Fig. 57Fig.57. Signs for the cycle showing coefficients above 13:a, From the Temple of the Inscriptions, Palenque;b, from Stela N, Copan.
Fig.57. Signs for the cycle showing coefficients above 13:a, From the Temple of the Inscriptions, Palenque;b, from Stela N, Copan.
The above points are simply positive evidence in support of this hypothesis, however, and in no way attempt to explain or otherwise account for the undoubtedly contradictory points given in the discussion of (1) on pages108-109. Furthermore, not until these contradictions have been cleared away can it be established that the great cycle in the inscriptions was of the same length as the great cycle in the codices. The writer believes the following explanation will satisfactorily dispose of these contradictions and make possible at the same time the acceptance of the theory that the great cycle in the inscriptions and in the codices was of equal length, being composed in each case of 20 cycles.
Assuming for the moment that there were 13 cycles in a great cycle; it is clear that if this were the case 13 cycles could never be recorded in the inscriptions, for the reason that, being equal to 1 great cycle, they would have to be recorded in terms of a great cycle. This is true because no period in the inscriptions is ever expressed, so far as now known, as the full number of the periods of which it was composed. For example, 1 uinal never appears as 20 kins; 1 tun is never written as its equivalent, 18 uinals; 1 katun is never recorded as 20 tuns, etc. Consequently, if a great cycle composed of 13 cycles had come to its end with the end of a Cycle 13, which fell on a day4 Ahau 8 Cumhu, such a Cycle 13 could never have been expressed, since in its place would have been recorded the end of the great cycle which fell on the same day. In other words, if there had been 13 cycles in a great cycle, the cycles would have been numbered from 0 to 12, inclusive, and the last, Cycle 13, would have been recorded instead as completing some great cycle. It is necessary toadmit this point or repudiate the numeration of all the other periods in the inscriptions. The writer believes, therefore, that, when the starting point of Maya chronology is declared to be a date4 Ahau 8 Cumhu, which an "ending sign" and a Cycle 13 further declare fell at the close of a Cycle 13, this does not indicate that there were 13 cycles in a great cycle, but that it is to be interpreted as a Period-ending date, pure and simple. Indeed, where this date is found in the inscriptions it occurs with a Cycle 13, and an "ending sign" which is practically identical with other undoubted "ending signs." Moreover, if we interpret these places as indicating that there were only 13 cycles in a great cycle, we have equal grounds for saying that the great cycle contained only 10 cycles. For example, on Zoömorph G at Quirigua the date7 Ahau 18 Zipis accompanied by an "ending sign" and Cycle 10, which on this basis of interpretation would signify that a great cycle had only 10 cycles. Similarly, it could be shown by such an interpretation that in some cases a cycle had 14 katuns, that is, where the end of a Katun 14 was recorded, or 17 katuns, where the end of a Katun 17 was recorded. All such places, including the date4 Ahau 8 Cumhu, which closed a Cycle 13 at the starting point of Maya chronology, are only Period-ending dates, the writer believes, and have no reference to the number of periods which any higher period contains whatsoever. They record merely the end of a particular period in the Long Count as the end of a certain Cycle 13, or a certain Cycle 10, or a certain Katun 14, or a certain Katun 17, as the case may be, and contain no reference to the beginning or the end of the period next higher.
There can be no doubt, however, as stated above, that the cycles were numbered from 1 to 13, inclusive, and then began again with 1. This sequence strikingly recalls that of the numerical coefficients of the days, and in the parallel which this latter sequence affords, the writer believes, lies the true explanation of the misconception concerning the length of the great cycle in the inscriptions.
Table XI.SEQUENCE OF TWENTY CONSECUTIVE DATES IN THE MONTH POP
The numerical coefficients of the days, as we have seen, were numbered from 1 to 13, inclusive, and then began again with 1. SeeTableXI, in which the 20 days of the monthPopare enumerated. Now it is evident from this table that, although the coefficients of the days themselves do not rise above 13, the numbers showing the positions of these days in the month continue up through 19. In other words, two different sets of numerals were used in describing the Maya days: (1) The numerals 1 to 13, inclusive, the coefficients of the days, and an integral part of their names; and (2) The numerals 0 to 19, inclusive, showing the positions of these days in the divisions of the year—the uinals, and the xma kaba kin. It is clear from the foregoing, moreover, that the number of possible day coefficients (13) has nothing whatever to do in determining the number of days in the period next higher. That is, although the coefficients of the days are numbered from 1 to 13, inclusive, it does not necessarily follow that the next higher period (the uinal) contained only 13 days. Similarly, the writer believes that while the cycles were undoubtedly numbered—that is, named—from 1 to 13, inclusive, like the coefficients of the days, it took 20 of them to make a great cycle, just as it took 20 kins to make a uinal. The two cases appear to be parallel. Confusion seems to have arisen through mistaking thenameof the period for itspositionin the period next higher—two entirely different things, as we have seen.
