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Arche (ἀρχή) is a Greek word with primary senses 'beginning', 'origin' or 'first cause' and 'power', 'sovereignty', 'domination' as extended meanings. This list is extended to 'ultimate underlying substance' and 'ultimate undemonstrable principle'. In the language of the archaic period (8th-6th century BC) arche (or archai) designates the source, origin or root of things that exist. If a thing is to be well established or founded, its arche or starting point must be secure, and the most secure foundations are those provided by the gods—the indestructible, immutable and eternal ordering of things. In ancient Greek Philosophy, Aristotle foregrounded the meaning of arche as the element or principle of a thing, which although undemonstrable and intangible in itself, provides the conditions of the possibility of that thing.
Mythical cosmogonies 
In the mythical Greek cosmogony of Hesiod (8th-7th century BC) the origin (arche) of the world is Chaos, an unlimited (formless) void considered as a divine primordial condition, from which everything else appeared. This is described as a large gap without bottom (yawning abyss) where are the roots and the ends of the earth, sky, sea and Tartarus. In the Orphic cosmogony the unageing Chronos produced Aether and Chaos and made in divine Aether a silvery egg, from which everything else appeared.
In the mythological cosmogonies of Near East, the universe is formless and empty and the only existing thing prior to creation was the water abyss. In the Babylonian creation story Enuma Elish the primordial world is described as a watery chaos from which everything else appeared. Something similar is described in Book of Genesis where the spirit of the God is moving upon the dark face of the waters.
In the Hindu cosmology which is similar to the Vedic cosmology in the beginning there was nothing in the Universe but only darkness. The self-manifested being created the primordial waters and established his seed into it. This turned to a golden egg (Hiranyagarbha) from which everything else appeared.
Arche in ancient Greek Philosophy 
The heritage of Greek mythology already embodied the desire to articulate reality as a whole and this universalizing impulse was fundamental for the first projects of speculative theorizing. It appears that the order of 'being' was first imaginatively visualized before it was abstractly thought. In the ancient Greek philosophy, arche is the element and the first principle of existing things. This is considered as a permanent substance or nature (physis) either one or more which is conserved in the generation of rest of it. From this all things first come to be and into this they are resolved in a final state. This source of entity is always preserved. (Aristotle-Metaph.A, 983, b6ff). Anaximander was the first philosopher that used arche for that which writers from Aristotle ownwards called 'the substratum" (Simplicius Phys. 150, 22). The Greek philosophers ascribed to arche divine attributes. It is the divine horizon of substance that encompasses and values all things.
Thales of Miletus (7th-6th century BC), the father of philosophy, claimed that the first principle of all things is water, and considered it as a substance that contains in it motion and change. His theory was supported by the observation of moisture throughout the world and coincided with his theory that the earth floated on water. His ideas were influenced by the Near-Eastern mythological cosmogony and probably by the Homeric statement that the surrounding Oceanus (ocean) is the source of all springs and rivers.
Thales' theory was refuted by his successor and esteemed pupil, Anaximander. Anaximander noted that water could not be the arche because it could not give rise to its opposite, fire. Anaximander claimed that none of the elements (earth, fire, air, water) could be arche for the same reason. Instead, he proposed the existence of the apeiron, an indefinite substance from which all things are born and to which all things will return. Apeiron (endless or boundless) is something completely indefinite and Anaximander was probably influenced by the original chaos of Hesiod (yawning abyss). He probably intended it to mean primarily 'indefinite in kind' but assumed it also to be 'of unlimited extent and duration'. The notion of temporal infinity was familiar to the Greek mind from remote antiquity in the religious conception of immortality and Anaximander's description was in terms appropriate to this conception. This arche is called "eternal and ageless". (Hippolitus I,6,I;DK B2)
Anaximenes, Anaximander's pupil, advanced yet another theory. He returns to the elemental theory, but this time posits air, rather than water, as the arche and ascribes to it divine attributes. He was the first recorded philosopher who provided a theory of change and supported it with observation. Using two contrary processes of rarefaction and condensation (thinning or thickening) he explains how air is part of a series of changes. Rarefied air becomes fire, condensed it becomes first wind, then cloud, water, earth, and stone in order. The arche is technically what underlies all of reality/appearances.
See also 
- Lidell and Scott Lexicon
- Peters Lexicon:1967:23
- Barry Sandywell (1996). Presocratic Philosophy.Vol 3. Routledge New York. p142-144
- The Theogony of Hesiod. Translation H.G.Evelyn White(1914): 116, 736-744 online
- G.S.Kirk,J.E.Raven and M.Schofield (2003). The Presocratic Philosophers. Cambridge University Press. p.24
- William Keith Chambers Guthrie (2000). A History of greek Philosophy. Cambridge University Press. p 58, 59
- Matsya Purana (2.25-30)online; The creation
- Barry Sandywell (1996). Precocratic Philosophy vol.3. Routledge New York. p.28,42
- William Keith Chambers Guthrie (2000). A History of Greek Philosophy. Cambridge University Press. p 55, 77
- <DK 7 B1a.>
- G.S.Kirk, J.E.Raven and M.Schofield (2003). The Pre-socratic Philosophers. Cambridge University Press. p 89, 93, 94
- Simplicius, Comments on Aristotle's Physics (24, 13).<DK 12 A9, B1>
- G.S.Kirk, J.E.Raven and M.Schofield (2003). The Pre-socratic Philosophers. Cambridge University Press. p 110
- William Keith Chambers Guthrie (2000). A History of Greek Philosophy. Cambridge University Press. p 83
- Daniel.W.Graham. The internet Encyclopedia of Philosophy. Anaximenesonline.
- C.S.Kirk, J.E.Raven and M.Schofield (2003). The Pre-socratic Philosophers. Cambridge University Press. p 144