Editing Hyperbolic angle (section)

==History==
The [[quadrature (mathematics)|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,
: [He made the] quadrature of a hyperbola to its [[asymptote]]s, and showed that as the [[area]] increased in [[arithmetic series]] the [[abscissa]]s increased in [[geometric series]].<ref>[[David Eugene Smith]] (1925) ''History of Mathematics'', pp. 424,5  v. 1</ref>

[[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 {{nowrap|1=''y'' = 1/''x''}} to the right of {{nowrap|1=''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 [[Squeeze mapping#Bridge to transcendentals|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''.<ref>[[Augustus De Morgan]] (1849) [https://books.google.com/books?id=7UwEAAAAQAAJ Trigonometry and Double Algebra], Chapter VI: "On the connection of common and hyperbolic trigonometry"</ref> In 1878 [[William Kingdon Clifford|W.K. Clifford]] used the hyperbolic angle to [[parametric equation|parametrize]] a [[unit hyperbola]], describing it as "quasi-[[harmonic oscillator|harmonic motion]]".

In 1894 [[Alexander Macfarlane]] circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generate [[versor#Hyperbolic versor|hyperbolic versors]], in his book ''Papers on Space Analysis''.<ref>[[Alexander Macfarlane]](1894) [https://archive.org/details/principlesalgeb01macfgoog ''Papers on Space Analysis''], B. Westerman, New York</ref> The following year [[Bulletin of the American Mathematical Society]] published [[Mellen W. Haskell]]'s outline of the [[hyperbolic function]]s.<ref>[[Mellen W. Haskell]] (1895) [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9</ref>

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 {{nowrap|1=[[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 {{nowrap|1=''v'' = (.7616)''c''}} for {{nowrap|1=''a'' = 1}}.
:[...] the rapidity {{nowrap|1=''a'' = 1}}, [...] consequently will represent the velocity .76&nbsp;''c'' which is a little above the velocity of light in water.

Silberstein also uses [[Nikolai Lobachevsky|Lobachevsky]]'s concept of [[angle of parallelism]] Π(''a'') to obtain {{nowrap|1=cos Π(''a'') = ''v''/''c''}}.<ref>[[Ludwik Silberstein]] (1914) [https://archive.org/details/theoryofrelativi00silbrich The Theory of Relativity], pp.&nbsp;180–1 via [[Internet Archive]]</ref>