discovery were often followed by centuries of stagnation.<ref>Eves, Howard (1990). ''History of Mathematics'', 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258.</ref> Beginning in [[Renaissance]] [[Italy]] in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an [[exponential growth|increasing pace]] that continues through the present day. This includes the groundbreaking work of both [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] in the development of infinitesimal [[calculus]] during the course of the 17th century and following discoveries of [[List of German mathematicians|German mathematicians]] like [[Carl Friedrich Gauss]] and [[David Hilbert]].

== Prehistoric{{anchor|Science_education#United_States}} ==
The origins of mathematical thought lie in the concepts of [[number]], [[patterns in nature]], [[magnitude (mathematics)|magnitude]], and [[Configuration (geometry)|form]].<ref name="Boyer 1991 loc=Origins p. 3">{{Harv|Boyer|1991|loc="Origins" p. 3}}</ref> Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in [[hunter-gatherer]] societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.<ref name="Boyer 1991 loc=Origins p. 3"/>

The use of yarn by [[Neanderthal|Neanderthals]] some 40,000 years ago at a site in Abri du Maras in the south of [[France]] suggests they knew basic concepts in mathematics.<ref>{{Cite journal |last1=Hardy |first1=B. L. |last2=Moncel |first2=M.-H. |last3=Kerfant |first3=C. |last4=Lebon |first4=M. |last5=Bellot-Gurlet |first5=L. |last6=Mélard |first6=N. |date=2020-04-09 |title=Direct evidence of Neanderthal fibre technology and its cognitive and behavioral implications |journal=Scientific Reports |language=en |volume=10 |issue=1 |page=4889 |doi=10.1038/s41598-020-61839-w |pmid=32273518 |issn=2045-2322|pmc=7145842 |bibcode=2020NatSR..10.4889H }}</ref><ref>{{Cite web |last=Rigby |first=Sara |date=2020-04-14 |title=40,000-year-old yarn suggests Neanderthals had basic maths skills |url=https://www.sciencefocus.com/news/40000-year-old-yarn-suggests-neanderthals-had-basic-maths-skills |access-date=2025-02-21 |website=BBC Science Focus Magazine |language=en}}</ref> The [[Ishango bone]], found near the headwaters of the [[Nile]] river (northeastern [[Democratic Republic of the Congo|Congo]]), may be more than [[Upper Paleolithic|20,000]] years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a ''tally'' of the earliest known demonstration of [[sequence]]s of [[prime number]]s<ref name="Diaspora">{{cite web | last = Williams | first = Scott W. | year = 2005 | url = http://www.math.buffalo.edu/mad/Ancient-Africa/lebombo.html | title = The Oldest Mathematical Object is in Swaziland | work = Mathematicians of the African Diaspora | publisher = SUNY Buffalo mathematics department | access-date = 2006-05-06}}</ref>{{Failed verification|date=April 2024}} or a six-month lunar calendar.<ref name=Marshack>Marshack, Alexander (1991). ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY.</ref> Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."<ref>{{cite book|last=Rudman|first=Peter Strom|title=How Mathematics Happened: The First 50,000 Years|year=2007|publisher=Prometheus Books|isbn=978-1-59102-477-4|page=[https://archive.org/details/howmathematicsha0000rudm/page/64 64]|url=https://archive.org/details/howmathematicsha0000rudm/page/64}}</ref> The Ishango bone, according to scholar [[Alexander Marshack]], may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.<ref>Marshack, A. (1972). ''The Roots of Civilization: the Cognitive Beginning of Man's First Art, Symbol and Notation''. New York: McGraw-Hill.</ref>

[[Predynastic Egypt]]ians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that [[megalith]]ic monuments in [[England]] and [[Scotland]], dating from the 3rd millennium BC, incorporate geometric ideas such as [[circle]]s, [[ellipse]]s, and [[Pythagorean triple]]s in their design.<ref>Thom, Alexander; Archie Thom (1988). "The metrology and geometry of Megalithic Man", pp. 132–51 in Ruggles, C. L. N. (ed.), ''Records in Stone: Papers in memory of Alexander Thom''. Cambridge University Press. {{ISBN|0-521-33381-4}}.</ref> All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.<ref>{{cite book|first=Peter|last=Damerow|date=1996 |chapter-url=https://books.google.com/books?id=c4yBmjnY1JIC&pg=PA199|title=Abstraction & Representation: Essays on the Cultural Evolution of Thinking (Boston Studies in the Philosophy & History of Science)|chapter=The Development of Arithmetical Thinking: On the Role of Calculating Aids in Ancient Egyptian & Babylonian Arithmetic|isbn=0792338162|publisher=Springer|access-date=2019-08-17}}</ref>

== Babylonian ==
{{Main|Babylonian mathematics}}
{{See also|Plimpton 322}}
[[Babylonia]]n mathematics refers to any mathematics of the peoples of [[Mesopotamia]] (modern [[Iraq]]) from the days of the early [[Sumer]]ians through the [[Hellenistic period]] almost to the dawn of [[Christianity]].<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 24}}</ref> The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC ([[Seleucid]] period).<ref name="Boyer 1991 loc=Mesopotamia p. 26">{{Harv|Boyer|1991|loc="Mesopotamia" p. 26}}</ref> It is named Babylonian mathematics due to the central role of [[Babylon]] as a place of study. Later under the [[Caliphate|Arab Empire]], Mesopotamia, especially [[Baghdad]], once again became an important center of study for [[Islamic mathematics]].

[[File:Geometry problem-Sb 13088-IMG 0593-white.jpg|thumb|Geometry problem on a clay tablet belonging to a school for scribes; [[Susa]], first half of the 2nd millennium BC]]
In contrast to the sparsity of sources in [[Egyptian mathematics]], knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<ref name="Boyer 1991 loc=Mesopotamia p. 25">{{Harv|Boyer|1991|loc="Mesopotamia" p. 25}}</ref> Written in [[Cuneiform script]], tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.<ref name="Boyer 1991 loc=Mesopotamia p. 41">{{Harv|Boyer|1991|loc="Mesopotamia" p. 41}}</ref>

The earliest evidence of written mathematics dates back to the ancient [[Sumer]]ians, who built the earliest civilization in Mesopotamia. They developed a complex system of [[metrology]] from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.<ref>{{Citation |last=Sharlach |first=Tonia |title=Calendars and Counting |url=http://dx.doi.org/10.4324/9780203096604.ch15 |work=The Sumerian World |year=2006 |pages=307–308 |access-date=2023-07-07 |publisher=Routledge |doi=10.4324/9780203096604.ch15 |isbn=978-0-203-09660-4}}</ref> From around 2500 BC onward, the Sumerians wrote [[multiplication table]]s on clay tablets and dealt with geometrical exercises and [[Division (mathematics)|division]] problems. The earliest traces of the Babylonian numerals also date back to this period.<ref>Melville, Duncan J. (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology] {{Webarchive|url=https://web.archive.org/web/20180707213616/http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html |date=2018-07-07 }}, ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>

[[Image:Plimpton 322.jpg|thumb|left|The Babylonian mathematical tablet [[Plimpton 322]], dated to 1800 BC.]]
Babylonian mathematics were written using a [[sexagesimal]] (base-60) [[numeral system]].<ref name="Boyer 1991 loc=Mesopotamia p. 25"/> From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30,<ref name="Boyer 1991 loc=Mesopotamia p. 25"/> and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc.) being able to easily calculate by hand was essential, and so a sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision.<ref name="Powell 1976 p. 418">{{Citation |last=Powell |first=M. |title=The Antecedents of Old Babylonian Place Notation and the Early History of Babylonian Mathematics |url=https://core.ac.uk/download/pdf/82557367.pdf |work=Historia Mathematica |volume=3 |pages=417–439 |year=1976 |access-date=July 6, 2023}}</ref> Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the [[decimal]] system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the [[Renaissance]], and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet [[YBC 7289]] gives an approximation of {{radic|2}} accurate to five decimal places.<ref name="Boyer 1991 loc=Mesopotamia p. 27">{{Harv|Boyer|1991|loc="Mesopotamia" p. 27}}</ref> The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/> By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/> This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.<ref name="Boyer 1991 loc=Mesopotamia p. 26"/>

Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of [[regular number]]s, and their [[Multiplicative inverse|reciprocal]] [[tuple|pairs]].<ref>{{cite book | author-link = Aaboe | last = Aaboe | first = Asger | title = Episodes from the Early History of Mathematics | year = 1998 | publisher = Random House | location = New York | pages = 30–31}}</ref> The tablets also include multiplication tables and methods for solving [[linear equation|linear]], [[quadratic equation]]s and [[cubic equation]]s, a remarkable achievement for the time.<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 33}}</ref> Tablets from the Old Babylonian period also contain the earliest known statement of the [[Pythagorean theorem]].<ref>{{Harv|Boyer|1991|loc="Mesopotamia" p. 39}}</ref> However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for [[mathematical proof|proofs]] or logical principles.<ref name="Boyer 1991 loc=Mesopotamia p. 41"/>

== Egyptian ==
{{Main|Egyptian mathematics}}
[[File:Moskou-papyrus.jpg|thumb|right|upright=1.5|Image of Problem 14 from the [[Moscow Mathematical Papyrus]]. The problem includes a diagram indicating the dimensions of the truncated pyramid.]]
[[Egypt]]ian mathematics refers to mathematics written in the [[Egyptian language]]. From the [[Hellenistic period]], [[Greek language|Greek]] replaced Egyptian as the written language of [[Egyptians|Egyptian]] scholars. Mathematical study in [[Egypt]] later continued under the [[Caliphate|Arab Empire]] as part of [[Islamic mathematics]], when [[Arabic]] became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.<ref>{{cite book |last1=Eglash |first1=Ron |title=African fractals : modern computing and indigenous design |date=1999 |publisher=Rutgers University Press |location=New Brunswick, N.J. |isbn=0813526140 |pages=89,141}}</ref> Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.<ref>{{cite journal |last1=Eglash, R. |title=Fractal Geometry in African Material Culture |journal=Symmetry: Culture and Science |date=1995 |volume=6-1 |pages=174–177}}</ref>

The most extensive Egyptian mathematical text is the [[Rhind papyrus]] (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the [[Middle Kingdom of Egypt|Middle Kingdom]] of about 2000–1800 BC.<ref name="Boyer 1991 loc=Egypt p. 11">{{Harv|Boyer|1991|loc="Egypt" p. 11}}</ref> It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,<ref>[http://www.mathpages.com/home/kmath340/kmath340.htm Egyptian Unit Fractions] at MathPages</ref> including [[composite number|composite]] and [[prime number]]s; [[arithmetic mean|arithmetic]], [[geometric mean|geometric]] and [[harmonic mean]]s; and simplistic understandings of both the [[Sieve of Eratosthenes]] and [[Perfect number|perfect number theory]] (namely, that of the number 6).<ref>[<!-- http://mathpages.com/home/rhind.htm -->http://mathpages.com/home/kmath340/kmath340.htm Egyptian Unit Fractions]</ref> It also shows how to solve first order [[linear equation]]s<ref>{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html|title=Egyptian Papyri|website=www-history.mcs.st-andrews.ac.uk}}</ref> as well as [[arithmetic series|arithmetic]] and [[geometric series]].<ref>{{Cite web|url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_algebra.html#areithmetic+series|title=Egyptian Algebra – Mathematicians of the African Diaspora|website=www.math.buffalo.edu}}</ref>

Another significant Egyptian mathematical text is the [[Moscow papyrus]], also from the [[Middle Kingdom of Egypt|Middle Kingdom]] period, dated to c. 1890 BC.<ref name="Boyer 1991 loc=Egypt p. 19">{{Harv|Boyer|1991|loc="Egypt" p. 19}}</ref> It consists of what are today called ''word problems'' or ''story problems'', which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a [[frustum]] (truncated pyramid).

Finally, the [[Berlin Papyrus 6619]] (c. 1800 BC) shows that ancient Egyptians could solve a second-order [[algebraic equation]].<ref>{{Cite web|url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin|title=Egyptian Mathematical Papyri – Mathematicians of the African Diaspora|website=www.math.buffalo.edu}}</ref>

== 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>

==Roman==
{{further|Roman abacus|Roman numerals}}
[[File:Aquinqum BHM IMG 0662 land surveyor equipment.jpg|thumb|Equipment used by an [[Ancient Rome|ancient Roman]] land [[surveyor]] (''[[gromatici]]''), found at the site of [[Aquincum]], modern [[Budapest]], [[Hungary]]]]
Although [[Greeks|ethnic Greek]] mathematicians continued under the rule of the late [[Roman Republic]] and subsequent [[Roman Empire]], there were no noteworthy [[Latins (Italic tribe)|native Latin]] mathematicians in comparison.<ref>{{Harv|Goodman|2016|p=119}}</ref><ref>{{Harv|Cuomo|2001|pp=194, 204–06}}</ref> [[Ancient Rome|Ancient Romans]] such as [[Cicero]] (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman [[surveyor]]s and [[Mental calculator|calculators]] were far more interested in [[applied mathematics]] than the [[theoretical mathematics]] and geometry that were prized by the Greeks.<ref>{{Harv|Cuomo|2001|pp=192–95}}</ref> It is unclear if the Romans first derived [[Roman numerals|their numerical system]] directly from [[Greek numerals|the Greek precedent]] or from [[Etruscan numerals]] used by the [[Etruscan civilization]] centered in what is now [[Tuscany]], [[central Italy]].<ref>{{Harv|Goodman|2016|pp=120–21}}</ref>

Using calculation, Romans were adept at both instigating and detecting financial [[fraud]], as well as [[List of Roman taxes|managing taxes]] for the [[treasury]].<ref>{{Harv|Cuomo|2001|p=196}}</ref> [[Siculus Flaccus]], one of the Roman ''[[gromatici]]'' (i.e. land surveyor), wrote the ''Categories of Fields'', which aided Roman surveyors in measuring the [[surface area]]s of allotted lands and territories.<ref>{{Harv|Cuomo|2001|pp=207–08}}</ref> Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in [[Roman engineering|engineering]], including the erection of [[Roman architecture|architecture]] such as [[Roman bridge|bridges]], [[Roman roads|road-building]], and [[Roman military engineering|preparation for military campaigns]].<ref>{{Harv|Goodman|2016|pp=119–20}}</ref> [[Roman art|Arts and crafts]] such as [[Roman mosaic]]s, inspired by previous [[Mosaics of Delos|Greek designs]], created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each [[tessera]] tile, the [[opus tessellatum]] pieces on average measuring eight millimeters square and the finer [[opus vermiculatum]] pieces having an average surface of four millimeters square.<ref>{{Harv|Tang|2005|pp=14–15, 45}}</ref><ref>{{Harv|Joyce|1979|p=256}}</ref>

The creation of the [[Roman calendar]] also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the [[Roman Kingdom]] and included 356 days plus a [[leap year]] every other year.<ref>{{Harv|Gullberg|1997|p=17}}</ref> In contrast, the [[lunar calendar]] of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the [[solar year]], a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February.<ref>{{Harv|Gullberg|1997|pp=17–18}}</ref> This calendar was supplanted by the [[Julian calendar]], a [[solar calendar]] organized by [[Julius Caesar]] (100–44 BC) and devised by [[Sosigenes of Alexandria]] to include a [[leap day]] every four years in a 365-day cycle.<ref>{{Harv|Gullberg|1997|p=18}}</ref> This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the [[Gregorian calendar]] organized by [[Pope Gregory XIII]] ({{reign|1572|1585}}), virtually the same solar calendar used in modern times as the international standard calendar.<ref>{{Harv|Gullberg|1997|pp=18–19}}</ref>

At roughly the same time, [[Science and technology of the Han dynasty|the Han Chinese]] and the Romans both invented the wheeled [[odometer]] device for measuring [[distance]]s traveled, the Roman model first described by the Roman civil engineer and architect [[Vitruvius]] ({{circa|80 BC|15 BC}}).<ref>{{Harv|Needham|Wang|2000|pp=281–85}}</ref> The device was used at least until the reign of emperor [[Commodus]] ({{reign|177|192 AD}}), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.<ref>{{Harv|Needham|Wang|2000|p=285}}</ref> Perhaps relying on similar gear-work and [[Roman technology|technology]] found in the [[Antikythera mechanism]], the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2&nbsp;m) in diameter turning four-hundred times in one [[Roman mile]] (roughly 4590&nbsp;ft/1400&nbsp;m). With each revolution, a pin-and-axle device engaged a 400-tooth [[cogwheel]] that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.<ref>{{Harv|Sleeswyk|1981|pp=188–200}}</ref>

== Chinese ==
{{Main|Chinese mathematics}}
{{further|Book on Numbers and Computation}}
{{see also|History of science#Chinese mathematics}}
[[File:Qinghuajian, Suan Biao.jpg|thumb|right|upright|The [[Tsinghua Bamboo Slips]], containing the world's earliest [[decimal]] multiplication table, dated 305 BC during the [[Warring States]] period]]
An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.<ref>{{Harv|Boyer|1991|loc="China and India" p. 201}}</ref> The oldest extant mathematical text from China is the ''[[Zhoubi Suanjing]]'' (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the [[Warring States Period]] appears reasonable.<ref name="Boyer 1991 loc=China and India p. 196">{{Harv|Boyer|1991|loc="China and India" p. 196}}</ref> However, the [[Tsinghua Bamboo Slips]], containing the earliest known [[decimal]] [[multiplication table]] (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.<ref name="Nature">{{cite journal | url =http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482| title =Ancient times table hidden in Chinese bamboo strips | journal =Nature |first=Jane|last=Qiu|author-link=Jane Qiu|date=7 January 2014| access-date =15 September 2014| doi =10.1038/nature.2014.14482 | s2cid =130132289 | doi-access =free}}</ref>

[[File:Chounumerals.svg|thumb|left|[[Counting rod numerals]]]]
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.<ref>{{Harvnb|Katz|2007|pp=194–99}}</ref> Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.<ref>{{Harv|Boyer|1991|loc="China and India" p. 198}}</ref> [[Counting rods|Rod numerals]] allowed the representation of numbers as large as desired and allowed calculations to be carried out on the ''[[suanpan|suan pan]]'', or Chinese abacus. The date of the invention of the ''suan pan'' is not certain, but the earliest written mention dates from AD 190, in [[Xu Yue (mathematician)|Xu Yue]]'s ''Supplementary Notes on the Art of Figures''.

The oldest extant work on geometry in China comes from the philosophical [[Mohism|Mohist]] canon {{circa|330 BC}}, compiled by the followers of [[Mozi]] (470–390 BC). The ''Mo Jing'' described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.<ref>{{Harv|Needham|Wang|1995|pp=91–92}}</ref> It also defined the concepts of [[circumference]], [[diameter]], [[radius]], and [[volume]].<ref>{{Harv|Needham|Wang|1995|p=94}}</ref>

[[File:九章算術.gif|thumb|upright|right|''[[The Nine Chapters on the Mathematical Art]]'', one of the earliest surviving mathematical texts from [[China]] (2nd century AD).]]
In 212 BC, the Emperor [[Qin Shi Huang]] commanded all books in the [[Qin Empire]] other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the [[Burning of books and burying of scholars|book burning]] of 212 BC, the [[Han dynasty]] (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is ''[[The Nine Chapters on the Mathematical Art]]'', the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for [[Chinese pagoda]] towers, engineering, [[surveying]], and includes material on [[right triangle]]s.<ref name="Boyer 1991 loc=China and India p. 196"/> It created mathematical proof for the [[Pythagorean theorem]],<ref>{{Harv|Needham|Wang|1995|p=22}}</ref> and a mathematical formula for [[Gaussian elimination]].<ref>{{Harv|Straffin|1998|p=164}}</ref> The treatise also provides values of [[Pi|π]],<ref name="Boyer 1991 loc=China and India p. 196"/> which Chinese mathematicians originally approximated as 3 until [[Liu Xin (scholar)|Liu Xin]] (d. 23 AD) provided a figure of 3.1457 and subsequently [[Zhang Heng]] (78–139) approximated pi as 3.1724,<ref>{{Harv|Needham|Wang|1995|pp=99–100}}</ref> as well as 3.162 by taking the [[square root]] of 10.<ref>{{Harv|Berggren|Borwein|Borwein|2004|p=27}}</ref><ref>{{Harv|de Crespigny|2007|p=1050}}</ref> [[Liu Hui]] commented on the ''Nine Chapters'' in the 3rd century AD and [[Liu Hui's π algorithm|gave a value of π]] accurate to 5 decimal places (i.e. 3.14159).<ref name="Boyer 1991 loc=China and India p. 202">{{Harv|Boyer|1991|loc="China and India" p. 202}}</ref><ref>{{Harv|Needham|Wang|1995|pp=100–01}}</ref> Though more of a matter of computational stamina than theoretical insight, in the 5th century AD [[Zu Chongzhi]] computed [[Milü|the value of π]] to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.<ref name="Boyer 1991 loc=China and India p. 202"/><ref>{{Harv|Berggren|Borwein|Borwein|2004|pp=20, 24–26}}</ref> He also established a method which would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].<ref>{{cite book
|title=Calculus: Early Transcendentals
|edition=3
|first1=Dennis G.
|last1=Zill
|first2=Scott
|last2=Wright
|first3=Warren S.
|last3=Wright
|publisher=Jones & Bartlett Learning
|year=2009
|isbn=978-0-7637-5995-7
|page=xxvii
|url=https://books.google.com/books?id=R3Hk4Uhb1Z0C}} [https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of p. 27]
</ref>

The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the [[Song dynasty]] (960–1279), with the development of Chinese algebra. The most important text from that period is the ''[[Jade Mirror of the Four Unknowns|Precious Mirror of the Four Elements]]'' by [[Zhu Shijie]] (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to [[Horner's method]].<ref name="Boyer 1991 loc=China and India p. 202"/> The ''Precious Mirror'' also contains a diagram of [[Pascal's triangle]] with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.<ref name="Boyer 1991 loc=China and India p. 205">{{Harv|Boyer|1991|loc="China and India" p. 205}}</ref> The Chinese also made use of the complex combinatorial diagram known as the [[magic square]] and [[Magic circle (mathematics)|magic circles]], described in ancient times and perfected by [[Yang Hui]] (AD 1238–1298).<ref name="Boyer 1991 loc=China and India p. 205" />

Even after European mathematics began to flourish during the [[Renaissance]], European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. [[Jesuit]] missionaries such as [[Matteo Ricci]] carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.<ref name="Boyer 1991 loc=China and India p. 205"/>

[[Japanese mathematics]], [[Korean numerals|Korean mathematics]], and [[Vietnamese numerals|Vietnamese mathematics]] are traditionally viewed as stemming from Chinese mathematics and belonging to the [[Confucian]]-based [[East Asian cultural sphere]].<ref>{{Harv|Volkov|2009|pp=153–56}}</ref> Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's [[Ming dynasty]] (1368–1644).<ref>{{Harv|Volkov|2009|pp=154–55}}</ref> For instance, although Vietnamese mathematical treatises were written in either [[Chinese characters|Chinese]] or the native Vietnamese [[Chữ Nôm]] script, all of them followed the Chinese format of presenting a collection of problems with [[algorithm]]s for solving them, followed by numerical answers.<ref>{{Harv|Volkov|2009|pp=156–57}}</ref> Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of [[History of astronomy|mathematicians and astronomers]], whereas in Japan it was more prevalent in the realm of [[private school]]s.<ref>{{Harv|Volkov|2009|p=155}}</ref>

