In the case assumed we should be dealing with a two-dimensional continuum or continuous manifold of sounds; for in order to locate a definite sound it would be necessary for us to designate it by two numbers, one specifying its pitch and the other its intensity. We may also notice that whereas in the first case, by removing one of the notes, we were able to cut the manifold of successive pitches into two parts, the removal of one particular note of definite pitch and intensity will now be incapable of effecting this separation. For instance, if we were to remove the note sounded with a definite intensity it would still be possible to pass in a continuous way from any one note of given intensity to any other note of our continuum by circumscribing the missing note; namely, by choosing some route of transfer which would pass through a differing in intensity from that of the we had removed. In the present case, if we wished to divide our two-dimensional manifold into two parts, it would be necessary to remove some one-dimensional continuum of sounds, for example the one-dimensional continuum formed by all the notes of a given pitch , but varying in intensity, or again of given intensity but varying in pitch. If this were done, it would be impossible for us to pass in a continuous way from a note of definite pitch and intensity of our two-dimensional continuum to a note of any other definite pitch and intensity; for we could never get past the removed line of sounds without our ear’s detecting a sudden change. We might complicate matters still further by taking into consideration variations in tonality, as for instance the variation which our ear can detect between two given notes of the same pitch and intensity sounded by two different instruments, such as a violin and an organ. Assuming that every one of our notes of given pitch and intensity in our two-dimensional manifold could also vary in tonality by imperceptible degrees, we should be dealing with a three-dimensional sensory continuum in which every note of given pitch, intensity and tonality could be defined unambiguously by the choice of three numbers. As before, we should find that the removal of a single note of given pitch, intensity and tonality, or even the removal of a one-dimensional continuum of notes such as all those of given intensity and pitch, but varying in tonality, was quite insufficient to effect a separation in our three-dimensional manifold. In the present case we should have [Pg 25] to remove some two-dimensional continuum—say, all notes of given intensity but varying in pitch and tonality. Only then should we have effected a separation between any given element and any other one, rendering it impossible for a continuity of sound impressions to extend between the two elements. By proceeding in this way indefinitely it is obvious that we can conceive of sensory continua of any number of dimensions; there is no need to limit ourselves to three.