19th century

Throughout the 19th century mathematics became increasingly abstract. In this century lived one of the greatest mathematicians of all time, Carl Friedrich Gauss (1777 - 1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on functions of complex variables, in geometry, and on the convergence of series. He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival the Hungarian mathematician Janos Bolyai both independently of each other discovered non-Euclidean geometry, their non-Euclidean geometry was called hyperbolic geometry and differed from traditional Euclidean geometry in that it rejected Euclid's fifth postulate, a rule of Euclidean geometry that states that parallel lines go on to infinity and never intersect, they replaced this with a postulate that allowed parallel lines to intersect, in hyperbolic geometry because parallel lines can intersect triangles have less than 180 degrees. The initial reaction of the mathematical and scientific communities to Lobachevsky's findings was hostile. It was very courageous of Lobachevsky to publish his findings in the face of this opposition. Lobachevsky's essay about non-Euclidean geometry A concise outline of the foundations of geometry was published by the Kazan Messenger but was rejected by Ostrogodski when the St. Petersberg Academy of Sciences submitted it for publication.

Another non-Euclidean geometry called elliptic geometry was developed later in the nineteenth century by the German mathematician George Friedrich Bernhard Riemmann. In elliptic geometry parallel lines do not exist and there are three dimensional triangles with more than 180 degrees. Despite the fact that the mathematical and scientific communities' initial reaction to non-Euclidean geometries was at first negative these new geometries and in particular elliptic geometry later turned out to be crucial to Albert Einstein's theory of relativity;e=mc squared, which is a theory about geometrical gravitational fields, although the theory of relativity also used Euclidean flat space. Also in the nineteenth century William Rowan Hamilton developed noncommutative algebra.

Lobachevsky also discovered a method for finding the approximations of the roots of algebraic equations which is still called the Lobachevsky method in Russia although it is called the Dandelin-Graffe method in the West after two western mathematicians who discovered it independently both of each other and of Lobachevsky.

In addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of calculus, in work by Augustin-Louis Cauchy and Karl Weierstrass.

A new form of algebra was developed in the nineteenth century called Boolean algebra. It was developed by a British mathematician named George Boole. It was a system that contained true and false statements, in it 1 meant true and 0 meant false. Boolean algebra later became important in the twentieth century because it was the math that would be used for computers.

Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four, and other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.

The 19th century also saw the founding of the first mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Mathematico di Palermo in 1884, the Edinburgh Mathematical Society in 1864, and the American Mathematical Society in 1888.

Before the 20th century, the number of creative mathematicians in the world at any one time was limited. For the most part, mathematicians were either born to wealth, like Napier, or supported by wealthy patrons, like Gauss. There were a few meager livelihoods to be had teaching at a university, like Fourier. Niels Henrik Abel, unable to obtain a position, died of tuberculosis.

source :en.wikipedia.org/wiki/History_of_mathematics