Chapter 12 of 31 · 12983 words · ~65 min read

Chapter VI

is reached, which treats of the codices exclusively. {96}

HEAD-VARIANT NUMERALS

Let us next turn to the consideration of the Maya "Arabic notation," that is, the head-variant numerals, which, like all other known head variants, are practically restricted to the inscriptions.[67] It should be noted here before proceeding further that the full-figure numerals found in connection with full-figure period, day, and month glyphs in a few inscriptions, have been classified with the head-variant numerals. As explained on page 67, the body-parts of such glyphs have no function in determining their meanings, and it is only the head-parts which present in each case the determining characteristics of the form intended.

In the "head" notation each of the numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13[68] is expressed by a distinctive type of head; each type has its own essential characteristic, by means of which it can be distinguished from all of the others. Above 13 and up to but _not including_ 20, the head numerals are expressed by the application of the essential characteristic of the head for 10 to the heads for 3 to 9, inclusive. No head forms for the numeral 20 have yet been discovered.

The identification of these head-variant numerals in some cases is not an easy matter, since their determining characteristics are not always presented clearly. Moreover, in the case of a few numerals, notably the heads for 2, 11, and 12, the essential elements have not yet been determined. Head forms for these numerals occur so rarely in the inscriptions that the comparative data are insufficient to enable us to fix on any particular element as the essential one. Another difficulty encountered in the identification of head-variant numerals is the apparent irregularity of the forms in the earlier inscriptions. The essential elements of these early head numerals in some cases seem to differ widely from those of the later forms, and consequently it is sometimes difficult, indeed even impossible, to determine their corresponding numerical values. {97}

[Illustration: FIG. 51. Head-variant numerals 1 to 7, inclusive.]

The head-variant numerals are shown in figures 51-53. Taking these up in their numerical order, let us commence with the head signifying 1; see figure 51, _a-e_. The essential element of this head is its forehead ornament, which, to signify the number 1, must be composed of more than one part (), in order to distinguish it from the forehead ornament (), which, as we shall see presently, is the essential element of the head for 8 (fig. 52, _a-f_). Except for their forehead ornaments the heads for 1 and 8 are almost identical, and great care must be exercised in order to avoid mistaking one for the other. {98}

The head for 2 (fig. 51, _f_, _g_) has been found only twice in the inscriptions--on Lintel 2 at Piedras Negras and on the tablet in the Temple of the Initial Series at Holactun. The oval at the top of the head seems to be the only element these two forms have in common, and the writer therefore accepts this element as the essential characteristic of the head for 2, admitting at the same time that the evidence is insufficient.

[Illustration: FIG. 52. Head-variant numerals 8 to 13, inclusive.]

The head for 3 is shown in figure 51, _h_, _i_. Its determining characteristic is the fillet, or headdress.

The head for 4 is shown in figure 51, _j-m_. It is to be distinguished by its large prominent eye and square irid (). (probably eroded in _l_), the snaglike front tooth, and the curling fang protruding from the back part of the mouth () (wanting in _l_ and _m_). {99}

The head for 5 (fig. 51, _n-s_) is always to be identified by its peculiar headdress (), which is the normal form of the tun sign. Compare figure 29, _a_, b. The same element appears also in the head for 15 (see fig. 53, _b-e_). The head for 5 is one of the most constant of all the head numerals.

[Illustration: FIG. 53. Head-variant numerals 14 to 19, inclusive, and 0.]

The head for 6 (fig. 51, _t-v_) is similarly unmistakable. It is always characterized by the so-called hatchet eye (), which appears also in the head for 16 (fig. 53, _f-i_).

The head for 7 (fig. 51, _w_) is found only once in the inscriptions--on the east side of Stela D at Quirigua. Its essential characteristic, {100} the large ornamental scroll passing under the eye and curling up in front of the forehead (), is better seen in the head for 17 (fig. 53, _j-m_).

