## Proposition 17

 Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Let there be two spheres about the same center A. It is required to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Cut the spheres by any plane through the center. Then the sections are circles, for as a sphere is produced by the diameter remaining fixed and the semicircle being carried round it, hence, in whatever position we conceive the semicircle to be, the plane carried through it produces a circle on the circumference of the sphere. XI.Def.14 And it is clear that this circle is the greatest possible, for the diameter of the sphere, which is of course the diameter both of the semicircle and of the circle, is greater than all the straight lines drawn across in the circle or the sphere.

### Corollary.

But if in another sphere a polyhedral solid is inscribed similar to the solid in the sphere BCDE, then the polyhedral solid in the sphere BCDE has to the polyhedral solid in the other sphere the ratio triplicate of that which the diameter of the sphere BCDE has to the diameter of the other sphere.
For, the solids being divided into their pyramids similar in multitude and arrangement, the pyramids will be similar.
But similar pyramids are to one another in the triplicate ratio of their corresponding sides, therefore the pyramid with the quadrilateral base KBPS and the vertex A has to the similarly arranged pyramid in the other sphere the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of that which the radius AB of the sphere about A as center has to the radius of the other sphere. XII.18,Cor.
Similarly each pyramid of those in the sphere about A as center has to each similarly arranged pyramid of those in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere.
And one of the antecedents is to one of the consequents as all the antecedents are to all the consequents, hence the whole polyhedral solid in the sphere about A as center has to the whole polyhedral solid in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere, that is, of that which the diameter BD has to the diameter of the other sphere. V.12
Q.E.F.

The purpose of this proposition and its corollary is to separate concentric spheres so that it can be proved in the next proposition XII.18 that spheres are to each other in triplicate ratios of their diameters.

The argument that the intersection of a sphere and a plane through its center is a circle is weak. It has not been shown that the sphere is generated by taking any of its diameters and rotating a semicircle on that diameter about the diameter. Even the very concept of rotation about an axis has not been formalized.

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 Select from Book XII Book XII intro XII.1 XII.2 XII.3 XII.4 XII.5 XII.6 XII.7 XII.8 XII.9 XII.10 XII.11 XII.12 XII.13 XII.14 XII.15 XII.16 XII.17 XII.18 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VI Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip