Epitome Astronomiae Copernicanae: Liber I, part 3
Part II. Of the figure of the heavens
What, then, is to be held concerning the figure of the heavens? Since we cannot mark the matter of the aethereal air with the eyes, nothing hinders us meanwhile from believing that it is poured through all the amplitude of the world and goes round the elementary sphere on every side. That the host of the stars stands round the Earth on every side, and so forms as it were a vault of an entire spherical figure, is plain from this: since the Earth is round, to whatever places men come, they see the stars above their heads in the same manner as we. And indeed, for those who stay in one place, within the space of a few days the whole army of the stars is made visible, so that those which we see last seem to join and follow on those with which we began. The order of the stars, therefore, returns upon itself, drawn round in a circle about the Earth.
Do you think, then, that the centres of the stars are placed in the same spherical surface?
This indeed is uncertain. For since some are small and others large, it is not unlikely that the small ones seem so because they have withdrawn far into the high aether, the large ones because they are nearer to us. Nor yet is it absurd that two fixed stars of unequal apparent magnitude should be at an equal distance from us.
But of the planets it is certain that they are not in the same spherical surface with the fixed stars, but are lower than the fixed stars; for they sometimes cover them, and are not in turn covered by them.
If nothing more certain is established of the fixed stars, that region seems to be infinite; and this Sun of ours will be nothing other than one of the fixed stars, seen by us as greater and brighter because nearer than the fixed stars; and so about any fixed star there may be such a world as there is about us; or, what comes to the same, among the innumerable places in that infinite heap of the fixed stars this world of ours with its Sun will be one, differing in nothing from the other places about single fixed stars, as in the figure below, letter M. So indeed Bruno and some of the ancients. But it does not follow, if the centres of the fixed stars are not in the same spherical surface, that therefore the region through which the fixed stars are scattered is on every side like itself.
For that region has altogether some empty hollow and vast cavity, shut in and bounded by the host of the fixed stars poured thickly round it, as by a wall or a vault; and in the embrace of this vast cavity our Earth, with the Sun and the movable stars, is contained. See the figure below.
What argument have you for this? If the region of the fixed stars were everywhere similarly set with stars, even in the neighbourhood of our movable world, so that the position of the world and of our Sun had no peculiar circumscription more than the position of any fixed star, then there would appear to us a few huge fixed stars, and not more than twelve (the number of angles the icosahedron has) could be all of the same distance and magnitude from us; those succeeding these, not many more, would have a distance already double that of the nearest, others above them triple, and so on, always more manifold.
And since the greatest of all appear so small that they can scarcely be marked or measured by instruments, those which were twice or three times and so on farther away would appear twice and three times smaller, their true magnitudes being supposed equal, and one would soon come to those that would become wholly insensible: so very few stars would be seen, and those in the greatest difference.
But the contrary appears. For we see fixed stars of the same apparent magnitude crowded very thick upon one another, so that the Greek astronomers count a thousand of the more notable, the Hebrews eleven thousand; nor is the difference between the apparent magnitudes of the fixed stars so great. So great a number of stars, therefore, of equal appearance must be at not very unequal intervals from us.
Wherefore, since the aspect of the fixed stars appears to us almost the same on every side, as regards the magnitude and multitude of the stars, it will be raised above us on every side at almost equal intervals. There is, therefore, a vast cavity in the middle of the region of the fixed stars, and an evident vaulting of the fixed stars round about us, who are in its embrace.
In the belt of Orion there are three great stars, which are distant from one another by an interval of 83′; suppose the semidiameter of one to appear of one minute only; it would then appear to the sight of one placed on another 83′, that is, nearly three Suns wide, eight times greater in surface than the Sun himself. Such, therefore, is not the prospect from any one fixed star toward the others as it is from this world of ours toward the fixed stars, and we are farther from the individual fixed stars than neighbouring fixed stars are from themselves.
Why do you use the figure of the icosahedron here? Because as far as angle is from angle in it, so far, or not much less, are all the angles from the centre; the figure is therefore apt to express this almost equal dispersion of the fixed stars on every side, so that the centre, equally with the angles, represents one place among the fixed stars.
But if we consider a figure of more angles, as the dodecahedron, which has 20 angles, these are now nearer to one another than to the common centre, wherefore 20 stars so arranged about some place or star would now represent some vaulting and circumscription of a notable cavity, which is the very thing we rely on our argument to confirm.
In the figure given above, no. 16, in place of the icosahedron, a solid figure, there is drawn a six-angled figure, equivalent to it here in the plane. In the later figure, no. 17, in place of the dodecahedron there is drawn the equivalent ten-angled figure in the plane.
The force of this argument seems to be weakened if one should hold that the higher the stars are from the Earth, the greater they are, as a rule. For if, of so many stars which are seen at almost equal angles, some are supposed to have small bodies, others huge, it follows that the former are near, the latter most remote; and consequently those which now seem to us nearest to one another might in this way be most remote from one another. Then, if not by emptiness, at least by the smallness of the stars neighbouring our movable world, that place will be marked, and thus the very slightness of the stars will present the appearance of a void, while the magnitude of the stars, growing continually toward the outer parts, will serve for a notable vaulting; and in general there will be less stellar matter in that cavity in which the movable world is placed, more matter in the circumference which encloses and bounds it: so it will nevertheless follow that this place is singular and notable beyond the other parts of the region of the fixed stars.
It is more probable, however, that those which are of nearly the same magnitude to the sense are at nearly equal intervals from us, and that from the crowding together of so many stars the appearance of a hollow sphere is formed.
Have you another argument by which you prove that this place, in whose embrace the Earth is with the planets, is peculiarly marked out above the other places in the region of the fixed stars? The Way, by the Greeks called the milky, by our people the path of St. James, is spread through the middle of the orb of the fixed stars (as that orb appears to us), dividing it into two apparent hemispheres; and its circle is indeed of unequal breadth, but yet round about not very unlike itself. Therefore the milky way notably marks the place of the Earth and of the movable world, as against all other places in the region of the fixed stars.
For suppose the Earth to stand to one side by one semidiameter of the milky way, then this milky way will appear to it a small circle or small ellipse, wholly inclining to the other side, and it will be at once visible in one view, whereas now only half can be seen at any moment. Again suppose the Earth to be in the plane of the milky way, but near its circumference on one side; then that part of the milky way will appear huge, the opposite part narrow.
Therefore the sphere of the fixed stars is bounded downward toward us not only by the orb of stars but also by the circle of milk.
Is the region of the fixed stars, then, infinite upward? Here astronomy pronounces nothing, for at so great a height it is deserted by the sense of the eyes. This only astronomy teaches, that as far as even the smallest stars are seen, the space is finite.
Could not one of the visible stars be distant from us by an actually infinite interval? No: for what is seen is seen by its extremities. A visible star, therefore, has bounds round about. But if the star had withdrawn into an actually infinite space, these bounds also would be distant from one another by infinite spaces;
for all of them together, that is, the whole body of the star, would be partakers of this infinite height; therefore, the angle of vision remaining the same, the diameter of the star, which is the line between its bounds, would be proportionally increased with its distance, so that, as the diameter of a body twice as remote becomes twice as long as the diameter of the nearer, so also, the diameter of a star at a finite distance being finite, when the body is supposed to take on a distance multiplied infinitely, the diameter itself would become infinitely greater than itself.
But to be infinite and to be bounded are contradictory; to be infinite and to have to another infinite a definite, that is a finite, proportion are contradictory. Nothing visible, therefore, is distant from us by an infinite interval.
