Epitome Astronomiae Copernicanae: Liber IV, part 2
How great, then, are the intervals of the several orbs? The Copernican demonstrations establish that the distance of Saturn is a little less than ten times that of the Earth from the Sun, that of Jupiter five times, of Mars one and a half, of Venus about three quarters, of Mercury about one third. So the diameter of the orb of Saturn is less than twice that of its neighbour Jupiter, that of Jupiter three times that of Mars below it, that of Mars one and a half times that of the terrestrial orb placed around the Sun, the terrestrial more than one and a third times the Venerean, and the Venerean about five thirds or eight fifths of the Mercurial. Where, however, it must be noted that the proportions of the distances are different in different parts of the orbits, especially in Mars and Mercury.
What, then, is the cause of the intervals themselves of the planets, from which the periodic times follow? The cause of the intervals in the archetype is the same as that of the number six of the primary planets.
I beg you, do you hope that causes of the number of the planets can be assigned? This care has succeeded, God being favourable, not badly. The reasons of geometry are coeternal with God; in them the first thing is the difference of the curved and the straight. The curved, it was said above in book one, bears in a certain manner a likeness of God, the straight represents creatures. And in the adornment of the world the outermost region of the fixed stars was first made spherical, after that geometrical likeness of God, because that, like some corporeal god (worshipped among the nations under the name of Jupiter), had to contain all the rest within itself. Straight quantities therefore belonged to the inner contents of the outermost sphere, the first and most beautiful of them to the primary ones. Of the straight figures the first, most perfect, most beautiful and most simple are those which are called the five regular bodies, which already two thousand years ago the Pythagoreans said were the figures of the world, supposing that the four elements and the heaven (the fifth essence) were formed after their archetype. But the truer account is that those five figures form as many intervals of orbs, enclosing one another. If, then, the spherical intervals are five, it is necessary that the orbs be six: as for four linear intervals it is necessary that there be five fingers.
What are those five regular figures? The cube, the tetrahedron, the dodecahedron, the icosahedron, the octahedron.
How are these figures distinguished, and into what kinds? The cube, tetrahedron and dodecahedron are primary, the octahedron and icosahedron secondary.
Why do you make those primary and these secondary? Those three have a prior origin and the simplest angle, that is, a three-lined one, and each a plane of its own; the two latter have their origin from the primary, and an angle of more lines and more compound, and a borrowed plane.
What is the order of the primary? These are called primary only with respect to the secondary, for among themselves they still have this order of priority: cube, tetrahedron, dodecahedron. For in these figures appears the first metaphysical opposition of all, between the same and the other or different. In the cube identity is seen, in the other two diversity. And between these there is the first geometrical contrariety, namely that which is between more and less than itself. For the cube is the thing itself, the tetrahedron is less than the cube, the dodecahedron more than the cube; or the cube is the first of the things generated, the tetrahedron the first of those cut out of the cube, the dodecahedron the first of the composites, the cube being increased and covered; which idea also prevails in their planes, the square, the triangle and the pentagon: for the square is generated first of all by the simplest and most uniform motions, as was said in book one; the same is resolved into two triangles, whereas the pentagon is composed of three suitable triangles.
Explain the genesis, the primacy and the form of the cube. Straight quantities have an origin visible to the mind; the spherical, as was said above, bears a certain character of eternity or eternal generation. The spherical being posited, a point is posited in its middle and infinite points on its surface. From the flow of point to point, therefore, arises a line; from the lateral flow of a line, a surface; from the lateral flow of a surface, a solid. If the flow is straight and also shortest, there arises hence a straight line bounded by two points; if the flow of a straight line is such that all its points flow equally, a parallelogram arises, bounded by four lines; if the parallelogram likewise flows, a parallelepiped arises, bounded by six planes. Again, if the flow of the line is equal to the line that flows, and the angle of the line along which the flow is made to the flowing line is anything but a right angle, a plane figure arises, called a rhombus, whose sides are equal to one another; if the angle was a right angle, it is a square that arises; if the square likewise flows, a cube arises, whose six planes are all squares and so equal to one another. Now the shortest is prior to the winding, the equal and like to itself to the unequal and unlike, the straight to the oblique. As, therefore, among generated lines the straight is prior (for the circle is posterior to the plane, the plane to the straight), among surfaces the square, so among quantities that which is perfect, that is, consists of three dimensions, namely among bodies, the cube is proved to be first.
