# How Tartaglia Solved the Cubic Equation - The First Solutions

Author(s):
Friedrich Katscher

The Bolognese university lecturer Scipione dal Ferro (1465-1526) found solutions to equations of types

A) x3+px=q, B) x3=px+q, and C) x3+q=px

in 1505 (by one report) or 1515 (by another report), but did not publish them. One of his pupils, the Venetian arithmetician Antoniomaria Fior, challenged the reckoning master Nicolo Tartaglia (1499/1500-1557) to a written mathematical contest. Each one made up 30 mathematical problems, with Fior's set consisting only of equations of the type we called A), and Tartaglia's 30 different algebraic and geometric questions.

Nicolo Tartaglia (1500-1557)

Luckily Tartaglia found the "general rule," as he called it, for type A) equations on February 12, 1535, eight days before the deadline to bring the solutions to a notary, and for type B) equations on the following day. He solved Fior's problems within two hours, whereas Fior was unable to answer Tartaglia's mathematical questions.

In the 1530s algebraic quantities were not yet represented by letters. The first who did this was the Frenchman François Viète (Latinized Vieta, 1540-1603) in 1591; he used the vowels A, E, I, O, U, Y for the unknowns, and the first Greek consonants - B(eta), G(amma), D(elta), Z(eta) - for constants. The letter x for the unknown was introduced by the Frenchman René Descartes (Latinized Cartesius, 1596-1650) in 1637. Therefore, in the 1530s, you had to either write out in full the terms used, or use abbreviations to save time and space.

Tartaglia denoted the three equations without the quadratic term thus:

A) Capitolo de cubo e cosa equal à numero,

B) Capitolo de cose e numero equal à cubo,

C) Capitolo de cubo e numero equal à cose.

The Italian word capitolo has the meaning of chapter or funny poem, but the mathematicians used it for "type of equation." Cosa, plural cose, means thing and was the term for the unknown, our x, but also for the whole linear term, our px, and even for our p alone. Cubo, of course, was our x3, and numero, number, was the designation of the absolute or constant term, our q. The word de means "of", e "and", and equal à "equal to". Now it is easy to understand the wording of Tartaglia although it is puzzling why in A) he used the singular cosa and in B) and C) the plural cose.

Friedrich Katscher, "How Tartaglia Solved the Cubic Equation - The First Solutions," Convergence (August 2011)