The 17th century is a famous epoch in the progress of science, and the mathematics in no way lagged behind. The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus (q.v..) Many new fields were opened up, but there was still continual progress in pure algebra. Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see CIRCLE), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions. Diophantine problems were revived by Gaspar Bachet, Pierre Fermat and Euler; the modern theory of numbers was founded by Fermat and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and Fermat, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and others. The germs of the theory of determinants are to be found in the works of Leibnitz; Etienne Bezout utilized them in 1764 for expressing the result obtained by the process of elimination known by his name, and since restated by Arthur Cayley.
In recent times many mathematicians have formulated other kinds of algebras, in which the operators do not obey the laws of ordinary algebra. This study was inaugurated by George Peacock, who was one of the earliest mathematicians to recognize the symbolic character of the fundamental principles of algebra. About the same time, D. F. Gregory published a paper "on the real nature of symbolical algebra." In Germany the work of Martin Ohm (System der Mathematik, 1822) marks a step forward. Notable service was also rendered by Augustus de Morgan, who applied logical analysis to the laws of mathematics.
The geometrical interpretation of imaginary quantities had a far-reaching influence on the development of symbolic algebras. The attempts to elucidate this question by H. Kuhn (1750-1751) and Jean Robert Argand (1806) were completed by Karl Friedrich Gauss, and the formulation of various systems of vector analysis by Sir William Rowan Hamilton, Hermann Grassmann and others, followed. These algebras were essentially geometrical, and it remained, more or less, for the American mathematician Benjamin Peirce to devise systems of pure symbolic algebras; in this work he was ably seconded by his son Charles S. Peirce. In England, multiple algebra was developed by James Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and QUATERNIONS.)
The preceding summary shows the specialized nature which algebra has assumed since the 17th century. To attempt a history of the development of the various topics in this article is inappropriate, and we refer the reader to the separate articles.
REFERENCES.---The history of algebra is treated in all
historical works on mathematics in general (see MATHEMATICS:
References.) Greek algebra can be specially studied in
T. L. Heath's Diophantus. See also John Wallis, Opera
Mathematica (1693-1699), and Charles Sutton, Mathematical and
Philosophical Dictionary (1815), article "Algebra." (C. E.*)
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