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83-554: 37°58′53″N 23°43′56″E  /  37.9813°N 23.7323°E  / 37.9813; 23.7323 This article about the history of mathematics is a stub . You can help Misplaced Pages by expanding it . This article about an organisation in Greece is a stub . You can help Misplaced Pages by expanding it . This article about a mathematics organization is a stub . You can help Misplaced Pages by expanding it . History of mathematics The history of mathematics deals with

166-528: A circle with approximately the same area as a given square , which imply several different approximations of the value of π. In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem . All of these results are present in Babylonian mathematics, indicating Mesopotamian influence. It is not known to what extent

249-601: A 365-day cycle. 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. At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled,

332-467: A base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China. 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. Thus, the number 123 would be written using the symbol for "1", followed by

415-403: 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, 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

498-485: A collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations . 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). Diophantus also made significant advances in notation,

581-636: A device corresponding to a binary numeral system . His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem . Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru ). 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. They are significant in that they contain

664-595: 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. 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). 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

747-529: A formula for obtaining Pythagorean triples bears his name. Eudoxus developed the method of exhaustion , a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes . The former allowed the calculations of areas and volumes of curvilinear figures, 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

830-578: A result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with

913-533: 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. 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

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996-544: A small number of geometrical theorems as well. It also defined the concepts of circumference , diameter , radius , and volume . In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After

1079-530: A tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10." 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. Predynastic Egyptians of

1162-463: Is also due the systematic use of the 360 degree circle. Heron of Alexandria ( c.  10 –70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria ( c.  100 AD ) pioneered spherical trigonometry through Menelaus' theorem . The most complete and influential trigonometric work of antiquity

1245-515: 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 . Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of

1328-644: Is found on a wax tablet dated to the 1st century AD (now found in the British Museum ). The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica . Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy , in Athens , became

1411-429: Is independent of Western mathematics; To this period belongs the mathematician Seki Takakazu , of great influence, for example, in the development of wasan (traditional Japanese mathematics), and whose discoveries (in areas such as integral calculus ), are almost simultaneous with contemporary European mathematicians such as Gottfried Leibniz . Japanese mathematics of this period is inspired by Chinese mathematics and

1494-635: Is named Babylonian mathematics due to the central role of Babylon as a place of study. 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 400 clay tablets unearthed since the 1850s. Written in Cuneiform script , tablets were inscribed whilst

1577-710: Is oriented towards essentially geometric problems. On wooden tablets called sangaku, "geometric enigmas" are proposed and solved; That's where, for example, Soddy's hexlet theorem comes from. The earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature second phase: 2600 to 1900 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

1660-528: Is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus , Simplicius and Eutocius . Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by

1743-458: Is the Almagest of Ptolemy ( c.  AD 90 –168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. 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. Following a period of stagnation after Ptolemy,

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1826-736: Is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC. It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of both

1909-625: 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

1992-571: The Academy of Athens in 529 AD. 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. Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show

2075-564: The Antikythera mechanism , the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-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

2158-462: The Arithmetica being the first instance of algebraic symbolism and syncopation. 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. Pappus is considered

2241-704: 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 . It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals . Islamic scholars carried knowledge of this number system to Europe by

2324-488: The Confucian -based East Asian cultural sphere . 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). For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed

2407-915: 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. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs. The most extensive Egyptian mathematical text

2490-487: 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 Pythagoreans , who coined the term "mathematics" from the ancient Greek μάθημα ( mathema ), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through

2573-468: The Pythagorean theorem , and a mathematical formula for Gaussian elimination . The treatise also provides values of π , 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, as well as 3.162 by taking the square root of 10. Liu Hui commented on the Nine Chapters in

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2656-569: The Sieve of Eratosthenes and perfect number theory (namely, that of the number 6). It also shows how to solve first order linear equations as well as arithmetic and geometric series . Another significant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period, dated to c. 1890 BC. It consists of what are today called word problems or story problems , which were apparently intended as entertainment. One problem

2739-402: 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 with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual. The Sulba Sutras give methods for constructing

2822-562: The axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today. In addition to

2905-702: The book burning of 212 BC, the Han dynasty (202 BC–220 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 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying , and includes material on right triangles . It created mathematical proof for

2988-456: 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,

3071-538: The lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year , a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February. This calendar was supplanted by the Julian calendar , a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in

3154-519: The opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square. 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 356 days plus a leap year every other year. In contrast,

3237-478: The spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious method of exponentiation for expressing very large numbers. 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. He regarded as his greatest achievement his finding of

3320-579: The theoretical mathematics and geometry that were prized by the Greeks. 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 . Using calculation, Romans were adept at both instigating and detecting financial fraud , as well as managing taxes for

3403-774: The treasury . Siculus Flaccus , one of the Roman gromatici (i.e. land surveyor), wrote the Categories of Fields , which aided Roman surveyors in measuring the surface areas of allotted lands and territories. Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering , including the erection of architecture such as bridges , road-building , and preparation for military campaigns . 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,

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3486-435: The "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry , largely to address the needs of astronomers. Hipparchus of Nicaea ( c.  190 –120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him

3569-452: 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. 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

3652-614: 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

3735-410: 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

3818-653: 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. The origins of mathematical thought lie in the concepts of number , patterns in nature , magnitude , and form . 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

3901-452: The 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159). 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. He also established a method which would later be called Cavalieri's principle to find

3984-539: 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. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources. Babylonian mathematics refers to any mathematics of

4067-528: The Babylonian tablet YBC 7289 gives an approximation of √ 2 accurate to five decimal places. 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. 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. This zero sign does not appear in terminal positions, thus

4150-412: The Babylonians came close but did not develop a true place value system. Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers , and their reciprocal pairs . The tablets also include multiplication tables and methods for solving linear , quadratic equations and cubic equations , a remarkable achievement for

4233-619: The Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers. 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 . The mathematics that developed in Japan during the Edo period (1603-1887)

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4316-479: The Roman model first described by the Roman civil engineer and architect Vitruvius ( c.  80 BC  – c.  15 BC ). The device was used at least until the reign of emperor Commodus ( r.  177 – 192 AD ), but its design seems to have been lost until experiments were made during the 15th century in Western Europe. Perhaps relying on similar gear-work and technology found in

4399-517: 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. Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar . His notation was similar to modern mathematical notation, and used metarules, transformations , and recursion . Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses

4482-463: 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-60) numeral system . From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in

4565-481: The bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. 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

4648-496: The centers of mathematical innovation were to be found elsewhere by this time. 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. Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than

4731-562: The clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient 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. From around 2500 BC onward,

4814-515: The development of mathematics by laying the foundations of logic . In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria . It was there that Euclid ( c.  300 BC ) taught, and wrote the Elements , widely considered the most successful and influential textbook of all time. The Elements introduced mathematical rigor through

4897-525: The emperor Justinian in 529 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 . Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and

4980-493: 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

5063-548: The familiar theorems of Euclidean geometry , the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory , algebra and solid geometry , including proofs that the square root of two is irrational and that there are infinitely many prime numbers. 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.  287 –212 BC) of Syracuse , widely considered

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5146-573: The field of astronomy to record time and formulate calendars . The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 ( Babylonian c.  2000 – 1900 BC), the Rhind Mathematical Papyrus ( Egyptian c. 1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples , so, by inference,

5229-581: 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. Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya". Around 500 AD, Aryabhata wrote the Aryabhatiya , a slim volume, written in verse, intended to supplement

5312-599: The first proof of the Pythagorean theorem , though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers . Although he was preceded by the Babylonians , Indians and the Chinese , the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables , whereas the oldest extant Greek multiplication table

5395-606: The first use of negative numbers . The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī . Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were

5478-570: The formation of a matrix . In the 12th century, Bhāskara II , 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. In the 14th century, Narayana Pandita completed his Ganita Kaumudi . Mu%E1%B8%A5ammad ibn M%C5%ABs%C4%81 al-Khw%C4%81rizm%C4%AB Too Many Requests If you report this error to

5561-493: The greatest mathematician of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 3+ ⁠ 10 / 71 ⁠ < π < 3+ ⁠ 10 / 70 ⁠ . He also studied

5644-413: The introduction of deductive reasoning and mathematical rigor in proofs ) and expanded the subject matter of mathematics. The ancient Romans used applied mathematics in surveying , structural engineering , mechanical engineering , bookkeeping , creation of lunar and solar calendars , and even arts and crafts . Chinese mathematics made early contributions, including a place value system and

5727-642: 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 Hypatia of Alexandria (AD 350–415). She succeeded her father ( Theon of Alexandria ) as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. Her death

5810-417: 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. Around the same time, Eratosthenes of Cyrene ( c.  276 –194 BC) devised the Sieve of Eratosthenes for finding prime numbers . The 3rd century BC is generally regarded as

5893-407: 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. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while

5976-742: The mathematics developed by the Maya civilization of Mexico and Central America , where the concept of zero was given a standard symbol in Maya numerals . 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

6059-624: The origin of discoveries in mathematics and the mathematical methods and notation of the past . Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer , Akkad and Assyria , followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic , algebra and geometry for purposes of taxation , commerce , trade and also in

6142-401: The peoples of Mesopotamia (modern Iraq ) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity . The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC ( Seleucid period). It

6225-413: The period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis , which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica ,

6308-519: The rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. It is in the Aryabhatiya that the decimal place-value system first appears. 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

6391-412: 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. Apollonius of Perga ( c.  262 –190 BC) made significant advances to the study of 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. He also coined

6474-417: 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. Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on

6557-499: The terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). 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. While neither Apollonius nor any other Greek mathematicians made

6640-510: The time. Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem . 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. Egyptian mathematics refers to mathematics written in

6723-423: 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. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although

6806-649: The volume of a sphere . 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 . The Precious Mirror also contains

6889-642: The world, leading scholars to assume an entirely independent development. The oldest extant mathematical text from China is the Zhoubi Suanjing (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the Warring States Period appears reasonable. However, the Tsinghua Bamboo Slips , containing the earliest known decimal multiplication table (although ancient Babylonians had ones with

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