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  A TIME LINE for WILLIAM ANDREWS, the Emigrant:  
Year Events Source #
1605 ca Estimated birth year for WILLIAM ANDREWS, based on children's birth years.  
1629 ca Estimated marriage year to his first wife, name unknown: based on children's births.  
1630 ca Son, WILLIAM, b. England, d. 3 Jan 1663, left no sons. 1
1630 June: Lord John Winthrop's arrival in Salem, with a fleet of 10 ships from Southampton, Eng.
Mr. Endecott held the office of governor until the arrival of Mr. John Winthrop, who had been chosen governor of the Company and also of the Plantation, in June, 1630 (See 1639).
8
1631 ca Daughter JANE born: ref. [2] has no date, but lists her between WILLIAM and SAMUEL.
WILLIAM had two other daughters, per ref. [1].
2
1632 ca Son SAMUEL, b. in England, 1632 per [1], [ref. 2 has ca 1635.], d. 6 Oct 1704 1
1635 WILLIAM left Southampton, England on 5 April, arrived in Massachusetts Bay on 3 June 1635. [Ship JAMES of LONDON] Likely landed at Salem, which had a thriving community. [4][6][9]
Early references [1][3] have departure from Hampton, 15 miles WSW of London. But Wikepedia [4][6] gives Southampton, England, which is more likely (see 1630).  Ref [6] does not include a wife and children with WILLIAM.
4,6,9
1635 WILLIAM is reportedly from Hampsworth, Eng. [1-1870][3-1900] A search for Hampsworth results in no town of that name. But ref. [7-1937] has a William Andrews which shows him coming to Salem, Mass. from Hampsthwaite in Yorkshire (Co.), England. And some of the early settlers of New Haven were from Yorkshire.  Ref. [6-1930] gives "WILLIAM ANDREWS of Hampsworth, county Yorkshire (?), [settled in] Salem.  The similarity between Hampsworth and Hampsthwaite leads one to support Hampsthwaite.  More research is needed, such as birth, marriage in Hampsthwaite. 
19 July 2012:  Found a Hamptworth in Wiltshire Co., England.  It lies within civil parishes Landford or Redlynch.  This region is very close to Southampton.  So more research needed.
1,3,7
1637 WILLIAM:  Juryman in Salem, Mass. 3
1637 There were two Eatons: Samuel Eaton, and Theophilus Eaton (eventually governor). They, along with Mr. John Davenport and Edward Hopkins, Esqrs., Mr. Thomas Gregson and many others.., arrived at Boston on the 26th of July, 1637. 5
1638 WILLIAM:  servant of Mr. Henry Coggan, before Gen. Court: 4 Oct 1638 3
1638 New Haven: On the 30th of March, 1638, Mr. Davenport, Mr. Prudden, Mr. Samuel Eaton and Theophilus Eaton, Esq., with the people of their company, sailed from Boston for Quinnipiac River (New Haven, Conn.). In about a fortnight they arrived at their desired port. On the 15th of April, they kept their first Sabbath in the place.
On the 24th of Nov., 1638, Theophilus Eaton, Esq., Mr. Davenport and other English planters, entered into an agreement with Momauguin, sachem of that part of the country…
5
1639 WILLIAM assigned to MR. ENDECOTT 3 JULY 1639.  Mr. ENDECOTT took the oath of Assistant Governor of the Mass. Bay Plantation on 7 Sept 1630, continuing to 1649. 3
1639 Son NATHAN, bp. 17 Nov 1639 NHC1 [= First Congregational Society, New Haven.
[Could William be in New Haven already?] [Birth could have been in Salem or New Haven.]
