Thomas Rogers, the eldest of Thomas Matthew Rogers and the 9th generation member of the Rogers family, was born circa 1586/7. He was married in England about 1606 probably in Wiltshire. It is thought he had moved from his birthplace to Dorset or Wiltshire where many of his relatives lived. With, or without their financial help, he went into business. He was said to have been a very successful businessman. He was known in London as a Camelot merchant. We know his first, and possibly only, marriage was to a woman named Grace but I cannot account for the fact that, in Holland, his wife was called Elsgen. At any rate, Thomas and Grace had five sons and, possibly, two daughters. The boys, we know, were all born in England before Thomas went to Leyden (Leiden), Holland in 1618 and became a citizen there. He was sponsored by two members of the Pilgrim community who had come from Worksop, Notts., and Sandwich, Kent in England. In April 1620, he sold his Leyden home in preparation for the trip across the Atlantic with the other Pilgrims. According to a 1622 Poll Tax in Leyden the Rogers family still in Leyden consisted of a widow Elsgen (Thomas had died in the first winter the Pilgrims were in Plymouth Colony and is buried on Coles Hill, Plymouth, MA), son John and daughters Lysbeth and Grietgen (or Elizabeth and Margaret). It is possible Elsgen was a second wife whom Thomas had married in Leyden after he arrived there in 1618 since the two daughters were considered infants at the time of the Poll Tax. It is also possible that Thomas actually had come to Leyden earlier than 1618. His youngest son, James was born in England in 1615 but only John went to Leyden with his father. Thomas, William and James were still quite young during these years so I can't help but wonder where they might have been at this time. Joseph, the eldest, would have been about 14 when he came on the Mayflower with his father but there is no evidence he had been in Leyden so he could have joined up with his father in England where he had been living with his younger brothers and his mother. Apparently, the Mayflower Society recognizes only Joseph and John as children of Pilgrim Thomas Rogers. Since John is our ancestor, I shall leave this whole puzzle of Thomas Rogers and his possible wives and children to some future generation that might like to try and solve it.
The English Ancestry of the Pilgrim Thomas Rogers and his wife Alice (Cosford) Rogers
The accepted lineage researched and published by Clifford Stott in The Genealogist
Thomas Rogers was one of the unfortunate Mayflower passengers who died during the “first sickness at Plymouth” in the winter of 1620/1. Nevertheless, an extensive posterity is attributed to him through at least two sons, Joseph and John.  Until recently little was known concerning Thomas Rogers’ children and parentage; this did not deter authors of genealogies and family historians from publishing many fictitious accounts of this emigrant and his family, some complete with illustrious ancestry and an ever-present coat of arms. One of the most common claims is that the Pilgrim was a great-grandson of (Rev.) John Rogers who was burnt at the stake in 1555, the first casualty of the purge of the English clergy by Queen Mary I. Many Rogers emigrants to New England and Virginia share this claim of descent from the Martyr John Rogers; these pedigrees should have been dead on arrival in the light of the well-documented study of the Martyr by (Col.) Joseph L. Chester (published as long ago as 1861  as well as a useful article by Henry F. Waters, another pioneer in New England research, in 1887. And yet the myth persists.
For two centuries and a half the only authentic evidence concerning Thomas Rogers and his family was a fragmentary and tantalizing account written in 1650 by a fellow passenger, (Governor) William Bradford: 
"Thomas Rogers, and Joseph, his sone (came). His other children came afterwards… Thomas Rogers dyed in the first sickness, but his sone Joseph is still living, and is married, and hath 6 children. The rest of Thomas Rogers (children) came over, and are maried, and have many children.”
Family historians speculated for many years concerning the identities of these “other children”. With little or no evidence, Thomas Rogers was made the father of a number of emigrants to New England, including James Rogers of New London, James Rogers of Newport, and William Rogers of Connecticut and Long Island. This last connection, at least, has been disproved. 
Stripping away the layers of assumptions and unproved claims that have accumulated for at least a century, what can actually be deduced from original sources? Only one of the “other children”, John Rogers, can be authenticated from New England records. He probably arrived at Plymouth about 1630 when the last of the Separatists arrived from Leiden. John was taxed in Plymouth on 25 March 1633. On 6 April 1640 Joseph Rogers and John Rogers “his brother” were granted fifty acres each at North River (Marshfield), thus proving John’s identity. 
Henry Martyn Dexter and Morton Dexter published evidence from Dutch records in 1905 showing that Thomas  Rogers was a camlet merchant. On 25 June 1618 he was admitted as a citizen of Leiden, a distinction not held by all the English Separatists. He sold his house on the Barbarasteeg to one Mordecai Cohen on 1 April 1620 for 300 guilders, probably in anticipation of his removal to America. Additional research in Leiden records by Jeremy Bangs showed that the house had been purchased in 1616 or 1617 from a baker, Jan Bloemsaet, and that Rogers had sued Blomsaet and his bondsman Gerrit Gerritsz, on 22 February 1619 to release a lien on the house.
Robert Wakefield’s examination of the 1622 Leiden poll tax lists reveal that when Thomas  Rogers left for America his wife Alice, two daughters Elizabeth and Margaret, and son John remained in Leiden. In the household of Antony Clements, apparently one of the English Separatists who did not emigrate to Plymouth, are found the following persons:
Jan Thomasz orphan from England without means
Elsgen (Alice)  Rogiers, widow of Thonis Rogiers an English woman in the back part of the house or in the kitchen
In the Dutch patronymic system Jan Thomasz is equivalent to John, son of Thomas. His placement above Thomas Rogers’ widow and his description as “orphan from England” suggest that he is John  Rogers, later of Plymouth. It has been speculated that the daughter Elizabeth came to Plymouth and married Samuel  Eddy, since the latter was granted land at Plymouth on 3 June 1662 reserved for “the firstborn children” of the colony or their parents. As Samuel did not in his own right qualify on either count, it is possible that his wife Elizabeth was a daughter of Thomas  Rogers. 
In summary, the primary evidence gathered from Dutch and New England sources establishes that Thomas Rogers had a wife named Alice, who survived him, and a least four children: Joseph, John, Elizabeth and Margaret, probably born in that order. Joseph was likely the eldest because he came with his father to Plymouth in 1620, while the birth orders of the others is suggest by the tax list. Thomas left England by 1616 or 1617, when he bought his house at Leiden.
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