A somewhat similar case is that of the katuns in the u kahlay katunob in TableIX. Assuming that a cycle commenced with the first katun there given, the name of this katun is Katun2 Ahau, although it occupied thefirstposition in the cycle. Again, the name of the second katun in the sequence is Katun13 Ahau, although it occupied the second position in the cycle. In other words, the katuns of the u kahlay katunob were named quite independently of their position in the period next higher (the cycle), and their names do not indicate the corresponding positions of the katun in the period next higher.
Applying the foregoing explanation to those passages in the inscriptions which show that the enumeration of the cycles was from 1 to 13, inclusive, we may interpret them as follows: When we find the date4 Ahau 8 Cumhuin the inscriptions, accompanied by an "ending sign" and a Cycle 13, that "Cycle 13," even granting that it stands at the end of some great cycle, does not signify that there were only 13 cycles in the great cycle of which it was a part. On the contrary, it records only the end of a particular Cycle 13, being a Period-ending date pure and simple. Such passages no more fix the length of the great cycle as containing 13 cycles than does the coefficient 13 of the day name13 Ixin TableXIlimit the number of days in a uinal to 13, or, again, the 13 of the katun name13 Ahauin TableIXlimit the number of katuns in a cycle to 13. This explanation not only accounts for the use of the 14 cycles or 17 cycles, asshown in figure57,a,b, but also satisfactorily provides for the enumeration of the cycles from 1 to 13, inclusive.
If the date "4 Ahau 8 Cumhuending Cycle 13" be regarded as a Period-ending date, not as indicating that the number of cycles in a great cycle was restricted to 13, the next question is—Did a great cycle also come to an end on the date4 Ahau 8 Cumhu—the starting point of Maya chronology and the closing date of a Cycle 13? That it did the writer is firmly convinced, although final proof of the point can not be presented until numerical series containing more than 5 terms shall have been considered. (See pp.114-127for this discussion.) The following points, however, which may be introduced here, tend to prove this condition:
1. In the natural course of affairs the Maya would have commenced their chronology with the beginning of some great cycle, and to have done this in the Maya system of counting time—that is, by elapsed periods—it was necessary to reckon from the end of the preceding great cycle as the starting point.
2. Moreover, it would seem as though the natural cycle with which to commence counting time would be aCycle 1, and if this were done time would have to be counted from aCycle 13, since a Cycle 1 could follow only a Cycle 13.
On these two probabilities, together with the discussion on pages114-127, the writer is inclined to believe that the Maya commenced their chronology with the beginning of a great cycle, whose first cycle was named Cycle 1, which was reckoned from the close of a great cycle whose ending cycle was a Cycle 13 and whose ending day fell on the date4 Ahau 8 Cumhu.
The second point (see p.108) on which rests the hypothesis of "13 cycles to a great cycle" in the inscriptions admits of no such plausible explanation as the first point. Indeed, it will probably never be known why in two inscriptions the Maya reckoned time from a starting point different from that used in all the others, one, moreover, which was 13 cycles in advance of the other, or more than 5,000 years earlier than the beginning of their chronology, and more than 8,000 years earlier than the beginning of their historic period. That this remoter starting point,4 Ahau 8 Zotz, from which proceed so far as known only two inscriptions throughout the whole Maya area, stood at theendof a great cycle the writer does not believe, in view of the evidence presented on pages114-127. On the contrary, the material given there tends to show that although the cycle which ended on the day4 Ahau 8 Zotzwas also named Cycle 13,[79]it was the 8th division of the grand cycle which ended on the day4 Ahau 8 Cumhu,the starting point of Maya chronology, and not the closing division of the preceding grand cycle. However, without attempting to settle this question at this time, the writer inclines to the belief, on the basis of the evidence at hand, that the great cycle in the inscriptions was of the same length as in the codices, where it is known to have contained 20 cycles.
Let us return to the discussion interrupted on page107, where the first method of expressing the higher numbers was being explained. We saw there how the higher numbers up to and including 1,872,000 were written, and the digression just concluded had for its purpose ascertaining how the numbers above this were expressed; that is, whether 13 or 20 units of the 5th order were equal to 1 unit of the 6th order. It was explained also that this number, 1,872,000, was perhaps the highest which has been found in the inscriptions. Three possible exceptions, however, to this statement should be noted here: (1) On the east side of Stela N at Copan six periods are recorded (see fig.58); (2) on the west panel from the Temple of the Inscriptions at Palenque six and probablysevenperiods occur (see fig.59); and (3) on Stela 10 at Tikal eight and perhapsnineperiods are found (see fig.60). If in any of these cases all of the periods belong to one and the same numerical series, the resulting numbers would be far higher than 1,872,000. Indeed, such numbers would exceed by many millions all others throughout the range of Maya writings, in either the codices or the inscriptions.
Before presenting these three numbers, however, a distinction should be drawn between them. The first and second (figs.58,59) are clearly not Initial Series. Probably they are Secondary Series, although this point can not be established with certainty, since they can not be connected with any known date the position of which is definitely fixed. The third number (fig.60), on the other hand, is an Initial Series, and the eight or nine periods of which it is composed may fix the initial date of Maya chronology (4 Ahau 8 Cumhu) in a much grander chronological scheme, as will appear presently.
Fig. 58Fig.58. Part of the inscription on Stela N, Copan, showing a number composed of six periods.
Fig.58. Part of the inscription on Stela N, Copan, showing a number composed of six periods.
Fig. 59Fig.59. Part of the inscription in the Temple of the Inscriptions, Palenque, showing a number composed of seven periods.
Fig.59. Part of the inscription in the Temple of the Inscriptions, Palenque, showing a number composed of seven periods.
Fig. 60Fig.60. Part of the inscription on Stela 10, Tikal (probably an Initial Series), showing a number composed of eight periods.
Fig.60. Part of the inscription on Stela 10, Tikal (probably an Initial Series), showing a number composed of eight periods.
The first of these three numbers (see fig.58), if all its six periods belong to the same series, equals 42,908,400. Although the order of the several periods is just the reverse of that in the numbers in figure56, this difference is unessential, as will shortly be explained, and in no way affects the value of the number recorded. Commencing at the bottom of figure58with the highest period involved and reading up, A6,[80]the 14 great cycles = 40,320,000 kins (see TableVIII, in which 1 great cycle = 2,880,000, and consequently 14 = 14 × 2,880,000 =40,320,000); A5, the 17 cycles = 2,448,000 kins (17 × 144,000); A4, the 19 katuns = 136,800 kins (19 × 7,200); A3, the 10 tuns = 3,600 kins (10 × 360); A2, the 0 uinals, 0 kins; and the 0 kins, 0 kins. The sum of these products = 40,320,000 + 2,448,000 + 136,800 + 3,600 + 0 + 0 = 42,908,400.
The second of these three numbers (see fig.59), if all of its seven terms belong to one and the same number, equals 455,393,401. Commencing at the bottom as in figure58, the first term A4, has the coefficient 7. Since this is the term following the sixth, or great cycle, we may call it the great-great cycle. But we have seen that thegreat cycle = 2,880,000; therefore the great-great cycle = twenty times this number, or 57,600,000. Our text shows, however, that seven of these great-great cycles are used in the number in question, therefore our first term = 403,200,000. The rest may be reduced by means of TableVIIIas follows: B3, 18 great cycles = 51,840,000; A3, 2 cycles = 288,000; B2, 9 katuns = 64,800; A2, 1 tun = 360; B1, 12 uinals = 240; B1, 1 kin = 1. The sum of these (403,200,000 + 51,840,000 + 288,000 + 64,800 + 360 + 240 +1) = 455,393,401.
The third of these numbers (see fig.60), if all of its terms belong to one and the same number, equals 1,841,639,800. Commencing with A2, this has a coefficient of 1. Since it immediately follows the great-great cycle, which we found above consisted of 57,600,000, we may assume that it is the great-great-great cycle, and that it consisted of 20 great-great cycles, or 1,152,000,000. Since its coefficient is only 1, this large number itself will be the first term in our series. The rest may readily be reduced as follows: A3, 11 great-great cycles = 633,600,000; A4, 19 great cycles = 54,720,000; A5, 9 cycles = 1,296,000; A6, 3 katuns = 21,600; A7, 6 tuns = 2,160; A8, 2 uinals = 40; A9, 0 kins = 0.[81]The sum of these (1,152,000,000 + 633,600,000 + 54,720,000 + 1,296,000 + 21,600 + 2,160 + 40 + 0) = 1,841,639,800, the highest number found anywhere in the Maya writings, equivalent to about 5,000,000 years.
Whether these three numbers are actually recorded in the inscriptions under discussion depends solely on the question whether or not the terms above the cycle in each belong to one and the same series. If it could be determined with certainty that these higher periods in each text were all parts of the same number, there would be no further doubt as to the accuracy of the figures given above; and more important still, the 17 cycles of the first number (see A5, fig.58) would then prove conclusively that more than 13 cycles were required to make a great cycle in the inscriptions as well as in the codices. And furthermore, the 14 great cycles in A6, figure58, the 18 in B3, figure59, and the 19 in A4, figure60, would also prove that more than 13 great cycles were required to make one of the period next higher—that is, the great-great cycle. It is needless to say that this point has not been universally admitted. Mr. Goodman (1897: p. 132) has suggested in the case of the Copan inscription (fig.58) that only the lowest four periods—the 19 katuns, the 10 tuns, the 0 uinals, and the 0 kins—A2, A3, and A4,[82]here form the number; and that if this number is counted backward from the Initial Series of the inscription, it will reach a Katun 17 of the preceding cycle. Finally, Mr. Goodmanbelieves this Katun 17 is declared in the glyph following the 19 katuns (A5), which the writer identifies as 17 cycles, and consequently according to the Goodman interpretation the whole passage is a Period-ending date. Mr. Bowditch (1910: p. 321) also offers the same interpretation as a possible reading of this passage. Even granting the truth of the above, this interpretation still leaves unexplained the lowest glyph of the number, which has a coefficient of 14 (A6).
