Many Greek and Arabic texts on mathematics weretranslated into Latin from the 12th century, leading to further development of mathematics inMedieval Europe. From ancient times through theMiddle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[11] Beginning inRenaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at anincreasing pace that continues through the present day. This includes the groundbreaking work of bothIsaac Newton andGottfried Wilhelm Leibniz in the development of infinitesimalcalculus during the 17th century and following discoveries ofGerman mathematicians likeCarl Friedrich Gauss andDavid Hilbert.
The origins of mathematical thought lie in the concepts ofnumber,patterns in nature,magnitude, andform.[12] Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life inhunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.[12]
The use of yarn byNeanderthals some 40,000 years ago at a site in Abri du Maras in the south of France suggests they knew basic concepts in mathematics.[13][14] TheIshango bone, found near the headwaters of theNile river (northeasternCongo), may be more than20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either atally of the earliest known demonstration ofsequences ofprime numbers[15] or a six-month lunar calendar.[16][17] Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[18] The Ishango bone, according to scholarAlexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[19]
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed thatmegalithic monuments inEngland andScotland, dating from the 3rd millennium BC, incorporate geometric ideas such ascircles,ellipses, andPythagorean triples in their design.[20] All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.[21]
Babylonian mathematics refers to any mathematics of the peoples ofMesopotamia (modernIraq) from the days of the earlySumerians through theHellenistic period almost to the dawn ofChristianity.[22] The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (Seleucid period).[23] It is named Babylonian mathematics due to the central role ofBabylon as a place of study. Later under theArab Empire, Mesopotamia, especiallyBaghdad, once again became an important center of study forIslamic mathematics.
Geometry problem on a clay tablet belonging to a school for scribes;Susa, first half of the 2nd millennium BC
In contrast to the sparsity of sources inEgyptian mathematics, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.[24] Written inCuneiform 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.[25]
The earliest evidence of written mathematics dates back to the ancientSumerians, who built the earliest civilization in Mesopotamia. They developed a complex system ofmetrology from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.[26] From around 2500 BC onward, the Sumerians wrotemultiplication tables on clay tablets and dealt with geometrical exercises anddivision problems. The earliest traces of the Babylonian numerals also date back to this period.[27]
The Babylonian mathematical tabletPlimpton 322, dated to 1800 BC.
Babylonian mathematics were written using asexagesimal (base-60)numeral system.[24] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30,[24] and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc.) being able to easily calculate by hand was essential, and so a sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision.[28] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in thedecimal system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until theRenaissance, and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tabletYBC 7289 gives an approximation of√2 accurate to five decimal places.[29] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.[23] By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.[23] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[23]
Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation ofregular numbers, and theirreciprocalpairs.[30] The tablets also include multiplication tables and methods for solvinglinear,quadratic equations andcubic equations, a remarkable achievement for the time.[31] Tablets from the Old Babylonian period also contain the earliest known statement of thePythagorean theorem.[32] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need forproofs or logical principles.[25]
Image of Problem 14 from theMoscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
Egyptian mathematics refers to mathematics written in theEgyptian language. From theHellenistic period,Greek replaced Egyptian as the written language ofEgyptian scholars. Mathematical study inEgypt later continued under theArab Empire as part ofIslamic mathematics, whenArabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[33] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[34]
The most extensive Egyptian mathematical text is theRhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from theMiddle Kingdom of about 2000–1800 BC.[35] It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,[36] includingcomposite andprime numbers;arithmetic,geometric andharmonic means; and simplistic understandings of both theSieve of Eratosthenes andperfect number theory (namely, that of the number 6).[37] It also shows how to solve first orderlinear equations[38] as well asarithmetic andgeometric series.[39]
Another significant Egyptian mathematical text is theMoscow papyrus, also from theMiddle Kingdom period, dated to c. 1890 BC.[40] It consists of what are today calledword problems orstory 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 afrustum (truncated pyramid).
