he result of
counting by the five fingers and the two hands, the numbers signified
being the products of continued multiplication by five and by two
alternately. The Romans adhered to their mode, nor is it entirely out of
use at the present day, being revered for its antiquity, admired for its
beauty, and practised for its convenience.

The ancient Greek series corresponded to that of the Romans, though
primarily the signs for 50, 500 and 5000 had no place. Ultimately,
however, those places were supplied by means of compound signs.

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 wel