Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 2

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In the words of Euclid:

Any cone is a third part of the cylinder which has the same base with it and equal height.

(The Elements: Book $\text{XII}$: Proposition $10$)


Let the cone be of height $h$.

Let the area of the base of the cone be $A$.

From Volume of Cylinder, the volume of a cylinder of base $A$ and height $h$ is $A h$.

Let the cone be divided by planes parallel to its base each positioned some small distance $d$ apart.

Let $d$ be sufficiently small that they can be approximated to cylinders in shape.

Let there be $n$ of these small cylinders in total.

Let the volumes of each of these small cylinders be:

$v_1, v_2, v_3, \ldots, v_n$

starting from the base of the cone and working up.

We have that:

$v_k = d a_k$

where $a_n$ is the

Let the areas of the bases of each of these small cylinders be:

$a_1, a_2, a_3, \ldots, a_n$

starting from the base of the cone and working up.

Historical Note

The technique of finding the Volume of Solid of Revolution by dividing up the solid of revolution into many thin disks and approximating them to cylinders was devised by Johannes Kepler sometime around or after $1612$, reportedly on the occasion of his wedding in $1613$.

His inspiration was in the problem of finding the volume of wine barrels accurately.

He published his technique in his $1615$ work Nova Stereometria Doliorum Vinariorum (New Stereometry of Wine Barrels).

Gottfried Wilhelm von Leibniz redefined the problem by applying the techniques of integral calculus around $1680$.