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30. puxamacauh = 20 + 10.
40. tipuxam = 2 20.
50. tipuxamacauh = 40 + 10.
60. totonpuxam = 3 20.
100. quitziz puxum = 5 20.
200. copuxam = 10 20.
1000. titamanacopuxam = 2 400 + 200.
The essential character of the vigesimal element is shown by the last two numerals. _Tontamen_, the square of 20, is a simple word, and 1000 is, as it should be, 2 times 400, plus 200. It is most unfortunate that the numeral for 8000, the cube of 20, is not given.
30. ceitevi apoan tamoamata = 20 + 10.
40. huapoa-tevi = 2 20.
60. huaeica-tevi = 3 20.
100. anxu-tevi = 5 20.
400. ceitevi-tevi = 20 20.
Closely allied with the Maya numerals and method of counting are those of the Quiches of Guatemala. The resemblance is so obvious that no detail in the Quiche scale calls for special mention.
20. hu-uinac = 1 man.
30. hu-uinac-lahuh = 20 + 10.
40. ca-uinac = 2 men.
50. lahu-r-ox-kal = -10 + 3 20.
60. ox-kal = 3 20.
70. lahu-u-humuch = -10 + 80.
90. lahu-r-ho-kal = -10 + 100.
Among South American vigesimal systems, the best known is that of the Chibchas or Muyscas of the Bogota region, which was obtained at an early date by the missionaries who laboured among them. This system is much less extensive than that of some of the more northern races; but it is as extensive as almost any other South American system with the exception of the Peruvian, which was, however, a pure decimal system. As has already been stated, the native races of South America were, as a rule, exceedingly deficient in regard to the number sense. Their scales are rude, and show great poverty, both in formation of numeral words and in the actual extent to which counting was carried. If extended as far as 20, these scales are likely to become vigesimal, but many stop far short of that limit, and no inconsiderable number of them fail to reach even 5. In this respect we are reminded of the Australian scales, which were so rudimentary as really to preclude any proper use of the word "system" in connection with them.
Counting among the South American tribes was often equally limited, and even less regular. Following are the significant numerals of the scale in question:
CHIBCHA, OR MUYSCA.
20. quihica ubchihica = thus says the foot, 10 = 10-10, or gueta = house.
30. guetas asaqui ubchihica = 20 + 10.
40. gue-bosa = 20 2.
60. gue-mica = 20 3.
80. gue-muyhica = 20 4.
100. gue-hisca = 20 5.
30. 'badinoguhanu = 20 + 10.
40. apudino = 2 20.
50. apudinoguhanu = 2 20 + 10.
60. asudino = 3 20.
70. asudinoguhanu = 3 20 + 10.
80. acudino = 4 20.
90. acudinoguhanu = 4 20 + 10.
100. huisudino = 5 20, or guhamba = great 10.
200. guahadino = 10 20.
400. dinoamba = great 20.
1000. guhaisudino = 10 5 20.
2000. hisudinoamba = 5 great 20's.
4000. guhadinoamba = 10 great 20's.
In considering the influence on the manners and customs of any people which could properly be ascribed to the use among them of any other base than 10, it must not be forgotten that no races, save those using that base, have ever attained any great degree of civilization, with the exception of the ancient Aztecs and their immediate neighbours, north and south. For reasons already pointed out, no highly civilized race has ever used an exclusively quinary system; and all that can be said of the influence of this mode of counting is that it gives rise to the habit of collecting objects in groups of five, rather than of ten, when any attempt is being made to ascertain their sum. In the case of the subsidiary base 12, for which the Teutonic races have always shown such a fondness, the dozen and gross of commerce, the divisions of English money, and of our common weights and measures are probably an outgrowth of this preference; and the Babylonian base, 60, has fastened upon the world forever a sexagesimal method of dividing time, and of measuring the circumference of the circle.
The advanced civilization attained by the races of Mexico and Central America render it possible to see some of the effects of vigesimal counting, just as a single thought will show how our entire lives are influenced by our habit of counting by tens. Among the Aztecs the universal unit was 20. A load of cloaks, of dresses, or other articles of convenient size, was 20. Time was divided into periods of 20 days each. The armies were numbered by divisions of 8000; and in countless other ways the vigesimal element of numbers entered into their lives, just as the decimal enters into ours; and it is to be supposed that they found it as useful and as convenient for all measuring purposes as we find our own system; as the tradesman of to-day finds the duodecimal system of commerce; or as the Babylonians of old found that singularly curious system, the sexagesimal.
Habituation, the laws which the habits and customs of every-day life impose upon us, are so powerful, that our instinctive readiness to make use of any concept depends, not on the intrinsic perfection or imperfection which pertains to it, but on the familiarity with which previous use has invested it. Hence, while one race may use a decimal, another a quinary-vigesimal, and another a sexagesimal scale, and while one system may actually be inherently superior to another, no user of one method of reckoning need ever think of any other method as possessing practical inconveniences, of which those employing it are ever conscious. And, to cite a single instance which illustrates the unconscious daily use of two modes of reckoning in one scale, we have only to think of the singular vigesimal fragment which remains to this day imbedded in the numeral scale of the French. In counting from 70 to 100, or in using any number which lies between those limits, no Frenchman is conscious of employing a method of numeration less simple or less convenient in any particular, than when he is at work with the strictly decimal portions of his scale. He passes from the one style of counting to the other, and from the second back to the first again, entirely unconscious of any break or change; entirely unconscious, in fact, that he is using any particular system, except that which the daily habit of years has made a part himself.
Deep regret must be felt by every student of philology, that the primitive meanings of simple numerals have been so generally lost. But, just as the pebble on the beach has been worn and rounded by the beating of the waves and by other pebbles, until no trace of its original form is left, and until we can say of it now only that it is quartz, or that it is diorite, so too the numerals of many languages have suffered from the attrition of the ages, until all semblance of their origin has been lost, and we can say of them only that they are numerals. Beyond a certain point we can carry the study neither of number nor of number words. At that point both the mathematician and the philologist must pause, and leave everything beyond to the speculations of those who delight in nothing else so much as in pure theory.