9 - Conclusion: Newtonians, Leibnizians and Eulerians

Summary

A common heritage

Despite the chauvinistic divisions originated by the Newton–Leibniz controversy, the Newtonian and the Leibnizian schools shared a common mathematical method. They adopted two algorithms, the analytical method of fluxions and the differential and integral calculus, which were translatable one into the other. As far as foundations are concerned, the situation was much more fragmented than is usually thought. The Channel did not divide the friends of infinitesimals from the adherents to limits. One finds British who refer to the infinitesimalist side of the Newtonian method (i.e. to ‘moments’) and equate moments with differentials; one finds Continentals who support a theory of limits.

Rather than seeing the Newtonians and the Leibnizians as two groups practising different mathematical methods, it is more fruitful, and more adherent to historical evidence, to focus on the amount of shared knowledge between the two schools. The British and the Continentals shared a vast quantity of common algorithmic techniques, notational devices and foundational ideas. Newton and Leibniz, however, tried, with partial success, to orient their disciples along different research lines. The difference between the Newtonians and the Leibnizians consisted in the manner in which they oriented themselves in this common mathematical heritage: it consisted in the values that they adopted, in the purposes they had in mind, and in the ways in which they placed this heritage in historical perspective.

Equivalence

Passing from the fluxional to the differential notation was a triviality.