维多·沃尔泰拉Vito Volterra

维多·沃尔泰拉的父母阿布拉莫沃尔泰拉，一位布商，和安杰利卡·阿尔马贾于1859年3月15日结婚。这个家庭有犹太血统。沃尔泰拉这个名字来自托斯卡纳小镇沃尔泰拉，沃尔泰拉的一位祖先在15世纪从博洛尼亚迁到那里。沃尔泰拉家族中有一人在佛罗伦萨开了一家银行，而其他家族成员则是作家、书籍和手稿收藏家以及旅行者。他们曾住在不同的城市，但从18世纪起，有沃尔泰拉家族的人住在安科纳。沃尔泰拉，他父母唯一的孩子，以外祖父的名字命名，出生时（1860年5月）正是朱塞佩·加里波第出发征服西西里和那不勒斯，从而创建统一意大利的时候。在西西里取得胜利后，加里波第于1860年8月迅速北上。安科纳当时处于教皇的直接统治下，但加里波第的军队围攻该镇，用投射物攻击它。沃尔泰拉，大约三个月大，正在摇篮里，一枚投射物落在附近，摧毁了摇篮，但幸运的是，婴儿沃尔泰拉活了下来。

在这次死里逃生之后，这个年幼孩子的麻烦当然并没有结束。他的父亲阿布拉莫在沃尔泰拉只有两岁时去世。这个家庭失去了任何经济支持手段，仅靠沃尔泰拉母亲的兄弟阿方索·阿尔马贾照顾他们才得以生存。1863年秋天，当阿方索在特尔尼的一家铁路公司工作时，他们全家从安科纳搬到了特尔尼，但在1865年1月，他成为都灵意大利银行的官员，此时沃尔泰拉和他的母亲回到了安科纳。在都灵只待了很短时间后，阿方索搬到了佛罗伦萨，继续为银行工作，沃尔泰拉和他的母亲也到那里与他团聚。正是在这座城市，沃尔泰拉度过了童年的大部分时光，尽管他曾回安科纳进行一些长时间的家庭探访，也是在佛罗伦萨，他接受了教育。他对数学的兴趣始于11岁，当时他开始学习阿德里安-马里·勒让德的Geometry。13岁时，他开始学习三体问题，并通过将时间划分为小间隔来取得一些进展，在这些间隔内他可以认为力是恒定的。小学毕业后，沃尔泰拉在但丁·阿利吉耶里初级中学学习，后来他在佛罗伦萨的伽利略技术学院学习了三年。

由于他的家庭极其贫困，沃尔泰拉面临压力，要在学习一年后离开伽利略技术学院去工作。然而，他继续学习的决心，在叔叔Edoardo的支持下，使他坚持了下来。Edoardo拥有应用数学和土木工程学位，拥有一家参与铁路建设的公司。Alfonso向沃尔泰拉施压，要他加入Edoardo的公司，但Edoardo一直拖延，以便让沃尔泰拉继续他的教育。沃尔泰拉在伽利略技术学院的一位数学老师是切萨雷·阿泽拉，他很快意识到沃尔泰拉是一位极具天赋的数学家。物理老师Antonio Ròiti（1843-1921）成为沃尔泰拉继续教育努力中的另一位重要盟友。Ròiti于1877年向沃尔泰拉提供了一个在佛罗伦萨高等研究与实践学院教授物理的职位，从而设法阻止了他被Alfonso强迫去意大利银行工作。沃尔泰拉得以在1878年秋季进入比萨大学，就读于理学院。次年秋季，他参加了比萨高等师范学校的入学竞争考试，并取得了最高分。这意味着他获得了奖学金以及免费食宿。在高等师范学校，他受到几位鼓舞人心的老师的教导，其中最有影响力的是乌利塞·迪尼和恩里科·贝蒂。沃尔泰拉在1881年毕业前发表了三篇论文，一篇关于数学物理Sul potenziale di un' elissoide eterogenea sopra sè stessa Ⓣ（论非均匀椭球体对其自身的势），两篇关于分析，Alcune osservazioni sulle funzioni punteggiate discontinue Ⓣ（关于间断点函数的一些注记），关于逐点不连续函数，以及Sui principii del calcolo integrale Ⓣ（论积分学原理），后者在积分历史的发展中变得重要。在比萨，他在恩里科·贝蒂的指导下撰写学位论文，并于1882年获得学位。他的关于流体力学的学位论文，Sopra alcuni problemi di idrodinami Ⓣ（论流体力学中的一些问题），包含了一些乔治·加布里埃尔·斯托克斯的结果，这些结果是沃尔泰拉后来独立发现的。

获得 laurea 毕业后，沃尔泰拉于1882年12月被任命为恩里科·贝蒂的助手。他于1883年被任命为比萨的理性力学教授，并立即赢得了严厉而要求高的教师的声誉。他深受同事喜爱，例如1887年到达比萨的Ernesto Pascal在给朋友的信中写道（见[4]）：-

沃尔泰拉是一个天使般的年轻人，具有典型的谦逊。

沃尔泰拉的母亲Angelica在沃尔泰拉于比萨求学期间一直与她的兄弟Alfonso同住，她继续住在佛罗伦萨，直到1887年她搬到比萨并与沃尔泰拉一起安家。他卓越的数学贡献很快得到认可，1887年他获得了国家科学院 of Italy（“XL科学院”）的金奖，1888年他成为Accademia dei Lincei的通讯院士和法国数学会的院士，三年后，他当选为Mathematical Circle of Palermo院士，并被授予意大利王冠骑士勋章。1892年恩里科·贝蒂去世后，他担任了比萨的数学物理讲席。同年，他成为比萨理学院院长，并接替恩里科·贝蒂担任Nuovo Cimento期刊的主编。在这些年里，在比萨担任讲席的同时，沃尔泰拉进行了多次出国旅行。他于1888年在巴黎拜访了儒勒·昂利·庞加莱，并在次年再次受邀前往巴黎参加数学科学国际书目大会。这被证明是一个重要事件，因为它见证了沃尔泰拉成为国际数学界的重要成员。1891年，由于这些新建立的联系，他前往德国，在哥廷根大学度过了一个月，在柏林大学度过了几天。在这次旅行中，他会见了许多德国顶尖数学家，包括赫尔曼·阿曼杜斯·施瓦茨和利奥波德·克罗内克。

尽管他从未正式申请该职位，但1893年7月，都灵大学向他提供了高等力学和理性力学讲席。看来恩里科·多维迪奥是这次邀请的幕后推手。几天后，沃尔泰拉写信给乌利塞·迪尼（见[4]）：-

最尊敬的教授，最近几天我多次来找您，但一直未能找到您。我想告诉您我前几天从Segre那里收到的一些消息，即都灵学院已邀请我去那里接替即将前往那不勒斯的Siacci。这个消息出乎意料地传到我这里，我已写信给都灵，说由于事关重要，我想等到有时间仔细考虑后再作决定。

比萨试图吸引他留下，而都灵则努力说服他接受他们的提议，沃尔泰拉最终接受了。他在都灵的新同事之一是朱塞佩·皮亚诺，从1895年开始，这两位数学家卷入了一场争执。这是一件不幸的事情，起初是优先权之争，但变得相当激烈。这场争执丝毫没有损害沃尔泰拉的声誉，如果说有什么影响的话，他反而从中获益，他继续因其贡献而获得荣誉。特别是，他于1894年当选为国家科学院 of Italy（“XL科学院”）院士，并于同年成为Mathematical Circle of Palermo董事会成员。他于1895年获得了Accademia dei Lincei的数学奖。1897年，他参加了在苏黎世举行的第一届国际数学家大会，在那里会见了保罗·潘勒韦和埃米尔·博雷尔，他们邀请他次年访问巴黎。他进行了这次访问，并与昂利·勒贝格进行了有益的讨论。下一届国际数学家大会于1900年8月在巴黎举行，沃尔泰拉受邀作四场全会报告之一；他作了Betti, Brioschi, Casorati - Trois analystes italiens et trois manières d'envisager les questions d'analyseⓉ（贝蒂，弗朗切斯科·布廖斯基，Casorati——三位意大利分析学家和看待分析问题的三种方式）的报告。

沃尔泰拉在巴黎作报告时已经结婚。经过短短一个月的订婚，他于1900年7月11日与他的远房表妹Virginia Almagià结婚，她是Edoardo Almagià的女儿。他们在瑞士度蜜月，游览了沃尔泰拉因1897年参加国际数学家大会而熟悉的地方。他还被任命为罗马大学数学物理讲席，接替于1900年2月去世的贝尔特拉米。沃尔泰拉于1900年11月在罗马就职，并于次年作了就职演讲Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali Ⓣ（论将数学应用于生物科学和社会科学的尝试），其中讨论了数学在生物科学和社会科学中的应用。他的第一个孩子是个儿子，生于1901年，但出生后不久便夭折。第二个孩子是个女儿Luisa，生于1902年，同年他的叔叔Alfonso去世。让我们继续介绍沃尔泰拉和Virginia的沃尔泰拉其他孩子：Edoardo（生于1904年）、Enrico（生于1905年）、Gustavo（生于1906年，但几个月后夭折）和Gustavo（生于1909年）。其中一些孩子出生时，沃尔泰拉不在罗马，因为这些年间他经常外出旅行。1901年夏天是在英格兰度过的（伦敦、牛津和剑桥），1902年他访问了德国（柏林）以及丹麦、瑞典和挪威。他在克里斯蒂安尼亚大学参加尼尔斯·阿贝尔诞辰百年纪念活动时，获得了该校的名誉学位。1904年，他再次来到英格兰，获剑桥大学授予名誉学位。他还出席了1904年8月在海德堡举行的国际数学家大会，四年后在罗马举行的国际数学家大会上作全会报告，作了题为 Le matematiche in Italia nella seconda metà del secolo XIX Ⓣ（十九世纪下半叶的意大利数学）的报告。1909年，他在美国马萨诸塞州克拉克大学作了一系列讲座，并作为该校20周年庆典的一部分获得该校名誉学位。1912年，他在伦敦参加皇家学会获得皇家特许状250周年庆典——他已于1910年当选为该会会士。同样在1912年，他前往美国各主要大学进行了长时间的巡回讲学。现在让我们来考察他的一些重要数学贡献。

沃尔泰拉于1883年提出了依赖于另一个函数的连续取值集合的函数理论的想法。雅克·阿达马后来引入了“泛函”一词，取代了沃尔泰拉最初的术语。1890年，沃尔泰拉通过他的泛函演算表明，威廉·哈密顿和卡尔·古斯塔夫·雅各布·雅可比关于动力学微分方程积分的理论可以推广到数学物理的其他问题。在1892年至1894年间，沃尔泰拉发表了关于偏微分方程的论文，特别是柱面波方程。他最著名的工作是在积分方程方面完成的。他于1884年开始这项研究，并于1896年发表了关于现在称为“沃尔泰拉型积分方程”的论文。他继续研究泛函分析在积分方程中的应用，发表了大量关于复合函数和可交换函数的论文。让我们引用[66]中关于沃尔泰拉这些贡献的论述：-

泛函理论作为多自变量函数概念的推广，是由沃尔泰拉在自1887年以来发表的一系列论文中发展起来的，其灵感来自变分法的问题。这些论文开创了现代泛函分析理论。它们立即引起了当时最杰出数学家的注意。实际上，“泛函”这个名称是后来由雅克·阿达马引入的，现在已经取代了沃尔泰拉最初的命名法。在发展这一理论时，沃尔泰拉已经遵循了一条指导他完成许多发现的原理，他称之为从离散到连续的过渡。正是这一原理，他将其应用于他对沃尔泰拉型积分方程的著名研究。他启发式地将积分方程视为线性代数方程组的极限情形，然后直接验证他的最终公式。他的方法为埃里克·伊瓦尔·弗雷德霍姆和大卫·希尔伯特开辟了道路，然而他们研究的是极限过程本身。

1914年第一次世界大战爆发，但敌对行动于8月3日开始后不久，意大利便宣布不会派兵参战。尽管它与德国和奥匈帝国结有同盟。沃尔泰拉强烈反对意大利履行与同盟国的条约。意大利于1915年5月3日废除了这一同盟，并于当月晚些时候对奥匈帝国宣战。在宣战的前一天，沃尔泰拉重申了他4月份提出的请求，要求入伍：-

...从事技术或实验室服务，或任何形式或性质的任何其他服务。

沃尔泰拉成为工程兵部队的中尉，并被派往中央航空研究所。他的任务是确定如何从飞艇上发射火炮而不使其着火。1917年，他被任命为发明局局长。该局协调军方、工业界和大学的贡献。1918年，它更名为发明与研究局。战后，他回到罗马大学，兴趣转向数学生物学。他研究了皮埃尔·弗朗索瓦·韦吕勒方程和逻辑斯谛曲线。他还撰写了关于捕食者-猎物方程的著作。Weinstein写道[66]：-

沃尔泰拉晚年工作的相当一部分致力于数学在生物学中的应用。这些研究的主题主要是研究不同物种动物共同生活的生物关联。换句话说，他对“适者生存”的数学理论感兴趣。尽管今天有其他随机性质的方法，沃尔泰拉的工作仍然对数学生物学的若干现代和相当近期的发展产生着主导影响。

1920年，他应邀在斯特拉斯堡国际数学家大会上作全会报告；这是他第三次受邀作全会报告。他作了题为Sur l'enseignement de la physique mathématique et de quelques points d'analyse Ⓣ（论数学物理的教学及分析的若干要点）的演讲。让我们注意到，他再次受邀在1928年博洛尼亚国际数学家大会上作全会报告，并在那里作了题为La teoria dei funzionali applicata ai fenomeni ereditari Ⓣ（应用于遗传现象的功能理论）的演讲。

意大利法西斯运动大约在1921年作为一场民族主义运动开始。在贝尼托·墨索里尼的领导下，法西斯分子于1923年上台，尽管此时他们并未采取种族政策。沃尔泰拉在意大利议会中与法西斯分子斗争，但继续是一位领军人物，于1923年当选为意大利国家研究委员会的第一任主席，并于同年当选为Accademia dei Lincei的主席。到1926年，法西斯分子已废除了所有反对党，沃尔泰拉感到自己无法继续担任这两个职务，于是辞去了主席一职。到1928年，议会选举已被废除，警方开始以“政治可疑”为由建立关于沃尔泰拉的档案。从那时起，他就受到警方的监视。当沃尔泰拉在1931年拒绝宣誓效忠法西斯政府时——这是所有终身教授和合同教授都必须做的——他被迫离开罗马大学。从第二年起，他大多住在国外，主要在巴黎，也在西班牙和其他国家。由于拒绝再次签署效忠法西斯的誓言，他于1934年被Accademia dei Lincei除名，并于1938年因是犹太人而被伦巴第科学与文学研究所除名。这是他极为成功的职业生涯的一个悲惨结局。墨索里尼于1938年7月颁布的《种族宣言》剥夺了犹太人的意大利公民身份，并禁止他们在银行、政府和教育部门任职。此时，沃尔泰拉的两个儿子已担任大学职位，他们因种族法律而被剥夺了这些职位。沃尔泰拉建议他的儿子们出国，在那里尝试开始新生活。恩里科移居美国，成为德克萨斯大学航空航天工程教授。

1938年，圣安德鲁斯大学授予沃尔泰拉荣誉学位，但尽管他希望前来，他的医生不允许他前往苏格兰接受该学位。埃德蒙·泰勒·惠特克写道[68]：-

1938年12月，他患了静脉炎：他的四肢功能再也没有恢复，但他的智力精力未受影响，正是在此之后，他的最后两篇论文《历史作用情形下生物冲突的一般方程》和《遗传弹性现象中的能量》分别由爱丁堡数学会和宗座科学院发表。1940年10月11日早晨，他在罗马的家中去世。按照他的愿望，他被安葬在阿里恰的小墓地，在一座小山上，靠近他深爱的乡间别墅，在那里他度过了他高贵而活跃一生中最宁静的时光。

Vito Volterra's parents Abramo Volterra, a cloth merchant, and Angelica Almagià were married on 15 March 1859. The family were of Jewish origins. The name Volterra comes from the Tuscan town of Volterra where one of Vito's ancestor moved in the 15th century, having originally come from Bologna. One of the Volterras had opened a bank in Florence while other members of the family were writers, collectors of books and manuscripts, and travellers. They had lived in various different cities but from the 18th century there were Volterras living in Ancona. Vito, his parents' only child, named after his maternal grandfather, was born at exactly the time (May 1860) when Giuseppe Garibaldi set out to conquer Sicily and Naples and so to create a unified Italy. After victories in Sicily, Garibaldi quickly moved north in August 1860. Ancona was at this time under the direct rule of the pope but Garibaldi's army besieged the town attacking it with missiles. Vito, aged about three months, was in his cradle when a missile fell close by destroying the cradle yet, by good fortune, baby Vito survived.

Surviving this close call certainly was not the end of the young child's problems. His father Abramo died when Vito was only two years old. The family was left without any means of financial support and only survived because Alfonso Almagià, a brother of Vito's mother, took care of them. They all moved from Ancona to Terni in the autumn of 1863 when Alfonso worked there for a railway company but in January 1865 he became an official with the Banco d'Italia in Turin and at this stage Vito and his mother returned to Ancona. After only a short stay in Turin, Alfonso moved to Florence where he continued to work for the bank, and Vito and his mother joined him there. It was in this city that Vito spent most of his childhood, although he returned to Ancona for some long family visits, and it was in Florence that he received his education. His interest in mathematics started at the age of 11 when he began to study Legendre's Geometry. At the age of 13 he began to study the Three Body Problem and made some progress by partitioning the time into small intervals over which he could consider the force constant. After elementary school, Vito studied at the lower secondary Dante Alighieri Technical School and later he spent three years at the Galileo Galilei Technical Institute in Florence.

Since his family were extremely poor, there was pressure on Vito to leave the Galileo Galilei Technical Institute after one year of study and take a job. However, his determination to continue studying, supported by his uncle Edoardo, saw him continue. Edoardo, who had a degree in applied mathematics and civil engineering, had a firm which was involved in railway construction. Alfonso put pressure on Vito to join Edoardo's firm but Edoardo kept delaying to allow Vito to continue his education. One of Vito's mathematics teachers at the Galileo Galilei Technical Institute was Cesare Arzelà and he quickly realised that Vito was an extremely talented mathematician. The physics teacher, Antonio Ròiti (1843-1921), became another important ally in Vito's bid to continue his education. Ròiti offered Vito a position teaching physics at the Istituto di Studi Superiori e di Perfezionamento in Florence in 1877, so managing to prevent him from being forced by Alfonso to take a job at the Banco d'Italia. Volterra was able to proceed to the University of Pisa in the autumn of 1878, enrolling in the faculty of Science. In the autumn of the following year he sat the competitive examination for entry to the Scuola Normale Superiore in Pisa and scored the highest possible marks. This meant he received a scholarship together with free board and lodgings. At the Scuola Normale he was taught by several inspiring teachers, the most influential being Ulisse Dini and Enrico Betti. Volterra published three papers in 1881 before graduating, one on mathematical physics Sul potenziale di un' elissoide eterogenea sopra sè stessa Ⓣ and two on analysis, Alcune osservazioni sulle funzioni punteggiate discontinue Ⓣ, on pointwise discontinuous functions, and Sui principii del calcolo integrale Ⓣ which became important in the development of the history of integration. At Pisa he worked for his thesis advised by Enrico Betti, graduating with his laurea in 1882. His thesis on hydrodynamics, Sopra alcuni problemi di idrodinami Ⓣ, included some results of George Stokes, which Volterra had discovered later but independently.

After graduating with his laurea, Volterra was appointed as Betti's assistant in December 1882. He was named Professor of Rational Mechanics at Pisa in 1883 and immediately gained a reputation as a severe and demanding teacher. He was well liked by his colleagues, for example Ernesto Pascal who arrived in Pisa in 1887 wrote in a letter to a friend (see [4]):-

Professor Volterra is an angelic young man, of characteristic modesty.

Volterra's mother Angelica had continued to live with her brother Alfonso while Volterra studied at Pisa and she continued to live in Florence until 1887 when she moved to Pisa and set up home with Volterra. His remarkable mathematical contributions were quickly recognised and he was awarded the gold medal of the National Academy of Sciences of Italy (the "Academy of XL") in 1887, he became a corresponding member of the Accademia dei Lincei and a member of the French Mathematical Society in 1888 and, three years later, he was elected to the Mathematical Circle of Palermo and also named a Knight of the Order of the Crown of Italy. After Betti's death in 1892, he occupied the Chair of Mathematical Physics at Pisa. In the same year he became dean of the Faculty of Science at Pisa and succeeded Betti at editor-in-chief of the journal Nuovo Cimento. During these years while holding a chair at Pisa, Volterra had made a number of trips abroad. He visited Henri Poincaré in Paris in 1888 and was invited to come to Paris again in the following year for the Congrés international de bibliographie des sciences mathématique. This proved an important event since it saw Volterra become a significant member of the international mathematical community. In 1891, as a consequence of these newly established connections, he travelled to Germany spending a month at the University of Göttingen and a few days at the University of Berlin. He met a number of leading German mathematicians on this trip including Hermann Schwarz and Leopold Kronecker.

Although he never formally applied for the position, he was offered the Chair of Higher Mechanics and Rational Mechanics at the University of Turin in July 1893. It would appear that Enrico D'Ovidio was behind the invitation. Volterra wrote to Ulisse Dini a few days later (see [4]):-

Most esteemed Professor, I have come to look for you several times in recent days but I have never succeeded in finding you. I would like to tell you some news that I received the other day from Segre, that is, that the Faculty of Turin has called me there to succeed Siacci, who is going to Naples. This news reached me unexpectedly, and I have written to Turin saying that since it concerns a matter of importance I wanted to wait until I had had time to reflect on it.

Pisa tried to entice him to stay, while Turin tried hard to persuade him to accept their offer which Volterra eventually did. One of his new colleagues at Turin was Giuseppe Peano and, beginning in 1895, the two mathematicians became involved in a dispute. This was an unfortunate affair which started as a priority dispute but became rather heated. The dispute did nothing to harm Volterra's reputation, if anything he came out of it rather well, and he continued to receive honours for his contributions. In particular, he was elected to the National Academy of Sciences of Italy (the "Academy of XL") in 1894 and, in the same year, became a member of the board of the Mathematical Circle of Palermo. He won the mathematics prize from the Accademia dei Lincei in 1895. In 1897 he attended the first International Congress of Mathematicians in Zürich and there met Paul Painlevé and Émile Borel who invited him to Paris in the following year. He made the visit and had useful discussions with Henri Lebesgue. The next International Congress of Mathematicians was held in Paris in August 1900 and Volterra was invited to give one of the four plenary lectures; he gave the lecture Betti, Brioschi, Casorati - Trois analystes italiens et trois manières d'envisager les questions d'analyse Ⓣ.

By the time Volterra gave his lecture in Paris he was married. After a short engagement of one month, he married his second cousin Virginia Almagià, the daughter of Edoardo Almagià, on 11 July 1900. They spent their honeymoon in Switzerland visiting places that Volterra knew from his visit for the International Congress in 1897. He had also been appointed to the Chair of Mathematical Physics at Rome where he succeeded Eugenio Beltrami who had died in February of 1900. Volterra took up the appointment in Rome in November 1900 and, in the following year, gave his inaugural lecture Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali Ⓣ which discussed applications of mathematics to the biological and social sciences. His first child, a son, was born in 1901 but died soon after birth. A second child, a daughter Luisa, was born in 1902 and in the same year his uncle Alfonso died. Let us continue to give Vito and Virginia Volterra's other children: Edoardo (born 1904), Enrico (born 1905), Gustavo (born 1906 but died after a few months) and Gustavo (born 1909). Volterra was not in Rome when some of these children were born since to travelled much over these years. The summer of 1901 was spent in England (London, Oxford and Cambridge), in 1902 he visited Germany (Berlin) as well as Denmark, Sweden and Norway. He received an honorary degree from the University of Christiana while there for the centenary of Niels Abel's birth. In 1904 he was back in England when awarded an honorary degree by the University of Cambridge. He also attended the International Congress of Mathematicians in Heidelberg in August 1904 and four years later was a plenary speaker at the International Congress of Mathematicians in Rome giving the address Le matematiche in Italia nella seconda metà del secolo XIX Ⓣ. In 1909 he gave a series of lectures at Clark University in Massachusetts, United States, and received an honorary degree from the university as part of its 20th anniversary celebrations. In 1912 he was in London for the 250th celebrations of the Royal Society receiving its Royal Charter - he had been elected as a fellow in 1910. Also in 1912, he went on a lengthy lecture tour of leading American universities. Let us now examine some of his important mathematical contributions.

Volterra conceived the idea of a theory of functions which depend on a continuous set of values of another function in 1883. Hadamard was later to introduce the word 'functional' which replaced Volterra's original terminology. In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics. During the years 1892 to 1894 Volterra published papers on partial differential equations, particularly the equation of cylindrical waves. His most famous work was done on integral equations. He began this study in 1884 and in 1896 he published papers on what is now called 'an integral equation of Volterra type'. He continued to study functional analysi applications to integral equations producing a large number of papers on composition and permutable functions. Let us quote from [66] regarding these contributions by Volterra:-

The theory of functionals as a generalization of the idea of a function of several independent variables was developed by Volterra in a series of papers published since 1887 and was inspired by the problems of the calculus of variations. These papers initiated the modern theory of functional analysis. They attracted attention at once from the foremost mathematicians of his time. Actually the name "functional" was introduced later by Hadamard and has now replaced Volterra's original nomenclature. In developing this theory Volterra already followed a principle which guided him through many discoveries and which he called the passage from the discrete to the continuous. It was this principle which he applied to his celebrated researches on integral equations of Volterra's type. He considered heuristically the integral equations as a limiting case of a system of linear algebraic equations and then checked his final formulae directly. His procedure opened the way for Fredholm and Hilbert who, however, investigated the limiting process itself.

World War I broke out in 1914 but, shortly after hostilities began on 3 August, Italy declared that it would not commit troops to the fighting. This was despite having an alliance with Germany and Austria-Hungary. Volterra spoke out strongly against Italy honouring its treaty with the Central Powers. Italy revoked this alliance on 3 May 1915 and later that month declared war on Austria-Hungary. The day before war was declared Volterra repeated a request he had made in April to enlist:-

... in technical or laboratory service or in another service of any form or nature whatsoever.

Volterra became a lieutenant in the corps of engineers and was sent to the Central Institute of Aeronautics. He was given the task of determining how guns could be fired from dirigibles without causing them to catch fire. He was appointed as head of the Office of Inventions in 1917. This coordinated contributions from the military, industry and the universities. It was renamed the Office of Inventions and Research in 1918. After the War he returned to the University of Rome and his interests moved to mathematical biology. He studied the Verhulst equation and the logistic curve. He also wrote on predator-prey equations. Weinstein writes [66]:-

A considerable part of the work of Volterra in the latter part of his life was devoted to the applications of mathematics to biology. The subject of these investigations was mainly the study of biological associations of animals of different species living together. In other words he was interested in a mathematical theory of the "survival of the fittest." While there are today other methods of a stochastic nature, the work of Volterra still exerts a dominant influence on several modern and quite recent developments in mathematical biology.

In 1920 he was invited to give a plenary address at the International Congress of Mathematicians in Strasbourg; this was his third invitation to be a plenary speaker. He gave the lecture Sur l'enseignement de la physique mathématique et de quelques points d'analyse Ⓣ. Let us note that he was again invited as a plenary speaker at the 1928 International Congress of Mathematicians in Bologna where he gave the lecture La teoria dei funzionali applicata ai fenomeni ereditari Ⓣ.

The Italian Fascist movement had started around 1921 as a nationalist movement. Led by Benito Mussolini, the Fascists came to power in 1923 although at this time they did not adopt racial policies. Volterra fought against the Fascists in the Italian Parliament but continued to be a leading figure being elected the first president of the Italian National Council for Research in 1923 and, in the same year, elected president of the Accademia dei Lincei. By 1926 the Fascists had abolished all opposition parties and Volterra felt he could not continue with these two roles, and resigned as president. By 1928 elections to Parliament had been abolished and the police began constructing a file on Volterra as being "politically suspect". He was subject to police surveillance from that time on. When Volterra refused to take the oath of allegiance to the Fascist Government in 1931, which was required of all tenured and contracted professors, he was forced to leave the University of Rome. From the following year he lived mostly abroad, mainly in Paris but also Spain and other countries. For refusing to sign another oath to Fascism he was removed from the Accademia dei Lincei in 1934 and removed from the Istituto Lombardo di Scienze e Lettere in 1938 because he was Jewish. It was a sad end to his highly successful career. The Manifesto della razza (Manifesto of Race) enacted by Mussolini in July 1938 stripped Jews of Italian citizenship and banned them from positions in banking, government, and education. By this time, two of Volterra's sons held university positions and they were deprived of these by the racial laws. Volterra advised his sons to go abroad and try to start a new life there. Enrico moved to the Untied States and became a professor of aerospace engineering at the University of Texas.

Volterra was offered an honorary degree by the University of St Andrews in 1938 but, although he wished to come, his doctor did not allow him to travel to Scotland to receive it. Edmund Whittaker writes [68]:-

In December 1938 he was affected by phlebitis: the use of his limbs was never recovered, but his intellectual energy was unaffected, and it was after this that his two last papers 'The general equations of biological strife in the case of historical actions' and 'Energia nei fenomeni elastici ereditarii' were published by the Edinburgh Mathematical Society and the Pontifical Academy of Sciences respectively. On the morning of 11 October 1940 he died at his house in Rome. In accordance with his wishes, he was buried in the small cemetery of Ariccia, on a little hill, near the country-house which he loved so much and where he had passed the serenest hours of his noble and active life.