拉扎尔·卡诺Lazare Carnot

拉扎尔·卡诺是Claude Abraham Jean Jacques Carnot（1719-1797）和Magdelaine Marguerite Pothier（1726-1788）的儿子。Claude Carnot是Nolay的一位主要公民，是Nolay侯爵领地的公证人、法官和法警。Nolay侯爵领地是在法国亨利四世（1553-1610）时期创建的。卡诺的祖父是Jean Carnot（1672-1735），一位公证人和财政检察官。Claude和Marguerite Carnot于1745年结婚，育有十七个孩子，包括Joseph-François-Claude Carnot（1852-1835）、卡诺（1753-1823，本传记的主人公）和Claude-Marie 卡诺 Feulins（1755-1836）。此时让我们注意，Joseph-François-Claude Carnot成为了一名法学家，是最高法院的成员（1801-1835），于1832年当选为道德与政治科学院院士，并成为荣誉军团军官。Claude-Marie 卡诺 Feulins是一名士兵。他是1791年立法议会的代表，1815年百日王朝期间的代表，以及1815年法国执行委员会的临时专员。这个比卡诺小两岁的兄弟，在他的兄弟姐妹中与他最亲近，他们变得形影不离。我们刚才提到的卡诺的两个兄弟都出版了几本书。我们提到的他的亲属中没有一个叫卡诺，但这是一个家族姓氏。他是[8]：-

……以一位叔叔的名字命名，这位叔叔是索邦大学的一位博学博士，也是Chalon教区的副主教。

卡诺的初等教育最初由M Masson负责，后由M Boisson负责，两人都是优秀的拉丁语学者，地点在Nolay的小学院。在他们的教导下，他达到了修辞学第二班。1767年，他十四岁时，卡诺和他的哥哥被送到Nolay正西约25公里的Autun。他在Autun寄宿在Mme Gondier家，并在奥拉托利会学院Collège d'Autun学习修辞学第三年，研习哲学和古典学。在Collège d'Autun学习一年后，他进入Autun的苏尔皮斯神学院。这所小型神学院由圣苏尔皮斯神父会管理，强调学术工作。在这里，卡诺学习数学、逻辑学和神学，他的老师是Abbé Bisson。他在Autun期间，当地爆发了天花疫情，卡诺也受到感染，但由于Mme Gondier和他的医生M Lhomme，他很快康复，学业几乎没有受到影响[8]：-

1768年，他在拉丁语的公开考试中表现出色，在论文答辩时拒绝接受老师们通常的协助，面对学识渊博的公开考官、他医生的妻子Madame Lhomme提出的反对意见。此时这个男孩非常虔诚，以至于他的某些亲戚建议他从事圣职。

卡诺的父亲Claude为Duke D'Aumont、Nolay侯爵工作，公爵对卡诺印象非常深刻，并觉得圣职不是他应该走的道路，于是向他的父亲建议，他的儿子会从军事生涯中受益。公爵提出联系战争部，允许卡诺进入工程兵部队，尽管最近有限制进入人数的举措。他开始在巴黎Marais地区的Louis-Siméon Mausserat de Longpré学校学习。当时巴黎有两所预备学校，为男孩们提供工程、炮兵和海军方面的培训。Mausserat de Longpré的学校是两所中较小的，但被认为学术上更强。学校校长与让·勒朗·达朗贝尔非常友好，正如卡诺的儿子Hippolyte Carnot（1801-1888）在[3]中所解释的：-

这位著名的院士会来坐在学生中间，喜欢通过向他们提出令人尴尬的问题来锻炼他们的智力。他注意到了年轻的卡诺……并预言他会取得巨大成功。我父亲终生怀着感激之情回忆与让·勒朗·达朗贝尔的谈话和鼓励，他将自己科学发展的显著影响归功于让·勒朗·达朗贝尔。

1771年2月，卡诺参加了进入工程学校的入学考试，在[8]中排名第三：-

……在参加考试的一百一十五人中，卡诺被任命为Mézières工程学校的少尉。他当时十八岁。考官这样评价他：一个极有前途的人，非常聪明，出色地掌握了他的课程。

卡诺在梅济耶尔的工程学院由加斯帕尔·蒙日等人教授。在梅济耶尔，他的课程包括几何学、力学、几何设计、地理学和水力学。大约在这个时候，他遇到了本杰明·富兰克林，后者于1772年访问法国，当时他是美国驻英国的特使，寻求法国对美国独立的支持。这段友谊对卡诺政治思想的发展很重要，几年后，他将这些思想付诸实践。

他于1773年从学校毕业，于2月13日离开。随后他开始了军事生涯，被任命为孔代亲王工程兵团的中尉。他很快以防御工事和飞艇方面的专家而闻名。他后来在这两个主题上都发表了重要著作。他还继续学习数学。

1778年，他写了Essai sur les machines en généralⓉ（《论一般机器》），并将其提交参加竞赛以争取奖项。他在1781年修订了它，最终于1783年出版。它涉及力学和工程领域。序言开头写道：-

尽管这里讨论的理论适用于所有关于运动传递的问题，但这本小册子被命名为“论一般机器”；首先，因为我们主要关注的是机器，作为力学最重要的对象；其次，因为不涉及任何特定的机器，而只涉及所有机器共有的性质。这一理论基于三个主要定义；第一个定义着眼于某些运动，我称之为几何运动，因为它们只能由几何学原理确定，并且绝对独立于动力学规则；我不认为我们能轻易地舍弃它，而不在主要比例的陈述中留下一些可疑之处，正如我特别为勒内·笛卡儿原理所展示的那样。通过我的第二个定义，我试图确定 soliciting force 和 resistant force 这两个术语的含义：在我看来，如果没有对这些不同力进行明确区分，就无法在机器中清晰地将原因与效果进行比较；而正是这种区分，在我看来，总是有些模糊和不确定。

1783年，他出版了Éloge de VaubanⓉ（《向沃邦致敬》）。在1778年12月3日的会议上，第戎科学院、艺术与文学学院为1784年的文学竞赛提出了“Éloge du Maréchal Vauban”。塞巴斯蒂安，沃邦侯爵（1633-1707），被认为是法国领先的工程师，其防御工事体系在他那个时代被认为是最好的。沃邦来自勃艮第，第戎科学院认为他应该得到比贝尔纳·勒布耶·德·丰特奈尔在巴黎科学院中给予他的更好的颂词。卡诺，也来自勃艮第，现在是防御工事专家，参加了第戎科学院的竞赛。卡诺，现在是阿拉斯驻军中皇家工程兵团的队长，于1784年8月2日获得了两枚金牌和科学院的奖项，颁奖由孔代亲王亲自进行。卡诺的Éloge de VaubanⓉ（《向沃邦致敬》）因其热情、表达的温暖和同情心而受到赞扬，他被描述为博学且雄辩。卡诺回复亲王[26]：-

殿下，被一位名叫孔代的英雄加冕是非常甜蜜的，您手中分发的桂冠，正如装饰您威严额头的那些，是永不凋谢的品种。

让我们引用Éloge de VaubanⓉ（《向沃邦致敬》）（例如，见[17]），因为我们在那里发现了关于卡诺的性格和他对国家的态度的事情：-

一个智者能收获自己作品的果实，这是多么罕见啊！他超越了自己的时代，他的话语只能被后人所理解；但这对他而言已是足够的支持；他的想象力穿透了错误的阴影；他是尚未出生之人的朋友，他在最深入的研究中与他们交谈；作为公民，他关注自己的民族，他的希望寄托于它，他为它的成功而喝彩；他参与它的胜利；作为哲学家，他已经跨越了分隔帝国的边界。他没有敌人，他是所有地方的公民，是所有时代的同代人；他从人类脆弱的起源一直陪伴到其最终的完善。

以下是Éloge de Vauban Ⓣ（向沃邦致敬）[43]中关于几何学的一段引文：-

有一种几何学比欧几里得的几何学更为精妙。这种[新的]自然几何学就是天才本身应用于度量科学。……正是通过这样一种自然几何学，人在任何计算之前就能看到——尽管如同在雾中——一个新假设的结果。这种自然几何学进行创造，另一种只是清理；没有前者，后者便毫无用处。

马克·勒内，蒙塔朗贝尔侯爵（1714-1800）曾改进了沃邦的防御工事，但如今卡诺因改进蒙塔朗贝尔的思想而迅速赢得声誉。1787年，他当选为第戎科学院院士。

1787年5月25日，卡诺在阿拉斯科学院发表演讲，我们再次从中了解到关于他的许多信息[17]：-

只有一种真正可行的道德，那就是教导我们从人类的共同利益中引出我们自身利益的道德。……通过服务于人类共同利益的习惯，通过美德的持续实践，公民达到了一种只有其实践本身才能给予的愉悦。……这是唯一一种不会因时间而陈旧、反而具有通过其实现而自我增长的独特优势的愉悦。当你行善时，你总是想做更多，你总是知道还有更多要做，你永远无法满足。

1789年爆发了法国大革命，同年5月法国成为君主立宪制国家，随后7月14日攻占巴士底狱。这见证了卡诺进入政治舞台。他于1791年当选为立法议会成员，并被任命为公共教育委员会成员。他致力于教育改革，基于他所有公民都应有权接受教育的信念。革命后的混乱动荡使得他的任何改革在当时都无法实施。

在此期间，卡诺于1791年5月17日与Marie Jacqueline Sophie Josèphe Dupont（1764-1813）结婚。这桩婚姻是由相当不寻常的一系列情况促成的。Claude-Marie 卡诺，即卡诺的弟弟，于1790年5月11日在加来海峡省的Saint Martin au Laërt与Marie Adélaïde Françoise Josèphe Dupont（1766-1854）结婚。她是Jacques Antoine Léonard Dupont，Moringhem领主（1731-1808）的女儿，他是一名律师，母亲是Marie Anne Françoise Sevault（1739-1807）。卡诺于1790年9月10日从马上摔下伤了腿，他的妻妹Marie Jacqueline Sophie Josèphe Dupon照料他直到康复，八个月后他们结婚了。他们有三个儿子：萨迪·卡诺，一位重要的数学家，本档案中有其传记，生于1794年7月19日；Nicolas Léonard Sadi Carnot生于1796年6月1日，他成为一名研究热力学的物理学家；以及卡诺 Hippolyte Carnot，生于1801年4月6日，他成为一名记者，后来成为公共教育部长和道德与政治科学院院士。在此我们注意到，Hippolyte Carnot撰写了文章[3]，而Hippolyte的儿子Marie François Sadi Carnot从1887年起担任法国总统，直到1894年遇刺身亡。

当立法议会解散后，卡诺于1792年当选为国民公会成员[44]：-

从1792年9月到11月，卡诺作为公会的特使在比利牛斯山。返回后，他提出了关于这个原本落后的山区经济发展的建议（发展其纺织工业），以便它能够摆脱英国的控制。

他在1793年4月后指挥北方军团，同年成为总防御委员会的主要成员和公共安全委员会成员，这意味着他负责规划军事行动。1793年9月，当冯·科堡将军率领的盟军威胁要向巴黎推进时，卡诺亲自前往前线[44]：-

卡诺作为公共安全委员会成员，从附近站着的士兵手中夺过一支步枪，穿着便服，仅凭帽子上插着的大羽毛辨认，他亲自率领部队发起进攻。格拉蒂安将军认为这次进攻过于大胆，因此卡诺在战场上当场下令逮捕他。这次进攻的心理效果是毁灭性的。科堡第二天向北撤退，巴黎得救了。

1794年9月，在卡诺和加斯帕尔·蒙日的指导下，成立了一所名为“中央公共工程学校”的“大学校”，但其名称在次年改为“巴黎综合理工学院”。

卡诺现在将他的军事战略付诸实践，导致法国击败了盟军。该战略的一部分涉及增强军事力量，但他也致力于其他方式以确保法国获胜，例如争取普鲁士中立以及破坏奥地利与英格兰之间的通信。法国取得了显著胜利，占领了荷兰。这种极为成功的军事领导使卡诺被称为“胜利的组织者”。1794年由保罗·巴拉斯领导的政变结束了恐怖统治，在卡诺的影响下，罗伯斯庇尔被罢免。然而，对卡诺来说，现在一切并不顺利，因为他强烈反对巴拉斯的政策，特别是在他镇压巴黎人民起义时。卡诺因反对巴拉斯而辞去公共安全委员会职务，但在1795年4月11日重新掌权，成为督政府成员，这是一个五人统治委员会。1796年4月30日，他当选为督政府主席。

1797年对卡诺来说是多变的一年。在这一年，他出版了他的著名著作Réflexions sur la métaphysique du calcul infinitésimal Ⓣ（关于微积分形而上学的思考）。这本书是卡诺在1786年提交给柏林科学院奖的论文的扩展版本，但未能获奖，该奖授予了Simon Lhuilier。这本书的广告以以下文字开头：-

然而，由于一切迹象表明数学文化将出现新的转向，作者认为出版这本专著是恰当的。

托瓦尔·尼古拉·蒂勒写道，卡诺的数学方法强烈地显示出他的工程背景。在[49]中，他写道，卡诺：-

... 只接受数学表达式，只要其中包含的量是实数且所涉及的操作有意义。... 对卡诺来说，负量是不可能的，而零，就像无穷大一样，是一个极限。... 无穷小量是真实的对象，可以表示为极限之间的差 ...

Jacques Harthong对卡诺的贡献[33]持更为积极的看法：-

本书的主要兴趣在于揭示一个非常普遍的观念的谬误：即无穷小卡尔·魏尔斯特拉斯只有借助卡尔·魏尔斯特拉斯才可能变得完全严格（也就是说，仅基于逻辑而不诉诸直觉或天赋）。诚然，卡尔·魏尔斯特拉斯的观点在今天占主导地位；它也的确提供了一种令人满意的方法；但同样真实的是，卡诺的观点（此外，他很大程度上得益于约瑟夫·拉格朗日）本可以提供一种同样令人满意的方法，而历史最终做出的选择必须另寻解释。事实上，卡诺提供了一种真正的公理化方法。

关于由牛津大学彭布罗克学院会士、文学硕士W R Browell牧师翻译、于1832年出版的英译本的摘录，见THIS LINK。

同年，即1797年，在皮埃尔·奥热罗发动政变之后，法国的政治局势变得使他无法再坚持其强烈的共和主义观点。1797年9月3日至4日夜间[35]：-

……一队士兵（包括炮兵），在巴拉斯窗下（紧邻卡诺在卢森堡宫的住所，督政官们在那里有他们的公寓）集结后，来到卡诺的前门，同时身份不明的“刺客”在后花园等待。卡诺命令另一位督政官的部队驱散刺客，让他的兄弟拖住前来逮捕他的卫兵，然后从卢森堡花园一个鲜为人知的小门溜走，流亡他乡。

