Gabriel's Horn - Apparent Paradox

Apparent Paradox

When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinite curve about the x-axis generates an object of finite volume was considered paradoxical. However, the explanation is that the bounding curve, is simply a special case–just like the simple harmonic series (Σx−1)–for which the successive area 'segments' do not decrease rapidly enough to allow for convergence to a limit. For volume segments however, and in fact for any generally constructed higher degree curve (e.g. y = 1/x1.001), the same is not true and the rate of decrease in the associated series is sufficiently rapid for convergence to a (finite) limiting sum.

The apparent paradox formed part of a great dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei.