As the numbers proceed from the monad, so he regarded the pure and simple essence of the Deity as the source of all the forms of nature.  Gods, demons, and heroes are emanations of the Supreme, and there is a fourth emanation, the human soul.  This is immortal, and when freed from the fetters of the body passes to the habitation of the dead, where it remains till it returns to the world, to dwell in some other human or animal body, and at last, when sufficiently purified, it returns to the source from which it proceeded.  This doctrine of the transmigration of souls (metempsychosis), which was originally Egyptian and connected with the doctrine of reward and punishment of human actions, was the chief cause why the Pythagoreans killed no animals.  Ovid represents Pythagoras addressing his disciples in these words:  “Souls never die, but always on quitting one abode pass to another.  I myself can remember that in the time of the Trojan war I was Euphorbus, the son of Panthus, and fell by the spear of Menelaus.  Lately being in the temple of Juno, at Argos, I recognized my shield hung up there among the trophies.  All things change, nothing perishes.  The soul passes hither and thither, occupying now this body, now that, passing from the body of a beast into that of a man, and thence to a beast’s again.  As wax is stamped with certain figures, then melted, then stamped anew with others, yet is always the same wax, so the soul, being always the same, yet wears, at different times, different forms.  Therefore, if the love of kindred is not extinct in your bosoms, forbear, I entreat you, to violate the life of those who may haply be your own relatives.”

Shakspeare, in the “Merchant of Venice,” makes Gratiano allude to the metempsychosis, where he says to Shylock: 

   “Thou almost mak’st me waver in my faith,
    To hold opinion with Pythagoras,
    That souls of animals infuse themselves
    Into the trunks of men; thy currish spirit
    Governed a wolf; who hanged for human slaughter
    Infused his soul in thee; for thy desires
    Are wolfish, bloody, starved and ravenous.”

The relation of the notes of the musical scale to numbers, whereby harmony results from vibrations in equal times, and discord from the reverse, led Pythagoras to apply the word “harmony” to the visible creation, meaning by it the just adaptation of parts to each other.  This is the idea which Dryden expresses in the beginning of his “Song for St. Cecilia’s Day”: 

   “From harmony, from heavenly harmony
    This everlasting frame began;
    From harmony to harmony
    Through all the compass of the notes it ran,
    The Diapason closing full in Man.”

In the centre of the universe (he taught) there was a central fire, the principle of life.  The central fire was surrounded by the earth, the moon, the sun, and the five planets.  The distances of the various heavenly bodies from one another were conceived to correspond to the proportions of the musical scale.  The heavenly bodies, with the gods who inhabited them, were supposed to perform a choral dance round the central fire, “not without song.”  It is this doctrine which Shakspeare alludes to when he makes Lorenzo teach astronomy to Jessica in this fashion: