For a traditional society, the concept of number is diametrically opposed to the one that might be held by a profane society such as ours. This calls for emphasis, since it was traditional societies that created numbers as concepts of relation that their sages and inspired men obtained by revelation. Modern society, meanwhile has pressed them into service exclusively for its material ends, ignoring their meaning-without any attention to their authentic sense, their true essence. In other words, modern society has demeaned numbers by taking account only of their quantitative values, rejecting the qualities of numbers, the ideas and concepts that they express. 

At the same time, we of the contemporary era take our numerical code simply as a given, without stopping to ask ourselves of what this system is a manifestation. Numbers expressed (and still express) ideas: metaphysical concepts concerning everything numbered, or participating in the categories of the numerable-that is, concerning what is "countable," finite, and successive. On the other hand, these "numerations" are the harmonious measure of all things, and the form in which these things related between each other. They are rhythmic standards, modules, and cycles that generate-in their quality as concepts-"proportion," and reveal the secret "ciphers" of the cosmos, of which they are active components. Obviously oneness does not correspond to the same idea as twoness or threeness, and does not manifest the same thing. Today, however, this is left out of consideration by the diminished, humdrum, horizontal view that we have of these concepts when we regard them as simple factors of quantitative multiplication. 

Let us further indicate that these numerations refer to different energies, and to the ordered intervention of these energies in the universe. After all, as we have said, they attest the interrelations of the creative elements-their waves, their vibrations-that join together in the numerical corpus. To give a very simple example: even persons of very little mental acumen know that it is not the same for a person to be alone (one) as to be a member of a pair (two) or to belong to a triangle (three). Number obviously alters our relations with others and our being in the world, since it actively intervenes in situations as a component of the same, as it signs or marks them with its conceptual and vital seal. And yet, generally speaking, even the simplest sense of the idea of number escapes men and women of this century. Indeed, most of them have never reflected on numbers, nor are they interested in the subject. But what is really arresting is the fact that not only have the common masses lost all notion of the fact that number is the sign of a quality that it represents and fixes-of a concept that it expresses in unequivocal fashion, a concept capable of articulating itself, and entering into interplay with, other concepts-but even today's mathematicians (who one supposes to be specialists) are ignorant, up to our very day, of the real conceptual (and emotional) charge carried by numbers. These persons proceed on quantitative criteria, those of the marketplace-fundamentally suited for the commercial and the material, but not for Knowledge. 

Traditional arithmetic corresponds to geometry, and numbers to geometrical figures, forming complementary symbolic codes that manifest identical concepts, correspondences, and analogies. At the same time, in the first three numbers, all of the others are synthesized. From the union of oneness and duality (which is its reflection), that is, from triad, proceed all of the other numbers, and from this primordial triangle all figures derive. 

There is also, for traditional civilizations, a direct relationship between numbers and letters of the alphabet, to the point where, with many alphabets, numbers were represented by letters, and had no special signs of their own. This is not the case with the early American cultures, which knew no alphabet, but we wish to call attention to this correspondence because not only the alphabetical code, but the numerical one, as well, describe all reality: that is, everything that is numerable or namable-in the sense of "ciphers," harmonious measures, "proportions"-in sum, the totality of the cosmos, of the knowable. 

This threeness or triad, to which we have referred earlier in our pages, has always been considered sacred-like oneness, duality, and all numbers-by virtue of its very properties and particular attributes. These properties and attributes are manifested in its threefold nature, which of itself is the inevitable expression of a principle, an archetypal fact, that solidifies in a series, as a representation of ideas and energies that materialize in magical, mysterious fashion while obeying precise, universal laws, which the numerical codes and their geometrical correspondences symbolize. 

However little these modules in their exterior expressive form be the same as we have today in our recent Arabic notation, the archetypes to which both refer are identical, the laws of the cosmos the same-for every time and place-and the model of the universe one only. We shall see, then, that Western numerology corresponds perfectly with the Indian, although the latter was commonly vigesimal (and, accordingly, decimal as well, both having the number five as a common base). We shall say something about these first five basic numbers, common to various peoples, but especially to the indigenous peoples and Christians, since this is the subject with which we are concerned. 

We have somewhat anticipated our discussion of the triad, as basic form or archetype, a concept present in all manifested things, which are generated by its multiplication.1 We have also asserted that it is produced by the amalgam of the primordial oneness with its own reflection. Now we shall add that this fact, whose design is successive (1, 2, 3), is actually simultaneous and eternal, and that from it proceed all numbers, or all manifested beings. Now let us see something about oneness and duality-concepts to be found at the foundation and origin of every traditional civilization or culture, among them those of America. 

We have called attention to duality, twoness, on various occasions, as the fundamental mover of the beliefs and cultures of the Precolumbians. This is especially clear among the Incas and the Aztecs, to take them as two examples of civilizations that were already developed by the time the Europeans arrived. In the former, Manco Capac and Mama Ocllo, equivalated to the sun and the moon, gold and silver, together establish Cuzco, which is divided at its center into two parts, one masculine and active, the other feminine and passive, which were denominated as the high part and the low part, and which we equivalate to the vertical and the horizontal. 

