PROPOSITION 18.
Spheres are to one another in the triplicate ratio of their respective diameters.
Let the spheres
ABC,
DEF be conceived, and let
BC,
EF be their diameters; I say that the sphere
ABC has to the sphere
DEF the ratio triplicate of that which
BC has to
EF.
For, if the sphere
ABC has not to the sphere
DEF the ratio triplicate of that which
BC has to
EF, then the sphere
ABC will have either to some less sphere than the sphere
DEF, or to a greater, the ratio triplicate of that which
BC has to
EF.
First, let it have that ratio to a less sphere
GHK, let
DEF be conceived about the same centre with
GHK, let there be inscribed in the greater sphere
DEF a polyhedral solid which does not touch the lesser sphere
GHK at its surface, [
XII. 17] and let there also be inscribed in the sphere
ABC a polyhedral solid similar to the polyhedral solid in the sphere
DEF; therefore the polyhedral solid in
ABC has to the polyhedral solid in
DEF the ratio triplicate of that which
BC has to
EF. [
XII. 17, Por.]
But the sphere
ABC also has to the sphere
GHK the ratio triplicate of that which
BC has to
EF; therefore, as the sphere
ABC is to the sphere
GHK, so is the polyhedral solid in the sphere
ABC to the polyhedral solid in the sphere
DEF; and, alternately, as the sphere
ABC is to the polyhedron in it, so is the sphere
GHK to the polyhedral solid in the sphere
DEF. [
V. 16]
But the sphere
ABC is greater than the polyhedron in it; therefore the sphere
GHK is also greater than the polyhedron in the sphere
DEF.
But it is also less, for it is enclosed by it.
Therefore the sphere
ABC has not to a less sphere than the sphere
DEF the ratio triplicate of that which the diameter
BC has to
EF.
Similarly we can prove that neither has the sphere
DEF to a less sphere than the sphere
ABC the ratio triplicate of that which
EF has to
BC.
I say next that neither has the sphere
ABC to any greater sphere than the sphere
DEF the ratio triplicate of that which
BC has to
EF.
For, if possible, let it have that ratio to a greater,
LMN; therefore, inversely, the sphere
LMN has to the sphere
ABC the ratio triplicate of that which the diameter
EF has to the diameter
BC.
But, inasmuch as
LMN is greater than
DEF, therefore, as the sphere
LMN is to the sphere
ABC, so is the sphere
DEF to some less sphere than the sphere
ABC, as was before proved. [
XII. 2, Lemma]
Therefore the sphere
DEF also has to some less sphere than the sphere
ABC the ratio triplicate of that which
EF has to
BC: which was proved impossible.
Therefore the sphere
ABC has not to any sphere greater than the sphere
DEF the ratio triplicate of that which
BC has to
EF.
But it was proved that neither has it that ratio to a less sphere.
Therefore the sphere
ABC has to the sphere
DEF the ratio triplicate of that which
BC has to
EF. Q. E. D.
