as first order logic (FOL). The precise meaning of the term is clarified in Section \[FOL\]. We first analyze several Leibnizian examples in Section \[one\].

This text continues a program of re-evaluation of the history of infinitesimal mathematics initiated in [@KK12], [@B11] and elsewhere.

Examples from Leibniz {#one}
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Let us examine some typical examples from Leibniz’s infinitesimal calculus so as to gauge their relationship to FOL.

Series presentation of $\pi/4$ {#s11}
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In his *De vera proportione* (1682), Leibniz represented $\frac{\pi}{4}$ in terms of the infinite series $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots$$ This is a remarkable result, but we wish to view it as a result concerning a specific real number, i.e., a *single case*, and in this sense involving no quantification, once we add a new function symbol for a black box procedure $\square$ called “evaluation of convergent series” (as well as a definition of $\pi$) (we will say a few words about the various implementations of $\square$ in modern frameworks in Section \[two\]).

Leibniz convergence criterion for alternating series {#s22}
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This refers to an arbitrary alternating series defined by an alternating sequence with terms of decreasing absolute value tending to zero, such as the series of Subsection \[s11\], or the 