Hermite's stability criterion of 1854 is proved using matrix algebra in a way which is much more direct than Hermite's own proof. A matrix algebraic interpretation of Hermite's original proof is also given. A by-product of the new proof is a. generalization of Orlando's formula, which relates the determinant of then×nHermite stability matrix to the roots of the underlying characteristic equation. Connections with Liapunov functions and other topics are pointed out.