In England the principles of fluxions were boldly attacked by Bishop Berkeley, the eminent metaphysician, who argued
with great acuteness, contending, among other things, that the fundamental idea of supposing a finite ratio to exist between terms absolutely evanescent–-"the ghosts of departed quantities," as he called them–-was absurd and unintelligible. The reply made by Jurin failed to remove all the
objections. Berkeley was the first to point out what was again shown later by Lazare Carnot, that correct answers were
reached by a "compensation of errors." Berkeley's attack was not devoid of good results, for it was the immediate cause of the work on fluxions by Maclaurin. In France Michel
Rolle rejected the differential calculus and had a controversy
with Varignon on the subject.
Among the most vigorous promoters of the calculus on the Continent were the Bernoullis. They and Euler made Basel in Switzerland famous as the cradle of great mathematicians. The family of Bernoullis furnished in course of a century eight members who distinguished themselves in mathematics. We subjoin the following genealogical table:–-
lll@ Jacob, 1654–1705 Nicolaus Johann, 1667–1748 Nicolaus, 1687–1759 Nicolaus, 1695–1726 Daniel, 1700–1782 Johann, 1710–1790 2c [c]$
ccc@ Daniel Johann, 1744–1807 Jacob, 1758–1789
^$
^Nicolaus Bernoulli, the Father$
Most celebrated were the two brothers Jacob (James) and Johann (John), and Daniel, the son of John. James and
John were staunch friends of Leibniz and worked hand in
hand with him. James Bernoulli (1654–1705) was born in
Basel. Becoming interested in the calculus, he mastered it without aid from a teacher. From 1687 until his death he occupied the mathematical chair at the University of Basel. He was the first to give a solution to Leibniz's problem of the isochronous curve. In his solution, published in the Acta Eruditorum, 1690, we meet for the first time with the word integral. Leibniz had called the integral calculus calculus summatorius, but in 1696 the term calculus integralis was agreed upon between Leibniz and John Bernoulli. James proposed the problem of the catenary, then proved the correctness
of Leibniz's construction of this curve, and solved the more complicated problems, supposing the string to be (1) of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre. Of these problems he published answers without explanations, while his brother John gave in addition their theory. He determined the shape of the "elastic curve" formed by an elastic plate or rod fixed