The strongest proof that this passage will not bear the construction placed on it by Mr. Goodman is afforded by the very glyph upon which his reading depends for its verification, namely, the glyph which he interprets Katun 17. This glyph (A5) bears no resemblance to the katun sign standing immediately above it, but on the contrary has for its lower jaw the clasping hand (*), which, as we have seen, is the determining characteristic of the cycle head. Indeed, this element is so clearly portrayed in the glyph in question that its identification as a head variant for the cycle follows almost of necessity. A comparison of this glyph with the head variant of the cycle given in figure25,d-f, shows that the two forms are practically identical. This correction deprives Mr. Goodman's reading of its chief support, and at the same time increases the probability that all the 6 terms here recorded belong to one and the same number. That is, since the first five are the kin, uinal, tun, katun, and cycle, respectively, it is probable that the sixth and last, which follows immediately the fifth, without a break or interruption of any kind, belongs to the same series also, in which event this glyph would be most likely to represent the units of the sixth order, or the so-called great cycles.
The passages in the Palenque and Tikal texts (figs.59and60, respectively) have never been satisfactorily explained. In default of calendric checks, as the known distance between two dates, for example, which may be applied to these three numbers to test their accuracy, the writer knows of no better check than to study the characteristics of this possible great-cycle glyph in all three, and of the possible great-great-cycle glyph in the last two.
Passing over the kins, the normal form of the uinal glyph appears in figures58, A2, and 59, B1 (see fig.31,a,b), and the head variant in figure60, A8. (See fig.31,d-f.) Below the uinal sign in A3, figure58, and A2, figure59, and above A7, in figure60the tuns are recorded as head variants, in all three of which the fleshless lower jaw, the determining characteristic of the tun head, appears. Compare these three head variants with the head variant for the tun in figure29,d-g. In the Copan inscription (fig.58) the katun glyph, A4, appears as a head variant, the essential elements of which seem to be the oval in the top part of the head and the curling fang protruding from the back part of the mouth. Compare this head with the head variant for the katun in figure27,e-h. In the Palenque and Tikal texts (seefigs.59, B2, and60, A6, respectively), on the other hand, the katun is expressed by its normal form, which is identical with the normal form shown in figure27,a,b. In figures58, A5, and59, A3, the cycle is expressed by its head variant, and the determining characteristic, the clasped hand, appears in both. Compare the cycle signs in figures58, A5, and59, A3, with the head variant for the cycle shown in figure25;d-f. The cycle glyph in the Tikal text (fig.60, A5) is clearly the normal form. (See fig.25,a-c.) The glyph following the cycle sign in these three texts (standing above the cycle sign in figure60at A4) probably stands for the period of the sixth order, the so-called great cycle. These three glyphs are redrawn in figure61,a-c, respectively. In the Copan inscription this glyph (fig.61,a) is a head variant, while in the Palenque and Tikal texts (aandbof the same figure, respectively) it is a normal form.
Inasmuch as these three inscriptions are the only ones in which numerical series composed of 6 or more consecutive terms are recorded, it is unfortunate that the sixth term in all three should not have been expressed by the same form, since this would have facilitated their comparison. Notwithstanding this handicap, however, the writer believes it will be possible to show clearly that the head variant in figure61,a, and the normal forms inbandcare only variants of one and the same sign, and that all three stand for one and the same thing, namely, the great cycle, or unit of the sixth order.
Fig. 61Fig.61. Signs for the great cycle (a-c), and the great-great cycle (d,e):a, Stela N, Copan;b,d, Temple of the Inscriptions, Palenque;c,e, Stela 10, Tikal.
Fig.61. Signs for the great cycle (a-c), and the great-great cycle (d,e):a, Stela N, Copan;b,d, Temple of the Inscriptions, Palenque;c,e, Stela 10, Tikal.
In the first place, it will be noted that each of the three glyphs just mentioned is composed in part of the cycle sign. For example, in figure61,a, the head variant has the same clasped hand as the head-variant cycle sign in the same text (see fig.58, A5), which, as we have seen elsewhere, is the determining characteristic of the head variant for the cycle. In figure61,b,c, the normal forms there presented contain the entire normal form for the cycle sign; compare figure25,a,c. Indeed, except for its superfix, the glyphs in figure61,b,c, are normal forms of the cycle sign; and the glyph inaof the same figure, except for its superfixial element, is similarly the head variant for the cycle. It would seem, therefore, that the determining characteristics of these three glyphs must be their superfixial elements. In the normal form in figure61,b, the superfix is very clear. Just inside the outline and parallel to it there is a line of smaller circles,and in the middle there are two infixes like shepherds' crooks facing away from the center (*). Incof the last-mentioned figure the superfix is of the same size and shape, and although it is partially destroyed the left-hand "shepherd's crook" can still be distinguished. A faint dot treatment around the edge can also still be traced. Although the superfix of the head variant inais somewhat weathered, enough remains to show that it was similar to, if indeed not identical with, the superfixes of the normal forms inbandc. The line of circles defining the left side of this superfix, as well as traces of the lower ends of the two "shepherd's crook" infixes, appears very clearly in the lower part of the superfix. Moreover, in general shape and proportions this element is so similar to the corresponding elements in figure61,b,c, that, taken together with the similarity of the other details pointed out above, it seems more than likely that all three of these superfixes are one and the same element. The points which have led the writer to identify glyphsa,b, andcin figure61as forms for the great cycle, or period of the sixth order, may be summarized as follows:
1. All three of these glyphs, head-variant as well as normal forms, are made up of the corresponding forms of the cycle sign plus another element, a superfix, which is probably the determining characteristic in each case.
