Old English slidan (intransitive, past tense slad, past participle sliden) "to glide, slip, fall, fall down;" figuratively "fail, lapse morally, err; be transitory or unstable," from Proto-Germanic *slidan "to slip, slide" (source also of Old High German slito, German Schlitten "sleigh, sled"), from PIE root *sleidh- "to slide, slip" (source also of Lithuanian slysti "to glide, slide," Old Church Slavonic sledu "track," Greek olisthos "slipperiness," olisthanein "to slip," Middle Irish sloet "slide").
Meaning "slip, lose one's footing" is from early 13c. Transitive sense from 1530s. Phrase let (something) slide "let it take its own course, take no consideration of" is in Chaucer (late 14c.) and Shakespeare. Sliding scale in reference to payments, etc., is from 1842.