Etymology
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authority (n.)
Origin and meaning of authority
c. 1200, autorite, auctorite "authoritative passage or statement, book or quotation that settles an argument, passage from Scripture," from Old French autorité, auctorité "authority, prestige, right, permission, dignity, gravity; the Scriptures" (12c.; Modern French autorité), from Latin auctoritatem (nominative auctoritas) "invention, advice, opinion, influence, command," from auctor "master, leader, author" (see author (n.)). Usually spelled with a -c- in English before 16c., when the letter was dropped in imitation of French, then with a -th-, probably by influence of authentic.

From c. 1300 in the general sense "legal validity," also "authoritative book; authoritative doctrine" (opposed to reason or experience); "author whose statements are regarded as correct." From mid-14c. as "right to rule or command, power to enforce obedience, power or right to command or act." In Middle English also "power derived from good reputation; power to convince people, capacity for inspiring trust." From c. 1400 as "official sanction, authorization." Meaning "persons in authority" is from 1610s; Authorities "those in charge, those with police powers" is recorded from mid-19c.
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prove (v.)

c. 1200, prēven, pruven, proven "to try by experience or by a test or standard; evaluate; demonstrate in practice," from Old French prover, pruver "show; convince; put to the test" (11c., Modern French prouver), from Latin probare "to make good; esteem, represent as good; make credible, show, demonstrate; test, inspect; judge by trial" (source also of Spanish probar, Italian probare, and English probe), from probus "worthy, good, upright, virtuous."

This is from PIE *pro-bhwo- "being in front," from *pro-, extended form of root *per- (1) "forward," hence "in front of," + root *bhu- "to be," source also of Latin fui "I have been," futurus "about to be;" Old English beon "to be;" see be.

From early 13c. as "render certain, put out of doubt," also "establish the validity or authenticity of a will, etc." By c. 1300 as "test and find worthy, virtuous, false, etc.," also "find out, discover, ascertain; prove by argument." By mid-14c. as "check the accuracy of." The meaning "be found to be (a hero, coward, etc.) by experience or trial" is by late 14c.

The word had many more senses and broader application in Middle English than Modern English: "to experience; to strive, endeavor; act, accomplish; thrive, succeed." Also in Middle English in a now-obsolete sense of "approve, sanction, praise" (c. 1300; compare approve). Related: Proved; proven; proving. Proving ground "place used for firing cannons for making ballistics tests and testing powder" is by 1837.

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beauty (n.)
Origin and meaning of beauty

early 14c., bealte, "physical attractiveness," also "goodness, courtesy," from Anglo-French beute, Old French biauté "beauty, seductiveness, beautiful person" (12c., Modern French beauté), earlier beltet, from Vulgar Latin *bellitatem (nominative bellitas) "state of being pleasing to the senses" (source also of Spanish beldad, Italian belta), from Latin bellus "pretty, handsome, charming," in classical Latin used especially of women and children, or ironically or insultingly of men, perhaps from PIE *dw-en-elo-, diminutive of root *deu- (2) "to do, perform; show favor, revere." Famously defined by Stendhal as la promesse de bonheur "the promise of happiness."

[I]t takes the one hundred men in ten million who understand beauty, which isn't imitation or an improvement on the beautiful as already understood by the common herd, twenty or thirty years to convince the twenty thousand next most sensitive souls after their own that this new beauty is truly beautiful. [Stendhal, "Life of Henry Brulard"]

Replaced Old English wlite. Concrete meaning "a beautiful woman" in English is first recorded late 14c. Beauty-sleep "sleep before midnight" (popularly regarded as the most refreshing) is attested by 1850. Beauty-spot "dark spot placed on the face formerly by women to heighten beauty" is from 1650s. Beauty-contest is from 1885; beauty-queen is from 1922 (earlier it was a show-name of cattle and hogs). Beauté du diable (literally "devil's beauty") is used as a French phrase in English from 1825.

But as it is hardly possible to define all the properties which constitute beauty, we may observe in general, that beauty consists in whatever pleases the eye of the beholder, whether in the human body, in a tree, in a landscape or in any other object. [from definition of BEAUTY in Noah Webster's "Dictionary of the English Language," 1828]
That pleasure which is at once the most intense, the most elevating, and the most pure is, I believe, found in the contemplation of the beautiful. When, indeed, men speak of Beauty, they mean, precisely, not a quality, as is supposed, but an effect—they refer, in short, just to that intense and pure elevation of soul—not of intellect, or of heart—upon which I have commented, and which is experienced in consequence of contemplating the “beautiful.” [Poe, "The Philosophy of Composition," 1846]
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