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Units of Measure

". . . Y empezó a venir Ah Ppisté, el Agrimensor, y entonces llegó Chacté Abán
a preparar las parcelas de tierra para ser cultivadas. Y vino Uac Habnal a
marcar los linderos con señales de hierbas, en tanto que Miscit Ahau limpiaba
las tierras deslindadas. Todo esto mientras Ah Ppisul trazaba sus amplias
medidas . . ." (Chilam Balam de Chumayel).
_________________________________________________________________

_Until now, it is ignored which system of weights and measures_ the
Mayas used. To clear this issue, this section has compiled an
statistic of the measures of length obtained from several
investigators in the archaeological zones of southeast of Mexico, as
well as Guatemala and Honduras. The main sources include Morley (1),
Rupert and Denison (2), Ricketson (3), Ruppert (4), Thomson and
Proskouriakoff (5), Pollock (6), Ruz Lhuillier (7) and many others.
The procedure used in this section consisted of creating a list of the
dimensions of structures, architectonic elements and stelas, as it has
been reported by numerous archaeologists who have measured them. This
list required a careful selection. For instance, in several occasions
it happens that stelas are fragmented and the informant reported as
its height the piece he discovered. In other circumstances, the
measured height is understood as the part of the stela serving as the
foundation, and with a high chance of probability, it did not enter in
the proportions that the artist designed. Another observation is that
the thickness of the stela can not help us to describe the scale used
by the Mayas to measure lengths for the simple explanation that that
thickness was usually the one acquired from the original stone. I can
not imagine that the Maya artist engaged in the reduction of the
natural thickness of the stone with the purpose of matching a round
number of units. Therefore, if we wish to discover a unit of length
that could relate statistically with the Maya linear units we need to
look for in the width of the stone. The width has more probabilities
of having being chosen by the artist.

_From the description of the measures_ of the buildings we need to
carefully select our data. Frequently, the reports informed us of
widths that had a consolidation and do not have the original
dimensions. The sinking of buildings are translated in collapses and
unevenness that also do not reveal the Mayan scales of lengths.
Besides, there is an unfortunate trend of archaeologists to report
approximate measures. Few of them carry out their calculations to the
centimeter and none reported to the milimeter. Thus, a length of 1.092
cm it is reported as 1.10 cm and another of 68.5 cm as 70 cm. Example
of these innacuracies are in the reports of Smith and Kidder on Nebaj
and in the Mexican Instituto Nacional de Antropologia e Historia in
which the dimension are asigned in multiples of 5. Obviously, this
data can not be taken into consideration for the statistics.

_It was compiled a total of 1.128 measures_ of monuments of the Old
Empire, covering the most important sites : Uaxactun, Tikal, Bonampak,
Piedras Negras, Yaaxchilán, Quiriguá, Copán, Lubaantin, Xultun,
Balakbal, Uolantun, Chichantun, Yaxha, Laguna Perdida, Itzimte, Polol,
Motul de San José, Tayasal, Ixlu, Naachtun y Palenque. All data was
processed independently and convergent tendencies were reunited in one
statistical group to attempt to find out a periodicity in the design
or otherwise an arbitrary proportion or different systems in the Maya
provinces. The result of the statistical analysis was definite and
revealing. (Figure 1 in construction.)

_It was discovered, using minimization of differences_ between the
statistical crests and several modules, that exist a periodicity with
an interval of 4.56 cm and its multiples. Also, there is no existing
variation in the area covered by the Old Empire. In fact, from
Quiriguá to Palenque, and from the most ancient structures
corresponding to the beginning of Baktun 8 (41 D.C.) to the last ones
of the end of the Great Classic Period (950 D.C.), the module remains
invariable.

_From these results,_ it emerges the curiosity to investigate if the
new unit of measures of lengths was continued in the New Empire.
Fortunately, we have the magnificent work of Ruppert about Chichén
Itzá from which 973 useful dimensions used to create an identical
statistic of the one compiled for the Old Empire. I consider, of great
transcendence, to have obtained a module of 4.55 cm that for practical
means can be considered identical to the Old Empire. The revealed
uniformity is an eloquent example of the great Unity and cultural
continuity that the Maya civilization had through several millenniums.
(Figure 2 in construction.)

_Landa (pp. 111) tells us that the Mayas_ used to ". . . crop with a
measure of 400 feet named _hun uinic_ measured with a stick of 20
feet, 20 in width and 20 in length." The Diccionario de Motul (pp.905)
confirms that the "uinic" was ". . . a measure of soil to till or
cultivation of 20 Kanes. . .," and in another page (pp. 494) the Kaan
is defined as ". . . a measure of a rope that the natives use to
measure their "milpas" called "mecate" among the Spaniards... they
have a measure of three sticks or ropes of three "brazas" each rope
that make 36 (4 x 9) "brazas." Don Juan Martínez Hernández who edited
the Diccionario de Motul explains ". . . kaan: mecate (from the
nahuatl mecatl): measure of 24 varas of Burgos of 838 milimeters; each
"mecate" is subdivided in 12 "brazas" of 2 varas of Burgos, that is,
20.112 linear meters.

_Now, "braza" was the designation_ that the Spaniards gave to the unit
of length that the Mayas coincided approximately with the European
unit. This measure was the "zap" but consulting to the Diccionario de
Motul, Pio Perez, Otero and Beltran de Santa Rosa, I don't see
anything different except the "zap" is a "braza" equivalent to 2
varas.

_Two hypothesis can be formulated._ Based in the vigesimal character
of the Maya arithmetic and in the high incidence of measures of 91 cm
in the monuments of the Old Empire as well as the 45.4 cm in Chichén
Itzá a conclusion is that the Maya meter had exactly 91.2 cm and
divided in 20 parts. In this case, the zap had 1.82 m and a kaan would
be equivalent to 12 zaps or 24 Mayan meters. The total length of the
archaic kaan would be 21.84 m which is equivalent to one "braza."

_The other theory would consist_ of assuming that the Kaan originally
had only 20 Mayan meters and its length is equivalent to 18.20 m
modified after the Spaniards introduced the unit "vara" in Yucatán. If
we reread the citation taken from the Diccionario of Motul we noticed
there are two measures and both designated with the name Kaan. One has
9 varas, that is 15.084 m while the other contains 24 varas reaching a
length of 20.112 m. The hypothesis that the kaan has 18.20 m would
place this fact in an intermediate position of both versions.

_To clear this point,_ a second statistic is made but this time we
select only the distances of one kaan of order of magnitude or greater
under the assumption that the Maya planner worked with kaanes instead
of minor units in the design of terraces, courtyards, and sidewalks.
The research concluded that the kaan has 24 Maya meters of 91 cm each
one and with the introduction of the Spanish "vara", this unit was
shortened to 1.73 m.

_Morley says_ ". . . in reality, these ropes of measure are, in the
north of Yucatán, of 21.50 m of length." The Mayans say these ropes
are somewhat longer because "of the length taken by the birds."
Ignoring this fine Yucatanian irony I conclude that the true reason
lies on the fact that the "mecate" of 21.50 m is closer in length to
the ancient Maya kaan. Moreover, I dare to say that the length of the
"mecates" in the north of Yucatán, cited by Morley, is not precisely
21.50 m because it is illogical to think that if the "mecate" was
redefined in the Colonial era as 24 varas, the Yucatanians of that
time would have anticipated the invention of the meter and had added
exactly one meter and a half to the rope. It is more reasonable to
think that they added a vara (1.67 m) and in this case, the "mecate"
has a length of 21.78 m differing by 6 cm only from the old kaan of
21.84 whose existence is postulated.

_None of these explanations_ make us assume that the Mayas knew the
mile or Spanish "legua." Instead, about the "sacbé" (white road) from
Cobá to Yaxuná - that, incidentally measures half kaan of width -
Alfonso Villa confirmed the presence of monuments placed approximately
within an 8 kilometers of distance between each other. I suggest this
is a good opportunity to verify or modify the two hypotheses presented
in this section. If the Mayas extended the vigesimal system to large
measures of length it is probably that the units of major length than
the kaan are the "kaalkan" (20 x 21.84 = 436.8 m.) and the "bakaan"
(400 x 21.84 = 8.736 m.) In this case, the trails of the road Cobá to
Yaxuná correspond to bakaanes.

_I return now to the Maya vara _of 91 cm, base of the investigation.
Beltrán y Pío Pérez concide in the measure of the vara called BETAN in
Maya. This name is highly significative because it is composed of two
roots BE (road) and TAN (a mean to produce something.) In consequence,
the word BETAN expresses the functional relation existing between the
unit of measure and the road construction. In the names KAAN and BETAN
are hidden a fascinant semantic concept: while the kaan, agricultural
measure by excellence, is associated with KAN (mature, crop, corn,
gold, etc.), BETAN establishes a conexion with engineering and
communications where the primary measure (the step) is a unit of
length and at the same time the act itself of transfer and
communication. Same as the "milla" originated from the distance of
1,000 steps of five Romans feet and the vara of Burgos of 83.8 cm as
the average length of the step, it is probably that the Maya betan
recognizes the same genesis.

_In a descending order of magnitude,_ half of a betán is 45.5 cm of
length. Our statistical data allow to know that in the constructions
of the Old Empire the dimensions are in the order of a betán
predominantly and in the New Empire dominates the half betán. This
fact is explained logically that the Old Empire' s geographical
domains are near the mountains where is easier to obtain stones of
larger sizes. Probably, the half betán mentioned in the Diccionario de
Motul is the kaan composed of 36 "brazas" of box. The word "box" is a
label that the Mayas gave to the black people of Campeche, imported by
the Spaniards, and is the despective name that the white Europeans
gave to the brown Yucatanians. In consequence, the "braza" of box may
be a designation given by the Spaniards to the original measure
corresponding to half betán.

_The module of 4.55 cm_ identified previously is exactly the 20th part
of a betán. We don't know the assigned names to the vigesimal
fractions judging from the microdimensions recognized in the codexes
and ornaments so I baptize them with the name of AZAB (middle)
equivalent to 4.55 cm and CHAN (small) to the 2.275 cm subunit, hoping
one day we discover the real names.

_As a summary, _I offer the following hypothetical metric scale.

_MAYA METRIC SCALE_
1 bakaan 20 kalkaanes 8.738 m
1 kalkaan 20 kaanes 436.8 m
1 kaan 24 betanes 21.84 m
1 betán 20 azabes 91 cm
1 azab 20 chanes 4.55 cm
1 chan 2.275 cm

The braza, codo, cuarta and geme are Spaniard units that seem to have
no relation with the Maya equivalents.

_Now, our curiosity takes us_ up to the geographical extension that
the betán could have had. Of course, I believe to have demonstrated
the uniform application in the Mayan zone of this unit. But, was it
known outside the geographical and cultural contour? Was it extended
to the Altiplano? Did it reach South America? These are questions of
great transcedence and I put ahead some data for others with the means
to investigate them.

_From the work of Javier Romero _(8) on the burials of the pyramid of
Cholula we extract 11 measures from a total of 19 samples that are
exact multiples of 4.55 cm. Pollock cites Cummings who inform that the
pyramid of Cuicuilco measures at the base 387 feet (118 m) that is
exactly 130 betanes. The top platform has a diameter of 291 feet (88.7
m) that is 97.5 betanes. The height is 74 feet (22.56 m) that concides
with 25 betanes and the ramp has a width of 70 feet (21.34 m) that is
23.5 betanes, which is almost a kaan.

_From Xochicalco_ we have compiled 86 measures from a total of 94
samples that are exact multiples of 4.55 cm. The pyramid of Tajín
measures at the base 36.63 m, that is, 40 betanes. The prints of the
seven main steps are 2.74 m that correspond to 3 betanes. Finally, the
height is 22.80 m or 25 betanes.

_In Quemada, Zacatecas,_ where Mohedano confirmed Mayan features and a
narrow parallelism with Xochicalco, 22 measures were obtained matching
the Mayan scale. It seems there are many coincidences that I do not
want to make premature conclusions until more data is available to
clear this issue.

_In regard to a geometric vocabulary,_ the Mayan language has precise
names and different for height (CAANALIL), width (COCH), thickness
(PIM, only for tabular forms), thickness (POLOC, for columnar bodies),
thickness (NUCUCH, for three dimensions), length (CHAUAC), depth
(TAM), inclined (CHINAN), flat (CHUEN), curved (PEPET), concave (LOP),
convex (BOZAN), vertical (UAOM), horizontal (TAX), etc. which confirm
the wide knowledge of geometry applied to buildings.

_The unit of volume, _for architectonic purposes, must have been the
cubic betán, but we know of an agricultural measure, the MUCUB,
containing half load of corn. The agricultural unit, as I explained
previously, is the "uinic" that consisted of 400 squared kaanes, that
is, 0.954 hectareas.

_For weighing,_ it was used the CUCH or "load" that must have
contained 43.4 kg. The scale was known to the Mayas using stones as
units of weights.

_In regard to time,_ we know of two equivalents of the hour. The LUB
that means "day's work" and KIN TZIL, that literally is translated as
"sun breaks." It is probably that the day was divided in 24 hours.
Besides the obvious relations with the 12 constellations of the Mayan
Zodiac of 12 signs the number 24 has desirable characteristics for the
use of the Mayan people. It is easy to appreciate that 24 can be
divided by half, by thirds, by fourths, by sixths, by eighths, by
twelveths and by twenty-fourths parts given entire numbers.

Few scholars have investigated about the way the Mayas measured the
hours of the day, in spite of being sure they used solar clocks. The
important units of time in one day (KIN) are well-known. In ascending
order, they are:

_Mayan Units of Time_
KIN One day
UINAL 20 kines
TZOLKIN 13 uinales = 260 kines
HAAB 18 uinales + 5 kines = 365 kines
TUN 18 uinales = 360 kines
KATUN 20 tunes (some authors comment the katunes are 24 in the New
Empire.)
BAKTUN 20 katunes
GREAT CYCLE 13 baktunes

It is inferred that other larger cycles are the PIKTUN, KINCHILTUN and
ALAUTUN, elevating the number of cycles to 3,200,000 years each one.
However, I have found a base to sustain that the cycle of 13 baktunes,
made evident in inscriptions, and when added is equal to 5,200 years
as the main cycle. Five of these cycles complete 26,000 years that the
terrestrial axis needs to describe a precesional circle that makes the
astronomical North shifts in the sky. If the Mayas observed this
precesion and calculated the period it is probably they chose it as
the Great Cycle. On the other hand, there is evidence that two
catastrophes occurred separately within a period of 13 baktunes.

To finalize this section, I must comment that the concept of time the
Mayas had was distinct from ours. These ideas will be included in the
section 9, [1]the Mathematical Concept of the Mayan Universe.

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References

1. Morley, S.G. The Inscriptions of Peten. Carnegie Institution. Publ.
No. 437. Washington, D.C. 1938.

2. Ruppert, K. and Denison, J.H. Archaeological Reconnaisance in
Campeche, Quintana Roo & Peten. Carnegie Institution. Publ. No. 543.
Washington, D. C. 1943.

3. Ricketson, E.B. Uaxactun, Guatemala. Carnegie Institution. Publ.
No. 477. Washington, D.C. 1937.

4. Ruppert, K. Chichen Itza. Carnegie Institution. Publ. No. 595.
Washington, D.C. 1952.

5. Ruppert, K., Thompson, E., and Proskouriakoff, T. Bonampak Chiapas,
Mexico. Carnegie Institution. Publ. No. 602. Washington, D.C. 1955.

6. Pollock, H.E.D. Round Structures of Aboriginal Middle America.
Carnegie Institution. Publ. No. 471. Washington, D.C. 1936.

7. Ruz Lhuiller. Exploraciones Arqueologicas en Palenque. Anales del
Instituto Nacional de Antropologia e Historia. S.E.P. Tomo X, No. 39.
pp. 69-299. Mexico. 1958.

8. Romero, Javier. Estudio de los Entierros de la Pirámide de Cholula.
Anales del Museo Nacional de Arqueología, Historia y Etnografía. Tomo
II. Mexico. 1935.

References

1. file://localhost/www/sat/files/chapter9.htm
2. file://localhost/www/sat/files/control.htm
3. file://localhost/www/sat/files/chapter8.htm