The Number Concept: Its Origin and Development Part 12

7. natolata inibacao-caini = 4-2 above.

8. natolata-natolata = 4-4.

There is probably no recorded instance of a number system formed on 6, 7, 8, or 9 as a base. No natural reason exists for the choice of any of these numbers for such a purpose; and it is hardly conceivable that any race should proceed beyond the unintelligent binary or quaternary stage, and then begin the formation of a scale for counting with any other base than one of the three natural bases to which allusion has already been made. Now and then some anomalous fragment is found imbedded in an otherwise regular system, which carries us back to the time when the savage was groping his way onward in his attempt to give expression to some number greater than any he had ever used before; and now and then one of these fragments is such as to lead us to the border land of the might-have-been, and to cause us to speculate on the possibility of so great a numerical curiosity as a senary or a septenary scale. The Bretons call 18 _triouec'h_, 3-6, but otherwise their language contains no hint of counting by sixes; and we are left at perfect liberty to theorize at will on the existence of so unusual a number word. Pott remarks[208] that the Bolans, of western Africa, appear to make some use of 6 as their number base, but their system, taken as a whole, is really a quinary-decimal. The language of the Sundas,[209] or mountaineers of Java, contains traces of senary counting. The Akra words for 7 and 8, _paggu_ and _paniu_, appear to mean 6-1 and 7-1, respectively; and the same is true of the corresponding Tambi words _pagu_ and _panjo_.[210] The Watji tribe[211] call 6 _andee_, and 7 _anderee_, which probably means 6-1. These words are to be regarded as accidental variations on the ordinary laws of formation, and are no more significant of a desire to count by sixes than is the Wallachian term _deu-maw_, which expresses 18 as 2-9, indicates the existence of a scale of which 9 is the base. One remarkably interesting number system is that exhibited by the Mosquito tribe[212] of Central America, who possess an extensive quinary-vigesimal scale containing one binary and three senary compounds. The first ten words of this singular scale, which has already been quoted, are:

1. k.u.mi.

2. wal.

3. niupa.

4. wal-wal = 2-2.

5. mata-sip = fingers of one hand.

6. matlalkabe.

7. matlalkabe pura k.u.mi = 6 + 1.

8. matlalkabe pura wal = 6 + 2.

9. matlalkabe pura niupa = 6 + 3.

10. mata-wal-sip = fingers of the second hand.

In pa.s.sing from 6 to 7, this tribe, also, has varied the almost universal law of progression, and has called 7 6-1. Their 8 and 9 are formed in a similar manner; but at 10 the ordinary method is resumed, and is continued from that point onward. Few number systems contain as many as three numerals which are a.s.sociated with 6 as their base. In nearly all instances we find such numerals singly, or at most in pairs; and in the structure of any system as a whole, they are of no importance whatever. For example, in the p.a.w.nee, a pure decimal scale, we find the following odd sequence:[213]

6. shekshabish.

7. petkoshekshabish = 2-6, _i.e._ 2d 6.

8. touwetshabish = 3-6, _i.e._ 3d 6.

9. loksherewa = 10 - 1.

In the Uainuma scale the expressions for 7 and 8 are obviously referred to 6, though the meaning of 7 is not given, and it is impossible to guess what it really does signify. The numerals in question are:[214]

6. aira-ettagapi.

7. aira-ettagapi-hairiwigani-apecapecapsi.

8. aira-ettagapi-matschahma = 6 + 2.

In the dialect of the Mille tribe a single trace of senary counting appears, as the numerals given below show:[215]

6. dildjidji.

7. dildjidji me djuun = 6 + 1.

Finally, in the numerals used by the natives of the Marshall Islands, the following curiously irregular sequence also contains a single senary numeral:[216]

6. thil thino = 3 + 3.

7. thilthilim-thuon = 6 + 1.

8. rua-li-dok = 10 - 2.

9. ruathim-thuon = 10 - 2 + 1.

