A milliradian (SI-symbol mrad, sometimes also abbreviated mil or mils) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel (up, down, left, or right). Milliradians are also used for comparing shot groupings, or to compare the difficulty of hitting different sized shooting targets at different distances. When using a scope with both mrad adjustment and a reticle with mrad markings (called an "mrad/mrad scope"), the shooter can use the reticle as a ruler to count the number of mrads a shot was off-target, which directly translates to the sight adjustment needed to hit the target with a follow-up shot. Optics with mrad markings in the reticle can also be used to make a range estimation of a known size target, or vice versa, to determine a target size if the distance is known, a practice called "milling". Milliradians are generally used for very small angles, which allows for very accurate mathematical approximations to more easily calculate with direct proportions, back and forth between the angular separation observed in an optic, linear subtension on target, and range. In such applications it is useful to use a unit for target size that is a thousandth of the unit for range, for instance by using the metric units millimeters for target size and meters for range. This coincides with the definition of the milliradian where the arc length is defined as 1/1000 of the radius. A common adjustment value in firearm sights is 1 cm at 100 meters which equals 10 mm/100 m = 1/10 mrad. The true definition of a milliradian is based on a unit circle with a radius of one and an arc divided into 1,000 mrad per radian, hence 2,000 π or approximately 6,283.185 milliradians in one turn, and rifle scope adjustments and reticles are calibrated to this definition. There are also other definitions used for land mapping and artillery which are rounded to more easily be divided into smaller parts for use with compasses, which are then often referred to as "mils", "lines", or similar. For instance there are artillery sights and compasses with 6,400 NATO mils, 6,000 Warsaw Pact mils or 6,300 Swedish "streck" per turn instead of 360° or 2π radians, achieving higher resolution than a 360° compass while also being easier to divide into parts than if true milliradians were used.
History
The milliradian (approximately 6,283.185 in a circle) was first used in the mid-19th century by Charles-Marc Dapples (1837–1920), a Swiss engineer and professor at the University of Lausanne. Degrees and minutes were the usual units of angular measurement but others were being proposed, with "grads" (400 gradians in a circle) under various names having considerable popularity in much of northern Europe. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles (60° per triangle, 6 triangles in a circle) and hence 600 units to a circle. Around the time of the start of World War I, France was experimenting with the use of millièmes or angular mils (6400 in a circle) for use with artillery sights instead of decigrades (4000 in a circle). The United Kingdom was also trialing them to replace degrees and minutes. They were adopted by France although decigrades also remained in use throughout World War I. Other nations also used decigrades. The United States, which copied many French artillery practices, adopted angular mils, later known as NATO mils. Before 2007 the Swedish defence forces used "streck" (6300 in a circle, streck meaning lines or marks) (together with degrees for some navigation) which is closer to the milliradian but then changed to NATO mils. After the Bolshevik Revolution and the adoption of the metric system of measurement (e.g. artillery replaced "units of base" with metres) the Red Army expanded the 600 unit circle into a 6000 mil circle. Hence the Russian mil has a somewhat different origin than those derived from French artillery practices. In the 1950s, NATO adopted metric units of measurement for land and general use. NATO mils, metres, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices.
Mathematical principle
Use of the milliradian is practical because it is concerned with small angles, and when using radians the small angle approximation shows that the angle approximates to the sine of the angle, that is sin θ ≃ θ {\displaystyle \sin \theta \simeq \theta } . This allows a user to dispense with trigonometry and use simple ratios to determine size and distance with high accuracy for rifle and short distance artillery calculations by using the handy property of subtension: One mrad approximately subtends one metre at a distance of one thousand metres. More in detail, because subtension ≃ arc length {\displaystyle {\text{subtension}}\simeq {\text{arc length}}} , instead of finding the angular distance denoted by θ (Greek letter theta) by using the tangent function
θ trig = arctan subtension range {\displaystyle \theta _{\text{trig}}=\arctan {\frac {\text{subtension}}{\text{range}}}} , one can instead make a good approximation by using the definition of a radian and the simplified formula:
θ rad ≃ subtension range {\displaystyle \theta _{\text{rad}}\simeq {\frac {\text{subtension}}{\text{range}}}}
Since a radian is mathematically defined as the angle formed when the length of a circular arc equals the radius of the circle, a milliradian, is the angle formed when the length of a circular arc equals 1/1000 of the radius of the circle. Just like the radian, the milliradian is dimensionless, but unlike the radian where the same unit must be used for radius and arc length, the milliradian needs to have a ratio between the units where the subtension is a thousandth of the radius when using the simplified formula.
… excerpt ends here. Continue reading the full article.
