Algebras

Lagrange's algebraic ambitions inspired some new algebras from the late eighteenth century onward. The names used below are modern.

Firstly, in differential operators, the process of differentiating a function in the calculus was symbolized by D, with the converse operation of integration taken as 1/ D, with 1 denoting the identity operation; similarly, finite differencing was symbolized by, with summation taken as 1/. Much success followed, especially in solving differential and difference equations, though the workings of the method remained mysterious. One earnest practitioner from the 1840s was George Boole (1815–1864), who then imitated it to form another one, today called Boolean algebra, to found logic.

Secondly, in functional equations, the "object" was the function f itself ("sine of," say) rather than its values. In this context F.-J. Servois (1767–1847) individuated two properties in 1814: "commutative" (fg gf) and "distributive" (f (gh) fg fh); they were to be important also in several other algebras.

As part of his effort to extend Lagrange's algebraization of applied mathematics, Hamilton introduced another new algebra in 1843. He enlarged complex numbers into quaternions q with four units 1, i, j and k:
q:= a + ib + jc + kd, where i2 = j2 = k2 = ijk = −1; and ij = k and ji = −k
and similar properties. He also individuated the property of associativity (his word), where i (jk) (ij) k.

At that time the German Hermann Grassmann (1809–1877) published Ausdehnungslehre (1844), a very general algebra for expressing relationships between geometrical magnitudes. It was capable of several other readings also; for example, later his brother Robert adapted it to rediscover parts of Boolean algebra. Reception of the Grassmanns was much slower than for Hamilton; but by the 1880s their theories were gaining much attention, with quaternions extended to, for example, the eight-unit "octaves," and boasting a supporting "International Association." However, the American J. W. Gibbs (1839–1903) was decomposing quaternions into separate theories of vector algebra and of vector analysis, and this revision came to prevail among mathematicians and physicists.

Another collection of algebras developed to refine means of handling systems of linear equations. The first step (1840s) was to introduce determinants, especially to express the formulae for the roots of systems of linear equations. The more profound move of inventing matrices as a manner of expressing and manipulating systems themselves dates from the 1860s. The Englishmen J. J. Sylvester (1814–1897) and Arthur Cayley (1821–1895) played important roles in developing matrices (Sylvester's word). An important inspiration was their study of quantics, homogeneous polynomials of some degree in any finite number of variables: the task was to find algebraic expressions that preserved their form under linear transformation of those variables. They and other figures also contributed to the important theory of the "latent roots and vectors" (Sylvester again) of matrices. Determinants and matrices together are known today as linear algebra; the analysis of quantics is part of invariant theory.

On polynomial equations, Lagrange's study of properties of functions of their roots led especially from the 1840s to a theory of substitution groups with Cauchy and others, where the operation of replacing one root by another one was treated as new algebra. Abel's even younger French contemporary Évariste Galois (1811–1832) found some remarkable properties of substitutions around 1830.

This theory of substitutions gradually generalized to group theory. In its abstract form, as pioneered by the German Richard Dedekind (1831–1916) in the 1850s, the theory was based upon a given collection of laws obeyed by objects that were not specified: substitutions provided one interpretation, but many others were found, such as their philological intrusion into projective and (non-)Euclidean geometries. The steady accumulation of these applications increased the importance of group theory.

Other algebras also appeared; for example, one to express the basic properties of probability theory. In analysis the Norwegian Sophus Lie (1842–1899) developed in the 1880s a theory of "infinitesimal transformations" as linear differential operators on functions, and formed it as an algebra that is now named after him, including a group version; it has become an important subject in its own right.