鲍耶·法卡斯Farkas Bolyai

鲍耶·法卡斯的母亲是Krisztina Pávai Vajna，她继承了Domáld的一个小农场，Domáld靠近特兰西瓦尼亚的Marosvásárhely。今天Marosvásárhely被称为Târgu-Mureș，是位于罗马尼亚中北部的Mureș judet（县）的首府。法卡斯的父亲Gáspár法卡斯在法卡什·久洛出生时43岁。他来自一个有着长期抗击土耳其入侵者历史的家族，他的祖先曾经富有，但尽管Gáspár仍然拥有Nagyszeben附近Bolya的一处小地产，这个家族此时已不再富有。今天Nagyszeben被称为Sibiu，是位于罗马尼亚中部的Sibiu judet（县）的首府。久洛出生时，Nagyszeben是特兰西瓦尼亚的军事中心，也是该地区的首府。

久洛在家中由父亲教导，直到六岁时被送到Nagyszeben的加尔文宗学校。他的老师们立刻认识到他在算术计算和学习语言方面的才能。十二岁时他离开学校，被任命为八岁的Simon Kemény的导师，Simon Kemény是Kemény男爵的儿子。这意味着法卡斯现在被当作这个国家主要家族之一的成员对待，他不仅成为Simon的导师，还成为他的密友。1790年，法卡斯和他的学生都进入了Kolozsvár的加尔文宗学院，在那里度过了五年。

在Kolozsvár度过的时光对法卡斯的发展很重要。启蒙运动已传遍欧洲，此时正在影响特兰西瓦尼亚。这意味着法卡斯接触到了其基本思想，即理性是理解宇宙和改善人类地位的途径。知识、自由和幸福应当是一个理性之人的目标。另一方面，匈牙利的民族主义情绪正在增长。这个国家在1699年摆脱了奥斯曼统治，在尝试获得独立之后，匈牙利一直受哈布斯堡家族控制。对哈布斯堡家族的不满日益增长，尤其是来自工人的不满，法卡斯也感到对匈牙利文化、语言和民族性的支持。还有相互冲突的宗教压力，因为基督教会的各分支相互争论，并反对启蒙运动的思想。

克洛日瓦尔学院的哲学教授是个令人印象深刻的人，他试图让法卡斯背离数学，转向宗教哲学。另一方面，法卡斯兴趣广泛，科学、数学和文学都令他感兴趣，1795年离开学院后，他花了几个星期考虑当演员。然而，他决定与Simon Kemény一起出国，进行由Kemény男爵资助的游学，在因意外生病而耽搁之后，他们于1796年春天出发。

他们首先到达耶拿，法卡斯在那里第一次开始系统地学习数学。他常常独自长时间散步，边走边思考数学。在耶拿待了六个月后，法卡斯和Kemény前往哥廷根。在那里，他受教于亚伯拉罕·哥特黑尔弗·凯斯特纳，并与卡尔·弗里德里希·高斯结为终身朋友，后者是哥廷根的同学。可以说，正是在这个时候，法卡斯真正成为了一名数学家。他开始思考欧几里得的几何公理，特别是第五公设的独立性。他与卡尔·弗里德里希·高斯讨论了这些问题，他后来的著作表明，他认为他们的友谊对他的数学发展有多么重要。

到1798年秋天，法卡斯和Kemény完成了学业，但回到匈牙利后，Kemény男爵在经济上陷入困境，尽管他提供了足够的钱让儿子回国，法卡斯却身无分文地留在哥廷根。他在那里待了一年，靠慈善和借来的钱买食物度日。这是极其艰难的一年，但也是在众多才华横溢的数学家环绕下，他在数学上继续发展的一年。一年后，一位朋友从匈牙利寄来足够的钱，偿还他所欠的债务，他于1799年7月徒步启程回国。

回到博利亚的家族庄园后，他从事数学研究。他前往克洛日瓦尔，在那里成为一名家庭教师。在那里，他遇到了Zsuzsanna Benkö，他们于1801年结婚。1802年12月15日，在他们的儿子鲍耶在Zsuzsanna父母家中出生。当法卡斯得到马里亚斯瓦萨海伊加尔文学院的一份工作时，他相当不愿接受，但他的父亲希望儿子有一份稳定的工作，便催促他接受。法卡斯此后一生都在马里亚斯瓦萨海伊教授数学、物理和化学。

