The Atlantic Monthly | Page 4

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