e shown how the Maya expressed their numbers and how they used their several time periods. In short, their arithmetica

in Roman notation stand for the values 1, 5, 10, 50, 100, 500, and 1,000, respectively. The head-variant numerals, on the other hand, more closely resemble our Arabic figures, since there was a special head form for each number up to and including 13, just as there are special characters for the first nine figures and zero in Arabic notation. Moreover, this parallel between our Arabic figures and the Maya head-variant numerals extends to the formation of the higher numbers. Thus, the Maya formed the head-variant

d Dot

ots for 2, three dots for 3, four dots for 4, one bar for 5, one bar and one dot for 6, one bar and two dots for 7, one bar and three dots for 8, one bar and four dots for 9, two bars for 10, and so on up to three bars and four dots for 19. The normal forms of the numerals, in the inscriptions (see fig. 40) are identical with those in the codices, excepting that they are more elaborate, the dots and bars both taking on various decorations. Some of the former contain a concentric circle (*) or cross-ha

f numerals 1 to 19, inc

numerals 1 to 19, inclu

showing the ornamentation which the bar unde

the same figure (the numbers 1, 6, 11, and 16, respectively) the single dot does not fill the space on the left-hand[60] side of the bar, or bars, as the case may be, and consequently the left-hand edge of the glyph block in each case is ragged. Similarly in b, d, f, and h, the numbers 2, 7, 12, and 17, respectively, the two dots at the left of the bar or bars are too far apart to fill in the left-hand edge of the glyph blocks neatly, and consequently in these cases also the left edge is ragged. The Maya were quick to note this discordant note in glyph design, and in the great majority of the places where these numbers (1, 2, 6, 7, 11, 12, 16, and 17) had to be recorded, other elements of a purely ornamental characte

he numerals 1, 2, 6, 7, 11, 12, 16, and 17 a

11, 12, 16, and 17 are used with period, day, or month signs. Note

merals 1 to 13, inclusive,

erical dots, and consequently the numerals 1, 2, 3 are frequently mistaken for one another, as are also 6, 7, and 8; 11, 12, and 13; and 16, 17, and 18. The student must exercise the greatest caution at all times in identifying these numerals in the inscriptions, or otherwise he will quickly find himself invo

native testimony is not lacking on this point. Doctor Brinton (1882 b: p. 48) gives this extract, accompanied by the draw

four dots for four, and so on; in addition to these they used a line; one line meant five years, two lines meant ten years; if one line and above it one dot, six years; if two d

d so extensively throughout both the inscriptions and the codices, that we are justified in identifying the bar and dot as t

distinguish one set of numerals from another, each of which has a different use. In such cases, however, bars of one color are never used

m to have had but one numerical sign, the dot, and they were obliged to resort to the clumsy makeshift of repeating this in order to represent all numbers abo

to 1 unit of the order next higher, and consequently 20 could not be attached to any period-glyph, since this number of periods (with the above exception) was always recorded as 1 period of the order next higher; and (2) although there were 20 positions

n for 20 in

in such a manner that the numbers of which they were composed could not be presented from bottom to top in the usual way, but had to be written horizontally from left to right. This destroyed the possibility of numeration by position,[62] according to the Maya point of view, and consequently some sign wa

gn for 0 in

ensable. Indeed, any numerical system which rises to a second order of units requires a character which will signify, when the need ari

presents an interior decoration which does not follow any fixed scheme.[63] Only a very few variants occur. The last one in figure 46 has clearly as one of its elements the normal form (lenticular). The remainin

for 0 in the

a, Outline of the days of the tonalamatl as represented graphically in the Codex

figure 48, a. Half of this (see fig. 48, b) is the sign which stands for zero (compare with fig. 47). The train of association by which half of the graphic representation of a tonalamatl could come to stand for zero is not clear. Perhaps a of figure 48 may have signified that a complete tonalamatl had passed with no additional days. From this the sign may have come to represent the idea of completeness as apart from

igure 49, a-e, are from the inscriptions and those in f-h from the Dresden Codex. They are all similar. The general outline of the sign has suggested the name "the spectacle" glyph. Its essential characteristic seems to be the division into two roughly circular parts, one above the other, best seen in the Dresden Codex forms (fig. 49, f-h) and a roughly circular infix in each. The lower infix is quite regular in all of the forms, being a circle or ring. The upper infix, however, varies considerably. In figure 49, a, b, this ring has degenerated into a loop. In c and d of the same figure it has

