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1870

Benjamin Peirce publishes

1871

Betti publishes a memoir on topology which contains the "Betti numbers".

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

Société Mathématique de France is founded.

1872

Méray publishes *Nouveau précis d'analyse infinitésimale* which aims to present the theory of functions of a complex variable using power series.

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.

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.

1873

Maxwell publishes *Electricity and Magnetism*. This work contains the four partial differential equations, now known as "Maxwell's equations".

1873

Hermite publishes *Sur la fonction exponentielle* (*On the Exponential Function*) in which he proves that *e* is a transcendental number.

1873

Gibbs publishes two important papers on diagrams in thermodynamics.

1873

Brocard produces his work on the triangle.

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.

1876

Gibbs publishes *On the Equilibrium of Heterogeneous Substances* which represents a major application of mathematics to chemistry.

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.

1878

Sylvester founds the *American Journal of Mathematics*.

1879

Kempe published his false proof of the Four Colour Theorem. (See this History Topic.)

1879

Lexis publishes *On the theory of the stability of statistical series* which begins the study of time series.

1879

Kharkov Mathematical Society is founded.

1880

Poincaré publishes important results on automorphic functions.

List of mathematicians alive in 1870.

List of mathematicians alive in 1880.

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