The term hydrostatic paradox is likely to suggest the classical experiment of “Pascal's vases.” In a demonstration with this apparatus the observer's attention is centered upon the vertical direction, specifically the common vertical depth of liquid in the different branches of the container, with their varying cross sections and volumes. But the term properly has a broader connotation. Fully as striking is the hydrostatic paradox seen in the fact that the horizontal thrust of the water against a dam is independent of the volume of water impounded, and dependent only on its depth at the barrier. To differentiate sharply between these two aspects of hydrostatic paradox, the terms “Phase I” and “Phase II” are proffered. The two aspects are discussed in the paper. A simple projection cell is described for demonstrating Phase II in a direct manner, corresponding to the demonstration of Phase I with Pascal's vases. Considered also is the involvement of hydrostatic paradox with Archimedes' principle, and in capillary tubes.