1755. He became known as one of the most expert epigrammatists in the

gay society of Louis XV.'s court, and by his verses won the friendship of Madame de Pompadour, the royal mistress, who obtained for him an apartment, furnished at her expense, in the Tuileries, and a yearly pension of 1500 livres (about L60). In 1751 he was appointed to the French embassy at Venice, where he acted, to the satisfaction of both parties, as mediator between the republic and Pope Benedict XIV. During his stay in Venice he received subdeacon's orders, and on his return to France in 1755 was made a papal councillor of state. He took an important part in the delicate negotiations between France and Austria which preceded the Seven Years' War. He regarded the alliance purely as a temporary expedient, and did not propose to employ the whole forces of France in a general war. But he was overruled by his colleagues. He became secretary for foreign affairs on the 27th of June 1757, but owing to his attempts to counteract the spendthrift policy of the marquise de Pompadour and her creatures, he fell into disgrace and was in December 1758 banished to Soissons by Louis XV., where he remained in retirement for six years. In the previous November he had been created cardinal by Clement XIII. On the death of the royal mistress in 1764, Bernis was recalled and once more offered the seals of office, but declined them, and was appointed archbishop of Albi. His occupancy of the see was not of long duration. In 1769 he went to Rome to assist at the conclave which resulted in the election of Clement XIV., and the talent which he displayed on that occasion procured him the appointment of ambassador in Rome, where he spent the remainder of his life. He was partly instrumental in bringing about the suppression of the Jesuits, and acted with greater moderation than is generally allowed. He lost his influence under Pius VI., who was friendly to the Jesuits, and the French Revolution, to which he was hostile, reduced him almost to penury; the court of Spain, however, mindful of the support he had given to their ambassador in obtaining the condemnation of the Jesuits, came to his relief with a handsome pension. He died at Rome on the 3rd of November 1794, and was buried in the church of S. Luigi de' Francesi. In 1803 his remains were transferred to the cathedral at Nimes. His poems, the longest of which is _La Religion vengee_ (Parma, 1794), have no merit; they were collected and published after his death (Paris, 1797, &c.); his _Memoires et lettres 1715-58_ (2 vols., Paris, 1878) are still interesting to the historian. See Frederic Masson's prefaces to the _Memoires et lettres_, and _Le Cardinal de Bernis depuis son ministere;_ (Paris, 1884); E. et J. de Goncourt, _Mme de Pompadour_ (Paris, 1888), and Sainte-Beuve, _Causeries du lundi_, t. viii. BERNKASTEL, a town of Germany, in the Prussian Rhine province, on the Mosel, in a deep and romantic valley, connected by a branch to Wengerohr with the main Trier-Coblenz railway. Pop. 2300. It has some unimportant manufactures; the chief industry is in wine, of which Berncastler Doctor enjoys great repute. Above the town lie the ruins of the castle Landshut. Bernkastel originally belonged to the chapter of Trier, and received its name from one of the provosts of the cathedral, Adalbero of Luxemburg (hence _Adalberonis castellum_). BERNOULLI, or BERNOUILLI, the name of an illustrious family in the annals of science, who came originally from Antwerp. Driven from their country during the oppressive government of Spain for their attachment to the Reformed religion, the Bernoullis sought first an asylum at Frankfort (1583), and afterwards at Basel, where they ultimately obtained the highest distinctions. In the course of a century eight of its members successfully cultivated various branches of mathematics, and contributed powerfully to the advance of science. The most celebrated were Jacques (James), Jean (John) and Daniel, the first, second and fourth as dealt with below; but, for the sake of perspicuity they may be considered as nearly as possible in the order of family succession. A complete summary of the great developments of mathematical learning, which the members of this family effected, lies outside the scope of this notice. More detailed accounts are to be found in the various mathematical articles. I. JACQUES BERNOULLI (1654-1705), mathematician, was born at Basel on the 27th of December 1654. He was educated at the public school of Basel, and also received private instruction from the learned Hoffmann, then professor of Greek. At the conclusion of his philosophical studies at the university, some geometrical figures, which fell in his way, excited in him a passion for mathematical pursuits, and in spite of the opposition of his father, who wished him to be a clergyman, he applied himself in secret to his favourite science. In 1676 he visited Geneva on his way to France, and subsequently travelled to England and Holland. While at Geneva he taught a blind girl several branches of science, and also how to write; and this led him to publish _A Method of Teaching Mathematics to the Blind_. At Bordeaux his _Universal Tables on Dialling_ were constructed; and in London he was admitted to the meetings of Robert Boyle, Robert Hooke and other learned and scientific men. On his final return to Basel in 1682, he devoted himself to physical and mathematical investigations, and opened a public seminary for experimental physics. In the same year he published his essay on comets, _Conamen Novi Systematis Cometarum_, which was occasioned by the appearance of the comet of 1680. This essay, and his next publication, entitled _De Gravitate Aetheris_, were deeply tinged with the philosophy of Rene Descartes, but they contain truths not unworthy of the philosophy of Sir Isaac Newton's _Principia_. Jacques Bernoulli cannot be strictly called an independent discoverer; but, from his extensive and successful application of the calculus and other mathematical methods, he is deserving of a place by the side of Newton and Leibnitz. As an additional claim to remembrance, he was the first to solve Leibnitz's problem of the isochronous curve (_Acta Eruditorum_, 1690). He proposed the problem of the catenary (q.v.) or curve formed by a chain suspended by its two extremities, accepted Leibnitz's construction of the curve and solved more complicated problems relating to it. He determined the "elastic curve," which is formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other, and which he showed to be the same as the curvature of an impervious sail filled with a liquid (_lintearia_). In his investigations respecting cycloidal lines and various spiral curves, his attention was directed to the loxodromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of reproducing itself under a variety of conditions. In 1696 he proposed the famous problem of isoperimetrical figures, and offered a reward for its solution. This problem engaged the attention of British as well as continental mathematicians; and its proposal gave rise to a painful quarrel with his brother Jean. Jean offered a solution of the problem; his brother pronounced it to be wrong. Jean then amended his solution, and again offered it, and claimed the reward. Jacques still declared it to be no solution, and soon after published his own. In 1701 he published also the demonstration of his solution, which was accepted by the marquis de l'Hopital and Leibnitz. Jean, however, held his peace for several years, and then dishonestly published, after the death of Jacques, another incorrect solution; and not until 1718 did he admit that he had been in error. Even then he set forth as his own his brother's solution purposely disguised. In 1687 the mathematical chair of the university of Basel was conferred upon Jacques. He was once made rector of his university, and had other distinctions bestowed on him. He and his brother Jean were the first two foreign associates of the Academy of Sciences of Paris; and, at the request of Leibnitz, they were both received as members of the academy of Berlin. In 1684 he had been offered a professorship at Heidelberg; but his marriage with a lady of his native city led him to decline the invitation. Intense application brought on infirmities and a slow fever, of which he died on the 16th of August 1705. Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, _Eadem mutata resurgo_. Jacques Bernoulli wrote elegant verses in Latin, German and French; but although these were held in high estimation in his own time, it is on his mathematical works that his fame now rests. These are:--_Jacobi Bernoulli Basiliensis Opera_ (Genevae, 1744), 2 tom. 4to; _Ars Conjectandi, opus posthumum: accedunt tractatus de Seriebus Infinitis, et epistola (Gallice scripta) de Ludo Pilae Reticularis_ (Basiliae, 1713), 1 tom. 4to. II. JEAN BERNOULLI (1667-1748), brother of the preceding, was born at Basel on the 27th of July 1667. After finishing his literary studies he was sent to Neuchatel to learn commerce and acquire the French language. But at the end of a year he renounced the pursuits of commerce, returned to the university of Basel, and was admitted to the degree of bachelor in philosophy, and a year later, at the age of 18, to that of master of arts. In his studies he was aided by his elder brother Jacques. Chemistry, as well as mathematics, seems to have been the object of his early attention; and in the year 1690 he published a dissertation on effervescence and fermentation. The same year he went to Geneva, where he gave instruction in the differential calculus to Nicolas Fatio de Duillier, and afterwards proceeded to Paris, where he enjoyed the society of N. Malebranche, J.D. Cassini, Philip de Lahire and Pierre Varignon. With the marquis de l'Hopital he spent four months studying higher geometry and the