加布里尔·克拉默Gabriel Cramer

加布里尔·克拉默的父亲是Jean Isaac Cramer，他是日内瓦的一名医生，而他的母亲是Anne Mallet。Jean和Anne有三个儿子，他们都取得了学术上的成功。除了克拉默，他们的另外两个儿子是Jean-Antione，他继承了父亲的职业，以及Jean，他成为了一名法学教授。

克拉默在日内瓦的教育进展迅速，1722年，当他只有十八岁时，他提交了一篇关于声音理论的论文，被授予博士学位。两年后，他竞争日内瓦Académie de Clavin的哲学讲席。

讲席的竞争在三个人之间进行；最年长的是Amédée de la Rive，而另外两个都是年轻人，Giovanni Ludovico Calandrini二十一岁，克拉默比他小一岁。负责任命的官员们倾向于选择更有经验的年长者，但他们对这两位才华横溢的年轻人印象深刻，于是想出了一个聪明的计划，使他们能够获得这三人的服务。显然，他们着眼于未来，在克拉默和Calandrini身上看到了两位将为科学院做出重要未来贡献的人。

地方法官提出的方案是把哲学讲席分成两个讲席，一个哲学讲席和一个数学讲席。德·拉·里夫被提供了哲学讲席，毕竟这正是他最初申请的，而克拉默和Calandrini则被提供了数学讲席，条件是两人分担职责并分享薪金。地方法官还对这项任命附加了另一个条件，即克拉默和Calandrini各自花两三年时间旅行，当一人外出时，另一人将承担全部职责并领取全部薪金。这是一个好计划，因为它不仅成功吸引了这三人来到科学院，而且还给了克拉默旅行并结识欧洲各地数学家的机会，他充分利用了这一点，这既使他受益，也使科学院受益。

克拉默和Calandrini划分了各自将教授的数学课程。克拉默教几何和力学，而Calandrini教代数和天文学。两人在这种安排中被配对，他们的朋友开玩笑地称他们为Castor和Pollux。如果他们的性格不同，这种安排可能会带来各种困难，但鉴于他们的天性，事情进展得非常好。据说克拉默是[1]：-

……友善、脾气好、声音和外表令人愉快，并且拥有良好的记忆力、判断力和健康。

我们不应给人这样的印象：克拉默只是适应了现有的教学模式。他提出了一项重大革新，并得到了科学院的采纳，那就是他用法语而非当时学者的传统语言拉丁语授课：——

……以便那些对这些科学有兴趣但不懂拉丁语的人能够受益。

克拉默于1724年被任命，他遵从任命条件，于1727年开始为期两年的旅行。他拜访了欧洲许多不同城市和国家的顶尖数学家。他直奔巴塞尔，那里有许多顶尖数学家在工作，与约翰·伯努利共事了五个月，也与莱昂哈德·欧拉共事，后者不久后前往圣彼得堡与丹尼尔·伯努利会合。随后克拉默访问了英格兰，在那里会见了爱德蒙·哈雷、亚伯拉罕·棣莫弗、斯特林和其他数学家。他与这些数学家的讨论以及他回到日内瓦后与他们的持续通信，对克拉默的工作产生了很大影响。

克拉默前往莱顿，在那里他会见了威廉·雅各布·葛洛夫桑德，然后他前往巴黎，在那里他与贝尔纳·勒布耶·德·丰特奈尔、皮埃尔·莫佩尔蒂、布丰、亚历克西斯·克劳德·克莱罗等人进行了讨论。这两年的旅行奠定了克拉默职业生涯的基调，因为他受到所遇到的所有数学家的高度评价，他一生都与他们通信，并且作为他们著作的编辑，他完成了许多极其有价值的重要任务。

1729年回到日内瓦后，克拉默正在为巴黎科学院为1730年设置的奖项撰写参赛作品，题目是“Quelle est la cause de la figure elliptique des planètes et de la mobilité de leurs aphélies?”Ⓣ（行星椭圆形状及其远日点移动的原因是什么？）克拉默的参赛作品被科学院评为收到的作品中第二好的，奖项由约翰·伯努利获得。1734年，当Calandrini被任命为哲学讲席时，“双胞胎”分开了，克拉默成为数学讲席的唯一持有者。

克拉默过着忙碌的生活，因为除了教学和与许多数学家通信外，他还撰写了相当有趣的文章，尽管这些文章不如与他通信的大多数顶尖数学家所写的文章重要。他在多个地方发表文章，包括1734年的Memoirs of the Paris Academy，以及1748年、1750年和1752年的柏林科学院。这些文章涵盖广泛的主题，包括几何问题的研究、数学史、哲学和复活节日期。他在Philosophical Transactions of the Royal Society of London上发表了一篇关于北极光的文章，还写了一篇关于法律的文章，其中他应用probability来证明拥有来自两三个证人的独立证词而非单一证人的重要性。

