加斯帕尔·蒙日Gaspard Monge

加斯帕尔·蒙日晚年成为佩吕斯伯爵，有时也以此名闻名。他的父亲是Jacques Monge，一位原籍法国东南部上萨瓦省的商人。蒙日的母亲，娘家姓让娜·鲁索，是勃艮第人，蒙日就是在勃艮第的博讷镇长大的。大约在蒙日出生时，博讷在经历了一段衰落期后，由于葡萄酒贸易的成功而再次繁荣起来。

蒙日就读于博讷的奥拉托利会学院。这所学校是为年轻贵族设立的，由神父管理。学校提供比其他宗教学校更自由的教育，不仅教授人文学科，还教授历史、数学和自然科学。正是在这所学校，蒙日首次展现了他的才华。1762年，16岁的蒙日前往里昂，在圣三一学院继续学业。尽管当时只有17岁，蒙日却被安排负责教授一门物理课程。1764年在那里完成学业后，蒙日回到博讷，绘制了该市的地图。

蒙日绘制的博讷地图对他职业生涯的方向产生了重大影响，因为这幅地图被梅济耶尔皇家工程学校的一名教职员看到。他对蒙日的工作印象深刻，1765年，蒙日被任命为皇家工程学校的绘图员。当然，在这个职位上，蒙日从事的任务并不完全合他的心意，因为他渴望在生活中得到一个更能发挥他数学才能的职位。然而，皇家工程学校使蒙日接触到了查尔斯夏尔·博叙，他是那里的数学教授。起初，蒙日的职位并不要求他运用数学才能，但蒙日利用自己的时间发展自己的几何学思想。

成为绘图员大约一年后，蒙日接到了一项任务，使他能够运用自己的数学技能来攻克所给的任务。他被要求绘制一份防御工事图，无论敌人处于什么位置，都能阻止敌人看到或射击军事阵地，蒙日没有使用当时可用的复杂方法，而是设计了自己的图解方法来建造这样的防御工事。这种方法充分利用了蒙日在自己时间中发展的几何技术。他的数学能力现在在皇家工程学校得到了认可，人们意识到蒙日在理论和实践科目上都具有非凡的能力。

夏尔·博叙于1768年当选为Académie des Sciences院士，他离开梅济耶尔的学校，成为卢浮宫流体动力学教授。1769年1月22日，蒙日写信给夏尔·博叙，解释说他正在写一部关于双曲率曲线evolutes的著作。他请夏尔·博叙对这部著作的原创性和实用性发表意见。夏尔·博叙一定以非常积极的方式回复了，因为6月Journal Encyclopédique上出现了一篇蒙日的出版物（他的第一篇出版物），概述了他所获得的结果。这篇论文中，蒙日推广了克里斯蒂安·惠更斯在空间曲线上获得的结果（作为克里斯蒂安·惠更斯对摆的研究的一部分），并增加了许多重要的新发现，在[19]中有详细描述。完成的著作于1770年10月提交给巴黎的Académie des Sciences，并于1771年8月在Académie前宣读（尽管直到1785年才由Académie出版）。

当夏尔·博叙离开梅济耶尔的皇家工程学校时，蒙日于1769年1月被任命接替他的职位。1770年，他又在皇家工程学校获得了一个额外职位，被任命为实验物理学讲师。尽管这对蒙日的职业生涯来说是向前迈出的一大步，但他更感兴趣的是在最高圈子里以数学家的身份扬名。意识到必须从顶尖数学家那里获得建议，蒙日在1771年初联系了让·勒朗·达朗贝尔和尼古拉·德·孔多塞。尼古拉·德·孔多塞一定对蒙日向他展示的数学深度印象深刻，因为他建议蒙日向他正在研究的四个数学领域中的每一个都向Académie des Sciences提交论文。

蒙日提交给Académie的四篇论文分别关于变分法的推广、无穷小几何、偏微分方程理论和组合学。在接下来的几年里，他向Académie提交了一系列关于偏微分方程的重要论文，他从几何角度研究了这些方程。他对数学以外学科的兴趣开始增长，并对物理学和化学中的问题产生了兴趣。

1777年，蒙日与Cathérine Huart结婚，由于妻子拥有一座锻造厂，除了广泛的数学和科学兴趣外，他还对冶金学产生了兴趣。他仍然深入参与梅济耶尔皇家工程学校的教学，并在那里组织建立了一个化学实验室。然而，从1780年起，他在梅济耶尔学校的工作上投入的时间减少了，因为那一年他被选为巴黎Académie des Sciences的adjoint géomètre。从那时起，他在巴黎度过了很长时间，代替夏尔·博叙教授流体动力学课程，并参与Académie在数学、物理学和化学方面开展的项目。不可能同时完成所有这些工作并在梅济耶尔教授所有课程，但他保留了那里的职位并领取全额薪水，从中支付其他人代替他教授一些课程。

在巴黎和梅济耶尔两地奔波三年后，蒙日又获得了一个职位，即接替艾蒂安·贝祖担任海军学员考官。蒙日本想保留所有这些职位，但在尝试安排一个不可能的时间表约一年后，他决定必须辞去在梅济耶尔的职位，并于1784年12月这样做了。在接下来的五年里，尽管考官职责繁重，蒙日仍进行了广泛科学课题的研究，向Académie提交了关于[1]的论文：-

...亚硝酸的组成、曲面的生成、有限差分方程、偏微分方程（1785年）；双折射与冰洲石的结构、铁、钢和铸铁的组成，以及电火花对二氧化碳气体的作用（1786年）；毛细现象（1787年）；某些气象现象的成因（1788年）；以及一项生理光学研究（1789年）。

当然，1789年是法国历史上多事之秋，1789年7月14日攻占巴士底狱标志着法国大革命的开始。这彻底改变了蒙日的人生轨迹。大革命爆发时，他是巴黎顶尖科学家之一，在广泛科学领域有着卓越的研究记录，拥有考官经验以及他在1786年作为考官职责一部分所进行的学校改革经验。在政治上，蒙日是大革命的坚定支持者，他的最初行动是通过加入各种支持大革命的社会团体来表明支持，但他继续履行海军学员考官的日常职责，并作为Académie工作中的重要人物。此时，他已是度量衡主要Académie委员会的成员。

路易十六于1791年6月20日试图逃离该国，但在瓦雷讷被拦截并带回巴黎，这终结了国王与议会分享政府的尝试。当国民议会宣布人民拥有自决权时，与欧洲的关系恶化。法国于1792年4月20日对奥地利和普鲁士宣战。法国的失败导致法国国内动荡，1792年8月10日，人民进一步革命，9月期间贵族和神职人员被谋杀。9月21日，法国废除君主制并宣布成立共和国。蒙日被国民公会提供政府海军部长职位。

并非对蒙日不敬，但不可能满足许多人的极端观点，蒙日担任海军部长的时期不能被视为成功。尽管他在困难环境中努力尝试，但他在这个职位上只坚持了八个月，就放弃了与周围人无休止的斗争，并于1793年4月10日提交了辞呈。几个月后，蒙日回到了Académie des Sciences的工作，但这并没有持续多久，因为1793年8月8日，Académie des Sciences被国民公会废除。

仍然是坚定的共和派和革命的支持者，蒙日从事与武器和炸药有关的各种军事项目。他撰写了关于这些主题的论文，也就这些军事主题开设了课程。他继续在度量衡委员会任职，该委员会尽管Académie des Sciences结束但仍然存在。他还向国民公会提出了教育改革建议，但尽管在1793年9月15日被接受，却在第二天被否决。这就是这个不稳定时期决策的多变性。

