Our next task is to determine how a metrics can be established in a sensory continuum. Consider, for example, a continuous stretch of shades of grey passing from white to black. What do we mean exactly when we say that some definite shade is twice as dark as another? Obviously no definite meaning can be assigned to this statement until we have posited some convention permitting us to establish comparisons. As another instance, take the case of a continuous stream of sounds varying in pitch. What do we mean by saying that some particular musical note is twice as high in pitch as some other, or that the interval between two notes is equal to the interval between two others? If we were to be guided solely by our ear we might assert that as certain musical notes, though differing in pitch, yet appear to present a certain undefinable similarity (the successive octaves), the intervals between these successive similar notes should be considered equal or congruent. We should thus define as equal the extension of notes subtending the successive octaves of a given musical note. But if now we had learnt to measure the frequencies of vibration of the various musical sounds, a new type of measurement would immediately suggest itself. Starting from any musical note—say, the middle of the piano, we should find that the octave of was vibrating twice as fast, the following three times as fast, and the second octave of our original four times as fast. It would then appear plausible to define equal intervals between musical notes by the differences in their rates of vibration, and we should infer that the distance between and its first octave was equal to the distance between this last note and the following , and equal again to the distance between this and the following superoctave . We should thus have obtained a definition of equal stretches of sounds which was at variance with our original definition, in which all octave intervals were regarded as equal or congruent. In view of these conflicting results, we could not well escape the conclusion that a sensory continuum of itself offers us no precise means of defining equal stretches, and that whatever definition we might finally select would be a mere matter of choice, an arbitrarily posited convention. And yet in the case of space, itself a sensory continuum, men have found no difficulty in agreeing on a common system of measurements. As we shall see in the following chapters, the definition of the equality of different stretches of space to which men were unavoidably led was imposed upon them by the behaviour of certain bodies located in space, bodies which were deemed to remain rigid, hence to occupy equal volumes and equal lengths of space wherever they were displaced. For the present, however, we may leave these metrical considerations aside and confine our attention to the general concept of mathematical or geometrical space, which is the subject of study of the pure mathematician.