Lecture 2 — The perfection of the universe both as body and as containing all

2 Now a continuum is that which is divisible into parts always capable of subdivision,

3 and a body is that which is divisible in every way.

4 A magnitude if divisible one way is a line, if two ways a surface, and if three a body.

5 Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all ways.

6 For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an 'all', and the number they give is the triad.

7 And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the gods.

8 Further, we use the terms in practice in this way. Of two things, or men, we say 'both', but not 'all': three is the first number to which the term 'all' has been appropriated. And in this, as we have said, we do but follow the lead which nature gives.

9 Therefore, since 'every' and 'all' and 'complete' do not differ from one another in respect of form, but only, if at all, in their matter and in that to which they are applied, body alone among magnitudes can be complete. For it alone is determined by the three dimensions, that is, is an 'all'. But if it is divisible in three dimensions it is every way divisible, while the other magnitudes are divisible in one dimension or in two alone: for the divisibility and continuity of magnitudes depend upon the number of the dimensions, one sort being continuous in one direction, another in two, another in all.

10 All magnitudes, then, which are divisible are also continuous. Whether we can also say that whatever is continuous is divisible does not yet, on our present grounds, appear. One thing, however, is clear. We cannot pass beyond body to a further kind, as we passed from length to surface, and from surface to body. For if we could, it would cease to be true that body is complete magnitude. We could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it has being in every respect.

11 Now bodies which are classed as parts of the whole are each complete according to our formula, since each possesses every dimension. But each is determined relatively to that part which is next to it by contact, for which reason each of them is in a sense many bodies.

12 But the whole of which they are parts must necessarily be complete, and thus, in accordance with the meaning of the word, have being, not in some respect only, but in every respect.

8. After showing by way of introduction that bodies and magnitudes are to be studied in natural science, the Philosopher here begins to carry out his main resolve. And because, as was said above, Aristotle in this book is mainly concerned with determining about the corporeal universe and its principal parts which are the simple bodies, among which the most important is the heavenly body, the book therefore is divided into three parts:

in the first he determines concerning the corporeal universe;

in the second concerning the heavenly body, in book II, at that the heaven as a whole neither came into being (283b26);

in the third about other simple bodies, i.e., heavy and light, in book III, at we have already discussed the first heaven and its parts (298a24).

With respect to the first he does two things:

first, he shows the perfection of the universe;

secondly, he determines certain of its conditions or properties, at this being clear (L. 9; 271b1).

About the first he does two things:

first, he shows the perfection of the universe;

secondly, he explains of what parts it is composed at the question as to the nature of the whole (L. 3; 268b11).

As to the first he does two things:

first, he shows the perfection which the universe has in virtue of the common notion of its genus, i.e., inasmuch as it is a body.

Secondly, he proves the perfection proper to it, at now bodies which are classed as parts (268b5; [17]).

About the first he does three things:

first he explains the definition of body, to be used in proving his proposition.

Secondly, he proves the proposition, at therefore, since 'every' and 'all' (268a21; [15]);

thirdly, he shows what could be clear from the foregoing, at all magnitudes, then, which are divisible (268a28; [16]).

As to the first he does two things:

first he defines "continuum," which is the genus of body;

secondly, he clarifies the definition of body, at and a body (268a7; [10]).

9. With regard to the first, we must consider that the continuum (268a6) is found defined in two ways by the Philosopher.

In one way with a formal definition, where it is said in the Predicaments (c. 4) that the continuum is that whose parts are joined at one common term; for the unity of a continuum is, as it were, its form.

In another way, with a material definition taken from the parts, which have the aspect of matter, as is said in Physics II—and it is thus that the continuum is defined here, namely, as what is divisible into parts always divisible. For no part of a continuum can be indivisible, because no continuum is composed of indivisibles, as is proved in Physics VI.

And it is fitting that this latter definition be used here, and the other one in the Predicaments, because the consideration of natural science is concerned with matter, while that of logic is concerned with notions and species.

10. Then, at and a body (268a7), he defines body.

First he proposes the definition, saying that body is a continuum which is divisible in every way (268a6), i.e., at every part or according to every dimension.

Secondly, at a magnitude if divisible (268a7), he proves the proposed definition with this argument: Body is divided according to three dimensions. But what is divided according to three dimensions is divided according to all. Therefore, body is divisible according to all the dimensions.

First, therefore, he explains the minor proposition as though by division. For among magnitudes there is one which is divided with respect to one dimension, and this is called line; another is divided with respect to two dimensions, and this is called plane, i.e., a surface; still another is divided according to three dimensions, and since such a magnitude is neither line nor surface, it follows that it is body.

The major proposition he gives at beyond these (268a9).

First, he mentions it and says that, besides these magnitudes or dimensions, there is no other magnitude or dimension, on the ground that "three" has the property of being all, because it implies a certain totality, and because whatever is thrice seems to be in all ways and "entirely," i.e., according to every mode.

11. Secondly, at for, as the Pythagoreans say (268a10), he proves what he had said in three ways.

First, according to the teaching of Pythagoras who said that what is called whole and all is determined by the number 3. For the beginning and the middle and the consummation, i.e., the end, have a number which befits what is "whole" and "all"—for in things divisible, the first part is not enough to complete the whole, which is completed by the ultimate that is reached by passing from the beginning through the middle. But these three, namely, beginning, middle, and end, have 3 as their number. Consequently, it is clear that the number 3 belongs to the "all" and "whole."

