Definition
Finite differences is a simple tabular method used to extend series of integer numbers forwards and backwards. Given, for example three numbers a, b, and c, where a < b < c, we can use a finite differences table to find the numbers following c in the series as well as the numbers preceding a in the series with the following table:
a 

b 

c 

d_{1} = b – a 

d_{2} = c – b 



base = d_{2} – d_{1} 


The series' base has special significance in gematria because it is considered the "driving force" of the series.
Many times we consider only the positive values in the resulting series of numbers.
When deriving series in this manner, of note are the 7th and 13th values (either the 7th or 13th overall positive values, or the 7th and 13th from the first number, a).
If we would start with 4 numbers, a, b, c, and d, then we would normally need to use a four row table to arrive at the base of the series. Some series (such as the Covenant Numbers) are overdetermined, i.e., though the Torah provides us with 4 initial values in the series, only 3 are needed to find the base (see Covenant Numbers for more).
External Resources
 Another description of the finitie differences method
 Finite Differences from Mathworld (a bit technical but rigorous)