恩里科·贝蒂Enrico Betti

恩里科·贝蒂的父亲Matteo Betti在贝蒂很小的时候就去世了，他的母亲皮耶罗·德拉·弗朗西斯卡 Dei不得不独自抚养和教育他。他有两个姐妹Luisa和Laura，她们都在年轻时就去世了。他的母亲弗朗西斯卡从皮斯托亚的两所小房子获得收入，并通过自己的工作补充收入来支持儿子的教育。贝蒂在皮斯托亚的Forteguerri学校接受教育，在那里他接受了古典教育。这所古老的学校由红衣主教Niccolò Forteguerri于1473年创立，旨在让贫困学生获得高等教育的机会。

贝蒂在比萨大学学习数学和物理，赢得了一个在大公学院之一作为学生的位置，在那里他通过私人辅导来维持生计。在大学里，他师从Ottaviano Fabrizio Mossotti（1791-1863）和Carlo Matteucci（1811-1868）。Mossotti因自由派观点而被逐出意大利，在瑞士和英国待了一段时间后，他曾任布宜诺斯艾利斯大学实验物理学教授，之后于1835年返回意大利。他从1840年起在比萨大学任教，在那里讲授数学物理、天体力学和大地测量学，贝蒂参加了这些课程。贝蒂从Matteucci那里学习了实验物理，Matteucci曾在巴黎师从弗朗索瓦·阿拉戈。正是弗朗索瓦·阿拉戈推荐他于1840年被任命到比萨。我们注意到，Matteucci于1844年被伦敦皇家学会授予科普利奖章。贝蒂于1846年毕业，获得纯粹与应用数学的laurea学位，其导师是代数教授Giuseppe Doveri（1792-1857）。Doveri的早期教育是在佛罗伦萨接受的，他从比萨大学获得了数学学位。

获得学位后，贝蒂被任命为比萨大学的助理。他在大学工作时，正值意大利的政治和军事事件随着国家接近统一而加剧。不仅有统一的内政问题，还有与奥地利和法国的问题，这两个国家都有自己的议程。Mossotti强烈支持争取独立的斗争，并领导了一个托斯卡纳大学营以实现这一目标。贝蒂加入了由Mossotti领导的这个营，以下士军衔，他参加了1848年5月29日的Curtatone和Montanara战役。这场战役是意大利独立战争的一部分，是在驻扎在设防城镇曼图亚的奥地利军队与由像托斯卡纳大学营这样的年轻志愿者支持的托斯卡纳士兵之间进行的。这些年轻人，像贝蒂一样，没有战斗经验，但对他们的事业充满热情，并被证明是优秀的战士。贝蒂和其他人仅仅为了这个场合而被提升为军官。他们在战役发生前进行了十五天的军事训练。当奥地利军队离开曼图亚攻击托斯卡纳人时，优势严重偏向奥地利军队，因为他们有20,000人对7,000托斯卡纳人。托斯卡纳大学营等待托斯卡纳军队的命令，但当没有命令到来，他们能听到大约2公里外的战斗声音时，他们冲入战斗，与正规托斯卡纳士兵并肩英勇作战。最终托斯卡纳人被迫撤退，损失惨重，但奥地利人自己也遭受了更大的损失，没有前进。贝蒂极其幸运地在战斗中幸存下来，这场战斗被证明是一场最终会成功的战役中的关键一战。这场战斗后，贝蒂回到了比萨大学。

在比萨大学担任助理之后，贝蒂回到了他的家乡皮斯托亚，并于1849年成为该镇Forteguerri中学的数学教师。当然，这正是贝蒂曾就读的学校。他接受这些教职并非因为想终身做一名中学教师，而是因为他必须在进行研究的同时谋生，他希望研究能为他赢得大学职位。Capecchi在[14]中写道：-

相对的文化隔绝决定了他关于用根式解代数方程的研究的独特性质。尽管埃瓦里斯特·伽罗瓦的著作起源于1820年代，但在19世纪下半叶，即使在法国，它们仍被认为难以理解。

1854年，他搬到佛罗伦萨，再次在一所中学任教。在这些年里，当贝蒂还是一名中学教师时，他正在进行研究，并以Mossotti为导师。在[21]中，发表了贝蒂与Mossotti在1847年至1857年间的现存通信。Mossotti不断给他的学生提供研究建议，但从通信中可以清楚地看出，两人之间的关系不仅仅是师生关系，他们也是朋友。贝蒂向Mossotti解释了他关于研究的想法，特别是他正在努力为埃瓦里斯特·伽罗瓦仅仅陈述而未给出任何证明的许多命题提供令人满意的证明。事实上，贝蒂成为第一个发表关于埃瓦里斯特·伽罗瓦理论的观察和论证的人，其论文发表于1851-1852年。这些论文发表在Annali di Scienze fisiche e matematiche上，它们是：Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo Ⓣ（论不可约一次代数方程可用根式解）(1851)；Un teorema sulle risolventi dell'equazioni risolubili per radicali Ⓣ（关于解可用根式解的方程的一个定理）(1851)；以及Sulla risoluzione dell'equazioni algebriche Ⓣ（论方程的代数解法）(1852)。然而，我们应该注意到，这些并不是贝蒂的第一批出版物，因为他已于1850年发表了关于数学物理的论文Sopra la determinazione analitica dell'efflusso dei liquidi per una piccolissima apertura Ⓣ（论流体通过极小孔口的流量的解析确定）。

贝蒂于1857年被任命为比萨大学高等代数教授。次年，他与弗朗切斯科·布廖斯基和Felice Casorati一起访问了欧洲主要的数学中心。他们访问了哥廷根、柏林和巴黎，建立了许多重要的数学联系。特别是在哥廷根，贝蒂结识了波恩哈德·黎曼并与之成为朋友。1859年回到比萨后，他转任分析与高等几何讲席。他于1860年发表了他的就职教授演讲，该演讲未出版，但其细节留存下来并在[11]中讨论。Mossotti持有数学物理讲席，于1863年去世，贝蒂被任命担任该讲席，同时兼任分析与高等几何讲席。

我们已经解释了贝蒂参与1858-59年对奥地利战争的情况，其中法国人起初加入意大利人对抗奥地利人。然而，到1861年3月17日，意大利王国正式成立。然而，罗马和威尼斯在这一阶段不属于意大利，随着政府结构的讨论，政治活动持续处于高水平。贝蒂在新国家的政府中任职，他于1862年成为议会议员，代表皮斯托亚，一直担任此职至1867年。

为了改善健康状况，波恩哈德·黎曼于1863年秋天访问了意大利，并重新与贝蒂建立了友谊。在1863年10月6日从佛罗伦萨写给他的朋友兼同事P Tardy的一封信中，贝蒂写道（见[4]）：-

我刚刚与波恩哈德·黎曼谈论了空间的连通性，并对这件事有了准确的想法。

多年来，贝蒂将政治服务与为大学服务结合在一起。他曾担任比萨大学校长一职，并于1864年成为其师范学院Scuola Normale Superiore的院长，担任此职直至去世。在他的领导下，比萨的Scuola Normale Superiore成为意大利领先的数学研究和数学教育中心。新成立的意大利王国引发了全国对数学及其教学的重新关注，贝蒂在其中发挥了重要作用。他对学校应如何教授数学有强烈的看法。他[1]：-

……热爱古典文化，并与弗朗切斯科·布廖斯基一起倡导在中学恢复欧几里得的教学，因为他将欧几里得的著作视为纪律与美的典范。

他与弗朗切斯科·布廖斯基合作翻译了欧几里得的Elements，以此推进这些目标。贝蒂在没有弗朗切斯科·布廖斯基协助的情况下，还翻译了另一部学校教材，即约瑟·伯特兰的Algebra elementareⓉ（初等代数）。我们已经注意到，1864年贝蒂接替Mossotti，被任命为数学物理讲席，并终身担任此讲席。他的另一位老师Carlo Matteucci于1844年创办了期刊Nuovo Cimento，1863年贝蒂成为其主编。他在该期刊上发表了十篇论文。1871年，贝蒂创办了Annali della Scuola Normale - Sezione della classe di scienze fisiche e matematiche，旨在为学生提供一个发表学位论文或教授资格论文（Habilitation）论文的场所。1870年，他从分析与几何讲席转到天体力学讲席。他也终身担任此讲席。

政治事件继续塑造着新的意大利国家，1866年《维也纳条约》将威尼斯并入意大利王国。次年，罗马遭到意大利军队进攻，但法国派兵保卫该城抵御进攻。由于对政府的不满，意大利各地普遍动荡，这个新统一的国家是否会再次分裂远未明朗。然而，1870年意大利军队攻占罗马，罗马成为意大利王国的首都。贝蒂继续在这个发展中的国家担任政治职务。他从1874年10月到1876年3月担任了十八个月的教育部副部长，但他[1]：-

……然而渴望学术生活、孤独的沉思以及与密友的讨论。

1884年他担任意大利议会参议员，但他再次怀念学术生活[1]：-

然而，他的主要目标始终是带有崇高哲学目的的纯粹科学研究。

正如乌利塞·迪尼在[15]中指出的，贝蒂在许多非常不同的数学领域工作，例如：代数方程理论；椭圆函数理论、复变量的代数函数、多维空间等分析及其在几何中的应用；他发表了许多关于数学物理和天体力学、牛顿力理论、热理论、电学理论、磁学、弹性、毛细现象、流体动力学、质点系统运动以及动力学原理的推广的著作。然而，他尤其以对代数和拓扑学的贡献而闻名。在他早期关于方程和代数领域的工作中，正如我们已经看到的，贝蒂扩展并给出了与伽罗瓦理论的代数概念相关的证明。这些概念此前发表时没有证明，在这项工作中，贝蒂因此为从古典代数向现代代数的过渡做出了重要贡献。他从1851年开始在几部著作中发表了这些重要贡献，并且是第一个证明埃瓦里斯特·伽罗瓦群在乘法下封闭的人。1854年，贝蒂表明五次方程可以用积分求解，从而得到椭圆函数。

然而，我们不应给人这样的印象：贝蒂是第一个澄清埃瓦里斯特·伽罗瓦工作中所有困难的人。尽管卡米耶·若尔当在其Traité des substitutions et des equations algebriquesⓉ（关于替换和代数方程的论文）（1870年）中称赞贝蒂填补了埃瓦里斯特·伽罗瓦论证中的空白，并第一个严格建立了埃瓦里斯特·伽罗瓦定理的序列，但事实是贝蒂的工作包含大量模糊和错误。Mammone在[19]中非常清楚地指出了这些点，然而论文[19]本身包含群论错误，正如Peter Neumann在评论它时指出的。贝蒂的错误似乎与群的正规子群有关，他做出了错误的假设（用现代术语来说）每个扩张都分裂。

我们上面已经提到，波恩哈德·黎曼于1863年在比萨拜访了贝蒂。在与朋友波恩哈德·黎曼的讨论影响下，贝蒂受到启发，在理论物理方面做了重要工作，特别是在势论和弹性方面。他还发表了关于函数理论的论文，专注于椭圆函数。事实上，贝蒂向数学物理的这种方向转变导致他如上所述于1870年在比萨接替了讲席。乌利塞·迪尼，贝蒂早些时候曾教过他，被任命填补他的分析和高等几何讲席。

让我们更详细地考察贝蒂在数学物理特别是弹性力学方面的工作。我们基于[14]中给出的描述稍作修改后的版本进行叙述：-

贝蒂 探讨了数学物理的若干方面；其中最重要的一个方面与经典力学有关。在他早期的工作中，他采取了一种机械论进路，其中力而非能量是基础概念，虚功是支配定律。在他关于毛细现象的工作《Memoria sopra la teoria della capillarità》Ⓣ（《毛细管理论论著》）发表于《Annali delle Università toscane (Pisa)》中，贝蒂 假定物体由分子构成，这些分子在短距离上相互吸引，在极短距离上相互排斥，而在更大但仍非常短的距离上实际上不发生相互作用。在他关于牛顿力的论著《La teorica delle forze che agiscono secondo la legge di 艾萨克·牛顿 e sue applicazioni all'elettrostatica》中，贝蒂 宣示了他的牛顿主义意识形态。贝蒂 在其关于毛细现象的第二篇论著《Teoria della capillarità》发表于《Nuovo Cimento》中改变了态度，在 开尔文 的研究基础上赋予势以能量的含义和基础性作用。这一改变在《Teoria della elasticità》Ⓣ（《弹性理论》）（1874）中是决定性的，其中没有提及内力，甚至避免明确提到应力。当 贝蒂 撰写《Teoria della elasticità》时，弹性理论已经成熟，其原理已为人所知，尽管并未被完全认同。那里展开的阐述，像现代手册一样，遵循公理化进路。贝蒂 的原理一方面是势能和应变的概念，另一方面是虚功原理。尽管这本书在理论方面并非特别原创，但它对于阐述一种评估三维弹性连续体上位移的一般程序以及解决具体问题具有重要意义。此外，各个论题的呈现方式成为大多数弹性理论手册的范式。

贝蒂发表了一篇关于椭圆函数论的论文La teorica delle funzioni ellitiche Ⓣ（椭圆函数论）（1860），其中包含的结果在几年后被卡尔·魏尔斯特拉斯进一步发展。他于1871年发表了一篇关于拓扑学的重要论文，其中包含了我们现在所称的“贝蒂数”。这就是他发表在Annali di matematica pura ed applicata上的著名论文Sopra gli spazi di un numero qualunque di dimensioni Ⓣ（论任意维空间）。贝蒂数是由儒勒·昂利·庞加莱命名的，他受到贝蒂在该主题上工作的启发而研究拓扑学。他的另一篇论文是Sopra una estensione della terza legge di Keplero Ⓣ（论约翰内斯·开普勒第三定律的推广）（1888）。这篇论文属于天体力学领域，推广了约瑟夫·拉格朗日关于三体问题的研究。

在此我们还应提到曾师从贝蒂的令人印象深刻的学生名单，包括：Ernesto Padova（1845-1896）、Eugenio Bertini、切萨雷·阿泽拉、Guido Ascoli、乌利塞·迪尼、格雷戈里奥·里奇-库尔巴斯托罗、维多·沃尔泰拉、Valentino Cerruti（1850-1909）、Giuseppe Lauricella（1867-1913）、Carlo Somigliana（1860-1955）、萨尔瓦托雷-平谢尔、玛利欧·派埃利、费代里戈·恩里克斯和路易吉·比安基。

1857年被任命为比萨的讲席后，他在每个学年都讲授广泛不同的数学主题，直到1890-91学年。他是一位热情而清晰的讲师，深受学生爱戴。然而到1890年11月，他已经难以承担教学任务，因为他逐渐瘫痪。这种瘫痪逐渐加重，使他越来越难以继续工作。然而，他在索亚纳的别墅中的去世来得很突然。尽管他是在暑假中期去世的，当时大多数教授都不在大学，但他仍然得到了许多荣誉的隆重葬礼。许多同事、以前的学生以及许多比萨市民参加了葬礼。他被安葬在比萨的Camposanto monumentale。

贝蒂获得了许多荣誉，包括当选罗马的Accademia dei Lincei（1851年）、National Academy of Sciences of Italy（“四十人科学院”）（1860年）、摩德纳科学、文学与艺术学院（1860年）、那不勒斯皇家学会（1863年）、米兰伦巴第科学与文学研究所（1864年）、Academy of Sciences of Turin（1864年）以及斯德哥尔摩的柏林科学院、Göttingen Academy of Sciences、Royal Society of London和瑞典皇家科学院。在[10]中，Bottazzini简要描述了比萨高等师范学校图书馆中十二个盒子的内容，其中包含贝蒂未发表的笔记和信件。似乎没有关于这些论文内容的进一步研究发表。

Enrico Betti's father, Matteo Betti, died when Enrico was very young and his mother, Francesca Dei, had to bring him up and educate him on her own. He had two sisters Luisa and Laura who both died at a young age. His mother Francesca had the income from two small houses in Pistoia and she supplemented this with work of her own to support the education of her son. Enrico's schooling was at the Forteguerri school in Pistoia where he had a classical education. This ancient school had been founded in 1473 by Cardinal Niccolò Forteguerri to allow poor students to get access to higher education.

Betti studied mathematics and physics at the University of Pisa, winning a place as a student in one of the grand-ducal colleges where he supported himself by private tutoring. At the university he was taught by Ottaviano Fabrizio Mossotti (1791-1863) and Carlo Matteucci (1811-1868). Mossotti had been exiled from Italy for his liberal views and, after a while in Switzerland and then England, he had been professor of experimental physics at the University of Buenos Aires before returning to Italy in 1835. He taught at the University of Pisa from 1840 where he gave courses on mathematical physics, celestial mechanics and geodesy which Betti attended. Betti learnt about experimental physics from Matteucci who had studied in Paris under François Arago. It was Arago who had recommended his appointment to Pisa in 1840. We note that Matteucci was awarded the Copley Medal by the Royal Society of London in 1844. Betti graduated with a laurea in pure and applied mathematics in 1846 having been advised by the professor of algebra Giuseppe Doveri (1792-1857). Doveri's early education had been in Florence and he had obtained a degree in mathematics from the University of Pisa.

Following the award of his degree, Betti was appointed as an assistant at the University of Pisa. He worked at the university at a time when political and military events in Italy were intensifying as the country came nearer to unification. There were not only the internal politics of unification but there were problems with Austria and France, both countries having their own agendas. Mossotti strongly supported the fight for independence and led a Tuscany University Battalion in an attempt to achieve this aim. Betti joined this battalion led by Mossotti and, with the rank of corporal, he fought in the battle of Curtatone and Montanara on 29 May 1848. This battle, part of the War of Independence of Italy, was fought between Austrian troops who had been stationed in the fortified town of Mantua, and Tuscan soldiers who were supported by young volunteers like those of the Tuscany University Battalion. These young men, like Betti, had no experience of battle but were filled with enthusiasm for their cause and proved to be excellent fighters. Betti and others had been promoted to officers merely for the occasion. They spent fifteen days carrying out military training before the battle took place. The odds were heavily in the favour of the Austrian troops when they left Mantua to attack the Tuscans for there were 20,000 of them against 7,000 Tuscans. The Tuscany University Battalion waited for orders from the Tuscan army but, when no orders arrived and they could hear the sounds of battle about 2 km away, they charged into the battle and fought with great bravery beside the regular Tuscan soldiers. Eventually the Tuscans were forced to retreat with heavy losses but the Austrians had themselves received even greater losses and did not advance. Betti was extremely fortunate to survive the battle which proved to be a vital one in a campaign which would eventually be successful. After this battle, Betti returned to the University of Pisa.

After working as an assistant at the University of Pisa, Betti returned to his home town of Pistoia where he became a teacher of mathematics at the Forteguerri secondary school in the town in 1849. This, of course, was the school at which Betti had studied. He took on these teaching positions not because he wanted to spend his life as a school teacher but rather because he had to earn his living while he undertook research which, he hoped, would gain him a university appointment. Capecchi writes in [14]:-

The relative cultural insulation determined the original character of his research on the solution by radicals of algebraic equations. Though Galois' works originated in the 1820s, still in the second half of the nineteenth century they were found hard to be understood even in France.

In 1854 he moved to Florence where again he taught in a secondary school. During these years when Betti was a secondary school teacher, he was undertaking research with Mossotti as his advisor. In [21] the extant correspondence between Betti and Mossotti between 1847 and 1857 is published. Mossotti gives continual research advice to his pupil but it is clear from the correspondence that the relation between the two is not simply that of teacher and pupil but the two are also friends. Betti explains his ideas about research to Mossotti, in particular he was working to give satisfactory proofs of many propositions which Galois had simply stated without giving any proof. In fact Betti became the first to publish observations and demonstrations on Galois theory with his papers of 1851-1852. These papers, published in the Annali di Scienze fisiche e matematiche, are: Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo Ⓣ (1851); Un teorema sulle risolventi dell'equazioni risolubili per radicali Ⓣ (1851); and Sulla risoluzione dell'equazioni algebriche Ⓣ (1852). We should note, however, that these are not Betti's first publications for he had published the paper on mathematical physics Sopra la determinazione analitica dell'efflusso dei liquidi per una piccolissima apertura Ⓣ in 1850.

Betti was appointed as professor of higher algebra at the University of Pisa in 1857. In the following year he, along with Francesco Brioschi and Felice Casorati, visited the leading mathematical centres of Europe. They visited Göttingen, Berlin and Paris making many important mathematical contacts. In particular in Göttingen Betti met and became friendly with Riemann. Back in Pisa in 1859 he moved to the chair of analysis and higher geometry. He gave his inaugural professorial address in 1860 which was not published but details of it survive and are discussed in [11]. Mossotti, who held the chair of mathematical physics, died in 1863 and Betti was appointed to that chair in addition to the chair of analysis and higher geometry.

We have already explained Betti's involvement in the 1858-59 war with Austria in which the French at first joined the Italians against the Austrians. However, by 17 March 1861, the Kingdom of Italy was formally created. Rome and Venice were not part of Italy at this stage, however, and there continued high levels of political activity as the government structure was discussed. Betti served in the government of the new country when he became a member of Parliament in 1862, representing Pistoia, continuing in this role until 1867.

In an attempt to improve his health, Riemann made an Italian visit in the autumn of 1863 and renewed his friendship with Betti. In a letter to his friend and colleague P Tardy, written from Florence on 6 October 1863, Betti writes (see [4]):-

I have newly talked with Riemann about the connectivity of spaces, and have formed an accurate idea of the matter.

Over quite a number of years Betti mixed political service with service for his university. He served a term as rector of the University of Pisa and he became director of its teachers' college, the Scuola Normale Superiore, in 1864 holding this post until his death. Under his leadership the Scuola Normale Superiore in Pisa became the leading Italian centre for mathematical research and mathematical education. The creation of the new Kingdom of Italy led to a renewed interest in mathematics and its teaching throughout the country and Betti played a major role in this. He had strong views on how mathematics should be taught in schools. He [1]:-

... loved classical culture, and with Brioschi he championed the return to the teaching of Euclid in secondary schools, for he regarded Euclid's work as a model of discipline and beauty.

He furthered these aims by collaborating with Brioschi in making a translation of Euclid's Elements. Betti, without Brioschi's assistance, also translated another school text, namely Joseph Bertrand's Algebra elementare Ⓣ. We have already noted that in 1864 Betti succeeded Mossotti when he was appointed to the chair of mathematical physics and he continued to hold this chair for the rest of his life. Carlo Matteucci, another of his teachers, had founded the journal Nuovo Cimento in 1844 and, in 1863 Betti became its editor-in-chief. He published ten papers in this journal. In 1871 Betti founded the Annali della Scuola Normale - Sezione della classe di scienze fisiche e matematiche, designed as a place where students could publish dissertations or habilitation theses. In 1870 he moved from the chair of analysis and geometry to the chair of celestial mechanics. He also held this chair for the rest of his life.

Political events continued to build the new country of Italy, with the Treaty of Vienna bringing Venice into the Italian Kingdom in 1866. Rome was attacked by Italian troops the following year but France defended the city with its troops against the attack. There was widespread unrest in Italy due to dissatisfaction with the government and it was far from clear that the newly unified country would not split apart again. In 1870, however, Italian troops captured Rome and it became the capital of the Kingdom of Italy. Betti continued to undertake political roles in the developing country. He served as an undersecretary of state for education for eighteen months from October 1874 to March 1876 but he [1]:-

... longed, however, for the academic life, solitary meditation, and discussion with close friends.

He served as a senator in the Italian Parliament in 1884 but again he missed the academic life [1]:-

His principal aim, however, was always pure scientific research with a noble philosophical purpose.

As Ulisse Dini notes in [15], Betti worked in many, very different, mathematical areas such as: the theory of algebraic equations; the theory of elliptic functions, algebraic functions of a complex variable, on spaces of many dimensions etc in analysis and on the applications of this to geometry; he published many works on mathematical physics and celestial mechanics, the theory of Newtonian forces, the theory of heat, the theory of electricity, magnetism, elasticity, capillary, hydrodynamics, the motion of systems of particles, and the extension of the principles of dynamics. However, he is particularly known for his contributions to algebra and topology. In his early work in the area of equations and algebra, as we have already seen, Betti extended and gave proofs relating to the algebraic concepts of Galois theory. These had been previously published without proofs and in this task Betti thus made an important contribution to the transition from classical to modern algebra. He published these important contributions in several works starting in 1851 and he was the first to give a proof that the Galois group is closed under multiplication. In 1854 Betti showed that the quintic equation could be solved in terms of integrals resulting in elliptic functions.

However, we should not give the impression that Betti was the first to clarify all the difficulties in Galois' work. Although Jordan, in his Traité des substitutions et des equations algebriques Ⓣ (1870) credits Betti with having filled the gaps in Galois' arguments and with having been the first to establish the sequence of Galois' theorems rigorously, the fact is that Betti's work contains substantial obscurities and errors. Mammone, in [19], brings these points out very clearly, yet the paper [19] itself contains group theoretical errors as was pointed out by Peter Neumann when he reviewed it. Betti's errors appear to relate to normal subgroups of groups and he makes the false assumption that (in modern terms) every extension splits.

We have already mentioned above that Riemann visited Betti in Pisa in 1863. Influenced by discussions with his friend Riemann, Betti was inspired to do important work in theoretical physics, in particular in potential theory and elasticity. He also published papers on the theory of functions, concentrating on elliptic functions. In fact this change of direction by Betti towards mathematical physics led to him substituting chairs at Pisa in 1870 as we remarked above. Dini, who Betti had taught earlier, was appointed to fill his chair of analysis and higher geometry.

Let us look in a little more detail at Betti's work on mathematical physics and in particular on elasticity. We give a description based on a somewhat modified version of that given in [14]:-

Betti explored several aspects of mathematical physics; one of the most important was that regarding classical mechanics. In his early work he assumed a mechanistic approach, where force and not energy is the funding concept and virtual work the regulating law. In his work on capillarity, 'Memoria sopra la teoria della capillarità' Ⓣ published in the 'Annali delle Università toscane (Pisa)', Betti assumes bodies as formed by molecules which attract each other at short distance and repel at very short distance, and which do not practically interact at larger, but still very short distances. In his memoirs 'La teorica delle forze che agiscono secondo la legge di Newton e sue applicazioni all'elettrostatica' on Newtonian forces, Betti declared his Newtonian ideaology. Betti changed his attitude in his second memoir on capillarity, 'Teoria della capillarità' published in 'Nuovo Cimento', by giving the potential an energetic meaning and a founding role, on the basis of William Thomson's studies. This change was definitive in the 'Teoria della elasticità' Ⓣ (1874), where no reference is made to internal forces, even avoiding the explicit mention of stress. When Betti wrote 'Teoria della elasticità', the theory of elasticity was already mature with known principles, though not completely shared. The exposition developed there, like modern handbooks, follows an axiomatic approach. Betti's principles are on the one hand the concepts of potential energy and strains, and on the other hand the principle of virtual work. Though the book is not particularly original for its theoretical aspects, it is important for the exposition of a general procedure to evaluate the displacements on a three-dimensional elastic continuum and for the solution of specific problems. Moreover, the manner in which the single arguments are presented became paradigmatic for most handbooks on the theory of elasticity.

Betti published a memoir on the theory of elliptic functions La teorica delle funzioni ellitiche Ⓣ (1860), containing results which were developed further by Weierstrass some years later. He published an important paper on topology in 1871 which contained what we now call the "Betti numbers". This was his famous Sopra gli spazi di un numero qualunque di dimensioni Ⓣ published in the Annali di matematica pura ed applicata. The Betti numbers were so named by Henri Poincaré who was inspired to study topology through Betti's work on the subject. Another of his papers is Sopra una estensione della terza legge di Keplero Ⓣ (1888). This paper, in the area of celestial mechanics, generalises Lagrange's studies on the three-body problem.

We should also mention at this point the impressive list of students that studied with Betti including: Ernesto Padova (1845-1896), Eugenio Bertini, Cesare Arzelà, Guido Ascoli, Ulisse Dini, Gregorio Ricci-Curbastro, Vito Volterra, Valentino Cerruti (1850-1909), Giuseppe Lauricella (1867-1913), Carlo Somigliana (1860-1955), Salvatore Pincherle, Mario Pieri, Federigo Enriques and Luigi Bianchi.

Following his appointment to the chair at Pisa in 1857 he gave lectures on a wide range of different mathematical topics in each academic year up to the academic year 1890-91. An enthusiastic and clear lecturer, he was greatly loved by his students. By November 1890, however, he was already struggling to undertake his teaching commitments as he slowly became paralysed. This paralysis gradually became more severe making it increasingly difficult for him to continue to work at all. However, his death in his villa at Soiana came unexpectedly. Although he died in the middle of the summer vacation when most of the professors were absent from the university, nevertheless he was given a solemn funeral with many honours. It was attended by many of his colleagues and former pupils as well as many citizens of Pisa. He was buried in the Camposanto monumentale of Pisa.

Betti received many honours including election to the Accademia dei Lincei in Rome (1851), the National Academy of Sciences of Italy (the "Academy of Forty") (1860), the Academy of Sciences, Letters and Arts of Modena (1860), the Royal Society of Naples (1863), the Lombard Institute of Science and Letters in Milan (1864), the Academy of Sciences of Turin (1864) as well as the Berlin Academy of Sciences, the Göttingen Academy of Sciences, the Royal Society of London and the Royal Swedish Academy of Sciences in Stockholm. In [10] Bottazzini briefly describes the contents of twelve boxes in the library of the Scuola Normale Superiore in Pisa that contain Betti's unpublished notes and letters. There does not appear to be any further research published on the contents of these papers.