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.
Logistics
The term "" is attested in English from 1846, and is from , where it was either coined or popularized by military officer and writer Antoine-Henri Jomini, who defined it in his "Summary of the Art of War" (). The term appears in the 1830 edition, then titled "Analytic Table" ("Tableau Analytique"), and Jomini explains that it is derived from (cognate to English ), in the terms and :
The term is credited to Jomini, and the term and its etymology criticized by in 1832, writing:
Chambray also notes that the term was present in the "" as a synonym for algebra.