"Riemann's fourth proposition is founded on a confusion between conceptual possibility and real or empirical possibility. Conceptual possibility is determined solely by the consistency or inconsistency of the elements of the concept to be formed—it is tested simply by the logical law of non-contradiction; while empirical possibility depends upon the consistency... with the various conditions of sensible reality or... laws of nature. ...Upon this distinction depend the utility and scope of the artifice not unfre quently resorted to in certain analytical investigations of supposing the existence of a fourth spatial dimension for the purpose of reducing certain functions to a symmetrical form and this distinction too is the basis of an observation made by Boole... "Space is presented to us, in perception, as possessing the three dimensions of length, breadth, and depth. But in a large class of problems relating to the properties of curved surfaces, the rotation of solid bodies around axes, the vibration of elastic media, etc., this limitation appears in the analytical investigation to be of an arbitrary character, and, if attention were paid to the processes of solution alone, no reason could be discovered why space should not exist in four, or in any greater number of, dimensions. The intellectual procedure in the imaginary world thus suggested can be apprehended by the clearest light of analogy." Upon the same ground... Hermann Grassmann, who is sometimes referred to as one of the founders of transcendental geometry, has developed the theory of extension in its general application to an indefinite number of dimensions, although he certainly did not cherish the delusion (as seems to be supposed by Victor Schlegel) that this could be the source of inferences respecting the number of actual or empirically possible dimensions of space. On this subject we have Grassmann's own explicit declaration: "It is clear," he says, "that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution.""