Chapter II

, the inscriptions generally do not appear to treat of tonalamatls.

[Illustration: FIG. 46. Sign for 0 in the codices.]

If the Maya numerical system had no vital need for a character to express the number 20, a sign to represent zero was absolutely {93} indispensable. Indeed, any numerical system which rises to a second order of units requires a character which will signify, when the need arises, that no units of a certain order are involved; as zero units and zero tens, for example, in writing 100 in our own Arabic notation.

The character zero seems to have played an important part in Maya calculations, and signs for it have been found in both the codices and the inscriptions. The form found in the codices (fig. 46) is lenticular; it 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 remaining two are different. It is noteworthy, however, that these last three forms all stand in the 2d, or uinal, place in the texts in which they occur, though whether this fact has influenced their variation is unknown.

[Illustration: FIG. 47. Sign for 0 in the inscriptions.]

[Illustration: FIG. 48. Figure showing possible derivation of the sign for 0 in the inscriptions: _a_, Outline of the days of the tonalamatl as represented graphically in the Codex Tro-Cortesiano; _b_, half of same outline, which is also sign for 0 shown in fig. 47.]

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[64] 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). 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 the tonalamatl, and finally the general idea of completeness {94} applicable to any period; for no period could be exactly complete without a fractional remainder unless all the lower periods were wanting; that is, represented by zero. 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,[65] is chiefly interesting because of its highly specialized function. Indeed, it was used for one purpose only, namely, to express the first, or zero, position in each of the 19 divisions of the haab, or year, and for no other. In other words, it denotes the positions 0 Pop, 0 Uo, 0 Zip, etc., which, as we have seen (pp. 47, 48), corresponded with our first days of the months. The forms shown in figure 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 become elaborated into a head. A simpler form is that in _f_ and _g_. Although comparatively rare, this glyph is so unusual in form that it can be readily recognized. Moreover, if the student will bear in mind the two following points concerning its use, he will never fail to identify it in the inscriptions: The "spectacle" sign (1) can be attached only to the glyphs for the 19 divisions of the haab, or year, that is, the 18 uinals and the xma kaba kin; in other words, it is found only with the glyphs shown in figures 19 and 20, the signs for the months in the inscriptions and codices, respectively.

[Illustration: FIG. 49. Special sign for 0 used exclusively as a month coefficient.]

(2) It can occur only in connection with one of the four day-signs, Ik, Manik, Eb, and Caban (see figs. 16, _c_, _j_, _s_, _t_, _u_, _a'_, _b'_, and 17, _c_, _d_, _k_, _r_, _x_, _y_, respectively), since these four alone, as appears in Table VII, can occupy the 0 (zero) positions in the several divisions of the haab. {95}

[Illustration: FIG. 50. Examples of the use of bar and dot numerals with period, day, or month signs. The translation of each glyph appears below it.]

Examples of the normal-form numerals as used with the day, month, and period glyphs in both the inscriptions and the codices are shown in figure 50. Under each is given its meaning in English.[66] The student is advised to familiarize himself 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 in figure 46. As these two forms never occur with day, month, or period glyphs, and as they have been found only in the codices, examples showing their use will not be given until

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