Explain the primacy of the tetrahedron among the cut figures, and the mode of cutting from the cube, and the form. When bodies are diminished so that a lesser exists, other solid figures are wont to arise, of which the first is to be reckoned that which arises if the first of the generated, namely the cube, is cut most simply and most equally. But there is no cutting (of those at least which designate a new plane figure) more uniform or simpler than if you cut off the four angles of the cube at the root: for you cut off as many tetrahedra, each with a right solid angle B and an equilateral triangular base ACD; there is left as it were a kind of belly of the cube, namely a fifth tetrahedron, everywhere like itself, contained of course by four equilateral triangles.
But if you use that cutting of the cube of which was said in book one, not five but six irregular tetrahedra will be made. Of the diminished figures, therefore, the tetrahedron is the first; it is a third part of the body of the cut cube, and each cut-off angle, such as BACD, is a sixth part of the same whole.
Explain also the origin of the dodecahedron from augmentation, and the reasons of its posteriority among the three primary, but priority among the augmented. As in the diminution of the cube, for the four angles of the cube cut off, four planes were constituted, and the remaining four angles of the cube stayed for the tetrahedron, but diminished and of the same kind, that is three-lined: so also, if we wish to constitute the first of the augmented, or those which are more than the cube, for the planes of the cube we constitute angles, and the angles of the cube we also transmit to the augmented, but clothed and increased the same leads, with twelve sides of the cube and as many planes laid over; AED is the plane of the cube, for which planes are to be laid down, just as before, for the six of the tetrahedron two angles are made, and square planes had been laid on the sides: there become also the angles of the cube A, E: for as the cube covers the tetrahedron, so this augmented pentagon covers the cube; and the figure which we here seek covers the cube. But if for each plane of the cube we were to set a single angle, we should set six four-lined angles, because the six planes of the cube are quadrangular, and eight three-lined angles of the cube would remain: the figure would therefore be mixed. In order, therefore, that the three-lined angle of augmentation may remain and all the solid angles be homogeneous among themselves, two angles are to be imposed on each cubic plane, not a single one, six prisms, such as the one before, not 6 pyramids, such as this one is, so that of two adjoining prisms there is always one common plane, laid on one side of the cube. And these 6 prisms make a little less than the cube itself on which they are placed.
So there will be made by the augmentation 12 angles, to which are added the eight angles of the cube; the sum is 20 angles.
How is the form of the dodecahedral plane built from this? The angles of the figure, as has been said, must be 20, each of three lines, of which any one meets at two angles; three ends twenty times make 60, but two ends close one line: therefore the lines or sides of the figure are 30, which are in power 60, with respect to the planes of the figure; for each side of the figure meets at two planes. The sixty lines or sides, divided into twelve planes, as necessary for this solid figure, indicate five for each. The planes, therefore, are five-sided. Of the augmented, therefore, the first again is the dodecahedron, having pentagonal planes.
What is the origin of the secondary figures, and why only two? To these three figures, the cube, tetrahedron and dodecahedron, three others indeed correspond, but one of them coincides with its primary; and they too are generated by diminution of the three primary, but by a diminution of a different kind, where not a side is left for a plane but an angle, for the surface, that is, of the primary figure not a line of the secondary but a point, the number of sides remaining; at the same time, however (as before), the plane of the secondary is generated for the angle of the primary, and a triangular plane, because the angle of its primary is three-lined, the three centres of the three planes of the primary, surrounding the solid angle, being joined. These second-born are, therefore, as it were certain entrails of the former. For whatever appears outwardly falls from the cube, there are left of it only the 6 centres, like certain navels of the 6 planes, and the angles of the new figure become six: and because the cube had 8 angles, the figure now receives for them 8 equilateral triangular planes, and so it is called the octahedron, and is the sixth part of its cube. So with the tetrahedron: for its 4 triangular planes 4 angles are constituted; for the 4 angles 4 triangles, and the same figure arises as its primary: so it is not counted as new. It is, however, the twenty-seventh part of the tetrahedron in which it is inscribed. So it is also with the dodecahedron, which from its 12 bases bestows on the new figure 12 angles, and for its 20 angles bestows on its secondary 20 triangular bases, whence the figure is called the icosahedron: and it is a little less than half of its dodecahedron.
One of the primary was generated by diminution of the cube, one by augmentation. Here, then, the octahedron: since the secondary are generated by diminution, is nothing of the secondary generated by augmentation? To this second diminution there corresponds indeed also a second augmentation of those three primary, an angle succeeding in the place of a plane, a plane in the place of an angle, but the figures that arise are the same as were made by this diminution. For as before the octahedron was inscribed in the cube, the icosahedron in the dodecahedron, so now conversely the cube is imagined inscribed in the octahedron, the dodecahedron in the icosahedron. All being thus surveyed, five first figures are found.
