Newton's method of differences

If one took the difference between the result of a ploynomial of degree n, and took the difference of the difference and so on for n+1 number of times, the result is 0. This means that the difference at the n level is constant. It is this that allows difference engines to compute large degree ploynomials using addition alone.

Illustrated below with p(x)=2x+1, ploynomial of degree 1:

• p(1)=3
• 5-3=2
  
• p(2)=5
• 7-5=2
  
• p(3)=7
• 9-7=2
  
• p(4)=9

Notice how the first difference is always 2. Simply by knowing the first 2 results, we can compute p(x) for anything integer x.

Cheers,
Steve