So I have to chuckle at the claims that people were filthy dirty back then. I think this notion comes from the modern assumption that if you don't have running water its difficult to wash and therefore people didn't bother. Its true that we don't have very many descriptions of people bathing back then, but there is plenty of written evidence that people did not ignore their hygiene and cleanliness, they just achieved it in different ways.
"Bathing" was swimming, or at least hanging out in a larger body of water. Washing was done as others have described above and I can tell you from personal experience that a basin and wash rag go a long was in keeping a body clean. I have used a tin basin and kettle of warm water at events a gotten as clean as you would in a modern shower. The trickiest part is finding the privacy. In my modern life when I have lived for weeks or months without indoor plumbing I have used the same practice with wash pans and buckets filled at the pump and heated on the stove (or left cold). So the practice has worked long term for me too. It works just fine in a modern house too and helps with water conservation if you are into that sort of thing.
If you piece together the many historical references it becomes clear that common hygiene practices involved a few commonalities.
The first is not getting dirt on you in the first place. They left sleeves rolled down and collar fixed to protect from sun and dirt. Women and men wore linen caps to keep their hair cleaner. Hair could also be cleaned with hair powder (either corn meal, flour or the proper kind- which I think was made from meal or flour anyway)
The second involves changing your body linen. This seems to be the big component for hygiene for 18th century people. Everyone but the most destitute would have owned more than one shirt and "shifted" their underclothes as needed and as their means allowed. Shirts and shifts were made for heavy washing, and laundered frequently.
Next is spot cleaning, sponge-bathing or washing up. Instead of full bathing, the common practice was washing just what was dirty... hands, face, and all the rest as required.
A lastly and most uncommonly the occasional full dip in the river or a good dousing with a bucket of water over the head. This was thought healthful because of the effects of the cold water on the skin and body.
So, one could spot wash, pare nails, wash hands, wash face, change into a clean shirt, powder and brush out your hair... and they're fresh as a daisy and have achieved it in a simple and authentic way.
I think we need to move away from the notion that people in the past were dirty and unwashed. It is perfectly possible to get and stay clean even without modern plumbing, we know early American cultural expectations valued cleanliness and good hygiene, and there are plenty of references from the historical record that describe washing practices.
