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The science of measurement, embracing both experimental and theoretical determinations at any level of uncertainty in any field of science and technology.

Simple Navigation via Distance Measuring

The distance from eye to measured object, and the height of the measured object, form two sides of a right triangle, and thus determine an angle:

D = S * d / s

where D is the distance to the object, S is the size of the object observed, d is the distance from the measurement device to the observer's eye, and s is the apparent size of the object observed.

Now, if you have a map, and know the dimensions of landmarks in your range of observation, you can calculate their distance and thus determine your position by way of triangulation.

Here is the Range Calculator (and the maths).


Since a radian is defined as the angle formed when the length of a circular arc equals the radius of the circle, a milliradian (mrad, sometimes called a mil), is the angle formed when the length of a circular arc equals 1/1000 of the radius of the circle. An object 5 meters high, for example, will cover 1 mrad at 5000 meters, or 5 mrad at 1000 meters, or 25 mrad at 200 meters. Since the radian expresses a ratio, it is independent of the units used; an object 6 feet high covering 1 mrad will be 6000 feet distant.

The relationship of the angular mil to the trigonometric radian gives rise to the handy property of subtension: One mil approximately subtends 1 m at a distance of 1000 m. (The small angle approximation for skinny triangles shows that the angle in radians approximates to the sine of the angle.)

Estimating mils with hands[1]

Approximate values for an outstretched arm with:

             distance  angle  mil
spread hand  x   3.5   17°    300
       fist  x   6     10°    175
      thumb  x  30      2°     33
  1/2 thumb  x  60      1°     17.5
  1/4 thumb  x 120      0.5°    9

For objects of known size the range is the size divided by the angle:

D = S / mil with mil = s / d

Celestial Navigation

A Kamal is an ancient nautical instrument based on that principle

A Kamal is a celestial navigation device that determines latitude. It consists of a rectangular wooden card of about 5.1 by 2.5 cm, to which a string with several equally spaced knots 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 card 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, typically 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 card, or a particular knot can be tied into the string if travelling to a known latitude.

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 is used for rough angle measurements, an angle known as issabah in Arabic, or a chih in Chinese. By modern measure, this is about 1 degree, 36 minutes, and 25 seconds, or just over 1.5 degrees. It is equal to the arcsine of the ratio of the width of the finger to the length of the arm.[2]


See also:


  1. Estimating mils with hands 
  2. Kamal Navigation