## Wndsn Quadrant Telemeter: The Telemeter Side

The Telemeter side hosts a set of scales to measure angular size as well as a nomogram to determine distance or altitude from the measured values.

## 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. (cf. GQT)

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.

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.

### 2. Nomography

The trigonometry embedded in Wndsn Telemeters provides the scale with which we measure our target object's angular size (or diameter) at a given distance, and re-uses that same scale to determine the object's absolute size together with the graphical computer completed by the other two scales on the instrument.

A

nomogram,derived from Greekνόμος nomos;law andγραμμή grammē;line, also called anomograph,alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d'Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.A nomogram consists of a set of

nscales, one for each variable in an equation. Knowing the values of $n-1$ variables, the value of the unknown variable can be found, or by fixing the values of some variables, the relationship between the unfixed ones can be studied. The result is obtained by laying a straightedge across the known values on the scales and reading the unknown value from where it crosses the scale for that variable. The virtual or drawn line created by the straightedge is called an index line or isopleth.Results from a nomogram are obtained very quickly and reliably by simply drawing one or more lines.

**The Wndsn Telemeters use the Kamal string after measuring the angular size as a means to create the index lines across the scales and read the result of the equation.**

The user does not have to know how to solve algebraic equations, look up data in tables, use a slide rule, or substitute numbers into equations to obtain results. The user does not even need to know the underlying equation the nomogram represents. In addition, nomograms naturally incorporate implicit or explicit domain knowledge into their design. For example, to create larger nomograms for greater accuracy the nomographer usually includes only scale ranges that are reasonable and of interest to the problem.

Like a slide rule, a nomogram is a graphical analog computation device, and like the slide rule, its accuracy is limited by the precision with which physical markings can be drawn, reproduced, viewed, and aligned. While the slide rule is intended to be a general-purpose device, a nomogram is designed to perform a specific calculation, with tables of values effectively built into the construction of the scales. (cf. GQT)

### 3. Slide Rules

**In addition to the principle of the Kamal and those of nomography, the Wndsn Telemeters borrow the treatment of the decimal point from the concept of slide rule calculation.**

The slide rule was invented around 1620-1630, shortly after John Napier's publication of the concept of the logarithm. In 1620, Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide. In c. 1622, William Oughtred of Cambridge combined two handheld Gunter rules to make a device that is recognizably the modern slide rule.

In slide rule calculations; the user determines the location of the decimal point in the result, based on mental estimation. A slide rule requires the user to separately compute the order of magnitude of the answer in order to position the decimal point in the results. Scientific notation is used to track the decimal point in more formal calculations. (Wikipedia)

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. Hence, a result of e.g. 5 on the nomograph may translate to 5, or 50, or 500 units measured (5×10^{0}, 5×10^{1}, 5×10^{2}), making the device useful to calculate arbitrary distances.