Chapter I

is based shows an absolutely uninterrupted sequence of katuns for more than 1,100 years. See Brinton (1882 b: pp. 152-164). It is necessary to note here a correction on p. 153 of that work. Doctor Brinton has omitted a Katun 8 Ahau from this u kahlay katunob, which is present in the Berendt copy, and he has incorrectly assigned the abandonment of Chichen Itza to the preceding katun, Katun 10 Ahau, whereas the Berendt copy shows this event took place during the katun omitted, Katun 8 Ahau.

[55] There are, of course, a few exceptions to this rule--that is, there are some monuments which indicate an interval of more than 3,000 years between the extreme dates. In such cases, however, this interval is not divided into katuns, nor in fact into any regularly recurring smaller unit, with the single exception mentioned in footnote 1, p. 84.

[56] On one monument, the tablet from the Temple of the Inscriptions at Palenque, there seems to be recorded a kind of u kahlay katunob; at least, there is a sequence of ten consecutive katuns.

[57] The word "numeral," as used here, has been restricted to the first twenty numbers, 0 to 19, inclusive.

[58] See p. 96, footnote 1.

[59] In one case, on the west side of Stela E at Quirigua, the number 14 is also shown with an ornamental element (). This is very unusual and, so far as the writer knows, is the only example of its kind. The four dots in the numbers 4, 9, 14, and 19 never appear thus separated in any other text known.

[60] In the examples given the numerical coefficients are attached as prefixes to the katun sign. Frequently, however, they occur as superfixes. In such cases, however, the above observations apply equally well.

[61] Care should be taken to distinguish the number or figure 20 from any period which contained 20 periods of the order next below it; otherwise the uinal, katun, and cycle glyphs could all be construed as signs for 20, since each of these periods contains 20 units of the period next lower.

[62] The Maya numbered by relative position from bottom to top, as will be presently explained.

[63] This form of zero is always red and is used with black bar and dot numerals as well as with red in the codices.

[64] It is interesting to note in this connection that the Zapotec made use of the same outline in graphic representations of the tonalamatl. On page 1 of the Zapotec Codex Féjerváry-Mayer an outline formed by the 260 days of the tonalamatl exactly like the one in fig. 48, _a_, is shown.

[65] This form of zero has been found only in the Dresden Codex. Its absence from the other two codices is doubtless due to the fact that the month glyphs are recorded only a very few times in them--but once in the Codex Tro-Cortesiano and three times in the Codex Peresianus.

[66] The forms shown attached to these numerals are those of the day and month signs (see figs. 16, 17, and 19, 20, respectively), and of the period glyphs (see figs. 25-35, inclusive). Reference to these figures will explain the English translation in the case of any form which the student may not remember.

[67] The following possible exceptions, however, should be noted: In the Codex Peresianus the normal form of the tun sign sometimes occurs attached to varying heads, as (). Whether these heads denote numerals is unknown, but the construction of this glyph in such cases (a head attached to the sign of a time period) absolutely parallels the use of head-variant numerals with time-period glyphs in the inscriptions. A much stronger example of the possible use of head numerals with period glyphs in the codices, however, is found in the Dresden Codex. Here the accompanying head () is almost surely that for the number 16, the hatchet eye denoting 6 and the fleshless lower jaw 10. Compare (+) with fig. 53, _f-i_, where the head for 16 is shown. The glyph () here shown is the normal form for the kin sign. Compare fig. 34, b. The meaning of these two forms would thus seem to be 16 kins. In the passage in which these glyphs occur the glyph next preceding the head for 16 is "8 tuns," the numerical coefficient 8 being expressed by one bar and three dots. It seems reasonably clear here, therefore, that the form in question is a head numeral. However, these cases are so very rare and the context where they occur is so little understood, that they have been excluded in the general consideration of head-variant numerals presented above.

[68] It will appear presently that the number 13 could be expressed in two different ways: (1) by a special head meaning 13, and (2) by the essential characteristic of the head for 10 applied to the head for 3 (i. e., 10 + 3 = 13).

[69] For the discussion of Initial Series in cycles other than Cycle 9, see pp. 194-207.

