鲍耶János Bolyai

鲍耶的父母是来自Kolozsvár的Zsuzsanna Benkö和来自Bolya（靠近Nagyszeben）的鲍耶·法卡斯。鲍耶出生在Zsuzsanna父母位于Kolozsvár（现罗马尼亚的Cluj）的家中，但很快搬到了Marosvásárhely，他的父亲法卡什·久洛在那里的一所加尔文学院教授数学、物理和化学。鲍耶·法卡斯一直希望儿子成为数学家，并以此为目标培养他。人们可能会认为这意味着鲍耶的教育在鲍耶家庭中被放在首位，但事实并非如此，因为久洛相信只有健康的身体才能承载健全的心智去成就伟业，所以在鲍耶早年，大部分注意力都放在了身体发育上。然而，从一开始就很清楚，鲍耶是一个极其聪明且善于观察的孩子[7]：-

……当他四岁时，他能分辨某些几何图形，了解正弦函数，并能识别最著名的星座。到五岁时，他几乎自学了阅读。他在语言和音乐学习方面远超平均水平。七岁时他开始拉小提琴，进步神速，很快就能演奏高难度的协奏曲。

重要的是要理解，尽管久洛有一个讲师职位，但薪水不高，即使他从各种不同来源赚取额外收入，鲍耶仍然在贫困的经济环境中长大。此外，鲍耶的母亲是一个相当难相处的人，这个家庭对男孩来说并不是一个特别快乐的成长环境。

直到鲍耶九岁，马罗斯瓦萨海伊学院最好的学生教他所有常规学校科目，除了数学，数学由他的父亲教。只有从九岁起，他才上学。到鲍耶13岁时，他已经掌握了微积分和其他形式的分析力学，他的父亲继续指导他。然而，此时他已经在马罗斯瓦萨海伊的加尔文学院就读，尽管他是从四年级开始的，并且经常参加为高年级学生准备的课程。

1816年，久洛写信给他的朋友卡尔·弗里德里希·高斯，询问他是否愿意让鲍耶住在他家并收他为学生，以便他能接受最好的数学教育。当然，这对鲍耶来说将是一种极好的教育，如果接受这个计划，数学界可能会获得什么好处，这很有趣。然而，卡尔·弗里德里希·高斯拒绝了这个想法。当鲍耶于1817年6月30日从马罗斯瓦萨海伊学院毕业时，尚不清楚他如何能获得良好的数学教育。当时佩斯或维也纳的大学都没有提供高质量的数学教育，而久洛也负担不起送儿子去国外更有声望的大学。决定鲍耶在维也纳工程学院的工程学院学习军事工程并非没有经过许多心痛和反省，但最终这条路线被选为最不坏的选择。这并不是说学院在数学教学方面不优秀，事实上整个课程都强调这门学科。鲍耶在马罗斯瓦萨海伊学院又留了一年，试图以尽可能高的水平进入维也纳学院，他做到了。

他于1818年至1822年在维也纳皇家工程学院学习，用四年时间完成了七年的课程。他是一名杰出的学生，从第二年开始，他在大多数科目中都名列前茅。他还有时间成为一名杰出的运动员，并继续认真对待小提琴演奏，在维也纳期间还进行表演。他的母亲于1821年9月18日去世，但他能够继续学业。当他在1822年9月6日从学院毕业时，他取得了如此杰出的成就，以至于他在维也纳又花了一年时间进行学术研究，然后才进入军队服役。当然，他在维也纳期间接受了军事训练，因为夏季月份专门用于此，但鲍耶的性格使他难以轻易接受严格的军事纪律。

1823年9月，他作为少尉进入陆军工程兵团，被派往Temesvár修建防御工事。他在军队服役共11年，被誉为奥匈帝国军队中最好的剑客和舞者。他不吸烟不喝酒，甚至不喝咖啡，据报道，23岁时他仍然保留着the modesty of innocence。他还是一位有造诣的语言学家，会说九种外语，包括汉语和藏语。

大约在1820年，当鲍耶还在维也纳学习时，他开始追随他父亲走过的同一条道路，试图用另一条可以从其他公理推导出来的公理来替换欧几里得的平行公理。事实上，他在一年之内就放弃了这种方法，因为正如他现在的笔记所显示的，在1820年他开始发展双曲几何的基本思想。1823年11月3日，他写信给父亲说他已经：-

