Studies on Two Supreme Principles of Plato’s Cosmos: The One and the Indefinite Dyad—The Division of a Straight Line into Extreme and Mean Ratio, and Pingala’s Matrameru

Abstract

The objective of this paper is to propose a mathematical interpretation of the continuous geometric proportion (Timaeus, 32a) with which Plato accomplishes the goal to unify, harmonically and symmetrically, the Two Opposite Elements of Timaeus Cosmos—Fire and Earth—through the Mean Ratio. As we know, from the algebraic point of view, it is possible to compose a continuous geometric proportion just starting from two different quantities a (Fire) and b (Earth); their sum would be the third term, so that we would obtain the continuous geometric proportion par excellence, which carries out the agreement of opposites most perfectly: (a + b)/a = a/b. This equal proportion, applied to linear geometry, corresponds to what Euclid called the Division into Extreme and Mean Ratio (DEMR) or The Golden Proportion. In fact, according to my mathematical interpretation, in the Timaeus 32b and in the Epinomis 991 a–b, Plato uses Pingala’s Matrameru or The First Analogy of the Double to mould the body of the Cosmos as a whole, to the point of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (\(\phi\) and1/\(\phi\))—with the DEMR. In effect, Fire and Earth are joined not by a single Mean Ratio but by two (namely, Air and Water). Moreover, using the Platonic approach to analyse the geometric properties of the shape of the Cosmos as a whole, I think that Timaeus constructed the 12 pentagonal faces of Dodecahedron by means of elementary Golden Triangles (a/b = \(\phi\)) and the Matrameru sequence. And, this would prove that my mathematical interpretation of the platonic texts is at least plausible. It is probable that Plato refers to the paradigm of the Line of the Horizon in his Republic when he speaks of the Divided Line to explain his cosmological doctrine of ideas.