His memoir, inserted in the collection of the Aca demy for 1768 (G. Mini. 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 fsom the principle of the preser vation 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 af forded sufficient grounds for the determination of the friction of fluids, that cases of this kind could be sub mitted to exact calculation.
In his Essay on the curve described by cannon balls, published among the Memoirs for 1769 (H. Mho. Ac. Par. 1769, p. 247), be has greatly simpli fied 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 eitations 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 ex periment.
In the meantime his talents were very actively em ployed in the naval service of his country, which he entered in 1767, by the nomination of M. Fresno. The time-keepers of Le Roy and Berthoud were be ginning 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 accu racy of these timekeepers. M. Borda was appoint ed a Lieutenant on board of the Flore, and acted jointly with M. Pingre 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 observa tions.; and the formula, which he has here given, for the correction of the effects of refraction and paral lax, is considered as equally elegant and convenient. He also presented to the Academy a separate Me moir on the results of the expedition. (1. Voyage pour iprouver lea ',moires de Leroy. 4. Paris. (K. Mina. Ac. Par. 1773, p. 258.) Atter an interval of six weeks, these watches were found capable of de termining the longitude within about fifteen minutes of the truth.
In order to supply some deficiencies in the obser vations 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 es a Major General, an appointment nearly similar to that of our Captains of the fleet. In this capacity, he ob served the inconvenience of too great a variety in the sizes of the vessels Constituting a fleet, and propos ed 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 refit ted. 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. Min,. Ac. Par. 1781), a mode of regulating elec tions, which was adopted by that body. Its pecu liarity consisted in having the names of the candi dates 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 unob jectionable.
M. Borda appears to have rendered an essential service, to the cultivators of Practical Astronomy, by the introduction and improvement of the repeat.. ing circle, although this instrument has probably been less employed in Great Britain than elsewhere, en account of the greater perfection of those which previously in common use. It had been sug gested 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 pro posal had been little noticed until ten years after wards, when Borda pursued the path pointed out by Mayer, and trained Lenoir, then a young and un licensed artist, to the execution of the improved in strument, 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 Ob servations of Nautical Astronomy ; but the officers of the French navy, for Whom this work was intend. ed, appear to have profited but little by his instruc tions. His instrument was, however, much employ ed 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 invent ed some very ingenious methods of overcoming the difficulties which present themselves in the pursuit of these objects ; but he was interrupted in his re searches 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 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. Berda thought it necessary to cancel a great num ber of the pages ; and in order to meet the expence 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 magnet ism, and in the calculation of astronomical refrac tion. 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 mere ly 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 lite rary 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; ena bling them, like a sort of instinct, to elude the in surmountable difficulties with which direct investiga tions 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, al though 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. Bor. da'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 Societe Phi loniathique. Vol. IV. 8. Par. 1800.) (AL.)