To see what’s wrong with differentials consider the circle and the differential triangle below.
Suppose that \(\dx{s}\) is one of the differentials that makes up the circle. Since \(\dx{s}\) is a non–zero increment it has two distinct endpoints so we can draw the two radii shown. Because the two (distinct) endpoints that lie on the circle are infinitely close together the two lines have the same slope. But all of the radii of a circle pass through the center of the circle so these two in particular must also intersect at the center, and we conclude that we have two parallel lines that intersect.
But the only way that can happen is if they are actually the same line, and if they are the same line then the points on the circle are not really distinct as we’ve drawn them. But if they are not distinct then \(\dx{x}\text{,}\)\(\dx{y}\text{,}\) and \(\dx{s}\) are all actually equal to zero. If \(\dx{x}\) =0 then \(\dfdx{y}{x}\) is meaningless (why?).
This simple argument appears to completely destroy the differential foundation upon which we’ve based everything we’ve done up until now. Try as we might we can’t escape the contradictions inherent in the very notion of infinitely small numbers.
