Fig. 37Fig.37. Ending signs and elements.
Fig.37. Ending signs and elements.
The hand-ending sign rarely appears as modifying period glyphs, although a few examples of such use have been found (see fig.37,j,k). This ending sign usually appears as the main element in a separate glyph, which precedes the sign of the period whose end is recorded (see fig.37,l-q). In these cases the subordinate elements differ somewhat, although the element (*) appears as the suffix inl,m,n,q, and the element (†) as a postfix therein, also inoandp. In a few cases the hand is combined with the other ending signs, sometimes with one and sometimes with the other.
The use of the hand as expressing the meaning "ending" is quite natural. The Aztec, we have seen, called their 52-year period thexiuhmolpilli, or "year bundle." This implies the concomitant idea of "tying up." As a period closed, metaphorically speaking, it was "tied up" or "bundled up." The Maya use of the hand to express the idea "ending" may be a graphic representation of the member by means of which this "tying up" was effected, the clasped hand indicating the closed period.
This method of describing a date may be called "dating by period endings." It was far less accurate than Initial-series or Secondary-series dating, since a date described as occurring at the end of a certain katun could recur after an interval of about 18,000 years in round numbers, as against 374,400 years in the other 2 methods. For all practical purposes, however, 18,000 years was as accurate as 374,400 years, since it far exceeds the range of time covered by the written records of mankind the world over.
Period-ending dates were not used much, and, as has been stated above, they are found only in connection with the larger periods—most frequently with the katun, next with the cycle, and but very rarely with the tun. Mr. Bowditch (1910: pp. 176 et seq.) has reviewed fully the use of ending signs, and students are referred to his work for further information on this subject.
U Kahlay Katunob
In addition to the foregoing methods of measuring time and recording dates, the Maya of Yucatan used still another, which, however, was probably derived directly from the application of Period-ending dating to the Long Count, and consequently introduces no new elements. This has been designated the Sequence of the Katuns, because in this method the katun, or 7,200-day period, was the unit used for measuring the passage of time. The Maya themselves called the Sequence of the Katunsu tzolan katun, "the series of the katuns"; oru kahlay uxocen katunob, "the record of the count of the katuns"; or even more simply,u kahlay katunob, "the record of the katuns." These names accurately describe this system, which is simply the record of the successive katuns, comprising in the aggregate the range of Maya chronology.
Each katun of the u kahlay katunob was named after the designation of its ending day, a practice derived no doubt from Period-ending dating, and the sequence of these ending days represented passed time, each ending day standing for the katun of which it was the close. The katun, as we have seen on page77, always ended with some dayAhau, consequently this day-name is the only one of the twenty which appears in the u kahlay katunob. In this method the katuns were distinguished from one another,notby the positionswhich they occupied in the cycle, as Katun 14, for example, but by the different daysAhauwith which they ended, as Katun2 Ahau, Katun13 Ahau, etc. See Table IX.
Table IX.—SEQUENCE OF KATUNS IN U KAHLAY KATUNOB
The peculiar retrograding sequence of the numerical coefficients in TableIX, decreasing by 2 from katun to katun, as 2, 13, 11, 9, 7, 5, 3, 1, 12, etc., results directly from the number of days which the katun contains. Since the 13 possible numerical coefficients, 1 to 13, inclusive, succeed each other in endless repetition, 1 following immediately after 13, it is clear that in counting forward any given number from any given numerical coefficient, the resulting numerical coefficient will not be affected if we first deduct all the 13s possible from the number to be counted forward. The mathematical demonstration of this fact follows. If we count forward 14 from any given coefficient, the same coefficient will be reached as if we had counted forward but 1. This is true because, (1) there are only 13 numerical coefficients, and (2) these follow each other without interruption, 1 following immediately after 13; hence, when 13 has been reached, the next coefficient is 1, not 14; therefore 13 or any multiple thereof may be counted forward or backward from any one of the 13 numerical coefficients without changing its value. This truth enables us to formulate the following rule for finding numerical coefficients: Deduct all the multiples of 13 possible from the number to be counted forward, and then count forward the remainder from the known coefficient, subtracting 13 if the resulting number is above 13, since 13 is the highest possible number which can be attached to a day sign. If we apply this rule to the sequence of the numerical coefficients in TableIX, we shall find that it accounts for the retrograding sequence there observed. The first katun in TableIX, Katun2 Ahau, is named after its ending day,2 Ahau. Now let us see whether the application of this rule will give us13 Ahauas the ending day of the next katun. The number to be counted forward from2 Ahauis 7,200, the number of days in one katun; therefore we must first deduct from 7,200 all the 13s possible. 