Latin & Mathematical Glossary

Lexicon of the 20 foundational Neo-Latin, Greek, and symbolic-geometry terms across Mysterium Cosmographicum, Strena, and Harmonices Mundi.

Epistemology & Euclidean Irrationals (Harmonices Mundi I)

  • scibilitas (knowability / demonstrabilitas) — Kepler’s criterion in Harmonices Mundi Book I (Defs. VII–VIII): a chord or polygon side is scibilis when constructible from the circle’s diameter in finitely many Euclidean steps and expressible within the quadratic radical hierarchy of Euclid Book X.

  • inscibilitas (geometric non-constructibility) — Impossibility of straightedge-and-compass determination from the diameter, exemplified by the regular heptagon (Septangulum, Prop. XLV), whose algebraic chord equation is an irreducible cubic.

  • effabilis longitudine (ῥητὴ μήκει) — Commensurable in length with the rational diameter $\mathbb{Q}$, such as the regular hexagon’s side $s_6 = R = \frac{1}{2}D$.

  • effabilis potentiâ tantùm (ῥητὴ δυνάμει μόνον) — Incommensurable in length, yet commensurable in square (quadrance), such as the inscribed square ($s_4^2 = 2R^2$) and equilateral triangle ($s_3^2 = 3R^2$).

  • mese (μέση / medial line) — Geometric mean of two lines commensurable in square only (Euclid X.21).

  • binominis & apotome (ἐκ δύο ὀνομάτων & ἀποτομή) — Sum ($a + \sqrt{b}$) and difference ($a - \sqrt{b}$) of lines commensurable in square only; the regular decagon’s side $s_{10} = \frac{\sqrt{5}-1}{2}R$ is an apotome.

  • mizon & elasson (μείζων & ἐλάσσων) — Major ($\sqrt{a + \sqrt{b}}$) and minor ($\sqrt{a - \sqrt{b}}$) nested radicals (Euclid X.39, X.76); the pentagram chord is a mizon line and the pentagon side $s_5 = \frac{R}{2}\sqrt{10 - 2\sqrt{5}}$ is an elasson line.

  • potentia / quadrance ($Q = r^2$) — Exact squared segment length (Euclid’s δύναμις / potentia; Rational Trigonometry’s quadrance), eliminating unnecessary nested square roots across polyhedral sphere ratios.

Divine Proportion, Polyhedra & Packing (Mysterium, Strena, Harmonices II)

  • sectio divina / proportio divina (extrema et media ratio) — Division $y : x = x : (x + y)$ yielding $\phi = \frac{1 + \sqrt{5}}{2}$ with $\phi^2 = \phi + 1$; celebrated by Kepler as the “precious jewel” of geometry and self-propagating archetype of pentamerous growth.

  • triangulum rectangulum Kepleri (Kepler Right Triangle) — Unique right triangle with sides $(1, \sqrt{\phi}, \phi)$ and quadrances $(1, \phi, \phi^2)$ in continuous geometric proportion over $\mathbb{Q}(\sqrt{\phi})$ (Epistola ad Mästlinum, Oct 1597).

  • corpora quinque regularia (Five Platonic Solids) — Cube $\{4,3\}$, Tetrahedron $\{3,3\}$, and Dodecahedron $\{5,3\}$ (primary outer solids), together with Icosahedron $\{3,5\}$ and Octahedron $\{3,4\}$ (secondary inner solids).

  • Terra est Circulus mensor omnium (Earth as Measuring Orb) — Kepler’s cosmological rule placing Earth’s unit orbit ($R = 1$) between the three outer primary solids and two inner secondary solids (Mysterium Cap. XIV).

  • dodecaedron rhombicum (Rhombic Dodecahedron) — Isohedral 12-rhombus space-filling Voronoi polyhedron of the FCC lattice with diagonal ratio $\sqrt{2}:1$ and dihedral angle $120^\circ$ (Strena, 1611).

  • triacontaedron rhombicum (Rhombic Triacontahedron) — Isohedral 30-rhombus golden zonohedron with diagonal ratio $\phi:1$, 32 vertices, 60 edges, and dihedral angle $144^\circ$ (Strena, 1611; Harmonices II & V, 1619).

  • coacervatio globulorum densissima (Kepler Sphere-Packing Conjecture) — Assertion in Strena (1611) that face-centered cubic and hexagonal close-packings (kissing number 12, density $\frac{\pi}{3\sqrt{2}} \approx 74.048\%$) achieve the maximum possible density of congruent spheres in $\mathbb{R}^3$.

  • congruentia figurarum (Planar & Solid Congruence) — Edge-to-edge fitting of regular and star polygons summing to $360^\circ$ in the plane (11 uniform Archimedean tilings) or closing around solid vertices ($< 360^\circ$) in 3-space (Harmonices Book II).

  • corpora semiregularia Archimedis (Thirteen Archimedean Solids) — The 13 convex vertex-transitive semi-regular polyhedra reconstructed and proved complete by Kepler in Harmonices Mundi Book II Prop. XXVIII.

  • stella quinquangula & corpora stellata (Pentagram & Stellated Dodecahedra) — Regular star pentagon $\{5/2\}$ and Kepler’s Small ($\{5/2, 5\}$) and Great ($\{5/2, 3\}$) Stellated Dodecahedra.

Celestial Harmonics (Harmonices Mundi V)

  • harmonia motuum extremorum (Aphelion–Perihelion Consonance) — Exact just-intonation musical intervals formed by planets’ extremal daily heliocentric angular speeds (Saturn $4:5$, Jupiter $5:6$, Mars $2:3$, Earth $15:16$, Venus $24:25$, Mercury $5:12$).

  • proportio sesquialtera / Lex Tertia (Kepler’s Harmonic Third Law) — Fundamental celestial law (Harmonices Mundi Book V, Cap. III, Prop. VIII): orbital periods stand in the exact $3/2$ power ratio of semi-major axes, $(T_1/T_2)^2 = (a_1/a_2)^3$.