1640s, "to evaporate through the pores," a back-formation from perspiration and in part from Latin perspirare "blow or breathe constantly," from per "through" (from PIE root *per- (1) "forward," hence "through") + spirare "to breathe, blow" (see spirit (n.)). Meaning "to sweat" is a polite usage attested from 1725. Medical men tried to maintain a distinction between "sensible" (sweat) and "insensible" perspiration:
[I]t is sufficient for common use to observe, that perspiration is that insensible discharge of vapour from the whole surface of the body and the lungs which is constantly going on in a healthy state; that it is always natural and always salutary; that sweat, on the contrary, is an evacuation, which never appears without some uncommon effort, or some disease to the system, that it weakens and relaxes, and is so far from coinciding with perspiration, that it obstructs and checks it. [Charles White, "A Treatise on the Management of Pregnant and Lying-in Women," London, 1791]
Related: Perspired; perspiring.