Wndsn Quadrant Telemeter Tutorials
Making the most out of our graphical telemetry computers.
Like with many complex instruments, there are multiple ways to solve certain problems and to measure the required inputs. Combining the various functions leads to a multitude of advanced uses.
The Quadrant Telemeter: An Equatorial Sundial
The Quadrant Telemeter can transform itself into a equatorial sundial, using only on-board features for the required calculations, namely determining local latitude and computing and measuring the necessary length of the gnomon.
The distinguishing characteristic of the equatorial dial (also called the equinoctial dial) is the planar surface that receives the shadow, which is exactly perpendicular to the gnomon's style. This plane is called equatorial, because it is parallel to the equator of the Earth and of the celestial sphere. If the gnomon is fixed and aligned with the Earth's rotational axis, the Sun's apparent rotation about the Earth casts a uniformly rotating sheet of shadow from the gnomon; this produces a uniformly rotating line of shadow on the equatorial plane. Since the Sun rotates 360° in 24 hours, the hour-lines on an equatorial dial are all spaced 15° apart (360/24).
In order for our quadrant to tell the time, we need to set it up parallel to the equator, meaning that when oriented southwards, it needs to be elevated by the exact angle of our latitude.
Of course, the Quadrant Telemeter can determine local latitude by sighting Polaris at night or by some more involved calculation as explained in the manual.
Now, the length of the gnomon, that is the stick we position in the vertex of the quadrant, determines that angle.
And guess what?
The length of the gnomon is a tan calculation just like determining the height of an object from a known distance.
We can't use the Telemeter side for the tan calculation this time, because it's set up for angles up to 7° only and we cannot employ scale jumps here since there is no linear relationship between 5.2° and 52°; 52° being our sample latitude for this tutorial.
Hence, we use the quadrant itself, by pulling the string across 52°. At that angle, the inner percent scale shows the value 130, which is:
tan(52°) ≈ 1.3
The triangle we need to solve for consists of the Telemeter (tm), which we know (adjacent) and the length of the gnomon (g), which we are looking for (opposite).
Since tan = opposite / adjacent or g / tm, we solve for g:
g = tm × tan(lat)
We have to do two calculations and rotate the quadrant in between (see above), once for the hours before noon (short gnomon) and once for after noon (long gnomon) since a quadrant only features 90° and not the 180° for a full day.
The short side of the Telemeter measures 49 mm from edge to vertex, the long side measures 79 mm.
49 × 1.3 = 63 mm (AM gnomon) 79 × 1.3 = 102 mm (PM gnomon)
With the respective setup for AM or PM, the time is easily read, since 24h ≙ 360° hence 1h = 15°.
Two modes of observation
During most of the year, the shadow is cast on the bottom of the setup, because the Sun's altitude is lower than the local latitude.
During the weeks around summer solstice, when the maximum Sun altitude H > latitude, the time is read on the top side of the equatorial sundial, which means that we have to turn our Quadrant around so the angle scale is facing up. This in turn means that our gnomon needs to be long enough to throw a shadow on the top side. How long? Again, the tangent is of help here.
H = 90° - latitude + declination
At a latitude of 52° and a maximum declination of 23.44°, the maximum Sun altitude H = 61°.
To find out the extra length of the gnomon, we look at the triangle (compare the picture above) between the side of the Quadrant and the angle between our latitude and the maximum Sun altitude, which is:
61° - 52° = 9°
tan(9°) ≈ 0.15
to determine the required extra length for a resulting set of gnomons:
49 × 0.15 = 7.4 + 63 + 3.2 = 73.6 mm (AM gnomon) 79 × 0.15 = 11.9 + 102 + 3.2 = 117.1 mm (PM gnomon)
Note that the extra length and the thickness of the device (3.2 mm) is added to the respective values determined above.