The Parents of William and Mathew McCauley
After realizing that both Mathew and William named their first two children John and Jane (Mathew also referring to his Jane as Jinney) and considering both the naming patterns common among the Scots at the time and that such naming would otherwise have been likely to cause some confusion among the cousins, it seems apparent that the father of Mathew and William was almost certainly named John. Whether Jane was their mother's name is just slightly less probable, the bit of doubt being due to the usual tendency to name the first daughter after the mother's mother rather than the father's mother (obviously not followed in Mathew's case).
The names used in this tree for the parents of William McCauley Esq and Mathew McCauley are not supported by primary evidence, and I have yet to see any such evidence. The names proposed here are "calculated" based upon the onomastic naming patterns used by the Scots and some Irish at the time and upon the fact that both William and Mathew named their first son and daughter, John and Jane respectively (see their wills and Martha McCauley's pension application for details). Without the use of such naming patterns, there would be little to explain the identical use of those two names in the families of these brothers who lived close lives.
Many trees give "Erin" as the name of the father of William and Mathew McCauley. As "Erin" is another name for Ireland, similar to Eire, it seems an odd name for the son of a Scotsman, albeit living in northern Ireland. The only source I have seen for such an assertion comes from a 1932 article, "The McCauley Apron" by J. Ray. Shute. H. P. M. in NOCALORE, Being The Transactions of the North Carolina Lodge of Research, No. 666, A.F.& A.M., Monroe, N.C., U.S.A., edited for the Lodge by John Raymond Shute II. Presumably, "NOCALORE" is a type of acronym for North Carolina Lore. In this article about the Masonic apron reputedly originally owned by William McCauley and worn by him at the dedication of the first building of the University of North Carolina in Chapel Hill in 1793, the author discusses what he thought was known of the origins of William and his brother Mathew. More than a little of it is disputed now. Among the comments made about the pair is this phrase, "These hardy sons of Erin were industrious and thrifty...."
What did the author mean by "Erin"? The article suggests no knowledge of William's and Mathew's actual family of origin. And where would such knowledge have come from? It seems to me that the phrase could more clearly have been written and read as "These hardy sons of Ireland", since they came from there to America. I do not take "Erin" to be the paternal name or to have been intended as such.
John Raymond Shute II (Jr.) of Monroe County NC, who edited his father's essay, 1904-1988, was the son of John Raymond Shute (1855-1945), who came to Monroe County NC from Lancaster County SC. When the essay might have been originally written is not stated in the publication, nor are sources listed in the (possibly partial) copy that I have.