Time as Motion and Motion as Time.
Gordon Fisher firstname.lastname@example.org
Aristotle says in the Physics: "Not only do we measure the movement by the time, but also the time by the movement, because they define each other. The time marks the movement, since it is its number, and the movement the time. We describe the time as much or little, measuring it by the movement, just as we know the number by what is numbered, e.g. the number of the horses by one horse as the unit. For we know how many horses there are by the use of the number; and again by using the one horse as unit we know the number of the horses itself. So it is with the time and the movement; for we measure the movement by the time and vice versa. It is natural that this should happen; for the movement goes with the distance and the time with the movement, because they are quanta and continuous and divisible. The movement has these attributes because the distance is of this nature, and the time has them because of the movement. And we measure both the distance by the movement and the movement by the distance; for we say that the road is long, if the [movement of the] journey is long, and that this [the movement of the journey] is long, if the road is long Ė the time [is long], too, if the movement [of the journey is long], and the movement [of the journey is long], if the time [is long]." (translated by R. P. Hardie and R. K. Gaye).
In talking about time measured (or "numbered") by motion, I will assume that changes of position of visible objects, i.e. distances between them, can be measured without paying any attention to questions about how long it may take to perform acts of measuring. I will take motion to be measured by changes of distances of a long hand of one common kind of clock. Since motion is in this case measured without taking into account how much some sort of "time" elapses, motion is measured "atemporally".
There are clocks which have circular faces near the circumference of which numerals from 1 to 12 are inscribed, and which have two hands attached to the center of the circular face which are rotated by some mechanism. The mechanisms are intended to move the hands with a regularity which approximates that of the sunís motion relative to other stars and to positions on the surface of the earth. Sundials may be taken to be ancestors of such clocks. The motions of celestial objects with respect to terrestrial observers can also be measured "atemporally". This is done by astronomers when they define right ascension and declination by setting an hour to be 15˚ of arc along certain circles on an imaginary celestial sphere. I will show that degrees of arc traversed along the circumference of a particular circle can be defined "atemporally".
A sector of a circular disk is the part of the disk bounded by two radii and an included arc, as shown in the figures below. The two endpoints of the included arc may be taken to mark positions of objects, whose distances apart measured along the included arc are different for different sized circles. For example, one of the disks may be taken to represent the face of a clock, and the other a disk determined by an observer at the center of the disk who measures the angle between two positions of a celestial object along an imaginary celestial sphere centered at the observerís position. This is a crude approximation to actual motion of, say, the sun, but it is consonant with the astronomical techniques known to Aristotle. For example, it doesnít take into account the elliptical motion of the earthís orbit (itself subject to perturbations), and it ignores the fact that the observer is not at the center of the earth but on the earthís rotating surface. However, my point is not to establish how time is measured nowadays, but to give an interpretation Aristotleís explanation of how "time" is the measure ("number") of motion, without considering the many mysteries which are said to be characteristic of "time". Aristotle did not have clocks of the sort illustrated above, but one may visualize instead a sundial, an ancestor of such clocks. A sundial may be thought of as a kind of clock without a mechanism to turn the hands, which instead makes use of shadows.
Given a circle with length of circumference C and length of any one of its radii r (same units of length as C), a radius may be rotated about the center of the circle, say counterclockwise, so that the new position and old position of the radius form an angle. The angle formed by two such positions of a radius will determine an arc of length s on the circumference. The angle is then said to be s/r radians in size, or more simply, since s/r is just a number without physical dimension, the measure of the angle is said to be s/r radians. The ratio C/r of the total length of the circumference C of a circle to the length of its radius r is 2π. Thus the angle formed by one complete revolution of a radius is 2π radians = 360˚. Given two circles with radii r1 and r2 which when rotated cut off arcs s1 and s2 in such a way that s1/r1 = s2/r2, the angles formed in the two circles will be equal in size, i.e. in the above figure θ1 = θ2.
This if the two circular disks in the figure represent a clock and an astronomical observation of the sunís motion from the earth, one may suppose that the mechanism of the clock is made in such a way that the clockís hands trace out the same size angles as the sunís observed positions do. More simply, if the smaller circle represents a sundial, then the tips of the shadows cast by the gnomon will necessarily trace out the same size angles. (The gnomon is the upright part of the sundial; see the illustration of a sundial above.)
Consider such a circular clock with length of circumference C and length of radius r. Successively rotate the longer hand of this clock 60 times through an angle of 2π/60 radians = 6˚. Let s denote the length of arc cut off on the circumference by a radius formed by the long hand during one of these rotations. Then for each such rotation there is an observable change of position of the hand, measured by the angle formed by the old and new positions of the hand. The size of this angle is s/r radians.
I will now introduce some terms which are commonly used to designate units of "time", but defined in terms of angular measure rather than in some other way. Set 1 H = 2π = 360˚, so that the angle for one complete rotation measures 1 H. Also define a unit M by 1 M = H/60 = 2π/60 = 6˚, so that 60 M = 1 H.
Suppose, for example, a vehicle travels 90 km, and during the trip the hand of a particular clock rotates through an angle measuring 3π radians = 540˚. Set H = 2π radians = 360˚, which measures the angle swept out during one revolution of a long hand or a gnomonís shadow on our clock. Then the average rate at which the vehicle traveled may be said to be 90 km/1.5H = 60 km/H = 60 km/2π radians = 30 km/π radians. This may be read as 30 km per each angular change of size π radians = 180˚, or if preferred, as 60 km per each complete rotation of the longer hand of the clock being used.
Suppose we call each revolution H, measured in radians, an hour, and each H/60 = M a minute. That is, suppose we call 2π radians or 360˚ an hour and 2π/60 radians or 6˚ a minute, so that there are 60 minutes in an hour. In the example above, we can then say that the average rate of change, or speed, of the vehicle is 60 km/hour.
Suppose further that we understand these hours and minutes only as referring to angular measures as determined in the way described above, and not as referring to something called "time", or "passage of time", or "consciousness of time", etc/. Then we have defined rates of change of distances, or speeds, not as so many units of distance per units of some entity called "time" which flows or passes or which people pass in or experience, but as so many units of distance per revolutions of a line segment, such as the long hand of a kind of clock.
Suppose we go further and define "time" in this way. That is, suppose we say that what "time" is, is a measure of fractions of revolutions of the radii of a circle. We then have, as Aristotle proposed, "time" as a measure or "number" of motion, where motion is itself measured with units of distance. Conversely, if we have "time" defined this way, and speeds defined by units of distance per units of this kind of "time" then we can calculate the corresponding changes in distance. For example, if we know that a vehicle travels at an average speed of 60 km/hour (i.e. 60 km per 2π radians) for 1.5 hours (i.e. for 3π radians), then it will traverse a distance of 90 km.
So "time," defined this way, is a number of motion, and motion is a number of "time," as Aristotle said.