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We already produce natural oils for our skin (sebum), and those oils accumulate skin cells, dirt and dust, sweat, etc. It was common for Romans to use bathhouses, where they would coat themselves with olive oil, sometimes with an abrasive like pumice or fine sand, then use a strigil to remove the oil, and everything that had accumulated on the skin, before soaking/rinsing in the bath. This is enough to get you "not greasy" and they likely had a different concept of what "clean skin" should feel like, given that before soap, your bodies natural oils would be fairly ever-present.

https://www.unrv.com/culture/strigil.php

https://www.getty.edu/art/collection/object/103SNR

https://maa.missouri.edu/education/museum-in-30-objects/stri...