4a. Let us return to the invitation to reflection that is extended to Glaucon by the sectioning of the realms “as if” they were a line; he must wonder why, as has been said, the Republic has no dialectical treatment either of the Good or of the eide under it. This missing logos is, however, absent in a different way for both of these dialectical objects. Let us begin with the Good.
The Good has no “place” within the realm of being, for it is “beyond being” (509b9). Since it is that which is “un-hypothesized” it cannot be traversed in the same way as the “hypotheses to being,” the stepping stones of the logos (511b8). Consequently, there is in this dialogue no power of the soul that corresponds to it, as is signified by the fact that it is off the top of the Divided Line. Although within the context of the imagery of sight the eye of the soul is said to look at it, a distinction between movement “among” and “through” the eide (510b8, 511c2) and movement “up unto” the Good (511b6, 533c8) is pretty generally maintained; the latter has about it something tangential and momentary; a glimpse of the Good is “scarcely” (517c1) achieved. Consequently, this beholding is not quite knowing in the dialectical sense at all, for the “idea of the Good” is the result not of a unitary act of sight but of “abstracting” and “determining in logos” (534b9). Socrates repeats this several times: The Good as the responsible source is known only after the eidetic vision; it is known on the downward return, so to speak, by a syllogismos or collection of logoi, a logos of logoi (516b9, 517c1). It is, in effect, the most comprehensive of all those “collections” (e.g., Sophist 267b1) that follow the “divisions” of dialectic. The Good, the “greatest study,” is a “learning matter” or a mathema only in a new and strange sense, for it is learned in the movement away from it—to confront the Whole as a knower is to step back among the parts.
4b. For those realms, however, that are on the Divided Line, the absence of logoi takes on a different significance and form. It is essential to the following discussion to recall that the word logos means not only account or reasoning but also the mathematical relation of ratio, a double meaning of great importance particularly in Pythagorean contexts (e.g., Epinomis 977c3). Now we are told that each of the unequal main sections of the line is again to be cut “in the same ratio” (509d7), but we are neither given the ratio itself—we are not even told whether it is numerical or irrational, i.e., a ratio of commensurable or incommensurable lines—nor are we definitely told whether the greater or the less of the unequal segments is to be the upper one (cf. Plutarch, Platonic Questions 1001d). We can conclude nothing except that the two middle segments must be equal, which means that pistis and dianoia are in some way coextensive. That is indeed necessary since the realm of the dianoia uses the realm of natural objects as corresponding images. 
4c. This absence of definite ratios is the more noteworthy because for the earlier tripartite soul the numerical ratios of the parts are, playfully, given: they form the musical progression of the “highest,” “middle,” and “lowest” place in the diapason (443d6; cf. 432a). If the “middle” is here taken non-technically to designate the mean through which the “first consonances,” the fifth and the fourth, are compounded (cf. Nicomachus, below) then these are as 6:4:3, the three terms of a harmony, where 4, which represents the spirited element, the thymos, is a “harmonic mean.” (If the mese understood as the string of the fourth, then, as Theon, Hiller, p. 62, shows, this ratio can be immediately obtained.) The use of the terms of the “harmonic proportion,” i.e., for b 4, a(6):c(3)::a-b(2):b- c(1), may have a special significance here. Nicomachus (Introduction to Arithmetic II, xxvi) says that Philolaos the Pythagorean regarded this proportion as the “geometric harmony,” expressing the cube, which has 12 sides, 8 angles, and 6 faces, so that its characteristics are given in the terms 6:4:3. The tripartite “embodied” soul is therefore here characterized as the basic solid. Now, Aristotle (On the Soul 404b16 ff.) reports a similar Platonic oral teaching about the soul. The noetic cosmos, called “the animal itself,” arises “from the idea of the one and primary length and breadth and depth:” so that in respect to the soul, which is similar to that animal, “intellect (nous) is one; knowledge (episteme) is two, since it is uniquely related to one; the number of the plane is opinion (doxa); and sense perception (aisthesis) belongs to the solid.” Evidently, the Academy too believed that the soul reaches some sort of solidity as it meets the body.
We may ask further how this dimensional structure is related to the quadripartite “knowing” soul of the central Republic. Although it is certainly not the same, there is an instructive relation between them which is best seen schematically:
The dimensional soul will be seen to be the more comprehensive of the two since it possesses elements for apprehending both extremities of dimensionality; namely, unit and body, where the former is the source of the whole soul, and the latter is its full-grown structure. The soul of the Republic has, as we have seen, no clear and separate capacities corresponding to these; furthermore, as the Divided Line shows, it has no dimensional progression, for the apprehension of solids occurs in one of the middle sections. On the other hand, it does have what one might call a certain reflective depth, which arises from the eikastic reduplication of episteme in dianoia and of pistis in eikasia that takes place within its two major parts, noesis and doxa (534a). In summary, it might be said that the dimensional soul is all-embracing or cosmic, which is why some can say that “the soul is the place of the eide” (On the Soul 429a27), while in respect to the soul that goes with the realms of the Divided Line, it might rather be said that “the eide are the place of the soul.” The soul ranges all over this place, sometimes settling in one spot and then moving again, remaining always somewhat a stranger—in accordance with the similarity between knowledge and the pervasive, piece-meal eidos of otherness described in the Sophist (257c7).
Later, we shall see the significance of the fact just pointed out; namely, that the soul of the Republic is not a cosmic harmony of number ratios.
But if the logoi themselves are absent, this much about them is given: They are the same throughout, for sameness of ratio defines a proportion, an ana-logia or “recurrence of logoi.” How is Glaucon to interpret the mathematical fact that is here presented to his dianoia?
4d. We should, first of all, keep firmly in mind that this mathematical presentation is itself only a simile; Glaucon is to cut the realms “just as” he would a line (509d6). He has no reason to think either that the realms of being literally have mathematical ratios to one another or that the inexplicitness of the logos in any way implies that the ratio of any two quantities is indeterminate (as are the greater and the less, a ratio technically known as “indefinite,” aoristos, Metaphysics 1021a4); on the contrary, the logos here stands for the possibility of articulate human language. Thus cautioned, let us see what the model will yield.
Immediately after the fundamental division of the line and the description of the lower subsections have been made, Socrates reads off a first proportion (510a9):
opined : known :: images : imaged object
This proportion announces that the internal relations of the two lowest realms are the same as those of the whole, that the relations connecting the whole are mirrored in even its lowest parts. At the very end of the Divided Line passage, he reads off yet another proportion (511e3):
segments of line : truth :: affections of soul : clarity
—which means in mathematical terms that the affections of the soul are the correspondents (Euclid V, Def. 11: Given a:b::c:d, a and c, as well as b and d, are said to correspond) of the realms of being that the line segments represent. Or, using analogical reasoning—that is, inferring the likeness of correspondents (cf. Metaphysics 1016b34, 1093b18; Topics 108a7)—we may conclude that known and knower are alike (cf. On the Soul, 404b18). Here the analogical method brings out the bond that “yokes together with the strongest yoke” (508a1), the linking of known and knower by the clarifying light of truth; this illumination can bind them because they are both like the Good” (509a3), that “ruling source” of the “community” of knowns and knowers (cf. Sophist 248a11).
And finally, in concluding the explication of both the sun and the cave age, Socrates forms two more proportions (534a3):
being : becoming :: thought : opinion
thought : opinion :: knowledge : trust :: thinking : image recognition
—the first of which signifies that the gradations of knowing are the same as the degrees of being. The last, more extended, proportion displays particularly well the force of the mathematical form Socrates has chosen. For, since the affections of the soul are coordinated with linear magnitudes, they may be “alternated” (Euclid, Elements V, Defs. 12, 3) so that the first is to be the third as the second to the fourth—and this is exactly what Socrates has done here; he has alternated the original proportion of the segments, so that now
knowledge : thinking :: trust : image recognition
This new form of the proportion draws attention to the close relation of each faculty in one main segment to the corresponding faculty in the other, a relation which mirrors that of the main faculties and again that of the realms of being. The last ratio, which links thinking (dianoia) with image recognition (eikasia), particularly justifies the notion of a “dianoetic eikasia,” a thinking use of images, while the preceding ratio shows a certain special relation between knowledge and trust, which we experience in that unassailable finality or incorrigibility, analogous to the self sufficiency of knowledge, which certain sense perceptions possess (523b1).
Obviously, by using the various Euclidean operations (Euclid V, Defs. 11-18) on these proportions, and by attending either to the sameness of the ratio relation or to the likeness of the correspondents in the new proportions, it is possible to obtain a variety of illuminating results. That this would be a legitimate enterprise is shown by the term Socrates uses when he dismisses further division of the line lest there be a surfeit of “multiplicate logoi” (534a7), a punning reference to the “duplicate” and “triplicate” ratios of the theory of proportion (Euclid V, Defs. 9-10).
All of these further results would be, however, only the expression of two fundamental similarities: First is that of the knower and the known, mentioned above, which Socrates has in mind when he tells Glaucon to “order them [the affections of the soul] analogously” (511e2) to the realms of being; and second is that—really prior—similarity of each degree of being to the next higher degree. It is by reason of this similarity that the successive realms of (1) images, (2) natural objects, (3) mathematicals, and (4) eide are described in turn as (1) “that which is made as something similar,” (2) “that to which it is made similar,” (3) “that which was before copied and is now treated as a likeness” (cf. 510a10, b4, 511a7); and (4) even the eide themselves are, as we learn from other sources, formed in the likeness of the Good understood as the One (see below), being themselves each one (e.g., Metaphysics 987b18 ff.—the formula is on/hen, being/one.)
