Editing Gudermannian function (section)

==History==
{{Broader| Mercator projection#History| Integral of the secant function}}
The function and its inverse are related to the [[Mercator projection]]. The vertical coordinate in the Mercator projection is called [[Latitude#Isometric latitude|isometric latitude]], and is often denoted <math display=inline>\psi.</math> In terms of [[latitude]] <math display=inline>\phi</math> on the sphere (expressed in [[radian]]s) the isometric latitude can be written

:<math>\psi = \operatorname{gd}^{-1} \phi = \int_0^\phi \sec t \,\mathrm{d}t.</math>

The inverse from the isometric latitude to spherical latitude is <math display=inline>\phi = \operatorname{gd} \psi.</math> (Note: on an [[ellipsoid of revolution]], the relation between geodetic latitude and isometric latitude is slightly more complicated.)

[[Gerardus Mercator]] plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599, [[Edward Wright (mathematician)|Edward Wright]] described a method for constructing a Mercator projection numerically from trigonometric tables, but did not produce a closed formula. The closed formula was published in 1668 by [[James Gregory (mathematician)|James Gregory]].

The Gudermannian function per se was introduced by [[Johann Heinrich Lambert]] in the 1760s at the same time as the [[hyperbolic functions]]. He called it the "transcendent angle", and it went by various names until 1862 when [[Arthur Cayley]] suggested it be given its current name as a tribute to [[Christoph Gudermann]]'s work in the 1830s on the theory of special functions.<ref>{{harvp|Becker|Van Orstrand|1909}}</ref>
Gudermann had published articles in ''[[Crelle's Journal]]'' that were later collected in a book<ref>{{harvp|Gudermann|1833}}</ref>
which expounded <math display=inline>\sinh</math> and <math display=inline>\cosh</math> to a wide audience (although represented by the symbols <math display=inline>\mathfrak{Sin}</math> and <math display=inline>\mathfrak{Cos}</math>).

The notation <math display=inline>\operatorname{gd}</math> was introduced by Cayley who starts by calling <math display=inline>\phi = \operatorname{gd} u</math> the [[Jacobi elliptic functions#am|Jacobi elliptic amplitude]] <math display=inline>\operatorname{am} u</math> in the degenerate case where the elliptic modulus is <math display=inline>m = 1,</math> so that <math display=inline>\sqrt{1 - m\sin\!^2\,\phi}</math> reduces to <math display=inline>\cos \phi.</math><ref>{{harvp|Cayley|1862}}</ref> This is the inverse of the [[integral of the secant function]]. Using Cayley's notation,

:<math>
u = \int_0 \frac{d\phi}{\cos \phi}
= {\log\, \tan}\bigl(\tfrac14\pi + \tfrac12 \phi\bigr).
</math>

He then derives "the definition of the transcendent",

:<math>
\operatorname{gd} u = {\frac1i \log\, \tan} \bigl(\tfrac14\pi + \tfrac12 ui\bigr),
</math>

observing that "although exhibited in an imaginary form, [it] is a real function of {{nobr|<math display=inline> u</math>".}}

The Gudermannian and its inverse were used to make [[trigonometric tables]] of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle <math display=inline>\psi</math>, hyperbolic functions could be found by first looking up <math display=inline>\phi = \operatorname{gd} \psi</math> in a Gudermannian table and then looking up the appropriate circular function of <math display=inline>\phi</math>, or by directly locating <math display=inline>\psi</math> in an auxiliary <math>\operatorname{gd}^{-1}</math> column of the trigonometric table.<ref>For example Hoüel labels the hyperbolic functions across the top in Table XIV of: {{cite book |last=Hoüel |first=Guillaume Jules |year=1885 |title=Recueil de formules et de tables numériques |publisher=Gauthier-Villars |page=36 |url=https://archive.org/details/recueildeformul00hogoog/page/n115/}} </ref>