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
+-------------+---------------------------------------------------------+
| Forms | Characteristics |
+-------------+---------------------------------------------------------+
| Head for 0 | Clasped hand across lower part of face. |
| Head for 1 | Forehead ornament composed of _more than one part_. |
| Head for 2 | Oval in upper part of head. (?) |
| Head for 3 | Banded headdress or fillet. |
| Head for 4 | Bulging eye with square irid, snaglike front tooth, |
| | curling fang from back of mouth. |
| Head for 5 | Normal form of tun sign as headdress. |
| Head for 6 | "Hatchet eye." |
| Head for 7 | Large scroll passing under eye and curling up in |
| | front of forehead. |
| Head for 8 | Forehead ornament composed of _one part_. |
| Head for 9 | Dots on lower cheek or around mouth and in some |
| | cases beard. |
| Head for 10 | Fleshless lower jaw and in some cases other |
| | death's-head characteristics, truncated nose, etc. |
| Head for 11 | Undetermined. |
| Head for 12 | Undetermined; type of head known, however. |
| Head for 13 | (_a_) Long pendulous nose, bulging eye, and curling |
| | fang from back of mouth. |
| | (_b_) Head for 3 with fleshless lower jaw of head |
| | for 10. |
| Head for 14 | Head for 4 with fleshless lower jaw of head for 10. |
| Head for 15 | Head for 5 with fleshless lower jaw of head for 10. |
| Head for 16 | Head for 6 with fleshless lower jaw of head for 10. |
| Head for 17 | Head for 7 with fleshless lower jaw of head for 10. |
| Head for 18 | Head for 8 with fleshless lower jaw of head for 10. |
| Head for 19 | Head for 9 with fleshless lower jaw of head for 10. |
+-------------+---------------------------------------------------------+
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 Chapter V that there are a few
Cycle-8 dates actually recorded, as well as a half a dozen Cycle-10 dates.
That is, the cycle, which changed its coefficient every 400 years, was a
period which they could _not_ regard as never changing within the range of
human experience. On the other hand, it was the shortest period of which
they were uncertain, since the great cycle could change its coefficient
only every 8,000 years--practically eternity so far as the Maya were
concerned. Therefore it could be omitted as well as the two higher periods
in a date without giving rise to confusion as to which great cycle was the
current one. The cycle, on the contrary, had to be given, as its
coefficient changed every 400 years, and the Maya are known to have
recorded dates in at least three cycles--Nos. 8, 9, and 10. Hence, it was
Great Cycle 19 for 8,000 years, Great-great Cycle 11 for 160,000, and
Great-great-great Cycle 1 for 3,200,000 years, whereas it was Cycle 9 for
only 400 years. This, not the fact that the Maya never had a period higher
than the cycle, the writer believes was the reason why the three higher
periods were omitted from Initial-series dates--they were unnecessary so
far as accuracy was concerned, since there could never be any doubt
concerning them. {127}
It is not necessary to press this point further, though it is believed the
foregoing conception of time had actually been worked out by the Maya. The
archaic date recorded by Stela 10 at Tikal (9.3.6.2.0) makes this monument
one of the very oldest in the Maya territory; indeed, there is only one
other stela which has an earlier Initial Series, Stela 3 at Tikal. In the
archaic period from which this monument dates the middle dot of the trinal
superfix in the Initial-series introducing glyph may still have retained
its numerical value, 1, but in later times this middle dot lost its
numerical characteristics and frequently appears as a scroll itself.
The early date of Stela 10 makes it not unlikely that this process of glyph
elaboration may not have set in at the time it was erected, and
consequently that we have in this simplified trinal element the genesis of
the later elaborated form; and, finally, that A1, figure 60, may have meant
"the present world-epoch" or something similar.
In concluding the presentation of these three numbers the writer may
express the opinion that a careful study of the period glyphs in figures
58-60 will lead to the following conclusions: (1) That the six periods
recorded in the first, the seven in the second, and the eight or nine in
the third, all belong to the same series in each case; and (2) that
throughout the six terms of the first, the seven of the second, and the
eight of the third, the series in each case conforms strictly to the
vigesimal system of numeration given in Table VIII.
As mentioned on page 116 (footnote 2), in this method of recording the
higher numbers the kin sign may sometimes be omitted without affecting the
numerical value of the series wherein the omission occurs. In such cases
the coefficient of the kin sign is usually prefixed to the uinal sign, the
coefficient of the uinal itself standing above the uinal sign. In figure
58, for example, the uinal and the kin coefficients are both 0. In this
case, however, the 0 on the left of the uinal sign is to be understood as
belonging to the kin sign, which is omitted, while the 0 above the uinal
sign is the uinal's own coefficient 0. Again in figure 59, the kin sign is
omitted and the kin coefficient 1 is prefixed to the uinal sign, while the
uinal's own coefficient 12 stands above the uinal sign. Similarly, the 12
uinals and 17 kins recorded in figure 56, _d_, might as well have been
written as in _o_ of the same figure, that is, with the kin sign omitted
and its coefficient 17 prefixed to the uinal sign, while the uinal's own
coefficient 12 appears above. Or again, the 9 uinals and 18 kins recorded
in _f_ also might have been written as in _p_, that is, with the kin sign
omitted and the kin coefficient 18 prefixed to the uinal sign while the
uinal's own coefficient 9 appears above.
