Origins
The word 'cure' descends from Latin 'cūra,' one of the most semantically productive words in the Roman vocabulary. In Classical Latin, 'cūra' meant 'care, concern, attention, anxiety, management, administration' — a broad range of meanings unified by the concept of taking responsibility for something. It was not a medical word. A Roman magistrate had the 'cūra' of public works; a worried lover suffered from 'cūra' (anxious care); a diligent craftsman worked 'cum cūrā' (with care, carefully). The medical meaning — healing a disease — developed in Late Latin and Old French, arising from the straightforward logic that to heal someone is to care for them.
The family of English words derived from 'cūra' is remarkably large and diverse. 'Curate' (a parish priest) comes from medieval Latin 'cūrātus,' one who has the 'cūra animārum' (cure of souls) — the spiritual care and responsibility for a parish. The French 'curé' preserves this ecclesiastical sense. 'Curator' (the keeper of a museum or collection) is one who takes care of objects. 'Curious' (from Latin 'cūriōsus,' full of care, eager to know) originally meant 'careful' or 'diligent' before shifting to 'eager to learn' and then to 'strange.' 'Accurate' (from 'ad-' + 'cūrāre,' done with care) means 'carefully made.' 'Secure' (from 'sē-' + 'cūra,' without care) means 'free from anxiety.' 'Procure' (to obtain with care) and 'sinecure' (a position without care, i.e., without duties) complete the set.
The transition from 'care' to 'heal' is visible in the dual meanings of Old French 'cure' and Middle English 'cure,' which retained both senses for centuries. A medieval English 'cure' could mean spiritual care (the cure of souls), medical treatment (the cure of a wound), or general concern. The exclusively medical sense — 'to eliminate a disease' — is a narrowing that became dominant only in the sixteenth and seventeenth centuries.
Scientific Usage
The verb 'to cure' also developed a set of non-medical technical meanings through the general sense of 'to take care of, to prepare, to preserve.' To cure meat (preserving it with salt or smoke), to cure leather (treating it), and to cure concrete (allowing it to harden properly) all use 'cure' in its older sense of 'to attend to carefully, to bring to a proper condition.' These technical uses preserve the pre-medical meaning of the word more faithfully than the medical use does.
The further etymology of Latin 'cūra' is uncertain. It has no widely accepted PIE source, though connections to Old Latin 'coera' and possible Italic cognates have been proposed. Some scholars have suggested a relationship to PIE *kois- (to lie down, to be at rest), which would make 'cūra' originally something like 'the act of attending to one who lies down' — a suggestive image for both nursing the sick and tending the dying.