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. The object under investigation (such as the top of a tower) is sighted alongside the edge of the instrument. The needed 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.

The first to refer to the term shadow and publish conversion tables (tangent) 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 is the basic element of the so-called geometrical square. It 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 an object, generally a gnomon, and its shadow. If the sun's ray 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 ray is at 45° its shadow falls exactly on the umbra media. To each value of the umbra recta corresponds a value of the umbra versa.

By holding the instrument on its side, we can use the same techniques 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

• From the umbra recta scale, we calculate:
distance = height × (12 / reading)
• From the umbra versa scale, we calculate:
height = distance × (reading / 12)

Note that 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

1. If your shadow is 4 feet long in your own feet, then what is the altitude of the sun? This problem can be solved through the use of the shadow square. 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. By moving the string to the 8 (8/12 = 4/6 = 4 ft shadow / 6 ft tall) and then reading the altitude scale we can see the sun is at an altitude of 56.3°.

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

How to use the shadow square to determine height or distance with the respective other value known. String set to 6° and 17°.

1. Let the known distance be 457 m and the angular size 6°. The equivalent of 6°, read on the shadow square equals approx. 1/10; and since we are measuring the height of the object, the formula is:
height = known distance × reading / 10
or 457 m × 1/10 = 46 m
The Telemeter nomograph check returns: approx. 50 m.
2. Let the known height be 46 m and the angular size 17°. The equivalent of 17°, read on the shadow square equals approx. 3/10; and since we are measuring the distance to the object, the formula is:
distance = known height × 10 / reading
or 46 m × 10/3 = 153 m
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 along the vertical scale, umbra versa, by holding the quadrant upright and aligning the string from vertex to eye in such a way that the line of sight points through the string 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.

• Lighthouse height = 30 m
• Umbra recta = 12
• Umbra versa = 7

Distance = 12/7 × 30 = 51.3 m