Читать книгу The Atlantic Monthly, Volume 05, No. 28, February, 1860 - Various - Страница 1

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 our own coins present a familiar example. For the reasons above stated, coins, to be practical, should represent the powers of two; yet, on examination, it will be found, that, of our twelve grades of coins, only one-half are obtained by binary division, and these not in a regular series. Do not these six grades, irregular as they are, give to our coins their principal convenience? Then why do we claim that our coins are decimal? Are not their gradations produced by the following multiplications: 1 x 5 x 2 x 2-1/2 x 2 x 2 x 2-1/2 x 2 x 2 x 2, and 1 x 3 x 100? Are any of these decimal? We might have decimal coins by dropping all but cents, dimes, dollars, and eagles; but the question is not, What we might have, but, What have we? Certainly we have not decimal coins. A purely decimal system of coins would be an intolerable nuisance, because it would require a greatly increased number of small coins. This may be illustrated by means of the ancient Greek notation, using the simple signs only, with the exception of the second sign, to make it purely decimal. To express $9.99 by such a notation, only three signs can be used; consequently nine repetitions of each are required, making a total of twenty-seven signs. To pay it in decimal coins, the same number of pieces are required. Including the second Greek sign, twenty-three signs are required; including the compound signs also, only fifteen. By Roman notation, without subtraction, fifteen; with subtraction, nine. By alphabetic notation, three signs without repetition. By the Arabic, one sign thrice repeated. By Federal coins, nine pieces, one of them being a repetition. By dual coins, six pieces without a repetition, a fraction remaining.

In the gradation of real weights, measures, and coins, it is important to adopt those grades which are most convenient, which require the least expense of capital, time, and labor, and which are least likely to be mistaken for each other. What, then, is the most convenient gradation? The base two gives a series of seven weights that may be used: 1, 2, 4, 8, 16, 32, 64 lbs. By these any weight from one to one hundred and twenty-seven pounds may be weighed. This is, perhaps, the smallest number of weights or of coins with which those several quantities of pounds or of dollars may be weighed or paid. With the same number of weights, representing the arithmetical series from one to seven, only from one to twenty-eight pounds may be weighed; and though a more extended series may be used, this will only add to their inconvenience; moreover, from similarity of size, such weights will be readily mistaken. The base ten gives only two weights that may be used. The base three gives a series of weights, 1, 3, 9, 27, etc., which has a great promise of convenience; but as only four may be used, the fifth being too heavy to handle, and as their use requires subtraction as well as addition, they have neither the convenience nor the capability of binary weights; moreover, the necessity for subtraction renders this series peculiarly unfit for coins.

The legitimate inference from the foregoing seems to be, that a perfectly practical system of weights, measures, and coins, one not practical only, but also agreeable and convenient, because requiring the smallest possible number of pieces, and these not readily mistaken for each other, and because agreeing with the natural division of things, and therefore commercially proper, and avoiding much fractional calculation, is that, and that only, the successive grades of which represent the successive powers of two.

That much fractional calculation may thus be avoided is evident from the fact that the system will be homogeneous. Thus, as binary gradation supplies one coin for every binary division of the dollar, down to the sixty-fourth part, and farther, if necessary, any of those divisions may be paid without a remainder. On the contrary, Federal gradation, though in part binary, gives one coin for each of the first two divisions only. Of the remaining four divisions, one requires two coins, and another three, and not one of them can be paid in full. Thus it appears there are four divisions of the dollar that cannot be paid in Federal coins, divisions that are constantly in use, and unavoidable, because resulting from the natural division of things, and from the popular division of the pound, gallon, yard, inch, etc., that has grown out of it. Those fractious that cannot be paid, the proper result of a heterogeneous system, are a constant source of jealousy, and often produce disputes, and sometimes bitter wrangling, between buyer and seller. The injury to public morals arising from this cause, like the destructive effect of the constant dropping of water, though too slow in its progress to be distinctly traced, is not the less certain. The economic value of binary gradation is, in the aggregate, immense; yet its moral value is not to be overlooked, when a full estimate of its worth is required.

