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We generalise Wesson's procedure, whereby vacuum $(4+1)-$dimensional fieldequations give rise to $(3+1)-$dimensional equations with sources, to arbitrarydimensions. We then employ this generalisation to relate the usual$(3+1)-$dimensional vacuum field equations to $(2+1)-$dimensional fieldequations with sources and derive the analogues of the classes of solutionsobtained by Ponce de Leon. This way of viewing lower dimensional gravitytheories can be of importance in establishing a relationship between suchtheories and the usual 4-dimensional general relativity, as well as giving away of producing exact solutions in $(2+1)$ dimensions that are naturallyrelated to the vacuum $(3+1)-$dimensional solutions. An outcome of thiscorrespondence, regarding the nature of lower dimensional gravity, is that theintuitions obtained in $(3+1)$ dimensions may not be automaticallytransportable to lower dimensions.  We also extend a number of physically motivated solutions studied by Wessonand Ponce de Leon to $(D+1)$ dimensions and employ the equivalence between the$(D+1)$ Kaluza-Klein theories with empty $D-$dimensional Brans-Dicke theories(with $\omega=0$) to throw some light on the solutions derived by theseauthors.Comment: 11 pages, latex, published in CQG vol. 12 no. 1

D -dimensional gravity from ( D + 1) dimensions