◊◊◊ Galileo Galilei (1564 – 1642) addressed these paradoxes of the continuum with great insight –
(1) the paradox of two concentric circles having the same number of infinite points, is covered in Chap.1 on Infinities.
(2) we cover here his paradox of the ‘point and the round razor’. The text is hyperlinked. Galileo constructed a cylinder with the same height as its radius and then embedded two geometrical figures in it as shown separately in Fig.1
The cone shown in red uses the same base as the cylinder and its peak rises to the centre-point of its top surface. The hemisphere shown in blue ‘cups’ into the cylinder touching its base at the centre-point of the bottom surface. If ‘ r ’ is the radius of the base, then by definition ‘ r ’ is also the height and the volume of the cylinder = π r3. That of the cone is (⅓ base x height) = ⅓ π r3 ; the hemisphere’s volume is ⅔ π r3.
Galileo compared the ‘cone’ shown in red with the ‘bowl’ i.e. the volume of the cylinder in the second figure with the blue hemisphere scooped out. Now, both these geometrical figures have the same volume viz. ⅓ π r3.