Many years ago a statement appeared which at once attracted attention and awakened curiosity. It was to the effect that the Maoris, the aboriginal inhabitants of New Zealand, used as the basis of their numeral system the number 11; and that the system was quite extensively developed, having simple words for 121 and 1331, _i.e._ for the square and cube of 11. No apparent reason existed for this anomaly, and the Maori scale was for a long time looked upon as something quite exceptional and outside all ordinary rules of number-system formation. But a closer and more accurate knowledge of the Maori language and customs served to correct the mistake, and to show that this system was a simple decimal system, and that the error arose from the following habit. Sometimes when counting a number of objects the Maoris would put aside 1 to represent each 10, and then those so set aside would afterward be counted to ascertain the number of tens in the heap. Early observers among this people, seeing them count 10 and then set aside 1, at the same time p.r.o.nouncing the word _tekau_, imagined that this word meant 11, and that the ignorant savage was making use of this number as his base. This misconception found its way into the early New Zealand dictionary, but was corrected in later editions. It is here mentioned only because of the wide diffusion of the error, and the interest it has always excited.[217]

Aside from our common decimal scale, there exist in the English language other methods of counting, some of them formal enough to be dignified by the term _system_--as the s.e.xagesimal method of measuring time and angular magnitude; and the duodecimal system of reckoning, so extensively used in buying and selling. Of these systems, other than decimal, two are noticed by Tylor,[218] and commented on at some length, as follows:

"One is the well-known dicing set, _ace_, _deuce_, _tray_, _cater_, _cinque_, _size_; thus _size-ace_ is 6-1, _cinques_ or _sinks_, double 5.

These came to us from France, and correspond with the common French numerals, except _ace_, which is Latin _as_, a word of great philological interest, meaning 'one.' The other borrowed set is to be found in the _Slang Dictionary_. It appears that the English street-folk have adopted as a means of secret communication a set of Italian numerals from the organ-grinders and image-sellers, or by other ways through which Italian or Lingua Franca is brought into the low neighbourhoods of London. In so doing they have performed a philological operation not only curious but instructive. By copying such expressions as _due soldi_, _tre soldi_, as equivalent to 'twopence,' 'threepence,' the word _saltee_ became a recognized slang term for 'penny'; and pence are reckoned as follows:

oney saltee 1d. uno soldo.

dooe saltee 2d. due soldi.

tray saltee 3d. tre soldi.

quarterer saltee 4d. quattro soldi.

c.h.i.n.ker saltee 5d. cinque soldi.

say saltee 6d. sei soldi.

say oney saltee, or setter saltee 7d. sette soldi.

say dooe saltee, or otter saltee 8d. otto soldi.

say tray saltee, or n.o.bba saltee 9d. nove soldi.

say quarterer saltee, or dacha saltee 10d. dieci soldi.

say c.h.i.n.ker saltee or dacha oney saltee 11d. undici soldi.

oney beong 1s.

a beong say saltee 1s. 6d.

dooe beong say saltee, or madza caroon 2s. 6d. (half-crown, mezza corona).

One of these series simply adopts Italian numerals decimally. But the other, when it has reached 6, having had enough of novelty, makes 7 by 6-1, and so forth. It is for no abstract reason that 6 is thus made the turning-point, but simply because the costermonger is adding pence up to the silver sixpence, and then adding pence again up to the s.h.i.+lling. Thus our duodecimal coinage has led to the practice of counting by sixes, and produced a philological curiosity, a real senary notation."

In addition to the two methods of counting here alluded to, another may be mentioned, which is equally instructive as showing how readily any special method of reckoning may be developed out of the needs arising in connection with any special line of work. As is well known, it is the custom in ocean, lake, and river navigation to measure soundings by the fathom. On the Mississippi River, where constant vigilance is needed because of the rapid s.h.i.+fting of sand-bars, a special sounding nomenclature has come into vogue,[219] which the following terms will ill.u.s.trate:

5 ft. = five feet.

6 ft. = six feet.

9 ft. = nine feet.

10-1/2 ft. = a quarter less twain; _i.e._ a quarter of a fathom less than 2.

