Etymology
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equidistant (adj.)
1560s, from French équidistant (14c.), from Late Latin aequidistantem (nominative aequidistans), from aequi- (see equal (adj.)) + distans (see distant). In reference to a type of map projection, from 1866. Related: Equidistance.
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circle (n.)

c. 1300, "figure of a circle, a plane figure whose periphery is everywhere equidistant from its center point," from Old French cercle "circle, ring (for the finger); hoop of a helmet or barrel" (12c.), from Latin circulus "circular figure; small ring, hoop; circular orbit" (also source of Italian cerchio), diminutive of circus "ring" (see circus).

Replaced Old English trendel and hring. Late Old English used circul, from Latin, but only in an astronomical sense. Also used of things felt to be analogous to a circle: The meaning "group of persons surrounding a center of interest" is from 1714 (it also was a secondary sense of Latin circulus); that of "coterie" is from 1640s (a sense also found in Latin circulus).

To come full circle is in Shakespeare. Sense in logic, "inconclusive argument in which unproved statements are used to prove each other" is from 1640s. Meaning "dark mark around or beneath the eyes" is from 1848.

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sphere (n.)
mid-15c., Latinized spelling of Middle English spere (c. 1300) "cosmos; space, conceived as a hollow globe about the world," from Anglo-French espiere, Old French espere (13c., Modern French sphère), from Latin sphaera "globe, ball, celestial sphere" (Medieval Latin spera), from Greek sphaira "globe, ball, playing ball, terrestrial globe," a word of unknown origin.

From late 14c. in reference to any of the supposed concentric, transparent, hollow, crystalline globes of the cosmos believed to revolve around the earth and contain the planets and the fixed stars; the supposed harmonious sound they made rubbing against one another was the music of the spheres (late 14c.). Also from late 14c. as "a globe; object of spherical form, a ball," and the geometric sense "solid figure with all points equidistant from the center." Meaning "range of something, place or scene of activity" is first recorded c. 1600 (as in sphere of influence, 1885, originally in reference to Anglo-German colonial rivalry in Africa).
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