BERNOULLI, or BERNOUILLI, a name illustrious in the annals of science, belonging to a family of respectability, originally of Antwerp. Driven from their country during the oppressive government of Spain for their attachment to the Reformed religion, the family 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 of the family were James, John, and Daniel; but, for the sake of perspicuity they may be considered as nearly as possible in the order of family succession.

I. JAMES BERNOULLI was born at Basel on the 27th 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 Boyle, Hooke, Stillingfleet, 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, Oonamen N~ovi Systematic Gometarum, which was occasioned by the appear-ance of the comet of 1680. This essay, and his next publication, entitled De Gravitate Jltheris, were deeply tinged with the philosophy of Descartes, but they contain truths not unworthy of the philosophy of the Principia.

James Bernoulli cannot be strictly called an independent discoverer ; but, from his extensive and successful application of the calculus, he is well 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, and to determine the catenary, or curve formed by a chain suspended by its two extremi-ties, which he also showed to be the same as the curva-ture of a sail filled with wind. This led him on to another curve, which, being formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other, he called the elastic curve, and which he showed to be the same as the curvature of an impervious sail filled with a liquid. In his investigations respecting cycloidal lines and various spiral curves, his attention was directed to the loxo-dromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of repro-ducing itself unde»a great variety of conditions.

In 1696 he proposed the famous problem of isoperi-metrical 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 between the brothers. John offered a solution of the problem ; his brother pronounced it to be wrong. John then amended his solution, and again offered it, and claimed the reward. James still declared it to be no solution, and soon after published his own. In 1701 he published also the demonstration of his solu-tion, which was accepted by De l'Hôpital and Leibnitz. John, however, held his peace for several years, and then dishonestly published, after the death of James, 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 James ; and in the discharge of its duties he was so successful as to attract students from other countries. Some of his pupils became afterwards pro-fessors in the universities of Germany. He was once made rector of his university, and had other distinctions bestowed on him. He and his brother John were the first two foreign associates of the Academy of Sciences at 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, with the resignation of a Christian and the firmness of a philosopher. Like another Archimedes, he requested that, as a monument of his labours and an emblem of his hope of a resurrection, the logarithmic spiral should be engraven on his tombstone, with these words, Modem mutata resurgo.

James 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—(1.) Jaeobi Bernoulli Basiliensis Opera, Genevœ, 1744, 2 torn. 4to ; (2.) Art Gonjectandi, opus posthumum : accedunt tractatus de Serie-bus Infinitis, et epistola (Gallice scripta) de Ludo Pilae Reticularis, Basiliae, 1713, 1 torn. 4to.





II. JOHN BERNOULLI, brother of the preceding, was born at Basel on the 7th August 1667. His education was begun at six years of age ; and after finishing his literary studies he was sent to Neufchâtel to learn commerce and acquire the French language. But at the end of a year he renounced the pursuits of commerce, returned to the Uni-versity 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 James. 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 Fatio de Duiller, and afterwards proceeded to Paris, where he en-joyed the society of Malebranche, Cassini, De Lahire, and Varignon. With the Marquis de l'Hôpital he spent four months at his country house in the study of the higher geometry and the resources of the new calculus. His inde-pendent 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 deter-mine, pointing out at the same time the beautiful relation which this curve bears to the path described by a ray or particle of light passing through strata of variable density, such as our atmosphere. On his return to his native city he studied medicine, and in 1694 took the degree of M.D. At this period he married into one of the oldest families in Basel ; and although he had declined a professorship in Germany, he now accepted an invitation to the chair of mathematics at Groningen (Commercium Philosophicum, epist. xL 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 experi-mental physics. During a residence of ten years in Gron-ingen, his controversies were almost as numerous as his discoveries. His dissertation on an electrical appearance of the barometer first observed by Picard, and discussed by John 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 Leibnitz he re-ceived from the king of Prussia a gold medal for his sup-posed discoveries ; but 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 Scotch mathe-maticians, and in his harsh and jealous bearing to his son Daniel, he showed a temper mean, unfair, and violent. He had declined, during his residence at Groningen, an invitation to Utrecht, but accepted in 1705 the mathe-matical chair in the university of his native city, vacant by the death of his brother James ; 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 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 les 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 Céleste, 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 Théorie de la Manœuvre des Vaisseaux. 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 Operi Omnia, Lausan. et Genev. 4 torn. 4to. His interesting correspondence with Leibnitz appeared under the title of Gui. Leibnitii et Johannis Bernoulli Commercium Philosophicum 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 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 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 Biccati, 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 of 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 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, the second son of John Bernoulli, was born 9th February 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 Nicholas. He afterwards studied in Italy under Michelotti and Mor-gagni. 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 botani-cal dissertations, besides his Exercitationes quaedam Mathe-maticce, containing a solution of the differential equation proposed by Biccati 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 of 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 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 "lam 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.
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 St Petersburg Memoirs; and bis separately published works are—(1.) Disaertatio Inaugur. Phys. Med. de Respiratione, Basil, 1721, 4to; (2.) Positiones Anatomico-Botanicee, Basil., 1721, 4to; (3.) Exercitationes qucedam Mathematicce, Venetiis, 1724, 4to; (4.) Hydrodynamica, Argentorati, 1738, 4to.

V. JOHN BERNOULLI, the youngest of the three sons of John Bernoulli, was born at Basel on the 18th 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 Maupertuis, who died under his roof while on his way to Berlin. He himself died in 1790. His two sons, John and James, are the last noted mathe-maticians of the family.

VI. NICHOLAS BERNOULLI, cousin of the three preceding, and son of Nicholas Bernoulli, one of the senators of Basel, was born in that city on the 10th October 1687. He visited England, where he was kindly received by Newton and Halley (Com. Phil. ep. 199), held for a time the mathematical chair at Padua, which Galileo had once filled, 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 James. His own works are contained in the Acta Eruditorum, the Qiornale di Letterati d'ltalia, and the Commercium Philosophicum.

VII. JOHN BERNOULLI, grandson of the first John Bernoulli, and son of the second of that name, was born at Basel on the 4th December 1744. He studied at Basel and at NeufcMtel, 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 subse-quently Italy, Russia, and Poland. On his return to Berlin he was appointed director of the mathematical de-partment of the academy. Here he died on the 10th July 1807. His writings consist of travels and astronomi-cal, geographical, and mathematical works. In 1774 he published a French translation of Euler's Elements of Algebra. He contributed several papers to the Academy of Berlin.

VIII. JAMES BERNOULLI, younger brother of the preceding, and the second of this name, was born at Basel on the 17th October 1759. Having finished his literary studies, he was, according to custom, sent to Neufchatel to learn French. On his return he studied law and took a degree. This study, however, did not check his heredi-tary taste for geometry. The early lessons which he had received from his father were continued by his uncle Daniel, and such was his progress in the exact sciences that at the age of twenty-one he was called to undertake the duties of tho 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 friend-ship with Lorgna, professor of mathematics at Verona, and one of the founders of the Italian society for the encouragement of the sciences. He was also made corre-sponding member of the Royal Society of Turin; and, while residing at Venice, he was, through the friendly representation of Fuss, admitted into the Academy of St Petersburg. In 1788 he was named one of its mathe-matical professors. In the following year he married a daughter of Albert Euler, son of the illustrious Euler.

This marriage was soon tragically dissolved by the death of the husband, who was drowned while bathing in the Neva in July 1789. Several of his papers are contained in the first six volumes of Nova Acta Acad. Scien. Imper. Petropol., in the Acta Helvetica, in the Memoirs of the Academies of Berlin and Turin, and in his brother John's publications. He also published separately some juridical and physical theses, and a German translation of Mémoires du Philosophe de Merian.