If now we ‘cut’ this cylinder at any place in between its height we cut the ‘cone’ in a smaller circle and the ‘bowl’ in an annular ring as shown in Fig.2 below :-Then geometrically the area of the smaller circle is the same as that of the annular ring. The simple geometrical proof is given in the attached article. In Galileo’s words :-
And as the cutting plane (here shown in green) moves upwards, coming near the top, the two solids (always equal) as well as their bases (areas which are also equal) finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely the ‘round razor’- the upper edge of the bowl and the ‘apex’ of the cone. Now, since as these solids diminish equality is maintained between them up to the very last, we are justified in saying that, at the extreme and final end of this diminution, they are still equal and that one is not infinitely greater than the other. It appears therefore that we may equate the circumference of a large circle to a single point. And this which is true of the solids is true of the surfaces which form their bases ; for these also preserve equality between themselves throughout their diminution and in the end vanish, the one into the circumference of a circle, the other into a single point. Shall we not call them equal seeing that they are the last traces and remnants of equal magnitudes ?