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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.

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 Greek νόμος nomos; law and γραμμή grammē; line, also called a nomograph, 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 n scales, 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. (Wikipedia)


Wndsn Telemeters are unitless, which means that you can input your data in the unit you prefer, and get the output in the same unit.

How to switch between units

Using arbitrary units with the Telemeter.