Arthur Plaisted in his "The manor and parish records of Medmenham, Bucks" (Longmans, 1925) gives a host of information about the origin of the Pooles.

He states that Geoffrey Poole was part of the Welsh contingent that followed Owen Tudor, peacefully, to England; Owen Tudor, of course, married Katherine de Valois, Henry V's widow.  Geoffrey then, over a period of thirty years, acquired all the three components of the manor of Medmenham that had been divided between the co-heirs, the sisters, of Thomas FitzAlan, earl of Arundel, who dsp in 1415.

Plaisted gives most of Geoffrey's will (PCC, 35, Wattys) dated 12th October 1478, which includes mention of his two sons Richard and Henry, his then unmarried daughter Eleanor and his second wife Bona (Danvers). Plaisted is the source of Geoffrey's death date of 4th Feb 1479; however his will and probate clearly indicates that he died in Feb/Mar 1474/5.

Plaisted gives a pedigree on page 68.  I failed to photocopy the list of references at the end of the book so I do not know where he got this information from; it may be that much of it was oral or, worse, conjectural.  So I only give it here without putting it in the main genealogy:

Blydhin ap Cynfyn, Prince of Wales, King of South Wales
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Meredyth, Prince of Powis [sic]
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Griffith who had Powis Gwenwynwyn
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Owen Cyfellog
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Owen the Baron = Joan, dau of Richard Clare, earl of Gloucester
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Sir Gilbert Poole "knighted by Richard I" (rectius Edward I)
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Owen Poole
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Eynon, Lord of Penyarth
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David
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Kew
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Madoc ap Kew
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David Vaux m. Margaret Griffith
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Geoffrey Poole d. 1479

However William Reitsweiner says that there is a better pedigree than the above in Bartrum’s Welsh Genealogies 2 at Gwenwys 2 (B); see <http://www.wargs.com/essays/welsh/pole.html>.  ;(Copy of Bartrum is held in SoG Library.)

Dec 2012, TFPL: I have found that all the pages of Bartrum have been scanned and are available at <http://cadair.aber.ac.uk/dspace/handle/2160/4026/browse>.  Further the Pole pedigree is on the sheet titled “Gwenwys 2(A,B)” and is:

6. Sir Richard Pool
   Kt of the Sepulchre Min Dep 243)
|
7. GILBERT POOL
   = Catrin f. Llw ap Gr
   8 Bl ap C 30
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8. Owain Pool
   = Constance f. Meurig
     Lord of Peniarth
|
9. Einion
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10. Dafydd
    = Mawd of Peniarth
    f. Ie4 ap M’ld
|
11. Llywelyn
    = Ann f. And Cradock
      of Buellt
|
12. Madog
    = Anne f. Gr ab Owain Goch
     12 Mael 2
|
13. Dafydd Fawr
|
14. Jeffry Pole, d. 1479   -   See NLW Journal XVII (1972) p. 277
    ?15 = Edith d. Oliver St John (2)
|
15. Richard Pole
    Kt of the Garter

    (H 823a)  [P]

The first thing to note is that there is no link from the top of this pedigree to any other of Bartrum’s sheets, even though it is on Gwenwys 2(A,B).  But also on this page are a couple of pedigrees which are connedted with Peniarth and as two of the Pool spouses are labelled as of Peniarth, perhaps this is why this pedigree has been put on this sheet.

However, Bartum does include three links to other sheets for the spouses:
  Bl.ap C. 30
  12 Mael 2
  St John (2)
But the last of these, St John (2), conflicts badly with Report 14 of the Friends of Lydiard Tregose and with the less scholary Beds Visitation of 1566.  One killer is that the gt-grandfather of Edith St John is shown as Alexander St John of Weobley and who is said to be living in 1499 - hardly likely when Edith’s son Richard Poole is known to have been born c.1459.  Bartrum does put in a question mark for the line to Edith and I think he has identified her family incorrectly, particularly with the connection with Margaret Beauchamp, grandmother to Henry VII.  In fact “St John 1” gives an account of the right family, includes Margaret the grandmother of Henry VII, but excludes Edith save as part of ‘daughters’.

“12 Mael 2” is in fact generation 12 on Mael Maelienydd 2 which is descended from Mael Maelienydd 1 and that from P.40 but frustratingly page 40 is omitted from that collection of files. of the “Tables _Early_Series”.  Intriguingly this Mael 2 also shows a Lewys Meifod which continues on Mael Maelienydd 2(C,D,E).

“Bl.ap C.30”, which is “Bleddyn ap Cynfyn 30” brings in a rich series of connections, including siblings of Catrin’s father Llewelyn who were surnamed “de la Pole” and Llewelyn was the son of Gruffudd ap Gwenwynwyn (c.1200 to 1286) and Hawis (poss m. 1292 and d. 1310)  d. of John le Strange of Knockin.
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Curiously my aunt, Ursula P-L, spent her World War II years at Medmenham, working on interpreting aerial photographs.  I cannot believe she had any knowledge that one of her ancestors once owned the manor!
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April 7th 2004: After a bit of a debate on the medieval newsgroup, it seems that CP's correction, in Vol XIV p. 567, that he was no knight, is correct.  There are two clinchers:
(a) In his will, less than five months before he died, he describes himself as an esquire (PRO Documents On Line),
(b) In his IPM, shortly after his death he is described as an armiger. (thanks to Rosie Bevan).

Curiously there are many documents after his death that refer to him as a knight, possibly because his eldest son was not merely a knight but also KG.
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In his will, dated October 12th 1474, he lists:
His late wife Editha,
His daughter Eleanor,
His wife Bona,
His son Henry,
His son Richard.