I got a surprise in a site about etymology:

Esculapius (Medicine) between Mercury (Merchants) and the Graces (Medicine, Hygiene and Panacea)
Esculapius dealt with Patients - Merchants make deals with Clients
Esculapius is linked with a Constellation of Idealistic Medical Ideas
Mercury or Hermes is linked with Hermaphroditism and Mercantile Mercenary views
Perhaps the single serpent on the Esculapian Caduceus represent one word or truth
Perhaps the two serpents on the Caduceus of Hermes represent the duality in "deals" or
"caviat emptor"

I had never before heard "Medicine" listed as one of the Graces. Now I've got to find out what responsibilities were assigned to each of the three.

After a lot of searching I found this, which appears to contradict the above material:

Aesculapius had 7 children who were all skilled in the art of healing like their father. Two of his
most celebrated daughters were Hygeia, the goddess of health in Greek mythology, and Panacea,
the Greek goddess of healing through herbs.(2) Modern readers will recognize the legacy of these
daughters in our words for hygiene and panacea.