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Interchangeability of space and time

The extent of a motion is measured in two ways: by its time (duration) and by its space (length). The relation between these two measures is the subject here.

Although a definition of uniform motion was given by Archimedes, Galileo was the first to give a complete definition:

Equal or uniform motion I understand to be that of which the parts run through by the moveable in any equal times whatever are equal to one another. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 148)

Archimedes first stated a proportion for uniform motion:

If a point move at a uniform rate along any line, and two lengths be taken on it, they will be proportional to the times of describing them. (Archimedes, The Works of Archimedes, ed. by T. L. Heath, Dover, p.155.)

In other words, given a uniform motion and on it any two lengths, L and M, then the lengths and the corresponding times of motion, T and U, will satisfy the proportion L : M :: T : U.

Galileo gave the converse proportion:

If a moveable equably carried [latum] with the same speed passes through two spaces, the times of motion will be to one another as the spaces passed through. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 149)

In other words, given a uniform motion and on it any two spaces, S1 and S2, then the times of motion, T1 and T2, for the corresponding lengths will satisfy the proportion T1 : T2 :: S1 : S2.

The proportion between times and spaces shows their interchangeability for uniform motion. Since uniform motion is the standard of comparison of all other motions, this indicates the interchangeability of times and spaces for all motions. The application of this to constantly accelerated motion was described by Galileo:

If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is are as the squares of those times. (Galileo, Two New Sciences, “On Naturally Accelerated Motion,” tr. by Stillman Drake, p.166).

This is illustrated by a figure on page 221:

Falling projectile

Figure 1

The projectile moves uniformly from the left on the horizontal line e-d-c-b-a. At b it is allowed to descend, and it follows the curve b-i-f-h, which Galileo shows is a semi-parabola.

Galileo describes a body moving horizontally at uniform speed, then descending, starting at b. In his explanation he states:

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.

Notice the shift of language: “the body moves from b to c” [i.e., a length-interval], then “the time-interval bc”. Galileo uses a length interval to measure a time-interval, which is justified since they are proportional for uniform motion.

Galileo takes the horizontal component of the motion along line abcde to represent space or time. The horizontal component of motion can represent a kind of linear clock. But the diagram also shows Galileo could just as well have let the horizontal component represent length as a kind of distance-clock.

Galileo takes the vertical component as the length of the body’s semi-parabolic descent, represented by horizontal lines with increasing gaps. By the same reasoning the y axis could just as well represent time, not elapsed time of descent but a measure with equal time intervals.

This would be as if by a uniform descending motion that could be expressed as space or time. This is a general principle that can be stated as follows:

Length and duration variables can be interchanged up to a factor since they are proportional measures of uniform motion.

Components can be interchanged separately since they are independent of one another.

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