56. To cut off from a sphere a segment by a plane such that the ratio of its volume to the volume of a cone having the base and height of the segment is a given ratio.

$$
\text { From the treatise "On Spirals". }
$$
56. $V_{\text {segm }}=V_{\text {sect }}-V_{\text {cone }}$,

$$
V_{ce2M}=\frac{2}{3} \pi r^{2} h-\frac{\pi r^{2}}{3}(R-h)=\pi r^{2}\left(h-\frac{R}{3}\right)
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-085.jpg?height=389&width=414&top_left_y=568&top_left_x=364)

Fig. 25.

Let the ratio be 2:

$$
\frac{\pi r^{2}\left(h-\frac{R}{3}\right)}{\left(\frac{\pi r^{2} h}{3}\right)}=2
$$

Then

$$
\frac{3 h-R}{h}=2 \text { and } h=R
$$

If the ratio is $n$, then $3 h-R=h n$ and $R=h(3-n)$, therefore, $n$ must be less than 3.