保罗·潘勒韦Paul Painlevé

保罗·潘勒韦的父亲莱昂Louis Painlevé（1832-1906）是一位石版画绘图员，生于巴黎，是欧弗罗西娜·塞莱斯蒂娜·勒鲁瓦和让Baptiste Painlevé的儿子。他后来成为一家印刷油墨厂的厂主，经济上相当宽裕。潘勒韦的母亲安托瓦内特·伊丽莎白·德唐（1835-1893）是伊丽莎白·艾蒂安内特·阿约和德尼·德唐的女儿，生于塞纳-马恩省的莫城。莱昂和安托瓦内特于1855年10月4日在巴黎的罗马天主教堂结婚。他们有三个孩子：玛丽·玛蒂尔德潘勒韦，生于1856年8月6日；布兰奇·朱莉潘勒韦，生于1861年9月18日；以及本传记的主人公潘勒韦。既然潘勒韦的两位姐妹后来会进入他的传记，让我们对她们稍作介绍。

潘勒韦从事花边和束身衣用品工作。她与Ferdinand Lamy结婚，育有四个孩子：Jeanne、Fernande、Pierre和Suzanne。Ferdinand Lamy于1901年前去世，其中一个孩子也早逝。Blanche Julie 潘勒韦与Maurice Louis Dainville（1856-1943）结婚，他是一位画家和建筑师，他们有一个女儿Jeanne Dainville（1889-1939），她嫁给了Pierre Appell（1887-1957），即保尔·阿佩尔的儿子。

三个潘勒韦家的孩子都受洗成为罗马天主教徒，但Léon 潘勒韦不再是一名实践的天主教徒，与教会保持距离。潘勒韦是[24]：-

……在法国熟练工匠家庭生活的朴素民主氛围中长大。

潘勒韦就读于圣路易中学，在那里他赢得了许多奖项，然后于1877年开始在路易大帝中学学习[23]：-

离开他始终深深依恋的小学后，他进入了中学，在那里一切似乎都同样容易……

在路易大帝中学，他在科学和文学方面都同样出色。他再次在整个教学大纲的范围内赢得了许多奖项。到他的中等教育完成时，他仍然没有决定自己想要在生活中走的方向，觉得自己想从事政治或工程，但最终选择开始研究生涯。

他于1883年进入巴黎高等师范学院，并立即在保尔·阿佩尔、让·加斯东·达布、夏尔·埃尔米特、埃米尔·皮卡、儒勒·昂利·庞加莱和于乐·达奈希等杰出教授的影响下被数学所吸引。他学习了微分和积分以及画法几何和力学。1885年，他获得了数学科学和物理科学的学士学位。1886年，他获得了数学教师资格。在完成由埃米尔·皮卡指导的博士学位论文期间，他前往哥廷根，在那里他受到了赫尔曼·阿曼杜斯·施瓦茨和菲利克斯·克莱因的影响，并参加了他们的课程；1887年，他凭借论文 Sur les lignes singulières des fonctions analytiques Ⓣ（《论解析函数的奇线》）从巴黎获得数学博士学位。在简要介绍他的论文之前，让我们叙述潘勒韦在哥廷根期间发生的政治发展中的一个重要事件。

Guillaume Schnaebelé是一名法国警探。他出生在阿尔萨斯的斯特拉斯堡附近，在1870-71年普法战争后，当斯特拉斯堡成为德国的一部分时，他选择住在法国。Schnaebelé于1887年4月在前往会见一名德国警探的途中被德国人逮捕。不清楚他是在法国领土上还是在德国领土上被捕，这一事件几乎导致法国和德国之间的战争。潘勒韦是一名和平主义者，他对双方表现出的民族主义力量以及军国主义力量感到震惊。他觉得这一事件被法国政府处理得非常糟糕，他也意识到法国在军事上处于弱势地位，这影响了他们的行动。

我们承诺回到潘勒韦的杰出论文[23]：-

潘勒韦的论文聚焦于一个重要而显著的性质：函数的个体性在某种程度上存在于其奇点中，并且只需知道这些奇点（其数量可能或多或少有限），就能了解整个函数。他特别处理了具有整条奇点线的函数；在这个主题上，他证明了几个新的重要定理，并开始将他的结果导向由微分方程定义的新超越函数的研究。

以下是潘勒韦引言中的摘录：-

这项工作的第一部分致力于研究奇线或割线附近的函数。割的概念出现在奥古斯丁·路易·柯西积分的讨论中；甚至可以引入它，如M 夏尔·埃尔米特使用积分变量为实数的定积分所展示的那样。布鲁克·泰勒级数在某个圆内收敛，也提供了一个割的简单例子，而MM 卡尔·魏尔斯特拉斯、于乐·达奈希、保尔·阿佩尔形成了许多级数，它们在平面的两个不同区域中表示两个不同的函数。我们在第一章中列出了符号（特别是级数）可能呈现的各种奇点，这些奇点在某个空间中定义了z的解析函数。...

