安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳Luigi Cremona

安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳在帕维亚的Ginnasio接受教育。克雷莫纳的父亲在克雷莫纳11岁时去世，这几乎使他无法接受适当的教育。然而，他的继父支持他完成学业，使他得以完成学校教育并进入帕维亚大学。

然而，政治事件对克雷莫纳的生活产生了重大影响，最初的冲击来自1848年的革命，这场革命试图实现一部新的、更自由的宪法。奥地利政府逮捕了威尼斯和米兰的革命反对派领袖，并镇压了帕多瓦和帕维亚的学生示威。克雷莫纳一生都是热忱的意大利民族主义者，立即加入了“自由意大利”营。

几天之内，奥地利军队几乎失去了整个伦巴第-威尼西亚并撤退。撒丁-皮埃蒙特对奥地利宣战。在意大利人取得一些初步胜利后，奥地利军队开始赢得胜利。在伦巴第，奥地利在1849年3月经过10天战斗重新征服布雷西亚，使威尼斯陷入孤立。克雷莫纳此时已是一名中士，随部队保卫威尼斯。该城抵抗奥地利军队直到1849年8月24日。守军的英勇使奥地利进攻者允许他们体面地离开该城。克雷莫纳得以返回帕维亚。

回到帕维亚后，克雷莫纳发现他的母亲在他为解放意大利而战斗期间已经去世。他再次因经济原因无法在没有家庭支持的情况下回到大学，而他的家人再次团结起来提供了必要的支持。他于1849年11月27日进入帕维亚大学攻读土木工程学位。在那里，他师从Bordoni、Casorati和弗朗切斯科·布廖斯基，但受弗朗切斯科·布廖斯基的影响最大。克雷莫纳后来写道，见[6]：-

我作为学生、后来作为同事与弗朗切斯科·布廖斯基一起度过的那些年，是我一生中重要的部分；在这些年的前期，我学会了热爱科学，而在后期，我学会了如何将它传授给广大听众。

1853年5月9日，克雷莫纳获得了帕维亚大学的土木工程博士学位。此时，Camillo Benso di Cavour已成为内阁首脑，意大利的统一已经开始。但奥地利仍然控制着该地区，这使得曾反抗奥地利占领的克雷莫纳生活艰难。正如Greitzer在[1]中所写：-

他反抗奥地利统治的兵役记录使他无法在教育系统中获得正式教职，因此他的第一份工作是在帕维亚的几个家庭担任私人教师。

此时克雷莫纳已从事数学研究，他的第一篇论文于1855年3月发表。这对他获得在帕维亚Ginnasio临时教授物理的许可一定起了一些作用，他本人曾在那里接受大学预科教育。1856年5月，他的第二篇数学论文发表，这连同他良好的教学，似乎帮助他在1856年12月获得了Ginnasio助理教师的职位。

1857年1月17日，克雷莫纳被任命为克雷莫纳Ginnasio的正式教师。克雷莫纳是伦巴第克雷莫纳地区的首府，位于米兰东南波河北岸。克雷莫纳在该校又待了三年，在此期间他撰写了许多数学文章。这些文章并不十分重要，只是其中一些使用射影方法研究曲线，这些技术将成为克雷莫纳后来重要数学工作的特征。

当数学家克雷莫纳在克雷莫纳城任教时，一些政治事件正在发生，这些事件将对他的未来产生重大影响。1858年7月，Cavour与法国拿破仑三世达成协议，以换取其对皮埃蒙特的支持。当奥地利在1859年4月26日对皮埃蒙特宣战时，法国履行了与皮埃蒙特的同盟义务。经过若干次战役后，奥军撤退，拿破仑三世在Villafranca签署条约，接受奥地利割让伦巴第，随后将其转交给皮埃蒙特。伦巴第从奥地利统治下解放后，人们很快认识到，不应再因政治原因而压制克雷莫纳。1859年11月28日，他被任命为米兰圣亚历山大中学的教师。

随着事态迅速走向在维克托·伊曼纽尔二世为国王的领导下统一意大利，克雷莫纳于1860年6月10日由皇家法令任命为博洛尼亚大学常任教授。意大利王国于1861年3月17日由在都灵召开的议会正式宣告成立。克雷莫纳一直留在博洛尼亚，直到1867年10月。

克雷莫纳的Complete Works中收录了他在博洛尼亚期间发表的45篇文章。其中16篇是对Nouvelles Annales中提出的问题的回答。有8篇书评和4篇旨在鼓励几何研究的史学文章。其中还包括克雷莫纳关于平面曲线变换的重要工作，这些工作于1863年和1865年期间发表。正是这项工作使他获得了1866年的雅各布·施泰纳奖，该奖由克雷莫纳和Rudolf Sturm共同获得。同样在博洛尼亚期间，克雷莫纳发展了双有理变换理论，后来被称为克雷莫纳变换，并撰写了一系列关于扭三次曲线曲面的论文。

White在[11]中描述了克雷莫纳工作的这一时期：-

米歇尔·沙勒是克雷莫纳最初模仿自己工作的那种人。卡尔·冯·施陶特的几何纯粹主义是后来的影响。因此，很自然地会为几何定义寻找度量基础，而不是射影基础，或者两者互换使用。代数几何的装置建立在极线之上，而极线又建立在距离之上。几何方法主要是使用术语或描述性关系，而不是方程。枚举几何的开端就在这里……

Greitzer在[1]中描述了克雷莫纳在博洛尼亚时期引入的变换的重要性：-

克雷莫纳变换已被用于研究有理曲面、解决平面曲线和空间曲线的奇点，以及研究elliptic integrals和波恩哈德·黎曼曲面。它们能有效地将曲线的奇点约化为具有不同切线的二重点。

1867年10月，根据弗朗切斯科·布廖斯基的推荐，又颁布了一项皇家法令，这次任命克雷莫纳到米兰理工学院。他在米兰期间于1872年获得了教授头衔。Greitzer在[1]中写道：-

他在米兰一直待到1873年，这段时期是克雷莫纳创造力最旺盛的时期。他撰写了关于扭曲三次曲线、可展曲面、圆锥曲线理论、平面曲线理论、三次和四次曲面、静力学和射影几何等多样化主题的文章。

克雷莫纳在静力学方面的工作非常重要，它以更清晰的形式给出了詹姆斯·克拉克·麦克斯韦的一些定理。在1872年的一篇论文中，克雷莫纳采纳了詹姆斯·克拉克·麦克斯韦在1867年一份工程期刊上发表的关于框架结构力的思想，并将詹姆斯·克拉克·麦克斯韦的互易图形概念解释为射影3-空间中的对偶。例如，这些互易图形中，一个图形里的三个平衡力由一个三角形表示，而在互易图形中它们由三条共点线表示。

1873年，克雷莫纳被提供了一个政治职位，担任新意大利政府的秘书长。这是对高度爱国的克雷莫纳的恰当致敬，然而他的数学研究如此引人入胜，以至于他拒绝接受这一政治职位。相反，他于1873年10月9日迁往罗马，再次由皇家法令任命为新成立的理工学院院长。此外，他还被任命为图解静力学教授。然而，如果他曾希望拒绝政治职位能让他继续数学研究，他很快发现，他的行政和教学职责实际上终结了他的研究。

1877年11月，克雷莫纳被任命为罗马大学高等数学讲席。然而政治压力使他觉得自己应当为新的意大利国家服务。1879年3月16日，他被任命为参议员，此时他的数学工作完全终止。他成为公共教育部长，并以参议院副议长的身份结束了自己的政治生涯。Greitzer在[1]中写道：-

1903年6月10日，克雷莫纳在离开病床去处理一些立法事务后，因心脏病发作去世。

克雷莫纳对意大利的几何学有很大影响。雅各布·施泰纳关于综合几何的许多证明由克雷莫纳修订和改进。在他整个职业生涯中几乎贯穿其所有工作的主题之一是射影几何。但是，Greitzer[1]指出：-

克雷莫纳在这一领域没有做出惊人的发现：但他确实推导出了射影相关图形的许多性质，并且他确实以旨在最简单地阐明和引出关系的方式向他的班级讲授这一主题。

事实上，克雷莫纳作为讲师享有很高的声誉[1]：-

克雷莫纳是一位出色的讲师：冷静、严谨，却又引人入胜，甚至令人兴奋。

White [11] 写道：-

……从他的著作中可以明显看出，正是对所选择科学的内在热爱促使他从事教学并编写那些使他的名字最为广为人知的书籍。

克雷莫纳在引用他人著作时极为公正，这在当时的许多数学家中并不太常见。我们再次引用White [11]：-

克雷莫纳本人对于搜寻他正在研究的问题上前人的工作，既认真又不知疲倦。

在他的一部著作中，经过两页的历史概述后，他加了一个脚注来致歉：-

……他此前对奥古斯特·费迪南德·莫比乌斯和Seydewitz的著作一无所知，也未能引用，而Schröter的论文现在引起了他的注意。

克雷莫纳有许多学生，他们后来对几何学做出了重大贡献，例如Bertini、朱塞佩·韦罗内塞和Guccia。

1879年，即他成为参议员的那一年，他获得了成为伦敦皇家学会通讯院士的科学荣誉。

Luigi Cremona was educated at the Ginnasio in Pavia. Luigi's father died when Luigi was 11 years old and this nearly prevented him from having a proper education. However, his stepfather supported him through school so that he was able to complete his school education and enter the University of Pavia.

Political events, however, were to have a major effect on Cremona's life and the first impact came from the revolution of 1848 which attempted to achieve a new and more liberal constitution. The Austrian government arrested opposition leaders of the revolution in Venice and Milan, and suppressed student demonstrations in Padua and Pavia. Cremona, all his life an ardent Italian nationalist, immediately joined the 'Free Italy' battalion.

Within a few days the Austrian army lost nearly all of Lombardy - Venetia and retreated. Sardinia - Piedmont declared war on Austria. After some initial successes by the Italians, the Austrian armies began to win victories. In Lombardy the Austrian reconquest of Brescia in March 1849, after 10 days of fighting, left Venice isolated. Cremona, by this time a sergeant, was with the troops defending Venice. The city resisted Austrian forces until 24 August 1849. The bravery of the defending forces had been such that the Austrian attackers allowed them to leave the city with honour. Cremona was able to return to Pavia.

Back in Pavia, Cremona discovered that his mother had died while he had been fighting to free Italy. Again he was financially unable to return to university without the support of his family and again his family rallied round and provided the necessary support. He entered the University of Pavia on 27 November 1849 to study for a degree in civil engineering. There he was taught by Bordoni, Casorati and Brioschi but he was most influenced by Brioschi. Cremona later wrote, see [6]:-

The years that I passed with Brioschi as pupil and later as colleague are a grand part of my life; in the first portion of these years I learned to love science and in the other how to transfer it to a large circle of auditors.

On 9 May 1853 Cremona was awarded his doctorate in civil engineering from the University of Pavia. By this time Camillo Benso di Cavour had become the head of the cabinet, and the unification of Italy had begun. But Austria still controlled the region and this made life hard for Cremona who had fought against the Austrian occupation. As Greitzer writes in [1]:-

His record of military service against Austrian rule prevented him from obtaining an official teaching post in the education system, so his first employment was as a private tutor to several families in Pavia.

Cremona was by this time engaged in mathematical research and his first paper appeared in March 1855. This must have had some effect in helping him to gain permission to teach physics on a temporary basis at the Ginnasio in Pavia where he had himself been trained for university. In May 1856 his second mathematical paper appeared and this, together with his good teaching, seems to have helped him secure the position of associate teacher at the Ginnasio in December 1856.

On 17 January 1857, Cremona was appointed a full teacher at the Ginnasio in Cremona. Cremona, the capital of the Cremona region of Lombardy was situated on the north bank of the Po River southeast of Milan. Cremona was to remain at the school there for three years and during this time he wrote a number of mathematical articles. They are not of great importance except that some of them were examining curves using projective methods, techniques which would be characteristic of Cremona's later important mathematics.

While the mathematician Cremona taught in the town of Cremona, political events were taking place which would have a large effect on his future. In July 1858 Cavour made an agreement with Napoleon III of France for his support of Piedmont. When Austria declared war on Piedmont on 26 April 1859, France honoured its alliance with Piedmont. After a number of battles the Austrians were in retreat and a treaty was signed at Villafranca by Napoleon III accepting the cession of Lombardy from Austria, which he then passed to Piedmont. Lombardy, liberated from Austrian rule, were quick to see that Cremona should no longer be held back for political reasons. On 28 November 1859 he was appointed as teacher at the Lycée St Alexandre in Milan.

With events moving quickly towards a unified Italy under Victor Emmanuel II as King, Cremona was appointed by Royal decree as an ordinary professor at the University of Bologna on 10 June 1860. The Kingdom of Italy was officially proclaimed on 17 March 1861, by a parliament sitting in Turin. Cremona was to remain in Bologna until October 1867.

In Cremona's Complete Works there appear 45 articles which he published while at Bologna. Sixteen of these articles are answers to questions posed in the Nouvelles Annales. There are eight book reviews and four historical articles intended to encourage research into geometry. Also included are Cremona's important work on transformations of plane curves, which were published during this period in 1863 and 1865. It was this work which won him the Steiner Prize for 1866, the prize being awarded jointly to Cremona and Rudolf Sturm. Also while at Bologna Cremona developed the theory of birational transformations, later known as Cremona transformations, and wrote a series of papers on twisted cubic surfaces.

White describes this period of Cremona's work in [11]:-

Chasles was the type on whom at first Cremona modelled his own work. The geometrical purism of von Staudt was a later influence. Hence it is natural to find metric foundations for geometric definitions instead of, or interchangeably with, projective. The apparatus of algebraic geometry is built upon polars, and these upon distances. The geometric method is principally a use of terms or descriptive relations instead of equations. And the beginnings of enumerative geometry are here...

Greitzer, writing in [1], describes the importance of Cremona transformations which he introduced in his Bologna period:-

Cremona transformations have been used for studying rational surfaces, for the resolution of singularities of plane and space curves, and for the study of elliptic integrals and Riemann surfaces. They are effective in the reduction of singularities of curves to double points with distinct tangents.

In October 1867, on Brioschi's recommendation, another Royal Decree was issued, this time appointing Cremona to the Polytechnic Institute of Milan. He received the title of Professor in 1872 while at Milan. Greitzer writes in [1]:-

The period in Milan, where he remained until 1873, was the time of Cremona's greatest creativity. He wrote articles on such diverse topics as twisted cubics, developable surfaces, the theory of conics, the theory of plane curves, third- and fourth-degree surfaces, statics and projective geometry.

Cremona's work in statics is of great importance and gives, in a clearer form, some theorems due to Maxwell. In a paper of 1872 Cremona took an idea of Maxwell's on forces in frame structures that had appeared in an engineering journal in 1867 and interpreted Maxwell's notion of reciprocal figures as duality in projective 3-space. These reciprocal figures, for example, have three forces in equilibrium in one figure represented by a triangle while in the reciprocal figure they are represented by three concurrent lines.

In 1873 Cremona was offered a political post as secretary general of the new Italian Government. This was a fitting tribute to the highly patriotic Cremona yet his mathematics researches were of such interest that he refused to accept the political post. Instead he moved to Rome on 9 October 1873, again appointed by Royal Decree, as director of the newly established Polytechnic School of Engineering. In addition he was appointed professor of graphic statics. However, if he had hoped that refusing the political post would leave him able to continue his mathematical research he soon found that his administration and teaching duties put an effective end to research.

In November 1877, Cremona was appointed to the chair of higher mathematics at the University of Rome. However political pressures made him feel that he should serve the new Italian State. On 16 March 1879 he was appointed a senator and at this point his mathematical work ended completely. He became Minister of Public Education and ended his political career as Vice-President of the Senate. Greitzer writes in [1]:-

On 10 June 1903, after leaving a sickbed to act on some legislation, Cremona succumbed to a heart attack.

Cremona had a large influence on geometry in Italy. Many of Steiner's proofs on synthetic geometry were revised and improved by Cremona. One of the themes which were present in almost all his work throughout his career was projective geometry. But, Greitzer [1], states:-

Cremona made no startling discoveries in this area: but he did derive many properties of projectively related figures, and he did present the subject to has classes in a manner calculated to clarify and bring out relationships most simply.

In fact Cremona had a fine reputation as a lecturer [1]:-

Cremona was an excellent lecturer: calm, rigorous, yet interesting and even exciting.

White [11] writes:-

... it is evident from his writings that it was the innate love of his chosen science that moved him to teaching and to the preparation of the books through which his name is most widely known.

Cremona was extremely fair in citing the works of others, something which was not too common among many mathematicians at that time. Again we quote White [11]:-

Cremona himself was conscientious and indefatigable in searching out the work of his predecessors upon matters that he himself was investigating.

In one of his works, after two pages of historical summary, he gives a footnote excusing:-

... his previous ignorance of, and failure to cite, the works of Möbius and Seydewitz to which the essay of Schröter had now directed his attention.

Cremona had many pupils who were to make major contributions to geometry, for example Bertini, Veronese and Guccia.

In 1879, the year he became a Senator, he received the scientific honour of becoming a corresponding member of the Royal Society of London.