卡诺逃往瑞士，随后前往德国的纽伦堡。

1800年，卡诺在拿破仑的允许下返回法国。拿破仑于1799年成为第一执政。卡诺成为拿破仑·波拿巴的战争部长，为期五个月，并被进一步提升为中将。然而，在1802年5月，他投票反对让拿破仑成为终身执政的提议。1804年12月2日，拿破仑在巴黎圣母院加冕为皇帝。卡诺的共和主义观点使他无法继续任职，他退出了公共生活。

卡诺最为人所知的是作为几何学家。1801年，他出版了De la correlation des figures de géométrieⓉ（几何图形的相关性），在其中他试图将纯几何置于一个普遍的框架中。他证明了欧几里得的Elements中的若干定理可以从一个单一定理建立起来。在这部1801年的著作中，卡诺就分析与综合之间的区别提出了一个独创性的想法。根据他的观点，两者的区别在于，在综合中：-

……人们只能对真实而有效的对象进行推理……

而在分析中，

……我们承认不存在的对象；我们用符号表示它们，也像表示真实的东西一样。我们把真实的存在与理性的存在混在一起；然后通过有条理的变换，我们设法在计算中消去或排除后者：于是公式中不可理解的东西消失了；剩下的东西，一种精细的综合必定能够揭示出来。但这一结果往往是通过一条更短、更容易、几乎纯机械的途径得到的，而若用别的方法去达到它，则要付出惊人的努力。这就是分析的优势，因而也是近代人胜过古代人的优势。

1803年，他出版了Géométrie de positionⓉ（位置几何学），其中首次在几何学中系统地使用了有向量。这部著作大大扩展了他1801年的工作，在其中卡诺再次表明量对他意味着什么，他写道：-

每一个量都是一个真实的对象，使得心灵能够把握它，或者至少能把握它在计算中的表示。

卡诺在导言中写道：-

这部著作的标题可能会让几何学家想起，著名的哥特弗里德·威廉·莱布尼茨曾构想过一种“位置分析”的思想；这一思想并未真正得到发展。……我关于位置几何学的工作不同于“位置分析”；尽管它与后者类似。哥特弗里德·威廉·莱布尼茨希望我们在表达一个几何问题的条件时，把所比较图形相应部分的多种位置纳入其中，以便通过用一种非常独特的特征把它们区分开来，我们就能在计算中更容易地把它们分离出来。如今，这种位置的多样性常常通过符号的简单变化来表达；而正是关于这些变化的理论，构成了我所设想的研究的基本主题，我把它称为“位置几何学”。

他还在1803年出版了Principes fondamentaux de l'équilibre et du mouvement Ⓣ（《平衡与运动基础》）。

卡诺的军事杰作De la défense des places fortes Ⓣ（防御要塞）于1809年出版。他后来担任安特卫普的军事总督，但在拿破仑于滑铁卢最终战败后，他流亡国外。他先前往华沙，随后逃往马格德堡，于1816年11月抵达马格德堡。

随着第一台蒸汽机于1818年运抵马格德堡，卡诺的兴趣转向了蒸汽机。他的儿子萨迪·卡诺于1821年在马格德堡探望了他，显然卡诺影响了他的儿子。萨迪·卡诺三年后出版了他关于蒸汽机热力学的杰作。

让我们以[34]中的这段话作为结束：-

当法国面临其最黑暗的危机、失败几乎已成定局之时，卡诺设法把失败转变为胜利。他做到这一点，是靠展现领导中的“卓越”，是靠对军事战争进行一场革命，动员法国最优秀的科学家，利用巴黎综合理工学院为法国公民的广泛教育奠定基础，同时把他自己在机床和机器制造领域对新科学原理的发现加以运用，从而为法国乃至整个欧洲的工业与技术革命奠定基础。

卡诺确信，组织民族和人民、提升公民使其成为真正建国者的关键，基于领导的道德品质，而这种卓越并非基于对理论和书籍的学术知识，而是尤其在危机、战争和胁迫的条件下显现出来。建国最大的资源和力量，是个体那自主而富有创造力的心灵；面对未知——面对挑战习俗意见的障碍和悖论——这种心灵被迫去寻找创造性的侧翼和大胆的解决方案。卡诺说：“环境有时会在我们身上发展出我们未曾想到其萌芽的才能，使我们的灵魂更伟大，并赋予我们的灵魂以能量。”在他最优秀的将军中，他选择25岁或30岁的人，把责任放在他们肩上，信任他们的想象力和大胆。

那种对于克服障碍和作出发现不可或缺的心灵的情感品质，就是卡诺所称的“热情”——激情。据他的儿子Hippolyte说——他写下了关于其父最有洞见、最精彩的传记——“伟大的激情是伟大整体的灵魂。”“激情是世界上一切美与伟大的唯一原则。”在一首题为《热情颂》的诗中，卡诺写道：

