History of mathematics

See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.

The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". Today, the term refers to a specific body of knowledge - the rigorous, deductive study of quantity, structure, space, and change.

One of the most striking facts about the history of mathematics is that there have been bursts of mathematical discovery, followed by centuries of silence. While almost all cultures use basic mathematics (counting and measuring) (there are recent reports of one culture in the Amazon rain forest that may be an exception), new mathematical discoveries have been reported in relatively few cultures and ages. Before the modern age, and the spread of knowledge to all parts of the globe, written examples of new mathematical discoveries have only come to light in a few locales. The most ancient mathematical texts come from ancient Egypt in the Middle Kingdom period circa 19th century BC (Berlin 6619), Mesopotamia in the 18th century BC (Plympton 322), and ancient India circa 800-600 BC (Sulba Sutras). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical discovery after basic arithmetic and geometry. Ancient Greece, and the Hellenistic culture in Egypt and on the island of Syracuse, incomparably increased mathematical knowledge. The Han Dynasty in ancient China circa 200 BC to AD 200 contributed the Sea Island Manual and The Nine Chapters on the Mathematical Art. Jaina mathematicians in ancient India circa 200 BC to AD 400 contributed the Bakshali Manuscript. Hindu mathematicians from 499 and Islamic mathematicians from 800 made major contributions to mathematics. Beginning in Renaissance Italy in the 16th century, new mathematical discoveries, interacting with new scientific discoveries, have been made at an ever increasing pace which continues to the present day. People throughout the world have contributed to modern mathematics.

Mathematics in prehistory

Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 BC and 20,000 BC, indicate early attempts to quantify time. Evidence exists that early counting involved women who kept records of their monthly biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a distinctive scratching on the bone or stone, for example. Moreover, hunters had the concepts of one, two, and many, as well as the idea of none or zero, when considering herds of animals. (references: [2], [3], [4]).

The Ishango Bone, found in the area of the headwaters of the Nile River in Egypt, is the earliest known demonstration of sequences of prime numbers, and some series of multiples, dating as early as 20,000 BC. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric spatial designs. Megalithic monuments from as early as the 5th millennium BC in Egypt, and then subsequently England and Scotland from the 3rd millenium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design, as well as a possible understanding of the measurement of time based on the movement of the stars.

The Indus Valley civilization in Pakistan/North India circa 3000 BC developed a system of uniform weights and measures, a surprisingly advanced brick technology which utilised ratios, and a number of geometrical shapes and designs, including circles, cuboids, barrels, cones, and cylinders.

Egyptian mathematics (2050 BC - 600 BC)

Main article: Egyptian mathematics

The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian Middle Kingdom papyrus dated circa 2050 BC - 1800 BC. Like many ancient mathematical texts, 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 pyramid: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. Your are to take 28 twice, result 56. See, it is 56. You will find it right."

The Rhind papyrus (circa 1650 BC) is another major Egyptian mathematical text, an instruction manual 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 (see [5]), 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)[6]. It also shows how to solve first order linear equations [7] as well as arithmetic and geometric series [8].

Also, three geometric elements contained in the Rhind papyrus are basic to simplistic analytical geometry: (1) first and foremost, how to obtain an approximation of $\pi$ accurate to within less than one per cent; (2) second, an ancient attempt at squaring the circle; and (3) third, the earliest known use of a kind of cotangent.

Finally, the Berlin Papyrus (circa 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation [9].

Babylonian mathematics (1900 BC - 539 BC)

Main article: Babylonian mathematics

Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. 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. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of Pythagorean triples (see Plimpton 322). The tablets also include multiplication tables, trigonometry tables, and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to nearly six decimal places.

The Babylonian system of mathematics was sexagesimal (base-60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a Highly composite number, having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1).

Ancient Indian mathematics (800 BC - 200 BC)

Main article: Indian mathematics

The earliest known mathematics in ancient India dates back to circa 3000 BC during the Indus Valley Civilization. After its collapse in 1700 BC however, writing was absent in South Asia for a long period. There is considerable controversy regarding the dates when writing was re-developed in India and when the Brahmi script was developed.[10] Some scholars, such as Georg Bühler, date the Brahmi script as early as the 8th century BC, others from the Maurya dynasty in the 4th century BC. Recent archeological evidence dates it to 600 BC (see Brāhmī), while some scholars even suggest 1000 BC.[11] If the earlier dates are correct, it is possible that Pythagoras traveled to India and learnt mathematics there, as some have claimed (such as Florian Cajori). If the later date is correct, then Indian mathematics may have benefited from contact with Greece following the invasion of Alexander. It is also likely that the two mathematical traditions developed independently.

After the Iron Age began, the progress of Indian mathematics has been fairly continuous until the 16th century AD, but it can be divided into roughly two periods of development. This section summarizes the period between 800 BC and 200 BC, when Indian mathematics wasn't studied for the sole purpose of science, but there are still advanced mathematics scattered throughout a large body of Indian texts from this period. Many of these however, are of uncertain date and authorship, and did not follow a serious mathematical tradition.

The Yajur-Veda composed by 900 BC, first explained the concept of numeric infinity. Yajnavalkya (circa 900-600 BC) computed the value of π to 2 decimal places. The Sulba Sutras (circa 800-600 BC) were geometry texts that first used irrational numbers, prime numbers, the rule of three, cube roots, computed the square root of 2 to five decimal places, gave the method of squaring the circle, solved linear equations and quadratic equations, discovered Pythagorean triples algebraically, and gave a numerical proof of the Pythagorean theorem.

The linguist Panini formulated the grammar rules for Sanskrit in the 5th century BC. His notation was similar to modern mathematical notation, and used metarules, transformations, and recursions with such sophistication that his grammar had the computing power equivalent to a Turing machine. Panini's work is also the forerunner to modern formal language theory (precursor to computing), while the Panini-Backus form used by most modern programming languages is also significantly similar to Panini's grammar rules. Pingala (4th-3rd century BC) invented the binary number system, Fibonacci series and Pascal's triangle, and also used a dot to denote zero and described the formation of a matrix. The works of Panini and Pingala were foundational to the development of computing.

Greek and Hellenistic mathematics (550 BC - 200 BC)

Main article: Greek mathematics

Greek mathematics refers to the mathematics of those who wrote in the Greek language before and during the Hellenistic period. Not all texts on Greek mathematics were written by Greeks however, as the Greek language was also spoken by non-Greek scholars throughout the Hellenistic world, which was spread across the Eastern end of the Mediterranean. Most mathematical texts written in Greek were found in Greece, Egypt, Mesopotamia, Asia Minor, North Africa, and Southern Italy.

Although the earliest found Greek texts on mathematics were written after the Hellenistic period, many of these are considered to be copies of works written during, and some before, the Hellenistic period. Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.

Greek mathematics is thought to have begun from the late 500s BC, when Thales and Pythagoras brought knowledge of Egyptian and Babylonian mathematics to Greece. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras stated the Pythagorean theorem and constructed Pythagorean triples algebraically, according to Proclus' commentary on Euclid.

Greek mathematics is characterized by its originality, its depth, its abstraction, and its reliance on logic. The Greeks were the first to give a proof for irrational numbers (due to the Pythagoreans), and the first to discover Eudoxus's method of exhaustion, and the Sieve of Eratosthenes for discovering prime numbers. They took the ad hoc methods of constructing a circle or an ellipse and developed a comprehensive theory of conics, they took the various formulas for areas and volumes and provided methods from separating correct formulas from incorrect, and for generating general formulas. The first recorded abstract proofs are in Greek, and all extant studies of logic proceed from the methods set down by Aristotle. Euclid, in the Elements, wrote a book that would be used as a mathematics textbook throughout Europe, the Near East, and North Africa for almost two thousand years. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, The Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Some say the greatest of Greek mathematicians was Archimedes 287 BC - 212 BC of Syracuse. At the age of 75, while drawing mathematical formulas in the dust, he was run through with a spear by a Roman soldier. The Romans had absolutely no interest in mathematics.

Chinese mathematics (200 BC - AD 1200)

Main article: Chinese mathematics

In China, in 212 BC, the Emperor Qin Shi Huang (Shi Huang-ti) commanded that all books be burned. While this order was not universally obeyed, it means that little is known with certainty about ancient Chinese mathematics. Another problem is that the Chinese wrote on bamboo, a perishable medium.

Dating from the Shang period (1500 BC - 1027 BC), the earliest extant Chinese mathematics consists of numbers scratched on tortoise shell. These numbers use a decimal system, so that the number 123 is written (from top to bottom) as the symbol for 1 followed by the symbol for a hundred, then the symbol for 2 followed by the symbol for ten, then the symbol for 3. This was the most advanced number system in the world at the time, and allowed calculations to be carried out on the suan pan or Chinese abacus. The date of the invention of the suan pan isn't certain but the earliest written reference was in AD 190 in the Supplementary Notes on the Art of Figures written by Xu Yue. The suan pan was most likely in use earlier than this date.

From the 12th century BC, the oldest mathematical work to survive the book burning is the I Ching, which uses the 64 permutations of a solid or broken line for philosophical or mystical purposes.

After the book burning, the Han dynasty (206 BC - AD 221) produced works of mathematics which presumably expand on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art. It consists of 246 word problems, involving agriculture, business, and engineering, and includes material on right triangles and on pi.

In the thousand years following the Han dynasty, starting in the Tang dynasty and ending in the Sung dynasty, Chinese mathematics thrived at a time when European mathematics did not exist. Discoveries first made in China, and only much later known in the West, include negative numbers, the binomial theorem, matrix methods for solving systems of linear equations, and the Chinese remainder theorem. They were also one of the first to discover Pascal's triangle and the rule of three.

Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with Chinese mathematics in decline, until the Jesuit missionaries in the 18th century carried mathematical ideas back and forth between the two cultures.

Classical Indian mathematics (200 BC - AD 1600)

Main article: Indian mathematics

From 200 BC, mathematicians in India began studying mathematics for the sole purpose of science, starting with Jaina mathematicians between 200 BC and AD 400. They discovered transfinite numbers, set theory, logarithms, fundamental laws of indices, cubic equations, quartic equations, sequences and progressions, permutations and combinations, squaring and extracting square roots, and finite and infinite powers. Discoveries written in the Bakshali Manuscript include solutions of linear equations with upto five unknowns, the solution of the quadratic equation, arithmetic and geometric progressions, compound series, quadratic indeterminate equations, simultaneous equations, and the use of zero and negative numbers. Accurate computations for irrational numbers could be found, which includes computing square roots of numbers as large as a million to at least 11 decimal places.

Aryabhata in AD 499 introduced a number of trigonometric functions and trigonometric tables, techniques and algorithms of algebra, and obtained whole number solutions to linear equations by a method equivalent to the modern method, along with astronomical calculations based on a heliocentric system of gravitation. An Arabic translation of his Aryabhatiya was available by the 10th century. He also computed the value of π to the fourth decimal place as 3.1416. Madhava later in the 14th century computed the value of π to the eleventh decimal place as 3.14159265359.

In the 7th century, Brahmagupta discovered the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and a decimal digit, and explained the Hindu-Arabic numeral system. It was from a translation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century and it has now displaced all older number systems throughout the world.

From the 12th century, Bhaskara, Madhava and a number of Kerala School mathematicians, first conceived differential calculus, mathematical analysis, floating point numbers, and concepts foundational to the overall development of calculus, including Rolle's theorem, term by term integration, tests of convergence, iterative methods for solutions of non-linear equations, the relationship between integrals and the area under a curve, and a number of infinite series and trigonometric series. In the 16th century, Jyeshtadeva consolidated many of the Kerala School's discoveries in the Yuktibhasa, the world's first differential calculus text, which also introduced concepts of integral calculus. Mathematical progress in India became stagnant from the late 16th century onwards due to subsequent political turmoil.

Arabic and Persian mathematics (650 - 1500)

Main article: Islamic mathematics

The Islamic Caliphate (Islamic Empire) established across the Middle East, North Africa, Iberia, and in parts of India (parts of Pakistan) in the 8th century preserved and translated much of the Greek mathematics largely forgotten in Europe at that time, along with texts on Indian mathematics that also had a major influence on Islamic mathematics, including the introduction of Hindu-Arabic numerals. Like the Indian mathematicians at the time, Islamic mathematicians were especially interested in astronomy. The works of Brahmagupta were translated into Arabic circa 766.

Although most Islamic texts on mathematics were written in Arabic, not all of them were written by Arabs, since much like the earlier Greek mathematics, Arabic was also spoken by non-Arab scholars throughout the Islamic world at the time. Some of the most important Islamic mathematicians were Persian.

Al-Khwarizmi, the 9th century Persian astronomer of the Caliph of Baghdad, wrote several important books, on the Hindu-Arabic numerals and on methods for solving equations. The word algorithm is derived from his name, and the word algebra from the title of one of his works, Al-Jabr wa-al-Muqabilah. Al-Khwarizmi is often considered to be the father of modern algebra and modern algorithms.

In the 10th century, Abul Wafa translated the works of Diophantus into Arabic and invented the tangent function.

Omar Khayyam, the 12th century poet, was also a mathematician, and wrote Discussions of the Difficulties in Euclid, a book about flaws in Euclid's Elements. He gave a geometric solution to cubic equations, one of the most original discoveries in Islamic mathematics. He was also very influential in calendar reform.

Spherical trigonometry was largely developed by the Persian mathematician Nasir al-Din Tusi (Nasireddin) in the 13th century. He also wrote influential work on Euclid's parallel postulate.

In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.

European Renaissance mathematics (1200 - 1600)

In Europe at the dawn of the Renaissance, most of what is now called school mathematics -- addition, subtraction, multiplication, division, and geometry -- was known to educated people, though the notation was cumbersome: Roman numerals and words were used, but no symbols: no plus sign, no equal sign, no zero, and no use of x as an unknown. Almost all of the mathematics now taught in college had yet to be discovered, or was known only to the small and isolated mathematical community in India.

Contact with Islamic scholars brought to Europe knowledge of the Hindu-Arabic numerals. In the 12th century Robert of Chester translated Al-Jabr wa-al-Muqabilah into Latin. The works of Aristotle were rediscovered, first in Arabic and later in Greek. Of particular importance to the development of mathematics was the rediscovery of a collection of Aristotle's logical writing, compiled in the 1st century, known as the Organon.

The reawakened desire for new knowledge sparked a renewed interest in mathematics. Fibonacci, in the early 13th century, produced the first original mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. But it was only from the late 16th century that European mathematicians began to make advances without precedent anywhere in the world, so far as is known today.

The first of these was the general solution of cubic equations, generally credited to Scipione del Ferro circa 1510, but first published in Cardan's Ars magna. It was quickly followed by Lodovico Ferrai's solution of the general quartic equation.

From this point on, mathematical discovery came swiftly, and combined with advances in science, to their mutual benefit. In the landmark year 1543, Copernicus published De revolutionibus, asserting that the Earth traveled around the Sun, and Vesalius published De humani corporis fabrica, treating the human body as a collection of organs.

By century's end, thanks to Regiomontanus (1436 - 1476) and François Vieta (1540 - 1603), among others, mathematics and science was written using Hindu-Arabic numerals and in a form not too different from the elegant symbolism used today.

17th century

The 17th century saw an unprecedented explosion of mathematical and scientific ideas that not only fascinated philosophers but had industrial applications that began to make major changes in the way people lived.

