Unit I · Mysterium Cosmographicum (1596 / 1621)

Original GEOMETOR English translation from Johannes Kepler’s Prodromus Dissertationum Cosmographicarum, continens Mysterium Cosmographicum (Tübingen, 1596; Editio altera cum notis authoris, Frankfurt, 1621; Frisch I, pp. 106–152).

1. Preface to the Reader — The Graz Discovery (July 19, 1595)

It is my intention, gentle Reader, to demonstrate in this little book that the Most Good and Great Creator, in founding this mobile world and arranging the order of the heavenly orbs, looked upon those five regular bodies that have earned such celebrated renown from the time of Pythagoras and Plato down to our own day, and that to their nature He accommodated the number of the heavens, their proportions, and the law of their motions.

When I was living at Tübingen six years ago and attending the lectures of the distinguished master Michael Mästlin, I perceived how encumbered with difficulties the customary way of conceiving the world was; and I took such delight in Copernicus, whom my teacher frequently mentioned in his lectures, that I not only defended his views repeatedly in physical disputations among the students, but even composed an accurate disputation attributing the first motion of the heavens to the rotation of the Earth. Already at that time I began to inquire why things stood precisely thus and not otherwise: namely, what caused there to be six mobile orbs around the immobile Sun—neither twenty nor a hundred—and what determined the magnitudes of the intervals separating Saturn, Jupiter, Mars, Earth, Venus, and Mercury.

At first I pursued numerical proportions, testing whether one orb might be double, triple, or quadruple another; when arithmetic failed, I inserted new invisible planets between Mars and Jupiter and between Venus and Mercury, yet still no finite law emerged. Almost an entire summer was spent in this arduous toil. At last, upon a quite slight occasion, I broke nearer to the very truth itself. On the ninth (or nineteenth, New Style) day of July in the year 1595, while showing my students at the chalkboard in Graz how the Great Conjunctions of Saturn and Jupiter leap successively across eight signs of the zodiac from one fiery trigon to the next, I inscribed a succession of equilateral triangles within the zodiacal circle and noticed that the intersection of their chords traced out an inner enveloped circle whose radius OM appeared visibly half the radius OA of the circumscribed outer circle cSAT—almost matching the true astronomical proportion between the orb of Saturn (cSAT) and the orb of Jupiter (cJUP)!

O A B C M Graz Conjunction Triangle (19 July 1595): r_in : R_out = OM : OA = 1 : 2 (Q = 1/4)

In the planar figure above, the equilateral triangle ABC inscribed within Saturn’s circle cSAT about center O touches Jupiter’s inscribed circle cJUP at midpoint M, yielding the exact radius ratio $OM : OA = 1 : 2$ and quadrance ratio $Q(O, M) / Q(O, A) = \frac{1}{4}$ ([XIII.12]).

Immediately I attempted to separate Mars and Jupiter with a square, Earth and Mars with a pentagon, and the remaining intervals with a hexagon, heptagon, and so forth. Yet I saw at once that of regular plane polygons there is an infinite multitude, whereas in the sky there are only five intervals between the six mobile Copernican orbs! Wherefore, cried I: “Why do I seek two-dimensional plane polygons for three-dimensional solid orbs? Behold, solid bodies must be interposed between solid spheres! And behold, reader, the invention and entire substance of this little treatise!” For by Euclid's final theorem ([XIII.18]), in all of three-dimensional geometry there exist five, and strictly no more than five, convex regular solid bodies!

2. Chapter II — Curved and Straight: Cusanus & the Five Solids

In this one matter above all, Nicholas of Cusa (De Docta Ignorantia, 1440) and others seem truly divine to me: namely, that they esteemed so highly the mutual relationship between the Straight (rectum) and the Curved (curvum), and dared to liken the Curved to God Himself, but the Straight to created beings. For in the outermost spherical surface of the fixed stars—centered upon the Sun and embracing the intermediate ethereal space—shines forth the image of the eternal Trinity; whereas the rectilinear corporeal boundaries of created magnitudes must be drawn within the sphere by inscription and circumscription.

