Popular Science Monthly/Volume 51/August 1897/The Origin and Development of Number Systems
|THE ORIGIN AND DEVELOPMENT OF NUMBER SYSTEMS.|
IT is generally acknowledged that we have in the number systems of the lower races to-day a means of studying the development of our own system. This is based upon the assumption that when the savage begins to count he does it always in essentially the same way. This is, in fact, more than an assumption. An analysis of the number systems of many races scattered all over the globe shows that such a similarity exists, and there is no reason to suppose that our own ancestors followed any other method. Indeed, such evidence as it is now possible to bring forward all goes to support this view.
Counting begins when man first forms the idea of two as distinct from one and more than two. We may perhaps go back even one step further, and say that it begins when the idea of one, as distinct from more than one, is formed. If this be taken as the starting point, the distinct conception of two forms the second step. It is difficult to realize that such ideas are not contemporaneous with the birth of intelligence, but there is evidence to show that such is not the case. According to Dr. Charles Letourneau, we have one example of a race which has not yet taken even this first step. He says in his book on Sociology (translated by Henry M. Trollope), page 582: "The Weddahs of Ceylon, who seem to be the least intelligent of men, have still no mathematical faculty whatever; they have no name for any number." To say that they have no name for any number probably does not imply that they are unable to realize that one group of objects contains more individuals than another group of the same objects. They could even determine which of two such groups contained the greater number of objects, by placing in succession one from each group in pairs until all in one group were exhausted. Such a process, however, is not counting, and the race which finds it necessary to resort to such an expedient may fairly be said to have no conception of number as such.
We find other races who have taken only the first two or three steps. These are chiefly the South American forest tribes and the bushmen of Australia. Speaking of these tribes, Edward B. Tylor says (Primitive Culture, London, 1871, Vol. I, page 220): "Five is actually found to be a number which the languages of some tribes do not know by a special word. Not only have travelers failed to get from them names for numbers above 2, 3, or 4, but the opinion that these are the real limits of their numeral series is strengthened by their use of the highest known number as an indefinite term for a great many. Spix and Martins say of the low tribes of Brazil: ‘They count commonly by their finger joints, and so up to three only. Any larger number they express by the word many.’ In a Puri vocabulary the numerals are given as—1, omi; 2, curiri; 3, prica, ‘many’: in a Botocudo vocabulary—1, mokenam; 2, uruhú, ‘many.’" It is needless to multiply these examples.
The next step in advance in the development of a number system is taken when the fingers and toes are brought in as aids in the art of counting. The advance is now comparatively rapid up to the point where, these means being exhausted, there is no further natural aid conveniently at hand. It is popularly believed that finger-counting represents the very earliest stage of the art, but the existence of tribes who can not count as far as five, as already cited, would seem to be conclusive evidence of a stage which antedates this. The etymological character of the numeral words in most of the known languages points to the same conclusion. Prof. Levi Leonard Conant, in a book recently published under the title The Number Concept, has collected and analyzed a great number of the numeral systems of savage and semicivilized tribes. He says (pages 98 and 99): "Collecting together and comparing with one another the great mass of terms by which we find any number expressed in different languages, and while admitting the great diversity of method practiced by different tribes, we observe certain resemblances which were not at first supposed to exist. The various meanings of 1, where they can be traced at all, cluster into a little group of significations with which at last we come to associate the idea of unity. Similarly of 2, or 5, or 10, or any of the little band which does picket duty for the advance guard of the great host of number words which are to follow. A careful examination of the first decade warrants the assertion that the probable meaning of any one of the units will be found in the list given below. The words selected are intended merely to serve as indications of the thought underlying the savage's choice, and not necessarily as the exact terms by means of which he describes his number. Only the commonest meanings are included in the tabulation here given:
|"1 = Existence, piece, group, beginning.|
|"2 = Repetition, division, natural pair.|
|"3 = Collection, many, two-one.|
|"4 = Two twos.|
|"5 = Hand, group, division.|
|"6 = Five-one, two threes, second one.|
|"7 = Five-two, second two, three from ten.|
|"8 = Five-three, second three, two fours, two from ten.|
|"9 = Five-four, three threes, one from ten.|
|"10 = One (group), two fives (hands), half-a-man, one man.|
|"15 = Ten-five, one foot, three fives.|
|"20 = Two tens, one man two feet."|
One of the most significant things to be observed in this table is the absence of any reference to the figures in the numerals for 1, 2, 3, and 4. This strongly confirms the view already expressed, that counting began before the use of the fingers as an aid was thought of. The higher numerals, on the contrary, are made up almost entirely of finger-words and their adjuncts. This does not appear in the English translations, but in the original words it is seen at once.
