Your inquiry caught my eye because of Harrison Beeman, a brother of my 3rd great grandfather, who I have been trying to research. While your Harrison is not the one I'm looking for, I found much information on your family line. From the 1860 census of Pike Co., OH, I found your 10 year old Harrison was the son of Monroe Beeman and Nancy. They were married in Pike Co. OH on 8 Apr 1849 and Nancy was the daughter of Matthew Current per Ohio marriage records.
Monroe Beeman was born 10 Apr 1828 in Jasper, OH to Reuben Beeman and Hannah Long.
Reuben Beeman was born 30 Sep 1771 in Kent, CT to Friend Beeman and Mary Logans.
Friend Beeman was born 6 Feb 1749/50 in Kent, CT to Thomas Beeman, Jr. and Bertha Tracy.
Thomas Beeman, Jr. was born Oct, 1715 to Thomas Beeman, Sr. and Phoebe Park probably in Preston, CT.

Once I had identified Monroe Beeman, it was easy to trace back further because his ancestry has been posted both in Genealogy.Com's World Family Tree and in Rootsweb's Family Tree Search. It is undoubtedly also in Ancestry.Com's family tree submissions, but I did not check that out. There is much more information in those sources than I outlined above which you will find interesting. If you don't have a paid subscription to Genealogy.Com or Ancestry.Com, go to Rootsweb.com since that is a free website
You might want to concentrate your search on the biological ancestry of William Speers Beeman since that would be your true ancestrial lineage.