at one end and bent by a weight applied to the other end; of the "lintearia," a flexible rectangular plate with two sides
fixed horizontally at the same height, filled with a liquid; of the "volaria," a rectangular sail filled with wind. He studied
the loxodromic and logarithmic spirals, in the last of which
he took particular delight from its remarkable property of reproducing itself under a variety of conditions. Following the example of Archimedes, he willed that the curve be engraved upon his tomb-stone with the inscription "eadem mutata resurgo." In 1696 he proposed the famous problem of isoperimetrical figures, and in 1701 published his own solution. He wrote a work on Ars Conjectandi, which is a development of the calculus of probabilities and contains the investigation now called "Bernoulli's theorem" and the so-called "numbers
of Bernoulli," which are in fact (though not so considered by
him) the coefficients of in the expansion of . Of his collected works, in three volumes, one was printed in 1713, the other two in 1744.
John Bernoulli (1667–1748) was initiated into mathematics by his brother. He afterwards visited France, where he met Malebranche, Cassini, De Lahire, Varignon, and de l'Hospital. For ten years he occupied the mathematical chair at Gröningen and then succeeded his brother at Basel. He was one of the most enthusiastic teachers and most successful original investigators of his time. He was a member of almost every learned society in Europe. His controversies were almost as numerous as his discoveries. He was ardent in his friendships, but unfair, mean, and violent toward all who incurred his dislike–-even his own brother and son. He had a bitter dispute with James on the isoperimetrical problem. James convicted him of several paralogisms. After his brother's death he attempted to substitute a disguised solution of the former for an incorrect one of his own. John admired the merits of Leibniz and Euler, but was blind to those of Newton. He
immensely enriched the integral calculus by his labours. Among his discoveries are the exponential calculus, the line of swiftest descent, and its beautiful relation to the path
described by a ray passing through strata of variable density. He treated trigonometry by the analytical method, studied
caustic curves and trajectories. Several times he was given
prizes by the Academy of Science in Paris.
Of his sons, Nicholas and Daniel were appointed professors of mathematics at the same time in the Academy of St. Petersburg. The former soon died in the prime of life; the latter returned to Basel in 1733, where he assumed the chair of experimental philosophy. His first mathematical publication
was the solution of a differential equation proposed by Riccati. He wrote a work on hydrodynamics. His investigations