The foregoing examples will show with considerable fulness the wide dispersion of the quinary scale. Every part of the world contributes its share except Europe, where the only exceptions to the universal use of the decimal system are the half-dozen languages, which still linger on its confines, whose number base is the vigesimal. Not only is there no living European tongue possessing a quinary number system, but no trace of this method of counting is found in any of the numerals of the earlier forms of speech, which have now become obsolete. The only possible exceptions of which I can think are the Greek πεμπάζειν, to count by fives, and a few kindred words which certainly do hint at a remote antiquity in which the ancestors of the Greeks counted on their fingers, and so grouped their units into fives. The Roman notation, the familiar I., II., III., IV. (originally IIII.), V., VI., etc., with equal certainty suggests quinary counting, but the Latin language contains no vestige of anything of the kind, and the whole range of Latin literature is silent on this point, though it contains numerous references to finger counting. It is quite within the bounds of possibility that the prehistoric nations of Europe possessed and used a quinary numeration. But of these races the modern world knows nothing save the few scanty facts that can be gathered from the stone implements which have now and then been brought to light. Their languages have perished as utterly as have the races themselves, and speculation concerning them is useless. Whatever their form of numeration may have been, it has left no perceptible trace on the languages by which they were succeeded. Even the languages of northern and central Europe which were contemporary with the Greek and Latin of classical times have, with the exception of the Celtic tongues of the extreme North-west, left behind them but meagre traces for the modern student to work on. We presume that the ancient Gauls and Goths, Huns and Scythians, and other barbarian tribes had the same method of numeration that their descendants now have; and it is a matter of certainty that the decimal scale was, at that time, not used with the universality which now obtains; but wherever the decimal was not used, the universal method was vigesimal; and that the quinary ever had anything of a foothold in Europe is only to be guessed from its presence to-day in almost all of the other corners of the world.

From the fact that the quinary is that one of the three natural scales with the smallest base, it has been conjectured that all tribes possess, at some time in their history, a quinary numeration, which at a later period merges into either the decimal or the vigesimal, and thus disappears or forms with one of the latter a mixed system.[323] In support of this theory it is urged that extensive regions which now show nothing but decimal counting were, beyond all reasonable doubt, quinary. It is well known, for example, that the decimal system of the Malays has spread over almost the entire Polynesian region, displacing whatever native scales it encountered. The same phenomenon has been observed in Africa, where the Arab traders have disseminated their own numeral system very widely, the native tribes adopting it or modifying their own scales in such a manner that the Arab influence is detected without difficulty.

In view of these facts, and of the extreme readiness with which a tribe would through its finger counting fall into the use of the quinary method, it does not at first seem improbable that the quinary was the original system. But an extended study of the methods of counting in vogue among the uncivilized races of all parts of the world has shown that this theory is entirely untenable. The decimal scale is no less simple in its structure than the quinary; and the savage, as he extends the limit of his scale from 5 to 6, may call his new number 5-1, or, with equal probability, give it an entirely new name, independent in all respects of any that have preceded it. With the use of this new name there may be associated the conception of “5 and 1 more”; but in such multitudes of instances the words employed show no trace of any such meaning, that it is impossible for any one to draw, with any degree of safety, the inference that the signification was originally there, but that the changes of time had wrought changes in verbal form so great as to bury it past the power of recovery. A full discussion of this question need not be entered upon here. But it will be of interest to notice two or three numeral scales in which the quinary influence is so faint as to be hardly discernible. They are found in considerable numbers among the North American Indian languages, as may be seen by consulting the vocabularies that have been prepared and published during the last half century.[324] From these I have selected the following, which are sufficient to illustrate the point in question:

1.milchtih.
2.nonnepah.
3.dahghenih.
4.tuah.
5.sattou.
6.schappeh.
7.pennapah.
8.pehdaghenih.
9.schunkkah.
10.gedeh bonah.
1.krara.
2.krowü.
3.krom miah.
4.krob king.
5.krasch kingde.
6.terdeh.
7.kogodeh.
8.kwongdeh.
9.schkawdeh.
10.dwowdeh.
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 kinship 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 relationship 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 passage 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 repetition 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 illustration 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?