让·加斯东·达布Jean Gaston Darboux

让·加斯东·达布是François 达布（1800-1849）和Alix Gourdoux（1811-1887）的儿子。François是一名服装商和缝纫用品商，是Antoine Darboux（1764-1803）和Magdelaine Amalric（卒于1812年）的儿子，他于1841年9月17日在尼姆与Alix Gourdoux结婚。达布出生在一座由大教堂小礼拜堂改建而成的房子里。各种资料给出的他的出生日期为8月13日或8月14日，这里存在真正的混淆，我们应该解释一下。他的出生证明明确写明8月14日凌晨1点，但达布始终坚称他出生于8月13日午夜。当被问及出生日期时，他总是给出8月13日，事实上，他的墓碑上写的是8月13日，因此我们在本传记中采用了这个日期。

达布有一个弟弟，达布 Louis Darboux，生于1844年5月15日，后来成为尼姆中学的数学教师。有资料提到还有第二个弟弟Paul Darboux，生于1845年11月28日，但这要么是错误，要么就是Paul在婴儿时期就夭折了，因为达布的所有讣告都提到达布只有一个弟弟Louis。1849年，这个家庭遭遇了不幸，因为那一年的12月14日，François 达布去世了。Alix接管了生意，并竭尽全力确保达布和Louis都接受良好的教育。达布和Louis都作为走读生就读于尼姆中学，这意味着他们从早上六点一直待到晚上八点。达布于1853年10月进入该校。达布的母亲Alix于1856年7月25日在尼姆与Auguste Sistre（1823-1887）结婚。他和Alix一样是丧偶者，他的妻子于1854年去世。

达布于1859年7月22日从尼姆的公立中学毕业，同年10月进入蒙彼利埃公立中学的特别数学班，那里设有为报考巴黎综合理工学院和巴黎高等师范学校的入学考试做准备的特别班。他在蒙彼利埃的老师是Charles Berger，此人曾在高等师范学校学习，1843年毕业[16]：-

教授Charles Berger清楚地讲解课程内容，晚上照看学生，把其中最优秀的带到图书馆，让他们阅读高等数学的书籍。

在蒙彼利埃公立中学学习一年后，达布参加了巴黎综合理工学院的入学考试，并非因为他想1860年入学，而只是因为他想取悦他的老师Charles Berger，后者热切希望他的明星学生参加考试。良好的表现使达布获得了录取资格，但他宁愿在蒙彼利埃再待一年，并在下一年参加两所大学校的入学考试。这一次，他在两所学校的考生中均排名第一，如果他遵循通常的模式，他会选择就读巴黎综合理工学院，这被认为最适合那些想成为各自领域领导者的人，尤其是工程师。然而，他选择了高等师范学校，这是能力稍逊、希望成为中学教师的人的通常路径。达布选择进入巴黎高等师范学校，因为他想成为一名教授。这是一个重大的决定，不仅对这位年轻人如此，对高等师范学校也是如此，因为其他顶尖数学家很快效仿了达布的榜样。

达布前往巴黎开始他的学业，他的母亲陪同他，并把他介绍给Louis Pasteur，后者是巴黎高等师范学院的院长。Pasteur当时正在进行一项改革计划，试图提高该校的科学工作水平，但他的做法并不总是受学生欢迎。他很高兴名列榜首的人选择了巴黎高等师范学院，他请求部长允许达布参加校外的任何他喜欢的课程。部长批准了，因此达布得以在法兰西学院聆听约瑟·伯特兰的讲座，并与之建立了深厚的友谊。达布于1863年7月9日获得数学科学学士学位，于1863年8月7日获得物理科学学士学位。

当他还是学生时，他在数学研究方面的巨大天赋就为周围人所清楚看到。还在巴黎高等师范学校读本科时，他就发表了他的第一篇关于正交曲面的论文，该文由约瑟夫·阿尔弗雷德·塞雷于1864年8月提交给科学院。一个月后，即9月20日，他在数学教师资格会考中名列第一，这使他获得了数学教师资格。

