248. That nearly all tables of numbers which follow any law,
however complicated, may be formed, to a greater or less extent,
solely by the proper arrangement of the successive addition and
subtraction of numbers befitting each table, is a general
principle which can be demonstrated to those only who are well
acquainted with mathematics; but the mind, even of the reader who
is but very slightly acquainted with that science, will readily
conceive that it is not impossible, by attending to the following
example.
The subjoined table is the beginning of one in very extensive
use, which has been printed and reprinted very frequently in many
countries, and is called a table of square numbers.
Terms of Table A Table B first Difference C second Difference
1 1
3
2 4 2
5
3 9 2
7
4 16 2
9
5 25 2
11
6 36 2
13
7 49
Any number in the table, column A, may be obtained, by
multiplying the number which expresses the distance of that term
from the commencement of the table by itself; thus, 25 is the
fifth term from the beginning of the table, and 5 multiplied by
itself, or by 5, is equal to 25.
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