The Bernoulli brothers
Not many people understood Newton's and Leibniz' ideas about infinitesimals. So that others would be able to understand it, Newton wrote his Principia (1687) deliberately without using his theory of fluxions. Instead, as far as possible, he tried to explain his theory using the old mathematics, with Euclid's Elements and Archimedes' mechanics as a basis. But Newton claimed that everything he had written was also confirmed by the theory of fluxions.
The brothers Jacob Bernoulli (1654–1705) and Johann Bernoulli (1667–1748) in Basel, Switzerland, were among the first interpreters of Leibniz' differential calculus. They were both critical of Newton's theories and maintained that the theory of fluxions was plagiarized from Leibniz' original theories. They put a great deal of effort into interpreting the propositions of Principia with the help of differential calculus and in this way uncovered errors in them. Their criticism, especially Johann's, prompted Newton to make important corrections in the second edition of Principia in 1714. Throughout his entire life Johann B. could not accept the theory, which Newton had proven, that the earth and the planets rotate around the sun in elliptical orbits, one of Kepler's laws. Instead, he tried to prove the so-called vortex theory, put forward by Descartes, which describes how the earth, the sun, and the planets move around each other in spirals.
Despite this suspicion of Newton, the Benoulli brothers had a tremendous impact on the development of the new mathematics in the early 1700s. For example, Jacob B. established the theory of probability in his posthumous Ars Conjectandi (The Art of Guessing; 1713) and Johann B. introduced a new direction in mathematics, the calculus of variations. Both of them made important contributions to the development of integral calculus, basically a theory to determine areas of surfaces bounded by curves, and to find other applications for it, especially in physics.
For six months in 1728–1729 Samuel Klingenstierna visited Basel. Johann Bernoulli was his teacher there. There are manuscripts where we can see that Bernoulli corrected Klingenstierna's text.