Lecture 6 — The heavens must be spherical, because this shape is most fitting

269 Again, since the whole revolves, palpably and by assumption, in a circle, and since it has been shown that outside the farthest circumference there is neither void nor place, from these grounds also it will follow necessarily that the heaven is spherical. For if it is to be rectilinear in shape, it will follow that there is place and body and void without it. For a rectilinear figure as it revolves never continues in the same room, but where formerly was body, is now none, and where now is none, body will be in a moment because of the projection at the corners. Similarly, if the world had some other figure with unequal radii, if, for instance, it were lentiform, or oviform, in every case we should have to admit space and void outside the moving body, because the whole body would not always occupy the same room.

270 Again, if the motion of the heaven is the measure of all movements whatever in virtue of being alone continuous and regular and eternal, and if, in each kind, the measure is the minimum, and the minimum movement is the swiftest, then, clearly, the movement of the heaven must be the swiftest of all movements. Now of lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line. Therefore, if the heaven moves in a circle and moves more swiftly than anything else, it must necessarily be spherical.

271 Corroborative evidence may be drawn from the bodies whose position is about the center. If earth is enclosed by water, water by air, air by fire, and these similarly by the upper bodies—which while not continuous are yet contiguous with them—and if the surface of water is spherical, and that which is continuous with or embraces the spherical must itself be spherical, then on these grounds also it is clear that the heavens are spherical.

272 But the surface of water is seen to be spherical if we take as our starting-point the fact that water naturally tends to collect in a hollow place—'hollow' meaning 'nearer the center'. Draw from the center the lines AB, AC, and let their extremities be joined by the straight line BC. The line AD, drawn to the base of the triangle, will be shorter than either of the radii. Therefore the place in which it terminates will be a hollow place. The water then will collect there until equality is established, that is until the line AE is equal to the two radii. Thus water forces its way to the ends of the radii, and there only will it rest: but the line which connects the extremities of the radii is circular: therefore the surface of the water BEC is spherical.

273 It is plain from the foregoing that the universe is spherical. It is plain, further, that it is turned (so to speak) with a finish which no manufactured thing nor anything else within the range of our observation can even approach. For the matter of which these are composed does not admit of anything like the same regularity and finish as the substance of the enveloping body; since with each step away from earth the matter manifestly becomes finer in the same proportion as water is finer than earth.

353. After showing that the heaven has a spherical shape on the ground that such a shape is the first of all figures, the Philosopher shows the same thing on the ground that this shape is most suitable to the heaven.

First from the fact that it is suitable to the heaven on account of its being the universal container of all bodies;

Secondly, on account of its motion being the universal measure of all motions, at again, if by complete (287a23; [356]).

354. With respect to the first (287a11) he puts forward two suppositions previously made clear.

The first is that the heaven is moved circularly—which is both evident to sense and is supposed from the proofs in book I. The second supposition is taken from what was proved in book I, in the chapter on the unity of the world, namely, that beyond the final circulation of the outermost sphere there is neither place nor void.

From these suppositions he necessarily concludes that the body of the heaven is spherical. For if it were not it would have to be either entirely rectilinear, or only partially circular without being a perfect sphere. But if the body of the heaven should be truly rectilinear—for example, a cube or a pyramid—it follows that beyond the heaven there would exist a place and a body and a void. That this consequence follows he shows on the ground that a rectilinear body in circular motion will not remain in the same place with respect to all its parts. As a matter of fact it will follow that where one part of it first was, no part of it now is; and again where no part of it now is, a part of it will later be. This is true on account of the interchange of corners. For in any body rectilinear in shape there must be certain bodily corners which extend beyond the other parts of the body, since a line drawn from the center of such a body [to a corner] is greater than a line drawn to some point on the plane surface of the body. Accordingly as the body turns, the line ending at the corner will reach the place previously occupied by the line drawn to a point between the corners and will occupy more space, and thus there will be a body where one was not before. The subsequent line [i.e., the line to the plane surface], arriving at the place formerly occupied by the corner will not be able to occupy the whole place formerly occupied by the corner, so that where there is now no body, there previously was body. Thus, therefore, outside the place in which the heaven now is, there can be existing a body, i.e., some part of the same heaven. Consequently, there is a place there, i.e., the receptacle of a body. Likewise there is a void, which is nothing more than a place not filled with the body of which it is capable.

355. But because there are certain figures that have no angles and yet are not spheres, he subsequently shows the same thing for these figures. And he says that a similar impossibility follows if we should attribute to the heaven some other shape, such that not all the lines from its center would be equal, as is proper to a sphere. And he says that there are two such figures, namely, lentil-shaped and oval-shaped. Now in an oval figure the line designating the length is greater than that designating the depth; indeed, an oval figure can be conceived as the result of two round pyramids joined at their bases. But the shape of a lentil is as though made in the fashion of a wheel whose width is greater than its thickness. Now in all such figures, it comes about in some way that beyond the ultimate motion of the outermost sphere a place and a void will exist, because of the fact that the whole is not always retaining the same place according to all its parts. This happens if one takes the poles about which the oval-shaped body is revolved on the part of its lesser diameter; for then the greater diameters must describe a circular motion and consequently one head of the moved oval will occupy a place in which previously no part of it was. But if, however, the length of the oval should be taken in its motion as the fixed axis, the revolutions would always take place according to circular parts, so that one part would succeed another. The same thing is also to be imagined with respect to the lentil-shaped figure; and likewise for a cylindrical figure and for any other such figure.

Hence, it is plain that only a spherical shape is such that, no matter how it be moved, no part will occupy some new place [not previously occupied by some other part]; rather one part of it always succeeds another part of it. Hence such a shape is most suitable to the heaven.

356. Then, at again, if by complete (287a23), he proves the same with another argument, based on the measurement of motions.

First he lays down the supposition that the motion of the heaven is the measure of all motions, as was maintained in Physics IV. And he gives the reason for this, namely, that only the motion of the heaven is continuous and regular and eternal; for in no other way could the quantity of other motions be certified through the motion of the heaven—and this is to measure them. For if the motion of the heaven were not continuous, but interrupted, there would not be equality of time between the measuring motion and the measured motion. If the motion were not regular, but now faster and now slower, it would not have within itself a determined certitude by which the quantity of other motions could be certified. If it were not eternal, there could not be measured through it motions that existed before and which will exist later, following the opinion of those who hold motion to be eternal by nature.

With these suppositions in mind, he argues to his point in the following manner: It is plain that whatever is the least in each genus is the measure of that genus, as is had in Metaphysics X, as a tone in melody, and the ounce among weights, and unity among numbers. It is also plain that the least motion is the one which is most rapid, i.e., the one having the least time, which is the measure of motion. Therefore the swiftest of all motions is the motion of the heaven. Here the swiftest of motions means the one which goes through its course sooner as to shortness of time, although there is not supposed equality on the part of the magnitude traversed by the motion, as is supposed in Physics IV, where it is stated that the swifter is that which in a lesser time traverse an equal, or even greater, distance. That is why he now adds here that the swiftest motion is considered with respect to the least magnitude. Now among all lines that return to their beginning the circle is the smallest—since in rectilinear figures there are angles such that lines drawn to them from the center are greater and consequently the corners of such figures jut out farther than a circular line. Accordingly, the motion of the heaven, which moves circularly, as though from the same to the same, and with the swiftest motion, traverses a circle in its motion. Therefore the heaven must be spherical.

357. Then, at corroborative evidence (287a30) he shows that the heaven has a spherical shape by using an argument taken from the lower bodies.

First he presents his argument;

secondly, he proves what he had supposed, at but the surface of water (287b4).

358. But it seems that this proof lacks necessity. For if it is assumed that water has a spherical shape, it will clearly follow that air on the concave side will be spherical, but this does not necessarily seem to follow for its convex side.

To this Alexander responds that from this demonstration it is proved that the bodies of the world are spherical as to their concave side, just as from the previous argument, which proceeded from the outermost heaven proceeding downward, it was proved that these bodies are spherical on their convex side. Hence according to this, neither of these arguments is complete without the other, but from the two arguments one demonstration results. But this seems to be contrary to Aristotle's intention, who presents each as a separate argument, as though each was complete in itself. And therefore it should be said, as Simplicius asserts, that this demonstration sufficiently proves that the bodies of the world are spherical both with respect to their concave and their convex. For since the convex surface of water is spherical, it is plain that the concave side of air is spherical. That the convex side of air is spherical is plain in the same way that it is plain for the water, namely, because all its parts equally concur to the same place. Accordingly it is also plain that the concave side of fire is also spherical. But that the convex surface of fire is spherical can be plain both from the fact that it is continuous with the sphere of the moon (for which reason it revolves together with it, as is clearly plain from the movement of a comet which moves from east to west in accordance with the motion of the heaven), as well as from the fact that the parts of fire are from all directions moved equally to their proper place.

On these suppositions let A be the center of the world, and B and G two designated points on the surface of the water that are equidistant from the center, and let there be drawn two lines AB and AG. Join the two points B and G so as to form the line BG, which will be a straight line, if the water's surface is flat. Let there be marked a point D in the line BG, which is the base of a triangle, and draw the line AD from the center. This line must be less than either of the other two lines AB and AG drawn from the center. For if it were equal, then all three lines proceeding from the same point would be equal, and thus the line BDG, passing through their extremities, would be circular, as is plain from Euclid III. But this would be against our assumption that the line BG is a straight line. Therefore, supposing that the line AD is shorter than the others, it will follow that the point D will be less distant from the center, and consequently its place will be deeper or lower. Hence, it will follow according to the foregoing supposition that the water in point G and in point B will flow to point D until the middle place is equated with the other two extremes. Let AE be the whole line equated to the two extremes from the meeting of the waters. Therefore the water must be along all the equal lines coming from the center; for water rests only when all the lines are equal. But the line which touches three equal lines proceeding from the center is circular, as is proved in Euclid III. It follows therefore that the surface of the water, on which the line BEG is described, is a spherical surface—and this is what he intended to demonstrate.

360. Then, at it is plain from the foregoing that the universe is spherical (287b14), he concludes from the foregoing that the world is manifestly spherical both on account of the first body which contains the entire world and also on account of the other bodies contained by it. But among us there are spherical bodies which nevertheless do not have a perfectly spherical shape; for example, the very body of the earth is said to be spherical, even though it has the great elevations of mountains and the depressions of valleys. Likewise, in our artificial bodies that are spherical, we find bumps and depressions, in spite of which such artifacts are said to be spherical in shape, since such additions or subtractions are reckoned as though nothing according to sense. Therefore lest anyone believe that the same thing happens in the heavenly body, Aristotle adds that it is turned with diligence, i.e., lacks all bumps and hollows, as do bodies carefully turned on a lathe, so much so that nothing chirocmeton, i.e., made by hand, is comparable to a heavenly body in this respect, nor any other natural body that our eyes behold. For the things from which such bodies are constituted cannot obtain, through the action of art or lower nature, that regularity, i.e., uniformity, and diligence for perfect spherical shape which a heavenly body possesses in virtue of its natural spherical shape. This he proves through the proportion of the parts of the world to one another. For it is plain that according to the same proportion by which water exceeds earth,. the containing elements always exceed the bodies contained—or even in greater proportion. But water, which contains the earth, does not have the swellings and depressions on its surface that earth has; rather its surface is more regular than the earth's surface. Similarly, the air's surface must be more regular than water s. Hence it follows that the surface of the outermost heavenly body is supremely regular, so that in it there is not even the slightest addition or subtraction.