The Cartesian Diver moves up and down due to changes in the balance of weight and buoyancy.

Buoyancy
The weight of an object pulls it down toward the Earth, but if the object is placed in a liquid, a force called buoyancy acts in the opposite direction. The buoyant force is equal to the weight of liquid that the object displaces, or pushes aside.

Similarities to Scuba Diving by Jonathan Dunlop Divers usually add lumps of lead to a belt worn around the waist until they just float, just like adding clay to the Cartesian Diver. There are 11kg (24 lb) of lead on my weightbelt if I am diving in the sea but occasionally I dive in fresh water, where I only need 9kg (20 lb). Fresh water is less buoyant than salt water so less weight is needed to overcome the buoyant force. Divers usually wear a buoyancy compensator, which contains a flexible bag that can hold air. The bag can be inflated with compressed air from the cylinder worn on the back, or deflated by letting bubbles escape into the water. The diver can therefore adjust buoyancy and choose to move up, down or stay at a certain depth, like the Cartesian Diver. Divers need fine control over their buoyancy so that they don’t kick the seabed. A kick could disturb clouds of sand or mud making it difficult to see, or could damage fragile objects on the seabed, such as coral.

When a well-made Cartesian Diver floats, only a small portion is above water. The part that is below water, along with its trapped air, displaces enough water to create a buoyant force exactly equal to the weight of the diver.

Compression
If you took a bicycle pump and blocked the end with your finger, you could push on the handle of the pump and squeeze the air inside so that it occupied a smaller volume. The harder you pushed, the more the air would be compressed. If you stopped pushing on the handle, the air would expand to its original volume again.

If you filled the pump with water, you wouldn’t be able to move the handle and decrease the volume of the water at all, because liquids are incompressible.

Squeezing on the sides of the bottle pushes on the water inside but does not compress it. The water pushes on the air in the diver and that does get compressed. If pen lid or medicine dropper is clear, you should be able to see the water level changing inside.

When the air trapped in the diver is compressed, its volume is reduced. The diver together with its trapped air then displaces less water. As a result, the upward buoyant force is reduced. When the decreasing buoyant force becomes less than the weight of the diver, the diver sinks.

If you let go of the bottle, the original pressure in the water is restored. The trapped air in the diver expands to its original volume, displacing more water as it does so. The increased amount of water displaced by the diver and its trapped air makes the buoyant force greater. When the buoyant force is greater than the weight of the diver, the diver rises. At the surface, the diver floats high enough out of the water for the buoyant force to balance its weight exactly.