Now if any rectilinear figures are to prescribe a finite limit to the number of mobile planetary orbs, they must share the perfection of uniformity, having all faces congruent regular polygons and all solid angles equal. Yet Euclid demonstrates in Book XIII (Elements XIII.18) that around a solid vertex at least three plane angles must meet with sum strictly less than four right angles ($360^\circ$), permitting solely:

  1. Three, four, or five equilateral triangles ($3 \times 60^\circ = 180^\circ$, Tetrahedron; $4 \times 60^\circ = 240^\circ$, Octahedron; $5 \times 60^\circ = 300^\circ$, Icosahedron);

  2. Three squares ($3 \times 90^\circ = 270^\circ$, Cube);

  3. Three regular pentagons ($3 \times 108^\circ = 324^\circ$, Dodecahedron);

while six triangles ($6 \times 60^\circ = 360^\circ$), four squares ($4 \times 90^\circ = 360^\circ$), or four pentagons ($4 \times 108^\circ = 432^\circ$), as well as three hexagons ($3 \times 120^\circ = 360^\circ$), either flatten into a plane or exceed solid closure altogether. Because five solid intervals require six bounding spherical orbs, the Creator established six mobile planets around the Sun.

3. Chapters XII–XIII — The Two Treasures of Geometry & Polar Duality

Geometry possesses two great treasures: the one is the ratio of the hypotenuse to the legs in a right-angled triangle ([I.47]), and the other is the straight line cut in extreme and mean ratio—the Divine Section ([II.11], [VI.30]). The first you may rightly compare to a measure of gold (massa auri); the second we may call a precious jewel (gemma). For from the first flows the construction of the Cube, the Tetrahedron (Pyramis), and the Octahedron; whereas from the second—self-propagating and ineffable—flows the construction of the Dodecahedron and the Icosahedron (Pacioli 1509).

When each regular solid is inscribed in a unit outer sphere ($R_{\text{out}} = 1$, $Q_{\text{out}} = 1$) so that its vertices touch the outer planet’s inner boundary while an inner sphere ($r_{\text{in}}$, $Q_{\text{in}} = r_{\text{in}}^2$) touches the centers of its faces, exact Euclidean demonstration yields three distinct quadrance ratios, because polar-dual solids share identical insphere-to-circumsphere proportions:

  • Cube & Octahedron ($\{4, 3\}$ and $\{3, 4\}$):

    $$\frac{r_{\text{in}}}{R_{\text{out}}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \approx 0.57735, \qquad \frac{Q_{\text{in}}}{Q_{\text{out}}} = \frac{1}{3}.$$

  • Tetrahedron ($\{3, 3\}$, self-dual):

    $$\frac{r_{\text{in}}}{R_{\text{out}}} = \frac{1}{3} \approx 0.33333, \qquad \frac{Q_{\text{in}}}{Q_{\text{out}}} = \frac{1}{9}.$$

  • Dodecahedron & Icosahedron ($\{5, 3\}$ and $\{3, 5\}$ over $\mathbb{Q}(\phi)$):

    $$\frac{r_{\text{in}}}{R_{\text{out}}} = \sqrt{\frac{5 + 2\sqrt{5}}{15}} = \sqrt{\frac{4\phi + 3}{15}} \approx 0.79465, \qquad \frac{Q_{\text{in}}}{Q_{\text{out}}} = \frac{5 + 2\sqrt{5}}{15}.$$

4. Chapter XIV — Earth as Measuring Orb (Circulus mensor omnium)

The five solids divide naturally into two noble ranks: three primary solids (primaria)—the Cube, Tetrahedron, and Dodecahedron, each having distinct polygonal faces (square, triangle, pentagon) and simple three-edge trihedral vertices—and two secondary solids (secundaria)—the Icosahedron and Octahedron, borrowing the triangular face and meeting four or five to a vertex around an axis. Fittingly, the dwelling place of contemplative humanity—the orb of Earth (cTERin)—was placed as a boundary wall between these two classes as the measuring circle of all:

“Earth is the Circle that measures all things: circumscribe about it a **Dodecahedron* (DODEC); the circle comprehending this will be Mars (cMARin). About Mars circumscribe a Tetrahedron; the circle comprehending this will be Jupiter (cJUP). About Jupiter circumscribe a Cube (CUBUS); the circle comprehending this will be Saturn (cSAT). Now inscribe within Earth an Icosahedron; the circle inscribed in it will be Venus (cVEN). Within Venus inscribe an Octahedron; the circle inscribed in it will be Mercury (cMER). There you have the reason for the number of the planets!”*

Sol Saturnus Iupiter Mars Mars (Dodecaedron) Terra (Mensor) Venus (Icosaedron) Mercurius I. Outer Primary Solids (Saturn -> Mars) II. Inner Secondary Solids around Terra (Mars -> Mercury)

In his 1621 Author’s Notes (Notae Authoris, Frankfurt, 1621), Kepler triumphantly observes that the small residual discrepancies between thin polyhedral spheres and Tycho Brahe's observations do not refute the five regular bodies, but rather correspond to the orbital eccentricities governed by harmonic consonances in Harmonices Mundi Book V (1619).