在第二章中，我们将具有奇点的函数理论中给出的和与积的分解形式推广到最一般的一致函数。1881年，在M 约斯塔·米塔格-莱弗勒定理之前，M 埃米尔·皮卡在圆割的情况下指出了一种乘积展开形式，适用于任何割线，不久之后，将一个以线段为割线的函数F(z)分解为n个只有一个割线的函数之和，然后将这些函数展开为级数。在约斯塔·米塔格-莱弗勒定理发现之后，M 爱徳华·古尔萨将该定理推广到具有任意奇点的一致函数。最后，M 约斯塔·米塔格-莱弗勒本人专门撰写了一篇回忆录来研究这些提议。我们给出这些定理的一个略有不同的证明，以及几种在平面上呈现单一割线的函数的级数展开方式，我们将回到这一点。第一个展开，类似于布鲁克·泰勒级数，基于共形表示；其他的推广了M 保尔·阿佩尔在圆弧围道内全纯函数情况下指出的展开。从这个定理可以得出，凸区域中的任何全纯函数都可以在该区域中展开为多项式级数。

因此，留数、阶、余项（定义为积分或系数）的概念及其附带的命题很容易推广。特别地，留数定理和约瑟夫·刘维尔的定理对于具有任意奇点的双周期函数仍然成立。

当时法国顶尖学者的标准职业路径是先在外省获得第一个职位，然后再设法回到巴黎。潘勒韦遵循了这条路线，于1887年被任命为里尔的有理力学与应用力学讲师，随后于1892年回到巴黎，在理学院和巴黎综合理工学院任教，并于7月23日就职。这是一次迅速的巴黎回归，显示出他所受到的高度重视。1895年，他受邀在斯德哥尔摩大学授课，于10月2日讲授了第一讲。这些讲义于1897年以Leçons sur l'intégration des équations différentielles professées à Stockholm Ⓣ（在斯德哥尔摩讲授的微分方程积分课程）为题出版。这些讲义[23]：-

……闪耀着格外生动的光辉……他将自己最好的部分倾注于这些讲义之中，阅读它们将在很长一段时间内成为我们青年一代的灵感源泉。

这些讲座的一个重要特点是关于 $n$ 体问题的 潘勒韦 猜想。我们引用 [15]：-

在约斯塔·米塔格-莱弗勒的建议下，瑞典和挪威国王奥斯卡二世——一位科学的保护者和支持者，尤其是对数学的支持者——于1887年设立了一项重要奖项，以解决三体问题。其表述非常精确：必须对任意选择的初始数据，获得一个将坐标表示为幂级数的解，该级数对时间变量的所有实值收敛。……出乎意料的是，没有人能够提供所要求的解。……1895年，32岁的潘勒韦已经是当时最著名的数学家之一，奥斯卡二世国王邀请他在当年9月至11月在斯德哥尔摩大学进行一系列讲座。这一事件被认为至关重要，甚至国王本人也出席了开场讲座。这些笔记于1897年以手写形式出版……最后几页包含将结果应用于三体问题，以及作者关于n体情形的意见，其表述后来被称为潘勒韦猜想。

从1896年起，潘勒韦在法兰西学院作为代理教授、在巴黎综合理工学院作为分析学辅导教师授课。从次年起，他成为巴黎高等师范学院的会议主讲。

潘勒韦在数学上最早感兴趣的领域是代数曲线和曲面的有理变换。在这一主题中，他引入了双一致变换的概念。他研究微分方程，特别是研究其奇点，并研究力学。他对力学的兴趣是自然的，因为这一学科为他已证明的微分方程结果的应用提供了自然的背景。他使用潘勒韦函数解决了儒勒·昂利·庞加莱和埃米尔·皮卡未能解决的微分方程，正如雅克·阿达马所写，这表明：-

……继续儒勒·昂利·庞加莱的工作并非人力所不能及。

由于他杰出的数学工作，潘勒韦获得了许多奖项。1890年，他被授予数学科学大奖，随后在1894年获得了著名的博尔丹奖，两年后又获得了让-维克托·彭赛列奖。1900年，他当选为Académie des Sciences几何部门的成员。为了他竞选科学院，潘勒韦写了一篇通告，该通告于1967年以Analyse des travaux scientifiques jusqu'en 1900 Ⓣ（截至1900年的科学工作分析）为题发表。

