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 IV. (1700–1782), the second son of Jean Bernoulli, was born on the 29th of January 1700, at Groningen. He studied medicine and became a physician, but his attention was early directed also to geometrical studies. The severity of his father’s manner was ill-calculated to encourage the first efforts of one so sensitive; but fortunately, at the age of eleven, he became the pupil of his brother Nicolas. He afterwards studied in Italy under Francesco Domenico Michelotti and Giambattista Morgagni. After his return, though only twenty-four years of age, he was invited to become president of an academy then projected at Genoa; but, declining this honour, he was, in the following year, appointed professor of mathematics at St Petersburg. In consequence of the state of his health, however, he returned to Basel in 1733, where he was appointed professor of anatomy and botany, and afterwards of experimental and speculative philosophy. In the labours of this office he spent the remaining years of his life. He had previously published some medical and botanical dissertations, besides his Exercitationes quaedam Mathematicae, containing a solution of the differential equation proposed by Riccati and now known by his name. In 1738 appeared his Hydrodynamica, in which the equilibrium, the pressure, the reaction and varied velocities of fluids are considered both theoretically and practically. One of these problems, illustrated by experiment, deals with an ingenious mode of propelling vessels by the reaction of water ejected from the stern. Some of his experiments on this subject were performed before Pierre Louis M. de Maupertuis and Alexis Claude Clairaut, whom the fame of the Bernoullis had attracted to Basel. With a success equalled only by Leonhard Euler, Daniel Bernoulli gained or shared no less than ten prizes of the Academy of Sciences of Paris. The first, for a memoir on the construction of a clepsydra for measuring time exactly at sea, he gained at the age of twenty-four; the second, for one on the physical cause of the inclination of the planetary orbits, he divided with his father; and the third, for a communication on the tides, he shared with Euler, Colin Maclaurin and another competitor. The problem of vibrating cords, which had been some time before resolved by Brook Taylor (1685–1731) and d’Alembert, became the subject of a long discussion conducted in a generous spirit between Bernoulli and his friend Euler. In one of his early investigations he gave an ingenious though indirect demonstration of the problem of the parallelogram of forces. His labours in the decline of life were chiefly directed to the doctrine of probabilities in reference to practical purposes, and in particular to economical subjects, as, for example, to inoculation, and to the duration of married life in the two sexes, as well as to the relative proportion of male and female births. He retained his usual vigour of understanding till near the age of eighty, when his nephew Jacques relieved him of his public duties. He was afflicted with asthma, and his retirement was relieved only by the society of a few chosen friends. He died on the 17th of March 1782 at Basel. Excluded by his professional character from the councils of the republic, he nevertheless received all the deference and honour due to a first magistrate. He was wont to mention the following as the two incidents in his life which had afforded him the greatest pleasure,—that a stranger, whom he had met as a travelling companion in his youth, made to his declaration “I am Daniel Bernoulli” the incredulous and mocking reply, “And I am Isaac Newton”; and that, while entertaining König and other guests, he solved without rising from table a problem which that mathematician had submitted as difficult and lengthy. Like his father, he was a member of almost every learned society of Europe, and he succeeded him as foreign associate of the Academy of Paris.

V. (1710–1790), the youngest of the three sons of Jean Bernoulli, was born at Basel on the 18th of May 1710. He studied law and mathematics, and, after travelling in France, was for five years professor of eloquence in the university of his native city. On the death of his father he succeeded him as professor of mathematics. He was thrice a successful competitor for the prizes of the Academy of Sciences of Paris. His prize subjects were, the capstan, the propagation of light, and the magnet. He enjoyed the friendship of P. L. M. de Maupertuis, who died under his roof while on his way to Berlin. He himself died in 1790. His two sons, Jean and Jacques, are the last noted mathematicians of the family.

VI. (1687–1759), cousin of the three preceding, and son of Nicolas Bernoulli, one of the senators of Basel, was born in that city on the 10th of October 1687. He visited England, where he was kindly received by Sir Isaac Newton and Edmund Halley (Com. Phil. ep. 199), held for a time the mathematical chair at Padua, and was successively professor of logic and of law at Basel, where he died on the 29th of November 1759. He was editor of the Ars Conjectandi of his uncle Jacques. His own works are contained in the Acta Eruditorum, the Giornale de’ letterati d’ Italia, and the Commercium Philosophicum.

VII. (1744–1807), grandson of the first Jean Bernoulli, and son of the second of that name, was born at Basel on the 4th of November 1744. He studied at Basel and at Neuchâtel, and when thirteen years of age took the degree of doctor in philosophy. At nineteen he was appointed astronomer royal of Berlin. Some years after, he visited Germany, France and England, and subsequently Italy, Russia and Poland. On his return to Berlin he was appointed director of the mathematical department of the academy. Here he died on the 13th of July 1807. His writings consist of travels and astronomical, geographical and mathematical works. In 1774 he published a French translation of Leonhard Euler’s Elements of Algebra. He contributed several papers to the Academy of Berlin.

VIII. (1759–1789), younger brother of the preceding, and the second of this name, was born at Basel on the 17th of October 1759. Having finished his literary studies, he was, according to custom, sent to Neuchâtel to learn French. On his return he graduated in law. This study, however, did not check his hereditary taste for geometry. The early lessons which he had received from his father were continued by his uncle Daniel, and such was his progress that at the age of twenty-one he was called to undertake the duties of the chair of experimental physics, which his uncle’s advanced years rendered him unable to discharge. He afterwards accepted the situation of secretary to count de Brenner, which afforded him an opportunity of seeing Germany and Italy. In Italy he formed a friendship with Lorgna, professor of mathematics at Verona, and one of the founders of the Società Italiana for the encouragement of the sciences. He was also made corresponding member of the royal society of Turin; and, while residing at Venice, he was, through the friendly representation of Nicolaus von Fuss, admitted into the academy of St Petersburg. In 1788 he was named one of its mathematical professors.

He was tragically drowned while bathing in the Neva in July 1789, a few months after his marriage with a daughter of Albert Euler, son of Leonhard Euler.

 BERNSTEIN, AARON (1812–1884), Jewish scientist, author and reformer. In the middle of the 19th century Bernstein took an active share in the movement for synagogue reform in Germany. He was the author of two delightful Ghetto stories, Vögele der Maggid and Mendel Gibbor, being one of the originators of this genre of modern fiction. He was also a publicist, and his History of Revolution and Reaction in Germany (3 vols., 1883–1884) was a collection of important political essays.