The independent quantity in a *proper rate* is the denominator. The independent quantity in an *improper rate* is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is *harmonic addition*:

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the *harmonic mean*:

*Time speed* is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. *Space speed* is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the space mean speed. The time mean speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added harmonically. Legerity normalized by an hypothesized maximum pace is proper, but if the legerity is divided by the pace of light, it is improper.