I am aware that through the early and high Middle Ages Europeans continued to bathe as their ancestors the Romans did, and that bathing only fell out of favor with the Black Death. Ironically, Charlemagine probably bathed more often than George Washington. But I know that the Romans bathed by coating themselves in oil, usually olive oil, then scraping themselves off with a metal scraper. When bathing became popular again in the 19th century everyone used soap. So when did the transition happen and why?
The differences between soap and olive oil is actually smaller than you think.
Rancid olive oil contains lots of free fatty acids, which have emulsive properties similiar to soap.
Furthermore, traditional soap is made from olive oil, and/or other triacylglycerides. If a base is added to olive oil it transforms into soap, through a process conveniently called saponification. Usually the base was ash. This process is relatively simple and was known to the Babylonian and Romans. You can buy this type of soap at any natural grocery store today, branded as glycerine or castille soap. However ancient soap was not as luxurious as these. Modern industrial techniques were needed to make it clear and pure. Roman Soap was probably more of a thick gloppy mess.
Since the mid 1800s soap quality has improve immensely, but its still fundamentally the same stuff the Romans, Carolingian, and Founding Fathers used.
Note: I am not a historian. I am a lipid chemist. I hope I didn't break any rules.