Copernicus, a Pole, had written that planets orbit the Sun. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, Lord Napier, in Scotland, invented natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry invented by Descartes, a Frenchman, allowed those orbits to be plotted on a graph. And Isaac Newton, and Englishman, discovered the laws of physics that explained planetary orbits and also the mathematics of calculus that could be used to deduce Kepler's laws from Newton's principle of universal gravitation. Science and mathematics had become an international endeavor. Soon this activity would spread over the entire world.

18th century

As we have seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older then any surviving written text. The earliest civilizations, in Mesopotamia, Egypt, India, and China, knew arithmetic.

One way to view the development of the various number systems of modern mathematics is to see new numbers invented to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1. In India and China, and much later in Germany, negative numbers were invented to answer the question: what do you get when you subtract a larger number from a smaller. The invention of the zero may have followed from similar question: what do you get when you subtract a number from itself.

Another natural question is: what kind of a number is the square root of two. The Greeks knew that it was not a fraction, and this question may have played a role in the development of continued fractions. But a better answer came with the invention of decimals, developed byLord Napier (1550 - 1617) and perfected 1655 by Simon Stevinis Using decimals, and an idea that anticipated the concept of the limit, Lord Napier also invented a new number, which Leonhard Euler (1707 - 1783) named $e$.

Euler asked the question: what kind of number is the square root of minus one. To answer the question, he invented what are now called imaginary numbers and complex numbers. He named the square root of minus 1 with the symbol $i$. He also popularized the use of the Greek letter $\pi$ to stand for the ratio of a circle's circumference to its diameter. He then discovered one of the most remarkable identities in all of mathematics.

e^πi = -1 .

Complex numbers

When the complex numbers were introduced, there were many who argued that they were imaginary constructs to solve the cubic, and that they should not be considered 'real'. This is the origin of the terms imaginary and real for the numbers. However, mathematicians found the new world of complex numbers to be elegant and compelling. To represent a solution to the equation shown above (i.e., $X*X+1=0$) mathematicians eventually settled on the letter i. However, in the early 19th century, one further extension of the real and complex numbers was found.

All of the numbers described above are algebraic; but Liouville showed how to construct transcendental numbers, which could not be expressed as the roots of any algebraic equation. In order to construct these transcendental numbers one needs a "completeness axiom". For this purpose, the supremum axiom has become a popular modern way to state the completeness of the reals: Any set of real numbers with an upper bound has a least upper bound. This fills in the real line with all of the irrational numbers that cannot be derived merely from algebraic equations. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of real or complex numbers is greater than that of the rationals.

The Fundamental Theorem of Algebra shows that all polynomial equations over the complex numbers can be solved; thus there is no need for any further extension on algebraic grounds-nevertheless, many further extensions of the complex numbers do exist, such as the quaternions, or the surreal numbers.

Mathematicians today rarely view the development of the complex numbers in this way (the preferred teaching method does not emphasize this stepwise development) but it demonstrates the tension in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.

Miscellaneous historical notes

The Maya calendar utilized a base-20 number system which included the 'number' zero (also see Maya numerals).

In China, Zu Chongzhi (5th century) of the Southern and Northern Dynasties was the first person to calculate the value of Pi to seven decimal places.

External links

• The MacTutor History of Mathematics Archive created by John J O'Connor and Edmund F Robertson, which contains biographies, timelines and historical articles about mathematical concepts.
• Earliest uses of various mathematical symbols by Jeff Miller
• Earliest known uses of some of the words of mathematics by Jeff Miller
• History of Indian mathematics by Ian Pearce
• History of Mathematics, public domain article
• Important publications in the history of mathematics
• History of calculus by Fred Rickey

References

• Boyer, C. B., A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
• Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0,
• Hoffman, Paul, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998 ISBN 0-7868-6362-5.
• van der Waerden, B. L., Geometry and Algebra in Ancient Civilizations, Springer, 1983, ISBN 3-387-12159-5.

Notes

Template:Ent Hoffman, p.187.