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 the necessary transposition. Some of them, in their autobiographies, have made, by the way, very suggestive remarks on the importance they attribute to sound; and, moreover, the musical world, with its notes, its intervals, and its orchestration, lives and develops in a manner totally independent of vibration.

Perhaps we can here quote one or two examples which may give us a lead. To measure the length of a body instead of applying to it a yard-wand, one might listen to its sound; for the pitch of the sound given by two cords allows us to deduce their difference of length, and even the absolute length of each. The chemical composition of a body might be noted by its electric resistance and the latter verified by the telephone; that is to say, by the ear. Or, to take a more subtle example. We might make calculations with sounds of which we have studied the harmonic relations as we do nowadays with figures. A sum in rule of three might even be solved sonorously; for, given three sounds, the ear can find a fourth which should have the same relation to the third as the second to the first. Every musical ear performs this operation easily; now, this fourth sound, what else is it but the fourth term in a rule of three? And by taking into consideration the number of its vibrations a numerical solution would be found to the problem. This