Etymology
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pad (n.)

1550s, "bundle of straw to lie on," a word of obscure origin (perhaps a merger of several separate words), possibly from or related to Low German or obsolete Flemish pad "sole of the foot," which is perhaps from PIE *pent- "to tread, go" (see find (v.)), but see path (n.).

Sense of "soft cushion" is from 1560s, originally a soft saddle. Generalized sense of "something soft" is from c. 1700. Meaning "cushion-like part on the sole of an animal foot" in English is from 1790. The sense of "a number of sheets fastened or glued together at the edge" (in writing-pad, drawing-pad, etc.) is from 1865.

Sense of "takeoff or landing place for a helicopter or missile" is from 1949; the notion is of something to prevent friction or jarring. The word persisted in underworld slang from early 18c. in the sense "sleeping place," and this was popularized again c. 1959, originally in beatnik speech (later hippie slang) in its original English sense of "place to sleep temporarily," also "a room to use drugs."

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pad (v.1)

"to walk, travel on foot, tramp slowly or wearily along," 1550s, probably from Middle Dutch paden "walk along a path, make a path," from pad, pat "path" (compare path). Originally a cant word among criminals and vagabonds, perhaps of imitative origin (sound of feet trudging on a dirt road). Related: Padded; padding. English also formerly had the noun pad meaning "path, foot path" (1560s), which might be from this verb, or from the Dutch noun, or a variant of path.

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pad (v.2)

"to stuff with pads or padding, increase the amount of," 1827, from pad (n.). Of writing, "expand by insertion of extraneous matter," 1831; transferred to expense accounts, etc. by 1890. Related: Padded; padding. The idea of a padded cell in an asylum or prison (1862, padded room) is to prevent those inside from injuring themselves by dashing against the walls.

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lily-pad (n.)
"broad leaf of the water-lily," 1834, American English, from lily (n.) + pad (n.).
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keypad (n.)
"handheld pad of labelled buttons to work electronics," 1975, from key (n.1) + pad (n.).
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kneepad (n.)
also knee-pad, 1858, from knee (n.) + pad (n.).
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padding (n.)

"material used in stuffing, stuffing used to keep a garment in the desired shape," 1828, verbal noun from pad (v.2).

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notepad (n.)

also note-pad, "pad of paper for writing notes," 1907, from note (n.) + pad (n.).

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paddle (v.1)

"to dabble, wade in water," 1520s, probably cognate with Low German paddeln "tramp about," frequentative of padjen "to tramp, to run in short steps," from the source of  pad (v.). Related: Paddled; paddling. Meaning "to move in water by means of paddles" is a different word (see paddle (v.3)).

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footpad (n.)

"highway robber," 1680s, from foot (n.) + pad "pathway, footpath" (1670s), from Middle Dutch pad "way, path," from Proto-Germanic *patha- "way, path" (see pad (v.1), and compare path). Pad was a cant word among thieves and vagabonds, in expressions such as stand pad "stand by the wayside begging."

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