Bernoulli. *Mensura Monochorde*; but his most celebrated work is a treatise *De Musica seu Tonis*. This latter tract is part of the Balioi manuscript, and follows the Enchiridion of Odo. Bernon was highly esteemed by the Emperor Henry II., and succeeded so well in his endeavours to promote learning, that the abbey of Reichenau was as famous in his time as those of St Gal and Cluni. He died in 1048.

a name illustrious in the annals of the exact sciences, and belonging to a family of respectability, originally of Antwerp. During the oppressive and bigoted government of Spain, when the Flemings, like the expatriated Greeks with their stores of ancient learning, carried their arts, industry, and genius, to the less enlightened portions of Europe, this exiled family, driven from their liberties and soil for their attachment to the reformed religion, sought first an asylum at Frankfort in 1583, and afterwards at Basel, where they ultimately obtained the highest professional and civic distinctions. In the course of a single century not less than eight of its members successfully cultivated different branches of the mathematics, and by their personal labours contributed much to the extension of science and to the diffusion of correct knowledge over a great portion of Europe. The most celebrated of these distinguished men were James, John, and Daniel; but, for the sake of perspicuity, it may be proper to consider them nearly in the order of family succession.

I. BERNOULLI, *James*, was born at Basel on the 27th December 1654. He was educated at the public school of Basel, and, preparatory to his philosophical course in the university of the same city, he received private instruction from the learned lexicographer Hoffman, then professor of Greek. At the conclusion of his philosophical studies, some geometrical figures, which fell in his way, excited in him so ardent a passion for mathematical pursuits, that it was not to be checked even by the opposition and entreaties of his father, who wished him to be a clergyman. Like Pascal, whom he also resembled in intensity of religious feeling, he applied himself in secret to his favourite science, and by his unaided exertions became one of the most distinguished mathematicians of his time. In these forbidden labours he chiefly cultivated astronomy; while, with classic propriety, he chose for his device Phaethon driving the chariot of the sun, with the motto, *Incito patre sidera versa*.

In the spring of 1676 he visited Geneva on his way to the south of France, and subsequently extended his travels to England and Holland, without however relinquishing the studies of his choice. While at Geneva he taught a blind young lady several branches of science, and also how to communicate her ideas in writing; and he afterwards gave to the public, under the title of *A Method of teaching Mathematics to the Blind*, an account of the plan he had followed, which consisted in having recourse to the sense of touch, instead of that of sight. At Bordeaux his Universal Tables on Dialling were constructed; and at London he had the happiness and advantage of being introduced into the philosophical meetings of Boyle, Hooke, Stillingfleet, and other learned and scientific men worthy of the age of Newton.

On his final return to Basel in 1682, he devoted himself to physical and mathematical investigations; and, in order to make the knowledge which he had acquired in travelling available, he opened a public seminary for experimental physics or mechanical philosophy, which, from the interest Bernoulli of the experiments and the eloquence of the lecturer, became a source of attraction to his fellow-citizens. In the same year he published his essay on comets, *Comenam novi systematis Cometarum*, occasioned by the appearance of the comet of 1680, which had excited the attention of all the astronomers of Europe. This essay, and his next publication, entitled *De Gravitate Aetheris*, though both deeply tinged with the philosophy of Descartes, contain some truths not unworthy of the more correct philosophy of the *Principia*. But these, and several interesting papers in the *Journal des Savans* and *Acta Eruditorum* of that period, were only the preludes to greater labours and higher achievements.

The pleasure which Pythagoras and Archimedes derived from their respective discoveries, was perhaps as vividly felt by the discoverers of the differential calculus. Newton and Leibnitz being independent discoverers, may be allowed to have had an equal share in that refined pleasure, as well as an equal claim to the merit of an invention which forms an epoch in mathematical science. But James Bernoulli, having had both the aid of his younger brother John, and the perusal of Leibnitz's new method of tangents and of maxima and minima, previously published in the Leipzig Transactions, 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 of merit, he was the first to solve Leibnitz's problem of the isochronous curve, and to determine a curve mistaken by Galileo for a parabola—the catenarian (*la chaînette*) or the curve formed by a chain suspended by its two extremities, and which he also showed to be the same as the curvature of a sail filled with wind. This curve, though proposed at first as a trial of Leibnitz's skill, is of much practical utility in regard to the nature of arches, and particularly of those of suspension.

These inquiries, like step following step, led to another interesting and useful curve, which, from 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 also showed to be the same as the curvature of an impervious sail filled with a fluid such as water. In his investigations respecting cycloidal lines and different spiral curves, his attention was particularly directed to the loxodromic and logarithmic spirals; and his discoveries regarding the last, which possesses the remarkable property of reproducing itself under a great variety of conditions, induced him afterwards to make it the monument of his labours and the emblem of his hopes.

Whether in the frailty of envying the scientific honours of his younger brother, or in the better spirit of noble emulation, which then urged the torch of truth into brighter animation, he proposed to geometers in general, and to his brother in particular, the famous problem of isoperimetrical figures, with a promised reward for its solution. This problem engaged the attention of the British as well as continental mathematicians; and however little we may be prepared to expect such a result in the calm researches of abstract truth, it gave rise to a most unfortunate family quarrel. His brother John's solution being too hasty, and not quite satisfactory, a discussion arose, which, after much altercation, became a lasting feud, to be ended only by death.

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1 *Journal des Savans*, 1669. Jacobi Bernoulli Opera. 2 "Collegium Experimentale Physico-Mechanicum publice aperuit." *Fils Jac. Bern. a Bat.* 3 Nova Methodus pro Maximis et Minimis, Remoque Tangentibus, qua nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus; per G. G. L. *Act. Erud.* 1684. As the reward of merit, the mathematical chair of the university of Basel, vacant by the death of the learned Megerlin, had been conferred upon James not long after his return from his travels; and, in the discharge of the duties of his office he was so successful as to attract students from foreign parts, and to have pupils who became afterwards professors in some of the most celebrated universities of Germany. He was once made rector of the distinguished university of which he was a member, and he had other honourable 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 earnest request of the discriminating Leibnitz, they were both made members of the academy of Berlin. His fame, too, was increased by his posthumous work De Arte Conjectandi, which embraces the application of the doctrine of probabilities to moral, political, and economical subjects.

Three years previous to his official appointment at Basel, he had been offered a professorship at Heidelberg; but his union with an accomplished lady of his native city led him to decline the invitation. In this marriage he was happy; but his family, consisting of a son and daughter, though highly respectable in life, possessed little of the characteristic genius of their father. Intense application brought on infirmities, and a slow fever, of which he died on the 16th of August 1705, with all the resignation of a Christian, and all the firmness of a philosopher. Like another Archimedes he requested, previous to his death, that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo; elegantly alluding, by the properties of the curve, to the Christian's hope of the resurrection of the body.

In James Bernoulli were united many of the highest powers of the understanding with some of the best qualities of the heart. From a regard to truth he chose rather to be silent than eloquently wrong; and in religion he was equally removed from superstition and from free-thinking. With a bilious temperament, inducing that "philosophic melancholy" which is so often the attendant of true genius, his mind was peculiarly marked by elegance and simplicity, and by retentiveness rather than quickness. Like Plato, he combined the elegant exercises of the imagination with the more severe labours of the understanding; poetry with mathematics, and eloquence with philosophy. He wrote elegant verses in Latin, in German, and in 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. Jacobi Bernoulli Basiliensis Opera, Geneva, 1744, 2 tom. 4to; 2. Jacobi Bernoulli Ars Conjectandi, opus posthumum: accedunt tractatus de Seriebus Infinitis, et epistola (Gallice scripta) de Ludo Pile Reticularis, Basiliae, 1713, 1 tom. 4to. (See also Mémoires de l'Acad. de Paris, from 1702 to 1705; Journal des Savans; Acta Eruditorum; Eloges de Fontenelle; and Biographie Universelle.)

II. Bernoulli, John, was born at Basel on the 7th August 1667. His education was begun at six years of age; and his progress was such as to give a fair promise of future greatness. Having finished his literary studies, he was sent to Neufchâtel to learn commerce and acquire the French language. But at the end of a year, renouncing the enriching pursuits of the merchant for the less lucrative investigations of the philosopher, he returned to the university of Basel, where he was admitted to the degree of bachelor in philosophy. In a year after, at the Bernoulli age of eighteen, he obtained that of master of arts. His thesis on the one occasion was De Igni lambente, written in Latin verse; and, on the other, Que le Prince est pour les Sujets, written in Greek verse.

In his mathematical education, it was his good fortune to have the able instruction of his elder brother James, whom he ultimately equalled in this congenial department of science. Chemistry, however, as well as mathematics, seems to have been the object of his early attention. In the year 1690 he published a chemical dissertation on effervescence and fermentation, in which he endeavours to explain the phenomena of chemical action by the different figures of what he calls active and passive particles. But to have made a juvenile failure where so many have erred, and where some uncertainty yet remains, can scarcely infer any disgrace.

In the same year he went to Geneva, where he gave some instructions in the differential calculus to Fatio de Duiller, and afterwards proceeded to Paris, where he enjoyed the society of Malebranche, Cassini, De la Hire, and Varignon. With the Marquis de l'Hôpital he spent four months at his country residence in the study of the higher geometry, and the resources of the new calculus, of which he was an ardent promoter, and which he had the honour of imparting to his distinguished host. But his independent discoveries in mathematics are numerous and important. Of these it may be proper to mention the exponential calculus, and the curve called by him the linea brachystochrona, or line of swiftest descent, which he was not only the first to determine, but at the same time to point out 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, a pursuit in some degree allied to his now favourite study of mathematics; and in 1694 he took the degree of M.D. On this occasion he maintained an ingenious thesis on muscular motion, in which, however questionable may be the theory which he adopts respecting the contraction of the muscles, he displays a happy application of mathematical principles to the investigation of their motive power.

At this period he formed a connection by marriage with one of the oldest and most respectable families in Basel; and although he had already declined a professorship in Germany, he now accepted an invitation to be professor of mathematics at Groningen. By his learned instructions he endeavoured to re-animate the then decayed state of mathematical science in that university; and in order to excite a taste for such pursuits, he gave a course of experimental physics to the public of that city. During a residence of ten years in Groningen, his controversies were almost as numerous as his discoveries. An incidental expression regarding the resurrection, founded on the anatomical doctrine of a constant change in the particles of the body during life, brought on him a charge of heresy, which was preferred by a theologian of more zeal than wisdom. Both his voice and pen, however, were exerted against his opponents in vindication of "his reputation, his religion, and his honour;" and in a medical dissertation De Nutritione, published at the time, he fully states his sentiments on this subject, and triumphs over his enemies with the resolution of living down such accusations. An electrical appearance of the barometer first observed by Picard, and considered by John Bernoulli under the name of mercurial phosphorus, or mercury shining in vacuo, procured for him the notice of royalty, and engaged him in controversy. Through Leibnitz he received from the king of Prussia a gold medal for his supposed discoveries; while Hartsecker and some of the French academicians disputed the fact. If we could suppose the calm breast of his brother James to have been in any respect ruffled by this mark of royal favour, it might perhaps be considered as the remote cause of the family quarrel about the problem of isoperimetrical figures above mentioned. But be this as it may, the problem was proposed on this occasion; and we have no desire to use any softening when truth forbids it. Indeed his most ardent admirers must allow that, 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 temper no less unfortunate for its possessor than calculated to impair the lustre of his character, and detract from the respect otherwise due to it. Having declined, during his residence at Groningen, an invitation to Utrecht, he however accepted, in 1765, the mathematical chair in the university of his native city, vacant by the death of his brother James; and here he remained till death, though flatteringly invited to higher emoluments in other distinguished universities. His inaugural discourse was De Paris Novae Analyseos et Geometriae Sublimis, which he himself so successfully applied in investigating the solid of least resistance, the problem of orthogonal trajectories, and various others both in pure and mixed mathematics. Successful in his official labours, at the request of the magistracy of Basel he applied himself to correct the relaxed discipline of the university, which was not indeed the least difficult task which he achieved.

