Wndsn Quadrant Telemeters

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. See also the printed manual.

Implementation and Use of the Azimuth Lines on the Quadrant

Level: Advanced

The use of azimuth lines on the Quadrant is not new. Azimuth lines can be found on Quadrants from the 17th century. Examples are Quadrants such as the Prophatius or the Ottoman Quadrant, used in the Ottoman Empire up to the beginning of the 20th century, as well as Gunter’s Quadrant.^[1]

What are azimuth lines used for?

With the help of the azimuth lines, the azimuth of a celestial body (such as the Sun) is determined by measuring its altitude above the horizon, thereby obtaining the cardinal direction for navigation. Azimuth lines are latitude dependent. A Quadrant with azimuth lines can thus be used as a main compass if a magnetic compass is not available, or as an auxiliary or backup compass to check the operation of the main magnetic compass. A true azimuth is determined, which is independent of the local magnetic declination and deviation, which can lead to magnetic variation with a magnetic compass.

Note the definition of the azimuth for astronomical observations:

“The azimuth of a star is the angle on the horizon plane between the meridian plane and the vertical plane of the star. The azimuth is counted in the traditional sense starting from south to west (south azimuth), so that a star in the south has an azimuth of 0° and a star in the west has an azimuth of 90°, or from north to east (north azimuth). This is the original astronomical counting method."^[2]

How are the azimuth lines calculated and how are they drawn on a Quadrant?

Compared to the horary lines, which are used as a function of the altitude of a celestial body above the horizon in the equatorial coordinate system, the azimuth lines are created as a function of the altitude of a celestial body above the horizon in the azimuthal (or horizontal) coordinate system.

The azimuth lines are to be treated in the same way as the horary lines; a celestial body with its declination (this remains constant) after rising gains altitude above the horizon until it culminates and thus wanders from one horary line to the other, as well as from one azimuth line to the other. After the culmination, the celestial body loses altitude until it sets and wanders back through the lines. The horary lines represent the horary meridians in the equatorial coordinate system and the azimuth lines represent the meridians in the horizontal coordinate system.

The relationship between the equatorial and azimuthal coordinate system is shown in Figure 3.

Where:

• α - azimuth of a celestial body in the azimuthal coordinate system.
• h - altitude of a celestial body above the horizon in the azimuthal coordinate system.
• τ - hour angle of a celestial body in the equatorial coordinate system.
• δ - declination of a celestial body from the celestial equator measured in the equatorial coordinate system.
• φ - latitude of the observation site.

The azimuth of a celestial body can be calculated with equation 36 from GQT5^[3] "Computed azimuth of celestial body":

cos(α) = (sin(δ) – sin(φ) * sin(h)) / (cos(φ) * cos(h))

In order to be able to draw the lines, the altitude of the Sun in relation to azimuth, declination and latitude is required. It is not possible to calculate the position of the Sun directly. Edmond Gunter has cleverly solved the problem and described it in detail in his work "The Description and Use of the Sector, Cross-Staff and Other Instruments"^[4].

First the altitude of the Sun is calculated for a given azimuth, for a declination of zero; i.e. the Sun is on the equator:

tan(h0) = cos(αs) / tan(φ)

then an auxiliary angle x is calculated:

sin(x) = (cos(h0) * sin(δ)) / sin(φ)

This yields the altitude of the Sun (h) for the azimuth and declination:

• If: αs < 90°, h = x + h0
• If: αs > 90°, h = x - h0

Since these arcs are not circular, a significant number of calculations is required. This can be done with a spreadsheet program or a simple computer program. For each individual arc, all values for declinations in the range -ε < δ < +ε for the Sun, or in the range equatorial altitude above the horizon < δ < +90° for the stars must be calculated.

Each calculated point of the arc, if plotted manually, is plotted on the Quadrant using the polar coordinates (h, δ); i.e. we set the cursor to the declination (δ) and rotate the string to the altitude (h) to apply the point.

The procedure (Figure 4):

1. Set the equator, φ starting from 90°.
2. Place the string on δ on the degree arc, measured from the equator.
3. Place the cursor on the sine arc.
4. Set the value of h as calculated above on the degree arc.
5. The position of the cursor is a point on the azimuth line.

The values with negative altitude are not plotted because the corresponding positions are located below the horizon (Figure 5, upper edge).

How are the azimuth lines used?

The determination of the azimuth of a celestial body is similar to the determination of the time with the help of the horary lines and is just as easy. To determine the azimuth of the Sun, the string is placed over the calendar on a given date and the cursor is moved to the point where the string crosses the 12-hour line.

1. To determine the azimuth of a star, the culmination altitude of the star must first be determined using the formula: hs = 90° - φ + δ
2. Then the string is set to the angle h_s on the angle scale and the cursor is moved to the point where the string crosses the 12-hour line.
3. Now the altitude of the celestial body is measured with the help of the Quadrant and the azimuth is read off from the azimuth lines on the cursor.
4. After the azimuth of the celestial body has been determined, the cardinal direction can be obtained with the method "Determining north and south via solar azimuth" from GQT5. It must be taken into account whether the measurement is made before or after noon, i.e. whether the celestial body is located before or after the meridian passage.

How are these azimuth lines to be interpreted? Figure 5 shows a latitude-specific Wndsn Quadrant with superimposed azimuth lines.

Celestial bodies with negative declination are located below the celestial equator or south of it and can be found in the upper part of the Quadrant above the line of the equinoxes. These celestial bodies rise south of east and set south of west. This can be read from the azimuth lines in this area of the Quadrant and on its right edge (horizon). These stars cross less azimuth lines with their orbit until the stars with the culmination altitude (h_s) closer to zero and with the declination δ = -(90° - φ) only briefly rise in the area of the local meridian and immediately set again.

Celestial bodies with positive declination are located above the celestial equator or north of it and can be found in the lower part of the Quadrant below the line of the equinoxes. These celestial bodies rise north of east and set north of west. This can be read from the azimuth lines in this area of the Quadrant and on its right edge (horizon). These stars cross more azimuth lines with their orbit until the stars with the culmination altitude (h_s) closer to 90° and with the declination δ > +(90° - φ) pass through the local zenith (90° mark on the degree scale of the Quadrant)

Using the azimuth lines is just as easy as using the horary lines and expands the possibilities of the Quadrant.

(Thanks to Telemeter user Andrej for this tutorial!)

Resources:

• James E. Morrison “The Astrolabe"

Footnotes:

1. "Gunter’s Quadrant" was described by Edmond Gunter (1581-1626) in "The Description and Use of the Sector, Cross-Staff and other Instruments". This Quadrant is easier to use to find the time than the Quadrans Novus, published in 1623, also called Sutton’s Quadrant. ↩
2. Azimuth ↩
3. Filiusventi, "GRIMOIRE QUADRANTIS TELEMETRUM", 5. Edition, 2021 ↩
4. Edmond Gunter, "The Description and Use of the Sector, Cross-Staff and other Instruments", 1619 ↩