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 The recurring practice of issuing challenge problems was inaugurated at this time by Leibniz. They were, at first, not intended as defiances, but merely as exercises in the new calculus. Such was the problem of the isochronous curve (to find the curve along which a body falls with uniform velocity), proposed by him to the Cartesians in 1687, and solved by James Bernoulli, himself, and John Bernoulli. James Bernoulli proposed in the Leipzig Journal the question to find the curve (the catenary) formed by a chain of uniform weight suspended freely from its ends. It was resolved by Huygens, Leibniz, and himself. In 1697 John Bernoulli challenged the best mathematicians in Europe to solve the difficult problem, to find the curve (the cycloid) along which a body falls from one point to another in the shortest possible time. Leibniz solved it the day he received it. Newton, de l'Hospital, and the two Bernoullis gave solutions. Newton's appeared anonymously in the Philosophical Transactions, but John Bernoulli recognised in it his powerful mind, "tanquam," he says, "ex ungue leonem." The problem of orthogonal trajectories (a system of curves described by a known law being given, to describe a curve which shall cut them all at right angles) had been long proposed in the Acta Eruditorum, but failed at first to receive much attention. It was again proposed in 1716 by Leibniz, to feel the pulse of the English mathematicians.

This may be considered as the first defiance problem professedly aimed at the English. Newton solved it the same evening on which it was delivered to him, although he was much fatigued by the day's work at the mint. His solution, as published, was a general plan of an investigation rather than an actual solution, and was, on that account, criticised by Bernoulli as being of no value. Brook Taylor undertook the defence of it, but ended by using very reprehensible language.