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HOW DID THEY DIG SO DEEP?
Peat  is  highly impermeable. If you dig a hole into wet peat to well below the water table,  water  will  only ooze out from the sides of the hole very slowly.  The deeper you go,  the  more  compressed is the peat by the weight of the ground above,  and the slower the water seeps out of it.
"The  possibility  of  digging  peat  to  a  depth of several feet even today is often underestimated . . . . .  practical experience has shown that a depth of several feet can be attained in places where the general water table is only a foot or two  below the surface.  For instance an ornamental swimming pool was  excavated  in  the  Hickling  marshes  to a depth of nearly three metres entirely  by  intermittent  hand labour  without  the use of elaborate pumps with  the  men  working  well  below  the level of the water in nearby dykes: indeed  the  landowner concerned suggests that a depth of twelve feet could easily have been attained."
Lambert
et al., 1960
There  is  plenty  more  evidence,  both  anecdotal  and  scientific,  which confirms the peculiar,  impermeable  properties  of  wet  peat.  All of which means that even today you  can  dig  up  peat  from  the  same  sort of depth as the broads, without using pumps and without even getting your feet very wet, always provided that you do it reasonably quickly.

In medieval times, that is exactly what they did.
"Turf digging then, as now, was probably a seasonal operation for the summer . . . ."
Smith with Lambert
et al.,1960.
All the evidence strongly suggests that they spent no more than two or three weeks each year digging up enough peat to last them for the next twelve months. Clifford Smith  reckoned  that  if twenty men from each village near to a broad worked for three  weeks  each  year,   the  broads could all have been created in three hundred years or so.   He also reckoned that one man could have dug up 1000 turves a day, that  a  medieval  turf  contained  one  quarter of a cubic foot of peat, and that the domestic  requirements  of a single household would have been 10,000 turves per year  -  five days' work for two men.

Dug out to a typical depth of ten feet, a pit with a volume of 2500 cubic feet would measure  ten  feet  wide  by  twenty  five feet long  -  not a bad size, you might well think, for an ornamental swimming pool.  I won't trouble you with any more sums but  there  is  every  reason to suppose that, digging down to the same depth as the broads, they could have extracted all the turves that they needed to last them for a year
even  if  water  levels  were  no  lower  than  they  are today. What is more they wouldn't have needed any sort of pump or baling device.

Which is  probably  just  as  well.  Apart from a bucket on the end of a rope, I can find  no  reference  to  any  manually  operated  baling device  which has ever been capable  of raising significant quantities of  water from the sort of depths to which most of the broads  were dug -
                               -  not the 'shadoof', not Archimedes' Screw, and certainly not the  ladle-and-gantry,   which  in  the 19th  century  was  used to good effect in the shallow peat diggings of the Somerset Levels.

If  they  had  invented  a pump,  then surely it would have been widely used at sea, yet  the  earliest  known  forms of  marine  bilge pump date from the 16th century. Not  without  reason  did  Joyce  Lambert  and her colleagues initially dismiss this sort of solution as "unlikely engineering".
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