Math History
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non-Math History
Prehistory and Ancient Times | Middle Ages | Renaissance | Reformation | Baroque Era | Enlightenment | Revolutions | Liberalism | |
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1850 | Chebyshev publishes On Primary Numbers in which he proves new results in the theory of prime numbers. He proves Bertrand's conjecture there is always at least one prime between n and 2n for n > 1. |
1850 | In his paper On a New Class of Theorems Sylvester first uses the word "matrix". |
1851 | Bolzano's book Paradoxien des Undendlichen (Paradoxes of the Infinite) is published three years after his death. It introduces his ideas about infinite sets. |
1851 | Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers". In particular he gave the example 0.1100010000000000000000010000... where there is a 1 in place n! and 0 elsewhere. |
1851 | Riemann's doctoral thesis contains ideas of exceptional importance, for example "Riemann surfaces" and their properties. |
1852 | Francis Guthrie poses the Four Colour Conjecture to De Morgan. |
1852 | Sylvester establishes the theory of algebraic invariants. |
1853 | Shanks gives p to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place). |
1854 | Boole publishes The Laws of Thought on Which are founded the Mathematical Theories of Logic and Probabilities. He reduces logic to algebra and this algebra of logic is now known as Boolean algebra. |
1854 | Cayley makes an important advance in group theory when he makes the first attempt, which is not completely successful, to define an abstract group. |
1854 | Riemann completes his Habilitation. In his dissertation he studied the representability of functions by trigonometric series. He gives the conditions for a function to have an integral, what we now call the condition of "Riemann integrability". In his lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854 he defines an n-dimensional space and gives a definition of what today is called a "Riemannian space". |
1856 | Weierstrass publishes his theory of inversion of hyperelliptic integrals in Theorie der Abelschen Functionen which appeared in Crelle's Journal. |
1857 | Riemann publishes Theory of abelian functions. It develops further the idea of Riemann surfaces and their topological properties, examines multi-valued functions as single valued over a special "Riemann surface", and solves general inversion problems special cases of which had been solved by Abel and Jacobi. |
1858 | Cayley gives an abstract definition of a matrix, a term introduced by Sylvester in 1850, and in A Memoir on the Theory of Matrices he studies its properties. |
1858 | Dedekind discovers a rigorous method to define irrational numbers with "Dedekind cuts". The idea comes to him while he is thinking how to teach differential and integral calculus. |
1858 | Möbius describes a strip of paper that has only one side and only one edge. Now known as the "Möbius strip", it has the surprising property that it remains in one piece when cut down the middle. Listing makes the same discovery in the same year. |
1859 | Mannheim invents the first modern slide rule that has a "cursor" or "indicator". |
1859 | Riemann makes a conjecture about the zeta function which involves prime numbers. It is still not known whether Riemann's hypothesis is true in general although it is known to be true in millions of cases. It is perhaps the most famous unsolved problem in mathematics in the 21st century. |
1861 | Weierstrass discovers a continuous curve that is not differentiable at any point. |
1863 | Weierstrass gives a proof in his lecture course that the complex numbers are the only commutative algebraic extension of the real numbers. |
1865 | Plücker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements. |
1868 | Beltrami publishes Essay on an Interpretation of Non-Euclidean Geometry which gives a concrete model for the non-euclidean geometry of Lobachevsky and Bolyai. |
1872 | Dedekind publishes his formal construction of real numbers and gives a rigorous definition of an integer. |
1872 | Heine publishes a paper which contains the theorem now known as the "Heine-Borel theorem". |
1872 | Klein gives his inaugural address at Erlanger. He defines geometry as the study of the properties of a space that are invariant under a given group of transformations. This became known as the "Erlanger programm" and profoundly influences mathematical development. |
1872 | Sylow publishes Théorèmes sur les groupes de substitutions which contains the famous three "Sylow theorems" about finite groups. He proves them for permutation groups. |
1873 | Hermite publishes Sur la fonction exponentielle (On the Exponential Function) in which he proves that e is a transcendental number. |
1874 | Cantor publishes his first paper on set theory. He rigorously describes the notion of infinity. He shows that infinities come in different sizes. He proves the controversial result that almost all numbers are transcendental. |
1877 | Cantor is surprised at his own discovery that there is a one-one correspondence between points on the interval [0, 1] and points in a square. |
1881 | Venn introduces his "Venn diagrams" which become a useful tools in set theory. |
