Part 2

1. _The Codices as Time-Counts._

In another work I have explained the numeral system in vogue among the ancient Mayas, as well as the etymology of the terms they employed.[14] It will be sufficient, therefore, to say here that their system was vigesimal, proceeding by multiples of twenty up to very large sums. In the same work I have quoted from original sources the information that the fives up to fifteen were represented by single straight lines and the intermediate numbers by dots. This has also been discovered independently by several students of the manuscripts.

The frequency and prominence of these elementary numerals in nearly every relic of Mayan writing, whether on paper, stone, or pottery, constitute a striking feature of such remains, and forcibly suggest that by far the majority of them have one and the same purpose, that is, _counting_; and when we find with almost equal frequency the signs for days and months associated with these numerals, we become certain that in these records we have before us _time-counts_—some sort of ephemerides or almanacs. This is true of all the Codices, and of nine out of ten of the inscriptions. Here, therefore, is a first and most important step gained toward the solution of the puzzle before us.

But did this incessant time-counting refer to the past or to the future? Was it history or was it prophecy? Or, passing beyond this world, was it astronomy? Was it mythology or ritual, the epochs and the eons of the gods? Perhaps the disposition, sequence, and values of the numbers themselves, once comprehended, will answer these vital questions.

2. _The Mayan Numeral System._

Unfortunately, the old writers, either Spanish or native, tell us little about Maya mathematics. They say the computation ran thus:—

20 units = one _kal_, 20. 20 _kal_ = one _bak_, 400. 20 _bak_ = one _pic_, 8000. 20 _pic_ = one _calab_, 160,000. 20 _calab_ = one _kinchil_ or _tzotzceh_, 3,200,000. 20 _kinchil_ = one _alau_, 64,000,000.

The Tzental system was the same, though the terms differed somewhat: 20 units = one _tab_ (cord or net-ful); 20 _tabs_ = one _bac_; 20 _bacs_ = one _bac-baquetic_ (bundle of bacs); 20 _bac-baquetics_ = one _mam_ (grandfather); 20 _mams_ = one _mechun_ (grandmother); 20 _mechuns_ = one _mucul mam_ (great-grandfather), 64,000,000.[15]

No doubt in the numerical notation there were special signs for each of these higher unities; but neither Bishop Landa nor the native writers who composed the singular “Books of Chilan Balam” have handed them down. Modern sagacity, however, has repaired ancient negligence, and we can, almost to a certainty, restore the numerical notation of the aboriginal arithmeticians.

The scholar who has worked most successfully in this field is Dr. Förstemann, the editor of the Codex of Dresden, and I shall introduce a condensed statement of his results, referring the student to his own writings for their demonstration.

3. _Numerical and Allied Signs._

The first important discovery of Dr. Förstemann in this direction was that of the sign for the naught or cipher, 0. It is given in Fig. 2.[16] It has a number of variants, some ornamental in design. Next, he discovered the system of notation of high numbers. This is not like ours, but resembles that in use in the arithmetic of ancient Babylonia and some parts of China. The numerals are arranged in columns, to be read from below upward, the value of each unit of a given number being that power of 20 which corresponds to the line on which it stands counted from the bottom. This will be readily understood from the following example:—

_Maya _Simple values._ _Composite numerals._ values._

An examination of the mural inscriptions showed that on them also the same plan for the expression of high numbers had been employed, and Dr. Förstemann was enabled to interpret with accuracy the computations on the monuments from Copan, Quirigua, and Palenque; developing incidentally the remarkable fact that the inscriptions of Copan contain as a rule higher numbers and are therefore presumably of later date than those of Palenque. The highest is that on “Stela N,” as catalogued by Mr. Maudslay, which ascends to 1,414,800 days, or 3930 years of 360 days.[17]

The next step was the identification of the graphic signs for the higher unities, 20, 360, and 7200,—corresponding to the native _kal_, _bak_, and _pic_.

That generally used for 20 was identified by several students. It is shown in Fig. 3, No. 3; another also employed under certain circumstances for 20 is shown Fig. 3, No. 2. This was identified independently, first by Pousse, later by Seler.[18] No. 4 is perhaps a variant of it.

The signs for the _bak_, 360, and the _pic_, 7200, are not so certainly established, but Dr. Förstemann has given cogent reasons for recognizing them respectively in the two shown Fig. 4, Nos. 6 and 7.

