A history of Cardano's formula

Summary

List of characters

History of the formula

Chronology

References

Proof of the formula

To understand the social context where the story developed, we need to revisit some basic assumptions about the role of research and intellectual discovery. Today we take for granted the notion that an author who makes some significant discovery (at least in mathematics) wants to publish or otherwise disseminate his discovery as soon as possible, in order to gain recognition for it. But in 16th century Italy, making an important mathematical discovery was considered a secret weapon, to be jealously guarded, and only used in order to gain advantage during public contests, that were common among academics in those times.

Bearing in mind this important point, the history of Cardano's formula developed as follows. In 1501-1502 Luca Pacioli (Franciscan friar, mathematician, accountant, collaborator of Leonardo da Vinci) visited the University of Bologna. In his book Summa de Arithmetica he had declared that a solution to the cubic was impossible, arising the interest of many mathematicians. Maybe because of this Scipione del Ferro (lecturer in Arithmetic and Geometry at the same university during Pacioli's tenure) worked on this problem and obtained a solution for the cubic equation in at least one special case. But del Ferro apparently guarded his secret well, and only revealed it to his student Antonio Maria del Fiore sometime before he died, or even on his deathbed in 1526. But he also had a notebook where he recorded his most important discoveries, and the notebook was inherited by his son-in-law Annibale della Nave.

Del Fiore was rather poor and he probably considered the solution of the cubic as a precious tool to gain employment or fame. In 1535 (nine years after del Ferro's death), he decided to make use of his valuable asset and challenged Niccolò Fontana Tartaglia (a brilliant mathematician, engineer, inventor and writer from Brescia) to a public contest, having del Ferro's formula as secret weapon. The contest consisted of thirty problems, that were apparently made public ahead of time. All 30 problems that del Fiore proposed were cubic equations. But he only knew how to solve the special case \(x^3+px=q\). Since negative numbers or zero were not used (or allowed), an equation such as \(x^3=px+q\) was considered different.

By a fantastic twist of events, the night before the contest, in February between the 12th and the 13th, Tartaglia independently discovered a more complete version of del Ferro's formula, that allowed him to solve more types of cubic equations. So on the day of the contest, Tartaglia was able to solve all thirty problems proposed by del Fiore, but del Fiore could only solve those proposed by Tartaglia that his limited version of the formula allowed. This resulted in del Fiore's public humiliation, and Tartaglia becoming famous throughout Italy.

When the well-established, relatively wealthy and powerful Girolamo Cardano in Milan heard of Tartaglia's success, he first tried to obtain the solution of the cubic from him through an intermediary, but with no apparent success. Then in 1539 he was able to convince Tartaglia to visit him in Milan. He lured him with the promise that he would introduce him to Milan's military commander (Alfonso d'Avalos), to whom Tartaglia wanted to show some of his military inventions. After some discussion, and under solemn oath never to disclose his secret, on March 29, 1539 Tartaglia made his formula known to Cardano. In the following years, Cardano, himself a talented scientist and mathematician, built on Tartaglia's work to find an even more complete formula to solve the cubic. And his household servant Lodovico Ferrari (who had quickly become his student and collaborator) was able to use the solution of the cubic to find a way to solve the quartic equation. In addition, in 1543 Cardano traveled to Bologna (hometown of the original discoverer del Ferro) and found the notebook that del Ferro's son-in-law Annibale della Nave had inherited where the original version of the formula (the one used by del Fiore in his failed attempt to challenge Tartaglia) was clearly stated, and pre-dating Tartaglia's discovery by many years.

At this point, considering he had clear evidence of the existence of the first version of the formula before Tartaglia's work, Cardano did not feel bound to the oath he had made to Tartaglia, and he published the formula in his Ars Magna book, that became famous in Italy (and beyond), and that resulted in his name being attached to the formula forever. He scrupulously gave credit to both del Ferro and Tartaglia for the formula.

Besides the fact that he had solved additional cases and wanted to publish Ferrari's solution to the quartic (that could not be explained without first revealing the solution of the cubic), he probably felt exonerated from the oath given to Tartaglia when he discovered the notebook inherited by della Nave. This is what Ferrari wrote on April 1, 1547:
But Tartaglia reacted bitterly to the publication of the book and he publicly accused Cardano of having stolen his formula. Cardano mostly did not respond, but his assistant Ferrari did, and in 1548 he challenged Tartaglia to a public contest. At this point Tartaglia was older and Ferrari had better knowledge of the formulas, including the one for the fourth degree equations. Ferrari won the contest so decisively that Tartaglia left before it was over.

Ferrari went on to become rich and famous. But he died young, in his 40's, and there were rumors that his sister poisoned him. Cardano too had serious tragedies in his life, with one son a gambler who stole from him and was disinherited, and another son being publicly executed for murder in spite of his powerful father doing all he could to try to save his life. Cardano was also arrested and imprisoned by the Inquisition for having published a horoscope of Jesus. He died in 1576, and he may have committed suicide.