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[[File:Euclid-proof.jpg|thumb|right|250px|A proof from [[Euclid|Euclid's]] ''[[Euclid's Elements|Elements]]'', 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>]]
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The area of study known as the '''history of mathematics''' is primarily an investigation into the origin of discoveries in [[mathematics]] and, to a lesser extent, an investigation into 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. The most ancient mathematical texts available are ''[[Plimpton 322]]'' ([[Babylonian mathematics]] c. 1900 BC),<ref>J. Friberg, "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.</ref> the ''[[Rhind Mathematical Papyrus]]'' (Egyptian mathematics c. 2000-1800 BC)<ref>{{Cite book | edition = 2 | publisher = [[Dover Publications]] | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = The Exact Sciences in Antiquity | origyear = 1957 | year = 1969 | isbn = 978-0-486-22332-2 | url = http://books.google.com/?id=JVhTtVA2zr8C}} Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.</ref> and the ''[[Moscow Mathematical Papyrus]]'' ([[Egyptian mathematics]] c. 1890 BC). All of these texts concern the so-called [[Pythagorean theorem]], which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
 
The study of mathematics as a subject in its own right begins in the 6th century BC with the [[Pythagoreans]], who coined the term "mathematics" from the ancient Greek ''μάθημα'' (''mathema''), meaning "subject of instruction".<ref>{{cite book|author=Heath|title=A Manual of Greek Mathematics|page=5}}</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>Sir Thomas L. Heath, ''A Manual of Greek Mathematics'', Dover, 1963, 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> [[Counting rods|Chinese mathematics]] made early contributions, including a [[place value system]].<ref>George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'',Penguin Books, London, 1991, pp.140—148</ref><ref>Georges Ifrah, ''Universalgeschichte der Zahlen'', Campus, Frankfurt/New York, 1986, pp.428—437</ref> The [[Hindu-Arabic numeral system]] and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in [[Indian mathematics|India]] and was transmitted to the west via Islamic mathematics.<ref>Robert Kaplan, "The Nothing That Is: A Natural History of Zero", Allen Lane/The Penguin Press, London, 1999</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|A.P. Juschkewitsch]], "Geschichte der Mathematik im Mittelalter", Teubner, Leipzig, 1964</ref> Many Greek and Arabic texts on mathematics were then [[Latin translations of the 12th century|translated into Latin]], which led to further development of mathematics in [[Middle Ages|medieval Europe]].
 
From ancient times through the [[postclassical age|Middle Ages]], bursts of mathematical creativity were often followed by centuries of stagnation.  Beginning in [[Renaissance]] [[Italy]] in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an [[exponential growth|increasing pace]] that continues through the present day.
 
== Prehistoric mathematics ==
The origin of mathematical thought lie in the concepts of [[number]], [[magnitude (mathematics)|magnitude]], and [[Modular form|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 oldest known possibly mathematical object is the [[Lebombo bone]], discovered in the Lebombo mountains of [[Swaziland]] and dated to approximately 35,000 BC.<ref>[http://mathworld.wolfram.com/LebomboBone.html Lebombo Bone - from Wolfram MathWorld<!-- Bot generated title -->]</ref> It consists of 29 distinct notches cut into a baboon's fibula.<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 | accessdate = 2006-05-06}}</ref> Also [[Prehistory|prehistoric]] [[artifact (archaeology)|artifact]]s discovered in Africa and [[France]], dated between [[35000 BC|35,000]] and [[Upper Paleolithic|20,000]] years old,<ref>[http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html An old mathematical object]</ref> suggest early attempts to [[quantification|quantify]] time.<ref>[http://etopia.sintlucas.be/3.14/Ishango_meeting/Mathematics_Africa.pdf Mathematics in (central) Africa before colonization]</ref>
 
The [[Ishango bone]], found near the headwaters of the [[Nile]] river (northeastern [[Democratic Republic of the Congo|Congo]]), may be as much as [[Upper Paleolithic|20,000]] years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of [[sequence]]s of [[prime number]]s<ref name="Diaspora"/> or a six-month lunar calendar.<ref name=Marshack>Marshack, Alexander (1991): ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY.</ref> In the book ''How Mathematics Happened: The First 50,000 Years'', 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=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-Hil</ref>
 
[[Predynastic Egypt]]ians of the 5th millennium BC pictorially represented [[Geometry|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, and Archie Thom, 1988, "The metrology and geometry of Megalithic Man", pp 132-151 in C.L.N. Ruggles, 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 usage is in Babylonian and dynastic Egyptian sources. Thus it took human beings at least 45,000 years from the attainment of [[behavioral modernity]] and language (generally thought to be a long time before that) to develop mathematics as such.
 
