罗巴切夫斯基Nikolai Ivanovich Lobachevsky

罗巴切夫斯基的父亲Ivan Maksimovich 罗巴切夫斯基在一家从事土地测量的办公室做职员，而罗巴切夫斯基 罗巴切夫斯基的母亲是Praskovia Alexandrovna Lobachevskaya。罗巴切夫斯基 罗巴切夫斯基是这个贫穷家庭三个儿子之一。当罗巴切夫斯基 罗巴切夫斯基七岁时，他的父亲去世了，1800年，他的母亲带着三个儿子搬到了俄罗斯西部、西伯利亚边缘的喀山市。在那里，男孩们就读于喀山文理中学，由政府奖学金资助，罗巴切夫斯基 罗巴切夫斯基于1802年入学。

1807年，罗巴切夫斯基从中学毕业，作为免费生进入喀山大学。喀山国立大学成立于1804年，是沙皇亚历山大一世众多改革之一的成果，并于次年开学，仅比罗巴切夫斯基开始其本科生涯早两年。他最初的打算是学医，但后来改为学习涉及数学和物理的广泛科学课程。Vinberg写道[44]：-

最初几年，系里的氛围相当有利。学生们充满热情。他们日夜学习以弥补知识的不足。教授们主要是从德国邀请来的，结果证明他们都是优秀的教师，这并不常见。罗巴切夫斯基在他所修的所有课程中都极为成功……

从德国邀请来的优秀教授之一是马丁·巴特尔斯（1769 - 1833），他被任命为数学教授。马丁·巴特尔斯是一位学校教师，也是卡尔·弗里德里希·高斯的朋友，两人有书信往来。我们稍后将回头讨论一些历史学家的观点，例如莫里斯·克莱因的观点，即卡尔·弗里德里希·高斯可能通过马丁·巴特尔斯与卡尔·弗里德里希·高斯之间的通信，就罗巴切夫斯基在数学工作中可能采取的方向给了罗巴切夫斯基一些提示。作为一位技艺精湛的教师，马丁·巴特尔斯很快使罗巴切夫斯基对数学产生了兴趣。我们确实知道马丁·巴特尔斯讲授过数学史，并且他开设了一门以J.É. 蒙蒂克拉的文本为基础的课程。由于欧几里得的Elements及其平行线理论在J.É. 蒙蒂克拉的书中得到了详细讨论，罗巴切夫斯基对第五公设的兴趣似乎很可能是由这些讲座激发的。Laptev，见[29]，已经确认罗巴切夫斯基参加了马丁·巴特尔斯开设的这门历史课程。

罗巴切夫斯基于1811年获得物理和数学硕士学位。1814年他被任命为讲师，1816年成为编外教授。1822年他被任命为正教授[1]：-

……同年，他开始了行政生涯，成为负责监督新大学建筑建设的委员会成员。

罗巴切夫斯基在喀山大学期间经历了困难。迪尔克·扬·斯特勒伊克在[2]中写道，由督学M L Magnitskii领导的行政当局：-

……反映了沙皇亚历山大一世晚年的精神，他不信任现代科学和哲学，尤其是德国哲学家Immanuel Kant的哲学，认为它们是法国大革命的邪恶产物，是对正统宗教的威胁。1819-26年间喀山的结果是派系斗争、学术标准衰败、解雇以及一些最优秀教授的离开，包括……马丁·巴特尔斯……

尽管有这些困难，其中许多困难按照罗巴切夫斯基在44中的说法是由他“正直而独立的性格”带来的，他还是取得了许多成就。除了我们将在本文后面谈到的他精力充沛的数学研究之外，他还教授包括数学、物理学和天文学在内的广泛课题。他的讲座[44]：-

……详尽而清晰，因此即使基础薄弱的学生也能理解。

罗巴切夫斯基为物理实验室购买设备，并为圣彼得堡的图书馆购置书籍。他在大学内被任命担任重要职务，例如1820年至1825年间担任数学与物理系主任，1825年至1835年间担任图书馆馆长。他还担任天文台台长，显然对大学内部的决策有着强烈影响。然而[30]：-

……与督学[马格尼茨基]的冲突仍在继续。

1826年，沙皇尼古拉一世成为统治者，并推行了一种更为宽容的制度。同年，马格尼茨基被解除喀山大学督学职务，新的督学M N穆辛-普希金被任命。此时气氛明显改变，穆辛-普希金发现罗巴切夫斯基是一个可以与他合作、为大学带来重要变革的人。1827年，罗巴切夫斯基成为喀山大学校长，此后他担任该职位达19年。次年，他发表了一篇演讲（于1832年出版）On the most important subjects of education，这清楚地展示了他教育哲学中的思想。拉普捷夫写道，罗巴切夫斯基[30]：-

……概述了人格和谐发展的理想，强调了教养和教育的社会意义，并讨论了科学的作用以及科学家对国家和人民的责任。

罗巴切夫斯基任校长期间，喀山大学蓬勃发展，这在很大程度上归功于他的影响。学校大力开展新建筑计划，建造了图书馆、天文台、新的医疗设施以及物理、化学和解剖学实验室。他极力推动提高科学研究水平，同时同样鼓励艺术领域的研究，尤其发展了一个领先的东方学研究中心。学生人数显著增加，罗巴切夫斯基投入了大量精力，不仅提高大学的教育水平，也提高当地学校的教育水平。

他担任喀山大学校长期间[44]，大学遭遇了两次自然灾害：-

……1830年的霍乱疫情和1842年的一场大火。由于罗巴切夫斯基采取了果断合理的措施，大学的损失被降到最低。由于他在霍乱疫情期间的活动，罗巴切夫斯基收到了皇帝的感谢信。

这本书[5]收录了罗巴切夫斯基作为喀山大学校长撰写的一些年度报告。已出版的这些只是从数百页手稿中选取的一小部分样本：-

……以[罗巴切夫斯基]饱满、有力的笔迹写成，几乎没有错误，更不用说涂改，这些报告无论当时还是现在都是所有学者真正工作道路上的障碍。

尽管行政负担如此繁重，罗巴切夫斯基仍继续教授各种不同的课题，如力学、流体动力学、积分、微分方程、变分法和数学物理。1838年至1840年间，他甚至抽出时间向公众讲授物理，但繁重的工作最终损害了他的健康。

1832年，罗巴切夫斯基与出身富裕家庭的瓦尔瓦拉·阿列克谢耶夫娜·莫伊谢耶娃女士结婚。结婚时，他的妻子还是个年轻姑娘，而罗巴切夫斯基已经四十岁了。这段婚姻使他们有了七个孩子，据[1]称，这些孩子：-

……而且他为庄园进行技术改良的花费，使他在退休时几乎没有什么钱。

在[44]中，温伯格写道：-

这对夫妇住在一栋三层的大房子里，热情好客，接待了许多客人。然而，罗巴切夫斯基的婚姻并不幸运。

罗巴切夫斯基于1846年退休（实际上是被喀山大学解聘），他的健康状况迅速恶化。马特维耶夫在其文章[34]中引用了许多关于罗巴切夫斯基在斯洛博德卡购得的庄园的记录。传记作者们有许多说法：-

罗巴切夫斯基是个不切实际的管理者，靠养老金生活却购买庄园，从而危及了自己的经济状况；他没有时间照管庄园，对庄园也兴趣寥寥；他陷入贫困，被地方官员忽视，等等。

但罗巴切夫斯基表明这些说法完全没有根据。然而，他退休后不久，他最喜爱的长子去世，罗巴切夫斯基深受这一悲剧的打击。他所患的疾病逐渐加重，最终导致失明。这些以及经济上的困难，加重了他晚年不得不承受的沉重负担。我们即将讨论的他那些伟大的数学成就，在他有生之年并未得到承认，他去世时对自己工作将获得的声誉和重要性毫无所知。

自欧几里得对几何学作出公理化表述以来，数学家们一直试图把他的第五公设作为从其他四条公理推出的定理来证明。第五公设说的是：给定一条直线和直线外一点，通过该点可以作唯一一条与给定直线平行的直线。罗巴切夫斯基没有试图把这个公设作为定理来证明。相反，他研究了第五公设不一定成立的几何学。罗巴切夫斯基把欧几里得几何归为这种更一般几何学的特例。

他的主要著作Geometriya于1823年完成，直到1909年才以其原始形式出版。1826年2月11日，在喀山大学物理数学科学系的会议上，罗巴切夫斯基请求听取他关于一种新几何学的工作，他的论文A concise outline of the foundations of geometry被送交评审人。这篇论文的文本没有留存下来，但其思想被纳入——也许是经过修改的形式——罗巴切夫斯基关于双曲几何的第一篇出版物中。他于1829年在non-euclidean geometry上发表了这项工作，这是该主题首次出现在印刷品中的论述。它发表在Kazan Messenger上，但当圣彼得堡科学院提交给米哈伊尔·奥斯特罗格拉德斯基要求发表时被拒绝了。

