La Habra High School
Math History Timeline

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Age of Liberalism
1848-1914 A.D.

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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