Addie,
I also have a research paper recently written by L. Ralph Mecham, son of L. Devon Mecham who complied the Mecham Book. He's trying to find the link from Jeremiah Meacham (in the colonies in 1636) back to England. When the black book was written there were some un-documented assumptions made as to the linage in England.
Ralph has not had much success in finding the documentation needed prior to 1636 to establish that Jeremiah was the son of Ambrose.
Also please note that some of our cousins made incorrect entries for other cousins in the black book.
My family has always (since Stephen Peabody Mecham, circa 1842) spelled the name Mecham. There was a move by one or two of the sons of Erastus Darwin Mecham to change the spelling to Meacham. So that shows up incorrectly on my father and grandfather - they however are adamant that their name is Mecham not Meacham.
Well - here are the links as promised. I think the links work - please let me know.
Paper from L Ralph Mecham:
https://drive.google.com/file/d/0BwLEE0cNztSqR1FxWHRrZTNSTVk...Mecham/Meacham Book of Rememberance - PDF:
https://drive.google.com/open?id=0BxG0Uq3ke3trQ2Y1VEQwMDFUeU...Cheers,
Darwin L Mecham