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.