This four-stepped ladder of similars is what makes the upward transition, i.e., the dialectical road, possible. It is, we should note, completely articulated first in the Divided Line; the sun image has only two undifferentiated realms, the intelligible and the visible. The Divided Line, in a certain way, preserves this original homogeneity of the larger realms; images and natural bodies are not found in differently constituted realms, for both are sensibles and either have their own body or use an alien one—reflections themselves are “in water” and “on smooth bodies” (510a1)—so that the difference is not really that between the plane and the solid dimension, or between visible and palpable things. Similarly, hypotheses and eide are equally intelligibles. What differentiates the realms internally is not, to use a latter-day expression, a different “material,” but rather the reflective distinction of like to likened and of genuine to counterfeit, which reflects in the parts of the line the imaging relation of the sun to the Good.
4e. Glaucon will, then, see that the logoi relating certain aspects of the whole are one and the same throughout, that on account of similarity or likeness (homoiotes, cf. Sophist 231a7, Statesman 285b6) there is one logos pervading the whole. In presenting this notion to Glaucon mathematically, Socrates is signifying that he is presenting him with such hypotheses about being and becoming as will make thinking itself possible—and by this he means thinking consistently, namely, “in such a manner that the sameness of logos is preserved” (homologoumenos, 510d2; cf. Aristotle, Topics 108b8). But if the characteristic dianoetic direction is downward to conclusions by deductions that win “agreement” (homologia) because the logoi in different souls have remained in concord, the discovering dianoia moves upward by an analogia. It is this latter use that is chiefly required in any search, and is therefore suggested to Glaucon in this part of the dialogue: “Make an analogy…” (524d9; cf. 509b2). An explication of this means of learning is given in the Statesman: When the teacher chooses something about which the learner has right opinion to “lead him up to” (anagein) as an example, that is, when the teacher shows the learner a paradeigma, i.e… “something to be shown beside” some unknown, which is able “to lead [the learner] to” (epagein) this unknown—then this unknown may become known to the learner by a recognition of the analogy (277d9; here Socrates, in the reflexive mode characteristic of him, explains “example” by giving an example of an example, just as in the Republic he explains “image” by an image). The sun, as an image of the Good, is just such an “example,” and since the Good is far above the sun, the epagoge, the “bringing up,” of Glaucon will be a true ascent. In fact, it will be an ascent—though only provisional—to the source of all examples, to that paradeigma which is no longer example but exemplar.
We might summarize this exposition from a different point of view by saying that the Divided Line tells the story of “recollection” mathematically, by presenting through proportions that “affinity” of all nature (cf. Meno 81d1) which makes it possible to move with a sense of recognition in unknown places. Aristotle will reduce this upward, or inward, journey to the “logical” procedure of epagoge or induction, of which he makes Socrates the inventor (Metaphysics 1078b28).
4f. The first and original affinity, the sun image implies, is that which the Good as progenitor has with the sun as the offspring made in its image. In other words, the Good itself possesses an image-making power that it passes down to the eide and that they pass on in turn (cf. Phaedrus 250a6). This “downward eikasia,” as it might be called, by making our world a progression of likenesses, is originally responsible for our own ability both to make ourselves like to the highest things by homoiosis (500c5, Theaetetus 176b1) and to recognize likenesses or to make analogies. It is, we might say, responsible for our “upward eikasia” and for the pleasure of recognition it gives us (cf. Aristotle, Poetics 1448b8); it is a power so unobtrusively indispensable that without it, we would never “know ourselves” even in the most superficial sense of having confronted in a mirror our own looks, the eidos of our own face!
4g. We can now see precisely why the criticism of poetry in Book III turns into that radical “ancient quarrel between philosophy and poetry” (607b5) in Book X. This quarrel, which already engaged Pythagoras, who descended to Hades to watch Homer and Hesiod suffer for their lies (Diogenes Laertius VIII, 21), is now given a precise cause. In the light of the sun image, poets are usurpers and perverters of the power of the Good. They are more despicable even than that charlatan who, having carried a mirror through the world, claims to have “made everything” (596c2), when he has really only borrowed the lowest effects of the power of the Good. For poets make artificial images, using a perverted power of eikasia, a “low” (603b4) generation called “mimetic” or imitative (602a11), which produces images of good and bad things indiscriminately (604e1, Sophist 233c; cf. also 267a) and distracts the listeners from true being (605a9). Such mimetic products are not natural likenesses, but are separated from the true source of images by the interposition of a human maker, who “makes images vilely” (Republic 377e1). Poetic mimesis makes artificial imitations; “artists,” to speak in modern terms, arrogate to themselves an unauthorized function of “creativity,” while Socratic eikasia makes likenesses in the sense of observing those that are already there by nature, clothing them in figures and putting the figures into words.
5a. We must now go on to see exactly what conjectures about the Good the sun image allows Glaucon to make on reflection, even though he cannot yet reach a full and sure logos.
