370 BCE
Eudoxus Develops Method of Exhaustion
Eudoxus of Cnidus formalizes the method of exhaustion, a precursor to integral calculus used to compute areas and volumes. #math #history
Chronological Storyline (40 Milestones)
250 BCE
Archimedes Computes Area Under Parabola
Archimedes uses the method of exhaustion to find the area bounded by a parabola and a line, an early example of integration. #math #history
By Domenico Fetti - http://archimedes2.mpiwg-berlin.mpg.de/archimedes_templates/popup.htm, Public domain, https://commons.wikimedia.org/w/index.php?curid=146592
1400 CE
Madhava Discovers Infinite Series for π
Madhava of Sangamagrama, of the Kerala school, develops infinite series expansions for trigonometric functions, anticipating calculus. #math #india
1635 CE
Cavalieri Introduces Method of Indivisibles
Bonaventura Cavalieri publishes Geometria Indivisibilibus, a geometric method for calculating integrals that influences later calculus. #math #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 creates his method of fluxions and infinite series, laying foundations for differential and integral calculus. #math #calculus
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 Notation
Gottfried Wilhelm Leibniz publishes his differential calculus, introducing the d-notation and integral sign that remain in use. #math #calculus
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum, online, Public domain, https://commons.wikimedia.org/w/index.php?curid=53159699
1817 CE
Bolzano Provides Rigorous Proof of Intermediate Value Theorem
Bernard Bolzano gives a purely analytic proof of the intermediate value theorem without geometric intuition, advancing rigorous analysis. #math #analysis
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=73300
1821 CE
Cauchy Formalizes Limits and Continuity
Augustin-Louis Cauchy in Cours d'Analyse defines limits, continuity, and convergence, providing a rigorous foundation for calculus. #math #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
1826 CE
Abel Proves Insolvability of Quintic and Studies Convergence
Niels Henrik Abel proves the impossibility of solving general quintic equations by radicals and contributes to convergence of series. #math #algebra
By Johan Gørbitz - Originally uploaded to English wikipedia by en:User:Pladask, http://www.math.uio.no/div/abelkonkurransen/, Public domain, https://commons.wikimedia.org/w/index.php?curid=90392
1829 CE
Dirichlet Gives Conditions for Fourier Series Convergence
Johann Peter Gustav Lejeune Dirichlet formulates sufficient conditions for pointwise convergence of Fourier series, advancing analysis. #math #analysis
1854 CE
Riemann Defines the Riemann Integral
Bernhard Riemann presents a rigorous definition of the definite integral, now called the Riemann integral, in his habilitation thesis. #math #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
1860 CE
Weierstrass Provides Epsilon-Delta Definition of Limit
Karl Weierstrass formalizes the rigorous (ε, δ)-definition of limits and continuous functions, standardizing real analysis. #math #analysis
By Eeyore22 - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=5075959
1874 CE
Cantor Establishes Set Theory and Uncountability
Georg Cantor publishes his work on infinite sets, proving that the real numbers are uncountable and founding set theory. #math #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
1878 CE
Dini Defines Uniform Convergence
Ulisse Dini publishes his theory of uniform convergence of functions, a critical concept in real analysis. #math #analysis
By Unknown author - http://biblio.unipi.it/content/image/ulisse-dini-1, Public domain, https://commons.wikimedia.org/w/index.php?curid=16036877
1881 CE
Volterra Introduces Functions of Bounded Variation
Vito Volterra defines functions of bounded variation and studies their differentiability properties, influencing integration theory. #math #analysis
By Unknown author - http://www.phys.uniroma1.it/DipWeb/dottorato/SCUO_VOLTERRA/scuola_volterra.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=16117839
1898 CE
Borel Publishes First Measure Theory
Émile Borel publishes his work on measure theory, introducing Borel sets and the concept of measure of a set. #math #measuretheory
By Agence de presse Mondial Photo-Presse - Bibliothèque nationale de France, Public domain, https://commons.wikimedia.org/w/index.php?curid=15265004
1899 CE
Baire Formulates Baire Category Theorem
René-Louis Baire proves the Baire category theorem, a fundamental result in general topology and functional analysis. #math #topology
1902 CE
Lebesgue Introduces Lebesgue Measure and Integral
Henri Lebesgue defines the Lebesgue measure and integral, revolutionizing integration theory with a more general approach. #math #measuretheory
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=336482
1906 CE
Fréchet Defines Metric Spaces
Maurice Fréchet introduces the concept of metric spaces, abstracting notions of distance and convergence. #math #topology
By Unknown author - MacTutor, Public domain, https://commons.wikimedia.org/w/index.php?curid=1980574
1907 CE
