Etymology
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Words related to infirm

in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
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*dher- 

Proto-Indo-European root meaning "to hold firmly, support." 

It forms all or part of: affirm; confirm; Darius; dharma; farm; fermata; firm (adj.); firm (n.); firmament; furl; infirm; infirmary; terra firma; throne.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit dharmah "custom, statute, law," dharayati "holds;" Prakrit dharaṇa "a holding firm;" Iranian dāra‑ "holding;" Greek thronos "seat;" Latin firmus "strong, steadfast, enduring, stable;" Lithuanian diržnas "strong;" Welsh dir "hard," Breton dir "steel."

infirmary (n.)
mid-15c., "sick bay in a monastery," formerly also enfermerie, also firmary, fermery, from Old French enfermerie "hospital" and directly from Medieval Latin infirmaria "a place for the infirm," from Latin infirmus "weak, frail," (see infirm). According to OED, the common name for a public hospital in 18c. England.
infirmity (n.)

late 14c., infirmite, "disease, sickness; lack of capability, weakness," from Old French infirmité, enfermete "illness, sickness, disease; moral weakness," and directly from Latin infirmitatem (nominative infirmitas) "want of strength, weakness, feebleness," also "the weaker sex" [Lewis], noun of quality from infirmus "weak, frail" (see infirm).