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Babbage, Charles, 1792-1871

"On the Economy of Machinery and Manufactures"

It calculates, in three
columns, a table with its first and second differences. Each
column can be expressed as far as five figures, so that these
fifteen figures constitute about one ninth part of the larger
engine. The ease and precision with which it works leave no room
to doubt its success in the more extended form. Besides tables of
squares, cubes, and portions of logarithmic tables, it possesses
the power of calculating certain series whose differences are not
constant; and it has already tabulated parts of series formed
from the following equations:
The third differential of ux = units figur of delta ux
The third differential of ux = nearest whole no. to (1/10,000
delta ux)
The subjoined is one amongst the series which it has calculated:
0 3,486 42,972
0 4,991 50,532
1 6,907 58,813
14 9,295 67,826
70 12,236 77,602
230 15,741 88,202
495 19,861 99,627
916 24,597 111,928
1,504 30,010 125,116
2,340 36,131 139,272
The general term of this is,
ux = (x(x-1)(x-2))/(1 X 2 X 3) + the whole number in x/10 +
10 Sigma^3 (units figure of (x(x-1)/2)

Chapter 21
On the Cost of Each Separate Process in a Manufacture
253.


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