Hydrostatics was one of the areas of “mixed mathematics” – including geometrical optics, positional astronomy, harmonics, and mechanics – developed by Alexandrian authors in the Hellenistic era. Until the late sixteenth century the canonical work on hydrostatics was “On Floating Bodies” by Archimedes (ca. 287–212 B.C.E.). It deals in a rigorous geometrical manner with the conditions under which fluids are at rest in statical equilibrium and with the equilibrium conditions of solid bodies floating in or upon fluids.

At the end of 1618, the twenty-two-year-old Descartes, working with Isaac Beeckman, addressed some problems in hydrostatics involving the “hydrostatic paradox.” In 1586 Simon Stevin, the leading exponent of the mixed mathematical sciences at the time, brilliantly extended Archimedean hydrostatics. He demonstrated that a fluid filling two vessels of equal base area and height exerts the same total pressure on the base, irrespective of the shape of the vessel and hence, paradoxically, independently of the amount of fluid it contains. Stevin's mathematically rigorous proof applied a condition of static equilibrium to various volumes and weights of portions of the water (Stevin 1955–66, 1:415–17).

In Descartes’ treatment of the hydrostatic paradox (AT X 67–74), the key problem involves vessels B and D, which have equal areas at their bases and equal height and are of equal weight when empty (see Figure 12). Descartes proposes to show that “the water in vessel B will weigh equally upon its base as the water in D upon its base” – Stevin's hydrostatic paradox (AT X 68–69).

First Descartes explicates the weight of the water on the bottom of a vessel as the total force of the water on the bottom, arising from the sum of the pressures exerted by the water on each unit area of the bottom. This “weighing down” is explained as “the force of motion by which a body is impelled in the first instant of its motion,” which, he insists, is not the same as the force of motion that “bears the body downward” during the actual course of its fall (AT X 68).