Admitting binary gradation to be proper to weights, measures, and coins, it follows that a corresponding base of numeration and notation must be provided, as that best suited to commerce. For this purpose, the number two immediately presents itself; but binary numeration and notation being too prolix for arithmetical practice, it becomes necessary to select for a base a power of two that will afford a more comprehensive notation: a power of two, because no other number will agree with binary gradation. It is scarcely proper to say the third power has been selected, for there was no alternative,—the second power being too small, and the fourth too large. Happily, the third is admirably suited to the purpose, combining, as it does, the comprehensiveness of eight with the simplicity of two.

It may be asked, how a number, hitherto almost entirely overlooked as a base of numeration, is suddenly found to be so well suited to the purpose. The fact is, the present base being accepted as proper for numeration, however erroneously, it is assumed to be proper for gradation also; and a very flattering assumption it is, promising a perfectly homogeneous system of weights, measures, coins, and numbers, than which nothing can be more desirable; but, siren-like, it draws the mind away from a proper investigation of the subject, and the basic qualities of numbers, being unquestioned, remain unknown. When the natural order is adopted, and the base of gradation is ascertained by its adaptation to things, and the base of numeration by its agreement with that of gradation, then, the basic qualities of numbers being questioned, two is found to be proper to the first use, and eight to the second.

The idea of changing the base of numeration will appear to most persons as absurd, and its realization as impossible; yet the probability is, it will be done. The question is one of time rather than of fact, and there is plenty of time. The diffusion of education will ultimately cause it to be demanded. A change of notation is not an impossible thing. The Greeks changed theirs, first for the alphabetic, and afterwards, with the rest of the civilized world, for the Arabic,—both greater changes than that now proposed. A change of numeration is truly a more serious matter, yet the difficulty may not be as great as our apprehensions paint it. Its inauguration must not be compared with that of French gradation, which, though theoretically perfect, is practically absurd.

Decimal numeration grew out of the fact that each person has ten fingers and thumbs, without reference to science, art, or commerce. Ultimately scientific men discovered that it was not the best for certain purposes, consequently that a change might be desirable; but as they were not disposed to accommodate themselves to popular practices, which they erroneously viewed, not as necessary consequences, but simply as bad habits, they suggested a base with reference not so much to commerce as to science. The suggestion was never acted on, however; indeed, it would have been in vain, as Delambre remarks, for the French commission to have made the attempt, not only for the reason he presents, but also because it does not agree with natural division, and is therefore not suited to commerce; neither is it suited to the average capacity of mankind for numbers; for, though some may be able to use duodecimal numeration and notation with ease, the great majority find themselves equal to decimal only, and some come short even of that, except in its simplest use. Theoretically, twelve should be preferred to ten, because it agrees with circle measure at least, and ten agrees with nothing; besides, it affords a more comprehensive notation, and is divisible by 6, 4, 3, and 2 without a fraction, qualities that are theoretically valuable.

At first sight, the universal use of decimal numeration seems to be an argument in its favor. It appears as though Nature had pointed directly to it, on account of some peculiar fitness. It is assumed, indeed, that this is the case, and habit confirms the assumption; yet, when reflection has overcome habit, it will be seen that its adoption was due to accident alone,—that it took place before any attention was paid to a general system, in short, without reflection,—and that its supposed perfection is a mere delusion; for, as a member of such a system, it presents disagreements on every hand; as has been said, it has no agreement with anything, unless it be allowable to say that it agrees with the Arabic mode of notation. This kind of agreement it has, in common with every other base. It is this that gives it character. On this account alone it is believed by many to be the perfection of harmony. They get the base of numeration and the mode of notation so mingled together, that they cannot separate them sufficiently to obtain a distinct idea of either; and some are not conscious that they are distinct, but see in the Arabic mode nothing save decimal notation, and attribute to it all those high qualities that belong to the mode only. The Arabic mode is an invention of the highest merit, not surpassed by any other; but the admiration that belongs to it is thus bestowed upon a quite commonplace idea, a misapplication, which, in this as in many other cases, arises from the fact, that it is much easier to admire than to investigate. This result of carelessness, if isolated, might be excused; but all errors are productive, and it should be remembered that this one has produced that extraordinary perversion of truth to be found in the reply to the question, How is all this confusion to be brought into harmony? It has produced it not only in words, but in deed. Was it not this reply that led the French commission to extend the use of the present base from numeration to gradation also, under the delusive hope of producing a perfectly homogeneous system, that would be practical also? Was it not under its influence, that, adhering to the base to which the world had been so long accustomed, instead of attempting to regulate ideal division by real, which might have led to the adoption of the true base and a practical system, they committed the one great error of endeavoring to reverse true order, by forcing real division into conformity with a preconceived ideal? This attempt was made at a time supposed by many to be peculiarly suited to the purpose, a time of changes. It was a time of changes, truly; but these were the result of high excitement, not of quiet thought, such as the subject requires,—a time for rushing forward, not for retracing misguided steps. Accordingly, a system was produced which from its magnitude and importance was truly imposing, and which, to the present day, is highly applauded by all those who, under the influence of the error alluded to, conceive decimal numeration to be a sacred truth: applauded, not because of its adaptation to commerce, but simply because of its beautiful proportions, its elegant symmetry, to say nothing of the array of learning and power engaged in its production and inauguration: imposing, truly, and alike on its authors and admirers; for the qualities they so much admire are not peculiar to the decimal base, but to the use of one and the same base for numeration, notation, and gradation. But if the base ten agrees with nothing, over, on, or under the earth, can it be the best for scientific use? can it be at all suited to commercial purposes? If true order is the object to be attained, and that for the sake of its utility, then agreement between real and ideal division is the one thing needful, the one essential change without which all other changes are vain, the only change that will yield the greatest good to the greatest number,—a change, which, as volition is with the ideal, and inertia with the real, can be attained only by adaptation of the ideal to the real.

