Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.