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.