From Plato’s Harmonic Cosmos to Kepler’s Harmonic Third Law

In Plato’s Timaeus (35b-36a), the character Timaeus of Locri promulgates a numerical harmonic cosmogony, wherein the Demiurge generates a series of proportions that give seven numbers: 1, 2, 3, 4, 9, 8, 27. At Plato’s Academy, Crantor arranged these numbers in the triangular shape of the letter Lambda, with even numbers down one side (2, 4, 8) and odd numbers down the other (3, 9, 27). These sequences include squares (2×2, 3×3) and cubes (2×2×2, 3×3×3). Plato links these seven numbers to the Circle of the Different (36d) that would indicate the course of the seven Wanderers (38c-e). Plutarch writes, in On The Generation of the Soul in Timaeus (c. 110 CE), about the Platonic numbers: ‘Yet certain people look for the prescribed proportions in the velocities of the planetary spheres, certain others rather in their distances...’ (tr. Cherniss, LCL 427, 1976: 321). And Macrobius writes, in Commentary on the Dream of Scipio (c. 400 CE): ‘Now we must ask ourselves whether these intervals, which in the incorporeal Soul are apprehended only in the mind and not by the senses, govern the distances between the planets poised in the corporeal universe.’ (tr. Stahl, 1952: 196). Macrobius gives a correlation between Plato’s numbers and planetary distances (Figure 1. Stephenson, The Music of the Heavens: Kepler’s Harmonic Astronomy, 1994: 40). Johannes Kepler at first sought to explain planetary orbits through Platonic solids (Mysterium Cosmographicum, 1596), but this was not entirely satisfactory. So Kepler turned to Plato’s seven numbers and, in Book 4 of Harmonices Mundi (Linz, 1619), he quotes Proclus’ commentary on Euclid: ‘And here we must follow Timaeus, who integrates and completes the whole source and structure from the mathematical types, and locates in it the causes of all things. For those seven terms of all numbers pre-existed in it, as far as cause is concerned.’ (tr. Aiton et al., 1997: 301). In Book 5, Kepler gives his sesquialterate (2:3) power formulation of the Third Law (Aiton, 1997: 411). Then Kepler uses Plato’s harmonic numbers to illustrate his Harmonic Law: ‘Let the periodic times of two planets be 27 and 8. Then the proportion of the mean daily motion of the former to the latter is as 8 to 27. 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). Ten years after he published his first two Laws, Kepler relates that, when he came upon this correlation, ‘at first I believed I was dreaming.’ (tr. Aiton, 1997: 411). The harmonic cosmic numbers in Plato’s Timaeus appear to have inspired Kepler’s Harmonic Third Law.