Lagrange and the Solution of Numerical Equations

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed a general systematic algorithm for detecting, isolating, and approximating all real and complex roots of a polynomial equation with real coeecients, with arbitrary precision. In contrast to Newton's Method, Lagrange's algorithm is guaranteed to converge. We discuss some lesser known aspects of Lagrange's work. In particular, some of his powerful ideas and techniques adumbrated methods developed much later in geometry and abstract algebra, such as MM obius transformations and quotient rings of polynomial rings. We also show that his techniques included accelerating both the convergence and calculation of his continued fraction expansions of the roots.