A Gabriel's horn (also called Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Judgment Day. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
These colourful informal names and the allusion to religion came along later. Torricelli's own name for it is to be found in the Latin title of his paper , written in 1643, a truncated acute hyperbolic solid, cut by a plane. Volume 1, part 1 of his published the following year included that paper and a second more orthodox (for the time) Archimedean proof of its theorem about the volume of a truncated acute hyperbolic solid. This name was used in mathematical dictionaries of the 18th century, including "Hyperbolicum Acutum" in Harris' 1704 dictionary and in Stone's 1726 one, and the French translation in d'Alembert's 1751 one.
Although credited with primacy by his contemporaries, Torricelli was not the first to describe an infinitely long shape with a finite volume or area. The work of Nicole Oresme in the 14th century had either been forgotten by, or was unknown to them. Oresme had posited such things as an infinitely long shape constructed by subdividing two squares of finite total area 2 using a geometric series and rearranging the parts into a figure, infinitely long in one dimension, comprising a series of rectangles.
Mathematical definition
Gabriel's horn is formed by taking the graph of
with the domain and rotating it in three dimensions about the axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today, calculus can be used to calculate the volume and surface area of the horn between and , where . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume and the surface area :
The value can be as large as required, but it can be seen from the equation that the volume of the part of the horn between and will never exceed ; however, it does gradually draw nearer to as increases. Mathematically, the volume approaches as approaches infinity. Using the limit notation of calculus,
The surface area formula above gives a lower bound for the area as 2 times the natural logarithm 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 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, .)