尼科洛·塔尔塔利亚Niccolò Tartaglia

尼科洛·塔尔塔利亚，人称塔尔塔利亚，1499年或1500年出生于布雷西亚，是一位诚实送信人Michele Fontana的儿子，此人被称为“骑手Micheletto”。Micheletto会骑马在布雷西亚和该地区其他城镇之间递送邮件。尽管贫穷，Micheletto仍为妻子、女儿和两个儿子尽了最大努力，而塔尔塔利亚从大约四岁起就上学。如果不是六岁时悲剧降临，塔尔塔利亚的人生也许会大不相同，因为那时他的父亲在外出送信时被谋杀。他从一个贫穷家庭的孩子，突然陷入了彻底的贫困。

塔尔塔利亚在1512年险些丧命，当时法国人攻占了他的家乡并进行了屠杀。法国军队由Gaston de Foix指挥，他们曾在一些坚定的布雷西亚民兵手中遭受羞辱。他们决定教训当地居民，在为期七天的战斗中重新夺回布雷西亚，期间有46,000名城市居民被杀作为报复。在大屠杀中，十二岁的塔尔塔利亚与母亲和妹妹一起躲在大教堂里，但被一名法国士兵用马刀砍伤面部，伤口可怕地切开了他的下巴和上颚。他被认为已死，即使他的母亲发现他还活着，也无力支付任何医疗费用。然而，他母亲的悉心照料确保了这个年轻人活了下来，但在后来的生活中，塔尔塔利亚总是留着胡须来掩盖毁容的疤痕，他说话也很困难，因此得了绰号塔尔塔利亚，即口吃者。

塔尔塔利亚在数学上是自学成才的，但由于他具有非凡的能力，他的母亲能够为他找到一位赞助人。卢多维科·巴尔比索尼奥带他去帕多瓦学习，但当他与赞助人一起回到布雷西亚时，他因对自己评价过高而不受欢迎。他离开布雷西亚，在维罗纳以教授数学为生，他在1516年至1518年间这样做。后来，仍在维罗纳，他在米赞蒂宫的一所学校任教，但据记载，当时他已婚并有家庭，却非常贫穷。他于1534年搬到威尼斯。作为威尼斯一位地位低下的数学教师，塔尔塔利亚通过成功参与大量辩论，逐渐获得了有前途数学家的声誉。

已知第一个用代数方法解出三次方程的人是希皮奥内·德尔·费罗，但他没有把自己的成就告诉任何人。然而，在临终之际，希皮奥内·德尔·费罗把这个秘密传给了他的（相当差的）学生Fior。对于当时的数学家来说，三次方程不止一种类型，而Fior只被希皮奥内·德尔·费罗告知如何解其中一种类型，即“未知数与立方等于数”，或者（用现代记号）$x^{3} + ax = b$。由于当时不使用负数，这又导致了许多其他情形，甚至对于没有平方项的方程也是如此。Fior开始吹嘘自己能够解三次方程，于是在1535年安排了他与塔尔塔利亚之间的一场挑战。事实上，塔尔塔利亚也已经发现了如何解一种类型的三次方程，因为他的朋友Zuanne da Coi提出了两个问题，使塔尔塔利亚得到了与Fior所能解的类型不同的一种一般解法，即“平方与立方等于数”，或者（用现代记号）$x^{3} + ax^{2} = b$。在塔尔塔利亚与Fior的竞赛中，每人都要提交三十个问题让对方解答。Fior极其自信，认为自己解三次方程的能力足以击败塔尔塔利亚，但塔尔塔利亚提交了各种不同的问题，暴露出Fior充其量不过是个平庸的数学家。另一方面，Fior给了塔尔塔利亚三十个解“未知数与立方”问题的机会，因为他相信塔尔塔利亚无法解出这种类型，而在竞赛设立时事实也确实如此。然而，在1535年2月13日凌晨，灵感降临到塔尔塔利亚身上，他发现了求解“平方与立方等于数”的方法。塔尔塔利亚随后在不到两小时内解出了Fior的全部三十个问题。由于Fior在塔尔塔利亚的问题上几乎没有进展，谁是胜者对所有的人来说都显而易见。然而，塔尔塔利亚没有从Fior那里拿走获胜的奖品，获胜的荣誉已经足够了。