他的工作并不局限于学术领域，因为他还对地方政府感兴趣，并于1734年担任二百人议会议员，1749年担任七十人议会议员。他在这些议会中的工作涉及运用他广泛的数学和科学知识，因为他承担了涉及火炮、防御工事、建筑重建、挖掘的任务，并担任档案管理员。他于1747年第二次出国旅行，这次只访问了巴黎，在那里他与贝尔纳·勒布耶·德·丰特奈尔重续友谊，并会见了让·勒朗·达朗贝尔。

我们应当强调克拉默数学工作的两个方面。一是他所承担的编辑工作，二是他于1750年出版的重要数学著作Introduction à l'analyse des lignes courbes algébriques Ⓣ（《代数曲线分析引论》）。

约翰·伯努利于1748年去世，仅比克拉默早三年左右，但他安排克拉默在他去世前出版了他的Complete Works。约翰·伯努利坚持他的著作除克拉默外不得由任何其他编者出版，这表明了他对克拉默的极大尊重。约翰·伯努利的Complete Works由克拉默于1742年分四卷出版。约翰·伯努利不仅安排克拉默出版他的Complete Works，还请求他编辑雅各布·伯努利的著作。雅各布·伯努利已于1705年去世，克拉默于1744年分两卷出版了他的Works。这些并不完整，因为Ars conjectandi Ⓣ（《猜想的艺术》）被遗漏了（该书曾由尼古拉·伯努利一世于1713年出版），但这些卷册确实包含了此前未发表的材料以及理解它们所必需的数学背景。1745年，克拉默与Johann Castillon联合出版了约翰·伯努利与哥特弗里德·威廉·莱布尼茨之间的通信。克拉默还编辑了Christian Wolff的五卷本著作，该书初版于1732年至1741年间，新版于1743年至1752年间问世。

最后，我们应当描述克拉默最著名的书Introduction à l'analyse des lignes courbes algébraiqueⓉ（代数曲线分析引论）。这是一部克拉默模仿艾萨克·牛顿关于三次曲线的论文而写的著作，他高度赞扬了斯特林为艾萨克·牛顿的论文所写的评论。他还评论说，如果他早些知道莱昂哈德·欧拉的Introductio in analysin infinitorumⓉ（无穷小分析引论），他会大量利用它。当然，莱昂哈德·欧拉的书直到1748年才出版，那时克拉默的书的大部分很可能已经写好了。Jones在[1]中写道：——

他对莱昂哈德·欧拉的工作利用甚少，这一点得到了一个相当令人惊讶的事实的支持：在整本书中，克拉默基本上没有使用哥特弗里德·威廉·莱布尼茨或艾萨克·牛顿形式的无穷小分析，尽管他处理了切线、极大值和极小值以及曲率等主题，并在脚注中引用了科林·麦克劳林和布鲁克·泰勒。人们推测他从未接受或掌握微积分。

认为克拉默从未掌握微积分的说法必定值得怀疑，尤其是考虑到约翰·伯努利对他评价很高。

在定义了曲线类型并讨论了绘制其图形技巧的导论性章节之后，克拉默接着进入第二章，研究简化曲线的变换。第三章考察曲线的分类，正是在这一章中给出了如今著名的“克拉默法则”。在给出$n$次方程中任意常数的数目为$\large\frac{1}{2}\normalsize n^{2} + \large\frac{3}{2}\normalsize n$之后，他推导出$n$次方程可以使其通过$n$个点。取$n = 5$，他给出了一个例子，说明如何求出使2次方程通过5个点所涉及的五个常数。这导致5个未知数的5个线性方程，他让读者参阅包含克拉默求解法则的附录。当然，我们应当指出，克拉默肯定不是第一个给出这一法则的人。

克拉默工作中另一个“众所周知”的部分是他对克拉默悖论的描述。他陈述了科林·麦克劳林的一个定理，该定理说$n$次方程与$m$次方程相交于$nm$个点。取$n = m = 3$，这就是说两条三次曲线相交于9个点，然而他自己的公式$\large\frac{1}{2}\normalsize n^{2} + \large\frac{3}{2}\normalsize n$在$n = 3$时给出9，因此一条三次曲线由9个点唯一确定。克拉默说，这是一个悖论，但他解释这个悖论的尝试是不正确的。

克拉默的名字有时与另一个问题联系在一起，即Castillon-克拉默问题。这个问题由克拉默向Castillon提出，问的是如何在一个圆内作一个三角形使其通过三个给定点。Castillon在克拉默去世25年后解决了这个问题，该问题后来发展为关于圆锥曲线内接多边形的各种推广。

克拉默在很长一段时间里极其努力地撰写他的Introduction à l'analyseⓉ（分析引论），并在所有日常工作之外承担了大量编辑工作。他向来身体健康，但这次过度劳累再加上从马车上摔下，使他的健康突然恶化。他在床上躺了两个月才恢复，随后医生建议他去法国南部安静休养一段时间，以完全恢复体力。1751年12月21日，他离开日内瓦开始旅程，但两周后仍在旅途中便去世了。

Gabriel Cramer's father was Jean Isaac Cramer, who was a medical doctor in Geneva, while his mother was Anne Mallet. Jean and Anne had three sons who all went on to academic success. Besides Gabriel, their other two sons were Jean-Antione who followed his father's profession and Jean who became a professor of law.

Gabriel certainly moved rapidly through his education in Geneva, and in 1722 while he was still only eighteen years old he was awarded a doctorate having submitted a thesis on the theory of sound. Two years later he was competing for the chair of philosophy at the Académie de Clavin in Geneva.

The competition for the chair was between three men; the eldest was Amédée de la Rive while the other two were both young men, Giovanni Ludovico Calandrini who was twenty-one years old and Cramer who was one year younger. The magistrates who were making the appointment favoured the older man with more experience but they were so impressed with brilliant two young men that they thought up a clever plan to enable them to acquire the services of all three. Clearly they were looking to the future and seeing in Cramer and Calandrini two men who would make important future contributions to the Academy.

The scheme the magistrates proposed was to split the chair of philosophy into two chairs, one chair of philosophy and one chair of mathematics. De la Rive was offered the philosophy chair, which after all was what he had applied for in the first place, while Cramer and Calandrini were offered the mathematics chair on the understanding that they shared the duties and shared the salary. The magistrates put another condition on the appointment too, namely that Cramer and Calandrini each spend two or three years travelling and while one was away the other would take on the full list of duties and the full salary. It was a good plan for not only did it successfully attract all three men to the Academy, but it also gave Cramer the opportunity to travel and meet mathematicians around Europe and he was to take full advantage of this which both benefited him and the Academy.

Cramer and Calandrini divided up the mathematics courses each would teach. Cramer taught geometry and mechanics while Calandrini taught algebra and astronomy. The two had been paired in the arrangement and their friends joking called them Castor and Pollux. Had their personalities been different the arrangement might have presented all sorts of difficulties, but given their natures things worked out remarkably well. Cramer is said to have been [1]:-

... friendly, good-humoured, pleasant in voice and appearance, and possessed of good memory, judgement and health.

We must not give the impression that Cramer just fitted into an existing pattern of teaching. He proposed a major innovation, which the Academy accepted, which was that he taught his courses in French instead of Latin, the traditional language of scholars at that time:-

... in order that persons who had a taste for these sciences but no Latin could profit.

Appointed in 1724, Cramer followed the conditions of his appointment and set out for two years of travelling in 1727. He visited leading mathematicians in many different cities and countries of Europe. He headed straight away for Basel where many leading mathematicians were working, spending five months working with Johann Bernoulli, and also Euler who soon afterwards headed off to St Petersburg to be with Daniel Bernoulli. Cramer then visited England where he met Halley, de Moivre, Stirling, and other mathematicians. His discussions with these mathematicians and the continuing correspondence with them after he returned to Geneva had a big influence on Cramer's work.

From England Cramer made his way to Leiden where he met 'sGravesande, then he moved on to Paris where he had discussions with Fontenelle, Maupertuis, Buffon, Clairaut, and others. These two years of travelling were to set the tone for Cramer's career for he was highly regarded by all the mathematicians he met, he corresponded with them throughout his life, and he was to perform many extremely valuable major tasks as an editor of their works.

Back in Geneva in 1729, Cramer was at work on an entry for the prize set by the Paris Academy for 1730, which was "Quelle est la cause de la figure elliptique des planètes et de la mobilité de leurs aphélies?" Ⓣ Cramer's entry was judged as the second best of those received by the Academy, the prize being won by Johann Bernoulli. In 1734 the "twins" split up when Calandrini was appointed to the chair of philosophy and Cramer became the sole holder of the Chair of Mathematics.

Cramer lived a busy life, for in addition to his teaching and correspondence with many mathematicians, he produced articles of considerable interest although these are not of the importance of the articles written by most of the top mathematicians with whom he corresponded. He published articles in various places including the Memoirs of the Paris Academy in 1734, and of the Berlin Academy in 1748, 1750 and 1752. The articles cover a wide range of subjects including the study of geometric problems, the history of mathematics, philosophy, and the date of Easter. He published an article on the aurora borealis in the Philosophical Transactions of the Royal Society of London and he also wrote an article on law where he applied probability to demonstrate the significance of having independent testimony from two or three witnesses rather than from a single witness.

His work was not confined to academic areas for he was also interested in local government and served as a member of the Council of Two Hundred in 1734 and of the Council of Seventy in 1749. His work on these councils involved him using his broad mathematical and scientific knowledge, for he undertook tasks involving artillery, fortification, reconstruction of buildings, excavations, and he acted as an archivist. He made a second trip abroad in 1747, this time only visiting Paris where he renewed his friendship with Fontenelle as well as meeting d'Alembert.

There are two areas of Cramer's mathematical work which we should highlight. This is the editorial work which he undertook and also his major mathematical work Introduction à l'analyse des lignes courbes algébriques Ⓣ published in 1750.

Johann Bernoulli died in 1748, only three or so years before Cramer, but he arranged for Cramer to publish his Complete Works before his death. It shows how much respect Bernoulli had for Cramer that he insisted that no other edition of his works be published by any editor other than Cramer. Johann Bernoulli's Complete Works was published by Cramer in four volumes in 1742. Not only did Johann Bernoulli arrange for Cramer to publish his Complete Works but he also requested that he edit Jacob Bernoulli's works. Jacob Bernoulli had died 1705 and Cramer published his Works in two volumes in 1744. These are not complete since Ars conjectandi Ⓣ is omitted (this had been published by Nicolaus Bernoulli in 1713), but the volumes do contain previously unpublished material and the mathematical background necessary to understand them. In 1745, jointly with Johann Castillon, Cramer published the correspondence between Johann Bernoulli and Leibniz. Cramer also edited the five volume work by Christian Wolff, first published between 1732 and 1741 with a new edition appearing between 1743 and 1752.

Finally we should describe Cramer's most famous book Introduction à l'analyse des lignes courbes algébraique Ⓣ. It is a work which Cramer modelled on Newton's memoir on cubic curves and he praises highly a commentary on Newton's memoir written by Stirling. He also comments that had he known of Euler's Introductio in analysin infinitorum Ⓣ earlier he would have made great use of it. Of course Euler's book was only published in 1748 at which time much of Cramer's book might well have been written. Jones writes in [1]:-

That he made little use of Euler's work is supported by the rather surprising fact that throughout his book Cramer makes essentially no use of the infinitesimal calculus in either Leibniz's or Newton's form, although he deals with such topics as tangents, maxima and minima, and curvature, and cites Maclaurin and Taylor in footnotes. One conjectures that he never accepted or mastered the calculus.

The suggestion that Cramer never mastered the calculus must be considered doubtful, particularly given the high regard that he was held in by Johann Bernoulli.

After an introductory chapter in which types of curves are defined and techniques for drawing their graphs are discussed, Cramer goes on to a second chapter in which transformations to simplify curves are studied. The third chapter looks at a classification of curves and it is in this chapter that the now famous "Cramer's rule" is given. After giving the number of arbitrary constants in an equation of degree $n$ as $\large\frac{1}{2}\normalsize n^{2} + \large\frac{3}{2}\normalsize n$, he deduces that an equation of degree $n$ can be made to pass through $n$ points. Taking $n = 5$ he gives an example of finding the five constants involved in making an equation of degree 2 pass through 5 points. This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution. We should remark, of course, that Cramer was certainly not the first to give this rule.

The other "well known" part of Cramer's work is his description of Cramer's paradox. He states a theorem by Maclaurin which says that an equation of degree $n$ intersects an equation of degree $m$ in $nm$ points. Taking $n = m = 3$ this says that two cubics intersect in 9 points, yet his own formula $\large\frac{1}{2}\normalsize n^{2} + \large\frac{3}{2}\normalsize n$ with $n = 3$ gives 9 so a cubic is uniquely determined by 9 points. This, says Cramer, is a paradox, but his attempt to explain the paradox is incorrect.

Cramer's name has sometimes been attached to another problem, namely the Castillon-Cramer problem. This problem, proposed by Cramer to Castillon, asked how to inscribe a triangle in a circle so that it passed through three given points. Castillon solved the problem 25 years after Cramer's death, and the problem went on to various generalisations about inscribed polygons in a conic section.

Cramer had worked extremely hard over a long period with writing his Introduction à l'analyse Ⓣ and undertaking the large amount of editorial work in addition to all his normal duties. Always of good health, this overwork coupled with a fall from his carriage, brought on a sudden decline. He spent two months in bed recovering, and his doctor then recommended that he spend a quiet period in the south of France to completely regain his strength. Leaving Geneva on 21 December 1751 he began his journey but he died two weeks later while still on the journey.