7,200 ÷ 13 = 55311⁄13. In other words, after we have deducted all the 13's possible, that is,553 of them, there is a remainder of 11. This the rule says is to be added (or counted forward) from the known coefficient (in this case 2) in order to reach the resulting coefficient. 2 + 11 = 13. Since this number is not above 13, 13 is not to be deducted from it; therefore the coefficient of the ending day of the second katun is 13, as shown in TableIX. Similarly we can prove that the coefficient of the ending day of the third katun in TableIXwill be 11. Again, we have 7,200 to count forward from the known coefficient, in this case 13 (the coefficient of the ending day of the second katun). But we have seen above that if we deduct all the 13s possible from 7,200 there will be a remainder of 11; consequently this remainder 11 must be added to 13, the known coefficient. 13 + 11 = 24; but since this number is above 13, we must deduct 13 from it in order to find out the resulting coefficient. 24-13 = 11, and 11 is the coefficient of the ending day of the third katun in TableIX. By applying the above rule, all of the coefficients of the ending days of the katuns could be shown to follow the sequence indicated in TableIX. And since the ending days of the katuns determined their names, this same sequence is also that of the katuns themselves.
The above table enables us to establish a constant by means of which we can always find the name of the next katun. Since 7,200 is always the number of days in any katun, after deducting all the 13s possible the remainder will always be 11, which has to be added to the known coefficient to find the unknown. But since 13 has to be deducted from the resulting number when it is above 13, subtracting 2 will always give us exactly the same coefficient as adding 11; consequently we may formulate for determining the numerical coefficients of the ending days of katuns the following simple rule: Subtract 2 from the coefficient of the ending day of the preceding katun in every case. A glance at TableIXwill demonstrate the truth of this rule.
In the names of the katuns given in TableIXit is noteworthy that the positions which the ending days occupied in the divisions of the haab, or 365-day year, are not mentioned. For example, the first katun was not called Katun2 Ahau 8 Zac, but simply Katun2 Ahau, the month part of the day, that is, its position in the year, was omitted. This omission of the month parts of the ending days of the katuns in the u kahlay katunob has rendered this method of dating far less accurate than any of the others previously described except Calendar-round Dating. For example, when a date was recorded as falling within a certain katun, as Katun2 Ahau, it might occur anywhere within a period of 7,200 days, or nearly 20 years, and yet fulfill the given conditions. In other words, no matter how accurately this Katun2 Ahauitself might be fixed in alongstretch of time, there was always the possibility of a maximum error of about 20 years insuch dating, since the statement of the katun did not fix a date any closer than as occurring somewhere within a certain 20-year period. When greater accuracy was desired the particular tun in which the date occurred was also given, as Tun 13 of Katun2 Ahau. This fixed a date as falling somewhere within a certain 360 days, which was accurately fixed in a much longer period of time. Very rarely, in the case of an extremely important event, the Calendar-round date was also given as9 Imix 19 Zipof Tun 9 of Katun13 Ahau. A date thus described satisfying all the given conditions could not recur until after the lapse of at least 7,000 years. The great majority of events, however, recorded by this method are described only as occurring in some particular katun, as Katun2 Ahau, for example, no attempt being made to refer them to any particular division (tun) of this period. Such accuracy doubtless was sufficient for recording the events of tribal history, since in no case could an event be more than 20 years out of the way.
Aside from this initial error, the accuracy of this method of dating has been challenged on the ground that since there were only thirteen possible numerical coefficients, any given katun, as Katun2 Ahau, for example, in TableIXwould recur in the sequence after the lapse of thirteen katuns, or about 256 years, thus paving the way for much confusion. While admitting that every thirteenth katun in the sequence had the same name (see TableIX), the writer believes, nevertheless, that when the sequence of the katuns was carefully kept, and the record of each entered immediately after its completion, so that there could be no chance of confusing it with an earlier katun of the same name in the sequence, accuracy in dating could be secured for as long a period as the sequence remained unbroken. Indeed, the u kahlay katunob[54]from which the synopsis of Maya history given in Chapter I was compiled, accurately fixes the date of events, ignoring the possible initial inaccuracy of 20 years, within a period of more than 1,100 years, a remarkable feat for any primitive chronology.
How early this method of recording dates was developed is uncertain. It has not yet been found (surely) in the inscriptions in either the south or the north; on the other hand, it is so closely connected with the Long Count and Period-ending dating, which occurs repeatedly throughout the inscriptions, that it seems as though the u kahlay katunob must have been developed while this system was still in use.
There should be noted here a possible exception to the above statement, namely, that the u kahlay katunob has not been found in the inscriptions. Mr. Bowditch (1910: pp. 192 et seq.) has pointed outwhat seem to be traces of another method of dating. This consists of some dayAhaumodified by one of the two elements shown in figure38(a-dande-h, respectively). In such cases the month part is sometimes recorded, though as frequently the dayAhaustands by itself. It is to be noted that in the great majority of these cases the daysAhauthus modified are the ending days of katuns, which are either expressed or at least indicated in adjacent glyphs. In other words, the dayAhauthus modified is usually the ending day of the next even katun after the last date recorded. The writer believes that this modification of certain daysAhauby either of the two elements shown in figure38may indicate that such days were the katun ending days nearest to the time when the inscriptions presenting them were engraved. The snake variants shown in figure38,a-d, are all from Palenque; the knot variants (e-hof the same figure) are found at both Copan and Quirigua.
Fig. 38Fig.38. "Snake" or "knot" element as used with day signAhau, possibly indicating presence of the u kahlay katunob in the inscriptions.
Fig.38. "Snake" or "knot" element as used with day signAhau, possibly indicating presence of the u kahlay katunob in the inscriptions.
It may be objected that one katun ending day in each inscription is far different from a sequence of katun ending days as shown in TableIX, and that one katun ending day by itself can not be construed as an u kahlay katunob, or sequence of katuns. The difference here, however, is apparent rather than real, and results from the different character of the monuments and the native chronicles. The u kahlay katunob in TableIXis but a part of a much longer sequence of katuns, which is shown in a number of native chronicles written shortly after the Spanish Conquest, and which record the events of Maya history for more than 1,100 years. They are in fact chronological synopses of Maya history, and from their very nature they have to do with long periods. This is not true of the monuments,[55]which, as we have seen, were probably set up to mark the passage of certain periods, not exceeding a katun in length in any case. Consequently, each monument would have inscribed upon it only one or twokatun ending days and the events which were connected more or less closely with it. In other words, the monuments were erected at short intervals[56]and probably recorded events contemporaneous with their erection, while the u kahlay katunob, on the other hand, were historical summaries reaching back to a remote time. The former were the periodicals of current events, the latter histories of the past. The former in the great majority of cases had no concern with the lapse of more than one or two katuns, while the latter measured centuries by the repetition of the same unit. The writer believes that from the very nature of the monuments—markers of current time—no u kahlay katunob will be found on them, but that the presence of the katun ending days above described indicates that the u kahlay katunob had been developed while the other system was still in use. If the foregoing be true, the signs in figure38,a-h, would have this meaning: "On this day came to an end the katun in which fall the accompanying dates," or some similar significance.
If we exclude the foregoing as indicating the u kahlay katunob, we have but one aboriginal source, that is one antedating the Spanish Conquest, which probably records a count of this kind. It has been stated (p.33) that the Codex Peresianus probably treats in part at least of historical matter. The basis for this assertion is that in this particular manuscript an u kahlay katunob is seemingly recorded; at least there is a sequence of the ending days of katuns shown, exactly like the one in TableIX, that is,13 Ahau, 11 Ahau, 9 Ahau, etc.
