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