Unit V · Historical & Mathematical Commentary

1. Mästlin’s 1597 Decimal 0.6180340 & the Quartic Kepler Field $\mathbb{Q}(\sqrt{\phi})$

In October 1597, while corresponding with Michael Mästlin (Frisch I, p. 377), Kepler posed the problem of constructing a right-angled triangle whose three sides form a continuous geometric progression $a : b : c$. Setting $a = 1$ and $b = \sqrt{c}$ forces $1 + c = c^2$, so $c = \phi = \frac{1 + \sqrt{5}}{2}$ and $b = \sqrt{\phi}$. On the letter itself, Mästlin wrote down for a hypotenuse of $10000000$ the greater segment $6180340$—historically the first known decimal approximation of the inverse golden ratio $\phi^{-1} \approx 0.6180339887\dots$, verified in geometor.kepler.harmonices_geometry.verify_maestlin_1597_decimal to have an exact error below $1.13 \times 10^{-8}$!

In geometor.model, coordinates involving $\sqrt{\phi}$ live naturally in the degree-4 algebraic number field get_kepler_field() generated by $\alpha = \sqrt{\phi}$ with minimal polynomial

$$\alpha^4 - \alpha^2 - 1 = 0.$$

Yet in Rational Trigonometry (quadrance), the squared side lengths $(1, \phi, \phi^2)$ descend cleanly to the quadratic golden field $\mathbb{Q}(\phi)$!

2. Angular Momentum & the Inverse-Square Ratio of Apsidal Angular Speeds

In Astronomia Nova (1609), Kepler discovered the Second Law (equal focal areas in equal times, $dA/dt = \frac{1}{2} r^2 \dot{\theta} = \text{const}$). At perihelion $r_{\text{per}} = a(1 - e)$ and aphelion $r_{\text{aph}} = a(1 + e)$, the velocity vector is perpendicular to the solar radius vector, so the heliocentric angular speeds $\omega = \dot{\theta}$ satisfy strictly

$$\frac{\omega_{\text{per}}}{\omega_{\text{aph}}} = \left(\frac{r_{\text{aph}}}{r_{\text{per}}}\right)^2 = \left(\frac{1 + e}{1 - e}\right)^2.$$

Solving $\left(\frac{1 + e}{1 - e}\right)^2 = \frac{q}{p}$ for a rational harmonic interval $p : q$ yields the algebraic eccentricity law $e = \frac{\sqrt{q} - \sqrt{p}}{\sqrt{q} + \sqrt{p}}$. For Mars ($p : q = 2 : 3$), this gives $e = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} = 5 - 2\sqrt{6} \approx 0.1010$, remarkably close to Mars’s true orbital eccentricity $e \approx 0.0934$!

3. From Proportio Sesquialtera (1618) to Universal Gravitation (1687) & the 1621 Second Edition

Kepler’s Third Law $(T_1 / T_2)^2 = (a_1 / a_2)^3$ (Book V, Ch. III, Prop. VIII) became the decisive quantitative touchstone from which Isaac Newton deduced in Principia (1687, Book I, Props. IV & XI) that a centripetal force holding planets in focal ellipses with $T^2 = \frac{4\pi^2}{GM} a^3$ must vary inversely as the square of the distance ($F \propto 1/r^2$). Two years after Harmonices Mundi, Kepler brought his lifework full circle by issuing the annotated second edition of Mysterium Cosmographicum (Frankfurt 1621), leaving his youthful 1596 text intact while supplying candid Notae Authoris explaining where pure polyhedral geometry required completion by elliptical dynamics and celestial harmony.