此时吉罗拉莫·卡尔达诺进入了这个故事。作为米兰Piatti基金会的公共数学讲师，他知道解三次方程的问题，但在那场竞赛之前，他一直相信卢卡·帕西奥利的话，并认为正如卢卡·帕西奥利在1494年出版的Summa Ⓣ（算术、几何、比例与比例性概要）中所说，解是不可能的。当Zuanne da Coi告诉他那场竞赛时，吉罗拉莫·卡尔达诺大为好奇，他立即着手试图自己发现塔尔塔利亚的方法，但没有成功。几年后，在1539年，他通过一位中间人联系了塔尔塔利亚，请求把该方法收录进他当年要出版的一本书中。塔尔塔利亚拒绝了这个机会，表示他打算在以后要写的一本自己的书中发表他的公式。吉罗拉莫·卡尔达诺接受了这一点，随后请求让他看看这个方法，并承诺保守秘密。然而，塔尔塔利亚拒绝了。

被激怒的吉罗拉莫·卡尔达诺现在直接写信给塔尔塔利亚，表达了他的怨恨，向他发起一场辩论挑战，但同时又暗示自己一直在与米兰总督Alfonso d'Avalos，即Marchese del Vasto谈论塔尔塔利亚的才华，而此人是吉罗拉莫·卡尔达诺有权势的赞助人之一。收到这封信后，塔尔塔利亚彻底改变了自己的态度，意识到结识这位有影响力的米兰总督可能非常有利，并且可能为他提供一条出路，使他摆脱当时所担任的卑微教师工作，进入米兰宫廷的一份收入丰厚的工作。他友好地回信给吉罗拉莫·卡尔达诺，谋求被引见给这位Signor Marchese。吉罗拉莫·卡尔达诺对塔尔塔利亚的新态度感到高兴，便邀请他到家中，并向塔尔塔利亚保证会安排与d'Avalos会面。

于是，在1539年3月，塔尔塔利亚离开威尼斯前往米兰。令塔尔塔利亚沮丧的是，总督暂时不在米兰，但吉罗拉莫·卡尔达诺照顾了客人的一切需要，很快谈话就转向了三次方程问题。塔尔塔利亚经过多方劝说，同意告诉吉罗拉莫·卡尔达诺他的方法，条件是吉罗拉莫·卡尔达诺要发誓永不泄露，而且此后只能以密码形式写下来，以便在他死后没有人能从他的文稿中发现这个秘密。吉罗拉莫·卡尔达诺欣然同意，而塔尔塔利亚以一首诗的形式透露了他的公式，以帮助保护这个秘密，以防纸张落入不该落入的人手中。他现在急于离开吉罗拉莫·卡尔达诺的家，便从主人那里取得了一封给Marchese的引见信，然后离开去寻找他。然而，他却转而返回威尼斯，怀疑自己决定透露公式是否是个错误。

当他到达威尼斯时，塔尔塔利亚确信自己信任吉罗拉莫·卡尔达诺是个错误，并开始对自己被诱使透露秘密公式感到非常愤怒。吉罗拉莫·卡尔达诺在那年晚些时候出版了两本数学书，塔尔塔利亚一拿到副本，就检查以确保自己的公式没有被收录。尽管他发现公式没有被收录在书中而感到稍微高兴一些，但当吉罗拉莫·卡尔达诺友好地写信给他时，塔尔塔利亚拒绝了他继续友好的表示，并毫不留情地就最微不足道的琐事嘲笑他的书。