== Babylonian mathematics ==
{{Main|Babylonian mathematics}}
{{See also|Plimpton 322}}
[[Image:Plimpton 322.jpg|thumb|250px|right|The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.]]
[[Babylonia]]n mathematics refers to any mathematics of the people 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> 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]].
 
In contrast to the sparsity of sources in [[Egyptian mathematics]], our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<ref>{{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.
 
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. From around 2500 BC onwards, the Sumerians wrote [[multiplication table]]s on clay tablets and dealt with [[geometry|geometrical]] exercises and [[Division (mathematics)|division]] problems. The earliest traces of the Babylonian numerals also date back to this period.<ref>Duncan J. Melville (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology], ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>
 
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of [[Regular number|regular]] [[Multiplicative inverse|reciprocal]] [[Twin prime|pairs]].<ref>{{cite book | authorlink = 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]] and [[quadratic equation]]s. The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.
 
Babylonian mathematics were written using a [[sexagesimal]] (base-60) [[numeral system]]. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the [[decimal]] system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context. On the other hand, this "defect" is equivalent to the modern-day usage of floating point arithmetic; moreover, the use of base 60 means that any reciprocal of an integer which is a multiple of divisors of 60 necessarily has a finite expansion to the base 60. (In decimal arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal expansions.) Accordingly, there is a strong argument that arithmetic Old Babylonian style is considerably more sophisticated than that of current usage.
 
The interpretation of Plimpton 322 was the source of controversy for many years after its significance in the context of Pythagorean triangles was realized. In historical context, inheritance problems involving equal-area subdivision of triangular and trapezoidal fields (with integer length sides) quickly convert into the need to calculate the square root of 2, or to solve the "Pythagorean equation" in integers.
 
Rather than considering a square as the sum of two squares, we can equivalently consider a square as a difference of two squares. Let a, b and c be integers that form a Pythagorean Triple: a^2 + b^2 = c^2. Then c^2 - a^2 = b^2, and using the expansion for the difference of two squares we get (c-a)(c+a)= b^2.
Dividing by b^2, it becomes the product of two rational numbers giving 1: (c/b - a/b)(c/b + a/b) = 1. We require two rational numbers which are reciprocals and which differ by 2(a/b). This is easily solved by consulting a table of reciprocal pairs. E.g., (1/2) (2) = 1 is a pair of reciprocals which differ by 3/2 = 2(a/b) Thus a/b = 3/4, giving a=3, b=4 and so c=5.
 
Solutions of the original equation are thus constructed by choosing a rational number x, from which Pythagorean-triples are 2x, x^2-1, x^2+1. Other triples are made by scaling these by an integer (the scaling integer being half the difference between the largest and one other side). All Pythagorean triples arise in this way, and the examples provided in Plimpton 322 involve some quite large numbers, by modern standards, such as (4601, 4800, 6649) in decimal notation.
 
== Egyptian mathematics ==
{{Main|Egyptian mathematics}}
[[File:Moskou-papyrus.jpg|thumb|right|300px|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.
 
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>[http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html Egyptian Papyri<!-- Bot generated title -->]</ref> as well as [[arithmetic series|arithmetic]] and [[geometric series]].<ref>[http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_algebra.html#areithmetic%20series Egyptian Algebra - Mathematicians of the African Diaspora<!-- Bot generated title -->]</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]]: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right."
 
Finally, the [[Berlin Papyrus 6619]] (c. 1800 BC) shows that ancient Egyptians could solve a second-order [[algebraic equation]].<ref>[http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin Egyptian Mathematical Papyri - Mathematicians of the African Diaspora<!-- Bot generated title -->]</ref>
 
== Greek mathematics ==
{{Main|Greek mathematics}}
[[File:Pythagorean.svg|thumb|left|200px|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>Howard Eves, ''An Introduction to the History of Mathematics'', Saunders, 1990, 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 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 [[mathematical proof|prove]] them.<ref>Martin Bernal, "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) 2000, 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.
 
[[File:Oxyrhynchus papyrus with Euclid's Elements.jpg|right|thumb|220px|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
|authorlink=
|coauthors=
|date=
|publisher=University of British Columbia
|accessdate=2008-09-26
}}</ref>]]
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, An Introduction to the History of Mathematics, Saunders, 1990, 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=harv}}</ref><ref>{{cite journal|title=The Pentagram and the Discovery of an Irrational Number|journal=The Two-Year College Mathematics Journal|author=James R. Choike|year=1980|ref=harv}}</ref>
 
[[File:Archimedes pi.svg|thumb|right|Archimedes used the [[method of exhaustion]] to approximate the value of [[pi]].]]
[[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]], 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, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 87}}</ref> The [[mathematical analysis|analytic method]] is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.<ref name="Boyer 1991 loc=The Age of Plato and Aristotle p. 88"/>
 
[[Eudoxus of Cnidus|Eudoxus]] (408–c.355 BC) 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—c.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>
 
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]] (c. 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 until the middle of the 20th century and its contents are still taught in geometry classes today.<ref>Howard Eves, ''An Introduction to the History of Mathematics'', Saunders, 1990, 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>
 
The first woman mathematician recorded by history was [[Hypatia]] of Alexandria (AD 350 - 415). She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles).<ref>[http://www.fordham.edu/halsall/source/hypatia.html Ecclesiastical History,Bk VI: Chap. 15]</ref>
 
[[File:Conic sections 2.png|thumb|left|250px|[[Apollonius of Perga]] made significant advances in the study of [[conic sections]].]]
[[Archimedes]] (c.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 [[pi|π]] with as much precision as desired, and obtained the most accurate value of π then known, 3{{frac|10|71}}&nbsp;<&nbsp;π&nbsp;<&nbsp;3{{frac|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 system 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>
 
[[Apollonius of Perga|Apollonius]] of [[Perga]] (c. 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]] (c. 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|Hipparchus]] of [[Nicaea]] (c. 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]] (c. 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 [http://books.google.com/books?id=J-2BRbFa5IkC&pg=PA1&dq=Heron+imaginary+numbers&hl=en&ei=UzjXToXwBMqhiALc9I2CCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDAQ6AEwAA#v=onepage&q=Heron%20imaginary%20numbers&f=false]. Harwood Publishing, 2007, 131 pages. ISBN 1-904275-25-7</ref> [[Menelaus of Alexandria]] (c. 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]] (c. 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>
 
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 />
 
== Chinese mathematics ==
{{Main|Chinese mathematics}}
[[File:Chounumerals.jpg|thumb|right|280px|Counting rod numerals]]
[[File:九章算術.gif|thumb|200px|right|''[[The Nine Chapters on the Mathematical Art]]'', one of the earliest surviving mathematical texts from [[China]] (2nd century AD).]]
 
Early Chinese mathematics is so different from that of other parts of the world that it is reasonable to assume independent development.<ref>{{Harv|Boyer|1991|loc="China and India" p. 201}}</ref> The oldest extant mathematical text from China is the ''[[Zhou Bi Suan Jing|Chou Pei Suan Ching]]'', variously dated to between 1200 BC and 100 BC, though a date of about 300 BC appears reasonable.<ref name="Boyer 1991 loc=China and India p. 196">{{Harv|Boyer|1991|loc="China and India" p. 196}}</ref>
 
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=194–199}}</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> 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's ''Supplementary Notes on the Art of Figures''.
 
