Chapter IV

and apply the several steps there given, by means of which Maya numbers may be solved. The first step on page 134 was 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 Table XIII, and the sum of the products arising from their multiplication by the coefficients recorded in the Initial Series in plate 6, A are given below:

A3 = 9 × 144,000 = 1,296,000 B3 = 18 × 7,200 = 129,600 A4 = 5 × 360 = 1,800 B4 = 0 × 20 = 0 A5 = 0 × 1 = 0 ---------- 1,427,400

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, page 136, if the number is an Initial Series the starting point, although never recorded, is practically always the date 4 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 date 4 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 page 137, the count is always forward, consequently 1,427,400 is to be counted _forward_ from 4 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 pages 139-143. Since our given number (1,427,400) is greater than 18,980, or 1 Calendar Round, the preliminary rule on page 143 applies 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 Table XVI, it appears that 1,427,400 contains 75 complete Calendar Rounds, or 1,423,500; hence, the latter number may be subtracted {160} from 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 from 4 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, page 139, 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, page 140, 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 Table I, from Ahau, the day sign of the starting point (4 Ahau 8 Cumhu), we reach Ahau as the day sign of the terminal date. In other words, in counting forward either 3,900 or 1,427,400 from 4 Ahau 8 Cumhu, the day reached will be 4 Ahau. It remains to show what position in the year this day 4 Ahau distant 1,427,400 from the date 4 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, page 141, to divide the given number or its equivalent by 365; 3,900 ÷ 365 = 10-250/365. 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 from 8 Cumhu, the year position of the day of the starting point 4 Ahau 8 Cumhu. It appears from Table XV, in which the 365 positions of the year are given, that after position 8 Cumhu there are only 16 positions in the year--11 more in Cumhu and 5 in Uayeb. 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 is Ceh (see Table XV); counting forward 14 positions in Ceh, we reach 13 Ceh, which is, therefore, the month glyph of our terminal date. In other words, counting 250 forward from 8 Cumhu, position 13 Ceh is reached. Assembling the above values, we find that by calculation we have determined the terminal date of the Initial Series in plate 6, _A_, to be 4 Ahau 13 Ceh. {161}

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 position 13 Ceh, position 8 Cumhu will be reached if our calculations were correct. This is true because there are only 365 positions in the year, and having reached 13 Ceh in counting forward 250 from 8 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 count _backward_ 250 from 13 Ceh; if our calculations have been correct, the starting point 8 Cumhu will be reached. Still another check, which may be applied is the following: From Table VII it is clear that the day sign Ahau can occupy only positions 3, 8, 13, or 18 in the divisions of the year;[119] hence, if in the above case the coefficient of Ceh had 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 page 152 that 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 day 4 Ahau by means of 4 dots prefixed to the sign shown in figure 16, _e'-g'_, which is the form for the day name Ahau, 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 figure 65. This glyph, which always has a coefficient of 9 or 10, was designated on page 152 the 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 part {162} of 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 for Ceh in figure 19, _u, v,_ to enable us to identify it as such.

Our complete Initial Series, therefore, reads: 9.18.5.0.0 4 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 9.18.5.0.0 4 Ahau 13 Ceh was 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 9.18.5.0.0 4 Ahau 13 Ceh of Maya chronology.[121]

In plate 6, _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 Table XIII this 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:

B1 = 9 × 144,000 = 1,296,000 A2 = 12 × 7,200 = 86,400 B2 = 15 × 360 = 5,400 A3 = 13 × 20 = 260 B3 = 7 × 1 = 7 --------- 1,388,067

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 date 4 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] {163}

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, page 143). According to Table XVI it 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 = 194-5/13, 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 = 126-7/20; and counting forward 7, the numerator of the fractional part of the quotient, from Ahau, the day sign of our starting point, 4 Ahau 8 Cumhu, in Table I, we reach Manik as the day sign of the terminal date. Therefore, the day of the terminal date will be 9 Manik.

Applying rule 3 (p. 141) to 2,527, we have: 2,527 ÷ 365 = 6-337/365; and counting forward 337, the numerator of the fractional part of the quotient, from 8 Cumhu, the year position of the starting point, 4 Ahau 8 Cumhu, in Table XV, we reach 0 Kayab as the year position of the terminal date. The calculations by means of which 0 Kayab is reached are as follows: After 8 Cumhu there 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 Table XV and 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 plate 6, _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 for Manik in figure 16, _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 day 9 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; {164} 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 be 0 Kayab, and by referring to figure 49 we 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 sign Kayab. Further, by referring to figure 19, _d'-f'_, we may fix in our minds the sign for the month Kayab, which should also appear in the text with one of the forms shown in figure 49.

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 figure 49 joined to figure 19, _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 with _d'-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 9.12.15.13.7 9 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 plate 6, 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 plate 6, 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 figure 54; in the present text, however, the 0 uinals and 0 kins are expressed by the form for 0 shown in figure 47, a new feature.

Reducing the above number to units of the 1st order by means of Table XIII, we have:

A3 = 9 × 144,000 = 1,296,000 B3a = 18 × 7,200 = 129,600 B3b = 10 × 360 = 3,600 A4a = 0 × 20 = 0 A4b = 0 × 1 = 0 --------- 1,429,200

Deducting from this number all the Calendar Rounds possible, 75 {165} (see Table XVI), it may be reduced to 5,700 without affecting its value in the present connection.

Applying rules 1 and 2 (pp. 139 and 140, respectively) to this number, the day reached will be found to be 10 Ahau; and by applying rule 3 (p. 141), the position of this day in the year will be found to be 8 Zac. Therefore, by calculation we have determined that the terminal date reached by this Initial Series is 10 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 day 10 Ahau, both the day sign and the coefficient being unusually clear, and practically unmistakable. Compare B4a with figure 16, _e'-g'_, the sign for the day name Ahau. Consequently the day recorded agrees with the day determined by calculation. The month glyph in this text, as mentioned on page 157, footnote 1, occurs out of its regular position, following immediately the day of the terminal date.

As mentioned on page 153, when the month glyph in Initial-series terminal dates is _not_ to be found in its usual position, it will be found in the regular position for the month glyphs in all other kinds of dates in the inscriptions, namely, immediately following the day glyph to which it belongs. In the present text we found that the day, 10 Ahau, was recorded in B4a; hence, since the month glyph was not recorded in its regular position, it must be in B4b, immediately following the day glyph. By comparing the glyph in B4b with the month signs in figure 19, it will be found exactly like the month sign for Zac (_s-t_), and we may therefore conclude that this is our month glyph and that it is Zac. The coefficient of B4b is quite clearly 8 and the month part therefore reads, 8 Zac. Combining this with the day recorded in B4a, we have the date 10 Ahau 8 Zac, which corresponds with the terminal date determined by calculation. The whole text therefore reads 9.18.10.0.0 10 Ahau 8 Zac.

[Illustration: FIG. 67. Signs representing the hotun, or 5-tun, period.]

It will be noted that this date 9.18.10.0.0 10 Ahau 8 Zac is just 5.0.0 (5 tuns) later than the date recorded by the Initial Series on Zoömorph P at Quirigua (see pl. 6, _A_). As explained in