约瑟·伯特兰Joseph Bertrand

约瑟·伯特兰的父亲是Alexandre Jacques François 伯特兰，母亲是Marie Caroline Belin。Alexandre曾在巴黎综合理工学院学习，在那里他与让-马里·迪阿梅尔成为朋友，后者后来娶了Alexandre Bertrand的妹妹。Alexandre是一位科普作家，但不幸英年早逝，在这一悲剧事件之后，九岁的伯特兰去与让-马里·迪阿梅尔及其妻子同住。当然，这一悲伤事件确实产生了有益的效果，即伯特兰得到了让-马里·迪阿梅尔的指导。在此阶段值得注意的是，伯特兰有一个妹妹Louise，她在1848年嫁给了夏尔·埃尔米特。

伯特兰在童年时就表现出非凡的才能，到九岁时他已理解代数和初等几何，还能流利地说拉丁语。两年后，当他十一岁时，他获准去巴黎综合理工学院听课，尽管当时他还不能正式入学，虽然他已经达到了所需的学业水平。他十六岁时获得第一个学位，次年因一篇关于热力学的学位论文而获得博士学位。1839年，也就是他获得博士学位的那一年，他正式进入巴黎综合理工学院，还发表了他的第一篇关于电的数学理论的论文。在那里学习两年后，他于1841年进入巴黎矿业学院，同年被任命为圣路易中学的初等数学教授。他在该中学担任此职直到1848年。

1842年发生了一件戏剧性的事件。伯特兰和他的兄弟与住在凡尔赛的Aclocque一家关系友好。他们经常乘火车从巴黎去拜访他们。在这次特别的拜访中，返程时火车撞毁，伯特兰受了重伤，鼻子被压碎，面部留下疤痕，这些疤痕伴随他一生。伯特兰于1844年与Louis Celine Aclocque结婚；他们的儿子Marcel Alexandre（1847年7月2日出生）、伯特兰 Désiré（1853年出生）和Léon Gratien（1858年出生）都在巴黎综合理工学院学习，Marcel后来成为一位著名的地质学家。

1843年，伯特兰发表了两篇关于Surfaces isothermes orthogonaleⓉ（等温正交曲面）的论文。1844年，他被任命为巴黎综合理工学院的分析学辅导教师，1856年雅克·夏尔·弗朗索瓦·施图姆去世后，他成为该学院的教授。1852年，他已被任命为亨利四世中学的特殊数学教授，同时也在巴黎高等师范学院任教。1862年，他成为法兰西公学院的分析学教授，接替于当年2月去世的让-巴蒂斯特·毕奥。1878年，伯特兰停止了在法兰西公学院的教学，但八年后，他又开始在那里任教。

1845年，伯特兰猜想对于每个$n > 3$，在$n$和$2n - 2$之间至少存在一个prime。这个猜想与莱昂哈德·欧拉一百年前提出的一个猜想类似，于1850年由巴夫尼提·列波维奇·切比雪夫证明。他对group theory做出了重大贡献，其文章Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renfermeⓉ（关于交换函数所含字母时函数可能取值数目的备忘录）于1845年3月提交给巴黎科学院。奥古斯丁·路易·柯西被要求对该工作进行报告，该工作研究了对称群中低指数的子群，这显然促使他重新研究置换群。伯特兰的重要论文最终发表在Journal de l'École Polytechnique上。

作为教科书作者而闻名，伯特兰出版的众多教科书中的前两部是Traité d'arithmetiqueⓉ（算术论）(1849)和Traité élémentaire d'algèbreⓉ（初等代数论）(1850)。这些书面向中学生，尽管后来的文本面向更高层次的学生，例如两卷本著作Traité de calcul différentiel et de calcul intégralⓉ（微积分与微积分论）(1864-70)、ThermodynamiqueⓉ（热力学）(1887)和Leçons sur la théorie mathématique de l'électricitéⓉ（电的数学理论讲义）(1890)。

伯特兰发表了许多关于微分几何和概率论的著作。他编辑了约瑟夫·拉格朗日的Mécanique analytique的一个版本，于1853年出版。1855年，他将卡尔·弗里德里希·高斯关于误差理论和最小二乘法的著作翻译成法文。他写了许多关于概率论和观测数据约化的笔记。他从大约1875年开始发表这些笔记，在1884年短暂中断三年后，他又开始发表关于概率的进一步笔记。

他的书Calcul des probabilitiésⓉ（概率计算）(1888)包含一个关于连续概率的悖论，现在称为Bertrand's paradox。它涉及一个任意圆的弦长于该圆内接等边三角形一边的概率。这个悖论的产生是因为“任意”一词没有被正确定义，见[12]。关于这项工作及伯特兰其他概率笔记的一般评论见[4]：-

1. Bertrand mentioned some of his predecessors （亚伯拉罕·棣莫弗，皮埃尔·西蒙·拉普拉斯，Bienaymé）, but did not refer to other scholars, notably to Chebyshev。
2. Bertrand's treatise contains mistakes and misprints. The conditions of many problems are stated carelessly and drawings are completely lacking. Verbal explanations, sometimes given instead of formulas, are irritating。
3. The treatise is badly organized。
4. Bertrand uses the term 'valeur probable' on a par with 'espérance mathématique'。
5. Bertrand's literary style is extremely attractive。

值得注意的是，伯特兰的这部论著深刻影响了儒勒·昂利·庞加莱的论著，后者同样名为Calcul des probabilitiés Ⓣ（《概率计算》），于1896年出版。事实上，儒勒·昂利·庞加莱在他的书中几乎只提到伯特兰。

当然，伯特兰在法国经历了政治困难的时期。1830年革命之后的十年里，国王屡遭挑战。然而从1840年起，有一段稳定的时期，伯特兰得以在此期间开始他的职业生涯。然而，1848年见证了君主制的推翻，在这场革命中，伯特兰担任国民自卫队上尉。经过多年相对稳定之后，普法战争于1870年7月爆发。法国遭受军事灾难，巴黎从1870年9月起被普鲁士军队围困四个月。对伯特兰及其家人来说，这是极其艰难的时期。法国战败后，巴黎人民于1871年3月成立了巴黎公社。对巴黎公社的镇压发生在5月底，经过一周的巷战，伯特兰的房子被烧毁，他的许多手稿在火灾中丢失，包括他打算写的Traité de calcul différentiel et de calcul intégralⓉ（微分与积分学论著）第三卷，他从未重写。伯特兰也在火灾中丢失了ThermodynamiqueⓉ（热力学）的手稿，但后来他重写了这部著作。家园被毁后，伯特兰搬到塞夫勒，后来又搬到维罗夫莱。

事实上，伯特兰的家对他特别重要，因为它是活跃的知识群体的中心。他的家庭成员增多，包括埃米尔·皮卡，他娶了夏尔·埃尔米特的一个女儿和伯特兰的妹妹路易丝。1881年，保尔·阿佩尔娶了阿梅莉，她是伯特兰和夏尔·埃尔米特的侄女，也是埃米尔·皮卡的表亲。让我们回来注意1870年法国军事失败的其他影响。它导致法国进行深刻的反思，特别是有人质疑科学教育，尤其是巴黎综合理工学院的课程，是否应负责任。伯特兰此时已是法国领先的数学家之一，不得不应对后果，因为有人质疑所教授的数学是否过于理论化，因而没有为法国顶尖年轻人做好军事准备。

伯特兰于1856年被任命为巴黎科学院成员，并从1874年起担任其终身秘书直至去世。他被授予荣誉军团大军官勋章。

迪尔克·扬·斯特勒伊克在[1]中总结了伯特兰在数学发展中的地位：-

1895年，他的学生们赠予他一枚奖章，以纪念他在巴黎综合理工学院执教五十年。然而，伯特兰著作的影响，几乎无法与他同时代的若干人及其学生相比。为避免被判定为昙花一现，必须将其置于十九世纪巴黎的背景中，以及伯特兰辉煌的学术生涯、显赫的社会地位，还有学生们给予他的爱与尊敬之中来审视。

Joseph Bertrand's father was Alexandre Jacques François Bertrand and his mother was Marie Caroline Belin. Alexandre had studied at the École Polytechnique where he had become friends with Jean-Marie Duhamel who later married Alexandre Bertrand's sister. Alexandre was a writer of popular science books but sadly he died young and after this tragic event Joseph, who was nine years old, went to live with Duhamel and his wife. Of course this sad event did have the beneficial effect that Joseph was guided by Duhamel. It is worth noting at this stage that Joseph had a sister Louise who married Hermite in 1848.

Joseph showed remarkable talents as a child and by the age of nine he understood algebra and elementary geometry, as well as being able to speak Latin fluently. Two years later, when he was eleven, he was given permission to attend lectures at the École Polytechnique although he could not officially enter at this time despite having gained the required academic level. He was awarded his first degree at the age sixteen and in following year he was awarded his doctorate for a thesis on thermodynamics. In 1839, the year he received his doctorate, he officially entered the École Polytechnique and also published his first paper on the mathematical theory of electricity. After two years of study there he entered the École des Mines in 1841 being appointed professor of elementary mathematics at the Lycée Saint-Louis in the same year. He held this post at the Lycée until 1848.

A dramatic event happened in 1842. Bertrand and his brother were friendly with the Aclocque family who lived in Versailles. They visited them frequently, travelling by train from Paris. On this particular occasion the train crashed on the return journey and Bertrand was badly injured, suffering a crushed nose and facial scars which he retained throughout his life. Bertrand married Louis Celine Aclocque in 1844; their sons Marcel Alexandre (born 2 July 1847), Joseph Désiré (born 1853) and Léon Gratien (born 1858) all studied at the École Polytechnique, and Marcel went on to become a famous geologist.

In 1843 Bertrand published two memoirs on Surfaces isothermes orthogonale Ⓣ. He was appointed répétiteur d'analyse at the École Polytechnique in 1844, being made professor there in 1856 following the death of François Sturm. He had been appointed professor of special mathematics at the Lycée Henry IV in 1852, also teaching at the École Normale Supérieure. In 1862, he became professor of analysis at the Collège de France, succeeding Biot who died in February of that year. In 1878 Bertrand stopped teaching at the Collège de France but, eight years later, he began teaching there again.

In 1845 Bertrand conjectured that there is at least one prime between $n$ and $2n - 2$ for every $n > 3$. This conjecture, similar to one stated by Euler one hundred years earlier, was proved by Chebyshev in 1850. He made a major contribution to group theory with his article Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme Ⓣ which he submitted to the Paris Academy in March 1845. Cauchy was asked to report on the work, which studied subgroups of low index in the symmetric group, and it clearly led him to return to study permutation groups himself. Bertrand's important paper was eventually published in the Journal de l'École Polytechnique.

Famed as an author of textbooks, the first two of many published by Bertrand were Traité d'arithmetique Ⓣ (1849) and Traité élémentaire d'algèbre Ⓣ (1850). These were aimed at pupils at secondary school, although later texts were aimed at students at more advanced levels, for example the two volume work Traité de calcul différentiel et de calcul intégral Ⓣ (1864-70), Thermodynamique Ⓣ (1887), and Leçons sur la théorie mathématique de l'électricité Ⓣ (1890).

Bertrand published many works on differential geometry and on probability theory. He edited an edition of Lagrange's Mécanique analytique which was published in 1853. In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the theory of probability and on the reduction of data from observations. He published these notes starting around 1875 and, after a short break of three years from 1884, he began publishing further notes on probability.

His book Calcul des probabilitiés Ⓣ (1888) contains a paradox on continuous probabilities now known as Bertrand's paradox. It concerns the probability that an arbitrary chord of a circle is longer than a side of an equilateral triangle inscribed in the circle. The paradox comes about since the word "arbitrary" is not properly defined, see [12]. General comments on this work and Bertrand's other notes on probability are made in [4]:-

1. Bertrand mentioned some of his predecessors (De Moivre, Laplace, Bienaymé), but did not refer to other scholars, notably to Chebyshev.
2. Bertrand's treatise contains mistakes and misprints. The conditions of many problems are stated carelessly and drawings are completely lacking. Verbal explanations, sometimes given instead of formulas, are irritating.
3. The treatise is badly organized.
4. Bertrand uses the term 'valeur probable' on a par with 'espérance mathématique'.
5. Bertrand's literary style is extremely attractive.

It is worth noting this treatise by Bertrand heavily influenced Poincaré's treatise also called Calcul des probabilitiés Ⓣ which he published in 1896. Indeed, Poincaré refers to almost nobody but Bertrand in his book.

Of course Bertrand lived through a time of political difficulties in France. The revolution of 1830 was followed by a decade marked by repeated challenges to the King. However from 1840 there was a stable period during which Bertrand was able to begin his career. The year 1848, however, saw the overthrow of the monarchy and during this revolution Bertrand served as a Captain in the National Guard. After years of comparative stability, the Franco-Prussian war broke out in July 1870. The French suffered a military disaster and Paris was under siege from the Prussian armies for four months from September 1870. These were extremely difficult times for Bertrand and his family. Following the French defeat the people of Paris formed the Commune of Paris in March 1871. The repression of the Paris Commune took place towards the end of May during a week of street fighting which saw Bertrand's house burned and many of his manuscripts were lost in the fire including an intended third volume of Traité de calcul différentiel et de calcul intégral Ⓣ which he never rewrote. Bertrand also lost his manuscript of Thermodynamique Ⓣ in the fire but he later rewrote the work. With his home destroyed Bertrand moved to Sèvres and later to Viroflay.

In fact Bertrand's home was particularly important to him for it provided the centre of a vigorous intellectual grouping. His family grew to include Émile Picard who married a daughter of Hermite and Bertrand's sister Louise. In 1881 Paul Appell married Amelie, a niece of Bertrand and of Hermite and a cousin of Émile Picard. Let us return to note other effects of the French military defeat of 1870. It led to considerable soul searching in France and, in particular, it was questioned whether science education, particularly the courses at the École Polytechnique, were responsible. Bertrand, by this time one of the leading mathematicians in France, had to deal with the consequences as it was questioned whether the mathematics being taught was too theoretical, and so not preparing the leading young Frenchmen for the military.

Bertrand was appointed a member of the Paris Academy of Sciences in 1856 and he served as its permanent secretary from 1874 to the end of his life. He was made a grand-officier of the Légion d'Honneur.

Struik, in [1], sums up Bertrand's place in the development of mathematics:-

In 1895 his pupils gave him a medal in commemoration of his fifty years of teaching at the École Polytechnique. The influence of Bertrand's work, however, is hardly comparable to that of several of his contemporaries and pupils. Lest it be judged ephemeral, it must be viewed in the context of nineteenth-century Paris and of Bertrand's brilliant academic career, his exalted social position, and the love and respect given him by his pupils.