Page:Euclid's Elements 1714 Barrow translation.djvu/514

Rh 3. To find the proportion of tie phere it elf (or of its olid content) to any determinate Cone or Cylinder; or to find a Cone or Cylinder equal to a given piere.

4. To find the proportion of a egment of a phere to any determinate Cone or Cylinder ; or to find a Cone or Cylinder equal to a given egment. Thee four Problems Archimedes proecutes eparately, and lays down Theorems immediately ubervient to their olution; but we reduce them to two: For ince an Hemiphere is the egment of a phere, and the method of finding out its relations, in repect to the uperficies and olid content, is comprehended in the general method of invetigating the proportion of the egments : And from the uperficies and olid content of an Hemiphere already found, the double of them, (that is, the uperficies and content of the whole phere) is at the ame time given. And indeed 'tis uperfluous and foreign from the Laws of good Method, to invetigate their relations ditinctly and eparately; o that if it were not a crime, I might on this account, blame even Archimedes himelf.

The whole matter therefore is reduc'd to thee two Problems.

1. To find the proportion of the uperficies of any egment of a phere, to a determinate circle; or to find a circle equal to the uperficies of a given egment.

2. To find the proportion of the olidity of any egment of a phere to any determinate Cone or Cylinder; or to find a Cone or Cylinder equal to an aign'd egment of a phere.

I hall reolve thee Problems by another much eaier and horter method: In which the order being inverted, firt, I hall eek the olidity of a egment, and from thence deduce