250 BCE
Archimedes' Exhaustion Method
Archimedes uses the method of exhaustion to compute areas and volumes, anticipating integral calculus. #analysis #history
Chronological Storyline (43 Milestones)
499 CE
Aryabhata's Sine and Cosine Series
Indian mathematician Aryabhata systematically treats trigonometric series, influencing later calculus. #india #math
By Cpjha13 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=49883192
600 CE
Brahmagupta's Calculus of Finite Differences
Brahmagupta presents interpolation formulas and the sum of squares, foundational for numerical analysis. #india #analysis
900 CE
Al-Kindi's Frequency Analysis
Al-Kindi applies discrete analysis to cryptography, early statistical and analytic thinking. #islamic #math
By Iraqi Post - Personal collection, Public domain, https://commons.wikimedia.org/w/index.php?curid=91832934
1021 CE
Alhazen's Integration of Parabolic Volume
Ibn al-Haytham (Alhazen) computes the volume of a paraboloid via summation, an early integral calculus technique. #islamic #analysis
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
1150 CE
Bhāskara II on Differential Calculus
Bhāskara II discusses the differential of sine and uses the concept of instantaneous motion. #india #calculus
By Unknown author - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=150275955
1350 CE
Madhava's Infinite Series
Madhava of Sangamagrama discovers infinite series for sine, cosine, and arctangent, a major precursor to calculus. #india #analysis
1500 CE
Kerala School of Astronomy and Mathematics
The Kerala school continues developing series expansions and calculus-like techniques independently of Europe. #india #math
By ChandlerMinh - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=92927231
1635 CE
Bonaventura Cavalieri's Method of Indivisibles
Cavalieri introduces geometry of indivisibles, an early integration method influencing real analysis. #analysis #history
By Christopher Grattoni - demonstrations.wolfram.com, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=145384849
1665 CE
Newton's Method of Fluxions
Isaac Newton develops fluxional calculus based on infinitesimals, founding classical analysis. #calculus #analysis
By Isaac Newton - File:The method of fluxions and infinite series.djvu, Public domain, https://commons.wikimedia.org/w/index.php?curid=54955387
1684 CE
Leibniz's First Publication on Differential Calculus
Gottfried Wilhelm Leibniz publishes Nova Methodus, introducing differential notation and laying foundations for real analysis. #calculus #history
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum, online, Public domain, https://commons.wikimedia.org/w/index.php?curid=53159699
1748 CE
Euler's Introductio in Analysin Infinitorum
Leonhard Euler systematizes real analysis, introducing functions and series expansions. #analysis #euler
By Cronholm144 at English Wikipedia - Transferred from en.wikipedia to Commons., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3128774
1817 CE
Bernard Bolzano's Purely Analytic Proof
Bolzano provides a rigorous proof of the intermediate value theorem using the least upper bound property, anticipating arithmetization of analysis. #analysis #rigor
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=73300
1821 CE
Cauchy's Cours d'Analyse
Augustin-Louis Cauchy publishes a rigorous treatment of limits, continuity, and convergence, redefining real analysis. #analysis #rigor
By Augustin Louis Cauchy - screenshot of https://archive.org/details/bub_gb_OlxT3B6EjykC/page/n5/mode/2up, CC0, https://commons.wikimedia.org/w/index.php?curid=94855343
1829 CE
Abel's Theorem on Series
Niels Henrik Abel proves that the binomial series converges for certain conditions, highlighting the need for rigorous convergence criteria. #analysis #series
1851 CE
Riemann's Concept of Definite Integral
Bernhard Riemann formalizes the Riemann integral in his habilitation thesis, a cornerstone of real analysis. #analysis #integration
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
1854 CE
Riemann's Conditions for Trigonometric Series
Riemann integrates discontinuous functions and develops conditions for Fourier series convergence, influencing measure theory. #analysis #fourier
By Unknown author - http://www.sil.si.edu/digitalcollections/hst/scientific-identity/explore.htm according to the German Wikipedia., Public domain, https://commons.wikimedia.org/w/index.php?curid=27383
1872 CE
Weierstrass's Continuous Everywhere, Differentiable Nowhere Function
Karl Weierstrass constructs a function that is continuous but nowhere differentiable, shocking the mathematical community and spurring rigor. #analysis #counterexample
By Eeyore22 - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=5075959
1872 CE
Dedekind Cuts and Real Numbers
Richard Dedekind defines real numbers via cuts, providing a rigorous arithmetical foundation for analysis. #analysis #numbers
By Melikamp - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=147735710
1874 CE
Cantor's Uncountability and Set Theory
Georg Cantor proves that real numbers are uncountable, launching set theory and its impact on measure theory. #settheory #analysis
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
1887 CE
Peano's Space-Filling Curve
