贝尔特拉米Eugenio Beltrami

贝尔特拉米的父亲也叫贝尔特拉米，是一位绘制细密画的艺术家。他的父亲出身于艺术世家，因为他的祖父曾雕刻宝石。年轻的贝尔特拉米无疑从家族继承了艺术天赋，但在他身上，除了后来获得的数学才能之外，在他生命中变得重要的却是音乐而非绘画。

贝尔特拉米于1853年至1856年在帕维亚学习，在那里他受教于弗朗切斯科·布廖斯基，后者在贝尔特拉米开始学业的前一年被任命为帕维亚大学应用数学教授。贝尔特拉米本想继续他的数学研究，但他经济困难，因此于1856年不得不中断学业并找了一份工作。他被聘为一名铁路工程师的秘书，这份工作先把他带到了维罗纳，然后又到了米兰。

贝尔特拉米在米兰时，意大利王国于1861年建立。这是一件重要的政治事件，极大地振兴了意大利的学术氛围，尽管意大利似乎不太可能取得其他欧洲国家那样的经济进步，因为超过四分之三的人口是文盲，且大多数人从事农业。

贝尔特拉米再次开始努力从事数学研究，并于1862年发表了他的第一篇论文。他于1862年被任命为博洛尼亚大学的代数学与解析几何学访问教授。在博洛尼亚两年后，贝尔特拉米接受了比萨大学的大地测量学讲席，从1864年到1866年担任该职。在比萨，他与恩里科·贝蒂交好。1866年他回到博洛尼亚，被任命为有理力学教授。

当意大利王国于1861年建立时，都灵是首都。1870年，意大利军队进入罗马。这座城市此前由教皇在法国的支持下占据，但在拿破仑三世战败并退位后，法国对占据罗马的支持便消失了。在新的意大利首都建立了一所新的罗马大学，贝尔特拉米于1873年被任命为那里的有理力学讲席。在罗马三年后，贝尔特拉米迁往帕维亚，担任那里的数学物理讲席。然而，贝尔特拉米于1891年回到罗马，并在那里度过了他最后几年的教学生涯。

受安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳、罗巴切夫斯基、卡尔·弗里德里希·高斯和波恩哈德·黎曼的影响，贝尔特拉米为微分几何中关于曲线和曲面的工作做出了贡献。他将卡尔·弗里德里希·高斯关于共形表示的工作译成意大利文。随后他考虑了曲面上的测地线何时能在平面上表示为直线的问题。贝尔特拉米表明并非所有测地线都能以这种方式表示，接着他继续考虑一个自然的问题：哪些曲面具有这样的性质，即曲面上的测地线能在平面上表示为直线。他的答案非常令人满意，因为他发现这些曲面恰恰就是常曲率曲面。贝尔特拉米随后考虑了常负曲率曲面，并由此得出了他1868年最著名的结果。

他1868年的论文Essay on an interpretation of non-euclidean geometry具体实现了罗巴切夫斯基和鲍耶的非欧几何，并将其与波恩哈德·黎曼的几何联系起来。这一具体实现使用了伪球面，即由曳物线绕其渐近线旋转生成的曲面。

贝尔特拉米在这篇1868年的论文中并未打算证明非欧几何的一致性，也未打算证明欧几里得平行公设的独立性。他所暗示的是，鲍耶和罗巴切夫斯基实际上根本没有引入新概念，而只是描述了负曲率曲面上的测地线理论。贝尔特拉米在这篇论文中写道：-

我们试图为这一学说找到一个真正的基础，而不是不得不承认它需要一种新的实体和概念秩序。

Houel于1870年将罗巴切夫斯基和贝尔特拉米的工作都翻译成了法文，并且他注意到贝尔特拉米的论文如何证明了欧几里得平行公设的独立性。

这篇1868年的论文本应更早发表，但它的出版被推迟了，因为安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳并不完全满意它并非基于循环论证。安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳担心欧几里得几何正被用来描述非欧几何，并从中看到了可能的逻辑困难。安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳错了，但他的担忧使贝尔特拉米把自己的工作搁置了一段时间，而波恩哈德·黎曼的工作使贝尔特拉米确信他的方法是可靠的。

贝尔特拉米还研究光学、热力学、弹性、电学和磁学。他对这些主题的贡献出现在四卷本著作Opere Matematiche（1902—20）中，该书在他去世后出版。他关于物理主题的一些工作与他的非欧几何有关，因为他考察了艾萨克·牛顿给出的引力势在负曲率空间中必须如何修改。他给出了皮埃尔·西蒙·拉普拉斯算子的广义形式。

在[11]中，Tazzioli考察了贝尔特拉米在考虑力学、弹性和势论问题时如何使用微分参数。他还在给出乔治·格林定理的一个推广时使用了这些参数。贝尔特拉米通过为格雷戈里奥·里奇-库尔巴斯托罗和图利奥·列维-齐维塔关于该主题的思想提供基础，间接影响了tensor analysis的发展。

贝尔特拉米最后的一些工作是关于詹姆斯·克拉克·麦克斯韦方程的力学解释。他与恩纳斯托·切萨罗的通信中包含了对贝尔特拉米关于这一主题思考的有趣见解，其中一部分收录在[7]中。其中一封信[7]：-

……（日期为1888年12月）致力于詹姆斯·克拉克·麦克斯韦方程的力学解释。在这里，贝尔特拉米给出了当六个给定函数是弹性变形的分量时条件的一个新证明。

最后，我们应当提到贝尔特拉米对数学史的一项重要贡献。这出现在1889年的一篇出版物中，其中贝尔特拉米使数学界注意到了Saccheri1733年对平行公设的研究。他将Saccheri的结果与吉奥万尼·阿方索·博雷利、约翰·沃利斯、克里斯托佛·克拉乌的结果以及罗巴切夫斯基和鲍耶的非欧几何进行了比较。

贝尔特拉米在意大利数学界发挥了重要作用，于1898年成为Accademia dei Lincei的主席。1899年，他成为意大利王国的参议员。

Eugenio Beltrami's father, also called Eugenio Beltrami, was an artist who painted miniatures. His father had come from an artistic family, for his own father had engraved precious stones. The young Eugenio certainly inherited artistic talents from his family, but in his case in addition to the mathematical talents he would acquire, it was music rather than painting that became important in his life.

