### Derivatives, Tangents, and Slopes

Fermat (born in 1601) took a slightly different approach than Kepler: In modern terms, he was interested in the tangent to a curve and the relationship between this tangent and the maximum (or minimum) of the function represented by the curve. Fermat's algebraic approach can be seen today as equivalent to studying the slope of the tangent to the graph of the function. Despite Kepler's intuition in this direction, Fermat is considered to have been the first to solve maximum-minimum problems by taking into account the characteristic behavior of a function near its extreme values. Newton and Leibniz understood even more clearly that a maximum or minimum was associated with a horizontal tangent.

Using our modern terminology, this is the geometric interpretation of the derivative of a function. We can see intuitively that if \(f(x)\) is a maximum (or minimum) value of the differentiable function \(f,\) then the value of \(f\) changes very slowly near \(x.\) Moreover, at the highest and lowest points on the graph of \(f,\) the tangent is horizontal; that is, its slope is \(0\). The derivative will be zero at extrema.

Returning to Kepler's problem of the proportions of a wine barrel, if \(V\) is the volume of the barrel (as a cylinder) with a fixed value of \(d,\) then \(V\) is a polynomial in \(h;\) namely, \[V=\frac{\pi}{4}d^2h-\frac{\pi}{16}h^3.\] Hence the derivative is easy to calculate: \[V^{\prime}(h) = \frac{{\pi}d^2}{4}-\frac{3\pi}{16}h^2.\]

For \(V\) to be a maximum, \(V^{\prime}\) must equal zero; hence \[3h^2=4d^2\,\,\,\,{\rm{or}}\,\,\,\,h=\frac{2d}{\sqrt{3}}.\]

And this was the result that Kepler found.

**Figure 10.** This applet shows the graphs of \(V\) and \(V^{\prime}\) as functions of \(h,\) along with the tangent line to the graph of \(V\) at any value of \(h,\) Here, it is assumed once again that, for a fixed value of \(d,\) the volume \(V\) of the wine barrel is a function of the height \(h\). In the diagram, the blue curve is the graph of \(y = V(h)\) and the red curve is the graph of \(y = V^{\prime}(h)\) for a fixed value of \(d.\) At the maximum volume of the barrel, the green line tangent to the graph of \(V\) is horizontal and its slope, the derivative \(V^{\prime},\) is zero. (Note that, while the animation permits negative heights and volumes for the barrel, in real life this would be impossible.) This diagram also appears in the Kepler's Barrel: Derivative, Tangents, and Slopes section of the copy of this article at MatematicasVisuales. (The applet here was re-created in Spring 2020 by Laura Turner.)

### Conclusion

Thus, the practical problem of measuring the volume of a wine barrel inspired Kepler to make important contributions to the development of both the integral and the differential calculus.