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Evangelista Torricelli interpreted the support of a Mercury column in terms of external atmospheric pressure. His 11 June 1644 letter to Michelangelo Ricci describes equal column heights in vessels with different upper spaces, testing whether the support came from outside the vessel. In modern hydrostatic pressure notation, $p_{\mathrm{atm}}-p_{\mathrm{top}}=\rho gh$, where $\rho$ is the liquid's density and $g$ is gravitational acceleration. When the upper pressure is small, the column height measures atmospheric pressure, as in a barometer. Comparing liquids of different densities explains why water requires a much taller column than Mercury.^[ref]

A physicist and mathematician associated here with Torricelli's law.^[ref]

Michelangelo Ricci investigated geometric problems of maxima and minima in the seventeenth century. A 1666 letter preserved by the Royal Society discusses his mathematical booklet and its argument concerning limits, providing contemporary evidence of its reception.^[ref]

A modern illustration of the relevant kind of optimization problem is to maximize $x^m(a-x{)}^n$ for $0<x<a$ and positive $m,n$. Differentiating its logarithm gives $m/x-n/(a-x)$, so the maximum occurs at $x=ma/(m+n)$. This calculation illustrates how a geometric extremum can become an algebraic relation; it does not attribute modern derivative notation to Ricci.