让-维克托·彭赛列Jean-Victor Poncelet

让-维克托·彭赛列的父亲是Claude Poncelet，一位富有的地主，也是梅斯议会的律师。他的母亲是Anne-Marie Perrein，但彭赛列是个私生子，尽管他出生在梅斯，但不到一岁时就被送走，由圣阿沃尔德（梅斯以东的一个城镇）的Olier家族抚养。我们还应补充，很久以后Claude Poncelet娶了Anne-Marie Perrein，使彭赛列从那时起成为合法子女。Olier家族充满爱意地照料他，他与他们一起生活直到1804年他15岁。对彭赛列来说这是一段快乐的时光，他对周围的一切都表现出极大的好奇心，尤其喜爱机械物件，他花了许多快乐的时光摆弄一个为他买的钟的机械装置。

他十五岁时，彭赛列回到梅斯，在当地的中学学习，参加为准备学生报考巴黎高等师范学校和巴黎综合理工学院入学考试而设的特别班。他于1807年进入巴黎综合理工学院，在那里遇到了杰出的教师，如数学家加斯帕尔·蒙日、拉扎尔·卡诺、夏尔·朱利安·布利安生、西尔维斯特·佛朗索瓦·拉克鲁瓦、安德烈-马里·安培、路易·普安索和Jean Hachette。然而他健康状况不佳，错过了第三年的大部分学习。他于1810年从巴黎综合理工学院毕业，时年22岁，由于健康问题多读了一年，比通常年龄大，并决定从事军事生涯。他加入了工程兵部队，前往梅斯在应用学校学习。经过两年的学习，他毕业了，获得了中尉军衔，并于1812年3月被分配的第一项任务是负责斯海尔德河（或埃斯科河）河口瓦尔赫伦岛上拉梅肯斯的防御工事[21]：

他在这里的第一项工程工作是在极短时间内，在一片泥炭土上，在无法获得合适地基材料的情况下，建造一座有掩体的堡垒。

然而，他于1812年6月被调离该任务，去参加拿破仑的俄国战役。

彭赛列在维捷布斯克镇加入了拿破仑的60万大军，当时军队正逼近俄国。到8月18日，军队已接近斯摩棱斯克，这是第一座真正的俄罗斯城市，彭赛列不顾守军炮火侦察了该城。当天晚些时候，他积极参与了战斗，随后在第二天，他负责在斯摩棱斯克下游的第聂伯河上架桥。他不仅要克服架桥的问题，还必须在对岸俄军炮火下进行。他设计了一个计划，将俄军的注意力引向某个特定的渡口，同时在不同的地点组织架桥。直到1812年9月，俄军才与法军全面交战，并在博罗季诺战役中被击败。彭赛列随军在莫斯科度过了令人沮丧的五周，然后于10月19日拿破仑下令军队撤退。随后俄军攻击了撤退中的法军，彭赛列在11月19日斯摩棱斯克不远的克拉斯诺伊战役后，被遗弃在战场上等死[21]：-

在这场战斗中，彭赛列率领一队工兵和地雷工兵冲向俄军炮台；他的坐骑在他身下被击毙……

他能活下来极其幸运，但更大的苦难还在后头[6]：-

他被敌军士兵救起，只是因为他们认为作为一名军官，他可能提供有用的情报。作为战俘，他被迫在冰冻的平原上行进了近五个月，前往伏尔加河畔的监狱[萨拉托夫]。起初，他疲惫、寒冷和饥饿得甚至无法思考；但当春天来临（“灿烂的四月阳光”），他决心利用时间回忆他所受数学教育的一切。后来他为此道歉说：“被剥夺了书籍和各种舒适，尤其因我的国家和我个人的不幸而痛苦，我未能使这些研究达到适当的完善。”

他从1813年3月被关押在监狱，直到1814年6月返回法国。在监禁期间，他回忆了几何学的基本原理，但忘记了从加斯帕尔·蒙日、拉扎尔·卡诺和夏尔·朱利安·布利安生那里学到的细节，进而发展了圆锥曲线的projective性质。他将所做的笔记称为“萨拉托夫笔记本”，但直到五十年后，他才将所写的大部分内容纳入其关于解析几何的论著Applications d'analyse et de géométrie Ⓣ（分析与几何的应用）（1862年）。他对与圆锥曲线相关的极点和极线的研究导致了对偶原理，但正如我们下面所解释的，这引发了一场优先权争议。他还发现了无穷远处的虚圆点。首先，我们看看朱利安·罗威尔·柯立芝关于彭赛列的启发性的工作所写的内容[15]：-

彭赛列为自己设定的基本问题是研究图形的图形性质，他将这些性质定义为不涉及距离或角度大小的性质。两点之间的距离不是射影不变的，但在寻找射影不变的构型时，他发现了调和构型，并对此进行了详细发展。……在第二章中，彭赛列以超越其前辈的勇气和彻底性，攻克了纯几何中虚点的问题。……他相当随意地做出了一个历史性陈述：同一平面上的两个圆不应被视为完全独立的图形，而应被视为共有两个虚无穷远点。这里首次宣告了度量几何的一条基本原理。后来，彭赛列在没有仔细定义的情况下允许了虚射影。

1814年5月30日，《巴黎条约》签署，法国与俄罗斯（以及卷入冲突的其他国家）实现和平。几天后，彭赛列从萨拉托夫监狱获释，但他直到当年9月才抵达法国。自1815年起，他在梅斯任教。他于1822年出版了Traité des propriétés projectives des figures Ⓣ（《图形的射影性质论》），这是一项关于在射影下保持不变的性质的研究。注意，该著作的副标题是“一部对研究描述几何应用及土地几何操作的人有用的著作”，在这里可以看到加斯帕尔·蒙日教学的影响。这部著作包含了射影几何的基本思想，如交比、透视、对合和无穷远圆点。在撰写此书期间，他与弗朗索瓦-约瑟夫·塞尔瓦进行了商讨，他在梅斯工作时就认识弗朗索瓦-约瑟夫·塞尔瓦，但后者已于1816年迁往巴黎。在Traité des propriétés projectives des figures Ⓣ（《图形的射影性质论》）中，彭赛列写道：-

如果一个图形由另一个图形经过连续变化得到，并且后者与前者同样一般，那么前者的任何性质都可以立即对后者断言。

他通过首先指出欧几里得几何中的定理来说明这一技巧，该定理指出圆内相交弦的线段乘积为常数。彭赛列随后运用他的原理表明，如果交点被认为在圆外，就得到割线及其外段乘积为常数的定理。彭赛列说，无需证明，因为只需使用欧几里得定理并援引他的连续性原理。值得指出的是，我们的术语“射影几何”来自这本书的标题，这相当恰当，因为彭赛列是与约瑟夫·热尔岗同时发现的现代射影几何的创始人之一。让我们简要看看安德雷·柯尔莫哥洛夫的描述[4]：-

彭赛列 证明了圆锥曲线（conic）是一个射影图形，并且要解决圆锥曲线中的一个困难问题，应当对圆锥曲线进行射影，将问题化为圆的问题求解，然后再进行逆射影。由于“映射平面”上平行线的“会聚点”并不对应于射影平面的实点，彭赛列 在所有平面上添加了“理想”点或“无穷远”点，这些点射影为“会聚点”。彭赛列 利用 拉扎尔·卡诺 的配极原理引入了无穷远点，他称之为“连续性原理”。彭赛列 发展了 拉扎尔·卡诺 关于“复配极”的思想，引入了平面的虚点，特别是虚无穷远点，例如“圆点”——属于平面上所有圆的点。两条圆锥曲线可以交于四个实点或虚点，而两个圆则交于两个实点和两个圆点。

此处一段未译出，以下为英文原文 The principle of continuity caused some disputes. In particular Augustin-Louis Cauchy, writing a report on Poncelet's work on 5 June 1820, claimed that the principle of continuity was "capable of leading to manifest errors". He gave an example to show that the principle was false, but his example was not correct. This was not the only dispute that Poncelet was involved in. Articles appearing in Joseph Gergonne's Annales des Mathématique which used the principle of duality gave Poncelet little credit. He protested his priority to Gergonne in December 1826 and his comments were published in March 1827 accompanied by critical remarks added by Gergonne. The priority dispute about duality lasted until May 1829 and also involved Julius Plücker. It pushed Poncelet away from his work on projective geometry and towards mechanics.

从1815年到1825年，他是梅斯的工兵上尉，负责监督梅斯兵工厂的机械建造，并在军事学院教授力学。在此期间，弗朗索瓦·阿拉戈 敦促他接受梅斯力学教授的职位，但他犹豫了一段时间。最终在1824年5月1日，他同意了，并于1825年1月上任。他担任这个职位十年。他应用力学来改进涡轮机和水车，使水车的效率提高了一倍多 [5]：-

彭赛列 熟悉 让-夏尔·德博尔达 的工作，以及高效水车必须让水无速度进入且无冲击离开的必要性。他在重新设计下射式水车时面临的基本问题是如何在保留传统结构的实用优势——简单、建造成本低、转速高——的同时实现这一点。彭赛列 宣称：“经过思考，我觉得我们可以通过用弯曲或圆柱形叶片替换普通轮子上的直叶片，使其凹面朝向水流，来满足这个双重条件。”因此，在1823年，彭赛列 取来旧的下射式水车，用弯曲叶片替换其平坦的径向叶片，并调整其闸门角度，使水尽可能接近下部叶片。这些改变产生了一个具有下射式水车所有优点且效率相对较高的轮子。

这些想法由 彭赛列 于1826年发表，并获得了法国政府的奖励。我们很难理解这项工作的重要性，因为当时许多工业都由水车驱动。从1831年5月开始，他还与 阿蒂尔·莫兰 合作进行摩擦实验。他们的工作证实并扩展了 夏尔·奥古斯丁·德·库仑 关于摩擦的工作，验证了他提出的三条一般定律。

彭赛列于1831年晋升为营长，然后于1834年搬到巴黎，同年3月他当选为Académie des Sciences力学部门的成员。他在射影几何方面的工作争议太大，尤其是在奥古斯丁·路易·柯西早些时候对其发起攻击之后，使他无法凭借这些贡献进入科学院。次年，他成为索邦大学的力学教授。他从1835年到1848年任职于巴黎防御工事委员会。1842年，彭赛列与Louise Palmyre Gaudin结婚，他的本意是过更安静的生活，但事件接连发生，使这一愿望多年未能实现。1841年他成为中校，三年后成为上校，并于1848年4月19日成为旅长。他还在1848年4月成为巴黎综合理工学院的院长，任职至1850年。在他担任此职期间，1848年6月23日至26日，巴黎发生了法国工人的“六月起义”。街垒被设置起来，军队攻击工人，造成大量人员伤亡。彭赛列作为巴黎综合理工学院的院长，带领他的学生穿过街垒前往卢森堡宫，在那里他们保护了临时政府。路易·欧仁·卡芬雅克，已成为法国国家元首并领导镇压起义，为表彰彭赛列的支持，任命他指挥塞纳省国民自卫军。他后来当选为制宪议会成员。1849年，彭赛列和阿蒂尔·莫兰发明了旋转测力计，加上后来的改进，成为研究功的基本工具。

1851年，万国工业产品大博览会在伦敦海德公园举行。彭赛列被任命为博览会科学委员会主席。回国后，他撰写了一份关于19世纪上半叶科学应用进展的报告，特别提到了他在展览中看到的英国机械和工具。受1851年大博览会的推动，法国组织了第一届世界博览会，于1855年5月在巴黎开幕。彭赛列也在这次博览会中发挥了重要作用。

