Everything That Changes Is Divisible

Everything that changes must be divisible.

Every change is from something to something.

When a thing is at the goal of its change, it is no longer changing.

• Before it starts to change it also does not change.

It follows that the changing must be between the starting-point and the goal.

The ‘goal of change’ means that which comes first in the process of change.

For example, in a process of change from white, the goal will be grey, not black. For it is not necessary that that which is changing should be at either of the extremes.

Therefore, everything that changes must be divisible.

Motion is divisible in 2 senses.

1. In virtue of the time that it occupies.

2. According to the motions of the several parts of that which is in motion

For example, if the whole AG is in motion, there will be a motion of AB and a motion of BG.

Let:

• DE be the motion of the part AB
• EZ be the motion of the part BG

Then the whole DZ must be the motion of AG. This is because DZ must constitute the motion of AG inasmuch as DE and EZ severally constitute the motions of each of its parts.

But the motion of a thing can never be constituted by the motion of something else.

Consequently, the whole motion is the motion of the whole magnitude.

Every motion is a motion of something.

Each of the parts DE, EZ is the motion of one of the parts AB, BG.

The whole motion DZ is not the motion of either of the parts or of anything else.

This is because:

• the whole motion is the motion of a whole.
• the parts of the motion are the motions of the parts of that whole

The parts of DZ are the motions of AB, BG and of nothing else.

A motion that is 1 cannot be the motion of more things than 1.

Since this is so, the whole motion will be the motion of the magnitude ABG.

If there is a motion of the whole other than DZ, say TI, the motion of each of the parts may be subtracted from it: and these motions will be equal to DE, EZ respectively: for the motion of that which is one must be one.

So if the whole motion TI may be divided into the motions of the parts, TI will be equal to DZ: if on the other hand there is any remainder, say KI, this will be a motion of nothing: for it can be the motion neither of the whole nor of the parts (as the motion of that which is one must be one) nor of anything else: for a motion that is continuous must be the motion of things that are continuous.

The same result follows if the division of TI reveals a surplus on the side of the motions of the parts.

Consequently, if this is impossible, the whole motion must be the same as and equal to DZ.

This then is what is meant by the division of motion according to the motions of the parts: and it must be applicable to everything that is divisible into parts.

Motion is also susceptible of another kind of division, that according to time. For since all motion is in time and all time is divisible, and in less time the motion is less, it follows that every motion must be divisible according to time.

Since everything that is in motion is in motion in a certain sphere and for a certain time and has a motion belonging to it, it follows that the time, the motion, the being-in-motion, the thing that is in motion, and the sphere of the motion must all be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus quantity is essentially, quality accidentally divisible).

For suppose that A is the time occupied by the motion B.

Then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity.

The time will be divisible similarly to the motion: for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time.

In the same way the being-in-motion will also be divisible. For let G be the whole being-in-motion. Then the being-in-motion that corresponds to half the motion will be less than the whole being-in-motion, that which corresponds to a quarter of the motion will be less again, and so on to infinity.

By setting out successively the being-in-motion corresponding to each of the 2 motions DG (say) and GE, we may argue that the whole being-in-motion will correspond to the whole motion.

For if it were some other being-in-motion that corresponded to the whole motion, there would be more than one being-in motion corresponding to the same motion.

The argument would be the same as that where we showed that the motion of a thing is divisible into the motions of the parts of the thing.

If we take separately the being-in motion corresponding to each of the two motions, we shall see that the whole being-in motion is continuous.

The same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change (though some of these are only accidentally divisible because that which changes is so): for the division of one term will involve the division of all.

So, too, in the matter of their being finite or infinite, they will all alike be either the one or the other.

The fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite.

The attributes ‘divisible’ and ‘infinite’ belong in the first instance to the thing that changes.

That divisibility does so we have already shown: that infinity does so will be made clear in what follows?