In all the above examples the coefficients of the omitted kin signs are on
the _left_ of the uinal signs, while the uinal coefficients are _above_ the
uinal signs. Sometimes, however, these positions are reversed, {128} and
the uinal coefficient stands _on the left_ of the uinal sign, while the kin
coefficient stands _above_. This interchange in certain cases probably
resulted from the needs of glyphic balance and symmetry. For example, in
figure 62, _a_, had the kin coefficient 19 been placed on the left of the
uinal sign, the uinal coefficient 4 would have been insufficient to fill
the space above the period glyph, and consequently the corner of the glyph
block would have appeared ragged. The use of the 19 _above_ and the 4 to
the left, on the other hand, properly fills this space, making a
symmetrical glyph. Such cases, however, are unusual, and the customary
position of the kin coefficient, when the kin sign is omitted, is on the
left of the uinal sign, not above it. This practice, namely, omitting the
kin sign in numerical series, seems to have prevailed extensively in
connection with both Initial Series and Secondary Series; indeed, in the
latter it is the rule to which there are but few exceptions.
[Illustration: FIG. 62. Glyphs showing misplacement of the kin coefficient
(_a_) or elimination of a period glyph (_b_, _c_): _a_, Stela E, Quirigua;
_b_, Altar U, Copan; _c_, Stela J, Copan.]
The omission of the kin sign, while by far the most common, is not the only
example of glyph omission found in numerical series in the inscriptions.
Sometimes, though very rarely, numbers occur in which periods other than
the kin are wanting. A case in point is figure 62, b. Here a tun sign
appears with the coefficient 13 above and 3 to the left. Since there are
only two coefficients (13 and 3) and three time periods (tun, uinal, and
kin), it is clear that the signs of both the lower periods have been
omitted as well as the coefficient of one of them. In _c_ of the
last-mentioned figure a somewhat different practice was followed. Here,
although three time periods are recorded--tuns, uinals and kins--one period
(the uinal) and its coefficient have been omitted, and there is nothing
between the 0 kins and 10 tuns. Such cases are exceedingly rare, however,
and may be disregarded by the beginner.
We have seen that the order of the periods in the numbers in figure 56 was
just the reverse of that in the numbers shown in figures 58 and 59; that in
one place the kins stand at the top and in the other at the bottom; and
finally, that this difference was not a vital one, since it had no effect
on the values of the numbers. This is true, because in the first method of
expressing the higher numbers, it matters not which end of the number comes
first, the highest or the {129} lowest period, so long as its several
periods always stand in the same relation to each other. For example, in
figure 56, _q_, 6 cycles, 17 katuns, 2 tuns, 10 uinals, and 0 kins
represent exactly the same number as 0 kins, 10 uinals, 2 tuns, 17 katuns,
and 6 cycles; that is, with the lowest term first.
It was explained on page 23 that the order in which the glyphs are to be
read is from top to bottom and from left to right. Applying this rule to
the inscriptions, the student will find that all Initial Series are
descending series; that in reading from top to bottom and left to right,
the cycles will be encountered first, the katuns next, the tuns next, the
uinals, and the kins last. Moreover, it will be found also that the great
majority of Secondary Series are ascending series, that is, in reading from
top to bottom and left to right, the kins will be encountered first, the
uinals next, the tuns next, the katuns next, and the cycles last. The
reason why Initial Series always should be presented as descending series,
and Secondary Series usually as ascending series is unknown; though as
stated above, the order in either case might have been reversed without
affecting in any way the numerical value of either series.
This concludes the discussion of the first method of expressing the higher
numbers, the only method which has been found in the inscriptions.
SECOND METHOD OF NUMERATION
The other method by means of which the Maya expressed their higher numbers
(the second method given on p. 103) may be called "numeration by position,"
since in this method the numerical value of the symbols depended solely on
position, just as in our own decimal system, in which the value of a figure
depends on its distance from the decimal point, whole numbers being written
to the left and fractions to the right. The ratio of increase, as the word
"decimal" implies, is 10 throughout, and the numerical values of the
consecutive positions increase as they recede from the decimal point in
each direction, according to the terms of a geometrical progression. For
example, in the number 8888.0, the second 8 from the decimal point,
counting from right to left, has a value ten times greater than the first
8, since it stands for 8 tens (80); the third 8 from the decimal point
similarly has a value ten times greater than the second 8, since it stands
for 8 hundreds (800); finally, the fourth 8 has a value ten times greater
than the third 8, since it stands for 8 thousands (8,000). Hence, although
the figures used are the same in each case, each has a different numerical
value, depending solely upon its position with reference to the decimal
point.
