History of mathematics

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A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.[1]
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The history of mathematics deals with the origin of discoveries in

calendars
.

The earliest mathematical texts available are from

Egyptian c. 1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem
seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the

zero was given a standard symbol in Maya numerals
.

Many Greek and Arabic texts on mathematics were

Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[11] Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus
during the course of the 17th century.

Table of numerals
European (descended from the West Arabic) 0 1 2 3 4 5 6 7 8 9
Arabic-Indic ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩
Eastern Arabic-Indic (Persian and Urdu) ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹
Devanagari (Hindi) ० १ २ ३ ४ ५ ६ ७ ८ ९
Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯
Chinese – Japanese 零 一 二 三 四 五 六 七 八 九
Tamil ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Prehistoric

The origins of mathematical thought lie in the concepts of number, patterns in nature, magnitude, and form.[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 in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.[12]

The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers[13][failed verification] or a six-month lunar calendar.[14] 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."[15] The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[16]

Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.[17] All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.[18]

Babylonian

Main article: Babylonian mathematics
See also: Plimpton 322

Islamic mathematics
.

Geometry problem on a clay tablet belonging to a school for scribes; Susa, first half of the 2nd millennium BCE

In contrast to the sparsity of sources 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.[22]

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.[23] From around 2500 BC onward, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[24]

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

Babylonian mathematics were written using a sexagesimal (base-60) numeral system.[21] 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,[21] 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.[25] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the Renaissance, and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.[26] 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.[20] 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.[20] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[20]

Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.[27] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.[28] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.[29] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.[22]

Egyptian

Main article:
Egyptian mathematics
Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.

Islamic mathematics, when Arabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[30] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[31]

The most extensive Egyptian mathematical text is the

arithmetic and geometric series.[36]

Another significant Egyptian mathematical text is the

Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.[37] It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum
(truncated pyramid).

Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation.[38]

Greek

Main article: Greek mathematics
Pythagoreans
are generally credited with the first proof of the theorem.

Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.[39] 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.[40]

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

mathematical rigor to prove them.[41]

Greek mathematics is thought to have begun with

Egyptian and Babylonian mathematics
. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.

Thales used

Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum).[48] The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica.[49]

Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.[50] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus (c. 390 - c. 340 BC), came.[51] Plato also discussed the foundations of mathematics,[52] clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.[53] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.[51]

Eudoxus developed the

incommensurable magnitudes.[55] The former allowed the calculations of areas and volumes of curvilinear figures,[56] while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384–c. 322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.[57]

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[58]

In the 3rd century BC, the premier center of mathematical education and research was the

conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[62]

Archimedes used the method of exhaustion to approximate the value of pi.

spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid),[64] and an ingenious method of exponentiation for expressing very large numbers.[66] 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.[67] 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.[68]

conic sections
.

conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[69] He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").[70] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[71] 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.[72]

Around the same time,

Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.[78] Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.[79]

Claude Gaspard Bachet de Méziriac
.

Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.

Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.[82] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).[83] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.[82]

The Hagia Sophia was designed by mathematicians Anthemius of Tralles and Isidore of Miletus.

Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.[84] Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.

The first woman mathematician recorded by history was

Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.[87] 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.[88]

Roman

Further information: Roman abacus and Roman numerals
surveyor (gromatici), found at the site of Aquincum, modern Budapest, Hungary

Although

theoretical mathematics and geometry that were prized by the Greeks.[91] It is unclear if the Romans first derived their numerical system directly from the Greek precedent or from Etruscan numerals used by the Etruscan civilization centered in what is now Tuscany, central Italy.[92]

Using calculation, Romans were adept at both instigating and detecting financial

architecture such as bridges, road-building, and preparation for military campaigns.[95] Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.[96][97]

The creation of the

leap day every four years in a 365-day cycle.[100] This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII (r. 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.[101]

At roughly the same time,

cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.[104]

Chinese

Main article: Chinese mathematics
Further information: Book on Numbers and Computation
See also: History of science § Chinese mathematics
Warring States
period

An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.

Warring States Period appears reasonable.[106] However, the Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.[47]

Counting rod numerals

Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[107] 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.[108] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's Supplementary Notes on the Art of Figures.

The oldest extant work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[109] It also defined the concepts of circumference, diameter, radius, and volume.[110]

The Nine Chapters on the Mathematical Art, one of the earliest surviving mathematical texts from China (2nd century AD).

In 212 BC, the Emperor

Chinese pagoda towers, engineering, surveying, and includes material on right triangles.[106] It created mathematical proof for the Pythagorean theorem,[111] and a mathematical formula for Gaussian elimination.[112] The treatise also provides values of π,[106] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,[113] as well as 3.162 by taking the square root of 10.[114][115] Liu Hui commented on the Nine Chapters in the 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159).[116][117] Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.[116][118] He also established a method which would later be called Cavalieri's principle to find the volume of a sphere.[119]

The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.[116] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[120] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).[120]

Even after European mathematics began to flourish during the

Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.[120]

East Asian cultural sphere.[121] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644).[122] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.[123] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.[124]

Indian

Main article: Indian mathematics
Further information: History of science § Indian mathematics
See also: History of the Hindu–Arabic numeral system
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Numerals evolution in India
Indian numerals in stone and copper inscriptions[125]
Brahmi numerals
Ancient Brahmi numerals in a part of India

The earliest civilization on the Indian subcontinent is the

Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.[126]

The oldest extant mathematical records from India are the

Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD),[127] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.[128] As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.[127] The Sulba Sutras give methods for constructing a circle with approximately the same area as a given square, which imply several different approximations of the value of π.[129][130][a] In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem.[130] All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.[127] 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.[127]

Fibonacci numbers (called mātrāmeru).[135]

The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (

Gupta period) showing strong Hellenistic influence.[136] 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.[137] Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".[137]

Explanation of the sine rule in Yuktibhāṣā

Around 500 AD,

Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".[139]

In the 7th century,

Brahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.[citation needed
]

In the 12th century, Bhāskara II,[141] 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.[142].[143]In the 14th century, Narayana Pandita completed his Ganita Kaumudi.[144]

Also in the 14th century,

Jesuit missionaries and traders who were active around the ancient port of Muziris at the time and, as a result, directly influenced later European developments in analysis and calculus.[148] However, other scholars argue that the Kerala School did not formulate a systematic theory of differentiation and integration, and that there is not any direct evidence of their results being transmitted outside Kerala.[149][150][151][152]

Islamic empires

Main article:
Mathematics in medieval Islam
See also: History of the Hindu–Arabic numeral system
Muhammad ibn Mūsā al-Khwārizmī
(c. AD 820)

The

Arabs, 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.[153]

In the 9th century, the Persian mathematician

exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[157]

In Egypt,

irrational numbers, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.[158]
His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.

Further developments in algebra were made by

fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[161]

In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[162]

In the 13th century,

Ruffini and Horner
.

Other achievements of Muslim mathematicians during this period include the addition of the

al-Qalasādī.[163]

During the time of the

Safavid Empire
from the 15th century, the development of Islamic mathematics became stagnant.

Maya

The Maya numerals for numbers 1 through 19, written in the Maya script

In the

zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[164]

Medieval European

Further information: List of medieval European scientists and European science in the Middle Ages
See also: Latin translations of the 12th century

Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight.[165]

Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.[166][167]

In the 12th century, European scholars traveled to Spain and Sicily

The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.[168][169]
These and other new sources sparked a renewal of mathematics.

Leonardo of Pisa, now known as

Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that)[170]
which Fibonacci used as an unremarkable example.

The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.[171] 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).

Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.[173]

Nicole Oresme (1323–1382), shown in this contemporary illuminated manuscript with an armillary sphere in the foreground, was the first to offer a mathematical proof for the divergence of the harmonic series.[174]
Adam Ries is known as the "father of modern calculating" because of his decisive contribution to the recognition that Roman numerals are unpractical and to their replacement by the considerably more practical Arabic numerals.[175]

One of the 14th-century

William Heytesbury, lacking differential calculus and the concept of 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".[176]

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]".[177]

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.[178] In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.[179]

Renaissance

Further information: Mathematics and art

During the Renaissance, the development of mathematics and of accounting were intertwined.[180] While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.

De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De quinque corporibus regularibus (On the Five Regular Solids).[181][182][183]

Portrait of Luca Pacioli, a painting traditionally attributed to Jacopo de' Barbari, 1495, (Museo di Capodimonte).

plus and minus
for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.

In Italy, during the first half of the 16th century,

Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities
that could appear in Cardano's formula for solving cubic equations.

real number system.[185][186]

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.[187]

During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that involved, were studied intensely.[188]

Mathematics during the Scientific Revolution

See also: Scientific Revolution

17th century

Gottfried Wilhelm Leibniz

The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe.

Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based Hans Lipperhey's. Tycho Brahe had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion.[189]
The
Cartesian coordinates
.

Building on earlier work by many predecessors,

Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, developed calculus and much of the calculus notation still in use today. He also refined the binary number system, which is the foundation of nearly all digital (electronic, solid-state, discrete logic) computers, including the Von Neumann architecture, which is the standard design paradigm, or "computer architecture", followed from the second half of the 20th century, and into the 21st. Leibniz has been called the "founder of computer science".[190]

Science and mathematics had become an international endeavor, which would soon spread over the entire world.[191]

In addition to the application of mathematics to the studies of the heavens,

utility theory
in the 18th–19th century.

18th century

Leonhard Euler

The most influential mathematician of the 18th century was arguably Leonhard Euler (1707–1783). His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol i, 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.

Other important European mathematicians of the 18th century included

Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Pierre-Simon Laplace, who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics
.

Modern

19th century

Carl Friedrich Gauss

Throughout the 19th century mathematics became increasingly abstract.

quadratic reciprocity law.[citation needed
]

Behavior of lines with a common perpendicular in each of the three types of geometry

This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician

Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces, and set the mathematical foundations for the theory of general relativity.[193]

The 19th century saw the beginning of a great deal of

noncommutative algebra.[citation needed] The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science.[citation needed
] Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.[citation needed]

Also, for the first time, the limits of mathematics were explored.

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.[citation needed] Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.[citation needed] On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.[citation needed
]

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

In the later 19th century,

A.N. Whitehead, initiated a long running debate on the foundations of mathematics.[citation needed
]

The 19th century saw the founding of a number of national mathematical societies: the

Société Mathématique de France in 1872,[196] the Circolo Matematico di Palermo in 1884,[197][198] the Edinburgh Mathematical Society in 1883,[199] and the American Mathematical Society in 1888.[200] The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.[201]

In 1897, Kurt Hensel introduced p-adic numbers.[202]

20th century

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.

Klein's encyclopedia.[204]

In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics.[205] These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.[citation needed]

Four Color Theorem

Notable historical conjectures were finally proven. In 1976,

standard axioms of set theory.[208] In 1998, Thomas Callister Hales proved the Kepler conjecture, also using a computer.[209]

Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.[210] A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.[211]

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with relativistic precession of apsides

Grothendieck and Serre recast algebraic geometry using sheaf theory.[citation needed] Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s.[citation needed
]
fractals.[citation needed] Lie theory with its Lie groups and Lie algebras became one of the major areas of study.[citation needed
]

combinatorial games.[citation needed
]

The development and continual improvement of

RSA algorithm of public-key cryptography.[citation needed
]

At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved[

cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.[citation needed
]

The absolute value of the Gamma function on the complex plane

One of the more colorful figures in 20th-century mathematics was

asymptotics,[214] and mock theta functions.[212] He also made major investigations in the areas of gamma functions,[215][216] modular forms,[212] divergent series,[212] hypergeometric series[212] and prime number theory.[212]

Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.[218][219]

Emmy Noether has been described by many as the most important woman in the history of mathematics.[220] She studied the theories of rings, fields, and algebras.[221]

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century, there were hundreds of specialized areas in mathematics, and the

mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.[citation needed
]

21st century

See also: List of unsolved problems in mathematics § Problems solved since 1995

In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems.[223] In 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award, as he was critical of the mathematics establishment).[224]

Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched.[

open access publishing, first made popular by arXiv.[citation needed
]

