Etymology
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Words related to overdose

over- 

word-forming element meaning variously "above; highest; across; higher in power or authority; too much; above normal; outer; beyond in time, too long," from Old English ofer (from PIE root *uper "over"). Over and its Germanic relations were widely used as prefixes, and sometimes could be used with negative force. This is rare in Modern English, but compare Gothic ufarmunnon "to forget," ufar-swaran "to swear falsely;" Old English ofercræft "fraud."

In some of its uses, moreover, over is a movable element, which can be prefixed at will to almost any verb or adjective of suitable sense, as freely as an adjective can be placed before a substantive or an adverb before an adjective. [OED]

Among the old words not now existing are Old English oferlufu (Middle English oferlufe), literally "over-love," hence "excessive or immoderate love." Over- in Middle English also could carry a sense of "too little, below normal," as in over-lyght "of too little weight" (c. 1400), overlitel "too small" (mid-14c.), overshort, etc.

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dose (n.)

early 15c., "the giving of medicine (in a specified amount or at a stated time)," from Old French dose (15c.) or directly from Medieval Latin dosis, from Greek dosis "a portion prescribed," literally "a giving," used by Galen and other Greek physicians to mean an amount of medicine, from stem of didonai "to give" (from PIE root *do- "to give").

Meaning "quantity of medicine given or prescribed" is from c. 1600. Extended sense, in reference to anything given to be "swallowed," literal or figurative, is from c. 1600. Slang meaning "a case of venereal disease" is by 1914.

dose (v.)

1650s, "give medicine to;" 1713, "administer in doses," from dose (n.). Related: Dosed; dosing.

O.D. 
abbreviation of overdose, attested from 1960.