As for the next number, namely, the quaternary, here too the author we are studying has a striking mixture of the finest dialectics with various kinds of numerical speculations, which can only obscure the essence of the matter. As we have seen above, underneath all such numerical speculations lies not a fantastic and arbitrarily speculative theory, but simply the conviction that all reality, in whatever form it may be taken, whether natural, human, or divine, and in general everything that is thought, is either in fact thought in a clear structural form, or at least should be thought in this way. When the quaternary is seen in the four seasons, in the form of the four ages of human life or the four winds, only one thing is meant everywhere, namely, the clarity of the structure, the distinct and chiseled figurery. Therefore, let's not look down on all this childish numerical fiction. Underneath it, we repeat, there is a sculpturally given and strictly numerically minted aesthetic objectivity. For this alone, one can forgive all this speculative naivety.

a) But here's the thing. In the first three numbers we found in the author of the treatise the most necessary and obvious dialectic of inseparable unity, separate multiplicity and undivided wholeness. The same precisely and very important dialectical category is concealed in this doctrine of the quaternary. It turns out that the first three numbers were still too abstract a construction. If the unity was an indivisible point, the double was the infinite becoming of this point, that is, a line, and the trinity gave us a third point already outside this line, that is, gave us a plane, then after this it becomes quite clear why it is necessary to look at the plane as a whole, that is, from the outside, for which it is already necessary to go beyond the plane and form the very same What in geometry is called a solid, that is, a three-dimensional structure. If a line required two points, and a plane requires three points not on one line, then a three-dimensional body requires four different points, that is, a quaternary is needed. And among all the fantastic assertions of the author of the treatise, one is at any rate not at all fantastic, but quite a real result of consistent thinking: if so far we have been able to obtain only a one-indivisible wholeness, then the question naturally arises: what exactly is this wholeness, and of what particular thing is it said to be whole? It seems to us that the dialectical course of reasoning of the author of the treatise is quite irreproachable. And it is after receiving wholeness that the author speaks about the body to which this wholeness is peculiar. And the author speaks about the quaternary as a principle of corporeality in this chapter of the treatise very expressively, he speaks several times. And here he has a really deep, and moreover, purely dialectical, construction.

b) Attention is drawn to the division of sciences made here, which differs from the four basic sciences of Plato - arithmetic, geometry, astronomy and music (R.P. VII 525 c - 531 c). In our treatise, arithmetic is in the foreground, which is understandable, since we are talking here about number. It is pointed out that the number can be talked about in general and can be spoken about in particular. This probably explains the fact that geometry is not separately marked in the treatise. Very interesting, although not very clear, is said about music. It is not in the fourth place here, as in Plato (R.P. VII 530e - 531c), in whom it is the doctrine of the harmony of the celestial spheres. As far as the obscure text allows us to judge, the author of the treatise does not understand music as numbers and quantities per se, but as their correlation, that is, we would say, the becoming of numbers, and already in the presence of becoming, the functioning of these relations, in particular, as harmonic intervals (octave, fifth, fourth), becomes clear. If this is really so, then the most essential feature of music as the art of pure time is captured here. The third science or art is geometry, which emphasizes movement in space and the result of this movement, rest, that is, the spatial figure in its construction and in its stable structure. And, finally, the spherical is understood in the treatise as what Plato would have called the harmony of the celestial spheres. This whole text (20, 21 - 21, 2) is very interesting, but in some ways controversial.

c) Finally, from this chapter on the quaternary we would point out the identification of the quaternary in one respect with the pyramid, and in another respect with the sphere. The basis for this identification is something that we are currently experiencing as a curiosity. But this is not a hundred percent curiosity at all. After all, here is expressed the tendency to think everything abstract bodily and figuratively. The body, taken as a body in general, is no more than an abstract concept. But the author of the treatise is afraid of just that. He needs the very concept of the body to be corporeal in its structure, that is, figurative. Therefore, all these arguments about the pyramid and the sphere need not be taken seriously; However, it is necessary to accept intellectual intuition, without which there are no abstract concepts at all for the author of the treatise.

3. Numbers Five - Nine

Further numbers after the quaternary we will neither consider in detail nor give a literal translation for them. They contain an unprecedented mixture of very serious and deep ideas with fantastic explanations and often amusing illustrations. Since, however, all this exposition is conducted in the treatise in the most serious tone, we will now try to dwell on the most important, but in the shortest form.

(a) The quintuple (pentas) is the "eidos of the integer" because it has a feminine even binary and a masculine odd trinity (30:17-19). Therefore, here for the first time we talk about marriage (as we know, marriage is already partly connected with the trinity). It is the quintuple that is a full-fledged symbol of marriage (30:19) and therefore Aphrodite (41:12). It is stated more clearly in the passage where the quintuple is declared to be the "nature of vitality" (physin dzootetos 32, 14) of the cosmos. We understand this in such a way that if the quaternary testified only to the body, then the quintary testifies to the living body, that is, to the organism. Other points seem to be of secondary importance, such as the combination of identity and otherness in the sphere (35, 1), "light" (35, 1), "justice" (35, 6), or "demigod" (41, 15).

