Plato’s Planetary Power Proportions Led to Kepler’s Third Law

Kepler articulated what became known as his Third Law of Planetary Motion in Harmonices Mundi (1619): ”But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances…’ (tr. Aiton et al., 1997: 411). The sesquialterate proportion is the ratio of one to one-and-a-half, or, of 2 to 3. Kepler formulated a sesquialterate power proportion (square to cube) between planetary speeds and distances. Such a planetary power relationship existed in antiquity. In Plato’s Timaeus (c. 360 BCE), the Demiurge fashions the cosmic Soul – invisible template of the universe – from an elemental mix of Same, Different, and Being. From this mixture, he takes various proportions that yield seven numbers: 1, 2, 3, 4, 9, 8, 27 (Timaeus, 35b-c), that are tied, through the Circle of the Different (Timaeus, 36d), to the celestial Wanderers (Timaeus, 38c). The Academy scholarch Crantor arranged these numbers in the shape of the letter Lambda, with 1 at the top, even numbers down one side (2, 4, 8), odd numbers down the other (3, 9, 27). This arrangement highlights the squares (2×2, 3×3) and cubes (2×2×2, 3×3×3) in Plato’s planetary ratios. According to Plutarch’s On the Generation of the Soul in Timaeus (c. 100 CE), scholars of his time sought Plato’s power proportions in planetary structure: ”Yet certain people look for the prescribed proportions in the velocities of the planetary spheres, certain others rather in their distances…’ (tr. Cherniss, 1976: 321). In Mathematics Useful for Understanding Plato (c. 120 CE), Theon of Smyrna discussed two numerical ‘quaternaries:’ the Pythagorean Tetraktys and the Platonist Lambda: ”There are then two quaternaries of numbers, one which is made by addition, the other by multiplication, and these quaternaries encompass the musical, geometric and arithmetic ratios of which the harmony of the universe is composed.’ (tr. Lawlor, 1979: 62-63). After stating his planetary sesquialterate power ratio in The Harmony of the World, Kepler uses Plato’s planetary power proportions to illustrate his discovery: ”Let the periodic times of two planets be 27 and 8… Hence the semidiameters of the orbits will be as 9 to 4. For the cube root of 27 is 3; that of 8 is 2; and the squares of these roots are 9 and 4…’ (tr. Aiton et al., 1997: 413). Planetary power proportions traveled, over about two thousand years, from Plato, to Crantor, to Cicero, to Plutarch, to Theon, to Calcidius, to Macrobius, to Proclus, and finally to Kepler, who illustrated his newly discovered planetary power law with Plato’s planetary power proportions from Timaeus.