*To cut a given sphere by a plane *so *that the
surfaces of the segments may have to one another a given ratio.*

Suppose the problem solved. Let *AA' *be a diameter
of a great circle of the sphere and suppose that a plane perpendicular
to *AA' *cuts the plane of the great circle in the straight line *BB',
*and *AA' *in M, and that it divides the sphere so that the surface
of the segment *BAB' *has to the surface of the segment *BA'B'
*the given ratio.

Now these surfaces are respectively equal to circles with
radii equal to *AB, A'B [I. 42, 43].*

Hence the ratio *AB ^{2 }: A'B^{2 }*is
equal to the given ratio, i.e.

Accordingly the synthesis proceeds as follows.

If H: K be the given ratio, divide *AA' *in M so
that

=(circle with radiusThus the ratio of the surfaces of the segments is equal to the ratio H:AB) :(circle with radiusA'B)

=(surface of segmentBA B')(surface of segmentBA 'B ').