Greek mathematics refers to the mathematics written in theGreek language from the time ofThales of Miletus (~600 BC) to the closure of theAcademy of Athens in 529 AD.[42] 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 followingAlexander the Great is sometimes calledHellenistic mathematics.[43]
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 ofinductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, useddeductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and usedmathematical rigor to prove them.[44]
Greek mathematics is thought to have begun withThales of Miletus (c. 624–c.546 BC) andPythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired byEgyptian andBabylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Thales usedgeometry to solve problems such as calculating the height ofpyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries toThales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[45] Pythagoras established thePythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".[46] It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of thePythagorean theorem,[47] though the statement of the theorem has a long history, and with the proof of the existence ofirrational numbers.[48][49] Although he was preceded by theBabylonians,Indians and theChinese,[50] theNeopythagorean mathematicianNicomachus (60–120 AD) provided one of the earliestGreco-Romanmultiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in theBritish Museum).[51] The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its laterMedieval name: themensa Pythagorica.[52]
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.[53] HisPlatonic Academy, inAthens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such asEudoxus of Cnidus (c. 390 - c. 340 BC), came.[54] Plato also discussed the foundations of mathematics,[55] clarified some of the definitions (e.g. that of a line as "breadthless length").
One of the oldest surviving fragments of Euclid'sElements, found atOxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[60]
In the 3rd century BC, the premier center of mathematical education and research was theMusaeum ofAlexandria.[61] It was there thatEuclid (c. 300 BC) taught, and wrote theElements, widely considered the most successful and influential textbook of all time.[1] TheElements introducedmathematical rigor through theaxiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of theElements were already known, Euclid arranged them into a single, coherent logical framework.[62] TheElements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.[63] In addition to the familiar theorems ofEuclidean geometry, theElements was meant as an introductory textbook to all mathematical subjects of the time, such asnumber theory,algebra andsolid geometry,[62] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid alsowrote extensively on other subjects, such asconic sections,optics,spherical geometry, and mechanics, but only half of his writings survive.[64]
Archimedes (c. 287–212 BC) ofSyracuse, widely considered the greatest mathematician of antiquity,[65] used themethod of exhaustion to calculate thearea under the arc of aparabola with thesummation of an infinite series, in a manner not too dissimilar from modern calculus.[66] He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known,3+10/71 < π < 3+10/70.[67] He also studied thespiral bearing his name, obtained formulas for thevolumes ofsurfaces of revolution (paraboloid, ellipsoid, hyperboloid),[66] and an ingenious method ofexponentiation for expressing very large numbers.[68] While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles.[69] He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.[70]
Apollonius of Perga (c. 262–190 BC) made significant advances to the study ofconic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[71] He also coined the terminology in use today for conic sections, namelyparabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").[72] His workConics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[73] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.[74]
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.[82] During this period,Diophantus made significant advances in algebra, particularlyindeterminate analysis, which is also known as "Diophantine analysis".[83] The study ofDiophantine equations andDiophantine approximations is a significant area of research to this day. His main work was theArithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate andindeterminate equations.[84] TheArithmetica had a significant influence on later mathematicians, such asPierre de Fermat, who arrived at his famousLast Theorem after trying to generalize a problem he had read in theArithmetica (that of dividing a square into two squares).[85] Diophantus also made significant advances in notation, theArithmetica being the first instance of algebraic symbolism and syncopation.[84]
Among the last great Greek mathematicians isPappus of Alexandria (4th century AD). He is known for hishexagon theorem andcentroid theorem, as well as thePappus configuration andPappus graph. HisCollection is a major source of knowledge on Greek mathematics as most of it has survived.[86] Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.
The first woman mathematician recorded wasHypatia of Alexandria (AD 350–415), who wrote many works on applied mathematics. Because of a political dispute, theChristian community in Alexandria had her stripped publicly and executed.[87] Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such asProclus,Simplicius andEutocius.[88] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-PlatonicAcademy of Athens by the emperorJustinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in theByzantine empire with mathematicians such asAnthemius of Tralles andIsidore of Miletus, the architects of theHagia Sophia.[89] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.[90]
The creation of theRoman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during theRoman Kingdom and included 356 days plus aleap year every other year.[100] In contrast, thelunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than thesolar year, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February.[101] This calendar was supplanted by theJulian calendar, asolar calendar organized byJulius Caesar (100–44 BC) and devised bySosigenes of Alexandria to include aleap day every four years in a 365-day cycle.[102] This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by theGregorian calendar organized byPope Gregory XIII (r. 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.[103]
At roughly the same time,the Han Chinese and the Romans both invented the wheeledodometer device for measuringdistances traveled, the Roman model first described by the Roman civil engineer and architectVitruvius (c. 80 BC – c. 15 BC).[104] The device was used at least until the reign of emperorCommodus (r. 177 – 192 AD), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.[105] Perhaps relying on similar gear-work andtechnology found in theAntikythera mechanism, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in oneRoman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-toothcogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.[106]
An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.[107] The oldest extant mathematical text from China is theZhoubi Suanjing (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during theWarring States Period appears reasonable.[108] However, theTsinghua Bamboo Slips, containing the earliest knowndecimalmultiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.[50]
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[109] Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.[110]Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on thesuan pan, or Chinese abacus. The date of the invention of thesuan pan is not certain, but the earliest written mention dates from AD 190, inXu Yue'sSupplementary Notes on the Art of Figures.
The oldest extant work on geometry in China comes from the philosophicalMohist canonc. 330 BC, compiled by the followers ofMozi (470–390 BC). TheMo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[111] It also defined the concepts ofcircumference,diameter,radius, andvolume.[112]
In 212 BC, the EmperorQin Shi Huang commanded all books in theQin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After thebook burning of 212 BC, theHan dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these isThe Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios forChinese pagoda towers, engineering,surveying, and includes material onright triangles.[108] It created mathematical proof for thePythagorean theorem,[113] and a mathematical formula forGaussian elimination.[114] The treatise also provides values ofπ,[108] which Chinese mathematicians originally approximated as 3 untilLiu Xin (d. 23 AD) provided a figure of 3.1457 and subsequentlyZhang Heng (78–139) approximated pi as 3.1724,[115] as well as 3.162 by taking thesquare root of 10.[116][117]Liu Hui commented on theNine Chapters in the 3rd century AD andgave a value of π accurate to 5 decimal places (i.e. 3.14159).[118][119] Though more of a matter of computational stamina than theoretical insight, in the 5th century ADZu Chongzhi computedthe value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.[118][120] He also established a method which would later be calledCavalieri's principle to find the volume of asphere.[121]
The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of theSong dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is thePrecious Mirror of the Four Elements byZhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar toHorner's method.[118] ThePrecious Mirror also contains a diagram ofPascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[122] The Chinese also made use of the complex combinatorial diagram known as themagic square andmagic circles, described in ancient times and perfected byYang Hui (AD 1238–1298).[122]
Even after European mathematics began to flourish during theRenaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards.Jesuit missionaries such asMatteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.[122]
Japanese mathematics,Korean mathematics, andVietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to theConfucian-basedEast Asian cultural sphere.[123] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China'sMing dynasty (1368–1644).[124] For instance, although Vietnamese mathematical treatises were written in eitherChinese or the native VietnameseChữ Nôm script, all of them followed the Chinese format of presenting a collection of problems withalgorithms for solving them, followed by numerical answers.[125] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy ofmathematicians and astronomers, whereas in Japan it was more prevalent in the realm ofprivate schools.[126]
The numerals used in theBakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Indian numerals in stone and copper inscriptions[127]
Ancient Brahmi numerals in a part of India
The earliest civilization on the Indian subcontinent is theIndus Valley civilization (mature second phase: 2600 to 1900 BC) that flourished in theIndus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.[128]
The oldest extant mathematical records from India are theSulba Sutras (dated variously between the 8th century BC and the 2nd century AD),[129] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.[130] As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.[129] The Sulba Sutras give methods for constructing acircle with approximately the same area as a given square, which imply several different approximations of the value of π.[131][132][a] In addition, they compute thesquare root of 2 to several decimal places, list Pythagorean triples, and give a statement of thePythagorean theorem.[132] All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.[129] It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.[129]
The next significant mathematical documents from India after theSulba Sutras are theSiddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence.[138] They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.[139] Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".[139]
Around 500 AD,Aryabhata wrote theAryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.[140] It is in theAryabhatiya that the decimal place-value system first appears. Several centuries later, theMuslim mathematicianAbu Rayhan Biruni described theAryabhatiya as a "mix of common pebbles and costly crystals".[141]
In the 7th century,Brahmagupta identified theBrahmagupta theorem,Brahmagupta's identity andBrahmagupta's formula, and for the first time, inBrahma-sphuta-siddhanta, he lucidly explained the use ofzero as both a placeholder anddecimal digit, and explained theHindu–Arabic numeral system.[142] 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 asArabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from theBrahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century,Halayudha's commentary onPingala's work contains a study of theFibonacci sequence[143] andPascal's triangle.[144]
In the 12th century,Bhāskara II,[145] who lived in southern India, wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals,the mean value theorem and the derivative of the sine function although he did not develop the notion of a derivative.[146][147] In the 14th century,Narayana Pandita completed hisGanita Kaumudi.[148]
Also in the 14th century,Madhava of Sangamagrama, the founder of theKerala School of Mathematics, found theMadhava–Leibniz series and obtained from it atransformed series, whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also foundthe Madhava-Gregory series to determine the arctangent, the Madhava-Newtonpower series to determine sine and cosine andthe Taylor approximation for sine and cosine functions.[149] In the 16th century,Jyesthadeva consolidated many of the Kerala School's developments and theorems in theYukti-bhāṣā.[150][151] It has been argued that certain ideas of calculus like infinite series and taylor series of some trigonometry functions, were transmitted to Europe in the 16th century[6] viaJesuit missionaries and traders who were active around the ancient port ofMuziris at the time and, as a result, directly influenced later European developments in analysis and calculus.[152] However, other scholars argue that the Kerala School did not formulate a systematic theory ofdifferentiation andintegration, and that there is not any direct evidence of their results being transmitted outside Kerala.[153][154][155][156]
TheIslamic Empire established across theMiddle East,Central Asia,North Africa,Iberia, and in parts ofIndia in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written inArabic, they were not all written byArabs, 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.[157]
In the 9th century, the Persian mathematicianMuḥammad ibn Mūsā al-Khwārizmī wrote an important book on theHindu–Arabic numerals and one on methods for solving equations. His bookOn the Calculation with Hindu Numerals, written about 825, along with the work ofAl-Kindi, were instrumental in spreadingIndian mathematics andIndian numerals to the West. The wordalgorithm is derived from the Latinization of his name, Algoritmi, and the wordalgebra from the title of one of his works,Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[158] and he was the first to teach algebra in anelementary form and for its own sake.[159] He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described asal-jabr.[160] His algebra was also no longer concerned "with a series of problems to be resolved, but anexposition 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."[161]
In Egypt,Abu Kamil extended algebra to the set ofirrational numbers, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.[162] His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.
Further developments in algebra were made byAl-Karaji in his treatiseal-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to aproof bymathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove thebinomial theorem,Pascal's triangle, and the sum of integralcubes.[163] Thehistorian of mathematics, F. Woepcke,[164] praised Al-Karaji for being "the first who introduced thetheory ofalgebraiccalculus." Also in the 10th century,Abul Wafa translated the works ofDiophantus into Arabic.Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of aparaboloid, and was able to generalize his result for the integrals ofpolynomials up to thefourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[165]
In the 13th century,Nasir al-Din Tusi (Nasireddin) made advances inspherical trigonometry. He also wrote influential work on Euclid'sparallel postulate. In the 15th century,Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculatingnth roots, which was a special case of the methods given many centuries later byRuffini andHorner.
In thePre-Columbian Americas, theMaya civilization that flourished inMexico andCentral America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.[168]Maya numerals used abase of twenty, thevigesimal system, instead of a base of ten that forms the basis of thedecimal system used by most modern cultures.[168] The Maya used mathematics to create theMaya calendar as well as to predict astronomical phenomena in their nativeMaya astronomy.[168] While the concept ofzero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[168]
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 byPlato'sTimaeus and the biblical passage (in theBook of Wisdom) that God hadordered all things in measure, and number, and weight.[169]
Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the termquadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wroteDe institutione arithmetica, a free translation from the Greek ofNicomachus'sIntroduction to Arithmetic;De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid'sElements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.[170][171]
Leonardo of Pisa, now known asFibonacci, serendipitously learned about theHindu–Arabic numerals on a trip to what is nowBéjaïa,Algeria with his merchant father. (Europe was still usingRoman numerals.) There, he observed a system ofarithmetic (specificallyalgorism) which due to thepositional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wroteLiber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as theFibonacci sequence (known to Indian mathematicians for hundreds of years before that)[174] which Fibonacci used as an unremarkable example.
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.[175] 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).[176] Bradwardine's analysis is an example of transferring a mathematical technique used byal-Kindi andArnald of Villanova to quantify the nature of compound medicines to a different physical problem.[177]
One of the 14th-centuryOxford Calculators,William of Heytesbury, lackingdifferential calculus and the concept oflimits, proposed to measure instantaneous speed "by the path thatwould 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".[179]
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".[180]
Nicole Oresme at theUniversity of Paris and the ItalianGiovanni 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.[181] In a later mathematical commentary on Euclid'sElements, 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.[182]
During theRenaissance, the development of mathematics and ofaccounting were intertwined.[183] While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (inFlanders andGermany) orabacus schools (known asabbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performingbookkeeping operations, but for complex bartering operations or the calculation ofcompound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.
Luca Pacioli'sSumma de Arithmetica, Geometria, Proportioni et Proportionalità (Italian: "Review ofArithmetic,Geometry,Ratio andProportion") was first printed and published inVenice in 1494. It included a 27-page treatise on bookkeeping,"Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from themathematical puzzles it contained, and to aid the education of their sons.[187] InSumma Arithmetica, Pacioli introduced symbols forplus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics.Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.
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 hisTrigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.[190]
During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that were involved, were studied intensely.[191]
The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe.Tycho Brahe had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant,Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention oflogarithms byJohn Napier andJost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion.[192]Theanalytic geometry developed byRené Descartes (1596–1650) allowed those orbits to be plotted on a graph, inCartesian coordinates.
Science and mathematics had become an international endeavor, which would soon spread over the entire world.[194]
In addition to the application of mathematics to the studies of the heavens,applied mathematics began to expand into new areas, with the correspondence ofPierre de Fermat andBlaise Pascal. Pascal and Fermat set the groundwork for the investigations ofprobability theory and the corresponding rules ofcombinatorics in their discussions over a game ofgambling. Pascal, with hiswager, 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 ofutility theory in the 18th and 19th centuries.
The most influential mathematician of the 18th century was arguablyLeonhard Euler (1707–83). His contributions range from founding the study ofgraph theory with theSeven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symboli, and he popularized the use of the Greek letter 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.[195]
Other important European mathematicians of the 18th century includedJoseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, andPierre-Simon Laplace, who, in the age ofNapoleon, did important work on the foundations ofcelestial mechanics and onstatistics.[196]
Behavior of lines with a common perpendicular in each of the three types of geometry
This century saw the development of the two forms ofnon-Euclidean geometry, where theparallel postulate of Euclidean geometry no longer holds.The Russian mathematicianNikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematicianJános Bolyai, independently defined and studiedhyperbolic 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 mathematicianBernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developedRiemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of amanifold, which generalizes the ideas ofcurves andsurfaces, and set the mathematical foundations for thetheory of general relativity.[199]
Also, for the first time, the limits of mathematics were explored.Niels Henrik Abel, a Norwegian, andÉvariste Galois, a Frenchman, proved there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem).[207] Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient totrisect 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.[208] Mathematicians had vainly attempted to solve these problems since the ancient Greeks. On the other hand, the limitation of threedimensions in geometry was surpassed in the 19th century through considerations ofparameter space andhypercomplex numbers.[209]
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments ofgroup theory, and the associated fields ofabstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to studysymmetry.[210]
The 20th century saw mathematics become a major profession. By the end of the century, thousands of new Ph.D.s in mathematics were being awarded every year, and jobs were available in both teaching and industry.[220] An effort to catalogue the areas and applications of mathematics was undertaken inKlein's encyclopedia.[221]
Mathematical collaborations of unprecedented size and scope took place. An example is theclassification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.[228] A group of French mathematicians, includingJean Dieudonné andAndré Weil, publishing under thepseudonym "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.[229]
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved byMojżesz Presburger, that the truth or falsity of all statements formulated about thenatural numbers plus either addition or multiplication (but not both), wasdecidable, i.e. could be determined by some algorithm.[244][245][246] In 1931,Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known asPeano arithmetic, was in factincomplete. (Peano arithmetic is adequate for a good deal ofnumber theory, including the notion ofprime number.) A consequence of Gödel's twoincompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all ofanalysis and geometry), truth necessarily outruns proof, i.e. there are true statements thatcannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, andDavid Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.[247]
Theabsolute value of the Gamma function on the complex plane
Paul Erdős published more papers than any other mathematician in history,[254] working with hundreds of collaborators. Mathematicians have a game equivalent to theKevin Bacon Game, which leads to theErdős number of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.[255][256]
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century, there were hundreds of specialized areas in mathematics, and theMathematics Subject Classification was dozens of pages long.[259] More and moremathematical journals were published and, by the end of the century, the development of theWorld Wide Web led to online publishing.
Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched.[262][263] There is an increasing drive towardopen access publishing, first made popular byarXiv.
There are many observable trends in mathematics, the most notable being that the subject is growing ever larger as computers are ever more important and powerful; the volume of data being produced by science and industry, facilitated by computers, continues expanding exponentially. As a result, there is a corresponding growth in the demand for mathematics to help process and understand thisbig data.[264] Math science careers are also expected to continue to grow, with the USBureau of Labor Statistics estimating (in 2018) that "employment of mathematical science occupations is projected to grow 27.9 percent from 2016 to 2026."[265]
^Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations",Historia Mathematica, 8, pp. 277–318.
^Heath, Thomas L. (1963).A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."
^abJoseph, George Gheverghese (1991).The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48.
^Ifrah, Georges (1986).Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37.
^Kaplan, Robert (1999).The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.
^"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplacehttp://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
^Juschkewitsch, A. P. (1964).Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.
^Eves, Howard (1990).History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258.
^Marshack, A. (1972).The Roots of Civilization: the Cognitive Beginning of Man's First Art, Symbol and Notation. New York: McGraw-Hill.
^Thom, Alexander; Archie Thom (1988). "The metrology and geometry of Megalithic Man", pp. 132–51 in Ruggles, C. L. N. (ed.),Records in Stone: Papers in memory of Alexander Thom. Cambridge University Press.ISBN0-521-33381-4.
^Eglash, Ron (1999).African fractals : modern computing and indigenous design. New Brunswick, N.J.: Rutgers University Press. pp. 89, 141.ISBN0813526140.
^Eglash, R. (1995). "Fractal Geometry in African Material Culture".Symmetry: Culture and Science.6–1:174–177.
^Eves, Howard (1990).An Introduction to the History of Mathematics, Saunders,ISBN0-03-029558-0
^(Boyer 1991, "The Age of Plato and Aristotle" p. 99)
^Bernal, Martin (2000). "Animadversions on the Origins of Western Science", pp. 72–83 in Michael H. Shank, ed.The Scientific Enterprise in Antiquity and the Middle Ages. Chicago: University of Chicago Press, p. 75.
^Eves, Howard (1990).An Introduction to the History of Mathematics, Saunders,ISBN0-03-029558-0.
^Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum".The Annals of Mathematics.
^Choike, James R. (1980). "The Pentagram and the Discovery of an Irrational Number".The Two-Year College Mathematics Journal.11 (5):312–316.doi:10.2307/3026893.JSTOR3026893.
^David E. Smith (1958),History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics, New York: Dover Publications (a reprint of the 1951 publication),ISBN0-486-20429-4, pp. 58, 129.
^Smith, David E. (1958).History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics, New York: Dover Publications (a reprint of the 1951 publication),ISBN0-486-20429-4, p. 129.
^(Boyer 1991, "The Age of Plato and Aristotle" p. 86)
^(Boyer 1991, "The Age of Plato and Aristotle" p. 88)
^Eves, Howard (1990).An Introduction to the History of Mathematics, Saunders.ISBN0-03-029558-0 p. 141: "No work, exceptThe Bible, has been more widely used..."
