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The Three-Shadowpoint method for finding an east-west line, based on an old Roman surveying method. This article was written by Thomas Stein, based on work by J.A.F. de Rijk, Th. de Vries, H. Janssen, and Ton vd Beld

**The Three-Shadowpoint Method: **

1.
In **Figure 1**, a gnomon “g” with foot “F”
and top “T” is erected.

2.
In **Fig. 2** at three different times, the
shadow of g is recorded on the ground.

Let the tip of top T be recorded in shadowpoints A,
B_{0} and C – the three shadowpoints.

Point A would be the first point, Point B_{0}
the second, and Point C latest of the three points.

3.
In **Fig. 3**, assuming B_{0} to be
closest to the foot of the gnomon F, then the distance from top T to

shadowpoint B_{0} will be the shortest of the distances. We measure the
distance TB

from T towards B_{0}, so obtaining the distance “ß”.

4.
In **Fig. 3**, we measure this
distance ß from T to shadowpoint A, and reach _{A}

Dropping a vertical from point B’_{A} to
the ground, we reach _{A}

5.
Likewise in **Fig. 3**, we measure this
distance ß from T to shadowpoint C, and reach _{C}

Dropping a vertical from point B’_{C} to
the ground, we reach _{C}

In **Fig. 4**
all of these sub-points are visible on the 3D representation.

6.
In **Fig. 5**, we then extend a line through sub-points
B_{A} and B_{C}. This
line generally runs along the ground. We
also extend a second line through sub-points B’_{A} and B’_{C}. Note that this second line is elevated
because both sub-points B’_{A} and B’_{C} are off the ground.

7.
In **Fig. 5**, we can see that these two lines
will intersect in _{AC}

8.
In **Fig. 6**, we then connect _{AC}_{0}.

This line is the local East-West direction!

The
whole procedure can be carried out with pegs and strings, and no calculations
are necessary.

**Why it works**

Suppose we have a plane **H** and a point B_{0}
in it. Above the plane two arbitrary
points B’_{A} and B’_{C} "float".

Call the plane through B_{0}, B’_{A}
and B’_{C} a plane **E**. Find
the intersection of planes **H** and **E**.

__Solution__

Drop perpendiculars from B’_{A}
and B’_{C} onto **H**, giving B_{A}
and B_{C}. Lines B’_{A} B’_{C} and _{A}_{C}_{AC}

The line through B_{0}
and _{AC}**H** and **E**.

In our
case, plane **H** is the surface of the
earth, and plane **E** is the shadowcone
periphery defined by the sun’s movement across the sky on a given day.

The
shadowcone is the cone traced on the celestial globe by the sun’s movement
across the sky on a given day. The
surface of the earth and the daily shadowcone intersect on an East-West line.

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