He was several times a competitor for the prizes given by the academy of sciences of Paris; and his essays crowned with success were, on the laws of motion, on the elliptical orbits of the planets, and on the inclination of the planetary orbits. This last honour his son Daniel, the pupil of a better philosophy than the Cartesian, had the merit to divide with him; though the ambitious sensibility of the old man seems to have been hurt with what in other circumstances would have afforded him pleasure.

Some years after his return to Basel he published a valuable essay, entitled Nouvelle Théorie de la Manœuvre des Vaissaux, which was occasioned by the questionable principles advanced by Renau on the same subject. But his works in pure mathematics are the permanent monuments of his fame, and must be studied to be fully appreciated. D'Alembert indeed acknowledges in gratitude, that "whatever he knew of mathematics he owed to the works of John Bernoulli." If his Hydraulica, one of the latest of his productions, was written, as has been supposed, in rivalry of the more original work of his son Daniel, it can only be considered as a proof of that infirmity from which great minds are not always free.

His health had at one period suffered much, as he says, from those midnight hours of mental labour stolen from sleep; yet 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 fell asleep, Bernoulli, and thus expired without agony, or even a discomposing pang, a death not unlike that of the illustrious chemist Black, who died with his usual meal of milk balanced in his hand, as if to show by an experiment how easy his exit from life had been.

He was a member of almost every learned society in Europe, and one of the first mathematicians of a mathematical age. With great warmth of temperament, he was as keen in his resentments as ardent in his friendships: a tender husband, though a severe brother; fondly attached to his family, yet disliking a deserving son; giving full praise to Leibnitz and Euler, yet blind to the surpassing excellence of Newton; sincere in his religion, though attacked by a theologian, and eulogised by a freethinker. The graphic if not faultless pencil of Voltaire has thus drawn his character:

Son esprit vit la vérité, Et son cœur connaît la justice. Il a fait l'honneur de la Suisse, Et celui de l'humanité.

His various writings were collected under his own eye by Cramer, professor of mathematics at Geneva, and published under the title of Johannis Bernoulli Opera Omnia, Lausam. et Genev. 4 tom. 4to; and his interesting correspondence with Leibnitz under the title of Gul. Leibnitii et Johannis Bernoulli Commercium Philosophicum et Mathematicum, Lausam. et Genev. 1745, 2 tom. 4to.

III. Bernoulli, Nicholas, the eldest of the three distinguished sons of John Bernoulli, was only a few months old when his father went to Groningen in the autumn of 1695. His early indications of genius were cherished with all the fond attention usually bestowed on a first-born. 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 after the highest degree in law.

During these pursuits the study of mathematics had not been 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.

Gladly embracing his father's permission to travel, he visited Italy and France. In the one he formed a friendship with Varignon, and in the other with Riccati, one of the first mathematicians of Italy, and the father of a family in some degree talented like the Bernoullis themselves; but ill health precluding him from reaping the advantages of more extended travelling, he returned to his native city.

