The purpose of the simile of the Sun is to show the difference between knowledge and opinion. In the world of opinion the sun gives light for us to gain empirical knowledge through sight. Objects (such as tables and mugs) and concepts (such as beauty and love) are part of the world of opinion because they are detected by our senses. The Form of beauty (or love) exists in the realm of the Forms. Plato believed that the Form of good gives philosophers knowledge of the Forms. The Form of Good gives the minds eye the power of knowledge and illuminates other Forms with truth and reality. In this way the Form of good allows us to see other forms and ultimately it allows us to see the form of good. The sun is also the source of all growth, and thus the Form of good is the source of all the Forms, and the cause of all existence. In the Sun, Platos idea is that visibility is opinion and the Form of good gives us the reality behind visibility knowledge of the Forms.
The divided line goes into more depth on Platos previous idea. Although an abstract example, Plato uses it to try and make his idea simpler. Plato believed that there were four different states of the mind, and four different states of reality, both progressing from a lower level (illusion / images and shadows) to a higher level (intelligence / Forms). The lowest level is the second hand images of physical things. As Plato thought that physical things were just in the mind anyway, a shadow of a physical thing is the lowest form of knowledge, which corresponds to illusion in the mind. The next step up is physical things, which also covers sciences as they are based on the study of physical things. Belief is the empirical knowledge of physical thing, but is still opinion.
The upper half of the divided line has mathematical reasoning, which is part of deductive reasoning. Deductive reasoning does not rely on empirical knowledge and so mathematical reasoning is not based on belief. The reasoning perceives mathematical ideas, which are based on some belief (such as seeing a square, you have a mathematical idea about the square), but these ideas do not rely on belief. The highest kind of knowledge in the mind is intelligence, which can only be reached by the philosopher. Mathematical reasoning leads to intelligence, but mathematical ideas (part of the intelligible world) lead to knowledge of the Forms.
In the idea of the Divided Line, Plato is showing us in more detail the differences between knowledge and opinion. He suggests progressions from both illusion to pure thought, and from the less real (images and shadows) to the truly real (the Forms). He also makes two clear distinctions between the world of knowledge and opinion (in the mind), and the world of appearances and the intelligible world (in reality). Platos idea of the Forms and the distinctions between the mind and reality have been coherent between the Sun and the Divided Line both showing very similar ideas but in different attentions to detail. There is a slight incoherence though. In the simile of the Sun, Plato tells us that the Form of good gives us power to see all forms, yet in the Divided Line it is unclear if the Form of good gives any power at all, as it appears to be mathematical ideas which allow us to perceive the Forms. It may be though that these mathematical ideas simply give way to the Form of good so it can allow us to see the Forms.
The Simile of the Cave moves into more specific detail on the accent of the mind from illusion to knowledge, shown by the prisoners accent from the cave to the outside world (intelligible world). The simile is also an example of the divided line, using ideas put forward such as shadows, physical things, illusion, the Forms (the Sun) etc. Like the Sun simile Plato depicts the Form of good as the greatest of all lights. There is a road between the fire and cave, maybe to portray a clear conversion from illusion to intelligence. There are people talking outside holding statues. Their voices and the images of the statues are projected into the cave, and the prisoners believe these are real. The people trapped in the cave are all looking at shadows of statues (images themselves) projected against the wall, and therefore are doubly ignorant.
Plato allows a prisoner to be released, and he walks towards the fire where he is dazzled. It hurts him, and he is told that the people on the road are real and that the shadows are nonsense. He returns to the cave: choosing illusion. Plato is showing a would-be philosopher failing his journey of knowledge, due to a flawed character. Later on, the prisoner is taken into the sunlight and held until his eyes grow accustomed to the sunlight. Once accustomed he can see shadows and reflections within time.
Plato here brings in the idea from the simile of the Sun, creating a coherent idea between the two similes, that the Form of good (the Sun) allows the philosopher to see other things until finally he can see the forms, and finally the Form of good itself (the Sun). Through duty the prisoner returns to the cave and is blinded by the darkness. This is an idea Plato has that the eyes can be blinded by the light and by the dark, as the mind can be confused by greater and lesser minds. This continues a coherent idea of Platos from the Sun simile, the idea of the Sun and the Mind, or if we were to bring the Divided Line into this, Reality and the Mind. So far Plato has created a seemingly coherent picture of knowledge and opinion.
After descending from the Forms the prisoner tells the other prisoners the statues. He has seen the Form of justice and they have only seen the shadow of justice, and so they hold different views and therefore he is put on trial. Plato here is showing societys view of the philosopher ruler in a similar way as he does with the simile of the Ship.
The different states of the Mind in the Divided Line are straight comparisons to features of the Cave. Illusion is the shadows cast by firelight; belief is the statues of things and the fire; mathematical reasoning is the shadows and reflections of real things of which the images are made; intelligence is the things themselves (moon and stars), and finally the Form of Good is the Sun. Plato uses a coherent idea throughout as appears to use similar / the same ideas in each of his similes (i.e., The Sun the Form of Good).
There are problems in each of the similes and they all have points where they break down and lose a certain amount of coherence. In the simile of the Sun, although Plato would not have known at the time, the eyes are not charged up by the Sun, like he wanted us to think. Within the simile, therefore, the Form of Good no longer charges the mind with the power of seeing the Forms, but rather casts light onto other Forms.
This is not what Plato wanted to portray. In the Sun, Plato explores the idea of the Sun being the ultimate Form in detail, but in the Cave, the sun itself is forgotten, and is seen more as a goal, or object, than as a metaphor for knowledge. Plato uses the fire and statues in more detail than the sun in the Cave simile, giving a better explanation of knowledge, which creates incoherence between the use of the sun in the Sun simile and its use in the Cave.
In the simile of the Cave, mathematical ideas are not mentioned at all, not even in the accent to the Forms. This breaks down the idea of the Divided Line. A solution would be though that in the Divided Line, mathematical ideas are implied rather than stated: Plato may not even have mentioned the idea of mathematical ideas. This may be a reason why Plato did not mention them in the Cave.
Plato says the prisoners only perceive shadows, which means the lowest level of knowledge illusion. This implies that Athenians cannot even see physical things, which is absurd because, unless all Athenians are blind, they will be able to visibly see objects such as tables, chairs, knifes etc. He also contradicts what he said earlier in the section on the Philosopher Ruler, that people in general only experience opinion / belief.
This is only a problem if we use a strict interpretation, and in that case it would not be a simile at all, as similes should be open to wide interpretation. Another problem occurring in the Cave is that the shadows are used to describe illusion once, and mathematical reasoning another time. If we take the Divided Line, shadows should show illusion, and mathematical reasoning should be based upon physical things and a certain amount of belief. Plato has misused his metaphor of shadows in one of these similes and created an inconsistency.
Although Plato has given us a clear picture of what his idea of knowledge and opinion is, and how the Form of Good fits into it, he has left many questions unanswered and a lot of gaps which need filling. His similes have been written and laid out in such a way that they can be interpreted in many varying, some contradictory fashions. This leads to the question of did Plato actually know himself where he was going with his arguments, or did he make them up as he went along? I personally think he thought them out well, but had to manipulate some things to fit in with his line of reasoning. This is perhaps a reason why these three similes are partly incoherent.