Numeral

“ That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system. ”

“ Wherever the numeral 20 is expressed by the term man, it may be expected that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually, though not always, carried as far as the system extends ”

“ And some, through the default of their memories, who cannot retain the several combinations of numbers, with their names, annexed in their distinct orders, and the dependence of so long a train of numeral progressions, and their relation one to another, are not able all their lifetime to reckon, or regularly go over any moderate series of numbers. ”

“ But the use of words in registering our thoughts is in nothing so evident as in numbering. A natural fool that could never learn by heart the order of numeral words, as one, two, and three, may observe every stroke of the clock, and nod to it, or say one, one, one, but can never know what hour it strikes. ”

“ It often happens that the composition of numeral words is less easily understood, and the original meanings more difficult to recover, than in the examples already given. But in searching for number systems which show in the formation of their words the influence of finger counting, it is not unusual to find those in which the derivation from native words signifying finger, hand, toe, foot, and man, is just as frankly obvious as in the case of the Zuñi, the Arawak, the Eskimo, or the Montagnais scale. ”

“ 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. ”

“ OBS. 18.—Since nouns and adjectives are different parts of speech, the suggestion, that, "Numeral adjectives are also names, or nouns," is, upon the very face of it, a flat absurdity; and the notion that "the name of a number" above unity, conveys only and always the idea of unity, like an ordinary "singular noun," is an other. A number in arithmetic is most commonly an adjective in grammar; and it is always, in form, an expression that tells how many, or—"denotes how many things are spoken of."—Chase, p. 11. But the name of a number is also a number, whenever it is not made plural in form. ”

“ 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? ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ One noteworthy and interesting fact relating to numeral nomenclature is the variation in form which words of this class undergo when applied to different classes of objects. To one accustomed as we are to absolute and unvarying forms for numerals, this seems at first a novel and almost unaccountable linguistic freak. But it is not uncommon among uncivilized races, and is extensively employed by so highly enlightened a people, even, as the Japanese. This variation in form is in no way analogous to that produced by inflectional changes, such as occur in Hebrew, Greek, Latin, etc. ”

“ The use of visible signs to represent numbers and aid reckoning is not only older than writing, but older than the development of numerical language on the denary system; we count by tens because our ancestors counted on their fingers and named numbers accordingly. So used, the fingers are really numerals, that is, visible numerical signs; and in antiquity the practice of counting by these natural signs prevailed in all classes of society. In the later times of antiquity the finger symbols were developed into a system capable of expressing all numbers below 10,000. ”

“ If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but that counting is carried beyond that point by the aid of fingers or other objects. ”

“ 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. ”

“ Both these modes of expressing what our own ancestors termed a “score,” are so common that one hesitates to say which is of the more frequent use. The following scale, from one of the Betoya dialects79 of South America, is quite remarkable among digital scales, making no use of either “man” or “foot,” but reckoning solely by fives, or hands, as the numerals indicate. ”

“ In savage languages it is sometimes possible to detect these meanings, and thus to obtain possession of the clue that leads to the development, in the barbarian's rude mind, of a count scale—a number system. But in languages like those of modern Europe, the pedigree claimed by numerals is so long that, in the successive changes through which they have passed, all trace of their origin seems to have been lost. ”

“ Francis Galton, respecting what he designated "visualized numerals," and the association of colors with various numbers. The term "visualized numerals" in its simplest sense means the conception, or mental vision, when any number is spoken or thought of, of that number as it is written, the power to form which may extend in some cases to lines of considerable length. ”

“ He resolves, indeed, that no one of the four words used, "is in construction with" any of the rest; for he thinks, "The meaning is, 'The number two taken twice is equal to four.'" Here, then, is a fourth opinion in relation to the subject of the verb: it must be "number" understood. Again, it is conceded by the same hand, that, "When numerals denoting more than one, are used as adjectives, with a substantive expressed or understood, they must have a plural construction." Now who can show that this is not the case in general with the numerals of multiplication? ”

“ OBS. 27.--The numeral words considered above, seem to have been originally adjectives, and such may be their most proper construction now; but all of them are susceptible of being construed as nouns, even if they are not such in the examples which have been cited. Dozen, or hundred, or thousand, when taken abstractly, is unquestionably a noun; for we often speak of dozens, hundreds, and thousands. Few and many never assume the plural form, because they have naturally a plural signification; and a few or a great many is not a collection so definite that we can well conceive of fews and manies ”

“ Another and much more common oddity is the tendency to visualize numerals in a peculiar manner, which characterizes, as I have roughly reckoned, about one woman in every fifteen, and one man in every thirty. Those who do so are never able to dissociate any single number from its own particular place in the field of their mental view, so that when they think of a series of numbers they always visualize them in a certain form. ”

“ Most of the terms employed are such as to defy any attempt to trace the process of reasoning which led to their adoption. It is not improbable that the choice was, in some of these cases at least, either accidental or arbitrary; but still, the changes in word meanings which occur with the lapse of time may have differentiated significations originally alike, until no trace of kinship would appear to the casual observer. Our numerals “score” and “gross” are never thought of as having any original relation to what is conveyed by the other meanings which attach to these words. ”

“ Numerals
(See also Chapter 13 "Tabular Work" and Chapter 14 "Leaderwork")
12.1.
Most rules for the use of numerals are based on the general principle that the reader comprehends numerals more readily than numerical word expressions, particularly in technical, scientific, or statistical matter. However, for special reasons, numbers are spelled out in certain instances, except in FIC & punc. and Fol. Lit. matter. ”

“ Normal forms of numerals 1 to 13, inclusive, in the Books of Chilan Balam.
The use of these purely ornamental elements, to fill the empty spaces in the normal forms of the numerals 1, 2, 6, 7, 11, 12, 16, and 17, is a fruitful source of error to the student of the inscriptions. Slight weathering of an inscription is often sufficient to make ornamental crescents look exactly like numerical dots, and consequently the numerals 1, 2, 3 are frequently mistaken for one another, as are also 6, 7, and 8; 11, 12, and 13; and 16, 17, and 18. ”

“ Such expressions, curious as they seem at first thought, are no more than the natural outgrowth of systems built up by the slow and tedious process which so often obtains among primitive races, where digit numerals are combined in an almost endless variety of ways, and where mere reduplication often serves in place of any independent names for higher units. ”

“ In the use of the duplicate forms for one and two, some care is required, for while we can say eh4 seh8, or ne7 seh8, for ten and twenty, we cannot say süò8 seh8, lang7 seh8, neither is it admissible to say sü`o͘8 pa4 eh4, süò8 pa4 ne7, for one hundred and ten and one hundred and twenty.
The system of abbreviated numerals used in business transactions is easily made to represent concrete numbers, as inches, feet, and poles ”

“ This fact, together with the method of numbering, is sufficient of itself to establish the correctness of the opinion I have advanced in reference to these red numerals. That they are here used to number the days is evident from the fact that they are applied to those days which are never used to name the years. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ We know of no language in which the suggestion of number does not appear, and we must admit that the words which give expression to the number sense would be among the early words to be formed in any language. They express ideas which are, at first, wholly concrete, which are of the greatest possible simplicity, and which seem in many ways to be clearly understood, even by the higher orders of the brute creation. The origin of number would in itself, then, appear to lie beyond the proper limits of inquiry; and the primitive conception of number to be fundamental with human thought. ”

“ Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance. The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,—or, in other words, of grasping the idea of number. A single illustration may be given which typifies all practical methods of numeration. ”

“ That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction of one unit : (2) That it retain in memory the names or marks of the several combinations, from an unit to that number ; and that not confusedly, and at random, but in that exact order that the numbers follow one another. In either of which, if it trips, the whole business of numbering will be disturbed, and there will remain only the confused idea of multitude, but the ideas necessary to distinct numeration will not be attained to. ”

“ I've analyzed both my home phone number and my office phone number in this fashion, figuring out what their components connote singly and as gestalts. And I can see why these fortuitous combinings command attention, why these numbers leap out of the directory pages right at you. Privately I call such a number a common denominator with a way of accreting its numerator.
I hope you're not laughing at me.
After all, when you remember what number is, what's happening follows naturally. Number's a language we use to blaze our way through the wood of reality. ”