Setup and operating instructions for Wndsn distance tools and related items.
Wndsn Distance Nomograph
A tool for makeshift navigation and rangefinding, the Wndsn Distance Nomograph enables you to do more than merely guessing distances. Find an object of known size at the distance you need to measure and let the baked-in trigonometry do the rest for you; all by aligning the provided string across the scales.
Inspired by both 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, the Wndsn XPD Distance Nomograph combines both techniques in an easy to use and handy distance measuring device.
How to set it up
To install, knot the string to the card through the provided hole. Measure a length of 57.3 cm (22.44 inches) from eye to card. You can make a knot at the end or a loop; for measuring distances, hold onto the string with your teeth.
Accuracy is determined by two elements, provided that the string length is respected:
- The reading of the scale and approximation of the corresponding mark
- The estimation of the height of the measured object
For training and reference purposes, you may want to create a table of the exact height of common objects.
Note that you can measure in any unit (cm, in, ft) or system (metric, imperial, custom), the factors are always the same and return your distance in the same unit you used to approximate the object measured.
Q: Why is the string length important, and what's special about the 57.3?
It says "baked-in trigonometry"; also, on the tool itself, there is a hint:
For d = 57.3 cm, arctan(s/d) in deg = s => 1° = 1 cm
The explanation is the definition of radian, or rad: The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3° (when the arc length is equal to the radius). The circumference subtends an angle of 2π radians.
Now, at a radius (hence the string length) of 57.3 cm, 1 cm in measured size subtends exactly 1° and makes for elegant calculations;
D = S * d / α where:
- D = distance from eye to observed object of known size
- S = known size of the object observed
- d = distance from tool to observer's eye
- α = the angular size of the object observed at the distance d
Q: How do I use the degree scale?
The degree scale measures the angular size of the object observed, in other words the measured height in degrees, (easily converted to multiplication factors at the distance of 57.3 cm) which can be used when the actual height of the object is unknown.
Example: the full moon viewed from Earth has a diameter of about 0.5° (30 arc minutes) which, knowing neither distance nor actual diameter, doesn't tell us anything about its actual size compared to other objects such as the sun, which, at a much bigger diameter and distance, has about the same angular size when observed from Earth. Now, knowing either distance or actual diameter, the angular size can easily converted to the respective other, absolute value.
Also note that:
α = (S * 57.3) / D
Relationships between angular size and multipliers (valid for d = 57.3 cm):
1° = x 57 2° = x57/2 = x 28.5 3° = x57/3 = x 19 4° = x57/4 = x 14.25 5° = x57/5 = x 11.4 6° = x57/6 = x 9.5
Q: Can I measure the width instead of the height of an object?
You can measure any dimension; width, height, etc., as long as it's on a plane that is perpendicular to you, the trigonometry doesn't care where in space the triangle is located.
Q: What if I need larger measurements, e.g. for celestial navigation?
For measurements above 7°, prepare the string with knots at 57.3/4 cm and 57.3/2 cm. Now, if you need to measure larger or closer objects, hold the instrument at the /2 knot and the engraved 7° becomes 14° (7 * 2); held at the /4 knot, the engraved 7° becomes 28° (7 * 4).
Q: What if I know the distance and want to determine the size of an object?
While the Distance Nomograph is designed to measure distances, a feature of nomograms is that any variable can be calculated from values of the other two, which means that if you happen to know the distance and need the size of the object, you can obtain that by aligning the string across left scale and center scale and read the object's size on the right-hand scale.