It is difficult to say who it is who first recognized the advantage of always equating to zero in the study of the general equation. It may very like… - 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.

[Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics <math>x^3 + ax^2 = c</math> and <math>x^3 + bx = c</math> could now be solved. The reduction of the general cubic <math>x^3 + ax^2 + bx = c</math> to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types <math>x^3 = ax^2 + c</math> and <math>x^3 + ax^2 = c</math> by substituting <math>x = y + \frac{1}{3}a</math> and <math>x = y - \frac{1}{3}a</math> respectively, and transformed the type <math>x^3 + c = ax^2</math> by the substitution <math>x = \sqrt {c^2/y},</math> thus freeing the equations of the term <math>x^2</math>. This completed the general solution, and he applied the method to the complete cubic in his later problems.

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.