In the image, the Good is presented in three successive capacities, a triplet proved to be fundamental by its recurrence in the Philebus (20b8). It is presented first as the father of the sun (508b12), then as that which is responsible for knowledge (e3), and last as the source of being (509b7). The first of these might be called its cosmogenic function, by which the potent male Good generates the sun as a male offspring to be lord of the visible world and an intermediate source of the world of becoming, analogous to the Good itself as ruler of the world of thought (508c4, 517c3); the obvious question that arises here is whether the sun also has a mother—the cave image will deal with that. However, although the sun resembles its maker in its brightness, its continual risings and settings clearly mark it as a part of the world of becoming and passing away, while at the same time they bear evidence that the Good is also a source of motion (cf. Alexander on Aristotle’s Metaphysics 988a11 ff., ed. Hayduck, p.59). In its second capacity, the Good is several times called the aitia, the “responsible cause” (508e3; 517c2), and aitos, “that which is to be called to account” (516c2) both for the passive state of the so-called nooumena, “beings known” (508e1, 509b6, d8) and for the active knower (508e2), that is, for the soul in its “act of knowing” (509b6)—this aitia is, however, yet more beautiful and more honorable than these effects. In its third capacity, the Good is called king and lord (509d2, 517c4) and arche, “ruling source” (510b7, 511b7) of the whole, or the “arche itself” (533c8), “in power and seniority exceeding the nature of being” (509b9); it gives things both their “state of being” (to einai) and their nature as beings or beingness” (ousia, 509b8). The latter two capacities are reduplicated by the sun as source of sight and becoming.
Socrates presents these functions in the order that will bring Glaucon up by analogy from the visible many to the invisible one (507b2). In the order of logical generation, however, the listing should clearly be reversed since being itself must somehow precede the confrontation of active and passive beings, and this split must, in turn, come before the birth of a world perceptible and perceived by sense. The grandest, most inclusive, most politically relevant, function of the Good is, therefore, its rule (arche) over being; next it acts as the “answerable cause” (aitia) for teachers and learners, while its most private function is that of a father. But in truth neither order holds, for the Good itself is not hierarchically ordered, being itself the source and beginning of all order—”the arche itself “(533c8).
5b. The diagram below shows the parts of this order; all its terms except one are taken from the text:
This scheme shows the Good as presiding over and bonding a kind of pervasive duplication: the Good as the reason for knowledge is responsible for the unifying confrontation of knower and known (right side) and so also for the agent of noesis, the soul (508d4; cf. Sophist 248c11, e ff.). As direct source, the Good also gives rise to ousia, beingness (left side), which, by reason of the soul’s presence, has a second aspect: It is the “place” (508c1, 509d2) provided for the soul, the topos noetos, which contains the “things for thought,” the noeta (the -tos ending signifying the capability of being thought; cf. Sophist 248d4). Finally, as the generating source, the Good puts forth the Sun, a sensible second source that reduplicates the whole structure of being on the lower level of sense and becoming.
At this point, the diagram brings out an aspect of the sun image that is of fundamental importance to the human place in the whole. The soul, which arises in the first instance as a knowing soul (508d6), is in some way also involved with the world of becoming: some aspects of sense “invite thought” (523b1). Furthermore, we do have opinions, that kind of set mental reaction significantly expressed in the phrase beginning “I feel that…” Human speech, too, can accommodate itself to becoming, since it is capable of the same admixture of non-being that gave rise to becoming (477a; cf. Sophist 260b10). In other words, the human soul as the moving agent of knowledge has a faculty, doxa, by which it ranges over becoming and has a place there. This place contains “that which is to be opined,” the doxaston, and the name of these things, as apprehended by the soul, has been added in the diagram—they are the dogmata, a word denoting both ordinary opinions (cf. Republic 538c6) and their political counterparts, the decrees and ordinances of the city.
Thus, from the point of view of the human soul, genesis, becoming, belongs to being as one of its gradations (just as the representative of the Good, the sun, is not above but within this world); hence, doxa has a certain kinship with knowledge (Meno 86a). But from the point of view of the body, becoming is the place of “sense perception,” aisthesis (507c4, e6), and, most characteristically, of “things to be seen,” the horata, whose organ of perception is the eye, which is “most like the sun” (508b3), just as the soul is like the Good. This is why Socrates has two names, doxaston and horaton, for the segment and realm of becoming (509d), just as each human being has two “organs:” that “by which” it sees, the soul; and that “through which” it sees, the eye (Theaetetus 184c6; cf. Timaeus 28a2). Becoming is then, in a sense, “within” being, not an external accretion to it (cf. M. Heidegger, Nietzsche [Pfullingen 1961] I, p. 207).
5c. However, seen in another way, the diagram brings out a certain downward doubling, affecting the differentiation, or rather proliferation, from one to many that Socrates had recalled when he introduced the image. Whence does it arise? Now although no source besides the Good is mentioned, the language of the image persistently implies that something already there is capable of “taking” the gifts of the Good in various ways. The Good “provides” (parechei, 503e1, 509a7; cf. b3, 517c4) what is known with that truth of which the things known “partake” (metechei, 511e3, 4); it “gives” this power to the knower (508e2), makes intelligibility “to be present” (pareinai, 509b7) in things, and causes being “to be added to them” (proseinai, b8). One might be tempted to think of some underlying “material” (with which Aristotle, speaking in his own terms, does indeed equate that in Plato’s teaching which “takes” the Good, Metaphysics 988a11), except that the Good does not actually differentiate some available stuff, but rather binds something disposed to come in a two-fold way. For instance, just as the “yoke” of light yokes two different things, vision and visibility (507c6), so the truth is the bond by which the Good binds the disparate knower and known. This dyadic disposition appears also in other ways: in the “double eide” (509d1, 4, 6) of the visible and the knowable (cf. the “two morphai of Parmenides’ “double philosophy,” Diels, Vorsokratiker, I. fr. 8, 53 and p. 218, 6) and their two-fold subdivisions (534a1), and in whatever makes “division into two,” the complement of making analogies (534a6), possible. We have here an intimation of that secondary dyadic principle, so often mentioned by Aristotle under the name of the Indefinite Dyad, which in Plato’s arithmological teaching is the secondary arche complementing the Good understood as the One.
Although it is only an intimation, it is one that must attend on any presentation of the whole, because the second principle has “a certain likeness to the whole” by reason of which it “contains all things” (Aristotle, Physics 207a19). As Speusippus explains: “For they held that the One is higher than being and is the source of being; and they delivered it even from the status of a principle. For they held that given the One in itself, conceived as separated and alone, without the other things, with no additional element, nothing else would come into existence. And so they introduced the Indefinite Duality as the principle of being.”
This second arche of Being is discovered and described in the Sophist, in pursuing the source that makes sophistical speaking (commonly known as “Double Tulk,” Dissoi Logo,) possible. A difficulty had arisen over the necessary two-ness of Being. This two-ness had been a consequence of the very fact that plays so pervasive a role in the sun image; namely, that there is both knowing and something knowable; therefore, Being, as known and knowing, possesses a knowing and living soul and is thus in motion, while as knowable it is steadfast and at rest (248d ff.). Being was thus both in motion and at rest, and this resulted in a quandary: Being kept cropping up in speech as “some third thing” beside these irreducibly separate two, motion and rest, so that neither of them could be said to be (250b-d). The solution to this quandary was found in the nature of the Other, which goes “through everything” (255e3), and has a correlative, the Same. For these make it possible to say that Being can be both motion and rest, since each of the two is just what it is by reason of being the same with itself and is not the other by reason of being other than that other, i.e., by participating in the Other. In this way, the deliverances of speech were justified and the logos was saved.
The eidos of the Other is described in the Sophist as always relative (255d), effecting Non-being, which is infinite (256e) and cut up into many parts (257c). But these are exactly the terms associated by Aristotle with the Indefinite Dyad (Metaphysics 987b19 ff., 1087b13 ff.). Plato, he says, made “the other nature” (cf. Sophist 256e1, 257c7) a dyad because the numbers “outside the first” (i.e., after the One and the Dyad) can be “begotten from her just as from a matrix” (Metaphysics 987b34) by the One as begetter. The first such definite arithmos is the eidetic Two (ibid. 1082a11), Being (cf. Note 35).
This “two-making” arche is precisely one of those “responsible causes of division” (Sophist 253c3) which is being sought, and, as has been said, it is the source of the possibility of that confrontation between knower and known, which the Good confirms with its illuminating truth. Clearly, then, such an arche is in the background of the sun image. We shall see, however, that in the context of the human good, namely that concerned with the embodied soul, it will appear not as a second source of being but as something opposed to that good, as the source of evil. But for the human knowing soul, Socrates has coined a special term—it is “like the Good,” agathoeides, or well-formed (Republic 509a3), and so are all its situations.
5d. We must now take a last look at the role of similarity or likeness in the sun image, not merely insofar as it permits ascent by analogy, but as a constitutional principle. In the conversation of the Parmenides—which is, incidentally, recounted to Adeimantus and Glaucon—young Socrates had tentatively presented “likeness” as a solution to the problem of methexis, the “participation” of the many sensible things in the one invisible eidos (132c12). He thought that the eide might be “patterns in nature” (d2), patterns to which “the sensible things become like and hence are things likened [i.e., copies];” therefore “their participation in the eide is nothing but this being made in their image” (d4). Parmenides shows him that his solution is impossible. The eide cannot be “in nature,” i.e., among beings, as patterns since, likeness being reciprocal, the pattern would be indistinguishable from its copy—both would be like and would require yet a third eidos above them to be like to (e7). This form of the “third man” argument (cf. Metaphysics 990b17) amounts to saying that patterns, as mere patterns, are not necessarily above their copies in the scale of being and need have no originating power, that they have no nature by and in themselves, nothing that permits them at once to retain their oneness as eide and to be a source of manyness—they are not sources. (This same criticism is in fact made by Aristotle of the Platonic eide, Metaphysics 1079b.) At this stage, young Socrates does not even suspect that the problem might have to be considered on a higher level; namely, that of the participation of the eide in each other, of the communities they form with each other, and, beyond that, of the whole they belong to altogether (129d6); hence, he does not see that his eidetic solution might after all be usable. Yet, as we shall see, there is a place for such a solution.
