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Incluso después de que las matemáticas europeas comenzasen a florecer durante el [[Renacimiento]], las matemáticas chinas y europeas mantuvieron tradiciones separadas, con un significativo declive de las chinas, hasta que misioneros [[jesuita]]s como [[Matteo Ricci]] intercambiaron las ideas matemáticas entre las dos culturas entre los siglos XVI y XVIII.
== Matemáticas en la Índia clásica (hacia 400–1600)==
{{AP|Matemáticas indias}}
{{Véase también|Historia del sistema de numeración hindo-arábico}}
[[Archivo:2064 aryabhata-crp.jpg|thumb|[[Aryabhata]].]]
El ''[[Surya Siddhanta]]'' (hacia el año 400) introdujo las [[funciones trigonométricas]] de [[seno]], [[coseno]] y arcoseno y estableció reglas para determinar las trayectorias de los astros que son conformes con sus posiciones actuales en el cielo. Los ciclos cosmológicos explicados en el texto, que eran una copia de trabajos anteriores, correspondían a un [[año sideral]] medio de 365.2563627 días, lo que sólo es 1,4 segundos máyor que el valor aceptado actualmente de 365.25636305 días. Este trabajo fue traducido del árabe al latín durante la Edad Media.
[[Aryabhata]], en 499, introdujo la función [[verseno]], produjo las primeras tablas [[Trigonometría|trigonométricas]] del seno, desarrolló técnicas y [[algoritmo]]s de [[álgebra]], [[infinitesimal]]es, [[Ecuación diferencial|ecuaciones diferenciales]] y obtuvo la solución completa de ecuaciones lineales por un método equivalente al actual, además de cálculos [[Astronomía|astronómicos]] basados en un sistema [[Heliocentrismo|heliocéntrico]] de [[Gravedad|gravitación]]. Desde el siglo VIII estuvo disponible una traducción al árabe de su ''Aryabhatiya'', seguida de una traducción al latín en el siglo XIII. También calculó el valor de π con once decimales (3,14159265359).
<!-- ARTÍCULO ORIGINAL EN INGLÉS
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]]. 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]].
In the 12th century, [[Bhaskara]] first conceived [[differential calculus]], along with the concepts of the [[derivative]], [[differential]] coefficient and [[differentiation]]. He also stated [[Rolle's theorem]] (a special case of the [[mean value theorem]]), studied [[Pell's equation]], and investigated the derivative of the sine function. From the 14th century, Madhava and other [[Kerala School]] mathematicians further developed his ideas. They developed the concepts of [[mathematical analysis]] and [[floating point]] numbers, and concepts fundamental to the overall development of [[calculus]], including the mean value theorem, term by term [[integral|integration]], the relationship of an area under a curve and its antiderivative or integral, the [[integral test for convergence]], [[iterative method]]s for solutions to [[non-linear]] equations, and a number of [[infinite series]], [[power series]], [[Taylor series]], and trigonometric series. In the 16th century, [[Jyeshtadeva]] consolidated many of the Kerala School's developments and theorems in the ''Yuktibhasa'', the world's first differential calculus text, which also introduced concepts of [[integral calculus]].
Mathematical progress in India stagnated from the late 16th century onwards due to political turmoil.
==Islamic mathematics (c. 800–1500)==
{{main|Mathematics in medieval Islam}}
{{see also|History of the Hindu-Arabic numeral system}}
[[Image:Abu Abdullah Muhammad bin Musa al-Khwarizmi.jpg|thumb|[[Muhammad ibn Mūsā al-Khwārizmī|Muḥammad ibn Mūsā al-Ḵwārizmī]] ]]
The [[Islam]]ic [[Arab Empire]] established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Iberian Peninsula|Iberia]], and in parts of [[History of India|India]] in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in [[Arabic language|Arabic]], they were not all written by [[Arab]]s, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Alongside Arabs, many important Islamic mathematicians were also [[Persian people|Persians]].
In the 9th century, {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī|Muḥammad ibn Mūsā al-Ḵwārizmī]]}} 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''). Al-Khwarizmi is often called the "father of algebra", for his fundamental contributions to the field.<ref>[http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra]. [[Louisiana State University]].</ref> 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-Khwarizmi'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 al-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 introduced 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-Khwarizmi 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>{{Citation | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0792325656 | 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. The first known [[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 and developed the [[tangent (trigonometry)|tangent]] function. [[Ibn al-Haytham]] was the first mathematician to derive the formula for the sum of the [[Quartic|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 flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]], and laid the foundations for [[analytic geometry]] and [[non-Euclidean geometry]].{{Fact|date=March 2009}} He was also the first to find the general geometric solution to [[cubic equation]]s. He was also very influential in [[calendar reform]].{{Fact|date=March 2009}}
In the late 12th century, [[Sharaf al-Dīn al-Tūsī]] introduced the concept of a [[Function (mathematics)|function]],<ref>{{Citation|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=[[Springer Science+Business Media|Springer Netherlands]]|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185–201 [192]}}</ref> and he was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]].<ref>J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", ''Journal of the American Oriental Society'' '''110''' (2), pp. 304–09.</ref> His ''Treatise on Equations'' developed concepts related to differential calculus, such as the derivative function and the [[maxima and minima]] of curves, in order to solve cubic equations which may not have positive solutions.<ref name=Sharaf>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref>
In the 13th century, [[Nasir al-Din Tusi]] (Nasireddin) made advances in [[spherical trigonometry]]. He also wrote influential work on [[Euclid]]'s [[parallel postulate]]. In the 15th century, [[Ghiyath al-Kashi]] computed the value of [[π]] to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by [[Ruffini]] and [[Horner]].