^Development Of Modern Numerals And Numeral Systems: The Hindu-Arabic system, Encyclopaedia Britannica, Quote: "The 1, 4, and 6 are found in the Ashoka inscriptions (3rd century BC); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st or 2nd century AD – all in forms that have considerable resemblance to today’s, 2 and 3 being well-recognized cursive derivations from the ancient = and ≡."
^Boyer (1991). "The Arabic Hegemony".History of Mathematics. Wiley. p. 226.ISBN9780471543978.By 766 we learn that an astronomical-mathematical work, known to the Arabs as theSindhind, was brought to Baghdad from India. It is generally thought that this was theBrahmasphuta Siddhanta, although it may have been theSurya Siddhanata. A few years later, perhaps about 775, thisSiddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrologicalTetrabiblos was translated into Arabic from the Greek.
^Plofker 2009 pp. 197–98; George Gheverghese Joseph,The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, "Indian Mathematics", pp. 118–30 inCompanion History of the History and Philosophy of the Mathematical Sciences, ed. I. Grattan. Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126.
^Divakaran, P. P. (2007). "The first textbook of calculus: Yukti-bhāṣā",Journal of Indian Philosophy 35, pp. 417–33.
^Almeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001). "Keralese mathematics: its possible transmission to Europe and the consequential educational implications".Journal of Natural Geometry.20 (1):77–104.
^Pingree, David (December 1992). "Hellenophilia versus the History of Science".Isis.83 (4):554–563.Bibcode:1992Isis...83..554P.doi:10.1086/356288.JSTOR234257.S2CID68570164.One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English byCharles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore theTransactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the serieswithout the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.
^Plofker, Kim (November 2001)."The 'Error' in the Indian "Taylor Series Approximation" to the Sine".Historia Mathematica.28 (4): 293.doi:10.1006/hmat.2001.2331.It is not unusual to encounter in discussions of Indian mathematics such assertions as that 'the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)' [Joseph 1991, 300], or that 'we may consider Madhava to have been the founder of mathematical analysis' (Joseph 1991, 293), or that Bhaskara II may claim to be 'the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus' (Bag 1979, 294).... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285))... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian 'discovery of the principle of the differential calculus' somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential 'principle' was not generalized to arbitrary functions – in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here
^Abdel Haleem, Muhammad A. S. "The Semitic Languages",https://doi.org/10.1515/9783110251586.811, "Arabic became the language of scholarship in science and philosophy in the 9th century when the ‘translation movement’ saw concerted work on translations of Greek, Indian, Persian and Chinese, medical, philosophical and scientific texts", p. 811.
^(Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions."
^Gandz and Saloman (1936). "The sources of Khwarizmi's algebra",Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
^(Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the termsal-jabr andmuqabalah mean, but the usual interpretation is similar to that implied in the translation above. The wordal-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 wordmuqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
^Caldwell, John (1981). "TheDe Institutione Arithmetica and theDe Institutione Musica", pp. 135–54 inMargaret Gibson, ed.,Boethius: His Life, Thought, and Influence, (Oxford: Basil Blackwell).
^Folkerts, Menso (1970)."Boethius" Geometrie II, Wiesbaden: Franz Steiner Verlag.
^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).
^Beaujouan, Guy. "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable,Renaissance and Renewal in the Twelfth Century. Cambridge: Harvard University Press, 1982.
^Singh, Parmanand (1985). "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7
^Grant, Edward and John E. Murdoch, eds. (1987).Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages. Cambridge: Cambridge University Press.ISBN0-521-32260-X.
^Clagett, Marshall (1961).The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 421–40.
^Murdoch, John E. (1969). "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", inArts libéraux et philosophie au Moyen Âge (Montréal: Institut d'Études Médiévales), pp. 224–27.
^Clagett, Marshall (1961).The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 210, 214–15, 236.
^Clagett, Marshall (1961).The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, p. 284.
^Clagett, Marshall (1961)The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 332–45, 382–91.
^Oresme, Nicole. "Questions on theGeometry 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.
