3. To find the proportion of tie ſphere it ſelf (or of its ſolid content) to any determinate Cone or Cylinder; or to find a Cone or Cylinder equal to a given ſpiere.
4. To find the proportion of a ſegment of a ſphere to any determinate Cone or Cylinder ; or to find a Cone or Cylinder equal to a given ſegment.
Theſe four Problems Archimedes proſecutes ſeparately, and lays down Theorems immediately ſubſervient to their ſolution; but we reduce them to two: For ſince an Hemiſphere is the ſegment of a ſphere, and the method of finding out its relations, in reſpect to the ſuperficies and ſolid content, is comprehended in the general method of inveſtigating the proportion of the ſegments : And from the ſuperficies and ſolid content of an Hemiſphere already found, the double of them, (that is, the ſuperficies and content of the whole ſphere) is at the ſame time given. And indeed 'tis ſuperfluous and foreign from the Laws of good Method, to inveſtigate their relations diſtinctly and ſeparately; ſo that if it were not a crime, I might on this account, blame even Archimedes himſelf.
The whole matter therefore is reduc'd to theſe two Problems.
1. To find the proportion of the ſuperficies of any ſegment of a ſphere, to a determinate circle; or to find a circle equal to the ſuperficies of a given ſegment.
2. To find the proportion of the ſolidity of any ſegment of a ſphere to any determinate Cone or Cylinder; or to find a Cone or Cylinder equal to an aſſign'd ſegment of a ſphere.
I ſhall reſolve theſe Problems by another much eaſier and ſhorter method: In which the order being inverted, firſt, I ſhall ſeek the ſolidity of a ſegment, and from thence deduce