Why do you call them the simplest figures? Because each is enclosed by planes of a single kind only, namely triangular or quadrangular or pentagonal, and also by a solid angle of a single kind, three-lined indeed in the three primary, four-lined in the octahedron, five-lined in the icosahedron. The other figures vary in one or the other. For there are some which have one kind of planes, like the rhombic mentioned, but not one kind of solid angles; for the rhombic dodecahedron has 6 four-lined angles and 8 three-lined, the rhombic triacontahedron has 12 five-lined and 20 three-lined. There are others which mix different planes, having uniform solid angles, like the 13 kinds of Archimedes. (Compare Vol. V, p. 123 ff.)
Why do you make those five the most beautiful and most perfect? Because they imitate the spherical, the image of God, as far as can be done by a straight figure, arranging all their angles in the same sphere and being inscribable in the sphere; and as the sphere is everywhere like itself, so here the planes of each figure are all alike among themselves, and are all inscribable in one and the same circle, with equal angles.
Could not more figures like these be constituted by some other method? By no means. For the solid angle of any figure is constituted by at least three planes. Therefore equilateral triangles meet three, four or five at a time, quadrangles three, pentagons likewise three, at the solid angle. But six triangles and three hexagons fill a plane and do not rise into a solid. And more than these, as also three heptagons and three of any others, exceed the sum of 4 right angles, which are arranged about the same point in a plane. See the last proposition of book XIII of Euclid, the scholion, and book II of my Harmonics.
How, then, from these figures are the number of the primary spheres and the intervals of the planetary orbs derived? Each figure is understood to have two spheres, one circumscribed about it and one touching the centres of its planes, so that the first sight of the figure as it were invites some architect to circumscribe and inscribe spheres: such, then, as is the proportion of the outer sphere to the inner, such also has been made the proportion of the sphere of the upper planet to the next lower, between which there is that interval.
What are these proportions of the orbs in the several figures? Let the semidiameter of the circumscribed sphere be 100000; the proportion of the inscribed will be this.
In the cube 57735. In power, a third part of the radius of the circumscribed.
In the tetrahedron 33333. A third part of the radius of the circumscribed.
In the dodecahedron 79465. A part inexpressible between two thirds and three fifths of the power of the radius of the circumscribed, namely the power of the apotome taken from eleven fifteenths In the icosahedron 79465, of the power of the radius.
In the octahedron 57735. In power a third part of the radius of the circumscribed.
The octahedron has also in its middle a square, formed by four middle sides, in which if a circle is inscribed, its semidiameter will be 70711, in power half of the circumscribed.
Show now what place belongs to the orb of the Earth among these figures. The five bodies were assigned above to two classes, to three first-born and two second-born, of which the former had a three-lined angle, the latter a many-lined. For as Adam is the first-born, Eve not his daughter but a part of him, both of whom are called the first-formed, whereas Cain and Abel and their sisters are now their offspring: so the cube is in the first place, from which in another and simpler way arose the tetrahedron, as it were a rib of the cube, and the dodecahedron, so that nevertheless all three remain among the primary; the octahedron and icosahedron, however, from the cube and dodecahedron as fathers, and from the triangular plane of the tetrahedron as a mother, two offspring have now been born, each bearing the likeness of its parent.
The three first figures of the same class, therefore, had to include the circuit of the centre of the Earth, the two second-born, as another class, had to be included by the orb in which the Earth revolves, and so that orb became the common boundary wall of both orders, because the chief of the movable globes was to be the Earth, the dwelling of the image of God. For in this way the nature of inscription is kept in the second class, of circumscription in the first: for it is more natural and more fitting that the octahedron be inscribed in the cube and the icosahedron in the dodecahedron than the cube in the octahedron and the dodecahedron in the icosahedron.
So, then, the circuit of the centre of the Earth became the middle of the planets; for outside it three had to be placed around, on account of the three primary figures, within its circuit two, on account of the two figures of the second class, to which a third was added, the Sun, in the very innermost embrace of the centre of the movables. So Saturn, Jupiter and Mars became the upper, Venus, Mercury and the Sun the lower; the Moon, however, encircling the Earth privately in the same common circuit around the Earth, is among the secondary planets, as was said above.