In the second method of writing their numbers the Maya had devised a
somewhat similar notation. Their ratio of increase was 20 in all positions
except the third. The value of these positions increased {130} with their
distance from the bottom, according to the terms of the vigesimal system
shown in Table VIII. This second method, or "numeration by position," as it
may be called, was a distinct advance over the first, since it required for
its expression only the signs for the numerals 0 to 19, inclusive, and did
not involve the use of any period glyphs, as did the first method. To its
greater brevity, no doubt, may be ascribed its use in the codices, where
numerical calculations running into numbers of 5 and 6 terms form a large
part of the subject matter. It should be remembered that in numeration by
position only the normal forms of the numbers--bar and dot numerals--are
used. This probably results from the fact that head-variant numerals never
occur independently, but are always prefixed to some other glyph, as
period, day, or month signs (see p. 104). Since no period glyphs are used
in numeration by position, only normal-form numerals, that is, bar and dot
numerals, can appear.
The numbers from 1 to 19, inclusive, are expressed in this method, as shown
in figure 39, and the number 0 as shown in figure 46. As all of these
numbers are below 20, they are expressed as units of the first place or
order, and consequently each should be regarded as having been multiplied
by 1, the numerical value of the first or lowest position.
The number 20 was expressed in two different ways: (1) By the sign shown in
figure 45; and (2) by the numeral 0 in the bottom place and the numeral 1
in the next place above it, as in figure 63, a. The first of these had only
a very restricted use in connection with the tonalamatl, wherein numeration
by position was impossible, and therefore a special character for 20 (see
fig. 45) was necessary. See Chapter VI.
The numbers from 21 to 359, inclusive, involved the use of two places--the
kin place and the uinal place--which, according to Table VIII, we saw had
numerical values of 1 and 20, respectively. For example, the number 37 was
expressed as shown in figure 63, b. The 17 in the kin place has a value of
17 (17 × 1) and the 1 in the uinal, or second, place a value of 20 (1 (the
numeral) × 20 (the fixed numerical value of the second place)). The sum of
these two products equals 37. Again, 300 was written as in figure 63, c.
The 0 in the kin place has the value 0 (0 × 1), and the 15 in the second
place has the value of 300 (15 × 20), and the sum of these products equals
300.
To express the numbers 360 to 7,199, inclusive, three places or terms were
necessary--kins, uinals, and tuns--of which the last had a numerical value
of 360. (See Table VIII.) For example, the number 360 is shown in figure
63, d. The 0 in the lowest place indicates that 0 kins are involved, the 0
in the second place indicates that 0 uinals or 20's are involved, while the
1 in the third place shows that there is 1 tun, or 360, kins recorded (1
(the numeral) × 360 (the fixed numerical value of the third position)); the
sum of these three products equals 360. Again, the number 7,113 is
expressed as shown in figure 63, e. {131} The 13 in the lowest place equals
13 (13 × 1); the 13 in the second place, 260 (13 × 20); and the 19 in the
third place, 6,840 (19 × 360). The sum of these three products equals 7,113
(13 + 260 + 6,840),
[Illustration: FIG. 63. Examples of the second method of numeration, used
exclusively in the codices.]
The numbers from 7,200 to 143,999, inclusive, involved the use of four
places or terms--kins, uinals, tuns, and katuns--the last of which (the
fourth place) had a numerical value of 7,200. (See Table VIII.) For
example, the number 7,202 is recorded in figure 63, _f_. {132} The 2 in the
first place equals 2 (2×1); the 0 in the second place, 0 (0×20); the 0 in
the third place, 0 (0×360); and the 1 in the fourth place, 7,200 (1×7,200).
The sum of these four products equals 7,202 (2+0+0+7,200). Again, the
number 100,932 is recorded in figure 63, _g_. Here the 12 in the first
place equals 12 (12×1); the 6 in the second place, 120 (6×20); the 0 in the
third place, 0 (0×360); and the 14 in the fourth place, 100,800 (14×7,200).
The sum of these four products equals 100,932 (12+120+0+100,800).
The numbers from 144,000 to 2,879,999, inclusive, involved the use of five
places or terms--kins, uinals, tuns, katuns, and cycles. The last of these
(the fifth place) had a numerical value of 144,000. (See Table VIII.) For
example, the number 169,200 is recorded in figure 63, _h_. The 0 in the
first place equals 0 (0×1); the 0 in the second place, 0 (0×20); the 10 in
the third place, 3,600 (10×360); the 3 in the fourth place, 21,600
(3×7,200); and the 1 in the fifth place, 144,000 (1×144,000). The sum of
these five products equals 169,200 (0+0+3,600+21,600+144,000). Again, the
number 2,577,301 is recorded in figure 63, _i_. The 1 in the first place
equals 1 (1×1); the 3 in the second place, 60 (3×20); the 19 in the third
place, 6,840 (19×360); the 17 in the fourth place, 122,400 (17×7,200); and
the 17 in the fifth place, 2,448,000 (17x144,000). The sum of these five
products equals 2,577,301 (1+60+6,480+122,400+2,448,000).
