As explained in the first podcast episode of A History Of Submarines – Submersibles: The First Submarines, the first serious description of a submarine is found in the work ‘Inventions or Devices’, published in 1578 by William Bourne. Bourne was a man of many trades. He was a mathematician and a legal advisor, but before that he was a gunner in the English Royal Navy under admiral Sir William Monson.
In one of his many Naval Tracts, Monson had described a kind of underwater cannon that would be brought to bear in some kind of submerged boat. It is unknown to what extent Bourne knew of his former admiral’s idea, but either way, Bourne is credited with drawing up the design of something that looked like a submersible rowboat, covered with a leather deck and a long tube protruding from it above sea level for air. It was an early kind of snorkel.
Bourne proposed propulsion by oars. This concept was not new – throughout the Middle Ages some thinkers had proposed simply using oars, like men in a rowboat.
What was new was that Bourne had thought about how such a boat could safely take on water so as to make it sink, and then export the water again. He basically invented proto-ballast tanks. These were in the sides of the boat and were to be big leather bellows with screws. The bellows would suck in water to submerge and, using the screws, the crew would push the water out again, making the submarine rise.

Bourne’s design would remain just that – a theoretical design. It would take a brilliant Dutchman from the town of Alkmaar, The Netherlands to design and construct the first, true and working submarine.
More excellent info on Bourne’s works here.
An easily accessible online copy of Bourne’s ‘Inventions or Devices’ is hard to come by. However, the texts have been gathered here.
For the ‘Submersibles: The First Submarines’ podcast episode and Bourne’s role in the historical development of submersibles, tune in on your favourite podcast app or listen to it here.