In one of the most disputed passages of Greek literature Plato in the Republic, 7. 528e–530c prescribes astronomy as the fourth study in the education of the Guardians. But what sort of astronomy? According to one school of thought it is a purely speculative study of bodies in motion having no relation to the celestial objects that we see. While this interpretation has rejoiced the hearts of Plato's detractors, who regard him as an obstacle to the progress of science, it (...) has dismayed his admirers. Another school of thought holds that what Plato meant was that astronomers must get to know the real motions of the heavenly bodies as opposed to their apparent motions as seen by us on earth. The opposed interpretations may be set out in the following representative citations from Sir Thomas Heath and John Burnet. (shrink)

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. (shrink)