At the time of the Spanish Conquest the Long Count seems to have been recorded entirely by the ending days of its katuns, that is, by the u kahlay katunob, and the use of Initial-series dating seems to have been discontinued, and perhaps even forgotten. Native as well as Spanish authorities state that at the time of the Conquest the Maya measured time by the passage of the katuns, and no mention is made of any system of dating which resembles in the least the Initial Series so prevalent in the southern and older cities. While the Spanish authorities do not mention the u kahlay katunob as do the native writers, they state very clearly that this was the system used in counting time. Says Bishop Landa (1864: p. 312) in this connection: "The Indians not only had a count by years and days ... but they had a certain method of counting time and their affairs by ages, which they made from twenty to twenty years ... these they call katunes." Cogolludo (1688: lib. iv, cap. v, p. 186) makes a similar statement: "They count their eras and ages, which they put in their books from twenty to twenty years ... [these] they call katun." Indeed, there can be but little doubt that the u kahlay katunob had entirely replaced the Initial Series in recording the Long Count centuries before the Spanish Conquest; and if the latter method of dating were knownat all, the knowledge of it came only from half-forgotten records the understanding of which was gradually passing from the minds of men.
It is clear from the foregoing that an important change in recording the passage of time took place sometime between the epoch of the great southern cities and the much later period when the northern cities flourished. In the former, time was reckoned and dates were recorded by Initial Series; in the latter, in so far as we can judge from post-Conquest sources, the u kahlay katunob and Calendar-round dating were the only systems used. As to when this change took place, we are not entirely in the dark. It is certain that the use of the Initial Series extended to Yucatan, since monuments presenting this method of dating have been found at a few of the northern cities, namely, at Chichen Itza, Holactun, and Tuluum. On the other hand, it is equally certain that Initial Series could not have been used very extensively in the north, since they have been discovered in only these three cities in Yucatan up to the present time. Moreover, the latest, that is, the most recent of these three, was probably contemporaneous with the rise of the Triple Alliance, a fairly early event of Northern Maya history. Taking these two points into consideration, the limited use of Initial Series in the north and the early dates recorded in the few Initial Series known, it seems likely that Initial-series dating did not long survive the transplanting of the Maya civilization in Yucatan.
Why this change came about is uncertain. It could hardly have been due to the desire for greater accuracy, since the u kahlay katunob was far less exact than Initial-series dating; not only could dates satisfying all given conditions recur much more frequently in the u kahlay katunob, but, as generally used, this method fixed a date merely as occurring somewhere within a period of about 20 years.
The writer believes the change under consideration arose from a very different cause; that it was in fact the result of a tendency toward greater brevity, which was present in the glyphic writing from the very earliest times, and which is to be noted on some of the earliest monuments that have survived the ravages of the passing centuries. At first, when but a single date was recorded on a monument, an Initial Series was used. Later, however, when the need or desire had arisen to inscribe more than one date on the same monument, additional dates werenotexpressed as Initial Series, each of which, as we have seen, involves the use of 8 glyphs, but as a Secondary Series, which for the record of short periods necessitated the use of fewer glyphs than were employed in Initial Series. It would seem almost as though Secondary Series had been invented to avoid the use of Initial Series when more than one date had to be recorded on the same monument. But this tendency toward brevity in dating did not cease with the invention of Secondary Series. Somewhat later, dating by period-endings was introduced, obviating thenecessity for the use of even one Initial Series on every monument, in order that one date might be fixed in the Long Count to which the others (Secondary Series) could be referred. For all practical purposes, as we have seen, Period-ending dating was as accurate as Initial-series dating for fixing dates in the Long Count, and its substitution for Initial-series dating resulted in a further saving of glyphs and a corresponding economy of space. Still later, probably after the Maya had colonized Yucatan, the u kahlay katunob, which was a direct application of Period-ending dating to the Long Count, came into general use. At this time a rich history lay behind the Maya people, and to have recorded all of its events by their corresponding Initial Series would have been far too cumbersome a practice. The u kahlay katunob offered a convenient and facile method by means of which long stretches of time could be recorded and events approximately dated; that is, within 20 years. This, together with the fact that the practice of setting up dated period-markers seems to have languished in the north, thus eliminating the greatest medium of all for the presentation of Initial Series, probably gave rise to the change from the one method of recording time to the other.