Giuseppe Peano constructs a continuous curve that fills a square, challenging intuitive notions of dimension and continuity. #analysis #topology
By No machine-readable author provided. Arbol01 assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1776029
1897 CE
Baire Category Theorem
René-Louis Baire formulates the Baire category theorem, a fundamental tool in functional analysis and real analysis. #analysis #topology
1901 CE
Lebesgue's Theory of Integration
Henri Lebesgue defines measure and the Lebesgue integral, revolutionizing real analysis and measure theory. #measuretheory #integration
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
1902 CE
Lebesgue's Thesis: Intégrale, Longueur, Aire
Lebesgue's doctoral dissertation fully develops measure theory and the Lebesgue integral, becoming a classic. #measuretheory #analysis
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=336482
1904 CE
Zermelo's Axiom of Choice
Ernst Zermelo introduces the axiom of choice, which later proves essential in analysis and measure theory (non-measurable sets). #settheory #analysis
By Wittiko - Own work (based on the original file), CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=189700520
1905 CE
Vitali's Non-Measurable Set
Giuseppe Vitali constructs a set of real numbers that is not Lebesgue measurable, using the axiom of choice. #measuretheory #counterexample
1910 CE
Fatou's Lemma
Pierre Fatou publishes his lemma on the integral of a limit of functions, a key tool in measure theory. #measuretheory #analysis
1913 CE
Hausdorff Measure and Dimension
Felix Hausdorff defines metric dimension and fractional measures, extending measure theory to fractals. #measuretheory #dimension
1920 CE
Banach Contraction Principle
Stefan Banach proves the fixed-point theorem, a cornerstone of functional analysis and real analysis. #analysis #fixedpoint
1923 CE
Wiener's Definition of Brownian Motion
Norbert Wiener constructs a rigorous measure on continuous paths, defining stochastic analysis. #measuretheory #stochastic
By Unknown author, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=146079160
1930 CE
Kolmogorov's Foundations of Probability
Andrey Kolmogorov axiomatizes probability using measure theory, creating modern probability theory. #measuretheory #probability
By Ainali - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3141713
1933 CE
Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung
Kolmogorov's monograph establishes measure-theoretic probability as a branch of analysis. #measuretheory #probability
By Konrad Jacobs - https://opc.mfo.de/detail?photoID=7493, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=11829175
1940 CE
Halmos' Measure Theory Monograph
Paul Halmos publishes 'Measure Theory', a classic textbook that systematizes the field. #measuretheory #education
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
Laurent Schwartz and Distribution Theory
Laurent Schwartz develops distribution theory, generalizing functions and connecting to measure theory. #analysis #distributions )
1963 CE
Lars Hörmander's Linear Partial Differential Operators
Hörmander uses measure theory and functional analysis to solve PDEs, influencing modern analysis. #analysis #pde
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
1966 CE
Robinson's Nonstandard Analysis
Abraham Robinson introduces nonstandard analysis using hyperreal numbers, providing a new foundation for calculus. #analysis #nonstandard
By Christoph Bernhard Francke - Herzog Anton Ulrich-Museum Braunschweig, Public domain, https://commons.wikimedia.org/w/index.php?curid=57268659
1970 CE
David Mumford's Algebraic Measure Theory
Mumford applies measure theory to algebraic geometry, integrating real and complex aspects. #measuretheory #algebraic
By George Bergman - https://opc.mfo.de/detail?photo_id=13740, GFDL 1.2, https://commons.wikimedia.org/w/index.php?curid=25005390
1974 CE
Fourier Analysis on Groups
Walter Rudin publishes 'Fourier Analysis on Groups', connecting measure theory with harmonic analysis. #analysis #fourier
1980 CE
Geometric Measure Theory Advances
Ennio De Giorgi, Herbert Federer, and colleagues develop geometric measure theory, refining minimal surfaces and rectifiable sets. #measuretheory #geometry
1982 CE
Mandelbrot's Fractal Geometry and Measure
Benoit Mandelbrot popularizes fractals, highlighting Hausdorff measure and non-integer dimensions. #measuretheory #fractal
By Steve Jurvetson - https://www.flickr.com/photos/jurvetson/4770047266/, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=155438846
1990 CE
Talagrand's Concentration of Measure
Michel Talagrand develops concentration of measure phenomena, crucial in functional analysis and probability. #measuretheory #probability
1999 CE
Tao's Quantitative Analysis
Terence Tao publishes 'Analysis I & II', modernizing the teaching of real analysis and integrating measure theory. #analysis #education
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
2002 CE
Villani's Optimal Transport and Measure Theory
Cédric Villani uses measure theory to solve the optimal transport problem, revolutionizing the field. #measuretheory #transport
By Marie-Lan Nguyen - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=45294074