Let us count forward the number 5,799 from the starting point2 Kan 7 Tzec. It is apparent at the outset that, since this number is less than 18,980, or 1 Calendar Round, the preliminary rule given on page143does not apply in this case. Therefore we may proceed with the first rule given on page139, by means of which the new day coefficient may be determined. Dividing the given number by 13 we have: 5,799 ÷ 13 = 4461⁄13. Counting forward the numerator of the fractional part of the resulting quotient (1) from the day coefficient of the starting point (2), we reach3as the day coefficient of the terminal date.
The second rule given on page140tells how to find the day sign of the terminal date. Dividing the given number by 20, we have: 5,799 ÷ 20 = 28919⁄20. Counting forward the numerator of the fractional part of the resulting quotient (19) from the day sign of the starting point,Kan, in the sequence of the twenty-day signs given in TableI, the day signAkbalwill be reached, which will be the day sign of the terminal date. Therefore the day of the terminal date will be3 Akbal.
The third rule, given on page141, tells how to find the position which the day of the terminal date occupied in the 365-day year. Dividing the given number by 365, we have: 5,799 ÷ 365 = 15324⁄365. Counting forward the numerator of the fractional part of the resulting quotient, 324, from the year position of the starting date,7 Tzec, in the sequence of the 365 year positions given in TableXV, the position6 Zipwill be reached as the position in the year of the day of the terminal date. The count by means of which the position6 Zipis determined is given in detail. After the year position of the starting point,7 Tzec, it requires 12 more positions (Nos. 8-19, inclusive) before the close of that month (see TableXV) will be reached. And after the close ofTzec, 13 uinals and the xma kaba kin must pass before the end of the year; 13 × 20 + 5 = 265, and 265 + 12 = 277. This latter number subtracted from 324, the total number of positions to be counted forward, will give the number of positions which remain to be counted in the next year following: 324-277 = 47. Counting forward 47 in the new year, we find that it will use up the monthsPopandUo(20 + 20 = 40) and extend 7 positions into the monthZip, or to6 Zip. Therefore, gathering together the values determined for the several parts of the terminal date, we may say that in counting forward 5,799 from the starting point2 Kan 7 Tzec, the terminal date reached will be3 Akbal 6 Zip.
For the next example let us select a much higher number, say 322,920, which we will assume is to be counted forward from the starting point13 Ik 0 Zip. Since this number is above 18,980, we may apply our preliminary rule (p.143) and deduct all the CalendarRounds possible. By turning to TableXVIwe see that 17 Calendar Rounds, or 322,660, may be deducted from our number: 322,920-322,660 = 260. In other words, we can use 260 exactly as though it were 322,920. Dividing by 13, we have 260 ÷ 13 = 20. Since there is no fraction in the quotient, the numerator of the fraction will be 0, and counting 0 forward from the day coefficient of the starting point,13, we have 13 as the day coefficient of the terminal date (rule 1, p.139). Dividing by 20 we have 260 ÷ 20 = 13. Since there is no fraction in the quotient, the numerator of the fraction will be 0, and counting forward 0 from the day sign of the starting point,Ikin TableI, the day signIkwill remain the day sign of the terminal date (rule 2, p.140). Combining the two values just determined, we see that the day of the terminal date will be13 Ik, or a day of the same name as the day of the starting point. This follows also from the fact that there are only 260 differently named days (see pp.41-44) and any given day will have to recur, therefore, after the lapse of 260 days.[101]Dividing by 365 we have: 260 ÷ 365 =260⁄365. Counting forward the numerator of the fraction, 260, from the year position of the starting point,0 Zip, in TableXV, the position in the year of the day of the terminal date will be found to be0 Pax. Since 260 days equal just 13 uinals, we have only to count forward from0 Zip13 uinals in order to reach the year position; that is,0 Zotzis 1 uinal; to0 Tzec2 uinals, to0 Xul3 uinals, and so on in TableXVto0 Pax, which will complete the last of the 13 uinals (rule 3, p.141).
Combining the above values, we find that in counting forward 322,920 (or 260) from the starting point13 Ik 0 Zip, the terminal date reached is13 Ik 0 Pax.
In order to illustrate the method of procedure when the count isbackward, let us assume an example of this kind. Suppose we count backward the number 9,663 from the starting point3 Imix 4 Uayeb. Since this number is below 18,980, no Calendar Round can be deducted from it. Dividing the given number by 13, we have: 9,663 ÷ 13 =7434⁄13. Counting the numerator of the fractional part of this quotient, 4,backwardfrom the day coefficient of the starting point,3, we reach 12 as the day coefficient of the terminal date, that is,2, 1, 13, 12(rule 1, p.139). Dividing the given number by 20, we have: 9,663 ÷ 20 = 4833⁄20. Counting the numerator of the fractional part of this quotient, 3,backwardfrom the day sign of the starting point,Imix, in TableI, we reachEznabas the day sign of the terminal date (Ahau, Cauac, Eznab); consequently the day reached in the count will be12 Eznab. Dividing the given number by 365, we have9,663 ÷ 365 = 26173⁄365. Countingbackwardthe numerator of the fractional part of this quotient, 173, from the year position of the starting point,4 Uayeb, the year position of the terminal date will be found to be11 Yax. Before position4 Uayeb(see TableXV) there are 4 positions in that division of the year (3, 2, 1, 0). Counting thesebackwardto the end of the monthCumhu(see TableXV), we have left 169 positions (173-4 = 169); this equals 8 uinals and 9 days extra. Therefore, beginning with the end ofCumhu, we may countbackward8 whole uinals, namely:Cumhu, Kayab, Pax, Muan, Kankin, Mac, Ceh, andZac, which will bring us to the end ofYax(since we are counting backward). As we have left still 9 days out of our original 173, these must be counted backward from position0 Zac, that is, beginning with position19 Yax: 19, 18, 17, 16, 15, 14, 13, 12, 11; so11 Yaxis the position in the year of the day of the terminal date. Assembling the above values, we find that in counting the number 9,663backwardfrom the starting point,2 Imix 4 Uayeb, the terminal date is12 Eznab 11 Yax. Whether the count be forward or backward, the method is the same, the only difference being in the direction of the counting.
