## 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.

## Direction of Sunrise

### Sine Quadrant

One of the many uses of a Sine Quadrant is to find the direction of sunrise and sunset for any place and date. You need to know the latitude of that place and also the solar declination for that day. This information can be found on the Quadrant, see Using the Obliquity Arc and Determining Local Latitude.

This picture shows the calculation for the direction of sunset in Baghdad, latitude 33°, on the winter solstice, December 22nd. Binder clips are useful for holding the strings in place. The horizontal string is set to the sine of -23.44°, which is the solar declination for the winter solstice. The obliquity arc in the corner conveniently shows where to position the string along the sine scale on the edge of the instrument, and the other end of the string is set to the 23.44° mark. The diagonal string has the cursor knot set at the cosine of 33°, the latitude of Baghdad. (For more on cosines, see the tutorial: Computing Sine (and Cosine)). The cursor knot is touching the horizontal string, while the diagonal string crosses the degree scale at ~28°. That means the Sun sets ~28° south of west or 270°-28° = 242°. This can also be calculated on Tycho.

Sunrise is the same, except we add our 28° to 90° east^{[1]} to get sunrise at 118°. The summer solstice is similar, except the declination is +23.44° instead of -23.44°. We add our result of 28° to 270° to get 298° for sunset and 90°-28° = 62° for sunrise on the summer solstice.

### Latitude Quadrant

A Latitude Quadrant, custom made for your latitude, has two features that make the calculation easier. These are the equinox hour line and the calendarium. This picture shows a latitude Quadrant for Berlin, latitude 52°30'.

One of the curves on the Quadrant is the equinox hour line. It is labeled 6 o'clock at one end and 12 o'clock at the other. It shows that on the equinoxes, March 21st and September 24th, the Sun rises at 6 o'clock, goes up to 37°30' of elevation over the horizon at noon, and sets at 6 in the evening. Solar declination for the equinoxes is 0°.

The equinox line is helpful for finding the direction of sunrise and sunset for any day, because it equals the cosine of the particular latitude the Quadrant is made for.

The calendarium runs along the edge of the instrument. Like the equinox hour line, it shows that the Sun rises 37°30' over the horizon at noon on the equinoxes, denoted by the Aries sign ♈. Remembering that declination is 0° at the equinoxes, we can use the calendarium for determining the declination of any date. For example, on November 4th, we see that the Sun will rise 22°30' over the horizon, or 15° lower than on the equinoxes. Thus the declination for November 4th is -15°.

Let's find the direction of sunset for November 4th, declination -15°. Like in our Baghdad example, we start by setting the string to sin(15°). We use binder clips to hold the string so it crosses both the 15° mark and the 15th mark from the vertex at the edge of the instrument.

Pull the remaining string through the slot in the vertex. Then move the string along the degree scale until the pivoting string, the equinox line and the horizontal string all intersect at the same point. Read where the string crosses the degree scale: 25°. Thus the Sun sets at 270°-25° = 245° and rises at 90°+25°=115°. You can also see this calculation on Tycho.

Our calculation can be expressed in the mathematical formula:

```
sin(direction) = sin(declination) · sec(latitude)
```

The formula can be adapted to the Quadrant like this:

```
sin(direction) = sin(declination) / cos(latitude)
```

(Tutorial written by Cute Puppy.)

**Footnotes:**

- The direction of sunrise is usually calculated from north. In the formula, the sunrise bearing, or 'ortive amplitude' is calculated from east and has to be adjusted accordingly. ↩