One is concerned here with the following problem: Let e3 be a normed linear vector space admitting real scalars. What further conditions placed on the norm of the vectors in e3 implies that the space is a Hilbert space? The type of condition which is envisaged must be specified further. What is desired is to invoke as an additional axiom some elementary metric property of euclidean geometry. In this way, one obtains information about the theorems of this geometry which characterize it. Such theorems will be mentioned presently. It should be pointed out that many other approaches to this problem are possible and that their formulation strongly involves the norm of the space though the manner of this involvement may be somewhat camouflaged. For example, one may require that it be possible to establish an isomorphism between e3 and its conjugate space * such that if f " f* under the isomorphism, then f*(f) = I f* I I f 1. Or one may demand that there be a projection of bound one on every closed linear manifold in B3. Up to the present, two principal results of the specific type which concern us here are known. The first invokes a property of parallelograms whose sides are represented by the vectors f and g while the diagonals are f + g and f g Suppose that