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, 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.
MY LAST LOVE
I had counted many more in my girlhood, in the first flush of blossoming,and a few, good men and true, whom I never meet even now without an added color; for, at one time or another, I thought I loved each of them.
"Why didn't I marry them, then?"
For the same reason that many another woman does not. We are afraid to trust our own likings. Too many of them are but sunrise vapors, very rosy to begin with, but by mid-day as dingy as any old dead cloud with the rain all shed out of it. I never see any of those old swains of mine, without feeling profoundly thankful that I don't belong to him. I shouldn't want to look over my husband's head in any sense. So they all got wives and children, and I lived an old maid,although I was scarcely conscious of the state; for, if my own eyes or other people's testimony were to be trusted, I didn't look old, and I'm quite sure I didn't feel so. But I came to myself on my thirty-second birthday, an old maid most truly, without benefit of clergy. And thereby hangs this tale; for on that birthday I first made acquaintance with my last love.
Something like a month before, there had come to Huntsville two gentlemen in search of game and quiet quarters for the summer. They soon found that a hotel in a country village affords little seclusion; but the woods were full of game, the mountain-brooks swarmed with trout too fine to be given up, and they decided to take a house of their own. After some search, they fixed on an old house, (I've forgotten whose "folly" it was called,) full a mile and a half from town, standing upon a mossy hill that bounded my fields, square and stiff and weather-beaten, and without any protection except a ragged pine-tree that thrust its huge limbs beneath the empty windows, as though it were running away with a stolen house under its arm. The place was musty, rat-eaten, and tenanted by a couple of ghosts, who thought a fever, once quite fatal within the walls, no suitable discharge from the property, and made themselves perfectly free of the quarters in properly weird seasons. But money and labor cleared out all the cobwebs, (for ghosts are but spiritual cobwebs, you know,) and the old house soon wore a charming air of rustic comfort.
I used to look over sometimes, for it was full in view from my chamber-windows, and see the sportsmen going off by sunrise with their guns or fishing-rods, or lying, after their late dinner, stretched upon the grass in front of the house, smoking and reading. Sometimes a fragment of a song would be dropped down from the lazy wings of the south wind, sometimes a long laugh filled all the summer air and frightened the pinewood into echoes, and, altogether, the new neighbors seemed to live an enviable life. They were very civil people, too; for, though their nearest path out lay across my fields, and close by the doorway, and they often stopped to buy fruit or cream or butter, we were never annoyed by an impertinent question or look. Once only I overheard a remark not altogether civil, and that was on the evening before my birthday. One of them, the elder, said, as he went away from my house with a basket of cherries, that he should like to get speech with that polyglot old maid, who read, and wrote, and made her own butter-pats. The other answered, that the butter was excellent at any rate, and perhaps she had a classical cow; and they went down the lane laughingly disputing about the matter, not knowing that I was behind the currant-bushes.
"Polyglot old maid!" I thought, very indignantly, as I went into the house. "I've a mind not to sell them another cake of my butter. But I wonder if people call me an old maid. I wonder if I am one."
I thought of it all the evening, and dreamt of it all night, waking the next morning with a new realization of the subject. That first sense of a lost youth! How sharp and strong it comes! That suddenly opened north door of middle life, through which the winter winds rush in, sweeping out of the southern windows all the splendors of the earlier time; it is like a sea-turn in late summer. It has seemed to be June all along, and we thought it was June, until the wind went round to the east, and the first red leaf admonished us. By-and-by we close, as well as we may, that open door, and look out again from the windows upon blooms, beautiful in their way, to which some birds yet sing; but, alas! the wind is still from the east, and blows as though, far away, it had lain among icebergs.
So I mused all the morning, watering the sentiment with a bit of a shower out of my cloud; and when the shadows turned themselves, I went out to see how old age would look to me in the fields and woods. It was a delicious afternoon, more like a warm dream of hay-making, odorous, misty, sleepily musical, than a waking reality, on which the sun shone. Tremulous blue clouds lay down all around upon the mountains, and lazy white ones lost themselves in the waters; and through the dozing air, the faint chirp of robin or cricket, and ding of bells in the woods, and mellow cut of scythe, melted into one song, as though the heart-beat of the luscious midsummer-time had set itself to tune.
I walked on to loiter through the woods. No dust-brush for brain or heart like the boughs of trees! There dwells a truth, and pure, strong health within them, an ever-returning youth, promising us a glorious leafage in some strange spring-time, and a symmetry and sweetness that possess us until our thoughts grow skyward like them, and wave and sing in some sunnier strata of soul-air. In the woods I was a girl again, and forgot the flow of the hours in their pleasant companionship. I must have grown tired and sat down by a thicket of pines to rest, though I have forgotten, and perhaps I had fallen asleep; for suddenly I became conscious of a sharp report, and a sharper pain in my shoulder, and, tearing off my cape, I found the blood was flowing from a wound just below the joint. I remember little more, for a sudden faintness came over me; but I have an indistinct remembrance of people coming up, of voices, of being carried home, and of the consternation there, and long delay in obtaining the surgeon. The pain of an operation brought me fully to my senses; and when that was over, I was left alone to sleep, or to think over my situation at leisure. I'm afraid I had but little of a Christian spirit then. All my plans of labor and pleasure spoiled by this one piece of carelessness! to call it by the mildest term. All those nice little fancies that should have grown into real flesh-and-blood articles for my publisher, hung up to dry and shrivel without shape or comeliness! The garden, the dairy, the new bit of carriage-way through the beeches,my pet scheme,the new music, the sewing, all laid upon the shelf for an indefinite time, and I with no better employment than to watch the wall-paper, and to wonder if it wasn't almost dinner- or supper-time, or nearly daylight! To be sure, I knew and thought of all the improving reflections of a sick-room; but it was much like a mild-spoken person making peace among twenty quarrelsome ones. You can see him making mouths, but you don't hear a word he says.