An Introduction to the Study of the Maya Hieroglyphs

We have recorded here four different dates, of which the last, 9.13.14.13.1 5 Imix 19 Zac, probably represents the actual date, or very near thereto, of this monument.^[190] The period covered between the first and last of these dates is about 32 years, within the range of a single lifetime or, indeed, of the tenure of some important office by a single individual. The unknown glyphs again probably set forth the nature of the events which occurred on the dates recorded.

In the two preceding texts the Secondary Series given are regular in every way. Not only was the count forward each time, but it also started in every case from the date immediately preceding the number counted. This regularity, however, is far from universal in Secondary-series texts, and the following examples comprise some of the more common departures from the usual practice.

In plate 18 is figured the Initial Series from Stela K at Quirigua.^[191] The text opens on the north side of this monument (see pl. 18, A) with the introducing glyph in A1-B2. This is followed by the Initial-series number 9.18.15.0.0 in A3-B4, which leads to the terminal date 3 Ahau 3 Yax. The day part of this date the student will find recorded in its regular position, A5a. Passing over A5b and B5, the meanings of which are unknown, we reach in A6 a Secondary-series number composed very clearly of 10 uinals and 10 kins (10.10), which reduces to the following number of units of the first order:

The first assumption is that this number is counted forward from the terminal date of the Initial Series, 3 Ahau 3 Yax, and performing the operations indicated in rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) the terminal date reached will be 5 Oc 8 Uo. Now, although the day sign in B6b is clearly Oc (see fig. 16, o-q), its coefficient is very clearly 1, not 5, and, moreover, the month in A7a is unmistakably 18 Kayab (see fig. 19, d'-f'). Here then instead of finding the date determined by calculation, 5 Oc 8 Uo, the date recorded is 1 Oc 18 Kayab, and consequently there is some departure from the practices heretofore encountered.

Since the association of the number 10.10 is so close with (1) the terminal date of the Initial Series, 3 Ahau 3 Yax, and (2) the date 1 Oc 18 Kayab almost immediately following it, it would almost seem as though these two dates must be the starting point and terminal date, respectively, of this number. If the count is forward, we have just proved that this can not be the case; so let us next count the number backward and see whether we can reach the date recorded in B6b-A7a (1 Oc 18 Kayab) in this way.

Counting 210 backward from 3 Ahau 3 Yax, according to rules 1, 2, and 3 (pp. 139, 140, and 141, respectively), the terminal date reached will be 1 Oc 18 Kayab, as recorded in B6b-A7. In other words, the Secondary Series in this text is counted backward from the Initial Series, and therefore precedes it in point of time. This will appear from the Initial-series value of 1 Oc 18 Kayab, which may be determined by calculation:

This text closes on the south side of the monument in a very unusual manner (see pl. 18, B). In B3a appears the month-sign indicator, here recorded as a head variant with a coefficient 10, and following immediately in B3b a Secondary-series number composed of 0 uinals and 0 kins, or, in other words, nothing. It is obvious that in counting this number 0.0, or nothing, either backward or forward from the date next preceding it in the text, 1 Oc 18 Kayab in B6b-A7a on the north side of the stela, the same date 1 Oc 18 Kayab will remain. But this date is not repeated in A4, where the terminal date of this Secondary Series, 0.0, seems to be recorded. However, if we count 0.0 from the terminal date of the Initial Series, 3 Ahau 3 Yax, we reach the date recorded in A4, 3 Ahau 3 Yax,^[192] and this whole text so far as deciphered will read:

The reason for recording a Secondary-series number equal to zero, the writer believes, was because the first Secondary-series date 1 Oc 18 Kayab precedes the Initial-series date, which in this case marks the time at which this monument was erected. Hence, in order to have the closing date on the monument record the contemporaneous time of the monument, it was necessary to repeat the Initial-series date; this was accomplished by adding to it a Secondary-series date denoting zero. Stela K is the next to the latest hotun-marker at Quirigua following immediately Stela I, the Initial series of which marks the hotun ending 9.18.10.0.0 10 Ahau 8 Zac (see pl. 6, C).

Mr. Bowditch (1910: p. 208) has advanced a very plausible explanation to account for the presence of the date 9.18.14.7.10 1 Oc 18 Kayab on this monument. He shows that at the time when Stela K was erected, namely, 9.18.15.0.0 3 Ahau 3 Yax, the official calendar had outrun the seasons by just 210 days, or exactly the number of days recorded in A6, plate 18, A (north side); and further, that instead of being the day 3 Yax, which occurred at Quirigua about the beginning of the dry season,^[193] in reality the season was 210 days behind, or at 18 Kayab, about the beginning of the rainy season. This very great discrepancy between calendar and season could not have escaped the notice of the priests, and the 210 days recorded in A6 may well represent the days actually needed on the date 9.18.15.0.0 3 Ahau 3 Yax to bring the calendar into harmony with the current season. If this be true, then the date 9.18.14.7.0 1 Oc 18 Kayab represented the day indicated by the sun when the calendar showed that the 3d hotun in Katun 18 of Cycle 9 had been completed. Mr. Bowditch suggests the following free interpretation of this passage: "The sun has just set at its northern point^[194] and we are counting the day 3 Yax—210 days from 18 Kayab—which is the true date in the calendar according to our traditions and records for the sun to set at this point on his course." As stated above, the writer believes this to be the true explanation of the record of 210 days on this monument.

In figures 78 and 79 are illustrated the Initial Series and Secondary Series from Stela J at Quirigua.^[195] For lack of space the introducing glyph in this text has been omitted; it occupies the position of six glyph-blocks, however, A1-B3, after which the Initial-series number 9.16.5.0.0 follows in A4-B8. This leads to the terminal date 8 Ahau 8 Zotz, which is recorded in A9, B9, B13, the glyph in A13 being the month-sign indicator here shown with the coefficient 9. Compare B9 with the second variant for Ahau in figure 16 h', i', and B13 with the sign for Zotz in figure 19, e, f. The Initial-series part of this text therefore in A1-B9, B13, is perfectly regular and reads as follows: 9.16.5.0.0 8 Ahau 8 Zotz. The Secondary Series, however, are unusual and differ in several respects from the ones heretofore presented.

The first Secondary Series inscribed on this monument (see fig. 79, A) is at B1-B2. This series the student should readily decipher as 3 kins, 13 uinals, 11 tuns, and 0 katuns, which we may write 0.11.13.3. This number presents one feature, which, so far as the writer knows, is unique in the whole range of Maya texts. The highest order of units actually involved in this number is the tun, but for some unknown reason the ancient scribe saw fit to add the katun sign also, B2, which, however, he proceeded to nullify at once by attaching to it the coefficient 0. For in so far as the numerical value is concerned, 11.13.3 and 0.11.13.3 are equal. The next peculiarity is that the date which follows this number in B3-A4 is not its terminal date, as we have every reason to expect, but, on the contrary, its starting point. In other words, in this Secondary Series the starting point follows instead of precedes the number counted from it. This date is very clearly 12 Caban 5 Kayab; compare B3 with the sign for Caban in figure 16, a', b', and A4 with the sign for Kayab in figure 19, d'-f'. So far as Stela J is concerned there is no record of the position which this date occupied in the Long Count; that is, there are no data by means of which its Initial Series may be calculated. Elsewhere at Quirigua, however, this date is recorded twice as an Initial Series and in each place it has the same value, 9.14.13.4.17. We may safely conclude, therefore, that the date in A3-B4 is 9.14.13.4.17 12 Caban 5 Kayab, and use it in our calculations as such. Reducing 0.11.13.3 to units of the first order, we have:

Applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) to this number, the terminal date reached will be 10 Ahau 8 Chen, which is nowhere recorded in the text (see fig. 79, A).

The Initial Series corresponding to this date, however, may be calculated from the Initial Series which we have assigned to the date 12 Caban 5 Kayab:

Although the date 9.15.5.0.0 10 Ahau 8 Chen is not actually recorded at Quirigua, it is reached on another monument by calculation just as here. It has a peculiar fitness here on Stela J in that it is just one katun earlier than the Initial Series on this monument (see fig. 78), 9.16.5.0.0 8 Ahau 8 Zotz.

The other Secondary Series on this monument (see fig. 79, B) appears at B1-A2, and records 18 tuns, 3 uinals, and 14 kins, which we may write thus: 18.3.14. As in the preceding case, the date following this number in B2-A3 is its starting point, not its terminal date, a very unusual feature, as has been explained. This date is 6 Cimi 4 Tzec—compare B2 with the sign for Cimi in figure 16, h, i, and A3 with the sign for Tzec in figure 19, g, h—and as far as Stela J is concerned it is not fixed in the Long Count. However, elsewhere at Quirigua this date is recorded in a Secondary Series, which is referred back to an Initial Series, and from this passage its corresponding Initial Series is found to be 9.15.6.14.6 6 Cimi 4 Tzec. Reducing the number recorded in B1-A2, 18.3.14, to units of the first order, we have:

Applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) to the number, the terminal date reached will be 8 Ahau 8 Zotz, which does not appear in figure 79, B. The Initial Series corresponding to this date may be calculated as follows:

But this was the Initial Series recorded on the reverse of this monument, consequently the Secondary-series dates, both of which have preceded the Initial-series date in point of time, bring this count up to the contemporaneous time of this monument, which was 9.16.5.0.0 8 Ahau 8 Zotz. In view of the fact that the Secondary Series on Stela J are both earlier than the Initial Series, the chronological sequence of the several dates is better preserved by regarding the Initial Series as being at the close of the inscription instead of at the beginning, thus:

By the above arrangement all the dates present in the text lead up to 9.16.5.0.0 8 Ahau 8 Zotz as the most important date, because it alone records the particular hotun-ending which Stela J marks. The importance of this date over the others is further emphasized by the fact that it alone appears as an Initial Series.

The text of Stela J illustrates two points in connection with Secondary Series which the student will do well to bear in mind: (1) The starting points of Secondary-series numbers do not always precede the numbers counted from them, and (2) the terminal dates and starting points are not always both recorded.

The former point will be illustrated in the following example:

In plate 19, A, is figured the Initial Series from the west side of Stela F at Quirigua.^[198] The introducing glyph appears in A1-B2 and is followed by the Initial-series number in A3-A5. This is expressed by head variants and reads as follows: 9.14.13.4.17. The terminal date reached by this number is 12 Caban 5 Kayab, which is recorded in B5-A6. The student will readily identify the numerals as above by comparing them with the forms in figures 51-53, and the day and month signs by comparing them with figures 16, a', b', and 19, d'-f', respectively. The Initial Series therefore reads 9.14.13.4.17 12 Caban 5 Kayab.^[199]

Passing over B6-A10, the meanings of which are unknown, we reach in B10 the Secondary-series number 13.9.9. Reducing this to units of the first order, we have:

Assuming that our starting point is the date next preceding this number in the text, that is, the Initial-series terminal date 12 Caban 5 Kayab in B5-A6, and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively), the terminal day reached will be 6 Cimi 4 Tzec. This date the student will find recorded in plate 19, B, B11b-A12a. Compare B11b with the sign for Cimi in figure 16, h, i, and A12a with the sign for Tzec in figure 19, g, h. Moreover, since the Initial-series value of the starting point 12 Caban 5 Kayab is known, the Initial-series value of the terminal date 6 Cimi 4 Tzec may be calculated from it:

In A15 is recorded the date 3 Ahau 3 Mol (compare A15a with fig. 16, k', i', and A15b with fig. 19, m, n) and in A17 the date 4 Ahau 13 Yax (compare A17a with fig. 16, e'-g' and A17b with fig. 19, q, r). This latter date, 4 Ahau 13 Yax, is recorded elsewhere at Quirigua in a Secondary Series attached to an Initial Series, where it has the Initial-series value 9.15.0.0.0. This value we may assume, therefore, belongs to it in the present case, giving us the full date 9.15.0.0.0 4 Ahau 13 Yax. For the present let us pass over the first of these two dates, namely, 3 Ahau 3 Mol, the Initial Series of which as well as the reason for its record here will better appear later.

