I've been doing some work on "neural" modeling based on some old
mathematics, the mathematics of finite differences.
In 1821, Charles Babbage used this mathematics to build his "Difference
Engine #2," a forerunner of the modern computer.
Bromley, Allan G. (1982) "Introduction" in BABBAGE'S CALCULATING
ENGINES (a collection of papers) Tomash Publishers, Los Angeles
and San Francisco.
Babbage based his machine on finite differences because "mathematicians
have discovered that all the Tables most important for practical purposes,
such as those related to Astronomy and Navigation can ... be calculated ...
by that method."
Babbage, Charles (1822) quoted by Cherfas, J. in Science, 7 June,
1991, page 1371.
Menabrea, L.F. (1842) 'Sketch of the Analytical Engine Invented by
Charles Babbage, Esq." in BABBAGE'S CALCULATING ENGINES: A COLLECTION
OF PAPERS Tomash Publishers, Los Angeles and San Francisco, 1982.
An example of Babbage's finite difference table organization is
shown below. (Legendre's organization was the same as Babbage's.)
This organization is embodied in the mechanical linkages and dials of
"Difference Engine #2."
Legendre-Babbage formulation:
x x4 D1 D2 D3 D4
1 1 15 50 60 24
2 16 65 110 84 24
3 81 175 194 108 24
4 256 369 302 132 24
5 625 671 434 156 24
6 1296 1105 590 180 24
7 2401 1695 770 204 24
8 4096 2465 974 228 24
Other array examples, generated by taking differences of the series
1, 2, 3, 4, 5, 6, 8 ... to the various integer powers, can be
constructed, and are useful. I've found that these arrays can be
modified to produce "signatures" of polynomial curves that can be
quickly encoded and decoded, that are adapted for various manipulations
that I hope may be of interest in explaining brain function. I set
that out in a biologically-cryptologically oriented draft
http://www.wisc.edu/rshowalt/pap2/ .
So I've traced these finite difference calculations back to the
Napoleonic era. I've seen a reference that says that this math was
used by teams under the great M. Legendre to construct many mathematical
tables of high accuracy in the time of Napoleon. (I'd love to know
more about these teams.)
I'm sure these finite difference arrays must have a history that
goes back farther. It seems likely that finite differencing arrays
for calculating polynomials must have been familiar objects for many
mathematicians in times past. How might I trace the matter back
farther? What resources might I use? Could someone please tell
me more about the history of these finite difference arrays?
Thank you,
Bob Showalter