The invitation of a Venetian nobleman, however, induced him again to visit Italy, where he resided two years, till his return to be a candidate for the chair of jurisprudence in the university of Basel. Disappointed in this object by the rather singular mode of election by lot, he was soon afterwards honourably appointed to a similar office in the university of Berne. Here he resided three years, respected and beloved, and with no other regret than what he felt on account of his separation from his brother Daniel; an attachment of kindred minds united

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1 Diss. Physica de Mercurio lucente in vacuo. Basil. 1719. 2 Commerc. Phil. ep. 214. 3 Essai d'une Nouvelle Physique Céleste. A Paris, 1735. 4 Hist. de l'Acad. 1748. 5 "Lampada nunc tradam filio meo natu maximo, juveni xxl annorum, ingenio mathematico aliisque dotibus satis instructo." (Com. Phil. ep. 223.) Bernoulli, in sentiment and pursuit, and deserving, in truth, the appellation of "par nobile fratum." Both seemed fortunate in being appointed at the same time professors of mathematics in the academy of Petersburg; but this important office Nicholas enjoyed for little more than eight months. In the end of July 1726 he was cut off by a lingering fever, in the prime of life, and in the opening career of useful labour.

Sensible of the loss which the nation had sustained by the death of Nicholas Bernoulli, the Empress Catherine ordered him a funeral at the public expense; and, waiting at the academy, she presented her condolence in person to his surviving brother.1

Some of his papers are published in his father's works, and the others in the Acta Eruditorum, and the Comment. Acad. Petrop.

IV. Bernoulli, Daniel, second son of John Bernoulli, was born 9th February 1700, during his father's residence in Groningen. Like his father, Daniel was intended for a mercantile life; but, preferring a learned profession, he studied medicine, became a physician, and, following with hereditary ability and ardour his paternal steps, he rose to an eminence and fame worthy of his first and favourite appellation, "Son of John Bernoulli."

Taught by his father to regard mathematical learning as the sure foundation of all science and all practical art, his attention was early directed to geometrical studies, though the severity of his father's manner was ill calculated to encourage the first efforts of one so sensitive a disposition; but fortunately, at the age of eleven, he became the pupil of his affectionate brother Nicholas.2 Mathematics and medicine were equally the studies of his youth, and these he had the means of cultivating with particular advantage in Italy under Michelotti and Morgagni; the former being an able geometer, and the latter well skilled in several branches of physical science, as well as in anatomy. In such society he breathed a kindred element; and when he left Italy, he returned to his native country with well-won literary and scientific honours.

About this time, 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 Petersburg. In this situation, although he enjoyed a competent salary, he still looked back with fond regard to the humble but equal lot of his native country. And on his resolving to leave Russia, the court of Petersburg not only augmented his salary, but granted him the half as a retiring pension, with the liberty of returning if he chose. This truly generous conduct made him for a time forego the gratification of a desire so peculiarly strong in a native of Switzerland. His health, however, allowing him only to remain three years longer, he returned to Basel, where in a short time he was appointed professor of anatomy and botany, and afterwards of experimental and speculative philosophy. In the useful and honourable labours of this office he spent the remaining years of his life, which are scarcely marked by any other events than the calm conquests of intellect, and the communication of valuable knowledge and good deeds to mankind.

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 which now bears the name of that celebrated geometer. In 1738 his Hydrodynamica appeared; and whatever may be its defects, it must be allowed to be at once an original and useful work. The equilibrium, the pressure, the re-action and varied velocity of fluids, are considered, not merely theoretically, but practically, with reference to the conveyance of water in pipes and canals, to machines for raising water, and also as a moving power in different mechanical operations. The concluding problem, illustrated by experiment, embraces an ingenious mode of propelling vessels by the reaction of water ejected from the stern.—Hydrodynamica, p. 50. Some of his experiments on this subject were performed in the presence of Maupertuis and Clairaut, whom the fame of the Bernoullis had attracted to Basel—a city famed for learning, and which had then, as D'Alembert remarks, become to the philosophers of France what Egypt had been to the philosophers of ancient Greece.

With a success equalled only by Euler, his fellow-citizen, his friend, and successor at Petersburg, Daniel Bernoulli gained or shared no less than ten prizes of the Academy of Sciences of Paris. The first, on the construction of a clepsydra for measuring time exactly at sea, he gained at the age of twenty-four; the second, on the physical cause of the inclination of the planetary orbits, he divided with his father; and the third, on the tides, he shared with Euler, Macaurin, and another competitor, whose only merit consisted in making a last effort to support the system of Descartes.

The problem of vibrating cords, which had been some time before resolved by Taylor and D'Alembert, became the subject of a long dispute between Daniel Bernoulli and his illustrious friend Euler. This discussion, however, was conducted with that generous spirit which soars above personalities, and with almost equal claims to praise; the one, in support of D'Alembert, displaying all the force of analysis; and the other, in defence of Taylor, using all the address of a mind fertile in resources.

In one of his early investigations on mechanics, he has given an ingenious though indirect demonstration of the problem of the parallelogram of forces; and in other papers, contained also in the Petersburg Memoirs, he has published researches of a higher character and of equal utility. 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; to inoculation as an object of national interest, and to the duration of married life in the different sexes, and the relative proportion of male and female births. But in such speculations the premises often do not embrace all the existing circumstances; so that we need not be much surprised to find the results sometimes at variance with nature. His great object however was, in the spirit of true philosophy, to employ theory only as a means to investigate the secrets of nature, and to apply mathematics to explain with accuracy and certainty the phenomena of the universe.

By a regular and calm tenor of life, though of a delicate constitution, he retained his usual vigour of understanding till near the age of eighty, when his nephew James relieved him of his public duties. Being afflicted with a troublesome asthma, his retirement was confined to the society of a few chosen friends. In the spring of 1782 all his complaints were increased, and, after some days illness, this excellent man and true philosopher died like his father, in the repose of sleep.3

He was never married, owing, it is said, to a misplaced affection in youth. His manners were unaffected, and his society agreeable. His house, his table, and his dress, had nothing inconsistent with simplicity. Frugal without parsimony, and benevolent without ostentation, the wealth which might have been expended in selfish pleasure or idle show was judiciously employed in forming an endowment for poor students at Basel. Excluded by his professional character from the councils of the republic, he however received all the deference and honour of a first magistrate. He indeed enjoyed an esteem and respect seldom accorded to living worth. One of the first lessons which a parent taught his child was to bow to Daniel Bernoulli. Grateful as he must have been by this flattering regard of his fellow-citizens, he was yet wont to mention two incidents in his life as having afforded him the greatest pleasure; the accidental meeting and indirect approbation of Newton, and the solving of Koenig's difficult problem, while he did the honours of the table for him.

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 interesting investigations are contained in the earlier volumes of the Petersburg Memoirs; and his separately published works are, 1. Dom. Bernoulli Dissertatio Inaugur. Phys. Med. de Respiratione. Basil. 1721, 4to. 2. Positiones Anatomico-Botanicae. Basil. 1721, 4to. 3. Exercitationes quaedam Mathematicae. Venetii, 1724, 4to. 4. Dom. Bernoulli Hydrodynamica. Argentorati, 1738, 4to.

V. Bernoulli, John, the youngest of the three distinguished sons of John Bernoulli, was born at Basel on the 18th May 1710. He studied law and mathematics, and, after travelling in France, was appointed professor of eloquence in the university of his native city. In this office he continued five years till the death of his father, whom he succeeded as professor of mathematics. He was thrice a successful competitor for the prizes of the academy of sciences of Paris. His prize subjects were, on the capstan, on the propagation of light, and on the magnet. The friendship of Maupertuis, who died under his roof while on his way to Berlin, was esteemed by him not the least of his honours. He himself died in 1790, at the age of eighty. His two sons, John and James, are the last noted mathematicians of the Bernoulli family.

VI. Bernoulli, Nicholas, 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 is frequently mentioned with approbation both by his uncle and Leibnitz in their epistolary correspondence. He visited England, where he was kindly received by Newton and Halley; previously to his appointment at Padua to the mathematical chair which Galileo had once filled. But neither the classical recollections, nor the superior wealth of Padua, were sufficient to prevent his return to the freedom and happy mediocrity of his native city. At Basel he was successively professor of logic and of law, where he died on the 29th of November 1759. He was editor of the Ars Conjectandi of his uncle James. His own works are not published separately, but are contained in the Acta Eruditorum, the Giornate de Letterati d'Italia, and the Commercium Philosophicum.

