History
The quadrature of the hyperbola is the evaluation of the area swept out by a radial segment from the origin as the terminus moves along the hyperbola, just the topic of hyperbolic angle. The quadrature of the hyperbola was first accomplished by Gregoire de Saint-Vincent in 1647 in his momentous Opus geometricum quadrature circuli et sectionum coni. As David Eugene Smith wrote in 1925:
- quadrature of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series the abscissas increased in geometric series.
- History of Mathematics, pp. 424,5 v. 1
The upshot was the logarithm function, as now 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 in Augustus De Morgan's 1849 textbook Trigonometry and Double Algebra (see references for link). In 1878 W.K. Clifford used 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.
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:
- It seems worth mentioning that to unit rapidity corresponds a huge velocity, amounting to 3/4 of the velocity of light; more accurately we have v = (.7616) c for a = 1.
- ... the rapidity a = 1, ... consequently will represent the velocity .76 c which is a little above the velocity of light in water.
Silberstein also uses Lobachevsky's concept of angle of parallelism Π(a) to obtain cos Π(a) = v/c.
Read more about this topic: Hyperbolic Angle
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