Is Archimedes's method for computing volumes equivalent to Cavalieri's? Encyclopaedia Britannica is unequivocal: "It turned out that Archimedes had used a method later known as Cavalieri’s principle, which involves slicing solids (whose volumes are to be compared) with a family of parallel planes." I suppose it's true that both are based on slicing, but Cavalieri's calculation for sphere and cylinder only needs a simple comparison of cross sections, whereas Archimedes's requires a diagram with a tangle of lines, and a lever with unequal arms to balance the pieces. Also in Cavalieri's case the sphere is inscribed into the cylinder, while in Archimedes's it has only half the cylinder's radius, and Cavalieri's cone is inverted compared to Archimedes's.
Can somebody explain the exact relationship between Archimedes's and Cavalieri's use of slicing? Is there a general way to rephrase any Archimedian calculation into a Cavalierian one and/or vice versa? Is one of them more general than the other?
 A: Archimedes used the optical properties of the parabola to prove that the area of the parabolic segment is $\frac{2}{3}$ of the circumscribed rectangle. In modern terms, the optical property is equivalent to $\frac{d}{dx}x^2 = 2x$, while the area depends on $\int x^2 = \frac{1}{3}x^3$, and the Archimedean argument for the parabolic segment can be viewed as the first attempt to provide an antiderivative for a polynomial.
With such a preliminary result, he proved that the volume of the sphere is $\frac{4\pi}{3}R^3$ by "slicing" it. Since the area of a section is proportional to the square radius of that section, to find the volume of a sphere is essentially the same problem as finding the area of a parabolic segment, since they both depend on $\int x^2$ - and not by chance, the coefficient $\frac{1}{3}$ appears in both of them.
So, in a quite vague (but not so much) sense, we can say that the Archimedean method was a precursor of the Cavalieri's method. And that Archimedes was a true genius.
A: The method used by Archimedes in that letter includes Cavalieri's method (and I don't know whether Cavalieri's method is really Cavalieri's), but Archimedes's method also involves centers of gravity. I think Archimedes may have been the first ever to conceive of centers of gravity. Archimedes used his method to show that the center of gravity of the interior of a hemisphere (not a sphere; a hemisphere) is $5/8$ of the way from the north pole (if we were to take it to be the northern hemisphere) to the center of the sphere.
