But who was Gudermann?  He's the guy they named the "gudermannian" 
after!  That's this function:

gd(u) = 2 arctan(exp(u)) - pi/2.

Now, if you're wondering why such a silly function deserves a name,
you should work out its inverse function:

gd^{-1}(x) = ln(sec(x) + tan(x)).

And if you don't recognize *this*, you probably haven't taught
freshman calculus lately!  It's the integral of sec(x), which is one
of the hardest of the basic integrals you teach in that kind of
course.  But it's not just hard, it's historically important: a point
at latitude gd(u) has distance u from the equator in a Mercator
projection map.  If you think about it a while, this is precisely
what's needed to make the projection be a conformal transformation -
that is, angle-preserving.  And that's just what you want if you're
sailing a ship in a constant direction according to a compass and you
want to know where you'll wind up.