infinite increase of the other", in the case of the acute hyperbolic solid by the equation of the Apollonian hyperbola . Painter's paradox Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its surface. However, this paradox is again only an apparent paradox caused by an incomplete definition of "paint", or by using contradictory definitions of paint for the actions of filling and painting. One could be postulating a "mathematical" paint that is infinitely divisible (or infinitely thinnable, or simply zero-width like the zero-width geometric lines that Hobbes took issue with) and capable of travelling at infinite speed, or a "physical" paint with the properties of paint in the real world. With either one, the apparent paradox vanishes: With "mathematical" paint, it does not follow in the first place that an infinite surface area requires an infinite volume of paint, as infinite surface area times zero-thickness paint is indeterminate. With physical paint, painting the outside of the solid would require an infinite amount of paint because physical paint has a non-zero thickness. Torricelli's theorem does not talk about a layer of finite width on the outside of the solid, which in fact would have infinite volume. Thus there is no contradiction between infinite volume of paint and infinite surface area to cover. It is also impossible to paint the interior of the solid, the finite volume of Torricelli's theorem, with physical paint, so no contradiction exists. This is because physical paint can only fill an approximation of the volume of the solid. The molecules do not completely tile 3-dimensional space and leave gaps, and there is a point where the "throat" of the solid becomes too narrow for paint molecules to flow down. Physical paint travels at a bounded speed and would take an infinite amount of time to flow down. This also applies to "mathematical" paint of zero thickness if one does not additionally postulate it flowing at infinite speed. Other different postulates of "mathematical" paint, such as infinite-speed paint that gets thinner at a fast enough rate, remove the paradox too. For volume of paint, as the surface area to be covered tends towards infinity, the thickness of the paint tends towards zero. Like with the solid itself, the infinite