In B17-A18a is recorded another Secondary-series number composed of 3 kins, 13 uinals, 16 tuns, and 1 katun, which we may write thus: 1.16.13.3. The student will note that the katun coefficient in A18a is expressed by an unusual form, the thumb. As explained on page 103, this has a numerical value of 1. Again, our text presents another irregular feature. Instead of being counted either forward or backward from the date next preceding it in the text; that is, 4 Ahau 13 Yax in A17, this number is counted from the date following it in the text, like the two Secondary-series numbers in Stela J, just discussed. This starting date recorded in A18b B18a is 12 Caban 5 Kayab, which, as we have seen, is also the date recorded by the Initial Series in plate 19, A, A1-A6. We are perfectly justified in assuming, therefore, that the 12 Caban 5 Kayab in A18b-B18a had the same Initial-series value as the 12 Caban 5 Kayab in plate 19, A, B5-A6, namely, 9.14.13.4.17. Reducing the number in B17-A18a, namely, 1.16.13.3, to units of the first order, we have:

Remembering that this number is to be counted forward from the date 12 Caban 5 Kayab, and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively), the terminal date reached will be 1 Ahau 3 Zip, which is recorded in A19. Compare the coefficient of the day sign in A19a with the coefficient of the katun sign in A18a, and the day sign itself with the profile variant for Ahau in figure 16, h', i'. For the month sign, compare A19b with figure 19, d. But since the Initial-series value of the starting point is known, we may calculate from it the Initial-series value of the new terminal date:

Passing over to the east side of this monument, the student will find recorded there the continuation of this inscription (see pl. 20).^[201] This side, like the other, opens with an introducing glyph A1-B2, which is followed by an Initial Series in A3-A5. Although this number is expressed by head variants, the forms are all familiar, and the student will have little difficulty in reading it as 9.16.10.0.0. The terminal date which this number reaches is recorded in B5-B8; that is, 1^[202] Ahau 3 Zip, the "month indicator" appearing as a head variant in A8 with the head-variant coefficient 10. But this date is identical with the date determined by calculation and actually recorded at the close of the inscription on the other side of this monument, and since no later date is recorded elsewhere in this text, we may conclude that 9.16.10.0.0 1 Ahau 3 Zip represents the contemporaneous time of Stela F, and hence that it was a regular hotun-marker. Here again, as in the case of Stela J at Quirigua, the importance of the "contemporaneous date" is emphasized not only by the fact that all the other dates lead up to it, but also by the fact that it is expressed as an Initial Series.

We have explained all the dates figured except 3 Ahau 3 Mol in plate 19, B, A15, the discussion of which was deferred until after the rest of the inscription had been considered. It will be remembered in connection with Stela J (figs. 78, 79) that one of the dates reached in the course of the calculations was just 1 katun earlier than the date recorded by the Initial Series on the same monument. Now, one of the Initial-series values corresponding to the date 3 Ahau 3 Mol here under discussion is 9.15.10.0.0, exactly 1 katun earlier than the Initial-series date on Stela F. In other words, if we give to the date 3 Ahau 3 Mol in A15 the value 9.15.10.0.0, the cases are exactly parallel. While it is impossible to prove that this particular Initial Series was the one which the ancient scribes had in mind when they recorded this date 3 Ahau 3 Mol, the writer believes that the coincidence and parallel here presented are sufficient to warrant the assumption that this is the case. The whole text reads as follows:

The student will note the close similarity between this inscription and that on Stela J (figured in figs. 78 and 79), a summary of which appears on page 239. Both commence with the same date, 9.14.13.4.17 12 Caban 5 Kayab; both show the date 9.15.6.14.6 6 Cimi 4 Tzec; both have dates which are just 1 katun in advance of the hotuns which they mark; and finally, both are hotun-markers, Stela J preceding Stela F by just 1 hotun. The date from which both proceed, 9.14.13.4.17 12 Caban 5 Kayab, is an important one at Quirigua, being the earliest date there. It appears on four monuments, namely, Stelæ J, F, and E, and Zoömorph G. Although the writer has not been able to prove the point, he is of the opinion that the glyph shown in figure 80, a, tells the meaning of the event which happened on this date, which is, moreover, the earliest date at Quirigua which it is possible to regard as being contemporaneous. Hence, it is not improbable that it might refer to the founding of the city or some similar event, though this is of course a matter of speculation. The fact, however, that 9.14.13.4.17 12 Caban 5 Kayab is the earliest date on four different hotun-markers shows that it was of supreme importance in the history of Quirigua. This concludes the discussion of texts showing the use of Secondary Series with Initial Series.

Texts Recording Period Endings

It was explained in Chapter III (p. 77) that in addition to Initial-series dating and Secondary-series dating, the Maya used still another method in fixing events, which was designated Period-ending dating. It was explained further that, although Period-ending dating was less exact than the other two methods, it served equally well for all practical purposes, since dates fixed by it could not recur until after a lapse of more than 18,000 years, a considerably longer period than that covered by the recorded history of mankind. Finally, the student will recall that the katun was said to be the period most commonly used in this method of dating.

The reason for this is near at hand. Practically all of the great southern cities rose, flourished, and fell within the period called Cycle 9 of Maya chronology. There could have been no doubt throughout the southern area which particular cycle was meant when the "current cycle" was spoken of. After the date 9.0.0.0.0 8 Ahau 13 Ceh had ushered in a new cycle there could be no change in the cycle coefficient until after a lapse of very nearly 400 (394.250 +) years. Consequently, after Cycle 9 had commenced many succeeding generations of men knew no other, and in time the term "current cycle" came to mean as much on a monument as "Cycle 9." Indeed, in Period-ending dating the Cycle 9 was taken for granted and scarcely ever recorded. The same practice obtains very generally to-day in regard to writing the current century, such expressions as July 4, '12, December 25, '13, being frequently seen in place of the full forms July 4, 1912, A. D., December 25, 1913, A. D.; or again, even more briefly, 7/4/12 and 12/25/13 to express the same dates, respectively. The desire for brevity, as has been explained, probably gave rise to Period-ending dating in the first place, and in this method the cycle was the first period to be eliminated as superfluous for all practical purposes. No one could have forgotten the number of the current cycle.

When we come to the next lower period, however, the katun, we find a different state of affairs. The numbers belonging to this period were changing every 20 (exactly, 19.71 +) years; that is, three or four times in the lifetime of many individuals; hence, there was plenty of opportunity for confusion about the number of the katun in which a particular event occurred. Consequently, in order to insure accuracy the katun is almost always the unit used in Period-ending dating.

In plate 21 are figured a number of Period-ending dates, the glyphs of which have been ranged in horizontal lines, and are numbered from left to right for convenience in reference. The true positions of these glyphs in the texts from which they have been taken are given in the footnotes in each case. In plate 21, A, is figured a Period-ending date from Stela 2 at Copan.^[203] The date 12 Ahau 8 Ceh appears very clearly in glyphs 1 and 2. Compare the month sign with figure 19, u, v. There follows in 3 a glyph the upper part of which probably represents the "ending sign" of this date. By comparing this form with the ending signs in figure 37 its resemblance to figure 37, o, will be evident. Indeed, figure 37, o, has precisely the same lower element as glyph 3. In glyph 4 follows the particular katun, 11, whose end fell on the date recorded in glyphs 1 and 2. The student can readily prove this for himself by reducing the Period-ending date here recorded to its corresponding Initial Series and counting the resulting number forward from the common starting point, 4 Ahau 8 Cumhu, as follows: Since the cycle glyph is not expressed, we may fill this omission as the Maya themselves filled it, by supplying Cycle 9. Moreover, since the end of a katun is recorded here, it is clear that all the lower periods—the tuns, uinals, and kins—will have to appear with the coefficient 0, as they are all brought to their respective ends with the ending of any katun. Therefore we may write the Initial-series number corresponding to the end of Katun 11, as 9.11.0.0.0. Treating this number as an Initial Series, that is, first reducing it to units of the first order, then deducting from it all the Calendar Rounds possible, and finally applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) to the remainder, the student will find that the terminal date reached will be the same as the date recorded in glyphs 1 and 2, namely, 12 Ahau 8 Ceh. In other words, the Katun 11, which ended on the date 12 Ahau 8 Ceh, was 9.11.0.0.0 12 Ahau 8 Ceh, and both indicate exactly the same position in the Long Count. The next example (pl. 21, B) is taken from the tablet in the Temple of the Foliated Cross at Palenque.^[204] In glyph 1 appears the date 8 Ahau 8 Uo (compare the month form with fig. 19, b, c) and in glyph 3 the "ending" of Katun 13. The ending sign here is the variant shown in figure 37, a-h, and it occurs just above the coefficient 13. These two glyphs therefore record the fact that Katun 13 ended with the day 8 Ahau 8 Uo. The student may again test the accuracy of the record by changing this Period-ending date to its corresponding Initial-series number, 9.13.0.0.0, and performing the various operations indicated in such cases. The resulting Initial-series terminal date will be the same as the date recorded in glyphs 1 and 2, 8 Ahau 8 Uo.

In plate 21, C, is figured a Period-ending date taken from Stela 23 at Naranjo.^[205] The date 6 Ahau 13 Muan appears very clearly in glyphs 1 and 2 (compare the month form with fig. 19, a', b'). Glyph 3 is the ending sign, here showing three common "ending elements," (1) the clasped hand; (2) the element with the curl infix; (3) the tassel-like postfix. Compare this form with the ending signs in figure 37, l-q, and with the zero signs in figure 54. In glyph 4 is recorded the particular katun, 14, which came to its end on the date recorded in 1 and 2. The element prefixed to the Katun 14 in glyph 4 is also an ending sign, though it always occurs as a prefix or superfix attached to the sign of the period whose close is recorded. Examples illustrating its use are shown in figure 37, a-h, with which the ending element in glyph 4 should be compared. The glyphs 1 to 4 in plate 21, C, therefore record that Katun 14 came to an end on the date 6 Ahau 13 Muan. As we have seen above, this could be shown to correspond with the Initial Series 9.14.0.0.0 6 Ahau 13 Muan.

This same date, 6 Ahau 13 Muan ending Katun 14, is also recorded on Stela 16 at Tikal (see pl. 21, D).^[206] The date itself appears in glyphs 1 and 2 and is followed in 3 by a sign which is almost exactly like the ending sign in glyph 3 just discussed (see pl. 21, C). The subfixes are identical in both cases, and it is possible to distinguish the lines of the hand element in the weathered upper part of the glyph in 3. Compare glyph 3 with the ending signs in figure 37, l-q, and with the zero signs in figure 54. As in the preceding example, glyph 4 shows the particular katun whose end is recorded here—Katun 14. The period glyph itself appears as a head variant to which is prefixed the same ending prefix or superfix shown with the period glyph in the preceding example. See also figure 37, a-h. As above stated, the Initial Series corresponding to this date is 9.14.0.0.0 6 Ahau 13 Muan.

One more example will suffice to illustrate the use of katun Period-ending dates. In plate 21, E, is figured a Period-ending date from Stela 4 at Copan.^[207] In glyphs 1 and 2 appears the date 4 Ahau 13 Yax (compare the month in glyph 2 with fig. 19, q, r), which is followed by the ending sign in 3. This is composed of the hand, a very common "ending" element (see fig. 37, j, k) with a grotesque head superfix, also another "ending sign" (see i, r, u, v of the plate just named). In glyph 4 follows the particular katun (Katun 15) whose end is here recorded. This date corresponds to the Initial Series 9.15.0.0.0 4 Ahau 13 Yax.

Cases where tun endings are recorded are exceedingly rare. The bare statement that a certain tun, as Tun 10, for example, had come to its end left much to be desired in the way of accuracy, since there was a Tun 10 in every katun, and consequently any given tun recurred after an interval of 20 years; in other words, there were three or four different Tun 10's to be distinguished from one another in the average lifetime. Indeed, to keep them apart at all it was necessary either to add the particular katun in which each fell or to add the date on which each closed. The former was a step away from the brevity which probably prompted the use of Period-ending dating in the first place, and the latter imposed too great a task on the memory, that is, keeping in mind the 60 or 70 various tun endings which the average lifetime included. For these reasons tun-ending dates occur but rarely, only when there was little or no doubt concerning the particular katun in which they fell.

In plate 21, F, is figured a tun-ending date from the tablet in the Temple of the Inscription at Palenque.^[208] In glyph 1 appears an ending sign showing the hand element and the grotesque flattened head (for the latter see fig. 37, i, r, u, v), both common ending signs. The remaining element, another grotesque head with a flaring postfix, is an unusual variant of the tun head found only at Palenque (see fig. 29, h). The presence of the tun sign with these two ending signs indicates probably that some tun ending follows. Glyphs 2 and 3 record the date 5 Ahau 18 Tzec, and glyph 4 records Tun 13. We have here then the record of a Tun 13, which ended on the date 5 Ahau 18 Tzec. But which of the many Tun 13s in the Long Count was the one that ended on this particular date? To begin with, we are perfectly justified in assuming that this particular tun occurred somewhere in Cycle 9, but this assumption does not aid us greatly, since there were twenty different Tun 13s in Cycle 9, one for each of the twenty katuns. However, in the full text of the inscription from which this example is taken, 5 Ahau 3 Chen is the date next preceding, and although the fact is not recorded, this latter date closed Katun 8 of Cycle 9. Moreover, shortly after the tun-ending date here under discussion, the date "3 Ahau 3 Zotz, end of Katun 9," is recorded. It seems likely, therefore, that this particular Tun 13, which ended on the date 5 Ahau 18 Tzec, was 9.8.13.0.0 of the Long Count, after 9.8.0.0.0 but before 9.9.0.0.0. Reducing this number to units of the first order, and applying the several rules given for solving Initial Series, the terminal date of 9.8.13.0.0 will be found to agree with the terminal date recorded in glyphs 2 and 3, namely, 5 Ahau 18 Tzec, and this tun ending corresponded, therefore, to the Initial Series 9.8.13.0.0 5 Ahau 18 Tzec.

Another tun-ending date from Stela 5 at Tikal is figured in plate 21, G.^[209] In glyphs 1 and 2 the date 4 Ahau 8 Yaxkin appears, the month sign being represented as a head variant, which has the essential elements of the sign for Yaxkin (see fig. 19, k, l). Following this in glyph 3 is Tun 13, to which is prefixed the same ending-sign variant as the prefixial or superfixial elements in figure 37, i, r, u, v. We have recorded here then "Tun 13 ending on 4 Ahau 8 Yaxkin," though there seems to be no mention elsewhere in this inscription of the number of the katun in which this particular tun fell. By referring to Great Cycle 54 of Goodman's Tables (Goodman, 1897), however, it appears that Tun 13 of Katun 15 of Cycle 9 closed with this date 4 Ahau 8 Yaxkin, and we may assume, therefore, that this is the correct position in the Long Count of the tun-ending date here recorded. This date corresponds to the Initial Series 9.15.13.0.0 4 Ahau 8 Yaxkin.

There is a very unusual Period-ending date on the west side of Stela C at Quirigua^[210] (see pl. 21, H). In glyphs 1 and 2 appears the number 0 kins, 0 uinals, 5 tuns, and 17 katuns, which we may write 17.5.0.0 and following this in glyphs 3 and 4 is the date 6 Ahau 13 Kayab. At first sight this would appear to be a Secondary Series, the number 17.5.0.0 being counted forward from some preceding date to reach the date 6 Ahau 13 Kayab recorded just after it. The next date preceding this on the west side of Stela C at Quirigua is the Initial-series terminal date 6 Ahau 13 Yaxkin, illustrated together with its corresponding Initial-series number in figure 68, A. However, all attempts to reach the date 6 Ahau 13 Kayab by counting either forward or backward the number 17.5.0.0 from the date 6 Ahau 13 Yaxkin will prove unsuccessful, and we must seek another explanation for the four glyphs here under discussion. If this were a Period-ending date it would mean that Tun 5 of Katun 17 came to an end on the date 6 Ahau 13 Kayab. Let us see whether this is true. Assuming that our cycle coefficient is 9, as we have done in all the other Period-ending dates presented, we may express glyphs 1 and 2 as the following Initial-series number, provided they represent a period ending, not a Secondary-series number: 9.17.5.0.0. Reducing this number to units of the 1st order, and applying the rules previously given for solving Initial Series, the terminal date reached will be 6 Ahau 13 Kayab, identical with the date recorded in glyphs 3 and 4. We may conclude, therefore, that this example records the fact that "Tun 5 of Katun 17 ended on the date 6 Ahau 13 Kayab," this being identical with the Initial Series 9.17.5.0.0 6 Ahau 13 Kayab.