The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Fur… - 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.

More than any of his predecessors Plato appreciated the scientific possibilities of geometry. .. By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is... clear. ...That Plato should hold the view... is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.

Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type <math>x^3 + bx = c</math>; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x²