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
In the first figure, as we have said, we have a value of unity: a quadrille ruling that, expanding in its potential directions, yields a cross (fig. 2). The latter, added to its center, is a quinary (1 + 4 = 5; 4 + 1 = 5), which generates another square, in which it is contained (fig. 3), and creates a limited plane, whose origin is in the unity and its radiation, and upon which this radiation can be resorbed, returning to its primordial source, once more to be projected in a dual process: ad extra, ad intra, diastole/systole, in which process the contraction, or limit of the indefinite, both marks the return to the original point, and shapes the quaternary plane upon which this phenomenon occurs (figs. 4, 5).5 
fig. 4
fig. 5
The number of quadrille squares on this first, archetypal plane, host of the unity and its radiation, is, as we readily observe, nine, which is equal to the one and its quaternary expansion (1 + 4 = 5) in all of the potential directions of space (5 + 4 = 9)-a number which, at the same time, has struck a relationship with the circumference and the cycle. Indeed, the perimeter of the circumference is 360o, or four right angles of ninety degrees, which-as we have shown-renders the circle and the square (also constituted by four right angles) equivalent when it comes to their symbolizing not only a limited-and, accordingly, ordered-field, but also a defined cycle, both of these being spatio-temporal images of the cosmos. 

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 360o 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
The so-called "magic squares" plotted on the basis of the quadrille pattern, not only were known by the ancient Mediterranean civilizations-by the Assyrians, the Chaldeans, the Babylonians, the Egyptians-in their astrological and cosmological investigations, but were in vogue in the Middle Ages and the European Renaissance. They endure to our own day in the West, just as in another of the countless forms of their use in the Nepohuatzintzin: a small Nahuatl calculator, or quadrille figure fashioned of the intersection of vertical and horizontal threads (twenty in number), a simple instrument with which the most complicated mathematical calculations can be carried out. It is similar to the abacus, which is still used by Islamic oil financiers. Its magic consists in the harmonious proportions and relationships that numbers, as incarnations of energies they symbolize, are capable of creating by way of the knowledge of their properties-that is, of the universal principles that they represent-in putting order into the chaos of the formless, the undifferentiated, the illusory, ineffective and nonexistent. According to M. Granet, these "squares"-especially the one with nine small squares that we have been considering-constituted the whole organization, including the social, of the Far Eastern Tradition, as he has reported it in the works he has devoted to Chinese civilization and thought. Nine in number were the provinces of the Empire, and they extended in the four cardinal directions and at their four intermediate points, the center being the place reserved for the Emperor. Upon the latter was bestowed the number five-the number of that region as well as the basis of the Chinese numerology and the sum total of its "material" elements8-as receptor of the vertical emanations received at this location, the heart of the Empire, and embodied by him as scion and heir of heaven through whom the divine energies were symbolically and actually expressed in the world.9 
fig. 8
Figure 8 represents the simplest magic square. (That of the sixty-four squares of the game of chess would be a different one) We should have been loath to omit its reproduction, given the number of speculative relations that it can occasion. The sum of any row of its numbers, whether horizontal, vertical, or diagonal, is fifteen, and many conclusions and transpositions can be performed, as a rather extensive bibliography on the subject will attest. In the Islamic tradition, these magic squares are called uifq, and in Judaism are regarded as powerful cosmic talismans. 

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
We shall notice that not only does the first cross, the internal one, consist of five small squares, and the second, the external-as a radiation of the original-of twenty,10 but that what we have before us is a symbolical, pyramidal motif which, in this form and in other, derived variants-such as the "Greek steps" or the "Greek hook," and so on-extends throughout the length and breadth of America. We find it everywhere-sculped in stone, painted on ceramics, embossed on seals, forming frets on textiles and basketwork, to cite a few examples-to the point that, while it is present in other civilizations, it can surely be considered prototypal of America.11 

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: 

Just as the twenty-day "month" was later divided into four parts (4 x 5 = 20), the primary month must have been divided, by the ancient sages, into four parts of seven days each (4 x 7 = 28). And just as-a familiar fact-they assigned the quarters of xiuhtlalpilli, the fifty-two-year period, and those of the tonalámatl, the 260-day period, to the four cosmic directions, so also the four weeks of the lunar month must have been ordered to the cardinal points. [Beyer, see bibliography]15 We have frequently emphasized, in the foregoing pages, the importance of the numbers four and five, along with nine, in the Precolumbian cultures-the last being basically cyclical or circular, and associated with repetition and movement. Here we wish to focus on the number seven, which, without being basic, or frequently cited, is nevertheless found to be linked to two grand, solemn myths (produced, therefore, in a primordial or earlier time). We refer to the origin, according to certain traditions, of the Aztecs and Incas. The former spring from seven caverns, to commence their pilgrimage through the world. The latter are born of a grotto, and begin their wanderings: they are the great chief Manco-Capac and his seven siblings. One of these latter is Mama Ocllo, who is assimilated to the moon, while Manco-Capac, who marries her, is equivalated with the sun. We have no intention of making use of these analogies in order to demonstrate any conclusions. But we do understand that an explicit numerical designation (in the context of a myth) is a symbolic key, a module, expressed in the proper manner of the ancient cultures. 
Chicomoztoc, the cavern of origins
Historia Tolteca Chichimeca
On the other hand, conquistador Coronado calls attention to a tribe, that of the Cibolanos, whose population distribution is nothing if not surprising. The tribe was constituted of seven cores or nuclei, and dwelt in seven villages (the "seven cities of Cibola"). The principal village bore the name of "The Middle," and the others those of "North," "South," "East," "West," "Above," and "Below," respectively. The native priests, chiefs, and functionaries of the six peripheral villages resided in "The Middle". All of the villages together formed a permanent confederation, or heptarchy. The Chilam Balam of Chumayel, as well, informs us: "Ah Uruk Cheknal [He who seven times fertilizes the maize] has been placed [in order to preside] over the whole world." Just so, this god proceeds from the "seven bosoms of the earth." The Cackchiqueles had organized their realm into seven tribes or provinces. "There are seven paradises. The firmament rests upon four columns," say the Guarani. And Fray Francisco Núñez, Bishop of Chiapas in the seventeenth century, indicates in his Constituciones Diocesanas del Obispado de Chiapas ("Constitutions of the Diocese of Chiapas"): "Among many peoples of the provinces of this diocese, painted in their Repertories or Calendars are seven Negritos, for purposes of divination and prognostication corresponding to the seven planets " (nº. 32 : 28). 

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.

1 The reality of the cosmogony, and not merely its description.
2 We have just read a special notice in a magazine, where it is stated: "Many do not know that the most illustrious French trickster was theologian and anthropologist Teilhard de Chardin. According to recent investigations, in order to play a practical joke on a colleague, archeologist Charles Dawson, Teilhard arranged the bones of a jaw a few thousand years old with the prehistoric skull that Dawson had discovered in his excavations. This led to the materialization of a sensational, and false, theory on the "missing link," the theory of the Piltdown man, which Teilhard, out of scientific mortification, never contradicted." For other unmaskings of this kind of scandal, see Gastón Georgel on bibliography. 
3 Let us only bring out one point. The Popol Vuh, and the Aztec Legend of the Suns, appear to speak of a progression of creations, which finally culminates in the creation of maize and man. That is, it speaks of an evolution. Non-Precolumbian traditions-with which, on the other hand, the aforementioned pair agree in all essentials-clearly report an "involution," symbolically expressed by the metals representing each period: gold, silver, bronze, iron, and the temporal duration of these periods, as well as that of the life of man. Just so, a great difference between the astronomies of the Old and the New World is that the reading of the native calendars must be done from right to left, in a retrograde direction, like the movement of the equinoctial precession.
4 Before being transformed into a bird, Quetzalcoatl spends four days in the hades of the North, just as do warriors after their death before becoming humming birds. Four, too, were the years of mourning, as it was considered that this was the time it took the soul to depart; four days was the term of the fasting of the chiefs before war and grand ceremonies; and women who had died in childbirth ascended to warriors' heaven in this same time. Four, as well, were the days that the gods had done penance before the creation of the world in Teotihuacan. Four is the number symbolically dividing any cycle. It is especially interesting to point out that, for many tribes of the North American redskins, the four cardinal virtues are: courage, patience, generosity, and wisdom. All American Indian creation myths include the idea of this sacred quaternary. The examples are countless.
* (Figures for printing).
5 Chac, the Mayan god of rain, and therefore a descending deity, split into four gods, took four different forms, as we have seen in the case of Itzam Na. We have also remarked that, in the myth of the foundation of Cuzco, the ancestral pair, Manco Capac and Mama Ocllo, direct descendants of the sun, embodiments of the divine energy, are able to radiate that energy throughout the four directions of the world in the quadrille network of their empire. Teotihuacan-perhaps the most magnificent city in Precolumbian America-orientated to the four paths of the world, contained a plane, based on the quadrille pattern, or network system, on which the spaces and the structures (the pyramids, temples, terraces, and all of the buildings and vacant areas) were perfectly and harmoniously distributed on modules of a common numerical base, which responded to cosmogonic "proportions," to the equilibrium of the divine economy, as has been amply demonstrated. On the plane, the quadrangle divided by a diagonal cedes to two reversed triangles joined at the base. Similarly, in the volumetric order, an octahedron is composed of two analogous and reversed pyramids.
6 Diccionario Maya Codermex, dir. Alfredo Barrera Vásquez (Mérida, Yucatan 1980).
7 As we know, the numbers 9, 4, and 13 (13 = 9 + 4) are sacred for these peoples, in strict accord with their calendar. Even today these numbers are fundamental for these peoples' magico-religious ceremonies. 
8 The decimal and vigesimal numeral codes correspond perfectly, in having the number five as their foundation, whether the latter be multiplied by two, three, or four. The Chinese, through the centuries, have used the decimal as well as the vigesimal, considering their common base, the number five.
9 Marcel Granet, La Pensée Chinoise (Albin Michel, Paris 1980).
10 The third cross is composed of 36 squares, and the fourth of, no less than, 52.
11 Hermann Beyer has called attention to this important point in "El origen, desarrollo y significado de la greca escalonada," in his Mito y Simbolismo del México Antiguo (Mexico City: Sociedad Alemana Mexicanista, 1965). Arthur Posnasky, as well, has emphasized this symbol as distinctive of the Precolumbian, attributing to these steps or grades the meaning of earth, and to the spiral that of heaven ("Puntos de Contacto Lingüístico y Dogmático en las Américas," in Actas del XXVII Congreso de Americanistas Mexico City, 1939.
12 Even today, the K'ekchi', of the Alta Verapaz, in Guatemala, follow a lunar agricultural calendar of 364 days, divided into 52 weeks and thirteen annual lunar months. Nor are they the only American people to do so.
13 The dates of movable feasts of the Catholic Church as important as Holy Week and Easter are reckoned according to the lunar calendar, Easter itself falling on the first Sunday following the first full moon after the spring equinox. Other feasts, however, are solar, as is obvious in the case of the two solstices.
14 Antiquity always performed its calculations on the basis of "round" and whole numbers, as it regarded these numbers as symbolical expressions of the universal harmony, the prototypal patterns. The exact synodical month of the moon is 29 days, 12 hours, 44 minutes, and 28 seconds. 
It is the same with the thirteen months, since at times there are only twelve moons in a year. The same is the case with the precession of the equinoxes, whose cycle, actually of 25,920 years, has been unanimously reckoned at 26,000 years. Venus completes its course not in 584 days, as the ancients held, but in 583.92 days. 
15 In curious fashion, the "Legend of the Suns" tells of an enigmatic 364 "years" (364 = 13 x 28), which was the duration of the second sun (Códice Chimalpopoca [Mexico City: U.N.A.M., 1975], p. 119). On the reverse of the Codex of Paris (or Peresiano), one can see the years divided into 364 days. The book Chilam Balam de Ixil contains, among other calendrical wheels, tables of relationship between the moon and the zodiac. It is well known that both the Codex Dresden, and various steles and hieroglyphics, contain tables of lunar eclipses. In the Chilam Balam of Chumayel, in the book denominated, "of the Spirits," the number seven is mentioned repeatedly, seemingly in relation to the stars and the moon-those "seven measures of the night." Pachamama and Ixmucane, Aymaran and Mayan goddesses of fertilization, associated with the moon and the earth, each have seven sons. Likewise, seven is the number representing fertility among the Maya-Quiches.