Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Arche
Arche (; ) is a Greek word with primary senses "beginning", "origin" or "source of action" (: from the beginning, οr : the original argument), and later "first principle" or "element", first so used by Anaximander (Simplicius in Ph. 150.23). By extension, it may mean "first place, power", "method of government", "empire, realm", "authorities" (in plural: ), "command". The first principle or element corresponds to the "ultimate underlying substance" and "ultimate undemonstrable principle". In the philosophical language of the archaic period (8th to 6th century BC), "arche" (or "archai") designates the source, origin or root of things that exist. 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.