兰伯特Johann Heinrich Lambert

兰伯特的家族原籍洛林，直到三十年战争之前，洛林一直是法国领土。罗马天主教、路德宗和加尔文宗这三个基督教教派都试图将自己强加于人，多个国家卷入了在欧洲肆虐多年的冲突。1635年，信奉加尔文宗的兰伯特家族作为宗教难民逃离洛林，定居于米卢斯。当时米卢斯是一个自由帝国城市，已与瑞士结成防御同盟。

兰伯特的父亲卢卡斯兰伯特像他自己的父亲一样是个裁缝。卢卡斯于1724年与伊丽莎白·施默伯结婚，兰伯特是他们五个儿子之一。这是一个大家庭，除了五个男孩外还有两个女孩，卢卡斯兰伯特的收入不足以使他舒适地供养和教育这个家庭。兰伯特在米卢斯上学，到十二岁时接受了相当不错的教育，除基础科目外还学习法语和拉丁语。然而，当他十二岁时，他不得不离开学校帮助父亲做裁缝。大多数小男孩会在那时结束教育，但年轻的兰伯特没有，他在自己的空闲时间继续学习。通常他白天忙于帮助父亲，但晚上他会自学科学科目。

这种学习模式变得越来越困难，因为在他十五岁时，他不得不做一名职员来为家庭赚更多的钱。事实上，这对兰伯特来说是一种自然的职业，因为他已经获得了高超的书法技能，他在塞普瓦的铁工厂得到了一份工作，该地位于米卢斯以南，几乎正位于巴塞尔以西。很快，他在学术科目上日益增长的技能帮助他获得了一份私人教师的工作。当他十七岁时，兰伯特离开了铁工厂的职位，担任兰伯特鲁道夫·伊塞林的秘书，后者是Basler Zeitung的编辑，这是一份保守的日报。这个职位对兰伯特来说很理想，他现在可以更深入地专注于自己对数学、天文学和哲学的研究。他在一封信中写道（见[1]）：-

我买了一些书来学习哲学的第一原理。我努力的首要目标是成为完美和幸福的手段。我明白，在心灵得到启迪之前，意志无法得到改善。我研究了Christian Wolff的《论人类心灵的力量》、尼古拉斯‧马勒伯朗士的《论真理的探究》和弗瑞兹·约翰 Locke的《人类理解论》。数学科学，特别是代数和力学，为我提供了清晰而深刻的例子来证实我所学到的规则。因此，我能够更容易、更深刻地渗透到其他科学中，并向他人解释它们。诚然，我意识到缺乏口头指导，但我试图以更加勤奋来弥补……

1748年，当他二十岁时，兰伯特获得了一个新职位，这次是在库尔的彼得·冯·萨利斯伯爵家中担任家庭教师。这个城镇位于格劳宾登，当时是瑞士联邦的一部分。他成为伯爵的孙子和他的表弟的家庭教师，他们两人都是十一岁，还有另一位七岁的家庭成员。兰伯特现在可以使用伯爵家中优秀的图书馆，并且处于更有利的地位来继续他对数学、天文学和哲学的研究。在库尔期间，他制作了自己的天文仪器，并深入钻研数学和物理课题。

正是在库尔，兰伯特首次引起了科学界的注意。他被选入库尔文学会和设在巴塞尔的瑞士科学学会。他为科学学会承担了诸如定期进行气象观测等工作，并开始发表科学文章，他的第一篇关于热质的文章于1755年发表在Acta Helvetica上。兰伯特这篇关于热理论的论文发表在1751年成立的赫尔维蒂学会所出版期刊的第2卷上（详情见[23]）。1756年，兰伯特带着他此前八年中辅导的两个较大的男孩离开了库尔；他们当时19岁。他带着这两个年轻人进行了一次欧洲“大旅行”，首先访问了哥廷根。在那里，兰伯特结识了亚伯拉罕·哥特黑尔弗·凯斯特纳和托比亚斯·梅耶，并当选为哥廷根博学会会员。

1756年并不是开始欧洲旅行的最佳时机。那一年，法奥联盟准备进攻普鲁士，但后者没有等待被进攻，而是于8月29日入侵萨克森，开始了敌对行动。法国和奥地利在1757年开始占据上风，并在兰伯特于哥廷根学习期间占领了该地。他带着两个学生离开，前往乌得勒支，并在接下来的两年里以此地为基地，游览了荷兰的大部分主要城市。

兰伯特的第一本书是关于光通过各种介质的传播，于1758年在海牙出版。在返回库尔之前，兰伯特带着他的学生去了巴黎，在那里他遇到了让·勒朗·达朗贝尔，还去了马赛、尼斯、都灵和米兰。三十岁的兰伯特现在决定是时候寻找一个科学职位了，并在彼得·冯·萨利斯一家完成他们的“壮游”后不久离开了他们。然而，这并不容易，他的第一个愿望，即在哥廷根获得一个职位，很快就被证明无法实现。在苏黎世花了几个月进行天文观测后，他回到了米卢斯的家中，又呆了几个月。1759年，他去了奥格斯堡，在那里他为他的另外两本书找到了出版商，Photometria Ⓣ（光度学）和Cosmologische Briefe Ⓣ（关于宇宙学的简短信件）。关于1760年出版的Photometria，斯克里巴写道：-

兰伯特用很少的原始仪器进行了实验，但他的结论导致了以他的名字命名的定律。光束通过均匀透明度的吸收介质时，光的指数衰减通常被称为“兰伯特吸收定律”，尽管皮埃尔·布格更早发现了它。“兰伯特余弦定律”指出，漫反射平面表面的亮度与视线和表面法线形成的角度的余弦成正比。

1760年，兰伯特被莱昂哈德·欧拉推荐担任圣彼得堡科学院天文学教授一职，以填补因科学院重组和政治变动而空缺数年的职位。兰伯特受邀在慕尼黑按照柏林科学院的模式组建巴伐利亚科学院，但他与项目其他成员发生争执，于1762年离开了新科学院。然而此时，他关于宇宙学的重要著作Cosmologische Briefe Ⓣ（《宇宙志简信》）（1761年）已经问世。这是首次科学地提出宇宙由恒星星系构成这一观念。Coffa在评论[22]时写道：-

兰伯特的有限宇宙由星系、超星系乃至更高层次的恒星系统组成，所有这些都围绕各自的中心旋转；每个中心都由一个“摄政者”占据，这是一个极其巨大、极其致密且不透明的天体。整体由最中心的物体或最高摄政者主宰，这个物体“驾驭着整个创造物围绕自己旋转”。兰伯特的著作还因其方法论立场的现代性而引人注目：他对事实、理论、预测和可能验证之间差异的系统考察，直到20世纪才在宇宙学文献中得到效仿。

从慕尼黑返回后，兰伯特参加了米兰与库尔之间边界的测量，还访问了莱比锡，在那里他为一部哲学著作Neues Organon（1764年出版）找到了出版商。

他希望获得柏林科学院的一个职位，从而成为莱昂哈德·欧拉和约瑟夫·拉格朗日的同事。因此，1764年兰伯特应莱昂哈德·欧拉之邀前往柏林时非常高兴。然而，尽管兰伯特加入了胡格诺教会——莱昂哈德·欧拉是该教会的坚定成员——两人之间很快产生了分歧，主要涉及科学院的收入，该收入依赖于其出售历书的特权。这些分歧很可能促成了莱昂哈德·欧拉在1766年离开柏林前往圣彼得堡的决定。然而，这些并不是兰伯特抵达柏林后面临的唯一问题，因为起初腓特烈二世因兰伯特不寻常的外貌、奇怪的衣着和古怪的行为而拒绝任命他进入科学院。这些部分归因于他卑微的出身，以及他刻意选择不遵从上层阶级的习俗。部分也归因于他虔诚的宗教态度。然而，一旦腓特烈二世了解了兰伯特，他发现他是一个具有非凡洞察力的人。Scriba写道[1]：-

作为物理学部的成员长达十二年，直到他四十九岁去世，兰伯特发表了150多部作品。他是科学院中唯一一位经常行使权利，不仅在自己所属的学部，也在任何其他学部宣读论文的成员。

1766年，兰伯特写了Theorie der Parallellinien Ⓣ（《平行线理论》），这是对平行公设的研究。通过假设平行公设不成立，他成功推导出大量非欧几里得结果。他注意到，在这种新几何中，三角形的内角和随着其面积的减小而增大。

关于他在几何学方面的工作，Folta 在14中写道：-

兰伯特试图从两条新原则出发构建几何学：测量与广延，在他的版本中，这两条原则作为更一般元理论的确定基石出现。最重要的是，兰伯特仔细考虑了这些在公理上稳固的原则的逻辑后果。他关于数的公理几乎无法与欧几里得的算术公理相比；在几何学中，他通过确立关联的性质，超越了先前所假定的空间概念。兰伯特的物理学学识还指出了另一条清晰的途径，通过函数对物理依赖性的类比，有可能消除三维几何学的传统神话。兰伯特在18世纪下半叶在其元理论中提出的一些问题，至今仍未失去其意义。

