Let us compare the motions of two bodies. Let the motion of one body be the *reference* motion. Let the motion of the other body be the *comparative* motion. Let the two bodies begin together at one place.

Definitions:

A *place* is the general term for an answer to *Where?* A *point-place*, or simply a *point*, is the smallest place. A *translation* is a vector from one point-place to another. Travel distance is the arc length of the trajectory of a motion, which includes any retracing of the trajectory.

Space and time refer to different perspectives of the universe of motion.

*Space* is the locus of all potential places for the *comparative* motion, which is said to be “in space.” *Displacement* is a translation vector from one point to another point of the *comparative* motion. The travel distance from the beginning point to the ending point of the *comparative* motion, is the *travel length* for a motion in space.

*Time* is the locus of all potential places for the *reference* motion, which is said to be “in time.” *Distimement* is a translation vector from one point to another point of the *reference* motion. The travel distance from the beginning point to the ending point of the *reference* motion, is the *travel time* for a motion in time.

There are two ways of proceeding:

(1) As the *reference* motion ends (physically or by marking), then stop or mark the end of the *comparative* motion. Measure each travel distance from beginning to end. This is the travel time.

(2) Or as the *comparative* motion ends (physically or by marking), then stop or mark the end of the *reference* motion. Measure each travel distance from beginning to end. This is the travel length.

*Note*: The *reference* motion could be a clock, in which case the travel distance would be the extent of angular motion.

In a ratio the independent variable is in the denominator and the dependent variable is in the numerator. That means there are two ways of forming a ratio of the *comparative* motion and the *reference* motion:

(A) The independent variable is the travel time of the *reference* motion. The dependent variable is the travel length or the displacement of the corresponding *comparative* motion. The *mean speed* is the travel time of the *reference* motion divided into the travel length of the corresponding *comparative* motion. The *mean velocity* is the travel time of the *reference* motion divided into the displacement of the corresponding *comparative* motion.

(B) The independent variable is the travel length of the *comparative* motion. The dependent variable is the travel time or the distimement of the corresponding *reference* motion. The *mean pace* is the travel length of the *comparative* motion divided into the travel time of the corresponding *reference* motion. The *mean legerity* is the travel length of the *comparative* motion divided into the distimement of the corresponding *reference* motion.

An example of (A) is setting the course of the *reference* motion to go a certain time, as in a time period, then measuring the travel length of the *comparative* motion. An example of (B) is setting the course of the *comparative* motion to go a certain length, as in a race, then measuring the travel time of the *reference* motion.

Space and time are completely exchangeable because they are the same concept (i.e., distances or vectors) except that one arises from the *comparative* motion (space) and the other arises from the *reference* motion (time).