Editing Squeeze mapping (section)

==Logarithm and hyperbolic angle==
The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the [[area]] bounded by a hyperbola (such as {{math|''xy'' {{=}} 1)}} is one of [[quadrature (mathematics)|quadrature]]. The solution, found by [[Grégoire de Saint-Vincent]] and [[Alphonse Antonio de Sarasa]] in 1647, required the [[natural logarithm]] function, a new concept. Some insight into logarithms comes through [[hyperbolic sector]]s that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a [[hyperbolic angle]] associated with the sector. The hyperbolic angle concept is quite independent of the [[angle|ordinary circular angle]], but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate [[invariant measure]]s but with respect to different transformation groups. The [[hyperbolic function]]s, which take hyperbolic angle as argument, perform the role that [[circular functions]] play with the circular angle argument.<ref>[[Mellen W. Haskell]] (1895) [http://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,particularly equation 12, page 159</ref>