60. oxkal = 3 20.
70. lahucankal = 80 - 10.
80. cankal = 4 20.
90. lahuyokal = 100 - 10.
100. hokal = 5 20.
110. lahu uackal = 120 - 10.
120. uackal = 6 20.
130. lahu uuckal = 140 - 10.
140. uuckal = 7 20.
200. lahuncal = 10 20.
300. holhukal = 15 20.
400. hunbak = 1 tying around.
500. hotubak.
600. lahutubak 800. calbak = 2 400.
900. hotu yoxbak.
1000. lahuyoxbak.
1200. oxbak = 3 400.
2000. capic (modern).
8000. hunpic = 1 sack.
16,000. ca pic (ancient).
160,000. calab = a filling full 3,200,000. kinchil.
64,000,000. hunalau.
In the Maya scale we have one of the best and most extended examples of vigesimal numeration ever developed by any race. To show in a more striking and forcible manner the perfect regularity of the system, the following tabulation is made of the various Maya units, which will correspond to the "10 units make one ten, 10 tens make one hundred, 10 hundreds make one thousand," etc., which old-fashioned arithmetic compelled us to learn in childhood. The scale is just as regular by twenties in Maya as by tens in English. It is[364]
20 hun = 1 kal = 20.
20 kal = 1 bak = 400.
20 bak = 1 pic = 8000.
20 pic = 1 calab = 160,000.
20 calab = 1 { kinchil } = 3,200,000.
{ tzotzceh } 20 kinchil = 1 alau = 64,000,000.
The original meaning of _pic_, given in the scale as "a sack," was rather "a short petticoat, somtimes used as a sack." The word _tzotzceh_ signified "deerskin." No reason can be given for the choice of this word as a numeral, though the appropriateness of the others is sufficiently manifest.
No evidence of digital numeration appears in the first 10 units, but, judging from the almost universal practice of the Indian tribes of both North and South America, such may readily have been the origin of Maya counting. Whatever its origin, it certainly expanded and grew into a system whose perfection challenges our admiration. It was worthy of the splendid civilization of this unfortunate race, and, through its simplicity and regularity, bears ample testimony to the intellectual capacity which originated it.
The only example of vigesimal reckoning which is comparable with that of the Mayas is the system employed by their northern neighbours, the Nahuatl, or, as they are more commonly designated, the Aztecs of Mexico. This system is quite as pure and quite as simple as the Maya, but differs from it in some important particulars. In its first 20 numerals it is quinary (see p.
141), and as a system must be regarded as quinary-vigesimal. The Maya scale is decimal through its first 20 numerals, and, if it is to be regarded as a mixed scale, must be characterized as decimal-vigesimal. But in both these instances the vigesimal element preponderates so strongly that these, in common with their kindred number systems of Mexico, Yucatan, and Central America, are always thought of and alluded to as vigesimal scales. On account of its importance, the Nahuatl system[365] is given in fuller detail than most of the other systems I have made use of.
10. matlactli = 2 hands.
20. cempoalli = 1 counting.
21. cempoalli once = 20-1.
22. cempoalli omome = 20-2.
30. cempoalli ommatlactli = 20-10.
31. cempoalli ommatlactli once = 20-10-1.
40. ompoalli = 2 20.
50. ompoalli ommatlactli = 40-10.
60. eipoalli, or epoalli, = 3 20.
70. epoalli ommatlactli = 60-10.
80. nauhpoalli = 4 20.
90. nauhpoalli ommatlactli = 90-10.
100. macuilpoalli = 5 20.
120. chiquacempoalli = 6 20.
140. chicompoalli = 7 20.
160. chicuepoalli = 8 20.
180. chiconauhpoalli = 9 20.
200. matlacpoalli = 10 20.
220. matlactli oncempoalli = 11 20.
240. matlactli omompoalli = 12 20.
260. matlactli omeipoalli = 13 20.
280. matlactli onnauhpoalli = 14 20.
300. caxtolpoalli = 15 20.
320. caxtolli oncempoalli.
399. caxtolli onnauhpoalli ipan caxtolli onnaui = 19 20 + 19.
400. centzontli = 1 bunch of gra.s.s, or 1 tuft of hair.
800. ometzontli = 2 400.
1200. eitzontli = 3 400.
7600. caxtolli onnauhtzontli = 19 400.
8000. cenxiquipilli, or cexiquipilli.
160,000. cempoalxiquipilli = 20 8000.
3,200,000. centzonxiquipilli = 400 8000.
64,000,000. cempoaltzonxiquipilli = 20 400 8000.
Up to 160,000 the Nahuatl system is as simple and regular in its construction as the English. But at this point it fails in the formation of a new unit, or rather in the expression of its new unit by a simple word; and in the expression of all higher numbers it is forced to resort in some measure to compound terms, just as the English might have done had it not been able to borrow from the Italian. The higher numeral terms, under such conditions, rapidly become complex and c.u.mbersome, as the following a.n.a.lysis of the number 1,279,999,999 shows.[366] The a.n.a.lysis will be readily understood when it is remembered that _ipan_ signifies plus.
_Caxtolli onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli ipan caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli ipan caxtolli onnauhtzontli ipan caxtolli onnauhpoalli ipan caxtolli onnaui;_ _i.e._ 1,216,000,000 + 60,800,000 + 3,040,000 + 152,000 + 7600 + 380 + 19. To show the compounding which takes place in the higher numerals, the a.n.a.lysis may be made more literally, thus: + (15 + 4) 400 800 + (15 + 4) 20 8000 + (15 + 4) 8000 + (15 + 4) 400 + (15 + 4) 20 + 15 + 4. Of course this resolution suffers from the fact that it is given in digits arranged in accordance with decimal notation, while the Nahuatl numerals express values by a base twice as great. This gives the effect of a complexity and awkwardness greater than really existed in the actual use of the scale. Except for the presence of the quinary element the number just given is really expressed with just as great simplicity as it could be in English words if our words "million" and "billion" were replaced by "thousand thousand" and "thousand thousand thousand." If Mexico had remained undisturbed by Europeans, and science and commerce had been left to their natural growth and development, uncompounded words would undoubtedly have been found for the higher units, 160,000, 3,200,000, etc., and the system thus rendered as simple as it is possible for a quinary-vigesimal system to be.
Other number scales of this region are given as follows:
HUASTECA.[367]
10. laluh.
20. hum-inic = 1 man.
30. hum-inic-lahu = 1 man 10.
40. tzab-inic = 2 men.
50. tzab-inic-lahu = 2 men 10.
60. ox-inic = 3 men.
70. ox-inic-lahu = 3 men 10.
80. tze-tnic = 4 men.
90. tze-ynic-kal-laluh = 4 men and 10.
100. bo-inic = 5 men.
200. tzab-bo-inic = 2 5 men.
300. ox-bo-inic = 3 5 men.
400. tsa-bo-inic = 4 5 men.
600. acac-bo-inic = 6 5 men.
800. huaxic-bo-inic = 8 5 men.
1000. xi.
8000. huaxic-xi = 8-1000.
The essentially vigesimal character of this system changes in the formation of some of the higher numerals, and a suspicion of the decimal enters. One hundred is _boinic_, 5 men; but 200, instead of being simply _lahuh-inic_, 10 men, is _tsa-bo-inic_, 2 100, or more strictly, 2 times 5 men.
Similarly, 300 is 3 100, 400 is 4 100, etc. The word for 1000 is simple instead of compound, and the thousands appear to be formed wholly on the decimal base. A comparison of this scale with that of the Nahuatl shows how much inferior it is to the latter, both in simplicity and consistency.