increase of the surface area to be painted in one dimension is compensated by the infinite decrease in another dimension, the thickness of the paint. Converse The converse of Torricelli's acute hyperbolic solid is a surface of revolution that has a finite surface area but an infinite volume. In response to Torricelli's theorem, after learning of it from Marin Mersenne, Christiaan Huygens and René-François de Sluse wrote letters to each other about extending the theorem to other infinitely long solids of revolution; which have been mistakenly identified as finding such a converse. Jan A. van Maanen, professor of mathematics at the University of Utrecht, reported in the 1990s that he once mis-stated in a conference at Kristiansand that de Sluse wrote to Huygens in 1658 that he had found such a shape: to be told in response (by Tony Gardiner and Man-Keung Siu of the University of Hong Kong) that any surface of rotation with a finite surface area would of necessity have a finite volume. Professor van Maanen realized that this was a misinterpretation of de Sluse's letter, and that what de Sluse was actually reporting that the solid "goblet" shape, formed by rotating the cissoid of Diocles and its asymptote about the y axis, had a finite volume (and hence "small weight") and enclosed a cavity of infinite volume. Huygens first showed that the area of the rotated two-dimensional shape (between the cissoid and its asymptote) was finite, calculating its area to be 3 times the area of the generating circle of the cissoid, and de Sluse applied Pappus's centroid theorem to show that the solid of revolution thus has finite volume, being a product of that finite area and the finite orbit of rotation. The area being rotated is finite; de Sluse did not actually say anything about the surface area of the resultant rotated volume. Such a converse cannot occur (assuming Euclidean geometry) when revolving a continuous function on a closed set. Theorem Let be a continuously differentiable function. Write for the solid of revolution of the graph about the axis. If the surface area of is finite, then so is the volume. Proof Since the lateral surface area is finite, the limit superior: Therefore, there exists a such that the supremum } is finite. Hence, must be finite, since is a continuous function, which implies that is bounded on the interval . Finally, the volume: