History of Mathematics

The history of mathematics starts with tally marks on bones more than 40,000 years ago and runs through the clay tablets of Babylon, the papyri of Egypt, the proofs of the Greeks, the numerals of India, the algebra of the Islamic world and the calculus of the 17th century to the computer-aided mathematics of today. No single country or person made it. It is a chain of ideas that were passed on, lost, found again and improved.

This page gives you a short overview, era by era, with the key people and results, followed by the stories of four big topics: algebra, geometry, calculus and zero. For the question of origins, see who invented math. The people are introduced in the list of famous mathematicians.

Timeline of the history of mathematics

1. about 44,000 to 20,000 years ago Tally bones: the Lebombo bone (South Africa) and the Ishango bone (Congo).
2. about 3300 BC The first number signs on clay tablets in Mesopotamia.
3. about 3100 BC Egyptian hieroglyphic numerals, up to a million.
4. about 1800 BC Babylonian tablets with base 60 arithmetic, quadratic equations and an approximation of √2.
5. about 1550 BC The scribe Ahmes copies the Rhind papyrus.
6. about 600 BC Thales and Pythagoras: the beginning of Greek proof.
7. about 300 BC Euclid writes the Elements.
8. about 250 BC Archimedes bounds π between 3 10/71 and 3 1/7.
9. 3rd century AD Diophantus in Alexandria, Liu Hui in China.
10. 628 Brahmagupta states rules for zero and negative numbers.
11. about 825 Al-Khwarizmi writes the book on algebra in Baghdad.
12. 1202 Fibonacci’s Liber abaci brings the Hindu-Arabic numerals to Europe.
13. 1545 Cardano publishes formulas for cubic and quartic equations.
14. 1637 Descartes joins algebra and geometry in coordinates.
15. 1660s to 1680s Newton and Leibniz develop calculus.
16. 1801 Gauss publishes the Disquisitiones Arithmeticae.
17. 1900 and 1931 Hilbert’s 23 problems; Gödel’s incompleteness theorems.
18. 1994 and 2003 Proofs of Fermat’s Last Theorem (Wiles) and the Poincaré conjecture (Perelman).
19. 2014 Maryam Mirzakhani is the first woman to win the Fields Medal.

Prehistory: counting before writing

The oldest evidence for counting is notched bones. The Lebombo bone from southern Africa is over 40,000 years old, and the Ishango bone from Central Africa about 20,000. What exactly was counted is not known. Our fingers shaped the bases 5, 10 and 20 that people still use. With farming, from about 8000 BC, clay tokens appeared as markers for goods, and from them the first number signs developed.

Babylon and Egypt

In Mesopotamia, scribes wrote on clay with a reed stylus. Around 2000 BC they used a place value system in base 60, still alive in our 60 minutes and 360 degrees, and solved problems that we write as quadratic equations. The tablet Plimpton 322 lists number triples that fit the Pythagorean theorem about a thousand years before Pythagoras.

In Egypt, numbers were written in hieroglyphs with a symbol for each power of ten (Egyptian numerals). The Rhind papyrus, copied by Ahmes around 1550 BC, contains 84 problems with unit fractions, areas and volumes. Egyptians multiplied by doubling and adding.

China

Chinese mathematics used counting rods on a board with decimal place value. Its main work, the Nine Chapters on the Mathematical Art with 246 problems, contains solutions of linear systems by elimination and the oldest known rules for negative numbers. Liu Hui commented on it in 263 and computed π to about 3.14 with polygons. Zu Chongzhi found 355/113 for π in the 5th century. Later, Qin Jiushao solved congruences and equations of higher degree, and Yang Hui wrote down the triangle that Europe calls Pascal’s triangle.

Chinese number words are decimal and very regular: the signs are 一 (1), 二 (2), 三 (3), 四 (4), 五 (5), 六 (6), 七 (7), 八 (8), 九 (9) and 十 (10), with 百 for 100, 千 for 1,000 and 万 for 10,000. Twenty-five is said “two-ten-five,” 二十五.

Greece and Rome

The Greeks made mathematics a science of proof. Thales and Pythagoras began, Eudoxus created a theory of proportions, and Euclid arranged geometry and number theory around 300 BC in the Elements, which was a textbook for more than two thousand years. Archimedes computed areas and volumes of curved shapes, Apollonius wrote about conic sections, Hipparchus and Ptolemy built early trigonometry, and Diophantus wrote the Arithmetica. More on this world is on ancient Greek mathematicians.

Roman mathematics was practical. The Romans surveyed land, built roads and aqueducts, reckoned on counting boards with stones and wrote numbers with Roman numerals, which have no zero and no place value. The theory stayed Greek, and the Latin texts were mostly handbooks and summaries. The Julian calendar and Boethius’ arithmetic, a school book of the Middle Ages, are part of the legacy.

India

Indian scholars built the system we use today. From the first centuries AD they used a place value system with ten digits, and by the 7th century zero was a number with rules (Brahmagupta, 628). Aryabhata (476–550) made a sine table and gave π as 3.1416, Bhaskara II (1114–1185) wrote the Lilavati and solved the Pell equation by the chakravala method, and Madhava (about 1340–1425) found infinite series for π, sine and cosine in Kerala. All of this is told on the page Hindu-Arabic numerals.

The Islamic world

From about 750 to 1450, scholars writing in Arabic and Persian translated Greek and Indian works, saved many that would otherwise be lost and developed algebra into a subject of its own. Al-Khwarizmi’s book on al-jabr (about 825) gave algebra its name. Al-Karaji worked with polynomials and used an early form of induction, Omar Khayyam solved cubic equations with conic sections, Ibn al-Haytham studied optics, and Nasir al-Din al-Tusi made trigonometry a field of its own and proved the law of sines for plane triangles.

