c. 1200, preove "evidence and argumentation to establish the fact of (something) beyond reasonable doubt," from Anglo-French prove, preove, Old French proeve, prueve "proof, test, experience" (13c., Modern French preuve), from Late Latin proba "a proof," a back-formation from Latin probare "to prove" (see prove). "The devocalization of v to f ensued upon the loss of final e; cf. the relation of v and f in believe, belief, relieve, relief, behove, behoof, etc." [OED].
The meaning "act of proving" is early 14c. The meaning "act of testing or making trial of anything" is from late 14c., from influence of prove. Meaning "standard of strength of distilled liquor" is from 1705, on the notion of "having been tested as to degree of strength." The use in photography is from 1855. The typographical sense of "trial impression to test type" is from c. 1600. The numismatic sense of "coin struck to test a die" is from 1762; now mostly in reference to coins struck from highly polished dies, mainly for collectors.
The adjectival sense "impenetrable, able to resist" (as in proof against) is recorded from 1590s, from the noun in expressions such as proof of (mid-15c.), hence extended senses involving "of tested power against" in compounds such as fireproof (1630s), rust-proof (1690s), waterproof (1725), fool-proof (1902), etc. Shakespeare has shame-proof. Expression the proof is in the pudding (1915) is a curious perversion of earlier proof of the pudding shall be in the eating (1708), with proof in the sense "quality of proving good or turning out well" (17c.).
waterproof outer coat or cloak, 1836, named for Charles Macintosh (1766-1843), inventor of a waterproofing process (patent #4804, June 17, 1823). The Mcintosh type of apple was named for John McIntosh of Upper Canada, who began selling them in 1835. The surname is from Gaelic Mac an toisich "Son of the chieftain." As a name of a type of computer it is attested from 1982.
In an interview with The Wall Street Journal, [President and CEO A.C.] Markkula said "more than three new products" are scheduled to be announced within the next year. Among them will be a high-end, personal business computer code-named "Lisa" and a limited, less expensive business computer called "Mackintosh." [Computerworld, Oct. 18, 1982]
1640s, "course, direction," from Latin ductus "a leading, a conduit pipe," noun use of past participle of ducere "to lead" (from PIE root *deuk- "to lead"). Anatomical sense "vessel of an animal body by which blood, lymph, etc., are conveyed" is from 1660s. Meaning "conduit, channel" is 1713; that of "air tube in a structure" is from 1884.
Duct tape originally was duck tape (1894), long, non-adhesive strips of plain cotton duck cloth used in various mechanical processes; from duck (n.2). The name was transferred to a plastic-coated adhesive tape used by U.S. soldiers in World War II (perhaps in part because it was waterproof). It continued in civilian use after the war, and the name shifted to duct tape by 1958, perhaps because it was frequently used on air ducts, which also accounts for its standard silver-gray color.