various problems botli in pure and mixed mathematics. At the request of the magistracy of Basel he applied himself
to correct the relaxed discipline of the university.He was several times a successful competitor for the prizes given by the Academy of Sciences of Paris; and the subjects of his essays were, the laws of motion (Discours sur Ics Lois de la Communication du Mouvement, 1727), the ellip tical orbits of the planets, and the inclinations of the plane tary orbits (Essai d une Nouvelle Physique Celeste, 1735). In the last case his son Daniel divided the prize with him. Some years after his return to Basel he published an essay, entitled Nouvelle Theorie de la Manoeuvre dcs Vaisseaiix. It is, however, his works in pure mathematics that are the permanent monuments of his fame. D Alembert acknow ledges with gratitude, that " whatever he knew of mathe matics he owed to the works of John Bernoulli" He was a member of almost every learned society in Europe, and one of the first mathematicians of a mathematical-age. He was as keen in his resentments as he was ardent in his friendships ; fondly attached to his family, he yet disliked a deserving son ; he gave full praise to Leibnitz and Euler, yet was blind to the excellence of Newton. Such was the vigour of his constitution that he continued to pursue his usual mathematical studies till the age of eighty. He was then attacked by a complaint at first apparently trifl ing ; but his strength daily and rapidly declined till the 1st of January 1748, when he died peacefully in his sleep.
His writings were collected under his own eye by Cramer, professor of mathematics at Geneva, and published under the title of Johannis Bernoulli Opcri Omnia, Lausan. et Genev. 4 torn. 4to. His interesting correspondence with Leibnitz appeared under the title of Gul. Leibnitii et Jo hannis Bernoulli Commercium Philosojrfiicum et Mathemati- cum, Lausan. et Genev. 1745, 2 torn. 4to.
III. Nicholas Bernoulli, the eldest of the three sons of John Bernoulli, was born in 1695. His early indications of genius were carefully cherished. At the age of eight lie could speak German, Dutch, French, and Latin. When his father returned to Basel he went to the university of that city, where, at the age of sixteen, he took the degree of doctor in philosophy, and four years later the highest degree in law. Meanwhile the study of mathematics was not neglected, as appears not only from his giving instructions in geometry to his younger brother Daniel, but from his writings on the differential, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands. ( " Lampada nunc tradam filio meo natu maximo, juveni xxi. annorum, ingenio mathe- matico aliisque dotibus satis instructo," Com. Phil. ep. 223). With his father s permission he visited Italy and France, and during his travels formed friendship with Varignon and with Pdccati, one of the first mathematicians of Italy. The invitation of a Venetian nobleman induced him again to visit Italy, where he resided two years, till his return to be a candidate for the chair of jurisprudence at Basel. He was unsuccessful, but was soon afterwards appointed to a similar office in the University of Bern. Here he resided three years, his happiness only marred by regret on account of his separation from his brother Daniel, with whom he was united in sentiment and pursuits. Both were appointed at the same time professors of mathe matics in the Academy qf St Petersburg ; but this office Nicholas enjoyed for little more than eight months. At the end of July 1726 he was cut off in the prime of life by a lingering fever. Sensible of the loss which the nation had sustained by his death, the Empress Catherine ordered him a fuueral at the public expense. Some of his papers are published in his father s works, and others in the Ada Eruditorum and the Comment. Acad. Petropol.
IV. Daniel Bernoulli, the second son of John Bernoulli, was born 9th February 1700, at Groningen. He studied medicine and became a physician, but his attention w;ia 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 Nicholas. He afterwards studied in Italy under Michelotti and Mor- g;igni. 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 after wards 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 botani cal dissertations, besides his Exercitationes qucedam Mathe matics, containing a solution of the differential equation proposed by Eiccati and now known by his name. In 1738 appeared his Hydrodynamica, in which the equi librium, 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 Maupertuis and Clairaut, whom the fame of the Bernoullis had attracted to Basel. With a success equalled only by Euler, Daniel Bernoulli gained or shared no less than ten prizes (>f the Academy of Sciences of Paris. The first, for a memoir on the con struction 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, Mac- laurin, and another competitor. The problem of vibrating cords, which had been some time before resolved by Taylor 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 James 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. In the spring of 1782, after some days illness, he died, like his father, in the repose of sleep. 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 Avont 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, mado to his declaration "I am Daniel Bernoulli" the incredulous and mocking reply, " And I am Isaac Newton ; and that, while entertaining Konig and other guests, he solved with out rising from table a problem which that mathematician had submitted as difficult and lengthy.
learned society of Europe, and he succeeded him as foreign associate of the Academy of Paris. Several of his investi
gations are contained in the earlier volumes of the St