This concludes the discussion of the actual arithmetical processes involved in counting forward or backward any given number from any given date; however, before explaining the fifth and final step in deciphering the Maya numbers, it is first necessary to show how this method of counting was applied to the Long Count.
The numbers used above in connection with dates merely express the difference in time between starting points and terminal dates, without assigning either set of dates to their proper positions in Maya chronology; that is, in the Long Count. Consequently, since any Maya date recurred at successive intervals of 52 years, by the time their historic period had been reached, more than 3,000 years after the starting point of their chronology, the Maya had upward of 70 distinct dates of exactly the same name to distinguish from one another.
It was stated on page61that the 0, or starting point of Maya chronology, was the date4 Ahau 8 Cumhu, from which all subsequent dates were reckoned; and further, on page63, that by recording the number of cycles, katuns, tuns, uinals, and kins which had elapsed in each case between this date and any subsequent dates in the Long Count, subsequent dates of the same name could be readily distinguished from one another and assigned at the same time to their proper positions in Maya chronology. This method of fixing a date in the Long Count has been designated Initial-series dating.
The generally accepted method of writing Initial Series is as follows:
9.0.0.0.0.8 Ahau 13 Ceh
9.0.0.0.0.8 Ahau 13 Ceh
9.0.0.0.0.8 Ahau 13 Ceh
The particular Initial-Series written here is to be interpreted thus: "Counting forward 9 cycles, 0 katuns, 0 tuns, 0 uinals, and 0 kinsfrom4 Ahau 8 Cumhu, the starting point of Maya chronology (always unexpressed in Initial Series), the terminal date reached will be8 Ahau 13 Ceh."[102]Or again:
9.14.13.4.17.12 Caban 5 Kayab
9.14.13.4.17.12 Caban 5 Kayab
9.14.13.4.17.12 Caban 5 Kayab
This Inital Series reads thus: "Counting forward 9 cycles, 14 katuns, 13 tuns, 4 uinals, and 17 kins from4 Ahau 8 Cumhu, the starting point of Maya chronology (unexpressed), the terminal date reached will be12 Caban 5 Kayab."
The time which separates any date from4 Ahau 8 Cumhumay be called that date's Initial-series value. For example, in the first of the above cases the number 9.0.0.0.0 is the Initial-series value of the date8 Ahau 13 Ceh, and in the second the number 9.14.13.4.17 is the Initial-series value of the date12 Caban 5 Kayab. It is clear from the foregoing that although the date8 Ahau 13 Ceh, for example, had recurred upward of 70 times since the beginning of their chronology, the Maya were able to distinguish any particular8 Ahau 13 Cehfrom all the others merely by recording its distance from the starting point; in other words, giving thereto its particular Initial-series value, as 9.0.0.0.0. in the present case. Similarly, any particular12 Caban 5 Kayab, by the addition of its corresponding Initial-series value, as 9.14.13.4.17 in the case above cited, was absolutely fixed in the Long Count—that is, in a period of 374,400 years.
Returning now to the question of how the counting of numbers was applied to the Long Count, it is evident thatevery date in Maya chronology, starting points as well as terminal dates, had its own particular Initial-series value, though in many cases these values are not recorded. However, in most of the cases in which the Initial-series values of dates are not recorded, they may be calculated by means of their distances from other dates, whose Initial-series values are known. This adding and subtracting of numbers to and from Initial Series[103]constitutes the application of the above-described arithmetical processes to the Long Count. Several examples of this use are given below.
Let us assume for the first case that the number 2.5.6.1 is to be counted forward from the Initial Series 9.0.0.0.0 8Ahau 13 Ceh. By multiplying the values of the katuns, tuns, uinals, and kins given in TableXIIIby their corresponding coefficients, in this case 2, 5, 6, and 1, respectively, and adding the resulting products together, we find that 2.5.6.1 reduces to 16,321 units of the first order.
Counting this forward from8 Ahau 13 Cehas indicated by the rules on pages138-143, the terminal date1 Imix 9 Yaxkinwill be reached.Moreover, since the Initial-series value of the starting point8 Ahau 13 Cehwas 9.0.0.0.0, the Initial-series value of1 Imix 9 Yaxkin, the terminal date, may be calculated by adding its distance from8 Ahau 13 Cehto the Initial-series value of that date:
9.0.0.0.0 (Initial-series value of starting point)8 Ahau 13 Ceh9.2.5.6.1 (distance from8 Ahau 13 Cehto1 Imix 9 Yaxkin)9.2.5.6.1 (Initial-series value of terminal date)1 Imix 9 Yaxkin
9.0.0.0.0 (Initial-series value of starting point)8 Ahau 13 Ceh9.2.5.6.1 (distance from8 Ahau 13 Cehto1 Imix 9 Yaxkin)9.2.5.6.1 (Initial-series value of terminal date)1 Imix 9 Yaxkin
9.0.0.0.0 (Initial-series value of starting point)8 Ahau 13 Ceh
9.2.5.6.1 (distance from8 Ahau 13 Cehto1 Imix 9 Yaxkin)
9.2.5.6.1 (Initial-series value of terminal date)1 Imix 9 Yaxkin
That is, by calculation we have determined the Initial-series value of the particular1 Imix 9 Yaxkin, which was distant 2.5.6.1 from 9.0.0.0.08 Ahau 13 Ceh, to be 9.2.5.6.1, notwithstanding that this fact was not recorded.
The student may prove the accuracy of this calculation by treating 9.2.5.6.11 Imix 9 Yaxkinas a new Initial Series and counting forward 9.2.5.6.1 from4 Ahau 8 Cumhu, the starting point of all Initial Series known except two. If our calculations are correct, the former date will be reached just as if we had counted forward only 2.5.6.1 from 9.0.0.0.08 Ahau 13 Ceh.
In the above example the distance number 2.5.6.1 and the date1 Imix 9 Yaxkinto which it reaches, together are called a Secondary Series. This method of dating already described (see pp.74-76et seq.) seems to have been used to avoid the repetition of the Initial-series values for all the dates in an inscription. For example, in the accompanying text—
the only parts actually recorded are the Initial Series 9.12.2.0.165 Cib 14 Yaxkin, and the Secondary Series 12.9.15 leading to9 Chuen 9 Kankin; the Secondary Series 5 leading to1 Cib 14 Kankin; and the Secondary Series 1.0.2.5 leading to5 Imix 19 Zac. The Initial-series values: 9.12.14.10.11; 9.12.14.10.16; and 9.13.14.13.1, belonging to the three dates of the Secondary Series, respectively, do not appear in the text at all (a fact indicated by the brackets), but are found only by calculation. Moreover, the student should note that in a succession of interdependent series like the ones just given the terminal date reached by one number, as9 Chuen 9 Kankin, becomes the starting point for the next number, 5. Again, the terminal date reached by counting 5 from9 Chuen 9 Kankin, that is,1 Cib 14 Kankin, becomes the starting point from which the next number, 1.0.2.5, is counted. In other words, these terms are only relative, since the terminal date of one number will be the starting point of the next.
Let us assume for the next example, that the number 3.2 is to be counted forward from the Initial Series 9.12.3.14.05 Ahau 8 Uo. Reducing 3 uinals and 2 kins to kins, we have 62 units of the first order. Counting forward 62 from5 Ahau 8 Uo, as indicated by the rules on pages138-143, it is found that the terminal date will be2 Ik 10 Tzec. Since the Initial-series value of the starting point5 Ahau 8 Uois known, namely, 9.12.3.14.0, the Initial Series corresponding to the terminal date may be calculated from it as before:
The bracketed 9.12.3.17.2 in the Initial-series value corresponding to the date2 Ik 10 Tzecdoes not appear in the record but was reached by calculation. The student may prove the accuracy of this result by treating 9.12.3.17.22 Ik 10 Tzecas a new Initial Series, and counting forward 9.12.3.17.2 from4 Ahau 8 Cumhu(the starting point of Maya chronology, unexpressed in Initial Series). If our calculations are correct, the same date,2 Ik 10 Tzec, will be reached, as though we had counted only 3.2 forward from the Initial Series 9.12.3.14.05 Ahau 8 Uo.
One more example presenting a "backward count" will suffice to illustrate this method. Let us count the number 14.13.4.17backwardfrom the Initial Series 9.14.13.4.1712 Caban 5 Kayab. Reducing 14.13.4.17 to units of the 1st order, we have 105,577. Counting this numberbackwardfrom12 Caban 5 Kayab, as indicated in the rules on pages138-143, we find that the terminal date will be8 Ahau 13 Ceh. Moreover, since the Initial-series value of the starting point12 Caban 5 Kayabis known, namely, 9.14.13.4.17, the Initial-series value ofthe terminal date may be calculated bysubtractingthe distance number 14.13.4.17 from the Initial Series of the starting point:
The bracketed parts are not expressed. We have seen elsewhere that the Initial Series 9.0.0.0.0 has for its terminal date8 Ahau 13 Ceh; therefore our calculation proves itself.
The foregoing examples make it sufficiently clear that the distance numbers of Secondary Series may be used to determine the Initial-series values of Secondary-series dates, either by their addition to or subtraction from known Initial-series dates.
We have come now to the final step in the consideration of Maya numbers, namely, the identification of the terminal dates determined by the calculations given under the fourth step, pages138-143. This step may be summed up as follows:
Fifth Step in Solving Maya Numbers
Find the terminal date to which the number leads.
As explained under the fourth step (pp.138-143), the terminal date may be found by calculation. The above direction, however, refers to the actual finding of the terminal dates in the texts; that is, where to look for them. It may be said at the outset in this connection that terminal dates in the great majority of cases follow immediately the numbers which lead to them. Indeed, the connection between distance numbers and their corresponding terminal dates is far closer than between distance numbers and their corresponding starting points. This probably results from the fact that the closing dates of Maya periods were of far more importance than their opening dates. Time was measured by elapsed periods and recorded in terms of the ending days of such periods. The great emphasis on the closing date of a period in comparison with its opening date probably caused the suppression and omission of the date4 Ahau 8 Cumhu, the starting point of Maya chronology, in all Initial Series. To the same cause also may probably be attributed the great uniformity in the positions of almost all terminal dates, i.e., immediately after the numbers leading to them.
