Cardano and the Solution of the Cubic - University of Kentucky
Cardano and the Solution
of the Cubic
Bryan Dorsey, Kerry-Lyn Downie,
and Marcus Huber
Pacioli
In 1494, the Italian Luca Pacioli produced his
volume titled Summa de Arithmetica.
In this a step was made in the direction of
symbolic algebra in which he treated the
standard mathematics of his day with emphasis
on solving both linear and quadratic equations.
He decided that the cubic was quite impossible
to solve, and thus laid out a challenge to the
Italian mathematical community to find a
solution.
Scipione del Ferro
del Ferro, of the University of Bologna,
decided to take up the challenge.
He discovered a formula that solved the so
called ¡°depressed cubic¡± of the form:
ax? + cx + d = 0
Instead of publishing his solution, del
Ferro kept it a secret until his deathbed,
telling his student Antonio Fior.
Niccolo Fontana - Tartaglia
Fior, with his new weapon, leveled a challenge against
the Brecian scholar Niccolo Fontana, also know as
Tartaglia in 1535.
Tartaglia claimed to know the solution to cubics of the
form: ax? + bx? + d = 0.
Tartaglia sent Fior a list of 30 various mathematical
problems and Fior in turn sent Tartaglia a list of 30
depressed cubics, placing Tartaglia in a bind.
He worked furiously trying to find a solution to these
depressed cubics and on the night of February 13, 1535,
he discovered the solution.
Tartaglia prevailed in the challenge.
Tartaglia¡¯s Poem
When the cube and its things near
Add to a new number, discrete,
Determine two new numbers different
By that one; this feat
Will be kept as a rule
Their product always equal, the same,
To the cube of a third
Of the number of things named.
Then, generally speaking,
The remaining amount
Of the cube roots subtracted
Will be our desired count.
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