"The achievement in pure mathematics for which Hamilton is now best remembered is the invention of quaternions, a linear algebra of rotations in space of three dimensions. Quaternions were the first non-commutative number system to be investigated in detail, and Hamilton's discovery that a consistent and useful system of algebra could be constructed without obeisance to the commutative law of multiplication was comparable in importance to the invention of non-Euclidean geometry." -- PMM 334.