This concludes the discussion of the five methods by means of which the Maya reckoned time and recorded dates: (1) Initial-series dating; (2) Secondary-series dating; (3) Calendar-round dating; (4) Period-ending dating; (5) Katun-ending dating, or the u kahlay katunob. While apparently differing considerably from one another, in reality all are expressions of the same fundamental idea, the combination of the numbers 13 and 20 (that is, 260) with the solar year conceived as containing 365 days, and all were recorded by the same vigesimal system of numeration; that is:
1. All used precisely the same dates, the 18,980 dates of the Calendar Round;
2. All may be reduced to the same fundamental unit, the day; and
3. All used the same time counters, those shown in TableVIII.
In conclusion, the student is strongly urged constantly to bear in mind two vital characteristics of Maya chronology:
1. The absolute continuity of all sequences which had to do with the counting of time: The 13 numerical coefficients of the day names, the 20 day names, the 260 days of the tonalamatl, the 365 positions of the haab, the 18,980 dates of the Calendar Round, and the kins, uinals, tuns, katuns, and cycles of the vigesimal system of numeration. When the conclusion of any one of these sequences had been reached, the sequence began anew without the interruption or omission of a single unit and continued repeating itself for all time.
2. All Maya periods expressed not current time, but passed time, as in the case of our hours, minutes, and seconds.
On these two facts rests the whole Maya conception of time.
ChapterIV
MAYA ARITHMETIC
The present chapter will be devoted to the consideration of Maya arithmetic in its relation to the calendar. It will be shown how the Maya expressed their numbers and how they used their several time periods. In short, their arithmetical processes will be explained, and the calculations resulting from their application to the calendar will be set forth.
The Maya had two different ways of writing their numerals,[57]namely: (1) With normal forms, and (2) with head variants; that is, each of the numerals up to and including 19 had two distinct characters which stood for it, just as in the case of the time periods and more rarely, the days and months. The normal forms of the numerals may be compared to our Roman figures, since they are built up by the combination of certain elements which had a fixed numerical value, like the letters I, V, X, L, C, D, and M, which in Roman notation stand for the values 1, 5, 10, 50, 100, 500, and 1,000, respectively. The head-variant numerals, on the other hand, more closely resemble our Arabic figures, since there was a special head form for each number up to and including 13, just as there are special characters for the first nine figures and zero in Arabic notation. Moreover, this parallel between our Arabic figures and the Maya head-variant numerals extends to the formation of the higher numbers. Thus, the Maya formed the head-variant numerals for 14, 15, 16, 17, 18, and 19 by applying the essential characteristic of the head variant for 10 to the head variants for 4, 5, 6, 7, 8, and 9, respectively, just as the sign for 10—that is, one in the tens place and zero in the units place—is used in connection with the signs for the first nine figures in Arabic notation to form the numbers 11 to 19, inclusive. Both of these notations occur in the inscriptions, but with very few exceptions[58]no head-variant numerals have yet been found in the codices.
Bar and Dot Numerals
The Maya "Roman numerals"—that is, the normal-form numerals, up to and including 19—were expressed by varying combinations of two elements, the dot (dot), which represented the numeral, or numerical value, 1, and the bar, or line (bar), which represented the numeral, or numerical value, 5. By various combinations of these twoelements alone the Maya expressed all the numerals from 1 to 19, inclusive. The normal forms of the numerals in the codices are shown in figure39, in which one dot stands for 1, two dots for 2, three dots for 3, four dots for 4, one bar for 5, one bar and one dot for 6, one bar and two dots for 7, one bar and three dots for 8, one bar and four dots for 9, two bars for 10, and so on up to three bars and four dots for 19. The normal forms of the numerals, in the inscriptions (see fig.40) are identical with those in the codices, excepting that they are more elaborate, the dots and bars both taking on various decorations. Some of the former contain a concentric circle (*) or cross-hatching (**); some appear as crescents (†) or curls (††), more rarely as (‡) or (‡‡). The bars show even a greater variety of treatment (see fig.41). All these decorations, however, in no way affect the numerical value of the bar and the dot, which remain 5 and 1, respectively, throughout the Maya writing. Such embellishments as those just described are found only in the inscriptions, and their use was probably due to the desire to make the bar and dot serve a decorative as well as a numerical function.
Fig. 39Fig.39. Normal forms of numerals 1 to 19, inclusive, in the codices.
Fig.39. Normal forms of numerals 1 to 19, inclusive, in the codices.
Fig. 40Fig.40. Normal forms of numerals 1 to 19, inclusive, in the inscriptions.