The head for 8 is shown in figure 52, _a-f_. It is very similar to the head for 1, as previously explained (compare figs. 51, _a-e_ and 52, _a-f_), and is to be distinguished from it only by the character of the forehead ornament, which is composed of but a single element (). In figure 52, _a_, _b_, this takes the form of a large curl. In _c_ of the same figure a flaring element is added above the curl and in _d_ and _e_ this element replaces the curl. In _f_ the tongue or tooth of a grotesque animal head forms the forehead ornament. The heads for 18 (fig. 53, _n-q_) follow the first variants (fig. 51, _a_, _b_), having the large curl, except _q_, which is similar to _d_ in having a flaring element instead.

The head for 9 occurs more frequently than all of the others with the exception of the zero head, because the great majority of all Initial Series record dates which fell after the completion of Cycle 9, but before the completion of Cycle 10. Consequently, 9 is the coefficient attached to the cycle glyph in almost all Initial Series.[69] The head for 9 is shown in figure 52, _g-l_. It has for its essential characteristic the dots on the lower cheek or around the mouth (). Sometimes these occur in a circle or again irregularly. Occasionally, as in _j-l_, the 9 head has a beard, though this is not a constant element as are the dots, which appear also in the head for 19. Compare figure 53, _r_.

The head for 10 (fig. 52, _m-r_) is extremely important since its essential element, the fleshless lower jaw (), stands for the numerical value 10, in composition with the heads for 3, 4, 5, 6, 7, 8, and 9, to form the heads for 13, 14, 15, 16, 17, 18, and 19, respectively. The 10 head is clearly the fleshless skull, having the truncated nose and fleshless jaws (see fig. 52, _m-p_). The fleshless lower jaw is shown in profile in all cases but one--Zoömorph B at Quirigua (see _r_ of the same figure). Here a full front view of a 10 head is shown in which the fleshless jaw extends clear across the lower part of the head, an interesting confirmation of the fact that this characteristic is the essential element of the head for 10.

The head for 11 (fig. 52, _s_) has been found only once in the inscriptions, namely, on Lintel 2 at Piedras Negras; hence comparative data are lacking for the determination of its essential element. This head has no fleshless lower jaw and consequently would seem, therefore, not to be built up of the heads for 1 and 10.

Similarly, the head for 12 (fig. 52, _t-v_) has no fleshless lower jaw, and consequently can not be composed of the heads for 10 and 2. It is to be noted, however, that all three of the faces are of the same type, even though their essential characteristic has not yet been determined. {101}

The head for 13 is shown in figure 52, _w-b'_. Only the first of these forms, _w_, however, is built on the 10 + 3 basis. Here we see the characteristic 3 head with its banded headdress or fillet (compare _h_ and _i_, fig. 51), to which has been added the essential element of the 10 head, the fleshless lower jaw, the combination of the two giving the head for 13. The other form for 13 seems to be a special character, and not a composition of the essential elements of the heads for 3 and 10, as in the preceding example. This form of the 13 head (fig. 52, _x-b'_) is grotesque. It seems to be characterized by its long pendulous nose surmounted by a curl (), its large bulging eye (**), and a curl () or fang (++) protruding from the back part of the mouth. Occurrences of the first type--the composite head--are very rare, there being only two examples of this kind known in all the inscriptions. The form given in _w_ is from the Temple of the Cross at Palenque, and the other is on the Hieroglyphic Stairway at Copan. The individual type, having the pendulous nose, bulging eye, and mouth curl is by far the more frequent.

The head for 14 (fig. 53, _a_) is found but once--in the inscriptions on the west side of Stela F at Quirigua. It has the fleshless lower jaw denoting 10, while the rest of the head shows the characteristics of 4--the bulging eye and snaglike tooth (compare fig. 51, _j-m_). The curl protruding from the back part of the mouth is wanting because the whole lower part of the 4 head has been replaced by the fleshless lower jaw.

The head for 15 (fig. 53, _b-e_) is composed of the essential element of the 5 head (the tun sign; see fig. 51, _n-s_) and the fleshless lower jaw of the head for 10.

