The 1880 census shows Adin A. Brooks, 13, living with his uncle and aunt Roscoe L. and Joanna Merrow in Smithfield. Roscoe Merrow and Joanna Perkins married in 1875. So let's assume that Adin's mother was either a Merrow or a Perkins. An Isaac R. Brooks married Charity J. Perkins in Fairfield in 1862. Charity Brooks lived in Fairfield in 1870 with son "Abram A." Brooks, aged 3. I believe that this was your great-grandfather.
This webpage gives some information on the Brooks family, and shows a son "Adrain," born 17 Mar. 1868 to Isaac R. and Jennie (Perkins) Brooks. The 1900 census for Belgrade gives the date as Mar. 1866, and other censuses support that date.
Line 24 may be of some interest.
Very interesting. Thanks for your help.
Nancy my grandfather was Carleton Brooks of Fairfield, Maine. His father was Adin Brooks, I have pictures and information on the Brooks side. Debra firstname.lastname@example.org