bodily senses altogether and to live entirely absorbed in |
religious meditation. Sleep he counted a loss, and com-
pared it to death. Food was only taken to keep him from
fainting. The most menial offices were his delight, and
even then his humility looked around for some lowlier
employment. Fortunately he loved nature, and found a
constant solace in her rocks and woods. "Trust one who
has tried it," he writes in one of his epistles, "you will
find more in woods than in books; trees and stones will
teach you what you can never learn from masters."
("Experto crede: aliquid amplius invenies in silvis quam
in libris; ligna et lapides docebunt te quod a magistris
audire non possis," Epist. 106.)

So ardent a nature soon found a sphere of ambition for itself. The monks of Citeaux, from being a poor and unknown company, began to attract attention after the accession of St Bernard and his friends. The fame of their self-denial was noised abroad, and out of their lowliness and abnegation came as usual distinction and success. The small monastery was unable to contain the inmates that gathered within it, and it began to send forth colonies in various directions. St Bernard had been two years an inmate, and the penetrating eye of the abbot had discovered beneath all his spiritual devotion a genius of rare power, and especially fitted to aid his measures of monastic reform. He was chosen accordingly to head a band of devotees who issued from Citeaux in 1115 in search of a new home. This band, with Bernard at their head, journeyed northwards till they reached a spot in the diocese of Langres a thick-wooded valley, wild and gloomy, but with a clear stream running through it. Here they settled and laid the foundations of the famous abbey of Clairvaux, with which St Bernard's name remains associated in history The hardships which the monks endured for a time in their new abode were such as to drive them almost to despair, and their leader fell seriously ill, and was only rescued from what seemed impending death by the kind compulsion of his friend William of Champeaux, the great doctor of the age, who besought and received the direction of Bernard for a year from his superior at Citeaux. Thanks to his considerate friend the abbot of Clairvaux was forced to abandon the cares of his new establishment, and in retirement and a healthful regimen to seek renewed health. The effect was all that could be desired, and in a few years Bernard had not only recovered his strength, but had begun that marvellous career of literary and ecclesiastical activity, of incessant correspondence and preaching, which was to make him in some respects the most influential man of

his age.

Gradually the influence of Bernard's character began to extend beyond his monastery. His friendship with William of Champeaux and others gave currency to his opinions, and from his simple retreat came by voice or pen an authority before which many bowed, not only within his own order but within the church at large. This influence was notably shown after the death of Pope Honorius II. in 1130. Two rival popes assumed the purple, each being able to appeal to his election by a section of the cardinals. Christendom was divided betwixt the claims of Anacletus II. and Innocent II. The former was backed by a strong Italian party, and drove his adversary from Rome and even from Italy. Innocent took refuge in France. The king, Louis the Fat, espoused his cause, and having summoned a council of archbishops and bishops, he laid his commands on the holy abbot of Clairvaux to be present also and give the benefit of his advice. With reluctance Bernard obeyed the call, and from the depths of seclusion was at once plunged into the heart of the great contest which was afflicting the Christian world. The king and prelates put the question before him in such a way as to invite his decision

and make him arbiter. After careful deliberation he gave his judgment in favour of Innocent, and not only so, but from that time forward threw himself with caaracteristic fervour and force into the cause for which he had declared. Not only France, but England, Spain, and Germany were won to the side of Innocent, who, banished from Rome, in the words of St Bernard, was "accepted by the world." He travelled from place to place with the powerful abbot by his side, who also received him in his humble cell at Clairvaux. Apparently, however, the meanness of the accommodation and the scantiness of the fare (one small fowl was all that could be got for the Pope's repast), left no wish on the part of Innocent or his retinue to continue their stay at Clairvaux. He found a more dainty reception elsewhere, but nowhere so powerful a friend. Through the persuasions of Bernard the emperor took up arms for Innocent; and Anacletus was driven to shut himself up in the impregnable castle of St Angelo, where his death opened the prospect of a united Christendom. A second anti-pope was elected, but after a few months retired from the field, owing also, it is said, to St Bernard's influence. A great triumph was gained not without a struggle, and the abbot of Clairvaux remained master of the ecclesiastical situation. No name stood higher in the Christian world.

The chief events which fill up his subsequent life attest the greatness of his influence. These were his contest with the famous Abelard, and his preaching of the second crusade.

Peter Abelard was twelve years older than Bernard, and had risen to eminence before Bernard had entered the gates of Citeaux. His first intellectual encounter had been with Bernard's aged friend William of Champeaux, whom he had driven from his scholastic throne at Paris by the superiority of his dialectics. His subsequent career, his ill-fated passion for Heloise, his misfortunes, his intellectual restlessness and audacity, his supposed heresies, had all shed additional renown on his name; and when a council was summoned at Sens in 1140, at which the French king and his nobles and all the prelates of the realm were to be present, Abelard dared his enemies to impugn his opinions. St Bernard had been amongst those most alarmed by Abelard's teaching, and had sought to stir up alike Pope, princes, and bishops to take measures against him. He did not readily, however, take up the gauntlet thrown down by the great hero of the schools. He professed himself a "stripling too unversed in logic to meet the giant practised in every kind of debate." But "all were come prepared for a spectacle," and he was forced into the field. To the amazement of all, when the combatants met and all seemed ready for the intellectual fray, Abelard refused to proceed with his defence. After several passages considered to be heretical had been read from his books he made no reply, but at once appealed to Rome and left the assembly. Probably he saw enough in the character of the meeting to assure him that it formed a very different audience from those which he had been accustomed to sway by his subtilty and eloquence, and had recourse to this expedient to gain time and foil his adversaries. Bernard followed up his assault by a letter of indictment to the Pope against the heretic. The Pope responded by a sentence of condemnation, and Abelard was silenced. Soon after he found refuge at Cluny with the kindly abbot, Peter the Venerable, who brought about something of a reconciliation betwixt him and Bernard. The latter, however, never heartily forgave the heretic. He was too zealous a churchman not to seo the danger there is in such a spirit as Abelard's, and the serious consequences to which it might lead.

