370 BCE
Eudoxus' Theory of Proportions
Eudoxus of Cnidus develops a theory of proportions that deals with incommensurable magnitudes, laying the groundwork for the concept of real numbers. #mathematics #history
Chronological Storyline (39 Milestones)
225 BCE
Archimedes' Method of Exhaustion
Archimedes uses the method of exhaustion to compute areas and volumes, a precursor to integral calculus. #mathematics #history
263 CE
Liu Hui's Approximation of Pi
Chinese mathematician Liu Hui computes π using inscribed polygons, achieving an accuracy of 3.1416. #mathematics #history
By Unknown - "Liu Hui (220 - 280) - Biography - MacTutor History of Mathematics" https://mathshistory.st-andrews.ac.uk/Biographies/Liu_Hui/, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=130481910
500 CE
Aryabhata's Approximation of Pi
Indian mathematician Aryabhata approximates π as 62832/20000 ≈ 3.1416, and provides sine tables. #mathematics #history
By Cpjha13 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=49883192
1000 CE
Alhazen's Quadrature of the Parabola
Alhazen (Ibn al-Haytham) uses integration-like methods to find the volume of a paraboloid. #mathematics #history
By Adolph Boÿ, engraved by Jeremias Falck. Used as the frontispiece to Johannes Hevelius, Selenographia, 1647 - https://www.loc.gov/resource/rbctos.2017rosen1321/?sp=9&r=0.059,0.617,0.327,0.182,0, Public domain, https://commons.wikimedia.org/w/index.php?curid=165125519
1635 CE
Cavalieri's Principle
Bonaventura Cavalieri introduces the principle that areas and volumes can be compared via cross-sections, a precursor to integral calculus. #mathematics #history
By Christopher Grattoni - demonstrations.wolfram.com, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=145384849
1665 CE
Newton Develops Calculus
Isaac Newton develops his method of fluxions, laying the foundation for differential and integral calculus. #mathematics #history
By Godfrey Kneller - File:Portrait of Sir Isaac Newton, 1689.jpg from https://exhibitions.lib.cam.ac.uk/linesofthought/artifacts, Public domain, https://commons.wikimedia.org/w/index.php?curid=132521185
1684 CE
Leibniz Publishes Calculus
Gottfried Wilhelm Leibniz publishes his first paper on differential calculus, introducing the notation still used today. #mathematics #history
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum, online, Public domain, https://commons.wikimedia.org/w/index.php?curid=53159699
1821 CE
Cauchy's Epsilon-Delta Definitions
Augustin-Louis Cauchy provides rigorous definitions of limits, continuity, and derivatives using epsilon-delta, founding modern analysis. #mathematics #analysis
By Public domain - Library of Congress Prints and Photographs Division. From an illustration in: Das neunzehnte Jahrhundert in Bildnissen / Karl Werckmeister, ed. Berlin : Kunstverlag der photographische gesellschaft, 1840, vol. V, no. 581., Public domain, https://commons.wikimedia.org/w/index.php?curid=7059486
1854 CE
Riemann Integral
Bernhard Riemann defines the Riemann integral, formalizing integration of functions on intervals. #mathematics #analysis
By Brad219 - Own work created using Inkscape 1.3. Redrawn from original PNG source authored by KSmrq, CC0, https://commons.wikimedia.org/w/index.php?curid=150414772
1872 CE
Weierstrass' Definition of Real Numbers
Karl Weierstrass provides a rigorous construction of real numbers using bounded monotone sequences, solidifying analysis foundations. #mathematics #analysis
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=324146
1874 CE
Cantor's Set Theory and Uncountability
Georg Cantor introduces set theory and proves the uncountability of real numbers, leading to the continuum hypothesis. #mathematics #settheory
By Unknown author - https://www.math.cmu.edu/~rcristof/pdf/Cantor_pubblicato.pdf and they had it from https://photos.aip.org/history-programs/niels-bohr-library/photos/cantor-georg-a1, Public domain, https://commons.wikimedia.org/w/index.php?curid=74820875
1892 CE
Jordan Measure
Camille Jordan introduces the Jordan measure, a precursor to the Lebesgue measure for subsets of Euclidean space. #mathematics #measuretheory
1894 CE
Stieltjes Integral
Thomas Joannes Stieltjes generalizes the Riemann integral to the Riemann–Stieltjes integral, integrating with respect to a function. #mathematics #analysis
1898 CE
Borel Measure Theory
Émile Borel introduces Borel sets and the concept of measure, laying groundwork for measure theory. #mathematics #measuretheory
By Agence de presse Mondial Photo-Presse - Bibliothèque nationale de France, Public domain, https://commons.wikimedia.org/w/index.php?curid=15265004
1901 CE
Lebesgue Measure and Integral
