Timeline of mathematics

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This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.


Rhetorical stage

Before 1000 BC

Syncopated stage

1st millennium BC

1st millennium AD

Symbolic stage


15th century

  • 1400 – Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places.
  • c. 1400 Jamshid al-Kashi "contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots, which is a special case of the methods given many centuries later by [Paolo] Ruffini and [William George] Horner." He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: "On whole number arithmetic", "On fractional arithmetic", "On astrology", "On areas", and "On finding the unknowns [unknown variables]". He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine.
  • 15th century Ibn al-Banna' al-Marrakushi and Abu'l-Hasan ibn Ali al-Qalasadi introduced symbolic notation for algebra and for mathematics in general. [12]
  • 15th century Nilakantha Somayaji, a Kerala school mathematician, writes the Aryabhatiya Bhasya, which contains work on infinite-series expansions, problems of algebra, and spherical geometry.
  • 1424 – Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons.
  • 1427 Jamshid al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
  • 1464 Regiomontanus writes De Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics.
  • 1478 – An anonymous author writes the Treviso Arithmetic .
  • 1494 Luca Pacioli writes Summa de arithmetica, geometria, proportioni et proportionalità ; introduces primitive symbolic algebra using "co" (cosa) for the unknown.


16th century

17th century

18th century

19th century


20th century


21st century

See also

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  1. Art Prehistory, Sean Henahan, January 10, 2002. Archived July 19, 2008, at the Wayback Machine
  2. How Menstruation Created Mathematics, Tacoma Community College, (archive link).
  3. "OLDEST Mathematical Object is in Swaziland" . Retrieved March 15, 2015.
  4. "an old Mathematical Object" . Retrieved March 15, 2015.
  5. 1 2 "Egyptian Mathematical Papyri - Mathematicians of the African Diaspora" . Retrieved March 15, 2015.
  6. Biggs, Norman; Keith Lloyd; Robin Wilson (1995). "44". In Ronald Graham; Martin Grötschel; László Lovász (eds.). Handbook of Combinatorics (Google book). MIT Press. pp. 2163–2188. ISBN   0-262-57172-2 . Retrieved March 8, 2008.
  7. Carl B. Boyer, A History of Mathematics, 2nd Ed.
  8. Corsi, Pietro; Weindling, Paul (1983). Information sources in the history of science and medicine. Butterworth Scientific. ISBN   9780408107648 . Retrieved July 6, 2014.
  9. Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255–259. Addison-Wesley. ISBN   0-321-01618-1.
  10. F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
  11. O'Connor, John J.; Robertson, Edmund F., "Abu l'Hasan Ali ibn Ahmad Al-Nasawi", MacTutor History of Mathematics Archive , University of St Andrews
  12. 1 2 3 Arabic mathematics, MacTutor History of Mathematics archive , University of St Andrews, Scotland
  13. 1 2 Various AP Lists and Statistics Archived July 28, 2012, at the Wayback Machine
  14. Weisstein, Eric W. "Taylor Series". mathworld.wolfram.com. Retrieved November 3, 2022.
  15. "The Taylor Series: an Introduction to the Theory of Functions of a Complex Variable". Nature. 130 (3275): 188. August 1932. Bibcode:1932Natur.130R.188.. doi: 10.1038/130188b0 . ISSN   1476-4687. S2CID   4088442.
  16. Saeed, Mehreen (August 19, 2021). "A Gentle Introduction to Taylor Series". Machine Learning Mastery. Retrieved November 3, 2022.
  17. D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord [string] forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214-219.
  18. "Sophie Germain and FLT".
  19. Paul Benacerraf and Hilary Putnam, Cambridge University Press, Philosophy of Mathematics: Selected Readings, ISBN   0-521-29648-X
  20. Laumon, G.; Ngô, B. C. (2004), Le lemme fondamental pour les groupes unitaires, arXiv: math/0404454 , Bibcode:2004math......4454L
  21. "UNH Mathematician's Proof Is Breakthrough Toward Centuries-Old Problem". University of New Hampshire. May 1, 2013. Retrieved May 20, 2013.
  22. Announcement of Completion. Project Flyspeck, Google Code.
  23. Team announces construction of a formal computer-verified proof of the Kepler conjecture. August 13, 2014 by Bob Yirk.
  24. Proof confirmed of 400-year-old fruit-stacking problem, 12 August 2014; New Scientist.
  25. A formal proof of the Kepler conjecture, arXiv.
  26. Solved: 400-Year-Old Maths Theory Finally Proven. Sky News, 16:39, UK, Tuesday 12 August 2014.