Etymology
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figure (n.)

c. 1200, "numeral;" mid-13c., "visible appearance of a person;" late 14c., "visible and tangible form of anything," from Old French figure "shape, body; form of a word; figure of speech; symbol, allegory" (10c), from Latin figura "a shape, form, figure; quality, kind, style; figure of speech," in Late Latin "a sketch, drawing" (from PIE root *dheigh- "to form, build").

Philosophical and scientific senses are from use of Latin figura to translate Greek skhema. Meaning "lines forming a shape" is from mid-14c. From mid-14c. as "human body as represented by art;" late 15c. as "a body, the human form as a whole."  From late 14c. as "a cut or diagram inserted in text."

The rhetorical use of figure, "peculiar use of words giving meaning different from usual," dates to late 14c.; hence figure of speech (1550s). Figure-skating is from 1835, so called for the circular patterns skaters formerly made on the ice to demonstrate control; they were dropped from international competition in 1990, but the name remains. Figure eight as a shape was originally figure of eight (c. 1600).

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figure (v.)

late 14c., "to represent" (in painting or sculpture), "make a likeness," also "to have a certain shape or appearance," from Old French figurer, from Latin figurare "to form, shape" (from PIE root *dheigh- "to form, build"). Meaning "to shape into" is c. 1400; from mid-15c. as "to cover or adorn with figures." Meaning "to picture in the mind" is from c. 1600. Intransitive meaning "make an appearance, make a figure, show oneself" is from c. 1600. Meaning "work out a sum" (by means of arithmetical figures) is from 1833, American English; hence colloquial sense "to calculate upon, expect" (1837). Related: Figured; figuring.

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shadow-figure (n.)

"silhouette," 1851, from shadow (n.) + figure (n.).

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salchow (n.)

type of skating jump, by 1921, named for Swedish figure skater Ulrich Salchow (1877-1949).

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lutz (n.)

type of skating jump, 1932, from the name Alois Lutz, "an obscure Austrian skater of the 1920s" [James R. Hines, "Historical Dictionary of Figure Skating," 2011], who is said to have first performed it in 1913. The surname is from a form of Ludwig.

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ice-skate (v.)

"to glide across a frozen surface on ice-skates," 1690s, from ice (n.) + skate (n.2). The verb usually was skate until the advent of roller-skating mid-18c. made distinction necessary. A run of severe winters that froze over the Thames in the late 17c. made ice-skating popular in England. Related: Ice-skates (1862).

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Choctaw 

native people formerly of southeastern U.S., 1722, from Choctaw (Muskogean) Chahta, of uncertain meaning, but also said to be from Spanish chato "flattened," for the tribe's custom of flattening the heads of male infants. As a figure skating step, first recorded 1892, probably based on Mohawk (it is a variation of that step). Sometimes used in 19c. American English as typical of a difficult or incomprehensible language (compare Greek in this sense from c. 1600).

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skate (v.)

1690s, "to ice-skate," from skate (n.2). U.S. slang sense of "to get away with something" is attested from 1945. Related: Skated; skating. A modern Latinate word for an ice-skating rink is glaciarium (1876).

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pigeon-wing (n.)

1807 as the name of a brisk, fancy step in dancing, skating, etc.; see pigeon + wing (n.).

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Zamboni (n.)

proprietary name of a machine used to resurface ice skating rinks, 1957, trademark of Frank J. Zamboni & Co., Paramount, Calif.

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