What is the order among the three outer figures, and what place belongs to each among the planets? The cube is the first of the figures, and is therefore placed between the two outermost orbs, Saturn and Jupiter; the tetrahedron follows in the genesis of figures, and so obtained the place between Jupiter and Mars; the last of the three was the dodecahedron, and so the last place was assigned to it, between the circular regions of Mars and the Earth.
Assign places also to the two inner ones. Although the octahedron has the nature of the cube, whose are the first parts, and the icosahedron that of the dodecahedron, whose are the last, yet the nearest place after the dodecahedron did not belong to the octahedron, for two causes. For first, the two classes of figures are in a manner opposed to one another: it was therefore fitting that the beginning of the placing should be made from opposite terms. But since, of the outer figures, that place was reckoned first which inclined more toward the outside, it followed that of the inner figures that place should be first which inclined more toward the inside, toward the centre. Secondly, it was more fitting to the nature of similar figures, the dodecahedron and icosahedron, and more apt to their mutual inscription, that they should succeed each other immediately, the circuit or orb of the Earth intervening, at which as at a common boundary wall each class of figures should end. So, then, it came about that the icosahedron was placed between the orbits of the Earth and Venus, and the octahedron between the innermost, those of Venus and Mercury. The Sun, however, has no orb in which his centre is carried round; he is therefore outside the count of the primary movables, but has in himself the fountain of motion, as externally the fixed stars have in themselves rest and give place to the movables and contain them.
Is the proportion of the figures also found between the orbs which you have given to each figure? The same proportion is so found that, although in the smallest things something is lacking, yet no interval of any two planets comes nearer to the proportions of the orbs of another figure than that which has hitherto been ascribed, by the best reasonings, to the two planets.
You see, in fact, that as Saturn above had less than twice the diameter of the orb of Jupiter, and Venus likewise less than twice the diameter of Mercury’s, namely five thirds or eight fifths, so also in the cube and octahedron 100000 is less than twice 57735. For if you take three fifths, you will have 60000; but if five eighths, then 62500 results. Again, as the Martian orb to the orb that carries the centre of the Earth had almost the least proportion, and nearly equal to the proportion of the orb of the Earth to the Venerean, so you see also in the dodecahedron and icosahedron the proportion of the orbs is the least, namely 100000 to 79465. You see thirdly that, as Jupiter to Mars constitutes the greatest proportion of orbs, namely threefold, so also in the tetrahedron the diameter of the circumscribed is three times that of the inscribed.
If the intervals come so near to the proportions of the figures, why, then, does some discrepancy remain? 1) Because the archetype of the movable world consists not only of the five regular figures, by which the courses of the planets and the number of the runners were defined, but also of harmonic proportions, by which the courses themselves had to be tempered to a kind of idea of celestial music, or of a harmonious concord of six voices. Since, however, this musical adornment required a distinction of motion in any one planet, of the slowest from the swiftest, which distinction is effected by variation of the interval between the planet and the Sun, and since the quantity or proportion of this variation was required to be different in different planets: hence it was necessary that from those figural intervals, which are exhibited by the figures uniform without variation, a very little should be taken away and left to the liberty of the harmonist for representing the harmonies of the motions.
2) Nor yet was the property of the regular figures neglected even in this so minute discrepancy. For as the proportion of the orbs of the tetrahedron is perfect, that is expressible simply, those of the cube and octahedron semiperfect, that is expressible in power, inexpressible in length, but those of the dodecahedron and icosahedron wholly imperfect, that is altogether inexpressible: so also the proportion of the tetrahedral planets imitates the figural most exactly, that is almost at the very extremities of the intervals; the proportions of the cubical and octahedral are less exactly figural, because the extreme intervals recede from them, but the intermediate ones square; but the whole spaces of the dodecahedral and icosahedral planets have deserted their figural proportions, although no others come nearer. For behold, how the longest distance of Mars from the least of Jupiter is most exactly a third part, as in the tetrahedron the inner orb of the outer, so that, the angles of the tetrahedron being placed in the innermost orb of Jupiter, the tetrahedral planes touch in a manner the outermost orb of Mars; behold again, how, the angles of the cube being placed in the innermost orb of Saturn and those of the octahedron in that of Venus, the planes of the figures are indeed sunk into the regions, those of Jupiter, these of Mercury, yet do not pass wholly through them, but penetrate only about to the middle; behold finally how, the angles of the dodecahedron being placed in the innermost orb of Mars and those of the icosahedron in that of the Earth, the planes of the figures by no means reach the regions beneath, those of the Earth, these of Venus: while meanwhile no intervals of the planets come nearer to these least proportions of all the figures. See on these things my Harmonies, book V, chapter 9, proposition 48 and throughout, where the causes are drawn out not only of the exact quantity of the proportions between pairs, but also of the extremes of the intervals of each one singly.