The head for 16 (fig. 53, _f-i_) is characterized by the fleshless lower jaw and the hatchet eye of the 6 head. Compare figures 51, _t-v_, and 52, _m-r_, which together form 16 (10 + 6).

The head for 17 (fig. 53, _j-m_) is composed of the essential element of the 7 head (the scroll projecting above the nose; see fig. 51, _w_) and the fleshless lower jaw of the head for 10.

The head for 18 (fig. 53, _n-q_) has the characteristic forehead ornament of the 8 head (compare fig. 52, _a-f_) and the fleshless lower jaw denoting 10.

Only one example (fig. 53, _r_) of the 19 head has been found in the inscriptions. This occurs on the Temple of the Cross at Palenque and seems to be formed regularly, both the dots of the 9 head and the fleshless lower jaw of the 10 head appearing.

The head for 0 (zero), figure 53, _s-w_, is always to be distinguished by the hand clasping the lower part of the face (). In this sign for zero, the hand probably represents the idea "ending" or "closing," just as it seems to have done in the ending signs used with {102} Period-ending dates. According to the Maya conception of time, when a period had ended or closed it was at zero, or at least no new period had commenced. Indeed, the normal form for zero in figure 47, the head variant for zero in figure 53, _s-w_, and the form for zero shown in figure 54 are used interchangeably in the same inscription to express the same idea--namely, that no periods thus modified are involved in the calculations and that consequently the end of some higher period is recorded; that is, no fractional parts of it are present.

That the hand in "ending signs" had exactly the same meaning as the hand in the head variants for zero (fig. 53, _s-w_) receives striking corroboration from the rather unusual sign for zero shown in figure 54, to which attention was called above. The essential elements of this sign are[70] (1) the clasped hand, identical with the hand in the head-variant forms for zero, and (2) the large element above it, containing a curling infix. This latter element also occurs though below the clasped hand, in the "ending signs" shown in figure 37, _l_, _m_, _n_, the first two of which accompany the closing date of Katun 14, and the last the closing date of Cycle 13. The resemblance of these three "ending signs" to the last three forms in figure 54 is so close that the conclusion is well-nigh inevitable that they represented one and the same idea. The writer is of the opinion that this meaning of the hand (ending or completion) will be found to explain its use throughout the inscriptions.

[Illustration: FIG. 54. A sign for 0, used also to express the idea "ending" or "end of" in Period-ending dates. (See figs. 47 and 53 _s-w_, for forms used interchangeably in the inscriptions to express the idea of 0 or of completion.)]

In order to familiarize the student with the head-variant numerals, their several essential characteristics have been gathered together in Table X, where they may be readily consulted. Examples covering their use with period, day, and month glyphs are given in figure 55 with the corresponding English translations below.

Head-variant numerals do not occur as frequently as the bar and dot forms, and they seem to have been developed at a much later period. At least, the earliest Initial Series recorded with bar and dot numerals antedates by nearly two hundred years the earliest Initial Series the numbers of which are expressed by head variants. This long priority in the use of the former would doubtless be considerably diminished if it were possible to read the earliest Initial Series which {103} have head-variant numerals; but that the earliest of these latter antedate the earnest bar and dot Initial Series may well be doubted.

TABLE X. CHARACTERISTICS OF HEAD-VARIANT NUMERALS 0 TO 19, INCLUSIVE

Mention should be made here of a numerical form which can not be classified either as a bar and dot numeral or a head variant. This is the thumb (), which has a numerical value of one.

We have seen in the foregoing pages the different characters which stood for the numerals 0 to 19, inclusive. The next point claiming our attention is, how were the higher numbers written, numbers which in the codices are in excess of 12,000,000, and in the inscriptions, in excess of 1,400,000? In short, how were numbers so large expressed by the foregoing twenty (0 to 19, inclusive) characters?