2. All three of these superfixes are probably identical, thus showing that the three glyphs in which they occur are probably variants of the same sign.
3. All three of these glyphs occur in numerical series, the preceding term of which in each case is a cycle sign, thus showing that by position they are the logical "next" term (the sixth) of the series.
Let us next examine the two texts in which great-great-cycle glyphs may occur. (See figs.59,60.) The two glyphs which may possibly be identified as the sign for this period are shown in figure61,d,e.
A comparison of these two forms shows that both are composed of the same elements: (1) The cycle sign; (2) a superfix in which the hand is the principal element.
The superfix in figure61,d, consists of a hand and a tassel-like postfix, not unlike the upper half of the ending signs in figure37,l-q. However, in the present case, if we accept the hypothesis thatdof figure61is the sign for the great-great cycle, we are obliged to see in its superfix alone the essentialelementof the great-great-cycle sign, since therestof this glyph (the lower part) is quite clearly the normal form for the cycle.
The superfix in figure61,e, consists of the same two elements as the above, with the slight difference that the hand ineholds a rod. Indeed, the similarity of the two forms is so close that in default ofany evidence to the contrary the writer believes they may be accepted as signs for one and the same period, namely, the great-great cycle.
The points on which this conclusion is based may be summarized as follows:
1. Both glyphs are made up of the same elements—(a) The normal form of the cycle sign; (b) a superfix composed of a hand with a tassel-like postfix.
2. Both glyphs occur in numerical series the next term but one of which is the cycle, showing that by position they are the logical next term but one, the seventh or great-great cycle, of the series.
3. Both of these glyphs stand next to glyphs which have been identified as great-cycle signs, that is, the sixth terms of the series in which they occur.
By this same line of reasoning it seems probable that A2 in figure60is the sign for the great-great-great cycle, although this fact can not be definitely established because of the lack of comparative evidence.
This possible sign for the great-great-great cycle, or period of the 8th order, is composed of two parts, just like the signs for the great cycle and the great-great cycle already described. These are: (1) The cycle sign; (2) a superfix composed of a hand and a semicircular postfix, quite distinct from the superfixes of the great cycle and great-great cycle signs.
However, since there is no other inscription known which presents a number composed of eight terms, we must lay aside this line of investigation and turn to another for further light on this point.
An examination of figure60shows that the glyphs which we have identified as the signs for the higher periods (A2, A3, A4, and A5,) contain one element common to all—the sign for the cycle, or period of 144,000 days. Indeed, A5 is composed of this sign alone with its usual coefficient of 9. Moreover, the next glyphs (A6, A7, A8, and A9[83]) are the signs for the katun, tun, uinal, and kin, respectively, and, together with A5, form a regular descending series of 5 terms, all of which are of known value.
The next question is, How is this glyph in the sixth place formed? We have seen that in the only three texts in which more than five periods are recorded this sign for the sixth period is composed of the same elements in each: (1) The cycle sign; (2) a superfix containing two "shepherd's crook" infixes and surrounded by dots.
Further, we have seen that in two cases in the inscriptions the cycle sign has a coefficient greater than 13, thus showing that in all probability 20, not 13, cycles made 1 great cycle.
Therefore, since the great-cycle signs in figure61,a-c, are composed of the cycle sign plus a superfix (*), this superfix must have the value of 20 in order to make the whole glyph have the value of20 cycles, or 1 great cycle (that is, 20 × 144,000 = 2,880,000). In other words, it may be accepted (1) that the glyphs in figure61,a-c, are signs for the great cycle, or period of the sixth place; and (2) that the great cycle was composed of 20 cycles shown graphically by two elements, one being the cycle sign itself and the other a superfix having the value of 20.
It has been shown that the last six glyphs in figure60(A4, A5, A6, A7, A8, and A9) all belong to the same series. Let us next examine the seventh glyph or term from the bottom (A3) and see how it is formed. We have seen that in the only two texts in which more than six periods are recorded the signs for the seventh period (see fig.61,d,e) are composed of the same elements in each: (1) The cycle sign; (2) a superfix having the hand as its principal element. We have seen, further, that in the only three places in which great cycles are recorded in the Maya writing (fig.61,a-c) the coefficient in every case is greater than 13, thus showing that in all probability 20, not 13, great cycles made 1 great-great cycle.
Therefore, since the great-great cycle signs in figure61,d,e, are composed of the cycle sign plus a superfix (*), this superfix must have the value of 400 (20 × 20) in order to make the whole glyph have the value of 20 great cycles, or 1 great-great cycle (20 × 2,880,000 = 57,600,000). In other words, it seems highly probable (1) that the glyphs in figure61,d,e, are signs for the great-great cycle or period of the seventh place, and (2) that the great-great cycle was composed of 20 great cycles, shown graphically by two elements, one being the cycle sign itself and the other a hand having the value of 400.
