p>The Atlantic Monthly, Volume 05, No. 28, February, 1860 / A Magazine of Literature, Art, and Politics
COUNTING AND MEASURING
Though, from the rapid action of the eye and the mind, grouping and counting by groups appear to be a single operation, yet, as things can be seen in succession only, however rapidly, the counting of things, whether ideal or real, is necessarily one by one. This is the first step of the art. The second step is grouping. The use of grouping is to economize speech in numeration, and writing in notation, by the exercise of the memory. The memorizing of groups is, therefore, a part of the primary education of every individual. Until this art is attained, to a certain extent, it is very convenient to use the fingers as representatives of the individuals of which the groups are composed. This practice led to the general adoption of a group derived from the fingers of the left hand. The adoption of this group was the first distinct step toward mental arithmetic. Previous groupings were for particular numerations; this for numeration in general; being, in fact, the first numeric base,—the quinary. As men advanced in the use of numbers, they adopted a group derived from the fingers of both hands; thus ten became the base of numeration.
Notation, like numeration, began with ones, advanced to fives, then to tens, etc. Roman notation consisted of a series of signs signifying 1, 5, 10, 50, 100, 500, 1000, etc.,—a series evidently the result of counting by the five fingers and the two hands, the numbers signified being the products of continued multiplication by five and by two alternately. The Romans adhered to their mode, nor is it entirely out of use at the present day, being revered for its antiquity, admired for its beauty, and practised for its convenience.
The ancient Greek series corresponded to that of the Romans, though primarily the signs for 50, 500 and 5000 had no place. Ultimately, however, those places were supplied by means of compound signs.
The Greeks abandoned their ancient mode in favor of the alphabetic, which, as it signified by a single letter each number of the arithmetical series from one to nine separately, and also in union by multiplication with the successive powers of the base of numeration, was a decided improvement; yet, as it consisted of signs which by their number were difficult to remember, and by their resemblance easy to mistake, it was far from being perfect.
Doubtless, strenuous efforts were made to remedy these defects, and, apparently as the result of those efforts, the Arabic or Indian mode appeared; which, signifying the powers of the base by position, reduced the number of signs to that of the arithmetical series, beginning with nought and ending with a number of the value of the base less one.
The peculiarity of the Arabic mode, therefore, in comparison with the Greek, the Roman, or the alphabetic, is place value; the value of a combination by either of these being simply equal to the sum of its elements. By that, the value of the successive places, counting from right to left, being equal to the successive powers of the base, beginning with the noughth power, each figure in the combination is multiplied in value by the power of the base proper to its place, and the value of the whole is equal to the sum of those products.
The Arabic mode is justly esteemed one of the happiest results of human intelligence; and though the most complex ever practised, its efficiency, as an arithmetical means, has obtained for it the reputation of great simplicity,—a reputation that extends even to the present base, which, from its intimate and habitual association with the mode, is taken to be a part of the mode itself.
With regard to this impression it may be remarked, that the qualities proper to a mode bear no resemblance to those proper to a base. The qualities of the present mode are well known and well accepted. Those of the present base are accepted with the mode, but those proper to a base remain to be determined. In attempting to ascertain these, it will be necessary to consider the uses of numeration and of notation.
These may be arranged in three divisions,—scientific, mechanical, and commercial. The first is limited, being confined to a few; the second is general, being common to many; the third is universal, being necessary to all. Commercial use, therefore, will govern the present inquiry.
Commerce, being the exchange of property, requires real quantity to be determined, and this in such proportions as are most readily obtained and most frequently required. This can be done only by the adoption of a unit of quantity that is both real and constant, and such multiples and divisions of it as are consistent with the nature of things and the requirements of use: real, because property, being real, can be measured by real measures only; constant, because the determination of quantity requires a standard of comparison that is invariable; conveniently proportioned, because both time and labor are precious. These rules being acted on, the result will be a system of real, constant, and convenient weights, measures, and coins. Consequently, the numeration and notation best suited to commerce will be those which agree best with such a system.
From the earliest periods, special attention has been paid to units of quantity, and, in the ignorance of more constant quantities, the governors of men have offered their own persons as measures; hence the fathom, yard, pace, cubit, foot, span, hand, digit, pound, and pint. It is quite probable that the Egyptians first gave to such measures the permanent form of government standards, and that copies of them were carried by commerce, and otherwise, to surrounding nations. In time, these became vitiated, and should have been verified by their originals; but for distant nations this was not convenient; moreover, the governors of those nations had a variety of reasons for preferring to verify them by their own persons. Thus they became doubly vitiated; yet, as they were not duly enforced, the people pleased themselves, so that almost every market-town and fair had its own weights and measures; and as, in the regulation of coins, governments, like the people, pleased themselves, so that almost every nation had a peculiar currency, the general result was, that with the laws and the practices of the governors and the governed, neither of whom pursued a legitimate course, confusion reigned supreme. Indeed, a system of weights, measures, and coins, with a constant and real standard, and corresponding multiples and divisions, though indulged in as a day-dream by a few, has never yet been presented to the world in a definite form; and as, in the absence of such a system, a corresponding system of numeration and notation can be of no real use, the probability is, that neither the one nor the other has ever been fully idealized. On the contrary, the present base is taken to be a fixed fact, of the order of the laws of the Medes and Persians; so much so, that, when the great question is asked, one of the leading questions of the age,—How is this mass of confusion to be brought into harmony?—the reply is,—It is only necessary to adopt one constant and real standard, with decimal multiples and divisions, and a corresponding nomenclature, and the work is done: a reply that is still persisted in, though the proposition has been fairly tried, and clearly proved to be impracticable.
Ever since commerce began, merchants, and governments for them, have, from time to time, established multiples and divisions of given standards; yet, for some reason, they have seldom chosen the number ten as a base. From the long-continued and intimate connection of decimal numeration and notation with the quantities commerce requires, may not the fact, that it has not been so used more frequently, be considered as sufficient evidence that this use is not proper to it? That it is not may be shown thus:—A thing may be divided directly into equal parts only by first dividing it into two, then dividing each of the parts into two, etc., producing 2, 4, 8, 16, etc., equal parts, but ten never. This results from the fact, that doubling or folding is the only direct mode of dividing real quantities into equal parts, and that balancing is the nearest indirect mode,—two facts that go far to prove binary division to be proper to weights, measures, and coins. Moreover, use evidently requires things to be divided by two more frequently than by any other number,—a fact apparently due to a natural agreement between men and things. Thus it appears the binary division of things is not only most readily obtained, but also most frequently required. Indeed, it is to some extent necessary; and though it may be set aside in part, with proportionate inconvenience, it can never be set aside entirely, as has been proved by experience. That men have set it aside in part, to their own loss, is sufficiently evidenced. Witness the heterogeneous mass of irregularities already pointed out. Of these