基于塔尔塔利亚的公式，吉罗拉莫·卡尔达诺和他的助手洛多维科·费拉里在寻找三次方程所有情形的证明方面取得了显著进展，更令人印象深刻的是，解决了四次方程。尽管此时人们已经知道存在这样一种方法，塔尔塔利亚却没有采取任何行动来发表他的公式。塔尔塔利亚可能希望将他的公式保留下来，以备将来任何辩论之需。

吉罗拉莫·卡尔达诺和洛多维科·费拉里于1543年前往博洛尼亚，从della Nave那里得知，最早解出三次方程的是希皮奥内·德尔·费罗，而非塔尔塔利亚。吉罗拉莫·卡尔达诺觉得，尽管他发誓不泄露塔尔塔利亚的方法，但显然没有什么能阻止他发表希皮奥内·德尔·费罗的公式。1545年，吉罗拉莫·卡尔达诺出版了Artis magnae sive de regulis algebraicis liber unus Ⓣ（《一本书中的代数规则之伟大艺术》），或者更常见的名称Ars magna Ⓣ（《伟大的艺术》），其中包含三次方程和四次方程的解法，以及他在塔尔塔利亚的公式上完成的全部额外工作。希皮奥内·德尔·费罗和塔尔塔利亚因其发现而受到赞誉，洛多维科·费拉里也是如此，这一故事被记载在文本中。

当塔尔塔利亚发现吉罗拉莫·卡尔达诺无视他的誓言时，他勃然大怒，他对吉罗拉莫·卡尔达诺的强烈厌恶变成了病态的仇恨。第二年，塔尔塔利亚出版了一本书New Problems and Inventions，其中清楚地陈述了他对这一事件的看法，以及他认为吉罗拉莫·卡尔达诺的行为极其不诚实的信念。此外，他还添加了几句针对吉罗拉莫·卡尔达诺的恶意人身侮辱。

Ars Magna Ⓣ（《伟大的艺术》）明确确立了吉罗拉莫·卡尔达诺作为世界领先数学家的地位，他并未因塔尔塔利亚的恶毒攻击而受到多大损害。然而，洛多维科·费拉里写信给塔尔塔利亚，无情地斥责他，并向他发起公开辩论的挑战。塔尔塔利亚极不愿意与洛多维科·费拉里争论，后者仍是一位相对不知名的数学家，即使战胜他也几乎不会带来什么实质性的好处。另一方面，与吉罗拉莫·卡尔达诺的辩论对塔尔塔利亚具有极大的吸引力。他不仅恨他，而且吉罗拉莫·卡尔达诺是数学界、医学界和文学界的领军人物，即使只是与他进行辩论，也会极大地提升塔尔塔利亚的地位。尽管塔尔塔利亚在解决三次方程问题上的发现光彩夺目，他仍然只是威尼斯一位相对贫穷的数学教师。

塔尔塔利亚回复了洛多维科·费拉里，试图将吉罗拉莫·卡尔达诺拉入辩论。然而，吉罗拉莫·卡尔达诺无意与塔尔塔利亚辩论。洛多维科·费拉里和塔尔塔利亚在大约一年的时间里互相写信，却毫无结果，交换着最具攻击性的人身侮辱，但在解决争端方面几乎毫无进展。1548年，塔尔塔利亚突然收到了一份令人印象深刻的邀请，请他到家乡布雷西亚担任讲师。为了明确确立他担任该职位的资历，塔尔塔利亚被要求前往米兰，参加与洛多维科·费拉里的竞赛。

1548年8月10日，竞赛在Frati Zoccolanti花园教堂举行。塔尔塔利亚在此类辩论中经验丰富，他期望获胜。然而，到第一天结束时，事情显然没有朝他期望的方向发展。洛多维科·费拉里显然更透彻地理解三次和四次方程，塔尔塔利亚决定当晚离开米兰，从而使竞赛悬而未决。随着塔尔塔利亚不光彩地离去，胜利留给了洛多维科·费拉里。