[70] The subfixial element in the first three forms of fig. 54 does not seem to be essential, since it is wanting in the last.

[71] As previously explained, the number 20 is used only in the codices and there only in connection with tonalamatls.

[72] Whether the Maya used their numerical system in the inscriptions and codices for counting anything besides time is not known. As used in the texts, the numbers occur only in connection with calendric matters, at least in so far as they have been deciphered. It is true many numbers are found in both the inscriptions and codices which are attached to signs of unknown meaning, and it is possible that these may have nothing to do with the calendar. An enumeration of cities or towns, or of tribute rolls, for example, may be recorded in some of these places. Both of these subjects are treated of in the Aztec manuscripts and may well be present in Maya texts.

[73] The numerals and periods given in fig. 56 are expressed by their normal forms in every case, since these may be more readily recognized than the corresponding head variants, and consequently entail less work for the student. It should be borne in mind, however, that any bar and dot numeral or any period in fig. 56 could be expressed equally well by its corresponding head form without affecting in the least the values of the resulting numbers.

[74] There may be three other numbers in the inscriptions which are considerably higher (see pp. 114-127).

[75] These are: (1) The tablet from the Temple of the Cross at Palenque; (2) Altar 1 at Piedras Negras; and (3) The east side of Stela C at Quirigua.

[76] This case occurs on the tablet from the Temple of the Foliated Cross at Palenque.

[77] It seems probable that the number on the north side of Stela C at Copan was not counted from the date 4 Ahau 8 Cumhu. The writer has not been able to satisfy himself, however, that this number is an Initial Series.

[78] Mr. Bowditch (1910: pp. 41-42) notes a seeming exception to this, not in the inscription, however, but in the Dresden Codex, in which, in a series of numbers on pp. 71-73, the number 390 is written 19 uinals and 10 kins, instead of 1 tun, 1 uinal, and 10 kins.

[79] That it was a Cycle 13 is shown from the fact that it was just 13 cycles in advance of Cycle 13 ending on the date 4 Ahau 8 Cumhu.

[80] See p. 156 and fig. 66 for method of designating the individual glyphs in a text.

[81] The kins are missing from this number (see A9, fig. 60). At the maximum, however, they could increase this large number only by 19. They have been used here as at 0.

[82] As will be explained presently, the kin sign is frequently omitted and its coefficient attached to the uinal glyph. See p. 127.

[83] Glyph A9 is missing but undoubtedly was the kin sign and coefficient.

[84] The lowest period, the kin, is missing. See A9, fig. 60.

[85] The use of the word "generally" seems reasonable here; these three texts come from widely separated centers--Copan in the extreme southeast, Palenque in the extreme west, and Tikal in the central part of the area.

[86] A few exceptions to this have been noted on pp. 127, 128.

[87] The Books of Chilan Balam have been included here as they are also expressions of the native Maya mind.

[88] This excludes, of course, the use of the numerals 1 to 13, inclusive, in the day names, and in the numeration of the cycles; also the numerals 0 to 19, inclusive, when used to denote the positions of the days in the divisions of the year, and the position of any period in the division next higher.

[89] Various methods and tables have been devised to avoid the necessity of reducing the higher terms of Maya numbers to units of the first order. Of the former, that suggested by Mr. Bowditch (1910: pp. 302-309) is probably the most serviceable. Of the tables Mr. 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. This method was first devised by Prof. Ernst Förstemann, and has the advantage of being the most readily understood by the beginner, sufficient reason for its use in this book.

[90] This number is formed on the basis of 20 cycles to a great cycle (20×144,000=2,880,000). The writer assumes that he has established the fact that 20 cycles were required to make 1 great cycle, in the inscriptions as well as in the codices.

[91] This is true in spite of the fact that in the codices the starting points frequently appear to follow--that is, they stand below--the numbers which are counted from them. In reality such cases are perfectly regular and conform to this rule, because there the order is not from top to bottom but from bottom to top, and, therefore, when read in this direction the dates come first.

[92] These intervening glyphs the writer believes, as stated in