The Mayan Numerical System

The Mayan Numerical System.

The Mayas had a mathematical turn, and possessed a developed numeral system . It counted by units and scores; in other words, it was a vigesimal system. The Mayan cardinal numbers were:—

Hun,	one.
Ca,	two.
Ox,	three.
Can,	four.
Ho,	five.
Uac,	six.
Uuc,	seven.
Uaxac,	eight.
Bolon,	nine.
Lahun,	ten.
Buluc,	eleven.
Lahca,	twelve.
Oxlahun,	thirteen.
Canlahun,	fourteen.
Holhun,	fifteen.
Uaclahun,	sixteen.
Uuclahun,	seventeen.
Uaxaclahun,	eighteen.
Bolonlahun,	nineteen.
Hunkal,	twenty.

The composition of these Mayan numerals from twelve to nineteen inclusive is easily seen. Lahun is apparently a compound of lah hun (sc. uinic), “it finishes one (man);” that is, in counting on the
fingers. Lah means the end, to end, and also the whole of anything. Kal, a score, is literally a fastening together, a shutting up, from the verb kal, to shut, to lock, to button up, etc.

The Mayan numerals from twenty upward, the scores are used:—

Hun tu kal,	one to the score, 21.
Ca tu kal,	two to the score, 22.
Ox tu kal,	three to the score, 23,

and so on up to

Ca kal,

two score, 40.

Above forty, three different methods can be used to continue the numeration.

1. We may continue the same employed between 20 and 40, thus:—

Hun tu cakal,	one to two score, 41.
Ca tu cakal,	two to two score, 42.
Ox tu cakal,	three to two score, 43,

and so on.

2. The Maya numeral copulative catac can be used, with the numeral particle tul; as:—

Cakal catac catul,	two score and two, 42.
Cakal catac oxtul,	two score and three, 43.

3. We may count upon the next score above, as:

Hun tu yoxkal,	one on the third score, 41.
Ca tu yoxkal,	two on the third score, 42.
Ox tu yoxkal,	three on the third score, 43.

The last mentioned Maya numeral system is that advanced by Father Beltran, and is the only one formally mentioned by him. It has recently been carefully analyzed by Prof. Leon de Rosny, who has shown that it is a consistent vigesimal method.

It might be asked, and the question is pertinent, and is left unanswered by Prof. Leon de Rosny, why hun tu kal means “one to the score,” and hun tu can kal is translated, “one on the fourth score.” This important shade of meaning may be given, I think, by the possessive u which originally belonged in the phrase, but suffered elision. Properly it should be,

Hun tu u can kal.

This seems apparent from other numbers where it has not suffered elision, but merely incorporation, as:—

Hun tu yox kal	=	hun tu u ox kal, 41.
Hu tu yokal	=	hun tu u ho kal, 81.

This Maya system of numeration, advanced by Beltran, appears to have been adopted by all of the later writers, who may have learned the Maya largely from his Grammar. Thus, in the transla]tion of the Gospel of St. John, published by the Baptist Bible Translation Society, chap. II, v. 20; Xupan uactuyoxkal hab utial u mental letile kulnaa, “forty and six years was this temple in building;” and in that of the Gospel of St. Luke, said to have been the work of Father Joaquin Ruz, the same system is followed.

Nevertheless, Beltran’s method has been severely criticised by Don Juan Pio Perez, who ranks among the ablest Yucatecan linguists of this century. He has pronounced it artificial, not in accordance with either the past or present use of the natives themselves, and built up out of an effort to assimilate the Maya to the Latin numeral system.

[I give his words in the original, from his unpublished essay on Maya grammar. and numbers

“Los Indios de Yucatan cuentan por veintenas, que llaman kal y en cierto modo tienen diez y nueve unidades hasta completar la primera veintena que es hunkal aunque en el curso de esta solo se encuentran once numeros simples, pues los nombres de los restantes se forman de los de la primera decena.

“Para contar de una à otra veintena los numeros fraccionarios ò las diez y nueve unidades, terminadas por la particula tul ò su sincopa tu,se juntan antepuestas à la veintena espresada; por exemplo, hunkal, 20; huntukal, 21; catukal, 22; y huntucakal, 41; catucakal, 42; oxtucankal, 83; cantuhokal, 140, etc.

“El Padre Fr. Beltran de Santa Rosa, como puede verse en su Arte de Lengua Maya, formó un sistema distinto à este desde la 2ª veintena hasta la ultima, pues para espresar las unidades entre este y la 3ª veintena pone à esta terminandolas y por consiguiente rebajandole su valor por solo su anteposicion à dichas unidades fraccionarias, y asi para espresar el numero 45 por ejemplo dice ho tu yoxkal, cuando oxkal ò yoxkal significa 60.

“No sé de donde tomó los fundamentos en que se apoya este sistema, quiza en el uso de su tiempo, que no ha llegado hasta este; aunque he visto en varios manuscritos antiguos, que los Indios de entonces como los de ahora, usaban el sistema que indico, y espresaban las unidades integras que numeraban, y para espresar el numero 65 dicen;Oxkal catac hotul ù hotu oxkal, que usa el Padre Beltran por 45.

“Mas el metodo que explico esta apoyado en el uso y aun en el curso que se advierte en la 1ª y 2ª veintena é indican que asi deben continuar las decenas hasta la 20ª y no formar sistemas confusos que por ser mas ô menos análogos à la numeracion romana lo juzgaban mas ô menos perfectos, porque la consideraban como un tipo a que debia arreglarse cualquiera otra lengua, cuando en ellas todo lo que no este conforme con el uso recibido y corriente, es construir castillos en el aire y hacer reformas que por mas ingeniosas que sean, no pasan de inoficiosas.”

In the face of this severe criticism of Father ]Beltran’s system, I cannot explain how it is that in Pio Perez’s own Dictionary of the Maya, the numerals above 40 are given according to Beltran’s system; and that this was not the work of the editors of that volume (which was published after his death), is shown by an autographic manuscript of his dictionary in my possession, written about 1846, in which also the numerals appear in Beltran’s form.

Three other manuscript dictionaries in my collection, all composed previous to 1690, affirm the system of Beltran, and I am therefore obliged to believe that it was authentic and current among the natives long before white scholars began to dress up their language in the ill-fitting garments of Aryan grammar.

Proceeding to higher numbers, it is interesting to note that they also proceed on the vigesimal system, although this has not heretofore been distinctly shown. The ancient computation was:

20 units	=	one kal	=	20
20 kal	=	one bak	=	400
20 bak	=	one pic	=	8,000
20 pic	=	one calab	=	160,000
20 calab	=	one kinchil or tzotzceh	=	3,200,000
20 kinchil	=	one alau	=	64,000,000

]This ancient system was obscured by the Spaniards using the word pic to mean 1000 and kinchil to mean 1,000,000, instead of their original significations.

The Maya meaning of kal, I have already explained to be a fastening together, a package, a bundle. Bak, as a verb, is to tie around and around with a network of cords; pic is the old word for the short petticoat worn by the women, which was occasionally used as a sac. If we remember that grains of corn or of cacao were what were generally employed as counters, then we may suppose these were measures of quantity. The word kal (qal), in Kiche means a score and also specifically 20 grains of cacao; bak in Cakchiquel means a corn-cob, and as a verb to shell an ear of corn, but I am not clear of any connection between this and the numeral. Other meanings of bak in Maya are “meat” and the partes pudendas of either sex.

Calab, seems to be an instrumental form from cal, to stuff, to fill full. The word calam is used in the sense of excessive, overmuch. In Cakchi]quel the phrase mani hu cala, not (merely) one cala, is synonymous with mani hu chuvi, not (merely) one bag or sack, both meaning a countless number. In that dialect the specific meaning of cala is 20 loads of cacao beans.

The term tzotzceh means deerskin, but for kinchil and alau, I have found no satisfactory derivation that does not strain the forms of the word too much. I would, however, suggest one possible connection of meaning.

In kinchil, we have the word kin, day; in alau, the word u month, and in the term for mathematical infinity, hunhablat, we find hun haab, one year, just as in the related expression, hunhablazic, which signifies that which lasts a whole year. If this suggestion is well grounded, then in these highest expressions of quantity (and I am inclined to think that originally hun hablat, one hablat=20 alau) we have applications of the three time periods, the day, the month, and the year, with the figurative sense that the increase of one over the other was as the relative lengths of these different periods.

[47]I think it worth while to go into these etymologies, as they may throw some light on the graphic representation of the numerals in the Maya hieroglyphics. It is quite likely that the figures chosen to represent the different higher units would resemble the objects which their names literally signify. The first nineteen numerals were written by a combination of dots and lines, examples of which we find in abundance in the Codex Troano and other manuscripts. The following explanation of it is from the pen of a native writer in the last century:—

“Yantac thun yetel paiche tu pachob, he hunppel thune hunppel bin haabe, uaix cappele cappel bin haabe, uaix oxppel thuun, ua canppel thuune, canppel binbe, uaix oxppel thuun baixan; he paichee yan yokol xane, ua hunppel paichee, hoppel haab bin; ua cappel paichee lahunppiz bin; uaix hunppel paichee yan yokol xane, ua yan hunppel thuune uacppel bin be; uaix cappel thuune yan yokol paichee uucppel bin be; ua oxppel thuun yan yokole, uaxppel binbe; uaixcanppel thun yan yokole paichee (bolonppel binbe); yanix thun yokol (cappel) paichee buluc [piz; uaix cappel thune lahcapiz; ua oxppel thuun, oxlahunpiz.”

“They (our ancestors) used (for numerals in their calendars) dots and lines back of them; one dot for one year, two dots for two years, three dots for three, four dots for four, and so on; in addition to these they used a line; one line meant five years, two lines 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; if four dots above the line, nine; a dot above two lines, eleven; if two dots, twelve; if three dots, thirteen.”

The plan of using the numerals in Maya differs somewhat from that in English.

In the first place, they are rarely named without the addition of a numeral particle, which is suffixed. These particles indicate the character or class of the objects which are, or are about to be, enumerated. When they are uttered, the hearer at once knows what kind of objects are to be spoken of. Many of them can be traced to a meaning which [49]has a definite application to a class, and they have analogues in European tongues. Thus I may say “seven head of”—and the hearer knows that I am going to speak of cattle, or sheep, or cabbages, or similar objects usually counted by heads. So in Maya ac means a turtle or a turtle shell; hence it is used as a particle in counting canoes, houses, stools, vases, pits, caves, altars, and troughs, and some general appropriateness can be seen; but when it is applied also to cornfields, the analogy seems remote.

Of these numeral particles, not less than seventy-six are given by Beltran, in his Grammar, and he does not exhaust the list. Of these piz and pel, both of which mean, single, singly, are used in counting years, and will frequently recur in the annals I present in this volume.

By their aid another method of numeration was in vogue for counting time. For “eighty-one years,” they did not say hutuyokal haab, but can kal haab catac hunpel haab, literally, “four score years and one year.” The copulative catac is also used in adding a smaller number to a bak, or 400, as for 450, hun bak catac lahuyoxkal, “one bak and ten toward the third score.” Catac is a compound of ca tac, ca meaning “then” or “and,” and tac, ]which Dr. Berendt considered to be an irregular future of talel, to come, “then will come fifty,” but which may be the imperative of tac (tacah, tace, third conjugation), which means to put something under another, as in the phrase tac ex che yalan cum, put you wood under the pot.

It will be seen that the latter method is by addition, the former by subtraction. Another variety of the latter is found in the annals. For instance, “ninety-nine years” is not expressed by bolonlahutuyokal haab, nor yet by cankal haab catac bolonlahunpel haab, but by hunpel haab minan ti hokal haab, “one single year lacking from five score years.”