Let V be a set of n points in the Euclidean space X of dimension d. The Johnson-Lindenstrauss theorem states that there is a projection between X a and Y, another Euclidean space of a smaller dimension k, with the property that images of points of X under projection do not differ by more that a multiplicative factor c arbitrarily close to 1. The theorem presents a relation among d, k and c, indicating the possibility of dramatic dimensional reduction of very faithful representations of V. The proof makes use of Grassmanians, the manifolds consisting of subspaces of dimension k in X. In the text, charts are presented, together with a measure which is homogeneous with respect to the natural action of the orthogonal group on the Grassmanian. The result follows by taking estimates using gaussians of certain integrals with a strong geometric flavor.