The writing of numbers above 2,880,000 up to and including 12,489,781 (the
highest number found in the codices) involves the use of six places, or
terms--kins, uinals, tuns, katuns, cycles, and great cycles--the last of
which (the sixth place) has the numerical value 2,880,000. It will be
remembered that some have held that the sixth place in the inscriptions
contained only 13 units of the fifth place, or 1,872,000 units of the first
place. In the codices, however, there are numerous calendric checks which
prove conclusively that in so far as the codices are concerned the sixth
place was composed of 20 units of the fifth place. For example, the number
5,832,060 is expressed as in figure 63, _j_. The 0 in the first place
equals 0 (0×1); the 3 in the second place, 60 (3×20); the 0 in the third
place, 0 (0×360); the 10 in the fourth place, 72,000 (10×7,200); the 0 in
the fifth place, 0 (0×144,000); and the 2 in the sixth place, 5,760,000
(2×2,880,000). The sum of these six terms equals 5,832,060
(0+60+0+72,000+0+5,760,000). The highest number in the codices, as
explained above, is 12,489,781, which is recorded on page 61 of the Dresden
Codex. This number is expressed as in figure 63, _k_. The 1 in the first
place equals 1 (1×1); the 15 in the second place, 300 (15×20); the 13 in
the third place, 4,680 (13×360); the 14 in the fourth place, 100,800
(14×7,200); the 6 in the fifth place, 864,000 (6×144,000); and the 4 in the
sixth place, 11,520,000 (4×2,880,000). The sum of these six products equals
12,489,781 (1+300+4,680+100,800+864,000+11,520,000). {133}
It is clear that in numeration by position the order of the units could not
be reversed as in the first method without seriously affecting their
numerical values. This must be true, since in the second method the
numerical values of the numerals depend entirely on their position--that
is, on their distance above the bottom or first term. In the first method,
the multiplicands--the period glyphs, each of which had a fixed numerical
value--are always expressed[86] with their corresponding multipliers--the
numerals 0 to 19, inclusive; in other words, the period glyphs themselves
show whether the series is an ascending or a descending one. But in the
second method the multiplicands are not expressed. Consequently, since
there is nothing about a column of bar and dot numerals which in itself
indicates whether the series is an ascending or a descending one, and since
in numeration by position a fixed starting point is absolutely essential,
in their second method the Maya were obliged not only to fix arbitrarily
the direction of reading, as from bottom to top, but also to confine
themselves exclusively to the presentation of one kind of series only--that
is, ascending series. Only by means of these two arbitrary rules was
confusion obviated in numeration by position.
However dissimilar these two methods of representing the numbers may appear
at first sight, fundamentally they are the same, since both have as their
basis the same vigesimal system of numeration. Indeed, it can not be too
strongly emphasized that throughout the range of the Maya writings,
codices, inscriptions, or Books of Chilam Balam[87] the several methods of
counting time and recording events found in each are all derived from the
same source, and all are expressions of the same numerical system.
That the student may better grasp the points of difference between the two
methods they are here contrasted:
TABLE XII. COMPARISON OF THE TWO METHODS OF NUMERATION
FIRST METHOD | SECOND METHOD
|
1. Use confined almost exclusively | 1. Use confined exclusively to
to the inscriptions. | the codices.
|
2. Numerals represented by both | 2. Numerals represented by
normal forms and head variants. | normal forms exclusively.
|
3. Numbers expressed by using | 3. Numbers expressed by using
the numerals 0 to 19, inclusive, | the numerals 0 to 19,
as multipliers with the period | inclusive, as multipliers in
glyphs as multiplicands. | certain positions the
| fixed numerical values of
| which served as
| multiplicands.
|
4. Numbers presented as | 4. Numbers presented as
ascending or descending series. | ascending series
| exclusively.
|
5. Direction of reading either | 5. Direction of reading from
from bottom to top, or vice | bottom to top exclusively.
versa.
We have seen in the foregoing pages (1) how the Maya wrote their 20 {134}
numerals, and (2) how these numerals were used to express the higher
numbers. The next question which concerns us is, How did they use these
numbers in their calculations; or in other words, how was their arithmetic
applied to their calendar? It may be said at the very outset in answer to
this question, that in so far as known, _numbers appear to have had but one
use throughout the Maya texts, namely, to express the time elapsing between
dates_.[88] In the codices and the inscriptions alike all the numbers whose
use is understood have been found to deal exclusively with the counting of
time.
This highly specialized use of the numbers in Maya texts has determined the
first step to be taken in the process of deciphering them. Since the
primary unit of the calendar was the day, all numbers should be reduced to
terms of this unit, or in other words, to units of the first order, or
place.[89] Hence, we may accept the following as the _first step_ in
ascertaining the meaning of any number:
FIRST STEP IN SOLVING MAYA NUMBERS
Reduce all the units of the higher orders to units of its first, or lowest,
order, and then add the resulting quantities together.
