Socrates, in the Republic (509 d–511 e), uses the symbol of a divided line to illustrate the distinction between the Visible and Intelligible Worlds, and between the kinds of perception appropriate to each. This paper will present a new hypothesis: that the proportions of the line are derived from optical theory.
The construction of the Divided Line is described as follows: Socrates asks his interlocutors to represent the Visible and Intelligible Worlds by a line divided into two unequal segments. (See Diagram I, below, where line AA' is divided at C.) The ratio in which the division is to be made is not specified, and it seems that any ratio is acceptable provided that one segment is longer than the other. Socrates then tells them to cut each part again according to the same ratio as the original division. (In Diagram I, below, AC is divided at F, and CA' is divided at F'.) After describing the division of the line thus into four parts, Socrates goes on to explain the philosophical significance of each part. For the purposes of this paper the following brief identification of each segment of the line will suffice.