It has been shown that the first seven glyphs (A3, A4, A5, A6, A7, A8, and A9) probably all belong to the same series. Let us next examine the eighth term (A2) and see how it is formed.
As stated above, comparative evidence can help us no further, since the text under discussion is the only one which presents a number composed of more than seven terms. Nevertheless, the writer believes it will be possible to show by the morphology of this, the only glyph which occupies the position of an eighth term, that it is 20 times the glyph in the seventh position, and consequently that the vigesimal system was perfect to the highest known unit found in the Maya writing.
We have seen (1) that the sixth term was composed of the fifth term plus a superfix which increased the fifth 20 times, and (2) that the seventh term was composed of the fifth term plus a superfix which increased the fifth 400 times, or the sixth 20 times.
Now let us examine the only known example of a sign for the eighth term (A2, fig.60). This glyph is composed of (1) the cycle sign; (2) a superfix of two elements, (a) the hand, and (b) a semicircular element in which dots appear.
But this same hand in the super-fix of the great-great cycle increased the cycle sign 400 times (20 × 20; see A3, fig.60). Therefore we must assume the same condition obtains here. And finally, since the eighth term = 20 × 20 × 20 × cycle, we must recognize in the second element of the superfix (*) a sign which means 20.
A close study of this element shows that it has two important points of resemblance to the superfix of the great-cycle glyph (see A4, fig.60), which was shown to have the value 20: (1) Both elements have the same outline, roughly semicircular; (2) both elements have the same chain of dots around their edges.
Compare this element in A2, figure60, with the superfixes in figure61,a,b, bearing in mind that there is more than 275 years' difference in time between the carving of A2, figure60, anda, figure61, and more than 200 years between the former and figure61,b. The writer believes both are variants of the same element, and consequently A2, figure60, is probably composed of elements which signify 20 × 400 (20 × 20) × the cycle, which equals one great-great-great cycle, or term of the eighth place.
Thus on the basis of the glyphs themselves it seems possible to show that all belong to one and the same numerical series, which progresses according to the terms of a vigesimal system of numeration.
The several points supporting this conclusion may be summarized as follows:
1. The eight periods[84]in figure60are consecutive, their sequence being uninterrupted throughout. Consequently it seems probable that all belong to one and the same number.
2. It has been shown that the highest three period glyphs are composed of elements which multiply the cycle sign by 20, 400, and 8,000, respectively, which has to be the case if they are the sixth, seventh, and eighth terms, respectively, of the Maya vigesimal system of numeration.
3. The highest three glyphs have numerical coefficients, just like the five lower ones; this tends to show that all eight are terms of the same numerical series.
4. In the two texts which alone can furnish comparative data for this sixth term, the sixth-period glyph in each is identical with A4, figure60, thus showing the existence of a sixth period in the inscriptions and a generally[85]accepted sign for it.
5. In the only other text which can furnish comparative data for the seventh term, the period glyph in its seventh place is identicalwith A3, figure60; thus showing the existence of a seventh period in the inscriptions and a generally accepted sign for it.
6. The one term higher than the cycle in the Copan text, the two terms higher in the Palenque text, and the three terms higher in this text, are all built on the same basic element, the cycle, thus showing that in each case the higher term or terms is a continuation of the same number, not a Period-ending date, as suggested by Mr. Goodman for the Copan text.
7. The other two texts, showing series composed of more than five terms, have all their period glyphs in an unbroken sequence in each, like the text under discussion, thus showing that in each of these other two texts all the terms present probably belong to one and the same number.
8. Finally, the two occurrences of the cycle sign with a coefficient above 13, and the three occurrences of the great-cycle sign with a coefficient above 13, indicate that 20, not 13, was the unit of progression in the higher numbers in the inscriptions just as it was in the codices.
Before closing the discussion of this unique inscription, there is one other important point in connection with it which must be considered, because of its possible bearing on the meaning of the Initial-series introducing glyph.
The first five glyphs on the east side of Stela 10 at Tikal are not illustrated in figure60. The sixth glyph is A1 in figure60, and the remaining glyphs in this figure carry the text to the bottom of this side of the monument. The first of these five unfigured glyphs is very clearly an Initial-series introducing glyph. Of this there can be no doubt. The second resembles the day8 Manik, though it is somewhat effaced. The remaining three are unknown. The next glyph, A1, figure60, is very clearly another Initial-series introducing glyph, having all of the five elements common to that sign. Compare A1 with the forms for the Initial series introducing glyph in figure24. This certainly would seem to indicate that an Initial Series is to follow. Moreover, the fourth glyph of the eight-term number following in A2-A9, inclusive (that is, A5), records "Cycle 9," the cycle in which practically all Initial-series dates fall. Indeed, if A2, A3, and A4 were omitted and A5, A6, A7, A8, and A9 were recorded immediately after A1, the record would be that of a regular Initial-series number (9.3.6.2.0). Can this be a matter of chance? If not, what effect can A2, A3, and A4 have on the Initial-series date in A1, A5-A9?