In all things Bernard was enthusiastically devoted to the church, and it was this enthusiasm which led him at last into the chief error of his career. Bad news reached

and the trade is chiefly in horses, grain, and flax. The | variety of form, and fluent diction, his verses are unsurtown, which was formerly fortified, was besieged by Duguesclin in 1378; it was taken by the English in 1418 and again in 1421, and by Admiral de Coligny in 1563. The fortress was razed in 1589. Population in 1872, 5806.

mutata resurgo.

geometrical figures, which fell in his way, excited in him a of a philosopher. Like another Archimedes, he requested passion for mathematical pursuits, and in spite of the that, as a monument of his labours and an emblem of his opposition of his father, who wished him to be a clergyman, hope of a resurrection, the logarithmic spiral should be he applied himself in secret to his favourite science. In engraven on his tombstone, with these words, Eadem 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 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, Conamen Novi Systematis Cometarum, which was occasioned by the appearance of the comet of 1680. This essay, and his next publication, entitled De Gravitate Etheris, 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 extremities, which he also showed to be the same as the curvature 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 loxodromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of reproducing itself under a great variety of conditions.

James Bernoulli wrote elegant verses in Latin, German, and French; but although these were held in high estimaAtion 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.) Ars Conjectandi, opus posthumum: accedunt tractatus de Seriebus Infinitis, et epistola (Gallice scripta) de Ludo Pila Reticularis, Basilia, 1713, 1 tom. 4to.

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 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 solution, 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 professors 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

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 ho 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 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 enjoyed 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 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 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. 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 an electrical appearance of the barometer first observed by Picard, and discussed by John Bernoulli under the name of mercurial phosphorus, of 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 received from the king of Prussia a gold medal for his supposed 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 mathematicians, 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 mathematical 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.

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 He was several times a successful competitor for the prizes early directed also to geometrical studies. The severity of given by the Academy of Sciences of Paris; and the subjects his father's manner was ill calculated to encourage the of his essays were, the laws of motion (Discours sur les first efforts of one so sensitive; but fortunately, at the ago Lois de la Communication du Mouvement, 1727), the ellip- of eleven, he became the pupil of his brother Nicholas tical orbits of the planets, and the inclinations of the plane- He afterwards studied in Italy under Michelotti and Mortary orbits (Essai d'une Nouvelle Physique Céleste, 1735). gagni. After his return, though only twenty-four years of In the last case his son Daniel divided the prize with him. age, he was invited to become president of an academy then Some years after his return to Basel he published an essay, projected at Genoa; but, declining this honour, he was, in entitled Nouvelle Théorie de la Manoeuvre des Vaisseaux. the following year, appointed professor of mathematics at It is, however, his works in pure mathematics that are the St Petersburg. In consequence of the state of his health, permanent monuments of his fame. D'Alembert acknow-however, he returned to Basel in 1733; where he was ledges with gratitude, that "whatever he knew of mathe- appointed professor of anatomy and botany, and aftermatics he owed to the works of John Bernoulli." He was wards of experimental and speculative philosophy. In tho a member of almost every learned society in Europe, and labours of this office he spent the remaining years of his one of the first mathematicians of a mathematical age. He life. He had previously published some medical and botaniwas as keen in his resentments as he was ardent in his cal dissertations, besides his Exercitationes quædam Mathefriendships; fondly attached to his family, he yet disliked a matica, containing a solution of the differential equation deserving son; he gave full praise to Leibnitz and Euler, proposed by Riccati and now known by his name. In yet was blind to the excellence of Newton. Such was 1738 appeared his Hydrodynamica, in which the equithe vigour of his constitution that he continued to pursue librium, the pressure, the reaction, and varied velocities of his usual mathematical studies till the age of eighty. He fluids are considered both theoretically and practically. One was then attacked by a complaint at first apparently trifl- of these problems, illustrated by experiment, deals with ing; but his strength daily and rapidly declined till the 1st an ingenious mode of propelling vessels by the reaction of of January 1748, when he died peacefully in his sleep. 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

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 tom. 4to. His interesting correspondence with Leibnitz appeared under the title of Gul. Leibnitii et Johannis Bernoulli Commercium Philosophicum et Mathemati-struction of a clepsydra for measuring time exactly at sea, cum, Lausan. et Genev. 1745, 2 tom. 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 mèo 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 Varignon and with Riccati, 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 mathematics 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.

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, Maclaurin, 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, mado to his declaration "I am Daniel Bernoulli" the incredulous and mocking reply, "And I am Isaac Newton ;" and that, while entertaining König 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 investi gations are contained in the earlier volumes of the St