In his letter of 6 January 1679/80 Hooke wrote to Newton ‘‘in truth I agree with You that the Explicating the Curve in which a body Descending to the Center of the Earth, would circumgyrate were a Speculation of noe Use yet’’ [TheCorrespondenceofIsaacNewton, edited by H. W. Turnbull (Cambridge U.P., Cambridge, 1960), Vol. 2, p. 309]. In these words, Hooke referred to a hypothetical problem which he was discussing in a correspondence with Newton. The problem was that of determining the orbit of a body moving inside a narrow cut made centrally across the Earth. This was perhaps the very first inverse central force problem to be attacked by Newton. Newton seems to have used his instantaneous impulse technique to provide a drawing of a solution that was almost exact. This paper explores this fascinating early solution of an inverse central force problem.