12. Secondly, at and so, having taken these three from nature (268a13), he proves the same by means of what is observed in divine worship. For we use this number 3 in the worship of the gods (whom, namely, the gentiles worshipped), i.e., in sacrifices and praises for them, as though we should receive from nature its laws and rules: just as nature completes all things with the number 3, so those who established the divine worship have, in their desire to attribute to God everything perfect, attributed to Him the number 3.

13. Thirdly, he proves the same, at further, we use the terms (268a15), by appealing to the general way we speak. And he says that we even assign names to things according to the aforementioned method, in which perfection agrees with the number 3. For when there are two things, we say both—thus we speak of two men as both—but we do not say all, which we use for the first time in the case of three. And we all in general use this way of speaking, because nature so inclines us. For whatever is peculiar to individuals in their way of speaking seems to arise from the particular conceptions of each, but what is generally observed among all would seem to arise from natural inclination.

14. Now, it should be noted that nowhere else does Aristotle either use the arguments of Pythagoras to explain a proposition, or from the properties of numbers conclude anything about things. And perhaps he does so here on account of the affinity of numbers to magnitudes, which he is now considering.

Be that as it may, the proof here given does not seem valid, for it does not seem, if 3 is the number corresponding to "whole" and all, that it follows there are three dimensions. Otherwise, it would follow according to the same reasoning that there would be only three elements or only three fingers on the hand. But it should be known that, as Simplicius says in his Commentary 13, Aristotle is not here proceeding demonstratively but according to probability, and this is sufficient after previous demonstrations or ones supposed from another science. Now, it is plain that the task of deciding about the dimensions of bodies as such pertains to mathematics; and whatever the natural philosopher considers with dimensions, he takes from mathematics. Therefore, to prove demonstratively that there are just three dimensions pertains to mathematics—thus, Ptolemy proves it by showing that it is impossible for more than three perpendicular lines to meet at the same point, while each dimension is measured according to a perpendicular line. Supposing such a demonstration from mathematics, Aristotle here uses testimony and signs, just as he customarily does after his own demonstrations.

15. Then at (268a21) he goes on to manifest the main proposition from what has been shown. And he says that these three, namely, all, whole, and perfect, do not differ from one another according to species, i.e., according to their formal notion, because all imply a certain completeness; but if they do differ in any way, it is in matter and subject, insofar as they are said of diverse things. For we use all in discrete things, as we say "all men"; we use it also with respect to continua which are easily divided, as we say "all water" and "all air." Whole, however, is used both with these and with all continua, as we say "the whole people" and "the whole world." But perfect is used with respect to these and forms: for we say "perfect whiteness" and "perfect virtue." Therefore, because all and perfect are the same, the consequence is that among magnitudes the perfect one is body, because only a body is determined by three dimensions, and this carries with it the notion of all, as has been shown above, for since it is divisible in three ways, it follows that it is divisible in every way, i.e., according to every dimension. But among other magnitudes, there is one divisible according to two dimensions, namely, a surface; and another according to one, namely, a line. Now according to the number that it has, i.e., the number of dimensions that a magnitude has, so is it divisible and continuous. Thus, one magnitude is continuous in one way, namely, a line; another in two ways, namely, a surface; but a body is continuous in every way. Hence, it is plain that body is a perfect magnitude, as possessing all ways of being continuous.

16. Then, at all magnitudes, then, which are divisibles (268a28), he shows what is or is not plain from the foregoing. And he mentions three things.

The first of these is plain in itself, namely, that any magnitude that is divisible is continuous; for if it were not continuous, it would not be a magnitude but a number.

The second is the converse of this, namely, that every continuum is divisible, as was indicated in the definition. And this is plain from what was proved in Physics VI, as was said above. But it is not plain from what was just said, however, because here he supposes, but does not prove, that a continuum is divisible.

The third thing is plain from the foregoing, namely, that unlike the passing from length to surface and from surface to body, there is no passing from body to another kind of magnitude. And he uses a way of speaking employed by geometers imagining that a point in motion makes a line, and a line in motion a surface, and a surface a body. But from body there is no transition to another magnitude, because such a passing, i.e., to another kind of magnitude is due to a defect in that from which the process beings—that is why natural motion is the act of an imperfect thing. But it is not possible that body, which is perfect magnitude, should be defective in this way, because it is continuous in every way. Consequently, no transition from body to another kind of magnitude is possible.

17. Then at now bodies which are classed as parts (265b5), he manifests the proper perfection of the universe based on its difference from particular bodies.

First, he mentions how particular bodies are related to perfection. And he says that each particular body, according to the common notion of body, is such, i.e., perfect, inasmuch as it has three dimensions; nevertheless, it is terminated at an adjacent body, inasmuch as it touches it. And thus every such body is in a certain way many, i.e., perfect, in having three dimensions, but imperfect in having another body outside it at which it is terminated. Or it is many according to contact with diverse bodies; or it is many because there are more than one in one species due to imperfection, whereas such is not the case with the universe.

18. Secondly, at but the whole (268b8), he shows how the universe is related to perfection. And he says that the whole, i.e., the universe, which has particular bodies as its parts, must be perfect in all ways, for the word "universe" signifies perfect in all ways, and not in one way to the exclusion of some other way, and it both has all the dimensions, and includes in itself all bodies.