……从无中创造了一个新的、另一个世界……

但几行之后他又补充说，它还没有被创造出来。然而到1824年，有证据表明他已经发展出了后来出现在他的论著中作为non-Euclidean geometry完整体系的大部分内容。1825年初，鲍耶前往马罗什瓦沙尔海伊，向父亲解释了他的发现。然而鲍耶·法卡斯并没有热情回应，这显然让鲍耶感到失望。鲍耶于1826年被派往阿拉德，在那里他发现沃尔特·冯·埃克维尔上尉——他在维也纳科学院的一位旧数学老师——也驻扎在那里。鲍耶把自己正在撰写的几何理论材料的草稿给了他，大概是因为他希望从他那里得到一些建设性的意见。然而从鲍耶后来的著作来看，他似乎从冯·埃克维尔那里什么也没得到，特别是他从未收到退回的手稿。

1830年，鲍耶得知他将被派往伦贝格任职。1831年初，他出发前往伦贝格，但在途中到马罗什瓦沙尔海伊看望了父亲。此时久洛已经理解了儿子所取得成就的全部意义，并极力鼓励他把这项工作写出来，作为TentamenⓉ（《试图向好学青年介绍纯数学基础》）的附录发表，该书即将出版。鲍耶后来写道：-

如果我的父亲没有在马罗什瓦沙尔海伊——在我前往伦贝格任职的途中——碰巧催促甚至强迫我立即把东西写下来，附录的内容可能永远不会问世。

这部数学杰作包含了什么内容？在建立了他自己对“平行”的定义，并证明了如果第五公设在一个空间区域成立则在整个空间都成立、反之亦然之后，他接着清楚地陈述了他将考虑的不同体系：-

……用Σ表示基于欧几里得第五公设成立的假设而建立起来的几何体系，用S表示基于相反假设而建立起来的体系。我们陈述的所有定理，如果没有明确说明该定理在Σ体系或S体系中成立，那么它们都是绝对的，也就是说，无论Σ还是S成立，它们都成立。

今天我们把这三种几何称为欧几里得几何、双曲几何和绝对几何。Appendix的大部分内容讨论的是绝对几何。到1831年6月20日，Appendix已经出版，因为在那一天鲍耶·法卡斯寄了一份抽印本给卡尔·弗里德里希·高斯，后者在读Appendix时写信给一位朋友说：-

我认为这位年轻的几何学家鲍耶是一流的天才。

然而，卡尔·弗里德里希·高斯写信给鲍耶·法卡斯说：-

称赞它就等于称赞我自己。因为这部作品的全部内容……几乎与我过去三十或三十五年间一直萦绕于心的思考完全吻合。

毫无疑问，卡尔·弗里德里希·高斯在这里只是在陈述事实。卡尔·弗里德里希·高斯书信中关于他研究非欧几何的最明确的提及，显示了他理解的深度，出现在他1824年11月8日写给Taurinus的一封信中，当时他写道：-

假设三角形的三个角之和小于180°，会引出一个奇特的几何学，它与我们的几何学[即欧几里得几何学]颇为不同，却完全相容；我已将其发展到令我完全满意的程度，因此除确定一个常数之外，我能解决其中的每一个问题，而这个常数无法先验地确定。……只要边取得足够大，三角形的三个角就可以小到任意程度，然而无论边有多大，三角形的面积都绝不会超过、甚至也达不到某个极限。

鲍耶 在伦贝格没有待很久，因为1832年他被派往奥尔米茨，此时他已是一名上尉。然而，发现 卡尔·弗里德里希·高斯 已经预见到他工作中的许多内容，使 鲍耶 大为不安，他视之为沉重的打击。他变得易怒，难以相处。他的健康开始恶化，并饱受发烧之苦，这常常使他无法工作，因此他越来越难以履行军职。他于1833年6月16日退役，请求领取养老金，并有一小段时间与父亲同住。

与父亲生活了一段时间后，鲍耶 前往家族从 鲍耶·法卡斯 的母亲那里继承来的多莫尔德庄园居住。他结识了罗扎莉亚·基贝迪·奥尔班，并从 1834 年起与她一起住在多莫尔德。然而，他们并未结婚，因为法律规定结婚前必须存入一笔钱，而 鲍耶 拿不出这笔钱。他与罗扎莉亚育有两个孩子，但他的养老金不足以让一家人过上舒适的生活。他似乎既没有管好自己手头的钱，也没有把庄园管理得井井有条。看来 鲍耶·法卡斯 不赞成罗扎莉亚，对儿子的经济状况感到不满，对多莫尔德庄园没有得到妥善照料感到不满，也对儿子损害自己的好名声感到不满，因为 久洛 是社区中极受尊敬的人。

鲍耶 住在多莫尔德期间继续发展数学理论，但由于与数学界其余部分隔绝，他尝试的许多工作价值不大。他的一项主要工作——试图基于公理系统发展全部数学——始于 1834 年，因为他在那一年写了序言，但他从未完成这项工作。他所写的部分涉及几何学，这部未发表的作品中有若干超前于时代的思想，例如拓扑不变性的概念。

除了在几何方面的工作外，鲍耶还将复数严格地发展为有序实数对，并给出了几何概念。他这样做是因为莱比锡的雅布沃诺夫斯基学会曾就该主题征集论文。鲍耶和他的父亲都提交了论文，但都没有得到好评。鲍耶的论文被称为Responsio，写作目的是回答几何中使用的虚量是否可以被构造的问题。鲍耶选择论证这个问题提法有误，这或许不是赢得评委青睐的最佳方式。他论证说，重要的不是它们的构造，而是它们的定义和在几何中的作用。

1846年，鲍耶 从多莫尔德庄园迁往马罗斯瓦沙尔海伊。这意味着他离父亲更近了，而这似乎使两人之间的关系更加紧张。1848年，鲍耶 发现 罗巴切夫斯基 已于1829年发表了一篇类似的作品。韦尼阿明·卡甘 写道[14]：-

鲍耶 仔细研究了 罗巴切夫斯基 的著作，逐行分析，甚至可以说是逐字分析，其仔细程度不亚于他撰写《附录》时所付出的心血。这部著作在他灵魂中激起了真正的风暴，他通过在《几何研究》中所加的评论来宣泄自己的苦恼。

《几何考察》的“评注”不仅仅是对该著作的批判性分析。它们表达了鲍耶在阅读此书时激发的思考与焦虑。其中包括他感到自己受到不公对待的抱怨，他怀疑罗巴切夫斯基根本不存在，以及一切都是卡尔·弗里德里希·高斯恶意的阴谋：这是一位天才几何学家的悲剧性哀叹，他意识到自己发现的重要性，却未能从唯一能赏识其价值的人那里获得支持。

尽管 鲍耶 是在精神激动之中把观察所得写到纸上的，他仍保持了足够的客观性，高度评价其对手的著作。在对定理35的评论中，他指出 罗巴切夫斯基 关于球面三角学的证明带有天才的印记，其著作应被视为一项大师级的成就。

1852年，鲍耶 离开了罗扎莉娅——他于1849年5月18日与她结婚，当时他相信由于匈牙利宣布独立，法律现在已经改变——而与罗扎莉娅分手至少有一个好处，那就是他与父亲的关系改善了。他在晚年放弃了数学研究，转而试图构建一种关于一切知识的理论。其中关于语言学和社会学的部分包含了一些有趣的思想。

尽管他生前只发表了附录的几页，但他在57岁死于肺炎时留下了超过20000页的数学手稿。这些手稿现在保存在特尔古穆列什的鲍耶-Teleki图书馆。1945年，克卢日的匈牙利大学以他的名字命名。曾名为斐迪南一世国王大学的罗马尼亚克卢日大学被更名为巴贝什大学（以罗马尼亚自然科学家Victor Babeș命名）。两所大学于1959年合并成为巴贝什-鲍耶大学。

János Bolyai's parents were Zsuzsanna Benkö, from Kolozsvár, and Farkas Bolyai, from Bolya (near Nagyszeben). János was born in Zsuzsanna's parents home in Kolozsvár (now renamed Cluj in Romania) but soon went to Marosvásárhely where his father Farkas had a job at the Calvinist College teaching mathematics, physics and chemistry. Farkas Bolyai always wanted his son to be a mathematician, and he brought him up with this in mind. One might suppose that this would mean that János's education was put first in the Bolyai household, but this was not so for Farkas believed that a sound mind could only achieve great things if it was in a sound healthy body, so in his early years most attention was paid to János's physical development. It was clear from early on, however, that János was an extremely bright and observant child [7]:-

... when he was four he could distinguish certain geometrical figures, knew about the sine function, and could identify the best known constellations. By the time he was five [he] had learnt, practically by himself, to read. He was well above the average at learning languages and music. At the age of seven he took up playing the violin and made such good progress that he was soon playing difficult concert pieces.

It is important to understand that although Farkas had a lecturing post he was not well paid and even although he earned extra money from a variety of different sources, János was still brought up in poor financial circumstances. Also János's mother was a rather difficult person and the household was not a particularly happy place for the boy to grow up.

Until János was nine years old the best students from the Marosvásárhely College taught him all the usual school subjects except mathematics, which he was taught by his father. Only from the age of nine did he attend school. By the time Bolyai was 13, he had mastered the calculus and other forms of analytical mechanics, his father continuing to give him instruction. By this time, however, he was attending the Calvinist College in Marosvásárhely although he had started in the fourth year and often attended lessons intended for the senior students.

In 1816 Farkas wrote to his friend Gauss asking him if he would let János live with him and take him on as a pupil so that he might receive the best possible mathematical education. Certainly it would have been a wonderful education for János and it is interesting to speculate what benefits might have come to the world of mathematics if he had accepted the plan. Gauss, however, rejected the idea. When János graduated from Marosvásárhely College on 30 June 1817 it was not clear how he might obtain a good mathematical education. Neither of the universities at Pest or Vienna offered a good quality mathematical education at this time, and Farkas could not afford to send his son to a more prestigious university abroad. The decision that János would study military engineering at the Academy of Engineering at Vienna was not taken without a lot of heartache and soul-searching but in the end this route was chosen as the least bad of the options. This is not to say that the Academy did not excel in teaching mathematics, for indeed the subject was emphasised throughout the course. János remained for one further year at Marosvásárhely College attempting to gain entry to the Academy at Vienna at the highest possible level which he achieved.

He studied at the Royal Engineering College in Vienna from 1818 to 1822 completing the seven year course in four years. He was an outstanding student and from his second year of study on he came top in most of the subjects he studied. He also had time to become an outstanding sportsman, and he continued to take his violin playing seriously and performed while in Vienna. His mother died on 18 September 1821 but he was able to continue his studies. When he graduated from the Academy on 6 September 1822 he had achieved such outstanding success that he spent a further year in Vienna on academic studies before entering military service. Of course he had received military training during his time in Vienna, for the summer months were devoted to this, but Bolyai's nature did not allow him to accept easily the strict military discipline.

In September 1823 he entered the army engineering corps as a sublieutenant and was sent to work on fortifications at Temesvár. He spent a total of 11 years in military service and was reputed to be the best swordsman and dancer in the Austro-Hungarian Imperial Army. He neither smoked nor drank, not even coffee, and at the age of 23 he was reported to still retain the modesty of innocence. He was also an accomplished linguist speaking nine foreign languages including Chinese and Tibetan.

Around 1820, when he was still studying in Vienna, Bolyai began to follow the same path that his father had taken in trying to replace Euclid's parallel axiom with another axiom which could be deduced from the others. In fact he gave up this approach within a year for still in 1820, as his notebooks now show, he began to develop the basic ideas of hyperbolic geometry. On 3 November 1823 he wrote to his father that he had:-

... created a new, another world out of nothing...

but he still added a few lines later that it was not created yet. By 1824, however, there is evidence to suggest that he had developed most of what would appear in his treatise as a complete system of non-Euclidean geometry. In early 1825 Bolyai travelled to Marosvásárhely and explained his discoveries to his father. However Farkas Bolyai did not react enthusiastically which clearly disappointed János. Bolyai was posted to Arad in 1826 and there he found that Captain Wolther von Eckwehr, one of his old teachers of mathematics from the Academy in Vienna, was also stationed. Bolyai gave him a draft of the materials which he was writing on the theory of geometry, probably because he hoped for some constructive comments from him. However it would appear from Bolyai's later writings that he got nothing from von Eckwehr, in particular he never received the manuscript back.

In 1830 Bolyai learnt that he was to be sent to a posting in Lemberg. Early in 1831 he set off for Lemberg but visited his father in Marosvásárhely on his way. By now Farkas had come to understand the full significance of what his son had accomplished and strongly encouraged him to write up the work for publication as an Appendix to the Tentamen Ⓣ which was close to publication. Bolyai later wrote:-

Had my father not happened to urge or even force me at Marosvásárhely, on my way to duty in Lemberg, to immediately put things to paper, possibly the contents of the Appendix would never have seen the light of day.

What was contained in this mathematical masterpiece? After setting up his own definitions of 'parallel' and showing that if the Fifth Postulate held in one region of space it held throughout, and vice versa, he then stated clearly the different systems he would consider:-

... denote by Σ the system of geometry based on the hypothesis that Euclid's Fifth Postulate is true, and by S the system based on the opposite hypothesis. All theorems we state without explicitly specifying the system Σ or S in which the theorem is valid are meant to be absolute, that is, valid independently of whether Σ or S is true.

Today we call these three geometries Euclidean, hyperbolic, and absolute. Most of the Appendix deals with absolute geometry. By 20 June 1831 the Appendix had been published for on that day Farkas Bolyai sent a reprint to Gauss who, on reading the Appendix, wrote to a friend saying:-

I regard this young geometer Bolyai as a genius of the first order .

To Farkas Bolyai, however, Gauss wrote:-

To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years .

There is no doubt that Gauss was simply stating facts here. The clearest reference in Gauss's letters to his work on non-euclidean geometry, which shows the depth of his understanding, occurs in a letter he wrote to Taurinus on 8 November 1824 when he wrote:-

The assumption that the sum of the three angles of a triangle is less than 180° leads to a curious geometry, quite different from ours [i.e. Euclidean geometry] but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it excepting the determination of a constant, which cannot be fixed a priori. .... the three angles of a triangle become as small as one wishes, if only the sides are taken large enough, yet the area of the triangle can never exceed, or even attain a certain limit, regardless of how great the sides are.

Bolyai did not remain long in Lemberg for in 1832 he was posted to Olmütz where he was now a captain. The discovery that Gauss had anticipated much of his work, however, greatly upset Bolyai who took it as a severe blow. He became irritable and a difficult person to get on with. His health began to deteriorate and he was plagued with a fever which frequently disabled him so he found it increasingly difficult to carry out his military duties. He retired on 16 June 1833, asking to be pensioned off, and for a short time went to live with his father.

After spending a while with his father, Bolyai went to live on the family estate at Domáld which the family had inherited from Farkas Bolyai's mother. He had met Rozália Kibédi Orbán and they lived together at Domáld from 1834. They did not marry, however, since the law insisted on money being deposited before a marriage could take place and Bolyai could not afford the money. He had two children with Rozália but his pension was insufficient to allow the family to live in comfort. He seems neither to have managed well what money he did have nor to have kept the estate in good order. It seems that Farkas Bolyai did not approve of Rozália, was unhappy about his son's financial position, was unhappy that the family estate at Domáld was not being properly cared for, and was unhappy that his son was damaging his good name for Farkas was a highly respected member of the community.

Certainly Bolyai continued to develop mathematical theories while he lived at Domáld, but being isolated from the rest of the world of mathematics much of what he attempted was of little value. His one major undertaking, to attempt to develop all of mathematics based on axiom systems, was begun in 1834, for he wrote the preface in that year, but he never completed the work. What he did write concerned geometry and there are several ideas in this unpublished work which were ahead of their time such as notions of topological invariance.

In addition to his work on geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers. This he did because the Jablonowsky Society in Leipzig had put out a call for papers on the topic. Both Bolyai and his father submitted papers but neither were well received. János's paper was called Responsio and it was written to answer the question of whether the imaginary quantities used in geometry could be constructed. Bolyai chose to argue that the question was wrongly formulated, not perhaps the best way to find favour with judges. He argued that it was not their construction that was important, rather it was their definition and role in geometry which were significant.

In 1846 Bolyai moved from the estate at Domáld to Marosvásárhely. This meant that he was closer to his father and this seemed to make relations between the two even more strained. In 1848 Bolyai discovered that Lobachevsky had published a similar piece of work in 1829. Kagan writes [14]:-

János studied Lobachevsky's work carefully and analysed it line by line, not to say word by word, with just as much care as he administered in working out the Appendix. The work stirred a real storm in his soul and he gave outlet to his tribulations in the comments added to the 'Geometrical Examinations'.

The 'Comments' to the 'Geometrical Examinations' are more than a critical analysis of the work. They express the thoughts and anxieties of János provoked by the perusal of the book. They include his complaint that he was wronged, his suspicion that Lobachevsky did not exist at all, and that everything was the spiteful machinations of Gauss: it is the tragic lament of an ingenious geometrician who was aware of the significance of his discovery but failed to get support from the only person who could have appreciated his merits.

In spite of his mental agitation amidst which János put observations to paper, he preserved enough objectivity to highly appreciate the work of his rival. In his comment to Theorem 35 he remarks that the proofs of Lobachevsky concerning spherical trigonometry bear the impress of genius and his work should be esteemed as a masterly achievement.

In 1852 Bolyai left Rozália, whom he had married on 18 May 1849 believing that the law had now changed due to the Hungarian declaration of independance, and splitting with Rozália at least had the advantage that relations with his father improved. He gave up working on mathematics in his last years and instead tried to construct a theory of all knowledge. There are interesting ideas contained in the sections on linguistics and sociology.

Although he never published more than the few pages of the Appendix he left more than 20000 pages of manuscript of mathematical work when he died of pneumonia at the age of 57. These are now in the Bolyai-Teleki library in Târgu-Mureș. In 1945 the Hungarian university in Cluj was named after him. The Romanian University of Cluj which had been the King Ferdinand I University, was renamed Babeș University (after the Romanian natural scientist Victor Babeș). The two universities combined to become the Babeș-Bolyai University in 1959.