1834年，罗巴切夫斯基找到了一种近似求解代数方程根的方法。这种代数方程数值解法由格拉夫独立发展出来，以回答柏林科学院的一道悬赏问题，如今是一种特别适合用计算机求解此类问题的方法。这种方法今天被称为当德兰-格拉夫方法，因为当德兰也独立地研究了它，但只有在俄罗斯，这种方法似乎以罗巴切夫斯基的名字命名，他是第三位独立发现者。关于这种方法及其三位发现者的讨论，见[24]。

1837年，罗巴切夫斯基发表了他的文章Géométrie imaginaireⓉ（《虚几何》），而他新几何学的摘要Geometrische Untersuchungen zur Theorie der ParellellinienⓉ（《关于平行线理论的几何研究》）于1840年在柏林出版。这后一篇出版物给卡尔·弗里德里希·高斯留下了深刻印象，但关于卡尔·弗里德里希·高斯在非欧几何发现中所起作用的许多论述完全是错误的。有一个巧合，源于我们知道卡尔·弗里德里希·高斯本人发现了非欧几何，但只告诉了极少数人，只有他最亲密的朋友。他的两位朋友是鲍耶·法卡斯——鲍耶（非欧几何的一位独立发现者）的父亲——和马丁·巴特尔斯，后者是罗巴切夫斯基的老师。这一巧合引发了猜测，认为罗巴切夫斯基和鲍耶都是由卡尔·弗里德里希·高斯引导作出他们的发现的。莫里斯·克莱因提出了这一理论，但已在多部著作中被驳斥；例如见[28]。此外，拉普捷夫在[29]中考察了马丁·巴特尔斯与卡尔·弗里德里希·高斯之间的通信，并表明马丁·巴特尔斯并不知道卡尔·弗里德里希·高斯在非欧几何方面的结果。

关于罗巴切夫斯基和非欧几何发现还有其他一些说法，最近已被驳斥。例如，[25]中声称罗巴切夫斯基与卡尔·弗里德里希·高斯有通信往来（卡尔·弗里德里希·高斯高度评价罗巴切夫斯基的工作，但与他没有私人通信），声称卡尔·弗里德里希·高斯学习俄语以阅读罗巴切夫斯基的俄文论文，如[1]中所说（实际上，卡尔·弗里德里希·高斯在听说罗巴切夫斯基之前就已经学过俄语），以及声称卡尔·弗里德里希·高斯是罗巴切夫斯基著作在德国的“好宣传者”（卡尔·弗里德里希·高斯从未公开评论过罗巴切夫斯基的工作），这些都被证明是假的。

罗巴切夫斯基的双曲几何如何被接受的故事很复杂，这本传记不适合深入细节，但我们将指出主要事件。1866年，即罗巴切夫斯基去世十年后，Hoüel出版了罗巴切夫斯基的Geometrische UntersuchungenⓉ（关于平行线理论的几何研究）的法文译本，以及卡尔·弗里德里希·高斯关于非欧几何的一些通信。贝尔特拉米在1868年给出了罗巴切夫斯基几何的具体实现。卡尔·魏尔斯特拉斯在1870年主持了一个关于罗巴切夫斯基几何的讨论班，菲利克斯·克莱因参加了，两年后，在菲利克斯·克莱因和索菲斯·李在巴黎讨论了这些新的几何推广之后，菲利克斯·克莱因提出了他对几何的一般观点，即几何是Erlanger Programm中某个群变换作用下不变的性质。1882年和1887年，儒勒·昂利·庞加莱对罗巴切夫斯基的几何做出了两项进一步的重大贡献。也许这些最终标志着罗巴切夫斯基思想的接受，这些思想最终将被视为解放数学家思维的关键步骤，使相对论有了自然的数学基础。

Nikolai Ivanovich Lobachevsky's father Ivan Maksimovich Lobachevsky, worked as a clerk in an office which was involved in land surveying while Nikolai Ivanovich's mother was Praskovia Alexandrovna Lobachevskaya. Nikolai Ivanovich was one of three sons in this poor family. When Nikolai Ivanovich was seven years of age his father died and, in 1800, his mother moved with her three sons to the city of Kazan in western Russia on the edge of Siberia. There the boys attended Kazan Gymnasium, financed by government scholarships, with Nikolai Ivanovich entering the school in 1802.

In 1807 Lobachevsky graduated from the Gymnasium and entered Kazan University as a free student. Kazan State University had been founded in 1804, the result of one of the many reforms of the emperor Alexander I, and it opened in the following year, only two years before Lobachevsky began his undergraduate career. His original intention was to study medicine but he changed to study a broad scientific course involving mathematics and physics. Vinberg writes [44]:-

In the first years the atmosphere in the Department was quite favourable. The students were full of enthusiasm. They studied day and night to compensate for lack of knowledge. The professors, mainly invited from Germany, turned out to be excellent teachers, which was not common. Lobachevsky was highly successful in all courses he took ...

