Tartaglia of the Depressed Cubic: A Mathematician's Duel

Prelude: The Mathematical Abyss

In the early 1500s, Europe was climbing out of the mathematical shadows of the Middle Ages. Algebra had been reborn through translations of Arabic and Greek works. Yet, one ancient problem still haunted every serious mathematician:

Solve the general cubic equation.

Specifically, the form of the equation we now call the "depressed cubic":

       x³ + ax = b

This was not just a technical curiosity — it was a badge of honor. Solving it meant ascending into the pantheon of mathematical greats.

Act I: Scipione del Ferro — The Keeper of the Secret

In the city of Bologna, a quiet, brilliant man named Scipione del Ferro held the position of mathematics professor at the University of Bologna. Around 1515, del Ferro made a breakthrough — he discovered how to solve the depressed cubic.

Del Ferro, cautious and soft-spoken, chose not to publish his discovery. He feared scrutiny, misappropriation, and possibly religious controversy. Instead, he passed the method down only to his student, Antonio Maria Fiore, instructing him to guard it closely — like a family heirloom.

Del Ferro died in 1526, taking the complete proof to his grave, leaving only his notes — and Fiore.

Act II: The Duel – Fiore vs. Tartaglia

Fiore, proud of his inheritance, believed he could capitalize on the secret. In 1535, he publicly challenged Niccolò Fontana, known as Tartaglia ("the stammerer" — a nickname from a childhood injury during war), to a mathematical duel.

These duels were intellectual gladiator matches. Each mathematician would pose 30 problems to the other. Whoever solved the most — won fame, money, and patronage.

Fiore, relying solely on problems in the form of the depressed cubic, believed his secret gave him an edge.

But Tartaglia — a self-taught genius from Brescia — had independently discovered a solution to the depressed cubic just days before the contest.

The result?

Tartaglia obliterated Fiore.

He solved all of Fiore’s problems within hours.

Fiore solved none of Tartaglia’s.

Tartaglia became a celebrity overnight.

Act III: The Oath and the Betrayal

Word of Tartaglia’s victory spread fast — and reached Gerolamo Cardano, a brilliant polymath in Milan, known for his flamboyant personality and thirst for glory.

Cardano desperately wanted Tartaglia’s solution, hoping to include it in a major mathematical work. But Tartaglia, still burned by Fiore’s arrogance, refused.

Cardano persisted, promising not to publish the solution if Tartaglia would share it in confidence.

Eventually, Tartaglia relented. In February 1539, he gave Cardano the method in the form of a cryptic poem, swearing him to secrecy. The method involved a clever substitution that transformed the cubic into a solvable form.

 “Don’t publish it,” Tartaglia said.

“I give this to you as a friend and fellow scholar.”

Cardano swore.

Act IV: The Broken Oath — Ars Magna

Years passed. In 1545, Cardano and his student Lodovico Ferrari made a discovery: del Ferro had found the solution before Tartaglia.

To Cardano, this nullified his oath — it was no longer just Tartaglia’s secret. And so, in his magnum opus, Ars Magna, Cardano published the solution to the depressed cubic.

He credited del Ferro, gave partial credit to Tartaglia, and never apologized.

“Truth,” Cardano would later write,

“is the daughter of time, not of authority.”

Tartaglia was furious. He accused Cardano of betrayal, plagiarism, and deception. He published scathing pamphlets, mocking Cardano. The two exchanged bitter attacks, each defending their honor.

Act V: Ferrari’s Revenge

Cardano remained largely silent. But his student Ferrari, fiercely loyal and mathematically gifted, took up the fight.

In 1548, Ferrari challenged Tartaglia to a public debate in Milan. By then, Ferrari had also mastered solutions to general cubics and quartics.

Tartaglia accepted.

In front of a roaring crowd, the debate turned into a humiliation for Tartaglia. Ferrari’s arguments were sharper, faster, and he had Cardano’s prestige behind him.

Tartaglia fled Milan that night. His star dimmed.

Epilogue: The Preserved Diary and Lost Notes

Cardano’s Ars Magna became one of the most influential works in the history of mathematics. But much of del Ferro’s original work — including his personal notes and student accounts — were lost or only partially preserved. Some claim Fiore kept a diary, now lost, describing del Ferro's proof and his own bitterness after defeat.

Tartaglia, later in life, faded into poverty and obscurity. He continued his work in military science and translation, but the mathematical world moved on.

Cardano, in a cruel twist of fate, was later imprisoned by the Inquisition for heresy. He died in 1576 — proud, unrepentant, and convinced of his place in history.

Legacy of the Depressed Cubic

This was the first real breakthrough in solving cubic equations, setting the stage for modern algebra.

It also marks the first time complex numbers (then called “imaginary” and feared) appeared naturally in solutions, even when the final answer was real.

The drama surrounding it became a foundational story of the ethics of knowledge, ego, collaboration, and betrayal.

                                                                                                                       

 

-- Pavitra Kanetkar