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Both France and Germany were caught up in the age of revolution which swept Europe in the late 18th Century, but the two countries treated mathematics quite differently. Joseph Fourier's study, at the beginning of the 19th Century, of infinite sums in which the terms are trigonometric functions were another important advance in mathematical analysis.

Periodic functions that can be expressed as the sum of an infinite series of sines and cosines are known today as Fourier Series, and they are still powerful tools in pure and applied mathematics.

Fourier following LeibnizEulerLagrange and others also contributed towards defining exactly what is meant by a function, although the definition that is found in texts today - defining it in terms of a correspondence between elements of the domain and the range - is usually attributed to the 19th Century German mathematician Peter Dirichlet.

Even though the Dane Caspar Wessel had produced a very similar paper at the end of the 18th Century, and even though it was Gauss who popularized the practice, they are still known today as Argand Diagrams. Galois ' work also laid the groundwork for further developments such as the beginnings of the field of abstract algebra, including areas like algebraic geometry, group theory, rings, fields, modules, vector spaces and non-commutative algebra.

Germany, on the other hand, under the influence of the great educationalist Wilhelm von Humboldt, took a rather different approach, supporting pure mathematics for its own sake, detached from the demands of the state and military. He is widely regarded as one of the three greatest mathematicians of all times, along with Archimedes and Newton.

Euclidean, hyperbolic and elliptic geometry Later in life, Gauss also claimed to have investigated a kind of non-Euclidean geometry using curved space but, unwilling to court controversy, he decided not to pursue or publish any of these avant-garde ideas.

The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called elliptic geometry, as well as on a generalized theory of all the different types of geometry. Riemannhowever, soon took this even further, breaking away completely from all the limitations of 2 and 3 dimensional geometry, whether flat or curved, and began to think in higher dimensions.

British mathematics also saw something of a resurgence in the early and midth century. Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th Century, it was Charles Babbage in 19th Century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape.

His large "difference engine" of was able to calculate logarithms and trigonometric functions, and was the true forerunner of the modern electronic computer. Although never actually built in his lifetime, a machine was built almost years later to his specifications and worked perfectly.

He also designed a much more sophisticated machine he called the "analytic engine", complete with punched cards, printer and computational abilities commensurate with modern computers. Another 19th Century Englishman, George Peacock, is usually credited with the invention of symbolic algebra, and the extension of the scope of algebra beyond the ordinary systems of numbers.

This recognition of the possible existence of non-arithmetical algebras was an important stepping stone toward future developments in abstract algebra.

In the midth Century, the British mathematician George Boole devised an algebra now called Boolean algebra or Boolean logicin which the only operators were AND, OR and NOT, and which could be applied to the solution of logical problems and mathematical functions.

Boolean algebra was the starting point of modern mathematical logic and ultimately led to the development of computer science. Quaternions, and its later generalization by Hermann Grassmann, provided the first example of a non-commutative algebra i.

The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions. But Cayley was one of the most prolific mathematicians in history, and was a pioneer of modern group theory, matrix algebra, the theory of higher singularities, and higher dimensional geometry anticipating the later ideas of Kleinas well as the theory of invariants.

Throughout the 19th Century, mathematics in general became ever more complex and abstract. But it also saw a re-visiting of some older methods and an emphasis on mathematical rigour.

In the first decades of the century, the Bohemian priest Bernhard Bolzano was one of the earliest mathematicians to begin instilling rigour into mathematical analysis, as well as giving the first purely analytic proof of both the fundamental theorem of algebra and the intermediate value theorem, and early consideration of sets collections of objects defined by a common property, such as "all the numbers greater than 7" or "all right triangles", etc.

Along with Riemann and, particularly, the Frenchman Augustin-Louis Cauchy, Weierstrass completely reformulated calculus in an even more rigorous fashion, leading to the development of mathematical analysis, a branch of pure mathematics largely concerned with the notion of limits whether it be the limit of a sequence or the limit of a function and with the theories of differentiation, integration, infinite series and analytic functions.

InCauchy also proved Cauchy's theorem, a fundamental theorem of group theory, which he discovered while examining permutation groups.St.

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