Riesz Representation Theorem
Frigyes Riesz proves the Riesz representation theorem linking linear functionals on continuous functions to measures. #math #analysis
1907 CE
Fubini's Theorem on Interchanging Integrals
Guido Fubini proves the theorem allowing double integrals to be computed as iterated integrals, a key tool in measure theory. #math #analysis
1910 CE
Vitali Convergence Theorem
Giuseppe Vitali proves the Vitali convergence theorem, giving conditions for interchange of limit and integral. #math #analysis
1914 CE
Carathéodory Develops Measure Extension
Constantin Carathéodory provides a general method for extending pre-measures to measures, now a standard construction. #math #measuretheory
1923 CE
Steinhaus on Almost Everywhere Convergence
Hugo Steinhaus studies almost everywhere convergence and related measure-theoretic concepts. #math #analysis
By Unknown author - Księga Pamiątkowa. Stulecie Gimnazjum i Liceum im. Króla Stanisława Leszczyńskiego w Jaśle 1868-1968, Public domain, https://commons.wikimedia.org/w/index.php?curid=2904128
1930 CE
Radon–Nikodym Theorem
Otto Nikodym and Johann Radon prove the Radon–Nikodym theorem concerning derivatives of measures. #math #measuretheory
1932 CE
von Neumann Formulates Spectral Theorem
John von Neumann proves the spectral theorem for unbounded self-adjoint operators, deepening operator theory. #math #analysis
1932 CE
Banach Publishes Théorie des Opérations Linéaires
Stefan Banach's monograph establishes normed vector spaces and uniform boundedness principle, founding functional analysis. #math #analysis
By nieznany/unknown - Warszawski Kalendarz Ilustrowany 1967, Wydawnictwo Warszawskiego Tygodnika "Stolica", Warszawa 1966, p. 152, Public domain, https://commons.wikimedia.org/w/index.php?curid=19058077
1933 CE
Bochner Integral Defined
Salomon Bochner introduces the Bochner integral for functions with values in Banach spaces, extending Lebesgue integration. #math #analysis
1933 CE
Kolmogorov Axiomatizes Probability Theory
Andrey Kolmogorov publishes Foundations of the Theory of Probability, basing probability on measure theory. #math #probability
By Ainali - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3141713
1950 CE
Halmos Publishes Measure Theory
Paul Halmos writes Measure Theory, a classic textbook that standardizes exposition of the subject. #math #books
By George Bergman - https://commons.wikimedia.org/wiki/File:Paul_Halmos_1986.jpg, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=180276883
1951 CE
Schwartz Develops Distribution Theory
Laurent Schwartz formalizes distributions (generalized functions), enabling rigorous treatment of singular phenomena in analysis. #math #analysis )
1953 CE
Rudin Publishes Principles of Mathematical Analysis
Walter Rudin's textbook becomes a standard introduction to real analysis for generations of students. #math #books
1960 CE
Nonstandard Analysis Introduced by Robinson
Abraham Robinson develops nonstandard analysis using hyperreal numbers to provide an alternative foundation for calculus. #math #logic
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum Braunschweig, Public domain, https://commons.wikimedia.org/w/index.php?curid=57268659
1963 CE
Hörmander Publishes on Linear Partial Differential Operators
Lars Hörmander's work on hypoellipticity and Fourier integral operators deepens analysis of PDEs. #math #analysis
By Konrad Jacobs - MFO: https://opc.mfo.de/detail?photoID=1777, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=3902616
1969 CE
Federer's Geometric Measure Theory
Herbert Federer publishes Geometric Measure Theory, developing area and coarea formulas for Lipschitz functions. #math #geometry
1971 CE
Fefferman Proves Pointwise Convergence of Fourier Series
Charles Fefferman resolves the problem of almost everywhere convergence of Fourier series for L² functions. #math #analysis
By Gert-Martin Greuel - MFO: https://opc.mfo.de/detail?photoID=8486, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=3917841
1977 CE
Nelson Formulates Internal Set Theory
Edward Nelson provides an axiomatic approach to nonstandard analysis called internal set theory, simplifying its usage. #math #logic
1982 CE
Freedman Solves the Four-Dimensional Poincaré Conjecture
Michael Freedman proves the topological Poincaré conjecture in dimension 4, using geometric and analytic techniques. #math #topology
By Søren Fuglede Jørgensen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=196091768
1993 CE
Bourgain's Work on Harmonic Analysis
Jean Bourgain makes fundamental contributions to harmonic analysis, including restriction theorems and nonlinear PDEs. #math #analysis
By George Bergman - This image has been extracted from another file, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=74881233
2000 CE
Tao Receives Fields Medal for Real Analysis
Terence Tao wins the Fields Medal for contributions to harmonic analysis, partial differential equations, and combinatorial number theory. #math #award
By Institute for Pure & Applied Mathematics - https://www.youtube.com/watch?v=ddTvK9nlquM, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=191601047