Algebras

René Descartes's (1596–1650) Géométrie (1637) was an important publication in the history algebra. While its title shows his main concern, in it he introduced analytic geometry, representing constants and also variable geometric magnitudes by letters. He even found an algebraic means of determining the normal to a curve. Both this method and the representation of variables were to help in the creation of the calculus by Isaac Newton (1642–1727) in the 1660s and Gottfried Wilhelm Leibniz (1647–1716) a decade later.

During the seventeenth century algebra came to be a staple part of mathematics, with textbooks beginning to be published. The binomial theorem was studied, with Newton extending it to non-integral exponents; and functions were given algebraic expression, including as power series. Algebraic number theory developed further, especially with Pierre de Fermat (1601–1665). Negative and complex numbers found friends, including Newton and Leonhard Euler (1707–1783); but some anxiety continued, especially in Britain.

The theory of polynomial equations and their roots remained prominent. In particular, in Descartes's time "the fundamental theorem of algebra" (a later name) was recognized though not proved: that for any positive integer n a polynomial equation of degree n has n roots, real and/or complex. The Italian mathematician J. L. Lagrange (1736–1813) and others tried to prove it during the eighteenth century, but the real breakthrough came from 1799 by the (young) C. F. Gauss (1777–1855), who was to produce three more difficult and not always rigorous proofs in 1816 and 1850. He and others also interpreted complex numbers geometrically instead of algebraically, a reading that gradually became popular.

Another major question concerning equations was finding the roots of a quintic: Lagrange tried various procedures, some elaborated by his compatriot Italian Paolo Ruffini (1765–1822). The suspicion developed that there was no algebraic formula for the roots: the young Norwegian Niels Henrik Abel (1802–1829) showed its correctness in 1826 with a proof that was independent of Lagrange's procedures.

Lagrange was the leading algebraist of the time: from the 1770s he not only worked on problems in algebra but also tried philologically to algebraize other branches of mathematics. He based the calculus upon an infinite power series (the Taylor series); however, his assumption was to be refuted by Augustin-Louis Cauchy (1789–1857) and W. R. Hamilton (1805–1865). He also grounded mechanics upon principles such as that of "least action" because they could be formulated exclusively in algebraic terms: while much mechanics was encompassed, Newtonian and energy mechanics were more pliable in many contexts.