Fig.40. Normal forms of numerals 1 to 19, inclusive, in the inscriptions.
Fig. 41Fig.41. Examples of bar and dot numeral 5, showing the ornamentation which the bar underwent without affecting its numerical value.
Fig.41. Examples of bar and dot numeral 5, showing the ornamentation which the bar underwent without affecting its numerical value.
An important exception to this statement should be noted here in connection with the normal forms for the numbers 1, 2, 6, 7, 11, 12, 16, and 17, that is, all which involve the use ofoneortwodots in their composition.[59]In the inscriptions, as we have seen in Chapter II, every glyph was a balanced picture, exactly fitting its allotted space, even at the cost of occasionally losing some of its elements. To have expressed the numbers 1, 2, 6, 7, 11, 12, 16, and 17 as in the codices, with just the proper number of bars and dots in each case, would have left unsightly gaps in the outlines of the glyph blocks (see fig.42,a-h, where these numbers are shown as the coefficients of the katun sign). Ina,c,e, andgof the same figure (the numbers 1, 6, 11, and 16, respectively) the single dot does not fill the space on the left-hand[60]side of the bar, or bars, as the case may be, and consequentlythe left-hand edge of the glyph block in each case is ragged. Similarly inb,d,f, andh, the numbers 2, 7, 12, and 17, respectively, the two dots at the left of the bar or bars are too far apart to fill in the left-hand edge of the glyph blocks neatly, and consequently in these cases also the left edge is ragged. The Maya were quick to note this discordant note in glyph design, and in the great majority of the places where these numbers (1, 2, 6, 7, 11, 12, 16, and 17) had to be recorded, other elements of a purely ornamental character were introduced to fill the empty spaces. In figure43,a,c,e,g, the spaces on each side of the single dot have been filled with ornamentalcrescents about the size of the dot, and these give the glyph in each case a final touch of balance and harmony, which is lacking without them. Inb,d,f, andhof the same figure a single crescent stands between the two numerical dots, and this again harmoniously fills in the glyph block. While the crescent (*) is the usual form taken by this purely decorative element, crossed lines (**) are found in places, as in (†); or, again, a pair of dotted elements (††), as in (‡). These variants, however, are of rare occurrence, the common form being the crescent shown in figure43.
Fig. 42Fig.42. Examples showing the way in which the numerals 1, 2, 6, 7, 11, 12, 16, and 17 arenotused with period, day, or month signs.
Fig.42. Examples showing the way in which the numerals 1, 2, 6, 7, 11, 12, 16, and 17 arenotused with period, day, or month signs.
Fig. 43Fig.43. Examples showing the way in which the numerals 1, 2, 6, 7, 11, 12, 16, and 17areused with period, day, or month signs. Note the filling of the otherwise vacant spaces with ornamental elements.
Fig.43. Examples showing the way in which the numerals 1, 2, 6, 7, 11, 12, 16, and 17areused with period, day, or month signs. Note the filling of the otherwise vacant spaces with ornamental elements.
Fig. 44Fig.44. Normal forms of numerals 1 to 13, inclusive, in the Books of Chilan Balam.
Fig.44. Normal forms of numerals 1 to 13, inclusive, in the Books of Chilan Balam.
The use of these purely ornamental elements, to fill the empty spaces in the normal forms of the numerals 1, 2, 6, 7, 11, 12, 16, and 17, is a fruitful source of error to the student of the inscriptions. Slight weathering of an inscription is often sufficient to make ornamental crescents look exactly like numerical dots, and consequently the numerals 1, 2, 3 are frequently mistaken for one another, as are also 6, 7, and 8; 11, 12, and 13; and 16, 17, and 18. The student must exercise the greatest caution at all times in identifying thesenumerals in the inscriptions, or otherwise he will quickly find himself involved in a tangle from which there seems to be no egress. Probably more errors in reading the inscriptions have been made through the incorrect identification of these numerals than through any other one cause, and the student is urged to be continually on his guard if he would avoid making this capital blunder.
