Lecture 5 — Spherical shape of the heaven shown from fact that it is the first of figures

260 The shape of the heaven is of necessity spherical; for that is the shape most appropriate to its substance and also by nature primary.

261 First, let us consider generally which shape is primary among planes and solids alike.

262 Every plane figure must be either rectilinear or curvilinear. Now the rectilinear is bounded by more than one line, the curvilinear by one only. But since in any kind the one is naturally prior to the many and the simple to the complex, the circle will be the first of plane figures.

263 Again, if by complete, as previously defined, we mean a thing outside which no part of itself can be found, and if addition is always possible to the straight line but never to the circular, clearly the line which embraces the circle is complete. If then the complete is prior to the incomplete, it follows on this ground also that the circle is primary among figures.

264 And the sphere holds the same position among solids. For it alone is embraced by a single surface, while rectilinear solids have several. The sphere is among solids what the circle is among plane figures.

265 Further, those who divide bodies into planes and generate them out of planes seem to bear witness to the truth of this. Alone among solids they leave the sphere undivided, as not possessing more than one surface: for the division into surfaces is not just dividing a whole by cutting it into its parts, but division of another fashion into parts different in form. It is clear, then, that the sphere is first of solid figures.

266 If, again, one orders figures according to their numbers, it is most natural to arrange them in this way. The circle corresponds to the number one, the triangle, being the sum of two right angles, to the number two. But if one is assigned to the triangle, the circle will not be a figure at all.

267 Now the first figure belongs to the first body, and the first body is that at the farthest circumference. It follows that the body which revolves with a circular movement must be spherical.

268 The same then will be true of the body continuous with it: for that which is continuous with the spherical is spherical. The same again holds of the bodies between these and the center. Bodies which are bounded by the spherical and in contact with it must be, as wholes, spherical; and the bodies below the sphere of the planets are contiguous with the sphere above them. The sphere then will be spherical throughout; for every body within it is contiguous and continuous with spheres.

344. Having discussed the eternity of the heaven and the diversity of its parts, the Philosopher here determines the question of the figure of the heaven.

First he shows that the heaven is spherical in shape;

secondly, that this shape exists in it perfectly, at it is plain from the foregoing that the universe is spherical (L. 6; 287b14).

Concerning the first he does two things:

first he proves with arguments based on the heaven itself that it is spherical in shape;

secondly, with an argument based on the lower bodies, at corroborative evidence may be drawn (L. 6; 287a30).

About the first he does two things.

First, he proposes what he intends (286b10) and says that the heaven has to be spherical in shape both because such a shape is most proper, i.e., appropriate, to the heavenly body, and because it is the first of all figures not only by nature, as the perfect is naturally prior to the imperfect, but by substance, i.e., according to its notion, as one is prior to many.

Secondly, at **first, let us consider generally, (286b11; [345]) he proves his proposition.

First, he shows that the heaven is spherical in shape because this figure is the first of figures;

secondly, because it is most appropriate to the heaven, at again, since the whole revolves (L. 6; 287a11).

Regarding the first he does two things:

first he shows that the supreme heaven has a spherical shape;

secondly, that the other heavenly bodies also are spherical in shape, at the same then will be true of the body continuous (287a6; [352]).

345. With respect to the first he gives the following argument.

The first figure should belong to the first body. But among bodily figures, the spherical is first. Therefore the heaven which is the first body is spherical in shape.

In regard to this argument he first proves the minor premise;

secondly, the major having been laid down, he draws the conclusion, at now the first figure (287a2; [351]).

As to the first he does two things:

first he proves with arguments that the spherical shape is the first of all bodily shapes;

secondly, by appealing to the opinions of others, at further, those who divide (286b27; [349]).

Regarding the first he does two things.

First, he proposes what he intends (286b11) and says that we must determine universally which is the first of all figures both in the realm of plane figures, i.e., surfaces, and in the realm of solid, i.e., bodily, figures. A plane figure is the shape of a plane and a bodily figure is the shape of a body.

346. Secondly, at every plane figure (286b13), he proves his proposition.

First, with respect to plane figures;

secondly, with respect to bodily figures, at and the sphere (286b23; [348]).