热情，对美的爱！
高贵火焰的原则。……
你不是狂乱的沉醉，
你不是冰冷的理性；
你走得比智慧更远，
却没有超出它的范围。
微妙的本能，它先于
审慎的劝告
和判断的计算。

没有“热情”，就永远不可能在科学或艺术中有创造性的发现。卡诺把心灵的那种创造能力称为“自然几何”，凭借一瞥（coup d'oeil），或一眼，凭借“艺术性的机巧”，心灵形成新的假设。

Lazare Carnot was the son of Claude Abraham Jean Jacques Carnot (1719-1797) and Magdelaine Marguerite Pothier (1726-1788). Claude Carnot was a leading citizen of Nolay, a notary, judge and bailiff for the Marquisate of Nolay. The Marquisate of Nolay had been created in the time of Henry IV of France (1553-1610). Lazare Carnot's paternal grandfather was Jean Carnot (1672-1735), a notary and procurator fiscal. Claude and Marguerite Carnot were married in 1745 and had seventeen children including Joseph-François-Claude Carnot (1852-1835), Lazare Carnot (1753-1823), the subject of this biography, and Claude-Marie Carnot Feulins (1755-1836). Let us note at this point that Joseph-François-Claude Carnot became a jurist, was a member of the Court of Cessation (1801-1835), was elected to the Académie des Sciences Morales et Politiques in 1832 and became an Officer of the Légion d'Honneur. Claude-Marie Carnot Feulins was a soldier. He was a deputy to the Legislative Assembly of 1791, a Representative during the Hundred Days of 1815 and a provisional commissioner in the French Executive Commission of 1815. This brother, two years younger that Lazare, was closest to him among his siblings and they became inseparable. Both of Lazare Carnot's two brothers we just mentioned published several books. None of his relatives that we have mentioned had the name Lazare, but it was a family name. He was [8]:-

... named after an uncle who was a learned doctor of the Sorbonne and a vicar general of the diocese of Chalon.

Lazare's elementary education was first from M Masson, then from M Boisson, both excellent Latinists, at the small college in Nolay. He reached the second rhetoric class under their instruction. In 1767, when he was fourteen years old, Lazare and his older brother were sent to Autun about 25 km due west of Nolay. He boarded with Mme Gondier in Autun and took the third year of rhetoric at the Collège d'Autun, an Oratorian college, studying philosophy and classics. After a year at the Collège d'Autun, he entered the Sulpician Seminary of Autun. This small Seminary was run by the Society of Priests of Saint Sulpice and emphasised academic work. Here Carnot studied mathematics, logic and theology, his teacher being Abbé Bisson. While he was in Autun there was a smallpox epidemic which affected Carnot but, thanks to Mme Gondier and his doctor M Lhomme, he made a quick recovery and his education suffered little [8]:-

In 1768 he passed a brilliant public examination in Latin, declining the usual assistance of the masters in the defence of his thesis, and in the face of learned objections presented by a public auditor, Madame Lhomme, the wife of his physician. At this time the boy was so devout that certain of his relatives suggested holy orders for him.

Lazare Carnot's father Claude worked for the Duke D'Aumont, the Marquis of Nolay, and the Duke, being very impressed with Lazare and feeling that holy orders was not the way he should proceed, recommended to his father that his son would benefit from a military career. The Duke offered to contact the Ministry of War to allow Carnot to be admitted to the engineering corps despite recent moves to limit numbers entering. He began studying at Louis-Siméon Mausserat de Longpré's school in the Marais area of Paris. At this time there were two preparatory schools in Paris which trained boys for engineering, artillery and the navy. Mausserat de Longpré's school was the smaller, and considered the stronger academically, of the two. The director of the school was very friendly with Jean d'Alembert, as Carnot's son Hippolyte Carnot (1801-1888) explained in [3]:-

The illustrious academician would come and sit among the students and enjoy exercising their intelligence by confronting them with embarrassing problems. He had noticed young Carnot ... and predicted great success for him. My father recalled all his life with gratitude the conversations and encouragement of D'Alembert, to whom he attributed a notable influence on his scientific development.

In February 1771 Carnot sat the entrance examination for entering an engineering school and was ranked third in the [8]:-

... class of one hundred and fifteen taking the examination, Carnot was appointed as a second lieutenant to the engineering school at Mézières. He was then eighteen years of age. The examiners said of him: An excellent prospect, very intelligent, knows his course excellently.

Carnot was taught at the School of Engineering in Mézières by Gaspard Monge among others. At Mézières his courses included geometry, mechanics, geometrical designing, geography, and hydraulics. It was around this time that he met Benjamin Franklin who visited France in 1772 while he was the American emissary in England seeking French support for American independence. This friendship was important in the development of Carnot's political ideas which, a few years later, he put into practice.

