The history of mathematics deals with the origin of discoveries in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and the mathematical methods and notation of the past. Before the

modern age The term modern period or modern era (sometimes also called modern history or modern times) is the period of history that succeeds the Middle Ages (which ended approximately 1500 AD). This terminology is a historical periodization that is applie ...

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 Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...

states of

Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...

Akkad Akkad may refer to: *Akkad (city), the capital of the Akkadian Empire *Akkadian Empire, the first ancient empire of Mesopotamia *Akkad SC, Iraqi football club People with the name *Abbas el-Akkad, Egyptian writer *Abdulrahman Akkad, Syrian LGBT act ...

and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and 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 The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...

'' ( 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, the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

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 Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Although they made virtually no contributions to

theoretical mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...

, the ancient Romans used applied mathematics in

surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...

, structural engineering, mechanical engineering,

bookkeeping Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business and other organizations. It involves preparing source documents for all transactions, operations, and other events of a business. Tr ...

, creation of lunar and

solar calendar A solar calendar is a calendar whose dates indicate the season or almost equivalently the apparent position of the Sun relative to the stars. The Gregorian calendar, widely accepted as a standard in the world, is an example of a solar calendar. T ...

s, and even

arts and crafts A handicraft, sometimes more precisely expressed as artisanal handicraft or handmade, is any of a wide variety of types of work where useful and decorative objects are made completely by one’s hand or by using only simple, non-automated re ...

Chinese mathematics Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...

made early contributions, including a place value system and 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 Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...

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 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 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.

Prehistoric

The origins of mathematical thought lie in the concepts of number, patterns in nature, magnitude, and

form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: * Form (document), a document (printed or electronic) with spaces in which to write or enter dat ...

. 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. The Ishango bone, found near the headwaters of the Nile river (northeastern

Congo Congo or The Congo may refer to either of two countries that border the Congo River in central Africa: * Democratic Republic of the Congo, the larger country to the southeast, capital Kinshasa, formerly known as Zaire, sometimes referred to a ...

), may be more than

20,000 20,000 (twenty thousand) is the natural number that comes after 19,999 and before 20,001. 20,000 is a round number, and is also in the title of Jules Verne's novel ''Twenty Thousand Leagues Under the Sea''. Selected numbers in the range 20001– ...

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 or a six-month lunar calendar.Marshack, Alexander (1991): ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY. 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." 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 the 5th millennium BC pictorially represented

geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles,

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

s, 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

Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...

n mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early

ians 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 is named Babylonian mathematics due to the central role of

Babylon ''Bābili(m)'' * sux, 𒆍𒀭𒊏𒆠 * arc, 𐡁𐡁𐡋 ''Bāḇel'' * syc, ܒܒܠ ''Bāḇel'' * grc-gre, Βαβυλών ''Babylṓn'' * he, בָּבֶל ''Bāvel'' * peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru'' * elx, 𒀸𒁀𒉿𒇷 ''Babi ...

as a place of study. Later under the

Arab Empire A caliphate or khilāfah ( ar, خِلَافَة, ) is an institution or public office under the leadership of an Islamic steward with the title of caliph (; ar, خَلِيفَة , ), a person considered a political-religious successor to th ...

, Mesopotamia, especially Baghdad, once again became an important center of study for

Geometry problem-Sb 13088-IMG 0593-white

In contrast to the sparsity of sources in

Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for count ...

, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in

Cuneiform script Cuneiform is a logo-syllabic script that was used to write several languages of the Ancient Middle East. The script was in active use from the early Bronze Age until the beginning of the Common Era. It is named for the characteristic wedge-sha ...

, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient

ians, who built the earliest civilization in Mesopotamia. They developed a complex system of

metrology Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fran ...

from 3000 BC. 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. 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 a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. 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 30. 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

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 YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest ...

gives an approximation of 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 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 the time. Tablets from the Old Babylonian period also contain the earliest known statement of the

. 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

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

as part of

, 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 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;

and

harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...

s; and simplistic understandings of both the

Sieve of Eratosthenes In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime n ...

and perfect number theory (namely, that of the number 6). It also shows how to solve first order

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...

s as well as

and geometric series. Another significant Egyptian mathematical text is the

Moscow papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...

, 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 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

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. 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 In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...

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 Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...

, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used

deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...

. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to

prove Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

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 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. Pythagoras established the

Pythagorean School Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

, 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 the first proof of the

, 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 Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...

and the Chinese, the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest

Greco-Roman The Greco-Roman civilization (; also Greco-Roman culture; spelled Graeco-Roman in the Commonwealth), as understood by modern scholars and writers, includes the geographical regions and countries that culturally—and so historically—were di ...

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). 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 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 Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

, 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 a formula for obtaining Pythagorean triples bears his name. Eudoxus (408–c. 355 BC) developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of

incommensurable magnitudes In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

. 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 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 Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of ...

'', widely considered the most successful and influential textbook of all time. The ''Elements'' introduced mathematical rigor through 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 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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...

, optics,

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, and mechanics, but only half of his writings survive. Archimedes pi

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

(c. 287–212 BC) of

Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana * Syracuse, Kansas *Syracuse, Miss ...

, widely considered 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, . He also studied the

spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: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 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. Conic sections 2

Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...

(c. 262–190 BC) made significant advances to the study of

, 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 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 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

for finding

prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. The 3rd century BC is generally regarded as 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 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 Menelaus of Alexandria (; grc-gre, Μενέλαος ὁ Ἀλεξανδρεύς, ''Menelaos ho Alexandreus''; c. 70 – 140 CE) was a Greek Encyclopædia Britannica "Greek mathematician and astronomer who first conceived and defined a sphe ...

(c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the ''

Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...

'' 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. Diophantus-cover

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. During this period,

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

made significant advances in algebra, particularly

indeterminate analysis In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates t ...

, 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'', 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, the ''Arithmetica'' being the first instance of algebraic symbolism and syncopation. Hagia Sophia Mars 2013

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 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 The Christian Community (german: Die Christengemeinschaft) is an esoteric Christian denomination. It was founded in 1922 in Switzerland by a group of ecumenically oriented, mainly Lutheran theologians and ministers led by liberal theologian Frie ...

in Alexandria had her stripped publicly and executed. Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...

, 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 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 the centers of mathematical innovation were to be found elsewhere by this time.

Roman

Aquinqum BHM IMG 0662 land surveyor equipment

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

surveyor Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...

s and

calculators An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-size ...

were far more interested in applied mathematics than the

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 In law, fraud is intentional deception to secure unfair or unlawful gain, or to deprive a victim of a legal right. Fraud can violate civil law (e.g., a fraud victim may sue the fraud perpetrator to avoid the fraud or recover monetary compens ...

, as well as managing taxes for 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 area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

s 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 A handicraft, sometimes more precisely expressed as artisanal handicraft or handmade, is any of a wide variety of types of work where useful and decorative objects are made completely by one’s hand or by using only simple, non-automated re ...

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. The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the

Roman Kingdom The Roman Kingdom (also referred to as the Roman monarchy, or the regal period of ancient Rome) was the earliest period of Roman history when the city and its territory were ruled by kings. According to oral accounts, the Roman Kingdom began wi ...

and included 356 days plus a leap year every other year. In contrast, 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

organized by

Julius Caesar Gaius Julius Caesar (; ; 12 July 100 BC – 15 March 44 BC), was a Roman general and statesman. A member of the First Triumvirate, Caesar led the Roman armies in the Gallic Wars before defeating his political rival Pompey in a civil war, and ...

(100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in 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 (), 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 An odometer or odograph is an instrument used for measuring the distance traveled by a vehicle, such as a bicycle or car. The device may be electronic, mechanical, or a combination of the two (electromechanical). The noun derives from ancient Gr ...

device for measuring distances traveled, 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 Commodus (; 31 August 161 – 31 December 192) was a Roman emperor who ruled from 177 to 192. He served jointly with his father Marcus Aurelius from 176 until the latter's death in 180, and thereafter he reigned alone until his assassination. ...

(), 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 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 A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic pr ...

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

Chinese

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. The oldest extant mathematical text from China is the '' Zhoubi Suanjing'', 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

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. Chounumerals

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 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 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 Xu Yue was a second-century mathematician, born in Donglai, in present-day Shandong province, China. Little is known of his life except that he was a student of Liu Hong, an astronomer and mathematician in second century China, and had frequent di ...

's ''Supplementary Notes on the Art of Figures''. The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of

Mozi Mozi (; ; Latinized as Micius ; – ), original name Mo Di (), was a Chinese philosopher who founded the school of Mohism during the Hundred Schools of Thought period (the early portion of the Warring States period, –221 BCE). The ancie ...

(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. It also defined the concepts of circumference, diameter, radius, and volume. 九章算術

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 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,

, and includes material on right triangles. It created mathematical proof for the

, and a mathematical formula for

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

. 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 Zhang Heng (; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, ma ...

(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 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 In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

to find 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 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 In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

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 the 13th century onwards.

Jesuit , image = Ihs-logo.svg , image_size = 175px , caption = ChristogramOfficial seal of the Jesuits , abbreviation = SJ , nickname = Jesuits , formation = , founders ...

missionaries such as

Matteo Ricci Matteo Ricci, SJ (; la, Mattheus Riccius; 6 October 1552 – 11 May 1610), was an Italians, Italian Society of Jesus, Jesuit Priesthood in the Catholic Church, priest and one of the founding figures of the Jesuit China missions. He create ...

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. 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 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.

Indian

The earliest civilization on the Indian subcontinent is the

Indus Valley civilization The Indus Valley Civilisation (IVC), also known as the Indus Civilisation was a Bronze Age civilisation in the northwestern regions of South Asia, lasting from 3300 BCE to 1300 BCE, and in its mature form 2600 BCE to 1900&n ...

(mature phase: 2600 to 1900 BC) that flourished in the

Indus river The Indus ( ) is a transboundary river of Asia and a trans-Himalayan river of South and Central Asia. The river rises in mountain springs northeast of Mount Kailash in Western Tibet, flows northwest through the disputed region of Kashmir, ...

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 with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual. 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. In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the

. All of these results are present in Babylonian mathematics, indicating Mesopotamian influence. 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. 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 a device corresponding to a

binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...

. His discussion of the

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

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 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 ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...

'', 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. Though about half of the entries are wrong, 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 Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...

identified the

Brahmagupta theorem In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. I ...

, 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 Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...

. 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 numeral The Brahmi numerals are a numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are a non positional decimal system. They are the direct graphic ancestors of the modern Hindu–Arabic numeral s ...

s. 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

. In the 12th century,

Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...

lived in southern India and wrote extensively on all then known branches of mathematics. 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. In the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and obtained from it a transformed series, whose first 21 terms he used to compute the value of π as 3.14159265359. 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. In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the ''Yukti-bhāṣā''. It has been argued that the advances of the Kerala school, which laid the foundations of the calculus, were transmitted to Europe in the 16th century via

Jesuit , image = Ihs-logo.svg , image_size = 175px , caption = ChristogramOfficial seal of the Jesuits , abbreviation = SJ , nickname = Jesuits , formation = , founders ...

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. 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.

Islamic empires

Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala

The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. 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. In the 9th century, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book ''On the Calculation with Hindu Numerals'', written about 825, along with the work of Al-Kindi, were instrumental in spreading

Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...

and Indian numerals to the West. The word '' algorithm'' is derived from the Latinization of his name, Algoritmi, and the word '' algebra'' from the title of one of his works, '' Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala'' (''The Compendious Book on Calculation by Completion and Balancing''). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and he was the first to teach algebra in an

elementary form In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...

and for its own sake. He also discussed the fundamental method of " reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as ''al-jabr''. "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." His algebra was also no longer concerned "with a series of problems to be resolved, but an 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." 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 2676 solutions. 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

Al-Karaji ( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal sur ...

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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes. The historian of mathematics, F. Woepcke, praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century,

Abul Wafa Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian mathematician a ...

translated the works of

into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of 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. In the 15th century,

Ghiyath al-Kashi Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer an ...

computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by Ruffini and

Horner Horner is an English and German surname that derives from the Middle English word for the occupation ''horner'', meaning horn-worker or horn-maker, or even horn-blower. People *Alison Horner (born 1966), British businesswoman *Arthur Horner (disa ...

. Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the

, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of

cryptanalysis Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic sec ...

and

frequency analysis In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers. Frequency analysis is based on t ...

, the development of

analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...

by Ibn al-Haytham, the beginning of

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

by Omar Khayyam and the development of an algebraic notation by al-Qalasādī. During the time of the Ottoman Empire and

Safavid Empire Safavid Iran or Safavid Persia (), also referred to as the Safavid Empire, '. was one of the greatest Iranian empires after the 7th-century Muslim conquest of Persia, which was ruled from 1501 to 1736 by the Safavid dynasty. It is often conside ...

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

Maya

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. Maya numerals used a base of twenty, the

vigesimal vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...

system, instead of a base of ten that forms the basis of the

system used by most modern cultures. The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy. While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.

Inca

The prevailing numeral system was the base-ten. In precolumbian americas the Inca empire had made various accomplishment in mathematical field.The Incas used their own numeral system which has the base ten. It is also confirmed that the Incas use of the decimal system by the interpretation of the quipus, which are organized in such a way that the knots — according to their location — can represent: units, tens, hundreds, etc. However, the main confirmation of the use of this system is expressed in the denomination of the numbers in Quechua, in which the numbers are developed in decimal form. This can be appreciated in the following table:

Japan

Seki Kowa Katsuyo Sampo Bernoulli numbers

Most of the mathematics in Japan had either imported from China although many of the mathematical accomplishment was also made by Japanese mathematicans. One prominent mathematician during Edo period was Seki Takakazu. He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. He had a significant impact on calculus. He discovered the Newton or Newton-Raphson method for solving equations (independent from Newton) and also had a version of the

Newton interpolation formula A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

. In 1674, Seki published Hatsubi Sanpō (発微算法), giving solutions to all 15 problems. The method he used is called ''bōsho-hō''. He introduced the use of kanji to represent unknowns and variables in equations. He also developed Bernoulli numbers independently from Jacob Bernoulli. They are also called as seki numbers. He studied equations treating both positive and negative roots. His manuscript and other treatises in 1683 show that the Japanese theory of determinate was fully applied to elimination method of arbitrary constant.It is assumed that he had invented the determinants. Seki pushed ahead with elimination theory, based on resultants, in the Kaifukudai no Hō (解伏題之法). To express the resultant, he developed the notion of the determinant. Mathematicans like Takebe Katahiro played and important role in developing Enri (" circle principle"), a crude analog to the Western calculus. Mathematical Society of Japan
/ref> He obtained power series expansion of

(arcsin(x))^2

in 1722, 15 years earlier than

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

. He used Richardson extrapolation in 1695, about 200 years earlier than Richardson. He also computed 41 digits of π, based on polygon approximation and Richardson extrapolation. Ajima Naonobu was credited for introducing calculus into Japanese mathematics.Ajima also posed the question of inscribing three mutually tangent circles in a triangle; circles are now known as Malfatti circles after the later work of Gian Francesco Malfatti, but two triangle centers derived from them, the Ajima–Malfatti points, are named after Ajima.

Medieval European

ππMedieval European interest in mathematiRichardson.Hey concerns quite different froπe icians. One driving element was the belief that mathematics .vided the key to understanding the created order of nature, frequently justified by Plato's ''

Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...

'' 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 ''

''. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works. In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's '' 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 Adelard of Bath ( la, Adelardus Bathensis; 1080? 1142–1152?) was a 12th-century English natural philosopher. He is known both for his original works and for translating many important Arabic and Greek scientific works of astrology, astronom ...

, Herman of Carinthia, and Gerard of Cremona. These and other new sources sparked a renewal of mathematics. Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

.) There, he observed a system of

(specifically algorism) which due to the

positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...

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) 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. One important contribution was development of mathematics of local motion.

Thomas Bradwardine Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often call ...

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). Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem. Oresme

One of the 14th-century

Oxford Calculators The Oxford Calculators were a group of 14th-century thinkers, almost all associated with Merton College, Oxford; for this reason they were dubbed "The Merton School". These men took a strikingly logical and mathematical approach to philosophical ...

