让-夏尔·德博尔达Jean-Charles de Borda

让-夏尔·德博尔达出生于法国西南部的达克斯镇。该镇是一个著名的温泉疗养地，位于阿杜尔河畔，在比利牛斯山脉和西班牙边境以北约80公里处。他的父母是德博尔达-路易·安托万 德博尔达，拉巴蒂领主，和Marie-Thérèse de la Croix。他们都出身贵族，家族有着可追溯许多代的强大军事联系。德博尔达-安托万和Marie-Thérèse有十六个孩子，八个男孩和八个女孩。Charles有五个哥哥和四个姐姐。Charles的大多数兄弟后来都从事军事职业，但至少有一个是教会中的教士。家族中还有另一位成员，他的表兄Jacques-François 德博尔达，他在Charles的成长过程中对他产生了重大影响。Jacques-François也出生在达克斯，在Charles出生时他十五岁。Jacques-François对数学和科学充满热情，并与他那个时代的顶尖数学家保持联系。他教导年幼的Charles，而Charles从很小的时候就表现出对学习科学的极大热情。

七岁时，Charles进入达克斯的巴拿巴会学院。巴拿巴会是16世纪创立的一个宗教修会，得名于米兰古老的圣巴拿巴教堂，他们致力于研究圣保罗书信。在学院里，他学习希腊语、拉丁语直到十一岁，但他从巴拿巴会士那里学到的数学或科学很少。在这个阶段，是Jacques-François鼓励Charles的父亲把他十一岁的儿子送到一所能学习数学和科学的学院。一个自然的选择是拉弗莱什的耶稣会学院，该校为男孩们从事军事工程、法律和公务员职业做准备。德博尔达被送到那里，学习古典学、科学、数学和形而上学，遵循一条通向军队职业生涯的课程。

1748年，当他十五岁时，德博尔达在拉弗莱什完成了学业。那里的耶稣会士极力鼓励他加入他们的修会，但这对他没有吸引力，因为他对宗教兴趣不大，他回到父母家中，试图说服父亲让他从事军事工程兵团的职业。尽管家族有着强大的军事联系，德博尔达的父亲曾希望他成为一名地方法官，但他允许儿子追随自己的意愿，在这个阶段，德博尔达开始了作为军队数学家的职业生涯。

1753年，二十岁的德博尔达完成了他第一篇关于几何学的论文，并寄给了让·勒朗·达朗贝尔。两年后，他获得了第一个数学家的职位，在轻骑兵团任职。在这个职位上，德博尔达进行了弹道学研究，并于1756年5月29日向巴黎的Académie des Sciences提交了一篇研究抛射体理论的论文。凭借这项工作，他于当年当选为Académie的通讯院士。后来，他当选为Académie的正式院士。

七年战争始于1756年，1757年7月法军在汉诺威的哈斯滕贝克战役中击败了坎伯兰公爵。德博尔达在这场战役中担任马耶布瓦元帅的副官。正是在这个时候，他的兴趣转向了海洋，并于1758年9月4日进入梅济耶尔的工程学校。他入学的课程本应持续两年，但德博尔达在一年内完成了它，然后继续作为海军军事工程师的职业生涯。他的数学工作进展顺利，1762年他证明了球形抛射体所受空气阻力仅为相同直径圆柱形物体的一半。他还证明了阻力大约与速度的平方成正比[1]：-

……他证明了艾萨克·牛顿的流体阻力理论是站不住脚的，并且阻力与流体速度的平方以及入射角的正弦成正比。他[还]……计算了从孔口流出的流体收缩系数。德博尔达对[能量]守恒原理的运用很重要，是拉扎尔·卡诺在力学中工作的先驱。