法卡斯在马里亚斯瓦萨海伊的生活并不轻松。他在学院教书所得甚微，不得不承担额外工作以赚取额外收入。他撰写并出版剧本，经营学院酒吧，还设计瓷砖和铸铁炉，这些产品被商业化生产。家里的生活也不轻松，因为法卡斯的妻子是个难以相处的人，随着她的健康逐年恶化，她变得越来越难相处。法卡斯教他的儿子鲍耶数学，因为这是他希望儿子从事的学科。直到János九岁，学院的学生教János其他科目，他直到这个年龄才上学。数学，这个法卡斯所热爱的学科，被降格为一种消遣。当然，他从学院的数学教学中几乎得不到满足感，因为他的学生水平很低。

鲍耶于1818年8月离家，前往维也纳的工程学院学习。三年后，1821年9月18日，法卡斯的妻子去世。他于1824年与Teréz Nagy再婚。在所有这些艰难的岁月里，法卡斯一直在撰写他于1832年出版的数学杰作：Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi Ⓣ（《试图引导勤奋青年进入纯数学基础》）。他艰难生活中唯一的数学乐趣，是他与卡尔·弗里德里希·高斯往来的信件，以及晚年他儿子的数学成就。

法卡斯 一生都对几何学的基础和 parallel axiom 感兴趣。他的主要著作 Tentamen Ⓣ（《试图引导勤奋的青年学习纯粹数学的基础》）试图为几何学、算术、代数和分析奠定严格而系统的基础。引言清楚地展示了他对世界的理性态度，他将世界比作一座钟或一座构造完美的教堂，科学家的目标在于：-

……理解钟的机制，熟悉教堂的平面图、柱子、建筑砌块和水泥，并通过破译最后一个字符，通过找到密码的钥匙，阅读整本自然之书。物理学所追求的就是这一点，物理学是使用数学的一个科学分支。我们借助数学架起雅各的天梯以到达天空……

Tentamen Ⓣ（《试图引导勤奋的青年学习纯粹数学的基础》）建立在 法卡斯 的信念之上，即数学由算术和几何学组成，算术是时间的数学，几何学是空间的数学。他试图将这两门学科建立在公理基础之上，而他坚定的信念之一是公理应当独立：-

……不应将任何从其他公理推出的东西列入公理之中。

法卡斯 还主张，公理应当显然是真实的：-

公理是常识不加进一步论证就理所当然地接受的判断……

在研究数学时最令法卡斯困惑的问题一直是欧几里得第五公设的独立性。1804年，他相信自己已经证明了它可以从其他公理推导出来，但他把证明寄给了卡尔·弗里德里希·高斯，后者发现了其中的错误。最终他放弃了证明其独立性的尝试，转而试图找到一个更容易被常识接受的等价版本。TentamenⓉ（《引导好学青年进入纯粹数学基础的一次尝试》）包含了八条与欧几里得第五公设等价的公理，例如：-

任何球体与任何其他球体在任何性质上都不能有所不同，除了其大小和位置。

不在同一直线上的三个点必定位于一个圆上

在TentamenⓉ（《引导好学青年进入纯粹数学基础的一次尝试》）中还有其他一些想法，显示了法卡斯作为数学家的品质。例如，他给出了求解方程的迭代程序，然后通过证明这些程序单调递增且有上界来证明其收敛性。他对级数收敛性的研究包括一个与约瑟夫·路德维希·拉比判别法等价的判别法，他独立发现了它，时间大约与拉比同时。该著作中的其他重要想法包括函数的一般定义，以及如果两个平面图形都能被分成有限个相等的两两全等的部分，则它们相等这一相等性的定义。

他阻止儿子研究平行公理的尝试幸好失败了！法卡斯写信给儿子说：-

把它当作淫秽的交往来憎恶吧，它会剥夺你所有的闲暇、健康、休息以及你一生的全部幸福。

在1820年4月4日写给维也纳的鲍耶的一封信中，法卡斯写道：-

不要以那种方式尝试平行线：那条路我早就知道。我已测量过那无底的夜晚，我生命中所有的光明和欢乐都在那里熄灭了。

然而，在1825年，法卡斯的儿子鲍耶向他展示了自己对non-euclidean geometry的发现。起初法卡斯并不热心，但到1830年他肯定已经变得热心了，因为在这个阶段他说服鲍耶把他的发现写出来。

法卡斯于1851年从教职退休，在几次中风之后，五年后去世。在遗嘱中，他对自己的一生作了这样颇具诗意的总结：-

直到我从德国回来之前，一直是早晨，有着美好日子的前景，而在一些充满火与冰的日子之后，变成了从永久阴沉的天空落下的雨，直到最近这场降雪。

Farkas Bolyai's mother was Krisztina Pávai Vajna and she inherited a small farm at Domáld which was near Marosvásárhely in Transylvania. Today Marosvásárhely is called Târgu-Mureș and is the capital of Mureș judet (county) situated in north-central Romania. Farkas Bolyai's father Gáspár Bolyai was 43 years old when Farkas was born. He came from a family with a long history of fighting against the Turkish invaders and his ancestors had been wealthy, but although Gáspár still owned a small estate at Bolya near Nagyszeben, the family were by this time no longer wealthy. Today Nagyszeben is called Sibiu and is the capital of Sibiu judet (county) situated in central Romania. When Farkas was born Nagyszeben was the military centre of Transylvania and the capital of the region.

Farkas was taught at home by his father until he reached the age of six years when he was sent to the Calvinist school in Nagyszeben. His teachers immediately recognised his talents both in arithmetical calculation and in learning languages. When he was twelve years old he left school and was appointed as a tutor to the eight year old Simon Kemény who was the son of Baron Kemény. This meant that Bolyai was now treated as a member of one of the leading families in the country, and he became not only a tutor to Simon but a close friend. In 1790 Bolyai and his pupil both entered the Calvinist College in Kolozsvár where they spent five years.

The time spent in Kolozsvár was important for Bolyai's development. The Enlightenment had spread across Europe and by this time was influencing Transylvania. This meant that Bolyai was presented with its fundamental idea that reason was the route to understanding the universe and to improving the position of man. Knowledge, freedom, and happiness should be the aims of a rational human being. On the other hand nationalist feeling were on the increase in Hungary. The country had been freed from Ottoman rule in 1699 and after an attempt at gaining independence, Hungary had been controlled by the Habsburgs. There was an increasing resentment against the Habsburgs, particularly from the workers, and Bolyai too felt support for Hungarian culture, language, and nationality. There were also conflicting religious pressures as branches of the Christian Church argued against each other and against the ideas of the Enlightenment.

The professor of philosophy at the College in Kolozsvár was an impressive person, and he tried to turn Bolyai against mathematics and towards religious philosophy. Bolyai on the other hand had wide ranging interests, science, mathematics, and literature all interested him and in 1795 after leaving the College he spent a few weeks considering a career as an actor. However, he decided to go abroad with Simon Kemény on an educational trip funded by Baron Kemény and, after a delay caused by an unexpected illness, they set off in the spring of 1796.

First they reached Jena where Bolyai for the first time began to study mathematics systematically. He would go for long walks on his own and think about mathematics as he walked. After six months in Jena Bolyai and Kemény went to Göttingen. There he was taught by Kästner and became a life long friend of Gauss, a fellow student at Göttingen. This was the time when one could say that Bolyai really became a mathematician. He began to think about Euclid's geometrical axioms and in particular the independence of the Fifth Postulate. He discussed these issues with Gauss and his later writing show how important he considered their friendship to be for his mathematical development.

By the autumn of 1798 Bolyai and Kemény had completed their studies but back in Hungary Baron Kemény had hit hard times financially and although he supplied enough money for his son to return, Bolyai was left penniless in Göttingen. He spent a year there relying on charity and borrowed money for food to survive. It was a year of great hardship, yet one where he continued to develop mathematically surrounded by other talented mathematicians. After a year a friend sent him enough money from Hungary to pay the debts he had incurred and he set off on foot to return in July 1799.