With respect to the first he gives two arguments:

the first of which is this (286b13): every plane figure is either rectilinear, as in the case of a triangle or square, or it is curved, as in the case of a circle. But every rectilinear figure is enclosed by a plurality of lines and not by just one (for a single straight line can be extended only in one direction, whereas the nature of a plane figure is that it be entirely closed in every direction). On the other hand, a circular figure is enclosed by but one single line extended in all directions. Now, in every genus, unity is prior to multiplicity, and simplicity to composition. Hence it remains that in the realm of plane figures, the circular is first.

347. The second argument is given at again, if by complete (286b18), namely, that the perfect is by definition that outside of which there exists nothing that could befit it; for example, a man is said to be perfect, if he lacks nothing that pertains to man. And this has been settled previously both in Physics III and in the beginning of this book. Now, we observe that, so far as the nature of a straight line is concerned, addition can always be made to it—although for some other reason it might be impossible to do so, as, for example, to add anything to the diameter of the world. And this is plain if the straight line is finite. Accordingly, every finite straight line is imperfect. But it is plain that an infinite line, too, is imperfect, for it lacks an end, which it should have. On the other hand, no addition can be made to a circular line, because its end meets its beginning. Consequently, the line encompassing a circular body is perfect. Now the perfect is prior to the imperfect, absolutely speaking, both by nature and in point of time. But in one and the same thing, the perfect is prior by nature but the imperfect is prior in point of time—for example, a man in point of time is a boy prior to being a perfect man; yet the perfect man is prior by nature, because this is what nature intends. Absolutely speaking, even in point of time the perfect is prior to the imperfect, because a boy is generated by some man. From this it is plain that the circle, too, is the first among plane figures.

348. Then, at and the sphere (286b23), he shows which is the first shape among bodily figures. And he says that in like manner the sphere is the first among solid, i.e., bodily, figures, since the spherical figure alone is bounded by just one surface which everywhere surrounds the spherical body. Rectilinear bodily figures, on the other hand, are bounded by several planes—as a cubic body by six, and a triangular pyramid by four. Therefore, as the circle is to planes, so the sphere is to solids, i.e., bodies.

349. Then, at further, those who divide (286b27), he shows his proposition by appealing to the opinions of others. And he gives two such opinions.

The first is held by those who resolve bodies into surfaces, and produce bodies from surfaces. For the only body that does not resolve into a number of surfaces is the sphere, because it is bounded by just one surface, while other figures are resolved into many surfaces, as, for example, a pyramid is into four triangular surfaces. However, such a division of bodies into surfaces is not according to the manner by which a body is divided into its bodily parts; for a sphere also is thus divided into its parts. But the division under discussion is one that, so to speak, separates into things that are different in kind from the whole that is divided. He concludes, therefore, that it is clear that the sphere is the first among solid figures.

350. The second opinion is given at if, again, one orders figures (286b33). And he says that some have assigned the order of figures according to the species of numbers, by adapting the figures to numbers. And he says that, according to this, it is most reasonable for the circle to be adapted to unity, on the ground that it is the first and simplest of shapes; but that triangle be adapted to 2, because its angles equal two right angles. Now, if unity were aligned with triangle, it would follow that the circle, which is naturally prior to the triangle, would be outside the genus of figures, supposing triangle to be the first of shapes.

351. Then, at now the first figure (287a2), having proved the minor premise, he syllogises to his proposition. And he says that, since the first figure is due to the first body, and the first body is that which is on the outer periphery of the whole world, it follows that such a body, which is moved circularly, will in itself be spherical.

352. Then, at the same then will be true of the body continuous (287a6), he shows that even the lower heavenly bodies are spherical. And he says that from the fact that the first body is spherical, it follows that the next body continuous to it, i.e., immediately joined to it, is spherical: for that body which is continuous to, i.e., immediately joined to, the spherical body, must itself be spherical. Now this is true if the first body is spherical not only on its convex, but also on its concave, side. Since the very same nature of first body is in both these sides, it must have the same figure on both sides. The same argument holds for the other bodies which are in the center of these and contained by them, namely, they too have to be spherical. For those bodies that are contained and touched by the body that is spherical according to its convex side must also be spherical on their convex side, and consequently spherical according to their concave side, if they are of one nature. Since, therefore, the spheres of the lower planets touch the higher sphere, it follows that the whole of what is carried, i.e., the whole body which is circularly moved, has a spherical shape, for all those bodies of the heavenly spheres mutually touch and are continuous, i.e., in immediate contact one with the other. And there is no intermediate body that fills up voids between spheres, as some say—for it would follow that those bodies would be idle, since they would not have a circular motion.