Karen, I don't know when the Giles County courthouse was built and I am not familiar with the Alexanders in early Giles County Tennessee but with some Alexanders who lived nearby with similar names.An Archibald Alexander (1765-1850s) lived in nearby Bedford County Tennessee in the 1810s.In Bedford County in 1810-15 he bought and sold lot 4 in Shelbyville and land on Flat and Sugar Creeks and the Duck River.A Joseph Alexander witnessed some of these deeds and this Joseph could have been Arch's father or some son/cousin; however, the Joseph who was Arch's father lived in Wayne County Kentucky after 1813 and he died there in 1816.(In 1805-10 Arch and his brother Andrew and their father Joseph seem to have lived in Rutherford County Tennessee.)Arch's brother Andrew Alexander died in 1815-16 in Lincoln County Tennessee and his wife Elizabeth administered his estate with help from Arch; I believe she later lived in Rutherford County.In 1816-20 this Arch Alexander appears to have moved to Monroe County Mississippi where he can be found in the 1820 and 1830 censuses.In the 1840 and 1850 censuses he appears in the household of his son Samuel in Tippah County Mississippi. A different, slightly younger, Archibald Alexander appears in the 1830 and 1840 censuses of Lincoln County Tennessee.I know little about him other than his census entries.However, I know that in an 1840 deed he sold 170 acres on Coffee Creek to a Doctor G. Alexander (DB M-1, p136). A George W. Alexander appears on Lincoln County tax lists of 1829-32 and he has no land listed, so (perhaps) he could have been the "doctor". An Arch Alexander also appears on the 1829-31 tax lists, with 75-100 acres of land somewhere in Lincoln County.This is fragmentary but I hope it helps!