在描述他对流体力学的贡献时，我们还应该注意到他对水轮和水泵研究的贡献。

1765年至1775年间，德博尔达多次横渡大西洋。他的工作既有军事性质，也有科学性质，常常将这两方面结合在他的水文地理学和制图学工作中，绘制了亚速尔群岛和加那利群岛的海图。1771年，他被派乘坐护卫舰“La Flore”号进行科学航行，进行研究以改进计算经度的方法，并测试某些航海天文钟。美国独立战争于1776年开始，两年后法国加入了对英国的冲突。法国和英国是主要的殖民大国，正是由于这些问题，他们在早期的七年战争中相互交战。法国和英国在美国独立战争期间为争夺海洋控制权而战，德博尔达深度参与了法国海军的行动。法国舰队由查尔斯-埃克托尔·德斯坦伯爵指挥，德博尔达担任“La Seine”号舰长并指挥几艘舰船。舰队在加勒比海和美国海岸附近航行，取得了一些显著胜利，但德斯坦伯爵在1779年10月进攻萨凡纳失败时受重伤，他带着他的中队返回法国。德格拉斯伯爵接管了法国舰队的指挥，并再次取得了重要胜利，但他在1782年于多米尼加附近的海圣斯战役中失利。德博尔达担任“la Solitaire”号舰长，并在这场战役中指挥六艘舰船。他和舰队领袖德格拉斯都被英国人俘虏。不久之后，德博尔达获准返回法国，但在此事件之后，他的健康状况每况愈下。

德博尔达很好地利用了微分学和实验方法来统一物理学的各个领域。他还结合他的测量技术开发了一系列三角函数表。他从事流体力学研究，研究了许多不同情况下的流体流动，如船舶、火炮、泵和科学仪器。他的一个仪器，德博尔达复测圆，在法国大革命时期被用来测量子午线弧长，作为引入十进制系统项目的一部分。这个仪器是由德博尔达在1785年左右提出的，它是从为船上使用而设计的仪器发展而来的。

德博尔达复测圆由两个小望远镜组成，每个望远镜都固定在一个环上，这些环可以独立地相对于刻度旋转。为了测量两点$A$和$B$之间的角度，仪器的设置使得旋转环的平面位于$A, B$和观察者所在的平面内。旋转一个望远镜直到它瞄准$A$，其圆相对于刻度固定，然后旋转第二个望远镜以瞄准$B$。在这个阶段，角度θ可以简单地读出，但现在，巧妙的部分来了。拧紧螺丝将旋转圆固定在一起，然后旋转它们，使第二个望远镜瞄准$A$。当然，第一个望远镜现在与$B$确定的角度相差2θ，因此，解开望远镜转动的环，并将其移回以瞄准$B$，就测量了角度2θ。重复这个过程继续将θ加倍，所以理论上误差可以小到任意程度，因为最终答案除以加倍的次数来找到θ。他在Description and use of the repeating circle（1787年）中描述了这个仪器，并给出了航海应用。

德博尔达的重复圆仪图片见THIS LINK。

德博尔达的重复圆仪的精度使得通过三角测量进行勘测来测定距离成为可能。当德博尔达被任命为度量衡委员会主席时，该委员会的成员包括尼古拉·德·孔多塞、拉瓦锡、皮埃尔·西蒙·拉普拉斯和阿德里安-马里·勒让德，他很快便将他精确的测量仪器派上了用场。该委员会成立于1790年，旨在引入统一的度量系统。它审议了一项已向法国政府提出的建议，即以每秒摆动一次的摆的长度作为米的基础。这项建议曾得到英国和美国的支持，他们认为这是一种真正国际性的度量。然而，德博尔达于1791年3月19日报告称，委员会已决定采用不同的标准，即一米应为从北极到赤道距离的一千万分之一。他反对摆标准的论据是，它将一个单位建立在另一个本身可能变化的单位之上，而且秒本身也是一个基于将一天分为12×60×60的任意单位。德博尔达主张应将一天分为10小时，每小时分为100分钟，每分钟分为100秒。在德博尔达的领导下，使用德博尔达重复圆仪精确测量从北极到赤道距离的项目得以实施。

1793年初，政治事件似乎将使该项目无法完成，德博尔达、约瑟夫·拉格朗日和皮埃尔·西蒙·拉普拉斯根据塞萨尔-弗朗索瓦·卡西尼·图里先前进行的勘测对米做出了临时估算。他们得出的数值实际上比让·巴蒂斯特·约瑟夫·德朗布尔-皮埃尔·梅尚对从敦刻尔克到巴塞罗那的子午线弧的勘测所得到的数值更为精确。在1793年9月至1794年7月的恐怖统治期间，德博尔达退隐到他的家族庄园，但随后又恢复了他在米制方面的工作。

