保尔·阿佩尔Paul Appell

保尔·阿佩尔的父母Jean-Pierre Appell和Elizabeth Müller是来自阿尔萨斯的天主教徒。他们忠于法国，生活在一个有争议的地区，该地区一直是德国领土，直到1648年《威斯特伐利亚和约》将阿尔萨斯-洛林的控制权交给法国。阿佩尔的父亲是里特胡斯的染工，整个家庭，包括两个同父异母的兄弟，都在这个行业中工作。

1866年，阿佩尔的家庭被迫离开里特胡斯，他的父亲于次年去世。阿佩尔拒绝了父母的天主教观点，但他保留了他们的强烈爱国法国观点。1869年，他坚持离开他所就读的天主教学校，转而在公立中学学习。

1870年7月14日，俾斯麦挑衅法国，五天后法国宣战。德国的进攻很有效率，而法国的动员则不然。阿佩尔最小的同父异母兄弟查尔斯，阿佩尔与他非常亲近，加入了法国军队。在四周内，德国人包围了梅斯的一支法国野战军。法国主力军队试图解围，但本身在色当被德国人包围并困住。9月2日，83,000名法国士兵投降。德国人在50天的围困后占领了斯特拉斯堡，阿佩尔在此期间遭受了巨大痛苦。

法国抵抗由一个新的法国政府继续进行。9月19日，德国人开始围攻巴黎。新的法国军队交战但无法击败德国军队。梅斯于10月27日投降，巴黎于1871年1月28日投降。德国吞并了阿尔萨斯和洛林的一半以及梅斯。斯特拉斯堡被德国人吞并，阿佩尔搬到南锡成为法国公民，并准备在巴黎的大学学习。

阿佩尔在南锡期间成为儒勒·昂利·庞加莱的终生朋友。他于1873年进入巴黎高等师范学校，1876年以第一名的成绩毕业，获得数学博士学位。K O May在[1]中写道：-

从这时起，阿佩尔在教学、研究、编辑和公共服务方面保持着惊人的活跃程度。

1881年，阿佩尔与Amelie结婚，她是约瑟·伯特兰和夏尔·埃尔米特的侄女，也是埃米尔·皮卡的表亲。他的三个女儿之一后来嫁给了埃米尔·博雷尔。1885年，他被任命为索邦大学力学讲席教授。

阿佩尔现在在巴黎工作，但每次假期都回到德国控制的阿尔萨斯。在那里，他的同父异母兄弟Charles向他提供信息，他回到巴黎后便将这些信息报告给法国陆军部。基本上，阿佩尔和他的兄弟Charles是在为法国充当间谍和线人，对抗德国。Charles为此付出了代价：1889年，他因反德活动而被监禁。

1892年，阿佩尔当选为Académie des Sciences院士。他于1903年至1920年担任巴黎大学理学院院长，在院长任期结束时，他被任命为巴黎大学校长。他于1920年至1925年担任校长。然而，他还担任了许多其他职务，他在这些职务中的活动在[1]中有描述：-

在多个政府职位上，包括担任公共教育最高委员会委员，他是教育改革的倡导者和众多大型项目的发起人……

此处一段未译出，以下为英文原文 One of the major political events which gripped France during much of the time that Appell held top posts was the Dreyfus Affair. Dreyfus, like Appell, came from Alsace. Born into a Jewish family, Dreyfus embarked on a military career. In 1894, when he was in the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. In 1898 the novelist Émile Zola wrote an open letter accusing the army of covering up its mistaken conviction of Dreyfus.

此案将法国分裂为两个对立阵营，导致的问题远远超出德雷福斯有罪或无罪本身。有人要求将左拉绳之以法，反犹骚乱爆发，还有人请愿要求重审德雷福斯。左拉被判处一年监禁并罚款3000法郎。到1899年，有人供认了伪造行为，随后自杀，德雷福斯被重审，再次被判有罪，但获得赦免。

阿佩尔与此案关系密切。他本人出身于类似背景，并经历了1870-71年普法战争，这场战争很大程度上以阿尔萨斯为中心。1904年德雷福斯获准重审时，阿佩尔在委员会中担任专家，该委员会到1906年7月已为德雷福斯洗清罪名并推翻了此前所有定罪。雅克·阿达马在为德雷福斯洗清名声方面也发挥了重要作用。

阿佩尔创立了Secours National，这是一个涉及所有政治和宗教团体的半官方组织，为战争中的平民受害者提供帮助。战后，当他的家乡阿尔萨斯回归法国时，阿佩尔实现了毕生的抱负。同样在战后，协约国在1919年巴黎和会上成立了国际联盟，阿佩尔在1920年代担任法国协会秘书长，当时国际联盟总部设在日内瓦。

阿佩尔在1876年的第一篇论文基于射影几何，延续了米歇尔·沙勒的工作。随后他撰写了关于代数函数、微分方程和复分析的文章。1878年，他指出了椭圆函数的虚周期在摆的解中的物理意义，此前这曾被认为是纯粹的数学奇趣。他表明，双周期性源于物理考虑。

1880年，阿佩尔定义了一系列函数，满足第$n$个函数的导数是$n$乘以第$(n - 1)$个函数这一条件。这些函数现在被称为阿佩尔多项式。1885年，他因解决加斯帕尔·蒙日的问题而获得博尔丹奖的一半：-

将一个给定的区域移动到另一个体积相等的区域，使得体积元与其新旧位置之间距离的积分最小。

1889年又设立了另一个奖项来解决这个问题：-

寻找一种有效的方法来计算两个复变量的四重周期函数展开中的约瑟夫·傅里叶系数

阿佩尔提交了一个解决方案，获得了第二名。该奖项的获得者是儒勒·昂利·庞加莱。

由阿佩尔本人撰写的文章[2]列出了140篇分析学著作、30篇几何学著作、87篇力学著作以及许多教科书、演讲、数学史讲座和数学教育讲座。这甚至不是阿佩尔出版物的完整列表，因为在[2]写成之后，他又发表了更多的著作。

关于阿佩尔，有一个显而易见的问题我们必须问自己。一个在其学科的每个领域都如此成功，并对数学之外的许多领域做出重大贡献的数学家，为什么今天不那么出名？也许线索来自[2]，其中阿佩尔自己写道：-

我一直不太喜欢发展一般理论，而更愿意研究有限而精确的问题，这些问题可能会开辟新的道路。

这是一个相当准确的评价：阿佩尔是数学史上最优秀的问题解决者之一。然而，他使用现有技术解决这些问题，因此除了问题本身被解决之外，他的工作几乎没有持久的影响。

May在[1]中写道，强调了这一点：-

……他的科学工作由一系列对特定问题的出色解决组成，其中一些问题极其困难。他是一位技术专家，使用他那个时代的经典方法来回答未解决的问题、推敲细节，并在19世纪后期的主流中进行自然的扩展；但他的工作并没有像他希望的那样打开新的大门。相反，他似乎没有探索任何通向一个无拘无束的抽象和概括时期的新路径。在他职业生涯的后半段，他是一个向后看的建制派的支柱，而这个建制派后来让位于尼古拉·布尔巴基，后者与一位将军同名，而这位将军是他童年时代的英雄之一。

Paul Appell's parents Jean-Pierre Appell and Elizabeth Müller were Catholics from Alsace. They were loyal to France, living in a disputed area which had been a German territory until the Peace of Westphalia in 1648 gave control of Alsace-Lorraine to France. Appell's father was a dyer at Ritterhus and the whole family, including two half-brothers, worked in the business.

In 1866 Appell's family were forced to leave Ritterhus and his father died the following year. Appell rejected the Catholic views of his parents but he retained their strong patriotic French views. He insisted, in 1869, on leaving the Catholic school which he was attending and studying instead at a lycée.

On 14 July 1870 Bismarck provoked France and they declared war five days later. The German offensive was efficient, the French mobilisation was not. Appell's youngest half-brother Charles, who Appell was very close to, joined the French forces. Within four weeks the Germans had surrounded a French field army at Metz. The main French army attempted to lift the siege but was itself surrounded and trapped by the Germans at Sedan. On 2 September 83,000 French troops surrendered. The Germans captured Strasbourg after a 50-day siege in which Appell suffered greatly.

French resistance was carried on by a new French government. On 19 September the Germans began to besiege Paris. New French armies engaged but could not defeat the German forces. Metz surrendered on 27 October and Paris surrendered on 28 January 1871. Germany annexed Alsace and half of Lorraine with Metz. Strasbourg was annexed by the Germans and Appell moved to Nancy to become a French citizen and prepare himself to study at university in Paris.

Appell became a life-long friend of Poincaré while at Nancy. He entered the École Normale Supérieure in Paris in 1873 and graduated in first place in 1876 with a doctorate in mathematics. K O May writes in [1]:-

From this time on, Appell maintained an amazing level of activity in teaching, research, editing and public service.

In 1881 Appell married Amelie, niece of Bertrand and of Hermite and a cousin of Émile Picard. One of his three daughters was to marry Borel. In 1885 he was appointed to the Chair of Mechanics at the Sorbonne.

Appell now worked in Paris but returned for each vacation to German held Alsace. There he was given information by his half-brother Charles, which he would report to the French War Office on his return to Paris. Basically Appell and his brother Charles were acting as spies and informants for France against Germany. Charles was to pay a price for this when, in 1889, he was imprisoned for anti-German activities.

In 1892 Appell was elected to the Académie des Sciences. He served as Dean of the Faculty of Science of the University of Paris from 1903 to 1920 and, at the end of his deanship, he was appointed Rector of the University of Paris. He was rector from 1920 to 1925. However he also served in many other roles and his activity in these is described in [1]:-

In various government posts, including membership in the Conseil Supérieure d'Instruction Publique, he was an exponent of educational reform and initiator of numerous large-scale projects ...

One of the major political events which gripped France during much of the time that Appell held top posts was the Dreyfus Affair. Dreyfus, like Appell, came from Alsace. Born into a Jewish family, Dreyfus embarked on a military career. In 1894, when he was in the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. In 1898 the novelist Émile Zola wrote an open letter accusing the army of covering up its mistaken conviction of Dreyfus.

The case split France into two opposing camps leading to issues far beyond the guilt or innocence of Dreyfus. There were demands to bring Zola to justice, anti-Semitic riots broke out, and there was a petition demanding that Dreyfus be retried. Zola was sentenced to a year in prison and fined 3,000 francs. By 1899 there had been a confession to the forgeries, followed by a suicide, and Dreyfus was retried, again found guilty, but pardoned.

Appell was very much involved with the case. He himself was from a similar background and had suffered through the Franco-Prussian war of 1870-71, so much centred on Alsace. When Dreyfus was granted a retrial in 1904, Appell served as an expert on the commission that, by July 1906, had cleared Dreyfus and reversed all previous convictions. Hadamard also played a major role in clearing Dreyfus's name.

During World War I Appell founded the Secours National, a semi-official organisation involving all political and religious groups, which gave help to civilian victims of the war. After the war Appell had the ambition of his life fulfilled when his homeland of Alsace was returned to France. Also following the war the League of Nations was set up by the Allies at the Paris Peace Conference in 1919, and Appell served as secretary-general for the French Association during the 1920s when the League had its headquarters at Geneva.

Appell's first paper in 1876 was based on projective geometry continuing work of Chasles. He then wrote on algebraic functions, differential equations and complex analysis. In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity. He showed that the double periodicity follows from physical considerations.

In 1880 Appell defined a series of functions satisfying the condition that the derivative of the $n$th function is $n$ times the $(n - 1)$th function. These are now called the Appell polynomials. In 1885 he was awarded half of the Bordin Prize for solving Monge's problem:-

To move a given region into another of equal volume so as to minimise the integral of the element of volume times the distance between its old and new positions.

When another prize was offered in 1889 to solve the problem:-

To find an effective method of calculating the Fourier coefficients in the expansion of quadruply periodic functions of two complex variables

Appell submitted a solution which won second place. The winner of this prize was Poincaré.

The article [2], written by Appell himself, lists 140 works in analysis, 30 works in geometry, 87 works in mechanics as well as many textbooks, addresses, lectures on the history of mathematics and lectures on mathematical education. This is not even the complete list of Appell's publications since he published further works after [2] was written.

There is an obvious question that we must ask ourselves about Appell. How is it that a mathematician who was so successful in every area of his subject and made significant contributions to many areas outside mathematics is not better known today? Perhaps a clue come from [2] where Appell himself writes:-

I always had little taste for developing general theories and preferred to study limited and precise questions that might open new paths.

This is a pretty accurate assessment: Appell was one of the finest problem solvers there has been in mathematics. However he solved these problems using existing techniques and therefore his work had little lasting impact other than that the problem itself had been solved.

May, writing in [1], emphasises this point:-

... his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty. He was a technician who used the classical methods of his time to answer open questions, work out details, and make natural extensions in the mainstream of the late nineteenth century; but his work did not open new doors as he hoped. On the contrary, he does not seem to have looked down any of the new paths that were leading to a period of unbridled abstraction and generalisation. During the last half of his career he was a pillar of a backward looking establishment that was to give way to Nicolas Bourbaki, a namesake of a general who was one of his boyhood heroes.