Surprisingly, the author of the treatise passes by a long line of philosophers who knew how to merge the limit and the infinite in what they called number. And such is Plato in the Philebus (16c-20e). In a surprising way, the logical construction is replaced here by a purely intuitive picture. However, for intuition in life, in nature, and in the whole world, there is an infinite number of such things and events "in which it is precisely this unity of stable being and the unstable becoming of this being that comes to the fore. And almost the entire chapter devoted to the trinity consists of pointing out examples of a deep synthesis of certain mutually contradictory moments.

When we say "peace" in the sense of tranquility, we certainly mean a certain kind of unity of opposites (19, 17). The same is meant in the use of such words as "good counsel" (16:18), "piety" (17:1), "like-mindedness" (19:18). Of course, it is impossible to imagine what "knowledge" is (16, 22) if one does not find in it the ability to distinguish and identify. The same must be said of "reasonableness" or "prudence" (16:19). In "friendship" (19, 17) and in "marriage" (19, 20), according to the author of the treatise, the trinity is also realized (although marriage finds its full expression here only in the five, as we shall see presently), not to mention "proportion" in general (15, 5) or "harmony" in general (19, 18). This trinity ensures for each thing its independent existence, "everyness" (16, 10 to hecaston), as well as everything in general to be everything, that is, to be "allness" for it (16, 11 to pan).

(b) A more precise definition of the trinity is the principle by virtue of which a beginning, a middle, and an end arise in all things and in the whole world, and this is the principle of "perfection" (17:17-18). In another way, it is said that the trinity is not just the middle, but the "middle" (15, 5) for everything that exists, that is, it is its semantic center, which comprehends it all. It is also clear in what sense "proportionality" is spoken of (15, 5), which the Greek term can also be translated as "proportionality".

c) Finally, from the point of view of the history of ancient aesthetics, it is very important to note that the author of the treatise himself connects the trinity with "beauty" and "splendor" (14, 14-15). If it is still possible to argue with regard to "beauty" whether it is an indication of the synthesis of the inner and the outer, then the term "splendor" used here, by virtue of this very Greek word-formation, testifies precisely to the fixation here of the identity of the inner and the outer. The impersonal Greek verb prepei means "fits," "fits," "fits," "fits," "fits," "fulfills its purpose." Another word included in this term, namely eu, also means in Greek "good", "important", "valuable". Therefore, this whole term "splendor" (eyprepeia) very accurately expresses the basic essence of all beauty, which requires, first of all, conformity to its purpose. This moment did not exist in either one or two, if we take them as two independent categories. And although the analysis of these two categories already leads to the need for an internal-external synthesis, in a special sense this synthesis is achieved only in the trinity. It can be said that the first three numbers, in the opinion of the author of the treatise, are a characteristic of beauty as a single whole. But in this aesthetic principle, unity is determined by unity, separateness is determined by two, and wholeness is determined by trinity.

It seems to us that if we stand from the point of view of Pythagorean Platonism, then it is difficult to imagine a better and in any case a more fundamental formula of beauty.

2. Quaternary (tetras)

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).