Try googling the "bows of the Mary Rose".
The longbows found on the sunken mary rose had a round back and belly. They were wider than they were deep, though, but when I first saw a photo i thought it was round. I think a width to depth ratio of 8:5 or 8:6.
I am currently making a heavy longbow with this type of cross section. I think that circular bows are really just a urban idea of longbows, when in truth being perfectly circular would probably not be a good design.
here is how I would do it: start with a stave about 1 1/4" wide, about 1" deep. first round the belly, then round the corners of the back, making sure that the belly is more crowned than the back. then round out any hard corners, but make sure to never cut directly into the back. round the corners of the back and the entire belly.
Remember that this design was for high quality bow wood, dense alpine yew. If you can't use suitable wood, I would make the bow shallower in cross section.
and make it long. not only does it look more "authentic" but it helps deal with the stress involved. I would'nt go less than 72".
when tillering, scrape off the belly and sides of the belly, keeping the round shape.