for 0 used exclusively

figs. 16, c, j, s, t, u, a', b', and 17, c, d, k, r, x, y, respectively), since these four alone

numerals with period, day, or month signs. T

lf with these forms, since on his ability to recognize them will largely depend his progress in reading the inscriptions. This figure illustrates the use of all the foregoing forms except the sign for 20 in figure 45 and the sign for zero i

riant N

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. A

ential 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 b

mined. 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 irregul

riant numerals 1

orehead 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 essentia

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

riant numerals 8

51, h, i. Its determining characte

minent eye and square irid (*). (probably eroded in l), the snaglike front tooth, and

the normal form of the tun sign. Compare figure 29, a, b. The same element appears also in the he

nt numerals 14 to 1

It is always characterized by the so-called hatchet eye (

at Quirigua. Its essential characteristic, the large ornamental scroll passing under the eye

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 o

f 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 t

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-Zo?morph B at Quirigua (see r of the s

Negras; hence comparative data are lacking for the determination of its essential element. This head has n

be composed of the heads for 10 and 2. It is to be noted, however, that all three of the face

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 t

enoting 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 pr

ntial element of the 5 head (the tun sign; see fig. 51

less lower jaw and the hatchet eye of the 6 head. Compare fi

lement of the 7 head (the scroll projecting above the nose; s

stic forehead ornament of the 8 head (compare fig

his occurs on the Temple of the Cross at Palenque and seems to be formed regularly

od-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

he 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.

" in Period-ending dates. (See figs. 47 and 53 s-w, for forms used inter

have been gathered together in Table X, where they may be readily consulted. Examples covering their use w

erals 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 di

CS OF HEAD-VARIANT NUM

haracte

ed hand across l

ornament composed of

l in upper par

anded headdr

are irid, snaglike front tooth,

l form of tun si

6 "Hatc

ssing under eye and curlin

ead ornament com

r cheek or around mouth

in some cases other death's-head c

11 Unde

rmined; type of he

us nose, bulging eye, and cu

h fleshless lower

4 with fleshless lowe

5 with fleshless lowe

6 with fleshless lowe

7 with fleshless lowe

8 with fleshless lowe

9 with fleshless lowe

be classified either as a bar and dot numeral or a head vari

g our attention is, how were the higher numbers written, numbers which in the codices are in excess of 12,000,000, and in the inscri

o ways, in both of which the numbers rise by

with the several periods of Table VIII (reduced in each c

f which had a fixed numerical value of its own, like the positions to

es. Moreover, although the first made use of both normal-form and head-variant numerals, the second could be expressed by nor

numerals with period, day, or month signs. Th

ign. Bar and dot numerals, on the other hand, frequently stand by themselves in the codices unattached t

thod of

ethod of numeration, used almost

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 figure. In other word

t 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

his 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 formin

atun = 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 katu

mber 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 numb

ycles in a

inions; and although its presentation will entail a somewhat lengthy digression from the subject under consideration it is so pertin

ct 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 word

d 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) incli

ts mainly on the tw

usive, and not from 0 to 19, inclusive, as in the case of all the othe

Ahau 8 Cumhu, the starting point of Maya chronology, are counted from a da

e 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 sev

ciding this point let us examine the two Initial Series mentioned above, as not proceeding from the

g 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 poi

ich seem to indicate that the great cycle in the inscription

der. This absolute uniformity in a strict vigesimal progression in the codices, so similar in other respects to the inscript

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 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 cycle

ients above 13: a, From the Temple of the In

hermore, 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

n 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 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.

This sequence strikingly recalls that of the numerical coefficients of the days, and in the parallel which this latter sequence affor

TWENTY CONSECUTIVE D

Pop 11

Pop 12 B

Pop 13 I

n 3 Pop 1

Pop 2 C

Pop 3 Ca

Pop 4 Ez

Pop 5 Ca

op 6 Aha

9 Pop 7 I

usive, 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 fo

ccupied 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

e 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

me 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 presente

g of some great cycle, and to have done this in the Maya system of counting time-that is, by elapsed

ce counting time would be a Cycle 1, and if this were done time would have

ya commenced their chronology with the beginning of a great cycle, whose first cycle was named Cycle 1, which was reckon

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

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 o

hough 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, i

n on Stela N, Copan, showing a

Temple of the Inscriptions, Palenque, sh

10, Tikal (probably an Initial Series),

n 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 = 40,320,

t the great-great cycle. But we have seen that the 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 = 403,200,0

t-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 cyc

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 ba

stic 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 probabilit

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 bett

ements 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 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

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 varia

at-great cycle (d, e): a, Stela N, Copan; b, d, Templ

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, 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

ade up of the corresponding forms of the cycle sign plus another element,

tical, thus showing that the three glyphs in which

ng term of which in each case is a cycle sign, thus showing that by

may occur. (See figs. 59, 60.) The two glyphs which may possibly be

omposed of the same elements: (1) The cycle sign; (2)

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 al

and in e holds a rod. Indeed, the similarity of the two forms is so close that in default of any evidence to the co

conclusion is based may

a) The normal form of the cycle sign; (b) a super

which is the cycle, showing that by position they are the logical

have been identified as great-cycle signs, that is,

is the sign for the great-great-great cycle, although this fact can no

gns for the great cycle and the great-great cycle already described. These are: (1) The cycle sign; (2) a superfix compose

a number composed of eight terms, we must lay aside this line of

e 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

in which more than five periods are recorded this sign for the sixth period is composed of the same elements

he cycle sign has a coefficient greater than 13, thus showing

e value of 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

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, furt

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 pe

, A7, A8, and A9) probably all belong to the same series. Let

n 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,

hich increased the fifth 20 times, and (2) that the seventh term was composed of the

fig. 60). This glyph is composed of (1) the cycle sign; (2) a superfix of two

ee A3, fig. 60). Therefore we must assume the same condition obtains here. And finally, since the eighth te

the great-cycle glyph (see A4, fig. 60), which was shown to have the value 20: (1) Both elements have t

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 consequ

ow that all belong to one and the same numerical series, which pro

ting this conclusion may

ir sequence being uninterrupted throughout. Consequently i

ply the cycle sign by 20, 400, and 8,000, respectively, which has to be the case if they are t

ts, just like the five lower ones; this tends to show

e sixth-period glyph in each is identical with A4, figure 60, thus showing the exist

period glyph in its seventh place is identical with A3, figure 60; thus showing the e

higher in this text, are all built on the same basic element, the cycle, thus showing that in each case the higher term

glyphs in an unbroken sequence in each, like the text under discussion, thus showing that in

ccurrences of the great-cycle sign with a coefficient above 13, indicate that 20, not 13, was

important point in connection with it which must be considered, because of

mewhat 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 followin

aya chronology in a far more comprehensive and elaborate chronological conception, a conceptio

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

ecause 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, furth

0. 0. 0. 0.

. 0.

0. 0. 0. 0.

18, the end of which (19. 0. 0. 0. 0. 0. 4 Ahau 8

that can be put on this text if we admit that the eight periods i

t historic period of the Maya civilization (Cycle 9) was itself the cl

9. 0. 0.

0. 0

. 0. 0.

cle 19 had completed itself, Great

Cycle 13, which was followed by Cycle 1, and the sequence repeated itself. We saw, however, that these names probably had nothing to d

at 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

cle = 1 × 20 × 20 × 20

es = 11 × 20 × 20 c

= 19 × 20 cycl

-

80 c

) we can find the name of the first cycle of Great-great-great Cycle 1, the highest Maya time period of which w

,7

,7

-

onception of time. In other words, the Maya conceived time to have commenced, in so far as we can judge from t

explained, in which the five components of that glyph are present in usual form: (1) Trinal superfix; (2) pair of comb-like lat

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 i

nting "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

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 coeffic

nument 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 t

he time it was erected, and consequently that we have in this simplified trinal element the genesis of the later

ions: (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

h 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 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.

cient (a) or elimination of a period glyph (b, c): a,

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 somewh

ot 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 lowest period, so long as its several peri

untered 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 katu

od of expressing the higher numbers, the only m

thod of N

he 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

e 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 n

in figure 46. As all of these numbers are below 20, they are expressed as units of the first place or order, and con

e 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 tonal

pressed 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 t

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 e

ond method of numeration, us

f. 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 recorde

s 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

usively 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 highe

merical 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 descen

esimal 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 Bal

e points of difference between the t

SON OF THE TWO MET

THOD SEC

y to the inscriptions. 1. Use con

forms and head variants. 2. Numerals

d glyphs as multiplicands. 3. Numbers expressed by using the numerals 0 to 19, inclusive, as

descending series. 4. Numbers prese

om to top, or vice versa. 5. Direction o

heir 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 throu

m. 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 un

in Solving

o units of its first, or lowest, order, and

ual number of primary units which it contains, and in this form it can be more conveniently utilized i

he student with this same information in a more condensed and accessible form, it is presented in the following tables, of which Table

