Abstract

In a well-known passage Aristotle ascribes to Plato, or as some think to his followers, the dictum, γρ ριθμóς στιν κ νòς κα τς ορίστον, ‘Number is from the one and the undetermined dyad ’, but what this apparently simple statement means has remained a mystery until modern times. In other passages Aristotle expands it to explain that the indefinite duality is a duality of the great and small, e.g., ς μν ον λην τò μγα κα τò μικρòν εναι ρχς, ς δ' οσίαν τò ν. ξ κείνων γρ κατ μθεξιν το νòς εναι τος ριθμος. ‘As the matter he posits the great and small for principles, as substance the one; for by the mixture of the one with them he says numbers.’ This identification of the dyad with the great and small, elsewhere called τò νισον and τò πειρον, gives a first clue to its nature. In a notable article in Mind 35, 419–40, continued in vol. 36,, 12–33, and amplified by D' Arcy Wentworth Thompson in vol. 38, 43–55, A. E. Taylor first suggested a connexion between the indefinite duality and the modern theory of continued fractions. In the light of subsequent research in the history of Greek mathematics it may now be asserted with a high degree of confidence that his conjecture was almost certainly correct; but it was then no more than a conjecture, and when he looked for confirmation he looked in the wrong direction.