Fig. 28. Maya symbols for zero: a, monumental; b, c, cursive. (From Bowditch.)
Without a zero sign and position values, two methods are open for the representation of higher numerical values. More and more signs can be added for the high values. This was done by the Greeks and Romans. MV means 1,005, and only that. This is simple enough; but 1,888 requires so cumbersome a denotation as MDCCCLXXXVIII—thirteen figures of six different kinds. A simple system of multiplying numbers expressed like this one is impossible. The unwieldiness is due to the fact that the Romans, not having hit upon the device of representing nothingness, employed the separate signs I, X, C, M for the quantities which we represent by the single symbol 1 with from no to three zeroes added.
The other method is that followed by the Chinese. Besides signs corresponding to our digits from 1 to 9, they developed symbols corresponding to “ten times,” “hundred times,” and so on. This was much as if we should use the asterisk, *, to denote tens, the dagger, †, for hundreds, the paragraph, ¶, for thousands. We could then represent 1,888 by 1 ¶ 8 † 8 * 8, and 1,005 by 1 ¶ 5, without any risk of being misunderstood. But the writing of the numbers would in most cases require more figures, and mathematical operations would be more awkward.
The only nation besides the Hindus to invent a zero sign and the representation of number values by position of the basic symbols, were the Mayas of Yucatan. Some forms of their zero are shown in [Figure 28]. This Maya development constitutes an indubitable parallel with the Hindu one. So far as the involved logical principle is concerned, the two inventions are identical. But again the concrete expressions of the principle are dissimilar. The Maya zero does not in the least have the form of our or the Hindus’ zero. Also, the Maya notation was vigesimal where ours is decimal. They worked with twenty fundamental digits instead of ten. Their “100” therefore stood for 400, their “1,000” for 8,000.[17] Accordingly, when they wrote, in their corresponding digits, 1,234, the value was not 1,234 but 8,864. Obviously there can be no question of a common origin for such a system and ours. They share an idea or a method, nothing more. As a matter of fact, these two notational systems, like all others, were preceded by numeral word counts. Our decimal word count is based on operations with the fingers, that of the Maya on operations with the fingers and toes. Twenty became their first higher unit because twenty finished a person.
It is interesting that of the two inventions of zero, the Maya one was the earlier. The arithmetical and calendrical system of which it formed part was developed and in use by the time of the birth of Christ. It may be older; it certainly required time to develop. The Hindus may have possessed the prototypes of our numerals as early as the second century after Christ, but as yet without the zero, which was added during the sixth or according to some authorities not until the ninth century. This priority of the Maya must weaken the arguments sometimes advanced that the ancient Americans derived their religion, zodiac, art, or writing from Asia. If the zero was their own product, why not the remainder of their progress also? The only recourse left the naïve migrationist would be to turn the tables and explain Egyptian and Babylonian civilization as due to a Maya invasion from Yucatan.
110. Exogamic Institutions
In many parts of the world nations live under institutions by which they are divided into hereditary social units that are exogamous to one another. That is, all persons born in a unit must take spouses born in some other unit, fellow members of one’s unit being regarded as kinsmen. The units are generally described as clans, gentes, or sibs; or, where there are only two, as moieties. In many cases the sibs or moieties are totemic; named after, or in some way associated with, an animal, plant, or other distinctive object that serves as a badge or symbol of the group. Often the association finds expression in magic or myth. Since under this system one is born into his social unit, cannot change it, and can belong to one only, it follows that descent is unilateral. It is impossible for a man to be a member of both his father’s and his mother’s sib or totem; custom has established everywhere a rigid choice between them. Some tribes follow descent from the mother or matrilinear reckoning, others are patrilinear.[18]
Institutions of this type have a wide and irregular distribution. They are frequent in Australia, New Guinea, and Melanesia; found in parts of the East Indies and southeastern Asia; quite rare or stunted in the remainder of Asia and Polynesia; fairly common in Africa, though they occur in scattered areas; characteristic again of a large part of North America, but confined to a few districts of South America. At a rough guess, it might be said that about as many savage peoples, the world over, possess totemic-exogamous clans or moieties as lack them. The patchiness on the map of exogamic institutions argues against their being all the result of a wave of culture transmission emanating from a single source. Had such a diffusion occurred, it should have left its marks among the numerous intervening tribes that are sibless. Further, both in the eastern and western hemispheres, the most primitive and backward tribes are, with fair regularity, sibless and non-totemic. If therefore a hypothetical totem-sib movement had encircled the planet, it could not have been at an extremely ancient date, else the primitive tribes would have been affected by it; and since records go back five thousand years in parts of the Mediterranean area, the movement, if relatively late, should have left some echo in history, which it has not.
Fig. 29. Distribution of types of exogamic institutions in Australia: 2M, two classes, matrilinear; 4M, four classes, matrilinear; 4P, four classes, patrilinear; 8P, eight classes, patrilinear; black areas, no classes, patrilinear exogamic totems; X, totems independent of classes; Y, totems replace sub-classes; Z, no organization; ?, uninhabited or unknown. (After Thomas and Graebner.)