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I descend from his youngest son, John Henry Beighley (1820-1893), whose grandson John Henry Beighley III (1890-1967) was my great-grandfather. Our immediate family did not carry on the naming tradition, but John Henry III's younger brother Loyal named his son John Henry IV, who has a son John Henry V. I don't know if there is a John Henry VI yet, but that is the numbering in our family.
Interestingly enough, the oldest son of Henry Beighley and Catherine Milliron was named John, also known as Miller John Beighley.
Something else should be taken into consideration: These were German Lutherans, and it has been my observation that many times in those early families, every son was given the name John, and often every daughter was given the name Mary, coupled with an additional name by which they would be known. Therefore, one family could have several sons named John, i.e. John Henry, John Jacob, John George, John Joseph, John Whoever...