Suppose we listen to the note emitted by a syren which is sounding with slowly increasing loudness but with a pitch which remains constant. We do not notice at first that the sound is becoming louder, but after a little time we do notice a difference. Let us call the amplitude of vibration of the air when the syren first sounds _E_, and then, when we notice a difference, let us call the amplitude Δ_E_ + _E_, Δ_E_ being the increment of amplitude. Let us call our sensation when the syren first sounds _S_, and our sensations when the sound has become louder _S_ + Δ_S_, Δ_S_ being the “increment of sensation.” Then the relation holds:-- Δ_E_/_E_ = constant. That is to say, the louder is the sound the greater must be the increase of loudness before we notice a difference. Let us assume now that the successive sensations of loudness that we receive as the syren blows louder and louder are, each of them, just the same amount louder than the preceding sound; that is to say, let us assume that what we experience are “minimal perceptible differences” of sensation--that they are “elements of loudness”--thus we construct a series of sounds each of which differs from that preceding it by an elemental increment of loudness. Now things that cannot be further decomposed are necessarily equal to each other; if, for instance, the atoms represent the ultimate units into which we break up the matter called oxygen, then these atoms are all equal to each other. Therefore the increments of loudness are equal to each other. [Illustration: FIG. 3.] If we plot these equal increments of loudness as the dependent variable _S_ in a graph, and the amplitude of the vibrations of the atmosphere as the independent variable _E_, we can obtain the following curve. If we investigate this we shall find that a certain relation exists between the “values” of the sensation and the values of the stimuli that correspond to them; a regular increase in the loudness of the sensation corresponds to a regular increase in the logarithms of the strength of the stimuli. Let S = the sensation, _E_ the stimulus, and _C_ and _Q_ constants; then _S_ = _C_ log(_E_/_Q_); so that we seem to establish a mathematical relation between the intensity of our sensations and the intensity of the stimuli that give rise to those sensations, but this relation depends on the assumption that what we call “minimal perceptible differences” of sensation are numerical differences that are equal to each other, and this is, of course, an assumption that cannot possibly be proved.