Medieval Europe and the Renaissance

For a long time Europe lagged behind. A turning point was Fibonacci’s Liber abaci of 1202, which taught merchants to calculate with the new digits (and posed the rabbit problem behind the Fibonacci sequence). Nicole Oresme drew graphs of changing quantities in the 14th century. In 1494 Luca Pacioli printed a summary of arithmetic and algebra, and in 1545 Cardano published the solutions of cubic and quartic equations. In the 16th century, books like those of Adam Ries in Germany taught ordinary people to calculate. In 1585 Simon Stevin spread decimal fractions, and in 1614 John Napier invented logarithms.

The 17th and 18th centuries

This is the century of coordinates and calculus. Descartes joined algebra and geometry in 1637. Fermat and Pascal founded probability in a correspondence of 1654 about a game of chance interrupted before the end. Newton and Leibniz developed calculus, and Jacob Bernoulli’s Ars Conjectandi (1713) proved the law of large numbers. Leonhard Euler then wrote on almost every field and fixed much of today’s notation.

The 19th century

Gauss founded modern number theory in 1801. Bolzano, Cauchy and Weierstrass put analysis on exact definitions of limits. Lobachevsky (1829) and Bolyai (1832) showed that a consistent geometry without the parallel postulate exists, and Riemann generalized the idea of space in 1854. Abel and Galois solved the question of which equations can be solved by roots. Boole and Frege made logic mathematical, Cantor created set theory with different sizes of infinity, Hamilton found the quaternions and Cayley the matrices (1858). Women such as Sophie Germain, Ada Lovelace and Sofya Kovalevskaya did pioneering work despite being excluded from universities.

The 20th and 21st centuries

In 1900 Hilbert presented 23 problems. Russell’s paradox and Gödel’s incompleteness theorems (1931) showed the limits of formal systems, and in 1936 Turing and Church proved that there is no general decision procedure, creating a model of the computer on the way. Emmy Noether, the Bourbaki group and Grothendieck made structures the main subject. Game theory, information theory, chaos theory and fractal geometry opened new fields. In 1976 the four color theorem was proved with a computer, which started a lasting debate about computer proofs.

Since 2000 the Clay Mathematics Institute has offered a prize for seven “Millennium Problems.” Only one has been solved so far, the Poincaré conjecture by Grigori Perelman in 2002 and 2003, and he declined the prize. In 2014 Maryam Mirzakhani became the first woman to win the Fields Medal. Computers now check proofs, such as the Kepler conjecture in 2014. The details are in the German pages on the 20th century and the 21st century.

History of algebra

Algebra moved from words to symbols. Babylonians solved quadratic equations around 1800 BC with recipes, Diophantus abbreviated the unknown about 250 AD, Brahmagupta gave general rules in 628, and al-Khwarizmi organized the subject around 825 and named it. Cardano published the formulas for degrees three and four in 1545, Viète used letters for known quantities in 1591 and Descartes wrote x, y, z in 1637. In the 19th century algebra became abstract with Galois, Hamilton and Cayley. The step-by-step story is in who invented algebra.

History of geometry

Geometry started with land measuring (the word means “earth measuring”). Egyptians and Babylonians knew rules for areas and volumes, for example the volume of a truncated pyramid in the Moscow papyrus. The Greeks turned it into a deductive science: Euclid built it from definitions and axioms. Descartes added coordinates in 1637 and Desargues projective geometry. The 19th century brought non-Euclidean geometries, and Riemann’s geometry of curved spaces later became the language of general relativity.

History of calculus

The roots of calculus are old. Archimedes computed areas by exhausting a shape with polygons, and Madhava’s school in Kerala had infinite series in the 14th century. In the 17th century Cavalieri, Fermat and Wallis found methods for areas and tangents. Newton developed calculus in the 1660s and Leibniz in the 1670s, independently. Leibniz published first, in 1684, and our notation dx and ∫ is his. A priority dispute followed. Cauchy and Weierstrass gave calculus an exact foundation with limits in the 19th century.

History of trigonometry

Trigonometry grew from astronomy. Hipparchus (2nd century BC) made the first known table of chords, and Ptolemy’s Almagest (about 150 AD) contains a full chord table. Indian scholars replaced chords by the sine, as in the Surya Siddhanta and Aryabhata’s table. Nasir al-Din al-Tusi turned trigonometry into a separate discipline in the 13th century.

History of zero

Zero first meant “nothing in this place” (Babylon, Maya, India) and became a number with calculating rules in India, where Brahmagupta wrote them down in 628. Fibonacci brought it to Europe in 1202. The Maya had a sign for zero, too, and the Roman system never did. See who invented zero and the section on the history of zero.

The history of pi and of symbols

The ratio of a circle’s circumference to its diameter has been chased for 4,000 years, from the 3 1/8 on a Babylonian tablet to a record of 314 trillion digits. Its story is on the page history of pi. How signs such as +, −, = and x were introduced is in who invented the x and the other symbols.

Frequently asked questions

Sources and further reading

• MacTutor History of Mathematics: History Topics
• MacTutor History of Mathematics: Biographies
• MacTutor: An overview of Indian mathematics and Arabic mathematics: forgotten brilliance?
• MacTutor: Mayan mathematics
• British Museum: The Rhind Mathematical Papyrus (EA 10058)
• Annette Imhausen: Mathematics in Ancient Egypt. A Contextual History, Princeton University Press, Princeton 2016.
• Eleanor Robson: Mathematics in Ancient Iraq. A Social History, Princeton University Press, Princeton 2008.
• Victor J. Katz: A History of Mathematics: An Introduction, 3rd edition, Pearson, Boston 2008.