Thanks for your information. I will copy below a section from my summary of my knowledge of the John and Catherine (Clark) Acheson family, which was compiled from several sources, but first I will comment on a few points of your note.
1. I took just a brief look at the Beers site that you referenced, and didn't find reference to John, and the Alexander mentioned there doesn't appear to be the right one (born too late). I'll go back and take a better look in case I overlooked something.
2. You mention John and his brother Matthew. I did know that John had a neighbor named Matthew Acheson, and assumed that they were related. Do you have any actual evidence that they were brothers?
3. You mention that three of John's daughters married Reed boys. According to my notes, two of the Acheson girls and one Acheson boy married Reeds (see below).
4. I have also seen various spellings of the surname in early references, and like you, have found that most of the later generations used the spelling ACHESON. Since I haven't seen any evidence to the contrary, I have used that spelling from the start. (A similar situation occurs with my Wherry family of Oxford, Chester Co., whose name appears as Wharry, Wharrey, and Warrey, among others.)
If you have more information on this family, or are interested in what I have on later generations, please contact me via e-mail at firstname.lastname@example.org. (This may change soon, as we are planning to move to another location.)
John Acheson and Catherine Clark
The earliest of our Acheson ancestors for whom we have records is John Acheson, who was born in northern Ireland in 1710. In 1733 he married Catherine Clark, who was born about 1710. About 1735 John, Catherine, and their infant son Matthew immigrated to America, settling in Lancaster County, PA. Although we know nothing of John’s parents or siblings, there was a Matthew Acheson who owned a farm adjacent to John’s, and may have been his brother.
On September 5, 1738, John warranted 253 acres in Little Britain Township, Lancaster Co. John lived here for over 40 years, and prospered in every way. He died in the early years of the Revolution. He sold 126 acres of his farm to his son John about August 1770. The invoice of the estate of John Sr. is on file at Lancaster, PA.
John died about March 1778. Catherine apparently then moved to Washington County with her sons Matthew and Humphrey, as she sold her land in Little Britain Township on November 30, 1779, and that same day bought 400 acres of land in Cross Creek Township, Washington County. On October 19, 1793, her children Catherine, Matthew and Humphrey and her son-in-law John Reed sold the farm on Cross Creek, indicating that Catherine (the mother) had died shortly before that. John and Catherine had seven children, all except Matthew born in Little Britain Township:
1. Matthew, born 1734 in Ireland, died 1814 in Mt. Pleasant Township, Washington County, PA; marriedJean Reed (6 chidren).
2. Humphrey, born 1740 in Little Britain Township, Lancaster County, PA, died 1814 in Washington County, PA; married (1) Rebecca Reed, sister of Jean (12 children), (2) a widow, Mrs. Hull.
3. Ann, born 1743, died November 15, 1823; married John Reed, brother of Jean and Rebecca.
4. Unknown daughter, married Long (first name unknown)
5. John, born 1745, died October, 1779; married Agnes Caldwell.
6. Christiana, born 1755, died 1820 in Casco, Cumberland County, ME; married (1) James Calvin, (2) Robert Tweed.
7. Catherine, born 1760, died March 29, 1822; married John Tannehill. 1 known child.