Henri Lebesgue defines the Lebesgue measure and integral, revolutionizing integration theory and enabling integration of more functions. #mathematics #measuretheory
By ~~helix84 15:43, 27 November 2006 (UTC) - own work, based on en:Image:Integral-area-under-curve.png, Public domain, https://commons.wikimedia.org/w/index.php?curid=1409775
1906 CE
Fréchet's Metric Spaces
Maurice Fréchet introduces metric spaces, providing a general framework for analysis and topology. #mathematics #analysis
By Unknown author - MacTutor, Public domain, https://commons.wikimedia.org/w/index.php?curid=1980574
1907 CE
Fubini's Theorem
Guido Fubini proves that the double integral of a measurable function can be computed as iterated integrals. #mathematics #analysis
1908 CE
Vitali Covering Theorem
Giuseppe Vitali proves a covering lemma essential for measure theory and differentiation. #mathematics #measuretheory
1913 CE
Radon-Nikodym Theorem
Johann Radon and Otto Nikodym prove the Radon-Nikodym theorem, characterizing absolutely continuous measures. #mathematics #measuretheory
1914 CE
Perron Integral
Oskar Perron introduces the Perron integral, a generalization of the Riemann integral. #mathematics #analysis
1914 CE
Carathéodory's Extension Theorem
Constantin Carathéodory proves a theorem extending pre-measures to measures, a cornerstone of measure theory. #mathematics #measuretheory
1915 CE
Luzin's Theorem
Nikolai Luzin proves that every measurable function is nearly continuous, linking measure and topology. #mathematics #analysis
1918 CE
Hausdorff Dimension
Felix Hausdorff introduces the Hausdorff dimension, a concept for measuring fractal sets. #mathematics #geometry
By Original: Chas_zzz_brown,Shibboleth Vector: The original uploader was Wxs at Wikimedia Commons. - Own work based on: KochFlake.png, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1898291
1919 CE
Littlewood's Three Principles
John Edensor Littlewood formulates three heuristic principles linking measure and topology. #mathematics #analysis
1923 CE
Wiener Measure
Norbert Wiener constructs a measure on the space of continuous paths, essential for stochastic analysis. #mathematics #probability
By Unknown author, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=146079160
1924 CE
Banach-Tarski Paradox
Stefan Banach and Alfred Tarski prove the paradoxical decomposition of a ball using the axiom of choice, challenging notions of measure. #mathematics #paradox
1930 CE
Hardy-Littlewood Maximal Function
Godfrey Harold Hardy and John Edensor Littlewood introduce the maximal function, a tool in harmonic analysis. #mathematics #analysis
1933 CE
Kolmogorov's Probability Axioms
Andrey Kolmogorov axiomatizes probability theory using measure theory, unifying the field. #mathematics #probability
By Ainali - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3141713
1944 CE
Itô Calculus
Kiyoshi Itô develops stochastic calculus, introducing the Itô integral for stochastic processes. #mathematics #stochastic
By Shiyu Ji - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=53067749
1950 CE
Calderón-Zygmund Theory
Alberto Calderón and Antoni Zygmund develop singular integral operator theory, fundamental for PDEs. #mathematics #analysis
1957 CE
De Giorgi's Regularity Theorem
Ennio de Giorgi proves regularity of minimal surfaces, a landmark in geometric measure theory. #mathematics #geometry
1957 CE
Henstock-Kurzweil Integral
Ralph Henstock and Jaroslav Kurzweil independently develop the gauge integral, a generalization of the Riemann and Lebesgue integrals. #mathematics #analysis
1960 CE
Geometric Measure Theory
Herbert Federer and Wendell Fleming lay foundations of geometric measure theory, generalizing integration on manifolds. #mathematics #geometry
1960 CE
Non-standard Analysis
Abraham Robinson develops non-standard analysis, introducing infinitesimals rigorously. #mathematics #analysis
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum Braunschweig, Public domain, https://commons.wikimedia.org/w/index.php?curid=57268659
1975 CE
Loeb Measure
Peter Loeb constructs a measure space from nonstandard inner models, linking nonstandard analysis to measure theory. #mathematics #measuretheory
1976 CE
Malliavin Calculus
Paul Malliavin introduces stochastic calculus of variations, extending Itô calculus. #mathematics #stochastic
1989 CE
Mumford-Shah Functional
David Mumford and Jayant Shah propose a variational model for image segmentation, applied in geometric measure theory. #mathematics #applied
2012 CE
Tao's Solution to Hilbert's Fifth Problem
Terence Tao (with others) resolves Hilbert's fifth problem concerning Lie groups, using analysis and measure theory. #mathematics #history