began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal.
Pascal and Fermat set the groundwork for the investigations of probability theory and the
corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with
his wager, attempted to use the newly developing probability theory to argue for a life devoted
to religion, on the grounds that even if the probability of success was small, the rewards were
infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th
century.
18th century
Leonhard Euler by Emanuel Handmann.
The most influential mathematician of the 18th century was arguably Leonhard Euler. His
contributions range from founding the study of graph theory with the Seven Bridges of
Königsberg problem to standardizing many modern mathematical terms and notations. For
example, he named the square root of minus 1 with the symbol i, and he popularized the use of
the Greek letter {\displaystyle \pi }\pi
to stand for the ratio of a circle's circumference to its
diameter. He made numerous contributions to the study of topology, graph theory, calculus,
combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations
named for him.
Other important European mathematicians of the 18th century included Joseph Louis Lagrange,
who did pioneering work in number theory, algebra, differential calculus, and the calculus of
variations, and Laplace who, in the age of Napoleon, did important work on the foundations of
celestial mechanics and on statistics.
Modern

19th century
Carl Friedrich Gauss.
Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss
(1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables,
in geometry, and on the convergence of series, leaving aside his many contributions to science.
He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the
quadratic reciprocity law.
Behavior of lines with a common perpendicular in each of the three types of geometry
This century saw the development of the two forms of non-Euclidean geometry, where the
parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai
Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently
defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this
geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was
developed later in the 19th century by the German mathematician Bernhard Riemann; here no
parallel can be found and the angles in a triangle add up to more than 180°. Riemann also
developed Riemannian geometry, which unifies and vastly generalizes the three types of
geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and
surfaces.