We know that numbers and their combinations have magical potencies, and that their anything but arbitrary relations activate energies in accordance with the actual laws of the universal cosmogony. The archaic cultures have always recognized the harmony of the numerical modules, as these cultures are based on the cycles and rhythms, and on their constant magical expression. Certainly the fact of language, for the primitives, has been a miracle that must be unanimously recognized. Pictography, ideography, and writing (whatever means be employed to set it down) have also been sacred, and all civilizations have expressed concepts in symbols, whose execution constituted authentic rituals that, along with being hierophanic, promoted communication and group cohesion. The Mayas attributed the invention of writing to their god Itzamna, as did the Egyptians to Thoth. 

The intimate relationships between thought, word (language), and their forms of graphic manifestation are almost too obvious to call for explicit examination. We must observe, however, that in no wise are contemporary expressions of the above-cited relationships the most perfect, let alone the only ones that man has known. Indeed, as we have already stated in this work, phonetic alphabets are far more limited than are other forms of writing, those expressing multiple associations. The former follow a logical linear sequence, more and more solidified, while the latter constantly re-create a world of analogies. The invention of the phonetic alphabet, then, despite the validity of its intended purposes, is representative of periods of increasing intellectual impoverishment, and of a preeminence of the practical and material, by contrast with the richness of the means of ideogrammatic expression-just as a simple commercial document, or a cluster of nontranscendent experiences such as today is denominated literature, are less than an authentic hierological language and an integrated conception of cosmic reality.¹ 

Hence, in investigations concerning the various forms of expression, or writing, of the archaic peoples, one must prescind as far as possible from everything relative to ideas like that to the effect that art is entertainment, an attitude patently modern and profane, along with any notions that seek-whether consciously or not-to degrade traditional thought to the level of literal or merely utilitarian matters. 

On the other hand, every expression is symbolic, and constitutes a language, whether the latter be a matter of smoke signals or different sounds for communicating at a distance, or codes of fixed gestures for the purpose of an understanding among various tribes, as we find among the North American natives. There likewise exist more complex media, such as paintings and pictographies writing-whether perishable or not, which, taken together, gradually come to constitute something similar to the media that nowadays are considered to be writing. This growing complexity becomes evident in Mesoamerica among the various Nahuatl societies, and reaches its highest degree of specialization, so to speak, among the peoples called Mayan. By no means, however, should we look down upon other methods of expressing thought (nor, therefore, on other expressions of language), such as the Incan quipus-sets of cords of different colors and sizes, knotted in various ways, genuine instruments of not only quantitative, but qualitative computation (for example, in sacred calculation), not to mention ritual tools of a mnemotechnical type, applicable to the whole of universal reality, and not only to partial fragments of that reality. 

With regard to the writing of the Mayans, we shall only indicate that it is found as a complete system on monuments antedating the fourth century B.C., albeit with certain later variations. Some writers hold that this writing has phonetic values, or better, phonetic roots, as has been demonstrated in the case of certain hieroglyphs (especially toponomical) of central Mexico. This position by no means seems to us to be unworthy of consideration, inasmuch as, in the course of our research, we have more and more frequently found the interrelationships among the peoples of Mesoamerica, be they denominated Olmec, Toltec, Zapotec, or Mayan. At the same time, Mayan iconography itself is very complex, and the glyphs clearly identified in the area may surpass eight hundred in number. This might afford the reader a picture of the thorny character of the question. We should likewise wish to indicate, as we did in the foregoing chapter, that the numerals and their functions have been established for years, and that, for various reasons, they continue to shed light not only on the investigation of Mesoamerican writing properly so-called, but also on all general topics referring to those cultures. Finally, let us observe that a considerable group of authors today believe that they are deciphering many texts of Mayan writing, although there also exist divers tendencies in this respect. In any event, we are dealing with a specific subject, within the study of the Precolumbian cultures, which joins, to the excitement of its own challenge, the broad, inviting field of the relationships prevailing between these expressions and other elements of the Precolumbian cosmographies, and provides an excellent approach to the study of the ancient American traditions, like the identification of many glyphs of places in the area, and other decipherings. 
  

The Great Eras 

A keen interest in the great eras has been common to all traditional peoples. These ages of the world and of humanity were studied and expressed by the Chaldean astronomers, the Greek philosophers, the Hindu and Chinese sages, the Hebrew cabalists, and so forth, all of whom, just as did the Precolumbians, entertained complex lucubrations on the subject. Generally, these "ages" are four or five in number (depending on whether or not the central point of the circle they follow is regarded spatially), are based on the period of the equinoctial precession of 25,920 years (26,000 or 24,000, in round numbers), and could be observed basically through analogical relationships. Each is usually known either by its association with an element (fire, air, water, earth), as in ancient America, or by a metal (gold, silver, bronze, iron), as in the tradition of the West. 

