Timeline of Mathematics - 19th Century

19th Century

• 1801 – Disquisitiones Arithmeticae, Carl Friedrich Gauss's number theory treatise, is published in Latin
• 1805 – Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations,
• 1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra.
• 1806 – Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram,
• 1807 – Joseph Fourier announces his discoveries about the trigonometric decomposition of functions,
• 1811 – Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration,
• 1815 – Siméon-Denis Poisson carries out integrations along paths in the complex plane,
• 1817 – Bernard Bolzano presents the intermediate value theorem---a continuous function which is negative at one point and positive at another point must be zero for at least one point in between,
• 1822 – Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane,
• 1824 – Niels Henrik Abel partially proves the Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots,
• 1825 – Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis,
• 1825 – Johann Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's Last Theorem for n = 5,
• 1825 – André-Marie Ampère discovers Stokes' theorem,
• 1828 – George Green proves Green's theorem,
• 1829 – Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry,
• 1831 – Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green,
• 1832 – Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory,
• 1832 – Peter Dirichlet proves Fermat's Last Theorem for n = 14,
• 1835 – Peter Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions,
• 1837 – Pierre Wantsel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons
• 1841 – Karl Weierstrass discovers but does not publish the Laurent expansion theorem,
• 1843 – Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
• 1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative,
• 1847 – George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what is now called Boolean algebra,
• 1849 – George Gabriel Stokes shows that solitary waves can arise from a combination of periodic waves,
• 1850 – Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points,
• 1850 – George Gabriel Stokes rediscovers and proves Stokes' theorem,
• 1854 – Bernhard Riemann introduces Riemannian geometry,
• 1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space,
• 1858 – August Ferdinand Möbius invents the Möbius strip,
• 1858 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions,
• 1859 – Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers,
• 1870 – Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate,
• 1872 – Richard Dedekind invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers,
• 1873 – Charles Hermite proves that e is transcendental,
• 1873 – Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,
• 1874 – Georg Cantor proves that the set of all real numbers is uncountably infinite but the set of all real algebraic numbers is countably infinite. His proof does not use his famous diagonal argument, which he published in 1891.
• 1882 – Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge,
• 1882 – Felix Klein invents the Klein bottle,
• 1895 – Diederik Korteweg and Gustav de Vries derive the Korteweg–de Vries equation to describe the development of long solitary water waves in a canal of rectangular cross section,
• 1895 – Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis,
• 1896 – Jacques Hadamard and Charles Jean de la Vallée-Poussin independently prove the prime number theorem,
• 1896 – Hermann Minkowski presents Geometry of numbers,
• 1899 – Georg Cantor discovers a contradiction in his set theory,
• 1899 – David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry,
• 1900 – David Hilbert states his list of 23 problems which show where some further mathematical work is needed.