== Indian ==
{{Main|Indian mathematics}}
{{Further|History of science#Indian mathematics}}
{{See also|History of the Hindu–Arabic numeral system}}
[[File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BC and the 2nd century AD.]]
{{multiple image
 | align = right
 | direction = vertical
 | width = 330
 | image1 = 1911 sketch of numerals script history ancient India, mathematical symbols shapes.jpg
 | alt1 = Numerals evolution in India
 | caption1 = Indian numerals in stone and copper inscriptions<ref name=britnanaghat>[https://www.britannica.com/topic/numeral#ref797082 Development Of Modern Numerals And Numeral Systems: The Hindu-Arabic system], Encyclopaedia Britannica, Quote: "The 1, 4, and 6 are found in the Ashoka inscriptions (3rd century BC); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st or 2nd century AD – all in forms that have considerable resemblance to today’s, 2 and 3 being well-recognized cursive derivations from the ancient = and ≡."</ref>
 | image2 = Indian numerals 100AD.svg
 | alt2 = Brahmi numerals
 | caption2 = Ancient Brahmi numerals in a part of India
}}
The earliest civilization on the Indian subcontinent is the [[Indus Valley civilization]] (mature second phase: 2600 to 1900 BC) that flourished in the [[Indus river]] basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.<ref>{{Harv|Boyer|1991|loc="China and India" p. 206}}</ref>

The oldest extant mathematical records from India are the [[Sulba Sutras]] (dated variously between the 8th century BC and the 2nd century AD),<ref name="Boyer 1991 loc=China and India p. 207">{{Harv|Boyer|1991|loc="China and India" p. 207}}</ref> appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.<ref>{{Cite book |first=T.K. |last=Puttaswamy |chapter=The Accomplishments of Ancient Indian Mathematicians |pages=411–12 |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-1-4020-0260-1 }}</ref> As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.<ref name="Boyer 1991 loc=China and India p. 207"/> The Sulba Sutras give methods for constructing a [[squaring the circle|circle with approximately the same area as a given square]], which imply several different approximations of the value of π.<ref>{{cite journal |first=R.P. |last=Kulkarni |url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_32.pdf |title=The Value of π known to Śulbasūtras |archive-url=https://web.archive.org/web/20120206150545/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_32.pdf |archive-date=2012-02-06 |journal=Indian Journal of History of Science |volume=13 |issue=1 |date=1978 |pages=32–41}}</ref><ref name="Indian_sulbasutras">{{cite web |first1=J.J. |last1=Connor |first2=E.F. |last2=Robertson |title=The Indian Sulbasutras |publisher=Univ. of St. Andrew, Scotland |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html}}</ref>{{efn|1=The approximate values for π are 4 x (13/15)<sup>2</sup> (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389)}} In addition, they compute the [[square root]] of 2 to several decimal places, list Pythagorean triples, and give a statement of the [[Pythagorean theorem]].<ref name="Indian_sulbasutras"/> All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.<ref name="Boyer 1991 loc=China and India p. 207"/> It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.<ref name="Boyer 1991 loc=China and India p. 207"/>

[[Pāṇini]] (c. 5th century BC) formulated the rules for [[Sanskrit grammar]].<ref>{{Cite journal
 | last=Bronkhorst
 | first=Johannes
 | author-link= Johannes Bronkhorst
 | title=Panini and Euclid: Reflections on Indian Geometry
 | journal=Journal of Indian Philosophy
 |volume=29
 |issue=1–2
 | year=2001
 | pages=43–80
 | doi=10.1023/A:1017506118885
 | s2cid=115779583
 }}</ref> His notation was similar to modern mathematical notation, and used metarules, [[Transformation (geometry)|transformations]], and [[recursion]].<ref>{{Cite journal|last=Kadvany|first=John|date=2008-02-08|title=Positional Value and Linguistic Recursion|journal=Journal of Indian Philosophy|language=en|volume=35|issue=5–6|pages=487–520|doi=10.1007/s10781-007-9025-5|issn=0022-1791|citeseerx=10.1.1.565.2083|s2cid=52885600}}</ref> [[Pingala]] (roughly 3rd–1st centuries BC) in his treatise of [[Prosody (poetry)|prosody]] uses a device corresponding to a [[binary numeral system]].<ref>{{Cite book
|last1=Sanchez
|first1=Julio
|last2=Canton
|first2=Maria P.
|title=Microcontroller programming : the microchip PIC
|year=2007
|publisher=CRC Press
|location=Boca Raton, Florida
|isbn=978-0-8493-7189-9
|page=37
}}</ref><ref>Anglin, W. S. and J. Lambek (1995). ''The Heritage of Thales'', Springer, {{ISBN|0-387-94544-X}}</ref> His discussion of the [[combinatorics]] of [[Metre (music)|meters]] corresponds to an elementary version of the [[binomial theorem]]. Pingala's work also contains the basic ideas of [[Fibonacci number]]s (called ''mātrāmeru'').<ref>{{cite journal | last1 = Hall | first1 = Rachel W. | year = 2008 | title = Math for poets and drummers | url = http://people.sju.edu/~rhall/mathforpoets.pdf | journal = Math Horizons | volume = 15 | issue = 3| pages = 10–11 | doi = 10.1080/10724117.2008.11974752 | s2cid = 3637061 }}</ref>

The next significant mathematical documents from India after the ''Sulba Sutras'' are the ''Siddhantas'', astronomical treatises from the 4th and 5th centuries AD ([[Gupta period]]) showing strong Hellenistic influence.<ref>{{Harv|Boyer|1991|loc="China and India" p. 208}}</ref> They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.<ref name=autogenerated2>{{Harv|Boyer|1991|loc="China and India" p. 209}}</ref> Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".<ref name=autogenerated2 />

[[Image:Yuktibhasa.svg|upright|left|thumb|Explanation of the [[Law of sines|sine rule]] in ''[[Yuktibhāṣā]]'']]
Around 500 AD, [[Aryabhata]] wrote the ''[[Aryabhatiya]]'', a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.<ref>{{Harv|Boyer|1991|loc="China and India" p. 210}}</ref> It is in the ''Aryabhatiya'' that the decimal place-value system first appears. Several centuries later, the [[Islamic mathematics|Muslim mathematician]] [[Abu Rayhan Biruni]] described the ''Aryabhatiya'' as a "mix of common pebbles and costly crystals".<ref>{{Harv|Boyer|1991|loc="China and India" p. 211}}</ref>

In the 7th century, [[Brahmagupta]] identified the [[Brahmagupta theorem]], [[Brahmagupta's identity]] and [[Brahmagupta's formula]], and for the first time, in ''[[Brahmasphutasiddhanta|Brahma-sphuta-siddhanta]]'', he lucidly explained the use of [[0 (number)|zero]] as both a placeholder and [[decimal digit]], and explained the [[Hindu–Arabic numeral system]].<ref name="Boyer Siddhanta">{{cite book|last=Boyer|ref=none|author-link=Carl Benjamin Boyer|title=History of Mathematics|url=https://archive.org/details/historyofmathema00boye|url-access=registration|year=1991|chapter=The Arabic Hegemony|page=[https://archive.org/details/historyofmathema00boye/page/226 226]|publisher=Wiley |isbn=9780471543978|quote=By 766 we learn that an astronomical-mathematical work, known to the Arabs as the ''Sindhind'', was brought to Baghdad from India. It is generally thought that this was the ''Brahmasphuta Siddhanta'', although it may have been the ''Surya Siddhanata''. A few years later, perhaps about 775, this ''Siddhanata'' was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological ''[[Tetrabiblos]]'' was translated into Arabic from the Greek.}}</ref> It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as [[Arabic numerals]]. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the [[Brahmi numeral]]s. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, [[Halayudha]]'s commentary on [[Pingala]]'s work contains a study of the [[Fibonacci sequence]]<ref>{{Cite journal |last=Singh |first=Parmanand |date=1985-08-01 |title=The so-called fibonacci numbers in ancient and medieval India |url=https://dx.doi.org/10.1016/0315-0860%2885%2990021-7 |journal=Historia Mathematica |volume=12 |issue=3 |pages=229–244 |doi=10.1016/0315-0860(85)90021-7 |issn=0315-0860}}</ref> and [[Pascal's triangle]],<ref>{{Cite book |last=Ramasubramanian |first=K. |url=https://books.google.com/books?id=AEe9DwAAQBAJ&dq=Selected+Works+of+Radha+Charan+Gupta+on+History+of+Mathematics+pascal&pg=PA289 |title=Gaṇitānanda: Selected Works of Radha Charan Gupta on History of Mathematics |date=2019-11-08 |publisher=Springer Nature |isbn=978-981-13-1229-8 |language=en}}</ref> and describes the formation of a [[matrix (mathematics)|matrix]].{{Citation needed|date=April 2010}}

In the 12th century, [[Bhāskara II]],<ref>Plofker 2009 182–207</ref> who lived in southern India, wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, [[Mean value theorem|the mean value theorem]] and the derivative of the sine function although he did not develop the notion of a derivative.<ref>{{cite book |first=Roger |last=Cooke |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |chapter=The Mathematics of the Hindus |pages=[https://archive.org/details/historyofmathema0000cook/page/213 213–215] |isbn=0-471-18082-3 |chapter-url=https://archive.org/details/historyofmathema0000cook/page/213}}</ref><ref>Plofker 2009 pp. 197–98; George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'', Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, "Indian Mathematics", pp. 118–30 in ''Companion History of the History and Philosophy of the Mathematical Sciences'', ed. I. Grattan. Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126.</ref> In the 14th century, [[Narayana Pandita (mathematician)|Narayana Pandita]] completed his ''[[Ganita Kaumudi]]''.<ref>{{Cite web |title=Narayana - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Narayana/ |access-date=2022-10-03 |website=Maths History |language=en}}</ref>

Also in the 14th century, [[Madhava of Sangamagrama]], the founder of the [[Kerala School of Astronomy and Mathematics|Kerala School of Mathematics]], found the [[Leibniz formula for pi|Madhava–Leibniz series]] and obtained from it a [[Approximations of π#Middle Ages|transformed series]], whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found [[Gregory's series|the Madhava-Gregory series]] to determine the arctangent, the Madhava-Newton [[power series]] to determine sine and cosine and [[Taylor series|the Taylor approximation]] for sine and cosine functions.<ref>Plofker 2009 pp. 217–53.</ref> In the 16th century, [[Jyesthadeva]] consolidated many of the Kerala School's developments and theorems in the ''Yukti-bhāṣā''.<ref name="rajujournal">
{{cite journal| author1=Raju, C. K. | title=Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā 
|url=http://ckraju.net/papers/Hawaii.pdf
| journal=Philosophy East & West 
| volume=51
| issue=3
| date=2001
| pages=325–362 
| doi=10.1353/pew.2001.0045 
| s2cid=170341845 
| access-date=2020-02-11
}}
</ref><ref>Divakaran, P. P. (2007). "The first textbook of calculus: Yukti-bhāṣā", ''Journal of Indian Philosophy'' 35, pp. 417–33.</ref> It has been argued that certain ideas of calculus like infinite series and taylor series of some trigonometry functions, were transmitted to Europe in the 16th century<ref name=":2" /> via [[Jesuit]] missionaries and traders who were active around the ancient port of [[Muziris]] at the time and, as a result, directly influenced later European developments in analysis and calculus.<ref name=almeida>{{cite journal
|author = Almeida, D. F.; J. K. John and A. Zadorozhnyy
|title = Keralese mathematics: its possible transmission to Europe and the consequential educational implications
| journal = Journal of Natural Geometry
|volume= 20
|year =2001
|pages=77–104
|issue=1
}}</ref> However, other scholars argue that the Kerala School did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], and that there is not any direct evidence of their results being transmitted outside Kerala.<ref>{{Cite journal | last = Pingree | first = David | author-link = David Pingree | title = Hellenophilia versus the History of Science | journal = Isis | volume = 83 | issue = 4 | pages = 554–563 |date=December 1992  | jstor = 234257 | doi = 10.1086/356288 | quote = One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by [[C. M. Whish|Charles Whish]], in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the ''Transactions of the Royal Asiatic Society'', in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series ''without'' the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.| bibcode = 1992Isis...83..554P | s2cid = 68570164 }}</ref><ref>{{Cite journal | last = Bressoud | first = David | author-link = David Bressoud | title = Was Calculus Invented in India? | journal = College Mathematics Journal | volume = 33 | issue = 1 | pages = 2–13 | year = 2002 | doi=10.2307/1558972| jstor = 1558972 }}</ref><ref>{{Cite journal | last = Plofker | first = Kim | author-link = Kim Plofker | title = The 'Error' in the Indian "Taylor Series Approximation" to the Sine | journal = Historia Mathematica | volume = 28 | issue = 4 | page = 293 |date=November 2001 | doi = 10.1006/hmat.2001.2331 | quote =It is not unusual to encounter in discussions of Indian mathematics such assertions as that 'the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)' [Joseph 1991, 300], or that 'we may consider Madhava to have been the founder of mathematical analysis' (Joseph 1991, 293), or that Bhaskara II may claim to be 'the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus' (Bag 1979, 294).... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285))... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian 'discovery of the principle of the differential calculus' somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential 'principle' was not generalized to arbitrary functions&nbsp;– in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here| doi-access = free }}</ref><ref>{{Cite journal | last = Katz | first = Victor J. | title = Ideas of Calculus in Islam and India | journal = Mathematics Magazine | volume = 68 | issue = 3 | pages = 163–74 |date=June 1995  | url = http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall12/128/CalcIslamIndia.pdf | jstor = 2691411 | doi=10.2307/2691411 }}</ref>