For the Precolumbians, these four great ages have a spatial image, as well, and dovetail by way of alternating cardinal courses. Each is in turn subdivided into four lesser orders; these great cycles thereby admit of subcycles, which are in turn divided until, in the year, each group of five days, having its spatial course, succeeds another, in discontinuity, but regularly. We shall expand on this subject in our next chapter, in the section devoted to the tonalámatl. 

But certain observations will first be in order. One is that enormous catastrophes must surely have occurred in America that caused entire cultures to disappear without a trace. As a result, no massive remains have been found that might be assigned a very ancient date. The floods to which the various native peoples refer seem to have been very near them in time, and not as remote as the biblical Flood-which, as tradition declares, was the one that destroyed Atlantis. For that matter, even today one perceives the degree of telluric ebullition proper to this continent, as expressed through constant earthquakes, inundations, hurricanes, and volcanic eruptions. 

Another observation will concern the question of method. The science of the Precolumbians is not inductive, as is science today. It is deductive, like that of all traditional peoples. From unity derive all other structures, which fine-tune themselves by responding to an invisible, unanimous plan. This articulation of the parts permits the action of the Principles in the entire complex, and hence their application to all particular forms. It goes without saying that this action manifests itself, indeed that it can be experienced and verified. Accordingly, far from conjuring up an image of some savage Indians who drew rudimentary-and surely false-conclusions regarding the multitude of phenomena, we ought to think just the opposite: we ought to think of beings who deduced these phenomena from universal principles revealed to them by their mythic forebears. 

At the same time, the problem with the modern sciences is that they attempt to apply laws that function at particular levels as if they were universal. They claim to weigh and measure all things by these coordinates, with the assistance of statistics, which has become indispensable as the legalizer of scientific validity. In fact, this occurs with manifestations which, precisely by virtue of their multi-level, or supraquantitative, nature, obviously do not lend themselves to computation, or to the rigidity of classification. Phenomena are always regarded as fixed, definite, constant, and invariable. And it is on these hypotheses that the modern sciences base their calculations, without taking into consideration the possibility that they are false (geometrical projection of error), and that what is supposed constant and uniform in space and time may not be such. We find the same with respect to the time-periods dealt with in Astronomy or Geology, or physical/chemical/biological concepts, or the various "specializations." To make this assertion means automatic blackballing at the hands of the scientists and their followers, who do not hesitate, if need be, to act as agents of repression of the very cultural system in which they have been begotten. 

To return to our theme: without departing from the "classical" Western tradition, let us note that these concepts bearing on the great eras are to be found in Hesiod, Plato, Ovid, Virgil, Plotinus, and so on. And yet, the modern world and its science regard these assertions-absolutely real for those who make them-as nothing but antiques, or fantasies, of a "mythic" type, having no current validity: on the contrary, they have developed a series of hypotheses certified by the pipe dream of progress, by totalitarian officialism, and by ever-present fashion. 

Some of today's disciplines for example, such as geology-which appeared only in the nineteenth century, the enterprise of a single inventor, Charles Lyell, and direct heir of the mathematical-mechanical thought of Descartes, in correspondence with the industrial "revolution"-is based entirely on the notion that the earth is like a piece of cake whose successive layers have been added since remotest antiquity. Amazingly, it fails to consider the constant movements, habitual or extraordinary, that are the pulsations of this being called earth, despite the fact that it studies those very phenomena. "Stratigraphy," to which many other disciplines, like archaeology, superstitiously and unhesitantly submit, is nothing but a logico-mechanical conception of life and the universe. Obviously if layers of the earth pile up on one another in temporal succession, the first will be the oldest. But to go on from there to take this statement as a dogma of faith, valid for all cases, is to reduce the universe to something "ideal," utterly dead and foreign to us, when it is actually undergoing daily transformations. 

We have neither the time nor the space here to address this subject as we really ought. But we do wish to register our disagreement with this manner of perceiving and apprehending reality, just as we have done with regard to the "theory of evolution" (or its "transformistic" equivalents), adopted in the same century and likewise a creator of modern thought. This doctrine is acknowledged as absolute truth by the most diverse branches of science, which ultimately converge in it and which derive their postulates from it. Neither can we approach this theme in extenso, as it deserves, in this book. We only wish to emphasize, for whoever may wish to grasp it, that those who rely on these two suppositions, or theories, are resting their conception of the world on very fragile foundations. 

Indeed, this last hypothesis that we are considering supposes that, over a certain length of time, indefinite, because of its immense duration, and through a series of transformations, which included an ascent from species to species, life gave rise to man. Traditional cultures, however, maintain that, in a series of steps, conditions were created for man (as in the initiatory processes), that the latter might arise of a sudden, whole and entire, and that, with this new being, the innumerable species might acquire meaning. Without the human being, creation's crowning work, that creation would be meaningless. 

These cultures likewise declare the reality of man to be nothing more than one of the modalities of a Universal Being that, in having likewise bestowed consciousness upon man, has made him a sharer in Its own integrity. In the cosmos, everything is a part of this gigantic Being, which embraces it wholly, nor are the waters, stones, plants, and animals, which are as alive as we are, lesser things than we. But man synthesizes all, and the complex of the world's things and beings is ordered to his service. 

For these traditions, there have been other creations-and also countless creations, in fact-various humanities and different creatures, manifold expressions of that same Universal Being. All of these expressions have engendered one another, finally always developing on their own. They have enjoyed the same spring and summer, and have suffered decline and death, to be reborn once more in another way. However, in order for this last to occur, a creation must live for a time in the blackest darkness. Thence the world arises anew-in an extended moment, it may be: "As the mist, as the cloud, and as the dust cloud was the creation" (Popol Vuh, book 1, chap. 1). 

Instead, relying on simple-and even very dubious-empirical arguments, our contemporaries suppose a hypothetical world, validated exclusively by the discovery of certain "humanoid" fossils and a theory of the quantitative progression applied to species which are conceived as having been transformed into others. The partisans of this view have never been able to prove it, so that we have the paradoxical, to say the least, phenomenon of a science that lays claim to the "scientific method" but clings tooth and nail to a mere supposition.² 

To conclude, let us observe that the great traditions have frequently taken the great year of the earth as one-half of the time of the equinoctial precession, which is calculated at 24,000 years. In other words, the great year would be 12,000 years in duration, as with the Chaldeans. More precisely, the Mesoamerican civilizations calculated the great year in 13,000 years, one-half of the precessional period (26,000 years), thereby more closely approximating the actual duration of that cycle, which is 25,920 years. The 13,000-year period admitted not only of a quaternary division, but of a quinary one, as well-into subperiods of 2,600 years. These great cycles, in interrelationship with the cycles of the stars and planets, shaped the calendars of these cultures, of which we shall treat in the following chapter.³ 
  

The Quadrille Pattern 

If the quaternary, and its symbolical geometrical expression, the square, are found to be present as a distinctive mark in any manifestation, the sum of these manifestations, beings, phenomena, and things would constitute the entire cosmos. Altogether naturally, then, it will be possible to represent the cosmos on a plane in a quadrille pattern, like a mesh or net, seizing and joining the scattered elements by the intersection of points-in correlation with the symbolism of knots, and of intertwining-which will maintain the cohesion and order of the structure. This figure is sacred, by virtue of the simple fact that all possibilities are inscribed within it. After all, it constitutes the warp and woof of which all things are woven or created. It constitutes their "universal standard." 

Accordingly, the very representation of this portentous phenomenon, the quadrille pattern, not only must have the same magical power as is attributed to Creation, but, in symbolizing that creation, will also express the same laws, the same interplay of numerical and geometrical possibilities, tensions and equivalencies, as does creation. After all, after its own fashion, this pattern manifests precisely that internal logic and constitutes an identical structure-a symmetrical product of multiplication-and that shapes a harmonious whole. The quadrille pattern, then, is an instrument of Knowledge and of work, and is a way of apprehending-as a network-the cosmic laws reproduced within it. After all, in that quadrille pattern-visible or invisible, tangible or intangible-the forms are manifested. All traditions have known this graphic representation of multiplication, have known the continuous intersection of the vertical with the horizontal-always joined at a point-and the forms and laws deriving from this reproduction proposed by the mesh plane. (The pantograph can create any projection, just as the "reticle" is one of the basic instruments in astronomy.) 

We shall not rehearse our examples of the quaternary, offered us, as we have seen throughout this work, from countless perspectives by the ancient American civilizations.⁴ But we shall indeed permit ourselves a simple examination of this matter as an exercise in the illustration of certain concepts or ways of working, particularly in connection with a certain type of numerical relation. We are especially interested in showing that, for the archaic mentality, this kind of speculation has an astonishing aspect. Again, the "concurrences" and analogies produced are the vehicles, at times, of a magical content. 

