The discovery of these harmonic numbers is due to Pythagoras. It is stated that when passing one day near a forge, he noticed that the hammers gave out very accurate musical concords. He had them weighed, and found that of those which sounded the octave, one weighed twice as much as the other; that of those which made a perfect fifth, one weighed one third more than the other, and in the case of a fourth, one quarter more. After having tried the hammers, he took a musical string stretched with weights, and found that when he had applied a given weight in the first instance to make any particular note, he had to double the weight to obtain the octave, to add one third extra only to obtain a fifth, a quarter for the fourth, and eight for one tone, and about an eighteenth for a half-tone; or more simply still, he stretched a cord once for all, and then when the whole length sounded any note, when stopped in the middle it gave the octave, at the third it gave the fifth, at the quarter the fourth, at the eighth the tone, and at the eighteenth the semi-tone. Since the ancients conceived of the soul by means of motion, the quantity of motion developed in anything was their measure of the quantity of its soul. Now the motion of the heavenly bodies seemed to them to depend on their distance from the centre of the universe, the fastest being those at the circumference of the whole. To determine the relative degrees of velocity, they imagined a straight line drawn outwards from the centre of the earth, as far as the empyreal heaven, and divided it according to the proportions of the musical scale, and these divisions they called the harmonic degrees of the soul of the universe. Taking the earth's radius for the first number, and calling it unity, or, in order to avoid fractions, denoting it by 384, the second degree, which is at the distance of an harmonic third, will be represented by 384 plus its eighth part, or 432. The third degree will be 432, plus its eighth part, or 486. The fourth, being a semitone, will be as 243 to 256, which will give 512; and so on. The eighth degree will in this way be the double of 384 or 768, and represents the first octave. They continued this series to 36 degrees, as in the following table:-- The Earth.