2
1640 WILLIAM made a freeman on 13 May 1640.  [Likely in Salem, but not stated.] 3
1640 WILLIAM likely went to New Haven, Conn. (But his son Nathan bp. in 1639.)  He was early at New Haven with Eaton and Rev. John Davenport.  New Haven had many people from Yorkshire, England. 1
1644 WILLIAM built the meeting house in New Haven. 1
1645 On page 76 is a “List of the Settlers in New Haven, from the years 1639 to 1645”.   It was first published in the New Haven Journal and Courier, April 24, 1851. WILLIAM ANDREWS is listed. 5
1646 On page 164-5 is the Seating of the First Meeting House, “a General Court” or Town Meeting, on the 10th of March, 1646. In the 5th seat is Bro. ANDREWS. 5
1663 Death of WILLIAM's first wife: d. 3 Jan 1663, per New Haven Vitals. 2
1665 WILLIAM married his second wife, ANNA GIBBANDS, on 7 Dec. 1665.  She was a daughter of William Gibbands, who was Colonial Secretary in 1657.  [Source: History of Wallingford.] [Source: Ancient Families of New Haven has:  Ann, daug. Edmund Tapp of M., widow of William Gibbard; Ann died 1701.] 1
1670 The town of Wallingford is established.  Sons SAMUEL and NATHAN move to Wallingford. 1
1676 MR. WILLIAM ANDREWS died at East Haven, 4 March 1676. 1
  CONCLUSION:
A “paper trail” may not exist that can prove that William Andrews, carpenter, on the ship “James of London” in 1635 is the same William who built the meeting house in 1644 in New Haven, Conn.  But it seems highly likely that William ended up in Salem, where he probably made contact with the Eatons and Rev. John Davenport (they were being “courted” by towns in Mass. to settle in their town), and so ended up in New Haven.
This “branch” of Andrews has already taken a Y-DNA test, and stands ready to discuss the same with any
other “branch” that may be interested. Email: waltbruetsch@usa.n
 
Sources / References
Item # Title Authors Publishers Dates
1 History of Wallingford, Conn.: From Its Settlement in 1670 to the Present. Charles Henry Stanley Davis An E-book. 1870
1 Early Families of Wallingford, Conn. Charles Henry Stanley Davis Excerpts reprinted by Genealogical Publishing Co., Inc., Baltimore 1979
2 Families of Ancient New Haven Donald Lines Jacobus Printer: Clarence D. Smith, Rome, NY.  Reprinted 9 volumes in 3 by Genealogical Publishing Co., Inc., Baltimore 1922-23;
1974, 1981, 1997
3 The Pioneers of Massachusetts, A Descriptive List Charles Henry Pope Originally published in Boston 1900
4 Wikipedia has: James 1635 Voyage The ship James of London sailed from Southampton on 5 April 1635, and arrived in Massachusetts Bay on 3 June 1635, with master William Cooper at the helm.  
5 History and Antiquities of New Haven, Conn.: from its earliest settlement to John Warner Barber, Lemuel Swift Punderson 1st edition in 1831, 2nd edition in 1856. 3rd edition 1870.
6 The Wikipedia Ref. 5 on 3 June 1635 was from: "The Planter of the Commonwealth" Charles Edward Banks Genealogical Publishing Co., Inc., Baltimore
pp.135-136
1961
7 Topographical Dictionsary of 2,005 English Emigrants to New England, 1620 - 1650 Charles Edward Banks   1937
8 The Winthrop Fleet of 1630 Charles Edward Banks,   1930
9 The Great Migration, Immigrants to New England, 1634-1635, Volume 1, A-B Robert Charles Anderson, George F. Sanborn, Jr, Melinde Lutz Sanborn Great Migration Study Project, New England Historical & Genealogical Society, Boston 1999
9 On page 62, WILLIAM ANDREWS:  On 5 April 1635, "William Andrews of Hampsworth, carpenter", was enrolled at Southampton in the "James of London" for New England (Drake's Founder 56).
9 This ref. book also states: Comments: This William Andrews cannot be connected with records for any of the men by that name in early New England. There was a William Andrews who in 1638 assaulted his master, Mr. Henry Coggan, and as a result was placed as servant to John Endicott [MBCR 1:246, 269]. These two records cannot be connected to any of the other known men by the name, but there is also nothing to connect the 1638 William Andrews with the passenger of 1635.