FACTUAL REFERENCE
FINITE DIFFERENCES IN A MECHANICAL ENGINE

The method of finite differences lets polynomial values be generated through
repeated addition once suitable starting differences are known.

That is mechanically attractive because addition can be implemented with columns
of number wheels and carry mechanisms more directly than general multiplication
or division.

Conceptually:

value <- value + first_difference
first_difference <- first_difference + second_difference
...

For a polynomial of fixed degree, a sufficiently high order difference becomes
constant. The machine repeatedly propagates additions through its columns.

Babbage's Difference Engine designs exploited this mathematical structure to turn
complex table generation into a regular mechanical sequence.