乌惹内·查尔斯·卡塔兰Eugène Charles Catalan

乌惹内·查尔斯·卡塔兰出生于布鲁日，尽管我们将其列为比利时，但在他出生时它实际上属于法国。比利时在1793年之前一直是奥地利尼德兰的一部分，但从1799年到1814年处于拿破仑执政府统治下并属于法兰西帝国的一部分。拿破仑战败后，列强决心不让比利时落入法国之手。荷兰王国于1815年6月由维也纳会议确认，但它的建立是为了欧洲的便利，而没有考虑比利时人和荷兰人的意愿。奥兰治亲王威廉于1815年3月16日登基，称威廉一世，并于9月27日加冕。因此卡塔兰出生于法国，因此理所当然地认为自己是法国人。然而他出生的城镇在他满一岁之前就成为了荷兰王国的一部分。他的出生证明上登记的名字是卡塔兰 卡塔兰 Bardin。他的母亲让娜·巴尔丹是让·巴尔丹十七岁的女儿。她出生于博讷，在儿子出生时与父母住在布鲁日。她未婚，她的儿子以她的姓氏巴尔丹登记。尽管让·巴尔丹在官方记录中显示为书商，但实际上他是一个从博讷搬到布鲁日的廉价书籍流动推销员。卡塔兰的母亲在幼子成长的岁月里靠做裁缝挣些钱。虽然起初她与父母同住，但她最终离开了他们的家，并于1820年在布鲁塞尔待了几个月。卡塔兰的父亲约瑟夫·维克多·艾蒂安·卡塔兰原籍巴黎。他于1821年与让娜·巴尔丹结婚，并于同年3月1日承认他是她将近七岁儿子的父亲。约瑟夫是一名珠宝商，但实际上他通过出售画作和香水等各种手段赚钱。1822年5月，约瑟夫和让娜·卡塔兰带着儿子住在里尔。卡塔兰肯定受过一些教育，到十岁时就能写出出色的法语。此时，即1824年，他成为一名珠宝学徒。然而，他只坚持了三个月的学徒期，就因为缺乏能力而放弃了。

目前不清楚这家人究竟何时搬到巴黎，但这一定发生在1825年左右。卡塔兰很快爱上了这座城市，他的父母允许他在街上游荡。他观看了1826年5月3日穿越巴黎的禧年游行，当时路易十六纪念碑的奠基石被安放并奉献，他在一封信中详细描述了这一情景。尽管他当时只有十二岁，卡塔兰对君主制的厌恶已清晰而有力地表达出来。在他余生中，他都认为自己是巴黎人。让娜·卡塔兰的确切去世时间不清楚，但一定是在这家人搬到巴黎后不久。Joseph Catalan此时已成为一名建筑师，于1828年8月2日在圣尼古拉·德·尚教堂与阿德莱德·维特里结婚。卡塔兰被安排追随父亲成为一名建筑师，尽管他没有上过中学，但他进入了皇家免费绘画与机械艺术数学学校。他在这所学校非常快乐，尤其对让·保罗·杜利奥（1788-1834）和让-巴蒂斯特-奥梅尔·拉维（1771-1836）两位教授印象深刻。除了在免费绘画学校上课外，卡塔兰还在美术学院上课。他一直是免费绘画学校的学生直到1831年，但在1829年，经过一次竞赛后，他被任命为该校同学的几何学教师。他一直担任这个教职直到1833年，但在学校期间，他变得更加政治活跃，受到1830年革命的激励。这场革命在整个欧洲产生了反响，特别是在比利时，对国王威廉一世统治的不满导致了反对君主制的行动，最终在当年早些时候巴黎七月革命之后，于1830年8月至9月爆发了“比利时革命”。1831年1月20日，伦敦的一次国际会议承认比利时为独立国家。

在免费绘画学校，卡塔兰由Louis Lefébure de Fourcy授课，后者于1826年被任命为巴黎理工学院的入学考试考官。他鼓励卡塔兰，后者展现出卓越的数学才能，准备参加这所著名大学的入学考试。他在沃吉拉尔街的德勒斯先生开办的机构学习。他还参加了路易·安托万 Delisle（Éléments de trigonométrie rectiligne et sphérique Ⓣ（直线和球面三角学基础）的作者）在圣路易中学附近的圣米歇尔大道上讲授的数学课程，以及同一大道上索邦大学的Louis Francoeur的课程。他在1833年夏天赢得了特别数学竞赛的第一名。卡塔兰没有继续学习，而是于当年8月和9月回到布鲁日，这是他自十一岁以来第一次回到那个城市。他回到那里是为了庆祝他祖母9月20日的生日。在参加了巴黎理工学院的入学考试后，他于1833年11月被录取，在录取学生名单中排名第53位，尽管如果不是因为学业中断，他几乎肯定会排名更高。在巴黎理工学院，卡塔兰参加了约瑟夫·刘维尔和加布里尔·拉梅讲授的数学课程。然而，他也学习了其他科目，包括法国文学和历史，阅读了伏尔泰、莫里哀和拉布吕耶尔等作家的作品。

巴黎理工学院的学生们深深卷入了当时法国正在发生的政治事件。那是一个政治极度不稳定的时期，1834年4月，在通过一项限制共和人权协会活动的法律后，巴黎发生了严重的骚乱。骚乱被军队残酷镇压。当局试图确保学生不卷入其中，并开除了一些宣扬共和同情的学生。尽管卡塔兰有强烈的共和信念，但他避免了麻烦。1834年11月，他完成了第一年的学习，在145名学生中排名第49位。他本应继续第二年的学习，但他的父亲收到一封日期为12月15日的信，通知他儿子已被巴黎理工学院开除并正在回家。卡塔兰并不是唯一受到这种惩罚的人，因为看来他整个年级都遭遇了同样的命运。事实上，在这个被迫的假期中，卡塔兰抓住机会与比他大两岁的夏洛特·奥古斯丁·蕾妮·佩兰（被称为欧仁妮）订婚。她出生在里尔，卡塔兰曾在那里度过几年。他们于1835年1月2日订婚，并共度余生。

