Bernoulli, Johann

▪ Swiss mathematician

born Aug. 6 [July 27, old style], 1667, Basel, Switz.

died Jan. 1, 1748, Basel

 major member of the Bernoulli family of Swiss mathematicians. He investigated the then new mathematical calculus, which he applied to the measurement of curves, to differential equations, and to mechanical problems.

      The son of a pharmacist, Johann studied medicine and obtained his doctor's degree in Basel in 1694, with a thesis on muscular contraction. However, he turned to mathematics despite his father's opposition. In 1691–92 he wrote two texts, not published until later, on differential and integral calculus. In 1692 he taught calculus to the mathematician Guillaume-François-Antoine de L'Hospital, who agreed to pay him for mathematical discoveries. From 1695 to 1705 he taught mathematics at Groningen, Neth., and, on the death of his elder brother, Jakob, assumed a professorship at Basel.

      Johann exceeded his brother in the number of contributions he made to mathematics. He applied calculus to the determination of lengths and areas of curves, such as the isochrone, along which a body will fall at constant speed, and the tautochrone, which was found to be important in clock construction. He also made contributions to the theory of differential equations, the mathematics of ship sails, and optics. Johann sent to L'Hospital in Paris a method or rule for solving problems involving limits that would apparently be expressed by the ratio of zero to zero, now called L'Hospital's rule on indeterminate forms because it was included in L'Hospital's influential textbook of 1696, Analyse des infiniment petits (“Analysis of the Infinitely Small”).

      The Bernoulli brothers often worked on the same problems, but not without friction. Their most bitter dispute concerned finding the equation for the path followed by a particle from one point to another in the shortest time, if the particle is acted upon by gravity alone, a problem originally discussed by Galileo. In 1697 Jakob offered a reward for its solution. Accepting the challenge, Johann proposed the cycloid, the path of a point on a moving wheel, pointing out at the same time the relation this curve bears to the path described by a ray of light passing through strata of variable density. A protracted, bitter dispute then arose when Jakob challenged the solution and proposed his own. The dispute marked the origin of a new discipline, the calculus of variations.

      Ardent in his friendships and keen in his resentments, Johann zealously defended the cause of G.W. Leibniz in the dispute with Isaac Newton over who had originated calculus. His text in integral calculus appeared in 1742 and his differential calculus shortly afterward. During his last years he worked mainly on the principles of mechanics. His works were published in Opera Johannis Bernoullii, 4 vol. (1742).

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