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 . Be 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, 169o) . 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 (tintearia) . 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 I 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), I torn . 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 1'Hbpital 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 mathetriatical 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 lucenle 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 Hartsoekerand 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 less 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 Gut .

Leibnitii et Johannis Bernoulli Commercium Philosophicum et Mathematicum (Lausan. et Genev . 1745), 2 tom . 4to . Some of his papers are published in his father's works, and others in the Acta Erudilorum 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 per-formed 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 (1710-179o), 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 loth of October 1687 . He visited England, where he was kindly received by Sir Isaac Newton and Edmund Halley (Corn . 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 Ada Eruditorum, the Giornale de' letterati d' Italia, and the Commercium Philosophicum . 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 . 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 Mimoires du philosophe de Merian . See generally M . Cantor, Geschichte der Mathemalik; J . C . Poggendorff, Biographisch-literarisches Handworterbuch (1863–1904) .