of . There is no upper bound for the natural logarithm of , as approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say, In Torricelli's original non-calculus proof used an object, slightly different to the aforegiven, that was constructed by truncating the acute hyperbolic solid with a plane perpendicular to the x axis and extending it from the opposite side of that plane with a cylinder of the same base. Whereas the calculus method proceeds by setting the plane of truncation at and integrating along the x axis, Torricelli proceeded by calculating the volume of this compound solid (with the added cylinder) by summing the surface areas of a series of concentric right cylinders within it along the y axis and showing that this was equivalent to summing areas within another solid whose (finite) volume was known. In modern terminology this solid was created by constructing a surface of revolution of the function (for strictly positive ) and Torricelli's theorem was that its volume is the same as the volume of the right cylinder with height and radius : Torricelli showed that the volume of the solid could be derived from the surface areas of this series of concentric right cylinders whose radii were and heights . Substituting in the formula for the surface areas of (just the sides of) these cylinders yields a constant surface area for all cylinders of . This is the also the area of a circle of radius and the nested surfaces of the cylinders (filling the volume of the solid) are thus equivalent to the stacked areas of the circles of radius stacked from 0 to , and hence the volume of the aforementioned right cylinder, which is known to be : (The volume of the added cylinder is of course and thus the volume of the truncated acute hyperbolic solid alone is . If , as in the modern calculus derivation, .) In the this is one of two proofs of the volume of the (truncated) acute hyperbolic solid. The use of Cavalieri's indivisibles in this proof was controversial at the time and the result shocking (Torricelli later recording that Gilles de Roberval had attempted to disprove it); so when the was published, the year after , Torricelli also supplied a second proof based upon orthodox Archimedean principles showing that the right cylinder