Galileo used the distance of uniform motion as a measure of the distime, i.e., time interval (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914):
Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time-interval bc finds itself at the point i. p.199
Without getting into the details of the Figure 108, notice the shift of language: “the body moves from b to c” [i.e., a distance], then “the time interval bc“. Galileo uses a distance to measure a time interval, which is justified since the motion is “with uniform speed” and so they are proportional.
The point to make here is that a distance and a distime can be interchanged if the motion is uniform. That is exactly the function of a clock: to provide a standard distime for a corresponding distance of motion. The change from distance to distime is basically a change of units. So, the line with a to e and beyond is a linear clock: it measures elapsed distime or “elapsed distance”.
Let there a ball be dropped out the window by a passenger on a train in uniform motion. Consider the following four scenarios, in which the distance or distime of a uniform motion is measured: (1) looking down above the moving ball, measuring the distance of fall; (2) looking down above the moving ball, measuring the (uniform) distime of fall; (3) looking from the side, measuring the distance of motion in two dimensions; and (4) looking from the side, measuring the (uniform) distime of motion in two dimensions.
Note that the distance traversed is not the same as the distance elapsed because they are in different dimensions. The distance elapsed is from an independent uniform motion. The distance traversed is from the dependent free fall motion. But this motion can be measured by distime as well as distance: the “distime traversed” is the measure of free fall motion by a uniform motion in the vertical dimension. It is proportional to the distance traversed and related quadratically with thw distance elapsed by a uniform motion.
In short, “elapsed” indicates a measure of independent uniform motion. “Traversed” indicates a measure of a dependent motion.
One could even have 2D space and 2D time, as in the figure above. Imagine a planar Earth with the Sun moving over in semi-circular fashion. In this context, space has two horizontal dimensions, and time has vertical and parametric dimensions. One could diagram this as a horizontal circle intersected by a vertical circle. Equations for spatial and temporal circular motion would be used. This 2 + 2 form of space and time is like an isochron map (see here) or a flat (2D) earth with celestial (2D) time.
But don’t space and time have different characters? Certainly, relativity recognizes this by using opposite signs for spatial and temporal coordinates in the invariant equation. The difference between space and time is also between the independent and dependent quantities.
Then switching from spatial to temporal coordinates inverts the independent and dependent coordinates. A temporal coordinate as the independent variable is changed into a spatial coordinate as the independent variable, or vice versa.
Revised 2020-09-04.