Moebius showed in Moebius (Der Barycentrische Calcul : ein neues Hulfsmittel zur analytis-chen Behandlung der Geometrie, 1827), of 1827, how to precisely define a projective plane as formed of points that can be represented as triples of real numbers. He developed homogeneous coordinates based on physical considerations. He observed that for three fixed points \(A, B, C,\) not collinear, then for any point P in the plane of those points, there are weights, possibly negative, a at A, b at B, and c at C, so the center of mass of that three-body configuration is at P. In other words, \(P = aA + bB + cC\) . And if the triple \((a, b, c)\) produces a center of mass at a point P, then so does \((ka, kb, kc)\) for any non-zero k.