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 shadow square is used for surveying tasks such as finding heights, depths, and distances. It can be used for calculations of known values as well as for sighting and thereby acquiring values. The object under investigation (such as the top of a tower) is sighted alongside the edge of the instrument. The unknown ratio of height or distance (the tangent of the angle) is read on one of the shadow scales.

The horizontal scales are called 'umbra recta', which is Latin for true, or straight shadow and describes the tangent. The vertical scales are named 'umbra versa', from Latin meaning turned, or upright shadow; the cotangent.

Note that cot(x) = 1/tan(x), so cotangent is the reciprocal of a tangent while arctan(x) is the angle whose tangent is x.

The first to refer to tangents as shadows and publish conversion tables was Al-Battani (858 - 929 CE), an Arabic astronomer and mathematician born in Harran (today's Turkey).

The shadow square, also called altitude scale, was generally placed on the back of astrolabes and quadrants and was used during the time of medieval astronomy to determine the height and track the movement of celestial bodies such as the sun, when more advanced measurement methods were not available. These methods can still be used today to determine the altitude, with reference to the horizon, of any visible celestial body.

To measure the linear height of an object using its shadow, the shadow square simulates the ratio between the object, generally a gnomon, and its shadow. If the sun's altitude is between 0° and 45°, the umbra versa (vertical axis) is used, between 45° and 90°, the umbra recta (horizontal axis) is used, and when the sun's altitude is at 45°, its shadow falls exactly on the umbra media. Each value of the umbra recta corresponds to a value of the umbra versa.

By holding the instrument on its side, the same techniques can be used to find the distance, for example across a river, using suitable objects on both banks.

The shadow square can be used to visualize Telemeter calculations and the corresponding triangles.

### Measurements

#### Distance formula

From the umbra recta scale as the base, we calculate:

``````distance = height × (12 / reading)
= height × cot(angle of ratio)
``````

#### Height formula

From the umbra versa scale as the base, we calculate:

``````height = distance × (reading / 12)
= distance × tan(angle of ratio)
``````

Note that whether using the 10- or 12-division scale depends entirely on which scale the values fall for a more precise reading.

### Example 1: Using actual shadows The height divided by the shadow length; 8/12 on the shadow square (umbra recta) results in the tangent value (on the slope scale) of 1.5 where arctan(1.5) ≈ 56°.

#### If you are 6 ft tall and your shadow is 4 ft long, what is the altitude of the sun?

The shadow square is divided in half, one half is calibrated by twelves, the other by tens. Because it is a shadow cast by the human body, the twelves are more convenient if we are using feet as a unit of measurement.

1. Using the technique of the scale jump, we multiply the 6 ft × 2 and do the same with the shadow; 4 ft × 2: 4 ft shadow / 6 ft tall = 4/6 = 8/12.
2. By moving the string to the 8, we can read on the degree scale that the sun is at an altitude of about 56°.

#### If your shadow is 18 feet long, what is the altitude of the sun?

Again, using the twelves side of the shadow square (because we are using a human body as measurement).

1. Using scale jumps the other way around this time: The longest shadow marked on a shadow square is twelve feet, this creates a problem any time the shadow is longer than the gnomon (you) that casts it.
2. The calculation is to figure out how tall a gnomon would be if it cast a 12 ft shadow in the same situation: in order to cast a twelve-foot shadow, the gnomon would be four feet tall: 12 ft shadow / 4 ft tall = 4/12 = 6/18.
3. If the shadow is longer than the gnomon, first rotate the instrument then set the string at 4, the height of the projected gnomon, then read off the altitude from the altitude scale.
4. It should read that the sun is at 19° above the horizon.

#### Getting the sun altitude with the help of a LEGO brick 10/12 on the shadow square (umbra versa; note the "overflow") for a tangent value (on the slope scale) of approx. 0.8 where arctan(0.8) = 39°.

Instead of using and measuring our own shadow with a tape measure, but we can go small and measure a shadow with the Shadow Square, on the Shadow square itself.

1. Take a LEGO brick (or ANY small object) and align it with the zero mark of the sexagesimal scale of the quadrant.
2. Rotate the instrument in such a way that it points towards the sun and read the length of the shadow on the scale.
3. In our example, we have a shadow length of 12 and measure the brick width as 10 (note that this is unitless, we are just after the ratio, the absolute dimensions don't matter).
4. We have a shadow ratio of 10/12, which means the shadow of a stick with a height of 1 unit has a length of 1.2 units.
5. Use the shadow square to project the string from 10/12 on the umbra versa (note how it translates to 0.8 on the tangent scale) and read the arctan on the angle scale (alternatively, the cotangent of 12/10 = 1.2 and we subtract the result from 90°).
6. ~39° is our sun altitude.

This goes back to Al-Battani in his Astronomical Tables:

Al-Battani gave a rule for finding the elevation θ of the Sun above the horizon in terms of the length s of the shadow cast by a vertical gnomon of height h. Al-Battani’s rule, s = h sin (90° − θ)/sin θ, is equivalent to the formula s = h cot θ. Based on this rule he constructed a "table of shadows", essentially a table of cotangents, for each degree from 1° to 90°.

### Example 2: Determine height or distance with the respective other value known

#### Let the known distance be 457 m and the angular size 6°.

1. The equivalent of 6°, read on the shadow square equals approx. 1/10; and since we are measuring the height of the object:
2. 457 m × 1/10 = 46 m
3. The Telemeter nomograph check returns: approx. 50 m.

#### Let the known height be 46 m and the angular size 17°.

1. The equivalent of 17°, read on the shadow square equals approx. 3/10; and since we are measuring the distance to the object:
2. 46 m × 10/3 = 153 m
3. The Telemeter nomograph check returns: approx. 147 m.

### Example 3: Sighting with the shadow square

We can use the shadow square to directly measure distances to or height of objects, given one dimension is known.

To determine the distance / height ratio, we use the shadow square to sight parallel to the vertical scale, umbra versa, alongside the integrated ridges on the edge of the instrument in such a way that the line of sight points to the top of the object to be measured. The umbra recta is divided by the value on the scale that is crossed by the string on the umbra versa and multiplied by the height of the object to get the distance.

#### How far is the lighthouse? Measuring the distance to a lighthouse (height known) from a boat.

Let the lighthouse height = 30 m

1. We sight with the quadrant and get umbra versa = 7
2. Umbra recta = 12
3. Hence, the distance = 12/7 × 30 = 51.3 m