This continues posts here and here.
The extent of a motion is measured by a reference motion, just as a length is measured by a reference length. The reference motion used to measure other motions is a uniform motion. Galileo’s definition of uniform motion is the following:
By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal. [Galileo’s Two New Sciences, Third Day]
Because equality is a symmetric relation, this could also be expressed as follows (cf. Archimedes):
A steady or uniform motion is one in which the travel times of the moving particle during any equal intervals of space, are themselves equal.
Another way of saying this is that for a uniform motion the intervals of space and the corresponding intervals of time are proportional. That is, a uniform rate of motion is constant.
There are two measures of the extent of a motion, length and duration. Applied to the reference motion, these two measures produce two scalars, a measure of length called stance, and a measure of duration called time. Given a reference starting placepoint, the length since the start point is called the stance, and the duration since the start timepoint is called the time. Because of the proportionality of uniform motion, given knowledge of stance or time along with the uniform rate, one may deduce the other measure.
The reference motion must also be an independent motion, not dependent on other motions, and it must continue indefinitely, so that any other motion would be at some point simultaneous or simulstanceous with the reference motion. Because of this, any motion may be a function of the reference motion.
This independent, uniform reference motion is commonly represented by a clock, which registers uniform motion continually. Even if the clock’s motion is a uniform circular motion, it represents a uniform linear motion as the numbers increase linearly. The reference motion may equally well be an odologe, which is a uniform motion that registers length continually. As a clock may be a uniform angular motion whose angles register durations, an odologe may be a uniform angular motion whose arcs register lengths.
Measures of the target motion may then be considered as a function of one of the reference measures, which acts as a parameter of the target motion. Parametric differential equations and geometry may then be used to represent the course of a motion.