The Cartesian diver experiment certainly occupies a place of honour in old physics textbooks as a vivid demonstration of Archimedes' buoyancy. The original experiment, as described in old textbooks, shows Archimedes buoyancy qualitatively: when the increased weight of the diver is not counterbalanced by Archimedes' buoyancy, the diver sinks. When the pressure on the diaphragm is relieved, compressed air in the diver expels water from the diver, so that the latter may rise up again. In this article, the authors present a simple mathematical analysis for deriving the pressure and temperature variations for complete sinking in the Cartesian diver experiment. Some additional remarks on the system, with respect to the existing literature, are also made. (Contains 5 figures.)