Higher signs than these in the direct numerical scale have not yet been ascertained; but such plausible reasons have been advanced by Dr. Förstemann for assigning calendar values to certain other signs that they should be added in this description of the numerals.

The first is that shown in Fig. 4, No. 8. It represents the katunic cycle of 52 years of 365 days each, = 18,980 days. The second is No. 9. This is taken to be the sign of the _ahau katun_, 24 years of 365 days, = 8760 days. The third is No. 10. This corresponds to one-third of an _ahau katun_, = 2920 days. The fourth, shown No. 11, is an old cycle of 20 years of 360 days, = 7200. No. 12 means an old katunic cycle of 52 years of 360 days, = 18,720 days, and No. 13 an old year of 360 days.[19]

There are also a series of other signs evidently connected with the numerals, the precise value of which is yet undetermined. One of these is a small right or oblique cross, or sometimes two arcs abutting against each other, connected or not. It is usually by the side of a single dot or unit, or between two such. In certain places, it seems to be a multiplier with the value 20; in others, it would indicate a change or alternation in the series presented of days or years. (See Fig. 5, Nos. 1–4.)

Of somewhat similar value are the calendar signs [Illustration], Fig. 4, Nos. 2, 3, 4, like an _S_ placed lengthwise. This is also understood to be a sign of alternation or change of series of years or cycles.

Of an opposite sense is the sign No. 5, the spiral, and also the sign No. 1, both of which are held to represent union.

This list exhausts the mathematical signs so far as they have been ascertained with probability. Those for high numbers brought forward by Brasseur,[20] have no evidence in their favor.

Mr. Maudslay has offered reasons for believing that the character in Fig. 6, _a_, stands for the numeral 20 in a certain class of mural inscriptions.[21] He further points out that the character _b_ is not unfrequently united with _a_, and that it (_b_) almost alone of the mural glyphs is found with a double set of numerals attached to it as in _c_. One or both these sets of numerals are at times replaced by the sign _a_, giving the composites _d_, _e_, and _f_. It is thus evident that _a_ has some numerical or calendar meaning. As a character itself, it is the “cosmic sign,” conveying the idea of the world or universe as a whole, as is seen by the examples to which Mr. Maudslay refers, from various Codices. The cross-hatching upon it means, as I shall show later, “strong, mighty,” and is merely a superlative. It may very well mean 20, as that is the number conveying completeness or perfection in this mythology.[22] That it appears on what Mr. Maudslay calls the “Initial Series” of glyphs (which I consider terminals), is explained by the nature of the computations they preserve. Another combination, belonging most likely to a similar class, is the following [Illustration] where the “cosmic sign” is united as a superfix to the _pax_ and the flint. It has usually been explained as a “phallic emblem,” and by Thomas as “tortillas.”[23]

4. _The Rhetorical and Symbolic Use of Numbers._

In the old Maya language we find that certain numbers were used in a rhetorical sense, and this explains their appearance in some non-mathematical portions of the Codices and inscriptions. The two most commonly employed were 9 and 13. These conveyed the ideas of indefinite greatness, of superlative excellence, of infinity, etc. A very lucky man was a “nine-souled man;” that which had existed forever was “thirteen generations old,” etc. The “demon with thirteen powers” was still prominent in Tzental mythology in the time of Nuñez de la Vega. Other numerals occasionally employed in a symbolic sense were 3, 4, and 7.[24]

All these occur in the Codices as prefixes in relations where they are not to be construed in their arithmetical values, but in those assigned them by the usages of the language or the customs of religious symbolism. Thus, “twenty,” owing to the vigesimal method of numeration, conveyed the associated ideas of completeness and perfection; and as the month of 20 days was divided into four equal parts of 5 days each, by which markets, etc., were assigned, these numbers also stood independently for other concepts than those of computation.

5. _The Mayan Methods of Counting Time._

Having ascertained the characters for the numerals, and having learned that these records are mainly time-counts, the next question which arises is: How did the Mayas count time?

About this we have considerable information from the works of the Spanish writers, Landa, Aguilar, Cogolludo, Pio Perez, etc., which has been supplemented by the researches of modern authors.

The Maya system was a complicated one, based on several originally distinct methods, which it was the duty and the aim of the astronomer-priests to bring into unison,—and the effort to accomplish this will chiefly explain their elaborate computations.

Undoubtedly their earliest time-count was that common to primitive tribes everywhere—a measurement of the solar year by lunations or “moons.” The exact lunar month is 29 days, 12 hours, 44 minutes, 3 seconds; but primitive peoples usually estimate it at 28 days, and allow 13 months to the solar year, as do yet many North Asiatic peoples, and as probably did the early Aryans;[25] or, they estimate the “moon” at 30 days, and allow 12 moons to the year. There are good grounds for believing that the Mayan tribes were at one time divided in custom about this, some using one, some the other method. At the time of the Conquest they had undoubtedly reached a knowledge of the length of the year as 365 days; and there is considerable probability that some of them at least made the correction arranged for in our bissextile or leap year.[26]

This is all familiar enough and would create no difficulty in deciphering these aboriginal almanacs; but a disturbing element enters. The real time-count by which they adjusted the important events of their lives, and which is most prominent in their records, had nothing to do with the motions of the sun, or the moon, or any other natural phenomenon. It was based on purely mythical relations supposed to exist between man and nature. As the number 20 (fingers and toes) completes the man, and as all the directions, that is, potencies, of the visible and invisible worlds were held to be 13, these two numbers, 13 and 20, formed the basis of an astrological and ritual calendar, by which auspicious and inauspicious days were assigned, future events foretold, the major feasts and festivals of religious worship dictated, and the like.

This singular time-count of 20 × 13 = 260 days was adopted with slight variations by every semi-civilized nation of Mexico and Central America, and even the names of the 20 days are practically of the same meaning in all these languages.[27] It constituted the _tonalamatl_ of the Nahuas, the “Book of Days,” used in divination.

This sacred period was subdivided into four equal parts of 65 days each, each of which was assigned to the rule of a special planet or star, and to a particular cardinal point with attendant divinities; and each was marked with a color of its own, white, black, red, or blue.

Each “month” of 20 days was subdivided into four periods of five days each, again each having its own divinity, assignment, etc.

But the importance to us of the _tonalamatl_ is that its computations underlay the measurement of long periods of time, the less and greater cycles. These were estimated by the methods of the sacred year, in groups of 13, 20, 24, 52, 104, 260 years, etc. These irregular numbers had to be brought into unison with the lunar and solar years, with the vigesimal system of counting by 20 and its multiples, and with the observed motions of the planets, who were divinities controlling the ritual divisions of time.

To devise a mathematical method of equalities and differences by which these conflicting numbers could be placed in harmonious relations, subsumed under common measures, and the ceremonies and forecasts which they controlled assigned by uniform laws—this is the arithmetical problem which fills the pages of the Mayan Codices, and in parts or at length is spread over the surface of the inscribed monuments and painted vases. We need not search for the facts of history, the names of mighty kings, or the dates of conquests. We shall not find them. Chronometry we shall find, but not chronicles; astronomy with astrological aims; rituals, but no records. Pre-Columbian history will not be reconstructed from them. This will be a disappointment to many; but it is the conclusion toward which tend all the soundest investigations of recent years.

Let us recapitulate the numbers which the Maya mathematician had to deal with and adjust under some scheme of uniformity:—

1. The “week” of 13 days, 13.

2. The “month” of 20 days, 20.

3. Its division into four parts (called _tzuc_), each, 5.

4. The complete _tonalamatl_, 13 × 20 days, 260.

5. Its divisions into four parts, each, 65.

6. The solar year, counted as 18 months of 20 days each, 360.

7. The solar year, counted as 12 months of 30 days each, 360.

8. The solar year, counted as 13 lunar months of 28 days each, 364.

9. The solar year, counted as 28 weeks of 13 days each, 364.

10. The true solar year, days, 365.

11. The bissextile year (?), 366.

12. The apparent revolution of Venus (_Noh-ek_, the Great Star), days, 584.

13. The apparent revolution of Mercury (?), days, 115.

14. The apparent revolution of Mars (?), days, 780.

15. The _kin katun_, or day-cycle of years, 13.

16. The older cycle of years, 20.

17. The newer cycle of years, 24.

18. The _katun_ cycle of years, 52.

19. The double cycle of years, 104.

20. The great cycle of years, 260.

6. _The Calculations in the Codices._

The Codices contain numerous calculations intended to bring these various quantities into definite relations as aliquot parts of some arithmetical whole, which might be taken as a general unit. The scribes appear to have begun by establishing a period of 14,040 days. This equals 39 years of 360 days each, and also 54 years of 260 days each, together, of course, with the divisors of these numbers, 13, 18, 20, 65, etc. Then followed the determination of the period of 18,980 days, = 73 _tonalamatl_, = 52 solar years, so prominent in the calendar and ritual of the Nahuas.

