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, "anquam," 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.
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 in geometry could be expressed as the ratio of finite quantities. In the interpretation of and Leibniz vacillated. At one time they appear in his
writings as finite lines; then they are called infinitely small
quantities, and again, quantitates inassignabiles, which spring from quantitates assignabiles by the law of continuity. In this last presentation Leibniz approached nearest to Newton.