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The planar figure of two triangles being in perspective position is associated with a theorem of Desargues, which is responsible for the fact that the coordinate set of the plane is a field. This well-known theorem of Desargues allows the obvious interpretation in space of a three-sided pyramid, which is intersected by two planes. The article is dedicated to generalizing this theorem of Desargues to perspective simplices in a projective n-space and their linear images in a subspace. Hereby one can find remarkable incidences and configurations. Starting point to this investigation is a planar figure of perspective quadrangles, where the second author discovered a remarkable coincidence figure. The proof of this incidence statement is based on an interpretation of the planar 'Ebisui figure' as the central projection of a (projective) cross-polytope in 4-space. This principle can also be extended to higher dimensions.