除了我们提到的那些，彭赛列 还发表了许多关于几何学和力学的文章，特别是在 约瑟夫·热尔岗 的 Annales des Mathématique 和 奥古斯都·利奥波德·克雷勒 的期刊上。他在梅斯讲授的课程最初以石印形式出现，经过一系列版本后，最终出版。例如，《应用于机器的力学》课程最初于1826年作为石印笔记出现，1832年再次作为第二版，然后在朋友 阿蒂尔·莫兰 的协助下于1836年出了第三版定稿。然而，这些笔记直到1874年才正式出版。课程笔记 Mécanique industrielle Ⓣ（工业力学）
经历了类似的过程。他还撰写了许多报告和回忆录，发表在《工兵纪念刊》和《防御工事委员会意见》上。例如，我们提到 Sustaining Walls; Geometrical Constructions to Determine Their Thickness Under Various Circumstances: Geometrical Constructions to Determine Their Thickness Under Various Circumstances（1845年）和 Memoir Upon the Stability of Revetments and of Their Foundations。1850年退休后的出版物包括两卷本的 Applications d'analyse et de géométrie Ⓣ（分析与几何的应用）：1862年和1864年。他还在1865-66年出版了 Traité des propriétés projectives des figures Ⓣ（图形射影性质论）的第二版，该版于1995年重印。以下是第二版的内容列表：

（1）一般原理（包括中心投影的初步概念；圆锥曲线的割线和理想弦的初步概念；以及与平面图形投影相关的原理）。

(2) 直线、圆与圆锥曲线的基本性质（包括直尺与截线几何；圆锥曲线的内接与外切图形；互反极点与极线；以及相似与位似、相似中心）。

(3) 圆锥曲线系统（包括同源图形、同源中心与轴，特别是对于圆锥曲线；圆锥曲线的完全系统；以及圆锥曲线的双重接触）。

(4) 关于角与多边形（本节包含圆锥曲线焦点的射影定义）。

(5) 中调和中心的一般理论。

(6) 互反极线的一般理论。

(7) 截线分析应用于几何曲线与曲面。

（8）任意阶几何曲线和曲面系统共有的性质。

除了当选为我们上面提到的科学院之外，彭赛列还获得了许多荣誉。他是荣誉军团军官和普鲁士勋章骑士。许多科学院和学术团体选举他为成员，包括伦敦皇家学会、Berlin Academy of Science、Imperial Academy of Sciences of St Petersburg以及都灵科学院。

经过长期而痛苦的疾病之后，彭赛列 于1867年12月去世。次年，他的妻子为执行 彭赛列 推进科学的遗愿而设立了 彭赛列 奖。该奖项后来又追加了一笔款项，从1876年起由 科学院 授予纯数学或力学方面的工作。他未发表的手稿一直保存到第一次世界大战，当时它们消失了，此后一直未被找到。遗憾的是，它们极有可能是在那时被毁掉了。

Jean-Victor Poncelet's father was Claude Poncelet, a rich landowner who was a lawyer at the Parliament of Metz. His mother was Anne-Marie Perrein, but Jean-Victor was an illegitimate child and, although he was born in Metz, he was sent away before he was a year old to be brought up by the Olier family in Saint-Avold, a town to the east of Metz. We should add that much later Claude Poncelet married Anne-Marie Perrein making Jean-Victor legitimate from that time. He was cared for with much love and affection by the Olier family and he lived with them until 1804 when he reached the age of 15. It was a happy time for Poncelet, who showed great curiosity for all things around him, particularly a love of mechanical objects and he spent many happy hours playing with the mechanism of a clock which had been bought for him.

When he was fifteen years old, Poncelet returned to Metz where he studied at the lycée taking the special classes designed to prepare students to take the entrance examinations for the École Normale and the École Polytechnique. He entered the École Polytechnique in 1807, and there he had outstanding teachers such as the mathematicians Gaspard Monge, Lazare Carnot, Charles Brianchon, Sylvestre Lacroix, André-Marie Ampère, Louis Poinsot, and Jean Hachette. However his health was poor and he missed most of his third year of study. He graduated from the École Polytechnique in 1810 at the age of 22, older than was usual due to taking an extra year because of his health problems, and decided on a military career. He joined the Engineering Corps and went to Metz to study at the École d'Application. After two years of study he graduated, having reached the rank of Lieutenant and, in March 1812, was given as a first assignment work on the fortifications of Ramekens on the island of Walcheren in the estuary of the river Scheldt (or Escaut) [21]:-

His first engineering work here was the erection of a casemated fort in a very limited time, on a peat soil, without having at his command proper materials for a foundation.

However he was called away from that assignment in June 1812 to take part in Napoleon's Russian campaign.

Poncelet joined Napoleon's army of 600,000 men at the town of Vitepsk as it was approaching Russia. By 18 August the army was nearing Smolensk, the first genuinely Russian city, and Poncelet reconnoitred the city despite being under fire from the defending garrison. He was actively involved in the fighting later that day, then on the following day he was responsible for constructing bridges over the Dnieper River below Smolensk. Not only did he have to overcome problems of constructing bridges but he had to do so while under fire from Russian guns on the opposite bank. He devised a plan to divert the attention of the Russians to a particular crossing point while he organised building bridges at a different location. It was not until September 1812 that the Russian army fully engaged with the French and were defeated at the Battle of Borodino. Poncelet spent five frustrating weeks with the army in Moscow, then on 19 October Napoleon ordered the army to withdraw. The Russians then attacked the retreating French army and Poncelet was left for dead on the battlefield following the Battle of Krasnoi, not far from Smolensk, on 19 November [21]:-

In this battle, Poncelet charged the Russian batteries at the head of a column of sappers and miners; his horse was killed under him ...

He was extremely fortunate to survive but had great hardships to come [6]:-

He was picked up by enemy soldiers only because they thought that being an officer he might be able to give useful information. As a prisoner of war, he was forced to march for nearly five months across frozen plains to his prison [Saratov] on the banks of the Volga. At first he was too exhausted, cold and hungry even to think; but when the spring came ("the splendid April sun"), he resolved to utilise his time by recalling all he could of his mathematical education. Later he was to apologise that "deprived of books and comforts of all sorts, distressed above all by the misfortune of my country and my own lot, I was not able to bring these studies to a proper perfection."

He was held in the prison from March 1813 to June 1814 when he returned to France. During his imprisonment he recalled the fundamental principles of geometry but, forgetting the details of what he had learnt from Monge, Carnot and Brianchon, he went on to develop projective properties of conics. He called the notes that he made the 'Saratov notebook,' but it was only fifty years later that he incorporated much of what he had written in his treatise on analytic geometry Applications d'analyse et de géométrie Ⓣ (1862). His development of the pole and polar lines associated with conics led to the principle of duality but this, as we explain below, led to a priority dispute. He also discovered circular points at infinity. First we look at what Julian Coolidge writes about Poncelet's inspired work [15]:-

The fundamental problem which Poncelet sets himself is to study the graphical properties of figures which he defines as those which do not involve the magnitude either of distances or of angles. The distance of two points is not projectively invariant, but in looking for projectively invariant configurations he finds the harmonic one, and this he develops at length. ... In his second chapter Poncelet attacks the problem of imaginary points in pure geometry with a courage and thoroughness ahead of anything shown by his predecessors. ... he makes quite casually the historic statement that two coplanar circles should not be looked upon as completely independent figures, but as having two imaginary infinite points in common. Here we have the first announcement of one of the basic principles of metrical geometry. Later Poncelet allows, without careful definition, imaginary projections.

On 30 May 1814 the Treaty of Paris was signed making peace between France and Russia (and the other countries involved in the conflict). A few days later Poncelet was released from Saratov prison but it took him until September of that year before he reached France. From 1815 he taught at Metz. He published Traité des propriétés projectives des figures Ⓣ in 1822, which is a study of those properties which remain invariant under projection. Note that the work was subtitled "A work of utility for those studying the applications of descriptive geometry and geometric operations on land" and here one can see the influence of Monge's teaching. This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity. While writing this book he consulted with François Servois whom he had known while he worked at Metz but who had moved to Paris in 1816. In Traité des propriétés projectives des figures Ⓣ, Poncelet wrote:-

If one figure is derived from another by a continuous change and the latter is as general as the former, then any property of the first figure can be asserted at once for the second figure.

