Squaring the Hyperbola
Grégoire Saint-Vincent is the greatest of circle-squarers … he found the property of the area of the hyperbola which led to Napier’s logarithms being called hyperbolic.—AUGUSTUS DE MORGAN, The Encyclopedia of Eccentrics (1915)
The problem of finding the area of a closed planar shape is known as quadrature, or squaring. The word refers to the very nature of the problem: to express the area in terms of units of area, that is, squares. To the Greeks this meant that the given shape had to be transformed into an equivalent one whose area could be found from fundamental principles. To give a simple example, suppose we want to find the area of a rectangle of sides a and b. If this rectangle is to have the same area as a square ...