Etymology
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interpolate (v.)
1610s, "to alter or enlarge (a writing) by inserting new material," from Latin interpolatus, past participle of interpolare "alter, freshen up, polish;" of writing, "falsify," from inter "among, between" (see inter-) + polare, which is related to polire "to smoothe, polish," from PIE root *pel- (5) "to thrust, strike, drive," the connecting notion being "to full cloth" [Watkins].

Sense evolved in Latin from "refurbish," to "alter appearance of," to "falsify (especially by adding new material)." Middle English had interpolen (early 15c.) in a similar sense. Related: Interpolated; interpolating.
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interpolator (n.)
1650s, from Late Latin interpolator "one who corrupts or spoils," agent noun from past participle stem of Latin interpolare "to polish; to alter; to falsify" (see interpolate).
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extrapolate (v.)

"make an approximate calculation by inferring unknown values from trends in the known data," 1862 (in a Harvard observatory account of the comet of 1858), from extra- + ending from interpolate. Said in early references to be a characteristic word of Sir George Airy (1801-1892), English mathematician and astronomer. Related: Extrapolated; extrapolating.

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interpolation (n.)
1610s, "act of interpolating;" 1670s, "that which is interpolated," from French interpolation (17c.) or directly from Latin interpolationem (nominative interpolatio), noun of action from past participle stem of interpolare "to alter; falsify" (see interpolate).
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*pel- (5)
Proto-Indo-European root meaning "to thrust, strike, drive."

It forms all or part of: anvil; appeal; catapult; compel; dispel; expel; felt (n.) "unwoven fabric matted together by rolling or beating;" filter; filtrate; impel; impulse; interpellation; interpolate; peal; pelt (v.) "to strike (with something);" polish; propel; pulsate; pulsation; pulse (n.1) "a throb, a beat;" push; rappel; repeal; repel; repousse.

It is the hypothetical source of/evidence for its existence is provided by: Greek pallein "to wield, brandish, swing," pelemizein "to shake, cause to tremble;" Latin pellere "to push, drive;" Old Church Slavonic plŭstĭ.
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