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.
"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.
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.