蒙日于1794年3月11日被国民公会任命到为建立中央公共工程学校（不久后成为巴黎综合理工学院）而设立的机构。他不仅利用在梅济耶尔的经验对建立这所学校产生了重大影响，而且于1794年11月9日被任命为画法几何讲师。他作为讲师的第一项任务是培训该校未来的教师，该校从1795年6月开始运作。蒙日关于无穷小几何的讲座构成了他的著作Application de l'analyse à la géométrie的基础。

另一个教育机构，巴黎高等师范学校，被设立来培训中学教师，蒙日讲授画法几何课程。他也是Académie des Sciences的坚定信仰者，并努力使其作为国家研究院恢复。国民公会于1795年10月26日批准了这个新机构。然而从1796年5月到1797年10月，蒙日在意大利担任委员会成员，为征服者挑选最好的艺术珍品并将它们带回法国。特别重要的是，他在意大利期间与拿破仑·波拿巴变得友好。拿破仑击败了奥地利，并于1797年10月17日签署了坎波福尔米奥条约，这对法国来说是一个异常有利的条约，保留了大部分法国征服的领土。蒙日带着坎波福尔米奥条约的文本回到了巴黎。

回到Paris Monge后，他重新担任了之前的角色，并被任命为巴黎综合理工学院院长这一有声望的新职位。到1798年2月，蒙日回到罗马，参与了罗马共和国的建立。在[17]中，作者利用蒙日当时从罗马寄给妻子的信件描述了这些事件。特别是蒙日提出了在罗马共和国建立高级学校的计划。拿破仑·波拿巴现在要求蒙日加入他的埃及远征，蒙日有些勉强地同意了。

蒙日于1798年5月26日离开意大利，加入了拿破仑的远征军。这次远征包括数学家约瑟夫·傅里叶和艾蒂安-路易·马吕斯以及蒙日，起初取得了巨大成功。1798年6月10日占领了马耳他，7月1日攻占了亚历山大，尼罗河三角洲也很快被占领。然而，1798年8月1日，法国舰队在尼罗河战役中被纳尔逊的舰队彻底摧毁，因此拿破仑发现自己被困在他所占领的土地上。蒙日于8月21日被任命为开罗埃及研究院院长。该研究院数学部有十二名成员，包括约瑟夫·傅里叶、蒙日、艾蒂安-路易·马吕斯和拿破仑·波拿巴。在拿破仑于埃及和叙利亚的困难时期，蒙日继续完善他的论著Application de l'analyse à la géométrie。

拿破仑于1799年放弃军队返回巴黎，很快在法国掌握了绝对权力。蒙日于1799年10月16日回到巴黎，担任巴黎综合理工学院院长。他发现他的论文Géométrie descriptiveⓉ（画法几何）已于1799年早些时候出版。这是应他妻子的要求完成的，由Hachette根据蒙日在巴黎高等师范学院的讲座整理而成。1799年11月9日，拿破仑与另外两人发动政变夺取政权，成立了新政府——执政府。拿破仑任命蒙日为终身执政府参议员。蒙日欣然接受，尽管他的共和观点本应意味着他反对拿破仑强加给法国的军事独裁。事实必然是蒙日[1]：-

...被拿破仑所迷惑...并接受了皇帝授予他的所有荣誉和礼物：1804年荣誉军团大军官，1806年参议院议长，1808年佩鲁斯伯爵，等等。

在接下来的几年里，蒙日继续从事一系列活动，履行参议员职责，同时保持对数学研究的兴趣，但他的数学工作大多涉及为巴黎综合理工学院的学生教学和编写教材。渐渐地，他越来越少参与数学研究，然后从1809年起，由于健康开始恶化，他放弃了在巴黎综合理工学院的教学。

1812年6月，拿破仑集结了约453,000人的大军团，其中包括被迫服役的普鲁士人和奥地利人，向俄罗斯进军。这场战役是一场灾难，但到9月，拿破仑的军队已进入一座空无一人的莫斯科。拿破仑撤退，普鲁士人和奥地利人脱离了大军团，而且在巴黎有人企图发动反对拿破仑的政变。蒙日对局势感到沮丧，他的健康突然崩溃。在拿破仑离开其残余军队返回巴黎以确立权威后，他的健康慢慢恢复。1813年拿破仑取得一些军事胜利后，反法联军得到加强。蒙日被派往列日组织该城的防御以抵御进攻。