Back on the family estate at Bolya he undertook research in mathematics. He went to Kolozsvár where he became a tutor. There he met Zsuzsanna Benkö and they married in 1801. In Zsuzsanna's parents home on 15 December 1802 their son János Bolyai was born. When Farkas Bolyai was offered a job at the Calvinist College in Marosvásárhely he was rather reluctant to accept but his father, wanting his son to have a secure job, pressed him to accept. Bolyai taught mathematics, physics and chemistry at Marosvásárhely for the rest of his life.

Life was not easy for Bolyai in Marosvásárhely. He was paid very little for his teaching at the College and had to take on extra work to bring in extra money. He wrote and published dramas, he ran the College pub, and he designed tiles and cast iron stoves which were produced commercially. Life was not easy at home too, for Bolyai's wife was a difficult person to live with and became increasingly difficult over the years as her health steadily deteriorated. Bolyai taught his son János mathematics, for this was the subject which he hoped that he would follow. Up until János was nine years old students from the College taught János other subjects, and only at this age did he attend school. Mathematics, the subject which Farkas Bolyai loved, was relegated to something to do for relaxation. Certainly he gained little satisfaction from his mathematics teaching at the College for the level of his students was low.

János left home in August 1818 to study at the Academy of Engineering at Vienna. Three years later, on 18 September 1821 Bolyai's wife died. He remarried in 1824 to Teréz Nagy. Through all these difficult years Bolyai was working on his mathematical masterpiece published in 1832: Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi Ⓣ. The only mathematical pleasures in his difficult life were the letters he exchanged with Gauss and, in later years, the mathematical achievements of his son.

All his life Bolyai was interested in the foundations of geometry and the parallel axiom. His main work, the Tentamen Ⓣ, was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis. The introduction shows clearly his rational approach to the world which he compares to a clock or a perfectly constructed church with the scientists aim:-

... to understand the mechanism of the clock, to become familiar with the plan, the columns, the building blocks, and cement of the church and, by deciphering the last character, by finding the key to the code, reading the whole book of nature. To this physics aspires, a branch of science which uses mathematics. We raise Jacob's ladder to reach the skies with the help of mathematics ...

The Tentamen Ⓣ is built on Bolyai's belief that mathematics consists of arithmetic and geometry with arithmetic as the mathematics of time and geometry as the mathematics of space. He tries to set both disciplines on an axiomatic basis and one his strong beliefs is that the axioms should be independent:-

... no thing should be included among the axioms which follows from the others.

Also the axioms should, argues Bolyai, be obviously true:-

An axiom is a judgement that common sense accepts without further arguments, as a matter of course ...

The problem which had perplexed Bolyai most in his study of mathematics had been the independence of Euclid's Fifth postulate. In 1804 he believed that he had a proof that it could be deduced from the other axims, but he sent his proof to Gauss who discovered the error. Eventually he gave up his attempts to prove its independence and tried instead to find an equivalent version which was more easily accepted by common sense. The Tentamen Ⓣ contains eight axioms equivalent to Euclid's Fifth Postulate such as:-

No sphere may differ from any other sphere in any property except its size and location.

Three points which do not lie on the same straight line must lie on a circle

There are other ideas in the Tentamen Ⓣ which show the quality of Bolyai as a mathematician. For example he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above. His study of the convergence of series includes a test equivalent to Raabe's test which he discovered independently and at about the same time as Raabe. Other important ideas in the work include a general definition of a function and a definition of an equality between two plane figures if they can both be divided into a finite equal number of pairwise congruent pieces.

His attempts to stop his son studying the parallel axiom fortunately failed! Farkas Bolyai wrote to his son:-

Detest it as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life.

In a letter written on 4 April 1820 written to János in Vienna, Bolyai wrote:-

Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.

However, in 1825 Bolyai's son János showed him his discovery of non-euclidean geometry. At first Farkas Bolyai was not enthusiastic but certainly by 1830 he had become enthusiastic for at this stage he persuaded János to write up his discoveries.

Bolyai retired from his teaching in 1851 and after a number of strokes, died five years later. In his will he gave this rather poetic summary of his life:-

Until I returned from Germany it had been morning with the prospect of beautiful days which, after some days laden with fire and ice, turned into raining from the permanently overcast sky until this recent snowfall.