One aspect of the higher methexis problem, the problem of the several “communities” that the eide have with each other, was, as we have seen, considered in the Sophist (254b7), where the solution to the question how both rest and motion can be was given in terms of the correlative archai of the Same and the Other. Both of these extend throughout Being, for by being one and the same with itself, each eidos remains integral and independent, while by being other than another it becomes so related to that other, namely as another “other,” as to be capable of being together with it in a “community” (256a10). That is why Otherness is the bond of Being.
Now if the point of view taken is not within but “beyond Being,” Likeness performs just such a function as Otherness did within Being, and, in a way, does so more plausibly. For within Being, the secondary, reflexive eidos of the Other was the source of community, the “bond” that ran “through everything” (255e3; cf. 253a5, c1), while the primary Same was responsible for the separate and independent oneness of each being (254d15). But the bonding of the whole is achieved precisely because of the Likeness of each thing within it to a pattern beyond and so to each other thing; it is exactly the failing power of “being like” that manifests itself within Being as the peculiar effect of Otherness called “being an image of” (Sophist 240a8). The fact is that Parmenides’ objection fails as soon as the pattern is really of a different order and sufficiently beyond reach, as the Good indeed is: “[and furthermore] the Good is not being but yet beyond Being in seniority and exceeding it in power” (Republic 509a3, b8). Note that while it was impossible for Being to be “some third thing” (Sophist 250b-d) beyond its constituent eide, the Good is to be imagined as precisely such a “third kind” of thing (Republic 507d1, e1). From the highest point of view, that of the whole, not otherness but likeness is the bond; in terms of the knowing soul, not logos but analogia is required. It is a token of this that the knowing part of the soul, to which the Other is compared in the Sophist (257c7), is said in the Republic to be “like the Good” (509a3).
It is precisely this bond by which the Good makes everything one whose mathematical image takes the form of a proportion: “And the most beautiful of bonds is that which makes itself and the things bound together as much as possible into one. Proportion accomplishes this most beautifully. For when the middle term of three numbers…is such that as the first is to it, so it itself is to the last…then necessarily all will turn out to be the same. They will all become one with each other” (Timaeus 31c2; cf. Metaphysics 1016b34). We can now see a second reason for the equality of the middle sections of the Divided Line: the three-term proportion (i.e., a:b::b:c) “makes one” or unifies by means of the power of the “in-between,” the metaxy—the Divided Line represents a harmonized Whole.
Socrates had already described and named this beautiful union, the true home of the philosopher, in that “persuasion to the rule of philosophy,” addressed to Adeimantus and the others, which precedes the presentation of the sun image to Glaucon (487b ff.). His language then was in terms of man and god; in keeping with his purpose he gave an anthropomorphic view of the realm of being, as it were: “For there is surely no leisure for him who has his thought truly set on beings to look down into the affairs of men and by fighting with them to be filled with envy and malice; but looking at such things as are ordered and always remain set, and observing that they neither do injustice nor have injustice done to them by one another, all being set in an ordered whole (kosmoi) and according to logos, he imitates these and makes himself as like [to them] as possible… And so the philosopher, conversing with what is divine and like an ordered whole, himself becomes as divine and ordered (kosmios) as is possible for man” (500c-d). Socrates is throughout employing the word used of the visible world, cosmos, and the man who becomes like a god is presently called a demiurge, who, like the divine artificer in the Timaeus, uses a “divine pattern” (Republic 500e3; cf. Timaeus 28a7) in making his work of art, the city. We see that the interior order of the world of being is to be imagined as analogous to a cosmos, an ordered visible world, having a taxis, a hierarchical order. The Good is to be understood as the comprehensive source of this order, which is here presented in the familiar language of Pythagorean cosmology: Justice is a reciprocal matter, the parts of the whole are related “according to logos,“ that is, as in ratios, and participation in the order is by imitation and likening (cf. Metaphysics 1075a12 ff.). We may conjecture that this is a popular presentation of that taxis (whose terms are also borrowed from the Pythagoreans) which supervenes when the Good is understood as the One, the articulating aitia of the eide, which makes them what they are (ibid. 988a12), and above all makes of the “greatest eide,” motion and rest—which together form being—the eidetic Two. The taxis that thus arises is that arithmological structure of the eide which is the prototype of all ordered associations, ordinal and cardinal (see ibid. 1080a ff.; also Note 35).
