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. 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.

249. Having thus, I hope, thrown some light upon
the theoretical part of the question, I shall endeavour
to shew that the mechanical execution of such an engine,
as would produce this series of numbers, is not so
far removed from that of ordinary machinery as might
be conceived.(3*) Let the reader imagine three clocks,
placed on a table side by side, each having only one
hand, and each having a thousand divisions instead
of twelve hours marked on the face; and every time
a string is pulled, let them strike on a bell the
numbers of the divisions to which their hands point.
Let him further suppose that two of the clocks, for
the sake of distinction called B and C, have some mechanism
by which the clock C advances the hand of the clock
B one division, for each stroke it makes upon its
own bell: and let the clock B by a similar contrivance
advance the hand of the clock A one division, for
each stroke it makes on its own bell. With such
an arrangement, having set the hand of the clock A
to the division I, that of B to *iii*, and that
of C to *ii*, let the reader imagine the repeating
parts of the clocks to be set in motion continually
in the following order: *viz*.—pull
the string of clock A; pull the string of clock B;
pull the string of clock C.