Differentials and differential coefficients in the Eulerian foundations of the calculus

In the 18th-century calculus the classical notion of quantity was understood as general quantity, which was expressed analytically and was subject to formal manipulation. Number was understood as the measure of quantity; however, only fractions and natural numbers were considered numbers in the true sense of term. The other types of numbers were fictitious entities, namely ideal entities firmly founded in the real world which could be operated upon as if they were numbers. In this context Eulerian infinitesimals should also be considered as fictitious numbers. They were symbols that represented a primordial and intuitive idea of limit, although they were manipulated in the same way as numbers. This conception allowed Euler to consider calculus as a calculus of functions (intended as analytical expressions of quantities) and, at the same time, to handle differentials formally. (C) 2003 Elsevier Inc. All rights reserved.