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For example, the Zhoubi Suanjing dates around � BC, yet many scholars believed it was written between and BC. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem a special case of the Pythagorean Theorem but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips , dated c. The abacus was first mentioned in the Mathematics 10th Standard Industrial Classification second century BC, alongside 'calculation with rods' suan zi in which small bamboo sticks are placed in successive squares of a checkerboard.
Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars , circa � BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering.
It is certain that one of the greatest feats of human history, the Great Wall of China , required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion. Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy , according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise. In the Han Dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called chousuan , consisting of only nine symbols with a blank space on the counting board representing zero.
From documentary evidence this tomb is known to have been closed in BC, early in the Western Han dynasty. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art.
Problems are set up with questions immediately followed by answers and procedure. The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of problems. The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from Yongle Encyclopedia , he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations.
The final version of Dai Zhen's work would come in , titled Ripple Pavilion , with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters. Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle.
This calculation would be discovered in Europe during the 16th century. There is no explicit method or record of how he calculated this estimate. Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty.
The Book of Computations is the first known text to solve systems of equations with two unknowns. Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns.
Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Liu Hui 's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available.
Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint. In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem already known by the 9 chapters , and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium.
He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE. In the fourth century, another influential mathematician named Zu Chongzhi , introduced the Da Ming Li.
This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui , we now know that Zu Chongzhi was one of the generations of mathematicians.
He used Liu Hui's pi-algorithm applied to a gon and obtained a value of pi to 7 accurate decimal places between 3. Along with his son, Zu Geng, Zu Chongzhi applied the Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi.
His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics.
A mathematical manual called Sunzi mathematical classic dated between and CE contained the most detailed step by step description of multiplication and division algorithm with counting rods.
Intriguingly, Sunzi may have influenced the development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi.
Khwarizmi's presentation is almost identical to the division algorithm in Sunzi , even regarding stylistic matters for example, using blank spaces to represent trailing zeros ; the similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China.
In the fifth century the manual called " Zhang Qiujian suanjing " discussed linear and quadratic equations. By this point the Chinese had the concept of negative numbers. By the Tang Dynasty study of mathematics was fairly standard in the great schools. The Sui dynasty and Tang dynasty ran the "School of Computations".
Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty , and he wrote a book: Jigu Suanjing Continuation of Ancient Mathematics , where numerical solutions which general cubic equations appear for the first time [30]. The Tibetans obtained their first knowledge of mathematics arithmetic from China during the reign of Nam-ri srong btsan , who died in The table of sines by the Indian mathematician , Aryabhata , were translated into the Chinese mathematical book of the Kaiyuan Zhanjing , compiled in AD during the Tang Dynasty.
Yi Xing , the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha , while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. Northern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner - Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations.
Yang Hui was also the first person in history to discover and prove " Pascal's Triangle ", along with its binomial proof although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD.
His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians.
Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie 's two books Suanxue qimeng and the Siyuan yujian. In one case he reportedly gave a method equivalent to Gauss 's pivotal condensation. Qin Jiushao c. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?
He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used fan fa , or Horner's method , to solve equations of degree as high as six, although he did not describe his method of solving equations.
Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner.
Others who used the Horner method were Ch'in Chiu-shao ca. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations.
It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa , today called Horner's method , to solve these equations. There are many summation series equations given without proof in the Mirror. A few of the summation series are: [44].
Shu-shu chiu-chang , or Mathematical Treatise in Nine Sections , was written by the wealthy governor and minister Ch'in Chiu-shao ca. The earliest known magic squares of order greater than three are attributed to Yang Hui fl. The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty � , where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.
Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until , with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi � and the Italian Jesuit Matteo Ricci � After the overthrow of the Yuan Dynasty , China became suspicious of Mongol-favored knowledge.
The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes:. At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science.
Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century.
Correspondingly, scholars paid less attention to mathematics; pre-eminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the Tian yuan shu Increase multiply method.
To the average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art , he omitted Tian yuan shu and the increase multiply method. Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its suan pan form.
Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. Zhusuan , the arithmetic calculation through abacus, inspired multiple new works. Suanfa Tongzong General Source of Computational Methods , a volume work published in by Cheng Dawei , remained in use for over years. Although this switch from counting rods to the abacus allowed for reduced computation times, it may have also led to the stagnation and decline of Chinese mathematics.
The pattern rich layout of counting rod numerals on counting boards inspired many Chinese inventions in mathematics, such as the cross multiplication principle of fractions and methods for solving linear equations. Similarly, Japanese mathematicians were influenced by the counting rod numeral layout in their definition of the concept of a matrix.
Algorithms for the abacus did not lead to similar conceptual advances. This distinction, of course, is a modern one: until the 20th century, Chinese mathematics was exclusively a computational science. In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi , he was able to translate Euclid's Elements using the same techniques used to teach classical Buddhist texts.
Under the Western-educated Kangxi Emperor , Chinese mathematics enjoyed a brief period of official support. Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending , Goucheng's grandfather. However, no sooner were the encyclopedias published than the Yongzheng Emperor acceded to the throne. Yongzheng introduced a sharply anti-Western turn to Chinese policy, and banished most missionaries from the Court.
With access to neither Western texts nor intelligible Chinese ones, Chinese mathematics stagnated. In , the First Opium War forced China to open its door and looked at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries.
In , the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements and 13 volumes on Algebra. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? The earliest mathematical texts available are from Mesopotamia and Egypt � Plimpton Babylonian c.
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. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe.
From ancient times through the Middle Ages , periods of mathematical discovery were often followed by centuries of stagnation. 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.
At the end of the 19th century the International Congress of Mathematicians was founded and continues to spearhead advances in the field. The origins of mathematical thought lie in the concepts of number , patterns in nature , magnitude , and form. 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. Prehistoric artifacts discovered in Africa, dated 20, years old or more suggest early attempts to quantify time. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers [12] or a six-month lunar calendar.
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 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.
Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia modern Iraq from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. Later under the Arab Empire , Mesopotamia, especially Baghdad , once again became an important center of study for Islamic mathematics. In contrast to the sparsity of sources in Egyptian mathematics , knowledge of Babylonian mathematics is derived from more than clay tablets unearthed since the s.
Some of these appear to be graded homework. 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 BC. From around 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.
Babylonian mathematics were written using a sexagesimal base numeral system. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers , and their reciprocal pairs.
Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period , Greek replaced Egyptian as the written language of Egyptian scholars.
Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics , when Arabic became the written language of Egyptian scholars. The most extensive Egyptian mathematical text is the Rhind papyrus sometimes also called the Ahmes Papyrus after its author , dated to c.
In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, [28] including composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory namely, that of the number 6.
Another significant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period, dated to c. 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 c. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning , that is, repeated observations used to establish rules of thumb.
Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them. Greek mathematics is thought to have begun with Thales of Miletus c. Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics.
According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.
He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. The Pythagoreans are credited with the first proof of the Pythagorean theorem , [39] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.
Eudoxus �c. Though he made no specific technical 10th Standard Mathematics Question Paper mathematical discoveries, Aristotle �c. In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.
Euclid also wrote extensively on other subjects, such as conic sections , optics , spherical geometry , and mechanics, but only half of his writings survive. Archimedes c. Apollonius of Perga c. Around the same time, Eratosthenes of Cyrene c. AD 90� , a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Following a period of stagnation after Ptolemy, the period between and AD is sometimes referred to as the "Silver Age" of Greek mathematics.
His main work was the Arithmetica , a collection of algebraic problems dealing with exact solutions to determinate and indeterminate equations. 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. The first woman mathematician recorded by history was Hypatia of Alexandria AD � She succeeded her father Theon of Alexandria as Librarian at the Great Library [ citation needed ] and wrote many works on applied mathematics.
Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus , the architects of the Hagia Sophia.
Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire , there were no noteworthy native Latin mathematicians in comparison. Using calculation, Romans were adept at both instigating and detecting financial fraud , as well as managing taxes for the treasury. The creation of the Roman calendar also necessitated basic mathematics.
The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included days plus a leap year every other year. At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius c. With each revolution, a pin-and-axle device engaged a tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.
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. 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.
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.
The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD , in Xu Yue 's Supplementary Notes on the Art of Figures.
The oldest existent work on geometry in China comes from the philosophical Mohist canon c. The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date.
After the book burning of BC, the Han dynasty BC� AD produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art , the full title of which appeared by AD , but existed in part under other titles beforehand. It consists of word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying , and includes material on right triangles.
The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty � , with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie � , dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method. Even after European mathematics began to flourish during the Renaissance , European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards.
Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.
Japanese mathematics , Korean mathematics , and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian -based East Asian cultural sphere. The earliest civilization on the Indian subcontinent is the Indus Valley Civilization mature phase: to BC that flourished in the Indus river basin.
Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization. The oldest extant mathematical records from India are the Sulba Sutras dated variously between the 8th century BC and the 2nd century AD , [] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.
As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity. 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. Around AD, Aryabhata wrote the Aryabhatiya , a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.
Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals". In the 7th century, Brahmagupta identified the Brahmagupta theorem , Brahmagupta's identity and Brahmagupta's formula , and for the first time, in Brahma-sphuta-siddhanta , he lucidly explained the use of zero as both a placeholder and decimal digit , and explained the Hindu�Arabic numeral system.
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 the 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. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function.
To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics. Madhava also found the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions.
Although most Islamic texts on mathematics were written in Arabic , most of them were not written by 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. Persians contributed to the world of Mathematics alongside Arabs. His book On the Calculation with Hindu Numerals , written about , along with the work of Al-Kindi , were instrumental in spreading Indian mathematics and Indian numerals to the West.
He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, [] and he was the first to teach algebra in an elementary form and for its own sake. In Egypt, Abu Kamil extended algebra to the set of 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 solutions.
Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri , where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around AD, who used it to prove the binomial theorem , Pascal's triangle , and the sum of integral cubes.
Woepcke, [] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid , and was able to generalize his result for the integrals of polynomials up to the 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. 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. In the 13th century, Nasir al-Din Tusi Nasireddin made advances in spherical trigonometry.
He also wrote influential work on Euclid 's parallel postulate. Kashi also had an algorithm for calculating n th roots, which was a special case of the methods given many centuries later by Ruffini and Horner. During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.
In the Pre-Columbian Americas , the Maya civilization that flourished in Mexico and Central 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. 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.
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.
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 updated in 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 which was used as an unremarkable example within the text.
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems. Thomas Bradwardine proposed that speed V increases in arithmetic proportion as the ratio of force F to resistance R increases in geometric proportion.
One of the 14th-century Oxford Calculators , 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 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]".
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. 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.
During the Renaissance , the development of mathematics and of accounting were intertwined. 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. Piero della Francesca c. It included a page treatise on bookkeeping , "Particularis de Computis et Scripturis" Italian: "Details of Calculation and Recording".
It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons. 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. Gerolamo Cardano published them in his book Ars Magna , together with a solution for the quartic equations , discovered by his student Lodovico Ferrari.
In 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. Simon Stevin 's book De Thiende 'the art of tenths' , first published in Dutch in , contained the first systematic treatment of decimal notation , which influenced all later work on the real number system.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in Regiomontanus's table of sines and cosines was published in 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. 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 on a toy imported from Holland. Tycho Brahe had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky.
By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler succeeded in formulating mathematical laws of planetary motion. Building on earlier work by many predecessors, Isaac Newton discovered the laws of physics explaining Kepler's Laws , and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz , developed calculus and much of the calculus notation still in use today.
Science and mathematics had become an international endeavor, which would soon spread over the entire world. In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal.
Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling.
Pascal, with his wager , attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th�19th century. The most influential mathematician of the 18th century was arguably Leonhard Euler He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
Other important European mathematicians of the 18th century included Joseph Louis Lagrange , who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon , did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss � epitomizes this trend. He did revolutionary work on functions of complex variables , in geometry , and on the convergence of series , leaving aside his many contributions to science.
He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry , where the parallel postulate of Euclidean geometry no longer holds. 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.
The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces , William Rowan Hamilton in Ireland developed noncommutative algebra.
The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra , in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science. Also, for the first time, the limits of mathematics were explored.
Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle , to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.
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