Fig. 48Fig.48. Figure showing possible derivation of the sign for 0 in the inscriptions:a, Outline of the days of the tonalamatl as represented graphically in the Codex Tro-Cortesiano;b, half of same outline, which is also sign for 0 shown in fig.47.
Fig.48. Figure showing possible derivation of the sign for 0 in the inscriptions:a, Outline of the days of the tonalamatl as represented graphically in the Codex Tro-Cortesiano;b, half of same outline, which is also sign for 0 shown in fig.47.
Both normal forms and head variants for zero, as indeed for all the numbers, have been found in the inscriptions. The normal forms for zero are shown in figure47. They are common and are unmistakable. An interesting origin for this sign has been suggested by Mr. A. P. Maudslay. On pages 75 and 76 of the Codex Tro-Cortesiano[64]the 260 days of a tonalamatl are graphically represented as forming the outline shown in figure48,a. Half of this (see fig.48,b) is the sign which stands for zero (compare with fig.47). The train of association by which half of the graphic representation of a tonalamatl could come to stand for zero is not clear. Perhapsaof figure48may have signified that a complete tonalamatl had passed with no additional days. From this the sign may have come to represent the idea of completeness as apart from the tonalamatl, and finally the general idea of completenessapplicable to any period; for no period could be exactly complete without a fractional remainder unless all the lower periods were wanting; that is, represented by zero. Whether this explains the connection between the outline of the tonalamatl and the zero sign, or whether indeed there be any connection between the two, is of course a matter of conjecture.
There is still one more normal form for zero not included in the examples given above, which must be described. This form (fig.49), which occurs throughout the inscriptions and in the Dresden Codex,[65]is chiefly interesting because of its highly specialized function. Indeed, it was used for one purpose only, namely, to express the first, or zero, position in each of the 19 divisions of the haab, or year, and for no other. In other words, it denotes the positions0 Pop,0 Uo,0 Zip, etc., which, as we have seen (pp.47,48), corresponded with our first days of the months. The forms shown in figure49,a-e, are from the inscriptions and those inf-hfrom the Dresden Codex. They are all similar. The general outline of the sign has suggested the name "the spectacle" glyph. Its essential characteristic seems to be the division into two roughly circular parts, one above the other, best seen in the Dresden Codex forms (fig.49,f-h) and a roughly circular infix in each. The lower infix is quite regular in all of the forms, being a circle or ring. The upper infix, however, varies considerably. In figure49,a,b, this ring has degenerated into a loop. Incanddof the same figure it has become elaborated into a head. A simpler form is that infandg. Although comparatively rare, this glyph is so unusual in form that it can be readily recognized. Moreover, if the student will bear in mind the two following points concerning its use, he will never fail to identify it in the inscriptions: The "spectacle" sign (1) can be attached only to the glyphs for the 19 divisions of the haab, or year, that is, the 18 uinals and the xma kaba kin; in other words, it is found only with the glyphs shown in figures19and20, the signs for the months in the inscriptions and codices, respectively.
Fig. 49Fig.49. Special sign for 0 used exclusively as a month coefficient.
Fig.49. Special sign for 0 used exclusively as a month coefficient.
(2) It can occur only in connection with one of the four day-signs,Ik,Manik,Eb, andCaban(see figs.16,c,j,s,t,u,a',b', and17,c,d,k,r,x,y, respectively), since these four alone, as appears in TableVII, can occupy the 0 (zero) positions in the several divisions of the haab.
Fig. 50Fig.50. Examples of the use of bar and dot numerals with period, day, or month signs. The translation of each glyph appears below it.
Fig.50. Examples of the use of bar and dot numerals with period, day, or month signs. The translation of each glyph appears below it.
Examples of the normal-form numerals as used with the day, month, and period glyphs in both the inscriptions and the codices are shown in figure50. Under each is given its meaning in English.[66]The student is advised to familiarize himself with these forms, since on his ability to recognize them will largely depend his progress in reading the inscriptions. This figure illustrates the use of all the foregoing forms except the sign for 20 in figure45and the sign for zero in figure46. As these two forms never occur with day, month, or period glyphs, and as they have been found only in the codices, examples showing their use will not be given until Chapter VI is reached, which treats of the codices exclusively.
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 page67, 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 butnot including20, 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.
Fig. 51Fig.51. Head-variant numerals 1 to 7, inclusive.
Fig.51. Head-variant numerals 1 to 7, inclusive.
The head-variant numerals are shown in figures51-53. Taking these up in their numerical order, let us commence with the head signifying 1; see figure51,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.
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.
Fig. 52Fig.52. Head-variant numerals 8 to 13, inclusive.
Fig.52. Head-variant numerals 8 to 13, inclusive.
The head for 3 is shown in figure51,h,i. Its determining characteristic is the fillet, or headdress.
The head for 4 is shown in figure51,j-m. It is to be distinguished by its large prominent eye and square irid (*). (probably eroded inl), the snaglike front tooth, and the curling fang protruding from the back part of the mouth (**) (wanting inlandm).
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 figure29,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.
Fig. 53Fig.53. Head-variant numerals 14 to 19, inclusive, and 0.
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,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 figure52,a-f. It is very similar to the head for 1, as previously explained (compare figs.51,a-eand52,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 figure52,a,b, this takes the form of a large curl. Incof the same figure a flaring element is added above the curl and indandethis element replaces the curl. Infthe 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, exceptq, which is similar todin 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 figure52,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 inj-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 figure53,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 (seerof 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.
The head for 13 is shown in figure52,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 (comparehandi, 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 inwis 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 figures51,t-v, and52,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), figure53,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 withPeriod-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 figure47, the head variant for zero in figure53,s-w, and the form for zero shown in figure54are 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 figure54, 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 figure37,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 figure54is 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.
Fig. 54Fig.54. A sign for 0, used also to express the idea "ending" or "end of" in Period-ending dates. (See figs.47and53s-w, for forms used interchangeably in the inscriptions to express the idea of 0 or of completion.)
Fig.54. A sign for 0, used also to express the idea "ending" or "end of" in Period-ending dates. (See figs.47and53s-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 TableX, where they may be readily consulted. Examples covering their use with period, day, and month glyphs are given in figure55with 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 whichhave 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 TableVIII(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.
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:
Fig. 55Fig.55. Examples of the use of head-variant numerals with period, day, or month signs. The translation of each glyph appears below it.
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.
First Method of Numeration
Fig. 56Fig.56. Examples of the first method of numeration, used almost exclusively in the inscriptions.
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 figure56,a, 12 days as shown inb, and 17 days as shown incof the samefigure. 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 figure56,d. We have seen in TableVIIIthat 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 figure56,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 figure56,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 figure56,g. We have seen in TableVIIIthat 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 figure56,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 figure56,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 kinsremaining. 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, figure56,j, shows the number 7,204. We have seen in TableVIIIthat 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 figure56,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 figure56,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 figure56,m. We have seen in TableVIIIthat 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 figure56,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, thisnumber 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 tosix placeswithout a break other than the one in the 2d place, just noted. See TableVIII.
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 date4 Ahau 8 Cumhu, the starting point of Maya chronology, are counted from a date4 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 date4 Ahau 8 Cumhuis 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 point4 Ahau 8 Cumhuis 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 thefollowinggreat 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.
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, asnotproceeding from the date4 Ahau 8 Cumhu, but from a date4 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 date4 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 day4 Ahau 8 Cumhuas 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 date4 Ahau 8 Cumhu.[77]However, by countingbackwardeach of these long numbers from their respective terminal days,8 Ahau 18 Tzec, in the case of the Palenque Initial Series, and4 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 date4 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.
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 figure57,a, but as 1 great cycle and 1 cycle; and 17 cycles would not have been recorded, as inbof 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.