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.
Organism
The term "organism" (from Greek ὀργανισμός, "organismos", from ὄργανον, "organon", i.e. "instrument, implement, tool, organ of sense or apprehension") first appeared in the English language in 1703 and took on its current definition by 1834 (Oxford English Dictionary). It is directly related to the term "organization". There is a long tradition of defining organisms as self-organizing beings, going back at least to Immanuel Kant's 1790 "Critique of Judgment".