The oldest existent work on [[geometry]] in China comes from the philosophical [[Mohism|Mohist]] canon c. 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>{{Cite journal|last=Needham |first=Joseph|authorlink=Joseph Needham|year=1986|title=[[Science and Civilisation in China]]|volume=3, ''Mathematics and the Sciences of the Heavens and the Earth''|ref=harv|location=Taipei|publisher=Caves Books Ltd.|postscript=<!--None-->}}</ref>
 
In 212 BC, the Emperor [[Qin Shi Huang]] (Shi Huang-ti) 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 and values of [[π]].<ref name="Boyer 1991 loc=China and India p. 196"/> It created mathematical proof for the [[Pythagorean theorem]], and a mathematical formula for [[Gaussian elimination]].{{Citation needed|date=April 2010}} [[Liu Hui]] commented on the work in the 3rd century AD, and gave a value of π accurate to 5 decimal places.<ref name="Boyer 1991 loc=China and India p. 202">{{Harv|Boyer|1991|loc="China and India" p. 202}}</ref> Though more of a matter of computational stamina than theoretical insight, in the 5th century AD [[Zu Chongzhi]] computed the value of π to seven decimal places, which remained the most accurate value of π for almost the next 1000 years.<ref name="Boyer 1991 loc=China and India p. 202"/> 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=0-7637-5995-3
|page=xxvii
|url=http://books.google.com/books?id=R3Hk4Uhb1Z0C}}, [http://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27]
</ref>
 
The high-water mark of Chinese mathematics occurs in the 13th century (latter part of the [[Sung dynasty|Sung period]]), with the development of Chinese algebra. The most important text from that period is the ''Precious Mirror of the Four Elements'' by Chu Shih-chieh (fl. 1280-1303), 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"/>
 
== Indian mathematics ==
{{Main|Indian mathematics}}
{{See also|History of the Hindu-Arabic numeral system}}
[[File:Bakhshali numerals 2.jpg|thumb|right|350px|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BCE and the 2nd century CE.]]
[[Image:Indian numerals 100AD.svg|frame|right|[[Brahmi numeral]]s (lower row) in [[India]] in the 1st century CE]]
The earliest civilization on the Indian subcontinent is the [[Indus Valley Civilization]] that flourished between 2600 and 1900 BC 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>T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, in {{Cite book|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=1-4020-0260-2|ref=harv|postscript=<!--None-->}}</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>R. P. Kulkarni, "[http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_32.pdf The Value of π known to Śulbasūtras]", ''Indian Journal for the History of Science'', 13 '''1''' (1978): 32-41</ref><ref>J.J. Connor, E.F. Robertson. ''The Indian Sulba Sutras'' Univ. of St. Andrew, Scotland [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html] The 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).</ref> 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>J.J. Connor, E.F. Robertson. ''The Indian Sulba Sutras'' Univ. of St. Andrew, Scotland [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html]</ref> 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"/>
 
{{Unicode|[[Pāṇini]]}} (c. 5th century BC) formulated the rules for [[Sanskrit grammar]].<ref>{{Cite journal
| last=Bronkhorst
| first=Johannes
| authorlink=
| title=Panini and Euclid: Reflections on Indian Geometry
| journal=Journal of Indian Philosophy,
| publisher=Springer Netherlands
|volume=29
|issue=1–2
| year=2001
| pages=43–80
| doi=10.1023/A:1017506118885
| ref=harv
| postscript=<!--None-->
}}</ref> His notation was similar to modern mathematical notation, and used metarules, [[Transformation (geometry)|transformation]]s, and [[recursion]].{{Citation needed|date=July 2011}} [[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
|month=
|publisher=CRC Press
|location=Boca Raton, Florida
|isbn=0-8493-7189-9
|page=37
|postscript=<!--None-->
}}</ref><ref>W. S. Anglin and J. Lambek, ''The Heritage of Thales'', Springer, 1995, 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>Rachel W. Hall. [http://www.sju.edu/~rhall/mathforpoets.pdf Math for poets and drummers]. ''Math Horizons'' '''15''' (2008) 10-11.</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 />
 
In the 5th century 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> Though about half of the entries are wrong, 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|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|page=226|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. In the 10th century, [[Halayudha]]'s commentary on [[Pingala]]'s work contains a study of the [[Fibonacci sequence]] and [[Pascal's triangle]], 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> lived in southern India and wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, [[Mean value theorem|the mean value theorem]] and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics.<ref>Plofker 2009 pp 197 - 198; 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 - 130 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>
[[Image:Yuktibhasa.gif|100px|right|thumb|Explanation of the [[Law of sines|sine rule]] in ''[[Yuktibhāṣā]]'']]
In the 14th century, [[Madhava of Sangamagrama]], the founder of the so-called [[Kerala school of astronomy and mathematics|Kerala School of Mathematics]], found the [[Leibniz formula for pi|Madhava–Leibniz series]], and, using 21 terms, computed 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 - 253</ref> In the 16th century, [[Jyesthadeva]] consolidated many of the Kerala School's developments and theorems in the ''Yukti-bhāṣā''.<ref>P. P. Divakaran, ''The first textbook of calculus: Yukti-bhāṣā'', ''Journal of Indian Philosophy'' 35, 2007, pp 417 - 433.</ref> However, the Kerala School did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], nor is there any direct evidence of their results being transmitted outside Kerala.<ref>{{Harv|Bressoud|2002|p=12}} Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."</ref><ref>{{Harvnb|Plofker|2001|p=293}} 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—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"</ref><ref>{{Harvnb|Pingree|1992|p=562}} 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 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."</ref><ref>{{Harvnb|Katz|1995|pp=173–174}} Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."</ref> Progress in mathematics along with other fields of science stagnated in India with the establishment of [[Muslim conquest in the Indian subcontinent|Muslim rule in India]].<ref>{{Cite book | last1 = Dutta| first1 = Sristidhar| last2 = Tripathy| first2 = Byomakesh| title = Martial traditions of North East India| publisher = Concept Publishing Company| year = 2006| page = 173| url = http://books.google.com/books?id=s_ttiCMvGH4C&pg=PA173| isbn = 978-81-8069-335-9}}</ref><ref>{{Cite book | last1 = Wickramasinghe| first1 = Nalin Chandra| last2 = Ikeda| first2 = Daisaku|title = Space and eternal life| publisher = Journeyman Press| year = 1998| page = 79| url = http://books.google.com/books? isbn = 978-1-85172-061-3}}</ref>
 
== Islamic mathematics ==
{{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 [[Persia]], 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]], most of them were not 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. [[Persian people|Persians]] contributed to the world of Mathematics alongside Arabs.
 
In the 9th century, the [[Persian people|Persian]] mathematician {{Unicode|[[Muḥammad ibn Mūsā al-Khwārizmī]]}} wrote several important books on the Hindu-Arabic numerals and 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 [[problem]]s 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=0-7923-2565-6 | oclc=29181926 | pages=11–12}}</ref>
 
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>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', pp. 255–59. [[Addison-Wesley]]. ISBN 0-321-01618-1.</ref> The [[historian]] of mathematics, F. Woepcke,<ref>F. Woepcke (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 [[integral]]s of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<ref name=Katz>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163–74.</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]].{{Citation needed|date=March 2009}}
 
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|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.
 
== Medieval European mathematics ==
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'', 11:21</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 ''[[Euclid's Elements|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, ed., ''Boethius: His Life, Thought, and Influence,'' (Oxford: Basil Blackwell).</ref><ref>Folkerts, Menso, ''"Boethius" Geometrie II'', (Wiesbaden: Franz Steiner Verlag, 1970).</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|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>Guy Beaujouan, "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>
 
{{see also|Latin translations of the 12th century}}
 
These new sources sparked a renewal of mathematics. [[Fibonacci]], writing in the ''[[Liber Abaci]]'', in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of [[Eratosthenes]], a gap of more than a thousand years. The work introduced [[Hindu-Arabic numerals]] to Europe, and discussed many other mathematical problems. <!-- Needs to spell out what Fibonacci did, not just praise it. -->
 
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.<ref>Grant, Edward and John E. Murdoch (1987), eds., ''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), at pp. 224–27.</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 [[Integral|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>Nicole Oresme, "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 mathematics ==
[[Image:Pacioli.jpg|thumb|right|250px|''[[Portrait of Luca Pacioli]]'', a painting traditionally attributed to [[Jacopo de' Barbari]], 1495, ([[Museo di Capodimonte]]).]]
 
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.
 
[[Luca Pacioli]]'s ''"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>Alan Sangster, 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"] (Accounting, Business & Financial History Conference, Cardiff, September 2007) p. 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]]. It is important to note that Pacioli himself had borrowed much of the work of [[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 book ''De Thiende'' ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of [[decimal notation]], which influenced all later work on the [[real number system]].
 
