Derive from Latin cura, meaning care, concern, or charge. What is another word for restoring the balance or healing or restoring to health. But it also means trouble, anxiety and sorrow. The etymology of the word suggests that cure is an active process of bringing care and concern to the trouble, anxiety, even sorrow that is the constant companion of illness. Sorrow - for the loss of health, loss of a future as one envisioned it, for many losses that are individualistic to each person - is essential to cure.
The root shows up in a number different words in English: accurate, secure, sure… The English word curious came in via Old French, and derives from Latin curiosis - careful, diligent, inquisitive - and in English has on mostly that last sense of inquisitive. Even if excessive “curiosity” sometime leads to insufficient “care.”