Indeed, if we regard two energies symbolized by the above-and-below, one rising and the other falling, we find that there is a neutral point, common to both, at which their oppositions do not exist. That center or middle, in which the contraries complement each other, creates a plane (or world) where that conjunction occurs-a reflection of the original metaphysical oneness, which gave rise to the manifestation of arithmetical singularity, represented by the number one or the geometrical point. It is this point or center that generates the plane (or world) in question-in this case the Incan civilization-acting on it as a reflection of the invisible axis. To put it another way: the center acts upon the world as a reflex of the active, vertical energy that conditions horizontal reception by coupling with it, and thus creating the said plane (or world). The limits of this world are constantly given by its own progression-which, although it can be considered indefinite, is marked by its own numerical laws that succeed one another ad infinitum. 

Thus the number four signs the first manifestation-the first action of the three ontological or primordial principles in the world (3 + 1 = 4)-the creational plan and its limitations, thanks to which latter any being or object can be constituted, and then assimilated to the world, especially to the earth.2 

We must explain that all of this dialectical production is successive, and that, in regard to it, the energy of oneness, constantly adding itself to the energy of the preceding number, transforms the latter into its own quality, while remaining ever-present and inalterable throughout the numerical series. 

Let us add that, in numerology, zero is a concept not only indicating lack of quantity or absence of numerical determination, but also serving as a mechanism of position and order in the tens, hundreds, thousands, and so on, which makes for great manageability when it comes to notation, and great facility in the calculation of large units. The Mayas knew the zero, and made use of the positional notation in their numerals, the only difference being that their system was vigesimal instead of decimal. Actually, they used the zero long before Europe did so: until the eighth century of our era, the system of position that we share today was not used. Of Hindu origin, our present system was propagated in the Middle East and Europe by the Arabs, although its dissemination occurred only between the tenth and twelfth centuries. That system has obvious advantages over Roman numerals. 

It is interesting to recall that the system of counting and calculating with pebbles (or grains of maize) of different colors or placed in different groups, common to the Precolumbian traditions and attested by various chroniclers, is basically the same as that with which the Pythagoreans effected their "measures," and their abstract "speculations." 

And so, let us return to our division of energies into above/below, ascending/descending, or of two forces mutually opposed but complementing each other at some point (and radiating outward in a horizontal plane in which these energies also project themselves in the same fashion, now standing in a two-to-two opposition and meeting at the point of intersection, the fifth, the quintessence, the center, the axis or heart of the figure). We see the validity of transposing the entire process and synthesizing it in the opposition of vertical/horizontal, high/low, heaven/earth, active/passive, male/female, since from this primordial opposition and of its complementarity spring all of the things of the universe. 

The binary is likewise revealed in the myth of the foundation of the Aztec city, and in the manifestations of that society. As we know, at the arrival of the Spaniards, the principal temple of Tenochtitlan was crowned with a double sanctuary. One of these (painted red) was dedicated to Huitzilopochtli, image of the ascending (from the earth to heaven) sun,-the sun of the zenith, the south, and noon. The other (painted blue) was consecrated to Tlaloc, god of rain, associated with thunder, lightning, thunderbolt and water, the deity who descends (from heaven to earth), and who is akin to the gods of fertility and the moon, the numina of vegetation and generation which are possible only when the energies of sun and rain-ascending and descending-of heaven and earth, of the eagle and the serpent, join together as one, to the exclusion of neither pole.3 

It would be superfluous to cite further examples of duality, as they are innumerable in the Precolumbian tradition, and readers can discover them for themselves. But we do wish to indicate the conception of the binary that is entertained by modern society-that is, the conception of duality with which it has outfitted us, this particular baggage of our convictions and the difference between it and the one held by a traditional society. In this respect, let us observe that the traditional conception does not reject evil (or the descending, passive, subterranean, or horizontal energy, in varying terminologies). Rather it accepts it, in conformity with the knowledge it possesses of cosmogony and theogony, which testifies to the continuous recycling of two universal energies, contrary forces that are not mutually exclusive but that this cosmogony and theogony incorporate as integral parts of reality and life. With each other (and in the mutual relations to which they give rise), these forces or principles constitute a complex of modules, measures, archetypal emanations, which, in their "coagulation," are even manifested as phenomena. The gods personify these opposed forces in many-faceted wise, which is the case, among many others, with the struggle of Tezcatlipoca as the nocturnal, dark deity and Quetzalcoatl as the diurnal, luminous god, just as between the latter and his twin, Xolotl, sometimes represented by a skull. The two forces are constantly at war with each other, and thereby balance each other, as we see so well in the perennial cosmic drama exemplified by the interplay of tensions prevailing in any quadrangle or any quaternary, where the forces are in opposition by pairs, two to two.4  

Just oppositely, we moderns have been educated in a milieu that always obliges us to choose between good and evil, and this is the principal cause, the root, of our conditioning. Worse, the only escape from the disjunction is the choice of an alleged goodness ascribed to one of the poles-monism-to the exclusion of the other, which is left entirely out of consideration, as it is ascribed a negative value, which need not be taken into account, but rather torn up by the roots. We fail to notice that the primacy we attribute to one of the factors of the duality good/evil is bestowed by evaluations that are altogether relative, circumstantial, or of purely personal or group interest, such as current society's "ideologies," usages, customs, phobias, and manias, channeled by means of nation, state, class, and even ethnic group, to all of which we necessarily belong. The same occurs with the attractive and the ugly, liking or disgust, the useful and the useless. All of these values are of a nature as variable as their contraries, with which they could be exchanged, and to which are attributed a supposed definitive, objective truth. 

The quaternary as a concept of creational manifestation-as the idea of generation and limit, or as the form of the earth (figured by the square or by the cross)-is basic in the early American cultures, and we should like to emphasize that this last geometrical form is equivalent to the circle (a rotating cross generates a circumference). The reason is that they each symbolize the same creational plane, alternately in its static and in its dynamic aspect, in its contraction and its dilation, in its crystallization and its expansion, and are assimilated respectively to the solid and the aerial, earth and sky. That is, they constitute complementary figures, just as do the world (horizontal plane) and man (vertical plane). 

In this sense, five being the number of the human being, as virtual center of the cosmic irradiation, this number, multiplied by that of the earth, or creational plane, shapes the ensemble of manifested possibilities. For we now have the number twenty, the "magic" measure or module common to various Precolumbian cultures and civilizations.5 

Let us recall what we declared in the foregoing chapter to the effect that the circle as well as the square are analogous and complementary symbols. These have been utilized by different societies with the same object, or in one and the same society by alternation or by conjunction, as connected to heaven and earth as a representation of the two halves of the cosmic model. 

At the same time, the symbols associated to the circle and the square, or derived from them, share their lot, and themselves entertain a mutual correspondence. This is the case with the circular spiral, as a representation of the evolution and emergence of the cosmos. It is also the case with the square. These two figures, in the volumetric, and in constructive symbolism, are the edifices, respectively, of the ziggurat (sig-gurat, literally, "mountain") and the pyramid as opportunities for a vertical ascent-via a succession of steps and revealed by the immutability of an axis, which is the center and origin of both monuments (temples). 

We only wish to emphasize-and on this note we shall bring this chapter to a close-that, for a traditional society, not only the stars, but also rocks, plants, animals, and human beings maintain an interplay of mutual relations, a dance of subtle possibilities. These possibilities complement one another, in the rhythmic cadence in which they all variously develop in reciprocal correspondence, establishing the guidelines, the measures of their interrelation, and conjoined in number as the synthesis of the archetypal meaning that these "modules," "measures," "ciphers," and "proportions" comport. And it is upon this conceptual base that one must study the Precolumbian arithmetical and geometrical symbolisms, just as any work on the subject will have to be orientated in this direction.

1 See René Guénon, The Great Triad (1991, trans. P. Kingsley).
2 The number four is equal to 2 x 2, or 22, that is, the total number of possibilities of the duality multiplied by itself. Let us observe that, in Mesoamerican civilizations, this progression is symbolized by the number 400, which is equal to 20 x 20, or the equivalent of an indefinite numerical series.
3 Among the Mayas of today, studies have been carried out, with respect to the health of the body, on the relationship between hot and cold, as the opposition of two contraries that are present throughout the cosmos. These contraries are also indicated as the dry and the humid-which must complement each other in order to reestablish the vital equilibrium. This way of measuring energy extends to various kinds of diseases, foods, herbs, etc., and is transferred to personages, events, and situations. It is autochthonous, and does not derive from Hippocratic or Arab medicine. Simply, as with so many other things, it coincides with other traditions in its archetypal concepts.
4 "The Canelas, of the plateau that rises to the south of the mouth of the Amazon, are composed of two exogamous matrilineal groups, one comprising the inhabitants of the eastern part of the circle of their habitat, the other those of the West. During the rainy season, a race is scheduled between the boys of these two groups, who are then painted red and black respectively. One of the groups represents the east, the sun, day, earth, red, and the season of drought; the other represents the west, the moon, night, water, black, and the rainy season. These two groups, who divide up all that exists with the same rigorous consistency as do the Iranian and Chinese religions, promote, with their ritual activity, the beneficent rhythm of the universe and nature" (Ake Hultkrantz, Histoire des Religions-3, Encyclopédie de la Pléiade, Gallimard, Paris 1976.
5 As for the number nine, we wish to emphasize that, by virtue of its intrinsic characteristics, and as an integral element of every series or group, it introduces into the latter the concept of circularity, or the cyclical. The same holds true for its multiples and exact submultiples.