He reviews how the

theory of equations evolved from ancient times to its completion by Galois around 1830.

If the [n.sup.2.sub.1] and [n.sup.2.sub.1] are the roots of the quadratic equation (51) in [n.sup.2], then from the

theory of equations we have

Geometrical

theory of equations with partial derivatives.

He explains Galois theory, including such topics as splitting fields and their automorphisms, the characteristics of a field, derivation of a polynomial (multiple roots), the degree of an extension field, group characters, fundamental theorems and finite fields, then moves to polynomials with integral coefficients, including irreducibility and primitive roots of unity, and the

theory of equations, including ruler and compass constructions, and the theorems of Steinitz and Abel.

It could only do some partial unifications, such as the geometry of conics and the

theory of equations. As the algebra of structures was yet to be created, mathematicians had no other choice but to find another way: the idea was to find a discipline that was logically prior to all of the other mathematical disciplines, but which did not predate them historically, and was necessarily posterior to all of them, so that it was able to actually provide them with the unifying principles.

The Unit Coefficient Reduction Algorithm, which is based on a similar algorithm generally found in textbooks involving the

theory of equations, is used to resolve the question (MacDuffee, 1954).

New First Course in the

Theory of Equations, John Wiley and Son, Inc., New York, p.

Mahoney details the evolution and significance of the enormous mathematical strides taken by Fermat: analytic geometry,

theory of equations, methods of finding maxima and minima and tangents of lines, the quadrature and rectification of curves.

More specifically, the paper demonstrates the applicability of Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the

theory of equations and rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).