plications in algebraical geometry and other parts of mathematics. For
further developments of the theory of determinants see ALGEBRAIC FORMS.
(A. CA.)

9. _History._--These functions were originally known as "resultants,"
a name applied to them by Pierre Simon Laplace, but now replaced by
the title "determinants," a name first applied to certain forms of
them by Carl Friedrich Gauss. The germ of the theory of determinants
is to be found in the writings of Gottfried Wilhelm Leibnitz (1693),
who incidentally discovered certain properties when reducing the
eliminant of a system of linear equations. Gabriel Cramer, in a note
to his _Analyse des lignes courbes algebriques_ (1750), gave the rule
which establishes the sign of a product as _plus_ or _minus_ according
as the number of displacements from the typical form has been even or
odd. Determinants were also employed by Etienne Bezout in 1764, but
the first connected account of these functions was published in 1772
by Charles Auguste Vandermonde. Laplace developed a theorem of
Vandermonde for the expansion of a determinant, and in 1773 Joseph
Louis Lagrange, in his memoir on _Pyramids_, used determinants of the
third order, and proved that the square of a determinant was also a
determinant. Although he obtained results now identified with
determinants, Lagrange did not discuss these functions systematically.
In 1801 Gauss published his _Disquisitiones arithmeticae_, which,
although written in an obscure form, gave a new impetus to
investigations on this and kindred subjects. To Gauss is due the
establishment of the important theorem, that the product of two
determinants both of the second and third orders is a determinant. The
formulation of the general theory is due to Augustin Louis Cauchy,
whose work was the forerunner of the brilliant discoveries made in the
following decades by Hoene-Wronski and J. Binet in France, Carl Gustav
Jacobi in Germany, and James Joseph Sylvester and Arthur Cayley in
England. Jacobi's researches were published in _Crelle's Journal_
(1826-1841). In these papers the subject was recast and enriched by
new and important theorems, through which the name of Jacobi is
indissolubly associated with this branch of science. The far-reaching
discoveries of Sylvester and Cayley rank as one of the most important
developments of