With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra... - David Eugene Smith

Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.

In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.

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>