Other notable Muslim mathematicians included [[al-Samawal]], [[Abu'l-Hasan al-Uqlidisi]], [[Jamshid al-Kashi]], [[Thabit ibn Qurra]], [[Abu Kamil]] and [[Abu Sahl al-Kuhi]].
Other achievements of Muslim mathematicians during this period include the development of [[algebra]] and [[algorithm]]s (see [[Muhammad ibn Mūsā al-Khwārizmī]]), the development of [[spherical trigonometry]],<ref>{{cite book |last=Syed |first=M. H. |title=Islam and Science |year=2005 |publisher=Anmol Publications PVT. LTD. |isbn=8-1261-1345-6 |page=71}}</ref> 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]], the first refutations of [[Euclidean geometry]] and the [[parallel postulate]] by [[Nasīr al-Dīn al-Tūsī]], the first attempt at a [[non-Euclidean geometry]] by Sadr al-Din, 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> and many other advances in algebra, [[arithmetic]], calculus, [[cryptography]], [[geometry]], [[number theory]] and [[trigonometry]].
During the time of the [[Ottoman Empire]] from the 15th century, the development of Islamic mathematics became stagnant.
==Medieval European mathematics (c. 500–1400)==
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]]'' and the biblical passage that God had "ordered all things in measure, and number, and weight" (''Wisdom'' 11:21).
===Early Middle Ages (c. 500–1100)===
[[Boethius]] provided a place for mathematics in the curriculum 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>
===Rebirth of mathematics in Europe (1100–1400)===
In the 12th century, European scholars traveled to Spain and Sicily [[Latin translations of the 12th century|seeking scientific Arabic texts]], including [[al-Khwarizmi]]'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>
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.ORIGINAL--> <!--COMENTARIO EN EL ORIGINAL Needs to spell out what Fibonacci did, not just praise it. -->
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The fourteenth 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 (mathematics)|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>
==Early modern European mathematics (c. 1400–1600)==
In Europe at the dawn of the [[Renaissance]], mathematics was still limited by the cumbersome notation using [[Roman numeral]]s and expressing relationships using words, rather than symbols: there was no plus sign, no equal sign, and no use of ''x'' as an unknown.{{Fact|date=November 2007}}
In 16th century European mathematicians began to make advances without precedent anywhere in the world, so far as is known today. The first of these was the general solution of [[cubic equation]]s, generally credited to [[Scipione del Ferro]] c. 1510, but first published by [[Johannes Petreius]] in [[Nuremberg]] in [[Gerolamo Cardano]]'s ''Ars magna'', which also included the solution of the general [[quartic equation]] from Cardano's student [[Lodovico Ferrari]] .
From this point on, mathematical developments came swiftly, contributing to and benefiting from contemporary advances in the [[physical sciences]]. This progress was greatly aided by advances in [[printing]]. The earliest [[antiquarian science book|mathematical books]] printed were [[Peurbach]]'s ''[[Theoricae nova planetarum]]'' (1472}, followed by a book on commercial arithmetic, the [[Treviso Arithmetic]] (1478), and then the first extant book on mathematics, Euclid's ''Elements'', printed and published by [[Ratdolt]] in 1482.
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>
By century's end, thanks to [[Regiomontanus]] (1436–76) and [[François Vieta]] (1540–1603), among others, mathematics was written using Hindu-Arabic numerals and in a form not too different from the notation used today.
==17th century==
The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. [[Galileo]], an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. [[Tycho Brahe]], a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, [[Johannes Kepler]], a German, began to work with this data. In part because he wanted to help Kepler in his calculations, [[John Napier]], in Scotland, was the first to investigate [[natural logarithm]]s. Kepler succeeded in formulating mathematical laws of planetary motion. The [[analytic geometry]] developed by [[René Descartes]] (1596–1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in [[Cartesian coordinates]].
Building on earlier work by many prececessors, [[Isaac Newton]], an Englishman, discovered the laws of physics explaining [[Kepler's Laws]], and brought together the concepts now known as [[calculus]]. Independently, [[Gottfried Wilhelm Leibniz]], in Germany, 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==
[[Image:Leonhard Euler.jpg|left|thumb|[[Leonhard Euler]] by [[Emanuel Handmann]].]]
The most influential mathematician of the 1700s was arguably [[Leonhard Euler]]. His contributions range from founding the study of [[graph theory]] with the [[Seven Bridges of K%C3%B6nigsberg]] problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol <font face="times new Roman">[[Imaginary unit|''i'']]</font>, 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]].
==19th century==
[[Image:noneuclid.svg|thumb|400px|Behavior of lines with a common perpendicular in each of the three types of geometry]]
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]].
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 [[Janos 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 generalize the ideas of [[curve]]s and [[surface]]s.
The 19th century saw the beginning of a great deal of [[abstract algebra]]. [[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 logic|Boolean algebra]], in which the only numbers were 0 and 1 and in which, famously, 1 + 1 = 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. 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.
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 Mathematico di Palermo]] in 1884, the [[Edinburgh Mathematical Society]] in 1883, and the [[American Mathematical Society]] in 1888.
==20th century==
[[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 are awarded, and jobs are available in both teaching and industry. In earlier centuries, there were few creative mathematicians in the world at any one time. For the most part, mathematicians were either born to wealth, like [[John Napier|Napier]], or supported by wealthy patrons, like [[Gauss]]. A few, like [[Joseph Fourier|Fourier]], derived meager livelihoods from teaching in universities. [[Niels Henrik Abel]], unable to obtain a position, died in poverty of malnutrition and tuberculosis at the age of twenty-six.
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.
Famous historical conjectures were finally proved. 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]].
Mathematical collaborations of unprecedented size and scope took place. A famous 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, ISBN13 978-0821839676.</ref>
Entire new areas of mathematics such as [[mathematical logic]], [[topology]], [[Computational complexity theory|complexity theory]], and [[game theory]] changed the kinds of questions that could be answered by mathematical methods.
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 [[decidable]], i.e., could be determined by 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; 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 died.
One of the more colorful figures in 20th century mathematics was [[Srinivasa Aiyangar Ramanujan]] (1887–1920) who, despite being largely self-educated, 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]], [[hypergeometric series]] and [[prime number theory]].
==See also==
*[[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 statistics]]
*[[History of trigonometry]]
*[[History of writing numbers]]
== References ==
{{reflist|2}}
== Further reading ==
<div class="references-2column">
*{{cite book
| last = Aaboe
| first = Asger
| year = 1964
| title = Episodes from the Early History of Mathematics
| publisher = Random House
| location = New York
}}
* Boyer, C. B., ''A History of Mathematics'', 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
* Eves, Howard, ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0-03-029558-0,
* [[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|first=Ivor|last=Grattan-Guinness|title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences|publisher=The Johns Hopkins University Press|year=2003|isbn=0801873975}}
* van der Waerden, B. L., ''Geometry and Algebra in Ancient Civilizations'', Springer, 1983, ISBN 0-387-12159-5.
* O'Connor, John J. and Robertson, Edmund F. ''[http://www-groups.dcs.st-andrews.ac.uk/~history/ The MacTutor History of Mathematics Archive]''. (See also [[MacTutor History of Mathematics archive]].) This website contains biographies, timelines and historical articles about mathematical concepts; at the School of Mathematics and Statistics, [[University of St. Andrews]], Scotland. (Or see the [http://www-gap.dcs.st-and.ac.uk/~history/Indexes/Hist_Topics_alph.html alphabetical list of history topics].)
*{{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 = Bell
| first = E.T.
| title = Men of Mathematics
| publisher = Simon and Schuster
| year = 1937
}}
*{{cite book
| last = Gillings
| first = Richard J.
| title = Mathematics in the time of the pharaohs
| publisher = M.I.T. Press
| location = Cambridge, MA
| year = 1972
}}
*{{cite book
| last = Heath
| first = Sir Thomas
| title = A History of Greek Mathematics
| publisher = Dover
| year = 1981
| isbn = 0-486-24073-8
}}
*{{cite book
| last = Menninger
| first = Karl W.
| year = 1969
| title = Number Words and Number Symbols: A Cultural History of Numbers
| publisher = MIT Press
| isbn = 0-262-13040-8
}}
* Burton, David M. ''The History of Mathematics: An Introduction''. McGraw Hill: 1997.
* Katz, Victor J. ''A History of Mathematics: An Introduction'', 2nd Edition. [[Addison-Wesley]]: 1998.
* Kline, Morris. ''Mathematical Thought from Ancient to Modern Times''.
</div>
==External links==
*[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 famous 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/histmath/ 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.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://astech.library.cornell.edu/ast/math/find/Collected-Works-of-Mathematicians.cfm A Bibliography of Collected Works and Correspondence of Mathematicians] (Steven W. Rockey; Cornell University Library).
*[http://www.mathourism.com Mathourism - Places with a mathematical historic interest]
;Journals
*[http://mathdl.maa.org/convergence/1/ 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://www.otterbein.edu/resources/library/libpages/subject/mathhis.htm History of Mathematics] (Courtright Memorial Library).
*[http://homepages.bw.edu/~dcalvis/history.html History of Mathematics Web Sites] (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://www.math.ilstu.edu/marshall/ Using History in Math Class]
*[http://www.abc.se/~m9847/matre/history.html Mathematical Resources: History of Mathematics] (Bruno Kevius)
*[http://www.dm.unipi.it/~tucci/index.html History of Mathematics] (Roberta Tucci)
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== Estructura, espacio y cambio ==
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