塔尔塔利亚因这场竞赛而遭受损失。在布雷西亚讲学一年后，他被告知他的津贴将不予支付。即使经过多次诉讼，塔尔塔利亚也未能得到任何报酬，只好回到威尼斯原来的工作岗位，经济上损失惨重，对吉罗拉莫·卡尔达诺怀有极大的怨恨。米兰的失败似乎是塔尔塔利亚未获报酬的原因。

塔尔塔利亚如今之所以被人铭记，是因为解三次方程的公式被称为吉罗拉莫·卡尔达诺-塔尔塔利亚公式。然而，塔尔塔利亚确实以许多其他方式对数学做出了贡献。在他职业生涯的相当早期，在卷入关于三次方程的争论之前，他写了Nova Scientia（1537年），内容是关于数学在炮火中的应用。在这部著作中，他描述了新的弹道方法和仪器，包括最早的射击表。他还写了一本流行的算术教科书，并于1543年成为欧几里得的Elements的第一位意大利翻译者和出版者。1546年，他出版了上文提到的Quesiti et Inventioni diverse de Nicolo Tartalea。

我们在塔尔塔利亚与卡尔达诺一文中给出了塔尔塔利亚这部著作的许多引文，其中上述事件是以这位数学家自己的话叙述的。

塔尔塔利亚还出版了阿基米德著作的拉丁文版本。他在威尼斯里亚托桥附近Calle del Sturion的家中贫困去世（不是现在这座桥，现在的桥大约30年后才建造）。

Niccolò Fontana, known as Tartaglia, was born in Brescia in 1499 or 1500, the son of an honest mail rider Michele Fontana who was known as 'Micheletto the Rider'. Micheletto would ride his horse between Brescia and other towns in the district making deliveries. Although he was poor, Micheletto did his best for his wife, daughter and two sons, and Niccolò attended school from the age of about four years. Life might have been very different for Niccolò had tragedy not come when he was six years old, for at that time his father was murdered while out making deliveries. From being a child in a poor family, he was suddenly plunged into total poverty.

Niccolò was nearly killed as a teenager when, in 1512, the French captured his home town and put it to the sword. The French army was commanded by Gaston de Foix and they had suffered humiliation at the hands of some determined Brescia militia. They decided to teach the local inhabitants a lesson and retook Brescia during seven days of fighting in which time 46,000 residents of the city were killed in an act of revenge. Amidst the general slaughter, the twelve year old Niccolò took refuge in the cathedral with his mother and younger sister, but was dealt horrific facial sabre wounds by a French soldier that cut his jaw and palate. He was left for dead and even when his mother discovered that he was still alive she could not afford to pay for any medical help. However, his mother's tender care ensured that the youngster did survive, but in later life Niccolò 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, his mother was able to find him a patron. Ludovico Balbisonio took him to Padua to study there, but when he returned with his patron to Brescia he made himself unpopular by having an inflated opinion of himself. He left Brescia to earn his living teaching mathematics at Verona which he did between 1516 and 1518. Later, still in Verona, he taught at a school in the Palazzo Mizzanti but it is recorded that at that time he was married with a family, yet was very poor. He moved to Venice in 1534. 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. For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) $x^{3} + ax = b$. As negative numbers were not used this led to a number of other cases, even for equations without a square term. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) $x^{3} + ax^{2} = b$. 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.

At this point Cardan enters the story. As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the problem of solving cubic equations, but, until the contest, he had taken Pacioli at his word and assumed that, as Pacioli stated in the Summa Ⓣ published in 1494, solutions were impossible. 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. Cardan, accepting this, then asked to be shown the method, promising 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 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. 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 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. 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 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 Ⓣ , or Ars magna Ⓣ 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. 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 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.

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 he expected to win. However, by the end of the first day, it was clear that things were not going his 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 be honoured. 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. 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. He 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.

We give many quotes from this work by Tartaglia in the article Tartaglia v Cardan where the events described above are recounted in the mathematicians own words.

Tartaglia also published Latin editions of Archimedes' works. He died in poverty in his house in the Calle del Sturion near the Rialto Bridge (not the present one which was constructed about 30 years later) in Venice.