He illustrated this technique by first noting the theorem from Euclidean geometry which states that the product of segments of intersecting chords in a circle is constant. Poncelet then used his principle to show that if the point of intersection is considered to be outside the circle, one obtains the theorem that the product of the secants and their external segments are constant. No proof is required, Poncelet says, for one simply uses the Euclidean theorem and invokes his principle of continuity. It is worth remarking that our term "projective geometry" comes from the title of this book, which is quite appropriate since Poncelet was one of the founders of modern projective geometry simultaneously discovered by Joseph Gergonne. Let us look briefly at Andrei Nikolaevich Kolmogorov's description [4]:-

Poncelet showed that a conic section (conic) is a projective figure and that to solve a difficult problem in conics, one should project the conic, solve the problem for the circle, and then carry out the inverse projection. Since the "points of convergence" of parallel lines on the "mapped plane" do not correspond to real points of the projective plane, Poncelet added "ideal" or "infinitely distant" points to all planes, points that project to "points of convergence." Poncelet introduced infinitely distant points using Carnot's principle of correlation, which he called "the principle of continuity." Developing an idea of Carnot on "complex correlation," Poncelet introduced imaginary points of the plane, and, in particular, imaginary infinitely distant points, such as, for example, "cyclic points" - points belonging to all circles in the plane. Two conics can intersect in four real or imaginary points, and two circles in two real and two cyclic points.

The principle of continuity caused some disputes. In particular Augustin-Louis Cauchy, writing a report on Poncelet's work on 5 June 1820, claimed that the principle of continuity was "capable of leading to manifest errors". He gave an example to show that the principle was false, but his example was not correct. This was not the only dispute that Poncelet was involved in. Articles appearing in Joseph Gergonne's Annales des Mathématique which used the principle of duality gave Poncelet little credit. He protested his priority to Gergonne in December 1826 and his comments were published in March 1827 accompanied by critical remarks added by Gergonne. The priority dispute about duality lasted until May 1829 and also involved Julius Plücker. It pushed Poncelet away from his work on projective geometry and towards mechanics.

From 1815 to 1825 he was a Captain of Engineers at Metz, overseeing the construction of machinery in the arsenal at Metz and teaching mechanics in the military college. During this time François Arago urged him to accept the position of Professor of Mechanics at Metz but for a while he hesitated. Finally on 1 May 1824 he agreed, taking up his duties in January 1825. He held this position for ten years. He applied mechanics to improve turbines and waterwheels more than doubling the efficiency of the waterwheel [5]:-

Poncelet was familiar with Borda's work and the necessity for an efficient water wheel of having water enter without velocity and leave without impact. The basic problem he faced in redesigning the undershot wheel was how to accomplish this while retaining the practical advantages of the traditional construction - simplicity, low construction costs, high rotational velocity. Poncelet declared: "After having reflected on this, it seemed to me that we could fulfil this double condition by replacing the straight blades on ordinary wheels with curved or cylindrical blades, presenting their concavity to the current." Thus, in 1823, Poncelet took the old undershot wheel and replaced its flat, radial blades with curved blades and angled its sluice gate to bring the water as close to the lower blades as possible. These changes produced a wheel with all of the advantages of the undershot wheel plus a relatively high efficiency.

These ideas were published by Poncelet in 1826 and were awarded a prize by the French government. It is hard for us to understand how important this work was for at this time much of industry was powered by waterwheels. He also collaborated with Arthur Morin on experiments on friction beginning in May 1831. Their work confirmed and extended Coulomb's work on friction, verifying the three general laws he had proposed.

