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Niccolo Fontana Tartaglia


Niccolo Fontana known as Tartaglia, was born in Brescia in 1499, the son of a humble mail rider. He was nearly killed as a teenager, when in 1512 the French captured his home town and put it to the sword. Amidst the general slaughter, the twelve year old boy was dealt horrific facial saber wounds that cut his jaw and palate and he was left for dead. His mother's tender care ensured that the youngster did survive, but in later life Niccolo always wore a beard to camouflage his disfiguring scars and he could only speak with difficulty, hence his nickname Tartaglia, or stammerer.

Tartaglia was self taught in mathematics but, having an extraordinary ability, was able to earn his living teaching at Verona and Venice. 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.

The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. Each man was to submit thirty questions for the other to complete. Fior was supremely confident that his ability to solve cubics would be enough to defeat Tartaglia but because negative numbers were not used there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type. 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 cosa and cube problem since he believed that he would be unable to solve this type. However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve both types of cubic. Tartaglia now knowing the method to solve the cosa and cube problems, quickly solved 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 the winner was.

At this point that Cardan enters the story. As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the cosa and cube problems, but, until the contest, he had taken Pacioli at his word and assumed as Pacioli stated in the Suma published in 1494, that solutions were impossible. Cardan was greatly intrigued when he learned of the contest and immediately set to work on 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. Cardan, accepting this, then asked to be shown the method, promising, if he was, to keep it secret. 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 the emperor's army in Milan, Alfonso d'Avalos, the Marchese del Vasto, who one of Cardan's powerful sponsors. On receipt of this letter, Tartaglia radically revised his attitude, realizing 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. He wrote back to Cardan in friendly terms, angling for an introduction to the Signor Marchese. 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 traveled 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 the cosa and cube problem. 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 a poem, to help protect the secret, should the paper fall into the wrong hands. Anxious now to leave Cardan's house, he obtained from his host, a letter of introduction to the Marchese and left to seek him out. Instead though, he turned back for Venice, wondering if his decision to part with his formula had been a mistake.

By the time he 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 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 traveled to Bologna and learned from della Nave that del Ferro and not Tartaglia 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 or Ars magna as it is more commonly known which contained solutions to the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula. Del Ferro and Tartaglia are fully credited with their discoveries, as is Ferrari, and the whole 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. For good measure, he added a few malicious personal insults directed against Cardan.

Ars Magna 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 cosa and cube 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 credential for the post, Tartaglia was asked to journey to Milan and take part in the contest with Ferrari.

On 10 August 1548 the contest took place in the Church in the Garden of the Frati Zoccolanti. Tartaglia was vastly experienced in such debates and expected to win. However, by the end of the first day, it was clear that things were not going Tartaglia's way. 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. After giving his lectures for a year in Brescia, he was informed that his stipend was not going to honored. 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 remembered 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. Fairly early in his career, before he became involved in the arguments about the cubic equation, he wrote Nova Scientia (1537) on the application of mathematics to artillery fire. In the work he described new ballistic methods and instruments, including the first firing tables.

Tartaglia also wrote a popular arithmetic text and was the first Italian translator and publisher of Euclid's Elements in 1543. In 1546 he published Quesiti et Inventioni diverse de Nicolo Tartalea referred to above. He also published Latin editions of Archimedes's works.