Wish I had a book with the lineage of that line.The furthest back my Cooke line goes is to Mary Cooke, she is a double Mayflower decedent.Her father was John Cooke who was on board the Mayflower with his father Francis Cooke who was the 17th signer. Her mother was Sarah Warren daughter of Richard Warren of the Mayflower.
I tried starting with Martha Missouri Cook and linked her to Lisbon Cook, I linked Lisbon to a Benjamin Cook. This could be done fairly easy because of the unusual names of the first two. But without dates I was stopped. I use rootsweb a lot. Many of the researchers post their sources.
I have 3 lines of Cook/Cooke
I have in my line James Cook, Abraham Cook Jr, Abraham Cook Sr, William Cook, Francis Cook.This seems to be of your line. My line divided with your with James. My line went on to a son James.
Then in another line I have Mary Cooke, John Cooke, Francis Cooke (the last two of the Mayflower.)
Then I have a Mary Cooke daughter of Thomas Cooke and Mary Borden.
Here is my tree that is online at rootsweb but give it a while because I just sent in an update.