Although the early Spanish authorities make no mention of the fact that the Maya expressed their numbers by bars and dots, native testimony is not lacking on this point. Doctor Brinton (1882 b: p. 48) gives this extract, accompanied by the drawing shown in figure44, from a native writer of the eighteenth century who clearly describes this system of writing numbers:
They [our ancestors] used [for numerals in their calendars] dots and lines [i. e., bars] back of them; one dot for one year, two dots for two years, three dots for three years, four dots for four, and so on; in addition to these they used a line; one line meant five years, two lines meant ten years; if one line and above it one dot, six years; if two dots above the line, seven years; if three dots above, eight years; if four dots above the line, nine; a dot above two lines, eleven; if two dots, twelve; if three dots, thirteen.
They [our ancestors] used [for numerals in their calendars] dots and lines [i. e., bars] back of them; one dot for one year, two dots for two years, three dots for three years, four dots for four, and so on; in addition to these they used a line; one line meant five years, two lines meant ten years; if one line and above it one dot, six years; if two dots above the line, seven years; if three dots above, eight years; if four dots above the line, nine; a dot above two lines, eleven; if two dots, twelve; if three dots, thirteen.
This description is so clear, and the values therein assigned to the several combinations of bars and dots have been verified so extensively throughout both the inscriptions and the codices, that we are justified in identifying the bar and dot as the signs for five and one, respectively, wherever they occur, whether they are found by themselves or in varying combinations.
In the codices, as will appear in Chapter VI, the bar and dot numerals were painted in two colors, black and red. These colors were used to distinguish one set of numerals from another, each of which has a different use. In such cases, however, bars of one color are never used with dots of the other color, each number being either all red or all black (see p.93, footnote 1, for the single exception to this rule).
By the development of a special character to represent the number 5 the Maya had far surpassed the Aztec in the science of mathematics; indeed, the latter seem to have had but one numerical sign, the dot, and they were obliged to resort to the clumsy makeshift of repeating this in order to represent all numbers above 1. It is clearly seen that such a system of notation has very definite limitations, which must have seriously retarded mathematical progress among the Aztec.
In the Maya system of numeration, which was vigesimal, there was no need for a special character to represent the number 20,[61]because(1) as we have seen in TableVIII, 20 units of any order (except the 2d, in which only 18 were required) were equal to 1 unit of the order next higher, and consequently 20 could not be attached to any period-glyph, since this number of periods (with the above exception) was always recorded as 1 period of the order next higher; and (2) although there were 20 positions in each period except the uinal, as 20 kins in each uinal, 20 tuns in each katun, 20 katuns in each cycle, these positions were numbered not from 1 to 20, but on the contrary from 0 to 19, a system which eliminated the need for a character expressing 20.
Fig. 45Fig.45. Sign for 20 in the codices.
Fig.45. Sign for 20 in the codices.
In spite of the foregoing fact, however, the number 20 has been found in the codices (see fig.45). A peculiar condition there, however, accounts satisfactorily for its presence. In the codices the sign for 20 occurs only in connection with tonalamatls, which, as we shall see later, were usually portrayed in such a manner that the numbers of which they were composed could not be presented from bottom to top in the usual way, but had to be written horizontally from left to right. This destroyed the possibility of numeration by position,[62]according to the Maya point of view, and consequently some sign was necessary which should stand for 20 regardless of its position or relation to others. The sign shown in figure45was used for this purpose. It has not yet been found in the inscriptions, perhaps because, as was pointed out in Chapter II, the inscriptions generally do not appear to treat of tonalamatls.
Fig. 46Fig.46. Sign for 0 in the codices.
Fig.46. Sign for 0 in the codices.
If the Maya numerical system had no vital need for a character to express the number 20, a sign to represent zero was absolutelyindispensable. Indeed, any numerical system which rises to a second order of units requires a character which will signify, when the need arises, that no units of a certain order are involved; as zero units and zero tens, for example, in writing 100 in our own Arabic notation.
The character zero seems to have played an important part in Maya calculations, and signs for it have been found in both the codices and the inscriptions. The form found in the codices (fig.46) is lenticular; it presents an interior decoration which does not follow any fixed scheme.[63]Only a very few variants occur. The last one in figure46has clearly as one of its elements the normal form (lenticular). The remaining two are different. It is noteworthy, however, that these last three forms all stand in the 2d, or uinal, place in the texts in which they occur, though whether this fact has influenced their variation is unknown.
Fig. 47Fig.47. Sign for 0 in the inscriptions.
Fig.47. Sign for 0 in the inscriptions.