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The most brilliant conquest in algebra during the sixteenth century had been the solution of cubic and bi-quadratic equations. All attempts at solving algebraically equations of higher degrees remaining fruitless, a new line of inquiry—the properties of equations and their roots—was gradually opened up. We have seen that Vieta had attained a partial knowledge of the relations between roots and coefficients. Peletarius, a Frenchman, had observed as early as 1558, that the root of an equation is a divisor of the last term. One who extended the theory of equations somewhat further than Vieta, was Albert Girard (1590–1634), a Flemish mathematician. Like Vieta, this ingenious author applied algebra to geometry, and was the first who understood the use of negative roots in the solution of geometric problems. He spoke of imaginary quantities; inferred by induction that every equation has as many roots as there are units in the number expressing its degree; and first showed how to express the sums of their powers in terms of the coefficients. Another algebraist of considerable power was the English Thomas Harriot (1560–1621). He accompanied the first colony sent out by Sir Walter Raleigh to Virginia. After having surveyed that country he returned to England. As a mathematician, he was the boast of his country. He brought the theory of equations under one comprehensive point of view by grasping that truth in its full extent to which Vieta and Girard only approximated; viz. that in an equation in its simplest form, the coefficient of the second term with its sign changed is equal to the sum of the roots; the coefficient of the third is equal to the sum of the products of every two of the roots; etc. He was the first to decompose equations into their simple factors; but, since he failed to recognise imaginary and even negative roots, he failed also to prove that every equation could be thus decomposed. Harriot made some changes in algebraic nota-