此时Pasteur意识到他有一个品质出众的学生，他非常希望达布留在巴黎，于是他在高等师范学校设立了一个“数学教师资格会考备考辅导员”的职位。以下是Pasteur请求为达布设立这一职位的信[18]（在[5]中也有引用）：-

他是一个其工作、品行、精神之卓越、性格与行为都无可挑剔的学生。这个年轻人很快就会跻身最杰出的数学家之列。发明精神是人们可以期望这位年轻大师仍需具备的一种品质。然而，他最近通过提交给科学院的一篇非常出色的著作，以及他在这一年中向教授和助教们提交的各种笔记，就各种主题展示了这一点，他能够致力于这些主题的研究，同时尽管有准备教师资格会考竞赛的种种事务，仍不间断地保持在其分组中的第一名。这个年轻人留在巴黎是绝对必要的。

你可以在THIS LINK阅读保尔·阿佩尔为国际计量委员会[5]撰写的达布讣告的英文翻译，本引文即取自该讣告。

达布研究过加布里尔·拉梅、夏尔·迪潘和Bonnet关于正交曲面系的工作。他推广了恩斯特·爱德华·库默尔的结果，给出了由一个方程定义的、具有许多有趣性质的曲面系。他于1864年8月1日向科学院宣布了他的结果，同一天Moutard也宣布他独立发现了同一个曲面系。这些结果被收入达布的博士学位论文Sur les surfaces orthogonales Ⓣ（《论正交曲面》），他于1866年7月14日通过论文答辩后获得博士学位。他的答辩委员是米歇尔·沙勒、约瑟夫·阿尔弗雷德·塞雷和让-克劳迪·波桂，他们对这篇论文给予了高度评价。由米歇尔·沙勒撰写的评阅报告如下（见[3]）：——

这篇学位论文是一部关于正交曲面的内容广泛且非常重要的著作。它由三部分组成。第一部分题为“爱德华·斯图迪一个值得注意的正交坐标系”，包含作者与M Moutard各自独立得到的那个三重正交系所形成的曲线坐标的各种性质。第二部分包含“关于一般正交曲面的研究”。达布先生以夏尔·迪潘的定理为出发点，该定理说：在任何三重正交曲面系中，曲面的交线都是它们的曲率线；他补充了如下命题作为补充：“当两个正交曲面系沿这些曲面的曲率线相交时，存在一个与这两个系正交的第三个系。”他首先给出了Ossian Bonnet定理的一个简单证明，即寻找所有正交系归结为完全积分一个含三个自变量的三阶偏微分方程。然后他公布了一种“研究正交系的新方法”，其基础是使用某个辅助函数V。第三部分包含第二部分所述方法的“应用”。作者首先考虑一类特殊的正交系，其中同一系的曲面是通过将其中一个曲面平行于自身作简单平移而不改变形状得到的。函数V的确定于是归结为积分一个含两个自变量的三阶偏微分方程。达布先生处理的第二种情形是三个系中曲率线均为平面的曲面。此时积分可以完全进行，结果形式非常简单，包含三个任意函数；这些曲面在某些情形下是上一段所研究的正交系的例子，也就是说，“三个系中的每一个都由一个形状不变、平行于自身移动的曲面构成。”第三个也是最后一个情形涉及每个曲面都能被其曲率线分成无穷小正方形的系。达布先生在第一部分中已经注意到，先前由M Moutard和他本人发现的那个三重正交系的曲面具有上述性质。通过一种巧妙且极其精巧的分析，他现在证明后一个系是唯一满足该问题的系。

达布曾被任命为约瑟·伯特兰的代课教师，于1864—65年在巴黎圣路易中学教授专门数学课程，同时仍在进行博士研究。获得博士学位后，他被任命接替约瑟·伯特兰在法兰西学院任教，于1866—67年教授数学物理课程。1867年10月31日，他被任命为让-克劳迪·波桂的代课教师，在路易大帝中学（埃瓦里斯特·伽罗瓦曾在此受教育）教授专门数学课程。一年后，他成为路易大帝中学的专门数学教授，在那里任教至1872年9月26日。1872年，他被任命到巴黎高等师范学校任教，直至1881年。从1873年到1878年，他在索邦大学担任约瑟夫·刘维尔理性力学讲席的代理教师。然后，在1878年，他成为同样在索邦大学的米歇尔·沙勒高等几何讲席的代理教师。两年后米歇尔·沙勒去世，达布接替他担任高等几何讲席，一直担任此讲席直至去世。他于1889年至1903年任理学院院长。

1872年7月14日，达布与Célina Amélie Carbonnier结婚，后者1850年3月29日出生于法国皮卡第大区瓦兹省博韦。我们注意到，他的兄弟达布 Louis Darboux已于1871年8月20日与Clare Marie Carbonnier结婚。我们推测达布家的两兄弟娶了Carbonnier家的两姐妹，但未能证实这一点。

达布对微分几何和分析做出了重要贡献。迪尔克·扬·斯特勒伊克在[1]中写道：——

……他秉承了加斯帕尔·蒙日的精神，而达布的精神在埃利·嘉当的工作中也能被察觉。

迪尔克·扬·斯特勒伊克再次写道[1]：-

依靠加斯帕尔·蒙日、卡尔·弗里德里希·高斯和夏尔·迪潘的经典结果，达布以自己富有创造性的方式充分运用了他的同事约瑟·伯特兰、Bonnet、Ribaucour等人的成果。

他现在最为人所知的可能是以他名字命名的达布积分。这个积分是在他1870年写的一篇关于二阶微分方程的论文中引入的。

1875年，他发表了自己看待波恩哈德·黎曼积分的方式，定义了上和与下和，并定义如果一个函数的上和与下和之差随着网格尺寸变小而趋于零，则该函数可积。

1873年，达布写了一篇关于四次圆纹曲面的论文，而在1887年至1896年间，他出版了四卷关于无穷小几何的著作，其中包含了他早期的大部分工作，书名为Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal

这部著作的第四卷中包含了关于一个曲面在另一个曲面上滚动的讨论。他特别研究了固定在滚动曲面上的点和线所生成的几何构型。路德·艾森哈特在[8]中谈到这部著作时说：-

他关于滚动曲面定理的几何证明……既纯粹又简洁而优美。

达布还研究了在曲面上求两点之间最短路径的问题。大约在同一时期，恩斯特·策梅洛和Kneser也在这一领域做了工作。

路德·艾森哈特在[8]中讨论了达布在研究上的成功：-

达布的能力基于几何想象力与分析能力的罕见结合。他既不赞同那些在解决几何问题时只使用几何推理的人，也不赞同那些认为严格遵循分析方法具有某种优点的人。……他将各种几何问题归结为共同的分析基础，并从共同的观点出发加以解决和发展，这些工作十分出色。

达布还作为一位杰出的教师、作家和管理者而闻名。路德·艾森哈特写道[8]：-

他的著作不仅内容充实，而且风格异常完美精致。在呈现结果时，阐述形式经过仔细推敲。达布多样的才能与他的个性相结合，使他成为一位伟大的教师，因此他身边总是围绕着一群有才华的学生。与加斯帕尔·蒙日一样，他不满足于发现，而是觉得培养弟子同样重要。

达布以其涉猎范围比上述更为广泛的数学而闻名。迪尔克·扬·斯特勒伊克在[1]中写道：-

达布还在函数论、代数、运动学和动力学方面做过研究。他对科学史的鉴赏体现在众多演讲中，其中许多是在科学院所作的颂词。他还编辑了约瑟夫·傅里叶的《文集》（1888-1890）。

关于达布的数学贡献的更多信息，请参见儒勒·昂利·庞加莱在THIS LINK的达布禧年庆典上的演讲。

当然，达布因其工作获得了许多荣誉。Lebon在[3]中列出了100多个选举达布为成员的科学学会。他于1902年当选为伦敦皇家学会，并于1916年获得其詹姆斯·约瑟夫·西尔维斯特奖章。1884年，他当选为Académie des Sciences，并于1900年成为其秘书。