We may formulate, therefore, the following general rule, which the student will do well to apply in every case, since exceptions to it are very rare:
Rule.The terminal date reached by a number or series almost invariably follows immediately the last term of the number or series leading to it.
This applies equally to all terminal dates, whether in Initial Series, Secondary Series, Calendar-round dating or Period-ending dating, though in the case of Initial Series a peculiar division or partition of the terminal date is to be noted.
Throughout the inscriptions, excepting in the case of Initial Series, the month parts of the dates almost invariably follow immediately the days whose positions in the year they designate, without any other glyphs standing between; as, for example,8 Ahau 13 Ceh,12 Caban 5 Kayab, etc. In Initial Series, on the other hand, the day parts of the dates, as8 Ahauand12 Caban, in the above examples, are almost invariably separated from their corresponding month parts,13 Cehor5 Kayab, by several intervening glyphs. The positions of the day parts in Initial-series terminal dates are quite regular according to the terms of the above rule; that is, they follow immediately the lowest period of the number which in each case shows their distance from the unexpressed starting point,4 Ahau 8 Cumhu. The positions of the corresponding month parts are, on the other hand, irregular. These, instead of standing immediately after the days whose positions in the year they designate, follow at the close of some six or seven intervening glyphs. These intervening glyphs have been called the Supplementary Series, though the count which they record has not as yet been deciphered.[105]The month glyph in the great majority of cases follows immediately the closing[106]glyph of the Supplementary Series. The form of this latter sign is always unmistakable (see fig.65), and it is further characterized by its numerical coefficient, which can never be anything but 9 or 10.[107]See examples of this sign in the figure just mentioned, where both normal formsa, c, e, g,andhand head variantsb, d,andfare included.
The student will find this glyph exceedingly helpful in locating the month parts of Initial-series terminal dates in the inscriptions. For example, let us suppose in deciphering the Initial Series 9.16.5.0.08 Ahau 8 Zotzthat the number 9.16.5.0.0 has been counted forwardfrom4 Ahau 8 Cumhu(the unexpressed starting point), and has been found by calculation to reach the terminal date8 Ahau 8 Zotz; and further, let us suppose that on inspecting the text the day part of this date (8 Ahau) has been found to be recorded immediately after the 0 kins of the number 9.16.5.0.0. Now, if the student will follow the next six or seven glyphs until he finds one like any of the forms in figure65, the glyph immediately following the latter sign will be in all probability the month part,8 Zotzin the above example, of an Initial-series' terminal date. In other words, although the meaning of the glyph shown in the last-mentioned figure is unknown, it is important for the student to recognize its form, since it is almost invariably the "indicator" of the month sign in Initial Series.
Fig. 65Fig. 65. Sign for the "month indicator":a, c, e, g, h, Normal forms;b, d, f, head variants.
Fig. 65. Sign for the "month indicator":a, c, e, g, h, Normal forms;b, d, f, head variants.
In all other cases in the inscriptions, including also the exceptions to the above rule, that is, where the month parts of Initial-series terminal dates do not immediately follow the closing glyph of the Supplementary Series, the month signs follow immediately the day signs whose positions in the year they severally designate.
In the codices the month signs when recorded[108]usually follow immediately the days signs to which they belong. The most notable exception[109]to this general rule occurs in connection with the Venus-solar periods represented on pages 46-50 of the Dresden Codex, where one set of day signs is used with three different sets of month signs to form three different sets of dates. For example, in one place the day2 Ahaustands above three different month signs—3 Cumhu, 3 Zotz,and13 Yax—with each of which it is used to form adifferent date—2 Ahau 3 Cumhu, 2 Ahau 3 Zotz,and2 Ahau 13 Yax. In these pages the month signs, with a few exceptions, do not follow immediately the days to which they belong, but on the contrary they are separated from them by several intervening glyphs. This abbreviation in the record of these dates was doubtless prompted by the desire or necessity for economizing space. In the above example, instead of repeating the2 Ahauwith each of the two lower month signs,3 Zotzand13 Yax, by writing it once above the upper month sign,3 Cumhu, the scribe intended that it should be used in turn with each one of the three month signs standing below it, to form three different dates, saving by this abbreviation the space of two glyphs, that is, double the space occupied by2 Ahau.
With the exception of the Initial-series dates in the inscriptions and the Venus-Solar dates on pages 46-50 of the Dresden Codex, we may say that the regular position of the month glyphs in Maya writing was immediately following the day glyphs whose positions in the year they severally designated.
In closing the presentation of this last step in the process of deciphering numbers in the texts, the great value of the terminal date as a final check for all the calculations involved under steps 1-4 (pp.134-151) should be pointed out. If after having worked out the terminal date of a given number according to these rules the terminal date thus found should differ from that actually recorded under step 5, we must accept one of the following alternatives:
1. There is an error in our own calculations; or2. There is an error in the original text; or3. The case in point lies without the operation of our rules.
1. There is an error in our own calculations; or
1. There is an error in our own calculations; or
2. There is an error in the original text; or
2. There is an error in the original text; or
3. The case in point lies without the operation of our rules.
3. The case in point lies without the operation of our rules.
It is always safe for the beginner to proceed on the assumption that the first of the above alternatives is the cause of the error; in other words, that his own calculations are at fault. If the terminal date as calculated does not agree with the terminal-date as recorded, the student should repeat his calculations several times, checking up each operation in order to eliminate the possibility of a purely arithmetical error, as a mistake in multiplication. After all attempts to reach the recorded terminal date by counting the given number from the starting point have failed, the process should be reversed and the attempt made to reach the starting point by counting backward the given number from its recorded terminal date. Sometimes this reverse process will work out correctly, showing that there must be some arithmetical error in our original calculations which we have failed to detect. However, when both processes have failed several times to connect the starting point with the recorded terminal date by use of the given number, there remains the possibility that either the starting point or the terminal date, or perhaps both, do not belong to the given number. The rules for determining this facthave been given under step 2, page135, and step 4, page138. If after applying these to the case in point it seems certain that the starting point and terminal date used in the calculations both belong to the given number, we have to fall back on the second of the above alternatives, that is, that there is an error in the original text.
Although very unusual, particularly in the inscriptions, errors in the original texts are by no means entirely unknown. These seem to be restricted chiefly to errors in numerals, as the record of 7 for 8, or 7 for 12 or 17, that is, the omission or insertion of one or more bars or dots. In a very few instances there seem to be errors in the month glyph. Such errors usually are obvious, as will be pointed out in connection with the texts in which they are found (see Chapters V and VI).
If both of the above alternatives are found not to apply, that is, if both our calculations and the original texts are free from error, we are obliged to accept the third alternative as the source of trouble, namely, that the case in point lies without the operation of our rules. In such cases it is obviously impossible to go further in the present state of our knowledge. Special conditions presented by glyphs whose meanings are unknown may govern such cases. At all events, the failure of the rules under 1-4 to reach the terminal dates recorded as under 5 introduces a new phase of glyph study—the meaning of unknown forms with which the beginner has no concern. Consequently, when a text falls without the operation of the rules given in this chapter—a very rare contingency—the beginner should turn his attention elsewhere.
ChapterV
THE INSCRIPTIONS
The present chapter will be devoted to the interpretation of texts drawn from monuments, a process which consists briefly in the application to the inscriptions[110]of the material presented in Chapters III and IV.
Fig. 66Fig. 66. Diagram showing the method of designating particular glyphs in a text.
Fig. 66. Diagram showing the method of designating particular glyphs in a text.
Before proceeding with this discussion it will first be necessary to explain the method followed in designating particular glyphs in a text. We have seen (p.23) that the Maya glyphs were presented in parallel columns, which are to be read two columns at a time, the order of the individual glyph-blocks[111]in each pair of columns being from left to right and from top to bottom. For convenience in referring to particular glyphs in the texts, the vertical columns of glyph-blocks are lettered from left to right, thus, A, B, C, D, etc., and the horizontal rows numbered from top to bottom, thus, 1, 2, 3, 4, etc. For example, in figure66the glyph-blocks in columns A and B are read together from left to right and top to bottom, thus, A1 B1, A2 B2, A3 B3, etc. When glyph-block B10 is reached the next in order is C1, which is followed by D1, C2 D2, C3 D3, etc. Again, when D10 is reached the next in order is E1, which is followed by F1, E2 F2, E3 F3, etc. In this way the order of reading proceeds from left to right and from top to bottom, in pairs of columns, that is, G H, I J, K L, and M N throughout the inscription, and usually closes with the glyph-block in the lower right-hand corner, as N10 in figure66. By this simple system of coordinates any particular glyph in a text may be readily referred to when the need arises. Thus, for example, in figure66glyphαis referred to as D3; glyphβas F6; glyphγas K4; glyphδas N10. In a few texts the glyph-blocks are so irregularly placed that it is impracticable to designate them by the above coordinates. In such cases the order of the glyph-blocks will be indicated by numerals, 1, 2, 3, etc. In two Copan texts, Altar S (fig.81) and Stela J (pl.15), made from the drawings of Mr. Maudslay, his numeration of the glyphs has been followed. This numeration appears in these two figures.
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 6
Plate 6GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR AND DOT NUMERALS AND NORMAL-FORM PERIOD GLYPHS
GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR AND DOT NUMERALS AND NORMAL-FORM PERIOD GLYPHS
Texts Recording Initial Series
Because of the fundamental importance of Initial Series in the Maya system of chronology, the first class of texts represented will illustrate this method of dating. Moreover, since the normal forms for the numerals and the period glyphs will be more easily recognised by the beginner than the corresponding head variants, the first Initial Series given will be found to have all the numerals and period glyphs expressed by normal forms.[112]
In plate6is figured the drawing of the Initial Series[113]from Zoömorph P at Quirigua, a monument which is said to be the finest piece of aboriginal sculpture in the western hemisphere. Our text opens with one large glyph, which occupies the space of four glyph-blocks, A1-B2.[114]Analysis of this form shows that it possesses all the elements mentioned on page65as belonging to the so-called Initial-series introducing glyph, without which Initial Series never seem to have been recorded in the inscriptions. These elements are: (1) the trinalsuperfix, (2) the pair of comblike lateral appendages, (3) the normal form of the tun sign, (4) the trinal subfix, and (5) the variable central element. As stated above, all these appear in the large glyph A1-B2. Moreover, a comparison of A1-B2 with the introducing glyphs given in figure24shows that these forms are variants of one and the same sign. Consequently, in A1-B2 we have recorded an Initial-series introducing glyph. The use of this sign is so highly specialized that, on the basis of its occurrence alone in a text, the student is perfectly justified in assuming that an Initial Series will immediately follow.[115]Exceptions to this rule are so very rare (see p.67) that the beginner will do well to disregard them altogether.
The next glyph after the introducing glyph in an Initial Series is the cycle sign, the highest period ever found in this kind of count[116]. The cycle sign in the present example appears in A3 with the coefficient 9 (1 bar and 4 dots). Although the period glyph is partially effaced in the original enough remains to trace its resemblance to the normal form of the cycle sign shown in figure25,a-c. The outline of the repeated Cauac sign appears in both places. We have then, in this glyph, the record of 9 cycles[117]. The glyph following the cycle sign in an Initial Series is always the katun sign, and this should appear in B3, the glyph next in order. This glyph is quite clearly the normal form of the katun sign, as a comparison of it with figure27,a, b, the normal form for the katun, will show. It has the normal-form numeral 18 (3 bars and 3 dots) prefixed to it, and this whole glyph therefore signifies 18 katuns. The next glyph should record the tuns, and a comparison of the glyph in A4 with the normal form of the tun sign in figure29,a, b, shows this to be the case. The numeral 5 (1 bar prefixed to the tun sign) shows that this period is to be used 5 times; that is, multiplied by 5. The next glyph (B4) should be the uinal sign, and a comparison of B4 with figure31,a-c, the normal form of the uinal sign, shows the identity of these two glyphs. The coefficient of the uinal sign contains as its most conspicuous element the clasped hand, which suggests that we may have 0 uinals recorded in B4. A comparison of this coefficient with the sign for zero in figure54proves this to be the case. The next glyph (A5) should be the kin sign, the lowest period involved in recording Initial Series. A comparison of A5 with the normal form of the kin sign in figure34,a, shows that these two forms are identical. The coefficient of A5 is, moreover, exactly like the coefficient of B4, which, we have seen, meant zero, hence glyph A5 stands for 0 kins. Summarizing the above, we may say that glyphs A3-A5 record an Initial-series number consisting of 6 cycles, 18 katuns, 5 tuns, 0 uinals, and 0 kins, which we may write thus: 9.18.5.0.0 (see p.138, footnote 1).
Now let us turn to Chapter IV and apply the several steps there given, by means of which Maya numbers may be solved. The first step on page134was to reduce the given number, in this case 9.18.5.0.0, to units of the first order; this may be done by multiplying the recorded coefficients by the numerical values of the periods to which they are respectively attached. These values are given in TableXIII, and the sum of the products arising from their multiplication by the coefficients recorded in the Initial Series in plate6, A are given below:
Therefore 1,427,400 will be the number used in the following calculations.
The second step (see step 2, p.135) is to determine the starting point from which this number is counted. According to rule 2, page136, if the number is an Initial Series the starting point, although never recorded, is practically always the date4 Ahau 8 Cumhu. Exceptions to this rule are so very rare that they may be disregarded by the beginner, and it may be taken for granted, therefore, in the present case, that our number 1,427,400 is to be counted from the date4 Ahau 8 Cumhu.
The third step (see step 3, p. £136) is to determine the direction of the count, whether forward or backward. In this connection it was stated that the general practice is to count forward, and that the student should always proceed upon this assumption. However, in the present case there is no room for uncertainty, since the direction of the count in an Initial Series is governed by an invariable rule. In Initial Series, according to the rule on page137, the count is always forward, consequently 1,427,400 is to be countedforwardfrom4 Ahau 8 Cumhu.
The fourth step (see step 4, p.138) is to count the given number from its starting point; and the rules governing this process will be found on pages139-143. Since our given number (1,427,400) is greater than 18,980, or 1 Calendar Round, the preliminary rule on page143applies in the present case, and we may therefore subtract from 1,427,400 all the Calendar Rounds possible before proceeding to count it from the starting point. By referring to TableXVI, it appears that 1,427,400 contains 75 complete Calendar Rounds, or 1,423,500; hence, the latter number may be subtractedfrom 1,427,400 without affecting the value of the resulting terminal date: 1,427,400-1,423,500 = 3,900. In other words, in counting forward 3,900 from4 Ahau 8 Cumhu, the same terminal date will be reached as though we had counted forward 1,427,400.[118]
In order to find the coefficient of the day of the terminal date, it is necessary, by rule 1, page139, to divide the given number or its equivalent by 13; 3,900 ÷ 13 = 300. Now since there is no fractional part in the resulting quotient, the numerator of an assumed fractional part will be 0; counting forward 0 from the coefficient of the day of the starting point, 4 (that is,4 Ahau 8 Cumhu), we reach 4 as the coefficient of the day of the terminal date.
In order to find the day sign of the terminal date, it is necessary, under rule 2, page140, to divide the given number or its equivalent by 20; 3,900 ÷ 20 = 195. Since there is no fractional part in the resulting quotient, the numerator of an assumed fractional part will be 0; counting forward 0 in TableI, fromAhau, the day sign of the starting point (4 Ahau 8 Cumhu), we reachAhauas the day sign of the terminal date. In other words, in counting forward either 3,900 or 1,427,400 from4 Ahau 8 Cumhu, the day reached will be4 Ahau. It remains to show what position in the year this day4 Ahaudistant 1,427,400 from the date4 Ahau 8 Cumhu, occupied.