The Maya expressed their higher numbers in two ways, in both of which the numbers rise by successive terms of the same vigesimal system:

1. By using the numbers 0 to 19, inclusive, as multipliers with the several periods of Table VIII (reduced in each case to units of the lowest order) as the multiplicands, and--

2. By using the same numbers[71] in certain relative positions, each of which had a fixed numerical value of its own, like the positions to the right and left of the decimal point in our own numerical notation. {104}

The first of these methods is rarely found outside of the inscriptions, while the second is confined exclusively to the codices. Moreover, although the first made use of both normal-form and head-variant numerals, the second could be expressed by normal forms only, that is, bar and dot numerals. This enables us to draw a comparison between these two forms of Maya numerals:

[Illustration: FIG. 55. Examples of the use of head-variant numerals with period, day, or month signs. The translation of each glyph appears below it.]

Head-variant numerals never occur independently, but are always prefixed to some period, day, or month sign. Bar and dot numerals, on the other hand, frequently stand by themselves in the codices unattached to other signs. In such cases, however, some sign was to be supplied mentally with the bar and dot numeral. {105}

FIRST METHOD OF NUMERATION

[Illustration: FIG. 56. Examples of the first method of numeration, used almost exclusively in the inscriptions.]

In the first of the above methods the numbers 0 to 19, inclusive, were expressed by multiplying the kin sign by the numerals[72] 0 to 19 in turn. Thus, for example, 6 days was written as shown in figure 56, _a_, 12 days as shown in _b_, and 17 days as shown in _c_ of the same {106} figure. In other words, up to and including 19 the numbers were expressed by prefixing the sign for the number desired to the kin sign, that is, the sign for 1 day.[73]

The numbers 20 to 359, inclusive, were expressed by multiplying both the kin and uinal signs by the numerical forms 0 to 19, and adding together the resulting products. For example, the number 257 was written as shown in figure 56, d. We have seen in Table VIII that 1 uinal = 20 kins, consequently 12 uinals (the 12 being indicated by 2 bars and 2 dots) = 240 kins. However, as this number falls short of 257 by 17 kins, it is necessary to express these by 17 kins, which are written immediately below the 12 uinals. The sum of these two products = 257. Again, the number 300 is written as in figure 56, e. The 15 uinals (three bars attached to the uinal sign) = 15 × 20 = 300 kins, exactly the number expressed. However, since no kins are required to complete the number, it is necessary to show that none were involved, and consequently 0 kins, or "no kins" is written immediately below the 15 uinals, and 300 + 0 = 300. One more example will suffice to show how the numbers 20 to 359 were expressed. In figure 56, _f_, the number 198 is shown. The 9 uinals = 9 × 20 = 180 kins. But this number falls short of 198 by 18, which is therefore expressed by 18 kins written immediately below the 9 uinals: and the sum of these two products is 198, the number to be recorded.

The numbers 360 to 7,199, inclusive, are indicated by multiplying the kin, uinal, and tun signs by the numerals 0 to 19, and adding together the resulting products. For example, the number 360 is shown in figure 56, _g_. We have seen in Table VIII that 1 tun = 18 uinals; but 18 uinals = 360 kins (18 × 20 = 360); therefore 1 tun also = 360 kins. However, in order to show that no uinals and kins are involved in forming this number, it is necessary to record this fact, which was done by writing 0 uinals immediately below the 1 tun, and 0 kins immediately below the 0 uinals. The sum of these three products equals 360 (360 + 0 + 0 = 360). Again, the number 3,602 is shown in figure 56, _h_. The 10 tuns = 10 × 360 = 3,600 kins. This falls short of 3,602 by only 2 units of the first order (2 kins), therefore no uinals are involved in forming this number, a fact which is shown by the use of 0 uinals between the 10 tuns and 2 kins. The sum of these three products = 3,602 (3,600 + 0 + 2). Again, in figure 56, _i_, the number 7,100 is recorded. The 19 tuns = 19 × 360 = 6,840 kins, which falls short of 7,100 kins by 7,100 - 6,840 = 260 kins. But 260 kins = 13 uinals with no kins {107} remaining. Consequently, the sum of these products equals 7,100 (6,840 + 260 + 0).

The numbers 7,200 to 143,999 were expressed by multiplying the kin, uinal, tun, and katun signs by the numerals 0 to 19, inclusive, and adding together the resulting products. For example, figure 56, _j_, shows the number 7,204. We have seen in Table VIII that 1 katun = 20 tuns, and we have seen that 20 tuns = 7,200 kins (20 × 360); therefore 1 katun = 7,200 kins. This number falls short of the number recorded by exactly 4 kins, or in other words, no tuns or uinals are involved in its composition, a fact shown by the 0 tuns and 0 uinals between the 1 katun and the 4 kins. The sum of these four products = 7,204 (7,200 + 0 + 0 + 4). The number 75,550 is shown in figure 56, _k_. The 10 katuns = 72,000; the 9 tuns, 3,240; the 15 uinals, 300; and the 10 kins, 10. The sum of these four products = 75,550 (72,000 + 3,240 + 300 + 10). Again, the number 143,567 is shown in figure 56, _l_. The 19 katuns = 136,800; the 18 tuns, 6,480; the 14 uinals, 280; and the 7 kins, 7. The sum of these four products = 143,567 (136,800 + 6,480 + 280 + 7).

The numbers 144,000 to 1,872,000 (the highest number, according to some authorities, which has been found[74] in the inscriptions) were expressed by multiplying the kin, uinal, tun, katun, and cycle signs by the numerals 0 to 19, inclusive, and adding together the resulting products. For example, the number 987,322 is shown in figure 56, _m_. We have seen in Table VIII that 1 cycle = 20 katuns, but 20 katuns = 144,000 kins; therefore 6 cycles = 864,000 kins; and 17 katuns = 122,400 kins; and 2 tuns, 720 kins; and 10 uinals, 200 kins; and the 2 kins, 2 kins. The sum of these five products equals the number recorded, 987,322 (864,000 + 122,400 + 720 + 200 + 2). The highest number in the inscriptions upon which all are agreed is 1,872,000, as shown in figure 56, _n_. It equals 13 cycles (13 × 144,000), and consequently all the periods below--the katun, tun, uinal, and kin--are indicated as being used 0 times.

NUMBER OF CYCLES IN A GREAT CYCLE

This brings us to the consideration of an extremely important point concerning which Maya students entertain two widely different opinions; and although its presentation will entail a somewhat lengthy digression from the subject under consideration it is so pertinent to the general question of the higher numbers and their formation, that the writer has thought best to discuss it at this point.

In a vigesimal system of numeration the unit of increase is 20, and so far as the codices are concerned, as we shall presently see, this {108} number was in fact the only unit of progression used, except in the 2d order, in which 18 instead of 20 units were required to make 1 unit of the 3d order. In other words, in the codices the Maya carried out their vigesimal system to _six places_ without a break other than the one in the 2d place, just noted. See Table VIII.

In the inscriptions, however, there is some ground for believing that only 13 units of the 5th order (cycles), not 20, were required to make 1 unit of the 6th order, or 1 great cycle. Both Mr. Bowditch (1910: App. IX, 319-321) and Mr. Goodman (1897: p. 25) incline to this opinion, and the former, in Appendix IX of his book, presents the evidence at some length for and against this hypothesis.

This hypothesis rests mainly on the two following points:

1. That the cycles in the inscriptions are numbered from 1 to 13, inclusive, and not from 0 to 19, inclusive, as in the case of all the other periods except the uinal, which is numbered from 0 to 17, inclusive.

2. That the only two Initial Series which are not counted from the date 4 Ahau 8 Cumhu, the starting point of Maya chronology, are counted from a date 4 Ahau 8 Zotz, which is exactly 13 cycles in advance of the former date.