关于这本1967年出版物以及潘勒韦其他书籍的评论摘录，见THIS LINK。

潘勒韦政治生涯的一步出现在1899年，当时他在雷恩的阿尔弗雷德·德雷福斯审判中作证。德雷福斯于1894年12月被判叛国罪，并被判处终身监禁。有证据表明他是无辜的，但出于反犹太主义而故意误判，文件被伪造。1899年8月在雷恩举行了新的军事法庭审判，德雷福斯再次被定罪。这一事件引起了极大的国家关注，大多数人反对德雷福斯，但他也有一些有权势的支持者。潘勒韦是雷恩第二次德雷福斯军事法庭审判的证人。他说他的陈述被操纵，成为对德雷福斯不利的起诉证据。他还谴责了笔迹专家阿尔方斯·贝蒂荣。潘勒韦将继续为德雷福斯争取正义，直到1906年他被平反。让·佩兰写道[28]：-

……德雷福斯事件导致少数罕见的知识分子反对几乎整个国家，起初他们信息不足，后来逐渐被说服。在这场法国可以引以为豪的戏剧中，因为冲突的两个群体都无私地为自己的信仰而战，也因为一旦犯错，没有其他国家会允许纠正错误，潘勒韦认识到正义在哪里，以及从那时起祖国的真正利益在哪里，我们祖国的利益。……这种对正义的炽热之爱，德雷福斯事件在潘勒韦身上揭示出来，也许是他道德品质中最引人注目的特征。

1901年11月8日，潘勒韦与朱莉·玛丽·玛格丽特·佩蒂·德·维勒讷沃（1868-1902）结婚，她被称为“加埃特”，是建筑师安德烈·朱尔斯·埃德蒙·佩蒂·德·维勒讷沃和玛丽·玛格丽特·莱奥迪·克莱兰的女儿。潘勒韦和Marguerite Painlevé的儿子让·玛丽·莱昂潘勒韦（1902-1989）于次年11月20日出生，不幸的是，玛格丽特在分娩六周后，于1902年12月31日因产褥热去世。让的一位朋友写道：-

很难找到一个观点更激进、更反对教会的年轻人。……Jean Painlevé，尽管他足智多谋且不负责任，却找到了一种方式参与一切哪怕只有最微弱社会抗议和混乱痕迹的事情。

让成为了一名电影制作人，今天有时被称为纪录片之父。在潘勒韦的妻子去世后，让曾短暂地由潘勒韦的妹妹布兰奇照顾，但很快潘勒韦和他守寡的妹妹玛丽决定与他们的四个孩子一起生活。他们先住在塞吉耶街18号，然后住在里尔街81号的一套有七个房间的公寓（包括三个女仆房）。玛丽·拉米希望独立，继续经营她的花边和紧身胸衣用品生意，而仆人则负责料理家务。

在1900年于巴黎举行的国际数学家大会上，潘勒韦 是分析分会的主席。1904年，潘勒韦 在海德堡国际数学家大会上作了全体会议报告。他于1904年8月11日发表了题为Le problème moderne de l'intégration des équations différentielles Ⓣ（微分方程积分的现代问题）的演讲。

您可以在THIS LINK 阅读讲座的第一部分。

潘勒韦对航空特别感兴趣，运用他的理论技能研究飞行理论。他于1907年向众议院提出，有必要建立一支涉及航空的军事分支；他成功了，军事航空服务得以建立。他是威尔伯·赖特的第一位乘客，于1908年在奥武尔创造了1小时10分钟的飞行记录，成为第一个乘坐两种不同飞机飞行的人，当时他是亨利·法尔曼的乘客，然后在1909年他创建了第一门航空力学大学课程。飞行后他写道[23]：-

信号发出；我们被发射到太空。愉悦和眩晕的感觉。我们飞翔，我们飞翔……但我们在渐浓的夜色中盘旋的不再是奥武尔营地，而是地球那无垠的面容，被这只大鸟主宰、征服。对空气的征服如今已完成。明天，在更宏大的飞行器上，安全而强大的引擎，摆脱重量限制，将带走原本沉重的负担。大自然带给人类的最大挑战终于被迎接。

1910年，潘勒韦和埃米尔·博雷尔出版了L'Aviation一书。他们的引言开头如下：-

大约120年前，人类先乘热气球，然后乘气球，冒险空中横渡海洋。超越鹰和秃鹫，气球在蔚蓝中划出崇高的洞。诗人歌颂它的荣耀和殉道者；他们颂扬那些敢于完成“用柳条篮，由泰坦之石发起的攻击”的高度征服者的胆量。但气球是风的玩具，它随风而去，而不是随飞行员而去。通过为它提供引擎，将其延长成鱼的形状，克雷布斯和勒纳尔在1884年赋予了它方向，或者至少，允许它在某种程度上移动。