resources of the new calculus. His independent discoveries in mathematics are numerous and important. Among these were the exponential calculus, and the curve called by him the _linea brachistochrona_, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density. On his return to his native city he studied medicine, and in 1694 took the degree of M.D. Although he had declined a professorship in Germany, he now accepted an invitation to the chair of mathematics at Groningen (_Commercium Philosophicum_, epist. xi. and xii.). There, in addition to the learned lectures by which he endeavoured to revive mathematical science in the university, he gave a public course of experimental physics. During a residence of ten years in Groningen, his controversies were almost as numerous as his discoveries. His dissertation on the "barometric light," first observed by Jean Picard, and discussed by Jean Bernoulli under the name of mercurial phosphorus, or mercury shining in vacuo (_Diss. physica de mercurio lucente in vacuo_), procured him the notice of royalty, and engaged him in controversy. Through the influence of Leibnitz he received from the king of Prussia a gold medal for his supposed discoveries; but Nicolaus Hartsoeker and some of the French academicians disputed the fact. The family quarrel about the problem of isoperimetrical figures above mentioned began about this time. In his dispute with his brother, in his controversies with the English and Scottish mathematicians, and in his harsh and jealous bearing to his son Daniel, he showed a mean, unfair and violent temper. He had declined, during his residence at Groningen, an invitation to Utrecht, but accepted in 1705 the mathematical chair in the university of his native city, vacant by the death of his brother Jacques; and here he remained till his death. His inaugural discourse was on the "new analysis," which he so successfully applied in investigating various problems both in pure and applied mathematics. He was several times a successful competitor for the prizes given by the Academy of Sciences of Paris; the subjects of his essays being:--the laws of motion (_Discours sur les lois de la communication du mouvement_, 1727), the elliptical orbits of the planets, and the inclinations of the planetary 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 des vaisseaux_. It is, however, his works in pure mathematics that are the permanent monuments of his fame. Jean le Rond d'Alembert acknowledges with gratitude, that "whatever he knew of mathematics he owed to the works of Jean 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 Leonhard Euler, yet was blind to the excellence of Sir Isaac 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 trifling; 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 Gabriel Cramer, professor of mathematics at Geneva, and published under the title of _Johannis Bernoulli Operi Omnia_ (Lausan. et Genev.), 4 tom. 4to; his interesting correspondence with Leibnitz appeared under the title of _Gul. Leibnitii et Johannis Bernoulli Commercium Philosophicum et Mathematicum_ (Lausan. et Genev. 1745), 2 tom. 4to. III. NICOLAS BERNOULLI (1695-1726), the eldest of the three sons of Jean Bernoulli, was born on the 27th of January 1695. At the age of eight he 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 instruction 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 mathematico 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 Pierre Varignon and Count Riccati. 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. Both were appointed at the same time professors of mathematics in the academy of St Petersburg; but this office Nicolas enjoyed for little more then eight months. He died on the 26th of July 1726 of a lingering fever. Sensible of the loss which the nation had sustained by his death, the empress Catherine ordered him a funeral at the public expense. Some of his papers are published in his father's works, and others in the _Acta Eruditorum_ and the _Comment. Acad. Petropol._ IV. DANIEL BERNOULLI (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 Konig 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. Several of his investigations are contained in the earlier volumes of the _Comment. Acad. Petropol._; and his separately published works are:--_Dissertatio Inaugur. Phys. Med. de Respiratione_ (Basil. 1721), 4to; _Positiones Anatomico-Botanicae_ (Basil. 1721), 4to; _Exercitationes quaedam Mathematicae_ (Venetiis, 1724), 4to; _Hydrodynamica_ (Argentorati, 1738), 4to. V. JEAN BERNOULLI (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. NICOLAS BERNOULLI (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. JEAN BERNOULLI (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 Neuchatel, 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. JACQUES BERNOULLI (1759-1789), younger brother of the preceding, and the second of this name, was born at Basel on the 17th of October