Let us examine the passages in the inscriptions upon which these points rest. In three places[75] in the inscriptions the date 4 Ahau 8 Cumhu is declared to have occurred at the end of a Cycle 13; that is, in these three places this date is accompanied by an "ending sign" and a Cycle 13. In another place in the inscriptions, although the starting point 4 Ahau 8 Cumhu is not itself expressed, the second cycle thereafter is declared to have been a Cycle 2, not a Cycle 15, as it would have been had the cycles been numbered from 0 to 19, inclusive, like all the other periods.[76] In still another place the ninth cycle after the starting point (that is, the end of a Cycle 13) is not a Cycle 2 in the _following_ great cycle, as would be the case if the cycles were numbered from 0 to 19, inclusive, but a Cycle 9, as if the cycles were numbered from 1 to 13. Again, the end of the tenth cycle after the starting point is recorded in several places, but not as Cycle 3 of the following great cycle, as if the cycles were numbered from 0 to 19, inclusive, but as Cycle 10, as would be the case if the cycles were numbered from 1 to 13. The above examples leave little doubt that the cycles were numbered from 1 to 13, inclusive, and not from 0 to 19, as in the case of the other periods. Thus, there can be no question concerning the truth of the first of the two above points on which this hypothesis rests. {109}

But because this is true it does not necessarily follow that 13 cycles made 1 great cycle. Before deciding this point let us examine the two Initial Series mentioned above, as _not_ proceeding from the date 4 Ahau 8 Cumhu, but from a date 4 Ahau 8 Zotz, exactly 13 cycles in advance of the former date.

These are in the Temple of the Cross at Palenque and on the east side of Stela C at Quirigua. In these two cases, if the long numbers expressed in terms of cycles, katuns, tuns, uinals, and kins are reduced to kins, and counted forward from the date 4 Ahau 8 Cumhu, the starting point of Maya chronology, in neither case will the recorded terminal day of the Initial Series be reached; hence these two Initial Series could not have had the day 4 Ahau 8 Cumhu as their starting point. It may be noted here that these two Initial Series are the only ones throughout the inscriptions known at the present time which are not counted from the date 4 Ahau 8 Cumhu.[77] However, by counting _backward_ each of these long numbers from their respective terminal days, 8 Ahau 18 Tzec, in the case of the Palenque Initial Series, and 4 Ahau 8 Cumhu, in the case of the Quirigua Initial Series, it will be found that both of them proceed from the same starting point, a date 4 Ahau 8 Zotz, exactly 13 cycles in advance of the starting point of Maya chronology. Or, in other words, the starting point of all Maya Initial Series save two, was exactly 13 cycles later than the starting point of these two. Because of this fact and the fact that the cycles were numbered from 1 to 13, inclusive, as shown above, Mr. Bowditch and Mr. Goodman have reached the conclusion that in the inscriptions only 13 cycles were required to make 1 great cycle.

It remains to present the points against this hypothesis, which seem to indicate that the great cycle in the inscriptions contained the same number of cycles (20) as in the codices:

1. In the codices where six orders (great cycles) are recorded it takes 20 of the 5th order (cycles) to make 1 of the 6th order. This absolute uniformity in a strict vigesimal progression in the codices, so similar in other respects to the inscriptions, gives presumptive support at least to the hypothesis that the 6th order in the inscriptions was formed in the same way.

2. The numerical system in both the codices and inscriptions is identical even to the slight irregularity in the second place, where only 18 instead of 20 units were required to make 1 of the third place. It would seem probable, therefore, that had there been any irregularity in the 5th place in the inscriptions (for such the use of 13 in a vigesimal system must be called), it would have been found also in the codices. {110}

3. Moreover, in the inscriptions themselves the cycle glyph occurs at least twice (see fig. 57, _a_, _b_) with a coefficient greater than 13, which would seem to imply that more than 13 cycles could be recorded, and consequently that it required more than 13 to make 1 of the period next higher. The writer knows of no place in the inscriptions where 20 kins, 18 uinals, 20 tuns, or 20 katuns are recorded, each of these being expressed as 1 uinal, 1 tun, 1 katun, and 1 cycle, respectively.[78] Therefore, if 13 cycles had made 1 great cycle, 14 cycles would not have been recorded, as in figure 57, _a_, but as 1 great cycle and 1 cycle; and 17 cycles would not have been recorded, as in _b_ of the same figure, but as 1 great cycle and 4 cycles. The fact that they were not recorded in this latter manner would seem to indicate, therefore, that more than 13 cycles were required to make a great cycle, or unit of the 6th place, in the inscriptions as well as in the codices.

[Illustration: 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 pages 108-109. 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 day 4 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 to {111} admit 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 date 4 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 date 7 Ahau 18 Zip is 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 date 4 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

1 Ik 0 Pop 2 Akbal 1 Pop 3 Kan 2 Pop 4 Chicchan 3 Pop 5 Cimi 4 Pop 6 Manik 5 Pop 7 Lamat 6 Pop 8 Muluc 7 Pop 9 Oc 8 Pop 10 Chuen 9 Pop 11 Eb 10 Pop 12 Ben 11 Pop 13 Ix 12 Pop 1 Men 13 Pop 2 Cib 14 Pop 3 Caban 15 Pop 4 Eznab 16 Pop 5 Cauac 17 Pop 6 Ahau 18 Pop 7 Imix 19 Pop

The numerical coefficients of the days, as we have seen, were numbered from 1 to 13, inclusive, and then began again with 1. See {112} Table XI, in which the 20 days of the month Pop are 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 the _name_ of the period for its _position_ in 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 Table IX. Assuming that a cycle commenced with the first katun there given, the name of this katun is Katun 2 Ahau, although it occupied the _first_ position in the cycle. Again, the name of the second katun in the sequence is Katun 13 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 date 4 Ahau 8 Cumhu in 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 name 13 Ix in Table XI limit the number of days in a uinal to 13, or, again, the 13 of the katun name 13 Ahau in Table IX limit the number of katuns in a cycle to 13. This explanation not only accounts for the use of the 14 cycles or 17 cycles, as {113} shown in figure 57, _a_, _b_, but also satisfactorily provides for the enumeration of the cycles from 1 to 13, inclusive.

If the date "4 Ahau 8 Cumhu ending 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 date 4 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. 114-127 for 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 a _Cycle 1_, and if this were done time would have to be counted from a _Cycle 13_, since a Cycle 1 could follow only a Cycle 13.

On these two probabilities, together with the discussion on pages 114-127, 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 date 4 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 the _end_ of a great cycle the writer does not believe, in view of the evidence presented on pages 114-127. On the contrary, the material given there tends to show that although the cycle which ended on the day 4 Ahau 8 Zotz was also named Cycle 13,[79] it was the 8th division of the grand cycle which ended on the day 4 Ahau 8 Cumhu, {114} 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 page 107, 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 probably _seven_ periods occur (see fig. 59); and (3) on Stela 10 at Tikal eight and perhaps _nine_ periods 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.

[Illustration: FIG. 58. Part of the inscription on Stela N, Copan, showing a number composed of six periods.]

[Illustration: FIG. 59. Part of the inscription in the Temple of the Inscriptions, Palenque, showing a number composed of seven periods.]

[Illustration: 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 figure 56, 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 figure 58 with the highest period involved and reading up, A6,[80] the 14 great cycles = 40,320,000 kins (see Table VIII, in which 1 great cycle = 2,880,000, and consequently 14 = 14 × 2,880,000 = {115} 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 figure 58, 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 the {116} great 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 Table VIII as 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, figure 58, the 18 in B3, figure 59, and the 19 in A4, figure 60, 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. Goodman {117} believes 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 figure 25, _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. 59 and 60, 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 figures 58, A2, and 59, B1 (see fig. 31, _a_, _b_), and the head variant in figure 60, A8. (See fig. 31, _d-f_.) Below the uinal sign in A3, figure 58, and A2, figure 59, and above A7, in figure 60 the 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 figure 29, _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 figure 27, _e-h_. In the Palenque and Tikal texts (see {118} figs. 59, B2, and 60, A6, respectively), on the other hand, the katun is expressed by its normal form, which is identical with the normal form shown in figure 27, _a_, b. In figures 58, A5, and 59, A3, the cycle is expressed by its head variant, and the determining characteristic, the clasped hand, appears in both. Compare the cycle signs in figures 58, A5, and 59, A3, with the head variant for the cycle shown in figure 25; _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 figure 60 at A4) probably stands for the period of the sixth order, the so-called great cycle. These three glyphs are redrawn in figure 61, _a-c_, respectively. In the Copan inscription this glyph (fig. 61, _a_) is a head variant, while in the Palenque and Tikal texts (_a_ and _b_ of 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 figure 61, _a_, and the normal forms in _b_ and _c_ are 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.

