(John Charles), a Mathematician and Nautical Astronomer, celebrated for his improvements in the theory of Hydraulics and Pneumatics, and in the construction of instruments for observation. He was born at Drax, the 4th of May 1733, and was originally destined for the bar, but abandoned the pursuit of the law in favour of a military life, which he considered as better calculated to afford him opportunities for the cultivation of his mathematical talents, and, for the application of the results of his studies to practice. His acquirements in science had very early attracted the attention of D'Alembert, who predicted his future eminence, and warmly recommended his turning his thoughts to the occupation of a place in the Academy. He obtained a commission in the Light Cavalry, and was appointed Teacher of Mathematics to the corps; and, in 1756, he presented to the Academy of Sciences (A.) A Memoir on the Paths of Bombs, which was ordered to be printed in the collection of the Savans Etrangers, but which has not excited much attention. He was elected in the same year a member of the Academy; and in the next he was present at the battle of Hastinbeck, in the capacity of Aide-de-Camp to the celebrated General Maillebois, to whom he looked up as a great master in the art of War.

He was afterwards admitted into the corps of Engineers, without the usual form of examination into his qualifications; and, being stationed at a seaport, the occurrences of the place naturally directed his attention anew to the phenomena of the resistance of fluids. He published, in 1763, a detailed memoir on this subject (B. Mém. Ac. Par. 1763, p. 358), in which he relates a variety of experiments, showing, that the resistance of the air is actually proportional to the square of the velocity, as had commonly been supposed from theoretical considerations. He also determines, by other experiments, the magnitude of the resistance to the motion of a sphere, and proves, that nothing can be more erroneous than the supposition, that the resistance to an oblique surface decreases as the square of the sine of the angle of incidence. He also finds, that the resistance to the motion of a cube, in the directions of the diagonal of its base and of one of the sides, are as 21 to 16, while the calculations of former theorists had made the resistance greatest in the direction of the side.

In 1766, he published an Essay on the discharge of fluids through the orifices of vessels (C. Mém. Ac. Par.1 766, p. 579), in which he first states the objections to considering the different strata of a fluid as descending in all cases very nearly in parallel directions; he examines the contraction of the jet after its escape from the orifice, and determines some of the effects of abrupt changes in the velocity of the fluid passing through pipes or apertures of different forms.

He contributed, in 1767, to the publications of the Academy, an important Memoir on Water Wheels (D. p. 270), which has escaped the notice of his able Biographer M. Lacroix. He observes, in this paper, that the simple hypothesis of a resistance varying as the square of the velocity, which is so near the truth in common cases, where a number of particles, proportional to the velocity, strikes, in a given time, upon a small exposed surface with a force also proportional to the velocity, is totally inapplicable to the action of a confined stream upon the floatboards of a wheel, since, in this instance, the number of particles concerned cannot vary materially with the velocity, the whole stream being supposed to operate in all cases upon the successive floatboards; so that the analogy would require us to suppose the force in this case nearly proportional to the simple relative velocity; a conclusion which agrees remarkably well with the experiments of some practical authors.

The same volume contains a continuation of M. Borda's researches relating to the resistance of oblique surfaces (E. Mém. Ac. Par. 1767, p. 495), with a statement of experiments still more conclusively confuting the received hypothesis, respecting oblique impulse, than his former investigations had done. We also find in it an Essay on isoperimetrical problems (F. p. 551), in which it is shown, that Euler's method of treating them, which had been in great measure abandoned by its equally profound and candid author, in favour of the more general and more elegant calculations of Lagrange, was still capable of affording all the results that had been derived from the method of variations; and he even pointed out some deficiencies in the first Memoir of Lagrange, which contained the detail of his ingenious invention. These investigations of M. Borda afford collateral evidence of the strict truth of the demonstrations of both his great predecessors; and though they have been little employed by later Mathematicians, yet it must be admitted to be of some importance, in enabling us to appreciate the value of a new mode of calculation, to determine whether its results are or are not such, as might be obtained, with almost equal convenience, by methods before in use.

His memoir, inserted in the collection of the Academy for 1768 (G. Mém. Ac. Par. 1738, p. 418), is devoted exclusively to the theory of pumps; and he considers especially the effect of the passage of the fluid through valves and other contracted parts, in diminishing the quantity of the discharge. His results are derived from the principle of the preservation of the living force or energy of a system of bodies, throughout all the vicissitudes of its motions, which had before been employed with success by Daniel Bernoulli in problems of a similar nature; but it was not until the experiments of Buat had afforded sufficient grounds for the determination of the friction of fluids, that cases of this kind could be submitted to exact calculation.

In his Essay on the curve described by cannon-balls, published among the Memoirs for 1769 (H. Mém. Ac. Par. 1769, p. 247), he has greatly simplified the practical theory of projectiles, which had been treated in a satisfactory, though very general manner by John Bernoulli, and had been reduced into a much more convenient form by Euler. M. Borda has substituted some approximate expressions for the true value of the density of the air, and has thus been enabled to integrate equations which, in their more strictly correct form, had resisted the powers of Euler himself; and he has justified the adoption of the formulas thus obtained by a comparison with experiment.

In the meantime his talents were very actively employed in the naval service of his country, which he entered in 1767, by the nomination of M. Praslin. The time-keepers of Le Roy and Berthoud were beginning to rival those of the English artists, and the French Government ordered several vessels to be fitted out for cruises, in order to examine the accuracy of these time-keepers. M. Borda was appointed a Lieutenant on board of the Flore, and acted jointly with M. Pingré as a delegate of the Academy of Sciences for the purposes of the expedition. The voyage occupied about a year, and extended to the Canaries, the West Indies, Newfoundland, Iceland, and Denmark. M. Borda had a considerable share in the account which was published of the observations; and the formula, which he has here given, for the correction of the effects of refraction and parallax, is considered as equally elegant and convenient. He also presented to the Academy a separate Memoir on the results of the expedition. (I. Voyage pour éprouver les montres de Leroy. 4. Paris. (K. Mém. Ac. Par. 1773, p. 258.) After an interval of six weeks, these watches were found capable of determining the longitude within about fifteen minutes of the truth.

In order to supply some deficiencies in the observations made at the Canaries, Borda was sent out a second time, with the Boussole and the Espiegle, and he published, after his return, (L.) a very correct and highly finished map of these islands. He was soon afterwards promoted to the rank of Captain, and served under the Count d'Estaing as a Major-General, an appointment nearly similar to that of our Captains of the fleet. In this capacity, he observed the inconvenience of too great a variety in the sizes of the vessels constituting a fleet, and proposed to abolish the class of 50 and of 64 gun ships, as too small for the line of battle, and to build ships of three rates only, the lowest carrying 74 guns, so that a smaller quantity of stores should require to be kept ready for use in the dock-yards, than when ships of more various dimensions were to be refitted. In 1780, he had the command of the Guerrier, and in 1781 of the Solitaire, which was taken, after a gallant resistance, by an English squadron. He was thus compelled to pay a visit to Great Britain, but was immediately set at liberty upon his parole.

He proposed to the Academy in this year (M. Mém. Ac. Par. 1781), a mode of regulating elections, which was adopted by that body. Its peculiarity consisted in having the names of the candidates arranged by each voter in a certain order, and collecting the numbers expressing the degrees of preference into separate results, so that the simple majority of voters did not necessarily establish the claim of any individual, if he was placed very low in the list by any considerable number of those who voted against him. But, it must be allowed, that this mode of election is by no means wholly unobjectionable.

M. Borda appears to have rendered an essential service to the cultivators of Practical Astronomy, by the introduction and improvement of the repeating circle, although this instrument has probably been less employed in Great Britain than elsewhere, on account of the greater perfection of those which were previously in common use. It had been suggested by Mayer, in 1767, that a circle with two moveable sights, would enable us to observe a given angle a great number of times in succession, and to add together the results, without any error in reading them off, and thus to obtain a degree of precision equal to that of much larger and better instruments of a different construction; but the proposal had been little noticed until ten years afterwards, when Borda pursued the path pointed out by Mayer, and trained Lenoir, then a young and unlicensed artist, to the execution of the improved instrument, notwithstanding the opposition of the rival opticians, and the want of encouragement from the opulent public. He published, in 1787 (N.), his Description and Use of the Reflecting Circle, with different Methods for Calculating the Principal Observations of Nautical Astronomy; but the officers of the French navy, for whom this work was intended, appear to have profited but little by his instructions. His instrument was, however, much employed in the operations for determining the length of the terrestrial meridian, and he himself took charge of the experiments required for ascertaining the length of the pendulum, and for the comparison of the different standards with each other. He invented some very ingenious methods of overcoming the difficulties which present themselves in the pursuit of these objects; but he was interrupted in his researches by the horrors of the Revolution, nor did he live to see the whole of the operations completed. He endeavoured, also, to promote the introduction of the new mode of subdividing the circle, by the laborious computation of Tables of Logarithms (O. 4to, Par. 1801), adapted to decimal parts of the quadrant,—a work in which he was assisted by M. Delambre. From the increasing indisposition of M. Callet, who had undertaken to correct the proofs of these tables, some very material errors had been committed in the first half of the tables, and M. Borda thought it necessary to cancel a great number of the pages; and in order to meet the expense thus entailed on him, he was obliged to dispose of an estate which he had lately acquired in his native place. He was also engaged, towards the close of his life, in the measurement of the force of magnetism, and in the calculation of astronomical refraction. His health had been threatened for several successive winters, and he died the 10th of March 1799.

In his manners he was animated and unaffected: he avoided those who sought his acquaintance merely from the vanity of being intimate with a man of talents, whatever pretensions to importance they might derive from their casual relations to general society. He never married; and he was too much absorbed in the pursuit of science, to associate with a very extensive circle even of private friends. Though not a man of learning, he was not deficient in literary taste, and he was, in particular, a passionate admirer of Homer. He seems to have possessed a considerable share of that natural tact and sagacity, which was so remarkable in Newton, and which we also discover in the works of Daniel Bernoulli; enabling them, like a sort of instinct, to elude the insurmountable difficulties with which direct investigations are often encumbered; while Euler, on the contrary, as M. Lacroix most truly observes, seems to have taken pleasure in searching for matter which would give scope to his analytical ingenuity, although wholly foreign to the physical investigations which had first led him to the difficulties in question. It would have been fortunate for the progress of science, if some of the most celebrated of M. Borda's countrymen had profited by his example, in studying to attain that unostentatious simplicity which is the last result of the highest cultivation. (Lacroix in Rapport des Travaux de la Société Philomathique. Vol. IV. 8. Par. 1800.)