  What did "algebra" mean before the development of the algebraic theories of
the 20th century ? This paper stresses the identities taken by the algebraic
practices developped during the century long discussion around the equation
around the equation of secular inequalities (1766- 1874). In 1874, a strong
controversy on the theory of bilinear and quadratic forms opposed Camille
Jordan and Leopold Kronecker. The arithmetical ideal of Kronecker faced
Jordan's claim for the simplicity of his algebraic canonical form. As the
controversy combined mathematical and historical arguments, it gave rise to the
writing of a history of the methods used by Lagrange, Laplace and Weierstrass
in a century long mathematical discussion around the "equation of secular
inequalities".
