**Aristotle and the objection to void on the grounds that natural motion is impossible (Physics D 8.215a24-216a21)**©2002- trans. by Henry Mendell, Cal. State U., L.A.

Return to Vignettes of Ancient Mathematics

A note on the verb
translated as 'to travel'. It actually means 'to be carried' as is clear from
the future passive at line 215^{b}1. It is convenient to use a verb
of locomotion that is not awkward, but the reader should be warned.

215^{a}24-29

Furthermore the claim that the void does not exist separately if there is motion (kinesis) is obvious from these matters. For we see that the same weight, i.e. body, travels faster for two reasons, either by there being a difference in the medium, e.g. through either water or earth or through either water or air, or by there being a difference in the traveller, if the others, the media, are the same, due to the excess of weight or lightness.

Notee: the principle here enunciated says that where the traveller is the same, we can compare the motions that result from different media, and where the media are the same we can compare the motions of travellers that are different in respect of weight or lightness. Aristotle does not tell us what to do where both the media are different and the weights or heaviness of the travellers are different.

215^{a}29-^{b}12
(Moving Diagram or
Fixed Diagramm)

And so, that through
which it travels is a cause since it most of all impedes by moving against it,
then too by remaining. That which is not more easily divided does this more,
and such is the denser. For __A__ will travel through B in time __G__,
but through D which is finer in time __E__, if the length of B is equal to
D, according to the proportion of the impeding body. For let B be water, and
D air. By as much as air is finer and less bodily than water, by so much will
A travel faster through D than through B. In fact, let the speed have the same
ratio to the speed which air differs with respect to water. Thus if it is doubly
fine, then it will traverse __B__ in double the time it
traverses __D__, and time __G__ will be double time __E__. And
by as much as what it travels through is less bodily and less impeding and more
easily divided, i t will travel faster.

Notes:

A = the traveller

medium | medium kind | time of travel E < G |
distance (not given) |

B | water | G | X |

D | air | E | X |

The principle might be put this way. The two media are the same distance, so that A is regarded as travelling exactly the same distance. the ratio of speeds is really the ratio of times of travel over the same distance. Aristotle says nothing about what happens in each medium.

air D : water B = time of travel through the air E : time of travel through water G

If D = 2 B, then E = 2 G.

Aristotle does not actually assert that this is the ratio. He says, "Let it have" this ratio. All that he requires is that there be some ratio.

215^{b}12-22

But the void has no ratio by which it is exceeded by the body, just as nothing doesn't have a ratio to a number. For if four exceeds three by one, but exceeds two by more, and exceeds one by more than it exceeds two, it will not longer have a ratio by which it exceeds nothing. For it is necessary that that which exceeds be divided into the excess and what is exceeded, so that four is be the amount by which it exceeds and nothing will be what is exceeded. Hence, a line does not exceed a point unless it is composed of points. Similarly too the void to the full cannot have any ratio, so that the motion will not have a ratio either, but if something travels through the finest in so much time over so much distance, it will exceed the ratio of motion through the void.

Note: 4 : 3 < 4 : 2 < 4 : 1, but 4 : 0 is not a ratio. The text seems to allude to the method of anthuphairesis. In a ratio a : b, one takes n such that n*b < a while (n+1) * b > a. The first term is n, and one gets the remainder c = a-n*b, and does the same with the ratio b : c, and so forth. Here, Aristotle merely says that one needs to get, in effect, 1*b and a-1*b, the first step in getting n*b.

In the previous case, the more fine was greater than the less fine. Since the void is less dense than air, but is compared to the denser, here Aristotle seems to think of the denser as greater than the more fine.

215^{b}22-216^{a}4
(Moving
Diagram or Fixed Diagram)

For let Z be void
and equal in magnitude to B and D. In fact, if A traverses and is moved in some
time __H__, in a time smaller than __E__, the void to the full will have
this ratio. But in as much time as __H__, A will traverse D over distance
__Q__. For if there is something, __Z__, differing in fineness from the
A in the proportion which time __E__ has to __H__, it will traverse it.
For if body __Z__ is finer than D by so much as E exceeds H, __A__ will
conversely traverse the distance __Z__ in it with its speed in as much time
H, if it travels. If then there is no body in Z, it will
traverse yet faster. But it went in time H. Thus in an equal time it
will traverse what is full as a void. But this is impossible.

Note:

A = the traveller

medium | medium kind |
time of travel |
distance (not given) |

B | water | G | X |

D | air | E | X |

Z | void | H | X |

D | air | H | Q < X (since H < E) |

Z2 | fine stuff | H | X |

This is the structure of the argument.

Let A travel through Z in H. Then since A travels over X in E > H, A will travel through D in H over a distance Q less than X. The implication is that Q is a distance (this is all that is important to the argument).

Let there be a medium Z2 such that A travels X in the time it takes A to travel Q. Then by the rule above

medium Z2 : air D = H : E

Hence, A must travel through the void Z in a time Y < H.

216^{a}4-11
(Moving
Diagram or Fixed Diagramm)

It is then obvious that if there is a time in which it travels any part of the void, this impossibility will follow. For something will be taken as traversing something that is full and a void in an equal time. For one body traversed is to another body traversed as the time is to the time. And to speak in sketch, the cause of the consequence is clear, that there is a ratio of any motion to a motion (for it occurs in time, and there is a ratio of any time to a time, given that they both are finite), but there is no ratio of a void to something full.

Note that the ratio of bodies in line 7 is the body through which A travels. Aristotle has not said anything about the nature of A other than that it travels through the different media. The principle is the same as the one enunciated. Hence, the consequence just is the one we expect, that the principle of motion entails that A will travel the void in an equal time.

The explanation takes it that if there is a ratio of motion to a motion there would have to be a ratio of media. It does not claim anything about the relation between them beyond this. In other words, the argument is that there is a ratio of times of motion. Hence, there is a ratio of motions. From this one can infer that there must be some ratio of media.

216^{a}11-21

And so, because what things travel through differ, these things follow, but these follow according to the excess of the things moved. For we see that things which have more impulse whether of weight or lightness move, given other things holding similarly [in their shapes--seclusit Ross], travel an equal area more quickly, and in the ratio which their magnitudes have to each other. Thus, this will also occur when they travel through the void. But is it impossible. For what reason will they travel more quickly? For this occurs of necessity in things that are full. For the larger divides more quickly with its strength. For either it will divide by its shape or by the impulse which the travelling or launched (or released (body has. Therefore, all things will move equally fast. But that is impossible.

Note: Aristotle does not state a theory of motion of bodies beyond two claims:

1. Size and shape affect motion, but couldn't affect motion through a void.

2. Let A and B be two bodies similar in shape and let their weight (or lightness) be proportional to their size. And let them travel the same distance. Then,

the size of A : the size of B = time of B's travel : time of A's travel

This is a claim that Galileo can rightly object to. But the law of motion does not say anything beyond that. I assume that the reason why Aristotle switches to size is that it is convenient. This assumption is consistent with Aristotle's mathematical practice (cf. De caelo A 6). Some might take the magnitude to be the magnitude of weight or lightness. That would be more general, but not what Aristotle says.

216^{a}21-26

And so it is clear from what's been said that if there is a void, the opposite occurs from the reason for which those who say that there is void establish it. And so, some people believe that if in fact there is motion in place it is separated out by itself. But it was said earlier that this is impossible.