The writer believes that the only possible effect they could have would be to fix Cycle 9 of Maya chronology in a far more comprehensive and elaborate chronological conception, a conception whichindeed staggers the imagination, dealing as it does with more than five million years.
If these eight terms all belong to one and the same numerical series, a fact the writer believes he has established in the foregoing pages, it means that Cycle 9, the first historic period of the Maya civilization, was Cycle 9 of Great Cycle 19 of Great-great Cycle 11 of Great-great-great Cycle 1. In other words, the starting point of Maya chronology, which we have seen was the date4 Ahau 8 Cumhu, 9 cycles before the close of a Cycle 9, was in reality 1. 11. 19. 0. 0. 0. 0. 0.4 Ahau 8 Cumhu, or simply a fixed point in a far vaster chronological conception.
Furthermore, it proves, as contended by the writer on page113, that a great cycle came to an end on this date,4 Ahau 8 Cumhu. This is true because on the above date (1. 11. 19. 0. 0. 0. 0. 0.4 Ahau 8 Cumhu) all the five periods lower than the great cycle are at 0. It proves, furthermore, as the writer also contended, that the date4 Ahau 8 Zotz, 13 cycles in advance of the date4 Ahau 8 Cumhu, did not end a great cycle—
but, on the contrary, was a Cycle 7 of Great Cycle 18, the end of which (19. 0. 0. 0. 0. 0.4 Ahau 8 Cumhu) was the starting point of Maya chronology.
It seems to the writer that the above construction is the only one that can be put on this text if we admit that the eight periods in A2-A9, figure60, all belong to one and the same numerical series.
Furthermore, it would show that the great cycle in which fell the first historic period of the Maya civilization (Cycle 9) was itself the closing great cycle of a great-great cycle, namely, Great-great Cycle 11:
That is to say, that when Great Cycle 19 had completed itself, Great-great Cycle 12 would be ushered in.
We have seen on pages108-113that the names of the cycles followed one another in this sequence: Cycle 1, Cycle 2, Cycle 3, etc., to Cycle 13, which was followed by Cycle 1, and the sequence repeated itself. We saw, however, that these names probably had nothing to do with the positions of the cycles in the great cycle; that on the contrary these numbers were names and not positions in a higher term.
Now we have seen that Maya chronology began with a Cycle 1; that is, it was counted from the end of a Cycle 13. Therefore, theclosing cycle of Great Cycle 19 of Great-great Cycle 11 of Great-great-great Cycle 1 was a Cycle 13, that is to say, 1. 11. 19. 0. 0. 0. 0. 0.4 Ahau 13 Cumhuconcluded a great cycle, the closing cycle of which was named Cycle 13. This large number, composed ofonegreat-great-great cycle,elevengreat-great cycles, andnineteengreat cycles, contains exactly 12,780 cycles, as below:
But the closing cycle of this number was named Cycle 13, and by deducting all the multiples of 13 possible (983) we can find the name of the first cycle of Great-great-great Cycle 1, the highest Maya time period of which we have any knowledge: 983 × 13 = 12,779. And deducting this from the number of cycles involved (12,780), we have—
This counted backward from Cycle 1, brings us again to a Cycle 13 as the name of the first cycle in the Maya conception of time. In other words, the Maya conceived time to have commenced, in so far as we can judge from the single record available, with a Cycle 13, not with the beginning of a Cycle 1, as they did their chronology.
We have still to explain A1, figure60. This glyph is quite clearly a form of the Initial-series introducing glyph, as already explained, in which the five components of that glyph are present in usual form: (1) Trinal superfix; (2) pair of comb-like lateral appendages; (3) the tun sign; (4) the trinal subfix; (5) the variable central element, here represented by a grotesque head.
Of these, the first only claims our attention here. The trinal superfix in A1 (fig.60), as its name signifies, is composed of three parts, but, unlike other forms of this element, the middle part seems to be nothing more nor less than a numerical dot or 1. The question at once arises, can the two flanking parts be merely ornamental and the whole element stand for the number 1? The introducing glyph at the beginning of this text (not figured here), so far as it can be made out, has a trinal superfix of exactly the same character—a dot with an ornamental scroll on each side. What can be the explanation of this element, and indeed of the whole glyph? Is it one great-great-great-great cycle—a period twenty times as great as the one recorded in A2, or is it not a term of the series in glyphs A2-A9?The writer believes that whatever it may be, it is at leastnota member of this series, and in support of his belief he suggests that if it were, why should it alone be retained in recordingallInitial-series dates, whereas the other three—the great-great-great cycle, the great-great cycle, and the great-cycle signs—have disappeared.
