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BER and scholars of the period, he passed some years in travelling through various parts of Europe, having in the course of the six years from 1676 to 1682 visited in succession Geneva, France, England, Germany, and Holland, and become acquainted with many of the learned and scientific men of those regions; amongst others with Stillingfleet, Boyle, Hooke, Voss, and Baxter. At Geneva he successfully put in practice, for the instruction of a young lady (Elizabeth von Waldkirch), a method which he had invented of teaching the blind by the sense of touch. In France he acted for about a year as chaplain and tutor in the family of the marquis de l'Estrange. On his return to Basle in 1682, he opened a college for instruction in experimental physics and mechanics. In 1684 he married Judith Stupan, a lady belonging to an eminent medical family of Bâle, by whom he left a son and a daughter. The son became an artist in painting; but little is known of his career. The daughter married Nicholas Rydimer, a merchant. In 1687 James Bernoulli was appointed professor of mathematics in the university of Bâle, in which office he continued till his death. After his marriage he devoted himself more assiduously than ever to the study of mathematics, and especially of the differential and integral calculus as set forth in the then recent publications of Leibnitz, its inventor, contemporaneously with Newton. That instrument of mathematical research Bernoulli enlarged and improved in many respects. He was the first to solve what is called a differential equation. He applied the calculus to the solution of many important problems, amongst which may be mentioned the following:—1, the curve of isochronous oscillation, that is, the cycloid, in which a pendulum swings in equal times, how great or how small soever may be the arc; 2, the catenary, or curve in which a chain of uniform cross section hangs, being also the curve of equilibrium of an arc of uniform cross section loaded with its weight only, and the key to the knowledge of a class of curves of equilibrium; 3, the elastic curve, formed by a bent spring or bow of uniform section, and identical with the curve of a sheet containing water; 4, many important properties of the logarithmic spiral, being that whose inclination to its radius-vector is uniform, and the loxodromic spiral, being the curve formed on the surface of a sphere by a line whose inclination to each meridian which it crosses is uniform; 5, the great , in which it is proposed amongst all the curves of a given length which can be drawn between two fixed points, to find that which shall have a given function of its figure a maximum or a minimum. This last problem may be regarded as being next to the discovery of the differential and integral calculus, the most important contribution to pure mathematics made at the period when Bernoulli flourished, for it led in after times to the invention of the calculus of variations. A posthumously published work of Bernoulli, entitled, "De Arte Conjectandi" (On the Art of Conjecture), is believed to be the first treatise in which the science of probabilities was applied to beneficial purposes. In 1699 he received along with his brother John the honour of being elected Foreign Associate of the Institute of France, an order then newly established, which is limited to eight members, and which, from the fewness and the eminence of those who have filled it, continues to be regarded as the highest mark of distinction attainable by a man of science. In 1700 he held the office of Lord Rector of the university of Bâle. James Bernoulli died on the 17th of August, 1705, at the early age of fifty-one. He is said to have been an accomplished orator and a pleasing poet, as well as a profound mathematician, and of an amiable and just character, as indeed his writings evince, especially those relative to a dispute with his brother John. His works are thus entitled—"Jacobi Bernoullii Basiliensis Opera," Geneva, 1744, 2 vols., 4to; "De Arte Conjectandi," Bâle, 1713, to which is appended an essay "De Seriebus Infinitis;" Detached Papers in the Memoirs of the Academy of Sciences from 1702 to 1705, the Journal des Savans and the Acta Eruditorum. His life, by Doctor John James Battier, professor of rhetoric in the university of Bâle, is prefixed to his works, and is the authority chiefly relied on in the present article.

, brother of the preceding, was born at Bâle on the 7th of August, 1667. Having studied humane letters in his native city, he went to Neufchâtel to learn the French language, and the business of a merchant; but preferring science to commerce, he returned to Bâle, where he entered the university, studied mathematics under his brother, and took the degree of bachelor of arts in 1684, and that of master of arts in 1685. Chemistry, physiology, and medicine, as well as mathematics and physics, engaged his attention. In 1690, or soon afterwards, he set out to travel, and visited Geneva and Paris, becoming personally acquainted with Malebranche, Cassini, De la Hire, Varignon, the marquis de l'Hôpital, and other distinguished men of the time. His labours for the advancement of mathematics, especially the differential and integral calculus, and of analytical mechanics, were incessant, and most important in their results. Amongst those results may be mentioned the discovery of the exponential calculus, of the method of integrating rational fractions, of the universality of the principle of virtual velocities, and of the property which a cycloid possesses of being the line of quickest descent between two points. He engaged with great ardour in the practice then common amongst mathematicians, of proposing to each other problems for solution, and was sometimes thereby involved in controversies, which he conducted with vigour. Amongst others he assailed his elder brother with a succession of problems, which James solved with much industry and patience; at length James turned the tables upon John, by proposing to him a problem that baffled his skill,—that of isoperimetric figures, already mentioned. John, having offered a solution which James showed to be erroneous, persisted in maintaining its accuracy, and conceived a lasting resentment against his brother. Having completed his medical studies at Bâle, he obtained in 1794 the degree of doctor of medicine, and read an inaugural thesis "on Muscular Motion," in which sound views of the mechanical action of the muscles are mingled with doubtful physiological hypotheses. He married a young lady of Bâle, and was soon afterwards, in 1695, appointed professor of mathematics at Groningen. In 1694 commenced that celebrated correspondence between Leibnitz and John Bernoulli, which continued until 1716, and was afterwards collected and published by Mark Michael Bousquet & Company, in two quarto volumes, under the title of "Gotofridi Gulielmi Leibnitii et Johannis Bernoullii Commercium Philosophicum et Mathematicum." In this collection of letters, unparalleled of its kind, there are discussed with consummate ability nearly all the mathematical and philosophical questions which arose during that period; in which the knowledge of the first principles of mechanics and mathematics made more rapid progress than it has ever done before or since. The letters of Bernoulli and Leibnitz have reference chiefly to the differential calculus, with its geometrical and mechanical applications; and occasionally to controversies such as that respecting the mode of stating the force (so called) of bodies in motion, and that which arose between the injudicious admirers of Newton and Leibnitz respecting the priority of invention of the method of fluxions or differential calculus, and the comparative merits of the two forms in which those two philosophers respectively set forth that branch of mathematics. In later times, mathematicians have recognized Newton and Leibnitz as independent discoverers, and have assigned to the special methods of each their peculiar merits, adopting Newton's demonstration of the fundamental principle of the calculus as the more philosophical, and Leibnitz's notation and forms of expression as the more convenient of application, and fertile in results. Bernoulli's letters betray in many cases a perversity of temper which constituted a serious blemish in his character, and led him to entertain an unworthy jealousy of the eminence, not only of his brother and instructor James, but even of his son Daniel. Such feelings, how high soever the intellectual powers with which they are combined, are the certain mark of a mind of the second order, and they seldom fail more or less to obscure the understanding itself, and to a certain extent to disqualify it for the discovery of truth. In the case of John Bernoulli there can be little doubt, that an obstinate and jealous temper led him to reject, to the end of his life, Newton's discovery of the law of gravitation, and to maintain the Cartesian hypothesis of celestial vortices. He continued from time to time to pursue his physiological studies. In an essay on nutrition, published at Groningen in 1669, he pointed out that the continual waste and repair of the particles of the human body must lead to an entire renewal of its substance in a period of a few years. This opinion was assailed as heretical, on the ground of its alleged inconsistency with the doctrine of the resurrection. Bernoulli refuted that objection in a paper which he afterwards refused to publish. In 1699 he was lord-rector of the university