In order to find the position in the year which the day of the terminal date occupied, it is necessary, under rule 3, page141, to divide the given number or its equivalent by 365; 3,900 ÷ 365 = 10250⁄365. Since the numerator of the fractional part of the resulting quotient is 250, to reach the year position of the day of the terminal date desired it is necessary to count 250 forward from8 Cumhu, the year position of the day of the starting point4 Ahau 8 Cumhu. It appears from TableXV, in which the 365 positions of the year are given, that after position8 Cumhuthere are only 16 positions in the year—11 more inCumhuand 5 inUayeb. These must be subtracted, therefore, from 250 in order to bring the count to the end of the year; 250-16 = 234, so 234 is the number of positions we must count forward in the new year. It is clear that the first 11 uinals in the year will use up exactly 220 of our 234 positions (11 × 20 = 220), and that 14 positions will be left, which must be counted in the next uinal, the 12th. But the 12th uinal of the year isCeh(see TableXV); counting forward 14 positions inCeh, we reach13 Ceh, which is, therefore, the month glyph of our terminal date. In other words, counting 250 forward from8 Cumhu, position13 Cehis reached. Assembling the above values, we find that by calculation we have determined the terminal date of the Initial Series in plate6,A, to be4 Ahau 13 Ceh.
At this point there are several checks which the student may apply to his result in order to test the accuracy of his calculations; for instance, in the present example if 115, the difference between 365 and 250 (115 + 250 = 365) is counted forward from position13 Ceh, position8 Cumhuwill be reached if our calculations were correct. This is true because there are only 365 positions in the year, and having reached13 Cehin counting forward 250 from8 Cumhu, counting the remaining 115 days forward from day reached by 250, that is,13 Ceh, we should reach our starting point (8 Cumhu) again. Another good check in the present case would be to countbackward250 from13 Ceh; if our calculations have been correct, the starting point8 Cumhuwill be reached. Still another check, which may be applied is the following: From TableVIIit is clear that the day signAhaucan occupy only positions 3, 8, 13, or 18 in the divisions of the year;[119]hence, if in the above case the coefficient ofCehhad been any other number but one of these four, our calculations would have been incorrect.
We come now to the final step (see step 5, p.151), the actual finding of the glyphs in our text which represent the two parts of the terminal date—the day and its corresponding position in the year. If we have made no arithmetical errors in calculations and if the text itself presents no irregular and unusual features, the terminal date recorded should agree with the terminal date obtained by calculation.
It was explained on page152that the two parts of an Initial-series terminal date are usually separated from each other by several intervening glyphs, and further that, although the day part follows immediately the last period glyph of the number (the kin glyph), the month part is not recorded until after the close of the Supplementary Series, usually a matter of six or seven glyphs. Returning to our text (pl.6,A), we find that the kins are recorded in A5, therefore the day part of the terminal date should appear in B5. The glyph in B5 quite clearly records the day4 Ahauby means of 4 dots prefixed to the sign shown in figure16,e'-g', which is the form for the day nameAhau, thereby agreeing with the value of the day part of the terminal date as determined by calculation. So far then we have read our text correctly. Following along the next six or seven glyphs, A6-C1a, which record the Supplementary Series,[120]we reach in C1a a sign similar to the forms shown in figure65. This glyph, which always has a coefficient of 9 or 10, was designated on page152the month-sign "indicator," since it usually immediately precedes the month sign in Initial-series terminal dates. In C1a it has the coefficient 9 (4 dots and 1 bar) and is followed in C1b by the month partof the terminal date,13 Ceh. The bar and dot numeral 13 appears very clearly above the month sign, which, though partially effaced, yet bears sufficient resemblance to the sign forCehin figure19,u, v,to enable us to identify it as such.
Our complete Initial Series, therefore, reads: 9.18.5.0.04 Ahau 13 Ceh, and since the terminal date recorded in B5, C1b agrees with the terminal date determined by calculation, we may conclude that this text is without error and, furthermore, that it records a date,4 Ahau 13 Ceh, which was distant 9.18.5.0.0 from the starting point of Maya chronology. The writer interprets this text as signifying that 9.18.5.0.04 Ahau 13 Cehwas the date on which Zoömorph P at Quirigua was formally consecrated or dedicated as a time-marker, or in other words, that Zoömorph P was the monument set up to mark the hotun, or 5-tun period, which came to a close on the date 9.18.5.0.04 Ahau 13 Cehof Maya chronology.[121]
In plate6,B, is figured a drawing of the Initial Series on Stela 22 at Naranjo.[122]The text opens in A1 with the Initial-series introducing glyph, which is followed in B1 B3 by the Initial-series number 9.12.15.13.7. The five period glyphs are all expressed by their corresponding normal forms, and the student will have no difficulty in identifying them and reading the number, as above recorded.
By means of TableXIIIthis number may be reduced to units of the 1st order, in which form it may be more conveniently used. This reduction, which forms the first step in the process of solving Maya numbers (see step 1, p.134), follows:
And 1,388,067 will be the number used in the following calculations.
The next step is to find the starting point from which 1,388,067 is counted (see step 2, p.135). Since this number is an Initial Series, in all probability its starting point will be the date4 Ahau 8 Cumhu; at least it is perfectly safe to proceed on that assumption.
The next step is to find the direction of the count (see step 3, p.136); since our number is an Initial Series, the count can only be forward (see rule 2, p.137).[123]
Having determined the number to be counted, the starting point from which the count commences, and the direction of the count, we may now proceed with the actual process of counting (see step 4, p.138).
