MAYA ARITHMETIC
The present chapter will be devoted to the consideration of Maya arithmetic
in its relation to the calendar. It will be shown how the Maya expressed
their numbers and how they used their several time periods. In short, their
arithmetical processes will be explained, and the calculations resulting
from their application to the calendar will be set forth.
The Maya had two different ways of writing their numerals,[57] namely: (1)
With normal forms, and (2) with head variants; that is, each of the
numerals up to and including 19 had two distinct characters which stood for
it, just as in the case of the time periods and more rarely, the days and
months. The normal forms of the numerals may be compared to our Roman
figures, since they are built up by the combination of certain elements
which had a fixed numerical value, like the letters I, V, X, L, C, D, and
M, which 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 numerals for 14, 15, 16, 17, 18, and 19 by
applying the essential characteristic of the head variant for 10 to the
head variants for 4, 5, 6, 7, 8, and 9, respectively, just as the sign for
10--that is, one in the tens place and zero in the units place--is used in
connection with the signs for the first nine figures in Arabic notation to
form the numbers 11 to 19, inclusive. Both of these notations occur in the
inscriptions, but with very few exceptions[58] no head-variant numerals
have yet been found in the codices.
BAR AND DOT NUMERALS
The Maya "Roman numerals"--that is, the normal-form numerals, up to and
including 19--were expressed by varying combinations of two elements, the
dot (.) which represented the numeral, or numerical value, 1, and the bar,
or line (--) which represented the numeral, or numerical value, 5. By
various combinations of these two {88} elements alone the Maya expressed
all the numerals from 1 to 19, inclusive. The normal forms of the numerals
in the codices are shown in figure 39, in which one dot stands for 1, two
dots 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-hatching ();
some appear as crescents (+) or curls (++), more rarely as (++) or (++++).
The bars show even a greater variety of treatment (see fig. 41). All these
decorations, however, in no way affect the numerical value of the bar and
the dot, which remain 5 and 1, respectively, throughout the Maya writing.
Such embellishments as those just described are found only in the
inscriptions, and their use was probably due to the desire to make the bar
and dot serve a decorative as well as a numerical function.
[Illustration: FIG. 39. Normal forms of numerals 1 to 19, inclusive, in the
codices.]
[Illustration: FIG. 40. Normal forms of numerals 1 to 19, inclusive, in the
inscriptions.]
[Illustration: FIG. 41. Examples of bar and dot numeral 5, showing the
ornamentation which the bar underwent without affecting its numerical
value.]
An important exception to this statement should be noted here in connection
with the normal forms for the numbers 1, 2, 6, 7, 11, 12, 16, and 17, that
is, all which involve the use of _one_ or _two_ dots in their
composition.[59] In the inscriptions, as we have seen in Chapter II, every
glyph was a balanced picture, exactly fitting its allotted space, even at
the cost of occasionally losing some of its elements. To have expressed the
numbers 1, 2, 6, 7, 11, 12, 16, and 17 as in the codices, with just the
proper number of bars and dots in each case, would have left unsightly gaps
in the outlines of the glyph blocks (see fig. 42, _a-h_, where these
numbers are shown as the coefficients of the katun sign). In _a_, _c_, _e_,
and _g_ of 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 {89} 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 character were introduced to fill the empty spaces. In figure
43, _a_, _c_, _e_, _g_, the spaces on each side of the single dot have been
filled with ornamental {90} crescents about the size of the dot, and these
give the glyph in each case a final touch of balance and harmony, which is
lacking without them. 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.
[Illustration: 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.]
[Illustration: FIG. 43. Examples showing the way in which the numerals 1,
2, 6, 7, 11, 12, 16, and 17 _are_ used with period, day, or month signs.
Note the filling of the otherwise vacant spaces with ornamental elements.]
[Illustration: FIG. 44. Normal forms of numerals 1 to 13, inclusive, in the
Books of Chilan Balam.]
The use of these purely ornamental elements, to fill the empty spaces in
the normal forms of the numerals 1, 2, 6, 7, 11, 12, 16, and 17, is a
fruitful source of error to the student of the inscriptions. Slight
weathering of an inscription is often sufficient to make ornamental
crescents look exactly like numerical 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 {91} numerals in the
inscriptions, or otherwise he will quickly find himself involved in a
tangle from which there seems to be no egress. Probably more errors in
reading the inscriptions have been made through the incorrect
identification of these numerals than through any other one cause, and the
student is urged to be continually on his guard if he would avoid making
this capital blunder.
Although the early Spanish authorities make no mention of the fact that the
Maya expressed their numbers by bars and dots, native testimony is not
lacking on this point. Doctor Brinton (1882 b: p. 48) gives this extract,
accompanied by the drawing shown in figure 44, from a native writer of the
eighteenth century who clearly describes this system of writing numbers:
They [our ancestors] used [for numerals in their calendars] dots and
lines [i. e., bars] back of them; one dot for one year, two dots for
two years, three dots for three years, 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 dots above the line, seven years; if three dots above, eight years;
if four dots above the line, nine; a dot above two lines, eleven; if
two dots, twelve; if three dots, thirteen.
This description is so clear, and the values therein assigned to the
several combinations of bars and dots have been verified so extensively
throughout both the inscriptions and the codices, that we are justified in
identifying the bar and dot as the signs for five and one, respectively,
wherever they occur, whether they are found by themselves or in varying
combinations.
In the codices, as will appear in Chapter VI, the bar and dot numerals were
painted in two colors, black and red. These colors were used to 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 with dots of the
other color, each number being either all red or all black (see p. 93,
footnote 1, for the single exception to this rule).
By the development of a special character to represent the number 5 the
Maya had far surpassed the Aztec in the science of mathematics; indeed, the
latter seem 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 above 1. It is clearly seen that such a system of
notation has very definite limitations, which must have seriously retarded
mathematical progress among the Aztec.
In the Maya system of numeration, which was vigesimal, there was no need
for a special character to represent the number 20,[61] because {92} (1) as
we have seen in Table VIII, 20 units of any order (except the 2d, in which
only 18 were required) were equal 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
in each period except the uinal, as 20 kins in each uinal, 20 tuns in each
katun, 20 katuns in each cycle, these positions were numbered not from 1 to
20, but on the contrary from 0 to 19, a system which eliminated the need
for a character expressing 20.
[Illustration: FIG. 45. Sign for 20 in the codices.]
In spite of the foregoing fact, however, the number 20 has been found in
the codices (see fig. 45). A peculiar condition there, however, accounts
satisfactorily for its presence. In the codices the sign for 20 occurs only
in connection with tonalamatls, which, as we shall see later, were usually
portrayed 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 was necessary which should stand for 20 regardless
of its position or relation to others. The sign shown in figure 45 was used
for this purpose. It has not yet been found in the inscriptions, perhaps
because, as was pointed out in 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