History of mathematics

A proof from Euclid's *Elements* (c. 300 BC), widely considered the most influential textbook of all time.^[1]

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in 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),^[2] the Rhind Mathematical Papyrus (Egyptian c. 1800 BC)^[3] 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 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".^[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.^[11] 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.

Table of numerals
European (descended from the West Arabic)	0	1	2	3	4	5	6	7	8	9
Arabic-Indic	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Eastern Arabic-Indic (Persian and Urdu)	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Devanagari (Hindi)	०	१	२	३	४	५	६	७	८	९
Bengali	০	১	২	৩	৪	৫	৬	৭	৮	৯
Chinese – Japanese	零	一	二	三	四	五	六	七	八	九
Tamil		௧	௨	௩	௪	௫	௬	௭	௮	௯

^ (Boyer 1991, "Euclid of Alexandria" p. 119)
^ Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, pp. 277–318.
^ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity. Vol. 9 (2 ed.). Dover Publications. pp. 1–191. ISBN 978-0-486-22332-2. PMID 14884919. {{cite book}}: |journal= ignored (help) Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.
^ Turnbull (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0. S2CID 3994109.
^ Heath, Thomas L. (1963). A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."
^ Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48.
^ Ifrah, Georges (1986). Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37.
^ Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.
^ "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
^ Juschkewitsch, A. P. (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.
^ Eves, Howard (1990). History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258.

[Boyer_1991_loc=Euclid_of_Alexandria_p._119-1] (Boyer 1991, "Euclid of Alexandria" p. 119)

[:0-2] Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, pp. 277–318.

[:1-3] Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity. Vol. 9 (2 ed.). Dover Publications. pp. 1–191. ISBN 978-0-486-22332-2. PMID 14884919. {{cite book}}: |journal= ignored (help) Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.

[4] Turnbull (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0. S2CID 3994109.

[5] Heath, Thomas L. (1963). A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."

[:2-6] Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48.

[7] Ifrah, Georges (1986). Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37.

[8] Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.

[9] "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html

[10] Juschkewitsch, A. P. (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.

[11] Eves, Howard (1990). History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258.

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History of mathematics

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