This is something that I came up with this morning and ran a couple of examples in the shower. I haven’t seen it before. Wikipedia notes in the history of Calculus the method of exhaustion and Cavalieri’s principle, but nothing quite like this. I feel like it must have appeared at some point, though, and wonder what it might be called.

Suppose I want to find the area of a circle. I’ll call it A, with radius r. I note that I can scale the circle up and down, which goes with the square of the radius. Consider a concentric circle with a slightly smaller radius (call the difference d): the area is A(r - d)^{2}/r^{2}.

The difference in the area between these two is a thin strip going around the circumference. We know the area of that, too: it’s 2*pi*r*d. Just the circumference times the width. Putting it all together, we have:

Drop the remaining d because it’s small:

And there we have it, the area of a circle. Let’s try another; the volume of a pyramid or cone. This time the volume shrinks with the cube of the height. Consider another cone with the points aligned. When it shrinks, there is a thin slice at the bottom that represents the difference, with volume Bd. So:

And again, the volume of a cone: area of the base times height over 3.

The technique does only have limited use; one needs some kind of similar shape, or at least a collection of shapes. It works the other way, too: given the volume of a sphere for instance, one could derive the surface area.

I guess one problem is that it depends on either limits or infinitesimals, depending on your interpretation of the “small d” I used. The ancient Greeks did use some techniques which were similar to limits but AFAIK nothing that quite resembled a small value that we can eventually take to be zero. Still, it would not shock me if it showed up in some form in Archimedes’ Palimpsest, say.