VII. Bernoulli, John, grandson of the celebrated 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 Neufchatel, and when thirteen years of age he took the degree of doctor in philosophy. His Thesis, De Variolarum Initio, is still preserved in Halle's Letters. It gives a short history of inoculation, as well as the method of operation and manner of treatment which had recently and successfully been practised on himself.

The kindred studies of mathematics and astronomy were his early pursuits. When only nineteen years of age he was appointed astronomer royal of Berlin. Some years after, obtaining leave to travel, he visited Germany, France, and Bernoulli, England; and subsequently Italy, Russia, and Poland. On his return to Berlin he was appointed director of the mathematical department of the academy of that city. Here he died on the 10th July 1807, being, at the time of his death, a member of several of the learned academies of Europe.

He is a very voluminous and varied writer. His writings consist of several volumes of travels, besides astronomical, geographical, and mathematical works. In 1774 he published a French translation of Euler's Elements of Algebra at Lyons, in 2 vols. 8vo. He was a conductor of some of the periodical works of Berlin, and in the academical memoirs of that city are several of his papers on astronomical subjects. Those for the years 1803 and 1804 contain his observations and experiments on the cultivation of zea mays or Indian corn in Switzerland and Germany, and on its advantageous application to various economical purposes.

VIII. Bernoulli, James, younger brother of the preceding, and the second of this name, was born at Basel on the 17th October 1759. He displayed early talents, which were cultivated with the greatest care. Having finished his literary studies, he was, according to custom, sent to Neufchatel to acquire the French language. On his return to his native city he studied law and took a degree.

The study of law, however, had neither alienated nor checked his hereditary taste for geometrical pursuits. The early mathematical lessons which he had received from his father were continued by his uncle Daniel, so as to give new vigour to his native predilection for the exact sciences. Such, indeed, was his progress in these studies, that at the age of twenty-one the directors of the university of Basel committed to him the duties of the chair of experimental physics, which his uncle's advanced years rendered him unable to discharge. But although he filled this important office with satisfaction, both to the directors and to his auditors, he did not succeed on his relative's death. The lot was capricious, and the voice of public opinion (by which almost all the important appointments at Basel were then decided) unfavourable. He had been formerly, too, an unsuccessful candidate for the chair of eloquence; so that, with perhaps a conscious feeling of unmerited neglect, he published on the present occasion a thesis De Sublimi.

This repeated disappointment, combined with the desire of travelling, so natural and so powerful in youth, induced him to accept the situation of secretary to Count de Brenner, which afforded him an opportunity of seeing a great part of Germany and Italy. In Italy he formed a friendship with Lorenzini, professor of mathematics at Verona, and one of the founders of the Italian society 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 Fuss, admitted into the academy of Petersburg, with the promise of promotion in the course of a year. In 1788 he was accordingly made one of the mathematical professors of that flourishing seminary, and in the following year he married a daughter of Albert Euler, son of the illustrious Euler. This marriage, which united names equally dear to science, and, from congeniality of sentiment, promised much happiness, was almost tragically dissolved in the space of two short months. Residence in the country, and the opportunity of bathing in the Neva, had given him a taste for that healthful but sometimes fatal exercise. Being an excellent swimmer, he was fearlessly enjoying this recreation, when at once he sunk without any appearance of danger or expression of pain. He was instantly taken out by his brother-in-law, and every means of resuscitation applied, but without success. The opinion of the physicians was that his death had been occasioned by a shock of apoplexy. His constitution indeed was naturally delicate, and two years before he had suffered much from a nervous fever. His premature death at the age of twenty-nine was a source of deep regret, not only to his relatives, but to many whom he had attached to him by a character open, obliging, gentle, and modest.

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.