No. T D^1 D^2 D^3 D^4
1 1 15 50 60 24
2 16 65 110 84 24
3 81 175 194 108 24
4 256 369 302 132 24
5 625 671 434 156 24
6 1296 1105 590 180 24
7 2401 1695 770 204 24
8 4096 2465 974 228 24
9 6561 3439 1202 252 24
10 10000 4641 1454 276
11 14641 6095 1730
12 20736 7825
13 28561
There are numerous tables in which, as already stated, to whatever order
of differences we may proceed, we should not obtain a series of
rigorously constant differences; but we should always obtain a certain
number of differences which to a given number of decimal places would
remain constant for a long succession of terms. It is plain that such a
table might be calculated by addition in the same manner as those which
have a difference rigorously and continuously constant; and if at every
point where the last difference requires an increase, that increase be
given to it, the same principle of addition may again be applied for a
like succession of terms, and so on.
By this principle it appears, that all tables in which each series of
differences continually increases, may be produced by the operation of
addition alone; provided the first terms of the table, and of each
series of differences, be given in the first instance. But it sometimes
happens, that while the table continually increases, one or more
serieses of differences may continually diminish. In this case, the
series of differences are found by subtracting each term of the series,
not from that which follows, but from that which precedes it; and
consequently, in the re-production of the several serieses, when their
first terms are given, it will be necessary in some cases to obtain them
by _addition_, and in others by _subtraction_. It is possible, however,
still to perform all the operations by addition alone: this is effected
in performing the operation of subtraction, by substituting for the
subtrahend its _arithmetical complement_, and adding that, omitting the
unit of the highest order in the result. This process, and its
principle, will be readily comprehended by an example. Let it be
required to subtract 357 from 768.