作为度量衡委员会主席，德博尔达提出了其他重要建议。其中之一是替代称量法，德博尔达对此的贡献由Jenemann在[4]中进行了评估。他写道：-

通过应用替代称量法，梁式天平的一些系统误差被消除。替代称量法在法国大革命期间出现并得到更广泛的应用，当时在德博尔达的指导下，引入了新的度量衡标准——米制和千克。

还有一个主题，德博尔达对此做出了重要贡献。尼古拉·德·孔多塞在1785年提出了一种在候选人超过两名时举行公平选举的方法。他的方法确保，如果要从$n$人中选出一人，那么当选者必须在与其余$n-1$名候选人的一对一竞争中全部获胜。德博尔达觉得尼古拉·德·孔多塞的提议是公平的，但他提出这在实践中不可行，因为可能选不出获胜者。德博尔达提出了按排名给候选人打分的制度：根据排名给每位候选人相应的分数。如果有$n$名候选人，选民应给他们最不喜欢的候选人1分，给下一位候选人2分，依此类推，直到他们最喜欢的候选人，给其$n$票。这个提议如今常用于选举中，其缺陷是当选者可能不是任何人的第一选择。德博尔达和尼古拉·德·孔多塞之间就这两种投票制度哪种最好展开了激烈争论，但当然，由于两种制度各有优缺点，这样的争论注定没有定论。

德博尔达的一项未获支持的建议是，应将直角分为100度，每度分为100分，每分分为100秒。这样的系统需要构建新的三角函数表，德博尔达组织了这项工作。在他1804年去世后，让·巴蒂斯特·约瑟夫·德朗布尔出版了Decimal trigonometrical tables，该书扩展了德博尔达的工作。

我们还应该提到德博尔达职业生涯中我们上面未提及的进一步成就。他于1767年当选为波尔多科学院院士，两年后当选为海军科学院院士。1795年，他成为经度局成员。

德博尔达在确定米长度的项目完成前不久去世。成立了一个国际委员会试图使米成为国际度量单位，但德博尔达反对这一做法，理由是测量基于地球，因此应同样为地球上每个国家所接受。他的葬礼在[3]中描述如下：-

……这位老指挥官在长期患病后去世。在大雨中，一队国际学者护送他的遗体沿着泥泞的道路上行，安葬于蒙马特之下。

Jean-Charles de Borda was born in the town of Dax in south west France. The town, which was a famous spa with thermal springs, was on the Adour River, about 80 km north of the Pyrenees and the border with Spain. His parents were Jean-Antoine de Borda, Lord of Labattut, and Marie-Thérèse de la Croix. They were both from the nobility and had families with strong military connections going back for many generations. Jean-Antoine and Marie-Thérèse has sixteen children, eight boys and eight girls. Charles had five older brothers and four older sisters. Most of Charles's brothers went on to have military careers, but at least one was a canon in the Church. There was another member of the family, his cousin Jacques-François de Borda, who would have a major influence on Charles as he grew up. Jacques-François was also born in Dax and was fifteen years old when Charles was born. Jacques-François had a passionate love of mathematics and science, and was in contact with the leading mathematicians of his day. He taught the young Charles who from the earliest age showed great enthusiasm for learning science.

At the age of seven, Charles entered the Collège des Barnabites at Dax. The Barnabites were a religious order founded in the 16th century, taking their name from the ancient church of St Barnabas in Milan, and they were devoted to the study of the Letters of St Paul. In the College he studied Greek, Latin until he reached the age of eleven but he learnt little of mathematics or science from the Barnabites. At this stage it was Jacques-François who encouraged Charles's father to send his eleven year old son to a college where he could learn mathematics and science. A natural choice was the Jesuit college at La Flèche which trained boys for careers in military engineering, law, and the civil service. Borda was sent there and studied classics, science, mathematics, and metaphysics, following a course which would lead to an career in the army.