联军开始向法国进军，蒙日逃亡。当拿破仑于1814年4月6日退位时，蒙日不在巴黎，但不久后他确实返回并试图重新开始生活。拿破仑从他曾被流放的厄尔巴岛逃脱，到1815年3月20日他回到巴黎。蒙日立即归附拿破仑并给予他全力支持。拿破仑在滑铁卢战败后，蒙日继续看望他，直到7月15日他被送上船。到10月，蒙日担心自己的生命安全，逃离了法国。

蒙日于1816年3月返回巴黎。返回两天后，他被法兰西学院开除，从那时起他的生活极其艰难，因为他在政治上受到骚扰，生命不断受到威胁。他去世时，巴黎综合理工学院的学生向他致敬，尽管法国政府坚持不应给予任何致敬。

在9中，蒙日的政治生涯得到了温和的对待，但G Jorland在对该文的评论中采取了更严厉的观点：-

蒙日在海军部的任职完全失败，他主持了对意大利和埃及的文化掠夺。如果拿破仑确实说过蒙日像情妇一样爱他，那证明最高的数学清晰度可以与政治盲目并存。

我们在上文已经相当频繁地评论了蒙日的科学工作。由于他在Application de l'analyse à la géométrie中的工作，他引入了三维空间中曲面曲率线的概念，因此被认为是微分几何之父。他发展了一种将几何应用于构造问题的一般方法。他还引入了两个相互垂直的投影平面，用于固体对象的图形描述。这些技术被推广为一个称为géométrie descriptive的系统，现在称为正投影，是现代机械制图中使用的图形方法。

蒙日数学方法背后的基本哲学在[13]中进行了讨论，作者指出蒙日的目标是：-

... 数学的几何化，基于：
(a) 分析中的运算与几何变换的类比或对应；
(b) 通过分析生成线的运动对曲面进行遗传分类和参数化。

蒙日将分析视为[13]：-

……不是一种自足的语言，而仅仅是构成现实的“运动几何景观”的“脚本”。

……[他的]新方法针对空间中最为深刻、密切和普遍的关系及其变换，使他能够以一种富有成果的、前所未闻的方式将几何与分析相互联系起来。实际关切促使蒙日以一种新的方式看待数学的对象和功能，违反了数学的公认赞助者所设定的形式主义（语言学）标准……

Gaspard Monge became the Comte de Péluse later in his life and he is sometimes known by this name. His father was Jacques Monge, a merchant who came originally from Haute-Savoie in southeastern France. Gaspard's mother, whose maiden name was Jeanne Rousseaux, was a native of Burgundy and it was in the town of Beaune in Burgundy that Gaspard was brought up. Around the time that Gaspard was born Beaune, after a period of decline, was becoming prosperous again due to the success of the wine trade.

Monge attended the Oratorian College in Beaune. This school was intended for young nobles and was run by priests. The school offered a more liberal education than other religious schools, providing instruction not only in the humanities but also in history, mathematics, and the natural sciences. It was at this school that Monge first showed his brilliance. In 1762, at the age of 16, Monge went to Lyon where he continued his education at the Collège de la Trinité. Despite being only 17 years of age at the time, Monge was put in charge of teaching a course in physics. Completing his education there in 1764, Monge returned to Beaune where he drew up a plan of the city.

The plan of Beaune that Monge constructed was to have a major influence in the direction that his career took, for the plan was seen by a member of staff at the École Royale du Génie at Mézières. He was very impressed by Monge's work and, in 1765, Monge was appointed to the École Royale du Génie as a draftsman. Of course, in this post Monge was undertaking tasks that were not entirely to his liking, for he aspired to a position in life which made far more use of his mathematical talents. However the École Royale du Génie brought Monge into contact with Charles Bossut who was the professor of mathematics there. At first Monge's post did not require him to use his mathematical talents, but Monge worked in his own time developing his own ideas of geometry.

