Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received i… - David Eugene Smith

In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation <math>x^2 + ax + b = 0</math> he placed <math>u + z</math> for <math>x</math>. He then had<math>u^2 + (2z + a)u +(z^2 + az + b) = 0.</math>He now let <math>2z + a = 0,</math> whence <math>z = -\frac{1}{2}a,</math>and this gave<math>u^2 - \frac{1}{4}(a^2 - 4b) = 0.</math>
<math>u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.</math>and<math>x = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}.</math>

The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.

He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam.