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… - David Eugene Smith

Although Cardan reduced his particular equations to forms lacking a term in <math>x^2</math>, it was Vieta who began with the general form<math>x^3 + px^2 + qx + r = 0</math>and made the substitution <math>x = y -\frac{1}{3}p,</math> thus reducing the equation to the form<math>y^3 + 3by = 2c.</math>He then made the substitution<math>z^3 + yz = b,</math> or <math>y = \frac{b - z^2}{z},</math>which led to the form<math>z^6 + 2cz^2 = b^2,</math>a sextic which he solved as a quadratic.

The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. ...This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri [of Deventer].

When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation <math>ax^2 + bx + c = 0</math>, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the science was known about 1800 B.C., and probably still earlier.<