Etymology
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tropic (n.)
late 14c., "either of the two circles in the celestial sphere which describe the northernmost and southernmost points of the ecliptic," from Late Latin tropicus "of or pertaining to the solstice" (as a noun, "one of the tropics"), from Latin tropicus "pertaining to a turn," from Greek tropikos "of or pertaining to a turn or change; of or pertaining to the solstice" (as a noun, "the solstice," short for tropikos kyklos), from trope "a turning" (from PIE root *trep- "to turn").

The notion is of the point at which the sun "turns back" after reaching its northernmost or southernmost point in the sky. Extended 1520s to the corresponding latitudes on the earth's surface (23 degrees 28 minutes north and south); meaning "region between the tropics of Cancer and Capricorn" is from 1837.
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entropy (n.)

1868, from German Entropie "measure of the disorder of a system," coined 1865 (on analogy of Energie) by German physicist Rudolph Clausius (1822-1888), in his work on the laws of thermodynamics, from Greek entropia "a turning toward," from en "in" (see en- (2)) + trope "a turning, a transformation" (from PIE root *trep- "to turn"). The notion is supposed to be "transformation contents." Related: Entropic.

It was not until 1865 that Clausius invented the word entropy as a suitable name for what he had been calling "the transformational content of the body." The new word made it possible to state the second law in the brief but portentous form: "The entropy of the universe tends toward a maximum," but Clausius did not view entropy as the basic concept for understanding that law. He preferred to express the physical meaning of the second law in terms of the concept of disgregation, another word that he coined, a concept that never became part of the accepted structure of thermodynamics. [Martin J. Klein, "The Scientific Style of Josiah Willard Gibbs," in "A Century of Mathematics in America," 1989]
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trophy (n.)

early 15c., trophe, "an overwhelming victory;" 1510s, "a spoil or prize of war," from Old French trophée (15c.) from Latin trophaeum "a sign of victory, monument," originally tropaeum, from Greek tropaion "monument of an enemy's defeat," noun use of neuter of adjective tropaios "of defeat, causing a rout," from trope "a rout," originally "a turning" (of the enemy); from PIE root *trep- "to turn."

In ancient Greece, spoils or arms taken in battle and set up on the field and dedicated to a god. Figurative extension to any token or memorial of victory is first recorded 1560s. As "a symbolic representation of a classical trophy" from 1630s.

Trophy wife "a second, attractive and generally younger, wife of a successful man who acquires her as a status symbol" was a trending phrase in media from 1988 ("Fortune" magazine did a cover story on it in 1989), but is older in isolated instances.

Variations on this theme ['convenience-wife'] include the HOSTESS-WIFE of a businessman who entertains extensively and seeks  a higher-level, home-branch version of his secretary; the TROPHY-WIFE — the woman who was hard to get because of birth or wealth or beauty — to be kept on exhibition like a mammoth tusk or prime Picasso ... [Phyllis I. Rosenteur, excerpt from "The Single Women," published in Philadelphia Daily News, Dec. 12, 1961]

The excerpt distinguishes the trophy wife from the "showcase wife," "chosen for her pulchritude and constantly displayed in public places, dripping mink and dangling diamonds," which seems more to suit the later use of trophy wife.

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