Is there a connection between frodo and sauron?

[Essex]

and the cirumstances in which each lost the Ring (and subsequently their fingers) were not similar.

But I think they WERE similar inasmuch as they both 'lost' the ring without voluntarily giving it up. This is why we see Frodo in such a state at the end of the book back in the Shire. I think Tolkien purposefully showed the similarity between the two events. Both tried to 'own' and use the ring and were ultimately thwarted. The only difference being that Sauron was defeated by the ring being lost, and Frodo, somewhat, was saved by Gollum biting it off. I think Tolkien is showing us Frodo's redemption in this case. If Frodo had allowed Gollum to be killed, then Sauron WOULD have retrieved the Ring after a short stuggle with Frodo and Middle-earth would have been lost. But he pitied Gollum and let him live, therefore the Quest was achieved.