然而，兰伯特最为人所知的是他关于π的工作。莱昂哈德·欧拉早在1737年就已确立$e$和$e^{2}$都是无理数。然而，兰伯特是第一个给出π是无理数的严格证明的人。在1768年提交给柏林科学院的一篇论文中，兰伯特表明，如果$x$是一个非零的rational number，那么$e^{x}$和$\tan x$都不可能是有理数。由于tan(π/4)=1，因此π/4必定是无理数。在[34]中讨论了这样一种说法：兰伯特的证明不完整，需要阿德里安-马里·勒让德的一个结果才能完成。Wallisser表明，兰伯特的证明不仅是完整的，而且是其时代的一项杰出数学成就。事实上，是Pringsheim在1898年首先指出，兰伯特的证明绝对正确，且在其时代堪称卓越，因为正切函数的展开不仅被形式地写下来，而且被证明是一个收敛的连分数。同样值得注意的是，兰伯特在这篇论文中猜想$e$和π是超越的。这一点又过了一个世纪才被证明，当时夏尔·埃尔米特证明了$e$是超越数，费迪南德·冯·林德曼证明了π是超越数。

兰伯特还首次系统地发展了双曲函数。几年前，Vincenzo Riccati已经研究过它们。兰伯特还因其对曲面上三角形的三角学研究、他在透视法和制图学方面的工作，以及他对theory of probability的贡献而重要。在后一主题中，他于1772年给出了死亡率法则的数学表述。他对概率论的贡献由Garibaldi和Penco在[15]中评价。他们写道：-

雅各布·伯努利在《推测术》中划出的路线被兰伯特以决定性的方式继承并发展，后者对测量误差理论的根本贡献近年来已被重新评价。兰伯特广泛而多方面的活动涵盖光学、宇宙学和大地测量学，并使他与他那个时代的主要科学家和哲学家（康德）有所接触。在其本人的根本性哲学著作《新工具》中，兰伯特发展了一种值得注意的逻辑概率理论，据我们所知，这一理论迄今尚未引起该领域杰出学者如约翰·梅纳德·凯恩斯的注意。逻辑概率是“第三种一般类型的概率”，在阐述顺序上紧随赌博中典型的“先验概率”和统计学的“后验概率”之后。

他还对哲学作出了重大贡献，并在Anlage zur ArchitectonicⓉ（《建筑体系》）（1771）中试图将哲学改造为一门演绎科学，以欧几里得处理几何学的方法为模型。在[3]中，Basso强调了兰伯特对演绎-几何方法论主要概念的理解，即公理、公设、定理、问题、作图和逻辑。

Johann Heinrich Lambert's family were originally from Lorraine which was a French territory up to the Thirty Years' War. Three Christian denominations, Roman Catholicism, Lutheranism, and Calvinism, each tried to impose themselves and various countries joined in the conflicts which raged across Europe for many years. In 1635 the Lambert family, who were Calvinists, fled from Lorraine as religious refugees and settled in Mulhouse. At that time Mulhouse was a free imperial city which had formed defensive alliances with the Swiss.

Lambert's father, Lukas Lambert, was a tailor as his own father had been. Lukas married Elizabeth Schmerber in 1724 and Johann Heinrich Lambert was one of their five sons. It was a large family, with two girls in addition to the five boys, and Lukas Lambert did not have sufficient income to enable him to support and educate the family in comfort. Heinrich attended school in Mulhouse, receiving a reasonably good education up to the age of twelve, studying French and Latin in addition to elementary subjects. However, when he was twelve years old he had to leave school to help his father with tailoring. Most young boys would have ended their education at that point, but not young Heinrich who continued to study in his own spare time. Usually his day was fully occupied in helping his father but in the evenings he would study scientific subjects on his own.

This pattern of study became increasingly difficult when, at the age of fifteen, he had to work as a clerk to earn more money for the family. In fact it was a natural occupation for Heinrich since he had acquired great skill in calligraphy and he was given a job at the ironworks at Seppois, which was south of Mulhouse and almost due west of Basel. Soon his increasing skill in academic subjects helped him to gain work as a private tutor. When he was seventeen years old Lambert left his position at the ironworks to take up a post as secretary to Johann Rudolf Iselin who was the editor of the Basler Zeitung, a conservative daily paper. This position was ideal for Lambert who could now concentrate even more deeply on his own study of mathematics, astronomy, and philosophy. He wrote in a letter (see [1]):-

I bought some books in order to learn the first principles of philosophy. The first object of my endeavours was the means to become perfect and happy. I understood that the will could not be improved before the mind had been enlightened. I studied Christian Wolff "On the power of the human mind", Nicolas Malebranche "On the investigation of truth" and John Locke "Essay concerning human understanding". The mathematical sciences, in particular algebra and mechanics, provided me with clear and profound examples to confirm the rules I had learned. Thereby I was able to penetrate into other sciences more easily and more profoundly, and to explain them to others, too. It is true that I was aware of the lack of oral instruction, but I tried to replace this by even more assiduity ...

In 1748, when he was twenty years old, Lambert took up a new position, this time as a tutor in the home of Count Peter von Salis in Chur. This town was in Graubünden which at that time was part of the Swiss Confederation. He became tutor to the Count's grandson and his cousin, who were both eleven years old, and another family member who was seven. Lambert could now use the excellent library in the Count's home and was in an even stronger position to continue his studies of mathematics, astronomy, and philosophy. While in Chur, he made his own astronomical instruments and delved deeply into mathematical and physical topics.

It was in Chur that Lambert first came to be noticed by the scientific community. He was elected to the Literary Society of Chur and to the Swiss Scientific Society based in Basel. He undertook work for the Scientific Society such as making regular meteorological observations and he began to publish scientific articles, his first being on caloric heat which appeared in Acta Helvetica in 1755. This paper by Lambert on the theory of heat appeared in Volume 2 of the journal published by the Societas Helvetica which had been founded in 1751 (see [23] for details). In 1756 Lambert left Chur with the two older boys whom he had tutored during the previous eight years; they were now 19 years old. He took the two young men on a "grand tour" of Europe, first visiting Göttingen. There Lambert met Kästner and Tobias Mayer, and was elected to the Learned Society of Göttingen.

Now 1756 was not the best time to begin a tour of Europe. In that year the French-Austrian alliance prepared to attack Prussia but the latter did not wait to be attacked and began hostilities by invading Saxony on 29 August. The French and Austrians began to get the upper hand in 1757 and occupied Göttingen while Lambert was studying there. He left with his two pupils and went to Utrecht which they used as a base for the next two years while they visited most of the main Dutch cities.

Lambert's first book, which was on the passage of light through various media, was published in The Hague in 1758. Before returning to Chur, Lambert took his pupils to Paris, where he met d'Alembert, and to Marseilles, Nice, Turin, and Milan. The thirty year old Lambert now decided that it was time to find a scientific position and left Peter von Salis's family soon after they had completed their "grand tour". However this did not prove to be an easy task and his first wish, namely to get a position in Göttingen, soon proved impossible to achieve. After a few months spent in Zürich making astronomical observations, he returned to his home in Mulhouse where he spent several further months. In 1759 he went to Augsburg where he found a publisher for two more of his books, Photometria Ⓣ and Cosmologische Briefe Ⓣ. Of Photometria, published in 1760, Scriba writes:-

Lambert carried out his experiments with few and primitive instruments, but his conclusions resulted in laws that bear his name. The exponential decrease of the light in a beam passing through an absorbing medium of uniform transparency is often called 'Lambert's law of absorption', although Bouguer discovered it earlier. 'Lambert's cosine law' states that the brightness of a diffusely radiating plane surface is proportional to the cosine of the angle formed by the line of sight and the normal to the surface.

In 1760 Euler recommended Lambert for the position of Professor of Astronomy at the St Petersburg Academy of Sciences to fill a vacancy which, due to a reorganization of the Academy and political changes, remained unfilled for several years. Lambert was asked to organise a Bavarian Academy of Sciences in Munich along the lines of the Berlin Academy, but he fell out with other members of the project and left the new Academy in 1762. By this time, however, his important work on cosmology Cosmologische Briefe Ⓣ (1761) had appeared. It is the first scientific presentation of the notion that the Universe is composed of galaxies of stars. Coffa, reviewing [22] writes:-

Lambert's finite Universe is composed of galaxies, supergalaxies and even higher systems of stars all of which rotate around their respective centres; each of these centres is occupied by a "regent", an immensely large, exceedingly dense and opaque body. The whole is dominated by the centralmost body or supreme regent, the body "which steers around itself the whole creation". Lambert's book is also remarkable for the modernity of its methodological stand: his systematic survey of the differences among facts, theories, predictions and possible verifications was not emulated in cosmological literature until the 20th century.

After returning from Munich, Lambert took part in a survey of the border between Milan and Chur, and also visited Leipzig where he was able to find a publisher for a work on philosophy Neues Organon (published in 1764).

