**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 16^{th} 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 2*θ* 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 2*θ*. 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.

Here is a picture of the Borda repeating circle.

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.

**Article by:** *J J O'Connor* and *E F Robertson*