This number, however, could not be adjusted to the cycle of the _ahau katun_, which was 24 years of 365 days each;[28] nor to the ceremonially prominent revolution of “the Great Star,” Venus, which coincides with the Earth’s revolutions in 2920 days, or eight solar years. To bring these into accord with the _tonalamatl_ required a period of 104 solar years, or 37,960 days; and to adjust under one number the _katuns_, the _ahau katuns_, the revolutions of Venus, the solar year, and the _tonalamatl_, three times that number of days are required, that is, 113,880, = 312 years.

This period had still to be brought into relation to the old year of 360 days, and this requires the estimation of a term covering 1,366,560 days, or 3744 years; and this extended era we find expressed in the Dresden Codex, page 24, in the following simple notation, the interpretation of which into our system of calculation, according to the method above explained, I add to the right.

This long period allowed all their important time-measures to be dealt with as aliquot parts of one whole, and would seem to be sufficient for the purposes they had in view. The credit of establishing it from their ancient writings is exclusively due to Dr. Förstemann, whose demonstrations of it appear to be conclusive.[29]

No doubt each of these periods of time had its appropriate name in the technical language of the Maya astronomers, and also its corresponding sign or character in their writing. None of them has been recorded by the Spanish writers; but from the analogy of the Nahuatl script and language, and from certain indications in the Maya writings, we may surmise that some of these technical terms were from one of the radicals meaning “to tie, or fasten together,” and that the corresponding signs would either directly, that is, pictorially; or ikonomatically, that is, by similarity of sound, express this idea.

Proceeding on the first of these suppositions, Dr. Förstemann has suggested that the character, Fig. 4, No. 8, signifies the period of 52 years, the Nahuatl _xiuhmolpilli_, “the tying together of the years,” represented in the Aztec pictographs by a bundle of faggots tied with cords. The Maya figure is explained as the day-sign _imix_, representing the first day of the calendar, and, by a kind of synecdoche, the whole calendar, with a superfix.

7. _Rules for Tracing the Tonalamatl, or Ritual Calendar._

That the computations of the _tonalamatl_ underlie most of the numerals in the Codices is shown by the rules for reading them, formulated by Pousse with reference to the red and black numerical signs. These rules are as follows:—[31]

1. If to a red number be added the black number immediately following it, the total less 13 (or its multiples, when the total is above 13) equals the next following red number.

2. When the red and black numbers are written alternately on the same line, they are to be read from left to right; when written one above the other, they are to be read from below upward; when in two vertical columns, they are to be read passing from one column to the other, beginning with the first black number on the left, passing to the first black number on the right, returning to the second black number on the left, and so on.

Sequences of this kind are governed by the following rules:—

1. In any of the above systems the beginning is always marked by one or more columns of days surmounted by a number.

2. This number is always the same as that which ends the series, and both are written in red.

3. The sum of the numbers written in black, multiplied by the number of days with different names represented by the hieroglyphs attached, always equals 260, that is, the number of days in the _tonalamatl_.

The above rules enable the student to recognize the relations of the different parts of the Codices. They prove, for instance, that the pages are not to be read from top to bottom, but that the separate parts or chapters are to be read in many instances from left to right in the section of the page in which they begin, without respect to the folds of the MSS.; and that evidently in reading these “books” they were unfolded and spread out. A good example of this is in Cod. Dresden, pages 4–10, on which one chapter covers all the upper thirds of the seven pages.

8. _The Codices as Astronomical Treatises._

A careful examination of Dr. Förstemann’s remarkable studies, as well as a number of other considerations drawn from the Codices themselves, have persuaded me that the general purpose of the Codices and the greater inscriptions, as those of Palenque, have been misunderstood and underrated by most writers. In one of his latest papers[32] Professor Cyrus Thomas says of the Codices: “These records are to a large extent only religious calendars;” and Dr. Seler has expressed his distrust in Dr. Förstemann’s opinions as to their astronomic contents. My own conviction is that they will prove to be much more astronomical than even the latter believes; that they are primarily and essentially records of the motions of the heavenly bodies; and that both figures and characters are to be interpreted as referring in the first instance to the sun and moon, the planets, and those constellations which are most prominent in the nightly sky in the latitude of Yucatan.