The unit is a quadrille ruling (fig. 1) that subsequently expands in the four directions of space (fig. 2), limiting the first frame, in which the original quadrille ruling is the center (fig. 3).^* 

fig. 1	fig. 2	fig. 3

fig. 4	fig. 5

The number nine is called, in Mayan, bolon, which has the sense of a complete thing, or a cycle. Bolon ts'akabil, according to the Diccionario Maya Codermex,⁶ means "eternity"-or, in another of its forms, "nine generations." It designates the "ninth"-which is implicitly to name the denary-and a closed order, as in bolon he ("ten days ago"), or bolon neh ("ten days from now"), which is the index of a numerical circularity and a defined temporal space.⁷ In the tradition of Pythagorean arithmetic and geometry, the number nine is the number of the circumference that, added to the central point or axis from which it has drawn its form, gives us the denary-that is to say, a totality-expressed by the entire geometrical figure of the circle. The number nine is intimately bound up with the 360^o in which a circumference is divided into its four natural parts: into four right angles of ninety degrees each, assimilable to the four gammas that integrate a cross, or to the four angles enclosing a square, provided these last two representations be inscribed in a plane of nine quadrille squares, respectively as centrifuge, ad extra, or as centripete, ad intra (figs. 6, 7). 

fig. 6	fig. 7

fig. 8

As we have seen, the number nine is considered irreducible, inasmuch as all of its multiples and submultiples always return to it (9 x 5 = 45, 4 + 5 = 9; 9 x 8 = 72, 7 + 2 = 9, and so forth), and for this reason was regarded as perfect and cyclical, a complete module equivalent to the circle or sphere-and to its corresponding quadrangle-an image of the cosmos and of totality. 

May we now be permitted to express ourselves through another simple illustration. Let us trace the first cross that can be sketched in squares, and then enclose it in another, larger cross, the second cross possible in the network. 

fig. 9

The Moon 

Among the heavenly bodies, the moon is without doubt the clearest sign of heaven, contrasting so powerfully as it does with the darkness of the night. Its monthly quarternary cycle (new, waxing, full, waning) is even more familiar than the annual cycle of the sun. At the same time, the moon's connection with the vital forms is obvious: regimen of the rains, influence on fishing, on menstruation, and on all generation, as for example on the fecundation and growth of certain plants and animals, so that it can be associated with the harvesting and planting of vegetable crops. The solar cycle, meanwhile, with its division into equinoxes and solstices, and especially the close connection of these latter with the rainy and dry seasons-in a succession common to all of Mesoamerica-must be considered rather with reference to the agricultural year, and therefore with the cultivation of the fields (agri-culture) and specifically with maize. It was the storage of maize that made it possible to create ever more complex structures, culminating in the high Precolumbian civilizations-although this process of the storing of grain that made foodstuffs available without the anxieties of the harvest, and instituted an orderly control of resources, is common to all of the great civilizations. 

It would be logical, then, that the lunar calendars should precede the solar, the latter being indicative of a sedentary and far more firmly crystallized way or life, which issues, by a cyclical process, in the construction in stone of great centers and cities, and a regular and more fixed and precise knowledge of the great cosmic laws that nomadic peoples perceive intuitively and directly.¹² Let us note, however, that these systems, although governed by different calendars, coexist, and are interrelated, within the societies that utilize them, in a "solilunar" calendar. This actually occurs, and has occurred throughout history.¹³ The moon effects its transformations over the course of twenty-eight days, in four phases of seven days each. This number, multiplied by thirteen, which is the number of times that the moon completes its cycle in a year, gives us a total of 364 days for a lunar year.¹⁴ 

But the interesting thing in this calculation is that 364 = 7 x 52: thus, fifty-two seven-day weeks equal this lunar calendar of thirteen months. This assumes a very particular importance in light of the fact that the number thirteen, like fifty-two (and of course four: 52 = 13 x 4) are key numbers in the native cosmogonic conception, as manifested in the Mesoamerican calendars. This hypothesis has been maintained by Ernest Förstemann, for the Mayan calendars, and by Hermann Beyer, for the Mexican. Beyer asserts: 

Chicomoztoc, the cavern of origins Historia Tolteca Chichimeca

It is very important to notice that the number fifty-two is a common property of the moon and the Pleiades: there are fifty-two weeks in the lunar year, and fifty-two years in the cycle of the Pleiades. We now invite our readers to make the investigations of this type that we shall present in the following chapter.