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 of John Bernoulli, has already been mentioned as taking part in the challenges issued by Leibniz and the Bernoullis. He helped powerfully in making the calculus of Leibniz better known to the mass of mathematicians by the publication of a treatise thereon in 1696. This contains for the first time the method of finding the limiting value of a fraction whose two terms tend toward zero at the same time.

Another zealous French advocate of the calculus was Pierre Varignon (1654–1722). Joseph Saurin (1659–1737) solved the delicate problem of how to determine the tangents at the multiple points of algebraic curves. François Nicole (1683–1758) in 1717 issued the first systematic treatise on finite differences, in which he finds the sums of a considerable number of interesting series. He wrote also on roulettes, particularly spherical epicycloids, and their rectification. Also interested in finite differences was Pierre Raymond de Montmort (1678–1719). His chief writings, on the theory of probability, served to stimulate his more distinguished successor, De Moivre. Jean Paul de Gua (1713–1785) gave the demonstration of Descartes' rule of signs, now given in books. This skilful geometer wrote in 1740 a work on analytical geometry, the object of which was to show that most investigations on curves could be carried on with the analysis of Descartes quite as easily as with the calculus. He shows how to find the tangents, asymptotes, and various singular points of curves of all degrees, and proved by perspective that several of these points can be at infinity. A mathematician who clung to the methods of the ancients was Philippe de Lahire (1640–1718), a pupil of Desargues. His work on conic sections is purely synthetic, but differs from ancient treatises in deducing the properties of conies from those of the circle in the same manner as did Desargues and Pascal. His innovations stand in close relation with modern synthetic geometry. He wrote on roulettes, on