## The Science behind Wndsn Telemeters

Kamal, nomography, and slide rules.

Inspired by the medieval Kamal, a celestial navigation tool that greatly facilitated latitude sailing, as well as nomography, an almost lost art and science invented in the late 19th century to provide engineers with fast graphical calculations of complicated formulas, and adding an important slide rule principle, the Wndsn Telemeter combines all three techniques in an easy to use and handy distance measuring device.

## 1. Kamal

The Wndsn Telemeter takes the principle of the Kamal and reverses it, adding a precision scale to the device itself and measuring distances in space like one would with a ruler on a map.

The string length ensures the correct distance from the eye of the observer and can be further manipulated to measure larger distances or objects. The values measured are the angular size and can thus be converted to the distance to the objects, knowing the actual dimensions.

The Kamal originated with Arab navigators of the late 9th century, and was employed in the Indian Ocean from the 10th century. It was adopted by Indian navigators soon after, and then adopted by Chinese navigators some time before the 16th century. (Wikipedia)

The Kamal consists of a rectangular piece of wood or horn ranging from about 1 by 2 inches (2.5 by 5.1 cm) up to 2 by 3 inches (5.1 by 7.6 cm), to which a string with several knots, either equally or spaced in increasing increments, is attached through a hole in the middle of the card. The Kamal is used by placing one end of the string in the teeth while the other end is held away from the body roughly parallel to the ground. The device is then moved along the string, positioned so the lower edge is even with the horizon, and the upper edge is occluding a target star, usually Polaris because its angle to the horizon does not change with longitude or time. The angle can then be measured by counting the number of knots from the teeth to the device, or a particular knot can be tied into the string if traveling to a known latitude. Note that with this construction, the smaller the device, the higher the range and the lower the resolution.

Kamal reconstruction. Includes the string with harmonically spaced knots.

Principle of Kamal graduation with derivation of knot spacing.

The knots were typically tied to measure angles of one finger-width. When held at arm's length, the width of a finger measures an angle that remains fairly similar from person to person. This was widely used (and still is today) for rough angle measurements, an angle known as issabah in Arabic. By modern measure, this is about 8/5 degrees; or 1 degree, 36 minutes, and 25 seconds, or just over 1.6 degrees. It is equal to the arcsine of the ratio of the width of the finger to the length of the arm. With a fixed-width device, to measure equal increments (of one issabah in this case), the knots have to be spaced out across the string in a non-linear way, that means that the graduation interval is increasing in one direction for a tangential division.