But what if there are some stars of finite bodies, scattered through infinite spaces upward, which on account of so great a distance are not seen by us? First, if they are not seen, they belong nothing to astronomy. Next, if the region of the fixed stars is bounded on the other side, downward, namely toward our movable world, why should it lack a bound upward? Third, although it cannot be denied that there may be many stars which, whether through slightness or through very great distance, are not seen, yet by these an infinite space cannot be obtained. For if each is finite in magnitude, all together must be finite in number. Otherwise, if infinite in number, however small, provided they have some quantity, they could constitute one infinite body, and there would be a body, open in three dimensions, nevertheless infinite, which implies a contradiction; for that is called infinite which lacks end and bound, and therefore dimension. So every number of things is actually finite by the very fact that it is a number; a finite number of finite bodies, therefore, does not posit an infinite space, as though piled up by the multiplication of spaces by a multitude of finite ones.
Yet will you not deny that the space at least is actually infinite upward? If the question is of empty space, that is, of a thing of nothing, which neither was created nor is, nor can resist another’s being there, the state of the question will be changed, and it will not be actual, being plainly nothing. But if space is for the sake of bodies to be placed in it, it has now been demonstrated that neither can any body capable of being placed be actually infinite, nor can bodies finite in magnitude be infinite in number. There is therefore no necessity that space for bodies to be placed should be infinite. Nor can there be, even between two bodies only, an actually infinite line. For again it is contradictory to be actually infinite and to be bounded on both sides by single finite bodies, like points, which are the bounds of a line.
What shall I hold of the infinite in potency, and can not infinite space or number at least be thought? It has a twofold sense when the infinite in potency is named. For either it is so taken as the division of a quantity is said to be infinite in power, namely that the sections are infinite, any one of which could be made in this quantity; yet whichever of them were made would be in itself finite, constituting finite parts. Or the infinite in potency is taken so that that potency is referred to the very amplitude of the whole infinite. In the former way it is true that no space, no number has ever been thought, nor will ever be thought, than which a greater cannot be thought; yet there will always remain infinite numbers, not yet even thought, any one of which could nevertheless afterward be thought. But in the latter way there is no infinite in power, namely that infinite space or infinite numbers should be thought actually all at once, that is, that the infinite be exhausted by thinking; for whatever is thought is by the very fact that it is thought finite; nor do we comprehend otherwise by the mind what the word infinite suggests to us, except as a thing of which only some part shines in the mind while the rest exceeds the mind’s thought; in which manner it is referred to the first sense and is a merely nominal concept.
If the world is finite, with what figure is it furnished on the outside? With what, if not the spherical?
What arguments have you for this? Astronomical, almost none; but two chiefly metaphysical; the first is drawn from the world itself, the other from its archetype.
Give the first. We dispute of the figure which encloses the world on the outside. All things, therefore, are within that figure, nothing outside. If it holds all things actually, it is very likely that it is also in form the most capacious. But a figure is more capacious when it is round than if the same quantity of surface were expressed in any other form that is not round, as the geometers teach, and Pappus in the fifth book of the Mathematical Collections. It is credible, therefore, that the world is bounded by a round surface.
Give the other. The archetype of the world is God himself, to whom no figure is more like (if any likeness has place) than the spherical surface. For as God is the being of beings, preceding all things, unbegotten, most simple, most perfect, immovable, most sufficient to himself and to all creatures, creating and sustaining all things, one in essence, in persons three, so the spherical also, in a certain rude manner, has the same properties among the other figures.
If so beautiful is the analogy between the spherical and divine things, it is worth the pains that the nature of the spherical be explained at greater length. Tell, therefore, why you make it the first of figures; I thought lines to be prior to surfaces, because they are simpler and consist of one dimension only, length; the spherical, as a surface, extends in length and breadth. First, lines are not themselves figures, but the bounds of figures. Next, lines are indeed prior to plane surfaces; but that is not on account of that simplicity in itself, for a surface is not composed of lines, but on account of the modes of generation, because a line generates a plane surface. But to the spherical surface lines are by no means prior, because they neither compose nor generate it, but rather arise out of it materially or formally. Finally, the simplicity of lines alleged is not of the essence of the figured thing, but rather its imperfection; since lines do not admit complete figuration, but only one element of it, in length. Such simplicity, however, which consists in participation, does not infer the priority of the participant over the participated.
Explain the difference of plane figures and of parallelepipeds, by the cause of their origin, that what has been said may be better understood. The figures generated in this metaphysical consideration are in a threefold difference. For either they are primigenial, or something less than the primigenial, or something more than they; that is, either arising by section from the primigenial ones, as parts, or by composition from the cut-off parts of the primigenial ones.
How are the primigenial ones generated? The primigenial ones are generated by flux or direct motion, first of a point, which is understood to hold a fixed position in another pre-existing figure; and from this flux straight lines are born. Then the straight line flows sideways, or toward a region which is outside its own stretch of length, and from this surfaces are born. If all the points of the line have flowed straight on to equal distances, which flux is uniform, what is then born is called a parallelogram, which is either a rhomboid, if the angle of the flux and of the flowing line was oblique, or a rectangle. Third, if the surface also so flows uniformly outside itself, a body is born, and indeed a parallelepiped; if the angle of the flux is right, also a columnar one.
If the length of the flux is equal to the flowing line, rhombic figures are born on both sides, with an oblique angle; but if the angle of the flux and of the flowing line is also right, in planes in place of the rhombus the square arises, in solids, the square flowing, the cube, the most perfect of the generated figures.
Explain also the origin by section. A line being drawn, which is called a diagonal, through the opposite angles of a parallelogram, the figure passes into two equal triangles, the rhomboid into scalene triangles of every kind, the rhombic into isosceles or equilateral. In solids, however, a plane surface being drawn through the diagonals of opposite parallel surfaces of a parallelepiped, the parallelepiped is resolved into two prisms. Then in a prism, a plane surface being drawn through three solid angles standing round some one angle of the prism, tetrahedra of every kind are cut off, and there remain quadrilateral pyramids, which by another plane drawn through the diagonal of the base and the vertex are soluble into two other tetrahedra, equal in bulk; so that every parallelepiped passes into six tetrahedra, equal in bulk.
How does the third kind arise by composition? All the remaining plane figures consist of triangles, which are parts of parallelograms, provided that single sides of pairs of triangles have the same length. So all many-sided bodies are composed of the aforesaid prisms or tetrahedra, parts of the primigenial ones, provided that single planes of pairs of tetrahedra coincide. And among composites the simplest are the quadrangular pyramids just mentioned, because they consist of only two tetrahedra each; prisms of three, the rest of more.
I understand the origin of the other figures: the spherical, then, seems to be among the primigenial, for it is born by the flux of a semicircular line about immovable poles and axis; and so the circular line will be prior to it. This mode of creating the spherical is merely geometrical, or rather mechanical, used by geometers to help the grasp of beginners, but is by no means suited to the nature of the spherical, or to its metaphysical consideration. For when the semicircle is carried round, certain points, as was said, rest; the nearest move slowly, the middle most swiftly, whereas the spherical is on every side like itself; whereas also in the origin of the primigenial figures, which are the viler, the flux of all the points was most uniform. And whereas the other primigenial figures have lineaments by which they are understood to have been so generated, in the spherical there is on the contrary no trace of this generation; for there is nothing in it which, above the other points, has the character of a pole. This, therefore, is not the genuine and metaphysical account of the generation of the spherical. Finally, the semicircular line, as generating, would be prior to the spherical, as generated, which is absurd and impossible, since it is posterior not only to the spherical but even to the plane figures themselves.