The application of this rule to any Maya number, no matter of how many
terms, will always give the actual number of primary units which it
contains, and in this form it can be more conveniently utilized in
connection with the calendar than if it were left as recorded, that is, in
terms of its higher orders.
The reduction of units of the higher orders to units of the first order has
been explained on pages 105-133, but in order to provide the student with
this same information in a more condensed and accessible form, it is
presented in the following tables, of which Table XIII is to be used for
reducing numbers to their primary units in the inscriptions, and Table XIV
for the same purpose in the codices. {135}
TABLE XIII. VALUES OF HIGHER PERIODS IN TERMS OF LOWEST, IN INSCRIPTIONS
1 great cycle = [90]2,880,000
1 cycle 144,000
1 katun 7,200
1 tun 360
1 uinal 20
1 kin 1
TABLE XIV. VALUES OF HIGHER PERIODS IN TERMS OF LOWEST, IN CODICES
1 unit of the 6th place = 2,880,000
1 unit of the 5th place 144,000
1 unit of the 4th place 7,200
1 unit of the 3d place 360
1 unit of the 2d place 20
1 unit of the 1st place 1
It should be remembered, in using these tables, that each of the signs for
the periods therein given has its own particular numerical value, and that
this value in each case is a multiplicand which is to be multiplied by the
numeral attached to it (not shown in Table XIII). For example, a 3 attached
to the katun sign reduces to 21,600 units of the first order (3×7,200).
Again, 5 attached to the uinal sign reduces to 100 units of the first order
(5×20). In using Table XIV, however, it should be remembered that the
position of a numeral multiplier determines at the same time that
multiplier's multiplicand. Thus a 5 in the third place indicates that the
5's multiplicand is 360, the numerical value of the third place, and such a
term reduces to 1,800 units of the first place (5×360=1,800). Again, a 10
in the fourth place indicates that the 10's multiplicand is 7,200, the
numerical value corresponding to the fourth place, and such a term reduces
to 72,000 units of the first place.
Having reduced all the terms of a number to units of the 1st order, the
next step in finding out its meaning is to discover the date from which it
is counted. This operation gives rise to the _second step_.
SECOND STEP IN SOLVING MAYA NUMBERS
Find the date from which the number is counted:
This is not always an easy matter, since the dates from which Maya numbers
are counted are frequently not expressed in the texts; consequently, it is
clear that no single rule can be formulated which will cover all cases.
There are, however, two general rules which will be found to apply to the
great majority of numbers in the texts:
_Rule 1._ When the starting point or date is expressed, usually, though not
invariably, it precedes[91] the number counted from it.
It should be noted, however, in connection with this rule, that the
starting date hardly ever immediately precedes the number from which it is
counted, but that several glyphs nearly always stand {136} between.[92]
Certain exceptions to the above rule are by no means rare, and the student
must be continually on the lookout for such reversals of the regular order.
These exceptions are cases in which the starting date (1) follows the
number counted from it, and (2) stands elsewhere in the text, entirely
disassociated from, and unattached to, the number counted from it.
The second of the above-mentioned general rules, covering the majority of
cases, follows:
_Rule 2_. When the starting point or date is not expressed, if the number
is an Initial Series the date from which it should be counted will be found
to be 4 Ahau 8 Cumhu.[93]
This rule is particularly useful in deciphering numbers in the
inscriptions. For example, when the student finds a number which he can
identify as an Initial Series,[94] he may assume at once that such a number
in all probability is counted from the date 4 Ahau 8 Cumhu, and proceed on
this assumption. The exceptions to this rule, that is, cases in which the
starting point is not expressed and the number is not an Initial Series,
are not numerous. No rule can be given covering all such cases, and the
starting points of such numbers can be determined only by means of the
calculations given under the third and fourth steps, below.
Having determined the starting point or date from which a given number is
to be counted (if this is possible), the next step is to find out which way
the count runs; that is, whether it is _forward_ from the starting point to
some _later date_, or whether it is _backward_ from the starting point to
some _earlier date_. This process may be called the _third step_.
THIRD STEP IN SOLVING MAYA NUMBERS
Ascertain whether the number is to be counted forward or backward from its
starting point.
It may be said at the very outset in this connection that the overwhelming
majority of Maya numbers are counted _forward_ from their starting points
and not backward. In other words, they proceed from _earlier to later
dates_ and not vice versa. Indeed, the preponderance of the former is so
great, and the exceptions are so rare, that the student should always
proceed on the postulate that the count is forward until proved definitely
to be otherwise. {137}
[Illustration: FIG. 64. Figure showing the use of the "minus" or "backward"
sign in the codices.]
In the codices, moreover, when the count is backward, or contrary to the
general practice, the fact is clearly indicated[95] by a special character.
This character, although attached only to the lowest term[96] of the number
which is to be counted backward, is to be interpreted as applying to all
the other terms as well, its effect extending to the number as a whole.
This "backward sign" (shown in fig. 64) is a circle drawn in red around the
lowest term of the number which it affects, and is surmounted by a knot of
the same color. An example covering the use of this sign is given in figure
64. Although the "backward sign" in this figure surrounds only the numeral
in the first place, 0, it is to be interpreted, as we have seen, as
applying to the 2 in the second place and the 6 in the third place. This
number, expressed as 6 tuns, 2 uinals, and 0 kins, reduces to 2,200 units
of the first place, and in this form may be more readily handled (first
step). Since the starting point usually precedes the number counted from it
and since in figure 64 the number is expressed by the second method, its
starting point will be found standing below it. This follows from the fact
that in numeration by position the order is from bottom to top. Therefore
the starting point from which the 2,200 recorded in figure 64 is counted
will be found to be below it, that is, the date 4 Ahau 8 Cumhu[97] (second
step). Finally, the red circle and knot surrounding the lowest (0) term of
this 2,200 indicates that this number is to be counted _backward_ from its
starting point, not forward (third step).
On the other hand, in the inscriptions no special character seems to have
been used with a number to indicate that it was to be counted backward; at
least no such sign has yet been discovered. In the inscriptions, therefore,
with the single exception[98] mentioned below, the student can only apply
the general rule given on page 136, that in the great majority of cases the
count is forward. This rule will be found to apply to at least nine out of
every ten numbers. The exception above noted, that is, where the practice
is so uniform as to render possible the formulation of an unfailing rule,
has to do with Initial Series. This rule, to which there are no known
exceptions, may be stated as follows:
_Rule 1_. In Initial Series the count is _always forward_, and, in general
throughout the inscriptions. The very few cases in which the count _is_
backward, are confined chiefly to Secondary Series, and it is in {138}
dealing with this kind of series that the student will find the greatest
number of exceptions to the general rule.
Having determined the direction of the count, whether it is forward or
backward, the next (_fourth_) step may be given.
FOURTH STEP IN SOLVING MAYA NUMBERS
To count the number from its starting point.
We have come now to a step that involves the consideration of actual
arithmetical processes, which it is thought can be set forth much more
clearly by the use of specific examples than by the statement of general
rules. Hence, we will formulate our rules after the processes which they
govern have been fully explained.
In counting any number, as 31,741, or 4.8.3.1 as it would be expressed in
Maya notation,[99] from any date, as 4 Ahau 8 Cumhu, there are four unknown
elements which have to be determined before we can write the date which the
count reaches. These are:
1. The day coefficient, which must be one of the numerals 1 to 13,
inclusive.
2. The day name, which must be one of the twenty given in Table I.
3. The position of the day in some division of the year, which must be one
of the numerals 0 to 19, inclusive.
4. The name of the division of the year, which must be one of the nineteen
given in Table III.
These four unknown elements all have to be determined from (1) the starting
date, and (2) the number which is to be counted from it.
If the student will constantly bear in mind that all Maya sequences,
whether the day coefficients, day signs, positions in the divisions of the
year, or what not, are absolutely continuous, repeating themselves without
any break or interruption whatsoever, he will better understand the
calculations which follow.
It was explained in the text (see pp. 41-44) and also shown graphically in
the tonalamatl wheel (pl. 5) that after the day coefficients had reached
the number 13 they returned to 1, following each other indefinitely in this
order without interruption. It is clear, therefore, that the highest
multiple of 13 which the given number contains may be subtracted from it
without affecting in any way the value of the day coefficient of the date
which the number will reach when counted from the starting point. This is
true, because no matter what the day coefficient of the starting point may
be, any multiple of 13 will always bring the count back to the same day
coefficient. {139}
Taking up the number, 31,741, which we have chosen for our first example,
let us deduct from it the highest multiple of 13 which it contains. This
will be found by dividing the number by 13, and multiplying the
_whole-number part_ of the resulting quotient by 13: 31,741 ÷ 13 =
2,441-8/13. Multiplying 2,441 by 13, we have 31,733, which is the highest
multiple of 13 that 31,741 contains; consequently it may be deducted from
31,741 without affecting the value of the resulting day coefficient: 31,741
- 31,733 = 8. In the example under consideration, therefore, 8 is the
number which, if counted from the day coefficient of the starting point,
will give the day coefficient of the resulting date. In other words, after
dividing by 13 the only part of the resulting quotient which is used in
determining the new day coefficient is the _numerator_ of the fractional
part.[100] Hence the following rule for determining the first unknown on
page 138 (the day coefficient):
_Rule 1._ To find the new day coefficient divide the given number by 13,
and count forward the numerator of the fractional part of the resulting
quotient from the starting point if the count is forward, and backward if
the count is backward, deducting 13 in either case from the resulting
number if it should exceed 13.
Applying this rule to 31,741, we have seen above that its division by 13
gives as the fractional part of the quotient 8/13. Assuming that the count
is forward from the starting point, 4 Ahau 8 Cumhu, if 8 (the numerator of
the fractional part of the quotient) be counted forward from 4, the day
coefficient of the starting point (4 Ahau 8 Cumhu), the day coefficient of
the resulting date will be 12 (4 + 8). Since this number is below 13, the
last sentence of the above rule has no application in this case. In
counting forward 31,741 from the date 4 Ahau 8 Cumhu, therefore, the day
coefficient of the resulting date will be 12; thus we have determined our
first unknown. Let us next find the second unknown, the day sign to which
this 12 is prefixed.
It was explained on page 37 that the twenty day signs given in Table I
succeed one another in endless rotation, the first following immediately
the twentieth no matter which one of the twenty was chosen as the first.
Consequently, it is clear that the highest multiple of 20 which the given
number contains may be deducted from it without affecting in any way the
name of the day sign of the date which the number will reach when counted
from the starting point. This is true because, no matter what the day sign
of the starting point may be, any multiple of 20 will always bring the
count back to the same day sign. {140}
Returning to the number 31,741, let us deduct from it the highest multiple
of 20 which it contains, found by dividing the number by 20 and multiplying
the whole number part of the resulting quotient by 20; 31,741 ÷ 20 =
1,587-1/20. Multiplying 1,587 by 20, we have 31,740, which is the highest
multiple of 20 that 31,741 contains, and which may be deducted from 31,741
without affecting the resulting day sign; 31,741 - 31,740 = 1. Therefore in
the present example 1 is the number which, if counted forward from the day
sign of the starting point in the sequence of the 20 day signs given in
Table I, will reach the day sign of the resulting date. In other words,
after dividing by 20 the only part of the resulting quotient which is used
in determining the new day sign is the numerator of the fractional part.
Thus we may formulate the rule for determining the second unknown on page
138 (the day sign):
_Rule 2._ To find the new day sign, divide the given number by 20, and
count forward the numerator of the fractional part of the resulting
quotient from the starting point in the sequence of the twenty day signs
given in Table I, if the count is forward, and backward if the count is
backward, and the sign reached will be the new day sign.
Applying this rule to 31,741, we have seen above that its division by 20
gives us as the fractional part of the quotient, 1/20. Since the count was
forward from the starting point, if 1 (the numerator of the fractional part
of the quotient) be counted forward in the sequence of the 20 day signs in
Table I from the day sign of the starting point, Ahau (4 Ahau 8 Cumhu), the
day sign reached will be the day sign of the resulting date. Counting
forward 1 from Ahau in Table I, the day sign Imix is reached, and Imix,
therefore, will be the new day sign. Thus our second unknown is determined.
By combining the above two values, the 12 for the first unknown and Imix
for the second, we can now say that in counting forward 31,741 from the
date 4 Ahau 8 Cumhu, the day reached will be 12 Imix. It remains to find
what position this particular day occupied in the 365-day year, or haab,
and thus to determine the third and fourth unknowns on page 138. Both of
these may be found at one time by the same operation.
It was explained on pages 44-51 that the Maya year, at least in so far as
the calendar was concerned, contained only 365 days, divided into 18 uinals
of 20 days each, and the _xma kaba kin_ of 5 days; and further, that when
the last position in the last division of the year (4 Uayeb) was reached,
it was followed without interruption by the first position of the first
division of the next year (0 Pop); and, finally, that this sequence was
continued indefinitely. Consequently it is clear that the highest multiple
of 365 which the given number contains may be subtracted from it without
affecting in any way the position in the year of the day which the number
will reach when {141} counted from the starting point. This is true,
because no matter what position in the year the day of the starting point
may occupy, any multiple of 365 will bring the count back again to the same
position in the year.
Returning again to the number 31,741, let us deduct from it the highest
multiple of 365 which it contains. This will be found by dividing the
number by 365 and multiplying the whole number part of the resulting
quotient by 365: 31,741 ÷ 365 = 86-351/365. Multiplying 86 by 365, we have
31,390, which is the highest multiple that 31,741 contains. Hence it may be
deducted from 31,741 without affecting the position in the year of the
resulting day; 31,741 - 31,390 = 351. Therefore, in the present example,
351 is the number which, if counted forward from the year position of the
starting date in the sequence of the 365 positions in the year, given in
Table XV, will reach the position in the year of the day of the resulting
date. This enables us to formulate the rule for determining the third and
fourth unknowns on page 138 (the position in the year of the day of the
resulting date):
_Rule 3._ To find the position in the year of the new day, divide the given
number by 365 and count forward the numerator of the fractional part of the
resulting quotient from the year position of the starting point in the
sequence of the 365 positions of the year shown in Table XV, if the count
is forward; and backward if the count is backward, and the position reached
will be the position in the year which the day of the resulting date will
occupy.
TABLE XV. THE 365 POSITIONS IN THE MAYA YEAR
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|Month|P |U |Z |Z |T |X |Y |M |C |Y |Z |C |M |K |M |P |K |C |U |
| |o |o |i |o |z |u |a |o |h |a |a |e |a |a |u |a |a |u |a |
| |p | |p |t |e |l |x |l |e |x |c |h |c |n |a |x |y |m |y |
| | | | |z |c | |k | |n | | | | |k |n | |a |h |e |
| | | | | | | |i | | | | | | |i | | |b |u |b |
| | | | | | | |n | | | | | | |n | | | | | |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|Posi-| | | | | | | | | | | | | | | | | | | |
|tion |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |1 |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |2 |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |3 |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |4 |
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |5 |--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |6 |--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |7 |--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |8 |--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |9 |--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |10|10|10|10|10|10|10|10|10|10|10|10|10|10|10|10|10|10|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |11|11|11|11|11|11|11|11|11|11|11|11|11|11|11|11|11|11|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |12|12|12|12|12|12|12|12|12|12|12|12|12|12|12|12|12|12|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |13|13|13|13|13|13|13|13|13|13|13|13|13|13|13|13|13|13|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |14|14|14|14|14|14|14|14|14|14|14|14|14|14|14|14|14|14|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |15|15|15|15|15|15|15|15|15|15|15|15|15|15|15|15|15|15|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |16|16|16|16|16|16|16|16|16|16|16|16|16|16|16|16|16|16|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |17|17|17|17|17|17|17|17|17|17|17|17|17|17|17|17|17|17|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |18|18|18|18|18|18|18|18|18|18|18|18|18|18|18|18|18|18|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| Do |19|19|19|19|19|19|19|19|19|19|19|19|19|19|19|19|19|19|--|
+-----+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
{142}
Applying this rule to the number 31,741, we have seen above that its
division by 365 gives 351 as the numerator of the fractional part of its
quotient. Assuming that the count is forward from the starting point, it
will be necessary, therefore, to count 351 forward in Table XV from the
position 8 Cumhu, the position of the day of the starting point, 4 Ahau 8
Cumhu.
A glance at the month of Cumhu in Table XV shows that after the position 8
Cumhu there are 11 positions in that month; adding to these the 5 in Uayeb,
the last division of the year, there will be in all 16 more positions
before the first of the next year. Subtracting these from 351, the total
number to be counted forward, there remains the number 335 (351-16), which
must be counted forward in Table XV from the beginning of the year. Since
each of the months has 20 positions, it is clear that 16 months will be
used before the month is reached in which will fall the 335th position from
the beginning of the year. In other words, 320 positions of our 335 will
exactly use up all the positions of the first 16 months, namely, Pop, Uo,
Zip, Zotz, Tzec, Xul, Yaxkin, Mol, Chen, Yax, Zac, Ceh, Mac, Kankin, Muan,
Pax, and will bring us to the beginning of the 17th month (Kayab) with
still 15 more positions to count forward. If the student will refer to this
month in Table XV he will see that 15 positions counted forward in this
month will reach the position 14 Kayab, which is also the position reached
by counting forward 31,741 positions from the starting position 8 Cumhu.
Having determined values for all of the unknowns on page 138, we can now
say that if the number 31,741 be counted forward from the date 4 Ahau 8
Cumhu, the date 12 Imix 14 Kayab will be reached. To this latter date,
i. e., the date reached by any count, the name "terminal date" has been
given. The rules indicating the processes by means of which this terminal
date is reached apply also to examples where the count is _backward_, not
forward, from the starting point. In such cases, as the rules say, the only
difference is that the numerators of the fractional parts of the quotients
resulting from the different divisions are to be counted backward from the
starting points, instead of forward as in the example above given.
Before proceeding to apply the rules by means of which our fourth step or
process (see p. 138) may be carried out, a modification may sometimes be
introduced which will considerably decrease the size of the number to be
counted without affecting the values of the several parts of its resulting
terminal date.
We have seen on pages 51-60 that in Maya chronology there were possible
only 18,980 different dates--that is, combinations of the 260 days and the
365 positions of the year--and further, that any given day of the 260 could
return to any given position of the 365 only after the lapse of 18,980
days, or 52 years. {143}
Since the foregoing is true, it follows, that this number 18,980 or any
multiple thereof, may be deducted from the number which is to be counted
without affecting in any way the terminal date which the number will reach
when counted from the starting point. It is obvious that this modification
applies only to numbers which are above 18,980, all others being divided by
13, 20, and 365 directly, as indicated in rules 1, 2, and 3, respectively.
This enables us to formulate another rule, which should be applied to the
number to be counted before proceeding with rules 1, 2, and 3 above, if
that number is above 18,980.
_Rule_. If the number to be counted is above 18,980, first deduct from it
the highest multiple of 18,980 which it contains.
This rule should be applied whenever possible, since it reduces the size of
the number to be handled, and consequently involves fewer calculations.
In Table XVI are given 80 Calendar Rounds, that is, 80 multiples of 18,980,
in terms of both the Maya notation and our own. These will be found
sufficient to cover most numbers.