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

[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.

The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.

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.