12 ft. = mark twain.

13-1/2 ft. = a quarter twain.

16-1/2 ft. = a quarter less three.

18 ft. = mark three.

19-1/2 ft. = a quarter three.

24 ft. = deep four.

As the soundings are taken, the readings are called off in the manner indicated in the table; 10-1/2 feet being "a quarter less twain," 12 feet "mark twain," etc. Any sounding above "deep four" is reported as "no bottom." In the Atlantic and Gulf waters on the coast of this country the same system prevails, only it is extended to meet the requirements of the deeper soundings there found, and instead of "six feet," "mark twain,"

etc., we find the fuller expressions, "by the mark one," "by the mark two,"

and so on, as far as the depth requires. This example also suggests the older and far more widely diffused method of reckoning time at sea by bells; a system in which "one bell," "two bells," "three bells," etc., mark the pa.s.sage of time for the sailor as distinctly as the hands of the clock could do it. Other examples of a similar nature will readily suggest themselves to the mind.

Two possible number systems that have, for purely theoretical reasons, attracted much attention, are the octonary and the duodecimal systems. In favour of the octonary system it is urged that 8 is an exact power of 2; or in other words, a large number of repeated halves can be taken with 8 as a starting-point, without producing a fractional result. With 8 as a base we should obtain by successive halvings, 4, 2, 1. A similar process in our decimal scale gives 5, 2-1/2, 1-1/4. All this is undeniably true, but, granting the argument up to this point, one is then tempted to ask "What of it?" A certain degree of simplicity would thereby be introduced into the Theory of Numbers; but the only persons sufficiently interested in this branch of mathematics to appreciate the benefit thus obtained are already trained mathematicians, who are concerned rather with the pure science involved, than with reckoning on any special base. A slightly increased simplicity would appear in the work of stockbrokers, and others who reckon extensively by quarters, eighths, and sixteenths. But such men experience no difficulty whatever in performing their mental computations in the decimal system; and they acquire through constant practice such quickness and accuracy of calculation, that it is difficult to see how octonary reckoning would materially a.s.sist them. Altogether, the reasons that have in the past been adduced in favour of this form of arithmetic seem trivial.

There is no record of any tribe that ever counted by eights, nor is there the slightest likelihood that such a system could ever meet with any general favour. It is said that the ancient Saxons used the octonary system,[220] but how, or for what purposes, is not stated. It is not to be supposed that this was the common system of counting, for it is well known that the decimal scale was in use as far back as the evidence of language will take us. But the field of speculation into which one is led by the octonary scale has proved most attractive to some, and the conclusion has been soberly reached, that in the history of the Aryan race the octonary was to be regarded as the predecessor of the decimal scale. In support of this theory no direct evidence is brought forward, but certain verbal resemblances. Those ignes fatuii of the philologist are made to perform the duty of supporting an hypothesis which would never have existed but for their own treacherous suggestions. Here is one of the most attractive of them:

Between the Latin words _novus_, new, and _novem_, nine, there exists a resemblance so close that it may well be more than accidental. Nine is, then, the _new_ number; that is, the first number on a new count, of which 8 must originally have been the base. Pursuing this thought by investigation into different languages, the same resemblance is found there. Hence the theory is strengthened by corroborative evidence. In language after language the same resemblance is found, until it seems impossible to doubt, that in prehistoric times, 9 _was_ the new number--the beginning of a second tale. The following table will show how widely spread is this coincidence:

Sanskrit, navan = 9. nava = new.

Persian, nuh = 9. nau = new.

Greek, [Greek: ennea] = 9. [Greek: neos] = new.

Latin, novem = 9. novus = new.

German, neun = 9. neu = new.

Swedish, nio = 9. ny = new.

Dutch, negen = 9. nieuw = new.

Danish, ni = 9. ny = new.

Icelandic, nyr = 9. niu = new.

English, nine = 9. new = new.

French, neuf = 9. nouveau = new.

Spanish, nueve = 9. neuvo = new.

Italian, nove = 9. nuovo = new.