It says he is the descendant of Mayflower passenger Thomas Rogers which is this profile Thomas Rogers, "Mayflower" Passenger who is his 4th Great grandfather.
His 3rd great grandfather Lt. Joseph Rogers, "Mayflower" Passenger was also on the Mayflower.
So this is how the relationship of Ephraim Joseph Butterfield, II is related to Thomas & Joseph Rogers of the Mayflower. Ephraim Joseph Butterfield, II born 5/13/1734 of Westford,Mass. is the child of Ephriam Butterfield & Elizabeth Butterfied (Davis) born 9/15/1718 in Barnstable, Mass. who is the daughter of Nathan Davis born 3/2/1690 in Barnstable, Mass. who married Elizabeth Phinney born 4/11/1690 in Barnstable, Mass.she is the daughter of John Phinney born 5/5/1665 in Barnstable, Mass. who married Mary Rogers born 9/22/1644 in Duxbury, Mass. and her father is Lt. Joseph Rogers born before 1/23/1603 in Watford, England whose father is Thomas Rogers of Watford, England. So it all came down to the relationship of the Butterfield line being connected to the Davis line to the Phinney line to the Rogers line. My husbands family also had similar connections also.
Hope this is what you were looking for.
Linda Batchelder