1650s, from hand in cap, a game whereby two bettors would engage a neutral umpire to determine the odds in an unequal contest. The bettors would put their hands holding forfeit money into a hat or cap. The umpire would announce the odds and the bettors would withdraw their hands — hands full meaning that they accepted the odds and the bet was on, hands empty meaning they did not accept the bet and were willing to forfeit the money. If one forfeited, then the money went to the other. If both agreed either on forfeiting or going ahead with the wager, then the umpire kept the money as payment. The custom, though not the name, is attested from 14c. ("Piers Plowman").
Reference to horse racing is 1754 (Handy-Cap Match), where the umpire decrees the superior horse should carry extra weight as a "handicap;" this led to sense of "encumbrance, disability" first recorded 1890. The main modern sense, "a mental or physical disability," is the last to develop, early 20c.
1580s, "a slight wound or laceration, slight tear in a skin or surface produced by something sharp or rough," from scratch (v.). Meaning "mark or slight furrow in metal, etc." is from 1660s.
The American English slang sense of "(paper) money" is from 1914, of uncertain signification. Many figurative senses (such as up to scratch, originally "ready to meet one's opponent") are from sporting use for "line or mark drawn as a starting place for contestants," attested from 1778 (but the earliest use is figurative). The meaning "nothing" (as in from scratch) is by 1918, generalized from specific 19c. sporting sense of "starting point of a competitor who receives no odds in a handicap match."
The use of the word in billiards is from 1850, originally "a stroke which is successful but not in the intended way." The meaning "pocket the cue ball" is by 1914. Scratch-pad, for hurried writing or drawing, is attested from 1883.
c. 1200, "given or granted in unusual circumstances, exceptional;" also "specific" as opposed to general or common; from Old French special, especial "special, particular, unusual" (12c., Modern French spécial) and directly from Latin specialis "individual, particular" (source also of Spanish especial, Italian speziale), from species "appearance, kind, sort" (see species).
Meaning "marked off from others by some distinguishing quality; dear, favored" is recorded from c. 1300. Also from c. 1300 is the sense of "selected for an important task; specially chosen." From mid-14c. as "extraordinary, distinguished, having a distinctive character," on the notion of "used for special occasions;" hence "excellent; precious."
From late 14c. as "individual, particular; characteristic." The meaning "limited as to function, operation, or purpose" is from 14c., but developed especially in the 19c. Special effects first attested 1951. Special interest in U.S. political sense is from 1910. Special pleading is recorded by 1680s, a term that had a sound legal meaning once but now is used generally and imprecisely. Special education in reference to those whose learning is impeded by some mental or physical handicap is from 1972.
Special pleading. (a) The allegation of special or new matter, as distinguished from a direct denial of matter previously alleged on the other side. ... (c) In popular use, the specious but unsound or unfair argumentation of one whose aim is victory rather than truth. [Century Dictionary]