Step 1: Frame the history. The question asks who first showed ether working as a surgical anaesthetic in public.
Step 2: Recall the milestone. In 1846 the dentist William T. G. Morton anaesthetised a patient with ether before an audience at Massachusetts General Hospital, the event that launched modern anaesthesia.
Step 3: Separate the others. Priestley is tied to discovering nitrous oxide, Wells to using nitrous oxide for dentistry, and Simpson to bringing chloroform into obstetric practice - none of them to the first ether demonstration.
Step 4: The correct name is Morton.
\[\boxed{\text{Morton}}\]