History of mathematics 14220 225434235 2008-07-13T17:56:54Z Pbroks13 2308181 /* 19th century */ SVG image [[Image:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|''[[The Compendious Book on Calculation by Completion and Balancing]]'']] The area of study known as the '''history of mathematics''' is primarily an investigation into the origin of new discoveries in [[mathematics]], to a lesser extent, an investigation into the standard 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]] ''ca.'' 1900 BC), the ''[[Moscow Mathematical Papyrus]]'' ([[Egyptian mathematics]] ''ca.'' 1850 BC), the ''[[Rhind Mathematical Papyrus]]'' (Egyptian mathematics ''ca.'' 1650 BC), and the ''[[Shulba Sutras]]'' ([[Indian mathematics]] ''ca.'' 800 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. Egyptian and Babylonian mathematics were then further developed in [[Greek mathematics|Greek and Hellenistic mathematics]], which is generally considered to be one of the most important for greatly expanding both the method and 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> The mathematics developed in these ancient civilizations were then further developed and greatly expanded in [[Islamic mathematics]]. Many Greek and Arabic texts on mathematics were then [[Latin translations of the 12th century|translated into Latin]] in medieval Europe and further developed there. One striking feature of the history of ancient and medieval mathematics is that bursts of mathematical development 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|ever increasing pace]], and this continues to the present day. ==Early mathematics== [[Image:Ishango bone.jpg|thumb|right|400px|The [[Ishango bone]], dating to perhaps 18000 to 20000 [[Anno Domini|B.C.]]]] Long before the earliest written records, there are drawings that do indicate a knowledge of mathematics and of measurement of time based on the stars. For example, [[Paleontology|paleontologists]] have discovered [[ochre]] rocks in a cave in [[South Africa]] adorned with scratched [[Geometry|geometric]] patterns dating back to c. 70,000 BC.<ref>{{cite web | last = Henahan | first = Sean | year = 2002 | url = http://www.accessexcellence.org/WN/SU/caveart.html | title = Art Prehistory | work = Science Updates | publisher = The National Health Museum | accessdate = 2006-05-06}}</ref> Also [[Prehistory|prehistoric]] [[artifact (archaeology)|artifact]]s discovered in Africa and [[France]], dated between [[35000 BC|35,000 BC]] and [[Upper Paleolithic|20,000 BC]],<ref>[http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html An old mathematical object]</ref> indicate 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> Evidence exists that early counting involved women who kept records of their monthly biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a distinctive marker, for example. Moreover, hunters had the concepts of ''one'', ''two'', and ''many'', as well as the idea of ''none'' or ''zero'', when considering herds of animals.<ref>{{cite web | last = Kellermeier | first = John | year = 2003 | url = http://www.tacomacc.edu/home/jkellerm/Papers/Menses/Menses.htm | title = How Menstruation Created Mathematics | work = Ethnomathematics | publisher = Tacoma Community College | accessdate = 2006-05-06}}</ref><ref>{{cite web | last = Williams | first = Scott W. | year = 2005 | url = http://www.math.buffalo.edu/mad/Ancient-Africa/lebombo.html | title = The Oledet Mathematical Object is in Swaziland | work = MATHEMATICIANS OF THE AFRICAN DIASPORA | publisher = SUNY Buffalo mathematics department | accessdate = 2006-05-06}}</ref> The [[Ishango Bone]], found in the area of the headwaters of the [[Nile River]] (northeastern [[Democratic Republic of the Congo|Congo]]), dates as early as [[Upper Paleolithic|20,000 BC]]. One common interpretation is that the bone is the earliest known demonstration<ref>{{cite web | last = Williams | first = Scott W. | year = 2005 | url = http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html | title = An Old Mathematical Object | work = MATHEMATICIANS OF THE AFRICAN DIASPORA | publisher = SUNY Buffalo mathematics department | accessdate = 2006-05-06}}</ref> of [[sequence]]s of [[prime number]]s and [[Ancient Egyptian multiplication]]. [[Predynastic Egypt]]ians of the 5th millennium BC pictorially represented [[Geometry|geometric]] [[spatial]] designs. It has been claimed that [[Megalith]]ic monuments in [[England]] and [[Scotland]] from the 3rd millennium BC, incorporate geometric ideas such as [[circle]]s, [[ellipse]]s, and [[Pythagorean triples]] in their design.<ref>Thom, Alexander and Archie Thom, "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: Cambridge Univ. Pr., 1988) ISBN 0-521-33381-4</ref> The earliest known mathematics in [[History of India|ancient India]] dates back to circa 3000-2600 BC in the [[Indus Valley Civilization]] ([[Harappan civilization]]) of [[North India]] and [[Pakistan]], which developed a system of [[Ancient Indus Valley units of measurement|uniform weights and measures]] that used the [[decimal]] system, a surprisingly advanced [[brick]] technology which utilised [[ratio]]s, streets laid out in perfect [[right angle]]s, and a number of geometrical shapes and designs, including [[cuboid]]s, [[barrel]]s, [[cone (geometry)|cones]], [[cylinder (geometry)|cylinders]], and drawings of concentric and intersecting [[circle]]s and [[triangle]]s. Mathematical instruments discovered include an accurate decimal ruler with small and precise subdivisions, a shell instrument that served as a [[compass]] to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees, a shell instrument used to measure 8–12 whole sections of the horizon and sky, and an instrument for measuring the positions of stars for navigational purposes. The [[Indus script]] has not yet been deciphered; hence very little is known about the written forms of [[Indian mathematics#Harappan Mathematics (3300 BC - 1500 BC)|Harappan mathematics]]. Archeological evidence has led some historians to believe that this civilization used a [[base 8]] [[numeral system]] and possessed knowledge of the ratio of the length of the [[circumference]] of the circle to its [[diameter]], thus a value of [[Pi|π]].<ref>{{cite web | last = Pearce | first = Ian G. | year = 2002 | url = http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch3.html | title = Early Indian culture - Indus civilisation | work = Indian Mathematics: Redressing the balance | publisher = School of Mathematical and Computational Sciences University of St Andrews | accessdate = 2006-05-06}}</ref> Dating from the [[Shang Dynasty]] (1600&mdash;1046 BC), the earliest extant Chinese mathematics consists of numbers scratched on tortoise shell [http://www.saxakali.com/COLOR_ASP/chinamh1.htm] [http://www.chinaculture.org/gb/en_madeinchina/2005-08/18/content_71974.htm]. These numbers use a decimal system, so that the number 123 is written (from top to bottom) as the symbol for 1 followed by the symbol for a hundred, then the symbol for 2 followed by the symbol for ten, then the symbol for 3. This was the most advanced number system in the world at the time 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 reference was in AD 190 in the ''Supplementary Notes on the Art of Figures'' written by Xu Yue. ==Ancient Near East (c. 1800-500 BC)== ===Mesopotamia=== {{Main|Babylonian mathematics}} [[Babylonia]]n mathematics refers to any mathematics of the people of [[Mesopotamia]] (modern [[Iraq]]) from the days of the early [[Sumer]]ians until the beginning of the [[Hellenistic period]]. It is named Babylonian mathematics due to the central role of [[Babylon]] as a place of study, which ceased to exist during the Hellenistic period. From this point, Babylonian mathematics merged with Greek and Egyptian mathematics to give rise to [[Greek mathematics|Hellenistic mathematics]]. Later under the [[Arab Empire]], Iraq/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. 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 [[Pythagorean triple]]s (see [[Plimpton 322]]).<ref>{{cite book | 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, [[trigonometry]] tables and methods for solving [[linear equation|linear]] and [[quadratic equation]]s. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to five decimal places. Babylonian mathematics were written using a [[sexagesimal]] (base-60) [[numeral system]]. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. 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. ===Egypt=== {{Main|Egyptian mathematics}} [[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, and from this point Egyptian mathematics merged with Greek and Babylonian mathematics to give rise to [[Greek mathematics|Hellenistic mathematics]]. Mathematical study in [[Egypt]] later continued under the [[Arab Empire]] as part of [[Islamic mathematics]], when [[Arabic]] became the written language of Egyptian scholars. The oldest mathematical text discovered so far is the [[Moscow papyrus]], which is an [[Ancient Egypt|Egyptian]] [[Middle Kingdom]] papyrus dated c. 2000&mdash;1800 BC.{{Facts|date=February 2007}} Like many ancient mathematical texts, 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." The [[Rhind papyrus]] (c. 1650 BC [http://www.cut-the-knot.org/arithmetic/RhindPapyrus.shtml]) is another major Egyptian mathematical text, an instruction manual 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 (see [http://mathpages.com/home/rhind.htm]), 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)[http://mathpages.com/home/rhind.htm]. It also shows how to solve first order [[linear equation]]s [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html] as well as [[arithmetic series|arithmetic]] and [[geometric series]] [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_algebra.html#areithmetic%20series]. Also, three geometric elements contained in the Rhind papyrus suggest the simplest of underpinnings to [[analytical geometry]]: (1) first and foremost, how to obtain an approximation of <math>\pi</math> accurate to within less than one percent; (2) second, an ancient attempt at [[squaring the circle]]; and (3) third, the earliest known use of a kind of [[cotangent]]. Finally, the [[Berlin papyrus]] (c. 1300 BC [http://www.hum.ku.dk/cni/papcoll/high008.html] [http://www.aams.org.au/contents.php?subdir=library/history/&filename=pharonic_egypt]) shows that ancient Egyptians could solve a second-order [[algebraic equation]] [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin]. ==Ancient Indian mathematics (c. 900 BC — AD 200)== {{main|Indian mathematics}} Vedic mathematics began in the early Iron Age, with the ''[[Shatapatha Brahmana]]'' (c. 9th century BC), which approximates the value of [[π]] to 2 decimal places.<sup>[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch4_1.html]</sup>, and the [[Sulba Sutras]] (c. 800-500 BC) were [[geometry]] texts that used [[irrational number]]s, [[prime number]]s, the [[rule of three (mathematics)|rule of three]] and [[cube root]]s; computed the [[square root]] of 2 to five decimal places; gave the method for [[squaring the circle]]; solved [[linear equation]]s and [[quadratic equation]]s; developed [[Pythagorean triple]]s algebraically and gave a statement and numerical [[mathematical proof|proof]] of the [[Pythagorean theorem]]. {{Unicode|[[Pāṇini]]}} (c. 5th century BC) formulated the [[Sanskrit grammar|grammar]] rules for [[Sanskrit]]. His notation was similar to modern mathematical notation, and used metarules, [[transformation]]s, and [[recursion]]s with such sophistication that his grammar had the [[computing]] power equivalent to a [[Turing machine]]. [[Pingala]] (roughly 3rd-1st centuries BC) in his treatise of [[prosody]] uses a device corresponding to a [[binary numeral system]]. His discussion of the [[combinatorics]] of [[Metre (music)|meters]], corresponds to the [[binomial theorem]]. Pingala's work also contains the basic ideas of [[Fibonacci number]]s (called ''mātrāmeru''). The [[Brāhmī]] script was developed at least from the [[Maurya dynasty]] in the 4th century BC, with recent archeological evidence appearing to push back that date to around 600 BC. The [[Brahmi numerals]] date to the 3rd century BC. Between 400 BC and AD 200, [[Indian mathematics#Jaina Mathematics (400 BCE - 200 CE)|Jaina mathematicians]] began studying mathematics for the sole purpose of mathematics. They were the first to develop [[transfinite numbers]], [[set theory]], [[logarithms]], fundamental laws of [[index (mathematics)|indices]], [[cubic equation]]s, [[quartic equation]]s, [[sequences]] and progressions, [[permutations and combinations]], squaring and extracting [[square root]]s, and finite and [[infinite]] [[Exponentiation|powers]]. The ''[[Bakhshali Manuscript]]'' written between 200 BC and AD 200 included solutions of linear equations with up to five unknowns, the solution of the quadratic equation, arithmetic and geometric progressions, compound series, quadratic indeterminate equations, [[simultaneous equation]]s, and the use of [[0 (number)|zero]] and [[negative numbers]]. Accurate computations for irrational numbers could be found, which includes computing square roots of numbers as large as a million to at least 11 decimal places. ==Greek and Hellenistic mathematics (c. 550 BC—AD 300)== {{main|Greek mathematics}} [[Image:Pythagoras von Samos.png|left|thumb|Pythagoras of Samos]] Greek mathematics refers to mathematics written in [[Greek language|Greek]] between about 600 BCE and 450 CE.<ref>Howard Eves, ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0030295580</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. [[Image:Thales.jpg|thumb|Thales of Miletus]] 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.<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: Univ. of Chicago Pr.) 2000, on mathematical proofs see p. 75.</ref> Greek mathematics is thought to have begun with [[Thales]] (c. 624&mdash;c.546 BC) and [[Pythagoras]] (c. 582—c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by the ideas of [[Egyptian mathematics|Egypt]], [[Babylonian mathematics|Mesopotamia]] and perhaps [[Indian mathematics|India]]. According to legend, Pythagoras travelled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used [[geometry]] to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras is credited with the first proof of the [[Pythagorean theorem]], though the statement of the theorem has a long history.<ref>Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0</ref> In his commentary on [[Euclid]], [[Proclus]] states that Pythagoras expressed the theorem that bears his name and constructed [[Pythagorean triples]] algebraically rather than geometrically. The [[Academy of Plato]] had the motto "let none unversed in geometry enter here". The [[Pythagoreans]] discovered the existence of irrational numbers. [[Eudoxus]] (408 &mdash;c.355 BC) developed the [[method of exhaustion]], a precursor of modern [[Integral|integration]]. [[Aristotle]] (384&mdash;c.322 BC) first wrote down the laws of [[logic]]. [[Euclid]] (c. 300 BC) is the earliest example of the format still used in mathematics today, definition, axiom, theorem, proof. He also studied [[conics]]. His book, [[Euclid's Elements|''Elements'']], was known to all educated people in the West until the middle of the 20th century.<ref>Howard Eves, ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0030295580 p. 141 "No work, except [[The Bible]], has been more widely used... ."</ref> In addition to the familiar theorems of geometry, such as the [[Pythagorean theorem]], ''Elements'' includes a proof that the square root of two is irrational and that there are infinitely many prime numbers. The [[Sieve of Eratosthenes]] (ca. 230 BC) was used to discover prime numbers. Some say the greatest of Greek mathematicians, if not of all time, was [[Archimedes]] (c.287&mdash;212 BC) of [[Syracuse, Italy|Syracuse]]. He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], and gave remarkably accurate approximations of [[Pi]].<ref>{{cite web | title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F. | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date= February 1996|accessdate= 2007-08-07}}</ref> He also defined the [[Archimedes spiral|spiral]] bearing his name, formulas for the [[volume]]s of [[surface of revolution|surfaces of revolution]] and an ingenious system for expressing very large numbers. ==Classical Chinese mathematics (c. 500 BC—AD 1300)== [[Image:九章算術.gif|thumb|200px|right|''The Nine Chapters on the Mathematical Art''.]] {{main|Chinese mathematics}} In [[China]](212 BC), the Emperor [[Qin Shi Huang]] (Shi Huang-ti) commanded that all books outside of Qin state to be burned. It was not universally obeyed, but as a consequence of this order little is known with certainty about ancient Chinese mathematics. From the [[Western Zhou Dynasty]] (from 1046 BC), the oldest mathematical work to survive the [[book burning]] is the ''[[I Ching]]'', which uses the 8 binary 3-[[tuple]]s (trigrams) and 64 binary 6-[[tuple]]s (hexagrams) for philosophical, mathematical, and/or mystical purposes. The binary tuples are composed of broken and solid lines, called yin 'female' and yang 'male' respectively (see [[King Wen sequence]]). The oldest existent work on [[geometry]] in China comes from the philosophical [[Mohism|Mohist]] canon of c. 330 BC, compiled by the followers of [[Mozi]] (470 BC-390 BC). The ''Mo Jing'' described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. After the book burning, the [[Han dynasty]] (202 BC&ndash;220 AD) produced works of mathematics which presumably expand 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 179 AD, 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 [[π]]. It also made use of [[Cavalieri's principle]] on volume more than a thousand years before Cavalieri would propose it in the West. It created mathematical proof for Pythagoras' [[Pythagorean theorem]], and mathematical formula for [[Gaussian elimination]]. The work was commented on by [[Liu Hui]] in the 3rd century AD. In addition, the mathematical works of the Han astronomer and inventor [[Zhang Heng]] (78-139 AD) had a formulation for [[pi]] as well, which differed from Liu Hui's calculation. Zhang Heng used his formula of pi to find spherical volume. There was also the written work of the mathematician and [[music theory|music theorist]] [[Jing Fang]] (78&ndash;37 BC); by using the [[Pythagorean comma]], Jing observed that 53 [[just fifth]]s approximates to 31 [[octave]]s. This would later lead to the discovery of [[53 equal temperament]], and was not calculated precisely elsewhere until the German [[Nicholas Mercator]] did so in the 17th century. The Chinese also made use of the complex combinatorial diagram known as the '[[magic square]]'' and ''[[Magic circle (mathematics)|magic circles]] which was described in ancient times and perfected by [[Yang Hui]] (1238&ndash;1398 AD). [[Image:Zhang Heng.jpg|thumb|left|180px|[[Zhang Heng]] (78-139)]] [[Zu Chongzhi]] (5th century) of the [[Southern and Northern Dynasties]] computed the value of π to seven decimal places, which remained the most accurate value of π for almost 1000 years. In the thousand years following the Han dynasty, starting in the [[Tang dynasty]] and ending in the [[Song dynasty]], Chinese mathematics thrived at a time when European mathematics did not exist. Developments first made in China, and only much later known in the [[Western world|West]], include [[negative number]]s, the [[binomial theorem]], [[Matrix (mathematics)|matrix]] methods for solving systems of [[linear equation]]s and the [[Chinese remainder theorem]]. The Chinese also developed [[Pascal's triangle]] and the [[rule of three (mathematics)|rule of three]] long before it was known in Europe. Besides Zu Chongzhi, some of the most important figures of Chinese mathematics during this period include [[Yi Xing]], [[Shen Kuo]], [[Qin Jiushao]], [[Zhu Shijie]], and others. The scientist Shen Kuo used problems involving [[calculus]], [[trigonometry]], [[metrology]], [[permutations]], and once computed the possible amount of terrain space that could be used with specific battle formations, as well as the longest possible military campaign given the amount of food carriers could bring for themselves and soldiers. Even after European mathematics began to flourish during the [[Renaissance]], European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline, until the [[Jesuit]] missionaries such as [[Matteo Ricci]] carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries. ==Classical Indian mathematics (c. 400—1600)== {{Main|Indian mathematics}} {{see also|History of the Hindu-Arabic numeral system}} [[Image:2064 aryabhata-crp.jpg|thumb|[[Aryabhata]] ]] The ''[[Surya Siddhanta]]'' (c. 400) introduced the [[trigonometric functions]] of [[sine]], [[cosine]], and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. The cosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average [[sidereal year]] of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work was translated to Arabic and [[Latin]] during the [[Middle Ages]]. [[Aryabhata]] in 499 introduced the [[versine]] function, produced the first [[trigonometry|trigonometric]] tables of sine, developed techniques and [[algorithm]]s of [[algebra]], [[infinitesimal]]s, [[differential equation]]s, and obtained whole number solutions to linear equations by a method equivalent to the modern method, along with accurate [[astronomy|astronomical]] calculations based on a [[heliocentrism|heliocentric]] system of [[gravity|gravitation]]. An [[Arabic]] translation of his ''Aryabhatiya'' was available from the 8th century, followed by a Latin translation in the 13th century. He also computed the value of [[π]] to the fourth decimal place as 3.1416. [[Madhava of Sangamagrama|Madhava]] later in the 14th century computed the value of π to the eleventh decimal place as 3.14159265359. 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 (around 770) that [[Islam]]ic 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 of Sangamagrama|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, [[Integral test for convergence|tests of convergence]], [[iterative method]]s for solutions of [[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 became stagnant from the late 16th century onwards due to subsequent political turmoil. ==Islamic mathematics (c. 800—1500)== {{main|Islamic mathematics}} {{see also|History of the Hindu-Arabic numeral system}} [[Image:Abu Abdullah Muhammad bin Musa al-Khwarizmi.jpg|thumb|[[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. Some of the most important Islamic mathematicians were [[Persian people|Persian]]. {{Unicode|[[Muḥammad ibn Mūsā al-Ḵwārizmī]]}}, a 9th century Persian mathematician and astronomer to the [[Caliph of Baghdad]], 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 the Arab mathematician 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, ''[[Al-Jabr wa-al-Muqabilah|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 preservation of ancient algebraic methods and for his original contributions to the field.<ref>[http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra]. [[Louisiana State University]].</ref> Further developments in [[algebra]] were made by [[Abu Bakr al-Karaji]] (953&mdash;1029) in his treatise ''al-Fakhri'', where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. In the [[10th century]], [[Abul Wafa]] translated the works of [[Diophantus]] into Arabic and developed the [[tangent (trigonometry)|tangent]] function. 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'', p. 255-259. [[Addison-Wesley]]. ISBN 0321016181.</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]]." [[Ibn al-Haytham]] was the first mathematician to derive the formula for the sum of the [[fourth power]]s, and using the method of induction, he developed a method for determining the general formula for the sum of any integral [[Exponentiation|powers]], which was fundamental to the development of [[integral]] [[calculus]].<ref>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 163-174.</ref> [[Omar Khayyam]], the 12th century [[Persian literature|poet]], was also a mathematician, and wrote ''Discussions of the Difficulties in Euclid'', a book about flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]], and thus he laid the foundations for [[analytic geometry]] and [[non-Euclidean geometry]]. He was also the first to find the general geometric solution to [[cubic equation]]s. He was also very influential in [[calendar reform]]. The Persian mathematician [[Nasir al-Din Tusi]] (Nasireddin) in the 13th century 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 |pages=71}}</ref> the addition of the [[decimal point]] notation to the [[Arabic numerals]], the discovery of all the modern [[trigonometric function]]s besides sine, [[al-Kindi]]'s introduction of [[cryptanalysis]] and [[frequency analysis]], [[al-Karaji]]'s introduction of algebraic [[calculus]] and [[Mathematical proof|proof]] by [[mathematical induction]], the development of [[analytic geometry]] and the earliest general formula for [[infinitesimal]] and [[integral]] calculus 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, and numerous 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. This parallels the stagnation of mathematics when the Romans conquered the Hellenistic world. John J. O'Connor and Edmund F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'': {{quote|"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Hellenistic mathematics."}} ==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|''Geometry'']]. 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. 135-154 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 travelled 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|al-Jabr wa-al-Muqabilah]]'', translated into Latin by [[Robert of Chester]], and the complete text of Euclid's [[Euclid's Elements|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. 421-462 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard Univ. Pr., 1982)</ref><ref>Guy Beaujouan, The Transformation of the Quadrivium", pp. 463-487 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard Univ. Pr., 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. <!-- Needs to spell out what Fibonacci did, not just praise it. -->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 area that contributed to the development of mathematics concerned the analysis 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: Univ. of Wisconsin Pr.), pp. 421-440.</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", pp. 215-254 in ''Arts libéraux et philosophie au Moyen Âge'' (Montréal: Institut d'Études Médiévales), at pp. 224-227.</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: Univ. of Wisconsin Pr.), pp. 210, 214-15, 236.</ref> Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (which we would solve by a simple [[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: Univ. of Wisconsin Pr.), 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: Univ. of Wisconsin Pr.), pp. 332-45, 382-91.</ref> In a later mathematical commentary on Euclid's ''Geometry'', 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. 560-5 in Marshall Clagett, ed., ''Nicole Oresme and the Medieval Geometry of Qualities and Motions,'' (Madison: Univ. of Wisconsin Pr., 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]] circa 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 mathematical books printed were [[Peurbach]]'s [[Theoricae nova planetarum]] 1472 followed by a book on commercial arithmetic, the 1478 [[Treviso Arithmetic]] and then the first real mathematics book [[Euclid]]'s Elements printed and published by [[Ratdolt]] 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' 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 | id = ISBN 0-393-32030-8}}</ref> By century's end, thanks to [[Regiomontanus]] (1436&mdash;1476) 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 mathematicians, [[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]].]] {{Cleanup-section|date=May 2008}} As we have seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older than any surviving written text. The earliest civilizations -- in Mesopotamia, Egypt, India and China -- knew arithmetic. One way to view the development of the various number systems of modern mathematics is to see new numbers studied and investigated to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1? In India and China, and much later in Germany, negative numbers were developed to answer the question: what do you get when you subtract a larger number from a smaller? Another natural question is: what kind of a number is the square root of two? The Greeks knew that it was not a fraction, and this question may have played a role in the development of [[continued fraction]]s. But a better answer came with the invention of decimals, developed by [[John Napier]] (1550 - 1617) and perfected later by [[Simon Stevin]]. Using decimals, and an idea that anticipated the concept of the [[limit (mathematics)|limit]], Napier also studied a new constant, which [[Leonhard Euler]] (1707 - 1783) named <font face="times new Roman">[[E (mathematical constant)|''e'']]</font>. Euler was very influential in the standardization of other mathematical terms and notations. He named the square root of minus 1 with the symbol <font face="times new Roman">[[Imaginary unit|''i'']]</font>. He also popularized the use of the Greek letter <math>\pi</math> to stand for the ratio of a circle's circumference to its diameter. He then derived one of the most remarkable identities in all of mathematics: :<math>e^{i \pi} +1 = 0 \,</math> (see [[Euler's Identity]].) ==19th century== [[Image:noneuclid.svg|right|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 this 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 discovered [[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. Also in the nineteenth century [[William Rowan Hamilton]] developed [[noncommutative algebra]]. In addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of [[calculus]], in work by [[Augustin-Louis Cauchy]] and [[Karl Weierstrass]]. A new form of algebra was developed in the nineteenth century called [[Boolean logic|Boolean algebra]], named after the British mathematician [[George Boole]]. It was a system in which the only numbers were 0 and 1, a system which today has important applications in [[computer science]]. 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]]. Towards the end of the 19th century, [[Georg Cantor]] invented the [[set theory]], which has become the common language of different mathematical branches. The introduction of [[infinite set]] set off a debate on [[foundations of mathematics]]. The 19th century also saw the founding of the first 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. Before the 20th century, there were very few creative mathematicians in the world at any one time. For the most part, mathematicians were either born to wealth, like Napier, or supported by wealthy patrons, like Gauss. There were a few who found meager livelihoods teaching at a university, like Fourier. [[Niels Henrik Abel]], unable to obtain a position, died in poverty of malnutrition and tuberculosis at the age of twenty-six. ==20th century== [[Image:Four Colour Map Example.svg|thumb|A map illustrating the [[Four Color Theorem]]]] The profession of mathematician became much more important in the 20th century. Every year, hundreds of new Ph.D.s in mathematics are awarded, and jobs are available both in teaching and industry. Mathematical development has grown at an exponential rate, with too many new developments for a survey to even touch on any but a few of the most profound. In 1900, [[David Hilbert]] presented a list of [[Hilbert's problems|23 unsolved problems in mathematics]] at the [[International Congress of Mathematicians]]. These problems spanned many areas of mathematics and have formed a central focus for much of 20th century mathematics. Today ten have been resolved, seven are partially resolved and two problems are still open. The remaining four are too loose to be stated as resolved or not. In the 1910s, [[Srinivasa Aiyangar Ramanujan]] (1887-1920) developed 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 breakthroughs and discoveries in the areas of [[gamma function]]s, [[modular form]]s, [[divergent series]], [[hypergeometric series]] and [[prime number theory]]. In 1931, [[Kurt Gödel]] published his two [[incompleteness theorem]]s which state the limit of mathematical logic. It put an end to David Hilbert's dream of a complete and consistent mathematical system. Famous conjectures of the past yielded to new and more powerful techniques. [[Wolfgang Haken]] and [[Kenneth Appel]] used a computer to prove the [[four color theorem]] in 1976. [[Andrew Wiles]], working alone in his office for years, proved [[Fermat's last theorem]] in 1995. Mathematical collaborations of unprecedented size and scope took place. The [[classification of finite simple groups]] (also called the "enormous theorem") spanned tens of thousands of pages in 500-odd journal articles written by about 100 authors, published mostly between 1955 and 1983. Entire new areas of mathematics such as [[mathematical logic]], [[topology]], [[complexity theory]], and [[game theory]] changed the kinds of questions that could be answered by mathematical methods. The French [[Nicolas Bourbaki|Bourbaki Group]] attempted to bring all of mathematics into a coherent rigorous whole, publishing under the [[pseudonym]] ''Nicolas Bourbaki''. Their extensive work had a controversial influence on mathematical education.<ref>Maurice Mashaal, ''Bourbaki: A Secret Society of Mathematicians'', American Mathematical Society, 2006, ISBN 0821839675, ISBN13 978-0821839676.</ref> There were also new investigations of limitations to mathematics. [[Kurt Gödel]] proved that in any mathematical system that includes the integers, there are true statements that [[Incompleteness theorem|cannot be proved]]. [[Paul Cohen (mathematician)|Paul Cohen]] proved the [[logical independence|independence]] of the [[continuum hypothesis]] from the [[ZFC|standard axioms of set theory]]. ==See also== *[[List of important publications in mathematics]] *[[History of algebra]] *[[History of calculus]] *[[History of geometry]] *[[History of logic]] *[[History of mathematical notation]] *[[History of statistics]] *[[History of trigonometric functions]] *[[History of writing numbers]] == References == {{reflist}} == 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|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|id=ISBN 0801873975}} * van der Waerden, B. L., ''Geometry and Algebra in Ancient Civilizations'', Springer, 1983, ISBN 0387121595. * 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 | id = 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 | id = 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 | id = 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://members.aol.com/jeff570/ Mathematics Pages] (Jeff Miller). Contains information on the earliest known uses of symbols and terms used in mathematics as well as a collection of postage stamps depicting mathematicians. *[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). ;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 Laguna) *[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) [[Category:History of mathematics| ]] {{Link FA|no}} [[ar:تاريخ الرياضيات]] [[bn:গণিতের ইতিহাস]] [[ca:Història de les matemàtiques]] [[cs:Dějiny matematiky]] [[da:Matematikkens historie]] [[de:Geschichte der Mathematik]] [[es:Historia de la matemática]] [[eo:Historio de matematiko]] [[fr:Histoire des mathématiques]] [[ko:수학의 역사]] [[it:Storia della matematica]] [[he:היסטוריה של המתמטיקה]] [[lt:Matematikos istorija]] [[hu:A matematika története]] [[nl:Geschiedenis van de wiskunde]] [[ja:数学史]] [[no:Matematikkens historie]] [[pl:Historia matematyki]] [[pt:História da matemática]] [[ru:История математики]] [[sq:Historia e matematikës]] [[sl:Zgodovina matematike]] [[sr:Историја математике]] [[su:Sajarah matematik]] [[fi:Matematiikan historia]] [[sv:Matematikens historia]] [[vi:Lịch sử toán học]] [[uk:Історія математики]] [[ur:تاریخ ریاضی]] [[zh:数学史]]