Prove this posteriority of the semicircular line. The circular line is understood to arise in two ways: the first, vulgar and mechanical, is this, that with one arm of the compass standing, the other goes round until it returns into its own tracks; although this mode is foreign to the nature of the circle, for it begins from a definite point, whereas in the circle there is nothing that has the character of a beginning. A circular line, therefore, cannot so exist without a plane surface in which it lies, a surface cannot be created without a straight line, a straight line without a point, a point cannot be without position in a place, a place without a figure that contains the place; and such a figure, circumscribing a place, is the spherical; from the spherical, therefore, is the beginning of the point, of the line, of the plane surface and of the circular line.
The other mode, metaphysical and suited to the nature of quantities, by which the circular line is generated, is by the section of the spherical made by a plane; again, therefore, both the plane and the spherical, as subjects and instruments of generating, must pre-exist the circular line. The spherical, therefore, is prior also to the circle.
Is the spherical, then, plainly an ungenerated figure? It is called ungenerated in respect of the other figures, because it is generated in none of those ways, but rather furnishes points and place for generating the others; but of itself it has a certain mode of generation, far most different, not from another, but from its own inmost point, of which below.
Prove that the spherical is also the simplest and most perfect of figures. All imperfection is in multiplicity, beauty in simplicity. The other figures, which are called perfect, are enclosed by planes, each by its own number, into which they are also resolved or divided; and these planes, since all are alike among themselves in each one, give rise to the name of perfection. The spherical is contained by a single surface, on every side most like itself, returning into itself and bounding itself; nor is there anywhere any trace that goes before to a division of the figure. That it is divisible into parts happens to it as being a quantity, not as being a figure.
Next, the other figures are not perfect unless they consist of three dimensions and are within full and as it were materiate, whence they are also called bodies. The spherical has the character of the immaterial, because it is understood without internal solidity. For the globe and the spherical differ. That is the globe which of a solid body draws off the spherical within and encloses it within itself.
But you will not call the spherical immovable, since no figure is more apt for motion? Motion is not rightly attributed in geometry to figures, since no figure is purely geometrically more movable than another. But if you consider physically, it is true in a certain respect. For if a round body is placed on a plane and an external mover approaches: these two conditions being posited, the figure is movable. So also angular figures are in a certain respect stable, if they are placed on a plane upon one of their planes. But if you place such a figure upon its angle, beside the spherical, which is wholly angle, the spherical will indeed rest, but the angular will move itself and fall over onto one of its planes. Therefore the spherical also is called immovable by this very fact, that in no part has it any cause within itself for motion, since it is on every side like itself. Nor does it find outside itself, in any other quantity, a moving cause, or one disposed to physical motion by geometrical properties, since it has been proved to be the first of all figures.
How will you prove that the spherical suffices for itself and for others? The other figures, as I said, are described, comprehended and bounded by planes, planes by lines, lines by points; but points require position in space, and space wants to be determined by some surrounding thing. The spherical, resting on itself, requires nothing outside itself that is not itself; for it is bounded by itself and by that very fact shuts within itself a place in which points, lines, surfaces and bodies can be constituted, a beginning being given.
But will you also set up this figure as the creator and sustainer of the others? Geometers indeed know that all the beauty of similarity, of reason and of perfection in the other figures is in them derived from the spherical. For the planes of one body must all square with the same circle, and the circle is from the spherical, as was said. Then all the solid angles of any perfect figure must be equally distant from the centre of the figure, that is, stand in the same sphere. The spherical, therefore, is for the other figures the cause of perfection and the norm, which has the character of form.
The comparison of the sides of a figure, and of the other diameters of a body, and of several bodies with one another, is made no otherwise than from a sphere posited, in which all bodies lie, divided by the diameter of that sphere in definite ratios. The spherical is, therefore, for the other figures the cause and norm of definition or scientific description, or of proportions. But the very beauties of these proportions are constituted and understood only by the benefit of the circle, and the geometers themselves call it the divine proportion.
Show that there is in the spherical an image of the Trinity to be adored. In the spherical there are three things: the centre, the surface and the equality of the interval; if one of them is denied, the rest fall, and they are distinct from one another, so that one is not the other.
