Consider the income compensation function Y = φ(p, y; p0) where Y, y ∈ R are income levels and p, p0 ∈ Rn are prices. By holding p0 fixed we may interpret this as an n-parameter transformation of R to R. With this interpretation we show that a system of demand functions is additively separable in income and prices if and only if φ is a Lie transformation group on R. Sophus Lie's 1888 classification of such groups into three fundamental types provides an alternative derivation of both the rank 3 condition of Gorman and the additively separable functional forms found by him.