He wanted to gain a position at the Berlin Academy of Sciences and so become a colleague of Euler and Lagrange. Lambert was therefore delighted to go to Berlin in 1764 at the invitation of Euler. However, although Lambert joined the Huguenot Church, of which Euler was a staunch member, differences between the two men soon arose, mainly concerning the income of the Academy, which depended on its privilege to sell calendars. It may well be that these differences contributed to Euler's decision to leave Berlin for St Petersburg in 1766. However these were not the only problems that Lambert faced after arriving in Berlin for at first Frederick II refused to appoint Lambert to the Academy on account of his unusual appearance, strange dress and eccentric behaviour. These were in part due to his humble background together with the fact that he deliberately chose not to conform to the conventions of the upper classes. Also it was in part due to his devout religious attitude. However, once Frederick II got to know Lambert, he discovered that he was a man of extraordinary insight. Scriba writes [1]:-

As a member of the physical class for twelve years, until his death at the age of forty-nine, Lambert produced more than 150 works for publication. He was the only member of the Academy to exercise regularly the right to read papers not only in his own class, but in any other class as well.

In 1766 Lambert wrote Theorie der Parallellinien Ⓣ which was a study of the parallel postulate. By assuming that the parallel postulate was false, he managed to deduce a large number of non-euclidean results. He noticed that in this new geometry the sum of the angles of a triangle increases as its area decreases.

Of his work on geometry, Folta writes in [14]:-

Lambert tried to build up geometry from two new principles: measurement and extent, which occurred in his version as definite building blocks of a more general metatheory. Above all, Lambert carefully considered the logical consequences of these axiomatically secure principles. His axioms concerning number can hardly be compared with Euclid's arithmetical axioms; in geometry he goes beyond the previously assumed concept of space, by establishing the properties of incidence. Lambert's physical erudition indicates yet another clear way in which it would be possible to eliminate the traditional myth of three-dimensional geometry through the parallels with the physical dependence of functions. A number of questions that were formulated by Lambert in his metatheory in the second half of the 18th century have not ceased to remain of interest today.

Lambert is best known, however, for his work on π. Euler had already established in 1737 that $e$ and $e^{2}$ are both irrational. However Lambert was the first to give a rigorous proof that π is irrational. In a paper presented to the Berlin Academy in 1768 Lambert showed that, if $x$ is a nonzero rational number, then neither $e^{x}$ nor $\tan x$ can be rational. Since tan (π/4) = 1 then π/4 must be irrational. In [34] there is discussion of the claim that Lambert's proof is incomplete and requires a result by Legendre to complete it. Wallisser shows that Lambert's proof is not only complete but is an outstanding mathematical achievement for its time. In fact it was Pringsheim in 1898 who first noted that Lambert's proof was absolutely correct and exceptional for its time, since the expansion of the tangent function was not only written down formally, but also proved to be a convergent continued fraction. Also, remarkably, Lambert conjectured in this paper that $e$ and π are transcendental. This was not proved for another century when Hermite proved that $e$ is transcendental and Lindemann proved that π is transcendental.

Lambert also made the first systematic development of hyperbolic functions. A few years earlier they had been studied by Vincenzo Riccati. Lambert is also important for his study of the trigonometry of triangles on surfaces, his work on perspective and cartography, as well as his contributions to the theory of probability. In this latter topic, he gave a mathematical formulation of the law of mortality in 1772. His contributions to probability are evaluated by Garibaldi and Penco in [15]. They write:-

The lines drawn by Jacob Bernoulli in the "Ars conjectandi" were taken up and developed in a decisive way by Lambert, whose fundamental contributions to the theory of errors in measurement have been re-evaluated in recent years. Lambert's vast, multifaceted activity covered optics, cosmology and geodesy and put him in contact with the major scientists and philosophers of his day (Kant). In his own fundamental philosophical opus, the "Neues Organon", Lambert developed a noteworthy theory of logical probability that, to our knowledge, has thus far escaped the attention of eminent scholars in the field such as Keynes. Logical probability is a 'third general type of probability' that follows in order of exposition the 'a priori probability' typical of games of change and the 'a posteriori probability' of statistics.

He also made a major contribution to philosophy and in Anlage zur Architectonic Ⓣ (1771) he attempted to transform philosophy into a deductive science, modelled on Euclid's approach to geometry. In [3] Basso highlights Lambert's understanding of the main concepts of the deductive-geometric methodology, namely axioms, postulates, theorems, problems, constructions, and logic.