Briggs was a human calculator, who worked on Napier's logarithm project. He generalized the method of antiquity for computing tables of single-variate dth degree Polynomials
f(x) = a0 + x(a1 + x(a2 + ...+ xad))))....)
Briggs's Idea: Difference Polynomials
f1(x) = f(x + e) - f(x) has degree no more than d - 1
fi(x) = fi-1(x + e) - fi-1(x) has degree no more than d - i. i = 1,...,d
fd(x) is a constant, possibly zero
x := xo repeat print(f(x)) -- d additions and d products using Horner's Rule x +:= e end => Invariants: ti = fi(x) for i = 0,...,d t0 := f(x0) t1 := f1(x0) ... td := fd(x0) repeat print(t0) t0 +:= t1 t1 +:= t2 ... td-1 +:= td -- d additions and 0 products end
Baggage's analytic difference engine was designed to implement the main loop of Briggs's method.
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