恩纳斯托·切萨罗Ernesto Cesaro

恩纳斯托·切萨罗的父亲是Luigi Cesàro，母亲是Fortunata Nunziante，她是Luigi的第二任妻子。Luigi是托雷安农齐亚塔的一位农民，也在商店里出售他的农产品。他是一个有远见的人，是意大利最早使用机械来提高农场产量的农民之一。1860年，即切萨罗出生后的那一年，由Giuseppe Garibaldi领导的旨在实现意大利统一的革命爆发了。事实上，1861年3月17日，几乎正好是切萨罗出生两年后，意大利王国正式成立。Luigi Cesàro强烈支持意大利统一的进程，但这对意大利的农民（对许多其他人也是如此）来说并不是一个容易的时期，切萨罗在困难的经济环境中长大。

新成立的意大利国饱受诸多问题困扰，但在教育方面也展现出新的信心，切萨罗早年便从中受益。他在那不勒斯的文理中学学习了一年，但完成第一年的课程后，他前往诺拉的一所神学院学习了两年。回到那不勒斯的文科中学后，他又在那里完成了一年的学业，于1872年从第四班毕业。他的哥哥Guiseppe自1867年起一直在列日。1873年，切萨罗的父亲将他送到Guiseppe那里，当时Guiseppe已是列日矿业学校的矿物学和结晶学讲师。切萨罗进入矿业学校学习，但他更愿意在意大利求学，于是申请了意大利的大学名额。他的申请未能成功，因此不得不留在列日的矿业学校，跟随乌惹内·查尔斯·卡塔兰学习数学。

1879年父亲去世后，切萨罗回到意大利的托雷安农齐亚塔待了数年。回到意大利后，他与近亲Angelina结婚。切萨罗父亲的去世给家庭带来了比以往更严重的财务问题，但最终切萨罗获得了一笔奖学金，使他能够在列日继续深造，1882年他回到比利时继续学业。乌惹内·查尔斯·卡塔兰帮助他发表了他的第一篇数学论文Sur diverses questions d'arithmétique Ⓣ（《论若干算术问题》），该文于1883年发表。

Sur diverses questions d'arithmétique Ⓣ（《论各种算术问题》）是切萨罗为theory of numbers撰写的一系列文章中的第一篇。到1885年，他又发表了九篇关于这一主题的论文。这些论文考察了与[1]有关的问题：-

……两个数的公约数的个数、它们平方和的总值的确定、三个任意数不可通约的概率，等等；对于这些，他试图把已获得的结果应用于Fourier series理论中。

切萨罗在列日学习期间访问了巴黎，并在那里听了夏尔·埃尔米特、让·加斯东·达布、约瑟夫·阿尔弗雷德·塞雷 夏尔·布里奥、让-克劳迪·波桂和米歇尔·沙勒在索邦大学的讲座。特别是夏尔·埃尔米特对切萨罗所获得的结果感兴趣，并在自己1883年的工作中引用了这些结果。切萨罗对让·加斯东·达布关于几何的讲座特别感兴趣，这促使他沿着类似的思路进行自己的内蕴几何研究。回到列日后，切萨罗与那里的一位教授发生争执，未完成学业便前往意大利。

他一直想在意大利学习，现在终于得到了机会。在安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳、Battaglini和乌利塞·迪尼的支持下，他获得了一笔奖学金，得以在罗马大学进行研究，并于1884年入学。在接下来的两年里，他写了八十篇关于[1]的著作：-

……无穷算术、等压问题、全纯函数、theory of probability，特别是内蕴几何。

人们可能会认为这一非凡的产出记录足以让他获得博士学位，但他又等了一年，直到1887年才被授予学位。此时他已经有了职位，在罗马的Lycée Terenzio Mamiani中学的讲席竞聘中获胜。然而，在Lycée Terenzio Mamiani中学一个月后，切萨罗被授予巴勒莫的数学讲席，Cremona建议他接受。他在巴勒莫待到1891年，然后搬到那不勒斯，在那里担任数学分析讲席直到去世。

切萨罗的主要贡献在于微分几何。在巴黎期间受让·加斯东·达布影响，他提出了“内蕴几何”。这是他最重要的贡献，他在Lezione di geometria intrinseca Ⓣ（《内蕴几何讲义》）（那不勒斯，1896年）中对此进行了描述。他出色地利用了让·加斯东·达布的一个想法，采用了一种适用于曲线的特殊坐标系。在曲线上一个可变点处，坐标由曲线的切线、主法线和副法线组成。Lezione di geometria intrinseca包含了对今天以切萨罗命名的曲线的描述。他后来将自己的方法推广到研究海里格·冯·科赫曲线，这类曲线处处连续但处处不可微。

Lezione di geometria intrinseca Ⓣ（《内蕴几何讲义》）还讨论了曲面和$n$维空间。切萨罗后来指出，事实上他的几何学并未使用平行公理，因此构成了对non-euclidean geometry的研究。

除了微分几何之外，切萨罗还研究了许多课题，例如数论，在这一领域，除了我们上面提到的课题外，他还研究素数的分布，试图改进巴夫尼提·列波维奇·切比雪夫在这一领域得到的结果。他还对发散级数的研究有所贡献，这是他在职业生涯早期就感兴趣的一个课题，我们应该注意到，在他关于数学物理的工作中，他是詹姆斯·克拉克·麦克斯韦的坚定追随者。这有助于将詹姆斯·克拉克·麦克斯韦的思想传播到欧洲大陆，这一点很重要，因为尽管现在很难意识到这一点，但科学家们花了很长时间才认识到他的理论的重要性。

切萨罗对数学物理的兴趣也体现在他写的两本非常成功的微积分教材中。随后他又写了更多关于数学物理的教材，完成了一本关于弹性的书。在他去世时，还有两部著作正在准备中，一部是关于热的数学理论，另一部是关于流体动力学。

切萨罗在悲惨的情况下去世。他十七岁的儿子在托雷安农齐亚塔附近的海里游泳，在汹涌的水中陷入困境。切萨罗去救他的儿子，但受了伤，导致了他的死亡。

Ernesto Cesàro's father was Luigi Cesàro and his mother was Fortunata Nunziante who was Luigi's second wife. Luigi was a farmer in Torre Annunziata who also sold his produce in a shop. He was a forward looking man being one of the first farmers in Italy to use machinery to improve production on his farm. In 1860, the year after Ernesto was born, there was a revolution led by Giuseppe Garibaldi aimed at achieving Italian unification. In fact on 17 March 1861, almost exactly two years after Ernesto's birth, the Kingdom of Italy was formally created. Luigi Cesàro strongly supported the move towards Italian unification but this was not an easy time for farmers in Italy (nor for many others) and Ernesto grew up in difficult financial circumstances.

The newly created country of Italy suffered many problems but it also had a new confidence in education from which Cesàro benefited in his early years. He studied at the Gymnasium in Naples for a year but after completing the first class he went to a seminary in Nola where he studied for two years. Returning to the Gymnasium in Naples he completed another year there graduating from the fourth class in 1872. His elder brother Guiseppe had been in Liège since 1867. In 1873 Cesàro's father sent him to join Guiseppe who was by that time a lecturer in mineralogy and crystallography at the École des Mines in Liège. Cesàro entered the École des Mines as a student but, preferring to study in Italy, made application for a university place there. His applications were unsuccessful so he had to remain at the École in Liège where he studied mathematics with Catalan.

Cesàro returned to Torre Annunziata in Italy for a number of years after the death of his father in 1879. Back in Italy he married Angelina, who was a close relation. The death of Cesàro's father had given the family even more financial problems than they had before, but eventually Cesàro won a scholarship to allow him to study further at Liège and in 1882 he returned to Belgium to continue his studies. Catalan helped him to publish his first mathematical paper Sur diverses questions d'arithmétique Ⓣ which was published in 1883.

Sur diverses questions d'arithmétique Ⓣ was the first of a series which Cesàro wrote on the theory of numbers. Nine further papers by him on this topic appeared by 1885. They looked at problems concerning [1]:-

... the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series.

Cesàro visited Paris during the period of his studies at Liège and there he attended lectures by Hermite, Darboux, Serret Briot, Bouquet and Chasles at the Sorbonne. Hermite in particular was interested in the results which Cesàro had obtained and he quoted these in his own work of 1883. Cesàro was particularly interested in lectures he attended given by Darboux on geometry and this led him to make his own studies of intrinsic geometry along similar lines. Back in Liège after the trip to Paris, Cesàro fell out with one of the professors there and left for Italy without completing his studies.

He had always wanted to study in Italy and now at last he was given the opportunity. Supported by Cremona, Battaglini and Dini, he was awarded a scholarship to allow him to undertake research at the University of Rome which he entered in 1884. Over the next two years wrote eighty works on [1]:-

... infinite arithmetics, isobaric problems, holomorphic functions, theory of probability, and, particularly, intrinsic geometry.

One might have thought that this remarkable record of productivity would have been sufficient to gain him his doctorate but he had to wait for a further year before this was awarded in 1887. By this time he already had a post, having won a competition for a chair at the Lycée Terenzio Mamiani in Rome. After one month at the Lycée Terenzio Mamiani, however, Cesàro was offered the chair of mathematics at Palermo and Cremona advised him to accept it. He remained at Palermo until 1891, moving then to Naples where he held the chair of mathematical analysis until his death.

Cesàro's main contribution was to differential geometry. Influenced by Darboux while in Paris he formulated 'intrinsic geometry'. This is his most important contribution which he described in Lezione di geometria intrinseca Ⓣ (Naples, 1896). He made excellent use of an idea due to Darboux which adopted a special coordinate system which applied to curves. At a variable point on the curve the coordinates consisted of the tangent to the curve, the principal normal and the binormal. The Lezione di geometria intrinseca contains descriptions of curves which today are named after Cesàro. He later extended his methods to study the Koch curves which are continuous everywhere but nowhere differentiable.

The Lezione di geometria intrinseca Ⓣ also deals with surfaces and $n$-dimensional spaces. Cesàro later pointed out that in fact his geometry did not use the parallel axiom so constituted a study of non-euclidean geometry.

In addition to differential geometry Cesàro worked on many topics such as number theory where, in addition to the topics we mentioned above, he studied the distribution of primes trying to improve on results obtained in this area by Chebyshev. He also contributed to the study of divergent series, a topic which interested him early in his career, and we should note that in his work on mathematical physics he was a staunch follower of Maxwell. This helped to spread Maxwell's ideas to the Continent which was important since, although it it hard to realise this now, it took a long time for scientists to realise the importance of his theories.

Cesàro's interest in mathematical physics is also evident in two very successful calculus texts which he wrote. He then went on to write further texts on mathematical physics, completing one on elasticity. Two further works, one on the mathematical theory of heat and the other on hydrodynamics, were in preparation at the time of his death.

Cesàro died in tragic circumstances. His seventeen year old son went swimming in the sea near Torre Annunziata and got into difficulties in rough water. Cesàro went to rescue his son but sustained injuries which led to his death.