16 § The Tetraktys and the Square of Four

We have often had occasion to allude to the Pythagorean Tetraktys with its numerical formula 1+2+3+4= 10, showing the relationship that unites ^[1] the number ten directly with the number four. The quite exceptional importance which the Pythagoreans attached to the _Tetraktys_ is well known and it can be measured especially by the fact that they swore by the 'Holy _Tetraktys_'. It is no doubt less well known that they also had another formula for their oaths, namely 'by the square of four'; between the two formulas there is an obvious relationship; for the number four is their common base. This implies, among other things, that the Pythagorean doctrine had to be put forward in a mode that was more 'cosmological' than purely metaphysical, nor is that an exceptional case when Western traditions are concerned, for we have already had cause to make an analogous remark about Hermetism. The reason for this deduction, which may seem strange at first to those who are unaccustomed to the use of numerical symbolism, is that the quaternary is always and everywhere considered as the number of universal manifestation. In this respect, it therefore marks the very starting point of cosmology, while the numbers that precede it, one, two and three, are strictly related to ontology. The particular emphasis on the quaternary _ipso facto_ corresponds to the cosmological perspective itself.

At the beginning of the _Rasā'il Ikhwān aş-Şafa'_, the four terms of the fundamental quaternary are enumerated as follows: 1 - the Principle, which is designated as _al-Bari'_, the Creator (which indicates that it is not the supreme Principle, but only Being, inasmuch as it is the first principle of manifestation which is, in fact, metaphysical Unity); 2 - the universal Spirit; 3 - the universal Soul; and 4 - the primordial _Hyle_. We will not develop just now the different points of view from which these terms can be understood; they could be said, for example, to correspond to the four 'worlds' of the Hebrew Kabbala, which also have their exact equivalents in Islamic esoterism. What concerns us for the moment is that the quaternary thus constituted is held to be presupposed by manifestation, in the sense that the presence of all its terms is necessary for the complete development of the possibilities which manifestation comprises; and this moreover is said to be why, in the order of manifested things, the mark of the quaternary, we might say its 'signature', is always especially noticeable —whence, for example, the four elements (Ether not being counted here, for it is a question only of the 'differentiated' elements), the four cardinal points (or the four regions of space which correspond to them, with the four 'pillars' of the world), the four phases into which each cycle is naturally divided (the ages of human life, the seasons in the yearly cycle, the lunar phases in the monthly cycle, etc), and so on. Any number of applications of the quaternary are there, all interconnected moreover by rigorous analogical correspondences, for basically they are just so many more or less specialised aspects of one same general 'schema' of manifestation.

This 'schema', in its geometric form, is one of the most widespread symbols, one of those which are truly common to all traditions: it is the circle divided into four equal parts by a cross formed from two diameters at right angles; and it can be noted at once that this figure expresses precisely the The relationship between the quaternary and the denary, as does the numerical formula we recalled at the beginning. In fact the quaternary is represented geometrically in its '_static_' aspect by the square, but in its '_dynamic_' aspect, as here, by the cross; and the cross, when it turns around its own centre, engenders the circumference which, with the centre, represents the denary which itself, as we have already said, is the complete numerical cycle. This is what is called the '_circling of the square_', the geometric representation of what is represented arithmetically by the formula 1+2+3+4=10. Inversely, the Hermetic problem of the '_squaring of the circle_' (an expression so often misunderstood) is nothing other than what is represented by the fourfold division of the circle implicit from the start by two diameters at right angles. This will be expressed numerically by the same formula, but written inversely: 10=1+2+3+4, to show that all the development of manifestation is thus brought back to the fundamental quaternary.

Let us now return to the relationship between the _Tetraktys_ and the square of four: the numbers 10 and 16 have the same rank, namely the fourth, in the two series of triangular numbers and square numbers respectively. The triangular numbers, of course, are those numbers obtained by adding the consecutive whole numbers from unity to each of the successive terms of the series. Unity itself is the first triangular number, as it is also the first square number; for being the principle and the origin of the series of whole numbers it must also be the principle and origin of all the other series that are derived from it. The second triangular number is 1 + 2 = 3, which shows, moreover, that once unity has produced the binary by its own polarisation, the immediate result is the ternary; and the geometric representation of this is obvious: 1 corresponds to the summit of the triangle, 2 to the extremities of the base, and the triangle itself taken as a whole is naturally the figure of the number 3. If we then consider the three terms of the ternary as having an independent existence, this sum gives the third triangular number: 1+2+3 = 6. This senary number, being the double of the ternary, can be said to imply a new ternary which is a reflection of the first, as in the well-known symbol of the '_seal of Solomon_'; but this could give rise to other considerations which would be outside our present subject. Continuing the series, the fourth triangular number is: 1+2+3+4=10, namely, the _Tetraktys_; and one sees by this, as we have already explained, that the quaternary in a sense contains all the numbers because it contains the denary—whence the formula of the _Tao-te-King_ that we have cited elsewhere, 'one has produced two, two has produced three, three has produced all the numbers', which amounts to saying once again that all manifestation is as it were enveloped within the quaternary, or, inversely, that the quaternary constitutes the whole basis of manifestation's integral development.

The _Tetraktys_, as a triangular number, was naturally represented by a symbol which, taken as a whole, was of ternary form, each of its outer sides comprising four elements; and this symbol was composed of ten elements in all, represented by as many points, of which nine were on the perimeter of the triangle and one at its centre. It will be noted that in this disposition, despite the difference of geometric forms, we have the equivalent of the already explained representation of the denary by the circle, as there also 1 corresponds to the centre and 9 to the circumference. In this connection, let us also note in passing, that it is because 9 and not 10 is the number of the circumference that its division is normally calculated in multiples of 9 (90 degrees for the quadrant and subsequently 360 degrees for the entire circumference) which is moreover directly related to the whole question of 'cyclic numbers'.

The square of four is, geometrically, a square of which the sides contain four elements, like those of the already mentioned triangle. If we consider the sides themselves as measured by the number of these elements, the result is that the sides of the triangle and those of the square will be equal. These two figures can then be united by making the base of the triangle and the upper side of the square coincide as in the figure below (where for greater clarity we have marked the points inside rather than on the sides themselves, so as to be able to count separately those which belong respectively to the triangle and to the square); and the whole thus obtained gives rise to several other considerations that are likewise not without importance. First, if we consider

Figure 10

only the triangle and the square as such, this combination is a geometrical figure of the septenary inasmuch as this is the sum of the ternary and the quaternary: 3+4=7. More precisely, and according to the actual arrangement of the figure, this septenary is formed from the union of an upper ternary and a lower quaternary, which lends itself to various applications. To limit ourselves to what especially concerns us here, suffice it to say that in the correspondence of triangular and square numbers, the first must be related to a higher domain than the second, from which it can be inferred that in Pythagorean symbolism the _Tetraktys_ must have had a function higher than that of the square of four; and in fact, all that is known of that symbolism would seem to indicate that this was indeed the case.

There is another point which is somewhat stranger and which, though it