This
唐 only imitates the sound of Down, after the English doctor John Langdon Down; it has nothing to do with the Tang dynasty.
氏 after a foreign doctor's name marks a disease named after him, as in
阿尔茨海默氏. Common forms:
唐氏综合征 (Down syndrome),
唐氏症, and
唐氏儿, a child with Down syndrome.