JAMES, a celebrated mathematician, born at Basle the 27th of December 1654. Having taken his degrees in the university of Basle, he applied himself to divinity, not so much from inclination as complaisance to his father. He gave very early proofs of his genius for mathematics, and soon became a geometrician, without any assistance from masters, and at first almost without books: for he was not allowed to have any books of this kind; and if one fell by chance into his hands, he was obliged to conceal it, that he might not incur the reprimands of his father, who desighed him for other studies. This severity made him Bernoulli. him choose for his device, Phaeton driving the chariot of the sun, with these words, Invito patre sidera verso, "I traverse the stars against my father's inclination." This had a particular reference to astronomy, the part of mathematics to which he at first applied himself. But the precautions of his father did not avail, for he pursued his favourite study with great application. In 1676 he began his travels. When he was at Geneva, he fell upon a method to teach a young girl to write, though she had lost her sight when she was but two months old. At Bourdeaux he composed universal gnomonic tables, but they were never published. He returned from France to his own country in 1680. About this time there appeared a comet, the return of which he foretold; and wrote a small treatise upon it, which he afterwards translated into Latin. He went soon after to Holland, where he applied himself to the study of the new philosophy. After having visited Flanders and Brabant, he went to Calais, and passed over from thence to England. At London he contracted an acquaintance with all the most eminent men in the several sciences; and had the honour of being frequently present at the philosophical societies held at the house of the famous Mr Boyle. He returned to his native country in 1682; and he exhibited at Basle a course of experiments in natural philosophy and mechanics, which consisted of a variety of new discoveries. In 1682, he published his essay of a new system of comets; and the year following, his dissertation on the weight of air. Mr Leibnitz, about this time, having published in the Acta Eruditorum at Leipzig some essay of his new Calculus differentialis, or infinimorum petit, but concealed the art and method of it; Mr Bernoulli, and one of his brothers, discovered, by the little which they saw, the beauty and extent of it: they endeavoured to unravel the secret; which they did with such success, that Mr Leibnitz declared, that the invention belonged to them as much as to himself. In 1687, the professorship of mathematics at Basle being vacant, Mr Bernoulli was appointed his successor. He discharged this trust with universal applause; and his reputation drew a great number of foreigners from all parts to hear his lectures. He had an admirable talent in teaching, and adapting himself to the different genius and capacity of his scholars. In 1699, he was admitted into the academy of sciences at Paris as a foreign member, and in 1701 the same honour was conferred upon him by the academy of Berlin. He wrote several pieces in the Acta Eruditorum of Leipzig, the Journal des Savans, and the Histoire de l'Academie des Sciences. His assiduous application to these studies brought upon him the gout, and by degrees, a flow fever, of which he died the 16th of August 1705, in the 58th year of his age.—Archimedes having found out the proportion of a sphere to a cylinder circumscribed about it, ordered it to be engraven upon his monument. In imitation of him Mr Bernoulli appointed, that a spiral logarithmical curve should be inscribed upon his tomb, with these words, Eadem mutata refurgo; in allusion to the hopes of the resurrection, which are represented in some measure by the properties of the curve which he had the honour of discovering.

Daniel, a celebrated physician and philosopher, was born at Groningen, February 9, 1700. He was intended by his parents for trade, but his genius led him to different pursuits. He passed some time in Italy, and at 24 refused to be president of an academy meant to have been established at Genoa. He spent several years at St Petersburg with great credit; and in 1733 returned to Basle, where he successively filled the chair of physic, natural and speculative philosophy. In his first work, Exercitationes Mathematicae, he took the only title he then had, viz. "Son of John Bernoulli," and never would suffer any other to be added to it. This work appeared in Italy with the great inquisitor's privilege added to it, and it clasped Bernoulli in the rank of inventors. He gained or divided nine prizes, which were contended for by the most illustrious mathematicians in Europe, from the academy of sciences. The only man who has had success of the same kind is Euler, his countryman, disciple, rival, and friend. His first prize he gained at 24 years of age. In 1734 he divided one with his father: but this hurt the family union: for the father confounded the contest itself into a want of respect; and the son did not sufficiently conceal that he thought (what was really the case) his own piece better than his father's. Besides this, he declared for Newton, against whom his father had contended all his life. In 1740, Mr Bernoulli divided the prize "On the Tides of the Sea" with Euler and Maclaurin. The academy at the same time crowned a fourth piece, whose only merit was that of being Cartesian; but this was the last public act of adoration paid by it to the authority of the author of the Vortices, which it had obeyed perhaps too long. In 1748, Mr Daniel Bernoulli succeeded his father in the academy of sciences, and was himself succeeded by his brother John; this place, since its first erection, i.e. 84 years, never having been without a Bernoulli to fill it. He was extremely respected at Basle; and to bow to Daniel Bernoulli, when they met him in the streets, was one of the first lessons which every father gave his child. He used to tell two little adventures, which he said had given him more pleasure than all the other honours he had received. He was travelling with a learned stranger, who, being pleased with his conversation, asked his name: "I am Daniel Bernoulli," answered he with great modesty; "And I," said the stranger (who thought he meant to laugh at him), "I am Isaac Newton." Another time he was giving a dinner to the famous Koenig the mathematician, who boasted, with a sufficient degree of self-complacency, of a difficult problem he had resolved with much trouble. Bernoulli went on doing the honours of his table; and, when they went to drink coffee, presented him with a solution of the problem more elegant than his own. He died in March 1782.