The Cartesian diver operates based on the principles of pressure and volume as described by the equation P1*V1=P2*V2, reflecting the behavior of gases in a closed system. When pressure is applied to the bottle, it compresses the air inside the diver, causing the volume of air to decrease and the density of the diver to increase. As the density surpasses that of water, the diver sinks. The discussion also raises questions about the relative air pressure inside the container compared to the external environment before pressure is applied. Understanding these mathematical relationships is key to explaining the mechanism of a Cartesian diver.