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 = 0-393-32030-8}}</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 involved, were studied intensely.<ref name="Kline">
{{cite book
| last = Kline
| first = Morris
| authorlink =
| title = Mathematics in Western Culture
| publisher = Pelican
| series =
| volume =
| edition =
| year = 1953
| location=Great Britain
| pages= 150–151
| language =
| url =
| doi =
| id =
| isbn =
| mr =
| zbl =
| jfm = }}</ref>
 
==Mathematics during the Scientific Revolution==
 
=== 17th century ===
[[Image:Gottfried Wilhelm von Leibniz.jpg|150px|thumb|right|[[Gottfried Wilhelm Leibniz]].]]
The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. [[Galileo]] observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. [[Tycho Brahe]] had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. Through 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 =89
| isbn =9780486602554
}}</ref>
The [[analytic geometry]] developed by [[René Descartes]] (1596–1650) allowed those orbits to be plotted on a graph, in [[Cartesian coordinates]]. [[Simon Stevin]] (1585) created the basis for modern decimal notation capable of describing all numbers, whether rational or irrational.
 
Building on earlier work by many predecessors, [[Isaac Newton]] discovered the laws of physics explaining [[Kepler's Laws]], and brought together the concepts now known as [[infinitesimal calculus]]. Independently, [[Gottfried Wilhelm Leibniz]], who is arguably one of the most important mathematicians of the 17th century, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.<ref>Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, 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–19th century.
 
=== 18th century ===
[[File:Leonhard Euler.jpg|right|thumb|150px|[[Leonhard Euler]] by [[Emanuel Handmann]].]]
 
The most influential mathematician of the 18th century was arguably [[Leonhard Euler]]. 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 [[Laplace]] who, in the age of [[Napoleon]], did important work on the foundations of [[celestial mechanics]] and on [[statistics]].
 
== Modern 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|100px|[[Carl Friedrich Gauss]].]]
Throughout the 19th century mathematics became increasingly abstract. In the 19th century lived [[Carl Friedrich Gauss]] (1777–1855). Leaving aside his many contributions to science, in [[pure mathematics]] he did revolutionary work on [[function (mathematics)|function]]s of [[complex variable]]s, in [[geometry]], and on the convergence of [[series (mathematics)|series]]. He gave the first satisfactory proofs of the [[fundamental theorem of algebra]] and of the [[quadratic reciprocity law]].
 
[[Image:noneuclid.svg|right|thumb|400px|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]]s.
 
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]]. 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 [[computer science]].
 
[[Augustin-Louis Cauchy]], [[Bernhard Riemann]], and [[Karl Weierstrass]] reformulated the calculus in a more rigorous fashion.
 
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]]). Other 19th-century mathematicians utilized this in their proofs that straightedge 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, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. 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.
 
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]].
 
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]].
 
The 19th century saw the founding of a number of national mathematical societies: the [[London Mathematical Society]] in 1865, the [[Société Mathématique de France]] in 1872, the [[Circolo Matematico di Palermo]] in 1884, the [[Edinburgh Mathematical Society]] in 1883, and the [[American Mathematical Society]] in 1888. The first international, special-interest society, the [[Quaternion Society]], was formed in 1899, in the context of a [[hyperbolic quaternion#Historical review|vector controversy]].
 
In 1897, Hensel introduced [[p-adic number]]s.
 
=== 20th century === <!-- Hibert's problems, foundational crisis, Bourbaki -->
[[Image:Four Colour Map Example.svg|thumb|A map illustrating the [[Four Color Theorem]]]]
 
The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were available in both teaching and industry. An effort to catalogue the areas and applications of mathematics was undertaken in [[Klein's encyclopedia]].
 
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]]. 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.
 
Notable historical conjectures were finally proven. In 1976, [[Wolfgang Haken]] and [[Kenneth Appel]] used a computer to prove the [[four color theorem]]. [[Andrew Wiles]], building on the work of others, proved [[Fermat's Last Theorem]] in 1995. [[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]]. In 1998 [[Thomas Callister Hales]] proved the [[Kepler conjecture]].
 
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 1983 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. 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, ISBN 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 [[Einstein]] used it in [[general relativity]]. Entire 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. All kinds of [[Mathematical structure|structures]] were abstracted using axioms and given names like [[metric space]]s, [[topological space]]s etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to [[category theory]]. [[Grothendieck]] and [[Jean-Pierre Serre|Serre]] recast [[algebraic geometry]] using [[Sheaf (mathematics)|sheaf theory]]. Large advances were made in the qualitative study of [[dynamical systems theory|dynamical systems]] that [[Henri Poincaré|Poincaré]] had begun in the 1890s.
[[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]]. [[Knot theory]] greatly expanded. [[Quantum mechanics]] led to the development of [[functional analysis]]. 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 [[Benoît Mandelbrot|Mandelbrot]]'s [[fractals]]. [[Lie theory]] with its [[Lie group]]s and [[Lie algebra]]s became one of the major areas of study.
 
[[Non-standard analysis]], introduced by [[Abraham Robinson]], rehabillitated 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. An even larger number system, the [[surreal number]]s were discovered by [[John Horton Conway]] in connection with [[combinatorial game]]s.
 
The development and continual improvement of [[computer]]s, at first mechanical analog machines and then digital electronic machines, allowed [[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 test]]; [[Claude Shannon]]'s [[information theory]]; [[signal processing]]; [[data analysis]]; [[Mathematical optimization|optimization]] and other areas of [[operations research]]. 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]]. 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]].
 
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the [[natural number]]s plus one of addition and multiplication, was [[Decidability (logic)|decidable]], i.e. could be determined by some algorithm. 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|incompletable]]. (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.
 
[[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]] who conjectured or proved over 3000 theorems, including properties of [[highly composite number]]s, the [[partition function (number theory)|partition function]] and its [[asymptotics]], and [[Ramanujan theta function|mock theta functions]]. He also made major investigations in the areas of [[gamma function]]s, [[modular form]]s, [[divergent series]], [[General hypergeometric function|hypergeometric series]] and [[prime number]] theory.
 
[[Paul Erdős]] published more papers than any other mathematician in history, 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 Paul Erdős, as measured by joint authorship of mathematical papers.
 
[[Emmy Noether]] has been described by many as the most important woman in the history of mathematics,<ref>{{citation|authorlink=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 = 0-8247-1550-0 |pages= 99–111}}.</ref> she revolutionized the theories of [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]].
 
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>[http://www.ams.org/mathscinet/msc/pdfs/classifications2000.pdf Mathematics Subject Classification 2000]</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.
 
=== 21st century ===
In 2000, the [[Clay Mathematics Institute]] announced the seven [[Millennium Prize Problems]], and in 2003 the [[Poincaré conjecture]] was solved by [[Grigori Perelman]] (who declined to accept an award on this point).
 
Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive towards [[Open access (publishing)|open access publishing]], first popularized by the [[arXiv]].
 
== Future of mathematics ==
{{Main|Future of mathematics}}
 
There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, the volume of data to be analyzed being produced by science and industry, facilitated by computers, is explosively expanding.{{citation needed|date=April 2013}}
 
== See also ==
{{Portal|Mathematics}}
*[[List of important publications in mathematics]]
*[[History of algebra]]
*[[History of calculus]]
*[[History of combinatorics]]
*[[History of geometry]]
*[[History of logic]]
*[[History of mathematical notation]]
*[[History of number theory]]
*[[History of statistics]]
*[[History of trigonometry]]
*[[History of writing numbers]]
*[[Kenneth O. May Prize]]
*[[Timeline of mathematics]]
*[[prime numbers]]
*[[irrational numbers]]
*[[Mathematics]]
*[[Mathematics education]]
 
== References ==
{{reflist|2}}
 
== External articles ==
* [[Howard Eves|Eves, Howard]], ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0-03-029558-0,
*{{cite book|first=Ivor|last=Grattan-Guinness|authorlink=Ivor Grattan-Guinness|title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences|publisher=The Johns Hopkins University Press|year=2003|isbn=0-8018-7397-5}}
*{{cite book
| last = Bell
| first = E. T.
| authorlink = Eric Temple Bell
| title = Men of Mathematics
| publisher = Simon and Schuster
| year = 1937
}}
* [[David M. Burton|Burton, David M.]] ''The History of Mathematics: An Introduction''. McGraw Hill: 1997.
* [[Victor J. Katz|Katz, Victor J.]] ''A History of Mathematics: An Introduction'', 2nd Edition. [[Addison-Wesley]]: 1998.
* Scimone, Aldo (2006). Talete, chi era costui? Vita e opere dei matematici incontrati a scuola. Palermo: Palumbo Pp.&nbsp;228.
 