In 1748, when he was fifteen years old, Borda completed his studies at La Flèche. The Jesuits there strongly encouraged him to join their Order but this did not appeal to him, for he had little interest in religion, and he returned to his parent's home to try to persuade his father to let him follow a career in the military engineering corps. Despite the strong military connections of the family, Borda's father had wanted him to become a magistrate but he allowed his son to follow his wishes and, at this stage, Borda began a career as a mathematician in the army.

In 1753, at the age of twenty, Borda produced his first memoir on geometry and sent it to d'Alembert. Two years later he received his first commission as a mathematician in the Light Cavalry Corps. While in this post Borda undertook research on ballistics and, on 29 May 1756, he submitted a memoir studying the theory of the projectiles to the Académie des Sciences in Paris. On the strength of this work he was elected an associé of the Académie in that year. He would later be elected a full member of the Académie.

The Seven Years' War began in 1756 and the French defeated the Duke of Cumberland at the Battle of Hastenbeck in Hanover in July 1757. Borda was aide-de-camp of Marshal Maillebois during this battle. It was at this time that his interests turned towards the sea and, on 4 September 1758, he entered to the École du Génie at Mézière. The course he entered should have lasted two years but Borda completed it in one year, then continued his career as a military engineer in the navy. His mathematical work progressed well and in 1762 he showed that a spherical projectile experiences only half the air a resistance of a cylindrical object of the same diameter. He also showed that the resistance was approximately proportional to the square of the velocity [1]:-

... he demonstrated that Newton's theory of fluid resistance was untenable and that the resistance is proportional to the square of the fluid velocity and to the sine of the angle of incidence. He [also] ... calculated the coefficient of fluid contraction from an orifice. Borda's use of the principle of conservation of [energy] was important as a precursor of Lazare Carnot's work in mechanics.

While describing his contributions to fluid mechanics we should also note the contributions he made to the study of waterwheels and pumps.

Between 1765 and 1775 Borda made several crossings of the Atlantic. His work was both of a military and of a scientific nature, often combining these two aspects in his work on hydrography and cartography, drawing up charts of the Azores and the Canary Islands. In 1771 he was sent on a scientific voyage on the frigate "La Flore" to undertake studies to improve methods of calculating the longitude and test certain marine chronometers. The American War of Independence began in 1776 and two years later France joined the conflict against Britain. France and Britain were major colonial powers and it was over such issues that they had fought against each other during the earlier Seven Years' War. It was for control of the seas that France and Britain fought during the American War of Independence, and Borda was heavily involved in the French naval actions. The French fleet was commanded by Charles-Hector, Count d'Estaing, with Borda as captain of the ship "La Seine" and in command of several of the ships. The fleet sailed in the Caribbean and off the American coast, won some notable victories, but Count d'Estaing was seriously wounded while unsuccessfully attacking Savannah in October 1779, and he returned to France with his squadron. Count de Grasse took over command of the French fleet and again won important victories but he lost the Battle of the Saints off Dominica in 1782. Borda captained "la Solitaire" and commanded six ships in this battle. He, and the leader of the fleet de Grasse, were both taken prisoner by the British. After a short period Borda was allowed to return to France but after this episode his health declined.

Borda made good use of the differential calculus and of experimental methods to unify areas of physics. He also developed a series of trigonometric tables in conjunction with his surveying techniques. He worked on fluid mechanics, studying fluid flow in many different situations such as ships, artillery, pumps and scientific instruments. One of his instruments, the Borda repeating circle, was used during the time of the French Revolution to measure an arc of a meridian as part of a project to introduce the decimal system. This instrument was proposed by Borda around 1785 and it had developed from instruments designed for use on ships.

The Borda repeating circle consisted of two small telescopes each fixed to rings which could rotate independenly against a scale. To measure the angle between two points $A$ and $B$, the instrument was set up so that the plane of the rotating rings was in the plane of $A, B$ and the observer. One scope was rotated until it sighted $A$, its circle fixed with respect to the scale and the second scope was then rotated to sight $B$. The angle θ could simply be read off at this stage but now, however, came the clever part. Screws were tightened to fix the rotating circles together, then they were rotated so that the second scope sighted $A$. Of course the first scope was now an angle of 2θ from the angle determined by $B$ so, decoupling the rings on which the scopes turned and moving it back to sight $B$ one measured the angle 2θ. Repeating the process continued to double θ, so in theory the error could be made as small as one desired since the final answer was divided by the number of doublings to find θ. He described this instrument in Description and use of the repeating circle (1787) giving nautical applications.