1907年12月，达布代表法国参加了在威斯敏斯特教堂举行的开尔文葬礼。更多细节见[15]；请参见THIS LINK上约瑟夫·拉莫尔为达布所写讣告的一个版本。

1908年，达布是1908年在罗马举行的国际数学家大会的全会演讲者。他于1908年4月7日星期二3:30作了The Origins, Methods and Problems of Infinitesimal Geometry演讲，该场会议由西蒙·纽康主持。我们在THIS LINK给出达布演讲部分的英文版本。

让我们以Ernest Lebon在1910年对达布的描述作为结束[16]：-

身材高大、严厉而冷漠的 M 达布 会让初次接近他的人感到畏惧。幸运的是，交谈几分钟后这种印象就迅速消失了。我们那时会认识到他是仁慈的，在粗糙的外表下隐藏着一颗慷慨的心。他多次证明了这两种品质，尤其是过去十年担任支持科学之友协会主席期间。他的谈话涉及最广泛的主题，既富有教益又引人入胜。他承认他的数学老师们发现、唤醒并保持了他对几何学的兴趣，他满怀感情和喜悦地重复他们的名字。他努力不带偏见地公正评判面前的问题。当他主持一个委员会时，他对同事有绝对的信任，如果他们受到攻击，他会为他们辩护。……在生活的各种情况下，M 达布 都按部就班地行事；因此，当他在制定课程计划并在黑板上按方程出现的顺序写下它们时，发现这一品质就不足为奇了。他天性非常认真，从不留下未完成的推理，并总是向听众呈现精心准备的课程。关于最后一点，他的图书馆里有无可辩驳的证据：那是十几本装订好的大笔记本，人们可以在其中找到他亲手清晰写下的、他在数学物理、分析力学和无穷小几何学中所教课程的展开。这些珍贵的手稿包含了他没有发表的方法和评论，但我们以后可以利用，因为他打算将它们交给研究所。M 达布 尽管身居高位，却依然朴实谦逊。必须指出，他仅凭自己的努力和才华才达到这一地位：他的祖先中没有一人在科学、行政或政治领域担任过哪怕微小的职位；如果学者们在他职业生涯初期保护他并为他打开荣耀之门，那是因为他们从他的著作中看到了可能推动科学进步之处，并认可他具有一流的品质。

达布 安葬在巴黎市的蒙帕纳斯公墓。

Gaston Darboux was the son of François Darboux (1800-1849) and Alix Gourdoux (1811-1887). François was a clothes merchant and haberdasher, the son of Antoine Darboux (1764-1803) and Magdelaine Amalric (died 1812), who married Alix Gourdoux in Nîmes on 17 September 1841. Gaston was born in a house which was a converted chapel of the cathedral. The date of his birth is given by various sources as 13 August or 14 August, and there is genuine confusion here which we should explain. His birth certificate clearly gives 1 a.m. on 14 August but Gaston always maintained that he was born at midnight on 13 August. He always gave 13 August when asked for his date of birth and, in fact, his gravestone gives 13 August, so we have given that date on this biography.

Gaston had a younger brother, Jean Louis Darboux, born on 15 May 1844, who became a mathematics teacher at the Lycée Nîmes. One source gives a second brother Paul Darboux, born 28 November 1845, but either this is an error or else Paul died as a baby since all of Gaston's obituaries refer to Gaston having only one brother Louis. Tragedy struck the family in 1849, for on 14 December of that year François Darboux died. Alix took over the business and made strenuous efforts to made sure that both Gaston and Louis received a good education. Gaston and Louis both attended the Lycée Nîmes as day pupils, meaning that they attended from six o'clock in the morning until eight o'clock in the evening. Gaston entered the lycée in October 1853. Alix, Gaston's mother, married Auguste Sistre (1823-1887) on 25 July 1856 in Nîmes. He, like Alix, was a widow, his wife having died in 1854.

Gaston Darboux graduated from the lycée in Nîmes on 22 July 1859 and later that year, in October, entered the special mathematics class at the lycée of Montpellier where there were special class to prepare pupils for the entrance examinations to the École Polytechnique and the École Normale Supérieure. His teacher at Montpellier was Charles Berger who had studied at the École Normale, graduating in 1843 [16]:-

The professor, Charles Berger, clearly explained the subjects of his course, took care of his students during the evenings, led the best of them to the library where he made them read books of Higher Mathematics.

After one year at lycée of Montpellier, Darboux sat the entrance examinations for the École Polytechnique, not because he wanted to enter in 1860, but simply because he wanted to please his teacher, Charles Berger, who was keen for his star pupil to sit the examinations. A good performance meant that Darboux gained admission but he preferred to spend another year at Montpellier and take the entrance examinations for both Grand École in the following year. This time he was ranked first from those taking the examinations for both and, if he had followed the usual pattern, he would have chosen to attend the École Polytechnique which was considered best for those wanting to be leaders in their fields, particularly engineers. He chose, however, the École Normale which was the usual route for those of somewhat lesser abilities wishing to become secondary school teachers. Darboux chose to enter the École Normale Supérieure, because he wanted to become a professor. It was a significant decision, not just for the young man, but for the École Normale since other top mathematicians soon followed Darboux's example.

When Darboux went to Paris to begin his studies, his mother accompanied him and introduced him to Louis Pasteur, who was the director of the École Normale. Pasteur was at this time in the middle of a reform programme trying to improve the standard of scientific work at the École but his approach was not always popular with students. He was delighted that the person topping the list had chosen the École Normale and he requested the Minister to allow Darboux to attend any of the courses which he liked outside the École. The Minister granted permission so Darboux was able to attend, at the Collège de France, the lectures of Joseph Bertrand, with whom he developed a deep friendship. Darboux was awarded the Licencié ès Sciences mathématiques on 9 July 1863 and the Licencié ès Sciences physiques on the 7 August 1863.

While he was a student his great talent for mathematical research became clear to those around him. While still an undergraduate student at the École Normale Supérieure, he published his first paper on orthogonal surfaces which had been presented to the Académie des Sciences by Joseph Serret in August 1864. A month later, on 20 September, he was placed first in the mathematics aggregation which qualified him to be a mathematics teacher.

Now Pasteur was aware that he had a student of outstanding qualities and he was very keen that Darboux remained in Paris so he created a position of "préparateur agrégé de Mathématiques" at the École Normale. Here is a letter by Pasteur requesting this position be set up for Darboux [18] (also quoted in [5]):-

He is a student whose work, conduct, distinction of spirit, character, and behaviour, nothing leaves to be desired. This young man will quickly be among the most eminent mathematicians. The spirit of invention was the one quality which one could expect this young master still had to achieve. However, he showed recently, with a very remarkable work presented to the Academy of Sciences and by various notes which he gave to the professors and assistants during the year, on various subjects the study of which he was able to devote himself without ceasing to retain the first rank in his division, in spite of the preoccupations of the preparation for the competition of the aggregation. It is absolutely essential that this young man stay in Paris.

You can read an English translation of Paul Appell's obituary of Darboux for the International Committee for Weights and Measures [5], from which this quote was taken, at THIS LINK.

Darboux had studied the work of Lamé, Dupin and Bonnet on orthogonal systems of surfaces. He generalised results of Kummer giving a system defined by a single equation with many interesting properties. He announced his results to the Académie des Sciences on 1 August 1864, and on the same day Moutard announced that he had also discovered the same system. These results were included in Darboux's doctoral thesis Sur les surfaces orthogonales Ⓣ for which he awarded his doctorate after defending his thesis on 14 July 1866. His examiners were Michel Chasles, Joseph Serret and Claude Bouquet and they gave high praise to the thesis. The report, written by Michel Chasles, is as follows (see [3]):-

This Thesis is an extensive and very important work on orthogonal surfaces. It consists of three Parts. The first, entitled: 'Study of a remarkable system of orthogonal coordinates', contains different properties of the curvilinear coordinates formed by the triple orthogonal system to which the author and M Moutard were led, independently of each other. The second part contains 'Research on orthogonal surfaces in general'. M Darboux, taking as a starting point the theorem of M Dupin, according to which in any triple system of orthogonal surfaces the intersection curves of the surfaces are their lines of curvature, to which he adds as a complement the following statement: "When two systems of orthogonal surfaces intersect along the lines of curvature of these surfaces, there is a third system orthogonal to the first two," first gives a simple demonstration of this theorem by M Ossian Bonnet, that the search for all orthogonal systems amounts to the complete integration of a third order partial difference equation with three independent variables. Then he makes known a 'New method of research of orthogonal systems', based on the use of a certain auxiliary function V. The third Part contains 'Applications' of the method exposed in the second Part. The author first considers a particular class of orthogonal systems in which the surfaces of the same system are obtained by moving one of them parallel to itself by a simple translation without alteration of shape. The determination of the function V then depends on the integration of a third order partial difference equation with two independent variables. The second case treated by M Darboux is that of surfaces for which the lines of curvature are plane in the three systems. The integrations then take place completely, and the result, in a very simple form, contains three arbitrary functions; these surfaces are, in certain cases, an example of the orthogonal systems studied in the preceding paragraph, that is to say that "each of the three systems is formed by a surface of invariant form which moves parallel to itself." The third and last case relates to systems for which each surface can be divided into infinitely small squares by its lines of curvature. M Darboux had already observed, in the first Part, that the surfaces of the triple orthogonal system previously discovered by M Moutard and by himself enjoy the property in question. By a clever and extremely ingenious analysis, he now shows that the latter system is the only one that answers the question.

Darboux had been appointed as a substitute for Joseph Bertrand to teach the special mathematics course at the Lycée Saint-Louis in Paris in 1864-65 while still undertaking research for his doctorate. After the award of his doctorate, he was appointed to replace Joseph Bertrand at the Collège de France where he taught the Mathematical Physics Course in 1866-67. He was appointed as a substitute for Jean-Claude Bouquet to teach the special mathematics course at the Lycée Louis le Grand (where Galois was educated) on 31 October 1867. A year later he became a Professor of Special Mathematics at the Lycée Louis le Grand where he taught until 26 September 1872. In 1872 he was appointed to the École Normale Supérieure where he taught until 1881. From 1873 to 1878 he was suppléant to Liouville in the chair of rational mechanics at the Sorbonne. Then, in 1878 he became suppléant to Chasles in the chair of higher geometry, also at the Sorbonne. Two years later Chasles died and Darboux succeeded him to the chair of higher geometry, holding this chair until his death. He was dean of the Faculty of Science from 1889 to 1903.

On 14 July 1872, Darboux married Célina Amélie Carbonnier, born 29 March 1850 in Beauvais, Oise, Picardie, France. We note that his brother Jean Louis Darboux had married Clare Marie Carbonnier on 20 August 1871. We assume that the two Darboux brothers married two Carbonnier sisters, but we have been unable to verify this.

Darboux made important contributions to differential geometry and analysis. D J Struik writes in [1]:-

... he followed in the spirit of Gaspard Monge, and Darboux's spirit can be detected in the work of Élie Cartan.

Again Struik writes [1]:-

Relying on the classical results of Monge, Gauss, and Dupin, Darboux fully used, in his own creative way, the results of his colleagues Bertrand, Bonnet, Ribaucour, and others.

He may now be best known for the Darboux integral which is named after him. This integral was introduced in a paper on differential equations of the second order which he wrote in 1870.

In 1875 he published his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.

In 1873 Darboux wrote a paper on cyclides and between 1887 and 1896 he produced four volumes on infinitesimal geometry which included most of his earlier work it was titled Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal

. Included in volume four of this work is a discussion of one surface rolling on another surface. In particular he studied the geometrical configuration generated by points and lines which are fixed on the rolling surface. Eisenhart says of this work in [8]:-

His geometrical proofs of the theorems dealing with rolling surfaces ... are as pure as they are simple and beautiful.

