Numeral Systems of Mexico and Central America

NUMERAL SYSTEMS OF MEXICO AND CENTRAL AMERICA BY CYRUS THOMAS 853 CONTENTS Page Primary numbers 859 Numbers above 10 882 Discussion and comparisons 919 Nmnbers in the Mexican codices 934 The mystical and ceremonial use of ninubers 948 855 . ILLUSTRATIONS Page Fu4URE 23. Symbols of tlie Mexican davis 937 2-1. Symbol for Atl (water) 938 25. Symbol for Calli (house) 938 26. Symbol for Itzquintli (dog). From Fejervary codex, plate 6 938 27. Symbol for Ocelotl (tiger). From Fejervary codex, plate 6 938 28. Symbol for 400. Mendoza codex, plate 20, figure 16 945 29. Symbol for 4,000. Mendoza codex, plate 28, figure 24 945 30. Symbol for 20 jars of honey. Mendoza codex, plate 38, figure 21 . 945 31. Symbol for 100 hatchets. Mendoza codex, plate 39, figure 20 946 32. Symbol for 20 baskets, ilendoza codex, plate 19, figure 2 946 33. Symbols for 20 days. Mendoza codex, plate 19, figures 10, 11, " 12, 13 946 34. Symbol for 8,000 sheets paper. Mendoza codex, plate 25, figure 1 1 946 35. Symbol for 8,000 pellets copal. Mendoza codex, plate 38, figure 35. 946 36. Symbol for 200 cacaxtles. Mendoza codex, plate 44, figure 34 947 37. Symbol for 1,800. Codex Telleriano-Eemensis, plate 25 947 38. Symlxil for 4,008. Vatican codex 3738, plate 7, figures 2 and 3 947 39. Symbol for 5,206. Vatican codex .3738, plate 10 947 40. Symbol for 19,600. Vatican codex 3738, plate 123 948 41. Diagram of figures on plates 11 and 12 of the Borgian codex 951 857 NUMERAL SYSTE3IS OF MEXICO AND CENTRAL AMERICA By Cykus Thomas PRIMARY NUMBERS It IS well known that the vigesimal system of nmneration prevailed among; the Mexican and Central American tribes, at least among- all which had adopted the so-called "native calendar"—that is, the calendar specially referred to in my paper entitled Mayan Calendar Systems, published in this volume. Niimerous short notices and inci- dental mentions of the g;euei'al system and completer notices of the systems of particular tribes are to be found in the early Spanish authorities and in the works of more recent writers. As. however, most if not all of them are limited in scope, relating to the system of but one tribe or people, or referring only to certain points, and as no paper devoted specially to the subject of numeral systems has appeared in English, it is deemed expedient to. present this paper as a supple- ment to those which have preceded it. Moreover, it is believed that a resume of the subject in the light of the recent advance in our knowledge of Mexican and Central American archaeology will be acceptable to those devoting attention to the study of prehistoric Mexico and Central America. As my paper on the calendar systems ^ related to the time system and symbols of the Mayan tribes, and incidentally to the numeral system as used by them in counting time, attention will here be paid to the numeral system in its more general application among the Nahu- atlan, Mayan, and other tribes of Mexico and Central America which used the vigesimal system. I have shown in the paper on calendar systems that in counting time 1 This expression will be used throughout to refer to the paper mentioned above, published in this volume. 859 !5()() NUMKKAL SYSTEMS [eth.ann.19 till' units used liy the Muyuii tiM))Os were as follow, the day being the priiiiiii'v unit: 1 unit III' the 1st order = 1 day. 1 unit of the I'd order = 20 units of the 1st order = 20 days. 1 unit of the ;^d order ^ 18 units of the 2d order = 300 days. 1 unit of the -ith order = 20 units of the 3d order = 7,200 days. , 1 unit of the 5th order = 20 units, of the 4th order = 144.000 days. 1 unit of the (Jth order = 20 units of the yth order = 2,880,000 days. As this notation has been fully explained and discussed in the pre- ceding paper, I pass at once to an examination of the general numeral system of the Mayan tribes. The notation given above dif- fered from that of general apj)lication in the; change of the second step from 20, as it should, be according to the regular vigesimal system, to 18, probably to facilitate counting with the month factor. As 20 is the basis of the higher counts, attention will be directed tirst to the steps leading up to this number. The oldest records to which we can appeal for knowledge of the system in use among the Ma)'an tribes are the inscriptions and codices. From these we can, however, learn only the method of vyrituu/ numbers, not the number names; 3'et the method of writing will indi(-ate to some extent the process in oral counts. Although the symbols conuuonly used for this ))ur])ose are now well known from the frequent notices of them which have been published, it is necessary for our present purpose that they be presented here. 11 ^^^^ 16 12 —1-1- 17 13 • • • 18 14 1-:-:-: i9 10 ^^^ 15 From these it is seen that the count as expressed in symbols is from 1 to 4 by sing dots, or the unit repeated; but that to indicate 5 the method is changed, and a single short line is used instead of five dots. Though fre([uently iiorizontal, it is not necessarily so, but is found both in the codices and inscriptions in a vertical position; oftener, even, in the latter than in the former. The next four numbers, G, 7, 8, and !*, are formed by adding to the single line one. two, three, and four dots or units, l)ut l(t is represented by two parallel lines. That these linens nnist be parallel, oi- sui)stantially so, whether horizontal or vertical, setMiis to be reiiuisite in the Mayan hiero- gly])hic writing. Dots are added to the two lines to indicate tiu^ numbers 11, 12. L3. and 14; three parallel lines arc used to rt'prcscnt 15, THOMAS] MAYAN NUMERALS 861 and dots are added to these to form the numbers 16, 17, 18, and 19, where the use of symbols of this form stops, 19 being the highest number for which they appear to have been used in Mayan writing. The higher numbers were, as has been shown in ni}' paper on calendar sj'stems, represented by othcu- symbols, or by relative position. Sub- stantially the same plan of writing numerals is seen in the Roman system, the line being used instead of the dot, thus: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, etc. , to XIX, 19. Attention is called to this because of another resemblance which will be noticed hereafter. Now it is apparent that if these symbols, taken in the order in which they stand, indicate the method followed in actual or oral counting, this method must have been as follows, from five upward: 5 and 1; 5 and 2; and so on to 2 fives; then 2 fives and 1; 2 fives and 2; and so on to 3 fives; then 3 fives and 1; 3 fives and 2, to 19. If this theory be true, we should expect to find terms in the language to correspond with the symbols; e\idence that these existed in Mayan count appears to be wanting, yet, as favoring the theory, we do find, as will appear, that the Nahuatl and some other surrounding languages contained terms corresponding precisely with this method of counting. It is, however, somewhat strange that the Borgian codex, which is proljably the oldest of the existing Mexican codices, does not use the short line for 5, but counts with single dots as high as 26, and in fact no one of these codices appears to use it in counting time from date to date, though it is used in them for other purposes. The ilayan terms from 10 to 20 follow not this cjuinai\^ s^'stem but the decimal order, as will be seen. The terms used for numbers up to 20 in the Maya (or Yucatec) dialect are, according to the usual orthography, as follow: 1 862 NUMERAL SYSTEMS [KTH. ANN. 19 1 THOMAS] MAYAN NUMERALS 863 Before commenting on the list, the names in some other dialects of this stock not included by Stoll and some variations from the orthography' of his list will be noted. Pupuluca' : 8(U NUMKRAL SYSTEMS [ETH. ANN.19 Quiche—As Brasseur's orthography (Gram. Lang. Quiche, p. lil) differ.s considerably from Stoll's. we give his li.st here: 1 hun. 4 oah, or cahil>. 7 vukub. 10 lahuh. 2 cab, or caib. 5 oo, or oob. 8 vahxakib. 20 huvinak. 3 ox, oroxib. C-, vakakib. f> beleb, or beleheb. Charencey i'ollow.s thi.s list, except in 8, which he writes vaxak. Quehchi {K'((Mc}ii, m' Cakxji)—Pinart (Vocabulario Castellano- K'ak'chi. page 7) gives for •!. htih; for 4, Irtath; for .5, jonh; for 6, gual'ih; foi' 7, (jukiih; and for 8, guajxakib. Charence}' (Melanges, page 64) gives for 1, hoon; for 2, cat/ for 3, oxi/ for 4, cagi; for 5, jooh,- for 6, 'wakk/'/ for 7, uukx; for 8, xvahshaki; for 9, helojem; and for 10, lajegem. Mam—As Stoll gives another list (Sprache dcr Ixil-lndianer, p. 146) which differs somewhat from that given abov(>. and as both vary from that given in Salmei'on's Arte y Vocabulario, page 156, this and Stoll's second list are given here (_/ being changed to A) THOMAsJ MAYAN NUMERALS 865 suffixes.

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