**Tartaglia** was an Italian mathematician who was famed for his algebraic solution of cubic equations which was eventually published in Cardan's *Ars Magna*.

- As a lowly mathematics teacher in Venice, Tartaglia gradually acquired a reputation as a promising mathematician by participating successfully in a large number of debates.
- Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535.
- For the contest between Tartaglia and Fior, each man was to submit thirty questions for the other to solve.
- Fior was supremely confident that his ability to solve cubics would be enough to defeat Tartaglia but Tartaglia submitted a variety of different questions, exposing Fior as an, at best, mediocre mathematician.
- Fior, on the other hand, offered Tartaglia thirty opportunities to solve the 'unknowns and cubes' problem since he believed that he would be unable to solve this type, as in fact had been the case when the contest was set up.
- However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve 'squares and cubes equal to numbers'.
- Tartaglia was then able to solve all thirty of Fior's problems in less than two hours.
- As Fior had made little headway with Tartaglia's questions, it was obvious to all who was the winner.
- Tartaglia did not take his prize for winning from Fior, however, the honour of winning was enough.
- Cardan was greatly intrigued when Zuanne da Coi told him about the contest and he immediately set to work trying to discover Tartaglia's method for himself, but was unsuccessful.
- A few years later, in 1539, he contacted Tartaglia, through an intermediary, requesting that the method could be included in a book he was publishing that year.
- Tartaglia declined this opportunity, stating his intention to publish his formula in a book of his own that he was going to write at a later date.
- Tartaglia, however, refused.
- An incensed Cardan now wrote to Tartaglia directly, expressing his bitterness, challenging him to a debate but, at the same time, hinting that he had been discussing Tartaglia's brilliance with the governor of Milan, Alfonso d'Avalos, the Marchese del Vasto, who was one of Cardan's powerful patrons.
- On receipt of this letter, Tartaglia radically revised his attitude, realising that acquaintance with the influential Milanese governor could be very rewarding and could provide a way out of the modest teacher's job he then held, and into a lucrative job at the Milanese court.
- Cardan was delighted at Tartaglia's new approach, and, inviting him to his house, assured Tartaglia that he would arrange a meeting with d'Avalos.
- So, in March 1539, Tartaglia left Venice and travelled to Milan.
- To Tartaglia's dismay, the governor was temporarily absent from Milan but Cardan attended to his guest's every need and soon the conversation turned to the problem of cubic equations.
- Tartaglia, after much persuasion, agreed to tell Cardan his method, if Cardan would swear never to reveal it and furthermore, to only ever write it down in code so that on his death, nobody would discover the secret from his papers.
- This Cardan readily agreed to, and Tartaglia divulged his formula in the form of a poem, to help protect the secret, should the paper fall into the wrong hands.
- By the time he had reached Venice, Tartaglia was sure he had made a mistake in trusting Cardan and began to feel very angry that he had been induced to reveal his secret formula.
- Cardan published two mathematical books later that year and, as soon as he could get copies, Tartaglia checked to make sure his formula was not included.
- Though he felt a little happier to find that the formula was not included in the texts, when Cardan wrote to him in a friendly manner Tartaglia rebuffed his offer of continued friendship and mercilessly ridiculed his books on the merest trivialities.
- Based on Tartaglia's formula, Cardan and Ferrari, his assistant, made remarkable progress finding proofs of all cases of the cubic and, even more impressively, solving the quartic equation.
- Tartaglia made no move to publish his formula despite the fact that, by now, it had become well known that such a method existed.
- Tartaglia probably wished to keep his formula in reserve for any upcoming debates.
- Cardan and Ferrari travelled to Bologna in 1543 and learnt from della Nave that it had been del Ferro, not Tartaglia, who had been the first to solve the cubic equation.
- Cardan felt that although he had sworn not to reveal Tartaglia's method surely nothing prevented him from publishing del Ferro's formula.
- In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus Ⓣ(The great art of algebraic rules in one book) , or Ars magna Ⓣ(The great art) as it is more commonly known, which contained solutions to both the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula.
- Del Ferro and Tartaglia are credited with their discoveries, as is Ferrari, and the story written down in the text.
- Tartaglia was furious when he discovered that Cardan had disregarded his oath and his intense dislike of Cardan turned into a pathological hatred.
- The following year Tartaglia published a book, New Problems and Inventions which clearly stated his side of the story and his belief that Cardan had acted in extreme bad faith.
- Ars Magna Ⓣ(The great art) had clearly established Cardan as the world's leading mathematician and he was not much damaged by Tartaglia's venomous attacks.
- Ferrari, however, wrote to Tartaglia, berating him mercilessly and challenged him to a public debate.
- Tartaglia was extremely reluctant to dispute with Ferrari, still a relatively unknown mathematician, against whom even a victory would do little material good.
- A debate with Cardan, on the other hand, held great appeal for Tartaglia.
- Not only did he hate him but Cardan was a leading figure in the mathematical, medical and literary worlds, and even to enter a debate with him would greatly enhance Tartaglia's standing.
- For all the brilliance of his discovery of the solution to the cubic equation problem, Tartaglia was still a relatively poor mathematics teacher in Venice.
- So Tartaglia replied to Ferrari, trying to bring Cardan into the debate.
- Cardan, however, had no intention of debating with Tartaglia.
- Ferrari and Tartaglia wrote fruitlessly to each other for about a year, trading the most offensive personal insults but achieving little in the way of resolving the dispute.
- Suddenly in 1548, Tartaglia received an impressive offer of a lectureship in his home town, Brescia.
- To clearly establish his credentials for the post, Tartaglia was asked to journey to Milan and take part in the contest with Ferrari.
- Tartaglia was vastly experienced in such debates and he expected to win.
- Ferrari clearly understood the cubic and quartic equations more thoroughly, and Tartaglia decided that he would leave Milan that night and thus leave the contest unresolved.
- With Tartaglia departing ignominiously, victory was left to Ferrari.
- Tartaglia suffered as a result of the contest.
- Even after numerous lawsuits, Tartaglia could not get any payment and returned, seriously out of pocket, to his previous job in Venice, nursing a huge resentment of Cardan.
- The defeat in Milan would appear to be responsible for Tartaglia's non-payment.
- Tartaglia is now remember in that the name of the formula for solving the cubic has been named the Cardan-Tartaglia formula.
- However, Tartaglia did contribute to mathematics in a number of other ways.
- Tartaglia also published Latin editions of Archimedes' works.

Born 1500, Brescia, Republic of Venice (now Italy). Died 13 December 1557, Venice, Republic of Venice (now Italy).

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive