Unit III · Harmonices Mundi Liber I: De Figuris Regularibus (1619)

Original GEOMETOR English translation from Johannes Kepler’s Harmonices Mundi Liber I: De ortu figurarum regularium harmonicarum (Linz: Tampach/Plancus, 1619, pp. 1–46; Frisch V, pp. 75–125).

1. Prooemium — Vindication of Euclid Book X Against Petrus Ramus

Since antiquity saw that the origins of harmonic proportions arise from the divisions of the circle into equal parts executed geometrically—that is, from knowable (scibiles) regular plane figures—it was necessary first of all to point out to geometers today those deepest mysteries which Euclid set forth in his Tenth Book concerning incommensurable lines (Elements X), corroborated by the commentary of Proclus Diadochus (In Euclidem).

Why did that great treatise of Euclid’s Tenth Book lie almost neglected down to our day? Because Petrus Ramus, an otherwise sharp and enterprising mind, stumbled upon its subtle distinctions without perceiving toward what goal Euclid was steering! Ramus reviled the Tenth Book so fiercely—claiming that among all human sciences no obscurity comparable to that labyrinth had ever appeared—that even my friend Willebrord Snellius, though republishing the enunciations, hesitated to defend their utility. I confess that I myself stood near Ramus’s camp until I entered upon the investigation of celestial harmonies: then at sudden daylight I beheld that Euclid had constructed the entire edifice of commensurable lengths, commensurable squares, medial lines, binomials, apotomes, majors, and minors for one supreme architectural purpose—to distinguish knowable and constructible regular figures (scibilia) from unknowable and non-constructible figures (inscibilia)!

2. Definitions I–XXV — The Eight Degrees of Geometric Scibilitas

  • Definition VII (*To Know in Geometry*): To know (scire) in geometric matters is to measure by a known measure; and in our problem of inscribing figures in a circle, that canonical known measure is the diameter of the circle ($D = 2R$).

  • Definition VIII (*Knowable / Demonstrable*): A line or chord is called knowable (scibilis, Greek γνώριμον) when it is either directly measurable by the diameter itself, or at least derived through a definite and geometrical chain of squares and square-roots descending from the diameter.

From the diameter $D = 2R$, geometric knowability descends across eight ordered degrees:

  1. Degree I (*Identity*): Equality with the diameter itself ($s_2 = D = 2R$).

  2. Degree II (*Expressible in Length*): Rational commensurability in length (effabilis longitudine), exemplified by the Regular Hexagon whose side equals the radius ($s_6 = R = \frac{1}{2}D$, [IV.15]).

  3. Degree III (*Expressible in Square Only*): Incommensurability in length paired with exact rational quadrance (effabilis potentiâ tantùm), exemplified by the Square ($s_4^2 = 2R^2 = \frac{1}{2}D^2$, [IV.6]) and the Equilateral Triangle ($s_3^2 = 3R^2 = \frac{3}{4}D^2$, [XIII.12]).

  4. Degree IV (*Medial / Mese*): Geometric mean of two lines commensurable in square only (mese, Euclid X.21), governing areas of the Regular Octagon and Regular Dodecagon.

  5. Degrees V & VI (*Binomial & Apotome*): Sum and difference $a \pm \sqrt{b}$ of lines commensurable in square only (binominis & apotome). Here shines forth the Regular Decagon (OG), whose side $s_{10} = \frac{\sqrt{5}-1}{2}R = \frac{R}{\phi}$ is an Apotome cut by the Divine Section ([XIII.9])!

  6. Degrees VII & VIII (*Major / Mizon & Minor / Elasson*): Compound nested quadratic radicals $\sqrt{a \pm \sqrt{b}}$ (mizon & elasson, Euclid X.39, X.76). In the left panel below, the Regular Pentagon (PENT5) has side AG equal to an Elasson (Minor) line $s_5 = \frac{R}{2}\sqrt{10 - 2\sqrt{5}}$ ([XIII.10]), while the five-pointed star chord (stella quinquangula) is its conjugate Mizon (Major) line $d_5 = \frac{R}{2}\sqrt{10 + 2\sqrt{5}}$, followed by the Pentadecagon ($n = 15$, [IV.16]) and iterated bisections ($n = 8, 12, 16, 20, 24\dots$).

O A M G H_1 H_2 H_3 H_4 H_5 H_6 H_7 I. Scibilis: OG = s_10 = R/phi, AG = s_5 (Quadratic) II. Inscibilis (Prop. XLV): Septangulum, x^3 + x^2 - 2x - 1 = 0

3. Proposition XXIX — Self-Propagation of the Divine Section

  • Definition XXVI & Proposition XXIX: A line is cut in Proportio Divina (Euclid’s extreme and mean ratio) when as the whole is to the greater segment, so is the greater segment to the lesser ($1 + \phi : \phi = \phi : 1$). Uniquely among all divisions of magnitude, once established it propagates itself forever: if the greater segment is laid off upon the whole, or if the whole is added to the greater segment ([XIII.5]), a new Divine Section springs forth immediately without fresh construction—just as Ptolemy's swing arc from midpoint M through apex A cuts off both the decagon side OG ($R/\phi$) and the pentagon chord AG ($\sqrt{R^2 + s_{10}^2}$) in a single stroke.

4. Proposition XLV — Non-Constructibility (Inscibilitas) of the Heptagon

  • Proposition XLV (*Enunciation*): The side of the Regular Heptagon (*Septangulum*, :g:`HEPT7`), and every figure whose number of sides is a prime number beyond five (such as 7, 11, 13, 17?, 19) or a multiple of nine, lacks geometric knowability and cannot be constructed from the diameter by Euclidean plane geometry.

For consider the heptagonal triangle H1H2H4 inscribed in cHEPT (right panel above), bounded by three unequal chords: the heptagon side $s_1 = H_1 H_2$, the short diagonal $s_2 = H_1 H_3$ (diags2), and the long diagonal $s_3 = H_1 H_4$ (diags3). Unlike the pentagon—where only two chord species exist ($s_5$ and $d_5$), locking their ratio immediately into a quadratic equation $x^2 - x - 1 = 0$ solvable by a circle and straight line—the heptagon entangles three chord species simultaneously!

My ingenious friend Jost Bürgi (Justus Byrgius) has indeed exhibited an algebraic (Cossic) equation relating the chord of one-seventh of the circumference to the radius:

$$y^6 - 7y^4 + 14y^2 - 7 = 0 \qquad \Longleftrightarrow \qquad x^3 + x^2 - 2x - 1 = 0 \quad \left(x = 2\cos\frac{2\pi}{7}\right).$$

Yet I reply that algebraic root-reckoning is not Euclidean geometric determination! First, this equation climbs to the cubic (third) solid degree in $x$ (or sixth in $y$), which no intersection of straight lines and circles in a plane can cut off; second, the very same algebraic equation admits three distinct real roots simultaneously—answering indifferently to the convex heptagon chord $\frac{1}{7}$, the first star heptagram $\frac{2}{7}$, and the second star heptagram $\frac{3}{7}$—without singling out one magnitude from the diameter by finite geometric necessity:

“Therefore no Regular Heptagon has ever been constructed by anyone knowingly, willingly, and acting from deliberate geometrical purpose; nor **can* it be constructed from deliberate purpose—though by blind chance its length might be stumbled upon, without anyone ever knowing whether it has been!”*

Thus the Creator excluded the heptagon, enneagon ($n = 9$), hendecagon ($n = 11$), and their kin from the archetypal council of cosmic harmonies.