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II.
O 47. I

Book XII angles to all the right lines drawn in that plain "; join BZ,

V ZK, then AZ will be perpendicular to BŻ, ZK. But the n def. 3. squares of AK, AB, are equal, and the squares of AZ, ZB, are

equal to the square of AB, and the squares of AZ, ZK, are equal to the iquare of AK"; therefore the squares of AZ, ZB, are equal to the squares of AZ, ZK. Take the common square of AZ from both, then the squares of EZ, ZK, are equal; that is, BZ equal to ZK. In like manner, lines drawn from Z to the points O, S, may be proved equal to BZ, ZK; therefore a circle described about the center Z, with either of these distances, will pass through the points O, S, K, B; and, because KBSO is a quadrilateral figure inscribed in a circle, and OB, BK, KS, are equal, and OS less than BK, the angle BZK will be obtule ; therefore BK is greater than BZ; but GL is greater than KB, and therefore much greater than BZ ; and the squares of AG, GL, are equal to the square of AL, AB, or AK ; therefore the squarcs of BZ, Zų, are equal to the square of AL; and the squares of AZ, ZB, are equal to the squares of AG, GL; but the square of GL was proved greater than the square of BZ; therefore the square of AZ is greater than the square of AG; that is, the right line AZ greater than AG; but AZ is perpen. dicular to one of the bases of the polyhedron ; and AG reaches the superficies of the lesser sphere; therefore the polyhedron does not touch the superficies of the lesser sphere. Wherefore, &c.

Cor. If a solid polyhedron is inscribed in another sphere, fimilar to that in BCDE, they shall be to one another in the triplicate ratio of the squares of their diameters; for, the folids being divided into pyramids, equal in number, and of the same order, they will be similar; and therefore to one another in the triplicate ratio of their homologous fides P; that is, as AB drawn from the center of the sphere BCDE, to the semidiameter of the other spheret; but the semidiameters of spheres are as their diameters?; and one of the antecedents is to one of the consequents as all the antecedents to all the confequents '; therefore, the polyhedron in the one fphere is to the fimilar polyhedron in the other fphere, in the triplicate ratio of their diameters.

6 12.

P 15. 5.

r 12. 5.

PRO P. XVIII. THEO R.

SPHERES are to one another in the triplicate ratio of their

dian.eters.

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Let ABC, DEF, be two spheres, and BC, EF, their diameters, Book XII the sphere ABC is to the sphere DEF, in the triplicate ratio of BC to EF. If not, let the fphere ABC be to a spheri GHK less than the sphere DEF, in the triplicate ratio of BC to EF. Let this sphere GHK be inscribed within the sphere DEF; likewise, in DEF, infcribe a polyhedron, which shall not touch the superficies of the lesser sphere GHK. In the sphere ABC infcribe a a 17. polyhedron, fimilar and alike situate to that in DEF; then these similar polyhedrons are to one another in the triplicate ratio of their dian.eters BC, EF b; but the sphere ABC, to the sphere b cor. 19. GHK, hath a triplicate ratio of BC to EFC; therefore the sphere c hyp. ABC is to the sphere GHK, as the polyhedron ABC to the fimilar polyhedron in DEF; but the sphere ABC is ĝr-åter than the polyhedron in it; therefore the 'sphere GHK is likewise greater than the polyhedron in DEF ; but it is less, as contained in it; which is absurd ; therefore the sphere ABC, to the sphere less than DEF, has not a triplicate ratio of BC to EF. For the same reason, the sphere DEF, to a sphere less than ABC, has not a triplicate ratio of EF to BC. Again, the fphere ABC, to a {phere of LMN, greater than DEF, has not a triplicate ratio of BC to EF. If it can, then, by invers. the sphere LMN, to the sphere ABC, shall have a triplicate ratio of the diameters EF to the diameter BC; but the sphere LMN is to the sphere ABC as the sphere DEF to fome Iphere less than ABC, because the sphere LMN is greater than DEF ; therefore the sphere DEF, to a sphere less than ABC, has a triplicate ratio of what EF has to BC ; which is proved absurd; therefore the sphere ABC, to ą, sphere greater or less than DEF, has not a triplicate ratio of what BC has to EF. Therefore ABC has to the !phere DEF triplicate ratio of what BC has to EF : Which was to be demonstrated.

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