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.
