Lecture 4 — Democritus' argument that natural bodies are composed out of indivisible bodies.

For to suppose that a body (i.e. a magnitude) is divisible through and through, and that this division is possible, involves a difficulty. What will there be in the body which escapes the division? If it is divisible through and through, and if this division is possible, then it might be, at one and the same moment, divided through and through, even though the dividings had not been effected simultaneously: and the actual occurrence of this result would involve no impossibility.

Hence the same principle will apply whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever: nothing impossible will have resulted if it has actually been divided—not even if it has been divided into innumerable parts, themselves divided innumerable times. Nothing impossible will have resulted, though perhaps nobody in fact could so divide it. Since, therefore, the body is divisible through and through, let it have been divided.

What, then, will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesis the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain, and yet division is to take place, the constituents of the body will either be points (i.e. without magnitude) or absolutely nothing. If its constituents are nothings, then it might both come-to-be out of nothings and exist as a composite of nothings: and thus presumably the whole body will be nothing but an appearance. But if it consists of points, a similar absurdity will result: it will not possess any magnitude. For when the points were in contact and coincided to form a single magnitude, they did not make the whole any bigger (since, when the body was divided into two or more parts, the whole was not a bit smaller or bigger than it was before the division): hence, even if all the points be put together, they will not make any magnitude.

But suppose that, as the body is being divided, a minute section—a piece of sawdust, as it were—is extracted, and that in this sense—a body 'comes away' from the magnitude, evading the division. Even then the same argument applies. For in what sense is that section divisible?

But if what came away was not a body but a separable form or quality, and if the magnitude is 'points or contacts thus qualified': it is paradoxical that a magnitude should consist of elements, which are not magnitudes.

Moreover, where will the points be? And are they motionless or moving? And every contact is always a contact of two somethings, i.e. there is always something besides the contact or the division or the point. These, then, are the difficulties resulting from the supposition that any and every body, whatever its size, is divisible through and through.

There is, besides, this further consideration. If, having divided a piece of wood or anything else, I put it together, it is again equal to what it was, and is one. Clearly this is so, whatever the point at which I cut the wood. The wood, therefore, has been divided potentially through and through. What, then, is there in the wood besides the division? For even if we suppose there is some quality, yet how is the wood dissolved into such constituents and how does it come-to-be out of them? Or how are such constituents separated so as to exist apart from one another?

Since, therefore, it is impossible for magnitudes to consist of contacts or points, there must be indivisible bodies and magnitudes. Yet, if we do postulate the latter, we are confronted with equally impossible consequences, which we have examined in other works.

26. After showing that, with regard to the matter at hand, the opinion of Democritus is stronger than that of Plato, and that his argument is more appropriate, to show this, the Philosopher presents Democritus' argument.

First, he cites it;

second, he answers it, at but we must try to disentangle (316b18; [34]).

About the first he does two things:

First, he presents the argument of Democritus, which leads to the impossibility that a body can be wholly divided, i.e., divided as much as it can be divided.

Second, he shows that this is impossible, at but suppose that (316a34; [30]).

Concerning the first he does two things:

first, he presents the argument that leads to this impossibility;

second, he shows the necessity existing in this argument, at what, then, will remain (316a24; [29]).

27. With respect to the first (316a14) it should be observed that one must hold either that a body is composed of indivisibles, or that it is wholly, i.e., totally, divisible, according to every sign [point]. And therefore Democritus, in order to show that a sensible body is composed of indivisible bodies, tries to show that it is impossible for a sensible body, such as wood or stone, to be wholly divisible, i.e., according to every given sign [i.e., point] in the body. Consequently, he says that if you posit that such a body (for example, a sensible body), or any magnitude whatsoever (for example, a surface or a line), is wholly divisible, i.e., with respect to every given sign, and this is considered possible, the question remains: what is it that escapes division, i.e., that remains after the division? For it is necessary that, when any divisible thing has been divided, there remain certain divisible parts, upon which division takes place.

Democritus says that such a question arises because if the body is wholly, i.e., in its entirety, divisible, and this is possible, then there will be nothing to prevent it from being at one time divided as far as it can, even though the division does not take place all at once but successively—just as, if a man can arrive at some particular place, there is nothing to prevent his having arrived there, even though he should not do so all at once, but successively. If this is granted, no impossibility should follow, because, when something possible is assumed, nothing impossible follows, according to the Philosopher in Prior Analytics I.

28. Then, at hence the same principle (316a19), he shows the necessity existing in the above argument. For if it should be granted that a body is divisible through the middle, and has been so divided, nothing impossible follows. And this is what he says: hence, since, something possible having been laid down, nothing impossible follows, it will be something similar to this if something is supposed as divisible and actually divided in the middle. And universally, if a body is such as to be apt to be wholly divided, i.e., according to every sign [point], then if it is divided; i.e., if it is assumed to have been so divided, no impossibility will result, i.e., from this nothing impossible should arise—any more than if something is divisible into a thousand times a thousand parts and we assume that it has been so divided, no impossibility follows, even if no one has actually divided it.

Consequently, it seems that whether a body is divisible into a few parts or into a great many parts or totally, no impossibility seems to follow upon the assumption that it has been divided as far as it is divisible. Therefore, since, according to those who maintain that a natural body is not composed of indivisible bodies, it is totally divisible, let it be considered as totally divided. But this is impossible; therefore the first, namely, the assumption that it is totally divisible, is also impossible. Therefore, it is composed of indivisibles.

29. Then, at what, then, will remain (316a24), he shows that it is impossible for a body to have been totally divided, on the ground that one cannot posit anything as remaining after the division.

first, therefore, he shows that one cannot give anything as remaining from the division as a principal part;

second, that one cannot give anything as remaining that might have fallen out of the division, at but if what came away (316b2; [31]).

