## Essay History Of Mathematics

INTRODUCTION

After the early mathematicians from Egypt, Babylon, and Greece, mathematics still continued to pave its way toward so many ideas and discoveries. The successors of the Greeks in the history of mathematics were the Hindus of India.

The Hindu civilization’s record in mathematics dates from about 800 B.C., but became significant only after influenced by the Greek’s achievements. They contributed a lot and most of them were very useful in the development of the field. Most of their mathematical works were motivated by algebra, astronomy, and geometry.

PRESENTATION OF DATA

(AD 630) BRAHMAGUPTA: ALGEBRA & ASTRONOMY

Born in AD 598 in Northwestern India, Brahmagupta was the most prominent Indian mathematician and astronomer of the seventh century. He is often referred to as Bhillamalacarya or ‘The Teacher from Bhillamala’. Immediately after the sixth century, the Hindu algebra experienced its ‘Golden Age’ through the work of Brahmagupta in the early seventh century. His major work was the Brahmasphutasiddhanta (‘The Opening of the Universe’) or simply the Siddhanta which was written in AD 628 when he was 30. A corrected and updated version of the Siddhanta, the Brahma Siddhanta (‘The System of Brahma in Astronomy’) was a comprehensive treatment of the astronomical knowledge of the time (Katz, 1998).

Similar to other mathematical works of Medieval India, the mathematical ideas of Brahmagupta was imbedded as chapters in astronomical works since he applied his mathematical techniques to various astronomical problems. Nevertheless, his description of mathematical techniques was generally fuller with some examples (Katz, 1998).

The Brahmasphutasiddhanta was divided into two chapters namely: Ganitad’ haya and Kutakhadyaka. The Ganitad’ haya (‘Lectures on Arithmetic’) identified a ganaca, a calculator which is competent enough to study astronomy, as one ‘who distinctly and severally knows addition and the rest of the twenty logistics and the eight determinations, including measurement by shadow” It discussed arithmetic progressions, the rule of three, simple interest, the mensuration of plane figures, and finding volumes (Nowlan, n.d.).

Meanwhile, the most influential advancement of the Hindu algebra was Brahmagupta’s big step toward operational symbolism. The Brahmasphutasiddhanta was considered the first textbook ‘to treat zero as a number in its own right.’ In its second chapter, the Kutakhadyaka (‘Lectures on Indeterminate Equations’) defined zero as the outcome of subtracting a number from itself and used dots underneath numbers to express a zero. He gave some properties of zero as follows: ‘When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.’ Not only that, he also gave rules for operating with zero and the ‘rules of sign’ stated in a marketplace language, using dhana (‘fortunes’) to denote positive numbers ad rina (‘debts’) to represent negative numbers (Nowlan, n.d.).

The following rules should be familiar except for the terms used although Brahmagupta incorrectly claimed that ‘zero divided by zero is zero’:

‘A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multiplied by zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt’ (O’Connor & Robertson, 2000).

Another mathematical work presented by Brahmagupta was his algorithm for computing square roots. His most remarkable work was his solution for the complete integer of the equation ax ?? by = c, where a, b, and c are constant integers. Furthermore, he discussed the indeterminate equation ax + c = by and the quadratic indeterminate equation. Perhaps, Brahmagupta used the method of continued fractions in finding the integral indeterminate equation of the type ax + c = by. He also gave the rules for summing series such as the sum of the squares of the first n natural numbers as (n(n+1)(2n+1))/6 and that of the cubes of the first n natural numbers as ((n(n+1))/2)2 although no proof was found (Bell, 1945).

In terms of astronomy, Brahmasphutasiddhanta dealt with solar and lunar eclipses, planet positions and conjunctions. He believed in a static Earth and presented the length of year as 365 days 6 hours 5 minutes 19 seconds but later changed it into 365 days 6 hours 12 minutes 36 seconds in his second book (O’Connor & Robertson, 2000).

Brahmagupta wrote his second book named Khandakhadyaka, which literally means ‘sweet meat’ when he was 67. With it, he became the first to use algebra in solving astronomical problems. Note that this is not an improvement since the true length of the years is less than 365 days 6 hours. Moreover, he gave the sidereal periods for many heavenly bodies and anticipated the gravitational theory, writing: ‘Bodies fall towards the earth as it is the nature of earth to attract bodies, just as it is the nature of water to flow’ (Nowlan, n.d.).

(AD 710) BEDE: CALENDAR & FINGER ARITHMETIC

In the early medieval Europe, there was a commonly-held belief that academic pursuits, particularly science and mathematics, had collapsed into a dark age. The majority of learned scholars were churchmen. One of them was Bede who was called as ‘Bede the Venerable’. He was born on AD 673 in Northumberland, England and became one of the greatest medieval church scholars because of his numerous writings, including his treatises on the calendar and on finger arithmetic (‘European Mathematicians’, n.d.).

In the early 8th century, Bede had a problem. Each year, Easter had to be foreseen with accuracy because all other moveable feasts in the annual cycle leaned on its date and the opinion on when exactly that date might be was divided. It was very critical that was why an entire division of mathematics was assigned to the subject named computus. However, computus have to respect the church rules including the relationships among the celebrations like Easter, linked to Passover (derived from lunar cycles) and to solar calendar (since if it could not be celebrated on Sunday, it had to be on the first full moon after the spring equinox which in turn is based to solar cycles). To make matters worse, the lunar and solar cycles did not match very well (Mulcare, 2013).

In order to solve the problem, a cyclical table was needed to be developed based on a common multiple m of solar and lunar periods. The general concept was that m years after some reference period, Easter would fall on the similar date as in the reference period itself because a whole number or rural months would have passed. However, there would be an extra month which was then added to the lunar calendar. The closest approximate cycles were 3/8, 4/11, 7/19, 31/84 but these cycles were not universally accepted. Because of these, Bede started reviewing, evaluating, and analyzing the tables of the age and thus, developed the calendar (Mulcare, 2013).

According to some sources, Indians could perform a meticulous method of counting using their fingers because of their three-joint thumbs. The first recorded method was made by Bede. He illustrated a finger arithmetic method and showed how to represent numbers with the aid of fingers, proceeding from left to right (Fink, 2007).

(AD 750) FIRST USE OF ZERO SYMBOL

Zero is the ‘number for none of the thing being counted’ and the ‘number that links positive and negative number lines’. However, no one had thought of having a symbol for zero given that it was only represented by a space in its use as a place-value indicator. Fortunately, the Hindus were the first to recognize a mathematical representation of concept of no quantity.

At first, the Indians represented zero as a dot but later replaced it with the symbol 0. It had been believed that the Hindu zero used by India was a round goose egg-like shape similar to the one used today. There were many beliefs about the origin of this characteristic form. It might have been a Hindu invention or it might have been suggested by the Greek use of omicron (??) for zero. (Boyer, 1944).

The first indubitable appearance of a circle symbol for zero appears in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth century AD (‘Numbers’, 2004).

(AD 810) MOHAMMED IBN MUSA AL-KHWARIZMI COINS TERM ‘ALGEBRA’

Al-Khwarizmi, or perhaps his ancestors, descended from Khwarizmi, the region south of the Aral Sea now part of Uzbekistan and Turkmenistan. He was an early member of the House of Wisdom and one of the astronomers called to cast a horoscope for the dying caliph al-Wathiq in 847 although he failed (Katz, 1998).

Al-Khwarizmi wrote the earliest available arithmetic text that discussed Hindu numbers. In this text, he introduced nine characters to designate the first nine numbers and a circle to denote zero. He demonstrated a process of writing any number using these characters in a place-value notation. Then, he described ‘the algorithms of addition, subtraction, multiplication, halving, doubling, and determining square roots and gave examples of their use’ (Katz, 1998).

