Let x₀, x₁, x₂,…, x_n be equally spaced values, so that x_i = x₀ + ih, for i = 1, 2,…, n. Suppose that the values f₀, f₁, f₂,…, f_n are known, where f_i = f(x_i), for some function f. The Gregory–Newton forward difference formula is a formula involving finite differences that gives an approximation for f(x), where x = x₀ + θ‎h, and 0<θ‎<1. It states that

the series terminating at some stage. The approximation f(x) ≈ f₀ + θ‎Δ‎ f₀ gives the result of linear interpolation. Terminating the series after one more term provides an example of quadratic interpolation.