The Number Concept: Its Origin and Development Part 32

MOHICAN

1. ngwitloh.

2. neesoh.

3. noghhoh.

4. nauwoh.

5. nunon.

6. ngwittus.

7. tupouwus.

8. ghusooh.

9. nauneeweh.

10. mtannit.

In the Quappa scale 7 and 8 appear to be derived from 2 and 3, while 6 and 9 show no visible trace of kins.h.i.+p with 1 and 4. In Mohican, on the other hand, 6 and 9 seem to be derived from 1 and 4, while 7 and 8 have little or no claim to relations.h.i.+p with 2 and 3. In some scales a single word only is found in the second quinate to indicate that 5 was originally the base on which the system rested. It is hardly to be doubted, even, that change might affect each and every one of the numerals from 5 to 10 or 6 to 9, so that a dependence which might once have been easily detected is now unrecognizable.

But if this is so, the natural and inevitable question follows--might not this have been the history of all numeral scales now purely decimal? May not the changes of time have altered the compounds which were once a clear indication of quinary counting, until no trace remains by which they can be followed back to their true origin? Perhaps so. It is not in the least degree probable, but its possibility may, of course, be admitted. But even then the universality of quinary counting for primitive peoples is by no means established. In Chapter II, examples were given of races which had no number base. Later on it was observed that in Australia and South America many tribes used 2 as their number base; in some cases counting on past 5 without showing any tendency to use that as a new unit. Again, through the habit of counting upon the finger joints, instead of the fingers themselves, the use of 3 as a base is brought into prominence, and 6 and 9 become 2 threes and 3 threes, respectively, instead of 5 + 1 and 5 + 4. The same may be noticed of 4. Counting by means of his fingers, without including the thumbs, the savage begins by dividing into fours instead of fives. Traces of this form of counting are somewhat numerous, especially among the North American aboriginal tribes. Hence the quinary form of counting, however widespread its use may be shown to be, can in no way be claimed as the universal method of any stage of development in the history of mankind.

In the vast majority of cases, the pa.s.sage from the base to the next succeeding number in any scale, is clearly defined. But among races whose intelligence is of a low order, or--if it be permissible to express it in this way--among races whose number sense is feeble, progression from one number to the next is not always in accordance with any well-defined law.

After one or two distinct numerals the count may, as in the case of the Veddas and the Andamans, proceed by finger pantomime and by the repet.i.tion of the same word. Occasionally the same word is used for two successive numbers, some gesture undoubtedly serving to distinguish the one from the other in the savage's mind. Examples of this are not infrequent among the forest tribes of South America. In the Tariana dialect 9 and 10 are expressed by the same word, _paihipawalianuda;_ in Cobeu, 8 and 9 by _pepelicoloblicouilini;_ in Barre, 4, 5, and 9 by _ualibucubi._[326] In other languages the change from one numeral to the next is so slight that one instinctively concludes that the savage is forming in his own mind another, to him new, numeral immediately from the last. In such cases the entire number system is scanty, and the creeping hesitancy with which progress is made is visible in the forms which the numerals are made to take. A single ill.u.s.tration or two of this must suffice; but the ones chosen are not isolated cases. The scale of the Macunis,[327] one of the numerous tribes of Brazil, is

1. pocchaenang.

2. haihg.

3. haigunhgnill.

4. haihgtschating.

5. haihgtschihating = another 4?

6. hathig-stchihathing = 2-4?

7. hathink-tschihathing = 2-5?

8. hathink-tschihating = 2 4?

The complete absence of--one is tempted to say--any rhyme or reason from this scale is more than enough to refute any argument which might tend to show that the quinary, or any other scale, was ever the sole number scale of primitive man. Irregular as this is, the system of the Montagnais fully matches it, as the subjoined numerals show:[328]

1. inl'are.

2. nak'e.

3. t'are.

4. dinri.

5. se-sunlare.

6. elkke-t'are = 2 3.

7. t'a-ye-oyertan = 10 - 3, or inl'as dinri = 4 + 3?

8. elkke-dinri = 2 4.

9. inl'a-ye-oyertan = 10 - 1.

10. onernan.

CHAPTER VII.

THE VIGESIMAL SYSTEM.

In its ordinary development the quinary system is almost sure to merge into either the decimal or the vigesimal system, and to form, with one or the other or both of these, a mixed system of counting. In Africa, Oceanica, and parts of North America, the union is almost always with the decimal scale; while in other parts of the world the quinary and the vigesimal systems have shown a decided affinity for each other. It is not to be understood that any geographical law of distribution has ever been observed which governs this, but merely that certain families of races have shown a preference for the one or the other method of counting. These families, disseminating their characteristics through their various branches, have produced certain groups of races which exhibit a well-marked tendency, here toward the decimal, and there toward the vigesimal form of numeration. As far as can be ascertained, the choice of the one or the other scale is determined by no external circ.u.mstances, but depends solely on the mental characteristics of the tribes themselves. Environment does not exert any appreciable influence either. Both decimal and vigesimal numeration are found indifferently in warm and in cold countries; in fruitful and in barren lands; in maritime and in inland regions; and among highly civilized or deeply degraded peoples.

Whether or not the princ.i.p.al number base of any tribe is to be 20 seems to depend entirely upon a single consideration; are the fingers alone used as an aid to counting, or are both fingers and toes used? If only the fingers are employed, the resulting scale must become decimal if sufficiently extended. If use is made of the toes in addition to the fingers, the outcome must inevitably be a vigesimal system. Subordinate to either one of these the quinary may and often does appear. It is never the princ.i.p.al base in any extended system.

To the statement just made respecting the origin of vigesimal counting, exception may, of course, be taken. In the case of numeral scales like the Welsh, the Nahuatl, and many others where the exact meanings of the numerals cannot be ascertained, no proof exists that the ancestors of these peoples ever used either finger or toe counting; and the sweeping statement that any vigesimal scale is the outgrowth of the use of these natural counters is not susceptible of proof. But so many examples are met with in which the origin is clearly of this nature, that no hesitation is felt in putting the above forward as a general explanation for the existence of this kind of counting. Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. Beginning again, he now repeats his count of 10, either on his own fingers or on the fingers of another. With the completion of the second 10 the result is announced, not in a new unit, but by means of a duplication of the term already used. It is scarcely credible that the unit unconsciously adopted at the termination of the first count should now be dropped, and a new one subst.i.tuted in its place. When the method here described is employed, 20 is not a natural unit to which higher numbers may be referred. It is wholly artificial; and it would be most surprising if it were adopted. But if the count of the second 10 is made on the toes in place of the fingers, the element of repet.i.tion which entered into the previous method is now wanting. Instead of referring each new number to the 10 already completed, the savage is still feeling his way along, designating his new terms by such phrases as "1 on the foot," "2 on the other foot," etc. And now, when 20 is reached, a single series is finished instead of a double series as before; and the result is expressed in one of the many methods already noticed--"one man," "hands and feet," "the feet finished," "all the fingers of hands and feet," or some equivalent formula. Ten is no longer the natural base. The number from which the new start is made is 20, and the resulting scale is inevitably vigesimal. If pebbles or sticks are used instead of fingers, the system will probably be decimal. But back of the stick and pebble counting the 10 natural counters always exist, and to them we must always look for the origin of this scale.

In any collection of the princ.i.p.al vigesimal number systems of the world, one would naturally begin with those possessed by the Celtic races of Europe. These races, the earliest European peoples of whom we have any exact knowledge, show a preference for counting by twenties, which is almost as decided as that manifested by Teutonic races for counting by tens. It has been conjectured by some writers that the explanation for this was to be found in the ancient commercial intercourse which existed between the Britons and the Carthaginians and Phoenicians, whose number systems showed traces of a vigesimal tendency. Considering the fact that the use of vigesimal counting was universal among Celtic races, this explanation is quite gratuitous. The reason why the Celts used this method is entirely unknown, and need not concern investigators in the least. But the fact that they did use it is important, and commands attention. The five Celtic languages, Breton, Irish, Welsh, Manx, and Gaelic, contain the following well-defined vigesimal scales. Only the princ.i.p.al or characteristic numerals are given, those being sufficient to enable the reader to follow intelligently the growth of the systems. Each contains the decimal element also, and is, therefore, to be regarded as a mixed decimal-vigesimal system.

IRISH.[329]

10. deic.

20. fice.

30. triocad = 3-10 40. da ficid = 2-20.

50. caogad = 5-10.

60. tri ficid = 3-20.

70. reactmoga = 7-10.

80. ceitqe ficid = 4-20.

90. nocad = 9-10.

100. cead.

1000. mile.

GAELIC.[330]

10. deich.

20. fichead.

30. deich ar fichead = 10 + 20.

40. da fhichead = 2-20.

50. da fhichead is deich = 40 + 10.

60. tri fichead = 3-20.

70. tri fichead is deich = 60 + 10.

80. ceithir fichead = 4-20.

90. ceithir fichead is deich = 80 + 10.

100. ceud.

1000. mile.

WELSH.[331]

10. deg.

20. ugain.

30. deg ar hugain = 10 + 20.

40. deugain = 2-20.

50. deg a deugain = 10 + 40.

60. trigain = 3-20.

70. deg a thrigain = 10 + 60.

80. pedwar ugain = 4-20.

90. deg a pedwar ugain = 80 + 10.