Among his <nowiki>[</nowiki>John Wallis'<nowiki>]</nowiki> interesting discoveries was the relation <math>\frac{4}{\pi} = \frac32\cdot\frac34\cdot\fr… - David Eugene Smith

Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants.

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>

Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations.
The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence.
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work... to appear in print was the General Trattato of Tartaglia...