One of the excellent professors who had been invited from Germany was Martin Bartels (1769 - 1833) who had been appointed as Professor of Mathematics. Bartels was a school teacher and friend of Gauss, and the two corresponded. We shall return later to discuss ideas of some historians, for example M Kline, that Gauss may have given Lobachevsky hints regarding directions that he might take in his mathematical work through the letters exchanged between Bartels and Gauss. A skilled teacher, Bartels soon interested Lobachevsky in mathematics. We do know that Bartels lectured on the history of mathematics and that he gave a course based on the text by Montucla. Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevsky's interest in the Fifth Postulate was stimulated by these lectures. Laptev, see [29], has established that Lobachevsky attended this history course given by Bartels.

Lobachevsky received a Master's Degree in physics and mathematics in 1811. In 1814 he was appointed to a lectureship and in 1816 he became an extraordinary professor. In 1822 he was appointed as a full professor [1]:-

... the same year in which he began an administrative career as a member of the committee formed to supervise the construction of new university buildings.

Lobachevsky had experienced difficulties during this period at the University of Kazan. Struik writes in [2] that the administration, led by the curator M L Magnitskii:-

... reflected the spirit of the later years of Tsar Alexander I, who was distrustful of modern science and philosophy, particularly that of the German philosopher Immanuel Kant, as evil products of the French Revolution and a menace to orthodox religion. The results at Kazan during the years 1819-26 were factionalism, decay of academic standards, dismissals, and departure of some of the best professors, including ... Bartels ...

Despite these difficulties, many brought on according to Vinberg in [44] by Lobachevsky's "upright and independent character", he achieved many things. As well as his vigorous mathematical research, which we shall talk about later in this article, he taught a wide range of topics including mathematics, physics and astronomy. His lectures [44]:-

... were detailed and clear, so that they could be understood even by poorly prepared students.

Lobachevsky bought equipment for the physics laboratory, and he purchased books for the library in St Petersburg. He was appointed to important positions within the university such as the dean of the Mathematics and Physics Department between 1820 and 1825 and head librarian from 1825 to 1835. He also served as Head of the Observatory and was clearly strongly influencing policy within the University. However [30]:-

... the clashes with the curator [Magnitskii] continued.

In 1826 Tsar Nicholas I became ruler and introduced a more tolerant regime. In that year Magnitskii was dismissed as curator of Kazan University and a new curator M N Musin-Pushkin was appointed. The atmosphere now changed markedly and Musin-Pushkin found in Lobachevsky someone who could work with him in bringing important changes to the university. In 1827 Lobachevsky became rector of Kazan University, a post he was to hold for the next 19 years. The following year he made a speech (which was published in 1832) On the most important subjects of education and this gives clearly what were the ideas in his educational philosophy. Laptev writes in that Lobachevsky [30]:-

... outlined the ideal of the harmonious development of the personality, emphasised the social significance of upbringing and education, and discussed the role of the sciences and the scientist's duty to his country and people.

The University of Kazan flourished while Lobachevsky was rector, and this was largely due to his influence. There was a vigorous programme of new building, with a library, an astronomical observatory, new medical facilities and physics, chemistry and anatomy laboratories being constructed. He pressed strongly for higher levels of scientific research and he equally encouraged research in the arts, particularly developing a leading centre for Oriental Studies. There was a marked increase in the number of students and Lobachevsky invested much effort in raising not only the standards of education in the university, but also in the local schools.

Two natural disasters struck the university while he was Rector of Kazan [44]:-

... a cholera epidemic in 1830 and a big fire in 1842. Owing to resolute and reasonable measures taken by Lobachevsky the damage to the University was reduced to a minimum. for his activity during the cholera epidemic Lobachevsky received a message of thanks from the Emperor.

The book [5] contains some yearly reports Lobachevsky wrote as rector of Kazan University. Those published are only a small sample taken from the hundreds of pages of manuscript:-

... written in [Lobachevsky's] full, firm hand, with hardly an error, let alone a crossing-out, reports which were an obstacle to real work in the path of all academics then as now.

Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics. He even found time to give lectures on physics to the general public during the years 1838 to 1840 but the heavy work-load was to eventually take its toll on his health.

In 1832 Lobachevsky married Lady Varvara Alexejevna Moisieva who came from a wealthy family. At the time of his marriage his wife was a young girl while Lobachevsky was forty years old. The marriage gave them seven children and it is claimed in [1] that the children:-

... and the cost of technological improvements for his estate left him with little money upon his retirement.

In [44] Vinberg writes:-

The couple lived in a big three-storey house and received a lot of guests with lavish hospitality. However Lobachevsky was not lucky in his marriage.

After Lobachevsky retired in 1846 (essentially dismissed by the University of Kazan), his health rapidly deteriorated. Matveev, in his article [34], quotes many records concerning Lobachevsky's estate which he purchased at Slobodka. There are many claims by biographers that:-

Lobachevsky was an impractical manager who jeopardised his financial position by purchasing the estate while living on a pension; that he had no time to look after the estate and took little interest in it; that he was left in poverty and ignored by the local officials, etc.

But Matveev shows that these claims are totally unjustified. Soon after he retired, however, his favourite eldest son died and Lobachevsky was hit hard by this tragedy. The illness that he suffered from became progressively worse and led to blindness. These and financial difficulties added to the heavy burdens he had to bear over his last years. His great mathematical achievements, which we shall now discuss, were not recognised in his lifetime and he died without having any notion of the fame and importance that his work would achieve.

Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms. The fifth postulate states that given a line and a point not on the line, a unique line can be drawn through the point parallel to the given line. Lobachevsky did not try to prove this postulate as a theorem. Instead he studied geometry in which the fifth postulate does not necessarily hold. Lobachevsky categorised euclidean as a special case of this more general geometry.

His major work, Geometriya completed in 1823, was not published in its original form until 1909. On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevsky requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees. The text of this paper has not survived but the ideas were incorporated, perhaps in a modified form, in Lobachevsky's first publication on hyperbolic geometry. He published this work on non-euclidean geometry, the first account of the subject to appear in print, in 1829. It was published in the Kazan Messenger but rejected by Ostrogradski when it was submitted for publication by the St Petersburg Academy of Sciences.

In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations. This method of numerical solution of algebraic equations, developed independently by Gräffe to answer a prize question of the Berlin Academy, is today a particularly suitable method for using computers to solve such problems. This method is today called the Dandelin-Gräffe method since Dandelin also independently investigated it, but only in Russia does the method appear to be named after Lobachevsky who is the third independent discoverer. See [24] for a discussion of the method and its three discoverers.

In 1837 Lobachevsky published his article Géométrie imaginaire Ⓣ and a summary of his new geometry Geometrische Untersuchungen zur Theorie der Parellellinien Ⓣ was published in Berlin in 1840. This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non-euclidean geometry which is just simply false. There is a coincidence which arises from the fact that we know that Gauss himself discovered non-euclidean geometry but told very few people, only his closest friends. Two of his friends were Farkas Bolyai, the father of János Bolyai (an independent discoverer of non-euclidean geometry), and Bartels who was Lobachevsky's teacher. This coincidence has prompted speculation that both Lobachevsky and Bolyai were led to their discoveries by Gauss. M Kline has put forward this theory but it has been refuted in several works; see for example [28]. Also Laptev in [29] has examined the correspondence between Bartels and Gauss and shown that Bartels did not know about Gauss's results in non-euclidean geometry.

There are other claims made about Lobachevsky and the discovery of non-euclidean geometry which have been recently refuted. For example in [25] the claims that Lobachevsky was in correspondence with Gauss ( Gauss appreciated Lobachevsky's works very highly but had no personal correspondence with him), that Gauss studied Russian to read Lobachevsky's Russian papers as claimed for example in [1] (actually, Gauss had studied Russian before he had even heard of Lobachevsky), and that Gauss was a "good propagandist" of Lobachevsky's works in Germany (Gauss never commented publicly on Lobachevsky's work) are shown to be false.

The story of how Lobachevsky's hyperbolic geometry came to be accepted is a complex one and this biography is not the place in which to go into details, but we shall note the main events. In 1866, ten years after Lobachevsky's death, Hoüel published a French translation of Lobachevsky's Geometrische Untersuchungen Ⓣ together with some of Gauss's correspondence on non-euclidean geometry. Beltrami, in 1868, gave a concrete realisation of Lobachevsky's geometry. Weierstrass led a seminar on Lobachevsky's geometry in 1870 which was attended by Klein and, two years later, after Klein and Lie had discussed these new generalisations of geometry in Paris, Klein produced his general view of geometry as the properties invariant under the action of some group of transformations in the Erlanger Programm. There were two further major contributions to Lobachevsky's geometry by Poincaré in 1882 and 1887. Perhaps these finally mark the acceptance of Lobachevsky's ideas which would eventually be seen as vital steps in freeing the thinking of mathematicians so that relativity theory had a natural mathematical foundation.