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BER of Groningen. In 1705, on the death of James Bernoulli, John was appointed his successor in the chair of mathematics in the university of Bâle, which office he held till his death, at the age of eighty, on the 1st of January, 1748. For forty-nine years he was a foreign associate of the French Academy of Sciences, which body conferred prizes on several of his memoirs. He was a fellow of the Royal Society of London, of the Academies of St. Petersburg and Berlin, and of other learned bodies. His works were collected and edited, under his own supervision, by Cramer, professor of mathematics at Geneva, and published in four volumes quarto. The best authorities on the events of his life are his beforementioned correspondence with Leibnitz, his éloge by d'Alembert, and a notice of his life and discoveries by Lacroix.

, ___, fourth son of the first-mentioned Nicholas, was an eminent physician at Bâle.

, son of the second-mentioned Nicholas, and nephew of James and John, was born at Bâle on the 10th of October, 1687. His mathematical attainments and labours, at an early age, are mentioned with much praise by Leibnitz, in his correspondence with John Bernoulli. From 1710 to 1713 he travelled in France, Holland, and England, and was treated with much kindness and distinction by men of science, especially by Newton, who proposed and obtained his election into the Royal Society of London. Soon afterwards he was elected a member of the Royal Academy of Berlin. Chiefly through the friendly exertions of Leibnitz, continued for several years, he obtained in 1716 the professorship of mathematics in the university of Padua, which he afterwards resigned for that of logic in the university of Bâle. In the latter university he held latterly the chair of law. He died at Basle on the 29th of November, 1759. He edited the posthumous work of his uncle James, De Arte Conjectandi, already mentioned. His original writings consist of an essay "On the Application of the Science of Probabilities to Legal Questions" (De Usu Artis Conjectandi in Jure), and of a great number of separate mathematical papers, published in Transactions and periodicals, especially the Acta Eruditorum, and the Giornale dei Letterati d'Italia. Many of his solutions of mathematical problems are mentioned in the correspondence of Leibnitz and John Bernoulli; and Lacroix considers that one of those solutions contained the germ of the theory of the conditions of integrability of differential equations.

, eldest and favourite son of John Bernoulli, was born at Bâle on the 27th of January, 1695. He is said, at the age of eight, to have spoken four languages correctly—German, Dutch, French, and Latin; and at sixteen to have obtained the degree of doctor of philosophy of the university of Bâle. From that time he began to assist his father in his correspondence with foreign mathematicians. He studied jurisprudence, and in 1715 took the degree of doctor of laws. He then spent about three years in travelling in France and Italy. About 1722 he was elected professor of jurisprudence at Berne. In 1725 he was appointed, along with his brother Daniel, professor of mathematics at St. Petersburg, where he died of fever in his thirty-second year, on the 26th of July, 1726. The Empress Catherine gave his remains the honour of a public funeral. His writings are to be found in the Acta Eruditorum, in the Transactions of the Academy of St. Petersburg, vol. i., and amongst the works of his father, John Bernoulli. A memoir of his life appeared in the second volume of the Transactions of the Academy of St. Petersburg.

, the second son of John Bernoulli, and the most distinguished member of the family, was born at Groningen on the 9th of February, 1700, and educated at Bâle. His father intended him to become a merchant; but his own preference led him to the study of medicine and of mathematics, in which latter branch of science he was instructed by his elder brother Nicholas. Having in 1721 taken the degree of doctor of medicine (on which occasion he read a thesis on respiration), he travelled to Italy to increase his knowledge of that art. He studied under Michelotti and Morgagni. Having meanwhile distinguished himself by some mathematical investigations (published in 1724 at Venice), he received an invitation, which he declined, to become president of a scientific academy at Genoa; and his reputation, as well as that of his brother Nicholas, having reached Peter the Great, that sovereign appointed them, in 1725, joint-professors of mathematics at St. Petersburg Here Daniel composed his celebrated treatise on hydrodynamics, in which the principle of conservation of the vis-viva (or energy) is applied to the phenomena of the motion of fluids, and which was published at Strasbourg in 1738. In this work it was first proposed to propel ships by the reaction of a stream of water thrown backwards; an invention which differs in detail only from the paddle and the screw. On the death, in 1726, of his brother, instructor, and colleague, Nicholas, to whom he was warmly attached, Daniel Bernoulli wished at first to return to the country of his forefathers, but was induced by the kind and generous conduct of the Empress Catherine to remain at St. Petersburg At length, however, finding the climate of Russia too severe for his health, he resigned his chair in 1733, and returned to Bâle, where he obtained at first the appointment of professor of anatomy and botany, and afterwards that of professor of physics and speculative philosophy. Continuing his labours for the advancement of science, he endeavoured to find more satisfactory demonstrations for the first principles of mechanics, especially of the law of the composition of forces, than had previously been known. In applying himself to the solution of special problems, he chose, by preference, those which were capable of useful application; and, in the opinion of Lacroix, his mathematical methods were characterized by a similar love of utility; for instead of investigating, as others had done, mechanical questions merely as means of exercising mathematical knowledge, he carefully limited the intricacy of his mathematical processes to that which was necessary for the solution of the problem, and showed remarkable skill in adopting such approximations as simplify calculation without sensibly affecting the accuracy of the result. He applied the theory of probabilities to questions of vital and social statistics. No fewer than ten of his memoirs were crowned by the French Academy of Sciences, of which he was a foreign associate. In some of these cases he shared the prizes with other competitors; his father amongst the number. The mortification of John Bernoulli on such occasions as this has been already referred to; it was possibly to a certain extent caused by the fact, that the son had adopted the doctrines of Newton as to the cause of the planetary motions, while the father adhered to those of Descartes. One of the most celebrated of Daniel Bernoulli's memoirs is that in which he wrote in French on the tides, "Sur le Flux et Reflux de la Mer," and which shared the prize offered by the Academy of Sciences for essays on that subject with three others, composed respectively by Euler, Maclaurin, and the jesuit, Father Cavallieri. The essay of Cavallieri was founded on the then expiring Cartesian hypothesis of vortices; those of Bernoulli, Euler, and Maclaurin, on the law of gravitation. (The last three are reprinted in the second volume of the Glasgow edition of Newton's Principia.) The memoir of Bernoulli, though not so profound or general in its mathematical reasoning as the other two, is considerably more clear and simple. The whole three memoirs, however, have the radical defect, first made evident by Laplace's more sound investigations, and afterwards clearly set forth in Mr. Airy's treatise on Tides and Waves, that they treat the rotation of the earth as having a merely secondary influence on the motions of the tides, capable of being allowed for by approximation, after the completion of the investigation of the effect of the attractions of the sun and moon; whereas the influence of that rotation is of primary importance, and the true effects of the attraction of the sun and moon cannot be determined except by taking the earth's rotation into account at the same time. Besides the Academy of Sciences, Daniel Bernoulli was a fellow of the Academies of St. Petersburg and Berlin, and of the Royal Society of London. He continued to perform his academical duties at Bâle with unabated vigour until he attained his seventy-seventh year, when increasing infirmity obliged him to have recourse to the assistance of his nephew James, son of his younger brother John. Owing, it is said, to an early disappointment in love, he never married; but in his old age, the want of the affection of a wife and children was supplied (as far as it is possible to supply that want) by the almost filial veneration with which the simple and benevolent character of a man so famous, inspired his fellow-townsmen and all who knew him. Out of his moderate emoluments he found means to practise much hospitality and more beneficence, and to bequeath an endowment for poor students. He died soon after the commencement of his eighty-third year, on the 17th March, 1782. His separate treatises have already been mentioned; his papers on various subjects may be found in the