Perhaps the English word originally is from a derivative of Old French garbe/jarbe "sheaf of wheat, bundle of sheaves," though the sense connection is difficult. This word is from Proto-Germanic *garba- (source also of Dutch garf, German garbe "sheaf"), from PIE *ghrebh- (1) "to seize, reach" (see grab (v.)).
"In modern American usage garbage is generally restricted to mean kitchen and vegetable wastes" [Craigie]. Used figuratively for "worthless, offensive stuff" from 1590s. Garbage can is from 1901. Garbage collector "trash man" is from 1872; Australian shortening garbo attested from 1953. Garbology "study of waste as a social science" is by 1976; garbologist is from 1965.
Old English heap "pile (of things); great number, crowd, multitude (of persons)," from West Germanic *haupaz (source also of Old Saxon hop, Old Frisian hap, Middle Low German hupe, Dutch hoop, German Haufe "heap"), of uncertain origin. The group is perhaps related to Old English heah "high" (see high), but OED suggests a common origin with Latin cubare "lie down," and Boutkan says it is probably not Indo-European at all.
Slang meaning "old car" is attested from 1924. Earlier it meant "slovenly woman" (1806). As a characteristic word in American Indian English speech, "a lot, a great deal," by 1832.
One grain of sand does not make a heap. A second grain of sand added to the first does not make a heap. Indeed each and every grain of sand, when added to the others, does not make a heap which was not a heap before. Therefore, all the grains of sand in existence can still not a heap make. [the fallacy of the heap, as described in Malcolm Murray and Nebojsa Kujundzic, "Critical Reflection," 2005]
1530s, "gather into a heap or mass" (transitive), from Latin cumulatus "heaped, increased, augmented," past participle of cumulare "to heap," from cumulus "mound, heap" (from suffixed form of PIE root *keue- "to swell"). Related: Cumulated; cumulating; cumulant.
"to heap up," 1610s, from Latin acervatus, past participle of acervare "to heap up," from acervus "heap," which is akin to acer "sharp," from PIE root *ak- "be sharp, rise (out) to a point, pierce." Related: Acervated; acervating; acerval; acervative; acervuline "occurring in clusters; clustered" (by 1859).
"brain-sand" (anatomical), 1806, medical Latin, literally "little heap," diminutive of Latin acervus "heap," which is akin to acer "sharp" (from PIE root *ak- "be sharp, rise (out) to a point, pierce").