Dozen

“ When the savage begins the process of counting he invents, one after another, names with which to designate the successive steps of his numerical journey. At first there is no attempt at definiteness in the description he gives of any considerable number. If he cannot show what he means by the use of his fingers, or perhaps by the fingers of a single hand, he unhesitatingly passes it by, calling it many, heap, innumerable, as many as the leaves on the trees, or something else equally expressive and equally indefinite. ”

“ As for their numbers, do not ask how many they are, that their resistance should be a possible thing; for if a thousand of them should take the field, they will meet thee in battle, and so will any number, be it less than this, or be it more. ”

“ Ten is the farthest number; if half ten
⁠Belongs to each woman, then
⁠Each woman may take half us men;
Or—if this will not serve their turn—since all
Numbers are odd, or even, and they fall
First into five, women may take us all. ”

“ Not only are the first distinct ideas of number co-ordinate with ideas of likeness and equality, but the first efforts at numeration displayed the same relationship. On reading the accounts of various savage tribes, we find that the method of counting by the fingers, still followed by many children, is the aboriginal method. Neglecting the several cases in which the ability to enumerate does not reach even to the number of fingers on one hand, there are many cases in which it does not extend beyond ten—the limit of the simple finger notation. ”

“ The fingers are, however, often employed in counting numbers far above the first decade. After giving the Il-Oigob numerals up to 60, Müller adds:8 “Above 60 all numbers, indicated by the proper figure pantomime, are expressed by means of the word ipi.” We know, moreover, that many of the American Indian tribes count one ten after another on their fingers; so that, whatever number they are endeavouring to indicate, we need feel no surprise if the savage continues to use his fingers throughout the entire extent of his counts. ”

“ Still, to find 10 expressed by the term man always conveys an impression of mental poverty; though it may, of course, be urged that this might arise from the fact that some races never use the toes in counting, but go over the fingers again, or perhaps bring into requisition the fingers of a second man to express the second 10. It is not safe to postulate an extremely low degree of civilization from the presence of certain peculiarities of numeral formation. ”

“ The knowledge of numbers exists more or less among all the peoples of the earth. We often say, “Such a people can only count up to three, because it has no special word to denote a higher number.” This reasoning is not always just, for, by adopting it, we might accuse the French of scarcely being able to count beyond sixty, since they have no special words for, say, seventy-five or eighty, and to express these fall back on words already employed in counting—sixty and fifteen or four score. ”

“ Many men might have been handsomer, but certainly there could be none whose appearance was more significant, if the expression may be used. Everything about the count seemed to have its meaning, for the constant habit of thought which he had acquired had given an ease and vigor to the expression of his face, and even to the most trifling gesture, scarcely to be understood. ”

“ A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. ”

“ In nearly all the examples given the decimal becomes the leading element in the formation of all units above 100, just as was the case in the Celtic scales already noticed.
Among the northern tribes of Siberia the numeral scales appear to be ruder and less simple than those just examined, and the counting to be more consistently vigesimal than in any scale we have thus far met with. The two following examples are exceedingly interesting, as being among the best illustrations of counting by twenties that are to be found anywhere in the Old World. ”

“ But races as fond of metaphor and of linguistic embellishment as are those of the East, or as are our American Indians even, might readily resort to some other source than that furnished by the members of the human body, when in want of a term with which to describe the 5, 10, or any other number of the numeral scale they were unconsciously forming. ”

“ This limit is often extended to 20, and not infrequently to 100. Again, we find 1000 as the limit; or perhaps 10,000; and sometimes the savage carries his number system on into the hundreds of thousands or millions. Indeed, the high limit to which some savage races carry their numeration is far more worthy of remark than the entire absence of the number sense exhibited by others of apparently equal intelligence. If the life of any tribe is such as to induce trade and barter with their neighbours, a considerable quickness in reckoning will be developed among them. ”

“ I noticed some of the company prepare their mouths and draw themselves up at the prospect of some agreeable forfeit. "Let us play at counting," said Charlotte. "Now, pay attention: I shall go round the circle from right to left; and each person is to count, one after the other, the number that comes to him, and must count fast; whoever stops or mistakes is to have a box on the ear, and so on, till we have counted a thousand." ”

“ How, then, is work perfected in the number forty? The reason, it may be, is, because the law was given in ten precepts, and was to be preached throughout the whole world: which whole world, we are to mark, is made up of four quarters, east and west, south and north, whence the number ten, multiplied by four, comes to forty. ”

“ The reason why ten is a favorite—perhaps the most favorite—number is obvious enough, namely, that we have ten fingers. "When we begin to count we almost of necessity do so with our fingers; if we have a large number of things to count, say a flock of sheep, [1] we instinctively divide them into tens, or perhaps into scores; if the number of things be very large, the collection of tens are naturally grouped again by tens, and so we have hundreds. ”

“ The theory that man’s primitive mode of counting was palpable reckoning on his hands, and the proof that many numerals in present use are actually derived from such a state of things, is a great step towards discovering the origin of numerals in general. Can we go farther, and state broadly the mental process by which savage men, having no numerals as yet in their language, came to invent them? ”

“ A student in a library of a thousand volumes, an officer before his regiment of a thousand men upon parade, may have a pretty clear idea of the units as well as of the totals; but when we come to a thousand times a thousand, or a thousand times a million, all view of the units, for most men, probably for every man, is lost: the million for the grown man is in a great degree like the hundred for the child. ”