Future

Main article: Future of mathematics

There are many observable trends in mathematics, the most notable being that the subject is growing ever larger as computers are ever more important and powerful; the volume of data being produced by science and industry, facilitated by computers, continues expanding exponentially. As a result, there is a corresponding growth in the demand for mathematics to help process and understand this big data.[225] Math science careers are also expected to continue to grow, with the US Bureau of Labor Statistics estimating (in 2018) that "employment of mathematical science occupations is projected to grow 27.9 percent from 2016 to 2026."[226]

See also

Notes

  1. ^ The approximate values for π are 4 x (13/15)2 (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389)
  1. ^ a b (Boyer 1991, "Euclid of Alexandria" p. 119)
  2. ^ 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.
  3. PMID 14884919. {{cite book}}: |journal= ignored (help
    )     Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.
  4. S2CID 3994109
    .
  5. ^ 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."
  6. ^ a b Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48.
  7. ^ Ifrah, Georges (1986). Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37.
  8. ^ Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.
  9. ^ "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
  10. Juschkewitsch, A. P.
    (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.
  11. ^ 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.
  12. ^ a b (Boyer 1991, "Origins" p. 3)
  13. ^ Williams, Scott W. (2005). "The Oldest Mathematical Object is in Swaziland". Mathematicians of the African Diaspora. SUNY Buffalo mathematics department. Retrieved 2006-05-06.
  14. ^ Marshack, Alexander (1991). The Roots of Civilization, Colonial Hill, Mount Kisco, NY.
  15. ISBN 978-1-59102-477-4
    .
  16. ^ Marshack, A. (1972). The Roots of Civilization: the Cognitive Beginning of Man's First Art, Symbol and Notation. New York: McGraw-Hill.
  17. ISBN 0-521-33381-4
    .
  18. ISBN 0792338162
    . Retrieved 2019-08-17.
  19. ^ (Boyer 1991, "Mesopotamia" p. 24)
  20. ^ a b c d (Boyer 1991, "Mesopotamia" p. 26)
  21. ^ a b c (Boyer 1991, "Mesopotamia" p. 25)
  22. ^ a b (Boyer 1991, "Mesopotamia" p. 41)
  23. ISBN 978-0-203-09660-4
    , retrieved 2023-07-07
  24. ^ Melville, Duncan J. (2003). Third Millennium Chronology Archived 2018-07-07 at the Wayback Machine, Third Millennium Mathematics. St. Lawrence University.
  25. ^ Powell, M. (1976), "The Antecedents of Old Babylonian Place Notation and the Early History of Babylonian Mathematics" (PDF), Historia Mathematica, vol. 3, pp. 417–439, retrieved July 6, 2023
  26. ^ (Boyer 1991, "Mesopotamia" p. 27)
  27. Aaboe, Asger
    (1998). Episodes from the Early History of Mathematics. New York: Random House. pp. 30–31.
  28. ^ (Boyer 1991, "Mesopotamia" p. 33)
  29. ^ (Boyer 1991, "Mesopotamia" p. 39)
  30. ISBN 0813526140
    .
  31. ^ Eglash, R. (1995). "Fractal Geometry in African Material Culture". Symmetry: Culture and Science. 6–1: 174–177.
  32. ^ (Boyer 1991, "Egypt" p. 11)
  33. ^ Egyptian Unit Fractions at MathPages
  34. ^ Egyptian Unit Fractions
  35. ^ "Egyptian Papyri". www-history.mcs.st-andrews.ac.uk.
  36. ^ "Egyptian Algebra – Mathematicians of the African Diaspora". www.math.buffalo.edu.
  37. ^ (Boyer 1991, "Egypt" p. 19)
  38. ^ "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". www.math.buffalo.edu.
  39. ISBN 0-03-029558-0
  40. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 99)
  41. ^ 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.
  42. ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
  43. ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 49)
  44. ISBN 0-03-029558-0
    .
  45. ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
  46. JSTOR 3026893
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  47. ^
    S2CID 130132289
    . Retrieved 15 September 2014.
  48. ISBN 0-486-20429-4
    , pp. 58, 129.
