# Cardano's Cubic Formula

The story of the invention and publication of the cubic formula is one of the most fascinating episodes in the history of mathematics. It first saw light of day in the great 1545 treatise Ars Magna by Gerolamo Cardano (1501-1576).

One formula for solving an equation x^3 + mx = n   was found around 1515 by Scipione del Ferro (1465-1526) who passed it on in secret to his pupil Antonio Maria Fior.

Why the secrecy? At the time it was common for mathematicians - in search of glory and sponsorship - to demonstrate their prowess in public contests.

About 1535, Nicolo Fontana (1499-1557), better known as Tartaglia (the stammerer), found a formula for solving an equation x^3 + px^2 = q   lacking the linear term. Being challenged by Fior, he also came up with a way of solving equations without the quadratic term. (Note that all four coefficients in the above equations - m, n, p, q  - were assumed to be positive integers; for, at the time, the negative numbers were not seen as valid quantities.)

Having a mastery of two types of equations against Fior's knowledge of a single formula, Tartaglia triumphed completely.

Later on, Cardano, under a solemn pledge of secrecy, managed to obtain the formulas from Tartaglia, but went on to publish both in his Ars Magna. The publication was followed by an acrimonious dispute in which Ludovico Ferrari (1522-1565), a pupil of Cardano, even accused Tartaglia of plagiarism from a source related to del Ferro.