About a year after becoming a draftsman, Monge was given a task which allowed him to use his mathematical skill to attack the task he was given. Asked to draw up a fortification plan which prevented an enemy from either seeing or firing at a military position no matter what the position of the enemy, Monge devised his own graphical method to construct such a fortification rather than use the complicated methods then available. This method made full use of the geometrical techniques which Monge was developing in his own time. His mathematical abilities were now recognised at the École Royale du Génie and it was realised that Monge was someone with exceptional abilities in both theoretical and practical subjects.

Bossut was elected to the Académie des Sciences in 1768 and he left the École in Mézières to become professor of hydrodynamics at the Louvre. On 22 January 1769 Monge wrote to Bossut explaining that he was writing a work on the evolutes of curves of double curvature. He asked Bossut to give an opinion on the originality and usefulness of the work. Bossut must have replied in a very positive fashion for in June a publication in the Journal Encyclopédique by Monge (his first publication) appeared giving a summary of the results which he had obtained. This paper, in which Monge generalised the results obtained by Huygens on space curves (as part of Huygens's investigation of the pendulum) and added many important new discoveries, is described in detail in [19]. The completed work was submitted to the Académie des Sciences in Paris in October 1770 and read before the Académie in August 1771 (although it was not published by the Académie until 1785).

When Bossut left the École Royale du Génie at Mézières, Monge was appointed to succeed him in January 1769. In 1770 he received an additional post at the École Royale du Génie when he was appointed as instructor in experimental physics. Although this was a large step forward for Monge's career, he was more interested in making his name as a mathematician in the highest circles. Realising that he had to obtain advice from the leading mathematicians, Monge approached d'Alembert and Condorcet early in 1771. Condorcet must have been impressed with the depth of the mathematics that Monge showed him, for he recommended that he present memoirs to the Académie des Sciences in each of the four areas of mathematics in which he was undertaking research.

The four memoirs that Monge submitted to the Académie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics. Over the next few years he submitted a series of important papers to the Académie on partial differential equations which he studied from a geometrical point of view. His interest in subjects other than mathematics began to grow and he became interested in problems in both physics and chemistry.

In 1777 Monge married Cathérine Huart and, since his wife had a forge, he became interested in metallurgy in addition to his wide range of mathematical and scientific interests. Still deeply involved in teaching at the École Royale du Génie at Mézières he organised the setting up of a chemistry laboratory there. From 1780, however, he devoted less time to his work at the École at Mézières since in that year he was elected as adjoint géomètre at the Académie des Sciences in Paris. From that time he spent long periods in Paris, teaching a course in hydrodynamics as a substitute for Bossut as well as participating in projects undertaken by the Académie in mathematics, physics and chemistry. It was not possible to do all this and to teach all his courses at Mézières but he kept his posts there and received his full salary out of which he paid others to teach some courses in his place.

After three years of dividing his time between Paris and Mézières, Monge was offered yet another post, namely to replace Bézout as examiner of naval cadets. Monge would have liked to keep all these positions, but after attempting to organise an impossible schedule for about a year, he decided that he would have to resign his posts in Mézières, which he did in December 1784. Over the next five years, despite heavy duties as an examiner, Monge undertook research in a wide range of scientific subjects presenting papers to the Académie on [1]:-

...the composition of nitrous acid, the generation of curved surfaces, finite difference equations, partial differential equations (1785); double refraction and the structure of Iceland spar, the composition of iron, steel, and cast iron, and the action of electricity sparks on carbon dioxide gas (1786); capillary phenomena (1787); and the causes of certain meteorological phenomena (1788); and a study in physiological optics (1789).

