“Now, think of a line that has been cut into two parts of different length; one of these represents the visible, another what can be attained by thought. Now cut them both again in the same proportion - - of the parts the first one [in my representation the ones below] are pictures - - second what the pictures are pictures from - - “ (The Republic, 509e-510a; my extremely free translation.)

We may illustrate this, taking into account other metaphors of The Republic as well:





Now, everybody can compare the different sections with each other and see what their relations are. One must take notice of the fact that the two middle parts of the line are of equal length, and it is apparent Plato was aware of this. (if the parts are given the names a, b, c, and d, from up to down, the equation goes: a/b = c/d = (a+b)/(c+d))