Archimedes, in "On the Sphere and Cylinder," employed the method of exhaustion to compare the volume of a sphere to that of a circumscribing cylinder, establishing the ratio as 2:3.
Archimedes compared the volume of a sphere to the volume of a cylinder in his work titled "On the Sphere and Cylinder." In this work, Archimedes used a geometric method known as the method of exhaustion. He considered a sphere and a cylinder that both shared the same radius and height.
Archimedes imagined inscribing the sphere within the cylinder in such a way that the sphere touches the cylinder at the top, bottom, and around its curved side. He then reasoned that the volume of the sphere must be two-thirds of the volume of the circumscribing cylinder.
Archimedes used this insight to establish that the volume (V) of a sphere is two-thirds the volume of a cylinder with the same radius (r) and height (h). This relationship is expressed mathematically as:
![\[V_{\text{sphere}} = (2)/(3) V_{\text{cylinder}}\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/jpdgpcl8yffu6dc11an2qfdbtzw7r5zr2q.png)
This discovery demonstrated Archimedes' profound understanding of geometry and his pioneering contributions to the method of exhaustion, a precursor to integral calculus.