== Islamic empires ==
{{Main|Mathematics in medieval Islam}}
{{See also|History of the Hindu–Arabic numeral system}}
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|Page from ''[[The Compendious Book on Calculation by Completion and Balancing]]'' by [[Muhammad ibn Mūsā al-Khwārizmī]] (c.&nbsp;AD 820)]]
The [[Caliphate|Islamic Empire]] established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Iberian Peninsula|Iberia]], and in parts of [[History of India|India]] in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in [[Arabic language|Arabic]], they were not all written by [[Arab]]s, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time.<ref>Abdel Haleem, Muhammad A. S. "The Semitic Languages", https://doi.org/10.1515/9783110251586.811, "Arabic became the language of scholarship in science and philosophy in the 9th century when the ‘translation movement’ saw concerted work on translations of Greek, Indian, Persian and Chinese, medical, philosophical and scientific texts", p. 811.</ref>

In the 9th century, the Persian mathematician [[Muḥammad ibn Mūsā al-Khwārizmī]] wrote an important book on the [[Hindu–Arabic numerals]] and one on methods for solving equations. His book ''On the Calculation with Hindu Numerals'', written about 825, along with the work of [[Al-Kindi]], were instrumental in spreading [[Indian mathematics]] and [[Hindu–Arabic numeral system|Indian numerals]] to the West. The word ''[[algorithm]]'' is derived from the Latinization of his name, Algoritmi, and the word ''algebra'' from the title of one of his works, ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala]]'' (''The Compendious Book on Calculation by Completion and Balancing''). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions."</ref> and he was the first to teach algebra in an [[Elementary algebra|elementary form]] and for its own sake.<ref>Gandz and Saloman (1936). "The sources of Khwarizmi's algebra", ''Osiris'' i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> He also discussed the fundamental method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as ''al-jabr''.<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."</ref> His algebra was also no longer concerned "with a series of problems to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=978-0-7923-2565-9 | oclc=29181926 | pages=11–12}}</ref>

In Egypt, [[Abu Kamil]] extended algebra to the set of [[irrational numbers]], accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.<ref name="HSTM">{{Cite encyclopedia | publisher = Springer|  pages = 4–5| last = Sesiano| first = Jacques| title = Abū Kāmil | encyclopedia = Encyclopaedia of the history of science, technology, and medicine in non-western cultures| year= 1997}}</ref> His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.

Further developments in algebra were made by [[Al-Karaji]] in his treatise ''al-Fakhri'', where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a [[Mathematical proof|proof]] by [[mathematical induction]] appears in a book written by Al-Karaji around 1000 AD, who used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of integral [[Cube (algebra)|cubes]].<ref>{{Harv|Katz|1998|loc=pp. 255–59}}</ref> The [[historian]] of mathematics, F. Woepcke,<ref>Woepcke, F. (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Also in the 10th century, [[Abul Wafa]] translated the works of [[Diophantus]] into Arabic. [[Ibn al-Haytham]] was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a [[paraboloid]], and was able to generalize his result for the integrals of [[polynomial]]s up to the [[Quartic polynomial|fourth degree]]. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<ref name=Katz>{{cite journal | last1 = Katz | first1 = Victor J. | year = 1995 | title = Ideas of Calculus in Islam and India | journal = Mathematics Magazine | volume = 68 | issue = 3| pages = 163–74 | doi=10.2307/2691411| jstor = 2691411 }}</ref>

In the late 11th century, [[Omar Khayyam]] wrote ''Discussions of the Difficulties in Euclid'', a book about what he perceived as flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]]. He was also the first to find the general geometric solution to [[cubic equation]]s. He was also very influential in [[calendar reform]].<ref>{{Cite journal|last=Alam|first=S|year=2015|title=Mathematics for All and Forever|url=http://www.iisrr.in/mainsite/wp-content/uploads/2015/01/IISRR-IJR-1-Mathematics-for-All-...-Syed-Samsul-Alam.pdf|journal=Indian Institute of Social Reform & Research International Journal of Research}}</ref>

In the 13th century, [[Nasir al-Din Tusi]] (Nasireddin) made advances in [[spherical trigonometry]]. He also wrote influential work on Euclid's [[parallel postulate]]. In the 15th century, [[Ghiyath al-Kashi]] computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by [[Paolo Ruffini (mathematician)|Ruffini]] and [[William George Horner|Horner]].

Other achievements of Muslim mathematicians during this period include the addition of the [[decimal point]] notation to the [[Arabic numerals]], the discovery of all the modern [[trigonometric function]]s besides the sine, [[al-Kindi]]'s introduction of [[cryptanalysis]] and [[frequency analysis]], the development of [[analytic geometry]] by [[Ibn al-Haytham]], the beginning of [[algebraic geometry]] by [[Omar Khayyam]] and the development of an [[Mathematical notation|algebraic notation]] by [[Abū al-Hasan ibn Alī al-Qalasādī|al-Qalasādī]].<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>

During the time of the [[Ottoman Empire]] and [[Safavid Empire]] from the 15th century, the development of Islamic mathematics became stagnant.

== Maya ==
[[File:Maya Hieroglyphs Fig 40.jpg|thumb|The [[Maya numerals]] for numbers 1 through 19, written in the [[Maya script]]]]
In the [[Pre-Columbian Americas]], the [[Maya civilization]] that flourished in [[Mexico]] and [[Central America]] during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.<ref name="Goodman 2016 p121">{{Harv|Goodman|2016|p=121}}</ref> [[Maya numerals]] used a [[Radix|base]] of twenty, the [[vigesimal]] system, instead of a base of ten that forms the basis of the [[decimal]] system used by most modern cultures.<ref name="Goodman 2016 p121"/> The Maya used mathematics to create the [[Maya calendar]] as well as to predict astronomical phenomena in their native [[Maya astronomy]].<ref name="Goodman 2016 p121"/> While the concept of [[zero]] had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.<ref name="Goodman 2016 p121"/>

== Medieval European ==
<!--Linked from [[Template:[Ancient] Greek mathematics]]-->
{{further|List of medieval European scientists|European science in the Middle Ages}}
{{see also|Latin translations of the 12th century}}
 	
Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by [[Plato]]'s ''[[Timaeus (dialogue)|Timaeus]]'' and the biblical passage (in the ''[[Book of Wisdom]]'') that God had ''ordered all things in measure, and number, and weight''.<ref>''Wisdom'', 11:20</ref>

[[Boethius]] provided a place for mathematics in the curriculum in the 6th century when he coined the term ''[[quadrivium]]'' to describe the study of arithmetic, geometry, astronomy, and music. He wrote ''De institutione arithmetica'', a free translation from the Greek of [[Nicomachus]]'s ''Introduction to Arithmetic''; ''De institutione musica'', also derived from Greek sources; and a series of excerpts from Euclid's ''Elements''. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.<ref>Caldwell, John (1981). "The ''De Institutione Arithmetica'' and the ''De Institutione Musica''", pp. 135–54 in [[Margaret Gibson (historian)|Margaret Gibson]], ed., ''Boethius: His Life, Thought, and Influence,'' (Oxford: Basil Blackwell).</ref><ref>Folkerts, Menso (1970). ''"Boethius" Geometrie II'', Wiesbaden: Franz Steiner Verlag.</ref>

In the 12th century, European scholars traveled to Spain and Sicily [[Latin translations of the 12th century|seeking scientific Arabic texts]], including [[al-Khwārizmī]]'s ''[[The Compendious Book on Calculation by Completion and Balancing]]'', translated into Latin by [[Robert of Chester]], and the complete text of Euclid's ''Elements'', translated in various versions by [[Adelard of Bath]], [[Herman of Carinthia]], and [[Gerard of Cremona]].<ref>[[Marie-Thérèse d'Alverny]], "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard University Press, 1982).</ref><ref>Beaujouan, Guy. "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century''. Cambridge: Harvard University Press, 1982.</ref> These and other new sources sparked a renewal of mathematics.

Leonardo of Pisa, now known as [[Fibonacci]], serendipitously learned about the [[Hindu–Arabic numerals]] on a trip to what is now [[Béjaïa]], [[Algeria]] with his merchant father. (Europe was still using [[Roman numerals]].) There, he observed a system of [[arithmetic]] (specifically [[algorism]]) which due to the [[positional notation]] of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote ''[[Liber Abaci]]'' in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the [[Fibonacci sequence]] (known to Indian mathematicians for hundreds of years before that)<ref>Singh, Parmanand (1985). "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7</ref> which Fibonacci used as an unremarkable example.

The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.<ref>Grant, Edward and John E. Murdoch, eds. (1987). ''Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages''. Cambridge: Cambridge University Press. {{ISBN|0-521-32260-X}}.</ref> One important contribution was development of mathematics of local motion.

[[Thomas Bradwardine]] proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing:
V = log (F/R).<ref>Clagett, Marshall (1961). ''The Science of Mechanics in the Middle Ages''. Madison: University of Wisconsin Press, pp. 421–40.</ref> Bradwardine's analysis is an example of transferring a mathematical technique used by [[al-Kindi]] and [[Arnald of Villanova]] to quantify the nature of compound medicines to a different physical problem.<ref>Murdoch, John E. (1969). "''Mathesis in Philosophiam Scholasticam Introducta:'' The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in ''Arts libéraux et philosophie au Moyen Âge'' (Montréal: Institut d'Études Médiévales), pp. 224–27.</ref>

{{multiple image
|align=right
|width1=156
|image1=Oresme.jpg
|caption1=[[Nicole Oresme]] (1323–1382), shown in this contemporary [[illuminated manuscript]] with an [[armillary sphere]] in the foreground, was the first to offer a mathematical proof for the [[divergent series|divergence]] of the [[harmonic series (mathematics)|harmonic series]].<ref>{{citation|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|first=Clifford A.|last=Pickover|author-link=Clifford A. Pickover|publisher=Sterling Publishing Company, Inc.|year=2009|isbn=978-1-4027-5796-9|page=104|url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA104|quotation=Nicole Oresme ... was the first to prove the divergence of the harmonic series (c. 1350). His results were lost for several centuries, and the result was proved again by Italian mathematician [[Pietro Mengoli]] in 1647 and by Swiss mathematician [[Johann Bernoulli]] in 1687.}}</ref>
|width2=155
|image2=Ries.PNG
|caption2=[[Adam Ries]] (1492–1559) is known as the "father of modern calculating" because of his decisive contribution to the recognition that [[Roman numerals]] are unpractical and to their replacement by the considerably more practical [[Arabic numerals]].<ref>extent.https://www.scientificlib.com/en/Mathematics/Biographies/AdamRies.html#:~:text=Adam%20Ries%20is%20generally%20considered%20to%20be%20the,more%20structured%20Arabic%20numerals%20to%20a%20large%20extent.</ref>
}}

One of the 14th-century [[Oxford Calculators]], [[William Heytesbury]], lacking [[differential calculus]] and the concept of [[Limit of a function|limits]], proposed to measure instantaneous speed "by the path that '''would''' be described by [a body] '''if'''... it were moved uniformly at the same degree of speed with which it is moved in that given instant".<ref>Clagett, Marshall (1961). ''The Science of Mechanics in the Middle Ages''. Madison: University of Wisconsin Press, pp. 210, 214–15, 236.</ref>

Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".<ref>Clagett, Marshall (1961). ''The Science of Mechanics in the Middle Ages''. Madison: University of Wisconsin Press, p. 284.</ref>

[[Nicole Oresme]] at the [[University of Paris]] and the Italian [[Giovanni di Casali]] independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages''. Madison: University of Wisconsin Press, pp. 332–45, 382–91.</ref> In a later mathematical commentary on Euclid's ''Elements'', Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.<ref>Oresme, Nicole. "Questions on the ''Geometry'' of Euclid" Q. 14, pp. 560–65, in Marshall Clagett, ed., ''Nicole Oresme and the Medieval Geometry of Qualities and Motions''. Madison: University of Wisconsin Press, 1968.</ref>

== Renaissance ==
{{further|Mathematics and art}}
During the [[Renaissance]], the development of mathematics and of [[accounting]] were intertwined.<ref>Heeffer, Albrecht: ''On the curious historical coincidence of algebra and double-entry bookkeeping'', Foundations of the Formal Sciences, [[Ghent University]], November 2009, p. 7 [http://logica.ugent.be/albrecht/thesis/FOTFS2008-Heeffer.pdf]</ref> While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in [[Flanders]] and [[Germany]]) or [[abacus school]]s (known as ''abbaco'' in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing [[bookkeeping]] operations, but for complex bartering operations or the calculation of [[compound interest]], a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.

[[Piero della Francesca]] (c. 1415–1492) wrote books on [[solid geometry]] and [[Perspective (graphical)|linear perspective]], including ''[[De Prospectiva Pingendi]] (On Perspective for Painting)'', ''Trattato d’Abaco (Abacus Treatise)'', and ''[[De quinque corporibus regularibus]] (On the Five Regular Solids)''.<ref>della Francesca, Piero (1942). ''De Prospectiva Pingendi'', ed. G. Nicco Fasola, 2 vols., Florence.</ref><ref>della Francesca, Piero. ''Trattato d'Abaco'', ed. G. Arrighi, Pisa (1970).</ref><ref>della Francesca, Piero (1916). ''L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli'', ed. G. Mancini, Rome.</ref>

[[Image:Pacioli.jpg|thumb|left|''[[Portrait of Luca Pacioli]]'', a painting traditionally attributed to [[Jacopo de' Barbari]], 1495, ([[Museo di Capodimonte]]).]]
[[Luca Pacioli]]'s ''[[Summa de arithmetica|Summa de Arithmetica, Geometria, Proportioni et Proportionalità]]'' (Italian: "Review of [[Arithmetic]], [[Geometry]], [[Ratio]] and [[Proportionality (mathematics)|Proportion]]") was first printed and published in [[Venice]] in 1494. It included a 27-page treatise on bookkeeping, ''"Particularis de Computis et Scripturis"'' (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the [[mathematical puzzles]] it contained, and to aid the education of their sons.<ref>Sangster, Alan; Greg Stoner & Patricia McCarthy: [http://eprints.mdx.ac.uk/3201/1/final_final_proof_Market_paper_050308.pdf "The market for Luca Pacioli’s Summa Arithmetica"] {{Webarchive|url=https://web.archive.org/web/20180126012523/http://eprints.mdx.ac.uk/3201/1/final_final_proof_Market_paper_050308.pdf |date=2018-01-26 }} (Accounting, Business & Financial History Conference, Cardiff, September 2007) pp. 1–2.</ref> In ''Summa Arithmetica'', Pacioli introduced symbols for [[plus and minus]] for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. ''Summa Arithmetica'' was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.

In Italy, during the first half of the 16th century, [[Scipione del Ferro]] and [[Niccolò Fontana Tartaglia]] discovered solutions for [[cubic equation]]s. [[Gerolamo Cardano]] published them in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]'', together with a solution for the [[quartic equation]]s, discovered by his student [[Lodovico Ferrari]]. In 1572 [[Rafael Bombelli]] published his ''L'Algebra'' in which he showed how to deal with the [[imaginary number|imaginary quantities]] that could appear in Cardano's formula for solving cubic equations.

[[Simon Stevin]]'s ''[[De Thiende]]'' ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of [[decimal notation]] in Europe, which influenced all later work on the [[real number system]].<ref>[[Roshdi Rashed]] (1996) ''Encyclopedia of the History of Arabic Science'', chapter 10: Numeration and Arithmetic, page 315, [[Routledge]] {{doi|10.4324/9780203403600}}</ref><ref name="GS35">{{Cite journal |last=Sarton |first=George |date=1935 |title=The First Explanation of Decimal Fractions and Measures (1585). Together with a History of the Decimal Idea and a Facsimile (No. XVII) of Stevin's Disme |url=https://www.jstor.org/stable/225223 |journal=Isis |volume=23 |issue=1 |pages=153–244 |doi=10.1086/346940 |jstor=225223 |s2cid=143395001 |issn=0021-1753}}</ref>

Driven by the demands of navigation and the growing need for accurate maps of large areas, [[trigonometry]] grew to be a major branch of mathematics. [[Bartholomaeus Pitiscus]] was the first to use the word, publishing his ''Trigonometria'' in 1595. Regiomontanus's table of sines and cosines was published in 1533.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 978-0-393-32030-5}}</ref>

During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that were involved, were studied intensely.<ref name="Kline">
{{cite book
 | last = Kline
 | first = Morris
 | author-link =Morris Kline
 | title = Mathematics in Western Culture
 | publisher = Pelican
 | year = 1953
 | location=Great Britain
 | pages= 150–51}}</ref>

==Mathematics during the Scientific Revolution==
{{See also|Scientific Revolution}}

===17th century===
[[File:JKepler.jpg|thumb|upright|left|[[Johannes Kepler]]]]
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright|right|[[Gottfried Wilhelm Leibniz]]]]
The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. [[Tycho Brahe]] had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, [[Johannes Kepler]] was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of [[logarithm]]s by [[John Napier]] and [[Jost Bürgi]]. Kepler succeeded in formulating mathematical laws of planetary motion.<ref>{{cite book
 | last =Struik
 | first =Dirk
 | title =A Concise History of Mathematics
 | publisher =Courier Dover Publications
 | edition =3rd.
 | year =1987
 | pages =[https://archive.org/details/concisehistoryof0000stru_m6j1/page/89 89]
 | isbn =978-0-486-60255-4
 | url =https://archive.org/details/concisehistoryof0000stru_m6j1/page/89
 }}</ref>
The [[analytic geometry]] developed by [[René Descartes]] (1596–1650) allowed those orbits to be plotted on a graph, in [[Cartesian coordinates]].

Building on earlier work by many predecessors, [[Isaac Newton]] discovered the laws of physics that explain [[Kepler's Laws]], and brought together the concepts now known as [[calculus]]. Independently, [[Gottfried Wilhelm Leibniz]], developed calculus and much of the calculus notation still in use today. He also refined the [[binary number]] system, which is the foundation of nearly all digital ([[Scientific calculator|electronic]], [[Solid-state electronics|solid-state]], [[Logic gate|discrete logic]]) [[computer]]s.<ref>{{cite web |title=2021: 375th birthday of Leibniz, father of computer science |url=https://people.idsia.ch/~juergen/leibniz-father-computer-science-375.html |website=people.idsia.ch}}</ref>

Science and mathematics had become an international endeavor, which would soon spread over the entire world.<ref>Eves, Howard (1990). ''An Introduction to the History of Mathematics'', Saunders. {{ISBN|0-03-029558-0}}, p. 379, "... the concepts of calculus... (are) so far reaching and have exercised such an impact on the modern world that it is perhaps correct to say that without some knowledge of them a person today can scarcely claim to be well educated."</ref>

In addition to the application of mathematics to the studies of the heavens, [[applied mathematics]] began to expand into new areas, with the correspondence of [[Pierre de Fermat]] and [[Blaise Pascal]]. Pascal and Fermat set the groundwork for the investigations of [[probability theory]] and the corresponding rules of [[combinatorics]] in their discussions over a game of [[gambling]]. Pascal, with his [[Pascal's Wager|wager]], attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of [[utility theory]] in the 18th and 19th centuries.

=== 18th century ===
[[File:Leonhard Euler - Jakob Emanuel Handmann (Kunstmuseum Basel).jpg|right|thumb|upright|[[Leonhard Euler]]]]

The most influential mathematician of the 18th century was arguably [[Leonhard Euler]] (1707–1783). His contributions range from founding the study of [[graph theory]] with the [[Seven Bridges of Königsberg]] problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol [[Imaginary unit|<span style="font-family:times new Roman;">''i''</span>]], and he popularized the use of the Greek letter <math>\pi</math> to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.

Other important European mathematicians of the 18th century included [[Joseph Louis Lagrange]], who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and [[Pierre-Simon Laplace]], who, in the age of [[Napoleon]], did important work on the foundations of [[celestial mechanics]] and on [[statistics]].

== Modern ==
{{more citations needed section|date=April 2021|find=History of mathematics}}

=== 19th century ===
<!-- Modern period stars here:
 * Mathematical analysis: Bolzano, Cauchy, Riemann, Weierstrass
 * "Purely existential" proofs by Dedekind and Hilbert
 * Dirichlet's "arbitrary function"
 * Cantor's different kinds of infinity
 * Concentration on structures instead of calculation (abstract algebra, non-Euclidean geometry)
 * Institutionalization
-->
[[Image:Carl Friedrich Gauss.jpg|thumb|right|upright|[[Carl Friedrich Gauss]]]]
Throughout the 19th century mathematics became increasingly abstract.<ref>Howard Eves, An Introduction to the History of Mathematics, 6th edition, 1990, "In the nineteenth century, mathematics underwent a great forward surge ... . The new mathematics began to free itself from its ties to mechanics and astronomy, and a purer outlook evolved." p. 493</ref> [[Carl Friedrich Gauss]] (1777–1855) epitomizes this trend.{{Citation needed|date=April 2023}} He did revolutionary work on [[function (mathematics)|functions]] of [[complex variable]]s, in [[geometry]], and on the convergence of [[series (mathematics)|series]], leaving aside his many contributions to science. He also gave the first satisfactory proofs of the [[fundamental theorem of algebra]] and of the [[quadratic reciprocity law]].{{Citation needed|date=January 2024}}

[[Image:noneuclid.svg|thumb|left|upright=1.5|Behavior of lines with a common perpendicular in each of the three types of geometry]]
This century saw the development of the two forms of [[non-Euclidean geometry]], where the [[parallel postulate]] of Euclidean geometry no longer holds.
The Russian mathematician [[Nikolai Ivanovich Lobachevsky]] and his rival, the Hungarian mathematician [[János Bolyai]], independently defined and studied [[hyperbolic geometry]], where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. [[Elliptic geometry]] was developed later in the 19th century by the German mathematician [[Bernhard Riemann]]; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed [[Riemannian geometry]], which unifies and vastly generalizes the three types of geometry, and he defined the concept of a [[manifold]], which generalizes the ideas of [[curve]]s and [[Surface (topology)|surfaces]], and set the mathematical foundations for the [[General relativity|theory of general relativity]].<ref>{{Cite web |last=Wendorf |first=Marcia |date=2020-09-23 |title=Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity |url=https://interestingengineering.com/science/bernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity |access-date=2023-10-14 |website=interestingengineering.com |language=en-US}}</ref>

The 19th century saw the beginning of a great deal of [[abstract algebra]]. [[Hermann Grassmann]] in Germany gave a first version of [[vector space]]s, [[William Rowan Hamilton]] in Ireland developed [[noncommutative algebra]].{{Citation needed|date=January 2024}} The British mathematician [[George Boole]] devised an algebra that soon evolved into what is now called [[Boolean algebra]], in which the only numbers were 0 and 1. Boolean algebra is the starting point of [[mathematical logic]] and has important applications in [[electrical engineering]] and [[computer science]].{{Citation needed|date=January 2024}}<ref>Mari, C. (2012). George Boole. ''Great Lives from History: Scientists & Science'', N.PAG. Salem Press. <nowiki>https://search.ebscohost.com/login.aspx?AN=</nowiki>

176953509</ref> 
[[Augustin-Louis Cauchy]], [[Bernhard Riemann]], and [[Karl Weierstrass]] reformulated the calculus in a more rigorous fashion.{{Citation needed|date=January 2024}}

Also, for the first time, the limits of mathematics were explored. [[Niels Henrik Abel]], a Norwegian, and [[Évariste Galois]], a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four ([[Abel–Ruffini theorem]]).<ref>{{Cite journal |last=Ayoub |first=Raymond G. |date=1980-09-01 |title=Paolo Ruffini's contributions to the quintic |url=https://doi.org/10.1007/BF00357046 |journal=Archive for History of Exact Sciences |language=en |volume=23 |issue=3 |pages=253–277 |doi=10.1007/BF00357046 |s2cid=123447349 |issn=1432-0657}}</ref> Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to [[trisect an arbitrary angle]], to construct the side of a cube twice the volume of a given cube, [[Squaring the circle|nor to construct a square equal in area to a given circle]].{{Citation needed|date=January 2024}} Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.{{Citation needed|date=January 2024}} On the other hand, the limitation of three [[dimension]]s in geometry was surpassed in the 19th century through considerations of [[parameter space]] and [[hypercomplex number]]s.{{Citation needed|date=January 2024}}

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of [[group theory]], and the associated fields of [[abstract algebra]]. In the 20th century physicists and other scientists have seen group theory as the ideal way to study [[symmetry]].{{Citation needed|date=January 2024}}

[[Image:Georg Cantor (Porträt).jpg|thumb|right|upright|[[Georg Cantor]]]]
In the later 19th century, [[Georg Cantor]] established the first foundations of [[set theory]], which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of [[mathematical logic]] in the hands of [[Peano]], [[L.E.J. Brouwer]], [[David Hilbert]], [[Bertrand Russell]], and [[A.N. Whitehead]], initiated a long running debate on the [[foundations of mathematics]].{{Citation needed|date=January 2024}}

The 19th century saw the founding of a number of national mathematical societies: the [[London Mathematical Society]] in 1865,<ref>{{Cite journal |last=Collingwood |first=E. F. |date=1966 |title=A Century of the London Mathematical Society |url=http://doi.wiley.com/10.1112/jlms/s1-41.1.577 |journal=Journal of the London Mathematical Society |language=en |volume=s1-41 |issue=1 |pages=577–594 |doi=10.1112/jlms/s1-41.1.577}}</ref> the [[Société Mathématique de France]] in 1872,<ref>{{Cite web |title=Nous connaître {{!}} Société Mathématique de France |url=https://smf.emath.fr/la-smf/connaitre-la-smf |access-date=2024-01-28 |website=smf.emath.fr}}</ref> the [[Circolo Matematico di Palermo]] in 1884,<ref>{{Cite web |title=Mathematical Circle of Palermo |url=https://mathshistory.st-andrews.ac.uk/Societies/Palermo/ |access-date=2024-01-28 |website=Maths History |language=en}}</ref><ref>{{Cite book |last1=Grattan-Guinness |first1=Ivor |url=https://books.google.com/books?id=mC9GcTdHqpcC&pg=PA656 |title=The Rainbow of Mathematics: A History of the Mathematical Sciences |last2=Grattan-Guinness |first2=I. |date=2000 |publisher=W. W. Norton & Company |isbn=978-0-393-32030-5 |language=en}}</ref> the [[Edinburgh Mathematical Society]] in 1883,<ref>{{Cite journal |last=Rankin |first=R. A. |date=June 1986 |title=The first hundred years (1883–1983) |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/23AAB4A7D96568FC8E9003DE64AA8EF3/S0013091500016849a.pdf/div-class-title-the-first-hundred-years-1883-1983-div.pdf |journal=Proceedings of the Edinburgh Mathematical Society |language=en |volume=26 |issue=2 |pages=135–150 |doi=10.1017/S0013091500016849 |issn=1464-3839}}</ref> and the [[American Mathematical Society]] in 1888.<ref>{{Cite journal |last=Archibald |first=Raymond Clare |date=January 1939 |title=History of the American Mathematical Society, 1888–1938 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-45/issue-1/History-of-the-American-Mathematical-Society-18881938/bams/1183501056.full |journal=Bulletin of the American Mathematical Society |volume=45 |issue=1 |pages=31–46 |doi=10.1090/S0002-9904-1939-06908-5 |issn=0002-9904|doi-access=free }}</ref> The first international, special-interest society, the [[Quaternion Society]], was formed in 1899, in the context of a [[hyperbolic quaternion#Historical review|vector controversy]].<ref>{{Cite journal |last1=Molenbroek |first1=P. |last2=Kimura |first2=Shunkichi |date=3 October 1895 |title=To Friends and Fellow Workers in Quaternions |url=https://www.nature.com/articles/052545a0.pdf |journal=Nature |language=en |volume=52 |issue=1353 |pages=545–546 |doi=10.1038/052545a0 |bibcode=1895Natur..52..545M |s2cid=4008586 |issn=1476-4687}}</ref>

In 1897, [[Kurt Hensel]] introduced [[p-adic number]]s.<ref>{{Cite book |last=Murty |first=M. Ram |url=https://books.google.com/books?id=SseFAwAAQBAJ&dq=p-adic+numbers+hensel+1897&pg=PR9 |title=Introduction to $p$-adic Analytic Number Theory |date=2009-02-09 |publisher=American Mathematical Soc. |isbn=978-0-8218-4774-9 |language=en}}</ref>

=== 20th century === <!-- Hibert's problems, foundational crisis, Bourbaki -->
The 20th century saw mathematics become a major profession. By the end of the century, thousands of new Ph.D.s in mathematics were being awarded every year, and jobs were available in both teaching and industry.<ref>{{cite web|url=https://dpcpsi.nih.gov/sites/default/files/opep/document/Final_Report_(03-517-OD-OER)%202006.pdf|title=U.S. Doctorates in the 20th Century|access-date=5 April 2023|website=nih.gov|date=June 2006|author1=Lori Thurgood|author2=Mary J. Golladay|author3=Susan T. Hill}}</ref> An effort to catalogue the areas and applications of mathematics was undertaken in [[Klein's encyclopedia]].<ref>{{Cite journal |last=Pitcher |first=A. D. |date=1922 |title=Encyklopâdie der Mathematischen Wissenschaften. |url=https://www.ams.org/journals/bull/1922-28-09/S0002-9904-1922-03635-X/S0002-9904-1922-03635-X.pdf |journal=[[Bulletin of the American Mathematical Society]] |volume=28 |issue=9 |pages=474 |doi=10.1090/s0002-9904-1922-03635-x}}</ref>

In a 1900 speech to the [[International Congress of Mathematicians]], [[David Hilbert]] set out a list of [[Hilbert's problems|23 unsolved problems in mathematics]].<ref>{{Cite journal |last=Hilbert |first=David |date=1902 |title=Mathematical problems |url=https://www.ams.org/bull/1902-08-10/S0002-9904-1902-00923-3/ |journal=Bulletin of the American Mathematical Society |language=en |volume=8 |issue=10 |pages=437–479 |doi=10.1090/S0002-9904-1902-00923-3 |issn=0002-9904|doi-access=free }}</ref> These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.<ref>{{Cite web |title=Hilbert's 23 problems {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/Hilberts-23-problems |access-date=2025-04-19 |website=www.britannica.com |language=en}}</ref>

[[Image:Four Colour Map Example.svg|thumb|left|upright|A map illustrating the [[Four Color Theorem]]]]
Notable historical conjectures were finally proven. In 1976, [[Wolfgang Haken]] and [[Kenneth Appel]] proved the [[four color theorem]], controversial at the time for the use of a computer to do so.<ref>{{Cite journal |last=Gonthier |first=Georges |date=December 2008 |title=Formal Proof—The Four-Color Theorem |url=https://www.ams.org/notices/200811/tx081101382p.pdf |journal=[[Notices of the AMS]] |volume=55 |issue=11 |pages=1382}}</ref> [[Andrew Wiles]], building on the work of others, proved [[Fermat's Last Theorem]] in 1995.<ref>{{Cite journal |last=Castelvecchi |first=Davide |date=2016-03-01 |title=Fermat's last theorem earns Andrew Wiles the Abel Prize |url=https://www.nature.com/articles/nature.2016.19552 |journal=Nature |language=en |volume=531 |issue=7594 |pages=287 |doi=10.1038/nature.2016.19552 |pmid=26983518 |bibcode=2016Natur.531..287C |issn=1476-4687}}</ref> [[Paul Cohen (mathematician)|Paul Cohen]] and [[Kurt Gödel]] proved that the [[continuum hypothesis]] is [[logical independence|independent]] of (could neither be proved nor disproved from) the [[ZFC|standard axioms of set theory]].<ref>{{Cite journal |last=Cohen |first=Paul |date=2002-12-01 |title=The Discovery of Forcing |url=https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-32/issue-4/The-Discovery-of-Forcing/10.1216/rmjm/1181070010.full |journal=Rocky Mountain Journal of Mathematics |volume=32 |issue=4 |doi=10.1216/rmjm/1181070010 |issn=0035-7596}}</ref> In 1998, [[Thomas Callister Hales]] proved the [[Kepler conjecture]], also using a computer.<ref>{{Cite news |last=Wolchover |first=Natalie |date=22 February 2013 |title=In Computers We Trust? |url=https://www.quantamagazine.org/in-computers-we-trust-20130222/ |access-date=28 January 2024 |work=[[Quanta Magazine]]}}</ref>

Mathematical collaborations of unprecedented size and scope took place. An example is the [[classification of finite simple groups]] (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.<ref>{{Cite web |title=An enormous theorem: the classification of finite simple groups |url=https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups |access-date=2024-01-28 |website=Plus Maths |language=en}}</ref> A group of French mathematicians, including [[Jean Dieudonné]] and [[André Weil]], publishing under the [[pseudonym]] "[[Nicolas Bourbaki]]", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.<ref>Maurice Mashaal, 2006. ''Bourbaki: A Secret Society of Mathematicians''. [[American Mathematical Society]]. {{ISBN|0-8218-3967-5|978-0-8218-3967-6}}.</ref>

[[File:Relativistic precession.svg|thumb|Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with [[General relativity#Orbital effects and the relativity of direction|relativistic precession of apsides]]]]
[[Differential geometry]] came into its own when [[Albert Einstein]] used it in [[general relativity]].{{Citation needed|date=January 2024}} Entirely new areas of mathematics such as [[mathematical logic]], [[topology]], and [[John von Neumann]]'s [[game theory]] changed the kinds of questions that could be answered by mathematical methods.{{Citation needed|date=January 2024}} All kinds of [[Mathematical structure|structures]] were abstracted using axioms and given names like [[metric space]]s, [[topological space]]s etc.{{Citation needed|date=January 2024}} As mathematicians do, the concept of an abstract structure was itself abstracted and led to [[category theory]].{{Citation needed|date=January 2024}} [[Grothendieck]] and [[Jean-Pierre Serre|Serre]] recast [[algebraic geometry]] using [[Sheaf (mathematics)|sheaf theory]].{{Citation needed|date=January 2024}} Large advances were made in the qualitative study of [[dynamical systems theory|dynamical systems]] that [[Henri Poincaré|Poincaré]] had begun in the 1890s.{{Citation needed|date=January 2024}}
[[Measure theory]] was developed in the late 19th and early 20th centuries. Applications of measures include the [[Lebesgue integral]], [[Kolmogorov]]'s axiomatisation of [[probability theory]], and [[ergodic theory]].{{Citation needed|date=January 2024}} [[Knot theory]] greatly expanded.{{Citation needed|date=January 2024}} [[Quantum mechanics]] led to the development of [[functional analysis]],{{Citation needed|date=January 2024}} a branch of mathematics that was greatly developed by [[Stefan Banach]] and his collaborators who formed the [[Lwów School of Mathematics]].<ref>{{cite web|url=https://www.britannica.com/biography/Stefan-Banach|title=Stefan Banach - Polish Mathematician|website=britannica.com|date=27 August 2023 }}</ref> Other new areas include [[Laurent Schwartz]]'s [[Distribution (mathematics)|distribution theory]], [[Fixed-point theorem|fixed point theory]], [[singularity theory]] and [[René Thom]]'s [[catastrophe theory]], [[model theory]], and [[Benoit Mandelbrot|Mandelbrot]]'s [[fractals]].{{Citation needed|date=January 2024}} [[Lie theory]] with its [[Lie group]]s and [[Lie algebra]]s became one of the major areas of study.<ref>*{{cite book |first=Thomas |last=Hawkins |authorlink=Thomas W. Hawkins Jr. |year=2000 |title=Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869–1926 |url=https://archive.org/details/emergenceoftheor0000hawk |url-access=registration |publisher=Springer |isbn=0-387-98963-3 }}</ref>

