You'll need to click on this image to see the 1200-pixel original, to make any sense of this. It is a clip from the Ancestry.com display of my family tree.
This shows four generations of ancestors of one Lydia Lamb, a great-great grandmother of mine, followed by four generations of ancestors of Dorcas Gayer, a great-great grandmother of hers—that is eight generations back from myself—and back to Dorcas Gayer's great-great grandfather Thomas Gayer, another four generations, which is twelve generations removed from me, and in ten more generations we reach Edward I Plantagenet, King of England.
The central figure in this diagram is William Gayer, Dorcas's father. A good many descendants of these Nantucketers claim Humphrey Gayer and Jane Sparks as William's parents. Their ancestry is well attested. It is known that they had a son William, but there is some controversy over whether this is the right William. There are two other William Gayers mentioned in documents of the middle to late 1600s, and some contend that one of them is the real father of Dorcas, while others say that the William Gayer, son of Humphrey, is further unknown and the antecendents of all other William Gayers are unknown. Makes life interesting.
The current evidence is, I'll admit, quite circumstantial. It is likely to remain so. This is one of four links in my family tree to English royalty of the 13th and 14th Centuries, and the only one that I can't pin down with certainty. But I do like to retain a link to Ed the First if I can. His engineers produced the first large-scale trebuchet, a weight-driven siege engine. Named War Wolf, it reduced several castles in Wales in wars to solidify the English claims therein. That makes Edward a bit of a bully, but I am quite fond of trebuchets. I've built a few smaller ones, including one that throws a golf practice ball no more than twenty feet. I have used it in a few speeches about the history of warfare.
As for William Gayer? I have a note in my tree that this branch may stop with him. For the nonce, I'll leave it at that. There is always a chance that letters between him and his parents will be found, clearing up the mystery.