A doctrine from Plato's Middle Period found in the Republic 6:509d-511c.
The divided line is a visual metaphor for Plato's ontological (and epistemological) view of the Universe. Reality is divided into two basic parts: the invisible, unchanging realm of universals (or Ideas also sometimes called Forms), and the visible, ever-changing realm of particulars (i.e., physical objects). Each of these two realms may be sub-divided giving us four realms of being and cognition. The lowest region is the realm of images (eikones) or reflections of physical objects which are cognized through the faculty of imagination (eikasia). Next is the realm of physical objects which are cognized through opinion or trust (pistis). The next level is the realm of mathematical objects (or what we would call abstract ideas) which are cognized through intellect (dianoia). Finally there is the realm of ideas which are cognized through reason (noesis). These four realms represent the ontological hierarchy of Plato's middle metaphysics.
In book 7:514a-517a Plato gives the famous Allegory of the Cave to further clarify the ontological and epistemological doctrines implied in the divided line.
For a graphical representation of the Divided Line click here.