尽管在政治上不如数学熟练，他于1906年开始了政治生涯，这导致他两度担任法国总理。说他政治上不如数学熟练似乎不公平，因为他达到了政治上的最高职位，但这句话更多是为了评论他真正杰出的数学贡献。尽管潘勒韦于1906年开始政治生涯，但这并不是他离开数学的年份。那一年他当选为政府成员，担任第五区拉丁区的巴黎代表[24]：-

他很快就因其演讲的出色内容以及对军事、海军和航空事务表现出的兴趣而声名鹊起，并在若干与国防力量有关的议会委员会中任职。

到1910年，他已放弃所有数学职位，成为全职政治家。他在军事事务上的专长意味着，1914年第一次世界大战爆发后，他主持了许多具有军事职责的委员会，例如为重组军需、海军和航空事务而设立的委员会。1915年，他作为公共教育与发明部长加入内阁。到1917年初，他被任命为战争部长，并违背自己的更好判断，接受了他的总司令关于对德军防线发动全面进攻的建议。这次进攻迅速失败，潘勒韦不得不更换他的总司令。

1917年5月20日星期日，一座纪念碑落成，以纪念Marcelin Berthelot，潘勒韦作为战争部长发表了讲话。他在讲话中高度赞扬了科学[23]：-

正是科学将确保人类社会获得公正而理性的法律和组织。它将通过成倍增加人的工业力量及其对自然的掌控来解决社会问题，不断创造新的财富，而这些财富不会从任何人那里夺走，然而它将通过其博爱的教训和智力的发展带来风俗的最终软化。它本质上集体性的努力已经把我们内心和思想深处带来了高度团结的赋予生命的教训。

1917年，亨利·弗雷德里克·贝克在写到潘勒韦时写道[11]：-

一方面，他目前是极少数其言论标志着我们西方文明命运的人之一；另一方面，不久之前，他还生活在现代解析函数理论最抽象的领域之一，而能够完全追随其思想精妙之处的人过去很少，将来也很少。

在与法国社会党人发生分歧后，总理里博被迫下台，1917年9月7日，潘勒韦成为总理。他在意大利拉帕洛举行的协约国会议上发挥了主导作用，但返回巴黎后遭到失败，于1917年11月13日辞去总理职务。此后直到1919年11月选举之前，他在政治事务中作用甚微，那次选举中他作为当选政府的强烈批评者崭露头角。在1924年5月的下一次选举中，潘勒韦是获胜联盟的一部分，并当选为众议院议长。该联盟是由潘勒韦和赫里欧先生缔造的，后者成为总理。

潘勒韦被提名为共和国总统候选人，但输给了加斯东·杜梅格先生。他继续担任众议院议长，直到1925年4月爱德华·玛丽·赫里欧先生因财政问题被击败。潘勒韦随后第二次成为总理[24]：-

他的新政府从一开始就很软弱。叙利亚的严重动乱进一步使其信誉受损。他的财政改革方案令人失望地未能达到人们对他这样能力的人所期望的水平，未能获得众议院的批准，1925年11月21日他不得不辞职。

然而，潘勒韦仍然保留着高级职位，因为他重新担任了陆军部长。1932年5月，他的名字在共和国总统选举中被提出，但他在投票前退出。此后，他担任航空部长，在这一职位上，他提议达成一项国际协议，终止所有国家的轰炸机生产，并建议建立一支国际空军，用于对抗任何侵略者。1933年1月政府倒台后，他的计划无法继续推进。这结束了他的政治生涯。

关于他的政治生涯的更详细考察，请参阅法国国民议会为庆祝潘勒韦2013年12月诞辰150周年而制作的一些文件，见THIS LINK。

在[24]中，他的个性被这样描述：-

潘勒韦天性朴实自然，具有一种独特的魅力，很少有人，甚至他的对手，能够抗拒。他精力充沛，不知疲倦……

Jean Perrin 在 [28] 中写道，是什么：-

……使他受人爱戴，他谈话的魅力，他的笑声，他非凡的青春活力，他欢乐中的力量，他的热情，他的生命力，他慷慨言辞的流淌，从他身上散发出的火焰，以及所有属于他灵魂的东西？