He graduated from the School in 1773, leaving on 13 February. He then embarked on his military career, appointed as a first lieutenant in the Prince of Condé's engineering corps. He quickly made a name for himself both as an expert on fortification and on dirigible balloons. He would later published important work on both these topics. He also continued to study mathematics.

In 1778 he wrote Essai sur les machines en général Ⓣ which he submitted for a prize in a competition. He revised it in 1781 and it was eventually published in 1783. It deals with mechanics and areas of engineering. The Preface begins:-

Although the theory in question here is applicable to all questions concerning the communication of movements, this pamphlet has been given the title of 'Essay on Machines in General'; first, because it is mainly the Machines that we have in view, as being the most important object of mechanics; and secondly, because there is no question of any particular Machine, but only of the properties which are common to all. This theory is based on three main definitions; the first looks at certain movements which I call geometrical, because they can be determined by only the principles of geometry, and are absolutely independent of the rules of Dynamics; I did not believe that we could easily do without it, without leaving something suspicious in the statement of the main proportions as I show in particular for the principle of Descartes. By the second of my definitions, I try to fix the meaning of the terms soliciting force and resistant force: one cannot, it seems to me, clearly compare the causes with the effects in the Machines, without a well-characterised distinction between these different forces; and it is this distinction on which it seems to me that something has always been left vague and indeterminate.

In 1783 he published Éloge de Vauban Ⓣ. At the session of 3 December 1778, the Academy of Sciences, Arts and Belles Lettres of Dijon had proposed the "Éloge du Maréchal Vauban" for the literary competition of 1784. Sébastien, Marquis de Vauban (1633-1707), was considered to have been the leading French engineer whose system of fortifications had, in his day, been considered the best. Vauban was from Burgundy and the Dijon Academy thought he deserved a better éloge than Bernard de Fontenelle had given him in the Paris Academy of Sciences. Carnot, also from Burgundy, and now an expert on fortifications had entered the Dijon Academy's competition. Carnot, now captain in the Corps du Royal Génie in the garrison at Arras, received two gold medals and the Academy's prize on 2 August 1784, the presentation being by the Prince of Condé himself. Carnot's Éloge de Vauban Ⓣ was praised for its enthusiasm, its warmth of expression, and its compassion and he was described as learned and eloquent. Carnot replied to the Prince [26]:-

Monseigneur, it is very sweet to be crowned by a hero with the name of Condé, and the laurels which your hand dispenses, just as those which decorate your august forehead, are of a species never to wither.

Let us quote from the Éloge de Vauban Ⓣ (see, for example, [17]) for we find there things about Carnot's character and his attitude towards his country:-

How rare it is for a wise man to reap the fruit of his works! He is ahead of his century, and his words can only be understood by posterity; but that is enough support for him; his imagination breaks through the shades of error; he is the friend of men yet to be born, he converses with them in his deepest research; as a citizen, he looks to his nation, his hopes are for it, he applauds its successes; he takes part in its triumphs; as philosopher, he has already crossed the boundaries separating empires. He has no enemies, he is a citizen of all places and contemporary of all times; he stays with man from his frail origin until his final perfection.

Here is a quote about geometry from the Éloge de Vauban Ⓣ [43]:-

There is a geometry more subtle than that of Euclid. This [new] natural geometry is genius itself applied to the science of measure. ... It is through such a natural geometry that man sees, although as in a fog, the results of a new hypothesis, before any calculation. This natural geometry creates, the other just cleans; without the first, the second is useless.

Marc René, Marquis de Montalembert (1714-1800) had improved on Vauban's fortifications but now Carnot was quickly gaining a reputation for improving on Montalembert's ideas. In 1787 he was elected a member of the Dijon Academy.

On 25 May 1787 Carnot addressed the Academy of Arras and again we learn much about him from it [17]:-

There is only one true practicable morality, it is the one that teaches us to draw our particular interest from the common interest of mankind. ... By the habit of serving the common interest of mankind, through the constant practice of virtue, the citizen arrives at a type of pleasure that only its very practice can give. ... It is the only pleasure that, rather than becoming time-worn, has the unique advantage of increasing itself through its fulfilment. When you do the good, you always want to do more, you always know that there is much more to do and you can never be satisfied.

The year 1789 saw the outbreak of the French Revolution when in May of that year France became a constitutional monarchy followed by the storming of the Bastille on 14 July. This saw Carnot enter the world of politics. He was elected to the Legislative Assembly in 1791 and was appointed to their Committee for Public Instruction. He worked on educational reforms based on his beliefs that all citizens should be entitled to an education. The chaotic upheavals following the Revolution prevented any of his reforms being implemented at this time.