Darboux also studied the problem of finding the shortest path between two points on a surface. Work in this area was also done at around the same time by Zermelo and by Kneser.

Darboux's success in research is discussed by Eisenhart in [8]:-

Darboux's ability was based on a rare combination of geometrical fancy and analytical power. He did not sympathise with those who use only geometrical reasoning in attacking geometrical problems, nor with those who feel that there is a certain virtue in adhering strictly to analytic processes. ... brilliant are his reductions of various geometrical problems to a common analytic basis, and their solution and development from a common point of view.

However Darboux was also renowned as an exceptional teacher, writer and administrator. Eisenhart writes [8]:-

His writings possess not only content but singular finish and refinement of style. In the presentation of results the form of exposition was carefully studied. Darboux's varied powers combined with his personality in making him a great teacher, so that he always had about him a group of able students. In common with Monge he was not content with discoveries, but he felt that it was equally important to make disciples.

Darboux is known for a wider range of mathematics than that described above. Struik writes in [1]:-

Darboux also did research in function theory, algebra, kinematics and dynamics. His appreciation of the history of science is shown in numerous addresses, many given as éloges before the Academy. He also edited Joseph Fourier's "Oeuvres" (1888-1890).

For more information about Darboux's mathematical contributions see Henri Poincaré's speech at the Darboux Jubilee at THIS LINK.

Of course Darboux received many honours for his work. Lebon in [3] lists over 100 Scientific Societies which elected Darboux as a member. He was elected to the Royal Society of London in 1902, winning its Sylvester Medal in 1916. In 1884 he was elected to the Académie des Sciences, becoming its secretary in 1900.

Darboux represented France at the funeral of Lord Kelvin in Westminster Abbey in December 1907. More details of this are given in [15]; see a version of Larmor's obituary of Darboux at THIS LINK.

In 1908 Darboux was a plenary speaker at the International Congress of Mathematicians held in Rome in 1908. He delivered the lecture The Origins, Methods and Problems of Infinitesimal Geometry at 3:30 on Tuesday 7 April 1908, the session being chaired by Simon Newcomb. We give an English version of part of Darboux's lecture at THIS LINK.

Let us end with the description of Darboux by Ernest Lebon written in 1910 [16]:-

Tall, severe and cold, M Darboux intimidates those who approach him for the first time. Fortunately, this impression disappears quickly after a few minutes of conversation. We recognize then that he is benevolent and that under a rough bark he hides a generous heart. He has given proofs of these two qualities several times, notably for the past ten years as president of the Society for the support of Friends of Science. His conversation, which ranges over the most diverse subjects, is both informative and attractive. He recognizes that his Mathematics teachers have discovered, awakened and maintained his taste for Geometry and he repeats their names with emotion and pleasure. He strives to judge fairly without bias the issues before him. When he chairs a committee, he has absolute confidence in his colleagues and defends them if they are attacked. ... In all circumstances of life, M Darboux proceeds methodically; one should therefore not be surprised to find this quality when he develops his course programme and writes the equations on the board in the order in which they appear. Very conscientious by nature, he leaves no reasoning unfinished and presents to his listeners lessons which are always carefully prepared. There is irrefutable proof in his library of this last fact: it consists of a dozen large bound notebooks, where one can find, clearly written by himself, the developments of the courses he taught in Mathematical Physics, in Analytical mechanics and infinitesimal geometry. These precious manuscripts contain methods and remarks that he did not publish, but which we can later take advantage of, because his intention is to give them to the Institute. M Darboux remained simple and modest, although he came to a very high position. It is important to point out that he owes it only to his own efforts and his talent: none of his ancestors has occupied even a modest position in the world of science, administration or politics; if scholars protected him at the beginning of his career and opened the doors of glory to him, it is because they had seen in his works points likely to advance Science and recognised in him qualities of first order.

Darboux is buried in Montparnasse Cemetery in the City of Paris.