[[Non-standard analysis]], introduced by [[Abraham Robinson]], rehabilitated the [[infinitesimal]] approach to calculus, which had fallen into disrepute in favour of the theory of [[Limit of a function|limits]], by extending the field of real numbers to the [[Hyperreal number]]s which include infinitesimal and infinite quantities.{{Citation needed|date=January 2024}} An even larger number system, the [[surreal number]]s were discovered by [[John Horton Conway]] in connection with [[combinatorial game]]s.{{Citation needed|date=January 2024}}

The development and continual improvement of [[computer]]s, at first mechanical analog machines and then digital electronic machines, allowed [[Private industry|industry]] to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: [[Alan Turing]]'s [[computability theory]]; [[Computational complexity theory|complexity theory]]; [[Derrick Henry Lehmer]]'s use of [[ENIAC]] to further number theory and the [[Lucas–Lehmer primality test]]; [[Rózsa Péter]]'s [[recursive function theory]]; [[Claude Shannon]]'s [[information theory]]; [[signal processing]]; [[data analysis]]; [[Mathematical optimization|optimization]] and other areas of [[operations research]].{{Citation needed|date=January 2024}} In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of [[discrete mathematics|discrete]] concepts and the expansion of [[combinatorics]] including [[graph theory]]. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as [[numerical analysis]] and [[symbolic computation]].{{Citation needed|date=January 2024}} Some of the most important methods and [[algorithm]]s of the 20th century are: the [[simplex algorithm]], the [[fast Fourier transform]], [[error-correcting code]]s, the [[Kalman filter]] from [[control theory]] and the [[RSA algorithm]] of [[public-key cryptography]].{{Citation needed|date=January 2024}}

At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved{{By whom|date=January 2024}} the truth or falsity of all statements formulated about the [[natural number]]s plus either addition or multiplication (but not both), was [[Decidability (logic)|decidable]], i.e. could be determined by some algorithm.{{Citation needed|date=January 2024}} In 1931, [[Kurt Gödel]] found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as [[Peano arithmetic]], was in fact [[incompleteness theorem|incomplete]]. (Peano arithmetic is adequate for a good deal of [[number theory]], including the notion of [[prime number]].) A consequence of Gödel's two [[incompleteness theorem]]s is that in any mathematical system that includes Peano arithmetic (including all of [[mathematical analysis|analysis]] and geometry), truth necessarily outruns proof, i.e. there are true statements that [[Incompleteness theorem|cannot be proved]] within the system. Hence mathematics cannot be reduced to mathematical logic, and [[David Hilbert]]'s dream of making all of mathematics complete and consistent needed to be reformulated.{{Citation needed|date=January 2024}}

[[Image:GammaAbsSmallPlot.png|thumb|right|The [[absolute value]] of the Gamma function on the complex plane]]
One of the more colorful figures in 20th-century mathematics was [[Srinivasa Aiyangar Ramanujan]] (1887–1920), an Indian [[autodidact]]<ref name=":3">{{Cite journal |last=Ono |first=Ken |date=2006 |title=Honoring a Gift from Kumbakonam |url=https://www.ams.org/notices/200606/fea-ono.pdf |journal=[[Notices of the AMS]] |volume=53 |issue=6 |pages=640–651}}</ref> {{Citation needed span|text=who conjectured or proved over 3000 theorems|date=January 2024|reason=theorem count not mentioned in the source}}, including properties of [[highly composite number]]s,<ref>{{Cite journal |last1=Alaoglu |first1=L. |author-link=Leonidas Alaoglu |last2=Erdős |first2=Paul |author-link2=Paul Erdős |date=14 February 1944 |title=On highly composite and similar numbers |url=https://community.ams.org/journals/tran/1944-056-00/S0002-9947-1944-0011087-2/S0002-9947-1944-0011087-2.pdf |journal=[[Transactions of the American Mathematical Society]] |volume=56 |pages=448–469|doi=10.1090/S0002-9947-1944-0011087-2 }}</ref> the [[partition function (number theory)|partition function]]<ref name=":3" /> and its [[asymptotics]],<ref>{{Cite journal |last=Murty |first=M. Ram |date=2013 |title=The Partition Function Revisited |url=https://mast.queensu.ca/~murty/partition.pd |journal=The Legacy of Srinivasa Ramanujan, RMS-Lecture Notes Series |volume=20 |pages=261–279}}</ref> and [[Ramanujan theta function|mock theta functions]].<ref name=":3" /> He also made major investigations in the areas of [[gamma function]]s,<ref>{{Citation |last=Bradley |first=David M. |title=Ramanujan's formula for the logarithmic derivative of the gamma function |date=2005-05-07 |arxiv=math/0505125 |bibcode=2005math......5125B }}</ref><ref>{{Cite journal |last=Askey |first=Richard |date=1980 |title=Ramanujan's Extensions of the Gamma and Beta Functions |url=https://www.jstor.org/stable/2321202 |journal=The American Mathematical Monthly |volume=87 |issue=5 |pages=346–359 |doi=10.2307/2321202 |jstor=2321202 |issn=0002-9890}}</ref> [[modular form]]s,<ref name=":3" /> [[divergent series]],<ref name=":3" /> [[General hypergeometric function|hypergeometric series]]<ref name=":3" /> and prime number theory.<ref name=":3" />

[[Paul Erdős]] published more papers than any other mathematician in history,<ref>{{cite web | url=http://oakland.edu/enp/trivia/ | title=Grossman – the Erdös Number Project }}</ref> working with hundreds of collaborators. Mathematicians have a game equivalent to the [[Kevin Bacon Game]], which leads to the [[Erdős number]] of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.<ref>{{Cite journal |last=Goffman |first=Casper |date=1969 |title=And What Is Your Erdos Number? |url=https://www.jstor.org/stable/2317868 |journal=The American Mathematical Monthly |volume=76 |issue=7 |pages=791 |doi=10.2307/2317868 |jstor=2317868 |issn=0002-9890}}</ref><ref>{{Cite web |title=grossman - The Erdös Number Project |url=https://sites.google.com/oakland.edu/grossman/home/the-erdoes-number-project |access-date=2024-01-28 |website=sites.google.com |language=en-US}}</ref>

[[Emmy Noether]] has been described by many as the most important woman in the history of mathematics.<ref>{{citation|author-link=Pavel Alexandrov|last=Alexandrov|first=Pavel S.|chapter=In Memory of Emmy Noether | title = Emmy Noether: A Tribute to Her Life and Work|editor1-first =James W | editor1-last = Brewer | editor2-first = Martha K | editor2-last = Smith | place = New York | publisher= Marcel Dekker | year= 1981 | isbn = 978-0-8247-1550-2 |pages= 99–111}}.</ref> She studied the theories of [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]].<ref>{{Cite news |last=Angier |first=Natalie |date=2012-03-26 |title=The Mighty Mathematician You've Never Heard Of |url=https://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html |access-date=2024-04-20 |work=The New York Times |language=en-US |issn=0362-4331}}</ref>

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century, there were hundreds of specialized areas in mathematics, and the [[Mathematics Subject Classification]] was dozens of pages long.<ref>{{Cite web|url=https://www.ams.org/mathscinet/msc/pdfs/classifications2000.pdf|title=Mathematics Subject Classification 2000|accessdate=5 April 2023}}</ref> More and more [[mathematical journal]]s were published and, by the end of the century, the development of the [[World Wide Web]] led to online publishing.{{Citation needed|date=January 2024}}

=== 21st century ===
{{See also|List of unsolved problems in mathematics#Problems solved since 1995}}
In 2000, the [[Clay Mathematics Institute]] announced the seven [[Millennium Prize Problems]].<ref>{{Cite journal |last=Dickson |first=David |date=2000-05-01 |title=Mathematicians chase the seven million-dollar proofs |url=https://www.nature.com/articles/35013216 |journal=Nature |language=en |volume=405 |issue=6785 |pages=383 |doi=10.1038/35013216 |pmid=10839504 |issn=1476-4687}}</ref> In 2003 the [[Poincaré conjecture]] was solved by [[Grigori Perelman]] (who declined to accept an award, as he was critical of the mathematics establishment).<ref>{{Cite news |date=22 August 2006 |title=Maths genius declines top prize |url=http://news.bbc.co.uk/2/hi/science/nature/5274040.stm |access-date=28 January 2024 |work=[[BBC News]] |language=en-GB}}</ref>

Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched.<ref>{{Cite web |title=Journal of Humanistic Mathematics - an online-only, open access, peer reviewed journal {{!}} Current Journals {{!}} Claremont Colleges |url=https://scholarship.claremont.edu/jhm/ |access-date=2024-08-05 |website=scholarship.claremont.edu}}</ref><ref>{{Cite web |title=Electronic Math Journals |url=https://www.stat.berkeley.edu/~mathsurv/ejournals.html |access-date=2024-08-05 |website=www.stat.berkeley.edu}}</ref> There is an increasing drive toward [[Open access (publishing)|open access publishing]], first made popular by [[arXiv]].{{Citation needed|date=January 2024}}

== Future ==
{{Main|Future of mathematics}}

There are many observable trends in mathematics, the most notable being that the subject is growing ever larger as computers are ever more important and powerful; the volume of data being produced by science and industry, facilitated by computers, continues expanding exponentially. As a result, there is a corresponding growth in the demand for mathematics to help process and understand this [[big data]].<ref>{{Cite web |last=Nations |first=United |title=Big Data for Sustainable Development |url=https://www.un.org/en/global-issues/big-data-for-sustainable-development |access-date=2023-11-28 |website=United Nations |language=en}}</ref> Math science careers are also expected to continue to grow, with the US [[Bureau of Labor Statistics]] estimating (in 2018) that "employment of mathematical science occupations is projected to grow 27.9 percent from 2016 to 2026."<ref>{{Cite web |last=Rieley |first=Michael |title=Big data adds up to opportunities in math careers : Beyond the Numbers: U.S. Bureau of Labor Statistics |url=https://www.bls.gov/opub/btn/volume-7/big-data-adds-up.htm |access-date=2023-11-28 |website=www.bls.gov |language=en}}</ref>

==See also==
{{Portal|Mathematics}}{{div col|colwidth=20em}}
* [[Archives of American Mathematics]]
* [[Ethnomathematics]]
* [[History of algebra]]
* [[History of arithmetic]]
* [[History of calculus]]
* [[History of combinatorics]]
* [[History of the function concept]]
* [[History of geometry]]
* [[History of group theory]]
* [[History of logic]]
* [[History of mathematicians]]
* [[History of mathematical notation]]
* [[History of measurement]]
* [[History of numbers]]
** [[History of ancient numeral systems]]
** [[Prehistoric counting]]
** [[List of books on history of number systems]]
* [[History of number theory]]
* [[History of statistics]]
* [[History of trigonometry]]
* [[History of writing numbers]]
* [[Kenneth O. May Prize]]
* [[List of important publications in mathematics]]
* [[Lists of mathematicians]]
* [[List of mathematics history topics]]
* [[Mathematical folklore]]
* [[Timeline of mathematics]]
{{div col end}}

