(height radius ) was both upper and lower bound for the volume. Ironically, this was an echo of Archimedes' own caution in supplying two proofs, mechanical and geometrical, in his Quadrature of the Parabola to Dositheus. Apparent paradox When the properties of Gabriel's horn were discovered, the fact that the rotation of an infinitely large section of the plane about the axis generates an object of finite volume was considered a paradox. While the section lying in the plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the "weighted sum" of sections, is finite. Another approach is to treat the solid as a stack of disks with diminishing radii. The sum of the radii produces a harmonic series that goes to infinity. However, the correct calculation is the sum of their squares. Every disk has a radius and an area or . The series diverges, but the series converges. In general, for any real , the series converges. The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time, including Thomas Hobbes, John Wallis, and Galileo Galilei. There is a similar phenomenon that applies to lengths and areas in the plane. The area between the curves and from 1 to infinity is finite, but the lengths of the two curves are clearly infinite. In lecture 16 of his 1666 , Isaac Barrow held that Torricelli's theorem had constrained Aristotle's general dictum (from De Caelo book 1, part 6) that "there is no proportion between the finite and the infinite". Aristotle had himself, strictly speaking, been making a case for the impossibility of the physical existence of an infinite body rather than a case for its impossibility as a geometrical abstract. Barrow had been adopting the contemporary 17th-century view that Aristotle's dictum and other geometrical axioms were (as he had said in lecture 7) from "some higher and universal science", underpinning both mathematics and physics. Thus Torricelli's demonstration of an object with a relation between a finite (volume) and an infinite (area) contradicted this dictum, at least in part. Barrow's explanation was that Aristotle's dictum still held, but only in a more limited fashion when comparing things of the same type, length with length, area with area, volume with volume, and so forth. It did not hold when