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I wish our Record Keeper had noted what the source was for the marriage of William Henry and Polly Johnson. It is so important to always give that detail. The birth certificate seems to tie in with the Census entries for his parents being William and Martha. I think I saw that there was a Cutts family living close by in the 1871 Census! Is it your line that connects with William Henry and Polly? Please will you at some time let the Society have that part of your tree? A photocopy of the birth certificate would always be welcome too.
I have just been looking at my Geographia, A1 Street Plan of Sheffield and found the Carterhall Lane, Ridgeway connection quite easily. It was very close to a route I took regularly in and out of Sheffield, via Gleadless. Carterlow must have been a corruption by the enumerator unless there has been a name change of a road.
If you are needing to come forward in time I found that the Sheffield Local Studies Library had an excellent set of Burgess Rolls (Electoral registers) as well as a good range of the Trade/Postal Directories.