Part 23

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 1759. Having finished his literary studies, he was, according to custom, sent to Neuchatel 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 _Societa 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.

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 _Memoires du philosophe de Merian_. See generally M. Cantor, _Geschichte der Mathematik_; J.C. Poggendorff, _Biographisch-literarisches Handworterbuch_ (1863-1904).

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, _Vogele 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.

BERNSTORFF, ANDREAS PETER, COUNT VON (1735-1797), Danish statesman, was born at Hanover on the 28th of August 1735. His career was determined by his uncle, Johann Hartwig Ernst Bernstorff, who early discerned the talents of his nephew and induced him to study in the German and Swiss universities and travel for some years in Italy, France, England and Holland, to prepare himself for a statesman's career. During these _Wanderjahre_ he made the acquaintance of the poets Gellert and Jacobi, the learned Jean-Jacques Barthelemy, the duc de Choiseul, and Gottfried Achenwall, the statistician. At his uncle's desire he rejected the Hanoverian for the Danish service, and in 1759 took his seat in the German chancery at Copenhagen. In 1767, at the same time as his uncle, he was created a count, and in 1769 was made a privy-councillor. He is described at this period as intellectual, upright and absolutely trustworthy, but obstinate and self-opinionated to the highest degree, arguing with antiquaries about coins, with equerries about horses, and with foreigners about their own countries, always certain that he was right and they wrong, whatever the discussion might be. He shared the disgrace of his uncle when Struensee came into power, but re-entered the Danish service after Struensee's fall at the end of 1772, working at first in the financial and economical departments, and taking an especial interest in agriculture. The improvements he introduced in the tenures of his peasantry anticipated in some respects the agricultural reforms of the next generation.

In April 1773 Bernstorff was transferred to the position for which he was especially fitted, the ministry of foreign affairs, with which he combined the presidency of the German chancery (for Schleswig-Holstein). His predecessor, Adolf Siegfried Osten, had been dismissed because he was not _persona grata_ at St Petersburg, and Bernstorff's first official act was to conclude the negotiations which had long been pending with the grand-duke Paul as duke of Holstein-Gottorp. The result was the exchange-treaty of the 1st of June (May 21 O.S.) 1773, confirming the previous treaty of 1767 (see BERNSTORFF, J.H.E.). This was followed by the treaty of alliance between Denmark and Russia of the 12th of August 1773, which was partly a mutually defensive league, and

## partly an engagement between the two states to upset the new

constitution recently established in Sweden by Gustavus III., when the right moment for doing so should arrive. For this mischievous and immoral alliance, which bound Denmark to the wheels of the Russian empress's chariot and sought to interfere in the internal affairs of a neighbouring state, Bernstorff was scarcely responsible, for the preliminaries had been definitely settled in his uncle's time and he merely concluded them. But there can be no doubt that he regarded this anti-Swedish policy as the correct one for Denmark, especially with a monarch like Gustavus III. on the Swedish throne. It is also pretty certain that the anti-Swedish alliance was Russia's price for compounding the Gottorp difficulty.

Starting from the hypothesis that Sweden was "Denmark-Norway's most

## active and irreconcilable enemy," Bernstorff logically included France,

the secular ally of Sweden, among the hostile powers with whom an alliance was to be avoided, and drew near to Great Britain as the natural foe of France, especially during the American War of Independence, and this too despite the irritation occasioned in Denmark-Norway by Great Britain's masterful interpretation of the expression "contraband." Bernstorff's sympathy with England grew stronger still when in 1779 Spain joined her enemies; and he was much inclined, the same winter, to join a triple alliance between Great Britain, Russia and Denmark-Norway, proposed by England for the purpose of compelling the Bourbon powers to accept reasonable terms of peace. But he was overruled by the crown prince Frederick, who thought such a policy too hazardous, when Russia declined to have anything to do with it. Instead of this the Russian chancellor Nikita Panin proposed an armed league to embrace all the neutral powers, for the purpose of protecting neutral shipping in time of war. This league was very similar to one proposed by Bernstorff himself in September 1778 for enforcing the principle "a free ship makes the cargo free"; but as now presented by Russia, he rightly regarded it as directed exclusively against England. He acceded to it indeed (9th of July 1780) because he could not help doing so; but he had previously, by a separate treaty with England, on the 4th of July, come to an understanding with that power as to the meaning of the expression "contraband of war." This independence caused great wrath at St Petersburg, where Bernstorff was accused of disloyalty, and ultimately sacrificed to the resentment of the Russian government (13th of November 1780), the more readily as he already disagreed on many important points of domestic administration with the prime minister Hoegh Guldberg. He retired to his Mecklenburg estates, but on the fall of Guldberg four years later, was recalled to office (April 1784). The ensuing thirteen years were perhaps the best days of the old Danish absolutism. The government, under the direction of such enlightened ministers as Bernstorff, Reventlow and others, held the mean between Struensee's extravagant cosmopolitanism and Guldberg's stiff conservatism. In such noble projects of reform as the emancipation of the serfs (see REVENTLOW) Bernstorff took a leading part, and so closely did he associate himself with everything Danish, so popular did he become in the Danish capital, that a Swedish diplomatist expressed the opinion that henceforth Bernstorff could not be removed without danger. Liberal-minded as he was, he held that "the will of the nation should be a law to the king," and he boldly upheld the freedom of the press as the surest of safety-valves.