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In ancient Greek, Daimôn (DYE-moan) can refer to any divinity from the High Gods on down, but Pythagoreans tend to restrict it to the Mediating Spirits between the Gods and us. Some Pythagoreans call the higher orders of Daimones the Angeloi (Messengers), because of Their special role as messengers of the Gods; the lower ranks are the Daimones proper and the Heroes. A fourth class comprises the Akhrantoi (the Immaculate or Undefiled Ones), who are Perfected Beings (including certain Sages), who choose to reincarnate so they can help humanity. All together, Pythagoreans refer to the Daimones as our "Betters"(Kreittones).
Daimones have an intermediate nature between humans and Gods. All three classes of beings are animate and possess reason. However, the Daimones are like the Gods in being immortal and like humans in having emotions. In contrast, the Gods are impassive, and humans are mortal.
Most of the Daimones reside in the Air, which, in cosmological terms, is the intermediate region between the earth, where we live, and the heavens, where the Gods reside. The Moon is the boundary between the aerial domain and the aetherial heavens, and so that is where Their Ruler, Hekate, has Her domain.
As a consequence of Their intermediate nature, Daimones serve as Mediators between the Gods and us, and They convey divine Providence (Pronoia) into the sublunary realm. In particular, They interpret the Gods' wishes for us and are the agents of divination, oracles, and rituals. They are ministering spirits who care for people, and are often called Savior (Sôtêr). (Recall that daimôn is from the Indo-European da-, to provide.) Many rituals are directed to Daimones (who have feelings and can be swayed), and They are thereby convinced to mediate with the Gods on our behalf.
angels-of-the-sunAccording to Pythagorean doctrine, the Gods reside in the Noetic Realm, the world of Platonic Ideas, so They cannot relate to us as individuals, but only as representatives of the Idea of humankind. The Daimones, however, being intermediate, participate in both the Ideal and material worlds, and so they interact with us as individuals. They know our personal and family histories, our personalities, and often our thoughts.
Each Daimôn is the offspring of a God, who is Their Arkhêgos (Leader, Progenitor, Originator), and thus a Daimôn combines the nature of a God with Its own individual characteristics. Therefore the Seira (Chord, Lineage) of a God includes His or Her Daimones, as well as the Successions of Sages already mentioned. (Nevertheless, all the Daimones are under the rule of Hekate, because Her station at the Gate of the Moon is above all the Daimones; She is Daimoniarkhês, Ruler of Daimones.) A God's Seira also includes various plants, animals, materials, words, etc. that are symbolically or sympathetically linked to the God; these are used in sacred magical operations directed toward the God.