Where can I find evidence for the contents of a sphere as Archimedes has shown?

Answer

Dear Karel,

Archimedes determined the content of the sphere by the so-called “method of exhaustion”. This is a predecessor of the classical integration method in which the body is divided into a series of smaller volumes, the contents of which could be determined. You can find more information on this at: http://www.britannica.com/eb/article-247681/analysis.

A variant of this proof goes as follows:

Consider the hemisphere and the cone (with the same radius and height) as shown in the first figure.
Archimedes already knew the formula for the volume of a cone: V(cone) = π.r².h/3 (in this case r and h are equal). He also knew that of a cylinder: V(cylinder) = π.r².h.
The second figure shows a side view of the cross-section through the center of the bodies, with the distances indicated. At the height z, one considers the area of ​​the horizontal section of the bodies (they are circles with respective radii given by √(r²-z²) and z; see figure 3).
The sum of the areas of these two circles is given by:

π.(r²-z²) + π.z² = π.r²,

so independent of the height z. In other words, the total area at any height is equal to that of a cylinder with radius r (circumscribed to the sphere). Therefore, the total volume of the hemisphere + the cone is also equal to that of the cylinder. In formulas, this yields the following:

V(hemisphere) + V(cone) = V(cylinder)

or:

V(sphere)/2 + π.r³/3 = π.r³

So: V(sphere) = 4.π.r³/3

Kind regards,