Thomas Greenwood 在 [20] 中写道关于 潘勒韦 的去世：-

在他过早去世前不久我去看他时，潘勒韦 教授正在一名助手的协助下编辑他最近在索邦大学讲授的著名讲座《Mecanique des Fluides》Ⓣ（流体力学）的第二部分。在他生命的暮色中，这位被朋友们亲切地称为“院长”的人，就这样回到了他最喜爱的研究，因为 M 潘勒韦 正是作为一名数学家开始了他非凡的职业生涯。他正从一次长期而危险的崩溃中慢慢恢复，并希望能在工艺美术学院的礼堂作一次就职演讲，该礼堂最近为纪念他的科学天才而以他的名字命名。这一希望未能实现：相反，被安放在那个礼堂里的却是他的棺材，随后它被送往先贤祠。最近有人听到他说：“我仍然坚持着生命；如果我不得不放手，我会尽量优雅地做到！”这些预言般的话成真了，10月29日，潘勒韦 教授因心力衰竭在自己家中去世。他的去世使法国失去了一位最杰出的儿子，也使世界失去了当时最伟大的数学家和政治家之一。

你可以在 THIS LINK 读到关于 潘勒韦 葬礼的报道。

Paul Painlevé's father, Léon Louis Painlevé (1832-1906), was a lithographic draughtsman, born in Paris, the son of Euphrosine Célestine Leroy and Jean Baptiste Painlevé. He became the owner of a printing ink factory and was quite well-off financially. Paul's mother Antoinette Élisabeth Détang (1835-1893), the daughter of Élisabeth Étiennette Hayot and Denis Détang, was born in Meaux, Seine-et-Marne. Léon and Antoinette were married in Paris on 4 October 1855 in the Roman Catholic Church. They had three children, Marie Mathilde Painlevé, born on 6 August 1856, Blanche Julie Painlevé, born on 18 September 1861, and Paul Painlevé, the subject of this biography. Let us say a little about Paul's two sisters since they enter his biography later.

Marie Mathilde Painlevé worked in lace and corset supplies. She married Ferdinand Lamy and had four children, Jeanne, Fernande, Pierre and Suzanne. Ferdinand Lamy died before 1901, as did one of the children. Blanche Julie Painlevé married Maurice Louis Dainville (1856-1943), a painter and architect, and had a daughter Jeanne Dainville (1889-1939) who married Pierre Appell (1887-1957), the son of Paul Appell.

The three Painlevé children were baptised into the Roman Catholic Church but Léon Painlevé stopped being a practising Catholic, distancing himself from the Church. Paul was [24]:-

... brought up in the simple democratic atmosphere of French skilled artisan family life.

Paul attended the lycée Saint-Louis, where he won many prizes, then in 1877 he began his studies at the lycée Louis-le-Grand [23]:-

Leaving primary school to which he would always remain so deeply attached, he went through high school where everything seemed equally easy ...

At Louis-le-Grand he showed himself to be equally outstanding at both sciences and literature. Again he won many prizes across the whole range of the syllabus. By the time his secondary education was completed he was still undecided on the direction that he wanted to take in life, feeling that he would like to take up politics or engineering but in the end chose to embark on the research career.

He entered the École Normale Supérieure in 1883, and was immediately drawn to mathematics through the influence of outstanding professors like Paul Appell, Gaston Darboux, Charles Hermite, Émile Picard, Henri Poincaré, and Jules Tannery. He studied differential and integral calculus as well as descriptive geometry and mechanics. In 1885 he obtained the licence in mathematical sciences and in physical sciences. He received his agrégation in mathematics in 1886. While completing the work for his doctoral dissertation, advised by Émile Picard, he went to Göttingen where he was influenced by Hermann Schwarz and Felix Klein whose courses he attended; he received a doctorate in mathematics from Paris in 1887 for his thesis Sur les lignes singulières des fonctions analytiques Ⓣ. Before giving a little information about his thesis, let us recount an important event in Painlevé's political development which occurred while he was in Göttingen.

Guillaume Schnaebelé was a French police inspector. He was born near Strasbourg in Alsace, and, after the Franco-Prussian war of 1870-71, opted to live in France after Strasbourg became part of Germany. Schnaebelé was arrested by the Germans in April 1887 when on his way to meet with a German police inspector. It was unclear whether he was arrested on French territory or on German territory and the incident nearly led to war between France and Germany. Painlevé was a pacifist and he was appalled both at the strength of nationalism shown by both sides and also the strength of militarism. He felt that the incident had been very badly handled by the French government, and he was also aware the France was in a weak position militarily, which had influenced their actions.

We promised to return to Painlevé's outstanding thesis [23]:-

Painlevé's thesis focused on the important and remarkable property that the individuality of a function resides, in a way, in its singularities, and that it suffices to know these, which can be in more or less limited number, to be informed about the entire function. He dealt more particularly with functions which have whole lines of singular points; on this subject he demonstrated several new and important theorems and began to orient his results towards the study of new transcendents defined by differential equations.

Here is an extract from Painlevé's introduction:-

The first part of this work is devoted to the study of a function in the vicinity of a singular line or cut. The notion of cut occurs in the discussion of the Cauchy integral; it can even be introduced, as shown by M Hermite using definite integrals where the integration variable is real. The Taylor series, converging in a certain circle, also offers a simple example of a cut, and MM Weierstrass, Jules Tannery, Appell have formed numerous series which represent two distinct functions in two different areas of the plane. We list, in the first Chapter, the various singularities that symbols (and particularly series) can present which define in a certain space, an analytical function of z. ...

In the second Chapter, we extend to the most general uniform functions the forms of decomposition into sums and products, given in the theory of functions with singular points. In 1881, before M Mittag-Leffler's theorem, M Émile Picard indicated, in the case of a circular cut, a form of development into a product, applicable to any cut, and, shortly after, decomposed a function F(z), having for cuts line segments, into a sum of n functions having only one cut, then developed these functions in series. After the discovery of the Mittag-Leffler theorem, M Goursat extended this theorem to uniform functions with any singularities. Finally, M Mittag-Leffler has himself devoted a Memoir to the study of these proposals. We give, with a slightly different demonstration of these theorems, several modes of development in series of functions presenting in the plane a single cut, to which we find ourselves returning. A first development, analogous to the Taylor series, is based on conformal representation; the others generalise the expansions, indicated by M Appell in the case of a holomorphic function inside a contour of arcs of circles. It follows from this theorem that any holomorphic function in a convex region can be developed in this region in a series of polynomials.

The notions of residue, order, remainder (defined either as integrals or as coefficients) are therefore easily generalised with the propositions to which they are attached. In particular, the residue theorem and Liouville's theorems remain for doubly periodic functions with arbitrary singularities.

The standard career path for a leading French academic at this time was to obtain a first post in the provinces, then later to attempt to return to Paris. Painlevé followed this route, being appointed as a lecturer in rational and applied mechanics at Lille in 1887, and then returning to Paris in 1892 where he taught both at the Faculty of Science and at the École Polytechnique taking up his appointment on 23 July. This was a rapid return to Paris and shows the high regard in which he was held. In 1895 he was invited to deliver lectures at the University of Stockholm giving the first on 2 October. The lectures were published in 1897 under the title Leçons sur l'intégration des équations différentielles professées à Stockholm Ⓣ. These lectures [23]:-

... shine with a particularly vivid brilliance ... He put the best of himself into these lessons and reading them will remain a source of inspiration for our youth for a long time.

An important feature of these lectures is the Painlevé Conjecture about the $n$-body problem. We quote from [15]:-

Advised by Gustav Mittag-Leffler, King Oscar II of Sweden and Norway, a protector and supporter of science and especially of mathematics, established in 1887 an important prize for solving the 3-body problem. The formulation was very precise: one must obtain, for any choice of the initial data, a solution expressing the coordinates as a power series, convergent for all real values of the time variable. ... Unexpectedly, nobody could provide the desired solution. ... In 1895, at 32 years of age, Paul Painlevé was already one of the most famous mathematicians of his time, and King Oscar II invited him to give a series of lectures at the University of Stockholm in September-November of that year. The event was considered of paramount importance, and even the King attended the introductory lecture. The notes were published in 1897 in handwritten form ... The last pages contain an application of the results to the 3-body problem and an opinion of the author concerning the n-body case, formulated as a statement which was known afterwards as the Conjecture of Painlevé.

From 1896 Painlevé taught courses at the Collége de France as a Professeur suppléant and at the École Polytechnique as a Répétiteur d'analyse. From the following year, he was a Maître de Conférences at the École Normale Supérieure.

Painlevé's first area of interest in mathematics was rational transformations of algebraic curves and surfaces. In this topic he introduced the notion of a biuniform transformation. He worked on differential equations, particularly studying their singular points, and on mechanics. His interest in mechanics was a natural one since this subject provided a natural setting for applications of the results which he had proved for differential equations. He solved, using Painlevé functions, differential equations which Henri Poincaré and Émile Picard had failed to solve, showing, as Jacques Hadamard wrote, that:-

... continuing the work of Henri Poincaré was not beyond human capacity.

For his outstanding mathematical work Painlevé received many awards. In 1890 he was awarded the Grand Prix des Sciences Mathématiques, then in 1894 he received the prestigious Prix Bordin followed two years later by the Prix Poncelet. In 1900 he was elected to the geometry section of the Académie des Sciences. For his candidacy to the Academy, Painlevé wrote a notice which was published in 1967 as Analyse des travaux scientifiques jusqu'en 1900 Ⓣ.

For extracts from reviews of this 1967 publication, and of other books by Painlevé, see THIS LINK.

