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 Bernoulli was not to be outdone in incivility, and made a bitter reply. Not long afterwards Taylor sent an open defiance to Continental mathematicians of a problem on the integration of a fluxion of complicated form which was known to very few geometers in England and supposed to be beyond the power of their adversaries. The selection was injudicious, for Bernoulli had long before explained the method of this and similar integrations. It served only to display the skill and augment the triumph of the followers of Leibniz. The last and most unskilful challenge was by John Keill. The problem was to find the path of a projectile in a medium which resists proportionally to the square of the velocity. Without first making sure that he himself could solve it, Keill boldly challenged Bernoulli to produce a solution. The latter resolved the question in very short time, not only for a resistance proportional to the square, but to any power of the velocity. Suspecting the weakness of the adversary, he repeatedly offered to send his solution to a confidential person in London, provided Keill would do the same. Keill never made a reply, and Bernoulli abused him and cruelly exulted over him.[26]

The explanations of the fundamental principles of the calculus, as given by Newton and Leibniz, lacked clearness and rigour. For that reason it met with opposition from several quarters. In 1694 Bernard Nieuwentyt of Holland denied the existence of differentials of higher orders and objected to the practice of neglecting infinitely small quantities. These objections Leibniz was not able to meet satisfactorily. In his reply he said the value of $$\scriptstyle{\frac{dy}{dx}}$$ in geometry could be expressed as the ratio of finite quantities. In the interpretation of dx and dy Leibniz vacillated. At one time they appear in his writings as finite lines; then they are called infinitely small