Leibniz grants that there is an infinitesimal straight stretch of the curve (a side of an infinilateral polygon coinciding with the curve) between abscissae 0 and e, which does not reduce to a single point.
The map (parametric description) itself, not the set of image points it traces; two curves are different if given by different maps even if their image sets are identical
grants(Leibniz is acknowledging a particular point about infinitesimals)
In philosophy, to 'grant' something means to acknowledge or accept it as true, even if you might disagree with it overall.
infinilateral polygon(Leibniz's geometric conception of curves as infinilateral polygons underlies his treatment of differentials.)
A polygon with infinitely many sides, each side being an infinitesimal straight stretch, conceived as coinciding with a curve.
infinitesimal(Nonstandard analysis)
A hyperreal a whose absolute value |a| is less than 1/(n+1) for every natural number n
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