However, he made no advancement except for exhibiting a positive and a negative root for a quadratic equation without explicitly rejecting the negative (Bell, 1945).

The most important contribution of Al-Khwarizmi in the world of mathematics was perhaps his arithmetic text which contributed to the important mathematical words used today. He was best known for coining the term ‘algebra’ from the name of his book ‘Al-jabr’ which demonstrated simple algebra and geometry. Since he was believed to have presented the first algebra text that demonstrated general methods, he was often called the ‘Father of Algebra’. Not only that, the word ‘algorithm’ originated from Al-Khwarizmi’s name. Moreover, he also derived the word ‘zero’ from the Arabic ‘sifr’, which was Latinized into ‘zephirum’ (Allen, n.d.).

Al-Khwarizmi’s texts on algebra and decimal arithmetic were considered to be among the most influential writings ever.

(AD 810) HINDU NUMERALS

Aside from the important mathematical terms coined by Al-Khwarizmi, he had also made another contribution with his strong advocacy of the Hindu numerical system. He wrote a book about Hindu numerals discussing numbers 1 to 9, spreading the use of Arabic numerals. He described the system as ‘having the power and efficiency needed to revolutionize Islamic mathematics. It was soon adopted by the entire Islamic world and by Europe as well (‘Islamic Mathematics’, 2010).

(AD 850) MAHAVIRA: ARITHMETIC & ALGEBRA

The Indian mathematicians also handled equations in several variables. One of them is Mahavira. Mahavira, also called Mahaviracharya (“Mahavira The Teacher’), was a mathematician from Mysore in Southern India (O’Connor & Robertson, 2000).

Mahavira wrote the earliest Indian text, the Ganita Sara Sarangha (‘Compendium of the Essence of Mathematics’), which was created as revised edition of Brahmagupta’s book. It was devoted completely to mathematics where he presented a version of the hundred fowls problem: ‘Doves are sold at the rate of 5 from 3 coins, cranes at the rate of 7 to 5, swans at the rate of 9 for 7, and peacocks at the rate of 3 for 9. A certain man was told to bring at these rates 100 birds, for 100 coins for the amusement of the king’s son and was sent to do so. What amount does he give for each’? In turn, he gave a rather complex rule for the solution (Katz, 1945).

Also, Mahavira was one of the first to indicate an awareness of the problem involving the square root of a negative number by writing that ‘a negative number cannot have a square root because a negative cannot be a square’ (Groza, 1968).

Although a place-value system with nine numerals was always used in his work, Mahavira became interested in developing a new place-value system with his description of the number 12345654321 which he had obtained after a calculation and described the number as starting with one and then increases until six, then decreases in reverse order. This description was a clear indication that Mahavira was open to the place-value system (O’Connor & Robertson, 2000).

Operations with fractions including the methods of decomposing integers and fractions into unit fractions were also discussed in his work. An example would be 2/17= 1/12+ 1/51+ 1/68. He also used a method called kuttaka in order to test integer solutions of first-degree indeterminate equations. The kuttaka (‘the pulveriser’) method was established based on Euclidean algorithm with the method of solution resembling the continued fraction process of Euler (O’Connor & Robertson, 2000).

Furthermore, Mahavira completely omitted addition and subtraction from his discussion of arithmetic. Instead, he took multiplication as the first eight fundamental operations and filled the gap with summation and subtraction of series (‘South Asian Mathematics’, n.d.).

Mahavira really had so many contributions like giving special rules for permutations and combinations, describing a process for the calculation of a sphere’s volume and of a number’s cube root. He also attempted to solve some unaccomplished mathematical problems of other Indian mathematicians (O’Connor & Robertson, 2000).

The area of study known as the **history of mathematics** is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.

The most ancient mathematical texts available are from Mesopotamia and Egypt - *Plimpton 322* (Babylonian c. 1900 BC),^{[2]} the *Rhind Mathematical Papyrus* (Egyptian c. 2000–1800 BC)^{[3]} and the *Moscow Mathematical Papyrus* (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

The study of mathematics as a "demonstrative discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek*μάθημα* (*mathema*), meaning "subject of instruction".^{[4]}Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.^{[5]} Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers.^{[6]}^{[7]} The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.^{[8]}^{[9]} Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.^{[10]} 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 RenaissanceItaly in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded and continues to spearhead advances in the field.

## Prehistoric[edit]

The origins of mathematical thought lie in the concepts of number, magnitude, and form.^{[11]} 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.^{[11]}

Prehistoricartifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time.^{[not in citation given]} The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a *tally* of the earliest known demonstration of sequences of prime numbers^{[12]} or a six-month lunar calendar.^{[13]} 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."^{[14]} 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.^{[15]}

Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.^{[16]} All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.

## Babylonian[edit]

Main article: Babylonian mathematics

See also: Plimpton 322

Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.^{[17]} 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).^{[18]} It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.

In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.^{[19]} Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.^{[20]}

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, 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.^{[21]}

Babylonian mathematics were written using a sexagesimal (base-60) numeral system.^{[19]} From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 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.^{[19]} Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system.^{[18]} 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 than multiplying integers, similar to our modern notation.^{[18]} The notational system of the Babylonians was the best of any civilization until the Renaissance,^{[22]} and its power allowed it to achieve remarkable computation accuracy and power; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.^{[22]} 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.^{[18]} 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.^{[18]} This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.^{[18]}

Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regularreciprocalpairs.^{[23]} The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.^{[24]} Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.^{[25]} 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.^{[20]}

## Egyptian[edit]

Main article: Egyptian mathematics

Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.

The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC.^{[26]} 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,^{[27]} including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6).^{[28]} It also shows how to solve first order linear equations^{[29]} as well as arithmetic and geometric series.^{[30]}

Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.^{[31]} 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.^{[32]}

## Greek[edit]

Main article: Greek mathematics

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.^{[33]} Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.^{[34]}

Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.^{[35]}

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.^{[36]} Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".^{[37]} 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 Pythagorean theorem,^{[38]} though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.^{[39]}^{[40]} Although he was preceded by the Babylonians and the Chinese,^{[41]} the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Romanmultiplication 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).^{[42]} The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the *mensa Pythagorica*.^{[43]}

Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.^{[45]} 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, came.^{[46]} Plato also discussed the foundations of mathematics, ^{[47]} clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.^{[48]} The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.^{[46]}

Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration^{[49]} and a theory of ratios that avoided the problem of incommensurable magnitudes.^{[50]} The former allowed the calculations of areas and volumes of curvilinear figures,^{[51]} 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.^{[52]}

In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.^{[53]} It was there that Euclid (c. 300 BC) taught, and wrote the *Elements*, widely considered the most successful and influential textbook of all time.^{[1]} 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.^{[54]} The *Elements* was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.^{[55]} 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,^{[54]} including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.^{[56]}

Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity,^{[57]} 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.^{[58]} 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, 310/71 < π < 310/70.^{[59]} He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid),^{[58]} and an ingenious method of exponentiation for expressing very large numbers.^{[60]} 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.^{[61]} 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.^{[62]}

Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.^{[63]} He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").^{[64]} 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.^{[65]} 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.^{[66]}

Around the same time, Eratosthenes of Cyrene (c. 276–194 BC) devised the Sieve of Eratosthenes for finding prime numbers.^{[67]} The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.^{[68]} Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.^{[68]}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.^{[69]}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.^{[70]}Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem.^{[71]} The most complete and influential trigonometric work of antiquity is the *Almagest* of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.^{[72]} 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.^{[73]}

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.^{[74]} During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis".^{[75]} 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.^{[76]} 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).^{[77]} Diophantus also made significant advances in notation, the *Arithmetica* being the first instance of algebraic symbolism and syncopation.^{[76]}