Beltrami studied at Pavia from 1853 to 1856, and there he was taught by Brioschi who had been appointed as professor of applied mathematics at the University of Pavia the year before Beltrami began his studies. Beltrami would have liked to continue his mathematical studies but he was suffering financial hardship so in 1856 he had to stop his studies and take up a job. He was employed as the secretary to a railway engineer and this job took him first to Verona and then to Milan.

While Beltrami was in Milan the Kingdom of Italy was established in 1861. It was an important political event which did much to invigorate the academic scene in Italy although it seemed unlikely that Italy could achieve the economic progress made by other European countries since over three quarters of the population was illiterate and most were engaged in agriculture.

At Milan Beltrami began to work hard at his mathematical studies again and in 1862 he published his first paper. He was appointed to the University of Bologna in 1862 as a visiting professor of algebra and analytic geometry. After two years in Bologna, Beltrami accepted the chair of geodesy at the University of Pisa, which he held from 1864 to 1866. At Pisa he became friendly with Betti. In 1866 he returned to Bologna where he was appointed professor of rational mechanics.

When the Kingdom of Italy was established in 1861 Turin was the capital. In 1870 Italian troops entered Rome. The city had been held by the Pope with support from the French, but after Napoleon III was defeated and abdicated, French support to hold Rome evaporated. A new University of Rome was set up in the new Italian capital and Beltrami was appointed to the chair of rational mechanics there in 1873. After three years in Rome, Beltrami moved to Pavia to take up the chair of mathematical physics there. However, Beltrami returned to Rome in 1891 and spent his last years teaching there.

Influenced by Cremona, Lobachevsky, Gauss and Riemann, Beltrami contributed to work in differential geometry on curves and surfaces. He translated Gauss's work on conformal representation into Italian. He then considered the problem of when the geodesics on a surface could be represented as straight lines on the plane. Beltrami showed that not all geodesics could be represented in this way and he then went on to consider the natural question of which surfaces had the property that geodesics on the surface could be represented as straight lines on the plane. His answer was very pleasing, for he discovered that they were precisely the surfaces of constant curvature. Beltrami then considered surfaces of constant negative curvature and was led to his most famous results of 1868.

His 1868 paper Essay on an interpretation of non-euclidean geometry which gives a concrete realisation of the non-euclidean geometry of Lobachevsky and Bolyai and connects it with Riemann's geometry. The concrete realisation uses the pseudosphere, a surface generated by the revolution of a tractrix about its asymptote.

Beltrami in this 1868 paper did not set out to prove the consistency of non-Euclidean geometry or the independence of the Euclidean parallel postulate. What he suggested was that Bolyai and Lobachevsky had not really introduced new concepts at all but had described the theory of geodesics on surfaces of negative curvature. Beltrami wrote in this paper:-

We have tried to find a real foundation to this doctrine, instead of having to admit for it the necessity of a new order of entities and concepts.

Houel translated both Lobachevsky's and Beltrami's work into French in 1870 and he noted how Beltrami's paper proved the independence of the Euclid's parallel postulate.

The 1868 paper should have appeared sooner but it was delayed in its publication because Cremona was not entirely happy that it was not based on a circular argument. Cremona worried that euclidean geometry was being used to describe non-euclidean geometry and he saw a possible logical difficulty in this. Cremona was wrong, but his worries caused Beltrami to put his work on one side for a while but the work of Riemann convinced Beltrami that his methods were sound.

Beltrami also worked on optics, thermodynamics, elasticity, electricity and magnetism. His contributions to these topics appeared in the four-volume work, Opere Matematiche (1902-20), published posthumously. Some of his work on physical topics relates to his non-euclidean geometry for he examined how the gravitational potential as given by Newton would have to be modified in a space of negative curvature. He gave a generalised form of the Laplace operator.

In [11] Tazzioli examines how Beltrami used differential parameters when considering problems in mechanics, elasticity, and potential theory. He also used them in giving a generalisation of Green's theorem. Beltrami indirectly influenced the development of tensor analysis by providing a basis for the ideas of Ricci-Curbastro and Levi-Civita on the topic.

Some of Beltrami's last work was on a mechanical interpretation of Maxwell's equations. There are interesting insights into Beltrami's thinking on this topic contained in his correspondence with Cesàro some of which is reproduced in [7]. One of these letters [7]:-

... (dated December, 1888) is devoted to the mechanical interpretation of Maxwell's equations. Here, Beltrami showed a new proof of the conditions when six given functions are the components of an elastic deformation.

Finally we should mention an important contribution by Beltrami to the history of mathematics. This appears in a 1889 publication in which Beltrami brought to the attention of the mathematical world Saccheri's 1733 study of the parallel postulate. He compared Saccheri's results with those of Borelli, Wallis, Clavius and the non-euclidean geometry of Lobachevsky and Bolyai.

Beltrami achieved an important role in Italian mathematics, becoming President of the Accademia dei Lincei in 1898. In 1899 he became a senator of the Kingdom of Italy.