Of course 1789 was an eventful year in French history with the storming of the Bastille on 14 July 1789 marking the start of the French Revolution. This was to completely change the course of Monge's life. At the onset of the Revolution he was one of the leading scientists in Paris with an outstanding research record in a wide variety of sciences, experience as an examiner and experience in school reforms which he had undertaken in 1786 as part of his duties as an examiner. Politically Monge was a strong supporter of the Revolution, and his first actions were to show his support by joining various societies supporting the Revolution, but he continued his normal duties as an examiner of naval cadets, and as a major figure in the work of the Académie. By this time he was on the major Académie Commission on Weights and Measures.

Louis XVI attempted to flee the country on 20 June 1791, but was stopped at Varennes and brought back to Paris, and this put an end to attempts to share government between the king and an assembly. Relations with Europe deteriorated when the National Assembly declared that a people had the right of self-determination. France declared war on Austria and Prussia on 20 April 1792. French defeats led to unrest in France and, on 10 August 1792, there was further revolutions by the people with nobles and clergy murdered during September. On 21 September the monarchy was abolished in France and a republic was declared. Monge was offered the post of Minister of the Navy in the government by the National Convention.

Without disrespect to Monge, it was impossible to satisfy the quite extreme views of many people, and Monge's period as Minister of the Navy cannot be viewed as a success. Although he tried hard in difficult circumstances, he survived only eight months in the post before he gave up the incessant battle with those around him, and he submitted his resignation on 10 April 1793. For a few months Monge returned to his work with the Académie des Sciences but this did not last long for, on 8 August 1793, the Académie des Sciences was abolished by the National Convention.

Still a strong republican and supporter of the Revolution, Monge worked on various military projects relating to arms and explosives. He wrote papers on the topics and also gave courses on these military topics. He continued to serve on the Commission on Weights and Measures which survived despite ending the Académie des Sciences. He also proposed educational reforms to the National Convention but, despite being accepted on 15 September 1793, it was rejected on the following day. Such was the volatile nature of decisions at this unstable time.

Monge was appointed by the National Convention on 11 March 1794 to the body that was put in place to establish the École Centrale des Travaux Publics (soon to become the École Polytechnique). Not only was he a major influence in setting up the École using his experience at Mézières to good effect, but he was appointed as an instructor in descriptive geometry on 9 November 1794. His first task as instructor was to train future teachers of the school which began to operate from June 1795. Monge's lectures on infinitesimal geometry were to form the basis of his book Application de l'analyse à la géométrie.

Another educational establishment, the École Normale, was set up to train secondary school teachers and Monge gave a course on descriptive geometry. He was also a strong believer in the Académie des Sciences and worked hard to see it reinstated as the Institut National. The National Convention approved the new body on 26 October 1795. However from May 1796 to October 1797, Monge was in Italy on a commission to select the best art treasures for the conquerors and bring them to France. Of particular significance was the fact that he became friendly with Napoleon Bonaparte during his time in Italy. Napoleon had defeated Austria and signed the Treaty of Campo Formio on 17 October 1797 which was an exceptionally good treaty for France, preserving most of the French conquests. Monge returned to Paris bringing the text of the Treaty of Campo Formio with him.

Back in Paris Monge slotted back into his previous roles and was appointed to the prestigious new one of Director of the École Polytechnique. By February 1798 Monge was back in Rome, involved with the setting up of the Republic of Rome. In [17] the author describes these events using letters which Monge sent to his wife from Rome at that time. In particular Monge proposed a project for advanced schools in the Republic of Rome. Napoleon Bonaparte now asked Monge to join him on his Egyptian expedition and, somewhat reluctantly, Monge agreed.

Monge left Italy on 26 May 1798 and joined Napoleon's expeditionary force. The expedition, which included the mathematicians Fourier and Malus as well as Monge, was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Monge was appointed president of the Institut d'Egypte in Cairo on 21 August. The Institut had twelve members of the mathematics division, including Fourier, Monge, Malus and Napoleon Bonaparte. During difficult times with Napoleon in Egypt and Syria, Monge continued to work on perfecting his treatise Application de l'analyse à la géométrie.

Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Monge was back in Paris on 16 October 1799 and took up his role as director of the École Polytechnique. He discovered that his memoir Géométrie descriptive Ⓣ had been published earlier in 1799. This had been done at his wife's request and had been put together by Hachette from Monge's lectures at the École Normale. On 9 November 1799 Napoleon and two others seized power in a coup and a new government, the Consulate, was set up. Napoleon named Monge a senator on the Consulate for life. Monge accepted with pleasure, although his republican views should have meant that he was opposed to the military dictatorship imposed by Napoleon on France. The truth must be that Monge was [1]:-

... dazzled by Napoleon ... and accepted all the honours and gifts the emperor bestowed upon him: grand officer of the Legion of Honour in 1804, president of the Senate in 1806, Count of Péluse in 1808, among others.

Over the next few years Monge continued a whole range of activities, undertaking his role as a senator while maintaining an interest in research in mathematics but mostly his mathematical work involved teaching and writing texts for the students at the École Polytechnique. Slowly he became less involved in mathematical research, then from 1809 he gave up his teaching at the École Polytechnique as his health began to fail.

In June 1812 Napoleon assembled his Grande Armée of about 453,000 men, including men from Prussia and from Austria who were forced to serve, and marched on Russia. The campaign was a disaster but by September Napoleon's army had entered a deserted Moscow. Napoleon withdrew, the Prussians and Austrians deserted the Grande Armée and in there were attempts at a coup against Napoleon in Paris. Monge was dismayed at the situation and his health suddenly collapsed. Slowly his health returned after Napoleon left the remains of his army and returned to Paris to assert his authority. After Napoleon had some military success in 1813, the allied armies against him strengthened. Monge was sent to Liège to organise the defence of the town against an attack.

The allied armies began to move against France and Monge fled. When Napoleon abdicated on 6 April 1814, Monge was not in Paris, but soon after he did return and tried to pick up his life again. Napoleon escaped from Elba, where he had been banished, and by 20 March 1815 he was back in Paris. Monge immediately rallied to Napoleon and gave him his full support. After Napoleon was defeated at Waterloo, Monge continued to see him until he was put on board a ship on 15 July. By October Monge feared for his life and fled from France.

Monge returned to Paris in March 1816. Two days after his return he was expelled from the Institut de France and from then on his life was desperately difficult as he was harassed politically and his life was continually threatened. On his death the students of the École Polytechnique paid tribute to him despite the insistence of the French Government that no tributes should be paid.

In [9] Monge's political career is treated kindly but G Jorland, in a review of that paper, takes a harder view:-

[Monge's] tenure at the Ministry of the Navy was a complete failure and he presided over the cultural pillage of Italy and Egypt. If Napoleon actually said that Monge loved him like a mistress, it proves that the utmost mathematical clarity can go hand in hand with political blindness.

We have commented quite frequently regarding Monge's scientific work above. He is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept of lines of curvature of a surface in 3-dimensional space. He developed a general method of applying geometry to problems of construction. He also introduced two planes of projection at right angles to each other for graphical description of solid objects. These techniques were generalised into a system called géométrie descriptive, which is now known as orthographic projection, the graphical method used in modern mechanical drawing.

The basic philosophy behind Monge's approach to mathematics is discussed in [13] where the author states that Monge's aims were the:-

... geometrisation of mathematics based on:
(a) the analogy or correspondence of operations in analysis with geometric transformations;
(b) the genetic classification and parametrisation of surfaces through analysis of the movement of generating lines.

Monge regarded analysis as being [13]:-

... not a self-contained language but merely the 'script' of the 'moving geometrical spectacle' that constitutes reality.

... [His] new approach addressed itself to the most profound, intimate and universal relations in space and their transformations, putting him in a position to interconnect geometry and analysis in a fertile, previously unheard-of fashion. Practical concerns induced Monge to perceive the object and function of mathematics in a new way, in violation of the formalistic (linguistic) standards set by the approved patrons of mathematics ...