William Heytesbury William of Heytesbury, or William Heytesbury, called in Latin Guglielmus Hentisberus or Tisberus (c. 1313 – 1372/1373), was an English philosopher and logician, best known as one of the Oxford Calculators of Merton College, Oxford, where he was ...

, lacking

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by bodyif... it were moved uniformly at the same degree of speed with which it is moved in that given instant". 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 f speedwill traverse in some given time a istancecompletely equal to that which it would traverse if it were moving continuously through the same time with the mean degree f speed. Nicole Oresme at the University of Paris and the Italian

Giovanni di Casali Giovanni (or Johannes) di Casali (or da Casale; c. 1320 – after 1374) was a friar in the Franciscan Order, a natural philosopher and a theologian, author of works on theology and science, and a papal legate. He was born in Casale Monferrato arou ...

independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled. 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.

Pacific

The residents of the island of Mangareva in

French Polynesia )Territorial motto: ( en, "Great Tahiti of the Golden Haze") , anthem = , song_type = Regional anthem , song = " Ia Ora 'O Tahiti Nui" , image_map = French Polynesia on the globe (French Polynesia centered).svg , map_alt = Location of Frenc ...

were using a hybrid binary-decimal system before 1450. They used it with base 2 rather than base 10 which most of the culture had adopted.

Renaissance

During the Renaissance, the development of mathematics and of

accounting Accounting, also known as accountancy, is the measurement, processing, and communication of financial and non financial information about economic entities such as businesses and corporations. Accounting, which has been called the "languag ...

were intertwined. 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 school Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra, among other subjects. These schools sprang up after the publicatio ...

s (known as ''abbaco'' in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing

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. 1415–1492) wrote books on solid geometry and linear perspective, including '' De Prospectiva Pingendi (On Perspective for Painting)'', ''Trattato d’Abaco (Abacus Treatise)'', and '' De quinque corporibus regularibus (On the Five Regular Solids)''.

Luca Pacioli Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...

's '' Summa de Arithmetica, Geometria, Proportioni et Proportionalità'' (Italian: "Review of

Arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

, Geometry, Ratio and Proportion") was first printed and published in

Venice Venice ( ; it, Venezia ; vec, Venesia or ) is a city in northeastern Italy and the capital of the Veneto Regions of Italy, region. It is built on a group of 118 small islands that are separated by canals and linked by over 400 ...

in 1494. It included a 27-page treatise on

, ''"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. In ''Summa Arithmetica'', Pacioli introduced symbols for

plus and minus The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulti ...

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, Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book '' 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.

Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...

's book ''De Thiende'' ('the art of tenths'), first published in Dutch in 1585, 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 1595. Regiomontanus's table of sines and cosines was published in 1533. 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.

Mathematics during the Scientific Revolution

17th century

Gottfried Wilhelm Leibniz, Bernhard Christoph Francke

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

Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

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 Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

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. The

developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

. 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 Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...

. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of

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.

18th century

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

\pi

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

Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

who, in the age of

Napoleon Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader who ...

, did important work on the foundations of celestial mechanics and on

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

Modern

19th century

Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) 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 In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

. 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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...

; 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

, which generalizes the ideas of

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

s 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 In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...

. 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 Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

and computer science.

Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

, and Karl Weierstrass reformulated the calculus in a more rigorous fashion. Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four ( Abel–Ruffini theorem). Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to

trisect an arbitrary angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

, 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. 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. 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

. In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L.E.J. Brouwer,

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics. The 19th century saw the founding of a number of national mathematical societies: the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

in 1865, the Société Mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy. In 1897, Kurt Hensel introduced p-adic numbers.

20th century

The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were available in both teaching and industry. An effort to catalogue the areas and applications of mathematics was undertaken in Klein's encyclopedia. In a 1900 speech to the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

set out a list of 23 unsolved problems in mathematics. 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. Notable historical conjectures were finally proven. In 1976, Wolfgang Haken and Kenneth Appel proved the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sh ...

, controversial at the time for the use of a computer to do so. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture. 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. A group of French mathematicians, including Jean Dieudonné and

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

, publishing under the pseudonym "

Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...

", 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.