[Illustration: 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 figure 61, _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 figure 61, _b_, _c_, the normal forms there presented contain the entire normal form for the cycle sign; compare figure 25, _a_, c. Indeed, except for its superfix, the glyphs in figure 61, _b_, _c_, are normal forms of the cycle sign; and the glyph in _a_ of 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 figure 61, _b_, the superfix is very clear. Just inside the outline and parallel to it there is a line of smaller circles, {119} and in the middle there are two infixes like shepherds' crooks facing away from the center (). In _c_ of 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 in _a_ is somewhat weathered, enough remains to show that it was similar to, if indeed not identical with, the superfixes of the normal forms in _b_ and c. 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 figure 61, _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 glyphs _a_, _b_, and _c_ in figure 61 as 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 figure 61, _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 figure 61, _d_, consists of a hand and a tassel-like postfix, not unlike the upper half of the ending signs in figure 37, _l-q_. However, in the present case, if we accept the hypothesis that _d_ of figure 61 is the sign for the great-great cycle, we are obliged to see in its superfix alone the essential _element_ of the great-great-cycle sign, since the _rest_ of this glyph (the lower part) is quite clearly the normal form for the cycle.

The superfix in figure 61, _e_, consists of the same two elements as the above, with the slight difference that the hand in _e_ holds a rod. Indeed, the similarity of the two forms is so close that in default of {120} any 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 figure 60 is 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 figure 60 shows 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 figure 61, _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 of {121} 20 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 figure 61, _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 figure 60 (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 figure 61, _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 figure 61, _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. {122}

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, figure 60, with the superfixes in figure 61, _a_, _b_, bearing in mind that there is more than 275 years' difference in time between the carving of A2, figure 60, and _a_, figure 61, and more than 200 years between the former and figure 61, b. The writer believes both are variants of the same element, and consequently A2, figure 60, 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 figure 60 are 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, figure 60, 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 identical {123} with A3, figure 60; 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 figure 60. The sixth glyph is A1 in figure 60, 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 day 8 Manik, though it is somewhat effaced. The remaining three are unknown. The next glyph, A1, figure 60, 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 figure 24. 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 which {124} indeed 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 date 4 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 page 113, 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 date 4 Ahau 8 Zotz, 13 cycles in advance of the date 4 Ahau 8 Cumhu, did not end a great cycle--

1. 11. 19. 0. 0. 0. 0. 0. 4 Ahau 8 Cumhu 13. 0. 0. 0. 0. 1. 11. 18. 7. 0. 0. 0. 0. 4 Ahau 8 Zotz

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, figure 60, 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:

1. 11. 19. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 1. 12. 0. 0. 0. 0. 0. 0.

That is to say, that when Great Cycle 19 had completed itself, Great-great Cycle 12 would be ushered in.

We have seen on pages 108-113 that 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, the {125} closing 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 Cumhu concluded a great cycle, the closing cycle of which was named Cycle 13. This large number, composed of _one_ great-great-great cycle, _eleven_ great-great cycles, and _nineteen_ great cycles, contains exactly 12,780 cycles, as below:

1 great-great-great cycle = 1 × 20 × 20 × 20 cycles = 8,000 cycles 11 great-great cycles = 11 × 20 × 20 cycles = 4,400 cycles 19 great cycles = 19 × 20 cycles = 380 cycles ----- 12,780 cycles

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--

12,780 12,779 ------ 1

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, figure 60. 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? {126} The writer believes that whatever it may be, it is at least _not_ a member of this series, and in support of his belief he suggests that if it were, why should it alone be retained in recording _all_ Initial-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