In checking Adam Beaver at familysearch.org I did find an Adam Beaver who served in the Civil War and received a pension. An Adam Beaver became an invalid, maybe from a would from the war.
I have not purchased the book. It does not seem to be available at a local library. There are three public library locations in New Philadelphia, Tuscarawas Co., OH. You might be able to get the book on an inter-library loan if it's at one of those libraries.
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