Since 1,388,067 is greater than 18,980 (1 Calendar Round), we may deduct from the former number all the Calendar Rounds possible (see preliminary rule, page143). According to TableXVIit appears that 1,388,067 contains 73 Calendar Rounds, or 1,385,540; after deducting this from the given number we have left 2,527 (1,388,067-1,385,540), a far more convenient number to handle than 1,388,067.
Applying rule 1 (p.139) to 2,527, we have: 2,527 ÷ 13 = 1945⁄13, and counting forward 5, the numerator of the fractional part of the quotient, from 4, the day coefficient of the starting point,4 Ahau 8 Cumhu, we reach 9 as the day coefficient of the terminal date.
Applying rule 2 (p.140) to 2,527, we have: 2,527 ÷ 20 = 1267⁄20; and counting forward 7, the numerator of the fractional part of the quotient, fromAhau, the day sign of our starting point,4 Ahau 8 Cumhu, in TableI, we reachManikas the day sign of the terminal date. Therefore, the day of the terminal date will be9 Manik.
Applying rule 3 (p.141) to 2,527, we have: 2,527 ÷ 365 = 6337⁄365; and counting forward 337, the numerator of the fractional part of the quotient, from8 Cumhu, the year position of the starting point,4 Ahau 8 Cumhu, in TableXV, we reach0 Kayabas the year position of the terminal date. The calculations by means of which0 Kayabis reached are as follows: After8 Cumhuthere are 16 positions in the year, which we must subtract from 337; 337-16 = 321, which is to be counted forward in the new year. This number contains just 1 more than 16 uinals, that is, 321 = (16 × 20) + 1; hence it will reach through the first 16 uinals in TableXVand to the first position in the 17th uinal,0 Kayab. Combining this with the day obtained above, we have for our terminal date determined by calculation,9 Manik 0 Kayab.
The next and last step (see step 5, p.151) is to find the above date in the text. In Initial Series (see p.152) the two parts of the terminal date are generally separated, the day part usually following immediately the last period glyph and the month part the closing glyph of the Supplementary Series. In plate6,B, the last period glyph, as we have seen, is recorded in B3; therefore the day should appear in A4. Comparing the glyph in A4 with the sign forManikin figure16,j, the two forms are seen to be identical. Moreover, A4 has the bar and dot coefficient 9 attached to it, that is, 4 dots and 1 bar; consequently it is clear that in A4 we have recorded the day9 Manik, the same day as reached by calculation. For some unknown reason, at Naranjo the month glyphs of the Initial-series terminal dates do not regularly follow the closing glyphs of the Supplementary Series;indeed, in the text here under discussion, so far as we can judge from the badly effaced glyphs, no Supplementary Series seems to have been recorded. However, reversing our operation, we know by calculation that the month part should be0 Kayab, and by referring to figure49we find the only form which can be used to express the 0 position with the month signs—the so-called "spectacles" glyph—which must be recorded somewhere in this text to express the idea 0 with the month signKayab. Further, by referring to figure19,d'-f', we may fix in our minds the sign for the monthKayab, which should also appear in the text with one of the forms shown in figure49.
Returning to our text once more and following along the glyphs after the day in A4, we pass over B4, A5, and B5 without finding a glyph resembling one of the forms in figure49joined to figure19,d'-f'; that is,0 Kayab. However, in A6 such a glyph is reached, and the student will have no difficulty in identifying the month sign withd'-f'in the above figure. Consequently, we have recorded in A4, A6 the same terminal date,9 Manik 0 Kayab, as determined by calculation, and may conclude, therefore, that our text records without error the date 9.12.15.13.79 Manik 0 Kayab[124]of Maya chronology.
The next text presented (pl.6, C) shows the Initial Series from Stela I at Quirigua.[125]Again, as in plate6, A, the introducing glyph occupies the space of four glyph-blocks, namely, A1-B2. Immediately after this, in A3-A4, is recorded the Initial-series number 9.18.10.0.0, all the period glyphs and coefficients of which are expressed by normal forms. The student's attention is called to the form for 0 used with the uinal and kin signs in A4a and A4b, respectively, which differs from the form for 0 recorded with the uinal and kin signs in plate6, A, B4, and A5, respectively. In the latter text the 0 uinals and 0 kins were expressed by the hand and curl form for zero shown in figure54; in the present text, however, the 0 uinals and 0 kins are expressed by the form for 0 shown in figure47, a new feature.
Reducing the above number to units of the 1st order by means of TableXIII, we have:
Deducting from this number all the Calendar Rounds possible, 75(see TableXVI), it may be reduced to 5,700 without affecting its value in the present connection.
Applying rules 1 and 2 (pp.139and140, respectively) to this number, the day reached will be found to be10 Ahau; and by applying rule 3 (p.141), the position of this day in the year will be found to be8 Zac. Therefore, by calculation we have determined that the terminal date reached by this Initial Series is10 Ahau 8 Zac. It remains to find this date in the text. The regular position for the day in Initial-series terminal dates is immediately following the last period glyph, which, as we have seen above, was in A4b. Therefore the day glyph should be B4a. An inspection of this latter glyph will show that it records the day10 Ahau, both the day sign and the coefficient being unusually clear, and practically unmistakable. Compare B4a with figure16,e'-g', the sign for the day nameAhau. Consequently the day recorded agrees with the day determined by calculation. The month glyph in this text, as mentioned on page157, footnote 1, occurs out of its regular position, following immediately the day of the terminal date.