Lecture 3 — Bodies not generated from surfaces—proved mathematically and naturally

412 Discussion of the other views may be postponed. But this last theory which composes every body of planes is, as the most superficial observation shows, in many respects in plain contradiction with mathematics. It is, however, wrong to remove the foundations of a science unless you can replace them with others more convincing.

413 And, secondly, the same theory which composes solids of planes clearly composes planes of lines and lines of points, so that a part of a line need not be a line. This matter has been already considered in our discussion of movement, where we have shown that an indivisible length is impossible.

414 But with respect to natural bodies there are impossibilities involved in the view which asserts indivisible lines, which we may briefly consider at this point. For the impossible consequences which result from this view in the mathematical sphere will reproduce themselves when it is applied to physical bodies, but there will be difficulties in physics which are not present in mathematics; for mathematics deals with an abstract and physics with a more concrete object.

415 There are many attributes necessarily present in physical bodies which are necessarily excluded by indivisibility; all attributes, in fact, which are divisible. There can be nothing divisible in an indivisible thing, but the attributes of bodies are all divisible in one of two ways. They are divisible into kinds, as colour is divided into white and black, and they are divisible per accidens when that which has them is divisible. In this latter sense attributes which are simple are nevertheless divisible. Attributes of this kind will serve, therefore, to illustrate the impossibility of the view.

416 It is impossible, if two parts of a thing have no weight, that the two together should have weight. But either all perceptible bodies or some, such as earth and water, have weight, as these thinkers would themselves admit. Now if the point has no weight, clearly the lines have not either, and, if they have not, neither have the planes. Therefore no body has weight.

417 It is, further, manifest that their point cannot have weight. For while a heavy thing may always be heavier than something and a light thing lighter than something, a thing which is heavier or lighter than something need not be itself heavy or light, just as a large thing is larger than others, but what is larger is not always large. A thing which, judged absolutely, is small may none the less be larger than other things. Whatever, then, is heavy and also heavier than something else, must exceed this by something which is heavy. A heavy thing therefore is always divisible. But it is common ground that a point is indivisible.

418 Again, suppose that what is heavy or weight is a dense body, and what is light rare. Dense differs from rare in containing more matter in the same cubic area. A point, then, if it may be heavy or light, may be dense or rare. But the dense is divisible while a point is indivisible.

419 And if what is heavy must be either hard or soft, an impossible consequence is easy to draw. For a thing is soft if its surface can be pressed in, hard if it cannot; and if it can be pressed in it is divisible.

420 Moreover, no weight can consist of parts not possessing weight. For how, except by the merest fiction, can they specify the number and character of the parts which will produce weight?

421 And, further, when one weight is greater than another, the difference is a third weight; from which it will follow that every indivisible part possesses weight. For suppose that a body of four points possesses weight. A body composed of more than four points will superior in weight to it, a thing which has weight. But the difference between weight and weight must be a weight, as the difference between white and whiter is white. Here the difference which makes the superior weight heavier is the single point which remains when the common number, four, is subtracted. A single point, therefore, has weight.

557. Having presented the opinions on the generation of things, he now inquires into the truth of the aforesaid opinions. And passing by the other opinions, which he discussed in other places, he makes inquiry especially about the last one, which is Plato's, both because it was more famous, and because in the order of inquiry it was prior. For the other opinions posited or removed generation of special bodies; but this opinion seemed to teach the generation of body insofar as it is body, since it posits body to be generated from surfaces. In regard to this he does two things:

first he disproves this opinion;

secondly, he shows that the same arguments can disprove the opinion of the Pythagoreans, who posit that bodies are generated from numbers, at the same consequences follow (300a15; [574]).

As to the first he does two things:

first he disproves the aforesaid opinion with mathematical arguments;

secondly, with natural arguments, at but with respect to natural (299a11; [560]).

558. As to the first he gives two arguments.

With respect to the first of these he says (299a1) that a discussion of the other opinions should be held elsewhere: partly, indeed in Physics I, partly in the book On Generation, and partly in a later section of the present book, But as to those who posit that all bodies are constituted of surfaces, it is possible at once to see that they are found to hold many things contrary to the disciplines, i.e., the mathematical sciences. These suppose that a point is indivisible, and that, consequently, a line, which is divisible, is not made of points. Mathematics supposes too, that a line is a length without breadth; consequently, a surface does not come to be from lines, because it has length and width without depth. Thus, too, neither does a body come from surfaces, since a body has length and width and depth. But it is not correct for anyone to reject these suppositions of the mathematicians, unless he can present arguments more probable than the suppositions. Consequently, the opinion of Plato seems to deserve to be disapproved, on the ground that he removes mathematical suppositions without a strong enough reason.

559. The second argument is at and, secondly (299a6). And he says the reason is the same for solids, i.e., bodies, to be composed of surfaces, and surfaces composed of lines, and lines of points, because, just as the point is the terminal and divider of the line, so the line is of the surface and the surface of the body. But if what Plato says is true, namely, that bodies are composed of surfaces, it will follow that a surface is composed of lines and lines of points. Consequently, it will not be necessary that a part of a line be a line. This matter, he says, was previously considered in the discussions on motion, i.e., in Physics VI, where it was proved that lines are not indivisible, and are not composed of indivisibles. There is also found a certain other short book in which it is proved that lines are not indivisible, said by some to be by Theophrastus.

560. Then at but with respect to natural (299a11) he disproves the aforesaid opinion with natural arguments.

First he states why it is necessary to present such arguments, i.e., not only mathematical, but also natural;

secondly, he pursues his plan, at there are many (299a17; [561]).

He says therefore first (299a11) that, because we have said that an impossibility in mathematics follows upon the assumption by some of indivisible lines, from which surfaces are composed, and consequently bodies, it is also necessary to consider now briefly the impossibilities affecting natural bodies that will follow from this assumption. The reason this is necessary is that whatever impossibilities affect mathematical bodies must, as a consequence, extend to natural bodies. And this is so because mathematical things are obtained by abstraction from natural things, but natural things are by apposition to mathematical things—for they add to mathematical objects a sensible nature and motion, from which mathematics abstracts. Thus it is clear that mathematical properties are saved in natural things, but not conversely. Therefore, whatever impossibilities are against mathematical things are also against natural things, but not vice versa.

561. Then at there are many (299a17) he shows what impossibilities affecting natural bodies flow from the aforesaid position.

First, he presents a certain general argument;

secondly, he explains it part by part, at it is impossible (299a25; [562]).

He says therefore first (299a17) that there are many things which must exist in natural bodies but cannot exist in indivisibles. We could take indivisibles here as "mathematical things" on the ground that they are said by abstraction; thus what is said here will be adduced to show what was just said, namely, that natural things are by apposition to mathematical, since many things must be present in natural things that cannot be present in mathematical things, such as all the passions [properties] that are divisible. But it is better to take "indivisibles" as surfaces with respect to bodies, and lines with respect to surfaces, and points to lines—which latter are indivisible absolutely. He says therefore that many things must be present in natural bodies that cannot be present in indivisible things: for example, if something is indivisible, such as a point or line or surface; or, for example, if something is divisible—because what is divisible is of necessity in a natural body, but not in things that are indivisible. For something divisible cannot exist at all in an indivisible thing—since what is in something is in a certain way comprehended by it: but the divisible cannot be comprehended by the indivisible according to quantity. All passions are divided in two ways: either according to species, or according to accident. This of course does not mean that each and every passion is divided in both ways, but that each is divided in one way or the other.

He then explains both ways of division. And he says that a passion is divided according to species in the way that the species of color are white and black. This can be understood in two senses. In one way, in the sense that this common thing which is color is divided into black and white, as though by its species; but this contributes nothing to our argument, because there is nothing to prevent something common to many from being predicated of some indivisible. Consequently, we must understand a passion divisible according to species in the sense of an intermediate color, which is a composite of two species of color, namely, black and white. Now, it does not seem that such a passion can exist in a thing that is entirely simple: because, since the proper passions are caused by the subject, the principles of a composite passion must be diverse, and diversity is repugnant to the simplicity of a subject. Then, he explains what is divisible according to accident. And he says that a passion is said to be divisible according to accident, if the subject of which it is an accident is divisible, as white is divided by dividing its subject. Hence, all passions that are simple according to species are divisible in this way, i.e., according to subject, insofar, namely, as they exist in a natural subject.

And therefore, regarding such passions which are divisible in one way or the other, it is to be understood that an impossibility follows upon those saying lines to be indivisible or surfaces, from which natural bodies are composed, namely, out of such things which cannot be the subject of the passions of natural bodies.

562. Then, at it is impossible (299a25), he gives special arguments which disprove the aforesaid position. In regard to the first argument he does two things:

first he presents the argument;

secondly, he proves what he had supposed, at it is, further, manifest that their point (299a30; [563]).

He says therefore first (299a25) that it is impossible, if neither of the things out of which something is composed has heaviness, that the composite of both have heaviness. But sensible bodies have heaviness: either all of them, as Democritus says, or some, namely, earth and water, as the Platonists said. Therefore, a sensible body cannot be composed of things not having heaviness. But a point has no heaviness; therefore, a thing with heaviness cannot be composed of points. Yet according to the aforesaid opinion, a line is composed of points. Therefore, a line also cannot have heaviness. Consequently, neither can a surface, which is composed of lines; furthermore, neither can a body, which is composed of surfaces—which is against the aforesaid. It should be noted that this argument holds for quantitative parts which are of the same nature and notion both in relation to one another and to the whole; but it does not hold in essential parts, which have a different notion with respect to one another and with respect to the whole. Hence, it does not follow, if matter is not heavy and form is not heavy, that the composite is not heavy—because matter is heavy in potency, but through form something is made actually heavy.

563. Then, at it is, further, manifest that their point (299a30) he proves what he had assumed in the preceding argument.

First he proves that a point is not heavy;

secondly, that something heavy cannot be from non-heavy things, at moreover, no weight can consist of (299b14; [566]).

The first he proves with three arguments.

The first of which is this: every heavy thing can be heavier than some other, and every light thing can be lighter than something else; but yet it is not necessary that everything heavier or lighter be heavy or light.