We may say, therefore, that the human race in learning to count passes through three stages. In the first stage the fingers are not used; progress is very slow; no distinct conception of numbers greater than two or three is formed; all beyond this is "many." Indeed, in this stage it is altogether probable that no conception of number, properly so called, is formed at all; that is, the idea of the number of things in a group is not distinctly abstracted from the objects themselves. In the second stage the fingers and toes are used, and counting can be carried as far as ten or twenty, or perhaps, by the use of more than one man, even a little further; but corresponding numeral words are not yet invented, so that counting is by gestures. In the third stage, words or expressions describing the gestures used in the second stage are assigned to do duty as numerals, and in the course of time they become pure numeral words—that is, they are used merely to indicate numbers, the mind no longer thinking of them as describing gestures that once served the same purpose.
The question now arises whether we can find any trace of finger-counting in our own numerals, and whether we can trace the origin of the lower numerals—those in which we should not naturally expect to find a finger origin. Mr. James Gow, of Cambridge, in his Short History of Greek Mathematics, Chapter I, gives some reasons that seem to show that our own Aryan ancestors, like other races, could not at first count beyond three or four, and afterward learned to count on their fingers. His reasons are three, as follows: 1. The words for 1, 2, 3, and 4 show a different grammatical character from the next six. He says (page 2): "The first three are adjectives, agreeing with only casual and partial exceptions (e. g., δύο) in gender and case with the substantives which they qualify. The same might be said of the fourth, but that in Latin quattuor is wholly indeclinable. The rest, from 5 to 10, are generally uninflected, and have, or had originally, the form of a neuter singular." 2. The existence of three grammatical numbers—singular, dual, and plural—probably points to a time when more than two was regarded indefinitely as many. 3. The names of the six numerals, from 5 to 10, may possibly be derived etymologically from a hand or finger source.
Mr. Gow's statements with regard to the grammatical character of the words is easily seen to be quite what we should expect. When man first begins to count, the numerals do not represent to him abstract ideas. He does not think of "2," for example, as a mere number, as we do. It represents nothing to him if separated from the group of objects which it is used to describe. The numeral words, therefore, naturally take on the form of adjectives. Later, when his ideas are further developed and he begins to use his fingers in counting, the real idea of number begins to assert itself, and the words used to designate different numbers appear in a more abstract form. The fingers become, in fact, when used thus, real numerical symbols, as much so as written ones, and the mind gradually becomes accustomed to thinking of the number of a set of objects as something which can be considered apart from the objects themselves, and which can be represented just as well by an equal number of other objects differing from the first set in all respects except that of number. Thus the abstract idea of number is formed.
Mr. Gow's ideas upon grammatical number seem to me, if they have the significance he claims, to point back still further. As soon as the idea of two or more as distinct from one is conceived, the necessity for a new grammatical form arises. Now, if the number sense were at all developed, the formation of grammatical number ought to stop here; for it would be apparent at once that to have a different grammatical form for every number is impracticable. But we find a distinct inflection set apart to express two, and a new inflection to designate three or more. The existence of this third or plural number would then indicate that the idea of three, or many, as distinct from two corresponds to another step in the development of the number sense. That the process of forming new grammatical numbers went no further then becomes an argument to show that the subsequent number development was more rapid, and the impossibility of making the former keep pace with the latter was realized.
The etymology of the first three or four of our numerals is probably quite beyond our reach. It has already been pointed out that in the selection of words to represent 1 or 2 the savage has such a wide range of objects to choose from that it is very much a matter of chance what he will select. Any concrete object that possesses the essential quality of unity or duality may be impressed into the service. For 3 and 4 the range of objects that will serve his purpose is more limited, but it is still sufficiently large to make it a mere accident what he will use. For example, the Abipones of the Paraguay region express 4 by "toes of an ostrich," and 5 by "neenhalek," a, five-colored, spotted hide. Nevertheless, some attempts have been made to discover these etymologies.
Below is given a list which includes, besides those for 3 and 4, some etymologies that have been suggested for the higher numerals. I quote from the work of Gow already referred to, page 3, footnote: "The common derivations, taken chiefly from Bopp, are set out in Morris's Historical Outlines of English Accidence, page 110, note. The following only need be cited:
|"Three = ‘what goes beyond’ (root tri, tar, to go beyond).|
|"Four (quattuor) = ‘and three’—i. e., 1 and 3.|
|"Five = ‘that which comes after’ (four), Sk. pashchút = after.|
|"Six; Sk. shash, is probably a compound of two and four.|
|"Seven = ‘that which follows’ (six).|
|"Eight, Sk. Ashtún = 1 + and + 3.|
|"Nine = new that which comes after 8 and begins a new quartette.|
|"Ten = two and eight."|
In commenting on these etymologies Gow says (pages 3 and 4): "When they say that pankan and saptan, ‘five’ and ‘seven,’ mean ‘following,’ because they follow ‘four’ and ‘six’ respectively, they suggest no reason why any other numeral above 1 should not have been called by either or both of these names; so when they say that navan, ‘nine,’ means new (νέος, etc.) because it begins a new quartette, they assume a primeval quaternary notation, and do not explain why ‘five’ was not called navan; so, again, when they say navan means ‘last’ (νέατος, etc.) because it is the last of the units, they evidently speak from the point of view of an arithmetician who has learned to use written symbols." The objections offered by Mr. Gow to these etymologies seem to me to be quite valid, with the exception of the last. It is not at all uncommon to find "9" expressed by some such phrase as "approaching completion," the fingers forming the natural scale, and serving the purpose of written numerals. The savage would be therefore in this respect in the position of an "arithmetician who has learned to use written symbols." In the Jiviro scale 9 is "hands next to complete" (Conant, page 61); in the Ewe scale it is "parting with the hands" (ibid., page 92), and in the Chippeway dialect the same numeral is shangosswoy, which is akin to chagissi, "used up" (ibid., page 162).
The derivations of the last six of the first ten numerals suggested by Gow are as follows: "Their original names appear to have been pankan or kankan (5), ksvaks or ksvaksva (6), saptan (7), aktan (8), navan (9), and dakan or dvakan (10). Some allusion to finger-counting may well underlie these words. Ever since A. von Humboldt first pointed out the resemblance between the Sanskrit pañk′an and the Persian penjeh, ‘the outspread hand,’ some connection between the two has always been admitted. . . . So also dvakan seems to be for dvakankan, meaning ‘twice five’ or ‘two hands’; dakan points to δεξνός, dexter, δέχομαμ, etc., or else to δάκτνλος, digitus, zehe, toe. Thus, whatever original forms we assume for these two numerals their roots appear again in some name or other for the hand or fingers. It is intrinsically probable, therefore, that pankan means ‘hand,’ and that dakan means ‘two hands’ or ‘right hand.’ It may be suggested here that the intervening numerals are the names of the little, third, middle, and fore fingers of the right hand. Thus, the little finger was called by the Greeks ώτίτης, by the Latins auricularis. This name is apparently explained by the Germans, who call this finger the ‘ear-cleaner.’ Now, ksvaks or ksvaksva seems to be a reduplicated form, containing the same root as ξέω, ξαίνω, ξνρέω, etc., and meaning ‘scraper.’ The name saptan seems to mean ‘follower’ (έπ–ομαμ, etc.), and the third finger might very well be so called because it follows and moves with the second, in the manner familiar to all musicians. The name aktan seems to contain the common root AK, and to mean, therefore, ‘projecting,’ a good enough name for the middle finger. Lastly, the first finger is known as άσπαστικός, index, salutatorius, demonstratorius (= ‘beck–oner,’ ‘pointer’), and the meaning probably underlies navan, which will then be connected with the root of novus, νεύω, new, etc., or that of νεύω, nuo, nod, etc., or both. Whatever be thought of these suggested etymologies, it must be admitted that there is no evidence whatever that our forefathers counted the fingers of the right hand in the order here assumed. They may have adopted the reverse order, from thumb to little finger, as many savages do, and as, in fact, the Greeks and Romans did with that later and more complicated system of finger-counting which we find in use in the first century of our era. If this reverse order be assumed, the numerals may still be explained in accordance with other finger-names in common use, besides those which have been cited. But, after all, the main support of these etymologies is their great a priori probability. The theory on which they are based brings the history of Aryan counting into accord with the history of counting everywhere else; and it explains the Aryan numerals in a way which is certainly correct for nearly all other languages. It is hardly to be expected that such a theory should be strictly provable at all points."
