Historians believe that the extant version of Problemata was not penned by Aristotle personally, but “while the Problemata is not the genuine Aristotelian work, it nevertheless contains an element derived from such a work”. Problemata XXXII.5 discusses breathing under water, including the oft quoted passage interpreted as referring to a diving bell:”Why do sponge-divers slit their ears and nostrils? Is it in order that the breath may pass more freely? For it is by this way that the breath seems to pass out; for it is said that they suffer more from difficulty of breathing by being unable to expel the breath, and they are relieved when they can as it were vomit the breath forth. It is strange, then, that they cannot achieve respiration for the sake of its cooling effect; this appears to be a greater necessity. Is it not quite natural that the strain should be greater when the breath is held, since then they are swollen and distended? But there appears to be a spontaneous passage of the breath outwards; and we must next consider whether breathing inwards is so also. Apparently it is; for they enable the divers to respire equally well by letting down a cauldron; for this does not fill with water, but retains the air, for it is forced down straight into the water; since, if it inclines at all from an upright position, the water flows in”.
The Alexander story does not appear in Problemata, and has all the hallmarks of a late anecdote. The siege of Tyre did happen in 332 BC, and divers were used to destroy underwater defenses, the same happened even earlier at the 413 BC siege of Syracuse according to Thucydides. But the "very fine barrel made entirely of white glass" called calimpha, the two companions, the bottom of the sea, the "large and powerful fish devour the small fry" are likely Alexandriad’s own fable. It got better as time went by, Alexander went 600 feet underwater in a glass case when he was 11, see Bachrach’s History of the Diving Bell. The conflation likely happened by associating Aristotle as Alexander’s mentor and the cauldron of Problemata with the calimpha fable.
Presumably sponge divers had some working understanding of the bell’s principle already in Aristotle’s time, and presumably it was reinvented multiple times in different places. A documented account of bell’s use does not appear until 16th century however, Bacon’s alleged contraption c. 1240 notwithstanding. Tasnier reported (p. 94) that in Toledo in 1538 he saw two Greeks lowering themselves under water in a large inverted kettle in the presence of Charles V, the inventor might have been de Lorena who is credited with "inventing" the bell in 1535 by some sources. Norwood took out a patent on diving devices in 1632, but by then Drebbel already built a working submersible (1620), and by the end of 1620s improved it enough to transport king James I.
Anything like modern understanding of water pressure etc. does not appear until Torricelli, even Galileo had trouble explaining siphons before that. In 1683 Sinclair published a book called Natural Philosophy Improven by New Experiments Touching the Mercurial Weather-Glass, the Hygroscope, Eclipsis, Conjunctions of Saturn and Jupiter, by New Experiments, Touching the Pressure of Fluids, the Diving-Bell, and All the Curiosities Thereof (sic!), which might have been the first modern work on physics of diving bells.