5e. Although Socrates had introduced the sun image with a reference to “the things said earlier [cf. 476] and often spoken of at other times” (507a7), namely the many and how they participate in the one idea that is “what is” in these many things (476a7, d1, 507b5), yet within the image he goes, as we have seen, beyond the oneness of each eidos to a still higher point of view, the way to, which is sung in the “hymn of dialectic.” There he says that “when someone leaves behind all sense perception to set out for that itself which each thing is (ep’ auto ho estin hekastos), and does not leave off before he grasps in thought that itself which is the Good (auto ho estin agathon), then he is at the very end of the knowable” (535a5; cf, 507b5, 7). Now the repetition of the phrase in which “the Good” is substituted for “each thing” is clearly meant to catch Glaucon’s attention and to convey to him something—actually the most pertinent thing in the dialogue—about the nature of the community governed by the Good itself. For upon having grasped what each thing is in itself, one would expect to learn what all things are together, and it is in place of this expected phrase that “the Good” occurs. This sentence then hints how the Good, as the “source of the whole” (511b7), will have to be understood: It is not simply a different being, but precisely the oneness of all beings (cf. 244e ff.), the All as that Whole which comprises what each partial whole is as well as what it is not, that within which different things are at one. It is “the source which is the Whole” (he tou pantos arche, 511b7; cf. the end of Note 35). As such, the Good is indeed the fit pattern of all community, and in the Republic especially of the political community: “using it as a pattern” (paradeigmati), the rulers are “to order the city and private men and themselves” (540a9); dialectic turns out to be the—eminently political—study of communities (cf. Sophist 253d). Socrates has, in his own efforts, not only composed the quarrel between philosophy and poetry, but he also composed the quarrel between philosophy and politics, for he has shown, in speech and in deed, that the lover of wisdom best knows and most desires genuine community.
5f. Nothing more direct is said here or in any other dialogue about that primary dialectical aspect of the Good—that it is the One (cf. Note 35). However, there is a dialogue in which Adeimantus and Glaucon are shown (on some occasion that must have occurred after Socrates’ death) the reason for Socrates’ silence. This is the Parmenides, in which Antiphon, their younger half-brother who has given up philosophy for horses (the dialogue itself shows why such things happen), recites from memory that is to say, without drawing any consequences—an old conversation. In it Parmenides performs a demonstration exercise for the then young Socrates, generously taking his own One (137b4), namely that One about which he himself says that “it is,” while others say that “it is not,” and showing Socrates what follows from either assertion. The dialogue ends—as some think, too abruptly—with the conclusion that “whether the One is or is not, it and the others, in relation to themselves as well as to one another, both are all in every way and are not, and both appear and do not appear—Most true” (166c). The dialogue has shown that when the One is conveyed in speech, as Parmenides’ One is capable of being conveyed (cf. Parmenides’ own poem: e.g., “it is necessary both to say and to think…” Diels, Vorsokratiker Fr. 6), such speech leads to its own denial (cf. Philebus 15d), while that denial itself cannot stand firm but leads again into its opposite. Everything possible to speech has been said about the One and has had the wrong consequences so that nothing remains for Parmenides but to fall silent. Socrates is listening; he knows that one hypothesis, the very hypothesis that is not possible to rational speech, has been omitted, namely that “the One is not one,” and that that is the crucial possibililty (cf. Sophist 258d, where the dialectical equivalent, “Non-being is,” is, with apologies to Parmenides, introduced). “Father” Parmenides, having confronted his own One in speech and having allowed its difficulties to emerge, is about to engender a new One in the rising generation. Socrates, recognizing the ultimate powerlessness of logos to convey this One, which as a true whole has “parts” (Sophist 244d14 ff.) and “rules” an order, remains silent; for, as far as the greatest matter is concerned, Plato thinks that “it can never be just said, as are other learning matters (mathemata)” but requires long and intimate intercourse (Seventh Letter 341c5).
5g. One last additional observation: What is characteristically Socratic about the sun image is that it is reflexive, an image of imaging. But more than that, in presenting the sun as an image of the whole, it shows not only how imaging itself is possible, but also how that particular kind of philosophical imaging which makes the whole reappear within itself is possible—how we can “see” the Good. This aspect of the imagery of the Republic is reflected in the central visual image in the closing myth.
The place in the Myth of Er where the souls choose their lives (616b) is not immediately easy to imagine. There seem to be two irreconcilable images; the first one consists of the whole of heaven, which has a shaft of light passing through it and the earth (b5); the second consists of Necessity sitting at the earth’s pole whirling a spindle tipped with a spindle-whorl that represents a planetary system hung on chains let down from the heavenly light encircling the whole (c4). Now if we recall what a spinning-woman actually looks like, these two disparate images become integrated: Necessity sits spinning. Between her knees she has a long distaff at the top of which a cloud of white wool is fastened; it feeds into the thread as it is spun. This thread is twisted into yarn by the spin of the whorl-weighted spindle, which is tied to the end of the thread; onto it the finished yarn is wound. In the figure of the myth, the axial shaft of light represents the distaff, the chain of heaven is the thread that is being spun, and the whorl of the spindle of Necessity is a miniature planetary system, an orrery, a model of the Whole, within whose sight the souls choose their lives.
Thus, finally, the same image, now taken as a mythical, magnified and other-worldly, complement of the cave image, makes evident the ultimate position of the human soul as an onlooker and therefore, strangely enough, an outsider, in respect to the whole. This placing of the soul is necessarily glossed in Socrates’ philosophical images, for every effort they are designed to elicit amounts to a de facto mending of this split.
This is the seventh essay in this series. The other essays may be found here: I, II, III, IV, V, VI, VIII, IX, and X. We highly recommend Eva Brann’s The Music of the Republic: Essays on Socrates’ Conversations and Plato’s Writings. This essay originally appeared in the St. John’s Review (Volume 39, Number 1 and 2, 1989 – 1990) and is republished here with gracious permission of the author. Miss Brann welcomes questions/comments via mail: Dr. Eva Brann, St. John’s College, 60 College Avenue, Annapolis, MD, 21401-1655 (she does not use computers).
 See Klein, A Commentary on Plato’s Meno (Chapel Hill 1965), pp. 119 and: Throughout the dialogue Socrates’ reiteration of themes, such as the “oft-told” tale of the one and the many, as well as the recapitulations he often elicits from Glaucon, have the effect of making Glaucon “recollect” (e.g., 507a7, 522b1, 544b4) from time to time the springs and the course of the argument. This is, obviously, not genuine Socratic recollection (see above, IV E 4d) but an exercise of that power of memory which philosophers must possess as part of their natural endowment (535cl). Such memory-recollection (anamnesis) was especially cultivated by the Pythagoreans: “A Pythagorean man does not arise from his bed before he has recollected what happened yesterday. And he performs the recollections in this way. He tries to recover by means of the dianoia what he first said or heard . . .” (Iamblichus, Life of Pythagoras 163, 20). The passage goes on to describe the discipline of completely recalling the logoi and erga of the previous day, a discipline that was considered part of the training needed for the acquisition of knowledge. It is obviously a technique Socrates himself had mastered.
 Klein, ibid. pp. 191-99 on the “solidity” of the soul in the dialogues. For further sources on the “dimensional soul” see Gaiser, op. cit. (supra, N. 28), pp. 545 ff.
 See Toeplitz, “Mathematik und Ideenlehre bei Platon,” Zur Geschichte der griechischen Mathematik (Darmstadt 1965), p. 59.
 The objects on the Divided Line are only twice referred to in terms of mimesis (510b4 and 532a2, cf. 507c6).
 In the Philebus, the Good as a human good comes to Socrates as a dream-like reminiscence of a “third thing,” other than and above both pleasure and human wisdom (20b8), e., it is a “one” above the other “two.” It has three characteristics: It is “perfect,” “adequate,” and “choiceworthy” (20d); its power, again, cannot be “caught” in one idea but must be captured in three: beauty, symmetry, and truth (65a1); their relation is not unlike that of the three effects of the power of the Good, namely world, knowledge, and being, in the Republic. Again Eudemus (Wehrli, Fr. 31; cited by Gaiser, op. cit. p. 480, note) says that Plato distinguished three ways the Good functions: as productive, as end, and as exemplary cause; these again correspond roughly to the Good as father, as end of learning, and as pattern of being. The whole complex is caught in a German word-play: the Good is the “Ursprung,” that is, the “Ur-sache” of all “Sachen” and their “Sachheit” (M. Heideg ger, Platons Lehre van der Wahrheit [Bern 1954], p. 40).
 Gaiser, op. cit. (supra, 28), p. 531; see F. Cornford, Plato and Parmenides (New York 1957), pp. 3-11 for further references. For one playful allusion to the Indefinite Dyad in the Platonic dialogues themselves see J. Klein, “A Note on Plato’s Parmenides,” in Lectures and Essays, ed. R. Williamson and E. Zuckerman (Annapolis 1985), pp. 285-88.
Further terms used of the Dyad in Metaphysics 989b19 ff.:material, the great and small, similar to the female, responsible for evil.
For Being as the eidetic Two, see J. Klein, Greek Mathematical Thought and the Origin of Algebra, trans. E. Brann (Cambridge 1968), p. 93.
The dialectical name of this or of any other eidetic number is not given explicitly in any ancient source. On the Good as the “First” and “the source which is the whole” see Klein, Meno, p. 123 and n. 39.
 The homoia and associated terms like paradeigma and analogia, reduced however to mere principles of classification, figure largely in the work of Speusippus, Plato’s successor in the Academy; see Pauly Wissowa, III, A. 2, pp. 1641-58. Themistius (Gaiser, op. cit., p. 535) says that Plato spoke of methexis in the Timaeus but called participation homoiosis in the Agrapha Dogmata.
 Adams op. cit. II, pp. 441, 470 ff.