At the end of Book VI of the Republic (509D-513E), Plato describes the visible world of perceived physical objects and the images we make of them (in our minds and in our drawings, for example). The sun, he said, not only provides the visibility of the objects, but also generates them and is the source of their growth and nurture. Many primitive religions identify the sun with God, for good reason.

Beyond this visible world, which later philosophers (esp. Immanuel Kant) would call the phenomenal world, lies an intelligible world (that Kant will call noumenal. The intelligible world is (metaphorically) illuminated by "the Good" (τον ἀγαθὸν), just as the visible world is illuminated by the sun.

The division of Plato's Line between Visible and Intelligible is then a divide between the Material and the Ideal, the foundation of most Dualisms. Plato may have coined the word "idea" (ἰδέα), using it somewhat interchangeably with the Greek word for shape or form (εἶδος ). The word idea derives from the Greek for "to have seen."

Plato's Line is also a division between Body and Mind. The upper half of the divided line is usually called Intelligible as opposed to Visible, meaning that it is "seen" by the mind (510E), by the Greek Nous (νοῦς), rather than by the eye.

In most modern Indo-European languages there are two words that correspond to the English "to know." One of these derives from "to be cognizant of" or "to be acquainted with," the other from "to have seen." The first is the cognate (sic) of English "know." e.g, Greek gnosis (γνῶσις), meaning knowledge. For knowledge the Greeks also used epistέme (ἐπιστήμη), the root for our word epistemology.

Examples of European words that we also translate "know," but which derive from "see," are savoir (Fr.) and wissen (Ger.).

In Republic, Book VI, 507C, Plato describes these two classes of things, those that can be seen but not thought, and those that can be thought but not seen:

καὶ τὰ μὲν δὴ ὁρᾶσθαί φαμεν, νοεῖσθαι δ᾽ οὔ, τὰς δ᾽ αὖ ἰδέας νοεῖσθαι μέν, ὁρᾶσθαι δ᾽ οὔ.

At Republic, Book VI, 508B-C, Plato makes an analogy between the role of the sun, whose light gives us our vision to see (ὄψις) and visible things to be seen (ὁρώμενα) and the role of the Good (τἀγαθὸν). The sun rules over our vision and the things we see. The Good rules over our (hypothetical) knowledge and the (real) objects of our knowledge (the forms, the ideas):

“This, then, you must understand that I meant by the offspring of the good which the good begot to stand in a proportion with itself: as the good is in the intelligible region to reason [CD] and the objects of reason [DE], so is this (sc. the sun) in the visible world to vision [AB] and the objects of vision [BC].”

φάναι με λέγειν τὸν τοῦ ἀγαθοῦ ἔκγονον, ὃν τἀγαθὸν ἐγέννησεν ἀνάλογον ἑαυτῷ, ὅτιπερ αὐτὸἐν τῷ νοητῷ τόπῳ πρός τε νοῦν καὶ τὰ νοούμενα, τοῦτο τοῦτον ἐν τῷ ὁρατῷ πρός τε ὄψιν καὶ τὰ ὁρώμενα.

At 509D-510A, Plato describes the line as divided into two sections that are not the same (ἄνισα) size. Most modern versions represent the Intelligible section as larger than the Visible. But there are strong reasons to think that for Plato the Intelligible (being unitary abstraction) is to the Visible (with its many concrete particulars) as the One is to the Many.

We don't know whether Plato imagined the Intelligible or the Visible to be the larger section, but it seems clear that he pictures the Intelligible section above the Visible, so we use a vertical line.

Plato then further divides each of the Intelligible and the Visible sections into two. He argues that the new divisions are in the same ratio as the fundamental division.

“Represent them then, as it were, by a line divided into two unequal sections and cut each section again in the same ratio (the section, that is, of the visible and that of the intelligible order), and then as an expression of the ratio of their comparative clearness and obscurity you will have, as one of the sections of the visible world, images. By images I mean, first, shadows, and then reflections in water and on surfaces of dense, smooth and bright texture, and everything of that kind, if you apprehend.”
“I do.”
“As the second section assume that of which this is a likeness or an image, that is, the animals about us and all plants and the whole class of objects made by man.”
“I so assume it,” he said.
“Would you be willing to say,” said I, “that the division in respect of reality and truth or the opposite is expressed by the proportion: as is the opinable (δόξα) to the knowable (γνῶσις) so is the likeness (ὁμοιωθὲν) to that of which it is a likeness (ὡμοιώθη)?”

ὥσπερ τοίνυν γραμμὴν δίχα τετμημένην λαβὼν ἄνισα τμήματα, πάλιν τέμνε ἑκάτερον τὸ τμῆμα ἀνὰ τὸν αὐτὸν λόγον, τό τε τοῦ ὁρωμένου γένους καὶ τὸ τοῦ νοουμένου, καί σοι ἔσται σαφηνείᾳ καὶ ἀσαφείᾳ πρὸς ἄλληλα ἐν μὲν τῷ ὁρωμένῳ τὸ μὲν ἕτερον τμῆμα εἰκόνες — λέγω δὲ τὰς εἰκόνας πρῶτον μὲν τὰς σκιάς, ἔπειτα τὰ ἐν τοῖς ὕδασι φαντάσματα καὶ ἐν τοῖς ὅσα πυκνά τε καὶ λεῖα καὶ φανὰ συνέστηκεν, καὶ πᾶν τὸ τοιοῦτον, εἰ κατανοεῖς.
ἀλλὰ κατανοῶ.
τὸ τοίνυν ἕτερον τίθει ᾧ τοῦτο ἔοικεν, τά τε περὶ ἡμᾶς ζῷς καὶ πᾶν τὸ φυτευτὸν καὶ τὸ σκευαστὸν ὅλον γένος.
τίθημι, ἔφη.
ἦ καὶ ἐθέλοις ἂν αὐτὸ φάναι, ἦν δ᾽ ἐγώ, διῃρῆσθαι ἀληθείᾳ τε καὶ μή, ὡς τὸ δοξαστὸν πρὸς τὸ γνωστόν, οὕτω τὸ ὁμοιωθὲν πρὸς τὸ ᾧ ὡμοιώθη;

Later, at 511D-E, Plato summarizes the four sections of the divided line

“Your interpretation is quite sufficient,” I said; “and now, answering to these four sections, assume these four affections occurring in the soul: intellection (νόησιν) or reason for the highest [DE], understanding (διάνοια) for the second [CD]; assign belief (πίστις) to the third [BC], and to the last picture-thinking or conjecture (εἰκασία) [AB], and arrange them in a proportion, considering that they participate in clearness and precision in the same degree as their objects partake of truth and reality.”

ἱκανώτατα, ἦν δ᾽ ἐγώ, ἀπεδέξω. καί μοι ἐπὶ τοῖς τέτταρσι τμήμασι τέτταρα ταῦτα παθήματα ἐν τῇ ψυχῇ γιγνόμενα λαβέ, νόησιν μὲν ἐπὶ τῷ ἀνωτάτω, διάνοιανδὲ ἐπὶ τῷ δευτέρῳ, τῷ τρίτῳ δὲ πίστιν ἀπόδος καὶ τῷ τελευταίῳ εἰκασίαν, καὶ τάξον αὐτὰ ἀνὰ λόγον, ὥσπερ ἐφ᾽ οἷς ἐστιν ἀληθείας μετέχει, οὕτω ταῦτα σαφηνείας ἡγησάμενος μετέχειν.

We can collect the various terms that Plato has used to describe the components of his divided line. Some terms are ontological, describing the contents of the four sections. Some are epistemological, describing how it is we know those contents.

509D-510A	likenesses, images, shadows, imitations, our vision (ὄψις, ὁμοιωθὲν)	the physical things that we see/perceive with our senses (ὁρώμενα, ὁμοιωθὲν)	opinion, beliefs (δόξα, νοῦν)	knowledge (γνῶσις, νοούμενα)
511D-E	conjectures, images, (εἰκασία)	trust, confidence, belief (πίστις)	understanding, hypothesis (διανόια)	intellection, the objects of reason (νόησις, ἰδέαι, ἐπιστήμην)

By insisting that the ratio or proportion of the division of the visibles (AB:BC) and the division of the intelligibles (CD:DE) are in the same ratio or proportion as the visibles to the intelligibles (AC:CE), Plato has made BC = CD. And indeed Plato at one point identifies the contents of these two sections. He says (510B) that in CD the soul is compelled to investigate by treating as images the things imitated in the former division (BC).

“Consider then again the way in which we are to make the division of the intelligible section.”
“In what way?”
“By the distinction that there is one section of it [CD] which the soul is compelled to investigate by treating as images the things imitated in the former division [BC], and by means of assumptions from which it proceeds not up to a first principle but down to a conclusion, while there is another section in which it advances from its assumption to a beginning or principle that transcends assumption (ἀρχὴν ἀνυπόθετον), and in which it makes no use of the images employed by the other section, relying on ideas only and progressing systematically through ideas.”

σκόπει δὴ αὖ καὶ τὴν τοῦ νοητοῦ τομὴν ᾗ τμητέον.
πῇ;
ἧι τὸ μὲν αὐτοῦ τοῖς τότε μιμηθεῖσιν ὡς εἰκόσιν χρωμένη ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ᾽ ἀρχὴν πορευομένη ἀλλ᾽ ἐπὶ τελευτήν, τὸ δ᾽ αὖ ἕτερον—τὸ ἐπ᾽ ἀρχὴν ἀνυπόθετον—ἐξ ὑποθέσεως ἰοῦσα καὶ ἄνευ τῶν περὶ ἐκεῖνο εἰκόνων, αὐτοῖς εἴδεσι δι᾽ αὐτῶν τὴν μέθοδον ποιουμένη.

Plato distinguishes two methods here. The first (the method of the mathematician or scientist) starts with assumptions or hypotheses (ὑποθέσεων) - Aristotle called them axioms - and proceeds to a conclusion (τελευτήν) which remains dependent on the hypotheses or axioms. The second (the dialectician or philosopher) advances from assumptions to a beginning or first principle (ἀρχὴν) that transcends the hypotheses (ἀνυπόθετον), relying on ideas only and progressing systematically through ideas.

The question whether and in what way such first principles exist is the subject of "first philosophy" that Aristotle's editors called Μetaphysics. Ιn the traditional order of Aristotle's works, these books came after the Physics and were so titled - ΤΟΝ ΜΕΤΑ ΤΑ ΦΥΣΙΚΑ. The Neoplatonist philosopher Porphyry is said to have asked the "fateful question" of whether Plato's ideas (forms or essences) really exist. Aristotle unequivocally thought that the Platonic ideas were simply abstractions from concrete existents. In the twentieth-century jargon of Existentialism, "existence precedes essence."

Platonism (sometimes called Platonic Realism or Platonic Idealism) is the view that the ideas exist, but since they are not physical, their existence is outside space and time. In the nineteenth and early twentieth centuries a view called psychologism (or conceptualism) was developed, arguing that the ideas exist as concepts only in human minds. Anti-Realism (also called Nominalism) denies the separate abstract existence of the ideas, claiming that the number 3, for example, exists only in all the concrete sets of three things. The abstract concept of redness exists in the set of all red things. This was Aristotle's view.

Plato claims that the dialectical method is more holistic and capable of reaching a higher form of knowledge, possibly related to his notion that the eternal soul has "seen" all these truths in past lives (incarnations) and before (we might call it the beforelife).

Aristotle's divided line!

Plato's idea that the Good (τον ἀγαθὸν) is the creator (δεμιύργος) of all the ideas (the forms of the Good) resembles our information philosophy view that ergodic processes are the source of everything of positive value (our Ergo).

Plato does not identify the Good with material things or even with the abstract ideas and forms. The Good is the creative process that generates the forms.