structural:expansion, information:filtering, structural:reformatting
History

The quadrature of the hyperbola is the evaluation of the area of a hyperbolic sector. It can be shown to be equal to the corresponding area against an asymptote. The quadrature was first accomplished by Gregoire de Saint-Vincent in 1647 in Opus geometricum quadrature circuli et sectionum coni. As expressed by a historian,

A. A. de Sarasa interpreted the quadrature as a logarithm and thus the geometrically defined natural logarithm (or "hyperbolic logarithm") is understood as the area under y = 1/ x to the right of x = 1. As an example of a transcendental function, the logarithm is more familiar than its motivator, the hyperbolic angle. Nevertheless, the hyperbolic angle plays a role when the theorem of Saint-Vincent is advanced with squeeze mapping.

Circular trigonometry was extended to the hyperbola by Augustus De Morgan in his textbook Trigonometry and Double Algebra. In 1878 W.K. Clifford used the hyperbolic angle to parametrize a unit hyperbola, describing it as "quasi- harmonic motion ".

In 1894 Alexander Macfarlane circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generate hyperbolic versors, in his book Papers on Space Analysis. The following year Bulletin of the American Mathematical Society published Mellen W. Haskell 's outline of the hyperbolic functions.

When Ludwik Silberstein penned his popular 1914 textbook on the new theory of relativity, he used the rapidity concept based on hyperbolic angle a, where tanh a = v / c, the ratio of velocity v to the speed of light. He wrote:

Silberstein also uses Lobachevsky 's concept of angle of parallelism Π(a) to obtain cos Π(a) = v / c.