Before leaving this part of the subject the writer wishes merely to add that the etymologies suggested above for six, seven, and eight appear to be quite plausible; for navan, or nine, however, it appears to him that the association with the idea of "last" is the more reasonable, and would fit in with the finger interpretation of the others just as well as the one suggested.
We return now to the general question of the development of number systems, which we left at the point where men were supposed to have learned to use their fingers and toes as a natural abacus, and to have reached, therefore, the number 20. Before any further progress can be made a scale of notation must be adopted. Of course, this is not done consciously. Within certain limits it is probably entirely a matter of chance what number will be selected as a base. I had better say what number will become the base; for the use of the word "selected" unconsciously implies that the savage exercises a choice, while in fact, as already stated, he is simply led by circumstances. In most cases he has adopted some kind of a base before he has counted as far as 20. We have already seen that one of the commonest forms for "6" is "hand-one" or five-one. When the savage expresses 6 in this way he is committed to a quinary scale. The chances are, however, overwhelmingly against his carrying out this system consistently in all higher numbers, and for very obvious reasons. A pure quinary system of numeration is therefore extremely rare. Nevertheless, at least one such exists, one that is purely quinary as far as it seems to be known; this is the scale of one of the Betoya dialects of South America. In this scale
|Six = teyente tey = hand + 1.|
|Eleven = caya ente-tey = 2 hands + 1.|
|Sixteen = toazumba-ente tey = 3 hands + 1.|
|Twenty = caesea ente = 4 hands.|
(Conant, pages 57 and 140.) It would be interesting to know whether this scale is carried on consistently—that is, whether 25, the square of the base, is recognized as a new starting point, or whether they call it simply "five hands," without any sign to mark it off distinctly from other multiples of the base.
What is generally found in these scales that introduce the quinary element at 6 is that "10" is designated by some expression other than "two fives"; and eleven then becomes 10 + 1; twelve, 10 + 2, etc.—that is, the quinary scale here merges into the decimal; and either we see no more of it, or it continues with the other in a subsidiary place. The latter is the more usual. Thus sixteen is 10 + 5 + 1; seventeen, 10 + 5 + 2, etc. Thus is formed a mixed decimal and quinary scale.
It is a question over which there has been considerable dispute whether all numeral systems were not originally quinary, and the adoption of a larger base came as, with the lapse of time, its superior advantages were recognized. I think that not only is the evidence in favor of the opposite view, but also that from a priori considerations we might expect to see the adoption of 10 as a base as readily as 5. It depends, I think, entirely upon whether 6 is called "five-one" or is designated in some other way. In speaking upon this point Prof. Conant says (pages 170, 171): "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. 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 limits 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 significance 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."
In support of this argument it may be said that at least in the languages of the most cultivated races to-day those elements of their numeral systems which are not homogeneous with the main characteristics of the system show great persistence. The "score" of English is a remnant of old vigesimal counting, and although it has lost its place in the ordinary number system, it is still retained as a semi-poetical form. Still more marked is the "quatre-vingts" of the French. In counting from 61 to 99 they use a purely vigesimal system. If these traces of vigesimal counting still remain, it would seem probable that if the quinary system had ever formed a part of the system it would also somewhere have left its marks, fainter, it is true, on account of its greater antiquity, but still discernible. Now the only indication from a philological source that such a system was ever employed by the Aryan peoples seems to be the Homeric πεμπάςειυ (literally to five), meaning "to count." It is sometimes stated also that the form of the Latin numerals I, II, III, IIII or IV, V, VI, etc., implies the existence of an early form of quinary counting. My own opinion is that evidence derived from written numerals, between which and the formation of the numeral system itself there can be no comparison as to dates, can be of very little weight in deciding what was the scale upon which the system was originally formed. If the Roman V for 5 and VI for 6 were adopted because of a quinary element in the Roman scale at the time these signs were first used, surely the spoken language would have retained some marks of the same system. The evidence all points, therefore, with the one exception quoted above, to the nonexistence of a quinary element in Aryan counting.
The third natural scale, besides the quinary and decimal, is the vigesimal. It is doubtful whether a pure vigesimal scale, unmixed with any quinary and decimal element, occurs in any part of the world. In certain regions, or with certain races, a strong tendency is found to make 20 the principal base of the numeral system. This is so with the Celtic peoples, with some Asiatic and a few African tribes, with some of the Eskimos, and with the peoples who formerly occupied the Central American regions. If a tribe counts up to 20, using their fingers and toes, and then continue their counting beyond this point in a consistent way, a vigesimal system will be the natural result; but on account of the practical difficulty of using the toes in any system of gesture-counting, which, as we have seen, is the second stage in the development of the number system, it seems plausible that most tribes confined themselves to the fingers alone. This would account for the greater predominance of 10 and 5 as number bases. It is true that in the case of many well-developed vigesimal scales we have no positive evidence that they originated in the custom of counting on the fingers and toes, but there is certainly great probability that they did all begin in this way. There seems to be no other good reason why 20 should have been adopted for a base. The most perfect examples of vigesimal scales are those of the Mayas of Yucatan and of the Aztecs of Mexico. It has already been mentioned that traces of this system are to be found in our own English numerals and in those of the French. Danish and some of the kindred languages show a strong tendency to vigesimal forms, although, as a whole, the Germanic systems of counting are purely decimal.
Among the important number systems of the world there is one which uses neither 5, 10, nor 20 as its base—namely, the sexagesimal scale of the ancient Babylonians. This system is of special interest to ourselves, for its influence is still felt in the division of our degree into 60 minutes, and the minute into 60 seconds. It seems to have arisen and continued in use side by side with a decimal system, for the monuments furnish examples of numbers which are wholly decimal, others wholly sexagesimal, and still others in which the two systems are combined. It is a question of great interest to know how such a system came to be adopted. It seems reasonable to suppose that it was formed artificially—that is, 60 did not come to be the base of this system by a process of natural development, as 5, 10, or 20 came to be the bases in the systems of other races. In all probability, therefore, it grew up after the decimal system, and may have been invented for the purposes of astronomical calculation, for the Babylonians were famous astronomers in their day. It is not impossible to suppose that its purpose originally was to render the calculations of the astronomers less intelligible to those who were acquainted with only the decimal scale. However that may have been, its use apparently became common. M. Cantor, the German writer on the History of Mathematics, seeks to explain its origin by saying that the Babylonians divided the circle of the heavens into 360 degrees, one degree for each of the 360 days into which they divided the year. They were probably also acquainted with the fact that the chord equal to the radius subtends exactly one sixth of the circumference, or 60 degrees. This may have led to the adoption of 60 as the number base. Prof. John P. Peters, in a letter published in the Proceedings of the Society of Biblical Archæology for May, 1883, pages 120, 121, says, in substance: The use of the fingers of one hand to count to 5 was in some cases extended to 6, by using the open hand with the fingers and thumb extended to express 5, and then indicating 6 by the closed hand. This method, if extended to both hands, gives rise ordinarily to a duodecimal system; and we have abundant evidence both in our own English and in some other languages of the presence of a duodecimal element, which may have arisen in the way suggested. The Babylonians, however, instead of developing a pure duodecimal system, combined the seximal with the decimal in a multiplicative manner, and so developed a sexagesimal system.
The reader may take his choice between these two attempts to explain what, in any case, must be regarded as a remarkable phenomenon. Without denying that either may possibly be the true explanation, the writer is of the opinion that much additional evidence will be required to finally solve the question. With the rapid advance that is now being made in the field of Babylonian antiquities it is not impossible that the needed information will be forthcoming.
- The order of counting from the little finger to the thumb is, however, the more usual method with savages. See a paper by Lieutenant F. H. Cushing, entitled Manual Concepts, in the American Anthropologist for October, 1892 (vol. v).
- An elaborate argument in support of this theory is to be found in Hervas's celebrated work, Arithmetica di quasi tutte Ie nazioni conosciute.