Homogeneity Applied to Angles

Extract

In all general equations in mechanics the fundamental units, length, time, mass, enter homogeneously; the result is that these equations are true whatever the units of length, time, and mass may be. Angles, however, occupy an anomalous position. An equation which involves angles, angular velocities or accelerations, is true only for a certain particular unit of angle, usually the radian. Thus the formula T = 2π/ω, for periodic time with angular velocity ω, is only true when the angular unit is the radian; so also for arc = rθ, or for . The reason is, that these equations are not homogeneous in angle; π, sin θ, and cos θ are pure numbers, arc and r are lengths. In explanation it is commonly said that angle is of zero dimensions in length, time, and mass. So it is, and yet it is not a pure number, but a quantity just as much as length or any other concrete magnitude. If π were the symbol for a certain angle, instead of being a pure number, equations like T — 27π/ω, α+β + γ=π, etc., would become homogeneous, and true for all angular units, π would then resemble the symbol g, which stands for a particular acceleration, 32.2 ft./sec.² or 981 cm./sec.², etc.; but π has too long been identified with the number 3.14159… , and is too useful in all sorts of questions not involving angles at all, to be appropriated in this way. If, however, we introduce another symbol, say P, for the angle π radians or the angle of half a revolution, we may make all our equations homogeneous in angles, and restore their generality; it will do equally well, and be generally simpler, if instead of P we use the symbol K, denoting an angle of one radian.