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.^{[78]} 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 as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed.^{[79]} 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, Simplicius and Eutocius.^{[80]} 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 Haghia Sophia.^{[81]} 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.^{[82]}

## Roman[edit]

Further information: Roman abacus and Roman numerals

Although ethnic Greek mathematicians continued to live under the rule of the late Roman Republic and subsequent Roman Empire, there were no noteworthy native Latin mathematicians in comparison.^{[83]}^{[84]}Ancient Romans such as Cicero (106-43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than the theoretical mathematics and geometry that were prized by the Greeks.^{[85]} 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.^{[86]}

Using calculation, Romans were adept at both instigating and detecting financial fraud, as well as managing taxes for the treasury.^{[87]}Siculus Flaccus, one of the Roman *gromatici* (i.e. land surveyor), wrote the *Categories of Fields*, which aided Roman surveyors in measuring the surface areas of allotted lands and territories.^{[88]} 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.^{[89]}Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.^{[90]}^{[91]}

The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included 356 days plus a leap year every other year.^{[92]} 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.^{[93]} This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100-44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle.^{[94]} This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII (r. 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.^{[95]}

At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius (c. 80 BC - c. 15 BC).^{[96]} The device was used at least until the reign of emperor Commodus (r. 177 – 192 AD), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.^{[97]} 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 that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.^{[98]}

## Chinese[edit]

Main article: Chinese mathematics

Further information: Book on Numbers and Computation

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.^{[99]} 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.^{[100]} However, the Tsinghua Bamboo Slips, containing the earliest known decimalmultiplication 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.^{[41]}

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.^{[101]} 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.^{[102]}Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the *suan pan*, or Chinese abacus. The date of the invention of the *suan pan* is not certain, but the earliest written mention dates from AD 190, in Xu Yue's *Supplementary Notes on the Art of Figures*.

The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The *Mo Jing* described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.^{[103]} It also defined the concepts of circumference, diameter, radius, and volume.^{[104]}

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, surveying, and includes material on right triangles.^{[100]} It created mathematical proof for the Pythagorean theorem,^{[105]} and a mathematical formula for Gaussian elimination.^{[106]} The treatise also provides values of π,^{[100]} which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78-139) approximated pi as 3.1724,^{[107]} as well as 3.162 by taking the square root of 10.^{[108]}^{[109]}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).^{[110]}^{[111]} 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 (i.e. 3.141592), which remained the most accurate value of π for almost the next 1000 years.^{[110]}^{[112]} He also established a method which would later be called Cavalieri's principle to find the volume of a sphere.^{[113]}

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.^{[110]} 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.^{[114]} The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).^{[114]}

Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.^{[114]}

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.^{[115]} 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).^{[116]} 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.^{[117]} 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.^{[118]}

## Indian[edit]

Main article: Indian mathematics

See also: History of the Hindu–Arabic numeral system

The earliest civilization on the Indian subcontinent is the Indus Valley Civilization (mature phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.^{[120]}

The oldest extant mathematical records from India are the Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD),^{[121]} appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.^{[122]} As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.^{[121]} 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.^{[123]}^{[124]} In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem.^{[125]} All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.^{[121]} 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.^{[121]}

Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar.^{[126]} His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion.^{[127]}Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system.^{[128]}^{[129]} His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called *mātrāmeru*).^{[130]}

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.^{[131]} 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.^{[132]} Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".^{[132]}

In the 5th century AD, Aryabhata wrote the *Aryabhatiya*, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.^{[133]} 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 mathematicianAbu Rayhan Biruni described the *Aryabhatiya* as a "mix of common pebbles and costly crystals".^{[134]}

In the 7th century,

*Elements*, widely considered the most influential textbook of all time.

^{[1]}

*Elements*, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.

^{[44]}

Indian numerals in stone and copper inscriptions^{[119]}

Ancient Brahmi numerals in a part of India

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