But it seems that what is said here is false, for the comparative presupposes the positive — everything whiter is white. Some therefore say that the comparative, if taken in its proper sense, does suppose and imply the positive; but that sometimes the comparison is "abusive." This occurs, for example, when a thing is said comparatively with respect to an opposite, as when a swan is said to be "whiter" than a raven, or even when a thing is described in the comparative degree because it possesses less of the opposite, as when an Ethiopian is said to be "whiter" than a raven because he is less black, or when it is said that some lesser evil is "more worthy of choice" than a greater evil—whereas evil is not worthy of choice, and an Ethiopian is not white. And it is in this way that the Philosopher here says that not everything heavier is heavy, nor everything lighter light. Hence, to designate an "abusive" comparison, he added, perchance.

But since it is not Aristotle's custom to argue from "abusive" terms, it therefore should be stated that there are some things which are described only absolutely, as in the case of white or sweet; and in such cases the comparative presupposes the positive, and implies it. But there are other things which are sometimes said absolutely and sometimes relatively—as in the case of heavy and light: for, as will be said in book IV, fire is said to be absolutely light and earth absolutely heavy; but air is heavy to fire, while light to water and earth. So too, water is light to earth but to air and fire heavy. Now, it is evident that what is absolutely heavy, is also heavy when compared to other things; and that which is absolutely light, is light compared to other things. In this sense, everything heavy is heavier and everything light is lighter. But it does not follow that everything heavier is heavy, and everything lighter is light—since it does not follow that, if something is light in relation to other things, that it is light absolutely; and the same goes for the heavy. That this is the explanation of the statement is plain from the example he gives. For "large," taken commonly, is relative, as is evident in the Predicaments, but as applied to some thing, that is called "large" absolutely which attains to the quantity due it — as a man is called "large" absolutely, who attains to the perfect quantity of a man. Thus it is plain that "large" is said absolutely and relatively. Hence everything "large" absolutely is said to be large in relation to something, and this is to be "larger"; but not everything described as "larger" is large absolutely, for there are many things which, absolutely considered, are small, and yet are larger than other things.

Therefore, if everything heavy is heavier than something else, then necessarily everything heavy must be greater in heaviness than something else. And so it follows that it is divisible: for everything that is greater is divided into what is equal and something more. But a point is indivisible, as is supposed from its definition. Therefore a point is not heavy.

564. The second argument is given at again, suppose that what is heavy (299b7), as follows: heavy and light follow upon rare and dense—for we see that according to rarity and density, elements differ in heaviness and lightness. But the dense differs from what is rare in that, in an equal mass, i.e., under the same dimensions, it contains more, because it has more matter, as is had in Physics IV. But since some bodies are heavy and some light, then if a point is posited as heavy, it is with equal reason posited as light; and if it is posited as dense, it is with equal reason posited as rare. But what is posited as dense must be divisible, insofar as it contains more under a smaller mass; similarly, what is rare must be divisible, insofar as it contains an equal amount under greater dimensions. But a point is indivisible. Therefore, it is neither thick nor rare and, consequently, neither heavy nor light.

565. The third argument is given at and if what is heavy (299b11), as follows: Whatever is heavy is either soft or hard: the season being that heaviness follows on two elements, namely, earth and water, one of which, namely, water, yields to the touch and is, accordingly, a principle of softness, while the other, namely, earth, does not yield and is a principle of hardness. Now, it is obvious that everything soft is divisible, because it yields within itself to what touches it. But this could not happen unless it had a number of parts, one of which is able in some way to move into the place of another. For the same reason, what is hard must be divisible, for it could not be said to be "unyielding," unless it had whence it could yield. Consequently, since a point is indivisible, it will be neither hard nor soft, and, therefore, not heavy.

566. Then, at moreover, no weight can consist of (299b14) he shows that nothing heavy can be composed of two or more things, none of which is heavy. But this must be understood of the composition by which something is composed of quantitative parts—for something heavy is composed of essential parts, for example, of matter and form, neither of which is heavy. To prove this he presents two arguments.

The first of these proceeds in accordance with the opinion of some who said that from certain things not heavy, when they were multiplied, something heavy was composed, but that when they were in a lesser number, nothing heavy was made of them. But those who present this opinion must determine how many such things are required to form heaviness; otherwise what is said without a sure reason is seen as fictitious.

567. The second argument is presented at and, further, when one weight (299b17), as follows: Every heaviness greater than another heaviness exceeds the lesser by a certain heaviness, because something is made greater by the addition of like things. From this it follows, according to the aforesaid position, that every indivisible has heaviness. For let us suppose a body constituted of four points, having heaviness; let us take another body made of more points, say five. Thus it will be heavier—and in such a way that that by which it is heavier, must be heavy. And although not everything heavier is heavy, as was said above, yet whatever is heavier than something heavy must be heavy, just as everything which is whiter than something white must be white. And therefore, since what is greater by one point is heavier than a body which is equal to it if that point be removed, it will follow that one point is heavy. But that is impossible, as is clear from what has gone before. Therefore, it remains that it is impossible for something heavy to be made from things that are not heavy.