I read in one article that "for any curve given by an algebraic equation, the hypotenuse of the differential triangle generated by an infinitesimal abscissal increment $\varepsilon$ coincides with the segment of the curve between $x$ and $x+\varepsilon$".
How to prove that hypotenuse of differential triangle coincide with the segment of the curve?
Thanks.
If one works in synthetic differential geometry, as the tag indicates, there is not much to prove here.
The Kock-Lawvere axiom literally says that all curves are linear on the infinitesimal scale. So the segment of the curve between $x$ and $x+\varepsilon$ is a line segment. The hypotenuse of the triangle between $x$ and $x+\varepsilon$ is also a line segment. Since these two line segments have the same endpoints, they coincide.
So the hypotenuse of the differential triangle coincides with the segment of the curve.
Of course, Nieuwentijdt's work precedes synthetic differential geometry by centuries, and I don't know whether his proof of this fact goes in a way comparable to the one outlined above: that's perhaps best left to some kind of StackExchange site where historians of science and mathematics congregate.