  49. ISBN 0-486-20429-4
    , p. 129.
  50. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 86)
  51. ^ a b (Boyer 1991, "The Age of Plato and Aristotle" p. 88)
  52. ^ Calian, George F. (2014). "One, Two, Three… A Discussion on the Generation of Numbers" (PDF). New Europe College. Archived from the original (PDF) on 2015-10-15.
  53. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 87)
  54. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 92)
  55. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 93)
  56. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 91)
  57. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 98)
  58. Bill Casselman. "One of the Oldest Extant Diagrams from Euclid"
    . University of British Columbia. Retrieved 2008-09-26.
  59. ^ (Boyer 1991, "Euclid of Alexandria" p. 100)
  60. ^ a b (Boyer 1991, "Euclid of Alexandria" p. 104)
  61. The Bible
    , has been more widely used..."
  62. ^ (Boyer 1991, "Euclid of Alexandria" p. 102)
  63. ^ (Boyer 1991, "Archimedes of Syracuse" p. 120)
  64. ^ a b (Boyer 1991, "Archimedes of Syracuse" p. 130)
  65. ^ (Boyer 1991, "Archimedes of Syracuse" p. 126)
  66. ^ (Boyer 1991, "Archimedes of Syracuse" p. 125)
  67. ^ (Boyer 1991, "Archimedes of Syracuse" p. 121)
  68. ^ (Boyer 1991, "Archimedes of Syracuse" p. 137)
  69. ^ (Boyer 1991, "Apollonius of Perga" p. 145)
  70. ^ (Boyer 1991, "Apollonius of Perga" p. 146)
  71. ^ (Boyer 1991, "Apollonius of Perga" p. 152)
  72. ^ (Boyer 1991, "Apollonius of Perga" p. 156)
  73. ^ (Boyer 1991, "Greek Trigonometry and Mensuration" p. 161)
  74. ^ a b (Boyer 1991, "Greek Trigonometry and Mensuration" p. 175)
  75. ^ (Boyer 1991, "Greek Trigonometry and Mensuration" p. 162)
  76. ISBN 1-904275-25-7
  77. ^ (Boyer 1991, "Greek Trigonometry and Mensuration" p. 163)
  78. ^ (Boyer 1991, "Greek Trigonometry and Mensuration" p. 164)
  79. ^ (Boyer 1991, "Greek Trigonometry and Mensuration" p. 168)
  80. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178)
  81. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180)
  82. ^ a b (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 181)
  83. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 183)
  84. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 183–90)
  85. ^ "Internet History Sourcebooks Project". sourcebooks.fordham.edu.
  86. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 190–94)
  87. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 193)
  88. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 194)
  89. ^ (Goodman 2016, p. 119)
  90. ^ (Cuomo 2001, pp. 194, 204–06)
  91. ^ (Cuomo 2001, pp. 192–95)
  92. ^ (Goodman 2016, pp. 120–21)
  93. ^ (Cuomo 2001, p. 196)
  94. ^ (Cuomo 2001, pp. 207–08)
  95. ^ (Goodman 2016, pp. 119–20)
  96. ^ (Tang 2005, pp. 14–15, 45)
  97. ^ (Joyce 1979, p. 256)
  98. ^ (Gullberg 1997, p. 17)
  99. ^ (Gullberg 1997, pp. 17–18)
  100. ^ (Gullberg 1997, p. 18)
  101. ^ (Gullberg 1997, pp. 18–19)
  102. ^ (Needham & Wang 2000, pp. 281–85)
  103. ^ (Needham & Wang 2000, p. 285)
  104. ^ (Sleeswyk 1981, pp. 188–200)
  105. ^ (Boyer 1991, "China and India" p. 201)
  106. ^ a b c (Boyer 1991, "China and India" p. 196)
  107. ^ Katz 2007, pp. 194–99
  108. ^ (Boyer 1991, "China and India" p. 198)
  109. ^ (Needham & Wang 1995, pp. 91–92)
  110. ^ (Needham & Wang 1995, p. 94)
  111. ^ (Needham & Wang 1995, p. 22)
  112. ^ (Straffin 1998, p. 164)
  113. ^ (Needham & Wang 1995, pp. 99–100)
  114. ^ (Berggren, Borwein & Borwein 2004, p. 27)
  115. ^ (de Crespigny 2007, p. 1050)
  116. ^ a b c (Boyer 1991, "China and India" p. 202)
  117. ^ (Needham & Wang 1995, pp. 100–01)
  118. ^ (Berggren, Borwein & Borwein 2004, pp. 20, 24–26)
  119. ISBN 978-0-7637-5995-7. Extract of p. 27
  120. ^ a b c (Boyer 1991, "China and India" p. 205)
  121. ^ (Volkov 2009, pp. 153–56)
  122. ^ (Volkov 2009, pp. 154–55)
  123. ^ (Volkov 2009, pp. 156–57)
  124. ^ (Volkov 2009, p. 155)
  125. ^ 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 ≡."
  126. ^ (Boyer 1991, "China and India" p. 206)
  127. ^ a b c d (Boyer 1991, "China and India" p. 207)
  128. ISBN 978-1-4020-0260-1
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  129. ^ Kulkarni, R.P. (1978). "The Value of π known to Śulbasūtras" (PDF). Indian Journal of History of Science. 13 (1): 32–41. Archived from the original (PDF) on 2012-02-06.
  130. ^ a b Connor, J.J.; Robertson, E.F. "The Indian Sulbasutras". Univ. of St. Andrew, Scotland.
  131. S2CID 115779583
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  132. S2CID 52885600
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  133. ISBN 978-0-8493-7189-9
    .
  134. ISBN 0-387-94544-X
  135. S2CID 3637061
    .
  136. ^ (Boyer 1991, "China and India" p. 208)
  137. ^ a b (Boyer 1991, "China and India" p. 209)
  138. ^ (Boyer 1991, "China and India" p. 210)
  139. ^ (Boyer 1991, "China and India" p. 211)
  140. ISBN 9780471543978. By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos
    was translated into Arabic from the Greek.
  141. ^ Plofker 2009 182–207
  142. ISBN 0-471-18082-3
    .
  143. ^ Plofker 2009 pp. 197–98; George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, "Indian Mathematics", pp. 118–30 in Companion History of the History and Philosophy of the Mathematical Sciences, ed. I. Grattan. Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126.
  144. ^ "Narayana - Biography". Maths History. Retrieved 2022-10-03.
  145. ^ Plofker 2009 pp. 217–53.
  146. ^ Raju, C. K. (2001). "Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā" (PDF). Philosophy East & West. 51 (3): 325–362.
    S2CID 170341845
    . Retrieved 2020-02-11.
  147. ^ Divakaran, P. P. (2007). "The first textbook of calculus: Yukti-bhāṣā", Journal of Indian Philosophy 35, pp. 417–33.
  148. ^ Almeida, D. F.; J. K. John and A. Zadorozhnyy (2001). "Keralese mathematics: its possible transmission to Europe and the consequential educational implications". Journal of Natural Geometry. 20 (1): 77–104.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  149. S2CID 68570164. One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish
    , in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.
  150. JSTOR 1558972
    .
  151. 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
  152. JSTOR 2691411
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  153. ^ 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.
  154. ^ (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."
  155. ^ 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".
  156. ^ (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
  157. OCLC 29181926
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  158. ^ Sesiano, Jacques (1997). "Abū Kāmil". Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. pp. 4–5.
  159. ^ (Katz 1998, pp. 255–59)
  160. ^ Woepcke, F. (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
  161. JSTOR 2691411
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  162. ^ Alam, S (2015). "Mathematics for All and Forever" (PDF). Indian Institute of Social Reform & Research International Journal of Research.
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  164. ^ a b c d (Goodman 2016, p. 121)
  165. ^ Wisdom, 11:20
  166. ^ Caldwell, John (1981). "The De Institutione Arithmetica and the De Institutione Musica", pp. 135–54 in Margaret Gibson, ed., Boethius: His Life, Thought, and Influence, (Oxford: Basil Blackwell).
  167. ^ Folkerts, Menso (1970). "Boethius" Geometrie II, Wiesbaden: Franz Steiner Verlag.
  168. ^ 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).
  169. ^ 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.
  170. ^ 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
  171. ISBN 0-521-32260-X
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  172. ^ Clagett, Marshall (1961). The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 421–40.
  173. ^ Murdoch, John E. (1969). "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in Arts libéraux et philosophie au Moyen Âge (Montréal: Institut d'Études Médiévales), pp. 224–27.
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