The concept of doubling, which was introduced around 1840 by Graves and Hamilton, associates with any quadratic algebraA over a eld k of characteristic not 2 its doubleV(A) =A A with multiplication (w;x)(y;z) = (wy zx;xy +zw). This yields an endofunctor on the category of all quadratic k-algebras which is faithful but not full. We study in which respect the division property of a quadratic k-algebra is preserved under doubling and, provided this is the case, whether the doubles of two non-isomorphic quadratic division algebras are again non-isomorphic. Generalizing a theorem of Dieterich (9) from R to arbitrary square-ordered ground elds k we prove that the division property of a quadratic k-algebra of dimension smaller than or equal to 4 is preserved under doubling. Generalizing an aspect of the celebrated (1,2,4,8)-theorem of Bott, Milnor (4) and Kervaire (21) from R to arbitrary ground elds k of characteristic not 2 we prove that the division property of an 8-dimensional doubled quadratic k-algebra is never preserved under doubling. Finally, we contribute to a solution of the still open problem of classifying all 8-dimensional real quadratic division algebras by extending an approach of Dieterich and Lindberg (12) and proving that, under a mild additional assumption, the doubles of two non-isomorphic 4-dimensional real quadratic division algebras are again non-isomorphic.