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1850

Chebyshev publishes

1850

In his paper *On a New Class of Theorems* Sylvester first uses the word "matrix". (See this History Topic.)

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

Sylvester establishes the theory of algebraic invariants.

1852

Francis Guthrie poses the Four Colour Conjecture to De Morgan. (See this History Topic.)

1852

Chasles publishes *Traité de géométrie* which discusses cross ratio, pencils and involutions, all notions which he introduced.

1853

Hamilton publishes *Lectures on Quaternions*.

1853

Shanks gives π to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place).

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".

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. (See this History Topic.)

1855

Maxwell publishes *On Faraday's lines of force* showing that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.

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

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.

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.

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.

1860

Delaunay publishes the first volume of *La Théorie du mouvement de la lune* which is the result of 20 years work. Delaunay solves the three-body problem by giving the longitude, latitude and parallax of the Moon as infinite series.

List of mathematicians alive in 1850.

List of mathematicians alive in 1860.

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