Are you just looking for Horatio's birth info? Or are you looking for anything from Newcomb as well? I ask because since it looks like Lucinda Newcomb is a line carrier to Bradford, hers will need to be rock solid. To prove Horatio's birth info, you might be able to show two consecutive census records showing his birth was in ~1809 in Pennsylvania.
Most locations didn't start keeping birth certificates this early, so, secondary sources will likely be the only things you will be able to use - but you will generally need multiple secondary sources to prove each fact without a primary source (with some caveats, though). It is much better to find as much documentation as you can to accompany your application.