Introduction
Harmonies and Solids
Chapters 3 and 4
Ellipse
Proposition 13
Prop 13 Calculator
Chapter 5
Harmony of the World
Chapter 9
A Dissonant Harmony

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Proposition 13 Calculator

Proposition 13 Calculator

Kepler uses proposition 13 of Chapter 3 later in the A Priori arguments of Chapter 9, when he is establishing the presence of harmonies based upon the proprortions of the spheres derived from the solids. He states that the smallest harmonies, the major and minor thirds could not exist among pairs of planets, nor can the fourth exist among the converging motions, because the proportions of the solids are larger than these harmonies. In Proposition 13, Kepler is able to calculate the proportion of the converging and diverging motions from the extreme distances of the orbits. Here you can verify what Kepler says yourself. See what happens when you try the various proportions of the spheres.

CUBE AND OCTAHEDRON 1000:577

TETRAHEDRON 1000:333

DODECAHEDRON AND ICOSAHEDRON 1000:795



Chapter 5