Sylvanus Griswold Morley

“ 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. ”

“ Goodman's Archæic Annual Calendar and Archæic Chronological Calendar (1897) are by far the best. By using either of the above the necessity of reducing the higher terms to units of the first order is obviated. On the other hand, the processes by means of which this is achieved in each case are far more complicated and less easy of comprehension than those of the method followed in this book, a method which from its simplicity might be termed perhaps the logical way, since it reduces all quantities to a primary unit, which is the same as the primary unit of the Maya calendar. ”

“ 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. ”

“ 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 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. ”

“ 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. ”

“ 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. ”

“ The numbers used above in connection with dates merely express the difference in time between starting points and terminal dates, without assigning either set of dates to their proper 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 been reached, more than 3,000 years after the starting point of their chronology, the Maya had upward of 70 distinct dates of exactly the same name to distinguish from one another. ”

“ For example, when the student finds a number which he can identify as an Initial Series, [38] 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. ”

“ 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 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. ”

“ 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. ”

“ 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. [13] 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. ”

“ 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 dealing with this kind of series that the student will find the greatest number of exceptions to the general rule. ”

“ 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. ”

“ 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. ”

“ 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.
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. ”

“ In b, d, f, and h of the same figure a single crescent stands between the two numerical dots, and this again harmoniously fills in the glyph block. While the crescent (
*) is the usual form taken by this purely decorative element, crossed lines (**) are found in places, as in (
†) ; or, again, a pair of dotted elements (††) , as in (‡) . These variants, however, are of rare occurrence, the common form being the crescent shown in figure 43.
Fig. 42. Examples showing the way in which the numerals 1, 2, 6, 7, 11, 12, 16, and 17 are not used with period, day, or month signs. ”

“ 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 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. ”

“ Both normal forms and head variants for zero, as indeed for all the numbers, have been found in the inscriptions. The normal forms for zero are shown in figure 47. They are common and are unmistakable. An interesting origin for this sign has been suggested by Mr. A. P. Maudslay. On pages 75 and 76 of the Codex Tro-Cortesiano [8] the 260 days of a tonalamatl are graphically represented as forming the outline shown in figure 48, a. Half of this (see fig. 48, b) is the sign which stands for zero (compare with fig. 47) . ”

“ 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 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. [22] 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. ”

“ Whether this explains the connection between the outline of the tonalamatl and the zero sign, or whether indeed there be any connection between the two, is of course a matter of conjecture.
There is still one more normal form for zero not included in the examples given above, which must be described. This form (fig. 49) , which occurs throughout the inscriptions and in the Dresden Codex, [9] is chiefly interesting because of its highly specialized function. ”

“ 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.
Fig. 53. Head-variant numerals 14 to 19, inclusive, and 0. ”

“ The case in point lies without the operation of our rules.
It is always safe for the beginner to proceed on the assumption that the first of the above alternatives is the cause of the error; in other words, that his own calculations are at fault. ”

“ 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) . ”

“ 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 ”

“ Commencing at the bottom of figure 58 with the highest period involved and reading up, A6, [24] 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 = 40,320,000) ; 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. ”