A very interesting drawing – for its shipwrighting aspects as well as explaining the pump.
I wonder why the pump ducts appear to change from round to square section part way up. What does "square chamber" mean, needing its special iron hoops to join it to the round ones?
On the face of it it might seem to have been easier to fit a square tube through the deck than round, but they did have lathes at the time so making a bush would not have been difficult. After all, they had to pass the much larger cylindrical masts through the decks to their feet on the ship's deeper structure.
Or is the square actually a trunking around the pump riser and return?
The pump well looks remarkably deep, but the drawing does not show the level of its inlets. I assume it was really a wall to protect the pump pipes from anything breaking loose in the hold.
The crew on the old wooden-walled sailing-ships often (routinely?) had to sound the depth of water in bilge. I don't know if this was done by physically going below or through a duct from main deck level.
I'd wondered in the past what the "orlop deck" was – evidently the lower deck forming the hold ceiling. I think it held such compartments as the sick-bay.
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Chian / rope pumps were just one of a quite a variety of water-supply (and as on the ship, drainage) pumps and equivalents invented since BP era.
One of the simplest, the shaduf, is still being used in parts of the Middle East and Africa some 3000 years after its first know evidence. It is not a "pump", but a simple counter-weighted derrick to lift water over a low height, such as a stream bank .
The shaduf has a sort of opposite: a gravity-driven mine drainage pump, conserved non-operably in the lead-mining museum site at Leadhills, SW Scotland. Water from a stream filled a bucket on a rocking beam to raise the mine pump-rods suspended from its other end. At the bottom of the stroke the bucket emptied to allow the pump-rods' weight to return the machine to its starting position.
While Archimedes could never have envisaged the Screw named after him being used over 2000 years later, in reverse: not to be turned by an energy-source (a person or donkey) to raise water; but to use the water as the energy-source – descending through it to run the machine in reverse, driving an alternator!
Try this:
https://www.mdpi.com/2073-4441/7/9/5031/htm
[Evolution of water-lifting devices over time.]
As a philosophical point, these very early machines could be seen as exemplifying the best principles of Engineering: creating a machine to solve a practical problem in the most efficient and reliable manner possible, with the most suitable materials and skills available in their place and time.