The centre is as it were the origin of the sphere, for the surface is understood not to go out from the centre, but to have gone out by straight lines infinite in number through the intermediate space into all regions, with no trace of itself left in the intermediate space, the point communicating itself into this amplitude as far as the equality of all the intervals; which is a far other manner of generation than that which geometers deliver for the sake of understanding, above, p. 141. And note that the creation of one straight line, delivered before, is the finite image of this infinite generation of the spherical surface from the centre. The centre is in itself invisible and past tracking out, but is shown on every side by the most uniform curve of the surface, by the mediation of the equality of the interval. Therefore the surface is the character and image of the centre, and as it were an effulgence from it and a way to it; and he who sees the surface thereby sees also the centre, and not otherwise.
The interval results from the comparison of the centre with the surface, and so proceeds from both and measures and searches the depth of this figure. But if any solid or plane figure is described within the spherical, it rests upon the surface by nothing else than certain ultimate bounds, that is, points, but in the whole amplitude, whether of body or of plane, it is extended through the interval and sustained in it. Therefore from the centre through the interval all the other regular bodies are made to rest upon the surface.
Have you no argument left for the roundness of the world, besides the two principal ones explained? Add this also from astronomy, that it is fitting that the whole be not bounded by a less noble figure than the parts. But the chief parts of the world are globular; this has already been proved of the Earth, so the Sun is always seen in the figure of a circle, the Moon for the most part, and the same they affirm of the other planets, who contemplate them more attentively with skilful glasses. Wherefore it is fitting that the whole world also be globular on the outside.
The Sun, Moon and planets appear in the form of a circular disc; they are not, then, solid globes. It does not follow that they are therefore truly plane discs. For the opticians demonstrate that even solid globes, if viewed from afar, are seen as circular discs.
But whence do you conjecture that they are rather solid globes than plane circular discs? First, from comparison with the Earth, that the heavenly bodies may not partake of circularity less perfectly than the Earth. Next, from the power of the bodies. For the Sun could not pour forth so much light and heat from himself if he lacked body, if he were a mere plane surface. Third, it will be proved in another part of astronomy that the Sun revolves about his own axis, and it is credible that the same is true also of the five planets. The opticians demonstrate, therefore, that if a revolving body always keeps the appearance of a circular disc, it is globular. Fourth, the light of the Moon, Venus and the other planets is borrowed from the Sun. If, then, the Moon or Venus were a disc, it would be illuminated at once on one surface, but this does not appear: for the illuminated part is first hollow, then takes on the aspect of a bisected disc, afterward the gibbous form arises, and finally the circle is perfectly filled. But these things happen about a globe illuminated on one half, if it is viewed from a distance, as we are taught in the optics.
But the figure of the heavens appears far other, namely, as you said at the beginning, like a cloth spread over us and gently blown out in the middle; since what is in the middle is near to the spectator, and round about it stands farther off. It is a deception of sight, which first lacks the aid of arguing the distances of the stars, the thing not being visible of itself; and next, even if the sky is overdrawn with clouds of equal height, the imagination of a vault, high in the middle, still follows, from the same principles of optics, so that when winds blow, all the clouds seem to tend to one point of it.
Every figure is in matter; what, then, is that matter in which the world is clothed on the outside? From sight nothing can be drawn in any direction; we rightly follow, therefore, the authority by which we are taught that the stars are all in the expanse, which in Hebrew is called raquia, and that the waters above it are the covering, that is, above the aethereal air attenuated to its extreme degree, and that the orb of the stars in it is made of water; which water, if anyone contends that it is congealed by frost and crystalline because of the Sun’s very long absence, is permitted him by the Copernican astronomy, which, content with the aspect of the stars themselves, does not use that orb, whatever it be.
For whatever figure of the world be posited, and however the stars be disposed at most unequal intervals, the eye of the spectator on the Earth will always imagine to itself that all those stars stand round it, as a centre, in the manner of a perfect spherical surface.