p. 46-47], Leibniz presents an alternative justification of the product rule (see [@Bos p. 58]). Here he divides by $dx$ and argues with differential quotients rather than differentials. Adjusting Leibniz’s notation, we obtain an equivalent calculation $$\begin{aligned}
\frac{d(uv)}{dx} &= \frac{(u+du)(v+dv)-uv}{dx} \\&=
\frac{udv+vdu+du\,dv}{dx} \\&= \frac{udv+vdu}{dx} + \frac{du\,dv}{dx}
\\&= \frac{udv+vdu}{dx}.
\end{aligned}$$ Under suitable conditions the term $\frac{du\,dv}{dx}$ is infinitesimal, and therefore the last step $$\label{72}
\frac{udv+vdu}{dx} + \frac{du\,dv}{dx} =
u\,\frac{dv}{dx}+v\,\frac{du}{dx},$$ relying on a generalized notion of equality, is legitimized as an instance of Leibniz’s *transcendental law of homogeneity*, which authorizes one to discard the higher-order terms in an expression containing infinitesimals of different orders.

Law of continuity
-----------------

Leibniz proposed a heuristic principle known as the *law of continuity* to the effect that

> … et il se trouve que les règles du fini réussissent dans l’infini … ; et que vice versa les règles de l’infini réussissent dans le fini, … [@Le02a p. 93-94],

cited by [@Kn02 p. 67], [@Ro66 p. 262], [@Lau92 p. 145], and other scholars.

On the face of it, one can find numerous counterexamples to such a principle. Thus, finite ordinal number addition is commutative, whereas for infinite ordinal numbers, the addition is no longer commutative: $1 