 Now Leibniz could retort that that this argument depends crucially on the assumption that the portion of the curve between abscissae 0 and \(\Dx\) is indeed straight. If this be denied, then of course it does not follow that \(\Dx ^2 = 0\). But if one grants, as Leibniz does, that that there is an infinitesimal straight stretch of the curve (a side, that is, of an infinilateral polygon coinciding with the curve) between abscissae 0 and \(e\), say, which does not reduce to a single point then \(e\) cannot be equated to 0 and yet the above argument shows that \(e^2 = 0\). It follows that, if curves are infinilateral polygons, then the “lengths” of the sides of these latter must be nilsquare infinitesimals. Accordingly, to do full justice to Leibniz’s (as well as Nieuwentijdt’s) conception, two sorts of infinitesimals are required: first, “differentials” obeying—as laid down by Leibniz—the same algebraic laws as finite quantities; and second the (necessarily smaller) nilsquare infinitesimals which measure the lengths of the sides of infinilateral polygons. It may be said that Leibniz recognized the need for the first, but not the second type of infinitesimal and Nieuwentijdt, vice-versa. It is of interest to note that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, and nilsquare infinitesimals in smooth infinitesimal analysis (for both types of analysis see below). In fact it has been shown to be possible to combine the two approaches, so creating an analytic framework realizing both Leibniz’s and Nieuwentijdt’s conceptions of infinitesimal.