Let us now subtract each term of
this table from the next succeeding term, and place the results
in another column (B), which may be called first difference
column. If we again subtract each term of this first difference
from the succeeding term, we find the result is always the number
2, (column C); and that the same number will always recur in that
column, which may be called the second difference, will appear to
any person who takes the trouble to carry on the table a few
terms further. Now when once this is admitted, it is quite clear
that, provided the first term (1) of the table, the first term
(3) of the first differences, and the first term (2) of the
second or constant difference, are originally given, we can
continue the table of square numbers to any extent, merely by
addition: for the series of first differences may be formed by
repeatedly adding the constant difference (2) to (3) the first
number in column B, and we then have the series of numbers, 3, 5,
6, etc.: and again, by successively adding each of these to the
first number (1) of the table, we produce the square numbers.
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