During this period, Carnot married Marie Jacqueline Sophie Josèphe Dupont (1764-1813) on 17 May 1791. This marriage came about due to rather unusual set of circumstances. Claude-Marie Carnot, Lazare's younger brother, married Marie Adélaïde Françoise Josèphe Dupont (1766-1854) at Saint Martin au Laërt, Pas de Calais, on 11 May 1790. She was the daughter of Jacques Antoine Léonard Dupont, Lord of Moringhem (1731-1808), who was a lawyer, and Marie Anne Françoise Sevault (1739-1807). Lazare Carnot injured his leg with a fall from his horse on 10 September 1790 and his sister-in-law, Marie Jacqueline Sophie Josèphe Dupon, nursed him until he recovered and eight months later they were married. They had three sons: Sadi Carnot, an important mathematician with a biography in this archive, was born on 19 July 1794; Nicolas Léonard Sadi Carnot was born on 1 June 1796, he became a physicist who worked on thermodynamics; and Lazare Hippolyte Carnot, born 6 April 1801, who became a journalist and then Minister of Public Instruction and a member of the Academy of Moral and Political Sciences. Let us note at this point that Hippolyte Carnot wrote the article [3] and Hippolyte's son Marie François Sadi Carnot was President of France from 1887 until his assassination in 1894.

When the Legislative Assembly was dissolved, Carnot was elected to the National Convention in 1792 [44]:-

From September to November 1792, Carnot was in the Pyrenees as a special emissary of the Convention. After his return, he presented proposals for the economic development of the otherwise backward mountain region (developing its textile industry) so that it could free itself from British domination.

He directed the Army of the North after April 1793 becoming in that year a leading member of the Committee of General Defence and a member of the Committee of Public Safety which meant he was responsible for planning military operations. In September 1793, with the allies under General von Coburg threatening to advance on Paris, Carnot went to the front himself [44]:-

Carnot, as a member of the Committee of Public Safety, grabbed a rifle from a soldier who was standing nearby, and, in his civilian clothes, recognisable only by the large feather he wore in his hat, he himself led the attack at the head of his troops. General Gratien thought this attack was too daring, so Carnot ordered his arrest right there on the battlefield. The psychological effect of the attack was devastating. Coburg withdrew to the north the next day, and Paris was saved.

In September 1794, under direction from Carnot and Monge, a 'Grande École' was set up called 'École Centrale des Travaux Publics' but its name was changed to 'École Polytechnique' in the following year.

Carnot now put into practice his military strategy which led to the French inflicting defeats on the allies. Part of the strategy involved building up the military forces but he also worked on other ways to see France victorious such as obtaining Prussian neutrality and disrupting communications between Austria and England. France won notable victories occupying the Netherlands. This highly successful military leadership led to Carnot being known as the 'Organiser of Victory'. A coup led by Paul Barras in 1794 led to the end of the Reign of Terror and under Carnot's influence Robespierre was removed. All was now not well for Carnot, however, for he strongly opposed the policies of Barras particularly when he put down a revolt of the people of Paris. Carnot resigned from the Committee of Public Safety because of his opposition to Barras but returned to power on 11 April 1795 when he became a member of the Directory, a five-man ruling committee. On 30 April 1796 he was elected President of the Directory.

The year 1797 was an eventful one for Carnot. In this year he published his famous text Réflexions sur la métaphysique du calcul infinitésimal Ⓣ. The book was an extended version of the essay which Carnot had submitted for the Berlin Academy Prize in 1786 but had failed to win, the prize being awarded to Simon Lhuilier. The advertisement for the book is introduced with the words:-

As however everything indicates that there will be a new turn in the culture of mathematics, the author deems it apposite to publish this monograph.

Thiele writes that Carnot's approach to mathematics shows strongly his engineering background. In [49] he writes that Carnot:-

... accepted mathematical expressions only insofar as the quantities contained in them were real and the operations involved held meaning. ... to Carnot negative quantities are impossible, and zero, just like infinity, is a limit. ... infinitely small quantities are real objects, being representable as differences between limits ...

Jacques Harthong takes a much more positive view of Carnot's contribution [33]:-

The main interest of this book is to reveal the fallacy of a very common idea: that the infinitesimal calculus would not have become fully rigorous (that is to say, based on logic alone without any recourse to intuition or flair) only with Weierstrass. It is true that Weierstrass's point of view is the dominant one today; it is also true that it gives a satisfactory approach; but it is no less true that that of Carnot (who, moreover, owes a lot to Lagrange) could have given a just as satisfactory approach, and another explanation must be found for the choice ultimately made by history. Indeed Carnot offers a true axiomatic approach.

For extracts from the English translation by the Rev W R Browell M.A., Fellow of Pembroke College, Oxford, published in 1832, see THIS LINK.

In the same year, 1797, following the coup by Pierre Augereau, the political situation in France became such that he could no longer remain with his strong republican views. On the night of 3-4 September 1797 [35]:-

... a body of troops (including artillery), having rallied under Barras' window (adjacent to Carnot's in the Luxembourg Palace where the Directors had their apartments), came to Carnot's front door while unidentified "assassins" waited at the back in the garden. Carnot ordered some other Director's troops to disperse the assassins, got his brother to slow up the guards coming to arrest him, and slipped through a little-known gate in the Luxembourg gardens away into exile.