A full investigation of the existing heterogeneous or fragmentary system will lead to the discovery that it contains two elements which are at variance with natural division and with each other, and that the unsuccessful issue of every attempt at regulation hitherto made has been the proper result of the mistake of supposing agreement between those elements to be a possible thing.

The first element of discord to be considered is the division of things by personal proportion, as by fathom, yard, cubit, foot, etc. It is obvious at a glance, that these do not agree with binary division, nor with decimal, nor yet with each other. It is this element that has suggested the duodecimal base, to which some adhere so tenaciously, apparently because they have not ascertained the essential quality of a base.

The second is the numeration of things by personal parts, as fingers, hands, etc.,—suggesting a base of numeration that has no agreement with the binary, nor with personal proportion, neither can it have with any proper general system. Are there any things in Nature that exist by tens, that associate by tens, that separate into tenths? Are there any things that are sold by tens, or by tenths? Even the fingers number eight, and, had there been any reflection used in the adoption of a base of numeration, the thumbs would not have been included. The ease with which the simplest arithmetical series may be continued led our fathers quietly to the adoption, first, of the quinary, and second, of the decimal group; and we have continued its use so quietly, that its propriety has rarely been questioned; indeed, most persons are both surprised and offended, when they hear it declared to be a purely artificial base, proper only to abstract numbers.

The binary base, on the contrary, is natural, real, simple, and accords with the tendency of the mind to simplify, to individualize. In business, who ever thinks of a half as two-fourths, or three-sixths, much less as two-and-a-half-fifths, or three-and-a-half-sevenths? For division by two produces a half at one operation; but with any other divisor, the reduction is too great, and must be followed by multiplication. Think of calling a half five-tenths, a quarter twenty-five-hundredths, an eighth one-hundred-and-twenty-five-thousandths! Arithmetic is seldom used as a plaything. It generally comes into use when the mind is too much occupied for sporting. Consequently, the smallest divisor that will serve the purpose is always preferred. A calculation is an appendage to a mercantile transaction, not a part of the transaction itself; it is, indeed, a hindrance, and in large business is performed by a distinct person. But even with him, simplicity, because necessary to speed, is second in merit only to correctness.

The binary base is not only simple, it is real. Accordingly, it has large agreement with the popular divisions of weights, etc. Grocers' weights, up to the four-pound piece, and all their measures, are binary; so are the divisions of the yard, the inch, etc.

It is not only simple and real, it is natural. On every hand, things may be found that are duplex in form, that associate in pairs, that separate into halves, that may be divided into two equal parts. Things are continually sold in pairs, in halves, and in quantities produced by halving.

The binary base, therefore, is here proposed, as the only proper base for gradation; and the octonal, as the true commercial base, for numeration and notation: two bases which in combination form a binoctonal system that is at once simple, comprehensive, and efficient.