Poncelet was promoted to Chef de Bataillon in 1831, and then moved to Paris in 1834 when he was elected in March of that year to the mechanics section of the Académie des Sciences. His work on projective geometry was too controversial, particularly following the attacks made on it earlier by Cauchy, for him to enter the Academy on the strength of these contributions. In the following year he become Professor of Mechanics at the Sorbonne. He served on the Committee for Fortifications of Paris from 1835 to 1848. In 1842, Poncelet married Louise Palmyre Gaudin and his intention was to have a quieter time, but events conspired to prevent this for several years. In 1841 he became a Lieutenant-Colonel, then three years later became Colonel and, on 19 April 1848, a General of Brigade. He also became director of the École Polytechnique in April 1848, holding the post until 1850. During his time in this role there occurred in Paris the "June Days Uprising" by French workers on 23-26 June 1848. Barricades were set up and the army attacked the workers with a large loss of life. Poncelet, as director of the École Polytechnique, led his students through the barricades to the Luxembourg Palace where they protected the Provisional Government. Louis Eugène Cavaignac, who had become French head of state and led the suppression of the revolt, honoured Poncelet for his support by appointing him to take command of the National Guards of the Department of the Seine. He was later elected to the governing assembly. In 1849 Poncelet and Arthur Morin invented the dynamometer of rotation, which together with later refinements, became the basic investigative tool in the study of work.

In 1851 the Great Exhibition of the Works of Industry of all Nations was held in Hyde Park, London. Poncelet was appointed as head of the Scientific Commission for the Exhibition. On his return he wrote a report on progress in the applications of science in the first half of the nineteenth century making particular mention of the English machinery and tools he had seen exhibited. Prompted by the Great Exhibition of 1851, the French organised the first Universal Exhibition which opened in Paris in May 1855. Poncelet also played an important role in this Exhibition.

Poncelet published many articles on geometry and mechanics in addition to those we have mentioned, particularly in Gergonne's Annales des Mathématique and Crelle's Journal. The lectures he gave at Metz were first produced in lithographed form then, after a series of versions, were eventually published. For example the course on Mechanics Applied to Machines appeared first as lithographed notes in 1826, again as a second version in 1832, then a third definitive version with the assistance of his friend Arthur Morin in 1836. The notes were not properly published, however, until 1874. The course notes Mécanique industrielle Ⓣ
went through a similar process. He also wrote many reports and memoirs which were published in the Mémorial du Génie and the Avis du Comité des Fortifications. For example we mention Sustaining Walls; Geometrical Constructions to Determine Their Thickness Under Various Circumstances: Geometrical Constructions to Determine Their Thickness Under Various Circumstances (1845), and Memoir Upon the Stability of Revetments and of Their Foundations. Publications following his retirement in 1850 include Applications d'analyse et de géométrie Ⓣ in two volumes: 1862 and 1864. He also published a second edition of Traité des propriétés projectives des figures Ⓣ in 1865-66 which was reprinted in 1995. Here is a list of the contents of this second edition:

(1) General principles (consisting of Preliminary notions of central projection; Preliminary notions on secants and ideal chords of conic sections; and Principles related to projection of plane figures).

(2) Fundamental properties of straight lines, circles, and conic sections (consisting of Geometry of ruler and transversals; Figures inscribed in and circumscribed around conic sections. Reciprocal poles and polars; and Similarity and homothety, centre of similarity).

(3) Systems of conic sections (consisting of Homologous figures, center and axis of homology, in particular for conic sections; Complete systems of conic sections; and Double contact of conic sections).

(4) On angles and polygons (this section contains the projective definition of foci of conic sections).

(5) General theory of centers of middle harmonics.

(6) General theory of reciprocal polars.

(7) Analysis of transversals applied to geometric curves and surfaces.

(8) Properties common to systems of geometric curves and surfaces of arbitrary order.

Poncelet received many honours in addition to being elected to the Academy which we mentioned above. He was an officer of the Legion of Honour, and Chevalier of the Prussian Order. Many academies and learned societies elected him to membership including the Royal Society of London, the Berlin Academy of Science, the Imperial Academy of Sciences of St Petersburg and Academy of Sciences in Turin.

After a long and painful illness, Poncelet died in December 1867. In the following year the Prix Poncelet was endowed by his wife in carrying out Poncelet's dying wish that the sciences be advanced. The prize, augmented by a further sum of money, was awarded for work in pure mathematics or mechanics by the Academy of Sciences from 1876. His unpublished manuscripts survived until World War I when they vanished and have not been traced since. Sadly it is highly likely that they were destroyed at this time.