Carnot fled to Switzerland going on to Nuremberg in Germany.

In 1800 Carnot returned to France with Napoleon's permission. Napoleon had became First Consul in 1799. Carnot became Napoleon Bonaparte's minister of war for a period of five months and was promoted further to the rank of lieutenant-general. In May 1802, however, he voted against the proposal to make Napoleon consul for life. On 2 December 1804 Napoleon was crowned emperor in Notre Dame. Carnot's republican views made further service impossible and he retired from public life.

Carnot is best known as a geometer. In 1801 he published De la correlation des figures de géométrie Ⓣ in which he tried to put pure geometry into a universal setting. He showed that several of the theorems of Euclid's Elements can be established from a single theorem. In this 1801 work, Carnot proposes an original idea regarding the difference between analysis and synthesis. According to him, they are distinguished from each other by the fact that, in synthesis:-

... one can never reason except on real and effective objects ...

while in analysis,

... we admit objects that do not exist; we represent them by symbols as well as what is real. We mix real beings with beings of reason; then by methodical transformations, we manage to eliminate or drive out the latter from the calculation: then what was unintelligible in the formulas disappears; there remains what a subtle synthesis would surely have been able to reveal. But this result has often been obtained by a shorter route, more easy, and almost in a purely mechanical way, when it would have taken prodigious efforts to achieve it otherwise. Such is the advantage of analysis, and consequently that of the moderns over the ancients.

In 1803 he published Géométrie de position Ⓣ in which sensed magnitudes were first used systematically in geometry. This work greatly extended his work of 1801 and in it Carnot again shows what quantities mean to him writing:-

Every quantity is a real object such that the mind can grasp it or at least its representation in calculation.

Carnot writes in the Introduction:-

The title of this work may remind Geometers that the illustrious Leibniz had conceived the idea of an 'Analyse de situation'; an idea which has not been really developed. ... My work on the geometry of position differs from that of 'Analyse de situation'; although it is analogous to it. Leibniz wanted us to include in the expression of the conditions of a geometric problem, the diversity of positions of the corresponding parts of the compared figures, so that by separating them by a very distinctive character, we could isolate them more easily in the calculation. Now this diversity of positions is often expressed by simple changes of signs; and it is precisely the theory of these changes that is the essential subject of the research that I have in view, and which I call 'Géométrie de position'.

He also published Principes fondamentaux de l'équilibre et du mouvement Ⓣ in 1803.

Carnot's military masterpiece De la défense des places fortes Ⓣ was published in 1809. He later served as military governor of Antwerp but after Napoleon's final defeat at Waterloo he went into exile. He fled to Magdeburg, after going first to Warsaw, arriving in Magdeburg in November 1816.

Carnot's interests turned toward the steam engine with the first steam engine coming to Magdeburg in 1818. His son Sadi Carnot visited him in Magdeburg in 1821 and it is clear that Lazare Carnot influenced his son. Sadi Carnot published his masterpiece on the thermodynamics of the steam engine three years later.

Let us end with this quote from [34]:-

At the moment when France faced its darkest crisis, when defeat was almost certain, Carnot managed to turn defeat into victory. He did it by showing "excellence" in leadership, by making a revolution in military warfare, mobilising the best scientists of France, making use of the École Polytechnique so as to lay the foundation for a broad-based education of French citizens, while using his own discoveries of new scientific principles in the field of machine-tools and machine building, so as to create the basis for an industrial and technological revolution in France and Europe as a whole.

Carnot was convinced that the key to organise nations and people, to elevate citizens to become true nation-builders, is based on the moral quality of leadership whose excellence is not based in the academic knowledge of theories and books, but which shows in particular under conditions of crisis, wars, and duress. The biggest resource and strength in building nations is the sovereign, creative mind of the individual who, faced with the unknown - with obstacles and paradoxes that challenge customary opinion - is forced to look for creative flanks and bold solutions. "Circumstances develop sometimes faculties in us whose germ we did not think of, making our souls greater and giving our souls energy," Carnot said. Among his most excellent generals, he chose people at the age of 25 or 30, upon whose shoulder he put responsibility, having confidence in their powers of imagination and boldness.

That emotional quality of the mind, indispensable for overcoming obstacles and making discoveries, is what Carnot calls "enthusiasm" - passion. According to his son Hippolyte, who wrote the most insightful and wonderful biography of his father, "A great passion is the soul of the great totality." "Passion is the unique principle of all that is beautiful and great in the world." In a poem called "Ode to Enthusiasm," Carnot writes:

Enthusiasm, love of beauty!
Principle of noble flames. ...
You are not raving drunkenness,
you are not cold reason;
you go further than wisdom,
without exceeding its extent.
Delicate instinct which precedes
both the counsel of prudence
and the calculations of judgment.

Without "enthusiasm," there can never be a creative discovery in science or art. Carnot calls that creative capacity of the mind the "natural geometry," where, with a 'coup d'oeil', or glance, with "artistic ingenuity" the mind forms new hypothesis.