“ Several nations in America have no other instruments of calculation. It was probably the same in the primitive ages. It is another strong presumption of the truth of what I now advance, that all civilized nations count by tens, tens of tens, or hundreds, tens of hundreds, thousands, and so on; still from ten to ten. We can discover no reason why the number ten should be chosen rather than any other for the term of numeration, except this primitive practice of counting by the fingers. ”

“ We experienced some little trouble in mastering the French numerals, until I tried a new scheme, and called out, "From the right, number, in French!" Then my merry convalescents began shouting gleefully, "Oon," "Doo," "Troy," "Catta," "Sink," etc.; but the French numerals stuck in their heads. Never did any one, I imagine, have such a set of jolly, cheery boys in blue as pupils, and the strong remnant of the child left in many of them made them the more attractive. ”

“ By adding together a black numeral and the preceding red one and casting out thirteen (or thirteens, as the case maybe) , when the sum exceeds this number, we obtain the following red one, thus: XIII + 9 - 13 = IX; IX + 11 - 13 = VII; VII + 20 - 13 - 13 = I; I + 10 = XI, and so on through the entire series. Attention is also called to the fact that the sum of the black (Arabic) numbers 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, 9, 11, 20, 10, 15, is 260, a multiple of 13. ”

“ The tens have a place of their own; if you see figure 6 in the ten-place, you know it means sixty. The tens are always put behind the units: when you see two figures standing side by side, thus, ‘55,’ the left-hand figure stands for so many tens ”

“ In the system of counting every place, or column, counting from the right, has a value ten times greater than the one in the place or column nearest on the right. Thus in the number 36,542 the first figure on [71] the right represents “ones,” the next ten times as much or “tens,” the next ten times as much again or “hundreds,” and so on. We really read this number backward when we name it, for in handling it in any way we have to start with the last figure, representing the “ones.” The number really means two ones, four tens, five hundreds, six thousands and three ten thousands. ”

“ Sometimes a number above ten, but not over twenty, is indicated with the hands only by counting up to ten in the ordinary way, and then opening all the fingers and counting again, until they reach the requisite amount in excess of ten. ”

“ But the one primitive method of counting which seems to have been almost universal throughout all time is the finger method. It is a matter of common experience and observation that every child, when he begins to count, turns instinctively to his fingers; and, with these convenient aids as counters, tallies off the little number he has in mind. This method is at once so natural and obvious that there can be no doubt that it has always been employed by savage tribes, since the first appearance of the human race in remote antiquity. ”

“ It is, however, correct to say, "The signers of the petition were less than a hundred," where the round number, a hundred, is something like a collective noun, and less is thought of as meaning a less quantity or amount. ”

“ Ten, according to some, is a perfect number, not in its composition like six, but because it contains inclosed in itself all the simple numbers and all the properties, as of even and odd, perfect and imperfect, and it is the beginning of all numbers composed of tens, and also for the foundation of our law. ”

“ SAPIENTIA. When numbers of any magnitude are set down in order. the first set down is called the "minor term" and the last the "major." When, in making a division, we say by how many the number is to be divided, we give the "divisor," but when we enumerate how many there are in each of the parts we set forth the"quotient." ”

“ If the strong man who had six fingers on each hand, and six toes on each foot, had invented a system of numeration, it would have gone in twelves, nearly like the duodecimals which our carpenters use; unless, indeed, he had been stupid after the manner of very strong men, and not gone beyond sixes. We see how the Romans, though they inherited from their Eastern ancestors a numeration by tens up to decem, and then beginning again undecim, &c., yet when they began to write a notation could get no farther than five—I., II., III., IV., V. ”

“ A consistent formation would require the expression of 10 by some phrase meaning “two fives,” 15 by “three fives,” etc. Such a scale is the one obtained from the Betoya language, already mentioned in Chapter III., where the formation of the numerals is purely quinary, as the following indicate:227
The same formation appears, with greater or less distinctness, in many of the quinary scales already quoted, and in many more of which mention might be made. ”

“ On the other hand, it would seem that, for most educated people, sixteen and seventeen or twenty-six and twenty-seven, and even eighty-six and eighty-seven, are single numbers, just as six and seven are, and are not made up of groups of tens and ones. In other words, the denary scale, though adopted in notation and in numeration, does not arise in the corresponding mental concept until we get beyond 100. ”

“ Allowing that one person could count a million in seven days, which is barely possible, it would have required that eighty thousand persons should have started at the creation of the world to complete the enumeration at the present time! ”

“ One number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the number-series. Thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4. To find how much greater, we compare two series, in one of which we go up to 9, while in the other we stop at 4 and then recommence our counting. ”

“ This method of counting, however, is not confined to the counting of ages, but it characterizes all their counting. If you ask a man how many days before a certain festival near at hand he will say ten where we would say but nine. ”

“ Method.—Multiplying by six and dividing by three gives twice the number; add forty we have twice the number, plus forty, divide by two we have once the number, plus twenty; hence, if I subtract twenty from the result he gives me I have the number thought of.
Let a person select a number greater than 1 and not exceeding 10. I will add to it a number not exceeding 10, alternately with himself; and, although he has the advantage in selecting the number to start with, I will reach the even hundred first. ”