## Overview: Mathematics 1800-1899

# Overview: Mathematics 1800-1899

As the eighteenth century drew to a close, mathematics was in a state of rapid change. New areas of mathematics remained wide open to research, while older, established areas of mathematics were finding new applications. Advances in analytic geometry, differential geometry, and algebra all played important roles in the development of mathematics in the eighteenth century. It was calculus, however, which commanded most of the attention of eighteenth-century mathematicians. Discovered by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) late in the seventeenth century, the theory and applications of calculus dominated the mathematical scene throughout the eighteenth century. New methods in calculus were developed by some of the greatest mathematicians in history: Newton, Leibniz, brothers Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748), Leonhard Euler (1707-1783), Joseph Louis Lagrange (1736-1813), and Pierre Simon Laplace (1749-1827), to name a few. However, as these techniques and applications of calculus were developed, the absence of rigor slowly began to become a more important question. Calculus worked: that much could not be argued. But what was the logical basis for the new techniques? Many mathematicians and philosophers of the eighteenth century addressed this question, but it was not finally answered until the nineteenth century. The development of calculus led to huge breakthroughs in the application of mathematics to the sciences and set the stage for much of the mathematical work of the nineteenth century.

Building upon the mathematical work of the eighteenth century and developing new areas of mathematics previously unknown, the nineteenth century witnessed some of the greatest discoveries in mathematics since Euclid (330?-260? b.c.) and the ancient Greeks. New areas of mathematics were discovered, important new applications of established mathematics were found, and advances in the understanding of the logical foundations of mathematics were made. Amidst these mathematical developments in the nineteenth century, the foundations of a professional mathematics community were being built.

Many new fields developed within mathematics in the nineteenth century. These emerging areas of mathematics had important ramifications for not only mathematics, but also for how man understood the world around him. For many centuries, since Euclid wrote *The Elements* around 300 B.C., Euclidean geometry was thought to be the only possible form of geometry. The development of non-Euclidean geometry by Wolfgang Bolyai (1775-1856), Nicolai Lobachevski (1793-1856), and Johann Gauss (1777-1855) radically changed how mathematics was viewed. Broken from the constraints of Euclid's ancient axioms, mathematicians began to think about how basic assumptions affected their work. In addition, the existence of non-Euclidean geometry caused doubts to arise concerning the very nature of human knowledge. If such an ancient and universal system of thought had alternative meanings, what other shrines of the human intellect would also be challenged?

The work of Georg Cantor (1845-1918) and others in the emerging field of set theory was another important development in nineteenth-century mathematics. This work led to new conceptions concerning the foundations of mathematics and new ideas about infinity, including the stunning conclusion that different sizes of infinity existed. Another development in mathematics that eventually changed our world was a new algebra invented by the English mathematician George Boole (1815-1864). Boolean algebra, developed by Boole as a logical symbolic language, became one of the foundations upon which computer science was built in the twentieth century. Other new developments in mathematics include the introduction of the Fourier series and the development of hypercomplex numbers. Also important was the work by Gauss, Georg Riemann (1826-1866), August Möbius (1790-1868) and others, which would form the basis of topology.

Many areas of mathematics, although not entirely new, received much attention during the nineteenth century. Euler had revived interest in number theory in the eighteenth century. New work in number theory by Gauss, Peter Dirichlet (1805-1859), Ernst Kummer (1810-1893), Riemann, and others in the nineteenth century led to renewed attempts to prove Pierre de Fermat's Last Theorem and the Prime Number Theorem. New concepts in probability and statistics revolutionized ideas concerning determinism in nature. Applications of these concepts led to revolutionary work by James Clerk Maxwell (1831-1879) and Laplace in the physical sciences and by Lambert Quetelet (1796-1874) and others in the social sciences. Calculus, which had been an indispensable tool for scientists for over a century, was finally given a rigorous foundation by Augustin Louis Cauchy (1789-1857), Riemann, and Karl Weierstrass (1815-1897), among others.

Other important advances in mathematics in the nineteenth century included work on descriptive geometry by Jean-Victor Poncelet (1788-1867), Gauss's proof of the Fundamental Theorem of Algebra, the work of Niels Abel (1802-1829) and Karl Jacobi (1804-1851) on elliptical functions, the codification of mathematical induction, advances in the theory of complex numbers, and the work of Jules Henri Poincaré (1854-1912), which became the seed for the twentieth-century discovery of chaos theory.

Without the developments in mathematics in the nineteenth century, many of the breathtaking advances in science would have been impossible. The work of George Green (1793-1841) involving electricity and magnetism, Maxwell's statistical theory of gases, and Laplace's extraordinary work in celestial mechanics all involved difficult mathematical concepts not available to scientists only a century earlier. Imaginary numbers, not even accepted by most mathematicians until the previous century, found application in electrical engineering, wave mechanics, and other areas of physical science.

Finally, the organization of the mathematical community underwent drastic changes in the nineteenth century. Prior to he nineteenth century, there was no "mathematical profession." Mathematics did not exist, or existed very tenuously at best, outside of its scientific applications. The nineteenth century, however, saw several changes that allowed for the development of a separate mathematics profession. Early in the century, a distinction began to appear between applied mathematics and pure mathematics. Mathematical research was accepted for its own sake, independent of its applications. This led to the establishment of new professorships in mathematics at universities throughout Europe, to new periodicals dedicated to pure mathematics, and to professional societies for mathematicians. All of these things were virtually nonexistent only a century before.

The new discoveries in mathematics, the new applications of mathematics to the sciences, and the professionalization of the field of mathematics also led to new ideas in mathematics education. Mathematics became required knowledge for the educated person who was interested in keeping up with the flurry of changes in the world. All in all, the developments of the nineteenth century had profound affects upon how the nature of mathematics was viewed, how mathematics was applied to science, and even how man viewed the world in which he lived. These developments impacted society in ways that continue to be studied by historians today.

The mathematical discoveries of the nineteenth century led to more breakthroughs in the twentieth century. New fields of mathematics such as abstract algebra and topology were developed. New applications of mathematics in such diverse areas as chaos theory and cryptography were found. The entire foundation of mathematics has come into question with work on set theory, logic, and with Kurt Gödel's incompleteness theorems. Finally, long-standing problems in mathematics have been solved. One of these problems, the four-color problem, introduced computers into mathematical proofs. Another, the proof of Fermat's Last Theorem, solved the most famous problem in modern mathematics. Each of these developments in mathematics in the twentieth century found their roots in the mathematics of the nineteenth century.

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## Overview: Mathematics 1800-1899

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**Overview: Mathematics 1800-1899**