GHER PERIODS IN TERMS OF

ycle = [9

le 14

tun

un

ina

ki

IGHER PERIODS IN TERMS

he 6th plac

the 5th pl

the 4th

f the 3d

f the 2d

f the 1s

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

the next step in finding out its meaning is to discover the date fro

in Solving

rom which the n

ssed in the texts; consequently, it is clear that no single rule can be formulated which will cover all cases.

expressed, usually, though not invariably,

stand 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 a

oned general rules, covering

if the number is an Initial Series the date from which i

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 Initi

t step is to find out which way the count runs; that is, whether it is forward from the starting point to some later d

in Solving

s to be counted forward or bac

d 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

use of the "minus" or "bac

olor. 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

ingle 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. T

ry few cases in which the count is backward, are confined chiefly to Secondary Series, and it is in dealin

count, whether it is forward or backwar

in Solving

umber from its

hought can be set forth much more clearly by the use of specific examples than by the statement of general

tion,[99] from any date, as 4 Ahau 8 Cumhu, there are four unknown elements which h

hich must be one of the n

must be one of the tw

ivision of the year, which must be o

the year, which must be one of

etermined from (1) the starting date, and (2

igns, positions in the divisions of the year, or what not, are absolutely continuous, repeating themselv

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 th

1,733, 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,

the fractional part of the resulting quotient from the starting point if the count is forward, and backw

uotient) 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

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 num

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, wi

al part of the resulting quotient from the starting point in the sequence of the twenty day signs given in Table

tor 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 wi

m 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 ye

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 affectin

ch 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

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 fo

365 POSITIONS

nt

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1

2 2 2 2 2 2 2

3 3 3 3 3 3 3

4 4 4 4 4 4 4

5 5 5 5 5 5 5

6 6 6 6 6 6 6

7 7 7 7 7 7 7

8 8 8 8 8 8 8

9 9 9 9 9 9 9

0 10 10 10 10 10 10

1 11 11 11 11 11 11

2 12 12 12 12 12 12

3 13 13 13 13 13 13

4 14 14 14 14 14 14

5 15 15 15 15 15 15

6 16 16 16 16 16 16

7 17 17 17 17 17 17

8 18 18 18 18 18 18

9 19 19 19 19 19 19

ctional part of its quotient. Assuming that the count is forward from the starting point, it will be necessary, therefore,

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, Za

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, a

arried out, a modification may sometimes be introduced which will considerably decrease the size of the

t is, combinations of the 260 days and the 365 positions of the year-and further, that any given day of

er 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

ve 18,980, first deduct from it the high

nce it reduces the size of the number to be handl

iples of 18,980, in terms of both the Maya notation and our

ay be deducted from it; 31,741 - 18,980 = 12,761. In other words, we can count the number 12,761 forward (or backward had the count been backward in our

proof of this

2,761 ÷ 20 = 6381?2012

dividing 31,741 by the same divisors, those indicated in rules 1, 2, and 3, respectively. Consequently, if these three numerators be counted forward from the corr

(see Table XVI), from the first, and 3 Calendar Rounds, 56,940 (see Table XVI), from the second, there would remain in each case 12,761. The student will find his calculations greatly faci

ROUNDS EXPRESSED IN A

le

Cycles, Et

ays Cycl

13. 0 41 778,1

8. 0 42 797,160

3. 0 43 816,14

16. 0 44 835,12

11. 0 45 854,100

6. 6. 0 46 873,

1. 0 47 892,060

. 14. 0 48 911,04

14. 9. 0 49 930,

7. 4. 0 50 949,00

19. 17. 0 51 967,

12. 12. 0 52 986,9

5. 7. 0 53 1,005,9

18. 2. 0 54 1,02

10. 15. 0 55 1,043

3. 10. 0 56 1,062

16. 5. 0 57 1,081

. 0. 0 58 1,100,84

1. 13. 0 59 1,119,

14. 8. 0 60 1,138,

7. 3. 0 61 1,157

19. 16. 0 62 1,176

12. 11. 0 63 1,19

5. 6. 0 64 1,214,

8. 1. 0 65 1,233,7

0. 14. 0 66 1,252,

3. 9. 0 67 1,271,6

16. 4. 0 68 1,290,

8. 17. 0 69 1,309,

1. 12. 0 70 1,328,

14. 7. 0 71 1,347

7. 2. 0 72 1,366,

9. 15. 0 73 1,385,

2. 10. 0 74 1,404,

5. 5. 0 75 1,423,5

18. 0. 0 76 1,442,

0. 13. 0 77 1,461,4

3. 8. 0 78 1,480,

16. 3. 0 79 1,499

. 16. 0 80 1,518,4

143 does not apply in this case. Therefore we may proceed with the first rule given on page 139, by means of which the new day coefficient may be determined. Dividing the given number by 13 we have: 5,799 ÷

forward the numerator of the fractional part of the resulting quotient (19) from the day sign of the starting point, Kan, in the sequence of the twenty-day sig

date. The count by means of which the position 6 Zip is determined is given in detail. After the year position of the starting point, 7 Tzec, it requires 12 more positions (Nos. 8-19, inclusive) before the close of that month (see Table XV) will be reached. And after the close of Tzec, 13 uinals and the xma kaba kin must pass before the end of the year; 13 × 20 + 5 = 265, and 265 + 12 = 277. This latter number subtracted from 324, the total number of positions to be counted forward,

fficient of the starting point, 13, we have 13 as the day coefficient of the terminal date (rule 1, p. 139). Dividing by 20 we have 260 ÷ 20 = 13. Since there is no fraction in the quotient, the numerator of the fraction will be 0, and counting forward 0 from the day sign of the starting point, Ik in Table I, the day sign Ik will remain the day sign of the terminal date (rule 2, p. 140). Combining the two values just determined, we see that the day of the terminal date will be 13 Ik, or a day of the same name as the day of the starting point. This follows also from the fact that there are only 260 differently named days (se

forward 322,920 (or 260) from the starting point 13

the starting point, Imix, in Table I, we reach Eznab as the day sign of the terminal date (Ahau, Cauac, Eznab); consequently the day reached in the count will be 12 Eznab. Dividing the given number by 365, we have 9,663 ÷ 365 = 26173?365. Counting backward the numerator of the fractional part of this quotient, 173, from the year position of the starting point, 4 Uayeb, the year position of the terminal date will be found to be 11 Yax. Before position 4 Uayeb (see Table XV) there are 4 positions in that division of the year (3, 2, 1, 0). Counting these backward to the end of the month Cumhu (see Table XV), we have left 169 positions (173 - 4 = 169); this equals 8 uinals and 9 days extra. Therefore, beginning with t

d any given number from any given date; however, before explaining the fifth and final step in deciphering

per positions in Maya chronology; that is, in the Long Count. Consequently, since any Maya date recurred at successive intervals of 52 years, by the time their historic period had

he number of cycles, katuns, tuns, uinals, and kins which had elapsed in each case between this date and any subsequent dates in the Long Count, subsequent dates of the same name could be readily

method of writing Initi

.0.8 Ah

katuns, 0 tuns, 0 uinals, and 0 kins from 4 Ahau 8 Cumhu, the starting point of Maya chronology (alw

.17.12 Ca

ns, 4 uinals, and 17 kins from 4 Ahau 8 Cumhu, the starting point of Maya c

5 Kayab. It is clear from the foregoing that although the date 8 Ahau 13 Ceh, for example, had recurred upward of 70 times since the beginning of their chronology, the Maya were able to distinguish any particular 8 Ahau 13 Ceh from all the others merely by recording its distance from the starting point; in other

though in many cases these values are not recorded. However, in most of the cases in which the Initial-series values of dates are not recorded, they may be calculated by means of their distances from other dates, whose Initial-ser

multiplying the values of the katuns, tuns, uinals, and kins given in Table XIII by their corresponding coefficients, in this case 2,

hed. Moreover, since the Initial-series value of the starting point 8 Ahau 13 Ceh was 9.0.0.0.0, the Initial-series value of 1 Imix

ries value of startin

from 8 Ahau 13 Ceh

ries value of termina

particular 1 Imix 9 Yaxkin, which was distant 2.5.6.1 from 9.0.0.0.0 8 Ah

unting forward 9.2.5.6.1 from 4 Ahau 8 Cumhu, the starting point of all Initial Series known except two. If our calculation

a Secondary Series. This method of dating already described (see pp. 74-76 et seq.) seems to have been used to avoid

0. 16 5 Ci

9.

. 11] [104] 9

10. 16] 1

0.

. 13. 1] 5

ng to the three dates of the Secondary Series, respectively, do not appear in the text at all (a fact indicated by the brackets), but are found only by calculation. Moreover, the student should note that in a succession of interdependent series like the ones just given the terminal date reached by one number, as 9 Chuen 9 Kankin, becomes

62 units of the first order. Counting forward 62 from 5 Ahau 8 Uo, as indicated by the rules on pages 138-143, it is found that the terminal date will be 2 Ik 10 Tzec. Since th

-series value of the sta

m 5 Ahau 8 Uo forw

al-series value of the t

cy of this result by treating 9.12.3.17.2 2 Ik 10 Tzec as a new Initial Series, and counting forward 9.12.3.17.2 from 4 Ahau 8 Cumhu (the starting point of Maya chronology, unexpressed in

st order, we have 105,577. Counting this number backward from 12 Caban 5 Kayab, as indicated in the rules on pages 138-143, we find that the terminal date will be 8 Ahau 13 Ceh. Moreover, since the Initial-series value of t

series value of the starti

from 12 Caban 5 Kayab ba

-series value of the ter

re that the Initial Series 9.0.0.0.0 has for its terminal d

ndary Series may be used to determine the Initial-series values of Secondary-series

y, the identification of the terminal dates determined by the calculations giv

in Solving

l date to which

connection between distance numbers and their corresponding terminal dates is far closer than between distance numbers and their corresponding starting points. This probably results from the fact that the closing dates of Maya periods were of far more importance than their opening dates. Time was measured by elapsed periods and recorded in terms of the ending days of such periods. The

l rule, which the student will do well to apply i

eries almost invariably follows immediately the

Series, Calendar-round dating or Period-ending dating, though in the case of In

e day parts in Initial-series terminal dates are quite regular according to the terms of the above rule; that is, they follow immediately the lowest period of the number which in each case shows their distance from the unexpressed starting point, 4 Ahau 8 Cumhu. The positions of the corresponding month parts are, on the other hand, irregular. These, instead of standing immediately after the days whose positions in the year they designate, follow at the close of some six or seven intervening glyphs. These intervening glyphs have been

to reach the terminal date 8 Ahau 8 Zotz; and further, let us suppose that on inspecting the text the day part of this date (8 Ahau) has been found to be recorded immediately after the 0 kins of the number 9.16.5.0.0. Now, if the student will follow the next six or seven glyphs until he finds one like any of the forms in figure 65, the glyph immediately follow

dicator": a, c, e, g, h, Normal

onth parts of Initial-series terminal dates do not immediately follow the closing glyph of the Supplementary

ee different month signs-3 Cumhu, 3 Zotz, and 13 Yax-with each of which it is used to form a different date-2 Ahau 3 Cumhu, 2 Ahau 3 Zotz, and 2 Ahau 13 Yax. In these pages the month signs, with a few exceptions, do not follow immediately the days to which they belong, but on the contrary they are separated from them by several intervening glyphs. This abbreviation in the record of these dates was doubtless prompted by the desire or n

es 46-50 of the Dresden Codex, we may say that the regular position of the month glyphs in Maya wri

or all the calculations involved under steps 1-4 (pp. 134-151) should be pointed out. If after having worked out the terminal date of a given number acco

rror in our own

error in the or

lies without the op

counting the given number from the starting point have failed, the process should be reversed and the attempt made to reach the starting point by counting backward the given number from its recorded terminal date. Sometimes this reverse process will work out correctly, showing that there must be some arithmetical error in our original calculations which we have failed to detect. However, when both processes have failed several times to connect the starting point with the recorded terminal date by use of the give

rs in numerals, as the record of 7 for 8, or 7 for 12 or 17, that is, the omission or insertion of one or more bars or dots. In a very few instances there seem to

es. In such cases it is obviously impossible to go further in the present state of our knowledge. Special conditions presented by glyphs whose meanings are unknown may govern such cases. At all events, the failure of the rules under 1-4 to reach the terminal dates recorded as