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.
Almanac
One etymology report says "The ultimate source of the word is obscure. Its first syllable, al-, and its general relevance to medieval science and technology, strongly suggest an Arabic origin, but no convincing candidate has been found". Another report similarly says of "almanac": "First seen in Roger Bacon. Apparently from Spanish Arabic, "al-manakh", but this is not an Arabic word...The word remains a puzzle." The "Oxford English Dictionary" similarly says "the word has no etymon in Arabic" but indirect circumstantial evidence "points to a Spanish Arabic "al-manākh"".