We present a unified derivation of affine invariant convergence results for Newton's method. Initially we derive affine invariant forms of the perturbation lemma and a mean value theorem. With their aid we obtain an optimal radius of convergence for Newton's method, from which further radius of convergence estimates follow. From the Newton-Kantorovitch theorem we derive other estimates of the radius of convergence. We discuss estimation of the parameters in the expressions we have derived.