Can any conjecture about the interposition of the figures also be drawn from the periodic times? All the proportions of times are indeed greater than the proportions of their orbits and so also than their figural proportions, as will be explained in the second part of this book: yet among them also the property of the figural can be recognised without difficulty. For as of the figural proportions there are three, the greatest solitary, but the middle and the least both double (the former from the single tetrahedron, the middle from both the cube and the octahedron, the least from both the dodecahedron and the icosahedron): so also between Jupiter and Mars the greatest and solitary proportion of times is nearly that of 6 to 1, namely 12 years to less than 2, a proof of the interposed tetrahedron; between Saturn and Jupiter, and between Venus and Mercury, the proportion of times is less and on both sides nearly the same, a proof of the interposed kindred bodies, there the cube, here the octahedron, which make the proportion of their orbs the same. For as 30 years of Saturn stand to 12 years of Jupiter, so very nearly do 225 days of Venus stand to 88 days of Mercury; finally between Mars and the Earth, and between this and Venus, the proportion of times is the least and again nearly the same on both sides, a proof of the interposed dodecahedron there, icosahedron here, bodies kindred and of the same proportion. For as 687 days of Mars stand to 365 and a quarter of the Earth, so do 365 and a quarter days stand to 194, whereas Venus has for these 225 days, namely somewhat more, making this temporal proportion the least of all. The causes of so slight a dissimilarity are explained in Harmonies, book V.
Have you any other proof, besides that from the two classes of figures, that the globe of the Earth was given chief account in the ordering? It is not fortuitous that the mean interval of the Earth, the middle planet, from the Sun is found quite precisely proportional in the middle place between the shortest interval of the lowest of the upper planets, Mars, and the longest of the highest of the lower, Venus. For as was said above, the space between Mars and Venus was left for the Earth by the figural inscriptions indefinite and loose and so free, in the dividing of which by the orb of the Earth either this or another proportion, if another had been better, could be expressed. This middle wall, therefore, of the figural classes, the middle wall of the upper and lower planets, had also to be mediated geometrically.
What, then, defined that space which the inscriptions did not define? Although there is a certain augmented figure, namely the spiked dodecahedron, which is found to define this space as accurately as the space between Jupiter and Mars is defined by the tetrahedron, and the association of that imperfect figure with its kindred, the dodecahedron and icosahedron, does not seem to lack its reason: yet neither this nor any other spaces are defined exactly by the figures alone, but this office was left to the harmonic adornment of the motions, which demanded for itself some liberty in defining these spaces exactly.
IV. Of the proportions of the chief bodies of the world among themselves.
Where do you think the inquiry into the proportions of the bodies should begin? From the Earth, 1) as the dwelling of the contemplative creature, 2) and likewise also the image of God the Creator, 3) for we read in divine Moses that in the beginning God created Heaven and Earth. 4) The orb of the Earth is also the figural middle among the planets and their common boundary wall, and between the limits of the upper and lower planets also the geometrical mean proportional. 5) Finally the very structure of these proportions cries aloud that God the Creator, in adapting the bodies and intervals to the body of the Sun, as to a measure prior in origin, made his beginning from the Earth.
What cause do you hold to be that of the magnitude of the body of the Sun? That the globe of the Sun is the first of all the bodies of the world in the order of creation, at least archetypal, if not also temporal, these things persuade: 1) Moses makes the work of the first day light, by which we can understand the body of the Sun. 2) The body of the Sun has obtained above, under very many names, the principate in natural things; therefore also in quantity and in the time at which it was created. Now the first body, by that very fact that it is first, received no proportion to those that follow, but those that follow rather to it as first. Wherefore there is no archetypal cause of the magnitude of the Sun, nor would any other globe have been twice as great as it now is: since together with it the whole remaining universe and man in it would have been twice as great as now.
By what medium, then, was the magnitude of the Earth adapted to the magnitude of the solar globe? By the medium of the vision of the Sun. For the Earth was to be the dwelling of the contemplative creature, for whose sake the whole universe was founded. Now contemplation has its origin in the vision of the stars; therefore also the quantity of the things to be contemplated had to have its origin in the quantity of the things to be seen. But the first visible is light or the Sun, since 1) it is the work of the first day and 2) the most excellent of all visibles, the most principal, the primary, and that which was to be for all the rest the cause of visibility. It follows, therefore, that from the vision of the Sun on the Earth the beginning was made of proportioning the bodies of the world, as also in the preceding the very spaces of the world were distinguished from one another by the proportional mediation of the orbit of the Earth.
How great is the apparent diameter of the Sun on the Earth? It is established by the most ancient observations of Aristarchus and the most recent of our time that, if the Earth has receded as far as possible from the Sun, then, a circle being described with the eye as centre, there is occupied and as it were determined by the diameter of the Sun, of that circle, most exactly the seven hundred and twentieth part, that is half a degree; or, which is the same, the angle between the lines touching either edge of the Sun is the seven hundred and twentieth part of four right angles.
What do you think the cause of this number? The cause of the first thing must be sought also archetypal among the first. Now the geometrical cause of the division of the circle into 720 from a bare figure of so many sides there is none. For this figure is derived by bisection from the figure of 45 sides, which has no demonstration, as was proved in book I of the Harmonics. It follows that this section of the circle has been taken from the composition of figures and so from harmonic reasons. And it seems to infer a necessity that the zodiac circle, in which both all the planets in truth and also the Sun in appearance had to exercise their harmonic motions, that this circle, I say, should be divided by the appearance of the first body into parts of harmonic numerosity. Now the least number which offers itself for determining all the parts of the monochord to constitute the double system of the diapason, that is, of both soft and hard song, this number, I say, is 720, as was demonstrated in book III of the Harmonies, chapter 6.
Wherefore, since the motions of all the planets, as I demonstrate in book V of the Harmonics, were to be adapted to this double system, it was consonant that the first body also, which would be the leader of this music, should by the appearance of its diameter on the Earth divide for the inhabitants of the Earth, that is for the contemplative creature, that circle, as an index and measure of the appearance of the harmonic motions, by the division of the monochord, that is into 720 parts, which is twice 360, three times 240, four times 180, five times 144, six times 120, eight times 90, nine times 80, ten times 72, twelve times 60, fifteen times 48, sixteen times 45, eighteen times 40, twenty times 36, twenty-four times 30, the most numerous form of division into aliquot parts.
What follows, then, for the interval of the Sun and the Earth from this assumed hypothesis, or how great is this ten-foot rod, hitherto used by us as the measure of the planetary orbs? If the diameter of the Sun had to occupy half a degree to the sight placed on the Earth, it is necessary that the sight, or in its place the centre of the terrestrial globe, should have receded from the centre of the Sun by 229 semidiameters of the round solar body, a little more, as we are taught in geometry.
I hold the interval; state also the quantity of the globe of the Earth through its causes. These things do not yet suffice to determine the quantity of the Earth, but another axiom is needed besides. Namely, because the Earth was to be the dwelling of the measuring creature, the Earth herself also had to become, by her body, the measure of the worldly bodies, and by her semidiameter, as a line, of lines, that is of intervals. Since, however, the measuring of bodies is distinct from the measuring of lines, and since the first proportion is between the bodies of the Earth and the Sun, the first also between the diameter of the Earth and the interval of the Earth from the Sun, nothing is more consonant with a right, fitting and ordered tempering than that an equality of both proportions be established, that as many times as the body of the Earth is contained in the body of the Sun, so many times also the semidiameter of the Earth be contained in the interval of the centres of the Sun and the Earth, so that as the body of the Earth is to the body of the Sun, so is the semidiameter of the Earth to the distance of the centres.
How now is the quantity of the semidiameter of the Earth drawn from these two axioms? The semidiameter of the Sun being set at 100000 parts, so that the interval of the centres of the Sun and the Earth is 22918166 such parts, the cube of 100000, that is 1000000000000000, is to be divided by the interval 22918166, and of the quotient (which is the sine of 0° 15′ 0″ continued) the root is to be sought, which will be 6606. So great will be the semidiameter of the Earth. For as 6606, the semidiameter of the Earth, is contained in 22918166, the interval of the Sun and the Earth (3469 times and a third), so also the cube of 6606, the semidiameter of the Earth, is contained in the cube of 100000, the semidiameter of the Sun, as many, namely 3469, times and a third. Now it is known from geometry that the proportion of cubes to one another is the same as that of the globes inscribed in the same cubes. So the semidiameter of the Sun will contain the semidiameter of the Earth fifteen times, a little more; but the body of the Sun will contain the body of the Earth about 3469 times.