A step towards Painlevé's political career came in 1899 when he gave evidence at the trial of Alfred Dreyfus in Rennes. Dreyfus had been convicted of treason in December 1894 and sentenced to life imprisonment. Evidence came forward that he was innocent but documents were forged in a deliberate miscarriage of justice prompted by anti-Semitism. A new court martial was held in Rennes beginning in August 1899 at which Dreyfus was again convicted. The affair became something of great national interest with most people against Dreyfus but he also had some high powered supporters. Painlevé was a witness at the second Dreyfus court martial in Rennes. He said that his statements had been manipulated to make them prosecution evidence against Dreyfus. He also denounced the hand writing expert Alphonse Bertillon. Painlevé would continue to fight for justice for Dreyfus until he was exonerated in 1906. Jean Perrin writes [28]:-

... the Dreyfus affair led a few rare intellectuals to oppose almost the whole of the Country, first ill-informed, then gradually won over. In this drama of which France can remain proud, because each of the two groups that clashed fought selflessly for its belief, because also no other Nation would have allowed the rectification of the error once committed, Painlevé recognised where Justice was, and where from then on was the true interest of the Fatherland, of our Fatherland. ... This passionate love of justice, which the Dreyfus affair revealed in Painlevé, was perhaps the most striking feature of his moral character.

On 8 November 1901 Painlevé married Julie Marie Marguerite Petit de Villeneuve (1868-1902), known as 'Gaette', the daughter of the architect André Jules Edmond Petit de Villeneuve and Marie Marguerite Léodie Clairin. Paul and Marguerite Painlevé's son Jean Marie Léon Painlevé (1902-1989) was born on the 20 November of the following year and, tragically, Marguerite died six weeks after the birth, on 31 December 1902, of puerperal fever. One of Jean's friends wrote:-

It would be hard to find a young man of more radical views and more opposed to the church. ... Jean Painlevé, resourceful and irresponsible as he was, found a way of taking part in everything that bore even the faintest trace of social protest and disorder.

Jean became a filmmaker and is sometimes known today as the father of the documentary film. After Painlevé's wife died, Jean was looked after for a short time by Painlevé's sister Blanche but quite soon Painlevé and his widowed sister Marie decided to live together with their four children. They lived first at 18 rue Séguier, then at 81 rue de Lille in a seven-room apartment (with three maids' rooms). Marie Lamy, who wished to be independent, continued to carry on her trade in lace and corset supplies while servants took care of the running of the household.

At the International Congress of Mathematicians held in Paris in 1900, Painlevé was chairman of the Analysis Section. In 1904 Painlevé was a plenary speaker at International Congress of Mathematicians in Heidelberg. He delivered his lecture Le problème moderne de l'intégration des équations différentielles Ⓣ on 11 August 1904.

You can read the first section of the lecture at THIS LINK.

Painlevé took a special interest in aviation, applying his theoretical skills to study the theory of flight. He approached the Chamber of Deputies in 1907 arguing that it was necessary to set up a branch of the military involved with aviation; he was successful and the military aviation service was set up. He was Wilbur Wright's first passenger making a record 1 hour 10 minute flight at Auvours in 1908, became the first person to fly on two different planes when he was a passenger with Henry Farman, then in 1909 he created the first university course in aeronautical mechanics. After his flight he wrote [23]:-

The signal is given; here we are launched into space. Sensation of delights and dizziness. We fly, we fly ... but it is no longer on the Auvours camp that we hover in the growing night, it is on the indefinite face of the Earth, dominated, conquered by the big bird. The conquest of the air is now accomplished. Tomorrow, on grander aircraft, safe and powerful engines, free from weight restrictions, will take away otherwise heavy burdens. The greatest challenge that nature had brought to man is finally met.

In 1910 Painlevé and Émile Borel published the book L'Aviation. They begin their Introduction as follows:-

It is some 120 years since, first on hot air balloons, then on balloons, man ventured across the ocean by air. Exceeding the eagle and the condor, the balloon made sublime holes in the azure. Poets sang of its glories and its martyrs; they celebrated the audacity of those conquerors of height who dared to complete: "by the wicker basket, the attack started by the rock of the Titans." But the balloon is a toy of the wind, it goes where the air leads it and not where the pilot leads it. By providing it with an engine, by lengthening it into the shape of a fish, Kerbs and Renard, in 1884, gave it direction, or at least, allowed it to move in a certain measure.

Although less skilled in politics than mathematics he began a political career in 1906 which led to two periods as French Prime Minister. It may seem unfair to say he was less skilled in politics than mathematics when he achieved the highest possible office in politics, but this statement is more meant to comment on his truly outstanding mathematical contributions. Although Painlevé began his political career in 1906, this was not the year he left mathematics. It was the year in which he was elected to the Government as a Paris Deputy for the fifth arrondissement, the Latin Quarter [24]:-

He was soon distinguished both by the excellent matter of his speeches and by the interest he displayed in military, naval, and aeronautical affairs, and served on several Parliamentary committees concerned with the national forces.

By 1910 he had given up all his mathematical posts and had become a full-time politician. His expertise in military affairs meant that after World War I started in 1914 he chaired many committees with a military remit, such as those set up to reorganise munitions, the navy, and aeronautics. He joined the Cabinet in 1915 as Minister of Public Instruction and Inventions. By early 1917 he was appointed as head of the Ministry of War and accepted, against his better judgement, the advice of his Commander-in-Chief to launch an all-out attack on the German lines. The attack rapidly failed and Painlevé had to replace his Commander-in-Chief.

On Sunday 20 May 1917 a monument was inaugurated to honour the memory of Marcelin Berthelot and Painlevé, as Minister of War, made a speech. In it he gave high praise to Science [23]:-

It is Science which will assure to human societies fair and rational laws and organisation. It will solve social problems by multiplying the industrial forces of man and his grip on nature, constantly creating new wealth that will not have been taken from anyone, however that it will bring the final softening of manners by its lessons of fraternity and by the development of intelligences. Already his essentially collective effort has brought to the bottom of our hearts and our minds the life-giving lesson of high solidarity.

Writing about Painlevé in 1917, Henry Baker wrote [11]:-

On the one hand, he is at present one of the very few men whose pronouncements mark the destiny of our Western civilisation; on the other, he lived, not so long ago, in one of the most abstract realms of the modern theory of analytic functions, and very few were, or will ever be, those who fully followed the subtlety of his thought.

After a disagreement with the French Socialists, Prime Minister Ribot was forced out and on 7 September 1917 and Painlevé became Prime Minister. He played a leading role in the Allied Conference at Rapallo in Italy, but was defeated after returning to Paris and he resigned as Prime Minister on 13 November 1917. He played little part in political affairs from this time until the election of November 1919 when he came to the fore as a strong critic of the elected Government. At the next election of May 1924 Painlevé was part of the winning alliance and was elected President of the Chamber of Deputies. The alliance had been forged by Painlevé and M Herriot and the latter became Prime Minister.

Painlevé was put forward for election as President of the Republic but lost out to M Gaston Doumergue. He remained President of the Chamber of Deputies until April 1925 when M Édouard Marie Herriot was defeated on a financial matter. Painlevé then became Prime Minister for a second time [24]:-

His new Government was weak from the first. Serious disorders in Syria further discredited it. His schemes for financial reform, which fell disappointingly short of what had been expected from a man of his ability, failed to meet with the approval of the Chamber, and on 21 November 1925 he had to resign.

Painlevé still retained high office, however, for he returned to his position as Minister of War. In May 1932 his name was put forward in the election for President of the Republic but he withdrew before voting took place. After this he held the position of Minister of Air and in this role he made proposals for an international agreement to end the production of bombers in all countries, and for an international air force to be set up to be used against any aggressor. His plans could be taken no further after the Government fell in January 1933. This ended his political career.

For a much more detailed look at his political career, see some of the documents produced by the French National Assembly to celebrate the 150th anniversary of Paul Painlevé's birth in December 2013 at THIS LINK.

In [24] his personality is described in these terms:-

Painlevé had a naturally simple and unaffected manner, and was possessed of a singular charm that few persons, even among his opponents, were able to resist. His energy was untiring ...

Jean Perrin writes in [28] about what:-

... made him loved, the charm of his conversation, his laughter, his extraordinary youth, the strength which was in his gaiety, his verve, his vitality, the flow of his generous words, the Flame which emanated from him, and all that was his soul?

Thomas Greenwood writes in [20] about Painlevé's death:-

When I went to see him shortly before his untimely death, Prof Paul Painlevé was editing, with the aid of an assistant, the second part of his famous lectures on the "Mecanique des Fluides" Ⓣ recently delivered at the Sorbonne. In the dusk of his life, the 'President', as he was familiarly called by his friends, was thus returning to his favourite studies, for it was as a mathematician that M Painlevé began his extraordinary career. He was slowly recovering from a long and dangerous breakdown and was hoping to give an inaugural lecture in the great hall of the Conservatoire des Arts et Metiers, which was recently named after him in honour of his scientific genius. The hope was not to be fulfilled: instead, it was his coffin which was placed in that very hall before it was borne to the Pantheon. "I am still holding on to life," he was heard to say recently; "and if I have to let go, I shall try to do it as elegantly as I can!" These prophetic words became true when on October 29, Prof Painlevé died in his own home from heart failure. In him, France loses one of her most distinguished sons, and the world one of the greatest mathematicians and statesmen of the day.

You can read a report of Painlevé's funeral at THIS LINK.