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Blaise Pascal was a mathematician and natural philosopher whose scientific work covered projective geometry, probability theory and hydrostatics. His correspondence with Pierre de Fermat in 1654 helped establish methods for calculating chances and dividing the stakes of an interrupted game. He also studied atmospheric pressure and developed a mechanical calculating machine.^[ref] The Pascal triangle connects these interests through binomial coefficients: its addition rule counts choices according to whether a specified object is selected. The same coefficients describe the distribution of the number of successes in repeated independent trials with a fixed success probability.

Hydrostatics describes fluids at rest under applied forces. For a fluid with mass density $\rho$ in uniform downward gravity, vertical force balance gives $dp/dz=-\rho g$, where $z$ increases upward and $p$ is pressure.

For constant $\rho$, the pressure at depth $h$ below a free surface is $p=p_0+\rho gh$. Integrating the pressure over a submerged body yields Archimedes' principle: the upward buoyant force equals the weight of displaced fluid. The same balance explains both a liquid column in a manometer and the increase of atmospheric pressure toward the ground, although atmospheric density cannot generally be treated as constant.

Simon Stevin was a mathematician and engineer who promoted the practical use of terminating decimals and made important contributions to statics and hydrostatics. His 1585 booklet on decimal arithmetic presented an accessible system for calculation and measurement; he did not invent terminating decimals, which had earlier precedents. His mechanical writings appeared the following year.^[ref] In hydrostatics, the pressure difference in a liquid of constant density depends on vertical depth through $\Delta p=\rho gh$. This explains why equal depths give equal pressures even in vessels of very different shapes, separating pressure from the total amount of liquid.