early 15c., maxime, "an axiom, statement of a self-evident truth," from Old French maxime, from Late Latin maxima, shortened from phrases such as maxima propositio (Boethius), maxima sententarium "axiom," literally "greatest or chief premise, greatest among propositions" (one which is general and absolute), from fem. of maximus "greatest," from PIE *mag-samo-, superlative form of root *meg- "great."
The modern meaning "summary statement of an established or accepted proposition serving as a rule or guide, a proposition ostensibly expressing some general truth" is from 1590s.
surname of the Buddha, from Sanskrit Gotamah, properly a patronymic, literally "descendant of the greatest ox," from superlative of gauh "ox, bull, cow."
"the greatest amount, quantity, or degree," 1740, from French maximum and directly from Latin maximum (plural maxima), neuter of maximus "greatest," which is superlative of magnus "great, large, big" (of size), "abundant" (of quantity), "great, considerable" (of value), "strong, powerful" (of force); of persons, "elder, aged," also, figuratively, "great, mighty, grand, important," from PIE *mag-samo-, superlative form of root *meg- "great."