卡塔兰不得不为一次“服从行为”道歉，才被允许于1835年1月中旬返回巴黎理工学院。他一定觉得这很难做，但他知道如果他想完成学业，就必须做这样的事情。他努力学习，于1835年夏天毕业，在140名有资格从事公共服务的学生中排名第16位。他在有资格进入桥梁公路学校的学生中排名第13位，但决定这不是他想走的路。他被任命为马恩河畔沙隆艺术与工艺学校的教授，于1835年秋天就职。1836年1月4日，卡塔兰的父亲去世，4月8日他的祖父让·巴尔丹去世，但他于4月28日与欧仁妮·佩兰结婚。尽管他不是虔诚的基督徒，他们还是在巴黎的圣厄斯塔什教堂举行了婚礼。他们的第一个孩子，女儿玛丽·阿德莱德，出生在马恩河畔沙隆。

约瑟夫·刘维尔于1836年开始出版Journal de Mathématiques Pures et Appliquées，卡塔兰的一篇论文Solution d'un problème de Probabilité relatif au jeu de rencontre Ⓣ（关于与相遇游戏相关的概率问题的解）于1837年发表在第二卷上。在同一期刊上，他于1838年发表了两篇论文：Note sur un Problème de combinaisons Ⓣ（关于组合问题的注记）和Note sur une Équation aux différences finies Ⓣ（关于有限差分方程的注记）。其中第二篇包含了“卡塔兰数”，它出现在通过不相交对角线将多边形分割成三角形的问题的解中。卡塔兰的四篇论文于1839年发表在第四卷上：Note sur la Théorie des Nombres Ⓣ（关于数论的注记）；Solution nouvelle de cette question: Un polygone étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales? Ⓣ（问题的新解：给定一个多边形，我们可以用对角线以多少种方式将其三角剖分？）；Addition à la Note sur une Équation aux différences finies Ⓣ（关于有限差分方程注记的补充）；以及Mémoire sur la réduction d'une classe d'intégrales multiples Ⓣ（关于一类多重积分简化的回忆录）。

你可以在THIS LINK上看到更多关于卡塔兰数的信息。

卡塔兰渴望回到巴黎，他申请了École Gratuite de Dessin的教授职位，该职位因其前任教授Jean-Baptiste-Omer Lavit于1836年去世而空缺。他未能成功。1837年1月，他写信给约瑟夫·刘维尔，后者回信鼓励他在复活节假期来巴黎与他讨论未来。他建议卡塔兰需要进一步的资格，如学士学位，以便取得成功的事业。那年晚些时候，卡塔兰辞去了Châlons-sur-Marne的职位，回到了巴黎。与雅克·夏尔·弗朗索瓦·施图姆和约瑟夫·刘维尔一起，他在索邦附近创办了École Sainte-Barbe。这所学校旨在培养学生进入École Polytechnique，于1838年开学。同年11月20日，他被任命为École Polytechnique的描述几何助理导师（répétiteur），协助教授该主题的卡塔兰 François 安托万 Leroy。卡塔兰在1840年讲授了描述几何课程的一部分，那一年他的第二个女儿Fanny出生。1839年，他被任命为École Polytechnique的副考官。一回到巴黎，除了这些其他任务外，卡塔兰还采纳了约瑟夫·刘维尔关于学士学位的建议，并开始努力学习。他于1839年获得双学士学位，并继续学习，在次年获得了“数学科学执照”。1841年，他因主要论文《力学》Attraction d'un ellipsoïde homogène sur un point extérieur ou sur un point intérieur Ⓣ（均匀椭球对外部点或内部点的吸引）而获得数学博士学位，并在天文学方面提交了另一篇论文Sur le mouvement des étoiles doubles Ⓣ（关于双星的运动）。1845年，他获得了“物理科学执照”。

奥古斯丁·路易·柯西是一位狂热的保皇党人，1830年离开巴黎，由于未能宣誓效忠新政权，他在那里失去了所有职位。然而，他于1838年10月返回巴黎，重新获得了在Académie des Sciences的职位，但没有恢复教学职位，因为他仍然拒绝宣誓效忠。1839年7月，奥古斯丁·路易·柯西邀请卡塔兰和约翰·彼得·古斯塔夫·勒热纳·狄利克雷共进晚餐。尽管这对年轻的卡塔兰来说是一种荣誉，但鉴于他们的政治观点处于光谱的两端，这永远不会发展成亲密的友谊。然而，卡塔兰仍然得到约瑟夫·刘维尔的支持，后者提名他加入哲学学会。在1839年未能成功后，卡塔兰于1840年5月当选为该学会会员，并开始在该学会的期刊上发表文章。1841年，他在Sur la transformation des variables dans les integrales multiples Ⓣ（《论多重积分中变量的变换》）中给出了n维积分的一般变量变换定理，该文发表在布鲁塞尔皇家科学与文学科学院的Mémoires Couronnés上。1843年，他首次在奥古斯都·利奥波德·克雷勒的Journal für die reine und angewandte Mathematik上发表文章。其中包含“卡塔兰猜想”：-

我恳求您，先生，请在您的期刊上宣布以下我认为真实但尚未完全证明的定理；也许其他人会更成功。两个连续整数，除了8和9之外，不可能是连续幂；换句话说，方程$x^{m} - y^{n} = 1$中未知数为正整数，只允许一个解。

Piotr Tchihatchef 是一位俄罗斯地理学家和地质学家，1842-43 年冬天他在巴黎度过。他找到 Louis Francoeur，请求辅导数学，Francoeur 建议他去找 卡塔兰。这对 卡塔兰 来说实际上是幸运的，因为 Tchihatchef 受俄罗斯数学家 巴夫尼提·列波维奇·切比雪夫 之托，要提交一篇论文在巴黎发表。Tchihatchef 把论文给 卡塔兰 看了，卡塔兰 开始与 巴夫尼提·列波维奇·切比雪夫 通信；这一通信持续了五十年。巴夫尼提·列波维奇·切比雪夫 的论文发表在 约瑟夫·刘维尔 的 Journal de mathématiques pures et appliquées 上，时间是 1843 年 5 月。同一期刊随后刊登了 卡塔兰 的一篇短文，证明了 巴夫尼提·列波维奇·切比雪夫 论文中未加证明陈述的一个多重积分变换公式。1844 年 1 月 6 日，卡塔兰 被任命为 Société Philomatique 的秘书。

鉴于他的成就，卡塔兰的大学教学生涯本应有所进展，但任何顺利发展的前景都因他持续的政治活动和强烈的左翼共和主义观点而受损。1844年11月，他在École Polytechnique的导师职位评选中被理事会排名第一，但Ossian Bonnet被任命为该职位，卡塔兰仍担任助理导师。阿道夫·凯特勒，这位比利时科学家，自1841年他在Academie Royale des Sciences et Belles-Lettres de Bruxelles的Mémoires couronnés Ⓣ（当前回忆录）中与他通信以来，鼓励他因在巴黎缺乏进展而寻求比利时的大学职位。1844年12月，他的职业生涯再次受挫，当时Francoeur表示希望卡塔兰作为他在索邦的副手，但他被皇家理事会拒绝。相反，他们任命了约瑟·伯特兰担任该职位。1846年，卡塔兰参加了Concours d'agrégation（成为大学教师的竞赛）并获得第一名。然而，他没有得到任何任命。卡塔兰为他强烈公开表达的共和主义观点付出了高昂的代价。

1846 年 9 月，卡塔兰 写信给 阿道夫·凯特勒，表示他开始后悔留在巴黎，而没有听从他的建议搬到比利时。然而，11 月他被任命负责查理曼学院的高等数学教学，但随后卷入了 1848 年法国的政治动乱。这场革命在没有任何非常明确原因的情况下爆发，尽管 1846 年以来的粮食短缺造成了大量经济困难和不满，共和派利用了这些。导火索似乎是政府取消了共和派为 1848 年 2 月安排的一场大型宴会。卡塔兰 在动乱中发挥了极其积极的作用，带领一群工人，在一群巴黎综合理工学院的学生之后立即进入市政厅。革命导致了第二共和国，选民选择路易-拿破仑·波拿巴担任总统。这与 卡塔兰 的共和主义观点非常契合，他报名参加国民议会选举。以下是他宣言的摘录：-

我从小就被教育要蔑视国王，我欣喜若狂地迎接 卡塔兰 X 的倒台。我屈服了，但带着愤怒抗议，屈服于我们当时强加给自己的那些老谋深算者和受骗者。我自己也曾一时受骗，但当正直的 Dupont（德·勒尔）退却时，我就不再受骗了，而且在 1830 年底之前，我已成为一名共和主义者。从那时起，我没有一天不贡献我的愿望、我的言语和行动，以推翻那个法国支持得太久的人！

你可以在 THIS LINK 看到完整的宣言。

他的职业生涯似乎又好转了，当他在1849年被任命到圣路易中学。然而，在巴黎综合理工学院，他继续感到失望。1848年任命了两名入学考官，尽管卡塔兰在提供给部的四人名单上，夏尔·埃尔米特和约瑟夫·阿尔弗雷德·塞雷被任命，尼尔斯·阿贝尔 Transon 和Pierre Bonnet作为他们的副手。尽管 Transon (1805-1876)，像卡塔兰一样，是一名社会主义者，但他在三十岁后很少参与政治。

1850年6月，一个委员会由于尔班·勒维耶主持，以重组巴黎综合理工学院的教学。让-维克托·彭赛列和让-马里·迪阿梅尔是委员会中的数学家。它在11月报告，卡塔兰既不喜欢提出的学术变革，特别是纯数学的大幅减少，也不喜欢导师将承担更多工作的事实。他在那个月晚些时候辞去了助理导师的职位。在他的辞职信中，他指出他担任助理导师已有十二年，并解释说他由于在圣路易中学的职责，无法承担所需的额外教学。卡塔兰不是唯一一个在于尔班·勒维耶委员会报告后离开巴黎综合理工学院的人；米歇尔·沙勒、约瑟夫·刘维尔和雅克·夏尔·弗朗索瓦·施图姆也立即辞职以示抗议。

第二共和国只持续了三年。1851年12月2日发生政变，路易-拿破仑·波拿巴掌握绝对权力并解散国民议会。整整一年后，他成为皇帝，称号拿破仑三世。这对卡塔兰不利，他讨厌波拿巴家族就像讨厌国王一样。他拒绝宣誓效忠，因此失去了工作。在接下来的几年里，他住在巴黎，教授数学，但没有正式工作。他曾在约瑟夫·刘维尔的Journal de mathématiques pures et appliquées上发表了许多文章，但在1854年他停止在那里发表，更喜欢在Académie des Sciences的Comptes rendus上发表。这几乎肯定是卡塔兰当选Académie des Sciences策略的一部分。例如，他发表了Sur des surfaces dont les rayons de courbure en chaque point sont égaux et de signes contraires Ⓣ（关于每点曲率半径相等且相反的曲面）(1855)，其中研究了与“卡塔兰的最小曲线”相关的“卡塔兰最小曲面”。他还在Comptes rendus上发表：Note de M Catalan à l'occasion d'un théorème de M Serret Ⓣ（M 卡塔兰关于M 约瑟夫·阿尔弗雷德·塞雷定理的注记）(1856)，Sur le calcul de la latitude par la méthode de M Babinet Ⓣ（关于用M Babinet方法计算纬度）(1856)，Sur quelques points de la théorie des séries Ⓣ（关于级数理论的某些点）(1856)，Sur un cas particulier de la formule du binôme Ⓣ（关于二项式定理的一个特例）(1857)，以及Sur la théorie des développées Ⓣ（关于展开理论）(1857)。到1860年，约瑟·伯特兰、夏尔·埃尔米特和约瑟夫·阿尔弗雷德·塞雷都已当选科学院。当约瑟夫·阿尔弗雷德·塞雷在1860年当选时，卡塔兰在考虑名单上，但仅排名并列第四，落后于约瑟夫·阿尔弗雷德·塞雷、Bonnet和维克托·皮瑟。Academy为1861年奖项提出了一个关于多面体的问题，卡塔兰提交了参赛作品。他被要求在1862年提交他的回忆录的新版本，由约瑟·伯特兰、米歇尔·沙勒、约瑟夫·刘维尔和约瑟夫·阿尔弗雷德·塞雷组成的评审团审议。米歇尔·沙勒和约瑟夫·刘维尔建议他获得奖项，但其他人提议不应颁发该奖项。

卡塔兰发现的多面体现在被称为Catalan solids。你可以在THIS LINK看到更多关于它们的信息。

1859年，卡塔兰曾试图说服公共教育部任命他为巴黎某所中学的数学教授。他没有成功。1865年1月，在13年没有固定职位之后，他被任命为列日大学的数学讲席。由于卡塔兰出生于布鲁日，人们可能会以为他在列日就职时会觉得自己回到了故乡。然而，卡塔兰始终认为自己是法国人，此前曾为恢复法国国籍付出过相当大的努力。他担任此讲席直到1884年退休，之后继续住在列日，直到十年后去世。然而，悲剧在卡塔兰一家抵达列日后不久便降临了。他带着妻子、两个女儿和岳母Louise Perin来到那里，后者在他们婚后于巴黎生活的整个期间一直与他们同住。一些传记称卡塔兰还有一个儿子，在全家搬到列日之前死于巴黎，但我们无法证实这是否属实。他们的长女Marie在全家搬到列日之前就已患病，并于1865年5月在那里去世。到这一年年底，他的岳母以79岁高龄去世，而他的次女Fanny也病倒了。卡塔兰的妻子把Fanny带回巴黎，希望这能使她的健康好转，但事与愿违，Fanny于1866年春天去世。

1870-71年的普法战争没有影响中立的比利时，但卡塔兰由于对法国的深厚热爱，对法国人遭受普鲁士人的羞辱深感悲痛。然而，由此导致的拿破仑倒台和法兰西第三共和国的建立无疑使他感到高兴。卡塔兰从未尊重过拿破仑·波拿巴于1802年创立的荣誉军团勋章。然而，当第三共和国于1887年12月30日授予他荣誉军团骑士十字勋章时，他对法国如今成为共和国所感到的喜悦超过了他对“波拿巴的荣誉”的反感，他欣然接受了这一荣誉。然而，Académie des Sciences在19世纪70年代曾长时间讨论选举卡塔兰，但未能实现。

Mélanges mathématiques Ⓣ（数学杂集）由列日皇家科学学会于1868年出版。它收录了卡塔兰的69篇论文，从Sur les combinaisons avec répétition Ⓣ（论有重复的组合）（1838年）开始，到他的论文Démonstration d'une formule de Poisson Ⓣ（一个西莫恩·德尼·泊松公式的证明）（1867年2月）结束。1887-88年，列日皇家科学学会出版了Mélanges mathématiques的三卷更新版。这部作品收录了卡塔兰的299篇论文，最后一篇是Sur une application du théorème de Bayes, faite par Laplace Ⓣ（论皮埃尔·西蒙·拉普拉斯对托马斯·贝叶斯定理的一个应用）（1888年8月）。第二卷包含卡塔兰的退休演讲，第三卷还列出了卡塔兰的出版物；该列表包含406个条目。

卡塔兰于1865年12月15日当选为Belgian Académie Royale des Sciences, des Lettres et des Beaux Arts。他还当选为图卢兹科学院和里尔科学学会。他是圣彼得堡科学院、Turin Academy of Sciences、Accademia Pontificia dei Nuovi Lincei、阿姆斯特丹数学学会、日内瓦国家研究院、勒阿弗尔多元研究学会和马恩省农业学会的通讯院士。他是列日科学学会、法国数学学会以及我们上文提到的巴黎博学学会的成员。比利时政府于1879年授予卡塔兰利奥波德骑士十字勋章以表彰他。卡塔兰于1884年6月1日退休，成为荣休教授。同年12月7日，许多朋友（包括专程从圣彼得堡赶来的巴夫尼提·列波维奇·切比雪夫）、同事和以前的学生参加了庆祝活动，以纪念他的退休。他获赠了一幅由一些朋友委托、Émile Delpérée绘制的肖像画。在这幅肖像画中，卡塔兰自豪地佩戴着利奥波德骑士十字勋章。1890年，比利时政府将卡塔兰晋升为利奥波德军官勋章。

你可以在THIS LINK看到卡塔兰退休时的演讲。

卡塔兰写了几部非常受欢迎的教材，许多都出了好几版。他写了：Elements de géométrie Ⓣ（几何学基础）（1843年）；两卷本著作Traité élémentaire de géométrie descriptive Ⓣ（初等画法几何论）（1850-52年），共出了5版，最后一版于1881年问世；Théorèmes et problèmes de géométrie élémentaire Ⓣ（定理与初等几何问题）（1852年），共出了6版，最后一版于1879年问世；Nouveau manuel des aspirants au baccalauréat ès sciences Ⓣ（理学士考生新手册）（1852年），共出了12版；Solutions des problèmes de mathématique et de physique donnés à la Sorbonne dans les compositions du baccalauréat ès sciences Ⓣ（索邦大学理学士考试中给出的数学与物理问题解答）（1855-56年）；两卷本Manuel des candidats à l'École Polytechnique Ⓣ（巴黎综合理工学院考生手册）（1857-58年）；Notions d'astronomie Ⓣ（天文学基础）（1860年），共出了6版；Traité élémentaire des séries Ⓣ（级数初等论）（1860年）；Histoire d'un concours Ⓣ（一场竞赛的历史）（1865年），第二版于1867年出版；以及Cours d'analyse de l'université de Liège Ⓣ（列日大学分析教程）（1870年），第二版于1880年出版。

1894年，巴黎综合理工学院计划庆祝其建校一百周年。卡塔兰，仍在世的最年长的前学生之一，计划最后一次访问巴黎以参加庆祝活动。然而，这未能实现。他的妻子在1894年2月初生病，然后，在2月9日，卡塔兰倒下并被送往医院，患有肺炎。他的妻子欧仁妮于2月11日去世，而卡塔兰本人三天后去世。

你可以在THIS LINK看到列日大学校长对卡塔兰的赞赏。

Eugène Catalan was born in Bruges and although we have given this as being in Belgium, it actually was in France at the time of his birth. Belgium had been part of the Austrian Netherlands until 1793 but was under the Napoleonic consulate and part of the French empire from 1799 to 1814. After the defeat of Napoleon, the Allied powers were determined not to leave Belgium in the possession of France. The Kingdom of the Netherlands was confirmed by the Congress of Vienna in June 1815 but it was established for the convenience of Europe without considering the desires of the Belgians and the Dutch. Prince William of Orange took the throne on 16 March 1815, taking the title William I, and he was crowned on 27 September. Catalan was therefore born in France, and so rightly considered himself French. However the town of his birth became part of the Kingdom of the Netherlands before he was one year old. His name was registered on his birth certificate as Eugène Charles Bardin. His mother, Jeanne Bardin, was the seventeen year old daughter of Jean Bardin. She had been born in Beaune and was living with her parents in Bruges at the time her son was born. She was unmarried, and her son was registered under her name of Bardin. Although Jean Bardin appears on official records as a bookseller, in reality he was a travelling salesman of cheap books who had moved from Beaune to Bruges. Eugène's mother earned some money as a dressmaker in the years when her young son was growing up. Although at first she lived with her parents, she eventually left their home and in 1820 she spent several months in Brussels. Eugène's father, Joseph Victor Étienne Catalan, was originally from Paris. He married Jeanne Bardin in 1821 and, on 1 March of that year, acknowledged that he was the father of her son who was nearly seven years old. Joseph was a jeweller but, in reality, he made money with various schemes such as selling pictures and perfume. In May 1822, Joseph and Jeanne Catalan were living in Lille with their son. Eugène certainly had had some education and was able to write excellent French by the age of ten. At this time, in 1824, he became an apprentice jeweller. However, he only kept up this apprenticeship for three months, giving up because he lacked ability.

It is unclear just when the family moved to Paris, but this must have occurred around 1825. Eugène rapidly fell in love with the city and his parents allowed him to roam the streets. He watched the jubilee procession through Paris which took place on 3 May 1826, when the corner-stone of a monument to Louis XVI was laid and dedicated, and he described it in detail in a letter. Although he was only twelve years old at the time, Eugène's dislike of the monarchy is clearly and forcibly expressed. For the rest of his life he considered himself a Parisian. It is unclear exactly when Jeanne Catalan died, but it must have been soon after the family moved to Paris. Joseph Catalan, who had now become an architect, married Adélaïde Vitry on 2 August 1828 in the church of Saint Nicolas des Champs. Eugène was set to follow his father and become an architect and, although he had notattended a lycée, he entered the École Royale Gratuite de Dessin et de Mathématiques en Faveur des Arts Mécaniques. He was very happy at this school, and was particularly impressed by professors Jean Paul Douliot (1788-1834) and Jean-Baptiste-Omer Lavit (1771-1836). In addition to attending courses at the École Gratuite de Dessin, Catalan attended courses at the École des Beaux-Arts. He remained a pupil at the École Gratuite de Dessin until 1831 but, in 1829, after a competition, he was appointed to teach geometry to his fellow pupils at the school. He retained this teaching post until 1833, but while he was at the school, he became more politically active, spurred on by the revolution of 1830. This revolution had repercussions across Europe, in particular in Belgium where dissatisfaction with King William I's rule led to moves against the monarchy culminating in the "Belgian Revolution" of August-September 1830, following the July Revolution in Paris earlier that year. On 20 January 1831, an international conference in London recognized Belgium as an independent state.

At the École Gratuite de Dessin, Catalan was taught by Louis Lefébure de Fourcy who had been appointed as an admissions examiner for the École Polytechnique in 1826. He encouraged Catalan, who showed an outstanding aptitude for mathematics, to prepare himself to take the admission examinations for the prestigious university. He studied at the institution run by Monsieur de Reusse on the Rue de Vaugirard. He also attended a mathematics course given by Antoine Delisle (author of Éléments de trigonométrie rectiligne et sphérique Ⓣ) at the Lycée Saint-Louis, nearby on the Boulevard Saint-Michel, as well as a course by Louis Francoeur at the Sorbonne on the same Boulevard. He won the first prize in the Concours Général de Mathématiques Spéciales in the summer of 1833. Rather than carry on studying, Catalan spent August and September of that year back in Bruges, the first time he had been in that city since he was eleven years old. He went back there to celebrate the birthday of his grandmother on 20 September. Having taken the entrance examinations for the École Polytechnique, he was admitted in November of 1833 ranked 53rd in the list of accepted students, although he almost certainly would have been ranked higher but for the break in his studies. At the École Polytechnique, Catalan attended mathematics courses given by Joseph Liouville and Gabriel Lamé. However, he also studied other topics including the literature and history of France reading authors such as Voltaire, Molière and La Bruyère.

The students at the École Polytechnique were much involved in the political events which were taking place in France at this time. It was a time of great political instability and, in April 1834, there were serious disturbances in Paris following the passing of a law to curtail the activities of the Republican Society of Human Rights. The disturbances were brutally put down by the army. The authorities tried to ensure that the students did not become involved and they send down a number of students who had trumpeted their Republican sympathies. Despite his strong Republican beliefs, Catalan kept out of trouble. In November 1834 he completed his first year of study, being ranked 49th out of 145 students. He was to proceed to his second year but, his father received a letter dated 15 December informing him that his son had been dismissed from the École Polytechnique and was returning home. Catalan had not been singled out for this punishment, for it appears that the whole of his year suffered the same fate. In fact during this forced vacation Catalan took the opportunity to become engaged to Charlotte Augustine Renée Perin (known as Eugénie) who was two years his elder. She had been born in Lille, where Catalan had spent several years. They were engaged to be married on 2 January 1835 and spent the rest of their lives together.

Catalan had to apologise for an "act of subordination" before he was allowed to return to the École Polytechnique in the middle of January 1835. He must have found that a difficult thing to do but he knew that he had to do such things if he was to complete his education. He studied hard and graduated in the summer of 1835, ranked 16th out of the 140 students who had qualified for public service. He was placed 13th among the students who were qualified to enter the École des Ponts et Chaussées but decided that this was not the route that he wanted to take. He was appointed as professor at the École des Arts et Métiers at Châlons-sur-Marne, taking up his appointment in the autumn of 1835. On 4 January 1836 Catalan's father died and on 8 April his grandfather Jean Bardin died, but he went ahead with his marriage to Eugénie Perin on 28 April. In spite of the fact that he was not a practising Christian, they was married in the church of Saint Eustache in Paris. Their first child, a daughter Marie Adélaïde, was born in Châlons-sur-Marne.

Liouville began publication of Journal de Mathématiques Pures et Appliquées in 1836 and a paper by Catalan, Solution d'un problème de Probabilité relatif au jeu de rencontre Ⓣ, was published in the second volume in 1837. In the same journal, he published two papers in 1838: Note sur un Problème de combinaisons Ⓣ, and Note sur une Équation aux différences finies Ⓣ. The second of these contains the 'Catalan numbers' which appears in the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. Four papers by Catalan are published in Volume 4 in 1839: Note sur la Théorie des Nombres Ⓣ; Solution nouvelle de cette question: Un polygone étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales? Ⓣ; Addition à la Note sur une Équation aux différences finies Ⓣ; and Mémoire sur la réduction d'une classe d'intégrales multiples Ⓣ.

You can see more about the Catalan numbers at THIS LINK.

Catalan was keen to return to Paris and he applied for the professorship at the École Gratuite de Dessin which had become vacant on the death of his former professor Jean-Baptiste-Omer Lavit in 1836. He was unsuccessful. He wrote to Liouville in January 1837 who replied encouraging him to come to Paris in the Easter holidays to discuss his future with him. He suggested that Catalan needed further qualifications, such as the baccalaureate, in order to make a successful career. Later that year, Catalan resigned his position at Châlons-sur-Marne and returned to Paris. Along with Charles-François Sturm and Joseph Liouville, he founded the École Sainte-Barbe near the Sorbonne. The school, whose aim was to train pupils for admission to the École Polytechnique, opened in 1838. In the same year, on 20 November, he was appointed assistant tutor (répétiteur) in descriptive geometry at the École Polytechnique, assisting Charles François Antoine Leroy who was teaching this topic. Catalan lectured on part of the descriptive geometry course in 1840, the year in which his second daughter Fanny was born. In 1839 he had been appointed as a deputy examiner at the École Polytechnique. As soon as he had returned to Paris, in addition to these other tasks, Catalan had taken Liouville's advice regarding his baccalaureate, and began studying hard. He was awarded a double baccalaureate in 1839 and continued to study, receiving his "licence in mathematical sciences" in the following year. He was awarded his doctorate in mathematics in 1841 for his main thesis in Mechanics Attraction d'un ellipsoïde homogène sur un point extérieur ou sur un point intérieur Ⓣ, and a further thesis Sur le mouvement des étoiles doubles Ⓣ in Astronomy. In 1845, he was awarded a "licence in physical sciences".

Augustin-Louis Cauchy, a fervent Royalist, had left Paris in 1830 and, failing to swear an oath of allegiance to the new regime, had lost all his positions there. However, he returned to Paris in October 1838 regaining his position at the Académie des Sciences but not his teaching positions because he still refused to take an oath of allegiance. In July 1839 Cauchy invited Catalan and Lejeune Dirichlet to dinner. Although this was an honour for the young Catalan, given that their political views were at the opposite ends of the spectrum, this was never going to develop into a close friendship. Catalan was, however, still supported by Liouville who proposed him for membership of the Société Philomatique. After being unsuccessful in 1839, Catalan was elected to the Society in May 1840 and began publishing articles in the Society's journal. In 1841 he gave the general change-of-variable theorem for n-dimensional integrals in Sur la transformation des variables dans les integrales multiples Ⓣ which was published in the Mémoires Couronnés of the Academie Royale des Sciences et Belles-Lettres de Bruxelles. In 1843 he published for the first time in Crelle's Journal für die reine und angewandte Mathematik. This contains the 'Catalan Conjecture':-

I beg you, sir, to please announce in your journal the following theorem that I believe true although I have not yet succeeded in completely proving it; perhaps others will be more successful. Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation $x^{m} - y^{n} = 1$ in which the unknowns are positive integers only admits a single solution.

Piotr Tchihatchef was a Russian geographer and geologist who was spending the winter of 1842-43 in Paris. He approached Louis Francoeur asking to be tutored in mathematics and Francoeur advised him to approach Catalan. This in fact proved fortunate for Catalan since Tchihatchef had been given a paper by the Russian mathematician Pafnuty Chebyshev to submit for publication in Paris. Tchihatchef showed Catalan the paper and Catalan began to correspond with Chebyshev; a correspondence which continued for fifty years. Chebyshev's paper appeared in Liouville's Journal de mathématiques pures et appliquées in May 1843. It was followed in the same journal by a short paper by Catalan proving a formula for the transformation of a multiple integral stated without proof in Chebyshev's paper. On 6 January 1844 Catalan was appointed as secretary of the Société Philomatique.

Given his achievements, Catalan's university teaching career should have been progressing but any prospects of a smooth path was damaged by his continuing political activities and his strong left-wing Republican views. In November 1844 he was ranked first by the Council for the position of tutor at the École Polytechnique but Ossian Bonnet was appointed to the position and Catalan remained as an assistant tutor. Adolphe Quetelet, the Belgium scientist with whom he had corresponded since his 1841 Mémoires couronnés Ⓣ in the Academie Royale des Sciences et Belles-Lettres de Bruxelles, encouraged him to seek a university position in Belgium because of his lack of progress in Paris. He had received another setback in his career in December 1844 when, after Francoeur had indicated that he wanted Catalan as his deputy at the Sorbonne, he had been rejected by the Royal Council. Instead, they appointed Joseph Bertrand to the position. In 1846 Catalan competed in the Concours d'agrégation (a contest to become a university teacher) and was placed first. However, he was given no appointment. Catalan was paying a high price for his strong, publicly expressed Republican views.

In September 1846 Catalan wrote to Quetelet indicating that he was beginning to regret having remained in Paris instead of following his suggestion to move to Belgium. However, in November he was appointed to take charge of teaching higher mathematics at the Collège de Charlemagne but then became involved in the political unrest in France in 1848. This revolution materialised without any very definite cause, although food shortages from 1846 onwards had caused much economic trouble and discontent of which the Republicans took advantage. The trigger appears to have been the cancellation by the government of a major banquet arranged for February 1848 by the Republicans. Catalan took a highly active role in the disturbances, leading a band of workers who entered the Hôtel-de-Ville immediately after a group of students from the École Polytechnique. The revolution led to the Second Republic, and the voters chose Louis-Napoleon Bonaparte to become president. This fitted well with Catalan's Republican views and he offered himself for election to the National Assembly. Here is an extract from his manifesto:-

I was brought up to be contemptuous of kings and I greeted the fall of Charles X rapturously. I submitted, but protesting with indignation, to the master schemers and dupes we imposed on ourselves at that time. I myself was fooled for a moment, but I stopped being fooled at the retreat of the virtuous Dupont (de l'Eure), and before the end of 1830, I was a Republican. Since then, I have not missed a single day without contributing my wishes, my words and deeds, to the overthrow of the man that France supported too long!

You can see the full manifesto at THIS LINK.

His career seemed to be going well again when he was appointed to the Lycée Saint Louis in 1849. At the École Polytechnique, however, he continued to be disappointed. Two admission examiners were appointed in 1848 and, although Catalan was on the list of four offered to the Ministry, Charles Hermite and Joseph Serret were appointed with Abel Transon and Pierre Bonnet as their deputies. Although Transon (1805-1876), like Catalan, was a socialist, he took little part in politics after the age of thirty.

In June 1850 a commission was chaired by Urbain Le Verrier to reorganise teaching at the École Polytechnique. Jean-Victor Poncelet and Jean-Marie Duhamel were the mathematicians on the commission. It reported in November and Catalan did not like either the academic changes that were proposed, particularly the considerable reduction in pure mathematics, nor did he like the fact that much more work was to be undertaken by the tutors. He resigned his position as assistant tutor later that month. In his letter of resignation he pointed out that he had been an assistant tutor for twelve years, and also explained that he would be unable to undertake the extra teaching required due to his duties at the Lycée Saint Louis. Catalan was not the only one to leave the École Polytechnique following the report of Le Verrier's commission; Michel Chasles, Joseph Liouville and Charles-François Sturm also immediately resigned in protest.

The Second Republic only lasted three years. On 2 December 1851 there was a coup d'état with Louis-Napoléon Bonaparte assuming absolute power and dissolving the National Assembly. Exactly one year later he became Emperor taking the title Napoleon III. This was bad for Catalan who disliked the Bonapartes as much as he disliked kings. He refused to take the required oath of allegiance and as a consequence lost his job. For the next few years he lived in Paris, teaching mathematics, but without any proper employment. He had published many articles in Liouville's Journal de mathématiques pures et appliquées but in 1854 he stopped publishing there, preferring to publish in the Comptes rendus of the Académie des Sciences. This was, almost certainly, part of Catalan's strategy to get elected to the Académie des Sciences. For example, he published Sur des surfaces dont les rayons de courbure en chaque point sont égaux et de signes contraires Ⓣ (1855) which looks at the 'Catalan minimal surface' associated with 'Catalan's minimal curve'. He also published in Comptes rendus: Note de M Catalan à l'occasion d'un théorème de M Serret Ⓣ (1856), Sur le calcul de la latitude par la méthode de M Babinet Ⓣ (1856), Sur quelques points de la théorie des séries Ⓣ (1856), Sur un cas particulier de la formule du binôme Ⓣ (1857), and Sur la théorie des développées Ⓣ (1857). By 1860 Bertrand, Hermite and Serret had all been elected to the Academy. When Serret was elected in 1860, Catalan was on the list to be considered but was only ranked fourth equal, behind Serret, Bonnet and Puiseux. The Academy proposed a question on polyhedra for the 1861 prize and Catalan submitted an entry. He was asked to submit a new version of his memoir in 1862 which was considered by a panel consisting of Bertrand, Chasles, Liouville and Serret. Chasles and Liouville recommended that he receive the prize, but the others proposed that the prize should not be awarded.

The polyhedra that Catalan discovered are now called the Catalan solids. You can see more about them at THIS LINK.

In 1859 Catalan had tried to persuade the Ministry of Public Instruction to name him as a Professor of Mathematics at one of the lycées in Paris. He was not successful. In January 1865, having had no permanent position for 13 years, he was appointed to the chair of mathematics at the University of Liège. Since Catalan was born in Bruges it might be supposed that he would feel as if he was coming back to his homeland when he took up the position in Liège. However, Catalan always considered himself French, having earlier undergone considerable efforts to reinstate his French citizenship. He held this chair until he retired in 1884, then continued to live in Liège until his death ten years later. However, tragedy struck Catalan's family soon after they arrived in Liège. He arrived there with his wife, two daughters, and his mother-in-law Louise Perin who had lived with the family throughout their married life in Paris. Some biographies state that Catalan also had a son who died in Paris before the family moved to Liège but we have been unable to confirm whether this is true. Marie, their eldest child, was already ill before the family moved to Liège and she died there in May 1865. By the end of this year, his mother-in-law had died at the age of 79, while his second daughter, Fanny, also became ill. Catalan's wife took Fanny back to Paris in the hope that this would see her health improve but it was not to be and Fanny died in the spring of 1866.

The Franco-Prussian war of 1870-71 did not affect neutral Belgium but Catalan, with his great love of France, was greatly saddened by the humiliation of the French by the Prussians. However, the resulting fall of Napoleon and the setting up of the Third Republic of France certainly gave him pleasure. Catalan had never respected the Légion d'Honneur which had been created by Napoleon Bonaparte in 1802. However, when the Third Republic awarded him the Knight's Cross of the Légion d'Honneur on 30 December 1887, his joy at France now being a Republic was greater than his dislike of "Bonaparte's honour" and he gladly accepted the honour. The Académie des Sciences, however, discussed at length electing Catalan during the 1870s but failed to do so.

Mélanges mathématiques Ⓣ was published by the Royal Society of Sciences of Liège in 1868. It contains 69 of Catalan's papers starting from Sur les combinaisons avec répétition Ⓣ (1838) and ending with his paper Démonstration d'une formule de Poisson Ⓣ (February 1867). In 1887-88 an updated version of Mélanges mathématiques in three volumes was published by the Royal Society of Sciences of Liège. This work contained 299 of Catalan's papers, the last paper being Sur une application du théorème de Bayes, faite par Laplace Ⓣ (August 1888). The second volume contains Catalan's retirement speech and the third volume also lists Catalan's publications; the list contains 406 items.

Catalan was elected to the Belgian Académie Royale des Sciences, des Lettres et des Beaux Arts on 15 December 1865. He was also elected to the Académie des Sciences de Toulouse and the Société des Sciences de Lille. He was a corresponding member of the St Petersburg Academy of Sciences, the Turin Academy of Sciences, the Accademia Pontificia dei Nuovi Lincei, the Mathematical Society of Amsterdam, the National Institute of Geneva, the Societé Havraise d'Études Divers and the Societé d'Agriculture de la Marne. He was a member of the Science Society of Liège, the Mathematical Society of France and, as we mentioned above, the Société Philomatique of Paris. The Belgium government honoured Catalan by awarding him the Cross of the Knight of the Order of Léopold in 1879. Catalan retired on 1 June 1884, becoming professor emeritus. His retirement was marked on 7 December of that year by a celebration attended by many of his friends (including Chebyshev who had come from St Petersburg especially for the occasion), colleagues and former students. He was presented with his portrait, commissioned by some of his friends, painted by Émile Delpérée. In this portrait, Catalan is proudly wearing the Knight's Cross of the Order of Léopold. In 1890 the Belgium government elevated Catalan to an Officer of the Order of Léopold.

You can see Catalan's speech at his retirement THIS LINK.

Catalan wrote several texts which were very popular and many ran into several editions. He wrote: Elements de géométrie Ⓣ (1843); the two volume work Traité élémentaire de géométrie descriptive Ⓣ (1850-52) which ran to 5 editions with the last appearing in 1881; Théorèmes et problèmes de géométrie élémentaire Ⓣ (1852) which ran to 6 editions with the last appearing in 1879; Nouveau manuel des aspirants au baccalauréat ès sciences Ⓣ (1852) which ran to 12 editions; Solutions des problèmes de mathématique et de physique donnés à la Sorbonne dans les compositions du baccalauréat ès sciences Ⓣ (1855-56); two volumes of Manuel des candidats à l'École Polytechnique Ⓣ (1857-58); Notions d'astronomie Ⓣ (1860) which ran to 6 editions; Traité élémentaire des séries Ⓣ (1860); Histoire d'un concours Ⓣ (1865) with a second edition published in 1867; and Cours d'analyse de l'université de Liège Ⓣ (1870) with a second edition published in 1880.

In 1894 the École Polytechnique planned to celebrate the centenary of its founding. Catalan, one of the oldest living former pupils, planned a final visit to Paris to take part in the celebrations. However, it was not to be. His wife became ill at the beginning of February 1894 then, on 9 February, Catalan collapsed and was taken to hospital suffering from pneumonia. His wife, Eugénie, died on 11 February and Catalan himself died three days later.

You can see an appreciation of Catalan by the rector of the University of Liège at THIS LINK.