598 VIII. ON THE STRUCTURE OF MATHEMATICS 









Fig. 14 There are two units of measurement of angles. In ordinary, or sexagesimal, measure, the unit angle is the degree, 1/360 of the entire angle about a point, 1/180 of a straight angle, or 1/90 of a right angle. The degree is divided into 60 equal parts called minutes. The minute is divided into 60 parts called seconds. In circular measure the unit angle is the radian, the angle at the centre of a circle whose arc is equal to the radius of the circle. This angle is a constant whether the circle be large or small, due to the fact that the circumferences of circles vary as their radii, and, in one circle, angles at the centre are proportional to their arcs. The constant ratio of the circumference of the circle to its radius is given by the number=3.14159 . . . , this number being 'incommensurable' with unity. As the length of the circumference of a circle with radius R, is 2R we see that the entire angle about the centre, which in degrees is 360, is in radians 2; that a straight angle equals 180 degrees or
radians; and that a right angle equals 90 degrees orradians.
... which, as it depends on the value
of is itself an 'irrational' number. The 'incommensurability' of the radian with right and straight angles makes its practical use inconvenient. One of the main uses of the radian is in theory as it introduces a marked simplification in that the ratio of the sine of an indefinitely small angle to the angle itself is 1, when the angle is measured in radians. In other words, the equivalence of an indefinitely small arc and chord becomes apparent numerically when the angle and sine are expressed in one unit. 
