Etymology
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let (v.)

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense would be "let go through weariness, neglect."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to." To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.

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die (v.)

mid-12c., dien, deighen, of sentient beings, "to cease to live," possibly from Old Danish døja or Old Norse deyja "to die, pass away," both from Proto-Germanic *dawjan (source also of Old Frisian deja "to kill," Old Saxon doian, Old High German touwen, Gothic diwans "mortal"), from PIE root *dheu- (3) "to pass away, die, become senseless" (source also of Old Irish dith "end, death," Old Church Slavonic daviti, Russian davit' "to choke, suffer").

It has been speculated that Old English had *diegan, from the same source, but it is not in any of the surviving texts and the preferred words were steorfan (see starve), sweltan (see swelter), wesan dead ("become dead"), also forðgan and other euphemisms.

Languages usually don't borrow words from abroad for central life experiences, but "die" words are an exception; they often are hidden or changed euphemistically out of superstitious dread. A Dutch euphemism translates as "to give the pipe to Maarten."

Regularly spelled dege through 15c., and still pronounced "dee" by some in Lancashire and Scotland. Of plants, "become devitalized, wither," late 14c.; in a general sense of "come to an end" from mid-13c. Meaning "be consumed with a great longing or yearning" (as in dying to go) is colloquial, from 1709. Used figuratively (of sounds, etc.) from 1580s; to die away "diminish gradually" is from 1670s. To die down "subside" is by 1834. Related: Died; dies.

To die out "become extinct" is from 1865. To die game "preserve a bold, resolute, and defiant spirit to the end" (especially of one facing the gallows) is from 1793. Phrase never say die "don't give up or in" is by 1822; the earliest contexts are in sailors' jargon.

"Never look so cloudy about it messmate," the latter continued in an unmoved tone—"Cheer up man, the rope is not twisted for your neck yet. Jack's alive; who's for a row? Never say die while there's a shot in the locker. Whup;" [Gerald Griffin, "Card Drawing," 1842]
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th 

A sound found chiefly in words of Old English, Old Norse or Greek origin, unpronounceable by Normans and many other Europeans. In Greek, the sound corresponds etymologically to Sanskrit -dh- and English -d-; and it was represented graphically by -TH- and at first pronounced as a true aspirate (as still in English outhouse, shithead, etc.).

But by 2c. B.C.E. the Greek letter theta was in universal use and had the modern "-th-" sound. Latin had neither the letter nor the sound, however, and the Romans represented Greek theta by -TH-, which they generally pronounced, at least in Late Latin, as simple "-t-" (passed down to Romanic languages, as in Spanish termal "thermal," teoria "theory," teatro "theater").

In Germanic languages it represents PIE *-t- and was common at the start of words or after stressed vowels. To represent it, Old English and Old Norse used the characters ð "eth" (a modified form of -d-) and þ "thorn," which originally was a rune. Old English, unlike Old Norse, seems never to have standardized which of the two versions of the sound ("hard" and "soft") was represented by which of the two letters.

The digraph -th- sometimes appears in early Old English, on the Roman model, and it returned in Middle English with the French scribes, driving out eth by c. 1250, but thorn persisted, especially in demonstratives (þat, þe, þis, etc.), even as other words were being spelled with -th-. The advent of printing dealt its death-blow, however, as types were imported from continental founders, who had no thorn. For a time y was used in its place (especially in Scotland), because it had a similar shape, hence ye for the in historical tourist trap Ye Olde _______ Shoppe (it never was pronounced "ye," only spelled that way).

The awareness that some Latin words in t- were from Greek th- encouraged over-correction in English and created unetymological forms such as Thames and author, while some words borrowed from Romanic languages preserve, on the Roman model, the Greek -th- spelling but the simple Latin "t" pronunciation (as in Thomas and thyme).

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