Etymology
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prime (n.)

"earliest canonical hour of the day" (6 a.m.), from Old English prim and Old French prime and directly from Medieval Latin prima "the first service," from Latin prima hora "the first hour" (of the Roman day), from Latin primus "first, the first, first part" (see prime (adj.)).  (In classical Latin, the noun uses of the adjective meant "first part, beginning; leading place.")

By extension, "the first division of the day" 6 a.m. to 9 a.m. (early 13c.). The sense of "beginning of a period or course of events" is from late 14c. From the notion of "the period or condition of greatest vigor in life" (by 1530s) comes the specific sense "springtime of human life" (often meaning ages roughly 21 to 28) is from 1590s. Also from 1590s as "that which is best in quality, highest or most perfect state of anything." As "a prime number," by 1530s.

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prime (v.)

"to fill, charge, load" (a weapon, before firing), 1510s, probably from prime (adj.). General sense of "perform the first operation on, prepare (something, especially wood, etc., for painting)" is from c. 1600. To prime a pump (1769) meant to pour water down the tube, which saturated the sucking mechanism and made it draw up water more readily. Related: Primed; priming.

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prime (adj.)

late 14c., "first, original, first in order of time," from Old French prime and directly from Latin primus "first, the first, first part," figuratively "chief, principal; excellent, distinguished, noble" (source also of Italian and Spanish primo), from Proto-Italic *prismos, superlative of PIE *preis- "before," from root *per- (1) "forward," hence "in front of, before, first, chief."

The meaning "of fine quality, of the first excellence" is from c. 1400. The meaning "first in rank, degree, or importance" is from 1610s in English. Arithmetical sense (as in prime number, one indivisible without a remainder except by 1) is from 1560s; prime meridian "the meridian of the earth from which longitude is measured, that of Greenwich, England," is from 1878. Prime time originally (c. 1500) meant "spring time;" the broadcasting sense of "peak tuning-in period" is attested by 1961.

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life (n.)

Old English life (dative lif) "animated corporeal existence; lifetime, period between birth and death; the history of an individual from birth to death, written account of a person's life; way of life (good or bad); condition of being a living thing, opposite of death; spiritual existence imparted by God, through Christ, to the believer," from Proto-Germanic *leiban (source also of Old Norse lif "life, body," Old Frisian, Old Saxon lif "life, person, body," Dutch lijf "body," Old High German lib "life," German Leib "body"), properly "continuance, perseverance," from PIE root *leip- "to stick, adhere."

The noun associated with live (v.) "to live," which is literally "to continue, remain." Extended 1703 to inanimate objects, "term of duration or existence." Sense of "vitality, energy in action, expression, etc." is from 1580s. Meaning "conspicuously active part of human existence, pleasures or pursuits of the world or society" is by 1770s. Meaning "cause or source of living" led to the sense "vivifying or animating principle," and thus "one who keeps things lively" in life of the party (1787). Meaning "imprisonment for life, a life sentence" is from 1903. Paired alliteratively with limb from 1640s. Not on your life "by no means" is attested from 1896.

In gaming, an additional turn at play for a character; this transferred use was prefigured by uses in card-playing (1806), billiards (1856), etc., in reference to a certain number of chances or required objects without which one's turn at the game fails. The life "the living form or model, semblance" is from 1590s. Life-and-death "of dire importance" is from 1822; life-or-death (adj.) is from 1897. Life-jacket is from 1840; life-preserver from 1630s of anything that is meant to save a life, 1803 of devices worn to prevent drowning. Life-saver is from 1883, figurative use from 1909, as a brand of hard sugar candy from 1912, so called for shape.

Life-form is from 1861; life-cycle is from 1855; life-expectancy from 1847; life-history in biology from 1870; life-science from 1935. Life-work "the labor to which one's life has been devoted" is from 1848. Expression this is the life is from 1919; verbal shrug that's life is from 1924 (earlier such is life, 1778).

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prime minister 

"leading minister of a government, the chief of the cabinet or ministry," 1640s, see prime (adj.) and minister (n.). Applied to the First Minister of State of Great Britain since 1694.

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still life (n.)
1690s, translating Dutch stilleven (17c); see still (adj.) + life (n.).
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half-life (n.)

also half life, 1864, "unsatisfactory way of living," from half + life; the sense in physics, "amount of time it takes half a given amount of radioactivity to decay" is first attested 1907.

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life-line (n.)
also lifeline, 1700, "rope used to save lives" in any way (especially for the safety of sailors on vessels in bad weather or on the yards), from life (n.) + line (n.); figurative sense first attested 1860. Sense in palmistry from 1890.
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life-saving (adj.)
1829, from life (n.) + present participle of save (v.).
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life of Riley (n.)

"life at ease," by 1902 (as Reilly), popularized in U.S. during World War I; it seems to have been military slang initially, sometimes said to trace to various songs but none of that title has been found.

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