^Heeffer, Albrecht:On the curious historical coincidence of algebra and double-entry bookkeeping, Foundations of the Formal Sciences,Ghent University, November 2009, p. 7[2]
^della Francesca, Piero.Trattato d'Abaco, ed. G. Arrighi, Pisa (1970).
^della Francesca, Piero (1916).L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini, Rome.
^Eves, Howard (1990).An Introduction to the History of Mathematics, Saunders.ISBN0-03-029558-0, 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."
^Howard Eves, An Introduction to the History of Mathematics, 6th edition, 1990, "In the nineteenth century, mathematics underwent a great forward surge ... . The new mathematics began to free itself from its ties to mechanics and astronomy, and a purer outlook evolved." p. 493
^Mari, C. (2012). George Boole.Great Lives from History: Scientists & Science, N.PAG. Salem Press. https://search.ebscohost.com/login.aspx?AN=176953509
^The Riemann integral was introduced in Riemann's paper "On the representability of a function by a trigonometric series". It was published in 1868 inAbhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, pages 87-132.
^Suppes, Patrick (1972).Axiomatic Set Theory. Dover. p. 1.ISBN9780486616308.With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.... As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.
^Marquis, Jean-Pierre (2023),"Category Theory", in Zalta, Edward N.; Nodelman, Uri (eds.),The Stanford Encyclopedia of Philosophy (Fall 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved2024-04-23
^Poincaré, Henri (2017).The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory. Translated by Popp, Bruce D. Cham, Switzerland: Springer International Publishing.ISBN978-3-319-52898-4.
^Weisstein, Eric W."Hyperreal Number".mathworld.wolfram.com. Retrieved2024-03-20.
^O'Connor, J.J.; Robertson, E.F. (June 2004),"John Horton Conway",School of Mathematics and Statistics, University of St Andrews, Scotland, archived fromthe original on 14 March 2008, retrieved2008-01-24
^Mojżesz Presburger and Dale Jacquette (1991). "On the Completeness of a Certain System of Arithmetic of Whole Numbers in Which Addition Occurs as the Only Operation".History and Philosophy of Logic.12 (2):225–33.doi:10.1080/014453409108837187.
^Wolfram, Stephen (2002).A New Kind of Science. Wolfram Media, Inc. p. 1152.ISBN1-57955-008-8.
^Douglas Hofstadter (1979).Gödel, Escher, Bach: an Eternal Golden Braid. New York: Basic Books.ISBN0-465-02656-7. Here:Introduction /Consistency, completeness, Hilbert's program; "Gödel published his work which in some sense completely destroyed Hilbert's program."
^Alexandrov, Pavel S. (1981), "In Memory of Emmy Noether", in Brewer, James W; Smith, Martha K (eds.),Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, pp. 99–111,ISBN978-0-8247-1550-2.
Joyce, Hetty (July 1979), "Form, Function and Technique in the Pavements of Delos and Pompeii",American Journal of Archaeology,83 (3):253–63,doi:10.2307/505056,JSTOR505056,S2CID191394716.
Katz, Victor J. (2007),The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press,ISBN978-0-691-11485-9
Needham, Joseph;Wang, Ling (1995) [1959],Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, Cambridge: Cambridge University Press,ISBN978-0-521-05801-8
Needham, Joseph; Wang, Ling (2000) [1965],Science and Civilization in China: Physics and Physical Technology: Mechanical Engineering, vol. 4 (reprint ed.), Cambridge: Cambridge University Press,ISBN978-0-521-05803-2
Straffin, Philip D. (1998), "Liu Hui and the First Golden Age of Chinese Mathematics",Mathematics Magazine,71 (3):163–81,doi:10.1080/0025570X.1998.11996627
Volkov, Alexei (2009), "Mathematics and Mathematics Education in Traditional Vietnam", in Robson, Eleanor; Stedall, Jacqueline (eds.),The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, pp. 153–76,ISBN978-0-19-921312-2
MacTutor History of Mathematics archive (John J. O'Connor and Edmund F. Robertson; University of St Andrews, Scotland). An award-winning website containing detailed biographies on many historical and contemporary mathematicians, as well as information on notable curves and various topics in the history of mathematics.
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