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 was the solution of a differential equation proposed by Riccati. He wrote a work on hydrodynamics. His investigations on probability are remarkable for their boldness and originality. He proposed the theory of moral expectation, which he thought would give results more in accordance with our ordinary notions than the theory of mathematical probability. His "moral expectation" has become classic, but no one ever makes use of it. He applies the theory of probability to insurance; to determine the mortality caused by small-pox at various stages of life; to determine the number of survivors at a given age from a given number of births; to determine how much inoculation lengthens the average duration of life. He showed how the differential calculus could be used in the theory of probability. He and Euler enjoyed the honour of having gained or shared no less than ten prizes from the Academy of Sciences in Paris.

Johann Bernoulli (born 1710) succeeded his father in the professorship of mathematics at Basel. He captured three prizes (on the capstan, the propagation of light, and the magnet) from the Academy of Sciences at Paris. Nicolaus Bernoulli (born 1687) held for a time the mathematical chair at Padua which Galileo had once filled. Johann Bernoulli (born 1744) at the age of nineteen was appointed astronomer royal at Berlin, and afterwards director of the mathematical department of the Academy. His brother Jacob took upon himself the duties of the chair of experimental physics at Basel, previously performed by his uncle Jacob, and later was appointed mathematical professor in the Academy at St Petersburg.

Brief mention will now be made of some other mathematicians belonging to the period of Newton, Leibniz, and the elder Bernoullis.

Guillaume François Antoine l'Hospital (1661–1704), a pupil