  This paper studies certain horocyclic orbits on
$\Gamma(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles
defined using the pencil of circles whose common point (in the words of the
Nielsen-Fenchel manuscript is $\infty$. The orbits involved in this case are
closed and long - judged by arc length between two points compared to the
hyperbolic distance between them.
  We are concerned with tracking the paths of individual horocycles. Using Ford
circles of Farey sequences we find lifts to the Standard Fundamental Region
(SFR) and find points of these lifts making given angles with a horizontal.
Next, we offer two methods, both involving continued fractions, of locating
points with such angles whose lifts are near any given point in the SFR. This
establishes in an effective manner a sort of transitivity, which necessarily
involves infinitely many such horocycles.
  Next, we study the homotopy classes of horizontal horocycles as we descend to
the real axis. We find these are stable during descent between encounters of
the horizontal with elliptic fixed points. Such encounters change - complicate
- the homotopy classes. We give these explicitly down to height
$1/(2\sqrt{3})$.
  Finally we do an initial study of the open (infinite length) horocycle path
with unit euclidean radius anchored at $\phi -1$, where $\phi$ is the Golden
Mean. Enough information is adduced to suggest that this path is itself
transitive. The methods resemble the Hardy-Littlewood Circle Method in a
certain regard, albeit without the exponential sums.
