exploit infinitesimals (rather than “exhaustion” methods) systematically in developing the calculus.[^1]

The series summation blackbox $\square$ (see Subsection \[s11\]) is handled differently in A and B. Framework A exploits a first-order “epsilontic” formulation that works in a complete Archimedean field. Thus, the convergence of a series $\sum_i u_i$ to $L$ would be expressed as follows: $$(\forall\epsilon>0)(\exists n\in{{\mathbb N}})(\forall m\in{{\mathbb N}})\;\left[m\geq
n\rightarrow \left|\sum_{i=1}^m u_i-L\right|<\epsilon\right].$$ [@Is90 Chapter 5] sought to interpret Leibnizian infinitesimal calculus by means of such quantified paraphrases, having apparently overlooked Leibniz’s remarks to the effect that his infinitesimals violate Euclid V.4 [@Le95b p. 322].

Meanwhile, framework B allows for an alternative interpretation in terms of the shadow (i.e., the standard part, closely related to Leibniz’s generalized notion of equality) and hyperfinite partial sums as follows: for each infinite hypernatural $H$ the partial sum $\sum_{i=1}^H u_i$ is infinitely close to $L$, i.e., $$(\forall H) \left[H \text{ infinite } \rightarrow \sum_{i=1}^H
u_i\approx L\right].$$ This is closer to the historical occurrences of the package $\square$ as found in Gregory, Leibniz, and Euler, as we argue below.

In pursuing modern interpretations of Leibniz’s work, a helpful distinction is that between ontological and procedural issues. More specifically, we seek to sidestep traditional questions concerning 