late 14c., reume, "watery fluid or humid matter in the eyes, nose, or mouth" (including tears, saliva, mucous discharge from the nostrils), from Old French reume "a head-cold" (13c., Modern French rhume) and directly from Latin rheuma, reuma, from Greek rheuma "discharge from the body, flux; a stream, current, flood, a flowing," literally "that which flows," from rhein "to flow" (from PIE root *sreu- "to flow").
In old medicine it was conceived as draining from the higher to lower parts of the body and causing ailments if out of balance. Also from late 14c. as "a head-cold, catarrh." The -h- was restored in early Modern English.
word-forming element indicating "branch of knowledge, science," now the usual form of -logy. Originally used c. 1800 in nonce formations (commonsensology, etc.), it gained legitimacy by influence of the proper formation in geology, mythology, etc., where the -o- is a stem vowel in the previous element.
The second element is prop[erly] -logy (-logue, etc.), the -o- belonging to the preceding element; but the accent makes the apparent element in E[nglish] to be -ology, which is hence often used as an independent word. [Century Dictionary]
<a href="https://www.etymonline.com/word/rheumatology">Etymology of rheumatology by etymonline</a>
Harper, D. (n.d.). Etymology of rheumatology. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/rheumatology