He says therefore first (316a24) that if a body is assumed to have been totally divided, it remains to ask what is left, i.e., what remains after the division—as in every division we are wont to see the things into which the divided object was resolved.

First, he shows that a magnitude will not remain. For such a thing is impossible—since it would follow that some divisible part not yet divided was remaining or that a magnitude was something not divisible. But it was said that body was wholly divisible. Consequently, what remains after division cannot be divisible in any way whatsoever, and it is being supposed by the adversary that the magnitude is wholly divisible.

Second, he concludes that, if what is left after division is neither a body nor a magnitude, and yet a total division was made, it remains that the division will either be out of points, in such a way that the body will be finally resolved into points, and consequently, the things from which a body is composed will lack dimensions, or else it follows that what remains after division is utterly nothing.

Third, he shows that the second alternative is impossible. For, since each thing is generated out of that into which it is resolved, then, if it is resolved into nothing, it will follow that it may be generated from nothing. But what is composed out of nothing, is nothing. It will follow, therefore, that the body in question is nothing, and for the same reason, the whole universe. Whatever there will be in nature will be there according to appearance only and not according to existence.

Fourth, he proves the first alternative set down above, namely, that there is not resolution into points. For then it would likewise follow that a body is composed of points and it would further follow that the body would not be quantified. For, before the body was divided, and the points were in contact, in the way that the extremities of two lines are together, so as to form a single continuous magnitude, and the points were all together and not yet set off from each other, they did not make a greater whole. For a point is nothing other than a certain division of the parts of a line. By the fact of a thing's being divided into two or more, the whole is not made either greater or less than it was previously, A small body, just as a large one, may be divided into two or more. Thus it is plain that points, which are nothing but divisions, do not make anything greater. Hence it remains that if the points are assembled, they do not make anything greater.

Consequently, it is seen to be impossible for a body to be divided through and through, because it is impossible to assign anything that remains as the residue of division, as a principal part of the divided body.

30. Then, at but suppose that (316a34), he shows that it is impossible to show some residue of division on the hypothesis of something falling out.

First, he shows that such a thing cannot be a body;

second, that it cannot be something incorporeal, at moreover, where will the points be (316b5; [32]).

He says therefore first (316a34) that if, the whole bodily magnitude of the divided body having been divided, something comparable to a piece of sawdust should be produced, which falls out by reason of the division apart from the principal parts into which the wood is divided, and one should say that out of the wholly divided bodily magnitude some body comes out as though a residue, there follows the same argument as above—namely, how can those hold that body to be still divisible who hold no body to be indivisible and one is holding a natural body to be wholly divided?

31. Then, at but if what came away (316a34), he shows that such a residue cannot be any incorporeal thing, for three reasons.

With regard to the first he says that if what slipped away from the totally divided magnitude is not a body but some species, i.e., a form segregatable, i.e., separable from the subject or also, as Anaxagoras says, some passion, since he taught that passions and habits can be separated and combined, and this passion which leaves a magnitude is after the manner of a point or contact, those who posit this fall first of all into this impossibility, namely, that a magnitude is composed of non-magnitudes. This is seen to be untenable, for each thing is constituted out of the things of its genus—colors are not composed from figures, nor conversely.

32. He presents the second reason at moreover, where will the points be (316b5), in regard to which it should be noted that certain ones maintained that a line is composed out of points. This can be in two ways: in one way, out of moved points, in the sense of those who suppose that a point in motion forms a line, and a line in motion forms a plane, and a plane in motion a body; in another way, out of points not moved, so that a magnitude is composed out of points as out of parts.

However, in whichever of these two ways a magnitude is supposed to be composed of points, one will have to designate where the points are, i.e., what position they occupy in the magnitude, as can be done for each part of which a magnitude is composed. But [in this case] this cannot be designated, for a point is seen to be nothing other in a magnitude than a certain contact of a continuous line, or the division of the parts of a line that has been divided. But contact is always one [contact] of some certain two which, namely, are parts of a magnitude, possessing definite positions in a magnitude—as though that which is a part of the magnitude, having a definite position among its parts, is something over and above the contact and the division, and consequently something over and above the point. It is not therefore seen as possible that a magnitude be divided into points or contacts or divisions. If, therefore, someone should posit any body, or any quantity, to be wholly divisible, there will occur this unacceptable consequence which has been stated.

33. He presents the third reason at there is, besides (316b9), and says that if after having divided a piece of wood or any other body, I put it together regain out of the same parts, an equal and single body will be produced, since the things into which something is divided, and out of which it is composed are the same. Hence the case seems to be the same if I divide the wood according to any sign [point] whatever (which he above referred to as wholly) so that, out of those things into which it is divided, it may be again composed. Let the wood then have been wholly divided in potency, i.e., into all that into which it can be divided. What then will remain beyond the division? For every division must terminate at something. If we should say that the residue of the division is some passion, it would follow that the body would have been divided into passions—and consequently, will be generated out of them, which is impossible. For neither substance nor quantity is generated out of passions. Or how are passions able to be separated?

He further concludes his principal proposition (316b14) saying that, if it is impossible that a magnitude be composed out of contacts or points, as the aforesaid arguments conclude, one must posit that there are certain indivisible bodies, and certain indivisible magnitudes—for, if a body should be wholly divisible, it would follow that it would be composed out of contacts or out of points, as is evident from what has been said.

Further still, after presenting the arguments of Democritus, Aristotle adds that, likewise for those who posit this, namely, the existence of indivisible bodies, no less an impossibility follows; and this was examined elsewhere, namely, in On the Heavens III.