皮埃尔·伐里农Pierre Varignon

皮埃尔·伐里农出生于一个天主教家庭，职业是承包石匠。除了他的父亲从事这种体力工作外，他的兄弟也成为了一名石匠。家庭贫穷，无法为伐里农提供经济支持。他自己评论说，他从家庭得到的唯一东西就是技术知识。虽然这似乎是伐里农说的，暗示他受到了相当不公平的对待，但实际上他获得的技术知识在他后来的生活中确实非常宝贵。他在卡昂的耶稣会学院接受神学和哲学教育，在那里他接受了神职培训。1676年12月19日，他接受了圣职，后来被接纳为神父。然后，作为今天会被描述为成熟学生的人，他在卡昂大学学习，并于1682年9月获得硕士学位。次年3月，他成为卡昂圣旺堂区的神父。

在卡昂，他与同学查尔斯·卡斯特尔，圣伐里农修道院院长成为朋友，后者安排伐里农获得300里弗尔的收入。两人合住，看来此时伐里农继续在卡昂大学学习。至此，他走了一条相当标准的通往神职的道路，但他的生活改变了方向，当他完全偶然地遇到欧几里得的Elements并开始阅读这部经典文本时。通过阅读欧几里得被引入数学后，他接着阅读了勒内·笛卡儿的Géométrie，此后致力于数学科学。当然，作为一名耶稣会士，他属于一个重视学术和教学的修会，因此伐里农能够将余生奉献给教学。1686年，他与朋友查尔斯·卡斯特尔，圣伐里农修道院院长一起，伐里农前往巴黎，并立即与那里的数学家和科学家取得了联系。

1687年，伐里农发表了Projet d'une nouvelle mécanique，该著作利用哥特弗里德·威廉·莱布尼茨的微分微积分研究力学中的力的合成。他将此著作献给科学院，显然它受到高度评价，因为同年他就被选入科学院。同样在1688年，他成为马扎林学院数学教授，担任一个新设立的讲席，在那里他开始以当时当前研究水平教授数学。1704年，除了马扎林学院的讲席外，他还成为皇家学院数学教授。他在马扎林学院的讲座出版为Élémens de mathématiques ... Ⓣ（数学要义...）（1731年），我们将在本文末尾讨论其内容。

伐里农的主要贡献在于图解静力学和力学。从他最早的出版物如1687年的Projet d'une nouvelle mécanique Ⓣ（新力学方案）中，可以清楚地看出他理解哥特弗里德·威廉·莱布尼茨微积分的价值。这令人惊讶地紧随哥特弗里德·威廉·莱布尼茨关于新微分微积分的两篇文章于1684年10月和1686年6月在Acta Eruditorum上发表之后。尽管伐里农没有做出重大数学贡献，他通过将哥特弗里德·威廉·莱布尼茨的微积分适应于艾萨克·牛顿的Principia的惯性力学，发展了分析动力学，是最早认识到微积分的力量和重要性的法国学者之一。虽然像克里斯蒂安·惠更斯这样的人作为数学家钦佩艾萨克·牛顿，但他们不接受基于超距作用的物理理论。伐里农抛开这些哲学顾虑，开始将Principia的大部分内容重新加工成哥特弗里德·威廉·莱布尼茨的微分和积分微积分方法。

1644年，埃万杰利斯塔·托里拆利在De motu aquarum中发表了Ⓣ（《论水的运动》），其中提出了现在被称为埃万杰利斯塔·托里拆利定律的内容。该定律指出，在重力作用下从水箱开口流出的液体的速度与液面和开口中心之间的垂直距离的平方根成正比，并与重力加速度的两倍的平方根成正比。已经发现了支持该定律的实验证据，但在1695年，伐里农试图推导它。F Sebastiani在评论[6]时写道：-

[伐里农]假设，在一个小时间间隔内，人们可以既忽略自由表面水平的变化，又认为整个液体的运动本质上是均匀的[永久的]。伐里农的证明基于这样的思想：每一时刻逸出的少量液体从其运动中获得全部运动，来自“底面积等于开口的液柱重量”所施加的压力。这个极其有趣的证明尽管如此，在关于力和运动量的使用的无穷小操作中仍然非常不确定。对于伐里农来说，从“原因总是与其结果成正比”的公理、力学原理和运动的一般定律共同推导出埃万杰利斯塔·托里拆利定律，就是“仅凭理性”证明了它。因此，他隐含地将欧几里得几何被认为拥有的同样的证明完美性归因于力学。

在他的其他工作中，有一篇1699年关于微分计算在流体流动和水钟中的应用的出版物。1702年，他将微积分应用于弹簧驱动的时钟。伐里农在1707年至1711年间向科学院提交的十一篇长篇论文中研究了抛射体在阻力介质中的运动问题。他对约翰·沃利斯、克里斯蒂安·惠更斯、哥特弗里德·威廉·莱布尼茨和艾萨克·牛顿已经研究过的特殊情况给出了新的统一处理。这项工作再次使用哥特弗里德·威廉·莱布尼茨的微积分方法进行。关于他关于这个主题的论文的更多细节，见[4]。1724年，伐里农的Nouvelle mécaniqueⓉ（《新力学》）出版，它给出了直到75年后路易·普安索的工作之前几何静力学的最佳方法。伐里农的工作对莱昂哈德·欧拉在其粒子动力学研究中产生了重大影响。

伐里农在捍卫微积分免受攻击方面发挥了重要作用。例如，1700年，米歇尔·罗尔反对微积分，理由既包括它没有坚实的基础，又包括它导致错误。伐里农在科学院面前论证，米歇尔·罗尔暗示微积分导致错误的论点是错误的。这两位数学家进行了长达五年的激烈交流，直到科学院裁定辩论结束。

1731年，即伐里农去世九年后，他用于学校数学教学的笔记出版为Élémens de mathématiques ...Ⓣ（《数学基础...》）。它[16]：-

……分为两部分，第一部分讲解算术与初等代数的概念，篇幅较大的第二部分涵盖欧几里得几何的若干主题。

书中包含今天所称的伐里农平行四边形定理：将四边形的各边中点依次连接所得到的图形是平行四边形。他给出了这一定理的完全严格的证明，是第一个这样做的人。

伐里农于1688年当选法国科学院，1713年当选柏林科学院，1718年当选伦敦皇家学会。

Pierre Varignon was born into a Catholic family who were by profession contracting masons. As well as his father undertaking this type of manual work, his brother also became a mason. The family was poor and could offer Pierre no financial support. He commented himself that the only thing he received from his family was technical knowledge. Although this seems to be said by Varignon with the suggestion that he was rather hard done by, indeed the technical knowledge he received did prove very valuable to him later in his life. He was educated in theology and philosophy at the Jesuit College in Caen where he trained for the priesthood. On 19 December 1676 he took holy orders and was later admitted into the priesthood. Then, being what today would be described as a mature student, he studied at the University of Caen where he received his M.A. in September 1682. In March of the following year he became a priest in the Saint Ouen parish in Caen.

In Caen he became friends with a fellow student Charles Castel, Abbé de Saint-Pierre, who arranged for Varignon to receive an income of 300 livres. The two shared lodgings and it appears that at this time Varignon continued his studies at the University of Caen. Up to this point he had taken a fairly standard route to the priesthood but his life changed course when, quite by chance, he came across Euclid's Elements and began reading the classic text. Led into mathematics by reading Euclid, he then read Descartes' Géométrie and thereafter devoted himself to the mathematical sciences. Of course as a Jesuit he belonged to an Order that valued scholarship and teaching so Varignon was able to devote the rest of his life to teaching. In 1686, together with his friend Charles Castel, Abbé de Saint-Pierre, Varignon went to Paris and immediately made contact with mathematicians and scientists there.

In 1687 Varignon published Projet d'une nouvelle mécanique which studied composition of forces using Leibniz's differential calculus in the study of mechanics. He dedicated this work to the Academy of Sciences and it was clearly highly thought of since he was elected to the Academy in the same year. Also in 1688 he became professor of mathematics at the Collège Mazarin, occupying a newly created chair, where he began to teach mathematics at the level of current research at the time. In 1704, in addition to the chair at Collège Mazarin, he became professor of mathematics at the Collège Royal. His lectures at the Collège Mazarin were published as Élémens de mathématiques ... Ⓣ (1731) and we discuss their content at the end of this article.

Varignon's chief contributions were to graphical statics and mechanics. From the earliest of his publications such as Projet d'une nouvelle mécanique Ⓣ in 1687, it was clear that he understood the value of Leibniz's calculus. This was surprisingly soon after Leibniz's two articles on the new differential calculus were published in the Acta Eruditorum in October 1684 and June 1686. Although Varignon made no major mathematical contributions, he developed analytic dynamics by adapting Leibniz's calculus to the inertial mechanics of Newton's Principia being one of the first French scholars to recognise the power and importance of the calculus. While those like Huygens admired Newton as a mathematician, they did not accept a physical theory based on action at a distance. Varignon put aside these philosophical worries and began to rework large sections of the Principia into the Leibniz's approach to the differential and integral calculus.

In 1644 Torricelli published in De motu aquarum Ⓣ what has now became known as Torricelli's law. This states that the speed of a liquid flowing under the force of gravity out of an opening in a tank is proportional to the square root of the vertical distance between the liquid surface and the centre of the opening, and to the square root of twice the acceleration due to gravity. Experimental evidence had been found to support the law but in 1695 Varignon tried to deduce it. F Sebastiani, reviewing [6], writes:-

[Varignon] assumed that, during a small time interval, one can both disregard the variation of the level of the free surface and consider the motion to be essentially uniform [permanent] for the whole liquid. Varignon's proof rests on the idea that the small quantity of liquid that escapes at each moment gets all of its motion from the pressure exercised by the "weight of the columns of liquid with base equal to the opening". This extremely interesting proof nevertheless remains very uncertain in infinitesimal manipulations with respect to the use of the power and quantity of motion. For Varignon, to have derived Torricelli's law jointly from the axiom according to which "causes are always proportional to their effects", from the principles of mechanics and from the general laws of motion was to have proved it "by reason alone". He thus implicitly attributed to mechanics the same demonstrative perfection that Euclidean geometry had been thought to possess.

Among his other work was a publication in 1699 on applications of the differential calculus to fluid flow and to water clocks. In 1702 he applied the calculus to clocks driven by a spring. Varignon studied the problem of the motion of projectiles in resisting media in eleven long memoirs which he presented to the Academy of Sciences between 1707 and 1711. He gave a new unified treatment of particular cases which had already been studied by Wallis, Huygens, Leibniz and Newton. Again this work is developed using Leibniz's approach to the calculus. See [4] for further details of his memoirs on this topic. In 1724 Varignon's Nouvelle mécanique Ⓣ was published which gave the best approach to geometrical statics until the work of Poinsot over 75 years later. Varignon's work was to have a major influence on Euler in his study on particle dynamics.

Varignon played a major role in defending the calculus from attacks. For example in 1700 Rolle argued against the calculus both on the grounds that it was without sound foundation and that it led to errors. Varignon argued before the Academy of Sciences that Rolle's arguments which suggested that the calculus led to errors was wrong. The two mathematicians maintained a vigorous exchange for five years until the Academy of Sciences decreed that the debate was ended.

In 1731, nine years after Varignon's death, his notes for teaching mathematics in schools was published as Élémens de mathématiques ... Ⓣ . It [16]:-

... is organised in two parts, the first part explaining concepts of arithmetic and elementary algebra and the larger second part covering topics in Euclidean geometry.

The book contains what is today known as Varignon's parallelogram theorem: The figure formed when the mid-points of the sides of a quadrilateral are joined in order is a parallelogram. He gives a completely rigorous proof of this theorem, being the first to do so.

Varignon was elected to the French Academy of Sciences in 1688, the Berlin Academy of Science in 1713 and the Royal Society of London in 1718.