Valerio De Angelis
A history of Cardano's formula
Chronology
- 1501-1502 Luca Pacioli visits the University of Bologna and lectures there. He had written in his published work that it was impossible to find a formula for the cubic equation.
- 1502-1526 Sometime after Pacioli's visit and before he died in 1526,
Scipione del Ferro discovers the formula to
solve the cubic equation of the form \( x^3+px=q\). It is not clear if he could also solve other forms.
- 1526 Shortly before he died, or even on his deathbed, del Ferro revealed his discovery to his student Antonio
Maria del Fiore.
- 1535 del Fiore, in hope of acquiring fame and employment, challenges Niccolò Tartaglia to a public contest, having as secret weapon del Ferro's formula for the cubic. But on the night before the contest, on February 12th
1535, Tartaglia independently discovers the solution of the cubic and wins the contest.
- 1539 Having heard of Tartagalia's success, Cardano invites him to Milan, and Tartaglia reveals his formula
to Cardano, under oath that he would never publish it.
- 1539-1545 Cardano extends Tartaglia's result, and Ferrari discovers the solution to the quartic equation.
- 1543 Cardano and Ferrari travel to Bologna and visit Annibale della Nave, who showed them the notebook he
had inherited from del Ferro, with the clearly written solution of the cubic pre-dating Tartagalia's discovery.
- 1545 Cardano publishes Ars Magna, in which he discloses the solution of the cubic, giving proper and
accurate credit to both Tartagalia and del Ferro.
- 1545-1548 Tartaglia and Ferrari spar bitterly and publicly over the published solution for the cubic
in Ars Magna,
Tartaglia accusing Cardano of having stolen his formula and broken his oath, and Ferrari defending his master.
- 1548 On August 10, 1548 Ferrari challenges Tartaglia to a public contest. Tartaglia loses the contest and
leaves before it is over.
List of dates
- Sep 24, 1501: Cardano is born
- Feb 12, 1535: Tartaglia discovers the formula to solve the cubic the night before his contest with del Fiore
- Jan 2, 1539: At Cardano's request, the bookseller Zuan Antonio da Bassano meets with Tartaglia in Venice
to obtain the formula to solve the cubic equation.
- Feb 12, 1539: Cardano sends Tartagalia comments about his book La nuova Scientia and repeats his request
for the formula.
- Mar 13, 1539: Cardano invites Tartaglia to visit him in Milan, promising him to introduce him to Marchese del
Vasto, the Spanish governor of Lombardy.
- March 29, 1539: Cardano swears on oath never to publish Tartaglia's formula.
- Feb 10, 1547: First answer by Ferrari to Tartaglia's attacks on Cardano after the publication
of Ars Magna.
- Feb 19, 1547: The questions for the public dispute between Tartaglia and Ferrari appear.
- Aug 10, 1548: Public dispute in Milan between Tartaglia and Ferrari, won by Ferrari.
- Apr 28, 1576: date of last item mentioned in Cardano's autobiography De Vita Propria Liber
- Sep 21, 1576: Cardano dies.