A picture of the Borda repeating circle is at THIS LINK.

The accuracy of Borda's repeating circle allowed distances to be found by surveying using triangulation. When Borda was made Chairman of the Commission of Weights and Measures, which had as its members Condorcet, Lavoisier, Laplace and Legendre, he soon put his accurate surveying instrument to good use. The Commission was set up in 1790 to bring in a uniform system of measurement. It considered a proposal which had already been made to the French government to base the metre on the length of a pendulum which beat at the rate of one second. This proposal had found favour with Britain and the United States who considered it a truly international measure. Borda, however, reported on the 19 March 1791 that the Commission had decided on a different standard, namely that one metre should be one ten millionth of the distance from the North Pole to the equator. His argument against the pendulum standard was that it based one unit on another, which might itself change, and also that the second itself was an arbitrary unit based on the division of a day by 12 × 60 × 60. Borda argued that the day should be divided into 10 hours with an hour divided into 100 minutes each of 100 seconds. Under Borda's leadership the project to accurately measure the distance from the North Pole to the equator using the Borda repeating circle was carried out.

In early 1793 it looked as though political events would prevent the project being completed and Borda, Lagrange and Laplace made a provisional estimate of the metre based on a survey previously carried our by Cassini de Thury. The value they came up with was actually more accurate than the one achieved by the Delambre-Méchain survey of the arc of the meridian from Dunkerque to Barcelona. Borda retired to his family estate during the Terror, which lasted from September 1793 to July 1794, but then resumed his work with the metric system.

As Chairman of the Commission of Weights and Measures, Borda made other important proposals. One was substitution weighing to which Borda's contribution is assessed by Jenemann in [4]. He writes:-

By application of substitution weighing some systematic errors of the beam balance are omitted. Substitution weighing had its advent and found wider application during the French Revolution, when, under the direction of Jean-Charles de Borda, new standards for measures and weights, the metre scale and the kilogram, were introduced.

There is another topic to which Borda made important contributions. Condorcet proposed in 1785 a method of holding fair elections where there was more than two candidates. His method ensures that if one person is to be elected from a collection of $n$ people then the person elected would have to have won in a head-to-head contest with every one of the other $n-1$ candidates. Borda felt that Condorcet's proposal was fair but he suggested that it was not workable in practice as no winner might result. Borda proposed the system of ranking candidates by giving each points corresponding to their rank. If there were $n$ candidates then voters should give the candidate they favoured least one point, the next candidate two points, and so on until they reached their most favoured candidate to whom they would give $n$ votes. This proposal, often used in elections today, has the deficiency that the candidate elected may not have been anyone's first choice. There was a vigorous argument between Borda and Condorcet as to which of the two voting systems was the best but of course since both systems had their strengths and weaknesses, such an argument was bound to be inconclusive.

One of Borda's proposals which has not found favour was that a right angle should be divided into 100 degrees, each degree into 100 minutes and each minute into 100 seconds. Such a system required new trigonometrical tables to be constructed and Borda organised this. After his death in 1804 Delambre published Decimal trigonometrical tables which extended Borda's work.

We should mention further achievements in Borda's career which we have not mentioned above. He was elected to the Académie de Bordeaux in 1767 and, two years later, to the Académie de Marine. In 1795 he was made a member of the Bureau des longitudes.

Borda died shortly before the project to determine the length of the metre was completed. An International Commission was set up to try to make the metre an international measure, but Borda argued against this on the grounds that the measurements were based on the Earth and should therefore be equally acceptable to every nation on Earth. His funeral is described in [3]:-

... the old commander, after a long illness, died. In pounding rain, a cortège of international savants bore his body up a muddy road for burial below Montmartre.