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I just came upon your post while searching Google for "Joseph Pinder Burnham." The Durham, NH, town history has this about Robert Burnham: "He married, about 1646 (in Boston), Frances -----. She was not daughter of Valentine Hill, as many have guessed." (There were many accounts of her name being Frances Hill.) We are intrigued by the Frances Pinder name which would support the history authors' statement and are wondering what is your source for this.
Joseph Pinder Burnham was born in Durham NH, originally Oyster River Parish of Dover NH, 19 April 1795, but we have no record of his parentage, and have thought, perhaps, his mother might have been a Pinder. It seems a stretch that after six generations the name, Pinder, would reappear in the Burnham lineage.
Robert Burnham lived the balance of his life in Oyster River Parish and died 12 June 1691 in Ipswich MA at the home of his brother, Thomas. Frances Burnham survived him and was named administrator of his will. He was also survived by sons, Samuel and Jeremiah. His son, Robert, b. 1647 in Boston, died in 1663, and a forth son and sixth child, b. 21 Feb 1664 was named Robert. One account has the second Robert living in 1683. An erroneous account has him married with a bunch of unnamed kids. Robert Burnham also had two daughters, Sarah and Elizabeth. My connection is to son, Jeremiah, and to daughter, Sarah.