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.
Evolutionary ideas of the Renaissance and Enlightenment
The word "evolution" (from the Latin "evolutio", meaning "to unroll like a scroll") appeared in English in the 17th century, referring to an orderly sequence of events, particularly one in which the outcome was somehow contained within it from the start. Notably, in 1677 Sir Matthew Hale, attacking the atheistic atomism of Democritus and Epicurus, used the term "evolution" to describe his opponent's ideas that vibrations and collisions of atoms in the void — without divine intervention — had formed "Primordial Seeds" (semina) which were the "immediate, primitive, productive Principles of Men, Animals, Birds and Fishes." For Hale, this mechanism was "absurd", because "it must have potentially at least the whole Systeme of Humane Nature, or at least that Ideal Principle or Configuration thereof, in the evolution whereof the complement and formation of the Humane Nature must consist ... and all this drawn from a fortuitous coalition of senseless and dead Atoms."