Proof: By Euclid
(related to Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
 For if the solids are divided into similarly numbered, and similarly situated, pyramids, then the pyramids will be similar.
 And similar pyramids are in the cubed ratio of corresponding sides [Prop. 12.8 corr.] .
 Thus, the pyramid whose base is quadrilateral $KBPS$, and apex the point $A$, will have to the similarly situated pyramid in the other sphere the cubed ratio that a corresponding side (has) to a corresponding side.
 That is to say, that of radius $AB$ of the sphere about center $A$ to the radius of the other sphere.
 And, similarly, each pyramid in the sphere about center $A$ will have to each similarly situated pyramid in the other sphere the cubed ratio that $AB$ (has) to the radius of the other sphere.
 And as one of the leading (magnitudes is) to one of the following (in two sets of proportional magnitudes), so (the sum of) all the leading (magnitudes is) to (the sum of) all of the following (magnitudes) [Prop. 5.12].
 Hence, the whole polyhedral solid in the sphere about center $A$ will have to the whole polyhedral solid in the other [sphere] the cubed ratio that (radius) $AB$ (has) to the radius of the other sphere.
 That is to say, that diameter $BD$ (has) to the diameter of the other sphere.
 (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"