; Books on a specific period
*{{cite book
| last = Gillings
| first = Richard J.
| authorlink = Richard J. Gillings
| title = Mathematics in the Time of the Pharaohs
| publisher = MIT Press
| location = Cambridge, MA
| year = 1972
}}
*{{cite book
| last = Heath
| first = Sir Thomas
| authorlink = Thomas Little Heath
| title = A History of Greek Mathematics
| publisher = Dover
| year = 1981
| isbn = 0-486-24073-8
}}
*{{Cite book
| year=2007
| editor1-last=Katz
| editor1-first=Victor J.
| title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
|volume=
| place=
| publisher=Princeton, NJ: Princeton University Press, 685 pages, pp 385-514
| isbn=0-691-11485-4
| ref=harv
| postscript=<!--None-->
}}.
* 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. 
*{{Cite book
| last1=Plofker
| first1=Kim
| authorlink1 = Kim Plofker
| year=2009
| title=Mathematics in India: 500 BCE–1800 CE
| place=
| publisher=Princeton, NJ: Princeton University Press. Pp. 384.
| isbn= 0-691-12067-6
| ref=harv
| postscript=<!--None-->
}}.
* [[Bartel Leendert van der Waerden|van der Waerden, B. L.]], ''Geometry and Algebra in Ancient Civilizations'', Springer, 1983, ISBN 0-387-12159-5.
; Books on a specific topic
* [[Paul Hoffman (science writer)|Hoffman, Paul]], ''The Man Who Loved Only Numbers: The Story of [[Paul Erdős]] and the Search for Mathematical Truth''. New York: Hyperion, 1998 ISBN 0-7868-6362-5.
*{{cite book| last = Stigler| first = Stephen M.| authorlink = Stephen Stigler| year = 1990| title = The History of Statistics: The Measurement of Uncertainty before 1900| publisher = Belknap Press | isbn = 0-674-40341-X}}
*{{cite book
| last = Menninger
| first = Karl W.
| authorlink = Karl W. Menninger
| year = 1969
| title = Number Words and Number Symbols: A Cultural History of Numbers
| publisher = MIT Press
| isbn = 0-262-13040-8
}}
 
; Documentaries
* [[BBC]] (2008). ''[[The Story of Maths]]''.
*[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; 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://www.math.sfu.ca/history_of_mathematics History of Mathematics] (Simon Fraser University).
*[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://www.dean.usma.edu/math/people/rickey/hm/ Fred Rickey's History of Mathematics Page]
*[http://mathematics.library.cornell.edu/additional/Collected-Works-of-Mathematicians A Bibliography of Collected Works and Correspondence of Mathematicians] [http://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
*[http://mathdl.maa.org/mathDL/46/ Convergence], the [[Mathematical Association of America]]'s online Math History Magazine
 
;Directories
*[http://www.dcs.warwick.ac.uk/bshm/resources.html Links to Web Sites on the History of Mathematics] (The British Society for the History of Mathematics)
*[http://archives.math.utk.edu/topics/history.html History of Mathematics] Math Archives (University of Tennessee, Knoxville)
*[http://mathforum.org/library/topics/history/ History/Biography] The Math Forum (Drexel University)
*[http://web.archive.org/web/20020716102307/http://www.otterbein.edu/resources/library/libpages/subject/mathhis.htm History of Mathematics]{{dead link|3/2012|date=March 2012}} (Courtright Memorial Library).
*[http://homepages.bw.edu/~dcalvis/history.html History of Mathematics Web Sites] (David Calvis; Baldwin-Wallace College)
*{{dmoz|Science/Math/History|History of mathematics}}
*[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)
*[http://math.illinoisstate.edu/marshall/ Using History in Math Class]
*[http://mathres.kevius.com/history.html Mathematical Resources: History of Mathematics] (Bruno Kevius)
*[http://www.dm.unipi.it/~tucci/index.html History of Mathematics] (Roberta Tucci)
 
[[Category:History of mathematics| ]]
[[Category:World Digital Library related]]
 
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[[as:গণিত#গণিতৰ ইতিহাস]]

Latest revision as of 03:05, 13 January 2015

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