"Nebraska State Census, 1885," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:X3F9-XMH : 11 March 2018), Arthur G Calkins in entry for W W Calkins, 1885; citing NARA microfilm publication M352 (Washington, D.C.: National Archives and Records Administration, n.d.); FHL microfilm 499,560.
"United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M3BD-DN4 : accessed 6 September 2020), Auther G Calkins in household of Wallace Calkins, Mill Precinct, Lancaster, Nebraska, United States; citing enumeration district (ED) 75, sheet 1A, family 2, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, 1972.); FHL microfilm 1,240,933.
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DNA Connections
It may be possible to confirm family relationships with Arthur by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: