Pythagoras taught lessons of morality through metaphors, he is said to have taught that mortals who during their earthly life had acted immorally had tendencies that were similar to certain animals.
Their punishment in the next life would be to return to earth in the form of the beasts they had grown to resemble. So a dishonest person would return in the form of a rat; an ignorant person in the form of a jackass and so on.
Though it is believed that Pythagoras gave this as an allegory to show metaphorically how human beings become beast like through desire and selfish tendencies. Pythagoras termed this transmigration but it is more commonly referred to as reincarnation, which is a concept that he learned in India.
“He was an important champion of what used to be called the doctrine of metempsychosis, understood as the soul’s transmigration into successive bodies. He himself had been (a) Aethalides, a son of Mercury; (b) Euphorbus, son of Panthus, who perished at the hands of Menelaus in the Trojan war; (c) Hermotimus, a prophet of Clazomenae, a city of Ionia; (d) a humble fisherman; and finally (e) the philosopher of Samos.” (Eliphas Levi, History of Magic, 1913)
Pythagoras taught an allegory of creation where God made each perfect form spiritual seal which left its impression upon the wax of physical form. Each seal was a stamp of dignity to its divine pattern. So he thought that man must claim his destiny by achieving a state where the lower nature of the material self would be discarded in a body of spiritualized ether. This subjection of the physical form would be in unity with the spiritual form of the Antichthon, or the eighth sphere. From this man would ascend into the realm of the immortals, or the astral realm refered to as Yesod in Kabbalistic teachings.
Pythagoras taught that everything in nature was triangular in nature so the wise man should view every problem as being in three parts. Pythagoras said, “Establish the triangle and the problem is two-thirds solved… All things consist of three.” Pythagoras also divided the universe into three parts, which he called the Supreme World, the Superior World, and the Inferior World.
The highest Supreme World exists as a spiritual essence, which contains all existence and manifests its presence into the lower planes of creation. In this world existed the Supreme Deity Monad which is omnipresent, omniactive, omnipotent, and omniscient. This world is based upon the rules of the infinite and transcends space and time.
The Superior World can be best understood in a Neo-Platonic sense as the space where the perfect Platonic forms exist. In a psychological sense the Supreme World can be viewed as Carl Jung’s “collected unconscious” the dwelling place of the archetypes, or in a Pythagorean sense the seals, which exist as casting their shadows upon the Inferior World, or material existence. This is the realm of the mind which the conscious mind interprets the spirit through the unconscious. This world is bound by space and time, but often as we daydream the conscious mind withdraws from the material world and time ceases except for that aspect of ourselves bound in the Inferior World.
The Inferior World is the natural realm based upon the finite existence of material substance. Pythagoras thought that this world is based upon the existence of the mortal gods or the Demiurgi. The Demiurgi are the angels who labor with men but also the demons who work against mankind; and finally mankind, animals, and plants. Pythagoras thought that man existed temporarily of the earth but was capable of rising above that sphere by reason and philosophy. The three worlds were viewed as receptacles to virtues from Monad as they manifested in the lower realms. The first was the receptacle of principles, the second was the receptacle of intelligences, and the third, or lowest, was the receptacle of quantities. It was through mastering these virtues that man could transcend his finite self.
“The Decad, number Ten, or Panteleia, which meant "All complete" or fully accomplished," is the grand summit of numbers, which once reached cannot be passed; to increase the sum we must retrograde to the Monad. The Pythagoreans were entranced with its virtues and called it Deity, Heaven, Eternity and the Sun. being the recipient or receptacle of all numbers was called Decad, from dechomai = to receive, and hence Heaven, which was ordained to receive all men. Like the Deity it is a Circle, with visible centre, but its circumference too vast for sight.
“The symmetrical solids were regarded by Pythagoras, and by the Greek thinkers after him, as of the greatest importance. To be perfectly symmetrical or regular, a solid must have an equal number of faces meeting at each of its angles, and these faces must be equal regular polygons, i. e., figures whose sides and angles are all equal. Pythagoras, perhaps, may be credited with the great discovery that there are only five such solids.” (H. Stanley Redgrove, in Bygone Beliefs.)
The spirit element was also held to be the correspondent between the elementals and to the symmetrical solids. Pythagoras found a more elaborate understanding of the ether element most likely by the Hindus, which refer to it as Akasa. Ether is the inner substance which infuses all of the other elements and acting as a common solvent common denominator and connector of them all.
It is legend that Pythagoras possessed hypnotic power not only over man but also over animals. It is said his mental powers would cause a bird to change the course of its flight, a bear to cease its ravages upon a community, and a bull to change its diet. He was also gifted with second sight, being able to see things at a distance and accurately describe incidents that had not yet come to pass. There is a legend where with his disciples drinking water from a spring and Pythagoras predicted an earthquake which was fulfilled and devastated the local communities. Pythagoras with his higher students practiced divination to predict future events. This divination was most likely a form of arithmomancy or hydromancy that he learned from the Egyptians.
Hydromancy is the divination through the observations of ripples in a bowl of water in a trance like state. It is likely that Pythagoras used a brass bowl; he believed that brass had psychic powers because of how rates of vibration were found in the metal. Pythagoras observed that even when everything was perfectly still and calm there was always a sound of vibration in his brass bowl. It is said that during a hydromancy session he addressed a prayer to the spirit of a river and out of the water arose a voice, which granted him the ability to cause demons to enter into the water and disturb its surface, and by means of the interpretations of the ripples certain things were predicted.