(b) The sixfold (hexas), as the author deduces from the presence in it of unity, two, and trinity, and from the multiplication of the double by the trinity, is also the masculine-feminine principle and also a marriage (43:5, 7), but only in a more special sense. It is marriage in the sense of the reproduction of offspring, that is, the generation of one's own kind. And that which is like itself is the whole and the parts (43:8). And since the whole governs the parts, the six are also the soul, that is, the soul that governs the body (45:8-13). But in this case, the six is also what has often been called the cosmos, since in the cosmos we find the unity of opposites, which arises "according to harmony" (48:18-20). Therefore the sixfold is also "the division of the whole" (45:11), and moreover not only in space but also in time (49:11, 17-18), and in this sense it is the "eidos of eidos" (45:7), and "far-reaching" (the epithet of Apollo) (49:11), "wholly caring" (holomeleia 48:6), "all-healing" (50:2), "perfect" (42:19), "health" (48:21), "love" (philosis 48:14). In short, all such definitions of the Six come down to the fact that it is a living and animate cosmos, soul and body, which are in eternal harmony. In other words, the Six is simply the entire cosmos, but as yet only at the level of the universal organism. If the five pointed to "vitality," that is, to the principle of the living organism, then the six no longer indicates the principle of the organism, but the organism itself. At the same time, a strict sequence of thought is observed here. Being not the principle of life, but life itself, the organism itself, the sixfold points to this organism in a global form. This is the cosmos, that is, as yet only a cosmic organism in general. The details necessary here follow immediately, in the next few days of the decade.

c) The week (heptas), after the amazingly eccentric numerical combinations, appears before our historical and philosophical gaze as an attempt to express not simply the unidivisibility of the cosmos, as we find in the six, but to express the rhythmic recurrence of the individual periods of the cosmos. Here again the point is not at all that a child can be born after seven months of pregnancy, and not that four sevens are equal to 28, and 28 is a perfect number equal to the sum of its factors. And the fact is that seven is the ubiquitous uniformity of periods of life development. If we do not draw this conclusion from this chapter of the treatise, then the individual epithets of the week have too general a meaning and say little. She is Athena (71:2), and "fate" (tyche 59:3; 70:24), and "leading away prey" (56:10), and "moment" (59:4; 70:24), and "purposeful" (55:6). Generally speaking, the septenary is also a cosmic organism, but already conceivable in all the mobile details of its organic structure.

d) The octal (octas), being the cube of the two, embraces the entire cosmos as a mother. Therefore she is not merely a mother (74:5), but is also a mother everywhere, which is why she is "all-harmonious" (panarmonios 73:5). She is Rhea, that is, the mother of all gods (74:7). After all, a cube in the geometric sense is a three-dimensional body. This means that motherhood here is also three-dimensional, that is, pancosmic, panharmonious. The treatise borrows from Philolaus (44A12) the definition of the eightfold as the culprit of "love, friendship, wisdom (metis) and ingenuity" (74, 14-15). However, this is a more general definition, as are "seat" and "security" (Q[75], A[2]), although all these general definitions still point to a solid ubiquity of harmony. For us, it is especially important to say that the eightfold "everywhere beautifully (pagcalos) and intercorrespondingly (parallelos) brought all harmonies into harmony" (76, 6-7). In other words, the eightfold is not just the principle of life (the five), nor the organic living cosmos (the six), nor the repetition of the universal organism in its individual organs or parts (the septenary). But this repetition is everywhere in agreement with itself and with the living cosmos as a universal wholeness, that is, cosmic organization is the omnipresent harmony of life. This is the essence of the eightfold.

In comparison, the other features of the eightfold indicated in the treatise have a tertiary meaning and are accidental. Thus, for example, it is indicated that there are eight stellar spheres, eight astronomical circles (zodiac, horizon, equinox circle, etc.), and that the octal is Euterpe or Cadmea. She is Euterpe because she is eytreptos ("agile"), which reminds one of the famous muse Euterpe. Cadmea is because Harmony is the wife of Cadmus. The most important thing in the eightfold, presumably, is cosmic all-harmony, while the septenary is simply the presence of harmony in the cosmos in general. The Eightfold is "all-harmonious" because of the "extraordinary harmonization" it produces (dia ten hyperphye catharmosin 73, 6).

e) The nine (enneas) is characterized in the treatise in such a contradictory and even contradictory way that it is not an easy task to find some main idea here and distinguish it from secondary thoughts. As far as can be judged, a novelty in comparison with the preceding numbers is the characterization of the ninefold as Hera, the sister and consort of Zeus (78, 3), which, according to the fantastic but constant etymology of the ancient Greeks, is air (aer), and this air is thought of in this case as the upper air, that is, as ether. Ether, on the other hand, is the finest substance, which permeates the entire cosmos, embraces it, and therefore makes it round (78:2). From here it is the Ocean (77, 4) and the horizon (77, 4). The fact that she is also Hephaestus (77:23) also indicates the fiery nature of ether. She is also Hyperion (78:9), because she stands above everything and embraces everything. The purposeful roundness of movement is indicated by the understanding of her as Terpsichore, the muse of dance (78, 11). She is also Prometheus (77:6), but instead of fire, thinking is indicated here.

If we understand the idea of the ninefold correctly, then of all the disparate features given in this characterization, the most important is probably the cosmic activity of the principle of harmony established above. In essence, this is still the same cosmic all-harmoniousness, but given in the aspect of its activity and general curvature of figures. It is an all-harmonious organism in the aspect of its active functioning.

The rest of the definitions and epithets of the ninefold seem to us too general, so that in a certain sense they can be applied to any number. Such, for example, are the epithets "blameless" (77, 14), "likeness" (77, 16), "like-mindedness" (77, 12), "perfect" (78, 16), "goal-setting" (78, 15), "far-sighted" (78, 4).