The U.S. War Department initially provided wooden headboards (see photo left) for the graves of the Union dead whose remains were removed from the battlefields and reinterred in the ever-expanding national military cemeteries. Quartermaster General Montgomery Meigs – whose boundless resentment over fellow West Pointer Robert E. Lee’s betrayal of his country resulted in his seizing Lee’s estate for Arlington Cemetery – lobbied hard for permanent, durable gravestones. But it was not until 1872 that the War Department adopted the slab marble design with the rounded top for the gravesites of the identified dead buried in the national cemeteries. 
In 1879 Congress, undoubtedly yielding to pressure from the members of the ever- growing Grand Army of the Republic, authorized the furnishing of stones for veterans buried in private cemeteries. 
The D. W. Whitney Company of Troy, NY was hired to provide the stones for many states including Connecticut.  Their work proved unsatisfactory and in 1881 the D. L. Kemp Company of East Dorset, VT was contracted to complete the work.
In 1882, the Connecticut General Assembly authorized funding for the State Quartermaster General to provide and supervise the installation of headstones for Union veterans buried in the state. The first contract was awarded to the Bridgeport Monument Company  who manufactured monuments made of “white bronze” (zinc). 193 of these stones were furnished until Quartermaster General Arthur Goodrich, displeased with the quality of the zinc monument, issued a request for proposals in May of 1885 for the manufacture and installation of headstones
to be of the best quality of American white marble, each stone to be not less than 5 feet 6 inches long, 1 foot 4 inches wide and four inches thick, and to stand 2 feet 6 inches above ground. That portion of the stone which will be above ground, when set (2 feet 6 inches) to be sand-rubbed; the top of the stone to be curved (convex). 
Hartford’s own Stephen Maslen won the bid and was paid $14.50 for each gravestone, including the setting at the gravesite. 
Thanks to the foresight of General Meigs, the rigid specifications of QM Goodrich, and the workmanship of Stephen Maslen, these grave markers have survived the vandalism inflicted in many cemeteries; and those with little means – like Private William Murphy – rest in marked graves, recognizable and identifiable 150 years later.
Sources:National Cemetery Association, “History of Government Furnished Headstones and Markers”
http://www.cem.va.gov/cem/history/hmhist.asp accessed Jul 1 2013  Ibid. ”Headstones for Soldiers’ Graves: Failure of the Government Contractor.” Hartford Courant, November 11, 1881, p. 2.  Ibid.  For more about the Bridgeport Monument Company see Rotundo, Barbara “Monumental Bronze: A Representative American Company.” Cemeteries & Gravemarkers: Voices of American Culture. Ed. Richard E. Meyer. (Logan, Utah: Utah State University Press, 1992) p. 263-291.  “Proposals for Soldier’s Headstones.” Hartford Courant, May 23, 1885, p. 2.  Record of Headstones for Civil War Veterans, 1882-1888, Record Group 13, Item 14, Connecticut State Library.