==Notes==
{{notelist}}
{{Reflist}}

==References==
*{{citation |surname=de Crespigny |given=Rafe |author-link=Rafe de Crespigny |title=A Biographical Dictionary of Later Han to the Three Kingdoms (23–220 AD) |location=Leiden |publisher=Koninklijke Brill |year=2007 |isbn=978-90-04-15605-0 |postscript=.}}
* {{citation |first1=Lennart |last1= Berggren | first2= Jonathan M. | last2= Borwein | first3= Peter B. | last3 = Borwein |title=Pi: A Source Book | place= New York |publisher= Springer |year= 2004 |isbn=978-0-387-20571-7}}
* {{citation |first=C.B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |place=New York |publisher=Wiley |year=1991 |orig-year=1989 |isbn=978-0-471-54397-8 |url=https://archive.org/details/historyofmathema00boye }}
* {{citation |first=Serafina |last=Cuomo |title=Ancient Mathematics| place= London |publisher= Routledge |year=2001 |isbn=978-0-415-16495-5}}
<!--UNUSED* {{citation |first=Howard |last=Eves |authorlink=Howard Eves|title=An Introduction to the History of Mathematics |publisher=Saunders |year=1990 |isbn=0-03-029558-0}}-->
* {{citation |first=Michael, K.J. |last=Goodman |title=An introduction of the Early Development of Mathematics| place= Hoboken |publisher= Wiley |year=2016 |isbn=978-1-119-10497-1}}
* {{citation |first=Jan |last=Gullberg |title=Mathematics: From the Birth of Numbers |place=New York |publisher=W.W. Norton and Company |year=1997 |isbn=978-0-393-04002-9 |url-access=registration |url=https://archive.org/details/mathematicsfromb1997gull }}
* {{citation|last=Joyce|first=Hetty|journal=American Journal of Archaeology|title=Form, Function and Technique in the Pavements of Delos and Pompeii|date=July 1979|volume=83|number=3|jstor=505056|doi=10.2307/505056|pages=253–63|s2cid=191394716|postscript=.}}
* {{citation |first=Victor J. |last=Katz |title=A History of Mathematics: An Introduction |edition=2nd |publisher=[[Addison-Wesley]] |year=1998 |isbn=978-0-321-01618-8 |url=https://archive.org/details/historyofmathema00katz }}
* {{citation | year=2007 | last=Katz | first=Victor J.  | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | place= Princeton, NJ | publisher=Princeton University Press | isbn=978-0-691-11485-9 }}
* {{citation | year= 1995 | orig-year= 1959 | last1=Needham | first1=Joseph | last2=Wang | first2=Ling | title= Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth | volume= 3 | place= Cambridge | publisher=Cambridge University Press | author-link1= Joseph Needham| author-link2= Wang Ling (historian)| isbn=978-0-521-05801-8}}
* {{citation | year= 2000 | orig-year= 1965 | last1=Needham | first1=Joseph | last2=Wang | first2=Ling | title= Science and Civilization in China: Physics and Physical Technology: Mechanical Engineering | volume= 4 | place= Cambridge 
| publisher=Cambridge University Press | edition = reprint| isbn=978-0-521-05803-2}}
* {{citation|last=Sleeswyk|first=Andre|journal=Scientific American|title=Vitruvius' odometer|date=October 1981|volume=252|number=4|pages=188–200|doi=10.1038/scientificamerican1081-188|bibcode=1981SciAm.245d.188S|postscript=.}}
* {{citation | year=1998 | last=Straffin | first=Philip D.  | title=Liu Hui and the First Golden Age of Chinese Mathematics | journal= Mathematics Magazine | volume= 71 | number= 3 | pages=163–81 | doi=10.1080/0025570X.1998.11996627 }}
* {{citation|last=Tang|first=Birgit|title=Delos, Carthage, Ampurias: the Housing of Three Mediterranean Trading Centres|year=2005|location=Rome|publisher=L'Erma di Bretschneider (Accademia di Danimarca)|isbn=978-88-8265-305-7|url=https://books.google.com/books?id=nw5eupvkvfEC|postscript=.}}
<!--UNUSED*{{citation | last=Plofker | first1=Kim | authorlink1 = Kim Plofker | year=2009 | title=Mathematics in India: 500 BCE–1800 CE | place=Princeton, NJ
 | publisher=Princeton University Press | isbn= 0-691-12067-6 }}-->
* {{citation | year=2009 | editor1= Robson, Eleanor | editor2= Stedall, Jacqueline | last=Volkov | first=Alexei  | title=The Oxford Handbook of the History of Mathematics | chapter=Mathematics and Mathematics Education in Traditional Vietnam | place= Oxford | publisher=Oxford University Press | pages=153–76 | isbn=978-0-19-921312-2}}

== Further reading ==

===General===
* {{cite book
 | last = Aaboe
 | first = Asger
 | author-link = Asger Aaboe
 | year = 1964
 | title = Episodes from the Early History of Mathematics
 | publisher = Random House
 | location = New York
}}
* {{cite book
 | last = Bell
 | first = E. T.
 | author-link = Eric Temple Bell
 | title = Men of Mathematics
 | url = https://archive.org/details/menofmathematics0041bell
 | url-access = registration
 | publisher = Simon and Schuster
 | year = 1937
}}
* [[David M. Burton|Burton, David M.]] (1997). ''The History of Mathematics: An Introduction''. McGraw Hill.
* {{cite book|first=Ivor|last=Grattan-Guinness|author-link=Ivor Grattan-Guinness|title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences|publisher=The Johns Hopkins University Press|year=2003|isbn=978-0-8018-7397-3}}
* [[Morris Kline|Kline, Morris]]. ''Mathematical Thought from Ancient to Modern Times''.
* [[Dirk Jan Struik|Struik, D. J.]] (1987). ''A Concise History of Mathematics'', fourth revised edition. Dover Publications, New York.

=== Books on a specific period ===
* {{cite book
 | last = Gillings
 | first = Richard J.
 | author-link = Richard J. Gillings
 | title = Mathematics in the Time of the Pharaohs
 | publisher = MIT Press
 | location = Cambridge, MA
 | year = 1972
}}
* {{cite book
 | last = Heath
 | first = Thomas Little
 | author-link = Thomas Little Heath
 | title = [[A History of Greek Mathematics]]
 | url = 
 | url-access = 
 | publisher = Oxford, Claredon Press
 | year = 1921
 | isbn = 
}}
<!--
* [[Annaliese Maier|Maier, Annaliese]] (1982). ''At the Threshold of Exact Science:  Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy'', edited by Steven Sargent, Philadelphia: University of Pennsylvania Press. -->
* [[Bartel Leendert van der Waerden|van der Waerden, B. L.]] (1983). ''Geometry and Algebra in Ancient Civilizations'', Springer, {{ISBN|0-387-12159-5}}.

=== Books on a specific topic ===
* {{citation |first=Leo |last=Corry |title=A Brief History of Numbers  |publisher=Oxford University Press |year=2015 |isbn=978-0198702597}}
* {{cite book| last = Hoffman| first = Paul| author-link = Paul Hoffman (science writer)| year = 1998| title = The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth| publisher = Hyperion| isbn = 0-7868-6362-5}}
* {{cite book| last = Menninger| first = Karl W.| author-link =  Karl Menninger (mathematics)| year = 1969| title = Number Words and Number Symbols: A Cultural History of Numbers| publisher = MIT Press| isbn = 978-0-262-13040-0}}
* {{cite book| last = Stigler| first = Stephen M.| author-link = Stephen Stigler| year = 1990| title = The History of Statistics: The Measurement of Uncertainty before 1900| publisher = Belknap Press | isbn = 978-0-674-40341-3}}

==External links==
{{wikiquote}}

=== Documentaries ===
* [[BBC]] (2008). ''[[The Story of Maths]]''.
* [https://www.bbc.co.uk/programmes/p003k9hq Renaissance Mathematics], BBC Radio 4 discussion with Robert Kaplan, Jim Bennett & Jackie Stedall (''In Our Time'', Jun 2, 2005)

=== Educational material ===
* [http://www-history.mcs.st-andrews.ac.uk/ MacTutor History of Mathematics archive] (John J. O'Connor and Edmund F. Robertson; University of St Andrews, Scotland). An award-winning website containing detailed biographies on many historical and contemporary mathematicians, as well as information on notable curves and various topics in the history of mathematics.
* [http://aleph0.clarku.edu/~djoyce/mathhist/ History of Mathematics Home Page] ([[David E. Joyce (mathematician)|David E. Joyce]]; Clark University). Articles on various topics in the history of mathematics with an extensive bibliography.
* [http://www.maths.tcd.ie/pub/HistMath/ The History of Mathematics] (David R. Wilkins; Trinity College, Dublin). Collections of material on the mathematics between the 17th and 19th century.
* [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics] (Jeff Miller). Contains information on the earliest known uses of terms used in mathematics.
* [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] (Jeff Miller). Contains information on the history of mathematical notations.
* [http://www.economics.soton.ac.uk/staff/aldrich/Mathematical%20Words.htm Mathematical Words: Origins and Sources] (John Aldrich, University of Southampton) Discusses the origins of the modern mathematical word stock.
* [http://www.agnesscott.edu/lriddle/women/women.htm Biographies of Women Mathematicians] (Larry Riddle; Agnes Scott College).
* [http://www.math.buffalo.edu/mad/ Mathematicians of the African Diaspora] (Scott W. Williams; University at Buffalo).
* [http://fredrickey.info/hm/mini/MinicourseDocuments-09.pdf Notes for MAA minicourse: teaching a course in the history of mathematics. (2009)] ([[V. Frederick Rickey]] & [[Victor J. Katz]]).
* [https://www.history-of-physics.com/2017/08/ancient-rome-odometer-of-vitruv.html#:~:text=What%20Was%20the%20Odometer%20of%20Vitruvius.%20The%20Odometer,wheel%20which%20was%20manually%20moved%20along%20by%20hand Ancient Rome: The Odometer Of Vitruv]. Pictorial (moving) re-construction of Vitusius' Roman ododmeter.

=== Bibliographies ===
* [http://mathematics.library.cornell.edu/additional/Collected-Works-of-Mathematicians A Bibliography of Collected Works and Correspondence of Mathematicians] [https://web.archive.org/web/20070317034718/http://astech.library.cornell.edu/ast/math/find/Collected-Works-of-Mathematicians.cfm archive dated 2007/3/17] (Steven W. Rockey; Cornell University Library).

===Organizations===
* [http://www.unizar.es/ichm/ International Commission for the History of Mathematics]

===Journals===
* ''[[Historia Mathematica]]''
* [http://www.maa.org/press/periodicals/convergence Convergence] {{Webarchive|url=https://web.archive.org/web/20200908223859/https://www.maa.org/press/periodicals/convergence |date=2020-09-08 }}, the [[Mathematical Association of America]]'s online ''Math History'' Magazine
* [http://archives.math.utk.edu/topics/history.html History of Mathematics] {{Webarchive|url=https://web.archive.org/web/20061004065105/http://archives.math.utk.edu/topics/history.html |date=2006-10-04 }} Math Archives (University of Tennessee, Knoxville)
* [http://mathforum.org/library/topics/history/ History/Biography] The Math Forum (Drexel University)
* [https://web.archive.org/web/20020716102307/http://www.otterbein.edu/resources/library/libpages/subject/mathhis.htm History of Mathematics] (Courtright Memorial Library).
* [http://homepages.bw.edu/~dcalvis/history.html History of Mathematics Web Sites] {{Webarchive|url=https://web.archive.org/web/20090525100502/http://homepages.bw.edu/~dcalvis/history.html |date=2009-05-25 }} (David Calvis; Baldwin-Wallace College)
* [https://web.archive.org/web/20030219004407/http://webpages.ull.es/users/jbarrios/hm/ Historia de las Matemáticas] (Universidad de La La guna)
* [http://www.mat.uc.pt/~jaimecs/indexhm.html História da Matemática] (Universidade de Coimbra)
* [https://web.archive.org/web/20110707053917/http://math.illinoisstate.edu/marshall Using History in Math Class]
* [http://mathres.kevius.com/history.html Mathematical Resources: History of Mathematics] (Bruno Kevius)
* [https://web.archive.org/web/20080615051823/http://www.dm.unipi.it/~tucci/index.html History of Mathematics] (Roberta Tucci)

{{Areas of mathematics}}
{{Indian mathematics}}
{{Islamic mathematics}}
{{History of science}}
{{History of mathematics}}

[[Category:History of mathematics| ]]
[[Category:History of science by discipline|Mathematics]]