Plato, in his dialogue The Republic Book 6 (509D–513E), has Socrates explain the literary device of a divided line to teach basic philosophical ideas about the four levels of existence (especially the intelligible world and the visible world) and the corresponding ways to which knowledge is acquired about what exists, or come to mere opinions about what exists.
The analogy of the divided line immediately follows another Platonic metaphor, that of the sun, and is in turn followed by the Allegory of the Cave.
Imagine a line divided into two parts
In The Republic, Plato describes the Divided Line this way:
Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness, and you will find that the first section in the sphere of the visible consists of images. And by images I mean, in the first place, shadows, and in the second place, reflections in water and in solid, smooth and polished bodies and the like: Do you understand?
Yes, I understand.
Imagine, now, the other section, of which this is only the resemblance, to include the animals which we see, and everything that grows or is made.
Thus AB represents shadows and reflections of physical things, and BC the physical things themselves. These correspond to two kinds of knowledge, namely belief (πίστις pistis) about physical things and illusion (εἰκασία eikasia) about "shadows," which do not really exist. In the Timaeus, Plato includes the natural sciences in the category of belief, while the category of illusion includes all the "opinions of which the minds of ordinary people are full."
The intelligible world
The segment CE, representing the intelligible world, is divided into the same ratio as AB, giving the subdivisions CD and DE (it can be readily verified that CD must have the same length as BC):
There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principle descends to the other end; in the higher of the two, the soul passes out of hypotheses, and goes up to a principle which is above hypotheses, making no use of images as in the former case, but proceeding only in and through the ideas themselves.
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