1882 | Lindemann proves that p is transcendental. This proves that it is impossible to construct a square with the same area as a given circle using a ruler and compass. The classic mathematical problem of squaring the circle dates back to ancient Greece and had proved a driving force for mathematical ideas through many centuries. |
1883 | Poincaré publishes a paper which initiates the study of the theory of analytic functions of several complex variables. |
1884 | Frobenius proves Sylow's theorems for abstract groups. |
1884 | Hölder discovers the "Hölder inequality". |
1885 | Weierstrass shows that a continuous function on a finite subinterval of the real line can be uniformly approximated arbitrarily closely by a polynomial. |
1886 | Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution. |
1887 | Levi-Civita publishes a paper developing the calculus of tensors. |
1888 | Dedekind publishes Was sind und was sollen die Zahlen? (The Nature and Meaning of Numbers). He puts arithmetic on a rigorous foundation giving what were later known as the "Peano axioms". |
1888 | Engel and Lie publish the first of three volumes of Theorie der Transformationsgruppen (Theory of Transformation Groups ) which is a major work on continuous groups of transformations. |
1889 | Peano publishes Arithmetices principia, nova methodo exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets. |
1893 | Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics. These papers contain contributions to regression analysis, the correlation coefficient and includes the chi-square test of statistical significance. |
1894 | Borel introduces "Borel measure". |
1895 | Cantor publishes the first of two major surveys on transfinite arithmetic. |
1896 | The prime number theorem is proved independently by Hadamard and de la Vallée-Poussin. This theorem gives an estimate of the number of primes there are up to a given number, showing that the number of primes less than n tends to infinity as n/log n. |
1897 | Burali-Forti is the first to discover a set theory paradox. |
1899 | Hilbert publishes Grundlagen der Geometrie (Foundations of Geometry) putting geometry in a formal axiomatic setting. |
1900 | Hilbert poses 23 problems at the Second International Congress of Mathematicians in Paris as a challenge for the 20th century. The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more. Many of the problems were solved during the 20th century, and each time one of the problems was solved it was a major event for mathematics. |
1901 | Dickson publishes Linear groups with an exposition of the Galois field theory. |
1901 | Lebesgue formulates the theory of measure. |
1901 | Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory. |
1902 | Beppo Levi states the axiom of choice for the first time. |
1902 | Lebesgue gives the definition of the "Lebesgue integral". |
1904 | Zermelo uses the axiom of choice to prove that every set can be well ordered. |
1905 | Einstein publishes the special theory of relativity. |
1906 | Fréchet, in his dissertation, investigated functionals on a metric space and formulated the abstract notion of compactness. |
1906 | Koch publishes Une methode geometrique elementaire pour l'etude de certaines questions de la theorie des courbes plane which contains the "Koch curve". It is a continuous curve which is of infinite length and nowhere differentiable. |
1906 | Markov studies random processes that are subsequently known as "Markov chains". |
1907 | Brouwer's doctoral thesis on the foundations of mathematics attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School. |
1907 | Einstein publishes his principle of equivalence, in which says that gravitational acceleration is indistinguishable from acceleration caused by mechanical forces. It is a key ingredient of general relativity. |
1908 | Hardy and Weinberg present a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. This establishes the mathematical basis for population genetics. |
1908 | Zermelo publishes Untersuchungen über die Grundlagen der Mengenlehre (Investigations on the Foundations of Set Theory). He bases set theory on seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity. This aims to overcome the difficulties with set theory encountered by Cantor. |
1910 | Russell and Whitehead publish the first volume of Principia Mathematica. They attempt to put the whole of mathematics on a logical foundation. They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. The third and final volume will appear three years later, while a fourth volume on geometry was planned but never completed. |
1910 | Steinitz gives the first abstract definition of a field in Algebraische Theorie der Körper. |
1911 | Sergi Bernstein introduces the "Bernstein polynomials" in giving a constructive proof of Weierstrass's theorem of 1885. |
1913 | Hardy receives a letter from Ramanujan. He brings Ramanujan to Cambridge and they go on to write five remarkable number theory papers together. |
1914 | Hausdorff publishes Grundzüge der Mengenlehre in which he creates a theory of topological and metric spaces. |
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1850 to 1860
1860 to 1870 1870 to 1880 1880 to 1890 1890 to 1900 1900 to 1910 1910 to 1920 on MacTutor History of Mathematics |