History of angles

History

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles.

Measuring angles

The angle θ is the quotient of s and r.

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

Units

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above.

With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to n units, for some whole number n (for example, in the case of degrees, n = 360). This is equivalent to setting k = n/2π in the formula above. (To see why, note that one full circle corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting, we get θ = ks/r = 2πk. But if one complete circle is to have a numerical angular value of n, then we need θ = n. This is achieved by setting k = n/2π.)

• The degree, denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. (The problem of having all "interesting" angles measured as whole numbers is of course insolvable.) Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics:
  • The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g., 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
  • The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

θ = s/r rad = 1 rad.

• The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system.

• The mil is approximately equal to a milliradian. There are several definitions.

• The full circle (or revolution, rotation, full turn or cycle) is one complete revolution. The revolution and rotation are abbreviated rev and rot, respectively, but just r in rpm (revolutions per minute). 1 full circle = 360° = 2π rad = 400 gon = 4 right angles.

• The right angle is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right angle = 90° = π/2 rad = 100 gon.

• The angle of the equilateral triangle is 1/6 of a full circle. It was the unit used by the Babylonians, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. One Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.

• The grad, also called grade, gradian, or gon is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.

• The point, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass." 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.

• The astronomical hour angle is 1/24 of a full circle. The sexagesimal subunits were called minute of time and second of time (even though they are units of angle). 1 hour = 15° = π/12 rad = 1/6 right angle ≈ 16.667 gon.

• The binary degree, also known as the binary radian (or brad), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single byte.

• The grade of a slope, or gradient, is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.