The following explanation, the writer believes, satisfactorily accounts for all of these points, though it is advanced here only by way of suggestion as a possible solution of the meaning of the Initial-series introducing glyph. It is suggested that in A1 we may have a sign representing "eternity," "this world," "time"; that is to say, a sign denoting the duration of the present world-epoch, the epoch of which the Maya civilization occupied only a small part. The middle dot of the upper element, being 1, denotes that this world-epoch is the first, or present, one, and the whole glyph itself might mean "the present world." The appropriateness of such a glyph ushering in every Initial-series date is apparent. It signified time in general, while the succeeding 7 glyphs denoted what particular day of time was designated in the inscription.
But why, even admitting the correctness of this interpretation of A1, should the great-great-great cycle, the great-great cycle, and the great cycle of their chronological scheme be omitted, and Initial-series dates always open with this glyph, which signifies time in general, followed by the current cycle? The answer to this question, the writer believes, is that the cycle was the greatest period with which the Maya could have had actual experience. It will be shown in Chapter V that there are a few Cycle-8 dates actually recorded, as well as a half a dozen Cycle-10 dates. That is, the cycle, which changed its coefficient every 400 years, was a period which they couldnotregard as never changing within the range of human experience. On the other hand, it was the shortest period of which they were uncertain, since the great cycle could change its coefficient only every 8,000 years—practically eternity so far as the Maya were concerned. Therefore it could be omitted as well as the two higher periods in a date without giving rise to confusion as to which great cycle was the current one. The cycle, on the contrary, had to be given, as its coefficient changed every 400 years, and the Maya are known to have recorded dates in at least three cycles—Nos. 8, 9, and 10. Hence, it was Great Cycle 19 for 8,000 years, Great-great Cycle 11 for 160,000, and Great-great-great Cycle 1 for 3,200,000 years, whereas it was Cycle 9 for only 400 years. This, not the fact that the Maya never had a period higher than the cycle, the writer believes was the reason why the three higher periods were omitted from Initial-series dates—they were unnecessary so far as accuracy was concerned, since there could never be any doubt concerning them.
It is not necessary to press this point further, though it is believed the foregoing conception of time had actually been worked out by the Maya. The archaic date recorded by Stela 10 at Tikal (9.3.6.2.0) makes this monument one of the very oldest in the Maya territory; indeed, there is only one other stela which has an earlier Initial Series, Stela 3 at Tikal. In the archaic period from which this monument dates the middle dot of the trinal superfix in the Initial-series introducing glyph may still have retained its numerical value, 1, but in later times this middle dot lost its numerical characteristics and frequently appears as a scroll itself.
The early date of Stela 10 makes it not unlikely that this process of glyph elaboration may not have set in at the time it was erected, and consequently that we have in this simplified trinal element the genesis of the later elaborated form; and, finally, that A1, figure60, may have meant "the present world-epoch" or something similar.
In concluding the presentation of these three numbers the writer may express the opinion that a careful study of the period glyphs in figures58-60will lead to the following conclusions: (1) That the six periods recorded in the first, the seven in the second, and the eight or nine in the third, all belong to the same series in each case; and (2) that throughout the six terms of the first, the seven of the second, and the eight of the third, the series in each case conforms strictly to the vigesimal system of numeration given in TableVIII.
As mentioned on page116(footnote 2), in this method of recording the higher numbers the kin sign may sometimes be omitted without affecting the numerical value of the series wherein the omission occurs. In such cases the coefficient of the kin sign is usually prefixed to the uinal sign, the coefficient of the uinal itself standing above the uinal sign. In figure58, for example, the uinal and the kin coefficients are both 0. In this case, however, the 0 on the left of the uinal sign is to be understood as belonging to the kin sign, which is omitted, while the 0 above the uinal sign is the uinal's own coefficient 0. Again in figure59, the kin sign is omitted and the kin coefficient 1 is prefixed to the uinal sign, while the uinal's own coefficient 12 stands above the uinal sign. Similarly, the 12 uinals and 17 kins recorded in figure56,d, might as well have been written as inoof the same figure, that is, with the kin sign omitted and its coefficient 17 prefixed to the uinal sign, while the uinal's own coefficient 12 appears above. Or again, the 9 uinals and 18 kins recorded infalso might have been written as inp, that is, with the kin sign omitted and the kin coefficient 18 prefixed to the uinal sign while the uinal's own coefficient 9 appears above.
In all the above examples the coefficients of the omitted kin signs are on theleftof the uinal signs, while the uinal coefficients areabovethe uinal signs. Sometimes, however, these positions are reversed,and the uinal coefficient standson the leftof the uinal sign, while the kin coefficient standsabove. This interchange in certain cases probably resulted from the needs of glyphic balance and symmetry. For example, in figure62,a, had the kin coefficient 19 been placed on the left of the uinal sign, the uinal coefficient 4 would have been insufficient to fill the space above the period glyph, and consequently the corner of the glyph block would have appeared ragged. The use of the 19aboveand the 4 to the left, on the other hand, properly fills this space, making a symmetrical glyph. Such cases, however, are unusual, and the customary position of the kin coefficient, when the kin sign is omitted, is on the left of the uinal sign, not above it. This practice, namely, omitting the kin sign in numerical series, seems to have prevailed extensively in connection with both Initial Series and Secondary Series; indeed, in the latter it is the rule to which there are but few exceptions.